From b677f64b473011a7bcc8c8e086b76ef3b55cc251 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Tue, 19 Nov 2024 14:47:03 +0000 Subject: [PATCH] build based on 08c2d54 --- previews/PR514/.documenter-siteinfo.json | 2 +- previews/PR514/assets/documenter.js | 370 ++++++----- .../PR514/assets/themes/catppuccin-frappe.css | 1 + .../PR514/assets/themes/catppuccin-latte.css | 1 + .../assets/themes/catppuccin-macchiato.css | 1 + .../PR514/assets/themes/catppuccin-mocha.css | 1 + .../PR514/assets/themes/documenter-dark.css | 4 +- .../PR514/assets/themes/documenter-light.css | 2 +- previews/PR514/authors/index.html | 2 +- previews/PR514/callbacks/index.html | 9 +- previews/PR514/code_of_conduct/index.html | 2 +- previews/PR514/contributing/index.html | 4 +- previews/PR514/development/index.html | 2 +- previews/PR514/examples/index.html | 2 +- .../general/density_calculators/index.html | 2 +- .../general/initial_condition/index.html | 24 +- .../PR514/general/interpolation/index.html | 12 +- 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a/previews/PR514/.documenter-siteinfo.json +++ b/previews/PR514/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-05T11:18:23","documenter_version":"1.4.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-19T14:46:55","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/previews/PR514/assets/documenter.js b/previews/PR514/assets/documenter.js index c6562b558..7d68cd808 100644 --- a/previews/PR514/assets/documenter.js +++ b/previews/PR514/assets/documenter.js @@ -77,30 +77,35 @@ require(['jquery'], function($) { let timer = 0; var isExpanded = true; -$(document).on("click", ".docstring header", function () { - let articleToggleTitle = "Expand docstring"; - - debounce(() => { - if ($(this).siblings("section").is(":visible")) { - $(this) - .find(".docstring-article-toggle-button") - .removeClass("fa-chevron-down") - .addClass("fa-chevron-right"); - } else { - $(this) - .find(".docstring-article-toggle-button") - .removeClass("fa-chevron-right") - .addClass("fa-chevron-down"); +$(document).on( + "click", + ".docstring .docstring-article-toggle-button", + function () { + let articleToggleTitle = "Expand docstring"; + const parent = $(this).parent(); + + debounce(() => { + if (parent.siblings("section").is(":visible")) { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); - articleToggleTitle = "Collapse docstring"; - } + articleToggleTitle = "Collapse docstring"; + } - $(this) - .find(".docstring-article-toggle-button") - .prop("title", articleToggleTitle); - $(this).siblings("section").slideToggle(); - }); -}); + parent + .children(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + parent.siblings("section").slideToggle(); + }); + } +); $(document).on("click", ".docs-article-toggle-button", function (event) { let articleToggleTitle = "Expand docstring"; @@ -110,7 +115,7 @@ $(document).on("click", ".docs-article-toggle-button", function (event) { debounce(() => { if (isExpanded) { $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); - $(".docstring-article-toggle-button") + $("a.docstring-article-toggle-button") .removeClass("fa-chevron-down") .addClass("fa-chevron-right"); @@ -119,7 +124,7 @@ $(document).on("click", ".docs-article-toggle-button", function (event) { $(".docstring section").slideUp(animationSpeed); } else { $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); - $(".docstring-article-toggle-button") + $("a.docstring-article-toggle-button") .removeClass("fa-chevron-right") .addClass("fa-chevron-down"); @@ -484,6 +489,14 @@ function worker_function(documenterSearchIndex, documenterBaseURL, filters) { : string || ""; } + /** + * RegX escape function from MDN + * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + } + /** * Make the result component given a minisearch result data object and the value * of the search input as queryString. To view the result object structure, refer: @@ -502,8 +515,8 @@ function worker_function(documenterSearchIndex, documenterBaseURL, filters) { if (result.page !== "") { display_link += ` (${result.page})`; } - - let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + searchstring = escapeRegExp(querystring); + let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); let text = textindex !== null ? result.text.slice( @@ -520,7 +533,7 @@ function worker_function(documenterSearchIndex, documenterBaseURL, filters) { let display_result = text.length ? "..." + text.replace( - new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence '$&' ) + "..." @@ -566,6 +579,7 @@ function worker_function(documenterSearchIndex, documenterBaseURL, filters) { // Only return relevant results return result.score >= 1; }, + combineWith: "AND", }); // Pre-filter to deduplicate and limit to 200 per category to the extent @@ -598,176 +612,194 @@ function worker_function(documenterSearchIndex, documenterBaseURL, filters) { }; } -// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! -const filters = [ - ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), -]; -const worker_str = - "(" + - worker_function.toString() + - ")(" + - JSON.stringify(documenterSearchIndex["docs"]) + - "," + - JSON.stringify(documenterBaseURL) + - "," + - JSON.stringify(filters) + - ")"; -const worker_blob = new Blob([worker_str], { type: "text/javascript" }); -const worker = new Worker(URL.createObjectURL(worker_blob)); - /////// SEARCH MAIN /////// -// Whether the worker is currently handling a search. This is a boolean -// as the worker only ever handles 1 or 0 searches at a time. -var worker_is_running = false; - -// The last search text that was sent to the worker. This is used to determine -// if the worker should be launched again when it reports back results. -var last_search_text = ""; - -// The results of the last search. This, in combination with the state of the filters -// in the DOM, is used compute the results to display on calls to update_search. -var unfiltered_results = []; - -// Which filter is currently selected -var selected_filter = ""; - -$(document).on("input", ".documenter-search-input", function (event) { - if (!worker_is_running) { - launch_search(); - } -}); - -function launch_search() { - worker_is_running = true; - last_search_text = $(".documenter-search-input").val(); - worker.postMessage(last_search_text); -} - -worker.onmessage = function (e) { - if (last_search_text !== $(".documenter-search-input").val()) { - launch_search(); - } else { - worker_is_running = false; - } - - unfiltered_results = e.data; - update_search(); -}; +function runSearchMainCode() { + // `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! + const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), + ]; + const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; + const worker_blob = new Blob([worker_str], { type: "text/javascript" }); + const worker = new Worker(URL.createObjectURL(worker_blob)); + + // Whether the worker is currently handling a search. This is a boolean + // as the worker only ever handles 1 or 0 searches at a time. + var worker_is_running = false; + + // The last search text that was sent to the worker. This is used to determine + // if the worker should be launched again when it reports back results. + var last_search_text = ""; + + // The results of the last search. This, in combination with the state of the filters + // in the DOM, is used compute the results to display on calls to update_search. + var unfiltered_results = []; + + // Which filter is currently selected + var selected_filter = ""; + + $(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } + }); -$(document).on("click", ".search-filter", function () { - if ($(this).hasClass("search-filter-selected")) { - selected_filter = ""; - } else { - selected_filter = $(this).text().toLowerCase(); + function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); } - // This updates search results and toggles classes for UI: - update_search(); -}); + worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } -/** - * Make/Update the search component - */ -function update_search() { - let querystring = $(".documenter-search-input").val(); + unfiltered_results = e.data; + update_search(); + }; - if (querystring.trim()) { - if (selected_filter == "") { - results = unfiltered_results; + $(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; } else { - results = unfiltered_results.filter((result) => { - return selected_filter == result.category.toLowerCase(); - }); + selected_filter = $(this).text().toLowerCase(); } - let search_result_container = ``; - let modal_filters = make_modal_body_filters(); - let search_divider = `
`; + // This updates search results and toggles classes for UI: + update_search(); + }); - if (results.length) { - let links = []; - let count = 0; - let search_results = ""; - - for (var i = 0, n = results.length; i < n && count < 200; ++i) { - let result = results[i]; - if (result.location && !links.includes(result.location)) { - search_results += result.div; - count++; - links.push(result.location); - } - } + /** + * Make/Update the search component + */ + function update_search() { + let querystring = $(".documenter-search-input").val(); - if (count == 1) { - count_str = "1 result"; - } else if (count == 200) { - count_str = "200+ results"; + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; } else { - count_str = count + " results"; + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); } - let result_count = `
${count_str}
`; - search_result_container = ` + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = `
${modal_filters} ${search_divider} - ${result_count} -
- ${search_results} -
-
+
0 result(s)
+ +
No result found!
`; - } else { - search_result_container = ` -
- ${modal_filters} - ${search_divider} -
0 result(s)
-
-
No result found!
- `; - } + } - if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { - $(".search-modal-card-body").removeClass("is-justify-content-center"); - } + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } - $(".search-modal-card-body").html(search_result_container); - } else { - if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { - $(".search-modal-card-body").addClass("is-justify-content-center"); + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); } + } - $(".search-modal-card-body").html(` -
Type something to get started!
- `); + /** + * Make the modal filter html + * + * @returns string + */ + function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; } } -/** - * Make the modal filter html - * - * @returns string - */ -function make_modal_body_filters() { - let str = filters - .map((val) => { - if (selected_filter == val.toLowerCase()) { - return `${val}`; - } else { - return `${val}`; - } - }) - .join(""); - - return ` -
- Filters: - ${str} -
`; +function waitUntilSearchIndexAvailable() { + // It is possible that the documenter.js script runs before the page + // has finished loading and documenterSearchIndex gets defined. + // So we need to wait until the search index actually loads before setting + // up all the search-related stuff. + if (typeof documenterSearchIndex !== "undefined") { + runSearchMainCode(); + } else { + console.warn("Search Index not available, waiting"); + setTimeout(waitUntilSearchIndexAvailable, 1000); + } } +// The actual entry point to the search code +waitUntilSearchIndexAvailable(); + }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { diff --git a/previews/PR514/assets/themes/catppuccin-frappe.css b/previews/PR514/assets/themes/catppuccin-frappe.css new file mode 100644 index 000000000..32e3f0082 --- /dev/null +++ b/previews/PR514/assets/themes/catppuccin-frappe.css @@ -0,0 +1 @@ +html.theme--catppuccin-frappe 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html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe 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.button.is-link.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link:focus:not(:active),html.theme--catppuccin-frappe .button.is-link.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(140,170,238,0.25)}html.theme--catppuccin-frappe .button.is-link:active,html.theme--catppuccin-frappe .button.is-link.is-active{background-color:#769aeb;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link{background-color:#8caaee;border-color:#8caaee;box-shadow:none}html.theme--catppuccin-frappe .button.is-link.is-inverted{background-color:#fff;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted:hover,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-link.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-focused{background-color:#8caaee;border-color:#8caaee;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-light{background-color:#edf2fc;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:hover,html.theme--catppuccin-frappe .button.is-link.is-light.is-hovered{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:active,html.theme--catppuccin-frappe .button.is-link.is-light.is-active{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:hover,html.theme--catppuccin-frappe .button.is-info.is-hovered{background-color:#78c4b9;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus,html.theme--catppuccin-frappe .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus:not(:active),html.theme--catppuccin-frappe .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe .button.is-info:active,html.theme--catppuccin-frappe .button.is-info.is-active{background-color:#6fc0b5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:#81c8be;box-shadow:none}html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-focused{background-color:#81c8be;border-color:#81c8be;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-success{background-color:#a6d189;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:hover,html.theme--catppuccin-frappe .button.is-success.is-hovered{background-color:#9fcd80;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:focus,html.theme--catppuccin-frappe .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:focus:not(:active),html.theme--catppuccin-frappe .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,209,137,0.25)}html.theme--catppuccin-frappe .button.is-success:active,html.theme--catppuccin-frappe .button.is-success.is-active{background-color:#98ca77;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-focused{background-color:#e5c890;border-color:#e5c890;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-frappe .content.is-normal{font-size:1rem}html.theme--catppuccin-frappe .content.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .content.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-frappe .icon.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-frappe .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-frappe .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-frappe .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-frappe .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-frappe .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-frappe div.icon-text{display:flex}html.theme--catppuccin-frappe .image,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-frappe .image img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-frappe .image img.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-frappe .image.is-fullwidth,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-frappe .image.is-square img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-frappe .image.is-square .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-frappe .image.is-1by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-frappe .image.is-1by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-frappe .image.is-5by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-frappe .image.is-5by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-frappe .image.is-4by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-frappe .image.is-4by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-frappe .image.is-3by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-frappe .image.is-5by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-frappe .image.is-5by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-frappe .image.is-16by9 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-frappe .image.is-16by9 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-frappe .image.is-2by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-frappe .image.is-2by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-frappe .image.is-3by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-frappe .image.is-3by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-frappe .image.is-4by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-frappe .image.is-4by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-frappe .image.is-3by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-frappe .image.is-3by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-frappe .image.is-2by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-frappe .image.is-2by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-frappe .image.is-3by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-frappe .image.is-9by16 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-frappe .image.is-9by16 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-frappe .image.is-1by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-frappe .image.is-1by2 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.modal-card{display:flex;flex-direction:column;max-height:calc(100vh - 40px);overflow:hidden;-ms-overflow-y:visible}html.theme--catppuccin-frappe .modal-card-head,html.theme--catppuccin-frappe .modal-card-foot{align-items:center;background-color:#292c3c;display:flex;flex-shrink:0;justify-content:flex-start;padding:20px;position:relative}html.theme--catppuccin-frappe .modal-card-head{border-bottom:1px solid #626880;border-top-left-radius:8px;border-top-right-radius:8px}html.theme--catppuccin-frappe .modal-card-title{color:#c6d0f5;flex-grow:1;flex-shrink:0;font-size:1.5rem;line-height:1}html.theme--catppuccin-frappe .modal-card-foot{border-bottom-left-radius:8px;border-bottom-right-radius:8px;border-top:1px solid #626880}html.theme--catppuccin-frappe .modal-card-foot .button:not(:last-child){margin-right:.5em}html.theme--catppuccin-frappe 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kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte 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.docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-primary.is-light,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-light.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-latte .button.is-primary.is-light.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:hover,html.theme--catppuccin-latte .button.is-link.is-hovered{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus,html.theme--catppuccin-latte .button.is-link.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus:not(:active),html.theme--catppuccin-latte .button.is-link.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-link:active,html.theme--catppuccin-latte .button.is-link.is-active{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:#1e66f5;box-shadow:none}html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-link.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-outlined.is-focused{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-link.is-light{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:hover,html.theme--catppuccin-latte .button.is-link.is-light.is-hovered{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:active,html.theme--catppuccin-latte .button.is-link.is-light.is-active{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:hover,html.theme--catppuccin-latte .button.is-info.is-hovered{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus,html.theme--catppuccin-latte .button.is-info.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus:not(:active),html.theme--catppuccin-latte .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(23,146,153,0.25)}html.theme--catppuccin-latte .button.is-info:active,html.theme--catppuccin-latte .button.is-info.is-active{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:#179299;box-shadow:none}html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;color:#179299}html.theme--catppuccin-latte .button.is-info.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-outlined.is-focused{background-color:#179299;border-color:#179299;color:#fff}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-info.is-light{background-color:#edfcfc;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:hover,html.theme--catppuccin-latte .button.is-info.is-light.is-hovered{background-color:#e2f9fb;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:active,html.theme--catppuccin-latte .button.is-info.is-light.is-active{background-color:#d7f7f9;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:hover,html.theme--catppuccin-latte .button.is-success.is-hovered{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus,html.theme--catppuccin-latte .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus:not(:active),html.theme--catppuccin-latte .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(64,160,43,0.25)}html.theme--catppuccin-latte .button.is-success:active,html.theme--catppuccin-latte .button.is-success.is-active{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:#40a02b;box-shadow:none}html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-outlined.is-focused{background-color:#40a02b;border-color:#40a02b;color:#fff}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table th:not([align]){text-align:inherit}html.theme--catppuccin-latte .content table thead td,html.theme--catppuccin-latte .content table thead th{border-width:0 0 2px;color:#41445a}html.theme--catppuccin-latte .content table tfoot td,html.theme--catppuccin-latte .content table tfoot th{border-width:2px 0 0;color:#41445a}html.theme--catppuccin-latte .content table tbody tr:last-child td,html.theme--catppuccin-latte .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-latte .content .tabs li+li{margin-top:0}html.theme--catppuccin-latte .content.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-latte .content.is-normal{font-size:1rem}html.theme--catppuccin-latte .content.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .content.is-large{font-size:1.5rem}html.theme--catppuccin-latte .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-latte .icon.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-latte .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-latte .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-latte .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-latte .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-latte .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-latte div.icon-text{display:flex}html.theme--catppuccin-latte .image,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-latte .image img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-latte .image img.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-latte .image.is-fullwidth,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-latte .image.is-square img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-latte .image.is-square .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-latte .image.is-1by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-latte .image.is-1by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-latte .image.is-5by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-latte .image.is-5by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-latte .image.is-4by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-latte .image.is-4by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-latte .image.is-3by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-latte .image.is-5by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-latte .image.is-5by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-latte .image.is-16by9 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-latte .image.is-16by9 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-latte .image.is-2by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-latte .image.is-2by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-latte .image.is-3by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-latte 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img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-latte .image.is-3by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-latte .image.is-9by16 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-latte .image.is-9by16 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-latte .image.is-1by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-latte .image.is-1by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-latte .image.is-1by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-latte .image.is-1by3 .has-ratio,html.theme--catppuccin-latte 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.button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:hover,html.theme--catppuccin-macchiato .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus,html.theme--catppuccin-macchiato .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus:not(:active),html.theme--catppuccin-macchiato .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-macchiato .button.is-black:active,html.theme--catppuccin-macchiato .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:hover,html.theme--catppuccin-macchiato .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus,html.theme--catppuccin-macchiato .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus:not(:active),html.theme--catppuccin-macchiato .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-macchiato .button.is-light:active,html.theme--catppuccin-macchiato .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato 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.content kbd.button{background-color:#363a4f;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:hover,html.theme--catppuccin-macchiato .content kbd.button:hover,html.theme--catppuccin-macchiato .button.is-dark.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-hovered{background-color:#313447;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus,html.theme--catppuccin-macchiato .content kbd.button:focus,html.theme--catppuccin-macchiato .button.is-dark.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus:not(:active),html.theme--catppuccin-macchiato .content kbd.button:focus:not(:active),html.theme--catppuccin-macchiato .button.is-dark.is-focused:not(:active),html.theme--catppuccin-macchiato .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(54,58,79,0.25)}html.theme--catppuccin-macchiato .button.is-dark:active,html.theme--catppuccin-macchiato .content kbd.button:active,html.theme--catppuccin-macchiato .button.is-dark.is-active,html.theme--catppuccin-macchiato .content kbd.button.is-active{background-color:#2c2f40;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark[disabled],html.theme--catppuccin-macchiato .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:#363a4f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-dark.is-inverted,html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted:hover,html.theme--catppuccin-macchiato 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.button.is-dark.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-focused{background-color:#363a4f;border-color:#363a4f;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content 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.button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato 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kbd.button.is-hovered{background-color:#2c2d3d;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus,html.theme--catppuccin-mocha .content kbd.button:focus,html.theme--catppuccin-mocha .button.is-dark.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus:not(:active),html.theme--catppuccin-mocha .content kbd.button:focus:not(:active),html.theme--catppuccin-mocha .button.is-dark.is-focused:not(:active),html.theme--catppuccin-mocha .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(49,50,68,0.25)}html.theme--catppuccin-mocha .button.is-dark:active,html.theme--catppuccin-mocha .content kbd.button:active,html.theme--catppuccin-mocha .button.is-dark.is-active,html.theme--catppuccin-mocha .content kbd.button.is-active{background-color:#262735;border-color:transparent;color:#fff}html.theme--catppuccin-mocha 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kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #313244 #313244 !important}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-primary,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink{background-color:#89b4fa;border-color:transparent;color:#fff}html.theme--catppuccin-mocha 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha 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.button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons 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.button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark 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.docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--documenter-dark .button.is-primary.is-outlined:focus,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-outlined.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#375a7f;border-color:#375a7f;color:#fff}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #375a7f #375a7f !important}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:focus::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-primary.is-outlined[disabled],html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-outlined,fieldset[disabled] html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink{background-color:transparent;border-color:#375a7f;box-shadow:none;color:#375a7f}html.theme--documenter-dark 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.button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter 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#documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--documenter-dark .image.is-3by4 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--documenter-dark .image.is-2by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--documenter-dark .image.is-2by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--documenter-dark .image.is-3by5 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--documenter-dark .image.is-3by5 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--documenter-dark .image.is-9by16 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--documenter-dark .image.is-9by16 .has-ratio,html.theme--documenter-dark #documenter 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.docs-logo>img.is-5by4{padding-top:80%}html.theme--documenter-dark .image.is-4by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--documenter-dark .image.is-3by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--documenter-dark .image.is-5by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--documenter-dark .image.is-16by9,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--documenter-dark .image.is-2by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--documenter-dark .image.is-3by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--documenter-dark .image.is-4by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--documenter-dark .image.is-3by4,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--documenter-dark .image.is-2by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--documenter-dark .image.is-3by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--documenter-dark .image.is-9by16,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--documenter-dark .image.is-1by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by2{padding-top:200%}html.theme--documenter-dark .image.is-1by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by3{padding-top:300%}html.theme--documenter-dark .image.is-16x16,html.theme--documenter-dark #documenter .docs-sidebar 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kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark 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rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark 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Theme: Default Description: Original highlight.js style Author: (c) Ivan Sagalaev diff --git a/previews/PR514/authors/index.html b/previews/PR514/authors/index.html index b431acade..7821c0644 100644 --- a/previews/PR514/authors/index.html +++ b/previews/PR514/authors/index.html @@ -1,2 +1,2 @@ -Authors · TrixiParticles.jl
GitHub

Authors

TrixiParticles.jl's development is coordinated by a group of principal developers, who are also its main contributors and who can be contacted in case of questions about TrixiParticles.jl. In addition, there are contributors who have provided substantial additions or modifications. Together, these two groups form "The TrixiParticles.jl Authors" as mentioned under License.

Principal Developers

Contributors

The following people contributed major additions or modifications to TrixiParticles.jl and are listed in alphabetical order:

  • Sven Berger
  • Erik Faulhaber
  • Gregor Gassner
  • Niklas Neher
  • Hendrik Ranocha
  • Michael Schlottke-Lakemper
+Authors · TrixiParticles.jl
GitHub

Authors

TrixiParticles.jl's development is coordinated by a group of principal developers, who are also its main contributors and who can be contacted in case of questions about TrixiParticles.jl. In addition, there are contributors who have provided substantial additions or modifications. Together, these two groups form "The TrixiParticles.jl Authors" as mentioned under License.

Principal Developers

Contributors

The following people contributed major additions or modifications to TrixiParticles.jl and are listed in alphabetical order:

  • Sven Berger
  • Erik Faulhaber
  • Gregor Gassner
  • Niklas Neher
  • Hendrik Ranocha
  • Michael Schlottke-Lakemper
diff --git a/previews/PR514/callbacks/index.html b/previews/PR514/callbacks/index.html index 4dd8fef74..e36a95f0c 100644 --- a/previews/PR514/callbacks/index.html +++ b/previews/PR514/callbacks/index.html @@ -1,11 +1,11 @@ -Callbacks · TrixiParticles.jl
GitHub

Callbacks

TrixiParticles.DensityReinitializationCallback — Type
DensityReinitializationCallback(; interval::Integer=0, dt=0.0)

Callback to reinitialize the density field when using ContinuityDensity.

Keywords

  • interval=0: Reinitialize the density every interval time steps.
  • dt: Reinitialize the density in regular intervals of dt in terms of integration time.
  • reinit_initial_solution: Reinitialize the initial solution (default=false)

References

  • Panizzo, Andrea, Giovanni Cuomo, and Robert A. Dalrymple. "3D-SPH simulation of landslide generated waves." In: Coastal Engineering 2006 (2007), pages 1503-1515. doi: 10.1142/9789812709554_0128
source
TrixiParticles.InfoCallback — Method
InfoCallback()

Create and return a callback that prints a human-readable summary of the simulation setup at the beginning of a simulation and then resets the timer. When the returned callback is executed directly, the current timer values are shown.

source
TrixiParticles.PostprocessCallback — Type
PostprocessCallback(; interval::Integer=0, dt=0.0, exclude_boundary=true, filename="values",
+Callbacks · TrixiParticles.jl
GitHub
Callbacks
TrixiParticles.DensityReinitializationCallback — Type
DensityReinitializationCallback(; interval::Integer=0, dt=0.0)
Callback to reinitialize the density field when using ContinuityDensity [42].
Keywords
  • interval=0:              Reinitialize the density every interval time steps.
  • dt:                      Reinitialize the density in regular intervals of dt in terms                            of integration time.
  • reinit_initial_solution: Reinitialize the initial solution (default=false)
source
TrixiParticles.InfoCallback — Method
InfoCallback()
Create and return a callback that prints a human-readable summary of the simulation setup at the beginning of a simulation and then resets the timer. When the returned callback is executed directly, the current timer values are shown.
source
TrixiParticles.PostprocessCallback — Type
PostprocessCallback(; interval::Integer=0, dt=0.0, exclude_boundary=true, filename="values",
                     output_directory="out", append_timestamp=false, write_csv=true,
                     write_json=true, write_file_interval=1, funcs...)
Create a callback to post-process simulation data at regular intervals. This callback allows for the execution of a user-defined function func at specified intervals during the simulation. The function is applied to the current state of the simulation, and its results can be saved or used for further analysis. The provided function cannot be anonymous as the function name will be used as part of the name of the value.
The callback can be triggered either by a fixed number of time steps (interval) or by a fixed interval of simulation time (dt).
Keywords
  • funcs...:   Functions to be executed at specified intervals during the simulation.               Each function must have the arguments (v, u, t, system),               and will be called for every system, where v and u are the               wrapped solution arrays for the corresponding system and t is               the current simulation time. Note that working with these v               and u arrays requires undocumented internal functions of               TrixiParticles. See Custom Quantities               for a list of pre-defined functions that can be used here.
  • interval=0: Specifies the number of time steps between each invocation of the callback.               If set to 0, the callback will not be triggered based on time steps.               Either interval or dt must be set to something larger than 0.
  • dt=0.0: Specifies the simulation time interval between each invocation of the callback.           If set to 0.0, the callback will not be triggered based on simulation time.           Either interval or dt must be set to something larger than 0.
  • exclude_boundary=true: If set to true, boundary particles will be excluded from the post-processing.
  • filename="values": The filename of the postprocessing files to be saved.
  • output_directory="out": The path where the results of the post-processing will be saved.
  • write_csv=true: If set to true, write a csv file.
  • write_json=true: If set to true, write a json file.
  • append_timestep=false: If set to true, the current timestamp will be added to the filename.
  • write_file_interval=1: Files will be written after every write_file_interval number of                          postprocessing execution steps. A value of 0 indicates that files                          are only written at the end of the simulation, eliminating I/O overhead.
Examples
# Create a callback that is triggered every 100 time steps
 postprocess_callback = PostprocessCallback(interval=100, example_quantity=kinetic_energy)
 
 # Create a callback that is triggered every 0.1 simulation time units
-postprocess_callback = PostprocessCallback(dt=0.1, example_quantity=kinetic_energy)
source
TrixiParticles.SolutionSavingCallback — Type
SolutionSavingCallback(; interval::Integer=0, dt=0.0, save_times=Array{Float64, 1}([]),
+postprocess_callback = PostprocessCallback(dt=0.1, example_quantity=kinetic_energy)
source
TrixiParticles.SolutionSavingCallback — Type
SolutionSavingCallback(; interval::Integer=0, dt=0.0, save_times=Array{Float64, 1}([]),
                        save_initial_solution=true, save_final_solution=true,
                        output_directory="out", append_timestamp=false, prefix="",
                        verbose=false, write_meta_data=true, max_coordinates=2^15,
@@ -16,5 +16,6 @@
 saving_callback = SolutionSavingCallback(dt=0.1)
 
 # Additionally store the kinetic energy of each system as "my_custom_quantity"
-saving_callback = SolutionSavingCallback(dt=0.1, my_custom_quantity=kinetic_energy)
source
TrixiParticles.StepsizeCallback — Method
StepsizeCallback(; cfl::Real)
Set the time step size according to a CFL condition if the time integration method isn't adaptive itself.
The current implementation is using the simplest form of CFL condition, which chooses a time step size that is constant during the simulation. The step size is therefore only applied once at the beginning of the simulation.
The step size $\Delta t$ is chosen as the minimum
\[    \Delta t = \min(\Delta t_\eta, \Delta t_a, \Delta t_c),\]
where
\[    \Delta t_\eta = 0.125 \, h^2 / \eta, \quad \Delta t_a = 0.25 \sqrt{h / \lVert g \rVert},
-    \quad \Delta t_c = \text{CFL} \, h / c,\]
with $\nu = \alpha h c / (2n + 4)$, where $\alpha$ is the parameter of the viscosity and $n$ is the number of dimensions.
Experimental implementation
This is an experimental feature and may change in future releases.
References
  • M. Antuono, A. Colagrossi, S. Marrone. "Numerical Diffusive Terms in Weakly-Compressible SPH Schemes." In: Computer Physics Communications 183, no. 12 (2012), pages 2570–80. doi: 10.1016/j.cpc.2012.07.006
  • S. Adami, X. Y. Hu, N. A. Adams. "A generalized wall boundary condition for smoothed particle hydrodynamics". In: Journal of Computational Physics 231, 21 (2012), pages 7057–7075. doi: 10.1016/J.JCP.2012.05.005
  • P. N. Sun, A. Colagrossi, S. Marrone, A. M. Zhang. "The δplus-SPH Model: Simple Procedures for a Further Improvement of the SPH Scheme." In: Computer Methods in Applied Mechanics and Engineering 315 (2017), pages 25–49. doi: 10.1016/j.cma.2016.10.028
  • M. Antuono, S. Marrone, A. Colagrossi, B. Bouscasse. "Energy Balance in the δ-SPH Scheme." In: Computer Methods in Applied Mechanics and Engineering 289 (2015), pages 209–26. doi: 10.1016/j.cma.2015.02.004
source
Custom Quantities
The following pre-defined custom quantities can be used with the SolutionSavingCallback and PostprocessCallback.

+saving_callback = SolutionSavingCallback(dt=0.1, my_custom_quantity=kinetic_energy)
source
TrixiParticles.SteadyStateReachedCallback — Type
SteadyStateReachedCallback(; interval::Integer=0, dt=0.0,
+                           interval_size::Integer=10, abstol=1.0e-8, reltol=1.0e-6)

Terminates the integration when the change of kinetic energy between time steps falls below the threshold specified by abstol + reltol * ekin, where ekin is the total kinetic energy of the simulation.

Keywords

  • interval=0: Check steady state condition every interval time steps.
  • dt=0.0: Check steady state condition in regular intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).
  • interval_size: The interval in which the change of the kinetic energy is considered. interval_size is a (integer) multiple of interval or dt.
  • abstol: Absolute tolerance.
  • reltol: Relative tolerance.
source
TrixiParticles.StepsizeCallback — Method
StepsizeCallback(; cfl::Real)

Set the time step size according to a CFL condition if the time integration method isn't adaptive itself.

The current implementation is using the simplest form of CFL condition, which chooses a time step size that is constant during the simulation. The step size is therefore only applied once at the beginning of the simulation.

The step size $\Delta t$ is chosen as the minimum

\[ \Delta t = \min(\Delta t_\eta, \Delta t_a, \Delta t_c),\]

where

\[ \Delta t_\eta = 0.125 \, h^2 / \eta, \quad \Delta t_a = 0.25 \sqrt{h / \lVert g \rVert}, + \quad \Delta t_c = \text{CFL} \, h / c,\]

with $\nu = \alpha h c / (2n + 4)$, where $\alpha$ is the parameter of the viscosity and $n$ is the number of dimensions.

Experimental implementation

This is an experimental feature and may change in future releases.

References

[21], [14], [43], [44]

source
TrixiParticles.UpdateCallback — Method
UpdateCallback(; interval::Integer, dt=0.0)

Callback to update quantities either at the end of every interval time steps or in intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).

Keywords

  • interval=1: Update quantities at the end of every interval time steps.
  • dt: Update quantities in regular intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).
source

Custom Quantities

The following pre-defined custom quantities can be used with the SolutionSavingCallback and PostprocessCallback.

diff --git a/previews/PR514/code_of_conduct/index.html b/previews/PR514/code_of_conduct/index.html index 2d2e0a829..900ccfab0 100644 --- a/previews/PR514/code_of_conduct/index.html +++ b/previews/PR514/code_of_conduct/index.html @@ -1,2 +1,2 @@ -Code of Conduct · TrixiParticles.jl
GitHub

Code of Conduct

Contributor Covenant Code of Conduct

Our Pledge

We as members, contributors, and leaders pledge to make participation in our community a harassment-free experience for everyone, regardless of age, body size, visible or invisible disability, ethnicity, sex characteristics, gender identity and expression, level of experience, education, socio-economic status, nationality, personal appearance, race, religion, or sexual identity and orientation.

We pledge to act and interact in ways that contribute to an open, welcoming, diverse, inclusive, and healthy community.

Our Standards

Examples of behavior that contributes to a positive environment for our community include:

  • Demonstrating empathy and kindness toward other people
  • Being respectful of differing opinions, viewpoints, and experiences
  • Giving and gracefully accepting constructive feedback
  • Accepting responsibility and apologizing to those affected by our mistakes, and learning from the experience
  • Focusing on what is best not just for us as individuals, but for the overall community

Examples of unacceptable behavior include:

  • The use of sexualized language or imagery, and sexual attention or advances of any kind
  • Trolling, insulting or derogatory comments, and personal or political attacks
  • Public or private harassment
  • Publishing others' private information, such as a physical or email address, without their explicit permission
  • Other conduct which could reasonably be considered inappropriate in a professional setting

Enforcement Responsibilities

Community leaders are responsible for clarifying and enforcing our standards of acceptable behavior and will take appropriate and fair corrective action in response to any behavior that they deem inappropriate, threatening, offensive, or harmful.

Community leaders have the right and responsibility to remove, edit, or reject comments, commits, code, wiki edits, issues, and other contributions that are not aligned to this Code of Conduct, and will communicate reasons for moderation decisions when appropriate.

Scope

This Code of Conduct applies within all community spaces, and also applies when an individual is officially representing the community in public spaces. Examples of representing our community include using an official e-mail address, posting via an official social media account, or acting as an appointed representative at an online or offline event.

Enforcement

Instances of abusive, harassing, or otherwise unacceptable behavior may be reported to Michael Schlottke-Lakemper, Sven Berger, or any other of the principal developers responsible for enforcement listed in Authors. All complaints will be reviewed and investigated promptly and fairly.

All community leaders are obligated to respect the privacy and security of the reporter of any incident.

Enforcement Guidelines

Community leaders will follow these Community Impact Guidelines in determining the consequences for any action they deem in violation of this Code of Conduct:

1. Correction

Community Impact: Use of inappropriate language or other behavior deemed unprofessional or unwelcome in the community.

Consequence: A private, written warning from community leaders, providing clarity around the nature of the violation and an explanation of why the behavior was inappropriate. A public apology may be requested.

2. Warning

Community Impact: A violation through a single incident or series of actions.

Consequence: A warning with consequences for continued behavior. No interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, for a specified period of time. This includes avoiding interactions in community spaces as well as external channels like social media. Violating these terms may lead to a temporary or permanent ban.

3. Temporary Ban

Community Impact: A serious violation of community standards, including sustained inappropriate behavior.

Consequence: A temporary ban from any sort of interaction or public communication with the community for a specified period of time. No public or private interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, is allowed during this period. Violating these terms may lead to a permanent ban.

4. Permanent Ban

Community Impact: Demonstrating a pattern of violation of community standards, including sustained inappropriate behavior, harassment of an individual, or aggression toward or disparagement of classes of individuals.

Consequence: A permanent ban from any sort of public interaction within the community.

Attribution

This Code of Conduct is adapted from the [Contributor Covenant][homepage], version 2.0, available at https://www.contributor-covenant.org/version/2/0/codeofconduct.html.

Community Impact Guidelines were inspired by Mozilla's code of conduct enforcement ladder.

[homepage]: https://www.contributor-covenant.org

For answers to common questions about this code of conduct, see the FAQ at https://www.contributor-covenant.org/faq. Translations are available at https://www.contributor-covenant.org/translations.

+Code of Conduct · TrixiParticles.jl
GitHub

Code of Conduct

Contributor Covenant Code of Conduct

Our Pledge

We as members, contributors, and leaders pledge to make participation in our community a harassment-free experience for everyone, regardless of age, body size, visible or invisible disability, ethnicity, sex characteristics, gender identity and expression, level of experience, education, socio-economic status, nationality, personal appearance, race, religion, or sexual identity and orientation.

We pledge to act and interact in ways that contribute to an open, welcoming, diverse, inclusive, and healthy community.

Our Standards

Examples of behavior that contributes to a positive environment for our community include:

  • Demonstrating empathy and kindness toward other people
  • Being respectful of differing opinions, viewpoints, and experiences
  • Giving and gracefully accepting constructive feedback
  • Accepting responsibility and apologizing to those affected by our mistakes, and learning from the experience
  • Focusing on what is best not just for us as individuals, but for the overall community

Examples of unacceptable behavior include:

  • The use of sexualized language or imagery, and sexual attention or advances of any kind
  • Trolling, insulting or derogatory comments, and personal or political attacks
  • Public or private harassment
  • Publishing others' private information, such as a physical or email address, without their explicit permission
  • Other conduct which could reasonably be considered inappropriate in a professional setting

Enforcement Responsibilities

Community leaders are responsible for clarifying and enforcing our standards of acceptable behavior and will take appropriate and fair corrective action in response to any behavior that they deem inappropriate, threatening, offensive, or harmful.

Community leaders have the right and responsibility to remove, edit, or reject comments, commits, code, wiki edits, issues, and other contributions that are not aligned to this Code of Conduct, and will communicate reasons for moderation decisions when appropriate.

Scope

This Code of Conduct applies within all community spaces, and also applies when an individual is officially representing the community in public spaces. Examples of representing our community include using an official e-mail address, posting via an official social media account, or acting as an appointed representative at an online or offline event.

Enforcement

Instances of abusive, harassing, or otherwise unacceptable behavior may be reported to Michael Schlottke-Lakemper, Sven Berger, or any other of the principal developers responsible for enforcement listed in Authors. All complaints will be reviewed and investigated promptly and fairly.

All community leaders are obligated to respect the privacy and security of the reporter of any incident.

Enforcement Guidelines

Community leaders will follow these Community Impact Guidelines in determining the consequences for any action they deem in violation of this Code of Conduct:

1. Correction

Community Impact: Use of inappropriate language or other behavior deemed unprofessional or unwelcome in the community.

Consequence: A private, written warning from community leaders, providing clarity around the nature of the violation and an explanation of why the behavior was inappropriate. A public apology may be requested.

2. Warning

Community Impact: A violation through a single incident or series of actions.

Consequence: A warning with consequences for continued behavior. No interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, for a specified period of time. This includes avoiding interactions in community spaces as well as external channels like social media. Violating these terms may lead to a temporary or permanent ban.

3. Temporary Ban

Community Impact: A serious violation of community standards, including sustained inappropriate behavior.

Consequence: A temporary ban from any sort of interaction or public communication with the community for a specified period of time. No public or private interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, is allowed during this period. Violating these terms may lead to a permanent ban.

4. Permanent Ban

Community Impact: Demonstrating a pattern of violation of community standards, including sustained inappropriate behavior, harassment of an individual, or aggression toward or disparagement of classes of individuals.

Consequence: A permanent ban from any sort of public interaction within the community.

Attribution

This Code of Conduct is adapted from the [Contributor Covenant][homepage], version 2.0, available at https://www.contributor-covenant.org/version/2/0/codeofconduct.html.

Community Impact Guidelines were inspired by Mozilla's code of conduct enforcement ladder.

[homepage]: https://www.contributor-covenant.org

For answers to common questions about this code of conduct, see the FAQ at https://www.contributor-covenant.org/faq. Translations are available at https://www.contributor-covenant.org/translations.

diff --git a/previews/PR514/contributing/index.html b/previews/PR514/contributing/index.html index 4aea13414..cd13610df 100644 --- a/previews/PR514/contributing/index.html +++ b/previews/PR514/contributing/index.html @@ -1,5 +1,5 @@ -Contributing · TrixiParticles.jl
GitHub

Contributing

TrixiParticles.jl is an open-source project and we are very happy to accept contributions from the community. Please feel free to open issues or submit patches (preferably as pull requests) any time. For planned larger contributions, it is often beneficial to get in contact with one of the principal developers first (see Authors).

TrixiParticles.jl and its contributions are licensed under the MIT license (see License). As a contributor, you certify that all your contributions are in conformance with the Developer Certificate of Origin (Version 1.1), which is reproduced below.

Developer Certificate of Origin (Version 1.1)

The following text was taken from https://developercertificate.org:

Developer Certificate of Origin
+Contributing · TrixiParticles.jl
GitHub
Contributing
TrixiParticles.jl is an open-source project and we are very happy to accept contributions from the community. Please feel free to open issues or submit patches (preferably as pull requests) any time. For planned larger contributions, it is often beneficial to get in contact with one of the principal developers first (see Authors).
TrixiParticles.jl and its contributions are licensed under the MIT license (see License). As a contributor, you certify that all your contributions are in conformance with the Developer Certificate of Origin (Version 1.1), which is reproduced below.
Developer Certificate of Origin (Version 1.1)
The following text was taken from https://developercertificate.org:
Developer Certificate of Origin
 Version 1.1
 
 Copyright (C) 2004, 2006 The Linux Foundation and its contributors.
@@ -35,4 +35,4 @@
     are public and that a record of the contribution (including all
     personal information I submit with it, including my sign-off) is
     maintained indefinitely and may be redistributed consistent with
-    this project or the open source license(s) involved.

+    this project or the open source license(s) involved.
diff --git a/previews/PR514/development/index.html b/previews/PR514/development/index.html index d983e2df3..21dfb3088 100644 --- a/previews/PR514/development/index.html +++ b/previews/PR514/development/index.html @@ -1,2 +1,2 @@ -Development · TrixiParticles.jl
GitHub

Development

Preview of the documentation

To generate the Documentation, first instantiate the docs environment by executing the following command from the TrixiParticles.jl root directory:

julia --project=docs -e "using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()"

This command only has to be run once. After that, maintain the docs environment as described under Installation.

With an instantiated docs environment, generate the docs with the following command (again from the TrixiParticles.jl root directory):

julia --project=docs --color=yes docs/make.jl

You can then open the generated files in docs/build with your webbrowser. Alternatively, run

python3 -m http.server -d docs/build

and open localhost:8000 in your webbrowser.

Release management

To create a new release for TrixiParticles.jl, perform the following steps:

  1. Make sure that all PRs and changes that you want to go into the release are merged to main and that the latest commit on main has passed all CI tests.
  2. Determine the currently released version of TrixiParticles.jl, e.g., on the release page. For this manual, we will assume that the latest release was v0.2.3.
  3. Decide on the next version number. We follow semantic versioning, thus each version is of the form vX.Y.Z where X is the major version, Y the minor version, and Z the patch version. In this manual, we assume that the major version is always 0, thus the decision process on the new version is as follows:
    • If the new release contains breaking changes (i.e., user code might not work as before without modifications), increase the minor version by one and set the patch version to zero. In our example, the new version should thus be v0.3.0.
    • If the new release only contains minor modifications and/or bug fixes, the minor version is kept as-is and the patch version is increased by one. In our example, the new version should thus be v0.2.4.
  4. Edit the version string in the Project.toml and set it to the new version. Push/merge this change to main.
  5. Go to GitHub and add a comment to the commit that you would like to become the new release (typically this will be the commit where you just updated the version). You can comment on a commit by going to the commit overview and clicking on the title of the commit. The comment should contain the following text:
    @JuliaRegistrator register
  6. Wait for the magic to happen! Specifically, JuliaRegistrator will create a new PR to the Julia registry with the new release information. After a grace period of ~15 minutes, this PR will be merged automatically. A short while after, TagBot will create a new release of TrixiParticles.jl in our GitHub repository.
  7. Once the new release has been created, the new version can be obtained through the Julia package manager as usual.
  8. To make sure people do not mistake the latest state of main as the latest release, we set the version in the Project.toml to a development version. The development version should be the latest released version, with the patch version incremented by one, and the -dev suffix added. For example, if you just released v0.3.0, the new development version should be v0.3.1-dev. If you just released v0.2.4, the new development version should be v0.2.5-dev.
+Development · TrixiParticles.jl
GitHub

Development

Preview of the documentation

To generate the Documentation, first instantiate the docs environment by executing the following command from the TrixiParticles.jl root directory:

julia --project=docs -e "using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()"

This command only has to be run once. After that, maintain the docs environment as described under Installation.

With an instantiated docs environment, generate the docs with the following command (again from the TrixiParticles.jl root directory):

julia --project=docs --color=yes docs/make.jl

You can then open the generated files in docs/build with your webbrowser. Alternatively, run

python3 -m http.server -d docs/build

and open localhost:8000 in your webbrowser.

Release management

To create a new release for TrixiParticles.jl, perform the following steps:

  1. Make sure that all PRs and changes that you want to go into the release are merged to main and that the latest commit on main has passed all CI tests.
  2. Determine the currently released version of TrixiParticles.jl, e.g., on the release page. For this manual, we will assume that the latest release was v0.2.3.
  3. Decide on the next version number. We follow semantic versioning, thus each version is of the form vX.Y.Z where X is the major version, Y the minor version, and Z the patch version. In this manual, we assume that the major version is always 0, thus the decision process on the new version is as follows:
    • If the new release contains breaking changes (i.e., user code might not work as before without modifications), increase the minor version by one and set the patch version to zero. In our example, the new version should thus be v0.3.0.
    • If the new release only contains minor modifications and/or bug fixes, the minor version is kept as-is and the patch version is increased by one. In our example, the new version should thus be v0.2.4.
  4. Edit the version string in the Project.toml and set it to the new version. Push/merge this change to main.
  5. Go to GitHub and add a comment to the commit that you would like to become the new release (typically this will be the commit where you just updated the version). You can comment on a commit by going to the commit overview and clicking on the title of the commit. The comment should contain the following text:
    @JuliaRegistrator register
  6. Wait for the magic to happen! Specifically, JuliaRegistrator will create a new PR to the Julia registry with the new release information. After a grace period of ~15 minutes, this PR will be merged automatically. A short while after, TagBot will create a new release of TrixiParticles.jl in our GitHub repository.
  7. Once the new release has been created, the new version can be obtained through the Julia package manager as usual.
  8. To make sure people do not mistake the latest state of main as the latest release, we set the version in the Project.toml to a development version. The development version should be the latest released version, with the patch version incremented by one, and the -dev suffix added. For example, if you just released v0.3.0, the new development version should be v0.3.1-dev. If you just released v0.2.4, the new development version should be v0.2.5-dev.
diff --git a/previews/PR514/examples/index.html b/previews/PR514/examples/index.html index a91a5a1b4..a15b89050 100644 --- a/previews/PR514/examples/index.html +++ b/previews/PR514/examples/index.html @@ -1,2 +1,2 @@ -Examples · TrixiParticles.jl +Examples · TrixiParticles.jl
GitHub

Examples

Fluid

Accelerated Tank 2D (fluid/accelerated_tank_2d.jl)

Dam Break 2D (fluid/dam_break_2d.jl)

Dam Break 3D (fluid/dam_break_3d.jl)

Falling Water Column (fluid/falling_water_column_2d.jl)

Hydrostatic Water Column (fluid/hydrostatic_water_column_*.jl)

Moving Wall (fluid/moving_wall_2d.jl)

Oscillating Drop (fluid/oscillating_drop_2d.jl)

Periodic Channel (fluid/periodic_channel_2d.jl)

Fluid Structure Interaction

Dam Break with Elastic Plate (fsi/dam_break_plate_2d.jl)

Falling Sphere 2D (fsi/falling_sphere_2d.jl)

Falling Spheres 2D (fsi/falling_spheres_2d.jl)

Structure Mechanics

Oscillating Beam (solid/oscillating_beam_2d.jl)

diff --git a/previews/PR514/general/density_calculators/index.html b/previews/PR514/general/density_calculators/index.html index 9abddc773..2d527307f 100644 --- a/previews/PR514/general/density_calculators/index.html +++ b/previews/PR514/general/density_calculators/index.html @@ -1,2 +1,2 @@ -Density Calculators · TrixiParticles.jl
GitHub

Density Calculators

TrixiParticles.ContinuityDensity — Type
ContinuityDensity()

Density calculator to integrate the density from the continuity equation

\[\frac{\mathrm{d}\rho_a}{\mathrm{d}t} = \sum_{b} m_b v_{ab} \cdot \nabla_{r_a} W(\Vert r_a - r_b \Vert, h),\]

where $\rho_a$ denotes the density of particle $a$ and $r_{ab} = r_a - r_b$ is the difference of the coordinates, $v_{ab} = v_a - v_b$ of the velocities of particles $a$ and $b$.

source
TrixiParticles.SummationDensity — Type
SummationDensity()

Density calculator to use the summation formula

\[\rho(r) = \sum_{b} m_b W(\Vert r - r_b \Vert, h),\]

for the density estimation, where $r_b$ denotes the coordinates and $m_b$ the mass of particle $b$.

source
+Density Calculators · TrixiParticles.jl
GitHub

Density Calculators

TrixiParticles.ContinuityDensity — Type
ContinuityDensity()

Density calculator to integrate the density from the continuity equation

\[\frac{\mathrm{d}\rho_a}{\mathrm{d}t} = \sum_{b} m_b v_{ab} \cdot \nabla_{r_a} W(\Vert r_a - r_b \Vert, h),\]

where $\rho_a$ denotes the density of particle $a$ and $r_{ab} = r_a - r_b$ is the difference of the coordinates, $v_{ab} = v_a - v_b$ of the velocities of particles $a$ and $b$.

source
TrixiParticles.SummationDensity — Type
SummationDensity()

Density calculator to use the summation formula

\[\rho(r) = \sum_{b} m_b W(\Vert r - r_b \Vert, h),\]

for the density estimation, where $r_b$ denotes the coordinates and $m_b$ the mass of particle $b$.

source
diff --git a/previews/PR514/general/initial_condition/index.html b/previews/PR514/general/initial_condition/index.html index 2f26a9ece..8909553c2 100644 --- a/previews/PR514/general/initial_condition/index.html +++ b/previews/PR514/general/initial_condition/index.html @@ -1,6 +1,6 @@ -Initial Condition and Setups · TrixiParticles.jl
GitHub

Initial Condition

TrixiParticles.InitialCondition — Type
InitialCondition(; coordinates, density, velocity=zeros(size(coordinates, 1)),
-                 mass=nothing, pressure=0.0, particle_spacing=-1.0)

Struct to hold the initial configuration of the particles.

The following setups return InitialConditions for commonly used setups:

InitialConditions support the set operations union, setdiff and intersect in order to build more complex geometries.

Arguments

  • coordinates: An array where the $i$-th column holds the coordinates of particle $i$.
  • density: Either a vector holding the density of each particle, or a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles.

Keywords

  • velocity: Either an array where the $i$-th column holds the velocity of particle $i$, or a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.
  • mass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a vector holding the mass of each particle, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.
  • pressure: Either a vector holding the pressure of each particle, or a function mapping each particle's coordinates to its pressure, or a scalar for a constant pressure over all particles. This is optional and only needed when using the EntropicallyDampedSPHSystem.
  • particle_spacing: The spacing between the particles. This is a scalar, as the spacing is assumed to be uniform. This is only needed when using set operations on the InitialCondition or for automatic mass calculation.

Examples

# Rectangle filled with particles
+Initial Condition and Setups · TrixiParticles.jl
GitHub
Initial Condition
TrixiParticles.InitialCondition — Type
InitialCondition(; coordinates, density, velocity=zeros(size(coordinates, 1)),
+                 mass=nothing, pressure=0.0, particle_spacing=-1.0)
Struct to hold the initial configuration of the particles.
The following setups return InitialConditions for commonly used setups:
InitialConditions support the set operations union, setdiff and intersect in order to build more complex geometries.
Arguments
  • coordinates: An array where the $i$-th column holds the coordinates of particle $i$.
  • density:     Either a vector holding the density of each particle,                or a function mapping each particle's coordinates to its density,                or a scalar for a constant density over all particles.
Keywords
  • velocity:   Either an array where the $i$-th column holds the velocity of particle $i$,               or a function mapping each particle's coordinates to its velocity,               or, for a constant fluid velocity, a vector holding this velocity.               Velocity is constant zero by default.
  • mass:       Either nothing (default) to automatically compute particle mass from particle               density and spacing, or a vector holding the mass of each particle,               or a function mapping each particle's coordinates to its mass,               or a scalar for a constant mass over all particles.
  • pressure:   Either a vector holding the pressure of each particle,               or a function mapping each particle's coordinates to its pressure,               or a scalar for a constant pressure over all particles. This is optional and               only needed when using the EntropicallyDampedSPHSystem.
  • particle_spacing: The spacing between the particles. This is a scalar, as the spacing                     is assumed to be uniform. This is only needed when using                     set operations on the InitialCondition or for automatic mass calculation.
Examples
# Rectangle filled with particles
 initial_condition = RectangularShape(0.1, (3, 4), (-1.0, 1.0), density=1.0)
 
 # Two spheres in one initial condition
@@ -28,9 +28,15 @@
 initial_condition = InitialCondition(; coordinates, velocity, mass, density)
 
 # With functions
-initial_condition = InitialCondition(; coordinates, velocity=x -> 2x, mass=1.0, density=1000.0)
source
Setups
TrixiParticles.extrude_geometry — Method
extrude_geometry(geometry; particle_spacing, direction, n_extrude::Integer,
+initial_condition = InitialCondition(; coordinates, velocity=x -> 2x, mass=1.0, density=1000.0)
source
Setups
TrixiParticles.ComplexShape — Method
ComplexShape(geometry::Union{TriangleMesh, Polygon}; particle_spacing, density,
+             pressure=0.0, mass=nothing, velocity=zeros(ndims(geometry)),
+             point_in_geometry_algorithm=WindingNumberJacobson(; geometry,
+                                                               hierarchical_winding=false,
+                                                               winding_number_factor=sqrt(eps())),
+             grid_offset::Real=0.0, max_nparticles=10^7,
+             pad_initial_particle_grid=2particle_spacing)
Sample a complex geometry with particles. Returns an InitialCondition. Note that an initial particle grid is generated inside the bounding box of the geometry. A point_in_geometry_algorithm checks if particles are inside the geometry or not. For more information about the method see WindingNumberJacobson or WindingNumberHormann.
Arguments
Keywords
  • particle_spacing:   Spacing between the particles.
  • density:            Either a function mapping each particle's coordinates to its density,                       or a scalar for a constant density over all particles.
  • velocity:           Either a function mapping each particle's coordinates to its velocity,                       or, for a constant fluid velocity, a vector holding this velocity.                       Velocity is constant zero by default.
  • mass:               Either nothing (default) to automatically compute particle mass from particle                       density and spacing, or a function mapping each particle's coordinates to its mass,                       or a scalar for a constant mass over all particles.
  • pressure:           Scalar to set the pressure of all particles to this value.                       This is only used by the EntropicallyDampedSPHSystem and                       will be overwritten when using an initial pressure function in the system.
  • point_in_geometry_algorithm: Algorithm for sampling the complex geometry with particles.                                It basically checks whether a particle is inside an object or not.                                For more information see WindingNumberJacobson or WindingNumberHormann
  • grid_offset: Offset of the initial particle grid of the bounding box of the geometry.
  • max_nparticles: Maximum number of particles in the initial particle grid.                   This is only used to avoid accidentally choosing a particle_spacing                   that is too small for the scale of the geometry.
  • pad_initial_particle_grid: Padding of the initial particle grid.
Experimental Implementation
This is an experimental feature and may change in any future releases.
source
TrixiParticles.extrude_geometry — Method
extrude_geometry(geometry; particle_spacing, direction, n_extrude::Integer,
                  velocity=zeros(length(direction)),
-                 mass=nothing, density=nothing, pressure=0.0)
Extrude either a line, a plane or a shape along a specific direction.
Arguments
  • geometry:           Either particle coordinates or an InitialCondition                       defining a 2D shape to extrude to a 3D volume, or two 2D points                       defining a line to extrude to a plane in 2D, or three 3D points defining                       a parallelogram to extrude to a parallelepiped.
Keywords
  • particle_spacing:   Spacing between the particles. Can be omitted when geometry is an                       InitialCondition (unless geometry.particle_spacing == -1).
  • direction:          A vector that specifies the direction in which to extrude.
  • n_extrude:          Number of layers of particles created in the direction of extrusion.
  • velocity:           Either a function mapping each particle's coordinates to its velocity,                       or, for a constant fluid velocity, a vector holding this velocity.                       Velocity is constant zero by default.
  • mass:               Either nothing (default) to automatically compute particle mass from particle                       density and spacing, or a function mapping each particle's coordinates to its mass,                       or a scalar for a constant mass over all particles.
  • density:            Either a function mapping each particle's coordinates to its density,                       or a scalar for a constant density over all particles.                       Obligatory when not using a state equation. Cannot be used together with                       state_equation.
  • pressure:           Scalar to set the pressure of all particles to this value.                       This is only used by the EntropicallyDampedSPHSystem and                       will be overwritten when using an initial pressure function in the system.                       Cannot be used together with hydrostatic pressure gradient.
  • tlsph:              With the TotalLagrangianSPHSystem, particles need to be placed                       on the boundary of the shape and not one particle radius away, as for fluids.                       When tlsph=true, particles will be placed on the boundary of the shape.
Examples
# Extrude a line in 2D to a plane in 2D
+                 mass=nothing, density=nothing, pressure=0.0)
Extrude either a line, a plane or a shape along a specific direction. Returns an InitialCondition.
Arguments
  • geometry:           Either particle coordinates or an InitialCondition                       defining a 2D shape to extrude to a 3D volume, or two 2D points                       $(A, B)$ defining the interval $[A, B]$ to extrude to a plane                       in 2D, or three 3D points $(A, B, C)$ defining the parallelogram                       spanned by the vectors $\widehat{AB}$ and $\widehat {AC}$ to extrude                       to a parallelepiped.
Keywords
  • particle_spacing:   Spacing between the particles. Can be omitted when geometry is an                       InitialCondition (unless geometry.particle_spacing == -1).
  • direction:          A vector that specifies the direction in which to extrude.
  • n_extrude:          Number of layers of particles created in the direction of extrusion.
  • velocity:           Either a function mapping each particle's coordinates to its velocity,                       or, for a constant fluid velocity, a vector holding this velocity.                       Velocity is constant zero by default.
  • mass:               Either nothing (default) to automatically compute particle mass from particle                       density and spacing, or a function mapping each particle's coordinates to its mass,                       or a scalar for a constant mass over all particles.
  • density:            Either a function mapping each particle's coordinates to its density,                       or a scalar for a constant density over all particles.
  • pressure:           Scalar to set the pressure of all particles to this value.                       This is only used by the EntropicallyDampedSPHSystem and                       will be overwritten when using an initial pressure function in the system.
  • tlsph:              With the TotalLagrangianSPHSystem, particles need to be placed                       on the boundary of the shape and not one particle radius away, as for fluids.                       When tlsph=true, particles will be placed on the boundary of the shape.
Examples
# Extrude a line in 2D to a plane in 2D
 p1 = [0.0, 0.0]
 p2 = [1.0, 1.0]
 
@@ -53,11 +59,11 @@
 
 direction = [0.0, 0.0, 1.0]
 
-shape = extrude_geometry(shape; direction, particle_spacing=0.1, n_extrude=4, density=1000.0)
Experimental Implementation
This is an experimental feature and may change in any future releases.
source
TrixiParticles.RectangularShape — Method
RectangularShape(particle_spacing, n_particles_per_dimension, min_coordinates;
+shape = extrude_geometry(shape; direction, particle_spacing=0.1, n_extrude=4, density=1000.0)
Experimental Implementation
This is an experimental feature and may change in any future releases.
source
TrixiParticles.RectangularShape — Method
RectangularShape(particle_spacing, n_particles_per_dimension, min_coordinates;
                  velocity=zeros(length(n_particles_per_dimension)),
                  mass=nothing, density=nothing, pressure=0.0,
                  acceleration=nothing, state_equation=nothing,
-                 tlsph=false, loop_order=nothing)
Rectangular shape filled with particles. Returns an InitialCondition.
Arguments
  • particle_spacing:           Spacing between the particles.
  • n_particles_per_dimension:  Tuple containing the number of particles in x, y and z                               (only 3D) direction, respectively.
  • min_coordinates:            Coordinates of the corner in negative coordinate directions.
Keywords
  • velocity:       Either a function mapping each particle's coordinates to its velocity,                   or, for a constant fluid velocity, a vector holding this velocity.                   Velocity is constant zero by default.
  • mass:           Either nothing (default) to automatically compute particle mass from particle                   density and spacing, or a function mapping each particle's coordinates to its mass,                   or a scalar for a constant mass over all particles.
  • density:        Either a function mapping each particle's coordinates to its density,                   or a scalar for a constant density over all particles.                   Obligatory when not using a state equation. Cannot be used together with                   state_equation.
  • pressure:       Scalar to set the pressure of all particles to this value.                   This is only used by the EntropicallyDampedSPHSystem and                   will be overwritten when using an initial pressure function in the system.                   Cannot be used together with hydrostatic pressure gradient.
  • acceleration:   In order to initialize particles with a hydrostatic pressure gradient,                   an acceleration vector can be passed. Note that only accelerations                   in one coordinate direction and no diagonal accelerations are supported.                   This will only change the pressure of the particles. When using the                   WeaklyCompressibleSPHSystem, pass a state_equation as well                   to initialize the particles with the corresponding density and mass.                   When using the EntropicallyDampedSPHSystem, the pressure                   will be overwritten when using an initial pressure function in the system.                   This cannot be used together with the pressure keyword argument.
  • state_equation: When calculating a hydrostatic pressure gradient by setting acceleration,                   the state_equation will be used to set the corresponding density.                   Cannot be used together with density.
  • tlsph:          With the TotalLagrangianSPHSystem, particles need to be placed                   on the boundary of the shape and not one particle radius away, as for fluids.                   When tlsph=true, particles will be placed on the boundary of the shape.
Examples
# 2D
+                 tlsph=false, loop_order=nothing)
Rectangular shape filled with particles. Returns an InitialCondition.
Arguments
  • particle_spacing:           Spacing between the particles.
  • n_particles_per_dimension:  Tuple containing the number of particles in x, y and z                               (only 3D) direction, respectively.
  • min_coordinates:            Coordinates of the corner in negative coordinate directions.
Keywords
  • velocity:       Either a function mapping each particle's coordinates to its velocity,                   or, for a constant fluid velocity, a vector holding this velocity.                   Velocity is constant zero by default.
  • mass:           Either nothing (default) to automatically compute particle mass from particle                   density and spacing, or a function mapping each particle's coordinates to its mass,                   or a scalar for a constant mass over all particles.
  • density:        Either a function mapping each particle's coordinates to its density,                   or a scalar for a constant density over all particles.                   Obligatory when not using a state equation. Cannot be used together with                   state_equation.
  • pressure:       Scalar to set the pressure of all particles to this value.                   This is only used by the EntropicallyDampedSPHSystem and                   will be overwritten when using an initial pressure function in the system.                   Cannot be used together with hydrostatic pressure gradient.
  • acceleration:   In order to initialize particles with a hydrostatic pressure gradient,                   an acceleration vector can be passed. Note that only accelerations                   in one coordinate direction and no diagonal accelerations are supported.                   This will only change the pressure of the particles. When using the                   WeaklyCompressibleSPHSystem, pass a state_equation as well                   to initialize the particles with the corresponding density and mass.                   When using the EntropicallyDampedSPHSystem, the pressure                   will be overwritten when using an initial pressure function in the system.                   This cannot be used together with the pressure keyword argument.
  • state_equation: When calculating a hydrostatic pressure gradient by setting acceleration,                   the state_equation will be used to set the corresponding density.                   Cannot be used together with density.
  • tlsph:          With the TotalLagrangianSPHSystem, particles need to be placed                   on the boundary of the shape and not one particle radius away, as for fluids.                   When tlsph=true, particles will be placed on the boundary of the shape.
  • coordinates_perturbation: Add a small random displacement to the particle positions,                             where the amplitude is coordinates_perturbation * particle_spacing.
Examples
# 2D
 rectangular = RectangularShape(particle_spacing, (5, 4), (1.0, 2.0), density=1000.0)
 
 # 2D with hydrostatic pressure gradient.
@@ -67,7 +73,7 @@
                                acceleration=(0.0, -9.81), state_equation=state_equation)
 
 # 3D
-rectangular = RectangularShape(particle_spacing, (5, 4, 7), (1.0, 2.0, 3.0), density=1000.0)
source
TrixiParticles.RectangularTank — Type
RectangularTank(particle_spacing, fluid_size, tank_size, fluid_density;
+rectangular = RectangularShape(particle_spacing, (5, 4, 7), (1.0, 2.0, 3.0), density=1000.0)
source
TrixiParticles.RectangularTank — Type
RectangularTank(particle_spacing, fluid_size, tank_size, fluid_density;
                 velocity=zeros(length(fluid_size)), fluid_mass=nothing,
                 pressure=0.0,
                 acceleration=nothing, state_equation=nothing,
@@ -89,7 +95,7 @@
 # 3D
 setup = RectangularTank(particle_spacing, (water_width, water_height, water_depth),
                         (container_width, container_height, container_depth), fluid_density,
-                        n_layers=2)
See also: reset_wall!.
source
TrixiParticles.reset_wall! — Method
reset_wall!(rectangular_tank::RectangularTank, reset_faces, positions)
The selected walls of the tank will be placed at the new positions.
Arguments
  • reset_faces: Boolean tuple of 4 (in 2D) or 6 (in 3D) dimensions, similar to faces in RectangularTank.
  • positions: Tuple of new positions
Warning
There are overlapping particles when adjacent walls are moved inwards simultaneously.
source
TrixiParticles.RoundSphere — Type
RoundSphere(; start_angle=0.0, end_angle=2π)
Construct a sphere (or sphere segment) by nesting perfectly round concentric spheres. The resulting ball will be perfectly round, but will not have a regular inner structure.
Keywords
  • start_angle: The starting angle of the sphere segment in radians. It determines the                beginning point of the segment. The default is set to 0.0 representing                the positive x-axis.
  • end_angle: The ending angle of the sphere segment in radians. It defines the termination              point of the segment. The default is set to 2pi, completing a full sphere.
Usage
See SphereShape on how to use this.
Warning
The sphere segment is intended for 2D geometries and hollow spheres. If used for filled spheres or in a 3D context, results may not be accurate.
source
TrixiParticles.VoxelSphere — Type
VoxelSphere()
Construct a sphere of voxels (where particles are placed in the voxel center) with a regular inner structure but corners on the surface. Essentially, a grid of particles is generated and all particles outside the sphere are removed. The resulting sphere will have a perfect inner structure, but is not perfectly round, as it will have corners (like a sphere in Minecraft).
Usage
See SphereShape on how to use this.
source
TrixiParticles.reset_wall! — Method
reset_wall!(rectangular_tank::RectangularTank, reset_faces, positions)
The selected walls of the tank will be placed at the new positions.
Arguments
  • reset_faces: Boolean tuple of 4 (in 2D) or 6 (in 3D) dimensions, similar to faces in RectangularTank.
  • positions: Tuple of new positions
Warning
There are overlapping particles when adjacent walls are moved inwards simultaneously.
source
TrixiParticles.RoundSphere — Type
RoundSphere(; start_angle=0.0, end_angle=2π)
Construct a sphere (or sphere segment) by nesting perfectly round concentric spheres. The resulting ball will be perfectly round, but will not have a regular inner structure.
Keywords
  • start_angle: The starting angle of the sphere segment in radians. It determines the                beginning point of the segment. The default is set to 0.0 representing                the positive x-axis.
  • end_angle: The ending angle of the sphere segment in radians. It defines the termination              point of the segment. The default is set to 2pi, completing a full sphere.
Usage
See SphereShape on how to use this.
Warning
The sphere segment is intended for 2D geometries and hollow spheres. If used for filled spheres or in a 3D context, results may not be accurate.
source
TrixiParticles.VoxelSphere — Type
VoxelSphere()
Construct a sphere of voxels (where particles are placed in the voxel center) with a regular inner structure but corners on the surface. Essentially, a grid of particles is generated and all particles outside the sphere are removed. The resulting sphere will have a perfect inner structure, but is not perfectly round, as it will have corners (like a sphere in Minecraft).
Usage
See SphereShape on how to use this.
source
TrixiParticles.SphereShape — Method
SphereShape(particle_spacing, radius, center_position, density;
             sphere_type=VoxelSphere(), n_layers=-1, layer_outwards=false,
             cutout_min=(0.0, 0.0), cutout_max=(0.0, 0.0), tlsph=false,
             velocity=zeros(length(center_position)), mass=nothing, pressure=0.0)
Generate a sphere that is either completely filled (by default) or hollow (by passing n_layers).
With the sphere type VoxelSphere, a sphere of voxels (where particles are placed in the voxel center) with a regular inner structure but corners on the surface is created. Essentially, a grid of particles is generated and all particles outside the sphere are removed. With the sphere type RoundSphere, a perfectly round sphere with an imperfect inner structure is created.
A cuboid can be cut out of the sphere by specifying the two corners in negative and positive coordinate directions as cutout_min and cutout_max.
Arguments
  • particle_spacing:   Spacing between the particles.
  • radius:             Radius of the sphere.
  • center_position:    The coordinates of the center of the sphere.
  • density:            Either a function mapping each particle's coordinates to its density,                       or a scalar for a constant density over all particles.
Keywords
  • sphere_type:    Either VoxelSphere or RoundSphere (see                   explanation above).
  • n_layers:       Set to an integer greater than zero to generate a hollow sphere,                   where the shell consists of n_layers layers.
  • layer_outwards: When set to false (by default), radius is the outer radius                   of the sphere. When set to true, radius is the inner radius                   of the sphere. This is only used when n_layers > 0.
  • cutout_min:     Corner in negative coordinate directions of a cuboid that is to be                   cut out of the sphere.
  • cutout_max:     Corner in positive coordinate directions of a cuboid that is to be                   cut out of the sphere.
  • tlsph:          With the TotalLagrangianSPHSystem, particles need to be placed                   on the boundary of the shape and not one particle radius away, as for fluids.                   When tlsph=true, particles will be placed on the boundary of the shape.
  • velocity:   Either a function mapping each particle's coordinates to its velocity,               or, for a constant fluid velocity, a vector holding this velocity.               Velocity is constant zero by default.
  • mass:       Either nothing (default) to automatically compute particle mass from particle               density and spacing, or a function mapping each particle's coordinates to its mass,               or a scalar for a constant mass over all particles.
  • pressure:   Either a function mapping each particle's coordinates to its pressure,               or a scalar for a constant pressure over all particles. This is optional and               only needed when using the EntropicallyDampedSPHSystem.
Examples
# Filled circle with radius 0.5, center in (0.2, 0.4) and a particle spacing of 0.1
@@ -117,4 +123,4 @@
 SphereShape(0.1, 0.5, (0.2, 0.4, 0.3), 1000.0)
 
 # Same as before, but perfectly round
-SphereShape(0.1, 0.5, (0.2, 0.4, 0.3), 1000.0, sphere_type=RoundSphere())
source

+SphereShape(0.1, 0.5, (0.2, 0.4, 0.3), 1000.0, sphere_type=RoundSphere())
source
diff --git a/previews/PR514/general/interpolation/index.html b/previews/PR514/general/interpolation/index.html index 0923f9ba0..a6f964243 100644 --- a/previews/PR514/general/interpolation/index.html +++ b/previews/PR514/general/interpolation/index.html @@ -1,18 +1,18 @@ -Interpolation · TrixiParticles.jl
GitHub

Interpolation

TrixiParticles.interpolate_line — Method
interpolate_line(start, end_, n_points, semi, ref_system, sol; endpoint=true,
+Interpolation · TrixiParticles.jl
GitHub
Interpolation
TrixiParticles.interpolate_line — Method
interpolate_line(start, end_, n_points, semi, ref_system, sol; endpoint=true,
                  smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,
                  clip_negative_pressure=false)
Interpolates properties along a line in a TrixiParticles simulation. The line interpolation is accomplished by generating a series of evenly spaced points between start and end_. If endpoint is false, the line is interpolated between the start and end points, but does not include these points.
See also: interpolate_point, interpolate_plane_2d,           interpolate_plane_2d_vtk, interpolate_plane_3d.
Arguments
  • start:      The starting point of the line.
  • end_:       The ending point of the line.
  • n_points:   The number of points to interpolate along the line.
  • semi:       The semidiscretization used for the simulation.
  • ref_system: The reference system for the interpolation.
  • sol:        The solution state from which the properties are interpolated.
Keywords
  • endpoint=true: A boolean to include (true) or exclude (false) the end point in the interpolation.
  • smoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.
  • cut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point                     is "closer" to the boundary than to the fluid in a kernel-weighted sense.                     Or, in more detail, when the boundary has more influence than the fluid                     on the density summation in this point, i.e., when the boundary particles                     add more kernel-weighted mass than the fluid particles.
  • clip_negative_pressure=false: One common approach in SPH models is to clip negative pressure                                 values, but this is unphysical. Instead we clip here during                                 interpolation thus only impacting the local interpolated value.
Returns
  • A NamedTuple of arrays containing interpolated properties at each point along the line.
Note
  • This function is particularly useful for analyzing gradients or creating visualizations along a specified line in the SPH simulation domain.
  • The interpolation accuracy is subject to the density of particles and the chosen smoothing length.
  • With cut_off_bnd, a density-based estimation of the surface is used which is not as accurate as a real surface reconstruction.
Examples
# Interpolating along a line from [1.0, 0.0] to [1.0, 1.0] with 5 points
-results = interpolate_line([1.0, 0.0], [1.0, 1.0], 5, semi, ref_system, sol)
source
TrixiParticles.interpolate_plane_2d — Method
interpolate_plane_2d(min_corner, max_corner, resolution, semi, ref_system, sol;
+results = interpolate_line([1.0, 0.0], [1.0, 1.0], 5, semi, ref_system, sol)
source
TrixiParticles.interpolate_plane_2d — Method
interpolate_plane_2d(min_corner, max_corner, resolution, semi, ref_system, sol;
                      smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,
                      clip_negative_pressure=false)
Interpolates properties along a plane in a TrixiParticles simulation. The region for interpolation is defined by its lower left and top right corners, with a specified resolution determining the density of the interpolation points.
The function generates a grid of points within the defined region, spaced uniformly according to the given resolution.
See also: interpolate_plane_2d_vtk, interpolate_plane_3d,           interpolate_line, interpolate_point.
Arguments
  • min_corner: The lower left corner of the interpolation region.
  • max_corner: The top right corner of the interpolation region.
  • resolution: The distance between adjacent interpolation points in the grid.
  • semi:       The semidiscretization used for the simulation.
  • ref_system: The reference system for the interpolation.
  • sol:        The solution state from which the properties are interpolated.
Keywords
  • smoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.
  • cut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point                     is "closer" to the boundary than to the fluid in a kernel-weighted sense.                     Or, in more detail, when the boundary has more influence than the fluid                     on the density summation in this point, i.e., when the boundary particles                     add more kernel-weighted mass than the fluid particles.
  • clip_negative_pressure=false: One common approach in SPH models is to clip negative pressure                                 values, but this is unphysical. Instead we clip here during                                 interpolation thus only impacting the local interpolated value.
Returns
  • A NamedTuple of arrays containing interpolated properties at each point within the plane.
Note
  • The interpolation accuracy is subject to the density of particles and the chosen smoothing length.
  • With cut_off_bnd, a density-based estimation of the surface is used, which is not as accurate as a real surface reconstruction.
Examples
# Interpolating across a plane from [0.0, 0.0] to [1.0, 1.0] with a resolution of 0.2
-results = interpolate_plane_2d([0.0, 0.0], [1.0, 1.0], 0.2, semi, ref_system, sol)
source
TrixiParticles.interpolate_plane_2d_vtk — Method
interpolate_plane_2d_vtk(min_corner, max_corner, resolution, semi, ref_system, sol;
                          smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,
                          clip_negative_pressure=false, output_directory="out", filename="plane")
Interpolates properties along a plane in a TrixiParticles simulation and exports the result as a VTI file. The region for interpolation is defined by its lower left and top right corners, with a specified resolution determining the density of the interpolation points.
The function generates a grid of points within the defined region, spaced uniformly according to the given resolution.
See also: interpolate_plane_2d, interpolate_plane_3d,           interpolate_line, interpolate_point.
Arguments
  • min_corner: The lower left corner of the interpolation region.
  • max_corner: The top right corner of the interpolation region.
  • resolution: The distance between adjacent interpolation points in the grid.
  • semi:       The semidiscretization used for the simulation.
  • ref_system: The reference system for the interpolation.
  • sol:        The solution state from which the properties are interpolated.
Keywords
  • smoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.
  • output_directory="out": Directory to save the VTI file.
  • filename="plane":       Name of the VTI file.
  • cut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point                     is "closer" to the boundary than to the fluid in a kernel-weighted sense.                     Or, in more detail, when the boundary has more influence than the fluid                     on the density summation in this point, i.e., when the boundary particles                     add more kernel-weighted mass than the fluid particles.
  • clip_negative_pressure=false: One common approach in SPH models is to clip negative pressure                                 values, but this is unphysical. Instead we clip here during                                 interpolation thus only impacting the local interpolated value.
Note
  • The interpolation accuracy is subject to the density of particles and the chosen smoothing length.
  • With cut_off_bnd, a density-based estimation of the surface is used, which is not as accurate as a real surface reconstruction.
Examples
# Interpolating across a plane from [0.0, 0.0] to [1.0, 1.0] with a resolution of 0.2
-results = interpolate_plane_2d([0.0, 0.0], [1.0, 1.0], 0.2, semi, ref_system, sol)
source
TrixiParticles.interpolate_plane_3d — Method
interpolate_plane_3d(point1, point2, point3, resolution, semi, ref_system, sol;
+results = interpolate_plane_2d([0.0, 0.0], [1.0, 1.0], 0.2, semi, ref_system, sol)
source
TrixiParticles.interpolate_plane_3d — Method
interpolate_plane_3d(point1, point2, point3, resolution, semi, ref_system, sol;
                      smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,
                      clip_negative_pressure=false)
Interpolates properties along a plane in a 3D space in a TrixiParticles simulation. The plane for interpolation is defined by three points in 3D space, with a specified resolution determining the density of the interpolation points.
The function generates a grid of points on a parallelogram within the plane defined by the three points, spaced uniformly according to the given resolution.
See also: interpolate_plane_2d, interpolate_plane_2d_vtk,           interpolate_line, interpolate_point.
Arguments
  • point1:     The first point defining the plane.
  • point2:     The second point defining the plane.
  • point3:     The third point defining the plane. The points must not be collinear.
  • resolution: The distance between adjacent interpolation points in the grid.
  • semi:       The semidiscretization used for the simulation.
  • ref_system: The reference system for the interpolation.
  • sol:        The solution state from which the properties are interpolated.
Keywords
  • smoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.
  • cut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point                     is "closer" to the boundary than to the fluid in a kernel-weighted sense.                     Or, in more detail, when the boundary has more influence than the fluid                     on the density summation in this point, i.e., when the boundary particles                     add more kernel-weighted mass than the fluid particles.
  • clip_negative_pressure=false: One common approach in SPH models is to clip negative pressure                                 values, but this is unphysical. Instead we clip here during                                 interpolation thus only impacting the local interpolated value.
Returns
  • A NamedTuple of arrays containing interpolated properties at each point within the plane.
Note
  • The interpolation accuracy is subject to the density of particles and the chosen smoothing length.
  • With cut_off_bnd, a density-based estimation of the surface is used which is not as accurate as a real surface reconstruction.
Examples
# Interpolating across a plane defined by points [0.0, 0.0, 0.0], [1.0, 0.0, 0.0], and [0.0, 1.0, 0.0]
 # with a resolution of 0.1
-results = interpolate_plane_3d([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 0.1, semi, ref_system, sol)
source
TrixiParticles.interpolate_point — Method
interpolate_point(points_coords::Array{Array{Float64,1},1}, semi, ref_system, sol;
+results = interpolate_plane_3d([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 0.1, semi, ref_system, sol)
source
TrixiParticles.interpolate_point — Method
interpolate_point(points_coords::Array{Array{Float64,1},1}, semi, ref_system, sol;
                   smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,
                   clip_negative_pressure=false)
 
@@ -23,4 +23,4 @@
 
 # For multiple points
 points = [[1.0, 0.5], [1.0, 0.6], [1.0, 0.7]]
-results = interpolate_point(points, semi, ref_system, sol)
Note
  • This function is particularly useful for analyzing gradients or creating visualizations along a specified line in the SPH simulation domain.
  • The interpolation accuracy is subject to the density of particles and the chosen smoothing length.
  • With cut_off_bnd, a density-based estimation of the surface is used which is not as
accurate as a real surface reconstruction.
source

+results = interpolate_point(points, semi, ref_system, sol)
Note
  • This function is particularly useful for analyzing gradients or creating visualizations along a specified line in the SPH simulation domain.
  • The interpolation accuracy is subject to the density of particles and the chosen smoothing length.
  • With cut_off_bnd, a density-based estimation of the surface is used which is not as

accurate as a real surface reconstruction.

source
diff --git a/previews/PR514/general/neighborhood_search/index.html b/previews/PR514/general/neighborhood_search/index.html index e8bd91204..036bd5254 100644 --- a/previews/PR514/general/neighborhood_search/index.html +++ b/previews/PR514/general/neighborhood_search/index.html @@ -1,12 +1,5 @@ -Neighborhood Search · TrixiParticles.jl
GitHub

Neighborhood Search

TrixiParticles.GridNeighborhoodSearch — Type
GridNeighborhoodSearch{NDIMS}(search_radius, n_particles; periodic_box_min_corner=nothing,
-                              periodic_box_max_corner=nothing, threaded_nhs_update=true)

Simple grid-based neighborhood search with uniform search radius. The domain is divided into a regular grid. For each (non-empty) grid cell, a list of particles in this cell is stored. Instead of representing a finite domain by an array of cells, a potentially infinite domain is represented by storing cell lists in a hash table (using Julia's Dict data structure), indexed by the cell index tuple

\[\left( \left\lfloor \frac{x}{d} \right\rfloor, \left\lfloor \frac{y}{d} \right\rfloor \right) \quad \text{or} \quad -\left( \left\lfloor \frac{x}{d} \right\rfloor, \left\lfloor \frac{y}{d} \right\rfloor, \left\lfloor \frac{z}{d} \right\rfloor \right),\]

where $x, y, z$ are the space coordinates and $d$ is the search radius.

To find particles within the search radius around a point, only particles in the neighboring cells are considered.

See also (Chalela et al., 2021), (Ihmsen et al. 2011, Section 4.4).

As opposed to (Ihmsen et al. 2011), we do not sort the particles in any way, since not sorting makes our implementation a lot faster (although less parallelizable).

Arguments

  • NDIMS: Number of dimensions.
  • search_radius: The uniform search radius.
  • n_particles: Total number of particles.

Keywords

  • periodic_box_min_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in negative coordinate directions.
  • periodic_box_max_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in positive coordinate directions.
  • threaded_nhs_update=true: Can be used to deactivate thread parallelization in the neighborhood search update. This can be one of the largest sources of variations between simulations with different thread numbers due to particle ordering changes.
Internal use only

Please note that this constructor is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.

To run a simulation with this neighborhood search, just pass the type to the constructor of Semidiscretization:

semi = Semidiscretization(system1, system2,
-                          neighborhood_search=GridNeighborhoodSearch)

The keyword arguments periodic_box_min_corner and periodic_box_max_corner explained above can also be passed to the Semidiscretization and will internally be forwarded to the neighborhood search:

semi = Semidiscretization(system1, system2,
-                          neighborhood_search=GridNeighborhoodSearch,
-                          periodic_box_min_corner=[0.0, -0.25],
-                          periodic_box_max_corner=[1.0, 0.75])

References

  • M. Chalela, E. Sillero, L. Pereyra, M.A. Garcia, J.B. Cabral, M. Lares, M. Merchán. "GriSPy: A Python package for fixed-radius nearest neighbors search". In: Astronomy and Computing 34 (2021). doi: 10.1016/j.ascom.2020.100443
  • Markus Ihmsen, Nadir Akinci, Markus Becker, Matthias Teschner. "A Parallel SPH Implementation on Multi-Core CPUs". In: Computer Graphics Forum 30.1 (2011), pages 99–112. doi: 10.1111/J.1467-8659.2010.01832.X
source
TrixiParticles.TrivialNeighborhoodSearch — Type
TrivialNeighborhoodSearch{NDIMS}(search_radius, eachparticle)

Trivial neighborhood search that simply loops over all particles. The search radius still needs to be passed in order to sort out particles outside the search radius in the internal function for_particle_neighbor, but it's not used in the internal function eachneighbor.

Arguments

  • NDIMS: Number of dimensions.
  • search_radius: The uniform search radius.
  • eachparticle: UnitRange of all particle indices. Usually just 1:n_particles.

Keywords

  • periodic_box_min_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in negative coordinate directions.
  • periodic_box_max_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in positive coordinate directions.
Internal use only

Please note that this constructor is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.

To run a simulation with this neighborhood search, just pass the type to the constructor of Semidiscretization:

semi = Semidiscretization(system1, system2,
-                          neighborhood_search=TrivialNeighborhoodSearch)

The keyword arguments periodic_box_min_corner and periodic_box_max_corner explained above can also be passed to the Semidiscretization and will internally be forwarded to the neighborhood search:

semi = Semidiscretization(system1, system2,
-                          neighborhood_search=TrivialNeighborhoodSearch,
-                          periodic_box_min_corner=[0.0, -0.25],
-                          periodic_box_max_corner=[1.0, 0.75])
source
+Neighborhood Search · TrixiParticles.jl
GitHub

Neighborhood Search

The neighborhood search is the most essential component for performance. We provide several implementations in the package PointNeighbors.jl. See the docs of this package for an overview and a comparison of different implementations.

Usage

To run a simulation with a neighborhood search implementation, pass a template of the neighborhood search to the constructor of the Semidiscretization. A template is just an empty neighborhood search with search radius 0.0. See copy_neighborhood_search and the examples below for more details.

semi = Semidiscretization(system1, system2,
+                          neighborhood_search=PrecomputedNeighborhoodSearch{2}())

The keyword argument periodic_box in the neighborhood search constructors can be used to define a periodic domain. See the PointNeighbors.jl docs for more details.

periodic_box = PeriodicBox(min_corner=[0.0, -0.25], max_corner=[1.0, 0.75])
+semi = Semidiscretization(system1, system2,
+                          neighborhood_search=GridNeighborhoodSearch{2}(; periodic_box))
diff --git a/previews/PR514/general/semidiscretization/index.html b/previews/PR514/general/semidiscretization/index.html index 15f110e35..c9f59f4a3 100644 --- a/previews/PR514/general/semidiscretization/index.html +++ b/previews/PR514/general/semidiscretization/index.html @@ -1,9 +1,17 @@ -Semidiscretization · TrixiParticles.jl
GitHub

Semidiscretization

TrixiParticles.Semidiscretization — Type
Semidiscretization(systems...; neighborhood_search=GridNeighborhoodSearch,
-                   periodic_box_min_corner=nothing, periodic_box_max_corner=nothing,
-                   threaded_nhs_update=true)

The semidiscretization couples the passed systems to one simulation.

The type of neighborhood search to be used in the simulation can be specified with the keyword argument neighborhood_search. A value of nothing means no neighborhood search.

Arguments

  • systems: Systems to be coupled in this semidiscretization

Keywords

  • neighborhood_search: The type of neighborhood search to be used in the simulation. By default, the GridNeighborhoodSearch is used. Use TrivialNeighborhoodSearch or nothing to loop over all particles (no neighborhood search).
  • periodic_box_min_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in negative coordinate directions.
  • periodic_box_max_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in positive coordinate directions.
  • threaded_nhs_update=true: Can be used to deactivate thread parallelization in the neighborhood search update. This can be one of the largest sources of variations between simulations with different thread numbers due to particle ordering changes.

Examples

semi = Semidiscretization(fluid_system, boundary_system)
+Semidiscretization · TrixiParticles.jl
GitHub
Semidiscretization
TrixiParticles.Semidiscretization — Type
Semidiscretization(systems...; neighborhood_search=GridNeighborhoodSearch{NDIMS}())
The semidiscretization couples the passed systems to one simulation.
Arguments
  • systems: Systems to be coupled in this semidiscretization
Keywords
  • neighborhood_search:    The neighborhood search to be used in the simulation.                           By default, the GridNeighborhoodSearch is used.                           Use nothing to loop over all particles (no neighborhood search).                           To use other neighborhood search implementations, pass a template                           of a neighborhood search. See copy_neighborhood_search                           and the examples below for more details.                           To use a periodic domain, pass a PeriodicBox to the                           neighborhood search.
  • threaded_nhs_update=true:   Can be used to deactivate thread parallelization in the neighborhood search update.                               This can be one of the largest sources of variations between simulations                               with different thread numbers due to particle ordering changes.
Examples
semi = Semidiscretization(fluid_system, boundary_system)
 
 semi = Semidiscretization(fluid_system, boundary_system,
-                          neighborhood_search=TrivialNeighborhoodSearch)
source
TrixiParticles.SourceTermDamping — Type
SourceTermDamping(; damping_coefficient)
A source term to be used when a damping step is required before running a full simulation. The term $-c \cdot v_a$ is added to the acceleration $\frac{\mathrm{d}v_a}{\mathrm{d}t}$ of particle $a$, where $c$ is the damping coefficient and $v_a$ is the velocity of particle $a$.
Keywords
  • damping_coefficient:    The coefficient $d$ above. A higher coefficient means more                           damping. A coefficient of 1e-4 is a good starting point for                           damping a fluid at rest.
Examples
source_terms = SourceTermDamping(; damping_coefficient=1e-4)
source
TrixiParticles.restart_with! — Method
restart_with!(semi, sol)
Set the initial coordinates and velocities of all systems in semi to the final values in the solution sol. semidiscretize has to be called again afterwards, or another Semidiscretization can be created with the updated systems.
Arguments
  • semi:   The semidiscretization
  • sol:    The ODESolution returned by solve of OrdinaryDiffEq
source
TrixiParticles.semidiscretize — Method
semidiscretize(semi, tspan; reset_threads=true)
Create an ODEProblem from the semidiscretization with the specified tspan.
Arguments
  • semi: A Semidiscretization holding the systems involved in the simulation.
  • tspan: The time span over which the simulation will be run.
Keywords
  • reset_threads: A boolean flag to reset Polyester.jl threads before the simulation (default: true). After an error within a threaded loop, threading might be disabled. Resetting the threads before the simulation ensures that threading is enabled again for the simulation. See also trixi-framework/Trixi.jl#1583.
Returns
A DynamicalODEProblem (see the OrdinaryDiffEq.jl docs) to be integrated with OrdinaryDiffEq.jl. Note that this is not a true DynamicalODEProblem where the acceleration does not depend on the velocity. Therefore, not all integrators designed for DynamicalODEProblems will work properly. However, all integrators designed for ODEProblems can be used.
Examples
semi = Semidiscretization(fluid_system, boundary_system)
+                          neighborhood_search=GridNeighborhoodSearch{2}(update_strategy=SerialUpdate()))
+
+periodic_box = PeriodicBox(min_corner = [0.0, 0.0], max_corner = [1.0, 1.0])
+semi = Semidiscretization(fluid_system, boundary_system,
+                          neighborhood_search=GridNeighborhoodSearch{2}(; periodic_box))
+
+semi = Semidiscretization(fluid_system, boundary_system,
+                          neighborhood_search=PrecomputedNeighborhoodSearch{2}())
+
+semi = Semidiscretization(fluid_system, boundary_system,
+                          neighborhood_search=nothing)
source
TrixiParticles.SourceTermDamping — Type
SourceTermDamping(; damping_coefficient)
A source term to be used when a damping step is required before running a full simulation. The term $-c \cdot v_a$ is added to the acceleration $\frac{\mathrm{d}v_a}{\mathrm{d}t}$ of particle $a$, where $c$ is the damping coefficient and $v_a$ is the velocity of particle $a$.
Keywords
  • damping_coefficient:    The coefficient $d$ above. A higher coefficient means more                           damping. A coefficient of 1e-4 is a good starting point for                           damping a fluid at rest.
Examples
source_terms = SourceTermDamping(; damping_coefficient=1e-4)
source
TrixiParticles.restart_with! — Method
restart_with!(semi, sol)
Set the initial coordinates and velocities of all systems in semi to the final values in the solution sol. semidiscretize has to be called again afterwards, or another Semidiscretization can be created with the updated systems.
Arguments
  • semi:   The semidiscretization
  • sol:    The ODESolution returned by solve of OrdinaryDiffEq
source
TrixiParticles.semidiscretize — Method
semidiscretize(semi, tspan; reset_threads=true)
Create an ODEProblem from the semidiscretization with the specified tspan.
Arguments
  • semi: A Semidiscretization holding the systems involved in the simulation.
  • tspan: The time span over which the simulation will be run.
Keywords
  • reset_threads: A boolean flag to reset Polyester.jl threads before the simulation (default: true). After an error within a threaded loop, threading might be disabled. Resetting the threads before the simulation ensures that threading is enabled again for the simulation. See also trixi-framework/Trixi.jl#1583.
Returns
A DynamicalODEProblem (see the OrdinaryDiffEq.jl docs) to be integrated with OrdinaryDiffEq.jl. Note that this is not a true DynamicalODEProblem where the acceleration does not depend on the velocity. Therefore, not all integrators designed for DynamicalODEProblems will work properly. However, all integrators designed for ODEProblems can be used.
Examples
semi = Semidiscretization(fluid_system, boundary_system)
 tspan = (0.0, 1.0)
-ode_problem = semidiscretize(semi, tspan)
source

+ode_problem = semidiscretize(semi, tspan)
source
diff --git a/previews/PR514/general/smoothing_kernels/index.html b/previews/PR514/general/smoothing_kernels/index.html index 898a771fd..e2eee63c7 100644 --- a/previews/PR514/general/smoothing_kernels/index.html +++ b/previews/PR514/general/smoothing_kernels/index.html @@ -1,33 +1,33 @@ -Smoothing Kernels · TrixiParticles.jl
GitHub

Smoothing Kernels

The following smoothing kernels are currently available:

Smoothing KernelCompact SupportTyp. Smoothing LengthRecommended ApplicationStability
SchoenbergCubicSplineKernel$[0, 2h]$$1.1$ to $1.3$General + sharp waves++
SchoenbergQuarticSplineKernel$[0, 2.5h]$$1.1$ to $1.5$General+++
SchoenbergQuinticSplineKernel$[0, 3h]$$1.1$ to $1.5$General++++
GaussianKernel$[0, 3h]$$1.0$ to $1.5$Literature+++++
WendlandC2Kernel$[0, 1h]$$2.5$ to $4.0$General (recommended)++++
WendlandC4Kernel$[0, 1h]$$3.0$ to $4.5$General+++++
WendlandC6Kernel$[0, 1h]$$3.5$ to $5.0$General+++++
Poly6Kernel$[0, 1h]$$1.5$ to $2.5$Literature+
SpikyKernel$[0, 1h]$$1.5$ to $3.0$Sharp corners + waves+

We recommend to use the WendlandC2Kernel for most applications. If less smoothing is needed, try SchoenbergCubicSplineKernel, for more smoothing try WendlandC6Kernel.

Usage

The kernel can be called as

TrixiParticles.kernel(smoothing_kernel, r, h)

The length of the compact support can be obtained as

TrixiParticles.compact_support(smoothing_kernel, h)

Note that $r$ has to be a scalar, so in the context of SPH, the kernel should be used as

\[W(\Vert r_a - r_b \Vert, h).\]

The gradient required in SPH,

\[ \nabla_{r_a} W(\Vert r_a - r_b \Vert, h)\]

can be called as

TrixiParticles.kernel_grad(smoothing_kernel, pos_diff, distance, h)

where pos_diff is $r_a - r_b$ and distance is $\Vert r_a - r_b \Vert$.

TrixiParticles.GaussianKernel — Type
GaussianKernel{NDIMS}()

Gaussian kernel given by

\[W(r, h) = \frac{\sigma_d}{h^d} e^{-r^2/h^2}\]

where $d$ is the number of dimensions and

  • $\sigma_2 = \frac{1}{\pi}$ for 2D,
  • $\sigma_3 = \frac{1}{\pi^{3/2}}$ for 3D.

This kernel function has an infinite support, but in practice, it's often truncated at a certain multiple of $h$, such as $3h$.

In this implementation, the kernel is truncated at $3h$, so this kernel function has a compact support of $[0, 3h]$.

The smoothing length is typically in the range $[1.0\delta, 1.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

Note: This truncation makes this Kernel not conservative, which is beneficial in regards to stability but makes it less accurate.

source
TrixiParticles.Poly6Kernel — Type
Poly6Kernel{NDIMS}()

Poly6 kernel, a commonly used kernel in SPH literature, especially in computer graphics contexts. It is defined as

\[W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} +Smoothing Kernels · TrixiParticles.jl

GitHub

Smoothing Kernels

The following smoothing kernels are currently available:

Smoothing KernelCompact SupportTyp. Smoothing LengthRecommended ApplicationStability
SchoenbergCubicSplineKernel$[0, 2h]$$1.1$ to $1.3$General + sharp waves++
SchoenbergQuarticSplineKernel$[0, 2.5h]$$1.1$ to $1.5$General+++
SchoenbergQuinticSplineKernel$[0, 3h]$$1.1$ to $1.5$General++++
GaussianKernel$[0, 3h]$$1.0$ to $1.5$Literature+++++
WendlandC2Kernel$[0, 1h]$$2.5$ to $4.0$General (recommended)++++
WendlandC4Kernel$[0, 1h]$$3.0$ to $4.5$General+++++
WendlandC6Kernel$[0, 1h]$$3.5$ to $5.0$General+++++
Poly6Kernel$[0, 1h]$$1.5$ to $2.5$Literature+
SpikyKernel$[0, 1h]$$1.5$ to $3.0$Sharp corners + waves+

We recommend to use the WendlandC2Kernel for most applications. If less smoothing is needed, try SchoenbergCubicSplineKernel, for more smoothing try WendlandC6Kernel.

Usage

The kernel can be called as

TrixiParticles.kernel(smoothing_kernel, r, h)

The length of the compact support can be obtained as

TrixiParticles.compact_support(smoothing_kernel, h)

Note that $r$ has to be a scalar, so in the context of SPH, the kernel should be used as

\[W(\Vert r_a - r_b \Vert, h).\]

The gradient required in SPH,

\[ \nabla_{r_a} W(\Vert r_a - r_b \Vert, h)\]

can be called as

TrixiParticles.kernel_grad(smoothing_kernel, pos_diff, distance, h)

where pos_diff is $r_a - r_b$ and distance is $\Vert r_a - r_b \Vert$.

TrixiParticles.GaussianKernel — Type
GaussianKernel{NDIMS}()

Gaussian kernel given by

\[W(r, h) = \frac{\sigma_d}{h^d} e^{-r^2/h^2}\]

where $d$ is the number of dimensions and

  • $\sigma_2 = \frac{1}{\pi}$ for 2D,
  • $\sigma_3 = \frac{1}{\pi^{3/2}}$ for 3D.

This kernel function has an infinite support, but in practice, it's often truncated at a certain multiple of $h$, such as $3h$.

In this implementation, the kernel is truncated at $3h$, so this kernel function has a compact support of $[0, 3h]$.

The smoothing length is typically in the range $[1.0\delta, 1.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

Note: This truncation makes this Kernel not conservative, which is beneficial in regards to stability but makes it less accurate.

source
TrixiParticles.Poly6Kernel — Type
Poly6Kernel{NDIMS}()

Poly6 kernel, a commonly used kernel in SPH literature [3], especially in computer graphics contexts. It is defined as

\[W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} (1 - q^2)^3 & \text{if } 0 \leq q < 1, \\ 0 & \text{if } q \geq 1, -\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor that depends on the dimension. The normalization factor $\sigma$ is $4 / \pi$ in two dimensions or $315 / 64\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

Poly6 is well-known for its computational simplicity, though it's worth noting that there are other kernels that might offer better accuracy for hydrodynamic simulations. Furthermore, its derivatives are not that smooth, which can lead to stability problems. It is also susceptible to clumping.

The smoothing length is typically in the range $[1.5\delta, 2.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Matthias Müller, David Charypar, and Markus Gross. "Particle-based fluid simulation for interactive applications". In: Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation. Eurographics Association. 2003, pages 154-159. doi: 10.5555/846276.846298
source
TrixiParticles.SchoenbergCubicSplineKernel — Type
SchoenbergCubicSplineKernel{NDIMS}()

Cubic spline kernel by Schoenberg (Schoenberg, 1946), given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} +\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor that depends on the dimension. The normalization factor $\sigma$ is $4 / \pi$ in two dimensions or $315 / 64\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

Poly6 is well-known for its computational simplicity, though it's worth noting that there are other kernels that might offer better accuracy for hydrodynamic simulations. Furthermore, its derivatives are not that smooth, which can lead to stability problems. It is also susceptible to clumping.

The smoothing length is typically in the range $[1.5\delta, 2.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
TrixiParticles.SchoenbergCubicSplineKernel — Type
SchoenbergCubicSplineKernel{NDIMS}()

Cubic spline kernel by Schoenberg (1946), given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} \frac{1}{4} (2 - q)^3 - (1 - q)^3 & \text{if } 0 \leq q < 1, \\ \frac{1}{4} (2 - q)^3 & \text{if } 1 \leq q < 2, \\ 0 & \text{if } q \geq 2, \\ -\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization constant given by $\sigma =[\frac{2}{3}, \frac{10}{7 \pi}, \frac{1}{\pi}]$ in $[1, 2, 3]$ dimensions.

This kernel function has a compact support of $[0, 2h]$.

For an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see (Price, 2012). For an analytic formula for higher order Schoenberg kernels, see (Monaghan, 1985). The largest disadvantage of Schoenberg Spline Kernel is the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.

The smoothing length is typically in the range $[1.1\delta, 1.3\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Daniel J. Price. "Smoothed particle hydrodynamics and magnetohydrodynamics". In: Journal of Computational Physics 231.3 (2012), pages 759-794. doi: 10.1016/j.jcp.2010.12.011
  • Joseph J. Monaghan. "Particle methods for hydrodynamics". In: Computer Physics Reports 3.2 (1985), pages 71–124. doi: 10.1016/0167-7977(85)90010-3
  • Isaac J. Schoenberg. "Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae." In: Quarterly of Applied Mathematics 4.2 (1946), pages 112–141. doi: 10.1090/QAM/16705
source
TrixiParticles.SchoenbergQuarticSplineKernel — Type
SchoenbergQuarticSplineKernel{NDIMS}()

Quartic spline kernel by Schoenberg (Schoenberg, 1946), given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} +\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization constant given by $\sigma =[\frac{2}{3}, \frac{10}{7 \pi}, \frac{1}{\pi}]$ in $[1, 2, 3]$ dimensions.

This kernel function has a compact support of $[0, 2h]$.

For an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see Price (2012). For an analytic formula for higher order Schoenberg kernels, see Monaghan (1985). The largest disadvantage of Schoenberg Spline Kernel is the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.

The smoothing length is typically in the range $[1.1\delta, 1.3\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
TrixiParticles.SchoenbergQuarticSplineKernel — Type
SchoenbergQuarticSplineKernel{NDIMS}()

Quartic spline kernel by Schoenberg (1946), given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} \left(5/2 - q \right)^4 - 5\left(3/2 - q \right)^4 + 10\left(1/2 - q \right)^4 & \text{if } 0 \leq q < \frac{1}{2}, \\ \left(5/2 - q \right)^4 - 5\left(3/2 - q \right)^4 & \text{if } \frac{1}{2} \leq q < \frac{3}{2}, \\ \left(5/2 - q \right)^4 & \text{if } \frac{3}{2} \leq q < \frac{5}{2}, \\ 0 & \text{if } q \geq \frac{5}{2}, -\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization constant given by $\sigma =[\frac{1}{24}, \frac{96}{1199 \pi}, \frac{1}{20\pi}]$ in $[1, 2, 3]$ dimensions.

This kernel function has a compact support of $[0, 2.5h]$.

For an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see (Price, 2012). For an analytic formula for higher order Schoenberg kernels, see (Monaghan, 1985).

The largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.

The smoothing length is typically in the range $[1.1\delta, 1.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Daniel J. Price. "Smoothed particle hydrodynamics and magnetohydrodynamics". In: Journal of Computational Physics 231.3 (2012), pages 759-794. doi: 10.1016/j.jcp.2010.12.011
  • Joseph J. Monaghan. "Particle methods for hydrodynamics". In: Computer Physics Reports 3.2 (1985), pages 71–124. doi: 10.1016/0167-7977(85)90010-3
  • Isaac J. Schoenberg. "Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae." In: Quarterly of Applied Mathematics 4.2 (1946), pages 112–141. doi: 10.1090/QAM/16705
source
TrixiParticles.SchoenbergQuinticSplineKernel — Type
SchoenbergQuinticSplineKernel{NDIMS}()

Quintic spline kernel by Schoenberg (Schoenberg, 1946), given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} +\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization constant given by $\sigma =[\frac{1}{24}, \frac{96}{1199 \pi}, \frac{1}{20\pi}]$ in $[1, 2, 3]$ dimensions.

This kernel function has a compact support of $[0, 2.5h]$.

For an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see Price (2012). For an analytic formula for higher order Schoenberg kernels, see Monaghan (1985).

The largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.

The smoothing length is typically in the range $[1.1\delta, 1.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
TrixiParticles.SchoenbergQuinticSplineKernel — Type
SchoenbergQuinticSplineKernel{NDIMS}()

Quintic spline kernel by Schoenberg (1946), given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} (3 - q)^5 - 6(2 - q)^5 + 15(1 - q)^5 & \text{if } 0 \leq q < 1, \\ (3 - q)^5 - 6(2 - q)^5 & \text{if } 1 \leq q < 2, \\ (3 - q)^5 & \text{if } 2 \leq q < 3, \\ 0 & \text{if } q \geq 3, -\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization constant given by $\sigma =[\frac{1}{120}, \frac{7}{478 \pi}, \frac{1}{120\pi}]$ in $[1, 2, 3]$ dimensions.

This kernel function has a compact support of $[0, 3h]$.

For an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see (Price, 2012). For an analytic formula for higher order Schoenberg kernels, see (Monaghan, 1985).

The largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.

The smoothing length is typically in the range $[1.1\delta, 1.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Daniel J. Price. "Smoothed particle hydrodynamics and magnetohydrodynamics". In: Journal of Computational Physics 231.3 (2012), pages 759-794. doi: 10.1016/j.jcp.2010.12.011
  • Joseph J. Monaghan. "Particle methods for hydrodynamics". In: Computer Physics Reports 3.2 (1985), pages 71–124. doi: 10.1016/0167-7977(85)90010-3
  • Isaac J. Schoenberg. "Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae." In: Quarterly of Applied Mathematics 4.2 (1946), pages 112–141. doi: 10.1090/QAM/16705
source
TrixiParticles.SpikyKernel — Type
SpikyKernel{NDIMS}()

The Spiky kernel is another frequently used kernel in SPH, especially due to its desirable properties in preserving features near boundaries in fluid simulations. It is defined as:

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with:

\[w(q) = \sigma \begin{cases} +\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization constant given by $\sigma =[\frac{1}{120}, \frac{7}{478 \pi}, \frac{1}{120\pi}]$ in $[1, 2, 3]$ dimensions.

This kernel function has a compact support of $[0, 3h]$.

For an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see Price (2012). For an analytic formula for higher order Schoenberg kernels, see Monaghan (1985).

The largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.

The smoothing length is typically in the range $[1.1\delta, 1.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
TrixiParticles.SpikyKernel — Type
SpikyKernel{NDIMS}()

The Spiky kernel is another frequently used kernel in SPH, especially due to its desirable properties in preserving features near boundaries in fluid simulations [3]. It is defined as:

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with:

\[w(q) = \sigma \begin{cases} (1 - q)^3 & \text{if } 0 \leq q < 1, \\ 0 & \text{if } q \geq 1, -\end{cases}\]

where $d$ is the number of dimensions and the normalization factor $\sigma$ is $10 / \pi$ in two dimensions or $15 / \pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

The Spiky kernel is particularly known for its sharp gradients, which can help to preserve sharp features in fluid simulations, especially near solid boundaries. These sharp gradients at the boundary are also the largest disadvantage as they can lead to instability.

The smoothing length is typically in the range $[1.5\delta, 3.0\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Matthias Müller, David Charypar, and Markus Gross. "Particle-based fluid simulation for interactive applications". In: Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation. Eurographics Association. 2003, pages 154-159. doi: 10.5555/846276.846298
source
TrixiParticles.WendlandC2Kernel — Type
WendlandC2Kernel{NDIMS}()

Wendland C2 kernel (Wendland, 1995), a piecewise polynomial function designed to have compact support and to be twice continuously differentiable everywhere. Given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} +\end{cases}\]

where $d$ is the number of dimensions and the normalization factor $\sigma$ is $10 / \pi$ in two dimensions or $15 / \pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

The Spiky kernel is particularly known for its sharp gradients, which can help to preserve sharp features in fluid simulations, especially near solid boundaries. These sharp gradients at the boundary are also the largest disadvantage as they can lead to instability.

The smoothing length is typically in the range $[1.5\delta, 3.0\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
TrixiParticles.WendlandC2Kernel — Type
WendlandC2Kernel{NDIMS}()

Wendland C2 kernel [7], a piecewise polynomial function designed to have compact support and to be twice continuously differentiable everywhere. Given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} (1 - q)^4 (4q + 1) & \text{if } 0 \leq q < 1, \\ 0 & \text{if } q \geq 1, -\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor dependent on the dimension. The normalization factor $\sigma$ is $40/7\pi$ in two dimensions or $21/2\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

For a detailed discussion on Wendland functions and their applications in SPH, see (Dehnen & Aly, 2012). The smoothness of these functions is also the largest disadvantage as they lose details at sharp corners.

The smoothing length is typically in the range $[2.5\delta, 4.0\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Walter Dehnen & Hassan Aly. "Improving convergence in smoothed particle hydrodynamics simulations without pairing instability". In: Monthly Notices of the Royal Astronomical Society 425.2 (2012), pages 1068-1082. doi: 10.1111/j.1365-2966.2012.21439.x

  • Holger Wendland. "Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree." In: Advances in computational Mathematics 4 (1995), pages 389-396. doi: 10.1007/BF02123482

source
TrixiParticles.WendlandC4Kernel — Type
WendlandC4Kernel{NDIMS}()

Wendland C4 kernel, a piecewise polynomial function designed to have compact support and to be four times continuously differentiable everywhere. Given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} +\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor dependent on the dimension. The normalization factor $\sigma$ is $40/7\pi$ in two dimensions or $21/2\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

For a detailed discussion on Wendland functions and their applications in SPH, see Dehnen (2012). The smoothness of these functions is also the largest disadvantage as they lose details at sharp corners.

The smoothing length is typically in the range $[2.5\delta, 4.0\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
TrixiParticles.WendlandC4Kernel — Type
WendlandC4Kernel{NDIMS}()

Wendland C4 kernel [7], a piecewise polynomial function designed to have compact support and to be four times continuously differentiable everywhere. Given by

\[ W(r, h) = \frac{1}{h^d} w(r/h)\]

with

\[w(q) = \sigma \begin{cases} (1 - q)^6 (35q^2 / 3 + 6q + 1) & \text{if } 0 \leq q < 1, \\ 0 & \text{if } q \geq 1, -\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor dependent on the dimension. The normalization factor $\sigma$ is $9 / \pi$ in two dimensions or $495 / 32\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

For a detailed discussion on Wendland functions and their applications in SPH, see (Dehnen & Aly, 2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.

The smoothing length is typically in the range $[3.0\delta, 4.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Walter Dehnen & Hassan Aly. "Improving convergence in smoothed particle hydrodynamics simulations without pairing instability". In: Monthly Notices of the Royal Astronomical Society 425.2 (2012), pages 1068-1082. doi: 10.1111/j.1365-2966.2012.21439.x

  • Holger Wendland. "Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree." In: Advances in computational Mathematics 4 (1995): 389-396. doi: 10.1007/BF02123482

source
TrixiParticles.WendlandC6Kernel — Type
WendlandC6Kernel{NDIMS}()

Wendland C6 kernel, a piecewise polynomial function designed to have compact support and to be six times continuously differentiable everywhere. Given by:

\[W(r, h) = \frac{1}{h^d} w(r/h)\]

with:

\[w(q) = \sigma \begin{cases} +\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor dependent on the dimension. The normalization factor $\sigma$ is $9 / \pi$ in two dimensions or $495 / 32\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

For a detailed discussion on Wendland functions and their applications in SPH, see Dehnen (2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.

The smoothing length is typically in the range $[3.0\delta, 4.5\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
TrixiParticles.WendlandC6Kernel — Type
WendlandC6Kernel{NDIMS}()

Wendland C6 kernel [7], a piecewise polynomial function designed to have compact support and to be six times continuously differentiable everywhere. Given by:

\[W(r, h) = \frac{1}{h^d} w(r/h)\]

with:

\[w(q) = \sigma \begin{cases} (1 - q)^8 (32q^3 + 25q^2 + 8q + 1) & \text{if } 0 \leq q < 1, \\ 0 & \text{if } q \geq 1, -\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor dependent on the dimension. The normalization factor $\sigma$ is $78 / 7 \pi$ in two dimensions or $1365 / 64\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

For a detailed discussion on Wendland functions and their applications in SPH, see (Dehnen & Aly, 2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.

The smoothing length is typically in the range $[3.5\delta, 5.0\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

References

  • Walter Dehnen & Hassan Aly. "Improving convergence in smoothed particle hydrodynamics simulations without pairing instability". In: Monthly Notices of the Royal Astronomical Society 425.2 (2012), pages 1068-1082. doi: 10.1111/j.1365-2966.2012.21439.x

  • Holger Wendland. "Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree." In: Advances in computational Mathematics 4 (1995): 389-396. doi: 10.1007/BF02123482

source
+\end{cases}\]

where $d$ is the number of dimensions and $\sigma$ is a normalization factor dependent on the dimension. The normalization factor $\sigma$ is $78 / 7 \pi$ in two dimensions or $1365 / 64\pi$ in three dimensions.

This kernel function has a compact support of $[0, h]$.

For a detailed discussion on Wendland functions and their applications in SPH, Dehnen (2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.

The smoothing length is typically in the range $[3.5\delta, 5.0\delta]$, where $\delta$ is the typical particle spacing.

For general information and usage see Smoothing Kernels.

source
diff --git a/previews/PR514/general/util/index.html b/previews/PR514/general/util/index.html index 11b1e0b45..be3ad17d5 100644 --- a/previews/PR514/general/util/index.html +++ b/previews/PR514/general/util/index.html @@ -1,3 +1,3 @@ -Util · TrixiParticles.jl
GitHub

Util

TrixiParticles.examples_dir — Method
examples_dir()

Return the directory where the example files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.

Copied from Trixi.jl.

Examples

readdir(examples_dir())
source
TrixiParticles.validation_dir — Method
validation_dir()

Return the directory where the validation files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.

Copied from Trixi.jl.

Examples

readdir(validation_dir())
source
TrixiParticles.@autoinfiltrate — Macro
@autoinfiltrate
-@autoinfiltrate condition::Bool

Invoke the @infiltrate macro of the package Infiltrator.jl to create a breakpoint for ad-hoc interactive debugging in the REPL. If the optional argument condition is given, the breakpoint is only enabled if condition evaluates to true.

As opposed to using Infiltrator.@infiltrate directly, this macro does not require Infiltrator.jl to be added as a dependency to TrixiParticles.jl. As a bonus, the macro will also attempt to load the Infiltrator module if it has not yet been loaded manually.

Note: For this macro to work, the Infiltrator.jl package needs to be installed in your current Julia environment stack.

See also: Infiltrator.jl

Internal use only

Please note that this macro is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.

source
TrixiParticles.@threaded — Macro
@threaded for ... end

Semantically the same as Threads.@threads when iterating over a AbstractUnitRange but without guarantee that the underlying implementation uses Threads.@threads or works for more general for loops. In particular, there may be an additional check whether only one thread is used to reduce the overhead of serial execution or the underlying threading capabilities might be provided by other packages such as Polyester.jl.

Warn

This macro does not necessarily work for general for loops. For example, it does not necessarily support general iterables such as eachline(filename).

Some discussion can be found at https://discourse.julialang.org/t/overhead-of-threads-threads/53964 and https://discourse.julialang.org/t/threads-threads-with-one-thread-how-to-remove-the-overhead/58435.

Copied from Trixi.jl.

source
+Util · TrixiParticles.jl
GitHub

Util

TrixiParticles.examples_dir — Method
examples_dir()

Return the directory where the example files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.

Copied from Trixi.jl.

Examples

readdir(examples_dir())
source
TrixiParticles.validation_dir — Method
validation_dir()

Return the directory where the validation files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.

Copied from Trixi.jl.

Examples

readdir(validation_dir())
source
TrixiParticles.@autoinfiltrate — Macro
@autoinfiltrate
+@autoinfiltrate condition::Bool

Invoke the @infiltrate macro of the package Infiltrator.jl to create a breakpoint for ad-hoc interactive debugging in the REPL. If the optional argument condition is given, the breakpoint is only enabled if condition evaluates to true.

As opposed to using Infiltrator.@infiltrate directly, this macro does not require Infiltrator.jl to be added as a dependency to TrixiParticles.jl. As a bonus, the macro will also attempt to load the Infiltrator module if it has not yet been loaded manually.

Note: For this macro to work, the Infiltrator.jl package needs to be installed in your current Julia environment stack.

See also: Infiltrator.jl

Internal use only

Please note that this macro is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.

source
diff --git a/previews/PR514/getting_started/index.html b/previews/PR514/getting_started/index.html index 010650eba..f3ccb5ad8 100644 --- a/previews/PR514/getting_started/index.html +++ b/previews/PR514/getting_started/index.html @@ -1,2 +1,2 @@ -Getting started · TrixiParticles.jl
GitHub

Getting started

If you have not installed TrixiParticles.jl, please follow the instructions given here.

In the following sections, we will give a short introduction. For a more thorough discussion, take a look at our Tutorials.

Running an Example

The easiest way to run a simulation is to run one of our predefined example files. We will run the file examples/fluid/hydrostatic_water_column_2d.jl, which simulates a fluid resting in a rectangular tank. Since TrixiParticles.jl uses multithreading, you should start Julia with the flag --threads auto (or, e.g. --threads 4 for 4 threads).

In the Julia REPL, first load the package TrixiParticles.jl.

julia> using TrixiParticles

Then start the simulation by executing

julia> trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"))

This will open a new window with a 2D visualization of the final solution: plot_hydrostatic_water_column

For more information about visualization, see Visualization.

Running other Examples

You can find a list of our other predefined examples under Examples. Execute them as follows from the Julia REPL by replacing subfolder and example_name

julia> trixi_include(joinpath(examples_dir(), "subfolder", "example_name.jl"))

Modifying an example

You can pass keyword arguments to the function trixi_include to overwrite assignments in the file.

With trixi_include, we can overwrite variables defined in the example file to run a different simulation without modifying the example file.

julia> trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"), initial_fluid_size=(1.0, 0.5))

This for example, will change the fluid size from $(0.9, 1.0)$ to $(1.0, 0.5)$.

To understand why, take a look into the file hydrostatic_water_column_2d.jl in the subfolder fluid inside the examples directory, which is the file that we executed earlier. You can see that the initial size of the fluid is defined in the variable initial_fluid_size, which we could overwrite with the trixi_include call above. Another variable that is worth experimenting with is fluid_particle_spacing, which controls the resolution of the simulation in this case. A lower value will increase the resolution and the runtime.

Set up you first simulation from scratch

See Set up your first simulation.

Find an overview over the available tutorials under Tutorials.

+Getting started · TrixiParticles.jl
GitHub

Getting started

If you have not installed TrixiParticles.jl, please follow the instructions given here.

In the following sections, we will give a short introduction. For a more thorough discussion, take a look at our Tutorials.

Running an Example

The easiest way to run a simulation is to run one of our predefined example files. We will run the file examples/fluid/hydrostatic_water_column_2d.jl, which simulates a fluid resting in a rectangular tank. Since TrixiParticles.jl uses multithreading, you should start Julia with the flag --threads auto (or, e.g. --threads 4 for 4 threads).

In the Julia REPL, first load the package TrixiParticles.jl.

julia> using TrixiParticles

Then start the simulation by executing

julia> trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"))

The easiest way to quickly visualize the result is to use Plots.jl:

julia> using Plots; plot(sol)

This will open a new window with a 2D visualization of the final solution: plot_hydrostatic_water_column

For more information about visualization, see Visualization.

Running other Examples

You can find a list of our other predefined examples under Examples. Execute them as follows from the Julia REPL by replacing subfolder and example_name

julia> trixi_include(joinpath(examples_dir(), "subfolder", "example_name.jl"))

Modifying an example

You can pass keyword arguments to the function trixi_include to overwrite assignments in the file.

With trixi_include, we can overwrite variables defined in the example file to run a different simulation without modifying the example file.

julia> trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"), initial_fluid_size=(1.0, 0.5))

This for example, will change the fluid size from $(0.9, 1.0)$ to $(1.0, 0.5)$.

To understand why, take a look into the file hydrostatic_water_column_2d.jl in the subfolder fluid inside the examples directory, which is the file that we executed earlier. You can see that the initial size of the fluid is defined in the variable initial_fluid_size, which we could overwrite with the trixi_include call above. Another variable that is worth experimenting with is fluid_particle_spacing, which controls the resolution of the simulation in this case. A lower value will increase the resolution and the runtime.

Set up you first simulation from scratch

See Set up your first simulation.

Find an overview over the available tutorials under Tutorials.

diff --git a/previews/PR514/gpu/index.html b/previews/PR514/gpu/index.html new file mode 100644 index 000000000..73414720e --- /dev/null +++ b/previews/PR514/gpu/index.html @@ -0,0 +1,8 @@ + +GPU Support · TrixiParticles.jl
GitHub

GPU Support

GPU support is still an experimental feature that is actively being worked on. As of now, the WeaklyCompressibleSPHSystem and the BoundarySPHSystem are supported on GPUs. We have tested this on GPUs by Nvidia and AMD.

To run a simulation on a GPU, we need to use the FullGridCellList as cell list for the GridNeighborhoodSearch. This cell list requires a bounding box for the domain, unlike the default cell list, which uses an unbounded domain. For simulations that are bounded by a closed tank, we can use the boundary of the tank to obtain the bounding box as follows.

search_radius = TrixiParticles.compact_support(smoothing_kernel, smoothing_length)
+min_corner = minimum(tank.boundary.coordinates, dims=2) .- search_radius
+max_corner = maximum(tank.boundary.coordinates, dims=2) .+ search_radius
+cell_list = TrixiParticles.PointNeighbors.FullGridCellList(; min_corner, max_corner)

We then need to pass this cell list to the neighborhood search and the neighborhood search to the Semidiscretization.

semi = Semidiscretization(fluid_system, boundary_system,
+                          neighborhood_search=GridNeighborhoodSearch{2}(; cell_list))

At this point, we should run the simulation and make sure that it still works and that the bounding box is large enough. For some simulations where particles move outside the initial tank coordinates, for example when the tank is not closed or when the tank is moving, an appropriate bounding box has to be specified.

Then, we only need to specify the data type that is used for the simulation. On an Nvidia GPU, we specify:

using CUDA
+ode = semidiscretize(semi, tspan, data_type=CuArray)

On an AMD GPU, we use:

using AMDGPU
+ode = semidiscretize(semi, tspan, data_type=ROCArray)

Then, we can run the simulation as usual. All data is transferred to the GPU during initialization and all loops over particles and their neighbors will be executed on the GPU as kernels generated by KernelAbstractions.jl. Data is only copied to the CPU for saving VTK files via the SolutionSavingCallback.

diff --git a/previews/PR514/index.html b/previews/PR514/index.html index f45b3cf0e..e2e9819fb 100644 --- a/previews/PR514/index.html +++ b/previews/PR514/index.html @@ -1,5 +1,5 @@ -Home · TrixiParticles.jl
GitHub

TrixiParticles.jl

TrixiParticles.jl is a numerical simulation framework designed for particle-based numerical methods, with an emphasis on multiphysics applications, written in Julia. A primary goal of the framework is to be user-friendly for engineering, science, and educational purposes. In addition to its extensible design and optimized implementation, we prioritize the user experience, including installation, pre- and postprocessing. Its features include:

Features

  • Incompressible Navier-Stokes
    • Methods: Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH), Entropically Damped Artificial Compressibility (EDAC)
  • Solid-body mechanics
    • Methods: Total Lagrangian SPH (TLSPH)
  • Fluid-Structure Interaction
  • Output formats:
    • VTK

Examples

+Home · TrixiParticles.jl
GitHub

TrixiParticles.jl

TrixiParticles.jl is a high-performance particle simulation framework designed to overcome challenges of particle-based numerical methods in multiphysics applications. Existing frameworks often lack user-friendliness, involve complex configuration, and are not easily extensible for development of new methods. In the future we also want to provide seamless scalability from CPU to Exascale-level computing with GPU support. TrixiParticles.jl addresses these limitations with an intuitive interface, straightforward configuration, and an extensible design, facilitating efficient simulation setup and execution.

TrixiParticles.jl focuses on the following use cases:

  • Development of new particle-based methods and models. By providing an extensible architecture to incorporate additional particle methods easily and not focusing on a single model or numerical method.
  • Accurate, reliable and efficient physics-based modelling of complex multiphysics problems by providing a flexible configuration system, tools, high performance and a wide range of validation and test cases.
  • Easy setup of accessible simulations for educational purposes, including student projects, coursework, and thesis work through extensive documentation, community engagement and readable configuration files.

Its features include:

Features

  • Incompressible Navier-Stokes
    • Methods: Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH), Entropically Damped Artificial Compressibility (EDAC)
    • Models: Surface Tension
  • Solid-body mechanics
    • Methods: Total Lagrangian SPH (TLSPH)
  • Fluid-Structure Interaction
  • Output formats:
    • VTK

Examples

-

2D Dam Break
@@ -16,4 +16,4 @@
Dam Break with Elastic Plate

Quickstart

  1. Installation
  2. Getting started

If you have any questions concerning TrixiParticles.jl you can join our community on Slack or open an issue with your question.

Start with development

To get started with development have a look at these pages:

  1. Installation
  2. Development
  3. Contributing
+

Quickstart

  1. Installation
  2. Getting started

If you have any questions concerning TrixiParticles.jl you can join our community on Slack or open an issue with your question.

Start with development

To get started with development have a look at these pages:

  1. Installation
  2. Development
  3. Contributing
diff --git a/previews/PR514/install/index.html b/previews/PR514/install/index.html index 7f1c167cd..56150b7e8 100644 --- a/previews/PR514/install/index.html +++ b/previews/PR514/install/index.html @@ -1,14 +1,14 @@ -Installation · TrixiParticles.jl
GitHub

Installation

Setting up Julia

If you have not yet installed Julia, please follow the instructions for your operating system. TrixiParticles.jl works with Julia v1.9 and newer. We recommend using the latest stable release of Julia.

For users

TrixiParticles.jl is a registered Julia package. You can install TrixiParticles.jl, OrdinaryDiffEq.jl (used for time integration) and Plots.jl by executing the following commands in the Julia REPL:

julia> using Pkg
+Installation · TrixiParticles.jl
GitHub
Installation
Setting up Julia
If you have not yet installed Julia, please follow the instructions on the official website. TrixiParticles.jl works with Julia v1.9 and newer. We recommend using the latest stable release of Julia.
For users
TrixiParticles.jl is a registered Julia package. You can install TrixiParticles.jl, OrdinaryDiffEq.jl (used for time integration) and Plots.jl by executing the following commands in the Julia REPL:
julia> using Pkg
 
 julia> Pkg.add(["TrixiParticles", "OrdinaryDiffEq", "Plots"])
For developers
If you plan on editing TrixiParticles.jl itself, you can download TrixiParticles.jl to a local folder and use the code from the cloned directory:
git clone git@github.com:trixi-framework/TrixiParticles.jl.git
 cd TrixiParticles.jl
 mkdir run
 julia --project=run -e 'using Pkg; Pkg.develop(PackageSpec(path="."))' # Add TrixiParticles.jl to `run` project
-julia --project=run -e 'using Pkg; Pkg.add("OrdinaryDiffEq", "Plots")' # Add additional packages
If you installed TrixiParticles.jl this way, you always have to start Julia with the --project flag set to your run directory, e.g.,
julia --project=run
from the TrixiParticles.jl root directory.
The advantage of using a separate run directory is that you can also add other related packages (e.g., OrdinaryDiffEq.jl, see above) to the project in the run folder and always have a reproducible environment at hand to share with others.
Optional software/packages
  • OrdinaryDiffEq.jl – A Julia package of ordinary differential equation solvers that is used in the examples
  • Plots.jl – Julia Plotting library that is used in some examples
  • PythonPlot.jl – Plotting library that can be used instead of Plots.jl
  • ParaView – Software that can be used for visualization of results
Common issues
If you followed the installation instructions for developers and you run into any problems with packages when pulling the latest version of TrixiParticles.jl, start Julia with the project in the run folder,
   julia --project=run
update all packages in that project, resolve all conflicts in the project, and install all new dependencies:
julia> using Pkg
+julia --project=run -e 'using Pkg; Pkg.add(["OrdinaryDiffEq", "Plots"])' # Add additional packages
If you installed TrixiParticles.jl this way, you always have to start Julia with the --project flag set to your run directory, e.g.,
julia --project=run
from the TrixiParticles.jl root directory.
The advantage of using a separate run directory is that you can also add other related packages (e.g., OrdinaryDiffEq.jl, see above) to the project in the run folder and always have a reproducible environment at hand to share with others.
Optional software/packages
  • OrdinaryDiffEq.jl – A Julia package of ordinary differential equation solvers that is used in the examples
  • Plots.jl – Julia Plotting library that is used in some examples
  • PythonPlot.jl – Plotting library that can be used instead of Plots.jl
  • ParaView – Software that can be used for visualization of results
Common issues
If you followed the installation instructions for developers and you run into any problems with packages when pulling the latest version of TrixiParticles.jl, start Julia with the project in the run folder,
   julia --project=run
update all packages in that project, resolve all conflicts in the project, and install all new dependencies:
julia> using Pkg
 
 julia> Pkg.update()
 
 julia> Pkg.resolve()
 
-julia> Pkg.instantiate()

+julia> Pkg.instantiate()
diff --git a/previews/PR514/license/index.html b/previews/PR514/license/index.html index ef80ea844..f89052a4c 100644 --- a/previews/PR514/license/index.html +++ b/previews/PR514/license/index.html @@ -1,2 +1,2 @@ -License · TrixiParticles.jl
GitHub

License

MIT License

Copyright (c) 2023-present The TrixiParticles.jl Authors (see Authors)
Copyright (c) 2023-present Helmholtz-Zentrum hereon GmbH, Institute of Surface Science

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

+License · TrixiParticles.jl
GitHub

License

MIT License

Copyright (c) 2023-present The TrixiParticles.jl Authors (see Authors)
Copyright (c) 2023-present Helmholtz-Zentrum hereon GmbH, Institute of Surface Science

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

diff --git a/previews/PR514/news/index.html b/previews/PR514/news/index.html index dee5fd338..a59dea1ae 100644 --- a/previews/PR514/news/index.html +++ b/previews/PR514/news/index.html @@ -1,2 +1,2 @@ -News · TrixiParticles.jl
GitHub

Changelog

TrixiParticles.jl follows the interpretation of semantic versioning (semver) used in the Julia ecosystem. Notable changes will be documented in this file for human readability. We aim at 3 to 4 month between major release versions and about 2 weeks between minor versions.

Version 0.2.x

Highlights

Added

Removed

Deprecated

Version 0.1.2

Highlights

Discrete Element Method

A basic implementation of the discrete element method was added.

Version 0.1.1

Added

A surface tension and adhesion model based on the work by Akinci et al., "Versatile Surface Tension and Adhesion for SPH Fluids", 2013 was added to WCSPH

Pre Initial Release (v0.1.0)

This section summarizes the initial features that TrixiParticles.jl was released with.

Highlights

EDAC

An implementation of EDAC (Entropically Damped Artificial Compressibility) was added, which allows for more stable simulations compared to basic WCSPH and reduces spurious pressure oscillations.

WCSPH

An implementation of WCSPH (Weakly Compressible Smoothed Particle Hydrodynamics), which is the classical SPH approach.

Features:

  • Correction schemes (Shepard (0. Order) ... MixedKernelGradient (1. Order))
  • Density reinitialization
  • Kernel summation and Continuity equation density formulations
  • Flexible boundary conditions e.g. dummy particles with Adami pressure extrapolation, pressure zeroing, pressure mirroring...
  • Moving boundaries
  • Density diffusion based on the models by Molteni & Colagrossi (2009), Ferrari et al. (2009) and Antuono et al. (2010).

TLSPH

An implementation of TLSPH (Total Lagrangian Smoothed Particle Hydrodynamics) for solid bodies enabling FSI (Fluid Structure Interactions).

+News · TrixiParticles.jl
GitHub

Changelog

TrixiParticles.jl follows the interpretation of semantic versioning (semver) used in the Julia ecosystem. Notable changes will be documented in this file for human readability.

Version 0.2.3

Highlights

Transport Velocity Formulation (TVF) based on the work of Ramachandran et al. "Entropically damped artificial compressibility for SPH" (2019) was added.

Version 0.2.2

Highlights

Hotfix for threaded sampling of complex geometries.

Version 0.2.1

Highlights

Particle sampling of complex geometries from .stl and .asc files.

Version 0.2.0

Removed

Use of the internal neighborhood search has been removed and replaced with PointNeighbors.jl.

Development Cycle 0.1

Highlights

Discrete Element Method

A basic implementation of the discrete element method was added.

Surface Tension and Adhesion Model

A surface tension and adhesion model based on the work by Akinci et al., "Versatile Surface Tension and Adhesion for SPH Fluids" (2013) was added to WCSPH.

Support for Open Boundaries

Open boundaries using the method of characteristics based on the work of Lastiwka et al., "Permeable and non-reflecting boundary conditions in SPH" (2009) were added for WCSPH and EDAC.

Pre Initial Release (v0.1.0)

This section summarizes the initial features that TrixiParticles.jl was released with.

Highlights

EDAC

An implementation of EDAC (Entropically Damped Artificial Compressibility) was added, which allows for more stable simulations compared to basic WCSPH and reduces spurious pressure oscillations.

WCSPH

An implementation of WCSPH (Weakly Compressible Smoothed Particle Hydrodynamics), which is the classical SPH approach.

Features:

  • Correction schemes (Shepard (0. Order) ... MixedKernelGradient (1. Order))
  • Density reinitialization
  • Kernel summation and Continuity equation density formulations
  • Flexible boundary conditions e.g. dummy particles with Adami pressure extrapolation, pressure zeroing, pressure mirroring...
  • Moving boundaries
  • Density diffusion based on the models by Molteni & Colagrossi (2009), Ferrari et al. (2009) and Antuono et al. (2010).

TLSPH

An implementation of TLSPH (Total Lagrangian Smoothed Particle Hydrodynamics) for solid bodies enabling FSI (Fluid Structure Interactions).

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GitHub

Overview

The following page gives a rough overview of important parts of the code.

Program flow

To initiate a simulation, the goal is to solve an ordinary differential equation, for example, by employing the time integration schemes provided by OrdinaryDiffEq.jl. These schemes are then utilized to integrate $\mathrm{d}u/\mathrm{d}t$ and $\mathrm{d}v/\mathrm{d}t$, where $u$ represents the particles' positions and $v$ their properties such as velocity and density. During a single time step or an intermediate step of the time integration scheme, the functions drift! and kick! are invoked, followed by the functions depicted in this diagram (with key parts highlighted in orange/yellow).

Main Program Flow

Structure

What we refer to as schemes are various models such as Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) or Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH). These schemes are categorized based on the applicable physical regimes, namely fluid, solid, gas, and others. Each scheme comprises at least two files: a system.jl file and an rhs.jl file. The system.jl file provides the data structure holding the particles of this scheme and some routines, particularly those for allocation and the main update routines, excluding system interactions. The interactions between particles of this scheme (and with particles of other schemes) are handled in the rhs.jl file.

+Overview · TrixiParticles.jl
GitHub

Overview

The actual API reference is not listed on a single page, like in most Julia packages, but instead is split into multiple sections that follow a similar structure as the code files themselves. In these sections, API docs are combined with explanations of the theoretical background of these methods.

The following page gives a rough overview of important parts of the code.

Program flow

To initiate a simulation, the goal is to solve an ordinary differential equation, for example, by employing the time integration schemes provided by OrdinaryDiffEq.jl. These schemes are then utilized to integrate $\mathrm{d}u/\mathrm{d}t$ and $\mathrm{d}v/\mathrm{d}t$, where $u$ represents the particles' positions and $v$ their properties such as velocity and density. During a single time step or an intermediate step of the time integration scheme, the functions drift! and kick! are invoked, followed by the functions depicted in this diagram (with key parts highlighted in orange/yellow).

Main Program Flow

Structure

What we refer to as schemes are various models such as Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) or Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH). These schemes are categorized based on the applicable physical regimes, namely fluid, solid, gas, and others. Each scheme comprises at least two files: a system.jl file and an rhs.jl file. The system.jl file provides the data structure holding the particles of this scheme and some routines, particularly those for allocation and the main update routines, excluding system interactions. The interactions between particles of this scheme (and with particles of other schemes) are handled in the rhs.jl file.

diff --git a/previews/PR514/preprocessing/preprocessing/index.html b/previews/PR514/preprocessing/preprocessing/index.html new file mode 100644 index 000000000..3873a4baf --- /dev/null +++ b/previews/PR514/preprocessing/preprocessing/index.html @@ -0,0 +1,52 @@ + +Sampling of Geometries · TrixiParticles.jl
GitHub

Sampling of Geometries

Generating the initial configuration of a simulation requires filling volumes (3D) or surfaces (2D) of complex geometries with particles. The algorithm to sample a complex geometry should be robust and fast, since for large problems (large numbers of particles) or complex geometries (many geometry faces), generating the initial configuration is not trivial and can be very expensive in terms of computational cost. We therefore use a winding number approach for an inside-outside segmentation of an object. The winding number $w(\mathbf{p})$ is a signed integer-valued function of a point $\mathbf{p}$ and is defined as

\[w(\mathbf{p}) = \frac{1}{2 \pi} \sum^n_{i=1} \Theta_i.\]

Here, $\Theta_i$ is the signed angle between $\mathbf{c}_i - \mathbf{p}$ and $\mathbf{c}_{i+1} - \mathbf{p}$ where $\mathbf{c}_i$ and $\mathbf{c}_{i+1}$ are two consecutive vertices on a curve. In 3D, we refer to the solid angle of an oriented triangle with respect to $\mathbf{p}$.

We provide the following methods to calculate $w(\mathbf{p})$:

Hierarchical Winding

According to Jacobson et al. (2013) the winding number with respect to a polygon (2D) or triangle mesh (3D) is the sum of the winding numbers with respect to each edge (2D) or face (3D). We can show this with the following example in which we determine the winding number for each edge of a triangle separately and sum them up:

using TrixiParticles
+using Plots
+
+triangle = [125.0 375.0 250.0 125.0;
+            175.0 175.0 350.0 175.0]
+
+# Delete all edges but one
+edge1 = deleteat!(TrixiParticles.Polygon(triangle), [2, 3])
+edge2 = deleteat!(TrixiParticles.Polygon(triangle), [1, 3])
+edge3 = deleteat!(TrixiParticles.Polygon(triangle), [1, 2])
+
+algorithm = WindingNumberJacobson()
+
+grid = hcat(([x, y] for x in 1:500, y in 1:500)...)
+
+_, w1 = algorithm(edge1, grid; store_winding_number=true)
+_, w2 = algorithm(edge2, grid; store_winding_number=true)
+_, w3 = algorithm(edge3, grid; store_winding_number=true)
+
+w = w1 + w2 + w3
+
+heatmap(1:500, 1:500, reshape(w1, 500, 500)', color=:coolwarm, showaxis=false,
+        tickfontsize=12, size=(570, 500), margin=6 * Plots.mm)
+heatmap(1:500, 1:500, reshape(w2, 500, 500)', color=:coolwarm, showaxis=false,
+        tickfontsize=12, size=(570, 500), margin=6 * Plots.mm)
+heatmap(1:500, 1:500, reshape(w3, 500, 500)', color=:coolwarm, showaxis=false,
+        tickfontsize=12, size=(570, 500), margin=6 * Plots.mm)
+heatmap(1:500, 1:500, reshape(w, 500, 500)', color=:coolwarm, showaxis=false,
+        tickfontsize=12, size=(570, 500), margin=6 * Plots.mm, clims=(-1, 1))
+
+ triangle +

This summation property has some interesting consequences that we can utilize for an efficient computation of the winding number. Let $\mathcal{S}$ be an open surface and $\bar{\mathcal{S}}$ an arbitrary closing surface, such that

\[\partial \bar{\mathcal{S}} = \partial \mathcal{S}\]

and $\mathcal{B} = \bar{\mathcal{S}} \cup \mathcal{S}$ is some closed oriented surface. For any query point $\mathbf{p}$ outside of $\mathcal{B}$, we know that

\[w_{\mathcal{S}}(\mathbf{p}) + w_{\bar{\mathcal{S}}}(\mathbf{p}) = w_{\mathcal{B}}(\mathbf{p}) = 0.\]

This means

\[w_{\mathcal{S}}(\mathbf{p}) = - w_{\bar{\mathcal{S}}}(\mathbf{p}),\]

regardless of how $\bar{\mathcal{S}}$ is constructed (as long as $\mathbf{p}$ is outside of $\mathcal{B}$).

We can use this property in the discrete case to efficiently compute the winding number of a query point by partitioning the polygon or mesh in a "small" part (as in consisting of a small number of edges/faces) and a "large" part. For the small part we just compute the winding number, and for the large part we construct a small closing and compute its winding number. The partitioning is based on a hierarchical construction of bounding boxes.

Bounding volume hierarchy

To efficiently find a "small part" and a "large part" as mentioned above, we construct a hierarchy of bounding boxes by starting with the whole domain and recursively splitting it in two equally sized boxes. The resulting hierarchy is a binary tree.

The algorithm by Jacobsen et al. (Algorithm 2, p. 5) traverses this binary tree recursively until we find the leaf in which the query point is located. The recursion stops with the following criteria:

  • if the bounding box $T$ is a leaf then $T.\mathcal{S} = \mathcal{S} \cap T$, the part of $\mathcal{S}$ that lies inside $T$, is the "small part" mentioned above, so evaluate the winding number naively as $w(\mathbf{p}, T.\mathcal{S})$.
  • else if $\mathbf{p}$ is outside $T$ then $T.\mathcal{S}$ is the "large part", so evaluate the winding number naively as $-w(\mathbf{p}, T.\bar{\mathcal{S}})$, where $T.\bar{\mathcal{S}}$ is the closing surface of $T.\mathcal{S}$.

Continuous example

Now consider the following continuous (not discretized to a polygon) 2D example. We compute the winding number of the point $\mathbf{p}$ with respect to $\mathcal{S}$ using the depicted hierarchy of bounding boxes.

+ continuous closing +

(1):

  • Recurse left: $w_{\text{left}} = \text{\texttt{hierarchical\_winding}} (\mathbf{p}, T.\text{left})$
  • Recurse right: $w_{\text{right}} = \text{\texttt{hierarchical\_winding}} (\mathbf{p},T.\text{right})$

(2):

  • Query point $\mathbf{p}$ is outside bounding box $T$, so don't recurse deeper.
  • Compute $w_{\mathcal{S}}(\mathbf{p}) = - w_{\bar{\mathcal{S}}}(\mathbf{p})$ with the closure $T.\bar{\mathcal{S}}$, which is generally much smaller (fewer edges in the discrete version) than $T.\mathcal{S}$:

\[w_{\text{left}} = -\text{\texttt{naive\_winding}} (\mathbf{p}, T.\bar{\mathcal{S}})\]

(3):

  • Bounding box $T$ is a leaf. Use open surface $T.\mathcal{S}$:

\[w_{\text{right}} = \text{\texttt{naive\_winding}} (\mathbf{p}, T.\mathcal{S})\]

The reconstructed surface will then look as in the following image.

+ reconstructed surface +

We finally sum up the winding numbers

\[w = w_{\text{left}} + w_{\text{right} } = -w_{T_{\text{left}}.\bar{\mathcal{S}}} + w_{T_{\text{right}}.\mathcal{S}}\]

Discrete example

We will now go through the discrete version of the example above.

+ discrete geometry +

To construct the hierarchy for the discrete piecewise-linear example in (1), we have to do the following.

(2): Each edge is distributed to the child whose box contains the edge's barycenter (red dots in (2)). Splitting stops when the number of a box's edges slips below a threshold (usually $\approx 100$ faces in 3D, here: 6 edges).

(3): For the closure, Jacobson et al. (2013) define exterior vertices (exterior edges in 3D) as boundary vertices of such a segmentation (red dots in (3)). To find them, we traverse around each edge (face in 3D) in order, and increment or decrement for each vertex (edge) a specific counter.

v1 = edge_vertices_ids[edge][1]
+v2 = edge_vertices_ids[edge][2]
+
+vertex_count[v1] += 1
+vertex_count[v2] -= 1

In 2D, a vertex is declared as exterior if vertex_count(vertex) != 0, so there is not the same amount of edges in this box going into versus out of the vertex. To construct the closing surface, the exterior vertices are then connected to one arbitrary exterior vertex using appropriately oriented line segments:

edge = vertex_count[v] > 0 ? (closing_vertex, v) : (v, closing_vertex)

The resulting closed surface $T.S \cup T.\bar{S}$ then has the same number of edges going into and out of each vertex.

Incorrect evaluation

If we follow the algorithm, we know that recursion stops if

  • the bounding box $T$ is a leaf or
  • the query point $\mathbf{p}$ is outside the box.
+ incorrect evaluation +

(1): The query point $\mathbf{p}$ is outside the box, so we calculate the winding number with the (red) closure of the box.

(2): The query point $\mathbf{p}$ is inside the box, so we use the (blue) edges distributed to the box.

(3): In this case, it leads to an incorrect evaluation of the winding number. The query point is clearly inside the box, but not inside the reconstructed surface. This is because the property $w_{\mathcal{S}}(\mathbf{p}) = - w_{\bar{\mathcal{S}}}(\mathbf{p})$ only holds when $\mathbf{p}$ is outside of $\mathcal{B}$, which is not the case here.

Correct evaluation

Jacobson et al. (2013) don't mention this problem or provide a solution to it. We contacted the authors and found that they know about this problem and solve it by resizing the bounding box to fully include the closing surface of the neighboring box, since it doesn't matter if the boxes overlap.

+ correct evaluation resizing +

To avoid resizing, we take a different approach and calculate the closure of the bounding box differently:

  • Exclude intersecting edges in the calculation of the exterior vertices.
  • This way, all exterior vertices are inside the bounding box, and so will be the closing surface.
  • The intersecting edges are later added with flipped orientation, so that the closing is actually a closing of the exterior plus intersecting edges.
+ correct evaluation intersecting +

The evaluation then looks as follows.

+ correct evaluation intersecting 2 +
TrixiParticles.WindingNumberHormann — Type
WindingNumberHormann()

Algorithm for inside-outside segmentation of a complex geometry proposed by Hormann (2001). It is only supported for 2D geometries. WindingNumberHormann might handle edge cases a bit better, since the winding number is an integer value.

Experimental Implementation

This is an experimental feature and may change in any future releases.

source
TrixiParticles.WindingNumberJacobson — Type
WindingNumberJacobson(; geometry=nothing, winding_number_factor=sqrt(eps()),
+                      hierarchical_winding=false)

Algorithm for inside-outside segmentation of a complex geometry proposed by [2].

Keywords

  • geometry: Complex geometry returned by load_geometry and is only required when using hierarchical_winding=true.
  • hierarchical_winding: If set to true, an optimized hierarchical approach will be used, which gives a significant speedup. For further information see Hierarchical Winding.
  • winding_number_factor: For leaky geometries, a factor of 0.4 will give a better inside-outside segmentation.
Experimental Implementation

This is an experimental feature and may change in any future releases.

source
TrixiParticles.load_geometry — Method
load_geometry(filename; element_type=Float64)

Load file and return corresponding type for ComplexShape. Supported file formats are .stl and .asc.

Arguments

  • filename: Name of the file to be loaded.

Keywords

  • element_type: Element type (default is Float64)
source
diff --git a/previews/PR514/reference-pointneighbors/index.html b/previews/PR514/reference-pointneighbors/index.html new file mode 100644 index 000000000..e36de314d --- /dev/null +++ b/previews/PR514/reference-pointneighbors/index.html @@ -0,0 +1,19 @@ + +PointNeighbors.jl API Reference · TrixiParticles.jl
GitHub

PointNeighbors.jl API

PointNeighbors.DictionaryCellList — Type
DictionaryCellList{NDIMS}()

A simple cell list implementation where a cell index (i, j) or (i, j, k) is mapped to a Vector{Int} by a Dict. By using a dictionary, which only stores non-empty cells, the domain is potentially infinite.

This implementation is very simple, but it neither uses an optimized hash function for integer tuples, nor does it use a contiguous memory layout. Consequently, this cell list is not GPU-compatible.

Arguments

  • NDIMS: Number of dimensions.
source
PointNeighbors.FullGridCellList — Type
FullGridCellList(; min_corner, max_corner, search_radius = 0.0,
+                 periodicity = false, backend = DynamicVectorOfVectors{Int32},
+                 max_points_per_cell = 100)

A simple cell list implementation where each (empty or non-empty) cell of a rectangular (axis-aligned) domain is assigned a list of points. This cell list only works when all points are inside the specified domain at all times.

Only set min_corner and max_corner and use the default values for the other arguments to create an empty "template" cell list that can be used to create an empty "template" neighborhood search. See copy_neighborhood_search for more details.

Keywords

  • min_corner: Coordinates of the domain corner in negative coordinate directions.
  • max_corner: Coordinates of the domain corner in positive coordinate directions.
  • search_radius = 0.0: Search radius of the neighborhood search, which will determine the cell size. Use the default of 0.0 to create a template (see above).
  • periodicity = false: Set to true when using a PeriodicBox with the neighborhood search. When using copy_neighborhood_search, this option can be ignored an will be set automatically depending on the periodicity of the neighborhood search.
  • backend = DynamicVectorOfVectors{Int32}: Type of the data structure to store the actual cell lists. Can be
    • Vector{Vector{Int32}}: Scattered memory, but very memory-efficient.
    • DynamicVectorOfVectors{Int32}: Contiguous memory, optimizing cache-hits.
  • max_points_per_cell = 100: Maximum number of points per cell. This will be used to allocate the DynamicVectorOfVectors. It is not used with the Vector{Vector{Int32}} backend.
source
PointNeighbors.GridNeighborhoodSearch — Type
GridNeighborhoodSearch{NDIMS}(; search_radius = 0.0, n_points = 0,
+                              periodic_box = nothing,
+                              cell_list = DictionaryCellList{NDIMS}(),
+                              update_strategy = nothing)

Simple grid-based neighborhood search with uniform search radius. The domain is divided into a regular grid. For each (non-empty) grid cell, a list of points in this cell is stored. Instead of representing a finite domain by an array of cells, a potentially infinite domain is represented by storing cell lists in a hash table (using Julia's Dict data structure), indexed by the cell index tuple

\[\left( \left\lfloor \frac{x}{d} \right\rfloor, \left\lfloor \frac{y}{d} \right\rfloor \right) \quad \text{or} \quad +\left( \left\lfloor \frac{x}{d} \right\rfloor, \left\lfloor \frac{y}{d} \right\rfloor, \left\lfloor \frac{z}{d} \right\rfloor \right),\]

where $x, y, z$ are the space coordinates and $d$ is the search radius.

To find points within the search radius around a position, only points in the neighboring cells are considered.

See also (Chalela et al., 2021), (Ihmsen et al. 2011, Section 4.4).

As opposed to (Ihmsen et al. 2011), we do not sort the points in any way, since not sorting makes our implementation a lot faster (although less parallelizable).

Arguments

  • NDIMS: Number of dimensions.

Keywords

  • search_radius = 0.0: The fixed search radius. The default of 0.0 is useful together with copy_neighborhood_search.
  • n_points = 0: Total number of points. The default of 0 is useful together with copy_neighborhood_search.
  • periodic_box = nothing: In order to use a (rectangular) periodic domain, pass a PeriodicBox.
  • cell_list: The cell list that maps a cell index to a list of points inside the cell. By default, a DictionaryCellList is used.
  • update_strategy = nothing: Strategy to parallelize update!. Available options are:

References

  • M. Chalela, E. Sillero, L. Pereyra, M.A. Garcia, J.B. Cabral, M. Lares, M. Merchán. "GriSPy: A Python package for fixed-radius nearest neighbors search". In: Astronomy and Computing 34 (2021). doi: 10.1016/j.ascom.2020.100443
  • Markus Ihmsen, Nadir Akinci, Markus Becker, Matthias Teschner. "A Parallel SPH Implementation on Multi-Core CPUs". In: Computer Graphics Forum 30.1 (2011), pages 99–112. doi: 10.1111/J.1467-8659.2010.01832.X
source
PointNeighbors.ParallelUpdate — Type
ParallelUpdate()

Fully parallel update by using atomic operations to avoid race conditions when adding points into the same cell. This is not available for all cell list implementations, but is the default when available.

See GridNeighborhoodSearch for usage information.

source
PointNeighbors.PeriodicBox — Type
PeriodicBox(; min_corner, max_corner)

Define a rectangular (axis-aligned) periodic domain.

Keywords

  • min_corner: Coordinates of the domain corner in negative coordinate directions.
  • max_corner: Coordinates of the domain corner in positive coordinate directions.
source
PointNeighbors.PrecomputedNeighborhoodSearch — Type
PrecomputedNeighborhoodSearch{NDIMS}(; search_radius = 0.0, n_points = 0,
+                                     periodic_box = nothing, update_strategy = nothing)

Neighborhood search with precomputed neighbor lists. A list of all neighbors is computed for each point during initialization and update. This neighborhood search maximizes the performance of neighbor loops at the cost of a much slower update!.

A GridNeighborhoodSearch is used internally to compute the neighbor lists during initialization and update.

Arguments

  • NDIMS: Number of dimensions.

Keywords

  • search_radius = 0.0: The fixed search radius. The default of 0.0 is useful together with copy_neighborhood_search.
  • n_points = 0: Total number of points. The default of 0 is useful together with copy_neighborhood_search.
  • periodic_box = nothing: In order to use a (rectangular) periodic domain, pass a PeriodicBox.
  • update_strategy: Strategy to parallelize update! of the internally used GridNeighborhoodSearch. See GridNeighborhoodSearch for available options.
source
PointNeighbors.SemiParallelUpdate — Type
SemiParallelUpdate()

Loop over all cells in parallel to mark cells with points that now belong to a different cell. Then, move points of affected cells serially to avoid race conditions. This is available for all cell list implementations and is the default when ParallelUpdate is not available.

See GridNeighborhoodSearch for usage information.

source
PointNeighbors.SerialUpdate — Type
SerialUpdate()

Deactivate parallelization in the neighborhood search update. Parallel neighborhood search update can be one of the largest sources of error variations between simulations with different thread numbers due to neighbor ordering changes.

See GridNeighborhoodSearch for usage information.

source
PointNeighbors.TrivialNeighborhoodSearch — Type
TrivialNeighborhoodSearch{NDIMS}(; search_radius = 0.0, eachpoint = 1:0,
+                                 periodic_box = nothing)

Trivial neighborhood search that simply loops over all points.

Arguments

  • NDIMS: Number of dimensions.

Keywords

  • search_radius = 0.0: The fixed search radius. The default of 0.0 is useful together with copy_neighborhood_search.
  • eachpoint = 1:0: Iterator for all point indices. Usually just 1:n_points. The default of 1:0 is useful together with copy_neighborhood_search.
  • periodic_box = nothing: In order to use a (rectangular) periodic domain, pass a PeriodicBox.
source
PointNeighbors.copy_neighborhood_search — Method
copy_neighborhood_search(search::AbstractNeighborhoodSearch, search_radius, n_points;
+                         eachpoint = 1:n_points)

Create a new uninitialized neighborhood search of the same type and with the same configuration options as search, but with a different search radius and number of points.

The TrivialNeighborhoodSearch also requires an iterator eachpoint, which most of the time will be 1:n_points. If the TrivialNeighborhoodSearch is never going to be used, the keyword argument eachpoint can be ignored.

This is useful when a simulation code requires multiple neighborhood searches of the same kind. One can then just pass an empty neighborhood search as a template and use this function inside the simulation code to generate similar neighborhood searches with different search radii and different numbers of points.

# Template
+nhs = GridNeighborhoodSearch{2}()
+
+# Inside the simulation code, generate similar neighborhood searches
+nhs1 = copy_neighborhood_search(nhs, 1.0, 100)
+
+# output
+GridNeighborhoodSearch{2, Float64, ...}(...)
source
PointNeighbors.foreach_point_neighbor — Method
foreach_point_neighbor(f, system_coords, neighbor_coords, neighborhood_search;
+                       points = axes(system_coords, 2), parallel = true)

Loop for each point in system_coords over all points in neighbor_coords whose distances to that point are smaller than the search radius and execute the function f(i, j, x, y, d), where

  • i is the column index of the point in system_coords,
  • j the column index of the neighbor in neighbor_coords,
  • x an SVector of the coordinates of the point (system_coords[:, i]),
  • y an SVector of the coordinates of the neighbor (neighbor_coords[:, j]),
  • d the distance between x and y.

The neighborhood_search must have been initialized or updated with system_coords as first coordinate array and neighbor_coords as second coordinate array.

Note that system_coords and neighbor_coords can be identical.

Arguments

  • f: The function explained above.
  • system_coords: A matrix where the i-th column contains the coordinates of point i.
  • neighbor_coords: A matrix where the j-th column contains the coordinates of point j.
  • neighborhood_search: A neighborhood search initialized or updated with system_coords as first coordinate array and neighbor_coords as second coordinate array.

Keywords

  • points: Loop over these point indices. By default all columns of system_coords.
  • parallel=true: Run the outer loop over points thread-parallel.

See also initialize!, update!.

source
PointNeighbors.initialize! — Method
initialize!(search::AbstractNeighborhoodSearch, x, y)

Initialize a neighborhood search with the two coordinate arrays x and y.

In general, the purpose of a neighborhood search is to find for one point in x all points in y whose distances to that point are smaller than the search radius. x and y are expected to be matrices, where the i-th column contains the coordinates of point i. Note that x and y can be identical.

See also update!.

source
PointNeighbors.update! — Method
update!(search::AbstractNeighborhoodSearch, x, y; points_moving = (true, true))

Update an already initialized neighborhood search with the two coordinate arrays x and y.

Like initialize!, but reusing the existing data structures of the already initialized neighborhood search. When the points only moved a small distance since the last update! or initialize!, this is significantly faster than initialize!.

Not all implementations support incremental updates. If incremental updates are not possible for an implementation, update! will fall back to a regular initialize!.

Some neighborhood searches might not need to update when only x changed since the last update! or initialize! and y did not change. Pass points_moving = (true, false) in this case to avoid unnecessary updates. The first flag in points_moving indicates if points in x are moving. The second flag indicates if points in y are moving.

Experimental Feature: Backend Specification

The keyword argument parallelization_backend allows users to specify the multithreading backend. This feature is currently considered experimental!

Possible parallelization backends are:

See also initialize!.

source
PointNeighbors.@threaded — Macro
@threaded x for ... end

Run either a threaded CPU loop or launch a kernel on the GPU, depending on the type of x. Semantically the same as Threads.@threads when iterating over a AbstractUnitRange but without guarantee that the underlying implementation uses Threads.@threads or works for more general for loops.

The first argument must either be a parallelization backend (see below) or an array from which the backend can be derived to determine if the loop must be run threaded on the CPU or launched as a kernel on the GPU. Passing KernelAbstractions.CPU() will run the GPU kernel on the CPU.

Possible parallelization backends are:

In particular, the underlying threading capabilities might be provided by other packages such as Polyester.jl.

Warning

This macro does not necessarily work for general for loops. For example, it does not necessarily support general iterables such as eachline(filename).

source
diff --git a/previews/PR514/reference-trixibase/index.html b/previews/PR514/reference-trixibase/index.html index 74aa4ae34..e95d2a9ce 100644 --- a/previews/PR514/reference-trixibase/index.html +++ b/previews/PR514/reference-trixibase/index.html @@ -1,5 +1,5 @@ -TrixiBase.jl API Reference · TrixiParticles.jl
GitHub

TrixiBase.jl API

TrixiBase.trixi_include — Method
trixi_include([mod::Module=Main,] elixir::AbstractString; kwargs...)

include the file elixir and evaluate its content in the global scope of module mod. You can override specific assignments in elixir by supplying keyword arguments. Its basic purpose is to make it easier to modify some parameters while running simulations from the REPL. Additionally, this is used in tests to reduce the computational burden for CI while still providing examples with sensible default values for users.

Before replacing assignments in elixir, the keyword argument maxiters is inserted into calls to solve with it's default value used in the SciML ecosystem for ODEs, see the "Miscellaneous" section of the documentation.

Examples

julia> using TrixiBase, Trixi
+TrixiBase.jl API Reference · TrixiParticles.jl
GitHub
TrixiBase.jl API
TrixiBase.trixi_include — Method
trixi_include([mod::Module=Main,] elixir::AbstractString; kwargs...)
include the file elixir and evaluate its content in the global scope of module mod. You can override specific assignments in elixir by supplying keyword arguments. Its basic purpose is to make it easier to modify some parameters while running simulations from the REPL. Additionally, this is used in tests to reduce the computational burden for CI while still providing examples with sensible default values for users.
Before replacing assignments in elixir, the keyword argument maxiters is inserted into calls to solve with it's default value used in the SciML ecosystem for ODEs, see the "Miscellaneous" section of the documentation.
Examples
julia> using TrixiBase, Trixi
 
 julia> redirect_stdout(devnull) do
          trixi_include(@__MODULE__, joinpath(examples_dir(), "tree_1d_dgsem", "elixir_advection_extended.jl"),
@@ -7,4 +7,4 @@
          sol.t[end]
        end
 [ Info: You just called `trixi_include`. Julia may now compile the code, please be patient.
-0.1
source

+0.1
source
TrixiBase.@trixi_timeit — Macro
@trixi_timeit timer() "some label" expression

Basically the same as a special case of @timeit_debug from TimerOutputs.jl, but without try ... finally ... end block. Thus, it's not exception-safe, but it also avoids some related performance problems. Since we do not use exception handling in Trixi.jl, that's not really an issue.

All @trixi_timeit timings can be disabled with disable_debug_timings. The timings should then be optimized away, allowing for truly zero-overhead.

See also disable_debug_timings, enable_debug_timings.

source
diff --git a/previews/PR514/references/index.html b/previews/PR514/references/index.html new file mode 100644 index 000000000..4065d809b --- /dev/null +++ b/previews/PR514/references/index.html @@ -0,0 +1,2 @@ + +References · TrixiParticles.jl
GitHub
[1]
K. Hormann and A. Agathos. The point in polygon problem for arbitrary polygons. Computational Geometry 20, 131–144 (2001).
[2]
A. Jacobson, L. Kavan and O. Sorkine-Hornung. Robust inside-outside segmentation using generalized winding numbers. ACM Transactions on Graphics 32, 1–12 (2013).
[3]
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J.}, + journal = {Annual Review of Astronomy and Astrophysics}, + title = {Smoothed Particle Hydrodynamics}, + year = {1992}, + issn = {1545-4282}, + month = sep, + number = {1}, + pages = {543--574}, + volume = {30}, + doi = {10.1146/annurev.aa.30.090192.002551}, + publisher = {Annual Reviews}, +} + + +@Article{Monaghan1994, + author = {Monaghan, J.J.}, + journal = {Journal of Computational Physics}, + title = {Simulating Free Surface Flows with SPH}, + year = {1994}, + issn = {0021-9991}, + month = feb, + number = {2}, + pages = {399--406}, + volume = {110}, + doi = {10.1006/jcph.1994.1034}, + publisher = {Elsevier BV}, +} + +@Article{Monaghan2005, + author = {Monaghan, J J}, + journal = {Reports on Progress in Physics}, + title = {Smoothed particle hydrodynamics}, + year = {2005}, + issn = {1361-6633}, + month = jul, + number = {8}, + pages = {1703--1759}, + volume = {68}, + doi = {10.1088/0034-4885/68/8/r01}, + publisher = {IOP Publishing}, +} + +@Article{Monaghan2009, + author = {Monaghan, J.J. and Kajtar, J.B.}, + journal = {Computer Physics Communications}, + title = {SPH particle boundary forces for arbitrary boundaries}, + year = {2009}, + issn = {0010-4655}, + month = oct, + number = {10}, + pages = {1811--1820}, + volume = {180}, + doi = {10.1016/j.cpc.2009.05.008}, + publisher = {Elsevier BV}, +} + +@Article{Morris1997, + author = {Morris, Joseph P. and Fox, Patrick J. and Zhu, Yi}, + journal = {Journal of Computational Physics}, + title = {Modeling Low Reynolds Number Incompressible Flows Using SPH}, + year = {1997}, + issn = {0021-9991}, + month = sep, + number = {1}, + pages = {214--226}, + volume = {136}, + doi = {10.1006/jcph.1997.5776}, + publisher = {Elsevier BV}, +} + +@InProceedings{Mueller2003, + author = {M{\"u}ller, Matthias and Charypar, David and Gross, Markus}, + booktitle = {Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation}, + title = {Particle-Based Fluid Simulation for Interactive Applications}, + year = {2003}, + month = {07}, + pages = {154-159}, + publisher = {Eurographics Association}, + doi = {10.2312/SCA/SCA03/154-159}, + isbn = {1581136595}, +} + +@Article{Negi2020, + author = {Negi, Pawan and Ramachandran, Prabhu and Haftu, Asmelash}, + journal = {Computer Methods in Applied Mechanics and Engineering}, + title = {An improved non-reflecting outlet boundary condition for weakly-compressible SPH}, + year = {2020}, + issn = {0045-7825}, + month = aug, + pages = {113119}, + volume = {367}, + doi = {10.1016/j.cma.2020.113119}, + publisher = {Elsevier BV}, +} + +@Article{O’Connor2021, + author = {O’Connor, Joseph and Rogers, Benedict D.}, + journal = {Journal of Fluids and Structures}, + title = {A fluid–structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU}, + year = {2021}, + issn = {0889-9746}, + month = jul, + pages = {103312}, + volume = {104}, + doi = {10.1016/j.jfluidstructs.2021.103312}, + publisher = {Elsevier BV}, +} + +@InProceedings{Panizzo2007, + author = {Panizzo, Andrea and Cuomo, Giovanni and Dalrymple, Robert A.}, + booktitle = {Coastal Engineering 2006}, + title = {3D-SPH SIMULATION OF LANDSLIDE GENERATED WAVES}, + year = {2007}, + month = apr, + publisher = {World Scientific Publishing Company}, + doi = {10.1142/9789812709554_0128}, +} + +@Article{Price2012, + author = {Price, Daniel J.}, + journal = {Journal of Computational Physics}, + title = {Smoothed particle hydrodynamics and magnetohydrodynamics}, + year = {2012}, + issn = {0021-9991}, + month = feb, + number = {3}, + pages = {759--794}, + volume = {231}, + doi = {10.1016/j.jcp.2010.12.011}, + publisher = {Elsevier BV}, +} + +@Article{Ramachandran2019, + author = {Ramachandran, Prabhu and Puri, Kunal}, + journal = {Computers \& Fluids}, + title = {Entropically damped artificial compressibility for SPH}, + year = {2019}, + issn = {0045-7930}, + month = jan, + pages = {579--594}, + volume = {179}, + doi = {10.1016/j.compfluid.2018.11.023}, + publisher = {Elsevier BV}, +} + +@Article{Schoenberg1946, + author = {Schoenberg, I. J.}, + journal = {Quarterly of Applied Mathematics}, + title = {Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae}, + year = {1946}, + issn = {1552-4485}, + number = {2}, + pages = {112--141}, + volume = {4}, + doi = {10.1090/qam/16705}, + publisher = {American Mathematical Society (AMS)}, +} + +@Article{Sun2017, + author = {Sun, P.N. and Colagrossi, A. and Marrone, S. and Zhang, A.M.}, + journal = {Computer Methods in Applied Mechanics and Engineering}, + title = {Delta-SPH model: Simple procedures for a further improvement of the SPH scheme}, + year = {2017}, + issn = {0045-7825}, + month = mar, + pages = {25--49}, + volume = {315}, + doi = {10.1016/j.cma.2016.10.028}, + publisher = {Elsevier BV}, +} + + +@Article{Valizadeh2015, + author = {Valizadeh, Alireza and Monaghan, Joseph J.}, + journal = {Journal of Computational Physics}, + title = {A study of solid wall models for weakly compressible SPH}, + year = {2015}, + issn = {0021-9991}, + month = nov, + pages = {5--19}, + volume = {300}, + doi = {10.1016/j.jcp.2015.07.033}, + publisher = {Elsevier BV}, +} + + +@Article{Wendland1995, + author = {Wendland, Holger}, + journal = {Advances in Computational Mathematics}, + title = {Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree}, + year = {1995}, + issn = {1572-9044}, + month = dec, + number = {1}, + pages = {389--396}, + volume = {4}, + doi = {10.1007/bf02123482}, + publisher = {Springer Science and Business Media LLC}, +} + +@Comment{jabref-meta: databaseType:bibtex;} diff --git a/previews/PR514/search_index.js b/previews/PR514/search_index.js index cda810ee4..bf7147eda 100644 --- a/previews/PR514/search_index.js +++ b/previews/PR514/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"systems/dem/#dem","page":"Discrete Element Method (Solid)","title":"Discrete Element Method","text":"","category":"section"},{"location":"systems/dem/","page":"Discrete Element Method (Solid)","title":"Discrete Element Method (Solid)","text":"The Discrete Element Method (DEM) is a computational technique widely used in physics, engineering, and applied mathematics for simulating the mechanical behavior of granular materials, such as powders, sand, soil, or rock, as well as other discontinua. Unlike continuum mechanics that treats materials as continuous, DEM considers individual particles or elements and their interactions. This approach provides detailed insights into the micro-mechanical behavior of materials, making it particularly valuable in fields such as geomechanics, material science, and mechanical engineering.","category":"page"},{"location":"systems/dem/#Fundamental-Principles","page":"Discrete Element Method (Solid)","title":"Fundamental Principles","text":"","category":"section"},{"location":"systems/dem/","page":"Discrete Element Method (Solid)","title":"Discrete Element Method (Solid)","text":"The core idea behind DEM is the discretization of a material system into a finite set of distinct, interacting mass elements (particles). These elements (particles) can vary in shape, size, and properties, and they interact with each other and possibly with their boundaries through contact forces and potential fields. The motion and behavior of each mass element are governed by Newton's laws of motion, accounting for the forces and moments acting upon them.","category":"page"},{"location":"systems/dem/","page":"Discrete Element Method (Solid)","title":"Discrete Element Method (Solid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"solid\", \"discrete_element_method\", \"system.jl\")]","category":"page"},{"location":"systems/dem/#TrixiParticles.DEMSystem","page":"Discrete Element Method (Solid)","title":"TrixiParticles.DEMSystem","text":"DEMSystem(initial_condition, normal_stiffness, elastic_modulus, poissons_ratio;\n damping_coefficient=0.0001, acceleration=ntuple(_ -> 0.0, NDIMS), source_terms=nothing)\n\nConstructs a Discrete Element Method (DEM) system for numerically simulating the dynamics of granular and particulate matter. DEM is employed to simulate and analyze the motion, interactions, and collective behavior of assemblies of discrete, solid particles, typically under mechanical loading. The model accounts for individual particle characteristics and implements interaction laws that govern contact forces (normal and tangential), based on specified material properties and contact mechanics.\n\nArguments\n\ninitial_condition: Initial condition of the system, encapsulating the initial positions, velocities, masses, and radii of particles.\nnormal_stiffness: Normal stiffness coefficient for particle-particle and particle-wall contacts.\nelastic_modulus: Elastic modulus for this particle system.\npoissons_ratio: Poisson ratio for this particle system.\n\nKeywords\n\nacceleration: Global acceleration vector applied to the system, such as gravity. Specified as an SVector of length NDIMS, with a default of zero in each dimension.\nsource_terms: Optional; additional forces or modifications to particle dynamics not captured by standard DEM interactions, such as electromagnetic forces or user-defined perturbations.\ndamping_coefficient=0.0001: Set a damping coefficient for the collision interactions.\n\n!!! warning \"Experimental Implementation\" This is an experimental feature and may change in a future releases.\n\n\n\n\n\n","category":"type"},{"location":"systems/dem/#References","page":"Discrete Element Method (Solid)","title":"References","text":"","category":"section"},{"location":"systems/dem/","page":"Discrete Element Method (Solid)","title":"Discrete Element Method (Solid)","text":"N. Bićanić. \"Discrete element methods\". In: Encyclopedia of Computational Mechanics (2007). doi: 10.1002/0470091355.ecm006.pub2\nP. Cundall and O. Strack. \"A discrete numerical model for granular assemblies\". In: Géotechnique 29.1 (1979), pages 47–65. doi: 10.1680/geot.1979.29.1.47\nA. Renzo and F. Maio. \"Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes\" In: Chemical Engineering Science 59.3 (2004), pages 525–541. doi: 10.1016/j.ces.2003.09.037","category":"page"},{"location":"tutorials/tut_beam_replaced/","page":"Example file","title":"Example file","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials/tut_beam.md\"","category":"page"},{"location":"tutorials/tut_beam_replaced/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_beam_replaced/","page":"Example file","title":"Example file","text":"using TrixiParticles\nusing OrdinaryDiffEq\n\n# ==========================================================================================\n# ==== Resolution\nn_particles_y = 5\n\n# ==========================================================================================\n# ==== Experiment Setup\ngravity = 2.0\ntspan = (0.0, 5.0)\n\nelastic_beam = (length=0.35, thickness=0.02)\nmaterial = (density=1000.0, E=1.4e6, nu=0.4)\nclamp_radius = 0.05\n\n# The structure starts at the position of the first particle and ends\n# at the position of the last particle.\nparticle_spacing = elastic_beam.thickness / (n_particles_y - 1)\n\n# Add particle_spacing/2 to the clamp_radius to ensure that particles are also placed on the radius\nfixed_particles = SphereShape(particle_spacing, clamp_radius + particle_spacing / 2,\n (0.0, elastic_beam.thickness / 2), material.density,\n cutout_min=(0.0, 0.0),\n cutout_max=(clamp_radius, elastic_beam.thickness),\n tlsph=true)\n\nn_particles_clamp_x = round(Int, clamp_radius / particle_spacing)\n\n# Beam and clamped particles\nn_particles_per_dimension = (round(Int, elastic_beam.length / particle_spacing) +\n n_particles_clamp_x + 1, n_particles_y)\n\n# Note that the `RectangularShape` puts the first particle half a particle spacing away\n# from the boundary, which is correct for fluids, but not for solids.\n# We therefore need to pass `tlsph=true`.\nbeam = RectangularShape(particle_spacing, n_particles_per_dimension,\n (0.0, 0.0), density=material.density, tlsph=true)\n\nsolid = union(beam, fixed_particles)\n\n# ==========================================================================================\n# ==== Solid\n# The kernel in the reference uses a differently scaled smoothing length,\n# so this is equivalent to the smoothing length of `sqrt(2) * particle_spacing` used in the paper.\nsmoothing_length = 2 * sqrt(2) * particle_spacing\nsmoothing_kernel = WendlandC2Kernel{2}()\n\nsolid_system = TotalLagrangianSPHSystem(solid, smoothing_kernel, smoothing_length,\n material.E, material.nu,\n n_fixed_particles=nparticles(fixed_particles),\n acceleration=(0.0, -gravity),\n penalty_force=nothing)\n\n# ==========================================================================================\n# ==== Simulation\nsemi = Semidiscretization(solid_system)\node = semidiscretize(semi, tspan)\n\ninfo_callback = InfoCallback(interval=100)\n\n# Track the position of the particle in the middle of the tip of the beam.\nmiddle_particle_id = Int(n_particles_per_dimension[1] * (n_particles_per_dimension[2] + 1) /\n 2)\nstartposition_x = beam.coordinates[1, middle_particle_id]\nstartposition_y = beam.coordinates[2, middle_particle_id]\n\nfunction deflection_x(v, u, t, system)\n return system.current_coordinates[1, middle_particle_id] - startposition_x\nend\n\nfunction deflection_y(v, u, t, system)\n return system.current_coordinates[2, middle_particle_id] - startposition_y\nend\n\nsaving_callback = SolutionSavingCallback(dt=0.02, prefix=\"\",\n deflection_x=deflection_x,\n deflection_y=deflection_y)\n\ncallbacks = CallbackSet(info_callback, saving_callback)\n\n# Use a Runge-Kutta method with automatic (error based) time step size control\nsol = solve(ode, RDPK3SpFSAL49(), save_everystep=false, callback=callbacks);\n\n","category":"page"},{"location":"general/smoothing_kernels/#smoothing_kernel","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"","category":"section"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"The following smoothing kernels are currently available:","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"Smoothing Kernel Compact Support Typ. Smoothing Length Recommended Application Stability\nSchoenbergCubicSplineKernel 0 2h 11 to 13 General + sharp waves ++\nSchoenbergQuarticSplineKernel 0 25h 11 to 15 General +++\nSchoenbergQuinticSplineKernel 0 3h 11 to 15 General ++++\nGaussianKernel 0 3h 10 to 15 Literature +++++\nWendlandC2Kernel 0 1h 25 to 40 General (recommended) ++++\nWendlandC4Kernel 0 1h 30 to 45 General +++++\nWendlandC6Kernel 0 1h 35 to 50 General +++++\nPoly6Kernel 0 1h 15 to 25 Literature +\nSpikyKernel 0 1h 15 to 30 Sharp corners + waves +","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"We recommend to use the WendlandC2Kernel for most applications. If less smoothing is needed, try SchoenbergCubicSplineKernel, for more smoothing try WendlandC6Kernel.","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"note: Usage\nThe kernel can be called asTrixiParticles.kernel(smoothing_kernel, r, h)The length of the compact support can be obtained asTrixiParticles.compact_support(smoothing_kernel, h)Note that r has to be a scalar, so in the context of SPH, the kernel should be used asW(Vert r_a - r_b Vert h)The gradient required in SPH, nabla_r_a W(Vert r_a - r_b Vert h)can be called asTrixiParticles.kernel_grad(smoothing_kernel, pos_diff, distance, h)where pos_diff is r_a - r_b and distance is Vert r_a - r_b Vert.","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"smoothing_kernels.jl\")]","category":"page"},{"location":"general/smoothing_kernels/#TrixiParticles.GaussianKernel","page":"Smoothing Kernels","title":"TrixiParticles.GaussianKernel","text":"GaussianKernel{NDIMS}()\n\nGaussian kernel given by\n\nW(r h) = fracsigma_dh^d e^-r^2h^2\n\nwhere d is the number of dimensions and\n\nsigma_2 = frac1pi for 2D,\nsigma_3 = frac1pi^32 for 3D.\n\nThis kernel function has an infinite support, but in practice, it's often truncated at a certain multiple of h, such as 3h.\n\nIn this implementation, the kernel is truncated at 3h, so this kernel function has a compact support of 0 3h.\n\nThe smoothing length is typically in the range 10delta 15delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nNote: This truncation makes this Kernel not conservative, which is beneficial in regards to stability but makes it less accurate.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.Poly6Kernel","page":"Smoothing Kernels","title":"TrixiParticles.Poly6Kernel","text":"Poly6Kernel{NDIMS}()\n\nPoly6 kernel, a commonly used kernel in SPH literature, especially in computer graphics contexts. It is defined as\n\nW(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (1 - q^2)^3 textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor that depends on the dimension. The normalization factor sigma is 4 pi in two dimensions or 315 64pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nPoly6 is well-known for its computational simplicity, though it's worth noting that there are other kernels that might offer better accuracy for hydrodynamic simulations. Furthermore, its derivatives are not that smooth, which can lead to stability problems. It is also susceptible to clumping.\n\nThe smoothing length is typically in the range 15delta 25delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nMatthias Müller, David Charypar, and Markus Gross. \"Particle-based fluid simulation for interactive applications\". In: Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation. Eurographics Association. 2003, pages 154-159. doi: 10.5555/846276.846298\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SchoenbergCubicSplineKernel","page":"Smoothing Kernels","title":"TrixiParticles.SchoenbergCubicSplineKernel","text":"SchoenbergCubicSplineKernel{NDIMS}()\n\nCubic spline kernel by Schoenberg (Schoenberg, 1946), given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n frac14 (2 - q)^3 - (1 - q)^3 textif 0 leq q 1 \n frac14 (2 - q)^3 textif 1 leq q 2 \n 0 textif q geq 2 \nendcases\n\nwhere d is the number of dimensions and sigma is a normalization constant given by sigma =frac23 frac107 pi frac1pi in 1 2 3 dimensions.\n\nThis kernel function has a compact support of 0 2h.\n\nFor an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see (Price, 2012). For an analytic formula for higher order Schoenberg kernels, see (Monaghan, 1985). The largest disadvantage of Schoenberg Spline Kernel is the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.\n\nThe smoothing length is typically in the range 11delta 13delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nDaniel J. Price. \"Smoothed particle hydrodynamics and magnetohydrodynamics\". In: Journal of Computational Physics 231.3 (2012), pages 759-794. doi: 10.1016/j.jcp.2010.12.011\nJoseph J. Monaghan. \"Particle methods for hydrodynamics\". In: Computer Physics Reports 3.2 (1985), pages 71–124. doi: 10.1016/0167-7977(85)90010-3\nIsaac J. Schoenberg. \"Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae.\" In: Quarterly of Applied Mathematics 4.2 (1946), pages 112–141. doi: 10.1090/QAM/16705\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SchoenbergQuarticSplineKernel","page":"Smoothing Kernels","title":"TrixiParticles.SchoenbergQuarticSplineKernel","text":"SchoenbergQuarticSplineKernel{NDIMS}()\n\nQuartic spline kernel by Schoenberg (Schoenberg, 1946), given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n left(52 - q right)^4 - 5left(32 - q right)^4\n + 10left(12 - q right)^4 textif 0 leq q frac12 \n left(52 - q right)^4 - 5left(32 - q right)^4\n textif frac12 leq q frac32 \n left(52 - q right)^4 textif frac32 leq q frac52 \n 0 textif q geq frac52\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization constant given by sigma =frac124 frac961199 pi frac120pi in 1 2 3 dimensions.\n\nThis kernel function has a compact support of 0 25h.\n\nFor an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see (Price, 2012). For an analytic formula for higher order Schoenberg kernels, see (Monaghan, 1985).\n\nThe largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.\n\nThe smoothing length is typically in the range 11delta 15delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nDaniel J. Price. \"Smoothed particle hydrodynamics and magnetohydrodynamics\". In: Journal of Computational Physics 231.3 (2012), pages 759-794. doi: 10.1016/j.jcp.2010.12.011\nJoseph J. Monaghan. \"Particle methods for hydrodynamics\". In: Computer Physics Reports 3.2 (1985), pages 71–124. doi: 10.1016/0167-7977(85)90010-3\nIsaac J. Schoenberg. \"Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae.\" In: Quarterly of Applied Mathematics 4.2 (1946), pages 112–141. doi: 10.1090/QAM/16705\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SchoenbergQuinticSplineKernel","page":"Smoothing Kernels","title":"TrixiParticles.SchoenbergQuinticSplineKernel","text":"SchoenbergQuinticSplineKernel{NDIMS}()\n\nQuintic spline kernel by Schoenberg (Schoenberg, 1946), given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (3 - q)^5 - 6(2 - q)^5 + 15(1 - q)^5 textif 0 leq q 1 \n (3 - q)^5 - 6(2 - q)^5 textif 1 leq q 2 \n (3 - q)^5 textif 2 leq q 3 \n 0 textif q geq 3\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization constant given by sigma =frac1120 frac7478 pi frac1120pi in 1 2 3 dimensions.\n\nThis kernel function has a compact support of 0 3h.\n\nFor an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see (Price, 2012). For an analytic formula for higher order Schoenberg kernels, see (Monaghan, 1985).\n\nThe largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.\n\nThe smoothing length is typically in the range 11delta 15delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nDaniel J. Price. \"Smoothed particle hydrodynamics and magnetohydrodynamics\". In: Journal of Computational Physics 231.3 (2012), pages 759-794. doi: 10.1016/j.jcp.2010.12.011\nJoseph J. Monaghan. \"Particle methods for hydrodynamics\". In: Computer Physics Reports 3.2 (1985), pages 71–124. doi: 10.1016/0167-7977(85)90010-3\nIsaac J. Schoenberg. \"Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae.\" In: Quarterly of Applied Mathematics 4.2 (1946), pages 112–141. doi: 10.1090/QAM/16705\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SpikyKernel","page":"Smoothing Kernels","title":"TrixiParticles.SpikyKernel","text":"SpikyKernel{NDIMS}()\n\nThe Spiky kernel is another frequently used kernel in SPH, especially due to its desirable properties in preserving features near boundaries in fluid simulations. It is defined as:\n\n W(r h) = frac1h^d w(rh)\n\nwith:\n\nw(q) = sigma begincases\n (1 - q)^3 textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and the normalization factor sigma is 10 pi in two dimensions or 15 pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nThe Spiky kernel is particularly known for its sharp gradients, which can help to preserve sharp features in fluid simulations, especially near solid boundaries. These sharp gradients at the boundary are also the largest disadvantage as they can lead to instability.\n\nThe smoothing length is typically in the range 15delta 30delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nMatthias Müller, David Charypar, and Markus Gross. \"Particle-based fluid simulation for interactive applications\". In: Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation. Eurographics Association. 2003, pages 154-159. doi: 10.5555/846276.846298\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.WendlandC2Kernel","page":"Smoothing Kernels","title":"TrixiParticles.WendlandC2Kernel","text":"WendlandC2Kernel{NDIMS}()\n\nWendland C2 kernel (Wendland, 1995), a piecewise polynomial function designed to have compact support and to be twice continuously differentiable everywhere. Given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (1 - q)^4 (4q + 1) textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor dependent on the dimension. The normalization factor sigma is 407pi in two dimensions or 212pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nFor a detailed discussion on Wendland functions and their applications in SPH, see (Dehnen & Aly, 2012). The smoothness of these functions is also the largest disadvantage as they lose details at sharp corners.\n\nThe smoothing length is typically in the range 25delta 40delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nWalter Dehnen & Hassan Aly. \"Improving convergence in smoothed particle hydrodynamics simulations without pairing instability\". In: Monthly Notices of the Royal Astronomical Society 425.2 (2012), pages 1068-1082. doi: 10.1111/j.1365-2966.2012.21439.x\nHolger Wendland. \"Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree.\" In: Advances in computational Mathematics 4 (1995), pages 389-396. doi: 10.1007/BF02123482\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.WendlandC4Kernel","page":"Smoothing Kernels","title":"TrixiParticles.WendlandC4Kernel","text":"WendlandC4Kernel{NDIMS}()\n\nWendland C4 kernel, a piecewise polynomial function designed to have compact support and to be four times continuously differentiable everywhere. Given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (1 - q)^6 (35q^2 3 + 6q + 1) textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor dependent on the dimension. The normalization factor sigma is 9 pi in two dimensions or 495 32pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nFor a detailed discussion on Wendland functions and their applications in SPH, see (Dehnen & Aly, 2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.\n\nThe smoothing length is typically in the range 30delta 45delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nWalter Dehnen & Hassan Aly. \"Improving convergence in smoothed particle hydrodynamics simulations without pairing instability\". In: Monthly Notices of the Royal Astronomical Society 425.2 (2012), pages 1068-1082. doi: 10.1111/j.1365-2966.2012.21439.x\nHolger Wendland. \"Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree.\" In: Advances in computational Mathematics 4 (1995): 389-396. doi: 10.1007/BF02123482\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.WendlandC6Kernel","page":"Smoothing Kernels","title":"TrixiParticles.WendlandC6Kernel","text":"WendlandC6Kernel{NDIMS}()\n\nWendland C6 kernel, a piecewise polynomial function designed to have compact support and to be six times continuously differentiable everywhere. Given by:\n\nW(r h) = frac1h^d w(rh)\n\nwith:\n\nw(q) = sigma begincases\n (1 - q)^8 (32q^3 + 25q^2 + 8q + 1) textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor dependent on the dimension. The normalization factor sigma is 78 7 pi in two dimensions or 1365 64pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nFor a detailed discussion on Wendland functions and their applications in SPH, see (Dehnen & Aly, 2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.\n\nThe smoothing length is typically in the range 35delta 50delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nReferences\n\nWalter Dehnen & Hassan Aly. \"Improving convergence in smoothed particle hydrodynamics simulations without pairing instability\". In: Monthly Notices of the Royal Astronomical Society 425.2 (2012), pages 1068-1082. doi: 10.1111/j.1365-2966.2012.21439.x\nHolger Wendland. \"Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree.\" In: Advances in computational Mathematics 4 (1995): 389-396. doi: 10.1007/BF02123482\n\n\n\n\n\n","category":"type"},{"location":"reference-trixibase/#TrixiBase.jl-API","page":"TrixiBase.jl API Reference","title":"TrixiBase.jl API","text":"","category":"section"},{"location":"reference-trixibase/","page":"TrixiBase.jl API Reference","title":"TrixiBase.jl API Reference","text":"CurrentModule = TrixiBase","category":"page"},{"location":"reference-trixibase/","page":"TrixiBase.jl API Reference","title":"TrixiBase.jl API Reference","text":"Modules = [TrixiBase]","category":"page"},{"location":"reference-trixibase/#TrixiBase.trixi_include-Tuple{Module, AbstractString}","page":"TrixiBase.jl API Reference","title":"TrixiBase.trixi_include","text":"trixi_include([mod::Module=Main,] elixir::AbstractString; kwargs...)\n\ninclude the file elixir and evaluate its content in the global scope of module mod. You can override specific assignments in elixir by supplying keyword arguments. Its basic purpose is to make it easier to modify some parameters while running simulations from the REPL. Additionally, this is used in tests to reduce the computational burden for CI while still providing examples with sensible default values for users.\n\nBefore replacing assignments in elixir, the keyword argument maxiters is inserted into calls to solve with it's default value used in the SciML ecosystem for ODEs, see the \"Miscellaneous\" section of the documentation.\n\nExamples\n\njulia> using TrixiBase, Trixi\n\njulia> redirect_stdout(devnull) do\n trixi_include(@__MODULE__, joinpath(examples_dir(), \"tree_1d_dgsem\", \"elixir_advection_extended.jl\"),\n tspan=(0.0, 0.1))\n sol.t[end]\n end\n[ Info: You just called `trixi_include`. Julia may now compile the code, please be patient.\n0.1\n\n\n\n\n\n","category":"method"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/CONTRIBUTING.md\"","category":"page"},{"location":"contributing/#Contributing","page":"Contributing","title":"Contributing","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"TrixiParticles.jl is an open-source project and we are very happy to accept contributions from the community. Please feel free to open issues or submit patches (preferably as pull requests) any time. For planned larger contributions, it is often beneficial to get in contact with one of the principal developers first (see Authors).","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"TrixiParticles.jl and its contributions are licensed under the MIT license (see License). As a contributor, you certify that all your contributions are in conformance with the Developer Certificate of Origin (Version 1.1), which is reproduced below.","category":"page"},{"location":"contributing/#Developer-Certificate-of-Origin-(Version-1.1)","page":"Contributing","title":"Developer Certificate of Origin (Version 1.1)","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The following text was taken from https://developercertificate.org:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Developer Certificate of Origin\nVersion 1.1\n\nCopyright (C) 2004, 2006 The Linux Foundation and its contributors.\n1 Letterman Drive\nSuite D4700\nSan Francisco, CA, 94129\n\nEveryone is permitted to copy and distribute verbatim copies of this\nlicense document, but changing it is not allowed.\n\n\nDeveloper's Certificate of Origin 1.1\n\nBy making a contribution to this project, I certify that:\n\n(a) The contribution was created in whole or in part by me and I\n have the right to submit it under the open source license\n indicated in the file; or\n\n(b) The contribution is based upon previous work that, to the best\n of my knowledge, is covered under an appropriate open source\n license and I have the right under that license to submit that\n work with modifications, whether created in whole or in part\n by me, under the same open source license (unless I am\n permitted to submit under a different license), as indicated\n in the file; or\n\n(c) The contribution was provided directly to me by some other\n person who certified (a), (b) or (c) and I have not modified\n it.\n\n(d) I understand and agree that this project and the contribution\n are public and that a record of the contribution (including all\n personal information I submit with it, including my sign-off) is\n maintained indefinitely and may be redistributed consistent with\n this project or the open source license(s) involved.","category":"page"},{"location":"development/#development","page":"Development","title":"Development","text":"","category":"section"},{"location":"development/#Preview-of-the-documentation","page":"Development","title":"Preview of the documentation","text":"","category":"section"},{"location":"development/","page":"Development","title":"Development","text":"To generate the Documentation, first instantiate the docs environment by executing the following command from the TrixiParticles.jl root directory:","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"julia --project=docs -e \"using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()\"","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"This command only has to be run once. After that, maintain the docs environment as described under Installation.","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"With an instantiated docs environment, generate the docs with the following command (again from the TrixiParticles.jl root directory):","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"julia --project=docs --color=yes docs/make.jl","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"You can then open the generated files in docs/build with your webbrowser. Alternatively, run","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"python3 -m http.server -d docs/build","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"and open localhost:8000 in your webbrowser.","category":"page"},{"location":"development/#Release-management","page":"Development","title":"Release management","text":"","category":"section"},{"location":"development/","page":"Development","title":"Development","text":"To create a new release for TrixiParticles.jl, perform the following steps:","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"Make sure that all PRs and changes that you want to go into the release are merged to main and that the latest commit on main has passed all CI tests.\nDetermine the currently released version of TrixiParticles.jl, e.g., on the release page. For this manual, we will assume that the latest release was v0.2.3.\nDecide on the next version number. We follow semantic versioning, thus each version is of the form vX.Y.Z where X is the major version, Y the minor version, and Z the patch version. In this manual, we assume that the major version is always 0, thus the decision process on the new version is as follows:\nIf the new release contains breaking changes (i.e., user code might not work as before without modifications), increase the minor version by one and set the patch version to zero. In our example, the new version should thus be v0.3.0.\nIf the new release only contains minor modifications and/or bug fixes, the minor version is kept as-is and the patch version is increased by one. In our example, the new version should thus be v0.2.4.\nEdit the version string in the Project.toml and set it to the new version. Push/merge this change to main.\nGo to GitHub and add a comment to the commit that you would like to become the new release (typically this will be the commit where you just updated the version). You can comment on a commit by going to the commit overview and clicking on the title of the commit. The comment should contain the following text:\n@JuliaRegistrator register\nWait for the magic to happen! Specifically, JuliaRegistrator will create a new PR to the Julia registry with the new release information. After a grace period of ~15 minutes, this PR will be merged automatically. A short while after, TagBot will create a new release of TrixiParticles.jl in our GitHub repository.\nOnce the new release has been created, the new version can be obtained through the Julia package manager as usual.\nTo make sure people do not mistake the latest state of main as the latest release, we set the version in the Project.toml to a development version. The development version should be the latest released version, with the patch version incremented by one, and the -dev suffix added. For example, if you just released v0.3.0, the new development version should be v0.3.1-dev. If you just released v0.2.4, the new development version should be v0.2.5-dev.","category":"page"},{"location":"install/#installation","page":"Installation","title":"Installation","text":"","category":"section"},{"location":"install/#Setting-up-Julia","page":"Installation","title":"Setting up Julia","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"If you have not yet installed Julia, please follow the instructions for your operating system. TrixiParticles.jl works with Julia v1.9 and newer. We recommend using the latest stable release of Julia.","category":"page"},{"location":"install/#For-users","page":"Installation","title":"For users","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"TrixiParticles.jl is a registered Julia package. You can install TrixiParticles.jl, OrdinaryDiffEq.jl (used for time integration) and Plots.jl by executing the following commands in the Julia REPL:","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"julia> using Pkg\n\njulia> Pkg.add([\"TrixiParticles\", \"OrdinaryDiffEq\", \"Plots\"])","category":"page"},{"location":"install/#for-developers","page":"Installation","title":"For developers","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"If you plan on editing TrixiParticles.jl itself, you can download TrixiParticles.jl to a local folder and use the code from the cloned directory:","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"git clone git@github.com:trixi-framework/TrixiParticles.jl.git\ncd TrixiParticles.jl\nmkdir run\njulia --project=run -e 'using Pkg; Pkg.develop(PackageSpec(path=\".\"))' # Add TrixiParticles.jl to `run` project\njulia --project=run -e 'using Pkg; Pkg.add(\"OrdinaryDiffEq\", \"Plots\")' # Add additional packages","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"If you installed TrixiParticles.jl this way, you always have to start Julia with the --project flag set to your run directory, e.g.,","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"julia --project=run","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"from the TrixiParticles.jl root directory.","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"The advantage of using a separate run directory is that you can also add other related packages (e.g., OrdinaryDiffEq.jl, see above) to the project in the run folder and always have a reproducible environment at hand to share with others.","category":"page"},{"location":"install/#Optional-software/packages","page":"Installation","title":"Optional software/packages","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"OrdinaryDiffEq.jl – A Julia package of ordinary differential equation solvers that is used in the examples\nPlots.jl – Julia Plotting library that is used in some examples\nPythonPlot.jl – Plotting library that can be used instead of Plots.jl\nParaView – Software that can be used for visualization of results","category":"page"},{"location":"install/#installation-issues","page":"Installation","title":"Common issues","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"If you followed the installation instructions for developers and you run into any problems with packages when pulling the latest version of TrixiParticles.jl, start Julia with the project in the run folder,","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":" julia --project=run","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"update all packages in that project, resolve all conflicts in the project, and install all new dependencies:","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"julia> using Pkg\n\njulia> Pkg.update()\n\njulia> Pkg.resolve()\n\njulia> Pkg.instantiate()","category":"page"},{"location":"visualization/#Visualization","page":"Visualization","title":"Visualization","text":"","category":"section"},{"location":"visualization/#Export-VTK-files","page":"Visualization","title":"Export VTK files","text":"","category":"section"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"You can export particle data as VTK files by using the SolutionSavingCallback. All our predefined examples are already using this callback to export VTK files to the out directory (relative to the directory that you are running Julia from). VTK files can be read by visualization tools like ParaView and VisIt.","category":"page"},{"location":"visualization/#ParaView","page":"Visualization","title":"ParaView","text":"","category":"section"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"Follow these steps to view the exported VTK files in ParaView:","category":"page"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"Click File -> Open.\nNavigate to the out directory (relative to the directory that you are running Julia from).\nOpen both boundary_1.pvd and fluid_1.pvd.\nClick \"Apply\", which by default is on the left pane below the \"Pipeline Browser\".\nHold the left mouse button to move the solution around.","category":"page"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"You will now see the following: (Image: image)","category":"page"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"To now view the result variables first make sure you have \"fluid_1.pvd\" highlighted in the \"Pipeline Browser\" then select them in the variable selection combo box (see picture below). Let's, for example, pick \"density\". To now view the time progression of the result hit the \"play button\" (see picture below). (Image: image)","category":"page"},{"location":"visualization/#API","page":"Visualization","title":"API","text":"","category":"section"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"Modules = [TrixiParticles]\nPages = map(file -> joinpath(\"visualization\", file), readdir(joinpath(\"..\", \"src\", \"visualization\")))","category":"page"},{"location":"visualization/#TrixiParticles.trixi2vtk-Tuple{Any, Any, Any}","page":"Visualization","title":"TrixiParticles.trixi2vtk","text":"trixi2vtk(vu_ode, semi, t; iter=nothing, output_directory=\"out\", prefix=\"\",\n write_meta_data=true, max_coordinates=Inf, custom_quantities...)\n\nConvert Trixi simulation data to VTK format.\n\nArguments\n\nvu_ode: Solution of the TrixiParticles ODE system at one time step. This expects an ArrayPartition as returned in the examples as sol.u[end].\nsemi: Semidiscretization of the TrixiParticles simulation.\nt: Current time of the simulation.\n\nKeywords\n\niter=nothing: Iteration number when multiple iterations are to be stored in separate files. This number is just appended to the filename.\noutput_directory=\"out\": Output directory path.\nprefix=\"\": Prefix for output files.\nwrite_meta_data=true: Write meta data.\nmax_coordinates=Inf The coordinates of particles will be clipped if their absolute values exceed this threshold.\ncustom_quantities...: Additional custom quantities to include in the VTK output. Each custom quantity must be a function of (v, u, t, system), which will be called for every system, where v and u are the wrapped solution arrays for the corresponding system and t is the current simulation time. Note that working with these v and u arrays requires undocumented internal functions of TrixiParticles. See Custom Quantities for a list of pre-defined custom quantities that can be used here.\n\nExample\n\ntrixi2vtk(sol.u[end], semi, 0.0, iter=1, output_directory=\"output\", prefix=\"solution\")\n\n# Additionally store the kinetic energy of each system as \"my_custom_quantity\"\ntrixi2vtk(sol.u[end], semi, 0.0, iter=1, my_custom_quantity=kinetic_energy)\n\n\n\n\n\n","category":"method"},{"location":"visualization/#TrixiParticles.trixi2vtk-Tuple{Any}","page":"Visualization","title":"TrixiParticles.trixi2vtk","text":"trixi2vtk(coordinates; output_directory=\"out\", prefix=\"\", filename=\"coordinates\")\n\nConvert coordinate data to VTK format.\n\nArguments\n\ncoordinates: Coordinates to be saved.\n\nKeywords\n\noutput_directory=\"out\": Output directory path.\nprefix=\"\": Prefix for the output file.\nfilename=\"coordinates\": Name of the output file.\n\nReturns\n\nfile::AbstractString: Path to the generated VTK file.\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#initial_condition","page":"Initial Condition and Setups","title":"Initial Condition","text":"","category":"section"},{"location":"general/initial_condition/","page":"Initial Condition and Setups","title":"Initial Condition and Setups","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"initial_condition.jl\")]","category":"page"},{"location":"general/initial_condition/#TrixiParticles.InitialCondition","page":"Initial Condition and Setups","title":"TrixiParticles.InitialCondition","text":"InitialCondition(; coordinates, density, velocity=zeros(size(coordinates, 1)),\n mass=nothing, pressure=0.0, particle_spacing=-1.0)\n\nStruct to hold the initial configuration of the particles.\n\nThe following setups return InitialConditions for commonly used setups:\n\nRectangularShape\nSphereShape\nRectangularTank\n\nInitialConditions support the set operations union, setdiff and intersect in order to build more complex geometries.\n\nArguments\n\ncoordinates: An array where the i-th column holds the coordinates of particle i.\ndensity: Either a vector holding the density of each particle, or a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles.\n\nKeywords\n\nvelocity: Either an array where the i-th column holds the velocity of particle i, or a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a vector holding the mass of each particle, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\npressure: Either a vector holding the pressure of each particle, or a function mapping each particle's coordinates to its pressure, or a scalar for a constant pressure over all particles. This is optional and only needed when using the EntropicallyDampedSPHSystem.\nparticle_spacing: The spacing between the particles. This is a scalar, as the spacing is assumed to be uniform. This is only needed when using set operations on the InitialCondition or for automatic mass calculation.\n\nExamples\n\n# Rectangle filled with particles\ninitial_condition = RectangularShape(0.1, (3, 4), (-1.0, 1.0), density=1.0)\n\n# Two spheres in one initial condition\ninitial_condition = union(SphereShape(0.15, 0.5, (-1.0, 1.0), 1.0),\n SphereShape(0.15, 0.2, (0.0, 1.0), 1.0))\n\n# Rectangle with a spherical hole\nshape1 = RectangularShape(0.1, (16, 13), (-0.8, 0.0), density=1.0)\nshape2 = SphereShape(0.1, 0.35, (0.0, 0.6), 1.0, sphere_type=RoundSphere())\ninitial_condition = setdiff(shape1, shape2)\n\n# Intersect of a rectangle with a sphere. Note that this keeps the particles of the\n# rectangle that are in the intersect, while `intersect(shape2, shape1)` would consist of\n# the particles of the sphere that are in the intersect.\nshape1 = RectangularShape(0.1, (16, 13), (-0.8, 0.0), density=1.0)\nshape2 = SphereShape(0.1, 0.35, (0.0, 0.6), 1.0, sphere_type=RoundSphere())\ninitial_condition = intersect(shape1, shape2)\n\n# Build `InitialCondition` manually\ncoordinates = [0.0 1.0 1.0\n 0.0 0.0 1.0]\nvelocity = zero(coordinates)\nmass = ones(3)\ndensity = 1000 * ones(3)\ninitial_condition = InitialCondition(; coordinates, velocity, mass, density)\n\n# With functions\ninitial_condition = InitialCondition(; coordinates, velocity=x -> 2x, mass=1.0, density=1000.0)\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#Setups","page":"Initial Condition and Setups","title":"Setups","text":"","category":"section"},{"location":"general/initial_condition/","page":"Initial Condition and Setups","title":"Initial Condition and Setups","text":"Modules = [TrixiParticles]\nPages = map(file -> joinpath(\"setups\", file), readdir(joinpath(\"..\", \"src\", \"setups\")))","category":"page"},{"location":"general/initial_condition/#TrixiParticles.extrude_geometry-Tuple{Any}","page":"Initial Condition and Setups","title":"TrixiParticles.extrude_geometry","text":"extrude_geometry(geometry; particle_spacing, direction, n_extrude::Integer,\n velocity=zeros(length(direction)),\n mass=nothing, density=nothing, pressure=0.0)\n\nExtrude either a line, a plane or a shape along a specific direction.\n\nArguments\n\ngeometry: Either particle coordinates or an InitialCondition defining a 2D shape to extrude to a 3D volume, or two 2D points defining a line to extrude to a plane in 2D, or three 3D points defining a parallelogram to extrude to a parallelepiped.\n\nKeywords\n\nparticle_spacing: Spacing between the particles. Can be omitted when geometry is an InitialCondition (unless geometry.particle_spacing == -1).\ndirection: A vector that specifies the direction in which to extrude.\nn_extrude: Number of layers of particles created in the direction of extrusion.\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\ndensity: Either a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles. Obligatory when not using a state equation. Cannot be used together with state_equation.\npressure: Scalar to set the pressure of all particles to this value. This is only used by the EntropicallyDampedSPHSystem and will be overwritten when using an initial pressure function in the system. Cannot be used together with hydrostatic pressure gradient.\ntlsph: With the TotalLagrangianSPHSystem, particles need to be placed on the boundary of the shape and not one particle radius away, as for fluids. When tlsph=true, particles will be placed on the boundary of the shape.\n\nExamples\n\n# Extrude a line in 2D to a plane in 2D\np1 = [0.0, 0.0]\np2 = [1.0, 1.0]\n\ndirection = [-1.0, 1.0]\n\nshape = extrude_geometry((p1, p2); direction, particle_spacing=0.1, n_extrude=4, density=1000.0)\n\n# Extrude a parallelogram in 3D to a parallelepiped in 3D\np1 = [0.0, 0.0, 0.0]\np2 = [0.5, 1.0, 0.0]\np3 = [1.0, 0.2, 0.0]\n\ndirection = [0.0, 0.0, 1.0]\n\nshape = extrude_geometry((p1, p2, p3); direction, particle_spacing=0.1, n_extrude=4, density=1000.0)\n\n# Extrude a 2D shape (here: a disc) to a 3D shape (here: a cylinder)\nshape = SphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3,\n sphere_type=RoundSphere(end_angle=pi))\n\ndirection = [0.0, 0.0, 1.0]\n\nshape = extrude_geometry(shape; direction, particle_spacing=0.1, n_extrude=4, density=1000.0)\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#TrixiParticles.RectangularShape-Tuple{Any, Any, Any}","page":"Initial Condition and Setups","title":"TrixiParticles.RectangularShape","text":"RectangularShape(particle_spacing, n_particles_per_dimension, min_coordinates;\n velocity=zeros(length(n_particles_per_dimension)),\n mass=nothing, density=nothing, pressure=0.0,\n acceleration=nothing, state_equation=nothing,\n tlsph=false, loop_order=nothing)\n\nRectangular shape filled with particles. Returns an InitialCondition.\n\nArguments\n\nparticle_spacing: Spacing between the particles.\nn_particles_per_dimension: Tuple containing the number of particles in x, y and z (only 3D) direction, respectively.\nmin_coordinates: Coordinates of the corner in negative coordinate directions.\n\nKeywords\n\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\ndensity: Either a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles. Obligatory when not using a state equation. Cannot be used together with state_equation.\npressure: Scalar to set the pressure of all particles to this value. This is only used by the EntropicallyDampedSPHSystem and will be overwritten when using an initial pressure function in the system. Cannot be used together with hydrostatic pressure gradient.\nacceleration: In order to initialize particles with a hydrostatic pressure gradient, an acceleration vector can be passed. Note that only accelerations in one coordinate direction and no diagonal accelerations are supported. This will only change the pressure of the particles. When using the WeaklyCompressibleSPHSystem, pass a state_equation as well to initialize the particles with the corresponding density and mass. When using the EntropicallyDampedSPHSystem, the pressure will be overwritten when using an initial pressure function in the system. This cannot be used together with the pressure keyword argument.\nstate_equation: When calculating a hydrostatic pressure gradient by setting acceleration, the state_equation will be used to set the corresponding density. Cannot be used together with density.\ntlsph: With the TotalLagrangianSPHSystem, particles need to be placed on the boundary of the shape and not one particle radius away, as for fluids. When tlsph=true, particles will be placed on the boundary of the shape.\n\nExamples\n\n# 2D\nrectangular = RectangularShape(particle_spacing, (5, 4), (1.0, 2.0), density=1000.0)\n\n# 2D with hydrostatic pressure gradient.\n# `state_equation` has to be the same as for the WCSPH system.\nstate_equation = StateEquationCole(sound_speed=20.0, exponent=7, reference_density=1000.0)\nrectangular = RectangularShape(particle_spacing, (5, 4), (1.0, 2.0),\n acceleration=(0.0, -9.81), state_equation=state_equation)\n\n# 3D\nrectangular = RectangularShape(particle_spacing, (5, 4, 7), (1.0, 2.0, 3.0), density=1000.0)\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#TrixiParticles.RectangularTank","page":"Initial Condition and Setups","title":"TrixiParticles.RectangularTank","text":"RectangularTank(particle_spacing, fluid_size, tank_size, fluid_density;\n velocity=zeros(length(fluid_size)), fluid_mass=nothing,\n pressure=0.0,\n acceleration=nothing, state_equation=nothing,\n boundary_density=fluid_density,\n n_layers=1, spacing_ratio=1.0,\n min_coordinates=zeros(length(fluid_size)),\n faces=Tuple(trues(2 * length(fluid_size))))\n\nRectangular tank filled with a fluid to set up dam-break-style simulations.\n\nArguments\n\nparticle_spacing: Spacing between the fluid particles.\nfluid_size: The dimensions of the fluid as (x, y) (or (x, y, z) in 3D).\ntank_size: The dimensions of the tank as (x, y) (or (x, y, z) in 3D).\nfluid_density: The rest density of the fluid. Will only be used as default for boundary_density when using a state equation.\n\nKeywords\n\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nfluid_mass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\npressure: Scalar to set the pressure of all particles to this value. This is only used by the EntropicallyDampedSPHSystem and will be overwritten when using an initial pressure function in the system. Cannot be used together with hydrostatic pressure gradient.\nacceleration: In order to initialize particles with a hydrostatic pressure gradient, an acceleration vector can be passed. Note that only accelerations in one coordinate direction and no diagonal accelerations are supported. This will only change the pressure of the particles. When using the WeaklyCompressibleSPHSystem, pass a state_equation as well to initialize the particles with the corresponding density and mass. When using the EntropicallyDampedSPHSystem, the pressure will be overwritten when using an initial pressure function in the system. This cannot be used together with the pressure keyword argument.\nstate_equation: When calculating a hydrostatic pressure gradient by setting acceleration, the state_equation will be used to set the corresponding density. Cannot be used together with density.\nboundary_density: Density of each boundary particle (by default set to the fluid density)\nn_layers: Number of boundary layers.\nspacing_ratio: Ratio of particle_spacing to boundary particle spacing. A value of 2 means that the boundary particle spacing will be half the fluid particle spacing.\nmin_coordinates: Coordinates of the corner in negative coordinate directions.\nfaces: By default all faces are generated. Set faces by passing a bit-array of length 4 (2D) or 6 (3D) to generate the faces in the normal direction: -x,+x,-y,+y,-z,+z.\n\nFields\n\nfluid::InitialCondition: InitialCondition for the fluid.\nboundary::InitialCondition: InitialCondition for the boundary.\nfluid_size::Tuple: Tuple containing the size of the fluid in each dimension after rounding.\ntank_size::Tuple: Tuple containing the size of the tank in each dimension after rounding.\n\nExamples\n\n# 2D\nsetup = RectangularTank(particle_spacing, (water_width, water_height),\n (container_width, container_height), fluid_density,\n n_layers=2, spacing_ratio=3)\n\n# 2D with hydrostatic pressure gradient.\n# `state_equation` has to be the same as for the WCSPH system.\nstate_equation = StateEquationCole(sound_speed=10.0, exponent=1, reference_density=1000.0)\nsetup = RectangularTank(particle_spacing, (water_width, water_height),\n (container_width, container_height), fluid_density,\n acceleration=(0.0, -9.81), state_equation=state_equation)\n\n# 3D\nsetup = RectangularTank(particle_spacing, (water_width, water_height, water_depth),\n (container_width, container_height, container_depth), fluid_density,\n n_layers=2)\n\nSee also: reset_wall!.\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#TrixiParticles.reset_wall!-Tuple{Any, Any, Any}","page":"Initial Condition and Setups","title":"TrixiParticles.reset_wall!","text":"reset_wall!(rectangular_tank::RectangularTank, reset_faces, positions)\n\nThe selected walls of the tank will be placed at the new positions.\n\nArguments\n\nreset_faces: Boolean tuple of 4 (in 2D) or 6 (in 3D) dimensions, similar to faces in RectangularTank.\npositions: Tuple of new positions\n\nwarning: Warning\nThere are overlapping particles when adjacent walls are moved inwards simultaneously.\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#TrixiParticles.RoundSphere","page":"Initial Condition and Setups","title":"TrixiParticles.RoundSphere","text":"RoundSphere(; start_angle=0.0, end_angle=2π)\n\nConstruct a sphere (or sphere segment) by nesting perfectly round concentric spheres. The resulting ball will be perfectly round, but will not have a regular inner structure.\n\nKeywords\n\nstart_angle: The starting angle of the sphere segment in radians. It determines the beginning point of the segment. The default is set to 0.0 representing the positive x-axis.\nend_angle: The ending angle of the sphere segment in radians. It defines the termination point of the segment. The default is set to 2pi, completing a full sphere.\n\nnote: Usage\nSee SphereShape on how to use this.\n\nwarning: Warning\nThe sphere segment is intended for 2D geometries and hollow spheres. If used for filled spheres or in a 3D context, results may not be accurate.\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#TrixiParticles.VoxelSphere","page":"Initial Condition and Setups","title":"TrixiParticles.VoxelSphere","text":"VoxelSphere()\n\nConstruct a sphere of voxels (where particles are placed in the voxel center) with a regular inner structure but corners on the surface. Essentially, a grid of particles is generated and all particles outside the sphere are removed. The resulting sphere will have a perfect inner structure, but is not perfectly round, as it will have corners (like a sphere in Minecraft).\n\nnote: Usage\nSee SphereShape on how to use this.\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#TrixiParticles.SphereShape-NTuple{4, Any}","page":"Initial Condition and Setups","title":"TrixiParticles.SphereShape","text":"SphereShape(particle_spacing, radius, center_position, density;\n sphere_type=VoxelSphere(), n_layers=-1, layer_outwards=false,\n cutout_min=(0.0, 0.0), cutout_max=(0.0, 0.0), tlsph=false,\n velocity=zeros(length(center_position)), mass=nothing, pressure=0.0)\n\nGenerate a sphere that is either completely filled (by default) or hollow (by passing n_layers).\n\nWith the sphere type VoxelSphere, a sphere of voxels (where particles are placed in the voxel center) with a regular inner structure but corners on the surface is created. Essentially, a grid of particles is generated and all particles outside the sphere are removed. With the sphere type RoundSphere, a perfectly round sphere with an imperfect inner structure is created.\n\nA cuboid can be cut out of the sphere by specifying the two corners in negative and positive coordinate directions as cutout_min and cutout_max.\n\nArguments\n\nparticle_spacing: Spacing between the particles.\nradius: Radius of the sphere.\ncenter_position: The coordinates of the center of the sphere.\ndensity: Either a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles.\n\nKeywords\n\nsphere_type: Either VoxelSphere or RoundSphere (see explanation above).\nn_layers: Set to an integer greater than zero to generate a hollow sphere, where the shell consists of n_layers layers.\nlayer_outwards: When set to false (by default), radius is the outer radius of the sphere. When set to true, radius is the inner radius of the sphere. This is only used when n_layers > 0.\ncutout_min: Corner in negative coordinate directions of a cuboid that is to be cut out of the sphere.\ncutout_max: Corner in positive coordinate directions of a cuboid that is to be cut out of the sphere.\ntlsph: With the TotalLagrangianSPHSystem, particles need to be placed on the boundary of the shape and not one particle radius away, as for fluids. When tlsph=true, particles will be placed on the boundary of the shape.\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\npressure: Either a function mapping each particle's coordinates to its pressure, or a scalar for a constant pressure over all particles. This is optional and only needed when using the EntropicallyDampedSPHSystem.\n\nExamples\n\n# Filled circle with radius 0.5, center in (0.2, 0.4) and a particle spacing of 0.1\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0)\n\n# Same as before, but perfectly round\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, sphere_type=RoundSphere())\n\n# Hollow circle with ~3 layers, outer radius 0.5, center in (0.2, 0.4) and a particle\n# spacing of 0.1.\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3)\n\n# Same as before, but perfectly round\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3, sphere_type=RoundSphere())\n\n# Hollow circle with 3 layers, inner radius 0.5, center in (0.2, 0.4) and a particle spacing\n# of 0.1.\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3, layer_outwards=true)\n\n# Filled circle with radius 0.1, center in (0.0, 0.0), particle spacing 0.1, but the\n# rectangle [0, 1] x [-0.2, 0.2] is cut out.\nSphereShape(0.1, 1.0, (0.0, 0.0), 1000.0, cutout_min=(0.0, -0.2), cutout_max=(1.0, 0.2))\n\n# Filled 3D sphere with radius 0.5, center in (0.2, 0.4, 0.3) and a particle spacing of 0.1\nSphereShape(0.1, 0.5, (0.2, 0.4, 0.3), 1000.0)\n\n# Same as before, but perfectly round\nSphereShape(0.1, 0.5, (0.2, 0.4, 0.3), 1000.0, sphere_type=RoundSphere())\n\n\n\n\n\n","category":"method"},{"location":"systems/weakly_compressible_sph/#wcsph","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Weakly compressible SPH as introduced by Monaghan (1994). This formulation relies on a stiff equation of state that generates large pressure changes for small density variations.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"weakly_compressible_sph\", \"system.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.WeaklyCompressibleSPHSystem","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.WeaklyCompressibleSPHSystem","text":"WeaklyCompressibleSPHSystem(initial_condition,\n density_calculator, state_equation,\n smoothing_kernel, smoothing_length;\n viscosity=nothing, density_diffusion=nothing,\n acceleration=ntuple(_ -> 0.0, NDIMS),\n correction=nothing, source_terms=nothing)\n\nSystem for particles of a fluid. The weakly compressible SPH (WCSPH) scheme is used, wherein a stiff equation of state generates large pressure changes for small density variations. See Weakly Compressible SPH for more details on the method.\n\nArguments\n\ninitial_condition: InitialCondition representing the system's particles.\ndensity_calculator: Density calculator for the system. See ContinuityDensity and SummationDensity.\nstate_equation: Equation of state for the system. See StateEquationCole.\nsmoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.\nsmoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.\n\nKeyword Arguments\n\nviscosity: Viscosity model for this system (default: no viscosity). See ArtificialViscosityMonaghan or ViscosityAdami.\ndensity_diffusion: Density diffusion terms for this system. See DensityDiffusion.\nacceleration: Acceleration vector for the system. (default: zero vector)\ncorrection: Correction method used for this system. (default: no correction, see Corrections)\nsource_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping. Note that these source terms will not be used in the calculation of the boundary pressure when using a boundary with BoundaryModelDummyParticles and AdamiPressureExtrapolation. The keyword argument acceleration should be used instead for gravity-like source terms.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#References","page":"Weakly Compressible SPH (Fluid)","title":"References","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Joseph J. Monaghan. \"Simulating Free Surface Flows in SPH\". In: Journal of Computational Physics 110 (1994), pages 399–406. doi: 10.1006/jcph.1994.1034","category":"page"},{"location":"systems/weakly_compressible_sph/#equation_of_state","page":"Weakly Compressible SPH (Fluid)","title":"Equation of State","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The equation of state is used to relate fluid density to pressure and thus allow an explicit simulation of the WCSPH system. The equation in the following formulation was introduced by Cole (Cole 1948, pp. 39 and 43). The pressure p is calculated as","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":" p = B left(left(fracrhorho_0right)^gamma - 1right) + p_textbackground","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"where rho denotes the density, rho_0 the reference density, and p_textbackground the background pressure, which is set to zero when applied to free-surface flows (Adami et al., 2012).","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The bulk modulus, B = fracrho_0 c^2gamma, is calculated from the artificial speed of sound c and the isentropic exponent gamma.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"An ideal gas equation of state with a linear relationship between pressure and density can be obtained by choosing exponent=1, i.e.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":" p = B left( fracrhorho_0 -1 right) = c^2(rho - rho_0)","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"For higher Reynolds numbers, exponent=7 is recommended, whereas at lower Reynolds numbers exponent=1 yields more accurate pressure estimates since pressure and density are proportional.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"When using SummationDensity (or DensityReinitializationCallback) and free surfaces, initializing particles with equal spacing will cause underestimated density and therefore strong attractive forces between particles at the free surface. Setting clip_negative_pressure=true can avoid this.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"weakly_compressible_sph\", \"state_equations.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.StateEquationCole","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.StateEquationCole","text":"StateEquationCole(; sound_speed, reference_density, exponent,\n background_pressure=0.0, clip_negative_pressure=false)\n\nEquation of state to describe the relationship between pressure and density of water up to high pressures.\n\nKeywords\n\nsound_speed: Artificial speed of sound.\nreference_density: Reference density of the fluid.\nexponent: A value of 7 is usually used for most simulations.\nbackground_pressure=0.0: Background pressure.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#References-2","page":"Weakly Compressible SPH (Fluid)","title":"References","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Robert H. Cole. \"Underwater Explosions\". Princeton University Press, 1948.\nJ. P. Morris, P. J. Fox, Y. Zhu \"Modeling Low Reynolds Number Incompressible Flows Using SPH \". In: Journal of Computational Physics , Vol. 136, No. 1, pages 214–226. doi: 10.1006/jcph.1997.5776\nS. Adami, X. Y. Hu, N. A. Adams. \"A generalized wall boundary condition for smoothed particle hydrodynamics\". In: Journal of Computational Physics 231, 21 (2012), pages 7057–7075. doi: 10.1016/J.JCP.2012.05.005","category":"page"},{"location":"systems/weakly_compressible_sph/#viscosity_wcsph","page":"Weakly Compressible SPH (Fluid)","title":"Viscosity","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"TODO: Explain viscosity.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"viscosity.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.ArtificialViscosityMonaghan","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.ArtificialViscosityMonaghan","text":"ArtificialViscosityMonaghan(; alpha, beta, epsilon=0.01)\n\nKeywords\n\nalpha: A value of 0.02 is usually used for most simulations. For a relation with the kinematic viscosity, see description below.\nbeta: A value of 0.0 works well for simulations with shocks of moderate strength. In simulations where the Mach number can be very high, eg. astrophysical calculation, good results can be obtained by choosing a value of beta=2 and alpha=1.\nepsilon=0.01: Parameter to prevent singularities.\n\nArtificial viscosity by Monaghan (Monaghan 1992, Monaghan 1989), given by\n\nPi_ab =\nbegincases\n -(alpha c mu_ab + beta mu_ab^2) barrho_ab textif v_ab cdot r_ab 0 \n 0 textotherwise\nendcases\n\nwith\n\nmu_ab = frach v_ab cdot r_abVert r_ab Vert^2 + epsilon h^2\n\nwhere alpha beta epsilon are parameters, c is the speed of sound, h is the smoothing length, r_ab = r_a - r_b is the difference of the coordinates of particles a and b, v_ab = v_a - v_b is the difference of their velocities, and barrho_ab is the arithmetic mean of their densities.\n\nNote that alpha needs to adjusted for different resolutions to maintain a specific Reynolds Number. To do so, Monaghan (Monaghan 2005) defined an equivalent effective physical kinematic viscosity nu by\n\n nu = fracalpha h c 2d + 4\n\nwhere d is the dimension.\n\nReferences\n\nJoseph J. Monaghan. \"Smoothed Particle Hydrodynamics\". In: Annual Review of Astronomy and Astrophysics 30.1 (1992), pages 543-574. doi: 10.1146/ANNUREV.AA.30.090192.002551\nJoseph J. Monaghan. \"Smoothed Particle Hydrodynamics\". In: Reports on Progress in Physics (2005), pages 1703-1759. doi: 10.1088/0034-4885/68/8/r01\nJoseph J. Monaghan. \"On the Problem of Penetration in Particle Methods\". In: Journal of Computational Physics 82.1, pages 1–15. doi: 10.1016/0021-9991(89)90032-6\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.ViscosityAdami","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.ViscosityAdami","text":"ViscosityAdami(; nu, epsilon=0.01)\n\nViscosity by Adami (Adami et al. 2012). The viscous interaction is calculated with the shear force for incompressible flows given by\n\nf_ab = sum_w bareta_ab left( V_a^2 + V_b^2 right) fracv_abr_ab^2+epsilon h_ab^2 nabla W_ab cdot r_ab\n\nwhere r_ab = r_a - r_b is the difference of the coordinates of particles a and b, v_ab = v_a - v_b is the difference of their velocities, h is the smoothing length and V is the particle volume. The parameter epsilon prevents singularities (see Ramachandran et al. 2019). The inter-particle-averaged shear stress is\n\n bareta_ab =frac2 eta_a eta_beta_a + eta_b\n\nwhere eta_a = rho_a nu_a with nu as the kinematic viscosity.\n\nKeywords\n\nnu: Kinematic viscosity\nepsilon=0.01: Parameter to prevent singularities\n\nReferences\n\nS. Adami et al. \"A generalized wall boundary condition for smoothed particle hydrodynamics\". In: Journal of Computational Physics 231 (2012), pages 7057-7075. doi: 10.1016/j.jcp.2012.05.005\nP. Ramachandran et al. \"Entropically damped artificial compressibility for SPH\". In: Journal of Computers and Fluids 179 (2019), pages 579-594. doi: 10.1016/j.compfluid.2018.11.023\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#Density-Diffusion","page":"Weakly Compressible SPH (Fluid)","title":"Density Diffusion","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Density diffusion can be used with ContinuityDensity to remove the noise in the pressure field. It is highly recommended to use density diffusion when using WCSPH.","category":"page"},{"location":"systems/weakly_compressible_sph/#Formulation","page":"Weakly Compressible SPH (Fluid)","title":"Formulation","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"All density diffusion terms extend the continuity equation (see ContinuityDensity) by an additional term","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"fracmathrmdrho_amathrmdt = sum_b m_b v_ab cdot nabla_r_a W(Vert r_ab Vert h)\n + delta h c sum_b V_b psi_ab cdot nabla_r_a W(Vert r_ab Vert h)","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"where V_b = m_b rho_b is the volume of particle b and psi_ab depends on the density diffusion method (see DensityDiffusion for available terms). Also, rho_a denotes the density of particle a and r_ab = r_a - r_b is the difference of the coordinates, v_ab = v_a - v_b of the velocities of particles a and b.","category":"page"},{"location":"systems/weakly_compressible_sph/#Numerical-Results","page":"Weakly Compressible SPH (Fluid)","title":"Numerical Results","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"All density diffusion terms remove numerical noise in the pressure field and produce more accurate results than weakly commpressible SPH without density diffusion. This can be demonstrated with dam break examples in 2D and 3D. Here, δ = 01 has been used for all terms. Note that, due to added stability, the adaptive time integration method that was used here can choose higher time steps in the simulations with density diffusion. For the cheap DensityDiffusionMolteniColagrossi, this results in reduced runtime.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"
\n \"density_diffusion_2d\"/\n
Dam break in 2D with different density diffusion terms
\n
","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"
\n \"density_diffusion_3d\"/\n
Dam break in 3D with different density diffusion terms
\n
","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The simpler terms DensityDiffusionMolteniColagrossi and DensityDiffusionFerrari do not solve the hydrostatic problem and lead to incorrect solutions in long-running steady-state hydrostatic simulations with free surfaces (Antuono et al., 2012). This can be seen when running the simple rectangular tank example until t = 40 (again using δ = 01):","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"
\n \"density_diffusion_tank\"/\n
Tank in rest under gravity in 3D with different density diffusion terms
\n
","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"DensityDiffusionAntuono adds a correction term to solve this problem, but this term is very expensive and adds about 40–50% of computational cost.","category":"page"},{"location":"systems/weakly_compressible_sph/#References-3","page":"Weakly Compressible SPH (Fluid)","title":"References","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"M. Antuono, A. Colagrossi, S. Marrone. \"Numerical Diffusive Terms in Weakly-Compressible SPH Schemes.\" In: Computer Physics Communications 183.12 (2012), pages 2570–2580. doi: 10.1016/j.cpc.2012.07.006","category":"page"},{"location":"systems/weakly_compressible_sph/#API","page":"Weakly Compressible SPH (Fluid)","title":"API","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"weakly_compressible_sph\", \"density_diffusion.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusion","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusion","text":"DensityDiffusion\n\nAn abstract supertype of all density diffusion formulations.\n\nCurrently, the following formulations are available:\n\nFormulation Suitable for Steady-State Simulations Low Computational Cost\nDensityDiffusionMolteniColagrossi ❌ ✅\nDensityDiffusionFerrari ❌ ✅\nDensityDiffusionAntuono ✅ ❌\n\nSee Density Diffusion for a comparison and more details.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusionAntuono","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusionAntuono","text":"DensityDiffusionAntuono(initial_condition; delta)\n\nThe commonly used density diffusion terms by Antuono et al. (2010), also referred to as δ-SPH. The density diffusion term by Molteni & Colagrossi (2009) is extended by a second term, which is nicely written down by Antuono et al. (2012).\n\nThe term psi_ab in the continuity equation in DensityDiffusion is defined by\n\npsi_ab = 2left(rho_a - rho_b - frac12big(nablarho^L_a + nablarho^L_bbig) cdot r_abright)\n fracr_abVert r_ab Vert^2\n\nwhere rho_a and rho_b denote the densities of particles a and b respectively and r_ab = r_a - r_b is the difference of the coordinates of particles a and b. The symbol nablarho^L_a denotes the renormalized density gradient defined as\n\nnablarho^L_a = -sum_b (rho_a - rho_b) V_b L_a nabla_r_a W(Vert r_ab Vert h)\n\nwith\n\nL_a = left( -sum_b V_b r_ab otimes nabla_r_a W(Vert r_ab Vert h) right)^-1 in R^d times d\n\nwhere d is the number of dimensions.\n\nSee DensityDiffusion for an overview and comparison of implemented density diffusion terms.\n\nReferences\n\nM. Antuono, A. Colagrossi, S. Marrone, D. Molteni. \"Free-Surface Flows Solved by Means of SPH Schemes with Numerical Diffusive Terms.\" In: Computer Physics Communications 181.3 (2010), pages 532–549. doi: 10.1016/j.cpc.2009.11.002\nM. Antuono, A. Colagrossi, S. Marrone. \"Numerical Diffusive Terms in Weakly-Compressible SPH Schemes.\" In: Computer Physics Communications 183.12 (2012), pages 2570–2580. doi: 10.1016/j.cpc.2012.07.006\nDiego Molteni, Andrea Colagrossi. \"A Simple Procedure to Improve the Pressure Evaluation in Hydrodynamic Context Using the SPH.\" In: Computer Physics Communications 180.6 (2009), pages 861–872. doi: 10.1016/j.cpc.2008.12.004\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusionFerrari","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusionFerrari","text":"DensityDiffusionFerrari()\n\nA density diffusion term by Ferrari et al. (2009).\n\nThe term psi_ab in the continuity equation in DensityDiffusion is defined by\n\npsi_ab = fracrho_a - rho_b2h fracr_abVert r_ab Vert\n\nwhere rho_a and rho_b denote the densities of particles a and b respectively, r_ab = r_a - r_b is the difference of the coordinates of particles a and b and h is the smoothing length.\n\nSee DensityDiffusion for an overview and comparison of implemented density diffusion terms.\n\nReferences\n\nAngela Ferrari, Michael Dumbser, Eleuterio F. Toro, Aronne Armanini. \"A New 3D Parallel SPH Scheme for Free Surface Flows.\" In: Computers & Fluids 38.6 (2009), pages 1203–1217. doi: 10.1016/j.compfluid.2008.11.012.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusionMolteniColagrossi","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusionMolteniColagrossi","text":"DensityDiffusionMolteniColagrossi(; delta)\n\nThe commonly used density diffusion term by Molteni & Colagrossi (2009).\n\nThe term psi_ab in the continuity equation in DensityDiffusion is defined by\n\npsi_ab = 2(rho_a - rho_b) fracr_abVert r_ab Vert^2\n\nwhere rho_a and rho_b denote the densities of particles a and b respectively and r_ab = r_a - r_b is the difference of the coordinates of particles a and b.\n\nSee DensityDiffusion for an overview and comparison of implemented density diffusion terms.\n\nReferences\n\nDiego Molteni, Andrea Colagrossi. \"A Simple Procedure to Improve the Pressure Evaluation in Hydrodynamic Context Using the SPH.\" In: Computer Physics Communications 180.6 (2009), pages 861–872. doi: 10.1016/j.cpc.2008.12.004\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#corrections","page":"Weakly Compressible SPH (Fluid)","title":"Corrections","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"corrections.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.AkinciFreeSurfaceCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.AkinciFreeSurfaceCorrection","text":"AkinciFreeSurfaceCorrection(rho0)\n\nFree surface correction according to Akinci et al. (2013). At a free surface, the mean density is typically lower than the reference density, resulting in reduced surface tension and viscosity forces. The free surface correction adjusts the viscosity, pressure, and surface tension forces near free surfaces to counter this effect. It's important to note that this correlation is unphysical and serves as an approximation. The computation time added by this method is about 2–3%.\n\nMathematically the idea is quite simple. If we have an SPH particle in the middle of a volume at rest, its density will be identical to the rest density rho_0. If we now consider an SPH particle at a free surface at rest, it will have neighbors missing in the direction normal to the surface, which will result in a lower density. If we calculate the correction factor\n\nk = rho_0rho_textmean\n\nthis value will be about ~1.5 for particles at the free surface and can then be used to increase the pressure and viscosity accordingly.\n\nArguments\n\nrho0: Rest density.\n\nReferences\n\nAkinci, N., Akinci, G., & Teschner, M. (2013). \"Versatile Surface Tension and Adhesion for SPH Fluids\". ACM Transactions on Graphics (TOG), 32(6), 182. doi: 10.1145/2508363.2508405\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.BlendedGradientCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.BlendedGradientCorrection","text":"BlendedGradientCorrection()\n\nCalculate a blended gradient to reduce the stability issues of the GradientCorrection.\n\nThis calculates the following,\n\ntildenabla A_i = (1-lambda) nabla A_i + lambda L_i nabla A_i\n\nwith 0 leq lambda leq 1 being the blending factor.\n\nArguments\n\nblending_factor: Blending factor between corrected and regular SPH gradient.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.GradientCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.GradientCorrection","text":"GradientCorrection()\n\nCompute the corrected gradient of particle interactions based on their relative positions.\n\nMathematical Details\n\nGiven the standard SPH representation, the gradient of a field A at particle a is given by\n\nnabla A_a = sum_b m_b fracA_b - A_arho_b nabla_r_a W(Vert r_a - r_b Vert h)\n\nwhere m_b is the mass of particle b and rho_b is the density of particle b.\n\nThe gradient correction, as commonly proposed, involves multiplying this gradient with a correction matrix L:\n\ntildenabla A_a = bmL_a nabla A_a\n\nThe correction matrix bmL_a is computed based on the provided particle configuration, aiming to make the corrected gradient more accurate, especially near domain boundaries.\n\nTo satisfy\n\nsum_b V_b r_ba otimes tildenablaW_b(r_a) = left( sum_b V_b r_ba otimes nabla W_b(r_a) right) bmL_a^T = bmI\n\nthe correction matrix bmL_a is evaluated explicitly as\n\nbmL_a = left( sum_b V_b nabla W_b(r_a) otimes r_ba right)^-1\n\nnote: Note\nStability issues arise, especially when particles separate into small clusters.\nDoubles the computational effort.\n\nBetter stability with smoother smoothing Kernels with larger support, e.g. SchoenbergQuinticSplineKernel or WendlandC6Kernel.\nSet dt_max =< 1e-3 for stability.\n\nReferences\n\nJ. Bonet, T.-S.L. Lok. \"Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations\". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97–115. doi: 10.1016/S0045-7825(99)00051-1\nMihai Basa, Nathan Quinlan, Martin Lastiwka. \"Robustness and accuracy of SPH formulations for viscous flow\". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.KernelCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.KernelCorrection","text":"KernelCorrection()\n\nKernel correction uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al. This can be further extended to obtain a kernel corrected gradient as shown by Basa et al.\n\nThe kernel correction coefficient is determined by\n\nc(x) = sum_b=1 V_b W_b(x)\n\nThe gradient of corrected kernel is determined by\n\nnabla tildeW_b(r) =fracnabla W_b(r) - W_b(r) gamma(r)sum_b=1 V_b W_b(r) quad textwhere quad\ngamma(r) = fracsum_b=1 V_b nabla W_b(r)sum_b=1 V_b W_b(r)\n\nThis correction can be applied with SummationDensity and ContinuityDensity, which leads to an improvement, especially at free surfaces.\n\nnote: Note\nThis only works when the boundary model uses SummationDensity (yet).\nIt is also referred to as \"0th order correction\".\nIn 2D, we can expect an increase of about 10–15% in computation time.\n\nReferences\n\nJ. Bonet, T.-S.L. Lok. \"Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations\". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97-115. doi: 10.1016/S0045-7825(99)00051-1\nMihai Basa, Nathan Quinlan, Martin Lastiwka. \"Robustness and accuracy of SPH formulations for viscous flow\". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927\nShaofan Li, Wing Kam Liu. \"Moving least-square reproducing kernel method Part II: Fourier analysis\". In: Computer Methods in Applied Mechanics and Engineering 139 (1996), pages 159-193. doi:10.1016/S0045-7825(96)01082-1\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.MixedKernelGradientCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.MixedKernelGradientCorrection","text":"MixedKernelGradientCorrection()\n\nCombines GradientCorrection and KernelCorrection, which results in a 1st-order-accurate SPH method.\n\nNotes:\n\nStability issues, especially when particles separate into small clusters.\nDoubles the computational effort.\n\nReferences\n\nJ. Bonet, T.-S.L. Lok. \"Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations\". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97–115. doi: 10.1016/S0045-7825(99)00051-1\nMihai Basa, Nathan Quinlan, Martin Lastiwka. \"Robustness and accuracy of SPH formulations for viscous flow\". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.ShepardKernelCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.ShepardKernelCorrection","text":"ShepardKernelCorrection()\n\nKernel correction uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al.\n\nThe kernel correction coefficient is determined by\n\nc(x) = sum_b=1 V_b W_b(x)\n\nwhere V_b = m_b rho_b is the volume of particle b.\n\nThis correction is applied with SummationDensity to correct the density and leads to an improvement, especially at free surfaces.\n\nnote: Note\nIt is also referred to as \"0th order correction\".\nIn 2D, we can expect an increase of about 5–6% in computation time.\n\nReferences\n\nJ. Bonet, T.-S.L. Lok. \"Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations\". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97-115. doi: 10.1016/S0045-7825(99)00051-1\nMihai Basa, Nathan Quinlan, Martin Lastiwka. \"Robustness and accuracy of SPH formulations for viscous flow\". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927\nShaofan Li, Wing Kam Liu. \"Moving least-square reproducing kernel method Part II: Fourier analysis\". In: Computer Methods in Applied Mechanics and Engineering 139 (1996), pages 159–193. doi:10.1016/S0045-7825(96)01082-1\n\n\n\n\n\n","category":"type"},{"location":"tutorials/tut_dam_break/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_dam_break/","page":"Example file","title":"Example file","text":"!!include:examples/fluid/dam_break_2d.jl!!\n","category":"page"},{"location":"general/neighborhood_search/#Neighborhood-Search","page":"Neighborhood Search","title":"Neighborhood Search","text":"","category":"section"},{"location":"general/neighborhood_search/","page":"Neighborhood Search","title":"Neighborhood Search","text":"Modules = [TrixiParticles]\nPages = map(file -> joinpath(\"neighborhood_search\", file), readdir(joinpath(\"..\", \"src\", \"neighborhood_search\")))","category":"page"},{"location":"general/neighborhood_search/#TrixiParticles.GridNeighborhoodSearch","page":"Neighborhood Search","title":"TrixiParticles.GridNeighborhoodSearch","text":"GridNeighborhoodSearch{NDIMS}(search_radius, n_particles; periodic_box_min_corner=nothing,\n periodic_box_max_corner=nothing, threaded_nhs_update=true)\n\nSimple grid-based neighborhood search with uniform search radius. The domain is divided into a regular grid. For each (non-empty) grid cell, a list of particles in this cell is stored. Instead of representing a finite domain by an array of cells, a potentially infinite domain is represented by storing cell lists in a hash table (using Julia's Dict data structure), indexed by the cell index tuple\n\nleft( leftlfloor fracxd rightrfloor leftlfloor fracyd rightrfloor right) quad textor quad\nleft( leftlfloor fracxd rightrfloor leftlfloor fracyd rightrfloor leftlfloor fraczd rightrfloor right)\n\nwhere x y z are the space coordinates and d is the search radius.\n\nTo find particles within the search radius around a point, only particles in the neighboring cells are considered.\n\nSee also (Chalela et al., 2021), (Ihmsen et al. 2011, Section 4.4).\n\nAs opposed to (Ihmsen et al. 2011), we do not sort the particles in any way, since not sorting makes our implementation a lot faster (although less parallelizable).\n\nArguments\n\nNDIMS: Number of dimensions.\nsearch_radius: The uniform search radius.\nn_particles: Total number of particles.\n\nKeywords\n\nperiodic_box_min_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in negative coordinate directions.\nperiodic_box_max_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in positive coordinate directions.\nthreaded_nhs_update=true: Can be used to deactivate thread parallelization in the neighborhood search update. This can be one of the largest sources of variations between simulations with different thread numbers due to particle ordering changes.\n\nwarning: Internal use only\nPlease note that this constructor is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.To run a simulation with this neighborhood search, just pass the type to the constructor of Semidiscretization:semi = Semidiscretization(system1, system2,\n neighborhood_search=GridNeighborhoodSearch)The keyword arguments periodic_box_min_corner and periodic_box_max_corner explained above can also be passed to the Semidiscretization and will internally be forwarded to the neighborhood search:semi = Semidiscretization(system1, system2,\n neighborhood_search=GridNeighborhoodSearch,\n periodic_box_min_corner=[0.0, -0.25],\n periodic_box_max_corner=[1.0, 0.75])\n\nReferences\n\nM. Chalela, E. Sillero, L. Pereyra, M.A. Garcia, J.B. Cabral, M. Lares, M. Merchán. \"GriSPy: A Python package for fixed-radius nearest neighbors search\". In: Astronomy and Computing 34 (2021). doi: 10.1016/j.ascom.2020.100443\nMarkus Ihmsen, Nadir Akinci, Markus Becker, Matthias Teschner. \"A Parallel SPH Implementation on Multi-Core CPUs\". In: Computer Graphics Forum 30.1 (2011), pages 99–112. doi: 10.1111/J.1467-8659.2010.01832.X\n\n\n\n\n\n","category":"type"},{"location":"general/neighborhood_search/#TrixiParticles.TrivialNeighborhoodSearch","page":"Neighborhood Search","title":"TrixiParticles.TrivialNeighborhoodSearch","text":"TrivialNeighborhoodSearch{NDIMS}(search_radius, eachparticle)\n\nTrivial neighborhood search that simply loops over all particles. The search radius still needs to be passed in order to sort out particles outside the search radius in the internal function for_particle_neighbor, but it's not used in the internal function eachneighbor.\n\nArguments\n\nNDIMS: Number of dimensions.\nsearch_radius: The uniform search radius.\neachparticle: UnitRange of all particle indices. Usually just 1:n_particles.\n\nKeywords\n\nperiodic_box_min_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in negative coordinate directions.\nperiodic_box_max_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in positive coordinate directions.\n\nwarning: Internal use only\nPlease note that this constructor is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.To run a simulation with this neighborhood search, just pass the type to the constructor of Semidiscretization:semi = Semidiscretization(system1, system2,\n neighborhood_search=TrivialNeighborhoodSearch)The keyword arguments periodic_box_min_corner and periodic_box_max_corner explained above can also be passed to the Semidiscretization and will internally be forwarded to the neighborhood search:semi = Semidiscretization(system1, system2,\n neighborhood_search=TrivialNeighborhoodSearch,\n periodic_box_min_corner=[0.0, -0.25],\n periodic_box_max_corner=[1.0, 0.75])\n\n\n\n\n\n","category":"type"},{"location":"tutorials/tut_falling/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_falling/","page":"Example file","title":"Example file","text":"!!include:examples/fsi/falling_spheres_2d.jl!!\n","category":"page"},{"location":"tutorials/tut_beam/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_beam/","page":"Example file","title":"Example file","text":"!!include:examples/solid/oscillating_beam_2d.jl!!\n","category":"page"},{"location":"general/density_calculators/#density_calculator","page":"Density Calculators","title":"Density Calculators","text":"","category":"section"},{"location":"general/density_calculators/","page":"Density Calculators","title":"Density Calculators","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"density_calculators.jl\")]","category":"page"},{"location":"general/density_calculators/#TrixiParticles.ContinuityDensity","page":"Density Calculators","title":"TrixiParticles.ContinuityDensity","text":"ContinuityDensity()\n\nDensity calculator to integrate the density from the continuity equation\n\nfracmathrmdrho_amathrmdt = sum_b m_b v_ab cdot nabla_r_a W(Vert r_a - r_b Vert h)\n\nwhere rho_a denotes the density of particle a and r_ab = r_a - r_b is the difference of the coordinates, v_ab = v_a - v_b of the velocities of particles a and b.\n\n\n\n\n\n","category":"type"},{"location":"general/density_calculators/#TrixiParticles.SummationDensity","page":"Density Calculators","title":"TrixiParticles.SummationDensity","text":"SummationDensity()\n\nDensity calculator to use the summation formula\n\nrho(r) = sum_b m_b W(Vert r - r_b Vert h)\n\nfor the density estimation, where r_b denotes the coordinates and m_b the mass of particle b.\n\n\n\n\n\n","category":"type"},{"location":"general/semidiscretization/#Semidiscretization","page":"Semidiscretization","title":"Semidiscretization","text":"","category":"section"},{"location":"general/semidiscretization/","page":"Semidiscretization","title":"Semidiscretization","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"semidiscretization.jl\")]","category":"page"},{"location":"general/semidiscretization/#TrixiParticles.Semidiscretization","page":"Semidiscretization","title":"TrixiParticles.Semidiscretization","text":"Semidiscretization(systems...; neighborhood_search=GridNeighborhoodSearch,\n periodic_box_min_corner=nothing, periodic_box_max_corner=nothing,\n threaded_nhs_update=true)\n\nThe semidiscretization couples the passed systems to one simulation.\n\nThe type of neighborhood search to be used in the simulation can be specified with the keyword argument neighborhood_search. A value of nothing means no neighborhood search.\n\nArguments\n\nsystems: Systems to be coupled in this semidiscretization\n\nKeywords\n\nneighborhood_search: The type of neighborhood search to be used in the simulation. By default, the GridNeighborhoodSearch is used. Use TrivialNeighborhoodSearch or nothing to loop over all particles (no neighborhood search).\nperiodic_box_min_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in negative coordinate directions.\nperiodic_box_max_corner: In order to use a (rectangular) periodic domain, pass the coordinates of the domain corner in positive coordinate directions.\nthreaded_nhs_update=true: Can be used to deactivate thread parallelization in the neighborhood search update. This can be one of the largest sources of variations between simulations with different thread numbers due to particle ordering changes.\n\nExamples\n\nsemi = Semidiscretization(fluid_system, boundary_system)\n\nsemi = Semidiscretization(fluid_system, boundary_system,\n neighborhood_search=TrivialNeighborhoodSearch)\n\n\n\n\n\n","category":"type"},{"location":"general/semidiscretization/#TrixiParticles.SourceTermDamping","page":"Semidiscretization","title":"TrixiParticles.SourceTermDamping","text":"SourceTermDamping(; damping_coefficient)\n\nA source term to be used when a damping step is required before running a full simulation. The term -c cdot v_a is added to the acceleration fracmathrmdv_amathrmdt of particle a, where c is the damping coefficient and v_a is the velocity of particle a.\n\nKeywords\n\ndamping_coefficient: The coefficient d above. A higher coefficient means more damping. A coefficient of 1e-4 is a good starting point for damping a fluid at rest.\n\nExamples\n\nsource_terms = SourceTermDamping(; damping_coefficient=1e-4)\n\n\n\n\n\n","category":"type"},{"location":"general/semidiscretization/#TrixiParticles.restart_with!-Tuple{Any, Any}","page":"Semidiscretization","title":"TrixiParticles.restart_with!","text":"restart_with!(semi, sol)\n\nSet the initial coordinates and velocities of all systems in semi to the final values in the solution sol. semidiscretize has to be called again afterwards, or another Semidiscretization can be created with the updated systems.\n\nArguments\n\nsemi: The semidiscretization\nsol: The ODESolution returned by solve of OrdinaryDiffEq\n\n\n\n\n\n","category":"method"},{"location":"general/semidiscretization/#TrixiParticles.semidiscretize-Tuple{Any, Any}","page":"Semidiscretization","title":"TrixiParticles.semidiscretize","text":"semidiscretize(semi, tspan; reset_threads=true)\n\nCreate an ODEProblem from the semidiscretization with the specified tspan.\n\nArguments\n\nsemi: A Semidiscretization holding the systems involved in the simulation.\ntspan: The time span over which the simulation will be run.\n\nKeywords\n\nreset_threads: A boolean flag to reset Polyester.jl threads before the simulation (default: true). After an error within a threaded loop, threading might be disabled. Resetting the threads before the simulation ensures that threading is enabled again for the simulation. See also trixi-framework/Trixi.jl#1583.\n\nReturns\n\nA DynamicalODEProblem (see the OrdinaryDiffEq.jl docs) to be integrated with OrdinaryDiffEq.jl. Note that this is not a true DynamicalODEProblem where the acceleration does not depend on the velocity. Therefore, not all integrators designed for DynamicalODEProblems will work properly. However, all integrators designed for ODEProblems can be used.\n\nExamples\n\nsemi = Semidiscretization(fluid_system, boundary_system)\ntspan = (0.0, 1.0)\node_problem = semidiscretize(semi, tspan)\n\n\n\n\n\n","category":"method"},{"location":"systems/boundary/#Boundary-System","page":"Boundary","title":"Boundary System","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundarySPHSystem","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundarySPHSystem","page":"Boundary","title":"TrixiParticles.BoundarySPHSystem","text":"BoundarySPHSystem(initial_condition, boundary_model; movement=nothing)\n\nSystem for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.\n\nArguments\n\ninitial_condition: Initial condition (see InitialCondition)\nboundary_model: Boundary model (see Boundary Models)\n\nKeyword Arguments\n\nmovement: For moving boundaries, a BoundaryMovement can be passed.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundaryDEMSystem","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryDEMSystem","page":"Boundary","title":"TrixiParticles.BoundaryDEMSystem","text":"BoundaryDEMSystem(initial_condition, normal_stiffness)\n\nSystem for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in a future releases.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundaryMovement","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryMovement","page":"Boundary","title":"TrixiParticles.BoundaryMovement","text":"BoundaryMovement(movement_function, is_moving; moving_particles=nothing)\n\nArguments\n\nmovement_function: Time-dependent function returning an SVector of d dimensions for a d-dimensional problem.\nis_moving: Function to determine in each timestep if the particles are moving or not. Its boolean return value is mandatory to determine if the neighborhood search will be updated.\n\nKeyword Arguments\n\nmoving_particles: Indices of moving particles. Default is each particle in BoundarySPHSystem.\n\nIn the example below, movement describes particles moving in a circle as long as the time is lower than 1.5.\n\nExamples\n\nmovement_function(t) = SVector(cos(2pi*t), sin(2pi*t))\nis_moving(t) = t < 1.5\n\nmovement = BoundaryMovement(movement_function, is_moving)\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#boundary_models","page":"Boundary","title":"Boundary Models","text":"","category":"section"},{"location":"systems/boundary/#Dummy-Particles","page":"Boundary","title":"Dummy Particles","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Boundaries modeled as dummy particles, which are treated like fluid particles, but their positions and velocities are not evolved in time. Since the force towards the fluid should not change with the material density when used with a TotalLagrangianSPHSystem, the dummy particles need to have a mass corresponding to the fluid's rest density, which we call \"hydrodynamic mass\", as opposed to mass corresponding to the material density of a TotalLagrangianSPHSystem.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Here, initial_density and hydrodynamic_mass are vectors that contains the initial density and the hydrodynamic mass respectively for each boundary particle. Note that when used with SummationDensity (see below), this is only used to determine the element type and the number of boundary particles.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"To establish a relationship between density and pressure, a state_equation has to be passed, which should be the same as for the adjacent fluid systems. To sum over neighboring particles, a smoothing_kernel and smoothing_length needs to be passed. This should be the same as for the adjacent fluid system with the largest smoothing length.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"In the literature, this kind of boundary particles is referred to as \"dummy particles\" (Adami et al., 2012 and Valizadeh & Monaghan, 2015), \"frozen fluid particles\" (Akinci et al., 2012) or \"dynamic boundaries (Crespo et al., 2007). The key detail of this boundary condition and the only difference between the boundary models in these references is the way the density and pressure of boundary particles is computed.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Since boundary particles are treated like fluid particles, the force on fluid particle a due to boundary particle b is given by","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"f_ab = m_a m_b left( fracp_arho_a^2 + fracp_brho_b^2 right) nabla_r_a W(Vert r_a - r_b Vert h)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The quantities to be defined here are the density rho_b and pressure p_b of the boundary particle b.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundaryModelDummyParticles","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryModelDummyParticles","page":"Boundary","title":"TrixiParticles.BoundaryModelDummyParticles","text":"BoundaryModelDummyParticles(initial_density, hydrodynamic_mass,\n density_calculator, smoothing_kernel,\n smoothing_length; viscosity=nothing,\n state_equation=nothing, correction=nothing)\n\nboundary_model for BoundarySPHSystem.\n\nArguments\n\ninitial_density: Vector holding the initial density of each boundary particle.\nhydrodynamic_mass: Vector holding the \"hydrodynamic mass\" of each boundary particle. See description above for more information.\ndensity_calculator: Strategy to compute the hydrodynamic density of the boundary particles. See description below for more information.\nsmoothing_kernel: Smoothing kernel should be the same as for the adjacent fluid system.\nsmoothing_length: Smoothing length should be the same as for the adjacent fluid system.\n\nKeywords\n\nstate_equation: This should be the same as for the adjacent fluid system (see e.g. StateEquationCole).\ncorrection: Correction method of the adjacent fluid system (see Corrections).\nviscosity: Slip (default) or no-slip condition. See description below for further information.\n\nExamples\n\n# Free-slip condition\nboundary_model = BoundaryModelDummyParticles(densities, masses, AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length)\n\n# No-slip condition\nboundary_model = BoundaryModelDummyParticles(densities, masses, AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length,\n viscosity=ViscosityAdami(nu=1e-6))\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#Hydrodynamic-density-of-dummy-particles","page":"Boundary","title":"Hydrodynamic density of dummy particles","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"We provide five options to compute the boundary density and pressure, determined by the density_calculator:","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"(Recommended) With AdamiPressureExtrapolation, the pressure is extrapolated from the pressure of the fluid according to (Adami et al., 2012), and the density is obtained by applying the inverse of the state equation. This option usually yields the best results of the options listed here.\nWith SummationDensity, the density is calculated by summation over the neighboring particles, and the pressure is computed from the density with the state equation.\nWith ContinuityDensity, the density is integrated from the continuity equation, and the pressure is computed from the density with the state equation. Note that this causes a gap between fluid and boundary where the boundary is initialized without any contact to the fluid. This is due to overestimation of the boundary density as soon as the fluid comes in contact with boundary particles that initially did not have contact to the fluid. Therefore, in dam break simulations, there is a visible \"step\", even though the boundary is supposed to be flat. See also dual.sphysics.org/faq/#Q_13.\nWith PressureZeroing, the density is set to the reference density and the pressure is computed from the density with the state equation. This option is not recommended. The other options yield significantly better results.\nWith PressureMirroring, the density is set to the reference density. The pressure is not used. Instead, the fluid pressure is mirrored as boundary pressure in the momentum equation. This option is not recommended due to stability issues. See PressureMirroring for more details.","category":"page"},{"location":"systems/boundary/#1.-[AdamiPressureExtrapolation](@ref)","page":"Boundary","title":"1. AdamiPressureExtrapolation","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The pressure of the boundary particles is obtained by extrapolating the pressure of the fluid according to (Adami et al., 2012). The pressure of a boundary particle b is given by","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"p_b = fracsum_f (p_f + rho_f (bmg - bma_b) cdot bmr_bf) W(Vert r_bf Vert h)sum_f W(Vert r_bf Vert h)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the sum is over all fluid particles, rho_f and p_f denote the density and pressure of fluid particle f, respectively, r_bf = r_b - r_f denotes the difference of the coordinates of particles b and f, bmg denotes the gravitational acceleration acting on the fluid, and bma_b denotes the acceleration of the boundary particle b.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" AdamiPressureExtrapolation","category":"page"},{"location":"systems/boundary/#TrixiParticles.AdamiPressureExtrapolation","page":"Boundary","title":"TrixiParticles.AdamiPressureExtrapolation","text":"AdamiPressureExtrapolation()\n\ndensity_calculator for BoundaryModelDummyParticles.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#4.-[PressureZeroing](@ref)","page":"Boundary","title":"4. PressureZeroing","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"This is the simplest way to implement dummy boundary particles. The density of each particle is set to the reference density and the pressure to the reference pressure (the corresponding pressure to the reference density by the state equation).","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" PressureZeroing","category":"page"},{"location":"systems/boundary/#TrixiParticles.PressureZeroing","page":"Boundary","title":"TrixiParticles.PressureZeroing","text":"PressureZeroing()\n\ndensity_calculator for BoundaryModelDummyParticles.\n\nnote: Note\nThis boundary model produces significantly worse results than all other models and is only included for research purposes.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#5.-[PressureMirroring](@ref)","page":"Boundary","title":"5. PressureMirroring","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Instead of calculating density and pressure for each boundary particle, we modify the momentum equation,","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"fracmathrmdv_amathrmdt = -sum_b m_b left( fracp_arho_a^2 + fracp_brho_b^2 right) nabla_a W_ab","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"to replace the unknown density rho_b if b is a boundary particle by the reference density and the unknown pressure p_b if b is a boundary particle by the pressure p_a of the interacting fluid particle. The momentum equation therefore becomes","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"fracmathrmdv_amathrmdt = -sum_f m_f left( fracp_arho_a^2 + fracp_frho_f^2 right) nabla_a W_af\n-sum_b m_b left( fracp_arho_a^2 + fracp_arho_0^2 right) nabla_a W_ab","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the first sum is over all fluid particles and the second over all boundary particles.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"This approach was first mentioned by Akinci et al. (2012) and written down in this form by Band et al. (2018).","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" PressureMirroring","category":"page"},{"location":"systems/boundary/#TrixiParticles.PressureMirroring","page":"Boundary","title":"TrixiParticles.PressureMirroring","text":"PressureMirroring()\n\ndensity_calculator for BoundaryModelDummyParticles.\n\nnote: Note\nThis boundary model requires high viscosity for stability with WCSPH. It also produces significantly worse results than AdamiPressureExtrapolation and is not more efficient because smaller time steps are required due to more noise in the pressure. We added this model only for research purposes and for comparison with SPlisHSPlasH.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#No-slip-conditions","page":"Boundary","title":"No-slip conditions","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"For the interaction of dummy particles and fluid particles, Adami et al. (2012) impose a no-slip boundary condition by assigning a wall velocity v_w to the dummy particle.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The wall velocity of particle a is calculated from the prescribed boundary particle velocity v_a and the smoothed velocity field","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"v_w = 2 v_a - fracsum_b v_b W_absum_b W_ab","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the sum is over all fluid particles.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"By choosing the viscosity model ViscosityAdami for viscosity, a no-slip condition is imposed. It is recommended to choose nu in the order of either the kinematic viscosity parameter of the adjacent fluid or the equivalent from the artificial parameter alpha of the adjacent fluid (nu = fracalpha h c 2d + 4). When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"warning: Warning\nThe viscosity model ArtificialViscosityMonaghan for BoundaryModelDummyParticles has not been verified yet.","category":"page"},{"location":"systems/boundary/#References","page":"Boundary","title":"References","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"S. Adami, X. Y. Hu, N. A. Adams. \"A generalized wall boundary condition for smoothed particle hydrodynamics\". In: Journal of Computational Physics 231, 21 (2012), pages 7057–7075. doi: 10.1016/J.JCP.2012.05.005\nAlireza Valizadeh, Joseph J. Monaghan. \"A study of solid wall models for weakly compressible SPH\". In: Journal of Computational Physics 300 (2015), pages 5–19. doi: 10.1016/J.JCP.2015.07.033\nNadir Akinci, Markus Ihmsen, Gizem Akinci, Barbara Solenthaler, Matthias Teschner. \"Versatile rigid-fluid coupling for incompressible SPH\". ACM Transactions on Graphics 31, 4 (2012), pages 1–8. doi: 10.1145/2185520.2185558\nA. J. C. Crespo, M. Gómez-Gesteira, R. A. Dalrymple. \"Boundary conditions generated by dynamic particles in SPH methods\" In: Computers, Materials and Continua 5 (2007), pages 173-184. doi: 10.3970/cmc.2007.005.173\nStefan Band, Christoph Gissler, Andreas Peer, and Matthias Teschner. \"MLS Pressure Boundaries for Divergence-Free and Viscous SPH Fluids.\" In: Computers & Graphics 76 (2018), pages 37–46. doi: 10.1016/j.cag.2018.08.001","category":"page"},{"location":"systems/boundary/#Repulsive-Particles","page":"Boundary","title":"Repulsive Particles","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Boundaries modeled as boundary particles which exert forces on the fluid particles (Monaghan, Kajtar, 2009). The force on fluid particle a due to boundary particle b is given by","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"f_ab = m_a left(tildef_ab - m_b Pi_ab nabla_r_a W(Vert r_a - r_b Vert h)right)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"with","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"tildef_ab = fracKbeta^n-1 fracr_abVert r_ab Vert (Vert r_ab Vert - d) Phi(Vert r_ab Vert h)\nfrac2 m_bm_a + m_b","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where m_a and m_b are the masses of fluid particle a and boundary particle b respectively, r_ab = r_a - r_b is the difference of the coordinates of particles a and b, d denotes the boundary particle spacing and n denotes the number of dimensions (see (Monaghan, Kajtar, 2009, Equation (3.1)) and (Valizadeh, Monaghan, 2015)). Note that the repulsive acceleration tildef_ab does not depend on the masses of the boundary particles. Here, Phi denotes the 1D Wendland C4 kernel, normalized to 177 for q=0 (Monaghan, Kajtar, 2009, Section 4), with Phi(r h) = w(rh) and","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"w(q) =\nbegincases\n (17732) (1 + (52)q + 2q^2)(2 - q)^5 textif 0 leq q 2 \n 0 textif q geq 2\nendcases","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The boundary particles are assumed to have uniform spacing by the factor beta smaller than the expected fluid particle spacing. For example, if the fluid particles have an expected spacing of 03 and the boundary particles have a uniform spacing of 01, then this parameter should be set to beta = 3. According to (Monaghan, Kajtar, 2009), a value of beta = 3 for the Wendland C4 that we use here is reasonable for most computing purposes.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The parameter K is used to scale the force exerted by the boundary particles. In (Monaghan, Kajtar, 2009), a value of gD is used for static tank simulations, where g is the gravitational acceleration and D is the depth of the fluid.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The viscosity Pi_ab is calculated according to the viscosity used in the simulation, where the density of the boundary particle if needed is assumed to be identical to the density of the fluid particle.","category":"page"},{"location":"systems/boundary/#No-slip-condition","page":"Boundary","title":"No-slip condition","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"By choosing the viscosity model ArtificialViscosityMonaghan for viscosity, a no-slip condition is imposed. When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"warning: Warning\nThe no-slip conditions for BoundaryModelMonaghanKajtar have not been verified yet.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"boundary\", \"monaghan_kajtar\", \"monaghan_kajtar.jl\")]","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryModelMonaghanKajtar","page":"Boundary","title":"TrixiParticles.BoundaryModelMonaghanKajtar","text":"BoundaryModelMonaghanKajtar(K, beta, boundary_particle_spacing, mass;\n viscosity=nothing)\n\nboundary_model for BoundarySPHSystem.\n\nArguments\n\nK: Scaling factor for repulsive force.\nbeta: Ratio of fluid particle spacing to boundary particle spacing.\nboundary_particle_spacing: Boundary particle spacing.\nmass: Vector holding the mass of each boundary particle.\n\nKeywords\n\nviscosity: Free-slip (default) or no-slip condition. See description above for further information.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#References-2","page":"Boundary","title":"References","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Joseph J. Monaghan, Jules B. Kajtar. \"SPH particle boundary forces for arbitrary boundaries\". In: Computer Physics Communications 180.10 (2009), pages 1811–1820. doi: 10.1016/j.cpc.2009.05.008\nAlireza Valizadeh, Joseph J. Monaghan. \"A study of solid wall models for weakly compressible SPH.\" In: Journal of Computational Physics 300 (2015), pages 5–19. doi: 10.1016/J.JCP.2015.07.033","category":"page"},{"location":"news/","page":"News","title":"News","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/NEWS.md\"","category":"page"},{"location":"news/#Changelog","page":"News","title":"Changelog","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"TrixiParticles.jl follows the interpretation of semantic versioning (semver) used in the Julia ecosystem. Notable changes will be documented in this file for human readability. We aim at 3 to 4 month between major release versions and about 2 weeks between minor versions. ","category":"page"},{"location":"news/#Version-0.2.x","page":"News","title":"Version 0.2.x","text":"","category":"section"},{"location":"news/#Highlights","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/#Added","page":"News","title":"Added","text":"","category":"section"},{"location":"news/#Removed","page":"News","title":"Removed","text":"","category":"section"},{"location":"news/#Deprecated","page":"News","title":"Deprecated","text":"","category":"section"},{"location":"news/#Version-0.1.2","page":"News","title":"Version 0.1.2","text":"","category":"section"},{"location":"news/#Highlights-2","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/#Discrete-Element-Method","page":"News","title":"Discrete Element Method","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"A basic implementation of the discrete element method was added.","category":"page"},{"location":"news/#Version-0.1.1","page":"News","title":"Version 0.1.1","text":"","category":"section"},{"location":"news/#Added-2","page":"News","title":"Added","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"A surface tension and adhesion model based on the work by Akinci et al., \"Versatile Surface Tension and Adhesion for SPH Fluids\", 2013 was added to WCSPH","category":"page"},{"location":"news/#Pre-Initial-Release-(v0.1.0)","page":"News","title":"Pre Initial Release (v0.1.0)","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"This section summarizes the initial features that TrixiParticles.jl was released with.","category":"page"},{"location":"news/#Highlights-3","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/#EDAC","page":"News","title":"EDAC","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"An implementation of EDAC (Entropically Damped Artificial Compressibility) was added, which allows for more stable simulations compared to basic WCSPH and reduces spurious pressure oscillations.","category":"page"},{"location":"news/#WCSPH","page":"News","title":"WCSPH","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"An implementation of WCSPH (Weakly Compressible Smoothed Particle Hydrodynamics), which is the classical SPH approach.","category":"page"},{"location":"news/","page":"News","title":"News","text":"Features:","category":"page"},{"location":"news/","page":"News","title":"News","text":"Correction schemes (Shepard (0. Order) ... MixedKernelGradient (1. Order))\nDensity reinitialization\nKernel summation and Continuity equation density formulations\nFlexible boundary conditions e.g. dummy particles with Adami pressure extrapolation, pressure zeroing, pressure mirroring...\nMoving boundaries\nDensity diffusion based on the models by Molteni & Colagrossi (2009), Ferrari et al. (2009) and Antuono et al. (2010).","category":"page"},{"location":"news/#TLSPH","page":"News","title":"TLSPH","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"An implementation of TLSPH (Total Lagrangian Smoothed Particle Hydrodynamics) for solid bodies enabling FSI (Fluid Structure Interactions).","category":"page"},{"location":"general/util/#Util","page":"Util","title":"Util","text":"","category":"section"},{"location":"general/util/","page":"Util","title":"Util","text":"Modules = [TrixiParticles]\nPages = [\"util.jl\"]","category":"page"},{"location":"general/util/#TrixiParticles.examples_dir-Tuple{}","page":"Util","title":"TrixiParticles.examples_dir","text":"examples_dir()\n\nReturn the directory where the example files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.\n\nCopied from Trixi.jl.\n\nExamples\n\nreaddir(examples_dir())\n\n\n\n\n\n","category":"method"},{"location":"general/util/#TrixiParticles.validation_dir-Tuple{}","page":"Util","title":"TrixiParticles.validation_dir","text":"validation_dir()\n\nReturn the directory where the validation files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.\n\nCopied from Trixi.jl.\n\nExamples\n\nreaddir(validation_dir())\n\n\n\n\n\n","category":"method"},{"location":"general/util/#TrixiParticles.@autoinfiltrate","page":"Util","title":"TrixiParticles.@autoinfiltrate","text":"@autoinfiltrate\n@autoinfiltrate condition::Bool\n\nInvoke the @infiltrate macro of the package Infiltrator.jl to create a breakpoint for ad-hoc interactive debugging in the REPL. If the optional argument condition is given, the breakpoint is only enabled if condition evaluates to true.\n\nAs opposed to using Infiltrator.@infiltrate directly, this macro does not require Infiltrator.jl to be added as a dependency to TrixiParticles.jl. As a bonus, the macro will also attempt to load the Infiltrator module if it has not yet been loaded manually.\n\nNote: For this macro to work, the Infiltrator.jl package needs to be installed in your current Julia environment stack.\n\nSee also: Infiltrator.jl\n\nwarning: Internal use only\nPlease note that this macro is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.\n\n\n\n\n\n","category":"macro"},{"location":"general/util/#TrixiParticles.@threaded-Tuple{Any}","page":"Util","title":"TrixiParticles.@threaded","text":"@threaded for ... end\n\nSemantically the same as Threads.@threads when iterating over a AbstractUnitRange but without guarantee that the underlying implementation uses Threads.@threads or works for more general for loops. In particular, there may be an additional check whether only one thread is used to reduce the overhead of serial execution or the underlying threading capabilities might be provided by other packages such as Polyester.jl.\n\nwarn: Warn\nThis macro does not necessarily work for general for loops. For example, it does not necessarily support general iterables such as eachline(filename).\n\nSome discussion can be found at https://discourse.julialang.org/t/overhead-of-threads-threads/53964 and https://discourse.julialang.org/t/threads-threads-with-one-thread-how-to-remove-the-overhead/58435.\n\nCopied from Trixi.jl.\n\n\n\n\n\n","category":"macro"},{"location":"code_of_conduct/","page":"Code of Conduct","title":"Code of Conduct","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/CODE_OF_CONDUCT.md\"","category":"page"},{"location":"code_of_conduct/#Code-of-Conduct","page":"Code of Conduct","title":"Code of Conduct","text":"","category":"section"},{"location":"code_of_conduct/","page":"Code of Conduct","title":"Code of Conduct","text":"Contributor Covenant Code of ConductOur PledgeWe as members, contributors, and leaders pledge to make participation in our community a harassment-free experience for everyone, regardless of age, body size, visible or invisible disability, ethnicity, sex characteristics, gender identity and expression, level of experience, education, socio-economic status, nationality, personal appearance, race, religion, or sexual identity and orientation.We pledge to act and interact in ways that contribute to an open, welcoming, diverse, inclusive, and healthy community.Our StandardsExamples of behavior that contributes to a positive environment for our community include:Demonstrating empathy and kindness toward other people\nBeing respectful of differing opinions, viewpoints, and experiences\nGiving and gracefully accepting constructive feedback\nAccepting responsibility and apologizing to those affected by our mistakes, and learning from the experience\nFocusing on what is best not just for us as individuals, but for the overall communityExamples of unacceptable behavior include:The use of sexualized language or imagery, and sexual attention or advances of any kind\nTrolling, insulting or derogatory comments, and personal or political attacks\nPublic or private harassment\nPublishing others' private information, such as a physical or email address, without their explicit permission\nOther conduct which could reasonably be considered inappropriate in a professional settingEnforcement ResponsibilitiesCommunity leaders are responsible for clarifying and enforcing our standards of acceptable behavior and will take appropriate and fair corrective action in response to any behavior that they deem inappropriate, threatening, offensive, or harmful.Community leaders have the right and responsibility to remove, edit, or reject comments, commits, code, wiki edits, issues, and other contributions that are not aligned to this Code of Conduct, and will communicate reasons for moderation decisions when appropriate.ScopeThis Code of Conduct applies within all community spaces, and also applies when an individual is officially representing the community in public spaces. Examples of representing our community include using an official e-mail address, posting via an official social media account, or acting as an appointed representative at an online or offline event.EnforcementInstances of abusive, harassing, or otherwise unacceptable behavior may be reported to Michael Schlottke-Lakemper, Sven Berger, or any other of the principal developers responsible for enforcement listed in Authors. All complaints will be reviewed and investigated promptly and fairly.All community leaders are obligated to respect the privacy and security of the reporter of any incident.Enforcement GuidelinesCommunity leaders will follow these Community Impact Guidelines in determining the consequences for any action they deem in violation of this Code of Conduct:1. CorrectionCommunity Impact: Use of inappropriate language or other behavior deemed unprofessional or unwelcome in the community.Consequence: A private, written warning from community leaders, providing clarity around the nature of the violation and an explanation of why the behavior was inappropriate. A public apology may be requested.2. WarningCommunity Impact: A violation through a single incident or series of actions.Consequence: A warning with consequences for continued behavior. No interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, for a specified period of time. This includes avoiding interactions in community spaces as well as external channels like social media. Violating these terms may lead to a temporary or permanent ban.3. Temporary BanCommunity Impact: A serious violation of community standards, including sustained inappropriate behavior.Consequence: A temporary ban from any sort of interaction or public communication with the community for a specified period of time. No public or private interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, is allowed during this period. Violating these terms may lead to a permanent ban.4. Permanent BanCommunity Impact: Demonstrating a pattern of violation of community standards, including sustained inappropriate behavior, harassment of an individual, or aggression toward or disparagement of classes of individuals.Consequence: A permanent ban from any sort of public interaction within the community.AttributionThis Code of Conduct is adapted from the [Contributor Covenant][homepage], version 2.0, available at https://www.contributor-covenant.org/version/2/0/codeofconduct.html.Community Impact Guidelines were inspired by Mozilla's code of conduct enforcement ladder.[homepage]: https://www.contributor-covenant.orgFor answers to common questions about this code of conduct, see the FAQ at https://www.contributor-covenant.org/faq. Translations are available at https://www.contributor-covenant.org/translations.","category":"page"},{"location":"tutorials/tut_dam_break_replaced/","page":"Example file","title":"Example file","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials/tut_dam_break.md\"","category":"page"},{"location":"tutorials/tut_dam_break_replaced/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_dam_break_replaced/","page":"Example file","title":"Example file","text":"# 2D dam break simulation based on\n#\n# S. Marrone, M. Antuono, A. Colagrossi, G. Colicchio, D. le Touzé, G. Graziani.\n# \"δ-SPH model for simulating violent impact flows\".\n# In: Computer Methods in Applied Mechanics and Engineering, Volume 200, Issues 13–16 (2011), pages 1526–1542.\n# https://doi.org/10.1016/J.CMA.2010.12.016\n\nusing TrixiParticles\nusing OrdinaryDiffEq\n\n# Size parameters\nH = 0.6\nW = 2 * H\n\n# ==========================================================================================\n# ==== Resolution\nfluid_particle_spacing = H / 40\n\n# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model\nboundary_layers = 4\nspacing_ratio = 1\n\nboundary_particle_spacing = fluid_particle_spacing / spacing_ratio\n\n# ==========================================================================================\n# ==== Experiment Setup\ngravity = 9.81\n\ntspan = (0.0, 5.7 / sqrt(gravity))\n\n# Boundary geometry and initial fluid particle positions\ninitial_fluid_size = (W, H)\ntank_size = (floor(5.366 * H / boundary_particle_spacing) * boundary_particle_spacing, 4.0)\n\nfluid_density = 1000.0\nsound_speed = 20 * sqrt(gravity * H)\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=1, clip_negative_pressure=false)\n\ntank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,\n n_layers=boundary_layers, spacing_ratio=spacing_ratio,\n acceleration=(0.0, -gravity), state_equation=state_equation)\n\n# ==========================================================================================\n# ==== Fluid\nsmoothing_length = 3.5 * fluid_particle_spacing\nsmoothing_kernel = WendlandC2Kernel{2}()\n\nfluid_density_calculator = ContinuityDensity()\nviscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)\n# Alternatively the density diffusion model by Molteni & Colagrossi can be used,\n# which will run faster.\n# density_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)\ndensity_diffusion = DensityDiffusionAntuono(tank.fluid, delta=0.1)\n\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, smoothing_kernel,\n smoothing_length, viscosity=viscosity,\n density_diffusion=density_diffusion,\n acceleration=(0.0, -gravity),\n correction=nothing)\n\n# ==========================================================================================\n# ==== Boundary\nboundary_density_calculator = AdamiPressureExtrapolation()\nboundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n boundary_density_calculator,\n smoothing_kernel, smoothing_length,\n correction=nothing)\n\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model)\n\n# ==========================================================================================\n# ==== Simulation\nsemi = Semidiscretization(fluid_system, boundary_system, threaded_nhs_update=true)\node = semidiscretize(semi, tspan)\n\ninfo_callback = InfoCallback(interval=100)\n\nsolution_prefix = \"\"\nsaving_callback = SolutionSavingCallback(dt=0.02, prefix=solution_prefix)\n\n# Save at certain timepoints which allows comparison to the results of Marrone et al.,\n# i.e. (1.5, 2.36, 3.0, 5.7, 6.45).\n# Please note that the images in Marrone et al. are obtained at a particle_spacing = H/320,\n# which takes between 2 and 4 hours.\nsaving_paper = SolutionSavingCallback(save_times=[0.0, 0.371, 0.584, 0.743, 1.411, 1.597],\n prefix=\"marrone_times\")\n\n# This can be overwritten with `trixi_include`\nextra_callback = nothing\n\nuse_reinit = false\ndensity_reinit_cb = use_reinit ?\n DensityReinitializationCallback(semi.systems[1], interval=10) :\n nothing\nstepsize_callback = StepsizeCallback(cfl=0.9)\n\ncallbacks = CallbackSet(info_callback, saving_callback, stepsize_callback, extra_callback,\n density_reinit_cb, saving_paper)\n\nsol = solve(ode, CarpenterKennedy2N54(williamson_condition=false),\n dt=1.0, # This is overwritten by the stepsize callback\n save_everystep=false, callback=callbacks);\n\n","category":"page"},{"location":"systems/entropically_damped_sph/#edac","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility (EDAC) for SPH","text":"","category":"section"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"fracmathrmd p_amathrmdt = - rho c_s^2 nabla cdot v + nu nabla^2 p","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"which is derived by Clausen (2013). This equation is similar to the continuity equation (first term, see ContinuityDensity), but also contains a pressure damping term (second term, similar to density diffusion see DensityDiffusion), which reduces acoustic pressure waves through an entropy-generation mechanism.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The pressure evolution is discretized with the SPH method by Ramachandran (2019) as following:","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The first term is equivalent to the classical artificial compressible methods, which are commonly motivated by assuming the artificial equation of state (StateEquationCole with exponent=1) and is discretized as","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"- rho c_s^2 nabla cdot v = sum_b m_b fracrho_arho_b c_s^2 v_ab cdot nabla_r_a W(Vert r_a - r_b Vert h)","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where rho_a, rho_b, r_a, r_b, denote the density and coordinates of particles a and b respectively, c_s is the speed of sound and v_ab = v_a - v_b is the difference in the velocity.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The second term smooths the pressure through the introduction of entropy and is discretized as","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"nu nabla^2 p = fracV_a^2 + V_b^2m_a tildeeta_ab fracp_abVert r_ab^2 Vert + eta h_ab^2 nabla_r_a\nW(Vert r_a - r_b Vert h) cdot r_ab","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where V_a, V_b denote the volume of particles a and b respectively and p_ab= p_a -p_b is the difference in the pressure.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The viscosity parameter eta_a for a particle a is given as","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"eta_a = rho_a fracalpha h c_s8","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where it is found in the numerical experiments of Ramachandran (2019) that alpha = 05 is a good choice for a wide range of Reynolds numbers (0.0125 to 10000).","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"note: Note\nThe EDAC formulation keeps the density constant and this eliminates the need for the continuity equation or the use of a summation density to find the pressure. However, in SPH discretizations, mrho is typically used as a proxy for the particle volume. The density of the fluids can therefore be computed using the summation density approach.Ramachandran (2019)","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"entropically_damped_sph\", \"system.jl\")]","category":"page"},{"location":"systems/entropically_damped_sph/#TrixiParticles.EntropicallyDampedSPHSystem","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"TrixiParticles.EntropicallyDampedSPHSystem","text":"EntropicallyDampedSPHSystem(initial_condition, smoothing_kernel,\n smoothing_length, sound_speed;\n pressure_acceleration=inter_particle_averaged_pressure,\n density_calculator=SummationDensity(),\n alpha=0.5, viscosity=nothing,\n acceleration=ntuple(_ -> 0.0, NDIMS),\n source_terms=nothing)\n\nSystem for particles of a fluid. As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure. See Entropically Damped Artificial Compressibility for SPH for more details on the method.\n\nArguments\n\ninitial_condition: Initial condition representing the system's particles.\nsound_speed: Speed of sound.\nsmoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.\nsmoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.\n\nKeyword Arguments\n\nviscosity: Viscosity model for this system (default: no viscosity). Recommended: ViscosityAdami.\nacceleration: Acceleration vector for the system. (default: zero vector)\npressure_acceleration: Pressure acceleration formulation (default: inter-particle averaged pressure). When set to nothing, the pressure acceleration formulation for the corresponding density calculator is chosen.\ndensity_calculator: Density calculator (default: SummationDensity)\nsource_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping. Note that these source terms will not be used in the calculation of the boundary pressure when using a boundary with BoundaryModelDummyParticles and AdamiPressureExtrapolation. The keyword argument acceleration should be used instead for gravity-like source terms.\n\n\n\n\n\n","category":"type"},{"location":"systems/entropically_damped_sph/#References","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"References","text":"","category":"section"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"Prabhu Ramachandran. \"Entropically damped artificial compressibility for SPH\". In: Computers and Fluids 179 (2019), pages 579–594. doi: 10.1016/j.compfluid.2018.11.023\nJonathan R. Clausen. \"Entropically damped form of artificial compressibility for explicit simulation of incompressible flow\". In: American Physical Society 87 (2013), page 13309. doi: 10.1103/PhysRevE.87.013309","category":"page"},{"location":"authors/","page":"Authors","title":"Authors","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/AUTHORS.md\"","category":"page"},{"location":"authors/#Authors","page":"Authors","title":"Authors","text":"","category":"section"},{"location":"authors/","page":"Authors","title":"Authors","text":"TrixiParticles.jl's development is coordinated by a group of principal developers, who are also its main contributors and who can be contacted in case of questions about TrixiParticles.jl. In addition, there are contributors who have provided substantial additions or modifications. Together, these two groups form \"The TrixiParticles.jl Authors\" as mentioned under License.","category":"page"},{"location":"authors/#Principal-Developers","page":"Authors","title":"Principal Developers","text":"","category":"section"},{"location":"authors/","page":"Authors","title":"Authors","text":"Erik Faulhaber, University of Cologne, Germany\nNiklas Neher, High-Performance Computing Center Stuttgart (HLRS), Germany\nSven Berger, Helmholtz Center Hereon, Germany","category":"page"},{"location":"authors/#Contributors","page":"Authors","title":"Contributors","text":"","category":"section"},{"location":"authors/","page":"Authors","title":"Authors","text":"The following people contributed major additions or modifications to TrixiParticles.jl and are listed in alphabetical order:","category":"page"},{"location":"authors/","page":"Authors","title":"Authors","text":"Sven Berger\nErik Faulhaber\nGregor Gassner\nNiklas Neher\nHendrik Ranocha\nMichael Schlottke-Lakemper","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials/tut_setup.md\"","category":"page"},{"location":"tutorials/tut_setup_replaced/#Setting-up-your-simulation-from-scratch","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For different setups and physics, have a look at our other example files.","category":"page"},{"location":"tutorials/tut_setup_replaced/#Resolution","page":"Setting up your simulation from scratch","title":"Resolution","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"At the beginning of most simulation files, we define the numerical resolution, so that it can easily be found and changed. First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"using TrixiParticles\nusing OrdinaryDiffEq","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Now, we define the particle spacing, which is our numerical resolution. We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_particle_spacing = 0.05\nboundary_layers = 3","category":"page"},{"location":"tutorials/tut_setup_replaced/#Experiment-setup","page":"Setting up your simulation from scratch","title":"Experiment setup","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. First, we define the physical parameters gravitational acceleration, initial fluid size, tank size, fluid density, and simulation time.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"gravity = 9.81\ntspan = (0.0, 1.0)\ninitial_fluid_size = (1.0, 0.9)\ntank_size = (1.0, 1.0)\nfluid_density = 1000.0","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sound_speed = 10.0\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=7)","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The speed of sound here is numerical and not physical. We artificially lower the speed of sound, since the physical speed of sound in water would lead to prohibitively small time steps. The speed of sound in Weakly Compressible SPH should be chosen as small as possible for numerical efficiency, but large enough to limit density fluctuations to about 1%.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"TrixiParticles.jl requires the initial particle positions and quantities in form of an InitialCondition. Instead of manually defining particle positions, you can work with our pre-defined setups. Among others, we provide setups for rectangular shapes, circles, and spheres. Initial conditions can also be combined with common set operations. See this page for a list of pre-defined setups and details on set operations on initial conditions.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Here, we use the RectangularTank setup, which generates a rectangular fluid inside a rectangular tank, and supports a hydrostatic pressure gradient by passing a gravitational acceleration and a state equation (see above).","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size,\n fluid_density, n_layers=boundary_layers,\n acceleration=(0.0, -gravity), state_equation=state_equation)","category":"page"},{"location":"tutorials/tut_setup_replaced/#Fluid-system","page":"Setting up your simulation from scratch","title":"Fluid system","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_length = 1.2 * fluid_particle_spacing\nsmoothing_kernel = SchoenbergCubicSplineKernel{2}()","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"You can find an overview over smoothing kernels and corresponding smoothing lengths here.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For stability, we need numerical dissipation in form of an artificial viscosity term.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We choose the parameters as small as possible to avoid visible viscosity, but as large as possible to stabilize the simulation.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The WCSPH method can either compute the particle density by a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating it over time. We choose the latter approach here by using the density calculator ContinuityDensity.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_density_calculator = ContinuityDensity()\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, smoothing_kernel,\n smoothing_length, viscosity=viscosity,\n acceleration=(0.0, -gravity))","category":"page"},{"location":"tutorials/tut_setup_replaced/#Boundary-system","page":"Setting up your simulation from scratch","title":"Boundary system","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation, which generally produces the best results of the implemented methods. See here for a comprehensive overview over boundary models.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length)\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model)","category":"page"},{"location":"tutorials/tut_setup_replaced/#Semidiscretization","page":"Setting up your simulation from scratch","title":"Semidiscretization","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The key component of every simulation is the Semidiscretization, which couples all systems of the simulation. All methods in TrixiParticles.jl are semidiscretizations, which discretize the equations in time to provide an ordinary differential equation that still has to be solved in time. By providing a simulation time span, we can call semidiscretize, which returns an ODEProblem that can be solved with a time integration method.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"semi = Semidiscretization(fluid_system, boundary_system)\node = semidiscretize(semi, tspan)","category":"page"},{"location":"tutorials/tut_setup_replaced/#Time-integration","page":"Setting up your simulation from scratch","title":"Time integration","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"info_callback = InfoCallback(interval=50)\nsaving_callback = SolutionSavingCallback(dt=0.02)\n\ncallbacks = CallbackSet(info_callback, saving_callback)","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"See Visualization for how to visualize the solution.","category":"page"},{"location":"tutorials/tut_setup_replaced/#Replacing-components-with-custom-implementations","page":"Setting up your simulation from scratch","title":"Replacing components with custom implementations","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.","category":"page"},{"location":"tutorials/tut_setup_replaced/#Custom-smoothing-kernel","page":"Setting up your simulation from scratch","title":"Custom smoothing kernel","text":"","category":"section"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)\n q = r / h\n\n if q < 2\n return 1 / (pi * h^2) * exp(-q^2)\n end\n\n return 0.0\nend\n\nfunction TrixiParticles.kernel_deriv(kernel::MyGaussianKernel, r, h)\n q = r * h\n\n if q < 2\n return 1 / (pi * h^2) * (-2 * q) * exp(-q^2) / h\n end\n\n return 0.0\nend\n\nTrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_kernel = MyGaussianKernel()","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"and run the simulation file again.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.","category":"page"},{"location":"tutorials/tut_setup_replaced/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"julia> trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"),\n smoothing_kernel=MyGaussianKernel());","category":"page"},{"location":"tutorials/tut_falling_replaced/","page":"Example file","title":"Example file","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials/tut_falling.md\"","category":"page"},{"location":"tutorials/tut_falling_replaced/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_falling_replaced/","page":"Example file","title":"Example file","text":"using TrixiParticles\nusing OrdinaryDiffEq\n\n# ==========================================================================================\n# ==== Resolution\nfluid_particle_spacing = 0.02\nsolid_particle_spacing = fluid_particle_spacing\n\n# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model\nboundary_layers = 3\nspacing_ratio = 1\n\n# ==========================================================================================\n# ==== Experiment Setup\ngravity = 9.81\ntspan = (0.0, 1.0)\n\n# Boundary geometry and initial fluid particle positions\ninitial_fluid_size = (2.0, 0.9)\ntank_size = (2.0, 1.0)\n\nfluid_density = 1000.0\nsound_speed = 10 * sqrt(gravity * initial_fluid_size[2])\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=1)\n\ntank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,\n n_layers=boundary_layers, spacing_ratio=spacing_ratio,\n faces=(true, true, true, false),\n acceleration=(0.0, -gravity), state_equation=state_equation)\n\nsphere1_radius = 0.3\nsphere2_radius = 0.2\nsphere1_density = 500.0\nsphere2_density = 1100.0\n\n# Young's modulus and Poisson ratio\nsphere1_E = 7e4\nsphere2_E = 1e5\nnu = 0.0\n\nsphere1_center = (0.5, 1.6)\nsphere2_center = (1.5, 1.6)\nsphere1 = SphereShape(solid_particle_spacing, sphere1_radius, sphere1_center,\n sphere1_density, sphere_type=VoxelSphere())\nsphere2 = SphereShape(solid_particle_spacing, sphere2_radius, sphere2_center,\n sphere2_density, sphere_type=VoxelSphere())\n\n# ==========================================================================================\n# ==== Fluid\nfluid_smoothing_length = 3.0 * fluid_particle_spacing\nfluid_smoothing_kernel = WendlandC2Kernel{2}()\n\nfluid_density_calculator = ContinuityDensity()\nviscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)\ndensity_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)\n\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, fluid_smoothing_kernel,\n fluid_smoothing_length, viscosity=viscosity,\n density_diffusion=density_diffusion,\n acceleration=(0.0, -gravity))\n\n# ==========================================================================================\n# ==== Boundary\nboundary_density_calculator = AdamiPressureExtrapolation()\nboundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n boundary_density_calculator,\n fluid_smoothing_kernel, fluid_smoothing_length)\n\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model)\n\n# ==========================================================================================\n# ==== Solid\nsolid_smoothing_length = 2 * sqrt(2) * solid_particle_spacing\nsolid_smoothing_kernel = WendlandC2Kernel{2}()\n\n# For the FSI we need the hydrodynamic masses and densities in the solid boundary model\nhydrodynamic_densites_1 = fluid_density * ones(size(sphere1.density))\nhydrodynamic_masses_1 = hydrodynamic_densites_1 * solid_particle_spacing^ndims(fluid_system)\n\nsolid_boundary_model_1 = BoundaryModelDummyParticles(hydrodynamic_densites_1,\n hydrodynamic_masses_1,\n state_equation=state_equation,\n boundary_density_calculator,\n fluid_smoothing_kernel,\n fluid_smoothing_length)\n\nhydrodynamic_densites_2 = fluid_density * ones(size(sphere2.density))\nhydrodynamic_masses_2 = hydrodynamic_densites_2 * solid_particle_spacing^ndims(fluid_system)\n\nsolid_boundary_model_2 = BoundaryModelDummyParticles(hydrodynamic_densites_2,\n hydrodynamic_masses_2,\n state_equation=state_equation,\n boundary_density_calculator,\n fluid_smoothing_kernel,\n fluid_smoothing_length)\n\nsolid_system_1 = TotalLagrangianSPHSystem(sphere1,\n solid_smoothing_kernel, solid_smoothing_length,\n sphere1_E, nu,\n acceleration=(0.0, -gravity),\n boundary_model=solid_boundary_model_1,\n penalty_force=PenaltyForceGanzenmueller(alpha=0.3))\n\nsolid_system_2 = TotalLagrangianSPHSystem(sphere2,\n solid_smoothing_kernel, solid_smoothing_length,\n sphere2_E, nu,\n acceleration=(0.0, -gravity),\n boundary_model=solid_boundary_model_2,\n penalty_force=PenaltyForceGanzenmueller(alpha=0.3))\n\n# ==========================================================================================\n# ==== Simulation\nsemi = Semidiscretization(fluid_system, boundary_system, solid_system_1, solid_system_2)\node = semidiscretize(semi, tspan)\n\ninfo_callback = InfoCallback(interval=10)\nsaving_callback = SolutionSavingCallback(dt=0.02, output_directory=\"out\", prefix=\"\",\n write_meta_data=true)\n\ncallbacks = CallbackSet(info_callback, saving_callback)\n\n# Use a Runge-Kutta method with automatic (error based) time step size control.\nsol = solve(ode, RDPK3SpFSAL49(),\n abstol=1e-6, # Default abstol is 1e-6\n reltol=1e-3, # Default reltol is 1e-3\n save_everystep=false, callback=callbacks);\n\n","category":"page"},{"location":"tutorials/tut_setup/#Setting-up-your-simulation-from-scratch","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For different setups and physics, have a look at our other example files.","category":"page"},{"location":"tutorials/tut_setup/#Resolution","page":"Setting up your simulation from scratch","title":"Resolution","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"At the beginning of most simulation files, we define the numerical resolution, so that it can easily be found and changed. First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"using TrixiParticles\nusing OrdinaryDiffEq","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Now, we define the particle spacing, which is our numerical resolution. We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_particle_spacing = 0.05\nboundary_layers = 3","category":"page"},{"location":"tutorials/tut_setup/#Experiment-setup","page":"Setting up your simulation from scratch","title":"Experiment setup","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. First, we define the physical parameters gravitational acceleration, initial fluid size, tank size, fluid density, and simulation time.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"gravity = 9.81\ntspan = (0.0, 1.0)\ninitial_fluid_size = (1.0, 0.9)\ntank_size = (1.0, 1.0)\nfluid_density = 1000.0","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sound_speed = 10.0\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=7)","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The speed of sound here is numerical and not physical. We artificially lower the speed of sound, since the physical speed of sound in water would lead to prohibitively small time steps. The speed of sound in Weakly Compressible SPH should be chosen as small as possible for numerical efficiency, but large enough to limit density fluctuations to about 1%.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"TrixiParticles.jl requires the initial particle positions and quantities in form of an InitialCondition. Instead of manually defining particle positions, you can work with our pre-defined setups. Among others, we provide setups for rectangular shapes, circles, and spheres. Initial conditions can also be combined with common set operations. See this page for a list of pre-defined setups and details on set operations on initial conditions.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Here, we use the RectangularTank setup, which generates a rectangular fluid inside a rectangular tank, and supports a hydrostatic pressure gradient by passing a gravitational acceleration and a state equation (see above).","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size,\n fluid_density, n_layers=boundary_layers,\n acceleration=(0.0, -gravity), state_equation=state_equation)","category":"page"},{"location":"tutorials/tut_setup/#Fluid-system","page":"Setting up your simulation from scratch","title":"Fluid system","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_length = 1.2 * fluid_particle_spacing\nsmoothing_kernel = SchoenbergCubicSplineKernel{2}()","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"You can find an overview over smoothing kernels and corresponding smoothing lengths here.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For stability, we need numerical dissipation in form of an artificial viscosity term.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We choose the parameters as small as possible to avoid visible viscosity, but as large as possible to stabilize the simulation.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The WCSPH method can either compute the particle density by a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating it over time. We choose the latter approach here by using the density calculator ContinuityDensity.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_density_calculator = ContinuityDensity()\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, smoothing_kernel,\n smoothing_length, viscosity=viscosity,\n acceleration=(0.0, -gravity))","category":"page"},{"location":"tutorials/tut_setup/#Boundary-system","page":"Setting up your simulation from scratch","title":"Boundary system","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation, which generally produces the best results of the implemented methods. See here for a comprehensive overview over boundary models.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length)\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model)","category":"page"},{"location":"tutorials/tut_setup/#Semidiscretization","page":"Setting up your simulation from scratch","title":"Semidiscretization","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The key component of every simulation is the Semidiscretization, which couples all systems of the simulation. All methods in TrixiParticles.jl are semidiscretizations, which discretize the equations in time to provide an ordinary differential equation that still has to be solved in time. By providing a simulation time span, we can call semidiscretize, which returns an ODEProblem that can be solved with a time integration method.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"semi = Semidiscretization(fluid_system, boundary_system)\node = semidiscretize(semi, tspan)","category":"page"},{"location":"tutorials/tut_setup/#Time-integration","page":"Setting up your simulation from scratch","title":"Time integration","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"info_callback = InfoCallback(interval=50)\nsaving_callback = SolutionSavingCallback(dt=0.02)\n\ncallbacks = CallbackSet(info_callback, saving_callback)","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"See Visualization for how to visualize the solution.","category":"page"},{"location":"tutorials/tut_setup/#Replacing-components-with-custom-implementations","page":"Setting up your simulation from scratch","title":"Replacing components with custom implementations","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.","category":"page"},{"location":"tutorials/tut_setup/#Custom-smoothing-kernel","page":"Setting up your simulation from scratch","title":"Custom smoothing kernel","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)\n q = r / h\n\n if q < 2\n return 1 / (pi * h^2) * exp(-q^2)\n end\n\n return 0.0\nend\n\nfunction TrixiParticles.kernel_deriv(kernel::MyGaussianKernel, r, h)\n q = r * h\n\n if q < 2\n return 1 / (pi * h^2) * (-2 * q) * exp(-q^2) / h\n end\n\n return 0.0\nend\n\nTrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_kernel = MyGaussianKernel()","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"and run the simulation file again.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"julia> trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"),\n smoothing_kernel=MyGaussianKernel());","category":"page"},{"location":"systems/total_lagrangian_sph/#tlsph","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH","text":"","category":"section"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"A Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers 2021, Belytschko et al. 2000).","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The governing equations with respect to the initial configuration are given by:","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"fracmathrmDbmvmathrmDt = frac1rho_0 nabla_0 cdot bmP + bmg","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"where the zero subscript denotes a derivative with respect to the initial configuration and bmP is the first Piola-Kirchhoff (PK1) stress tensor.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The discretized version of this equation is given by O’Connor & Rogers (2021):","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"fracmathrmdbmv_amathrmdt = sum_b m_0b\n left( fracbmP_a bmL_0arho_0a^2 + fracbmP_b bmL_0brho_0b^2 right)\n nabla_0a W(bmX_ab) + fracbmf_a^PFm_0a + bmg","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"with the correction matrix (see also GradientCorrection)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmL_0a = left( -sum_b fracm_0brho_0b nabla_0a W(bmX_ab) bmX_ab^T right)^-1 in R^d times d","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The subscripts a and b denote quantities of particle a and b, respectively. The zero subscript on quantities denotes that the quantity is to be measured in the initial configuration. The difference in the initial coordinates is denoted by bmX_ab = bmX_a - bmX_b, the difference in the current coordinates is denoted by bmx_ab = bmx_a - bmx_b.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"For the computation of the PK1 stress tensor, the deformation gradient bmF is computed per particle as","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmF_a = sum_b fracm_0brho_0b bmx_ba (bmL_0anabla_0a W(bmX_ab))^T \n qquad = -left(sum_b fracm_0brho_0b bmx_ab (nabla_0a W(bmX_ab))^T right) bmL_0a^T","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"with 1 leq ij leq d. From the deformation gradient, the Green-Lagrange strain","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmE = frac12(bmF^TbmF - bmI)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"and the second Piola-Kirchhoff stress tensor","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmS = lambda operatornametr(bmE) bmI + 2mu bmE","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"are computed to obtain the PK1 stress tensor as","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmP = bmFbmS","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Here,","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"mu = fracE2(1 + nu)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"and","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"lambda = fracEnu(1 + nu)(1 - 2nu)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"are the Lamé coefficients, where E is the Young's modulus and nu is the Poisson ratio.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The term bmf_a^PF is an optional penalty force. See e.g. PenaltyForceGanzenmueller.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"solid\", \"total_lagrangian_sph\", \"system.jl\")]","category":"page"},{"location":"systems/total_lagrangian_sph/#TrixiParticles.TotalLagrangianSPHSystem","page":"Total Lagrangian SPH (Elastic Structure)","title":"TrixiParticles.TotalLagrangianSPHSystem","text":"TotalLagrangianSPHSystem(initial_condition,\n smoothing_kernel, smoothing_length,\n young_modulus, poisson_ratio;\n n_fixed_particles=0, boundary_model=nothing,\n acceleration=ntuple(_ -> 0.0, NDIMS),\n penalty_force=nothing, source_terms=nothing)\n\nSystem for particles of an elastic structure.\n\nA Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers 2021, Belytschko et al. 2000). See Total Lagrangian SPH for more details on the method.\n\nArguments\n\ninitial_condition: Initial condition representing the system's particles.\nyoung_modulus: Young's modulus.\npoisson_ratio: Poisson ratio.\nsmoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.\nsmoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.\n\nKeyword Arguments\n\nn_fixed_particles: Number of fixed particles which are used to clamp the structure particles. Note that the fixed particles must be the last particles in the InitialCondition. See the info box below.\nboundary_model: Boundary model to compute the hydrodynamic density and pressure for fluid-structure interaction (see Boundary Models).\npenalty_force: Penalty force to ensure regular particle position under large deformations (see PenaltyForceGanzenmueller).\nacceleration: Acceleration vector for the system. (default: zero vector)\nsource_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping.\n\nnote: Note\nThe fixed particles must be the last particles in the InitialCondition. To do so, e.g. use the union function:solid = union(beam, fixed_particles)where beam and fixed_particles are of type InitialCondition.\n\n\n\n\n\n","category":"type"},{"location":"systems/total_lagrangian_sph/#References","page":"Total Lagrangian SPH (Elastic Structure)","title":"References","text":"","category":"section"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Joseph O’Connor, Benedict D. Rogers. \"A fluid-structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU\". In: Journal of Fluids and Structures 104 (2021). doi: 10.1016/J.JFLUIDSTRUCTS.2021.103312\nTed Belytschko, Yong Guo, Wing Kam Liu, Shao Ping Xiao. \"A unified stability analysis of meshless particle methods\". In: International Journal for Numerical Methods in Engineering 48 (2000), pages 1359–1400. doi: 10.1002/1097-0207","category":"page"},{"location":"systems/total_lagrangian_sph/#Penalty-Force","page":"Total Lagrangian SPH (Elastic Structure)","title":"Penalty Force","text":"","category":"section"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"In FEM, underintegrated elements can deform without an associated increase of energy. This is caused by the stiffness matrix having zero eigenvalues (so-called hourglass modes). The name \"hourglass modes\" comes from the fact that elements can deform into an hourglass shape.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Similar effects can occur in SPH as well. Particles can change positions without changing the SPH approximation of the deformation gradient bmF, thus, without causing an increase of energy. To ensure regular particle positions, we can apply similar correction forces as are used in FEM.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Ganzenmüller (2015) introduced a so-called hourglass correction force or penalty force f^PF, which is given by","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmf_a^PF = frac12 alpha sum_b fracm_0a m_0b W_0abrho_0arho_0b bmX_ab^2\n left( E delta_ab^a + E delta_ba^b right) fracbmx_abbmx_ab","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The subscripts a and b denote quantities of particle a and b, respectively. The zero subscript on quantities denotes that the quantity is to be measured in the initial configuration. The difference in the initial coordinates is denoted by bmX_ab = bmX_a - bmX_b, the difference in the current coordinates is denoted by bmx_ab = bmx_a - bmx_b. Note that Ganzenmüller (2015) has a flipped sign here because they define bmx_ab the other way around.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"This correction force is based on the potential energy density of a Hookean material. Thus, E is the Young's modulus and alpha is a dimensionless coefficient that controls the amplitude of hourglass correction. The separation vector delta_ab^a indicates the change of distance which the particle separation should attain in order to minimize the error and is given by","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":" delta_ab^a = fracbmepsilon_ab^a cdot bmx_abbmx_ab","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"where the error vector is defined as","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":" bmepsilon_ab^a = bmF_a bmX_ab - bmx_ab","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"solid\", \"total_lagrangian_sph\", \"penalty_force.jl\")]","category":"page"},{"location":"systems/total_lagrangian_sph/#TrixiParticles.PenaltyForceGanzenmueller","page":"Total Lagrangian SPH (Elastic Structure)","title":"TrixiParticles.PenaltyForceGanzenmueller","text":"PenaltyForceGanzenmueller(; alpha=0.1)\n\nPenalty force to ensure regular particle positions under large deformations.\n\nKeywords\n\nalpha: Coefficient to control the amplitude of hourglass correction.\n\n\n\n\n\n","category":"type"},{"location":"systems/total_lagrangian_sph/#References-2","page":"Total Lagrangian SPH (Elastic Structure)","title":"References","text":"","category":"section"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Georg C. Ganzenmüller. \"An hourglass control algorithm for Lagrangian Smooth Particle Hydrodynamics\". In: Computer Methods in Applied Mechanics and Engineering 286 (2015). doi: 10.1016/j.cma.2014.12.005","category":"page"},{"location":"license/","page":"License","title":"License","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/LICENSE.md\"","category":"page"},{"location":"license/#License","page":"License","title":"License","text":"","category":"section"},{"location":"license/","page":"License","title":"License","text":"MIT LicenseCopyright (c) 2023-present The TrixiParticles.jl Authors (see Authors) \nCopyright (c) 2023-present Helmholtz-Zentrum hereon GmbH, Institute of Surface Science \n \nPermission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the \"Software\"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.","category":"page"},{"location":"getting_started/#getting_started","page":"Getting started","title":"Getting started","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"If you have not installed TrixiParticles.jl, please follow the instructions given here.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"In the following sections, we will give a short introduction. For a more thorough discussion, take a look at our Tutorials.","category":"page"},{"location":"getting_started/#Running-an-Example","page":"Getting started","title":"Running an Example","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"The easiest way to run a simulation is to run one of our predefined example files. We will run the file examples/fluid/hydrostatic_water_column_2d.jl, which simulates a fluid resting in a rectangular tank. Since TrixiParticles.jl uses multithreading, you should start Julia with the flag --threads auto (or, e.g. --threads 4 for 4 threads).","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"In the Julia REPL, first load the package TrixiParticles.jl.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> using TrixiParticles","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"Then start the simulation by executing","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"))","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"This will open a new window with a 2D visualization of the final solution: (Image: plot_hydrostatic_water_column)","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"For more information about visualization, see Visualization.","category":"page"},{"location":"getting_started/#Running-other-Examples","page":"Getting started","title":"Running other Examples","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"You can find a list of our other predefined examples under Examples. Execute them as follows from the Julia REPL by replacing subfolder and example_name","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> trixi_include(joinpath(examples_dir(), \"subfolder\", \"example_name.jl\"))","category":"page"},{"location":"getting_started/#Modifying-an-example","page":"Getting started","title":"Modifying an example","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"You can pass keyword arguments to the function trixi_include to overwrite assignments in the file.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"With trixi_include, we can overwrite variables defined in the example file to run a different simulation without modifying the example file.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"), initial_fluid_size=(1.0, 0.5))","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"This for example, will change the fluid size from (09 10) to (10 05).","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"To understand why, take a look into the file hydrostatic_water_column_2d.jl in the subfolder fluid inside the examples directory, which is the file that we executed earlier. You can see that the initial size of the fluid is defined in the variable initial_fluid_size, which we could overwrite with the trixi_include call above. Another variable that is worth experimenting with is fluid_particle_spacing, which controls the resolution of the simulation in this case. A lower value will increase the resolution and the runtime.","category":"page"},{"location":"getting_started/#Set-up-you-first-simulation-from-scratch","page":"Getting started","title":"Set up you first simulation from scratch","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"See Set up your first simulation.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"Find an overview over the available tutorials under Tutorials.","category":"page"},{"location":"examples/#examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/#Fluid","page":"Examples","title":"Fluid","text":"","category":"section"},{"location":"examples/#Accelerated-Tank-2D-(fluid/accelerated_tank_2d.jl)","page":"Examples","title":"Accelerated Tank 2D (fluid/accelerated_tank_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Dam-Break-2D-(fluid/dam_break_2d.jl)","page":"Examples","title":"Dam Break 2D (fluid/dam_break_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Dam-Break-3D-(fluid/dam_break_3d.jl)","page":"Examples","title":"Dam Break 3D (fluid/dam_break_3d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Falling-Water-Column-(fluid/falling_water_column_2d.jl)","page":"Examples","title":"Falling Water Column (fluid/falling_water_column_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Hydrostatic-Water-Column-(fluid/hydrostatic_water_column_*.jl)","page":"Examples","title":"Hydrostatic Water Column (fluid/hydrostatic_water_column_*.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Moving-Wall-(fluid/moving_wall_2d.jl)","page":"Examples","title":"Moving Wall (fluid/moving_wall_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Oscillating-Drop-(fluid/oscillating_drop_2d.jl)","page":"Examples","title":"Oscillating Drop (fluid/oscillating_drop_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Periodic-Channel-(fluid/periodic_channel_2d.jl)","page":"Examples","title":"Periodic Channel (fluid/periodic_channel_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Fluid-Structure-Interaction","page":"Examples","title":"Fluid Structure Interaction","text":"","category":"section"},{"location":"examples/#Dam-Break-with-Elastic-Plate-(fsi/dam_break_plate_2d.jl)","page":"Examples","title":"Dam Break with Elastic Plate (fsi/dam_break_plate_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Falling-Sphere-2D-(fsi/falling_sphere_2d.jl)","page":"Examples","title":"Falling Sphere 2D (fsi/falling_sphere_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Falling-Spheres-2D-(fsi/falling_spheres_2d.jl)","page":"Examples","title":"Falling Spheres 2D (fsi/falling_spheres_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Structure-Mechanics","page":"Examples","title":"Structure Mechanics","text":"","category":"section"},{"location":"examples/#Oscillating-Beam-(solid/oscillating_beam_2d.jl)","page":"Examples","title":"Oscillating Beam (solid/oscillating_beam_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"overview/#Overview","page":"Overview","title":"Overview","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"The following page gives a rough overview of important parts of the code.","category":"page"},{"location":"overview/#Program-flow","page":"Overview","title":"Program flow","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"To initiate a simulation, the goal is to solve an ordinary differential equation, for example, by employing the time integration schemes provided by OrdinaryDiffEq.jl. These schemes are then utilized to integrate mathrmdumathrmdt and mathrmdvmathrmdt, where u represents the particles' positions and v their properties such as velocity and density. During a single time step or an intermediate step of the time integration scheme, the functions drift! and kick! are invoked, followed by the functions depicted in this diagram (with key parts highlighted in orange/yellow).","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: Main Program Flow)","category":"page"},{"location":"overview/#Structure","page":"Overview","title":"Structure","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"What we refer to as schemes are various models such as Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) or Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH). These schemes are categorized based on the applicable physical regimes, namely fluid, solid, gas, and others. Each scheme comprises at least two files: a system.jl file and an rhs.jl file. The system.jl file provides the data structure holding the particles of this scheme and some routines, particularly those for allocation and the main update routines, excluding system interactions. The interactions between particles of this scheme (and with particles of other schemes) are handled in the rhs.jl file.","category":"page"},{"location":"callbacks/#Callbacks","page":"Callbacks","title":"Callbacks","text":"","category":"section"},{"location":"callbacks/","page":"Callbacks","title":"Callbacks","text":"Modules = [TrixiParticles]\nPages = map(file -> joinpath(\"callbacks\", file), readdir(joinpath(\"..\", \"src\", \"callbacks\")))","category":"page"},{"location":"callbacks/#TrixiParticles.DensityReinitializationCallback","page":"Callbacks","title":"TrixiParticles.DensityReinitializationCallback","text":"DensityReinitializationCallback(; interval::Integer=0, dt=0.0)\n\nCallback to reinitialize the density field when using ContinuityDensity.\n\nKeywords\n\ninterval=0: Reinitialize the density every interval time steps.\ndt: Reinitialize the density in regular intervals of dt in terms of integration time.\nreinit_initial_solution: Reinitialize the initial solution (default=false)\n\nReferences\n\nPanizzo, Andrea, Giovanni Cuomo, and Robert A. Dalrymple. \"3D-SPH simulation of landslide generated waves.\" In: Coastal Engineering 2006 (2007), pages 1503-1515. doi: 10.1142/9789812709554_0128\n\n\n\n\n\n","category":"type"},{"location":"callbacks/#TrixiParticles.InfoCallback-Tuple{}","page":"Callbacks","title":"TrixiParticles.InfoCallback","text":"InfoCallback()\n\nCreate and return a callback that prints a human-readable summary of the simulation setup at the beginning of a simulation and then resets the timer. When the returned callback is executed directly, the current timer values are shown.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.PostprocessCallback","page":"Callbacks","title":"TrixiParticles.PostprocessCallback","text":"PostprocessCallback(; interval::Integer=0, dt=0.0, exclude_boundary=true, filename=\"values\",\n output_directory=\"out\", append_timestamp=false, write_csv=true,\n write_json=true, write_file_interval=1, funcs...)\n\nCreate a callback to post-process simulation data at regular intervals. This callback allows for the execution of a user-defined function func at specified intervals during the simulation. The function is applied to the current state of the simulation, and its results can be saved or used for further analysis. The provided function cannot be anonymous as the function name will be used as part of the name of the value.\n\nThe callback can be triggered either by a fixed number of time steps (interval) or by a fixed interval of simulation time (dt).\n\nKeywords\n\nfuncs...: Functions to be executed at specified intervals during the simulation. Each function must have the arguments (v, u, t, system), and will be called for every system, where v and u are the wrapped solution arrays for the corresponding system and t is the current simulation time. Note that working with these v and u arrays requires undocumented internal functions of TrixiParticles. See Custom Quantities for a list of pre-defined functions that can be used here.\ninterval=0: Specifies the number of time steps between each invocation of the callback. If set to 0, the callback will not be triggered based on time steps. Either interval or dt must be set to something larger than 0.\ndt=0.0: Specifies the simulation time interval between each invocation of the callback. If set to 0.0, the callback will not be triggered based on simulation time. Either interval or dt must be set to something larger than 0.\nexclude_boundary=true: If set to true, boundary particles will be excluded from the post-processing.\nfilename=\"values\": The filename of the postprocessing files to be saved.\noutput_directory=\"out\": The path where the results of the post-processing will be saved.\nwrite_csv=true: If set to true, write a csv file.\nwrite_json=true: If set to true, write a json file.\nappend_timestep=false: If set to true, the current timestamp will be added to the filename.\nwrite_file_interval=1: Files will be written after every write_file_interval number of postprocessing execution steps. A value of 0 indicates that files are only written at the end of the simulation, eliminating I/O overhead.\n\nExamples\n\n# Create a callback that is triggered every 100 time steps\npostprocess_callback = PostprocessCallback(interval=100, example_quantity=kinetic_energy)\n\n# Create a callback that is triggered every 0.1 simulation time units\npostprocess_callback = PostprocessCallback(dt=0.1, example_quantity=kinetic_energy)\n\n\n\n\n\n","category":"type"},{"location":"callbacks/#TrixiParticles.SolutionSavingCallback","page":"Callbacks","title":"TrixiParticles.SolutionSavingCallback","text":"SolutionSavingCallback(; interval::Integer=0, dt=0.0, save_times=Array{Float64, 1}([]),\n save_initial_solution=true, save_final_solution=true,\n output_directory=\"out\", append_timestamp=false, prefix=\"\",\n verbose=false, write_meta_data=true, max_coordinates=2^15,\n custom_quantities...)\n\nCallback to save the current numerical solution in VTK format in regular intervals. Either pass interval to save every interval time steps, or pass dt to save in intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).\n\nAdditional user-defined quantities can be saved by passing functions as keyword arguments, which map (v, u, t, system) to an Array where the columns represent the particles in the same order as in u. To ignore a custom quantity for a specific system, return nothing.\n\nKeywords\n\ninterval=0: Save the solution every interval time steps.\ndt: Save the solution in regular intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).\nsave_times=[] List of times at which to save a solution.\nsave_initial_solution=true: Save the initial solution.\nsave_final_solution=true: Save the final solution.\noutput_directory=\"out\": Directory to save the VTK files.\nappend_timestamp=false: Append current timestamp to the output directory.\n'prefix=\"\"': Prefix added to the filename.\ncustom_quantities...: Additional user-defined quantities.\nwrite_meta_data=true: Write meta data.\nverbose=false: Print to standard IO when a file is written.\nmax_coordinates=2^15: The coordinates of particles will be clipped if their absolute values exceed this threshold.\ncustom_quantities...: Additional custom quantities to include in the VTK output. Each custom quantity must be a function of (v, u, t, system), which will be called for every system, where v and u are the wrapped solution arrays for the corresponding system and t is the current simulation time. Note that working with these v and u arrays requires undocumented internal functions of TrixiParticles. See Custom Quantities for a list of pre-defined custom quantities that can be used here.\n\nExamples\n\n# Save every 100 time steps\nsaving_callback = SolutionSavingCallback(interval=100)\n\n# Save in intervals of 0.1 in terms of simulation time\nsaving_callback = SolutionSavingCallback(dt=0.1)\n\n# Additionally store the kinetic energy of each system as \"my_custom_quantity\"\nsaving_callback = SolutionSavingCallback(dt=0.1, my_custom_quantity=kinetic_energy)\n\n\n\n\n\n","category":"type"},{"location":"callbacks/#TrixiParticles.StepsizeCallback-Tuple{}","page":"Callbacks","title":"TrixiParticles.StepsizeCallback","text":"StepsizeCallback(; cfl::Real)\n\nSet the time step size according to a CFL condition if the time integration method isn't adaptive itself.\n\nThe current implementation is using the simplest form of CFL condition, which chooses a time step size that is constant during the simulation. The step size is therefore only applied once at the beginning of the simulation.\n\nThe step size Delta t is chosen as the minimum\n\n Delta t = min(Delta t_eta Delta t_a Delta t_c)\n\nwhere\n\n Delta t_eta = 0125 h^2 eta quad Delta t_a = 025 sqrth lVert g rVert\n quad Delta t_c = textCFL h c\n\nwith nu = alpha h c (2n + 4), where alpha is the parameter of the viscosity and n is the number of dimensions.\n\nwarning: Experimental implementation\nThis is an experimental feature and may change in future releases.\n\nReferences\n\nM. Antuono, A. Colagrossi, S. Marrone. \"Numerical Diffusive Terms in Weakly-Compressible SPH Schemes.\" In: Computer Physics Communications 183, no. 12 (2012), pages 2570–80. doi: 10.1016/j.cpc.2012.07.006\nS. Adami, X. Y. Hu, N. A. Adams. \"A generalized wall boundary condition for smoothed particle hydrodynamics\". In: Journal of Computational Physics 231, 21 (2012), pages 7057–7075. doi: 10.1016/J.JCP.2012.05.005\nP. N. Sun, A. Colagrossi, S. Marrone, A. M. Zhang. \"The δplus-SPH Model: Simple Procedures for a Further Improvement of the SPH Scheme.\" In: Computer Methods in Applied Mechanics and Engineering 315 (2017), pages 25–49. doi: 10.1016/j.cma.2016.10.028\nM. Antuono, S. Marrone, A. Colagrossi, B. Bouscasse. \"Energy Balance in the δ-SPH Scheme.\" In: Computer Methods in Applied Mechanics and Engineering 289 (2015), pages 209–26. doi: 10.1016/j.cma.2015.02.004\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#custom_quantities","page":"Callbacks","title":"Custom Quantities","text":"","category":"section"},{"location":"callbacks/","page":"Callbacks","title":"Callbacks","text":"The following pre-defined custom quantities can be used with the SolutionSavingCallback and PostprocessCallback.","category":"page"},{"location":"callbacks/","page":"Callbacks","title":"Callbacks","text":"Modules = [TrixiParticles]\nPages = [\"general/custom_quantities.jl\"]","category":"page"},{"location":"callbacks/#TrixiParticles.avg_density-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.avg_density","text":"avg_density\n\nReturns the average_density over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.avg_pressure-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.avg_pressure","text":"avg_pressure\n\nReturns the average pressure over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.kinetic_energy-NTuple{4, Any}","page":"Callbacks","title":"TrixiParticles.kinetic_energy","text":"kinetic_energy\n\nReturns the total kinetic energy of all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.max_density-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.max_density","text":"max_density\n\nReturns the maximum density over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.max_pressure-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.max_pressure","text":"max_pressure\n\nReturns the maximum pressure over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.min_density-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.min_density","text":"min_density\n\nReturns the minimum density over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.min_pressure-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.min_pressure","text":"min_pressure\n\nReturns the minimum pressure over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.total_mass-NTuple{4, Any}","page":"Callbacks","title":"TrixiParticles.total_mass","text":"total_mass\n\nReturns the total mass of all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#Interpolation","page":"Interpolation","title":"Interpolation","text":"","category":"section"},{"location":"general/interpolation/","page":"Interpolation","title":"Interpolation","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"interpolation.jl\")]","category":"page"},{"location":"general/interpolation/#TrixiParticles.interpolate_line-Tuple{Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_line","text":"interpolate_line(start, end_, n_points, semi, ref_system, sol; endpoint=true,\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nInterpolates properties along a line in a TrixiParticles simulation. The line interpolation is accomplished by generating a series of evenly spaced points between start and end_. If endpoint is false, the line is interpolated between the start and end points, but does not include these points.\n\nSee also: interpolate_point, interpolate_plane_2d, interpolate_plane_2d_vtk, interpolate_plane_3d.\n\nArguments\n\nstart: The starting point of the line.\nend_: The ending point of the line.\nn_points: The number of points to interpolate along the line.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nendpoint=true: A boolean to include (true) or exclude (false) the end point in the interpolation.\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nA NamedTuple of arrays containing interpolated properties at each point along the line.\n\nnote: Note\nThis function is particularly useful for analyzing gradients or creating visualizations along a specified line in the SPH simulation domain.\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating along a line from [1.0, 0.0] to [1.0, 1.0] with 5 points\nresults = interpolate_line([1.0, 0.0], [1.0, 1.0], 5, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_plane_2d-Tuple{Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_plane_2d","text":"interpolate_plane_2d(min_corner, max_corner, resolution, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nInterpolates properties along a plane in a TrixiParticles simulation. The region for interpolation is defined by its lower left and top right corners, with a specified resolution determining the density of the interpolation points.\n\nThe function generates a grid of points within the defined region, spaced uniformly according to the given resolution.\n\nSee also: interpolate_plane_2d_vtk, interpolate_plane_3d, interpolate_line, interpolate_point.\n\nArguments\n\nmin_corner: The lower left corner of the interpolation region.\nmax_corner: The top right corner of the interpolation region.\nresolution: The distance between adjacent interpolation points in the grid.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nA NamedTuple of arrays containing interpolated properties at each point within the plane.\n\nnote: Note\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used, which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating across a plane from [0.0, 0.0] to [1.0, 1.0] with a resolution of 0.2\nresults = interpolate_plane_2d([0.0, 0.0], [1.0, 1.0], 0.2, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_plane_2d_vtk-Tuple{Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_plane_2d_vtk","text":"interpolate_plane_2d_vtk(min_corner, max_corner, resolution, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false, output_directory=\"out\", filename=\"plane\")\n\nInterpolates properties along a plane in a TrixiParticles simulation and exports the result as a VTI file. The region for interpolation is defined by its lower left and top right corners, with a specified resolution determining the density of the interpolation points.\n\nThe function generates a grid of points within the defined region, spaced uniformly according to the given resolution.\n\nSee also: interpolate_plane_2d, interpolate_plane_3d, interpolate_line, interpolate_point.\n\nArguments\n\nmin_corner: The lower left corner of the interpolation region.\nmax_corner: The top right corner of the interpolation region.\nresolution: The distance between adjacent interpolation points in the grid.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\noutput_directory=\"out\": Directory to save the VTI file.\nfilename=\"plane\": Name of the VTI file.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nnote: Note\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used, which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating across a plane from [0.0, 0.0] to [1.0, 1.0] with a resolution of 0.2\nresults = interpolate_plane_2d([0.0, 0.0], [1.0, 1.0], 0.2, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_plane_3d-Tuple{Any, Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_plane_3d","text":"interpolate_plane_3d(point1, point2, point3, resolution, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nInterpolates properties along a plane in a 3D space in a TrixiParticles simulation. The plane for interpolation is defined by three points in 3D space, with a specified resolution determining the density of the interpolation points.\n\nThe function generates a grid of points on a parallelogram within the plane defined by the three points, spaced uniformly according to the given resolution.\n\nSee also: interpolate_plane_2d, interpolate_plane_2d_vtk, interpolate_line, interpolate_point.\n\nArguments\n\npoint1: The first point defining the plane.\npoint2: The second point defining the plane.\npoint3: The third point defining the plane. The points must not be collinear.\nresolution: The distance between adjacent interpolation points in the grid.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nA NamedTuple of arrays containing interpolated properties at each point within the plane.\n\nnote: Note\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating across a plane defined by points [0.0, 0.0, 0.0], [1.0, 0.0, 0.0], and [0.0, 1.0, 0.0]\n# with a resolution of 0.1\nresults = interpolate_plane_3d([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 0.1, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_point-Tuple{Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_point","text":"interpolate_point(points_coords::Array{Array{Float64,1},1}, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\ninterpolate_point(point_coords, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nPerforms interpolation of properties at specified points or an array of points in a TrixiParticles simulation.\n\nWhen given an array of points (points_coords), it iterates over each point and applies interpolation individually. For a single point (point_coords), it performs the interpolation at that specific location. The interpolation utilizes the same kernel function of the SPH simulation to weigh contributions from nearby particles.\n\nSee also: interpolate_line, interpolate_plane_2d, interpolate_plane_2d_vtk, interpolate_plane_3d, .\n\nArguments\n\npoints_coords: An array of point coordinates, for which to interpolate properties.\npoint_coords: The coordinates of a single point for interpolation.\nsemi: The semidiscretization used in the SPH simulation.\nref_system: The reference system defining the properties of the SPH particles.\nsol: The current solution state from which properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nFor multiple points: A NamedTuple of arrays containing interpolated properties at each point.\nFor a single point: A NamedTuple of interpolated properties at the point.\n\nExamples\n\n# For a single point\nresult = interpolate_point([1.0, 0.5], semi, ref_system, sol)\n\n# For multiple points\npoints = [[1.0, 0.5], [1.0, 0.6], [1.0, 0.7]]\nresults = interpolate_point(points, semi, ref_system, sol)\n\nnote: Note\nThis function is particularly useful for analyzing gradients or creating visualizations along a specified line in the SPH simulation domain.\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used which is not asaccurate as a real surface reconstruction.\n\n\n\n\n\n","category":"method"},{"location":"#TrixiParticles.jl","page":"Home","title":"TrixiParticles.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TrixiParticles.jl is a numerical simulation framework designed for particle-based numerical methods, with an emphasis on multiphysics applications, written in Julia. A primary goal of the framework is to be user-friendly for engineering, science, and educational purposes. In addition to its extensible design and optimized implementation, we prioritize the user experience, including installation, pre- and postprocessing. Its features include:","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Incompressible Navier-Stokes\nMethods: Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH), Entropically Damped Artificial Compressibility (EDAC)\nSolid-body mechanics\nMethods: Total Lagrangian SPH (TLSPH)\nFluid-Structure Interaction\nOutput formats:\nVTK","category":"page"},{"location":"#Examples","page":"Home","title":"Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"\n \n \n \n \n \n \n \n \n
\n
2D Dam Break
\n 		\n
Moving Wall
\n
\n
Oscillating Beam
\n 		\n
Dam Break with Elastic Plate
\n
","category":"page"},{"location":"#Quickstart","page":"Home","title":"Quickstart","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Installation\nGetting started","category":"page"},{"location":"","page":"Home","title":"Home","text":"If you have any questions concerning TrixiParticles.jl you can join our community on Slack or open an issue with your question.","category":"page"},{"location":"#Start-with-development","page":"Home","title":"Start with development","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To get started with development have a look at these pages:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Installation\nDevelopment\nContributing","category":"page"},{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/#Fluid","page":"Tutorial","title":"Fluid","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Setting up your simulation from scratch\nSetting up a dam break simulation","category":"page"},{"location":"tutorial/#Mechanics","page":"Tutorial","title":"Mechanics","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Deforming a beam","category":"page"},{"location":"tutorial/#Fluid-Structure-Interaction","page":"Tutorial","title":"Fluid-Structure Interaction","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Setting up a falling structure","category":"page"}] +[{"location":"references/","page":"References","title":"References","text":"K. Hormann and A. Agathos. The point in polygon problem for arbitrary polygons. Computational Geometry 20, 131–144 (2001).\n\n\n\nA. Jacobson, L. Kavan and O. Sorkine-Hornung. Robust inside-outside segmentation using generalized winding numbers. ACM Transactions on Graphics 32, 1–12 (2013).\n\n\n\nM. Müller, D. Charypar and M. Gross. Particle-Based Fluid Simulation for Interactive Applications. In: Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (Eurographics Association, 07 2003); pp. 154–159.\n\n\n\nI. J. Schoenberg. Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae. Quarterly of Applied Mathematics 4, 112–141 (1946).\n\n\n\nD. J. Price. Smoothed particle hydrodynamics and magnetohydrodynamics. Journal of Computational Physics 231, 759–794 (2012).\n\n\n\nJ. Monaghan. Particle methods for hydrodynamics. Computer Physics Reports 3, 71–124 (1985).\n\n\n\nH. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics 4, 389–396 (1995).\n\n\n\nW. Dehnen and H. Aly. Improving convergence in smoothed particle hydrodynamics simulations without pairing instability: SPH without pairing instability. Monthly Notices of the Royal Astronomical Society 425, 1068–1082 (2012).\n\n\n\nN. Bićanić. Discrete Element Methods. In: Discrete Element Methods (Wiley, 2004).\n\n\n\nP. A. Cundall and O. D. Strack. A discrete numerical model for granular assemblies. Géotechnique 29, 47–65 (1979).\n\n\n\nA. Di Renzo and F. P. Di Maio. Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chemical Engineering Science 59, 525–541 (2004).\n\n\n\nJ. Monaghan. Simulating Free Surface Flows with SPH. Journal of Computational Physics 110, 399–406 (1994).\n\n\n\nR. H. Cole and R. Weller. Underwater Explosions. Physics Today 1, 35–35 (1948).\n\n\n\nS. Adami, X. Hu and N. Adams. A generalized wall boundary condition for smoothed particle hydrodynamics. Journal of Computational Physics 231, 7057–7075 (2012).\n\n\n\nJ. P. Morris, P. J. Fox and Y. Zhu. Modeling Low Reynolds Number Incompressible Flows Using SPH. Journal of Computational Physics 136, 214–226 (1997).\n\n\n\nJ. J. Monaghan. Smoothed Particle Hydrodynamics. Annual Review of Astronomy and Astrophysics 30, 543–574 (1992).\n\n\n\nJ. Monaghan. On the problem of penetration in particle methods. Journal of Computational Physics 82, 1–15 (1989).\n\n\n\nJ. J. Monaghan. Smoothed particle hydrodynamics. Reports on Progress in Physics 68, 1703–1759 (2005).\n\n\n\nP. Ramachandran and K. Puri. Entropically damped artificial compressibility for SPH. Computers & Fluids 179, 579–594 (2019).\n\n\n\nG. Fourtakas, J. M. Dominguez, R. Vacondio and B. D. Rogers. Local uniform stencil (LUST) boundary condition for arbitrary 3-D boundaries in parallel smoothed particle hydrodynamics (SPH) models. Computers & Fluids 190, 346–361 (2019).\n\n\n\nM. Antuono, A. Colagrossi and S. Marrone. Numerical diffusive terms in weakly-compressible SPH schemes. Computer Physics Communications 183, 2570–2580 (2012).\n\n\n\nM. Antuono, A. Colagrossi, S. Marrone and D. Molteni. Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Computer Physics Communications 181, 532–549 (2010).\n\n\n\nD. Molteni and A. Colagrossi. A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Computer Physics Communications 180, 861–872 (2009).\n\n\n\nA. Ferrari, M. Dumbser, E. F. Toro and A. Armanini. A new 3D parallel SPH scheme for free surface flows. Computers & Fluids 38, 1203–1217 (2009).\n\n\n\nN. Akinci, G. Akinci and M. Teschner. Versatile surface tension and adhesion for SPH fluids. ACM Transactions on Graphics 32, 1–8 (2013).\n\n\n\nJ. Bonet and T.-S. Lok. Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations. Computer Methods in Applied Mechanics and Engineering 180, 97–115 (1999).\n\n\n\nM. Basa, N. J. Quinlan and M. Lastiwka. Robustness and accuracy of SPH formulations for viscous flow. International Journal for Numerical Methods in Fluids 60, 1127–1148 (2008).\n\n\n\nS. Li and W. K. Liu. Moving least-square reproducing kernel method Part II: Fourier analysis. Computer Methods in Applied Mechanics and Engineering 139, 159–193 (1996).\n\n\n\nJ. R. Clausen. Entropically damped form of artificial compressibility for explicit simulation of incompressible flow. Physical Review E 87, 013309 (2013).\n\n\n\nS. Adami, X. Hu and N. Adams. A transport-velocity formulation for smoothed particle hydrodynamics. Journal of Computational Physics 241, 292–307 (2013).\n\n\n\nJ. O’Connor and B. D. Rogers. A fluid–structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU. Journal of Fluids and Structures 104, 103312 (2021).\n\n\n\nT. Belytschko, Y. Guo, W. K. Liu and S. P. Xiao. A unified stability analysis of meshless particle methods. International Journal for Numerical Methods in Engineering 48, 1359–1400 (2000).\n\n\n\nG. C. Ganzenmüller. An hourglass control algorithm for Lagrangian Smooth Particle Hydrodynamics. Computer Methods in Applied Mechanics and Engineering 286, 87–106 (2015).\n\n\n\nA. Valizadeh and J. J. Monaghan. A study of solid wall models for weakly compressible SPH. Journal of Computational Physics 300, 5–19 (2015).\n\n\n\nN. Akinci, M. Ihmsen, G. Akinci, B. Solenthaler and M. Teschner. Versatile rigid-fluid coupling for incompressible SPH. ACM Transactions on Graphics 31, 1–8 (2012).\n\n\n\nA. J. Crespo, M. Gomez-Gesteira and R. A. Dalrymple. Boundary Conditions Generated by Dynamic Particles in SPH Methods. Computers, Materials and Continua 5, 173–184 (2007).\n\n\n\nS. Band, C. Gissler, A. Peer and M. Teschner. MLS pressure boundaries for divergence-free and viscous SPH fluids. Computers & Graphics 76, 37–46 (2018).\n\n\n\nJ. Monaghan and J. Kajtar. SPH particle boundary forces for arbitrary boundaries. Computer Physics Communications 180, 1811–1820 (2009).\n\n\n\nM. B. Giles. Nonreflecting boundary conditions for Euler equation calculations. AIAA Journal 28, 2050–2058 (1990).\n\n\n\nM. Lastiwka, M. Basa and N. J. Quinlan. Permeable and non‐reflecting boundary conditions in SPH. International Journal for Numerical Methods in Fluids 61, 709–724 (2008).\n\n\n\nP. Negi, P. Ramachandran and A. Haftu. An improved non-reflecting outlet boundary condition for weakly-compressible SPH. Computer Methods in Applied Mechanics and Engineering 367, 113119 (2020).\n\n\n\nA. Panizzo, G. Cuomo and R. A. Dalrymple. 3D-SPH SIMULATION OF LANDSLIDE GENERATED WAVES. In: Coastal Engineering 2006 (World Scientific Publishing Company, Apr 2007).\n\n\n\nP. Sun, A. Colagrossi, S. Marrone and A. Zhang. Delta-SPH model: Simple procedures for a further improvement of the SPH scheme. Computer Methods in Applied Mechanics and Engineering 315, 25–49 (2017).\n\n\n\nM. Antuono, S. Marrone, A. Colagrossi and B. Bouscasse. Energy balance in the Delta-SPH scheme. Computer Methods in Applied Mechanics and Engineering 289, 209–226 (2015).\n\n\n\n","category":"page"},{"location":"preprocessing/preprocessing/#Sampling-of-Geometries","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"","category":"section"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Generating the initial configuration of a simulation requires filling volumes (3D) or surfaces (2D) of complex geometries with particles. The algorithm to sample a complex geometry should be robust and fast, since for large problems (large numbers of particles) or complex geometries (many geometry faces), generating the initial configuration is not trivial and can be very expensive in terms of computational cost. We therefore use a winding number approach for an inside-outside segmentation of an object. The winding number w(mathbfp) is a signed integer-valued function of a point mathbfp and is defined as","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"w(mathbfp) = frac12 pi sum^n_i=1 Theta_i","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Here, Theta_i is the signed angle between mathbfc_i - mathbfp and mathbfc_i+1 - mathbfp where mathbfc_i and mathbfc_i+1 are two consecutive vertices on a curve. In 3D, we refer to the solid angle of an oriented triangle with respect to mathbfp.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"We provide the following methods to calculate w(mathbfp):","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Hormann et al. (2001) evaluate the winding number combined with an even-odd rule, but only for 2D polygons (see WindingNumberHormann).\nNaive winding: Jacobson et al. (2013) generalized the winding number so that the algorithm can be applied for both 2D and 3D geometries (see WindingNumberJacobson).\nHierarchical winding: Jacobson et al. (2013) also introduced a fast hierarchical evaluation of the winding number. For further information see the description below.","category":"page"},{"location":"preprocessing/preprocessing/#hierarchical_winding","page":"Sampling of Geometries","title":"Hierarchical Winding","text":"","category":"section"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"According to Jacobson et al. (2013) the winding number with respect to a polygon (2D) or triangle mesh (3D) is the sum of the winding numbers with respect to each edge (2D) or face (3D). We can show this with the following example in which we determine the winding number for each edge of a triangle separately and sum them up:","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"using TrixiParticles\nusing Plots\n\ntriangle = [125.0 375.0 250.0 125.0;\n 175.0 175.0 350.0 175.0]\n\n# Delete all edges but one\nedge1 = deleteat!(TrixiParticles.Polygon(triangle), [2, 3])\nedge2 = deleteat!(TrixiParticles.Polygon(triangle), [1, 3])\nedge3 = deleteat!(TrixiParticles.Polygon(triangle), [1, 2])\n\nalgorithm = WindingNumberJacobson()\n\ngrid = hcat(([x, y] for x in 1:500, y in 1:500)...)\n\n_, w1 = algorithm(edge1, grid; store_winding_number=true)\n_, w2 = algorithm(edge2, grid; store_winding_number=true)\n_, w3 = algorithm(edge3, grid; store_winding_number=true)\n\nw = w1 + w2 + w3\n\nheatmap(1:500, 1:500, reshape(w1, 500, 500)', color=:coolwarm, showaxis=false,\n tickfontsize=12, size=(570, 500), margin=6 * Plots.mm)\nheatmap(1:500, 1:500, reshape(w2, 500, 500)', color=:coolwarm, showaxis=false,\n tickfontsize=12, size=(570, 500), margin=6 * Plots.mm)\nheatmap(1:500, 1:500, reshape(w3, 500, 500)', color=:coolwarm, showaxis=false,\n tickfontsize=12, size=(570, 500), margin=6 * Plots.mm)\nheatmap(1:500, 1:500, reshape(w, 500, 500)', color=:coolwarm, showaxis=false,\n tickfontsize=12, size=(570, 500), margin=6 * Plots.mm, clims=(-1, 1))\n","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"triangle\"/\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"This summation property has some interesting consequences that we can utilize for an efficient computation of the winding number. Let mathcalS be an open surface and barmathcalS an arbitrary closing surface, such that","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"partial barmathcalS = partial mathcalS","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"and mathcalB = barmathcalS cup mathcalS is some closed oriented surface. For any query point mathbfp outside of mathcalB, we know that","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"w_mathcalS(mathbfp) + w_barmathcalS(mathbfp) = w_mathcalB(mathbfp) = 0","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"This means","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"w_mathcalS(mathbfp) = - w_barmathcalS(mathbfp)","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"regardless of how barmathcalS is constructed (as long as mathbfp is outside of mathcalB).","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"We can use this property in the discrete case to efficiently compute the winding number of a query point by partitioning the polygon or mesh in a \"small\" part (as in consisting of a small number of edges/faces) and a \"large\" part. For the small part we just compute the winding number, and for the large part we construct a small closing and compute its winding number. The partitioning is based on a hierarchical construction of bounding boxes.","category":"page"},{"location":"preprocessing/preprocessing/#Bounding-volume-hierarchy","page":"Sampling of Geometries","title":"Bounding volume hierarchy","text":"","category":"section"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"To efficiently find a \"small part\" and a \"large part\" as mentioned above, we construct a hierarchy of bounding boxes by starting with the whole domain and recursively splitting it in two equally sized boxes. The resulting hierarchy is a binary tree.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"The algorithm by Jacobsen et al. (Algorithm 2, p. 5) traverses this binary tree recursively until we find the leaf in which the query point is located. The recursion stops with the following criteria:","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"if the bounding box T is a leaf then TmathcalS = mathcalS cap T, the part of mathcalS that lies inside T, is the \"small part\" mentioned above, so evaluate the winding number naively as w(mathbfp TmathcalS).\nelse if mathbfp is outside T then TmathcalS is the \"large part\", so evaluate the winding number naively as -w(mathbfp TbarmathcalS), where TbarmathcalS is the closing surface of TmathcalS.","category":"page"},{"location":"preprocessing/preprocessing/#Continuous-example","page":"Sampling of Geometries","title":"Continuous example","text":"","category":"section"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Now consider the following continuous (not discretized to a polygon) 2D example. We compute the winding number of the point mathbfp with respect to mathcalS using the depicted hierarchy of bounding boxes.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"continuous\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(1):","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Recurse left: w_textleft = texttexttthierarchical_winding (mathbfp Ttextleft)\nRecurse right: w_textright = texttexttthierarchical_winding (mathbfpTtextright)","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(2):","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Query point mathbfp is outside bounding box T, so don't recurse deeper.\nCompute w_mathcalS(mathbfp) = - w_barmathcalS(mathbfp) with the closure TbarmathcalS, which is generally much smaller (fewer edges in the discrete version) than TmathcalS:","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"w_textleft = -texttextttnaive_winding (mathbfp TbarmathcalS)","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(3):","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Bounding box T is a leaf. Use open surface TmathcalS:","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"w_textright = texttextttnaive_winding (mathbfp TmathcalS)","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"The reconstructed surface will then look as in the following image.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"reconstructed\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"We finally sum up the winding numbers","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"w = w_textleft + w_textright = -w_T_textleftbarmathcalS + w_T_textrightmathcalS","category":"page"},{"location":"preprocessing/preprocessing/#Discrete-example","page":"Sampling of Geometries","title":"Discrete example","text":"","category":"section"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"We will now go through the discrete version of the example above.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"discrete\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"To construct the hierarchy for the discrete piecewise-linear example in (1), we have to do the following.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(2): Each edge is distributed to the child whose box contains the edge's barycenter (red dots in (2)). Splitting stops when the number of a box's edges slips below a threshold (usually approx 100 faces in 3D, here: 6 edges).","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(3): For the closure, Jacobson et al. (2013) define exterior vertices (exterior edges in 3D) as boundary vertices of such a segmentation (red dots in (3)). To find them, we traverse around each edge (face in 3D) in order, and increment or decrement for each vertex (edge) a specific counter.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"v1 = edge_vertices_ids[edge][1]\nv2 = edge_vertices_ids[edge][2]\n\nvertex_count[v1] += 1\nvertex_count[v2] -= 1","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"In 2D, a vertex is declared as exterior if vertex_count(vertex) != 0, so there is not the same amount of edges in this box going into versus out of the vertex. To construct the closing surface, the exterior vertices are then connected to one arbitrary exterior vertex using appropriately oriented line segments:","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"edge = vertex_count[v] > 0 ? (closing_vertex, v) : (v, closing_vertex)","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"The resulting closed surface TS cup TbarS then has the same number of edges going into and out of each vertex.","category":"page"},{"location":"preprocessing/preprocessing/#Incorrect-evaluation","page":"Sampling of Geometries","title":"Incorrect evaluation","text":"","category":"section"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"If we follow the algorithm, we know that recursion stops if","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"the bounding box T is a leaf or\nthe query point mathbfp is outside the box.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"incorrect\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(1): The query point mathbfp is outside the box, so we calculate the winding number with the (red) closure of the box.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(2): The query point mathbfp is inside the box, so we use the (blue) edges distributed to the box.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"(3): In this case, it leads to an incorrect evaluation of the winding number. The query point is clearly inside the box, but not inside the reconstructed surface. This is because the property w_mathcalS(mathbfp) = - w_barmathcalS(mathbfp) only holds when mathbfp is outside of mathcalB, which is not the case here.","category":"page"},{"location":"preprocessing/preprocessing/#Correct-evaluation","page":"Sampling of Geometries","title":"Correct evaluation","text":"","category":"section"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Jacobson et al. (2013) don't mention this problem or provide a solution to it. We contacted the authors and found that they know about this problem and solve it by resizing the bounding box to fully include the closing surface of the neighboring box, since it doesn't matter if the boxes overlap.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"correct\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"To avoid resizing, we take a different approach and calculate the closure of the bounding box differently:","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Exclude intersecting edges in the calculation of the exterior vertices.\nThis way, all exterior vertices are inside the bounding box, and so will be the closing surface.\nThe intersecting edges are later added with flipped orientation, so that the closing is actually a closing of the exterior plus intersecting edges.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"correct\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"The evaluation then looks as follows.","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"
\n \"correct\n
","category":"page"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"preprocessing\", \"point_in_poly\", \"winding_number_hormann.jl\")]","category":"page"},{"location":"preprocessing/preprocessing/#TrixiParticles.WindingNumberHormann","page":"Sampling of Geometries","title":"TrixiParticles.WindingNumberHormann","text":"WindingNumberHormann()\n\nAlgorithm for inside-outside segmentation of a complex geometry proposed by Hormann (2001). It is only supported for 2D geometries. WindingNumberHormann might handle edge cases a bit better, since the winding number is an integer value.\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"type"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"preprocessing\", \"point_in_poly\", \"winding_number_jacobson.jl\")]","category":"page"},{"location":"preprocessing/preprocessing/#TrixiParticles.WindingNumberJacobson","page":"Sampling of Geometries","title":"TrixiParticles.WindingNumberJacobson","text":"WindingNumberJacobson(; geometry=nothing, winding_number_factor=sqrt(eps()),\n hierarchical_winding=false)\n\nAlgorithm for inside-outside segmentation of a complex geometry proposed by [2].\n\nKeywords\n\ngeometry: Complex geometry returned by load_geometry and is only required when using hierarchical_winding=true.\nhierarchical_winding: If set to true, an optimized hierarchical approach will be used, which gives a significant speedup. For further information see Hierarchical Winding.\nwinding_number_factor: For leaky geometries, a factor of 0.4 will give a better inside-outside segmentation.\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"type"},{"location":"preprocessing/preprocessing/","page":"Sampling of Geometries","title":"Sampling of Geometries","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"preprocessing\", \"geometries\", \"io.jl\")]","category":"page"},{"location":"preprocessing/preprocessing/#TrixiParticles.load_geometry-Tuple{Any}","page":"Sampling of Geometries","title":"TrixiParticles.load_geometry","text":"load_geometry(filename; element_type=Float64)\n\nLoad file and return corresponding type for ComplexShape. Supported file formats are .stl and .asc.\n\nArguments\n\nfilename: Name of the file to be loaded.\n\nKeywords\n\nelement_type: Element type (default is Float64)\n\n\n\n\n\n","category":"method"},{"location":"systems/dem/#dem","page":"Discrete Element Method (Solid)","title":"Discrete Element Method","text":"","category":"section"},{"location":"systems/dem/","page":"Discrete Element Method (Solid)","title":"Discrete Element Method (Solid)","text":"The Discrete Element Method (DEM) is a computational technique widely used in physics, engineering, and applied mathematics for simulating the mechanical behavior of granular materials, such as powders, sand, soil, or rock, as well as other discontinua. Unlike continuum mechanics that treats materials as continuous, DEM considers individual particles or elements and their interactions. This approach provides detailed insights into the micro-mechanical behavior of materials, making it particularly valuable in fields such as geomechanics, material science, and mechanical engineering.","category":"page"},{"location":"systems/dem/#Fundamental-Principles","page":"Discrete Element Method (Solid)","title":"Fundamental Principles","text":"","category":"section"},{"location":"systems/dem/","page":"Discrete Element Method (Solid)","title":"Discrete Element Method (Solid)","text":"The core idea behind DEM is the discretization of a material system into a finite set of distinct, interacting mass elements (particles). These elements (particles) can vary in shape, size, and properties, and they interact with each other and possibly with their boundaries through contact forces and potential fields. The motion and behavior of each mass element are governed by Newton's laws of motion, accounting for the forces and moments acting upon them.","category":"page"},{"location":"systems/dem/","page":"Discrete Element Method (Solid)","title":"Discrete Element Method (Solid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"solid\", \"discrete_element_method\", \"system.jl\")]","category":"page"},{"location":"systems/dem/#TrixiParticles.DEMSystem","page":"Discrete Element Method (Solid)","title":"TrixiParticles.DEMSystem","text":"DEMSystem(initial_condition, normal_stiffness, elastic_modulus, poissons_ratio;\n damping_coefficient=0.0001, acceleration=ntuple(_ -> 0.0, NDIMS), source_terms=nothing)\n\nConstructs a Discrete Element Method (DEM) system for numerically simulating the dynamics of granular and particulate matter. DEM is employed to simulate and analyze the motion, interactions, and collective behavior of assemblies of discrete, solid particles, typically under mechanical loading. The model accounts for individual particle characteristics and implements interaction laws that govern contact forces (normal and tangential), based on specified material properties and contact mechanics.\n\nArguments\n\ninitial_condition: Initial condition of the system, encapsulating the initial positions, velocities, masses, and radii of particles.\nnormal_stiffness: Normal stiffness coefficient for particle-particle and particle-wall contacts.\nelastic_modulus: Elastic modulus for this particle system.\npoissons_ratio: Poisson ratio for this particle system.\n\nKeywords\n\nacceleration: Global acceleration vector applied to the system, such as gravity. Specified as an SVector of length NDIMS, with a default of zero in each dimension.\nsource_terms: Optional; additional forces or modifications to particle dynamics not captured by standard DEM interactions, such as electromagnetic forces or user-defined perturbations.\ndamping_coefficient=0.0001: Set a damping coefficient for the collision interactions.\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in a future releases.\n\nReferences\n\n[9], [10], [11]\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#smoothing_kernel","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"","category":"section"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"The following smoothing kernels are currently available:","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"Smoothing Kernel Compact Support Typ. Smoothing Length Recommended Application Stability\nSchoenbergCubicSplineKernel 0 2h 11 to 13 General + sharp waves ++\nSchoenbergQuarticSplineKernel 0 25h 11 to 15 General +++\nSchoenbergQuinticSplineKernel 0 3h 11 to 15 General ++++\nGaussianKernel 0 3h 10 to 15 Literature +++++\nWendlandC2Kernel 0 1h 25 to 40 General (recommended) ++++\nWendlandC4Kernel 0 1h 30 to 45 General +++++\nWendlandC6Kernel 0 1h 35 to 50 General +++++\nPoly6Kernel 0 1h 15 to 25 Literature +\nSpikyKernel 0 1h 15 to 30 Sharp corners + waves +","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"We recommend to use the WendlandC2Kernel for most applications. If less smoothing is needed, try SchoenbergCubicSplineKernel, for more smoothing try WendlandC6Kernel.","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"note: Usage\nThe kernel can be called asTrixiParticles.kernel(smoothing_kernel, r, h)The length of the compact support can be obtained asTrixiParticles.compact_support(smoothing_kernel, h)Note that r has to be a scalar, so in the context of SPH, the kernel should be used asW(Vert r_a - r_b Vert h)The gradient required in SPH, nabla_r_a W(Vert r_a - r_b Vert h)can be called asTrixiParticles.kernel_grad(smoothing_kernel, pos_diff, distance, h)where pos_diff is r_a - r_b and distance is Vert r_a - r_b Vert.","category":"page"},{"location":"general/smoothing_kernels/","page":"Smoothing Kernels","title":"Smoothing Kernels","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"smoothing_kernels.jl\")]","category":"page"},{"location":"general/smoothing_kernels/#TrixiParticles.GaussianKernel","page":"Smoothing Kernels","title":"TrixiParticles.GaussianKernel","text":"GaussianKernel{NDIMS}()\n\nGaussian kernel given by\n\nW(r h) = fracsigma_dh^d e^-r^2h^2\n\nwhere d is the number of dimensions and\n\nsigma_2 = frac1pi for 2D,\nsigma_3 = frac1pi^32 for 3D.\n\nThis kernel function has an infinite support, but in practice, it's often truncated at a certain multiple of h, such as 3h.\n\nIn this implementation, the kernel is truncated at 3h, so this kernel function has a compact support of 0 3h.\n\nThe smoothing length is typically in the range 10delta 15delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\nNote: This truncation makes this Kernel not conservative, which is beneficial in regards to stability but makes it less accurate.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.Poly6Kernel","page":"Smoothing Kernels","title":"TrixiParticles.Poly6Kernel","text":"Poly6Kernel{NDIMS}()\n\nPoly6 kernel, a commonly used kernel in SPH literature [3], especially in computer graphics contexts. It is defined as\n\nW(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (1 - q^2)^3 textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor that depends on the dimension. The normalization factor sigma is 4 pi in two dimensions or 315 64pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nPoly6 is well-known for its computational simplicity, though it's worth noting that there are other kernels that might offer better accuracy for hydrodynamic simulations. Furthermore, its derivatives are not that smooth, which can lead to stability problems. It is also susceptible to clumping.\n\nThe smoothing length is typically in the range 15delta 25delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SchoenbergCubicSplineKernel","page":"Smoothing Kernels","title":"TrixiParticles.SchoenbergCubicSplineKernel","text":"SchoenbergCubicSplineKernel{NDIMS}()\n\nCubic spline kernel by Schoenberg (1946), given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n frac14 (2 - q)^3 - (1 - q)^3 textif 0 leq q 1 \n frac14 (2 - q)^3 textif 1 leq q 2 \n 0 textif q geq 2 \nendcases\n\nwhere d is the number of dimensions and sigma is a normalization constant given by sigma =frac23 frac107 pi frac1pi in 1 2 3 dimensions.\n\nThis kernel function has a compact support of 0 2h.\n\nFor an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see Price (2012). For an analytic formula for higher order Schoenberg kernels, see Monaghan (1985). The largest disadvantage of Schoenberg Spline Kernel is the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.\n\nThe smoothing length is typically in the range 11delta 13delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SchoenbergQuarticSplineKernel","page":"Smoothing Kernels","title":"TrixiParticles.SchoenbergQuarticSplineKernel","text":"SchoenbergQuarticSplineKernel{NDIMS}()\n\nQuartic spline kernel by Schoenberg (1946), given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n left(52 - q right)^4 - 5left(32 - q right)^4\n + 10left(12 - q right)^4 textif 0 leq q frac12 \n left(52 - q right)^4 - 5left(32 - q right)^4\n textif frac12 leq q frac32 \n left(52 - q right)^4 textif frac32 leq q frac52 \n 0 textif q geq frac52\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization constant given by sigma =frac124 frac961199 pi frac120pi in 1 2 3 dimensions.\n\nThis kernel function has a compact support of 0 25h.\n\nFor an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see Price (2012). For an analytic formula for higher order Schoenberg kernels, see Monaghan (1985).\n\nThe largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.\n\nThe smoothing length is typically in the range 11delta 15delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SchoenbergQuinticSplineKernel","page":"Smoothing Kernels","title":"TrixiParticles.SchoenbergQuinticSplineKernel","text":"SchoenbergQuinticSplineKernel{NDIMS}()\n\nQuintic spline kernel by Schoenberg (1946), given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (3 - q)^5 - 6(2 - q)^5 + 15(1 - q)^5 textif 0 leq q 1 \n (3 - q)^5 - 6(2 - q)^5 textif 1 leq q 2 \n (3 - q)^5 textif 2 leq q 3 \n 0 textif q geq 3\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization constant given by sigma =frac1120 frac7478 pi frac1120pi in 1 2 3 dimensions.\n\nThis kernel function has a compact support of 0 3h.\n\nFor an overview of Schoenberg cubic, quartic and quintic spline kernels including normalization factors, see Price (2012). For an analytic formula for higher order Schoenberg kernels, see Monaghan (1985).\n\nThe largest disadvantage of Schoenberg Spline Kernel are the rather non-smooth first derivative, which can lead to increased noise compared to other kernel variants.\n\nThe smoothing length is typically in the range 11delta 15delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.SpikyKernel","page":"Smoothing Kernels","title":"TrixiParticles.SpikyKernel","text":"SpikyKernel{NDIMS}()\n\nThe Spiky kernel is another frequently used kernel in SPH, especially due to its desirable properties in preserving features near boundaries in fluid simulations [3]. It is defined as:\n\n W(r h) = frac1h^d w(rh)\n\nwith:\n\nw(q) = sigma begincases\n (1 - q)^3 textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and the normalization factor sigma is 10 pi in two dimensions or 15 pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nThe Spiky kernel is particularly known for its sharp gradients, which can help to preserve sharp features in fluid simulations, especially near solid boundaries. These sharp gradients at the boundary are also the largest disadvantage as they can lead to instability.\n\nThe smoothing length is typically in the range 15delta 30delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.WendlandC2Kernel","page":"Smoothing Kernels","title":"TrixiParticles.WendlandC2Kernel","text":"WendlandC2Kernel{NDIMS}()\n\nWendland C2 kernel [7], a piecewise polynomial function designed to have compact support and to be twice continuously differentiable everywhere. Given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (1 - q)^4 (4q + 1) textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor dependent on the dimension. The normalization factor sigma is 407pi in two dimensions or 212pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nFor a detailed discussion on Wendland functions and their applications in SPH, see Dehnen (2012). The smoothness of these functions is also the largest disadvantage as they lose details at sharp corners.\n\nThe smoothing length is typically in the range 25delta 40delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.WendlandC4Kernel","page":"Smoothing Kernels","title":"TrixiParticles.WendlandC4Kernel","text":"WendlandC4Kernel{NDIMS}()\n\nWendland C4 kernel [7], a piecewise polynomial function designed to have compact support and to be four times continuously differentiable everywhere. Given by\n\n W(r h) = frac1h^d w(rh)\n\nwith\n\nw(q) = sigma begincases\n (1 - q)^6 (35q^2 3 + 6q + 1) textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor dependent on the dimension. The normalization factor sigma is 9 pi in two dimensions or 495 32pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nFor a detailed discussion on Wendland functions and their applications in SPH, see Dehnen (2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.\n\nThe smoothing length is typically in the range 30delta 45delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"general/smoothing_kernels/#TrixiParticles.WendlandC6Kernel","page":"Smoothing Kernels","title":"TrixiParticles.WendlandC6Kernel","text":"WendlandC6Kernel{NDIMS}()\n\nWendland C6 kernel [7], a piecewise polynomial function designed to have compact support and to be six times continuously differentiable everywhere. Given by:\n\nW(r h) = frac1h^d w(rh)\n\nwith:\n\nw(q) = sigma begincases\n (1 - q)^8 (32q^3 + 25q^2 + 8q + 1) textif 0 leq q 1 \n 0 textif q geq 1\nendcases\n\nwhere d is the number of dimensions and sigma is a normalization factor dependent on the dimension. The normalization factor sigma is 78 7 pi in two dimensions or 1365 64pi in three dimensions.\n\nThis kernel function has a compact support of 0 h.\n\nFor a detailed discussion on Wendland functions and their applications in SPH, Dehnen (2012). The smoothness of these functions is also the largest disadvantage as they loose details at sharp corners.\n\nThe smoothing length is typically in the range 35delta 50delta, where delta is the typical particle spacing.\n\nFor general information and usage see Smoothing Kernels.\n\n\n\n\n\n","category":"type"},{"location":"reference-trixibase/#TrixiBase.jl-API","page":"TrixiBase.jl API Reference","title":"TrixiBase.jl API","text":"","category":"section"},{"location":"reference-trixibase/","page":"TrixiBase.jl API Reference","title":"TrixiBase.jl API Reference","text":"CurrentModule = TrixiBase","category":"page"},{"location":"reference-trixibase/","page":"TrixiBase.jl API Reference","title":"TrixiBase.jl API Reference","text":"Modules = [TrixiBase]","category":"page"},{"location":"reference-trixibase/#TrixiBase.disable_debug_timings-Tuple{}","page":"TrixiBase.jl API Reference","title":"TrixiBase.disable_debug_timings","text":"disable_debug_timings()\n\nDisable all @trixi_timeit timings. The timings should be optimized away, allowing for truly zero-overhead. Enable timings again with enable_debug_timings.\n\nSee also enable_debug_timings, @trixi_timeit.\n\n\n\n\n\n","category":"method"},{"location":"reference-trixibase/#TrixiBase.enable_debug_timings-Tuple{}","page":"TrixiBase.jl API Reference","title":"TrixiBase.enable_debug_timings","text":"enable_debug_timings()\n\nEnable all @trixi_timeit timings (default behavior).\n\nSee also disable_debug_timings, @trixi_timeit.\n\n\n\n\n\n","category":"method"},{"location":"reference-trixibase/#TrixiBase.timer-Tuple{}","page":"TrixiBase.jl API Reference","title":"TrixiBase.timer","text":"timer()\n\nMain timer for global timing, e.g., to be used with @trixi_timeit.\n\n\n\n\n\n","category":"method"},{"location":"reference-trixibase/#TrixiBase.trixi_include-Tuple{Module, AbstractString}","page":"TrixiBase.jl API Reference","title":"TrixiBase.trixi_include","text":"trixi_include([mod::Module=Main,] elixir::AbstractString; kwargs...)\n\ninclude the file elixir and evaluate its content in the global scope of module mod. You can override specific assignments in elixir by supplying keyword arguments. Its basic purpose is to make it easier to modify some parameters while running simulations from the REPL. Additionally, this is used in tests to reduce the computational burden for CI while still providing examples with sensible default values for users.\n\nBefore replacing assignments in elixir, the keyword argument maxiters is inserted into calls to solve with it's default value used in the SciML ecosystem for ODEs, see the \"Miscellaneous\" section of the documentation.\n\nExamples\n\njulia> using TrixiBase, Trixi\n\njulia> redirect_stdout(devnull) do\n trixi_include(@__MODULE__, joinpath(examples_dir(), \"tree_1d_dgsem\", \"elixir_advection_extended.jl\"),\n tspan=(0.0, 0.1))\n sol.t[end]\n end\n[ Info: You just called `trixi_include`. Julia may now compile the code, please be patient.\n0.1\n\n\n\n\n\n","category":"method"},{"location":"reference-trixibase/#TrixiBase.@trixi_timeit-Tuple{Any, Any, Any}","page":"TrixiBase.jl API Reference","title":"TrixiBase.@trixi_timeit","text":"@trixi_timeit timer() \"some label\" expression\n\nBasically the same as a special case of @timeit_debug from TimerOutputs.jl, but without try ... finally ... end block. Thus, it's not exception-safe, but it also avoids some related performance problems. Since we do not use exception handling in Trixi.jl, that's not really an issue.\n\nAll @trixi_timeit timings can be disabled with disable_debug_timings. The timings should then be optimized away, allowing for truly zero-overhead.\n\nSee also disable_debug_timings, enable_debug_timings.\n\n\n\n\n\n","category":"macro"},{"location":"tutorials_template/tut_beam/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials_template/tut_beam/","page":"Example file","title":"Example file","text":"!!include:examples/solid/oscillating_beam_2d.jl!!\n","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/CONTRIBUTING.md\"","category":"page"},{"location":"contributing/#Contributing","page":"Contributing","title":"Contributing","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"TrixiParticles.jl is an open-source project and we are very happy to accept contributions from the community. Please feel free to open issues or submit patches (preferably as pull requests) any time. For planned larger contributions, it is often beneficial to get in contact with one of the principal developers first (see Authors).","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"TrixiParticles.jl and its contributions are licensed under the MIT license (see License). As a contributor, you certify that all your contributions are in conformance with the Developer Certificate of Origin (Version 1.1), which is reproduced below.","category":"page"},{"location":"contributing/#Developer-Certificate-of-Origin-(Version-1.1)","page":"Contributing","title":"Developer Certificate of Origin (Version 1.1)","text":"","category":"section"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"The following text was taken from https://developercertificate.org:","category":"page"},{"location":"contributing/","page":"Contributing","title":"Contributing","text":"Developer Certificate of Origin\nVersion 1.1\n\nCopyright (C) 2004, 2006 The Linux Foundation and its contributors.\n1 Letterman Drive\nSuite D4700\nSan Francisco, CA, 94129\n\nEveryone is permitted to copy and distribute verbatim copies of this\nlicense document, but changing it is not allowed.\n\n\nDeveloper's Certificate of Origin 1.1\n\nBy making a contribution to this project, I certify that:\n\n(a) The contribution was created in whole or in part by me and I\n have the right to submit it under the open source license\n indicated in the file; or\n\n(b) The contribution is based upon previous work that, to the best\n of my knowledge, is covered under an appropriate open source\n license and I have the right under that license to submit that\n work with modifications, whether created in whole or in part\n by me, under the same open source license (unless I am\n permitted to submit under a different license), as indicated\n in the file; or\n\n(c) The contribution was provided directly to me by some other\n person who certified (a), (b) or (c) and I have not modified\n it.\n\n(d) I understand and agree that this project and the contribution\n are public and that a record of the contribution (including all\n personal information I submit with it, including my sign-off) is\n maintained indefinitely and may be redistributed consistent with\n this project or the open source license(s) involved.","category":"page"},{"location":"development/#development","page":"Development","title":"Development","text":"","category":"section"},{"location":"development/#Preview-of-the-documentation","page":"Development","title":"Preview of the documentation","text":"","category":"section"},{"location":"development/","page":"Development","title":"Development","text":"To generate the Documentation, first instantiate the docs environment by executing the following command from the TrixiParticles.jl root directory:","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"julia --project=docs -e \"using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()\"","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"This command only has to be run once. After that, maintain the docs environment as described under Installation.","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"With an instantiated docs environment, generate the docs with the following command (again from the TrixiParticles.jl root directory):","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"julia --project=docs --color=yes docs/make.jl","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"You can then open the generated files in docs/build with your webbrowser. Alternatively, run","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"python3 -m http.server -d docs/build","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"and open localhost:8000 in your webbrowser.","category":"page"},{"location":"development/#Release-management","page":"Development","title":"Release management","text":"","category":"section"},{"location":"development/","page":"Development","title":"Development","text":"To create a new release for TrixiParticles.jl, perform the following steps:","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"Make sure that all PRs and changes that you want to go into the release are merged to main and that the latest commit on main has passed all CI tests.\nDetermine the currently released version of TrixiParticles.jl, e.g., on the release page. For this manual, we will assume that the latest release was v0.2.3.\nDecide on the next version number. We follow semantic versioning, thus each version is of the form vX.Y.Z where X is the major version, Y the minor version, and Z the patch version. In this manual, we assume that the major version is always 0, thus the decision process on the new version is as follows:\nIf the new release contains breaking changes (i.e., user code might not work as before without modifications), increase the minor version by one and set the patch version to zero. In our example, the new version should thus be v0.3.0.\nIf the new release only contains minor modifications and/or bug fixes, the minor version is kept as-is and the patch version is increased by one. In our example, the new version should thus be v0.2.4.\nEdit the version string in the Project.toml and set it to the new version. Push/merge this change to main.\nGo to GitHub and add a comment to the commit that you would like to become the new release (typically this will be the commit where you just updated the version). You can comment on a commit by going to the commit overview and clicking on the title of the commit. The comment should contain the following text:\n@JuliaRegistrator register\nWait for the magic to happen! Specifically, JuliaRegistrator will create a new PR to the Julia registry with the new release information. After a grace period of ~15 minutes, this PR will be merged automatically. A short while after, TagBot will create a new release of TrixiParticles.jl in our GitHub repository.\nOnce the new release has been created, the new version can be obtained through the Julia package manager as usual.\nTo make sure people do not mistake the latest state of main as the latest release, we set the version in the Project.toml to a development version. The development version should be the latest released version, with the patch version incremented by one, and the -dev suffix added. For example, if you just released v0.3.0, the new development version should be v0.3.1-dev. If you just released v0.2.4, the new development version should be v0.2.5-dev.","category":"page"},{"location":"install/#installation","page":"Installation","title":"Installation","text":"","category":"section"},{"location":"install/#Setting-up-Julia","page":"Installation","title":"Setting up Julia","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"If you have not yet installed Julia, please follow the instructions on the official website. TrixiParticles.jl works with Julia v1.9 and newer. We recommend using the latest stable release of Julia.","category":"page"},{"location":"install/#For-users","page":"Installation","title":"For users","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"TrixiParticles.jl is a registered Julia package. You can install TrixiParticles.jl, OrdinaryDiffEq.jl (used for time integration) and Plots.jl by executing the following commands in the Julia REPL:","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"julia> using Pkg\n\njulia> Pkg.add([\"TrixiParticles\", \"OrdinaryDiffEq\", \"Plots\"])","category":"page"},{"location":"install/#for-developers","page":"Installation","title":"For developers","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"If you plan on editing TrixiParticles.jl itself, you can download TrixiParticles.jl to a local folder and use the code from the cloned directory:","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"git clone git@github.com:trixi-framework/TrixiParticles.jl.git\ncd TrixiParticles.jl\nmkdir run\njulia --project=run -e 'using Pkg; Pkg.develop(PackageSpec(path=\".\"))' # Add TrixiParticles.jl to `run` project\njulia --project=run -e 'using Pkg; Pkg.add([\"OrdinaryDiffEq\", \"Plots\"])' # Add additional packages","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"If you installed TrixiParticles.jl this way, you always have to start Julia with the --project flag set to your run directory, e.g.,","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"julia --project=run","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"from the TrixiParticles.jl root directory.","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"The advantage of using a separate run directory is that you can also add other related packages (e.g., OrdinaryDiffEq.jl, see above) to the project in the run folder and always have a reproducible environment at hand to share with others.","category":"page"},{"location":"install/#Optional-software/packages","page":"Installation","title":"Optional software/packages","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"OrdinaryDiffEq.jl – A Julia package of ordinary differential equation solvers that is used in the examples\nPlots.jl – Julia Plotting library that is used in some examples\nPythonPlot.jl – Plotting library that can be used instead of Plots.jl\nParaView – Software that can be used for visualization of results","category":"page"},{"location":"install/#installation-issues","page":"Installation","title":"Common issues","text":"","category":"section"},{"location":"install/","page":"Installation","title":"Installation","text":"If you followed the installation instructions for developers and you run into any problems with packages when pulling the latest version of TrixiParticles.jl, start Julia with the project in the run folder,","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":" julia --project=run","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"update all packages in that project, resolve all conflicts in the project, and install all new dependencies:","category":"page"},{"location":"install/","page":"Installation","title":"Installation","text":"julia> using Pkg\n\njulia> Pkg.update()\n\njulia> Pkg.resolve()\n\njulia> Pkg.instantiate()","category":"page"},{"location":"visualization/#Visualization","page":"Visualization","title":"Visualization","text":"","category":"section"},{"location":"visualization/#Export-VTK-files","page":"Visualization","title":"Export VTK files","text":"","category":"section"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"You can export particle data as VTK files by using the SolutionSavingCallback. All our predefined examples are already using this callback to export VTK files to the out directory (relative to the directory that you are running Julia from). VTK files can be read by visualization tools like ParaView and VisIt.","category":"page"},{"location":"visualization/#ParaView","page":"Visualization","title":"ParaView","text":"","category":"section"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"Follow these steps to view the exported VTK files in ParaView:","category":"page"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"Click File -> Open.\nNavigate to the out directory (relative to the directory that you are running Julia from).\nOpen both boundary_1.pvd and fluid_1.pvd.\nClick \"Apply\", which by default is on the left pane below the \"Pipeline Browser\".\nHold the left mouse button to move the solution around.","category":"page"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"You will now see the following: (Image: image)","category":"page"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"It is useful to make the particles larger. For this, first make sure you have \"fluid_1.pvd\" highlighted in the \"Pipeline Browser\" then in the \"Properties\" window in the bottom left change \"Point Size\" to a larger value. (Image: image)","category":"page"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"To now view the result variables first make sure you have \"fluid_1.pvd\" highlighted in the \"Pipeline Browser\" then select them in the variable selection combo box (see picture below). Let's, for example, pick \"density\". To now view the time progression of the result hit the \"play button\" (see picture below). (Image: image)","category":"page"},{"location":"visualization/#API","page":"Visualization","title":"API","text":"","category":"section"},{"location":"visualization/","page":"Visualization","title":"Visualization","text":"Modules = [TrixiParticles]\nPages = map(file -> joinpath(\"visualization\", file), readdir(joinpath(\"..\", \"src\", \"visualization\")))","category":"page"},{"location":"visualization/#TrixiParticles.trixi2vtk-Tuple{Any, Any, Any}","page":"Visualization","title":"TrixiParticles.trixi2vtk","text":"trixi2vtk(vu_ode, semi, t; iter=nothing, output_directory=\"out\", prefix=\"\",\n write_meta_data=true, max_coordinates=Inf, custom_quantities...)\n\nConvert Trixi simulation data to VTK format.\n\nArguments\n\nvu_ode: Solution of the TrixiParticles ODE system at one time step. This expects an ArrayPartition as returned in the examples as sol.u[end].\nsemi: Semidiscretization of the TrixiParticles simulation.\nt: Current time of the simulation.\n\nKeywords\n\niter=nothing: Iteration number when multiple iterations are to be stored in separate files. This number is just appended to the filename.\noutput_directory=\"out\": Output directory path.\nprefix=\"\": Prefix for output files.\nwrite_meta_data=true: Write meta data.\nmax_coordinates=Inf The coordinates of particles will be clipped if their absolute values exceed this threshold.\ncustom_quantities...: Additional custom quantities to include in the VTK output. Each custom quantity must be a function of (v, u, t, system), which will be called for every system, where v and u are the wrapped solution arrays for the corresponding system and t is the current simulation time. Note that working with these v and u arrays requires undocumented internal functions of TrixiParticles. See Custom Quantities for a list of pre-defined custom quantities that can be used here.\n\nExample\n\ntrixi2vtk(sol.u[end], semi, 0.0, iter=1, output_directory=\"output\", prefix=\"solution\")\n\n# Additionally store the kinetic energy of each system as \"my_custom_quantity\"\ntrixi2vtk(sol.u[end], semi, 0.0, iter=1, my_custom_quantity=kinetic_energy)\n\n\n\n\n\n","category":"method"},{"location":"visualization/#TrixiParticles.trixi2vtk-Tuple{Any}","page":"Visualization","title":"TrixiParticles.trixi2vtk","text":"trixi2vtk(coordinates; output_directory=\"out\", prefix=\"\", filename=\"coordinates\",\n custom_quantities...)\n\nConvert coordinate data to VTK format.\n\nArguments\n\ncoordinates: Coordinates to be saved.\n\nKeywords\n\noutput_directory=\"out\": Output directory path.\nprefix=\"\": Prefix for the output file.\nfilename=\"coordinates\": Name of the output file.\ncustom_quantities...: Additional custom quantities to include in the VTK output.\n\nReturns\n\nfile::AbstractString: Path to the generated VTK file.\n\n\n\n\n\n","category":"method"},{"location":"visualization/#TrixiParticles.trixi2vtk-Tuple{InitialCondition}","page":"Visualization","title":"TrixiParticles.trixi2vtk","text":"trixi2vtk(initial_condition::InitialCondition; output_directory=\"out\",\n prefix=\"\", filename=\"initial_condition\", custom_quantities...)\n\nConvert InitialCondition data to VTK format.\n\nArguments\n\ninitial_condition: InitialCondition to be saved.\n\nKeywords\n\noutput_directory=\"out\": Output directory path.\nprefix=\"\": Prefix for the output file.\nfilename=\"coordinates\": Name of the output file.\ncustom_quantities...: Additional custom quantities to include in the VTK output.\n\nReturns\n\nfile::AbstractString: Path to the generated VTK file.\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#initial_condition","page":"Initial Condition and Setups","title":"Initial Condition","text":"","category":"section"},{"location":"general/initial_condition/","page":"Initial Condition and Setups","title":"Initial Condition and Setups","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"initial_condition.jl\")]","category":"page"},{"location":"general/initial_condition/#TrixiParticles.InitialCondition","page":"Initial Condition and Setups","title":"TrixiParticles.InitialCondition","text":"InitialCondition(; coordinates, density, velocity=zeros(size(coordinates, 1)),\n mass=nothing, pressure=0.0, particle_spacing=-1.0)\n\nStruct to hold the initial configuration of the particles.\n\nThe following setups return InitialConditions for commonly used setups:\n\nRectangularShape\nSphereShape\nRectangularTank\nComplexShape\nextrude_geometry\n\nInitialConditions support the set operations union, setdiff and intersect in order to build more complex geometries.\n\nArguments\n\ncoordinates: An array where the i-th column holds the coordinates of particle i.\ndensity: Either a vector holding the density of each particle, or a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles.\n\nKeywords\n\nvelocity: Either an array where the i-th column holds the velocity of particle i, or a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a vector holding the mass of each particle, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\npressure: Either a vector holding the pressure of each particle, or a function mapping each particle's coordinates to its pressure, or a scalar for a constant pressure over all particles. This is optional and only needed when using the EntropicallyDampedSPHSystem.\nparticle_spacing: The spacing between the particles. This is a scalar, as the spacing is assumed to be uniform. This is only needed when using set operations on the InitialCondition or for automatic mass calculation.\n\nExamples\n\n# Rectangle filled with particles\ninitial_condition = RectangularShape(0.1, (3, 4), (-1.0, 1.0), density=1.0)\n\n# Two spheres in one initial condition\ninitial_condition = union(SphereShape(0.15, 0.5, (-1.0, 1.0), 1.0),\n SphereShape(0.15, 0.2, (0.0, 1.0), 1.0))\n\n# Rectangle with a spherical hole\nshape1 = RectangularShape(0.1, (16, 13), (-0.8, 0.0), density=1.0)\nshape2 = SphereShape(0.1, 0.35, (0.0, 0.6), 1.0, sphere_type=RoundSphere())\ninitial_condition = setdiff(shape1, shape2)\n\n# Intersect of a rectangle with a sphere. Note that this keeps the particles of the\n# rectangle that are in the intersect, while `intersect(shape2, shape1)` would consist of\n# the particles of the sphere that are in the intersect.\nshape1 = RectangularShape(0.1, (16, 13), (-0.8, 0.0), density=1.0)\nshape2 = SphereShape(0.1, 0.35, (0.0, 0.6), 1.0, sphere_type=RoundSphere())\ninitial_condition = intersect(shape1, shape2)\n\n# Build `InitialCondition` manually\ncoordinates = [0.0 1.0 1.0\n 0.0 0.0 1.0]\nvelocity = zero(coordinates)\nmass = ones(3)\ndensity = 1000 * ones(3)\ninitial_condition = InitialCondition(; coordinates, velocity, mass, density)\n\n# With functions\ninitial_condition = InitialCondition(; coordinates, velocity=x -> 2x, mass=1.0, density=1000.0)\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#Setups","page":"Initial Condition and Setups","title":"Setups","text":"","category":"section"},{"location":"general/initial_condition/","page":"Initial Condition and Setups","title":"Initial Condition and Setups","text":"Modules = [TrixiParticles]\nPages = map(file -> joinpath(\"setups\", file), readdir(joinpath(\"..\", \"src\", \"setups\")))","category":"page"},{"location":"general/initial_condition/#TrixiParticles.ComplexShape-Tuple{Union{TrixiParticles.Polygon, TrixiParticles.TriangleMesh}}","page":"Initial Condition and Setups","title":"TrixiParticles.ComplexShape","text":"ComplexShape(geometry::Union{TriangleMesh, Polygon}; particle_spacing, density,\n pressure=0.0, mass=nothing, velocity=zeros(ndims(geometry)),\n point_in_geometry_algorithm=WindingNumberJacobson(; geometry,\n hierarchical_winding=false,\n winding_number_factor=sqrt(eps())),\n grid_offset::Real=0.0, max_nparticles=10^7,\n pad_initial_particle_grid=2particle_spacing)\n\nSample a complex geometry with particles. Returns an InitialCondition. Note that an initial particle grid is generated inside the bounding box of the geometry. A point_in_geometry_algorithm checks if particles are inside the geometry or not. For more information about the method see WindingNumberJacobson or WindingNumberHormann.\n\nArguments\n\ngeometry: Geometry returned by load_geometry.\n\nKeywords\n\nparticle_spacing: Spacing between the particles.\ndensity: Either a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles.\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\npressure: Scalar to set the pressure of all particles to this value. This is only used by the EntropicallyDampedSPHSystem and will be overwritten when using an initial pressure function in the system.\npoint_in_geometry_algorithm: Algorithm for sampling the complex geometry with particles. It basically checks whether a particle is inside an object or not. For more information see WindingNumberJacobson or WindingNumberHormann\ngrid_offset: Offset of the initial particle grid of the bounding box of the geometry.\nmax_nparticles: Maximum number of particles in the initial particle grid. This is only used to avoid accidentally choosing a particle_spacing that is too small for the scale of the geometry.\npad_initial_particle_grid: Padding of the initial particle grid.\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#TrixiParticles.extrude_geometry-Tuple{Any}","page":"Initial Condition and Setups","title":"TrixiParticles.extrude_geometry","text":"extrude_geometry(geometry; particle_spacing, direction, n_extrude::Integer,\n velocity=zeros(length(direction)),\n mass=nothing, density=nothing, pressure=0.0)\n\nExtrude either a line, a plane or a shape along a specific direction. Returns an InitialCondition.\n\nArguments\n\ngeometry: Either particle coordinates or an InitialCondition defining a 2D shape to extrude to a 3D volume, or two 2D points (A B) defining the interval A B to extrude to a plane in 2D, or three 3D points (A B C) defining the parallelogram spanned by the vectors widehatAB and widehat AC to extrude to a parallelepiped.\n\nKeywords\n\nparticle_spacing: Spacing between the particles. Can be omitted when geometry is an InitialCondition (unless geometry.particle_spacing == -1).\ndirection: A vector that specifies the direction in which to extrude.\nn_extrude: Number of layers of particles created in the direction of extrusion.\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\ndensity: Either a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles.\npressure: Scalar to set the pressure of all particles to this value. This is only used by the EntropicallyDampedSPHSystem and will be overwritten when using an initial pressure function in the system.\ntlsph: With the TotalLagrangianSPHSystem, particles need to be placed on the boundary of the shape and not one particle radius away, as for fluids. When tlsph=true, particles will be placed on the boundary of the shape.\n\nExamples\n\n# Extrude a line in 2D to a plane in 2D\np1 = [0.0, 0.0]\np2 = [1.0, 1.0]\n\ndirection = [-1.0, 1.0]\n\nshape = extrude_geometry((p1, p2); direction, particle_spacing=0.1, n_extrude=4, density=1000.0)\n\n# Extrude a parallelogram in 3D to a parallelepiped in 3D\np1 = [0.0, 0.0, 0.0]\np2 = [0.5, 1.0, 0.0]\np3 = [1.0, 0.2, 0.0]\n\ndirection = [0.0, 0.0, 1.0]\n\nshape = extrude_geometry((p1, p2, p3); direction, particle_spacing=0.1, n_extrude=4, density=1000.0)\n\n# Extrude a 2D shape (here: a disc) to a 3D shape (here: a cylinder)\nshape = SphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3,\n sphere_type=RoundSphere(end_angle=pi))\n\ndirection = [0.0, 0.0, 1.0]\n\nshape = extrude_geometry(shape; direction, particle_spacing=0.1, n_extrude=4, density=1000.0)\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#TrixiParticles.RectangularShape-Tuple{Any, Any, Any}","page":"Initial Condition and Setups","title":"TrixiParticles.RectangularShape","text":"RectangularShape(particle_spacing, n_particles_per_dimension, min_coordinates;\n velocity=zeros(length(n_particles_per_dimension)),\n mass=nothing, density=nothing, pressure=0.0,\n acceleration=nothing, state_equation=nothing,\n tlsph=false, loop_order=nothing)\n\nRectangular shape filled with particles. Returns an InitialCondition.\n\nArguments\n\nparticle_spacing: Spacing between the particles.\nn_particles_per_dimension: Tuple containing the number of particles in x, y and z (only 3D) direction, respectively.\nmin_coordinates: Coordinates of the corner in negative coordinate directions.\n\nKeywords\n\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\ndensity: Either a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles. Obligatory when not using a state equation. Cannot be used together with state_equation.\npressure: Scalar to set the pressure of all particles to this value. This is only used by the EntropicallyDampedSPHSystem and will be overwritten when using an initial pressure function in the system. Cannot be used together with hydrostatic pressure gradient.\nacceleration: In order to initialize particles with a hydrostatic pressure gradient, an acceleration vector can be passed. Note that only accelerations in one coordinate direction and no diagonal accelerations are supported. This will only change the pressure of the particles. When using the WeaklyCompressibleSPHSystem, pass a state_equation as well to initialize the particles with the corresponding density and mass. When using the EntropicallyDampedSPHSystem, the pressure will be overwritten when using an initial pressure function in the system. This cannot be used together with the pressure keyword argument.\nstate_equation: When calculating a hydrostatic pressure gradient by setting acceleration, the state_equation will be used to set the corresponding density. Cannot be used together with density.\ntlsph: With the TotalLagrangianSPHSystem, particles need to be placed on the boundary of the shape and not one particle radius away, as for fluids. When tlsph=true, particles will be placed on the boundary of the shape.\ncoordinates_perturbation: Add a small random displacement to the particle positions, where the amplitude is coordinates_perturbation * particle_spacing.\n\nExamples\n\n# 2D\nrectangular = RectangularShape(particle_spacing, (5, 4), (1.0, 2.0), density=1000.0)\n\n# 2D with hydrostatic pressure gradient.\n# `state_equation` has to be the same as for the WCSPH system.\nstate_equation = StateEquationCole(sound_speed=20.0, exponent=7, reference_density=1000.0)\nrectangular = RectangularShape(particle_spacing, (5, 4), (1.0, 2.0),\n acceleration=(0.0, -9.81), state_equation=state_equation)\n\n# 3D\nrectangular = RectangularShape(particle_spacing, (5, 4, 7), (1.0, 2.0, 3.0), density=1000.0)\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#TrixiParticles.RectangularTank","page":"Initial Condition and Setups","title":"TrixiParticles.RectangularTank","text":"RectangularTank(particle_spacing, fluid_size, tank_size, fluid_density;\n velocity=zeros(length(fluid_size)), fluid_mass=nothing,\n pressure=0.0,\n acceleration=nothing, state_equation=nothing,\n boundary_density=fluid_density,\n n_layers=1, spacing_ratio=1.0,\n min_coordinates=zeros(length(fluid_size)),\n faces=Tuple(trues(2 * length(fluid_size))))\n\nRectangular tank filled with a fluid to set up dam-break-style simulations.\n\nArguments\n\nparticle_spacing: Spacing between the fluid particles.\nfluid_size: The dimensions of the fluid as (x, y) (or (x, y, z) in 3D).\ntank_size: The dimensions of the tank as (x, y) (or (x, y, z) in 3D).\nfluid_density: The rest density of the fluid. Will only be used as default for boundary_density when using a state equation.\n\nKeywords\n\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nfluid_mass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\npressure: Scalar to set the pressure of all particles to this value. This is only used by the EntropicallyDampedSPHSystem and will be overwritten when using an initial pressure function in the system. Cannot be used together with hydrostatic pressure gradient.\nacceleration: In order to initialize particles with a hydrostatic pressure gradient, an acceleration vector can be passed. Note that only accelerations in one coordinate direction and no diagonal accelerations are supported. This will only change the pressure of the particles. When using the WeaklyCompressibleSPHSystem, pass a state_equation as well to initialize the particles with the corresponding density and mass. When using the EntropicallyDampedSPHSystem, the pressure will be overwritten when using an initial pressure function in the system. This cannot be used together with the pressure keyword argument.\nstate_equation: When calculating a hydrostatic pressure gradient by setting acceleration, the state_equation will be used to set the corresponding density. Cannot be used together with density.\nboundary_density: Density of each boundary particle (by default set to the fluid density)\nn_layers: Number of boundary layers.\nspacing_ratio: Ratio of particle_spacing to boundary particle spacing. A value of 2 means that the boundary particle spacing will be half the fluid particle spacing.\nmin_coordinates: Coordinates of the corner in negative coordinate directions.\nfaces: By default all faces are generated. Set faces by passing a bit-array of length 4 (2D) or 6 (3D) to generate the faces in the normal direction: -x,+x,-y,+y,-z,+z.\n\nFields\n\nfluid::InitialCondition: InitialCondition for the fluid.\nboundary::InitialCondition: InitialCondition for the boundary.\nfluid_size::Tuple: Tuple containing the size of the fluid in each dimension after rounding.\ntank_size::Tuple: Tuple containing the size of the tank in each dimension after rounding.\n\nExamples\n\n# 2D\nsetup = RectangularTank(particle_spacing, (water_width, water_height),\n (container_width, container_height), fluid_density,\n n_layers=2, spacing_ratio=3)\n\n# 2D with hydrostatic pressure gradient.\n# `state_equation` has to be the same as for the WCSPH system.\nstate_equation = StateEquationCole(sound_speed=10.0, exponent=1, reference_density=1000.0)\nsetup = RectangularTank(particle_spacing, (water_width, water_height),\n (container_width, container_height), fluid_density,\n acceleration=(0.0, -9.81), state_equation=state_equation)\n\n# 3D\nsetup = RectangularTank(particle_spacing, (water_width, water_height, water_depth),\n (container_width, container_height, container_depth), fluid_density,\n n_layers=2)\n\nSee also: reset_wall!.\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#TrixiParticles.reset_wall!-Tuple{Any, Any, Any}","page":"Initial Condition and Setups","title":"TrixiParticles.reset_wall!","text":"reset_wall!(rectangular_tank::RectangularTank, reset_faces, positions)\n\nThe selected walls of the tank will be placed at the new positions.\n\nArguments\n\nreset_faces: Boolean tuple of 4 (in 2D) or 6 (in 3D) dimensions, similar to faces in RectangularTank.\npositions: Tuple of new positions\n\nwarning: Warning\nThere are overlapping particles when adjacent walls are moved inwards simultaneously.\n\n\n\n\n\n","category":"method"},{"location":"general/initial_condition/#TrixiParticles.RoundSphere","page":"Initial Condition and Setups","title":"TrixiParticles.RoundSphere","text":"RoundSphere(; start_angle=0.0, end_angle=2π)\n\nConstruct a sphere (or sphere segment) by nesting perfectly round concentric spheres. The resulting ball will be perfectly round, but will not have a regular inner structure.\n\nKeywords\n\nstart_angle: The starting angle of the sphere segment in radians. It determines the beginning point of the segment. The default is set to 0.0 representing the positive x-axis.\nend_angle: The ending angle of the sphere segment in radians. It defines the termination point of the segment. The default is set to 2pi, completing a full sphere.\n\nnote: Usage\nSee SphereShape on how to use this.\n\nwarning: Warning\nThe sphere segment is intended for 2D geometries and hollow spheres. If used for filled spheres or in a 3D context, results may not be accurate.\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#TrixiParticles.VoxelSphere","page":"Initial Condition and Setups","title":"TrixiParticles.VoxelSphere","text":"VoxelSphere()\n\nConstruct a sphere of voxels (where particles are placed in the voxel center) with a regular inner structure but corners on the surface. Essentially, a grid of particles is generated and all particles outside the sphere are removed. The resulting sphere will have a perfect inner structure, but is not perfectly round, as it will have corners (like a sphere in Minecraft).\n\nnote: Usage\nSee SphereShape on how to use this.\n\n\n\n\n\n","category":"type"},{"location":"general/initial_condition/#TrixiParticles.SphereShape-NTuple{4, Any}","page":"Initial Condition and Setups","title":"TrixiParticles.SphereShape","text":"SphereShape(particle_spacing, radius, center_position, density;\n sphere_type=VoxelSphere(), n_layers=-1, layer_outwards=false,\n cutout_min=(0.0, 0.0), cutout_max=(0.0, 0.0), tlsph=false,\n velocity=zeros(length(center_position)), mass=nothing, pressure=0.0)\n\nGenerate a sphere that is either completely filled (by default) or hollow (by passing n_layers).\n\nWith the sphere type VoxelSphere, a sphere of voxels (where particles are placed in the voxel center) with a regular inner structure but corners on the surface is created. Essentially, a grid of particles is generated and all particles outside the sphere are removed. With the sphere type RoundSphere, a perfectly round sphere with an imperfect inner structure is created.\n\nA cuboid can be cut out of the sphere by specifying the two corners in negative and positive coordinate directions as cutout_min and cutout_max.\n\nArguments\n\nparticle_spacing: Spacing between the particles.\nradius: Radius of the sphere.\ncenter_position: The coordinates of the center of the sphere.\ndensity: Either a function mapping each particle's coordinates to its density, or a scalar for a constant density over all particles.\n\nKeywords\n\nsphere_type: Either VoxelSphere or RoundSphere (see explanation above).\nn_layers: Set to an integer greater than zero to generate a hollow sphere, where the shell consists of n_layers layers.\nlayer_outwards: When set to false (by default), radius is the outer radius of the sphere. When set to true, radius is the inner radius of the sphere. This is only used when n_layers > 0.\ncutout_min: Corner in negative coordinate directions of a cuboid that is to be cut out of the sphere.\ncutout_max: Corner in positive coordinate directions of a cuboid that is to be cut out of the sphere.\ntlsph: With the TotalLagrangianSPHSystem, particles need to be placed on the boundary of the shape and not one particle radius away, as for fluids. When tlsph=true, particles will be placed on the boundary of the shape.\nvelocity: Either a function mapping each particle's coordinates to its velocity, or, for a constant fluid velocity, a vector holding this velocity. Velocity is constant zero by default.\nmass: Either nothing (default) to automatically compute particle mass from particle density and spacing, or a function mapping each particle's coordinates to its mass, or a scalar for a constant mass over all particles.\npressure: Either a function mapping each particle's coordinates to its pressure, or a scalar for a constant pressure over all particles. This is optional and only needed when using the EntropicallyDampedSPHSystem.\n\nExamples\n\n# Filled circle with radius 0.5, center in (0.2, 0.4) and a particle spacing of 0.1\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0)\n\n# Same as before, but perfectly round\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, sphere_type=RoundSphere())\n\n# Hollow circle with ~3 layers, outer radius 0.5, center in (0.2, 0.4) and a particle\n# spacing of 0.1.\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3)\n\n# Same as before, but perfectly round\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3, sphere_type=RoundSphere())\n\n# Hollow circle with 3 layers, inner radius 0.5, center in (0.2, 0.4) and a particle spacing\n# of 0.1.\nSphereShape(0.1, 0.5, (0.2, 0.4), 1000.0, n_layers=3, layer_outwards=true)\n\n# Filled circle with radius 0.1, center in (0.0, 0.0), particle spacing 0.1, but the\n# rectangle [0, 1] x [-0.2, 0.2] is cut out.\nSphereShape(0.1, 1.0, (0.0, 0.0), 1000.0, cutout_min=(0.0, -0.2), cutout_max=(1.0, 0.2))\n\n# Filled 3D sphere with radius 0.5, center in (0.2, 0.4, 0.3) and a particle spacing of 0.1\nSphereShape(0.1, 0.5, (0.2, 0.4, 0.3), 1000.0)\n\n# Same as before, but perfectly round\nSphereShape(0.1, 0.5, (0.2, 0.4, 0.3), 1000.0, sphere_type=RoundSphere())\n\n\n\n\n\n","category":"method"},{"location":"systems/weakly_compressible_sph/#wcsph","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Weakly compressible SPH as introduced by Monaghan (1994). This formulation relies on a stiff equation of state that generates large pressure changes for small density variations.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"weakly_compressible_sph\", \"system.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.WeaklyCompressibleSPHSystem","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.WeaklyCompressibleSPHSystem","text":"WeaklyCompressibleSPHSystem(initial_condition,\n density_calculator, state_equation,\n smoothing_kernel, smoothing_length;\n viscosity=nothing, density_diffusion=nothing,\n acceleration=ntuple(_ -> 0.0, NDIMS),\n buffer_size=nothing,\n correction=nothing, source_terms=nothing)\n\nSystem for particles of a fluid. The weakly compressible SPH (WCSPH) scheme is used, wherein a stiff equation of state generates large pressure changes for small density variations. See Weakly Compressible SPH for more details on the method.\n\nArguments\n\ninitial_condition: InitialCondition representing the system's particles.\ndensity_calculator: Density calculator for the system. See ContinuityDensity and SummationDensity.\nstate_equation: Equation of state for the system. See StateEquationCole.\nsmoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.\nsmoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.\n\nKeyword Arguments\n\nviscosity: Viscosity model for this system (default: no viscosity). See ArtificialViscosityMonaghan or ViscosityAdami.\ndensity_diffusion: Density diffusion terms for this system. See DensityDiffusion.\nacceleration: Acceleration vector for the system. (default: zero vector)\nbuffer_size: Number of buffer particles. This is needed when simulating with OpenBoundarySPHSystem.\ncorrection: Correction method used for this system. (default: no correction, see Corrections)\nsource_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure, t) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping. Note that these source terms will not be used in the calculation of the boundary pressure when using a boundary with BoundaryModelDummyParticles and AdamiPressureExtrapolation. The keyword argument acceleration should be used instead for gravity-like source terms.\nsurface_tension: Surface tension model used for this SPH system. (default: no surface tension)\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#equation_of_state","page":"Weakly Compressible SPH (Fluid)","title":"Equation of State","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The equation of state is used to relate fluid density to pressure and thus allow an explicit simulation of the WCSPH system. The equation in the following formulation was introduced by Cole (1948) (pp. 39 and 43). The pressure p is calculated as","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":" p = B left(left(fracrhorho_0right)^gamma - 1right) + p_textbackground","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"where rho denotes the density, rho_0 the reference density, and p_textbackground the background pressure, which is set to zero when applied to free-surface flows (Adami et al., 2012).","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The bulk modulus, B = fracrho_0 c^2gamma, is calculated from the artificial speed of sound c and the isentropic exponent gamma.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"An ideal gas equation of state with a linear relationship between pressure and density can be obtained by choosing exponent=1, i.e.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":" p = B left( fracrhorho_0 -1 right) = c^2(rho - rho_0)","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"For higher Reynolds numbers, exponent=7 is recommended, whereas at lower Reynolds numbers exponent=1 yields more accurate pressure estimates since pressure and density are proportional (see Morris, 1997).","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"When using SummationDensity (or DensityReinitializationCallback) and free surfaces, initializing particles with equal spacing will cause underestimated density and therefore strong attractive forces between particles at the free surface. Setting clip_negative_pressure=true can avoid this.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"weakly_compressible_sph\", \"state_equations.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.StateEquationCole","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.StateEquationCole","text":"StateEquationCole(; sound_speed, reference_density, exponent,\n background_pressure=0.0, clip_negative_pressure=false)\n\nEquation of state to describe the relationship between pressure and density of water up to high pressures.\n\nKeywords\n\nsound_speed: Artificial speed of sound.\nreference_density: Reference density of the fluid.\nexponent: A value of 7 is usually used for most simulations.\nbackground_pressure=0.0: Background pressure.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#viscosity_wcsph","page":"Weakly Compressible SPH (Fluid)","title":"Viscosity","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"TODO: Explain viscosity.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"viscosity.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.ArtificialViscosityMonaghan","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.ArtificialViscosityMonaghan","text":"ArtificialViscosityMonaghan(; alpha, beta=0.0, epsilon=0.01)\n\nArtificial viscosity by Monaghan ([16], [17]), given by\n\nPi_ab =\nbegincases\n -(alpha c mu_ab + beta mu_ab^2) barrho_ab textif v_ab cdot r_ab 0 \n 0 textotherwise\nendcases\n\nwith\n\nmu_ab = frach v_ab cdot r_abVert r_ab Vert^2 + epsilon h^2\n\nwhere alpha beta epsilon are parameters, c is the speed of sound, h is the smoothing length, r_ab = r_a - r_b is the difference of the coordinates of particles a and b, v_ab = v_a - v_b is the difference of their velocities, and barrho_ab is the arithmetic mean of their densities.\n\nNote that alpha needs to adjusted for different resolutions to maintain a specific Reynolds Number. To do so, Monaghan (2005) defined an equivalent effective physical kinematic viscosity nu by\n\n nu = fracalpha h c 2d + 4\n\nwhere d is the dimension.\n\nKeywords\n\nalpha: A value of 0.02 is usually used for most simulations. For a relation with the kinematic viscosity, see description above.\nbeta=0.0: A value of 0.0 works well for most fluid simulations and simulations with shocks of moderate strength. In simulations where the Mach number can be very high, eg. astrophysical calculation, good results can be obtained by choosing a value of beta=2.0 and alpha=1.0.\nepsilon=0.01: Parameter to prevent singularities.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.ViscosityAdami","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.ViscosityAdami","text":"ViscosityAdami(; nu, epsilon=0.01)\n\nViscosity by Adami (2012). The viscous interaction is calculated with the shear force for incompressible flows given by\n\nf_ab = sum_w bareta_ab left( V_a^2 + V_b^2 right) fracv_abr_ab^2+epsilon h_ab^2 nabla W_ab cdot r_ab\n\nwhere r_ab = r_a - r_b is the difference of the coordinates of particles a and b, v_ab = v_a - v_b is the difference of their velocities, h is the smoothing length and V is the particle volume. The parameter epsilon prevents singularities (see Ramachandran (2019)). The inter-particle-averaged shear stress is\n\n bareta_ab =frac2 eta_a eta_beta_a + eta_b\n\nwhere eta_a = rho_a nu_a with nu as the kinematic viscosity.\n\nKeywords\n\nnu: Kinematic viscosity\nepsilon=0.01: Parameter to prevent singularities\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.ViscosityMorris","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.ViscosityMorris","text":"ViscosityMorris(; nu, epsilon=0.01)\n\nViscosity by Morris (1997) also used by Fourtakas (2019).\n\nTo the force f_ab between two particles a and b due to pressure gradients, an additional force term tildef_ab is added with\n\ntildef_ab = m_a m_b frac(mu_a + mu_b) r_ab cdot nabla W_abrho_a rho_b (Vert r_ab Vert^2 + epsilon h^2) v_ab\n\nwhere mu_a = rho_a nu and mu_b = rho_b nu denote the dynamic viscosity of particle a and b respectively, and nu is the kinematic viscosity.\n\nKeywords\n\nnu: Kinematic viscosity\nepsilon=0.01: Parameter to prevent singularities\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#Density-Diffusion","page":"Weakly Compressible SPH (Fluid)","title":"Density Diffusion","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Density diffusion can be used with ContinuityDensity to remove the noise in the pressure field. It is highly recommended to use density diffusion when using WCSPH.","category":"page"},{"location":"systems/weakly_compressible_sph/#Formulation","page":"Weakly Compressible SPH (Fluid)","title":"Formulation","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"All density diffusion terms extend the continuity equation (see ContinuityDensity) by an additional term","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"fracmathrmdrho_amathrmdt = sum_b m_b v_ab cdot nabla_r_a W(Vert r_ab Vert h)\n + delta h c sum_b V_b psi_ab cdot nabla_r_a W(Vert r_ab Vert h)","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"where V_b = m_b rho_b is the volume of particle b and psi_ab depends on the density diffusion method (see DensityDiffusion for available terms). Also, rho_a denotes the density of particle a and r_ab = r_a - r_b is the difference of the coordinates, v_ab = v_a - v_b of the velocities of particles a and b.","category":"page"},{"location":"systems/weakly_compressible_sph/#Numerical-Results","page":"Weakly Compressible SPH (Fluid)","title":"Numerical Results","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"All density diffusion terms remove numerical noise in the pressure field and produce more accurate results than weakly commpressible SPH without density diffusion. This can be demonstrated with dam break examples in 2D and 3D. Here, δ = 01 has been used for all terms. Note that, due to added stability, the adaptive time integration method that was used here can choose higher time steps in the simulations with density diffusion. For the cheap DensityDiffusionMolteniColagrossi, this results in reduced runtime.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"
\n \"density_diffusion_2d\"/\n
Dam break in 2D with different density diffusion terms
\n
","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"
\n \"density_diffusion_3d\"/\n
Dam break in 3D with different density diffusion terms
\n
","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The simpler terms DensityDiffusionMolteniColagrossi and DensityDiffusionFerrari do not solve the hydrostatic problem and lead to incorrect solutions in long-running steady-state hydrostatic simulations with free surfaces (Antuono et al., 2012). This can be seen when running the simple rectangular tank example until t = 40 (again using δ = 01):","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"
\n \"density_diffusion_tank\"/\n
Tank in rest under gravity in 3D with different density diffusion terms
\n
","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"DensityDiffusionAntuono adds a correction term to solve this problem, but this term is very expensive and adds about 40–50% of computational cost.","category":"page"},{"location":"systems/weakly_compressible_sph/#API","page":"Weakly Compressible SPH (Fluid)","title":"API","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"weakly_compressible_sph\", \"density_diffusion.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusion","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusion","text":"DensityDiffusion\n\nAn abstract supertype of all density diffusion formulations.\n\nCurrently, the following formulations are available:\n\nFormulation Suitable for Steady-State Simulations Low Computational Cost\nDensityDiffusionMolteniColagrossi ❌ ✅\nDensityDiffusionFerrari ❌ ✅\nDensityDiffusionAntuono ✅ ❌\n\nSee Density Diffusion for a comparison and more details.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusionAntuono","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusionAntuono","text":"DensityDiffusionAntuono(initial_condition; delta)\n\nThe commonly used density diffusion terms by Antuono (2010), also referred to as δ-SPH. The density diffusion term by Molteni (2009) is extended by a second term, which is nicely written down by Antuono (2012).\n\nThe term psi_ab in the continuity equation in DensityDiffusion is defined by\n\npsi_ab = 2left(rho_a - rho_b - frac12big(nablarho^L_a + nablarho^L_bbig) cdot r_abright)\n fracr_abVert r_ab Vert^2\n\nwhere rho_a and rho_b denote the densities of particles a and b respectively and r_ab = r_a - r_b is the difference of the coordinates of particles a and b. The symbol nablarho^L_a denotes the renormalized density gradient defined as\n\nnablarho^L_a = -sum_b (rho_a - rho_b) V_b L_a nabla_r_a W(Vert r_ab Vert h)\n\nwith\n\nL_a = left( -sum_b V_b r_ab otimes nabla_r_a W(Vert r_ab Vert h) right)^-1 in R^d times d\n\nwhere d is the number of dimensions.\n\nSee DensityDiffusion for an overview and comparison of implemented density diffusion terms.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusionFerrari","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusionFerrari","text":"DensityDiffusionFerrari()\n\nA density diffusion term by Ferrari (2009).\n\nThe term psi_ab in the continuity equation in DensityDiffusion is defined by\n\npsi_ab = fracrho_a - rho_b2h fracr_abVert r_ab Vert\n\nwhere rho_a and rho_b denote the densities of particles a and b respectively, r_ab = r_a - r_b is the difference of the coordinates of particles a and b and h is the smoothing length.\n\nSee DensityDiffusion for an overview and comparison of implemented density diffusion terms.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.DensityDiffusionMolteniColagrossi","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.DensityDiffusionMolteniColagrossi","text":"DensityDiffusionMolteniColagrossi(; delta)\n\nThe commonly used density diffusion term by Molteni (2009).\n\nThe term psi_ab in the continuity equation in DensityDiffusion is defined by\n\npsi_ab = 2(rho_a - rho_b) fracr_abVert r_ab Vert^2\n\nwhere rho_a and rho_b denote the densities of particles a and b respectively and r_ab = r_a - r_b is the difference of the coordinates of particles a and b.\n\nSee DensityDiffusion for an overview and comparison of implemented density diffusion terms.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#corrections","page":"Weakly Compressible SPH (Fluid)","title":"Corrections","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"corrections.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.AkinciFreeSurfaceCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.AkinciFreeSurfaceCorrection","text":"AkinciFreeSurfaceCorrection(rho0)\n\nFree surface correction according to Akinci et al. (2013). At a free surface, the mean density is typically lower than the reference density, resulting in reduced surface tension and viscosity forces. The free surface correction adjusts the viscosity, pressure, and surface tension forces near free surfaces to counter this effect. It's important to note that this correlation is unphysical and serves as an approximation. The computation time added by this method is about 2–3%.\n\nMathematically the idea is quite simple. If we have an SPH particle in the middle of a volume at rest, its density will be identical to the rest density rho_0. If we now consider an SPH particle at a free surface at rest, it will have neighbors missing in the direction normal to the surface, which will result in a lower density. If we calculate the correction factor\n\nk = rho_0rho_textmean\n\nthis value will be about ~1.5 for particles at the free surface and can then be used to increase the pressure and viscosity accordingly.\n\nArguments\n\nrho0: Rest density.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.BlendedGradientCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.BlendedGradientCorrection","text":"BlendedGradientCorrection()\n\nCalculate a blended gradient to reduce the stability issues of the GradientCorrection as explained by Bonet (1999).\n\nThis calculates the following,\n\ntildenabla A_i = (1-lambda) nabla A_i + lambda L_i nabla A_i\n\nwith 0 leq lambda leq 1 being the blending factor.\n\nArguments\n\nblending_factor: Blending factor between corrected and regular SPH gradient.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.GradientCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.GradientCorrection","text":"GradientCorrection()\n\nCompute the corrected gradient of particle interactions based on their relative positions (see Bonet, 1999).\n\nMathematical Details\n\nGiven the standard SPH representation, the gradient of a field A at particle a is given by\n\nnabla A_a = sum_b m_b fracA_b - A_arho_b nabla_r_a W(Vert r_a - r_b Vert h)\n\nwhere m_b is the mass of particle b and rho_b is the density of particle b.\n\nThe gradient correction, as commonly proposed, involves multiplying this gradient with a correction matrix L:\n\ntildenabla A_a = bmL_a nabla A_a\n\nThe correction matrix bmL_a is computed based on the provided particle configuration, aiming to make the corrected gradient more accurate, especially near domain boundaries.\n\nTo satisfy\n\nsum_b V_b r_ba otimes tildenablaW_b(r_a) = left( sum_b V_b r_ba otimes nabla W_b(r_a) right) bmL_a^T = bmI\n\nthe correction matrix bmL_a is evaluated explicitly as\n\nbmL_a = left( sum_b V_b nabla W_b(r_a) otimes r_ba right)^-1\n\nnote: Note\nStability issues arise, especially when particles separate into small clusters.\nDoubles the computational effort.\nBetter stability with smoother smoothing Kernels with larger support, e.g. SchoenbergQuinticSplineKernel or WendlandC6Kernel.\nSet dt_max =< 1e-3 for stability.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.KernelCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.KernelCorrection","text":"KernelCorrection()\n\nKernel correction, as explained by Bonet (1999), uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al. This can be further extended to obtain a kernel corrected gradient as shown by Basa et al. (2008).\n\nThe kernel correction coefficient is determined by\n\nc(x) = sum_b=1 V_b W_b(x)\n\nThe gradient of corrected kernel is determined by\n\nnabla tildeW_b(r) =fracnabla W_b(r) - W_b(r) gamma(r)sum_b=1 V_b W_b(r) quad textwhere quad\ngamma(r) = fracsum_b=1 V_b nabla W_b(r)sum_b=1 V_b W_b(r)\n\nThis correction can be applied with SummationDensity and ContinuityDensity, which leads to an improvement, especially at free surfaces.\n\nnote: Note\nThis only works when the boundary model uses SummationDensity (yet).\nIt is also referred to as \"0th order correction\".\nIn 2D, we can expect an increase of about 10–15% in computation time.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.MixedKernelGradientCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.MixedKernelGradientCorrection","text":"MixedKernelGradientCorrection()\n\nCombines GradientCorrection and KernelCorrection, which results in a 1st-order-accurate SPH method (see Bonet, 1999).\n\nNotes:\n\nStability issues, especially when particles separate into small clusters.\nDoubles the computational effort.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.ShepardKernelCorrection","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.ShepardKernelCorrection","text":"ShepardKernelCorrection()\n\nKernel correction, as explained by Bonet (1999), uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al. (1996).\n\nThe kernel correction coefficient is determined by\n\nc(x) = sum_b=1 V_b W_b(x)\n\nwhere V_b = m_b rho_b is the volume of particle b.\n\nThis correction is applied with SummationDensity to correct the density and leads to an improvement, especially at free surfaces.\n\nnote: Note\nIt is also referred to as \"0th order correction\".\nIn 2D, we can expect an increase of about 5–6% in computation time.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#surface_tension","page":"Weakly Compressible SPH (Fluid)","title":"Surface Tension","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/#Akinci-based-intra-particle-force-surface-tension-and-wall-adhesion-model","page":"Weakly Compressible SPH (Fluid)","title":"Akinci-based intra-particle force surface tension and wall adhesion model","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The work by Akinci proposes three forces:","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"a cohesion force\na surface area minimization force\na wall adhesion force","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The classical model is composed of the curvature minimization and cohesion force.","category":"page"},{"location":"systems/weakly_compressible_sph/#Cohesion-force","page":"Weakly Compressible SPH (Fluid)","title":"Cohesion force","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The model calculates the cohesion force based on the distance between particles and the support radius h_c. This force is determined using two distinct regimes within the support radius:","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"For particles closer than half the support radius, a repulsive force is calculated to prevent particle clustering too tightly, enhancing the simulation's stability and realism.\nBeyond half the support radius and within the full support radius, an attractive force is computed, simulating the effects of surface tension that draw particles together.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The cohesion force, F_textcohesion, for a pair of particles is given by:","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"F_textcohesion = -sigma m_b C(r) fracrVert r Vert","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"where:","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"sigma represents the surface tension coefficient, adjusting the overall strength of the cohesion effect.\nC is a scalar function of the distance between particles.","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The cohesion kernel C is defined as","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"C(r)=frac32pi h_c^9\nbegincases\n(h_c-r)^3 r^3 textif 2r h_c \n2(h_c-r)^3 r^3 - frach^664 textif r 0 text and 2r leq h_c \n0 textotherwise\nendcases","category":"page"},{"location":"systems/weakly_compressible_sph/#Surface-area-minimization-force","page":"Weakly Compressible SPH (Fluid)","title":"Surface area minimization force","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"To model the minimization of the surface area and curvature of the fluid, a curvature force is used, which is calculated as","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"F_textcurvature = -sigma (n_a - n_b)","category":"page"},{"location":"systems/weakly_compressible_sph/#Wall-adhesion-force","page":"Weakly Compressible SPH (Fluid)","title":"Wall adhesion force","text":"","category":"section"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"The wall adhesion model proposed by Akinci et al. is based on a kernel function which is 0 from 0.0 to 0.5 support radiia with a maximum at 0.75. With the force calculated with an adhesion coefficient beta as","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"F_textadhesion = -beta m_b A(r) fracrVert r Vert","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"with A being the adhesion kernel defined as","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"A(r)= frac0007h_c^325\nbegincases\nsqrt4- frac4r^2h_c + 6r - 2h_c textif 2r h_c text and r leq h_c \n0 textotherwise\nendcases","category":"page"},{"location":"systems/weakly_compressible_sph/","page":"Weakly Compressible SPH (Fluid)","title":"Weakly Compressible SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"surface_tension.jl\")]","category":"page"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.CohesionForceAkinci","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.CohesionForceAkinci","text":"CohesionForceAkinci(surface_tension_coefficient=1.0)\n\nThis model only implements the cohesion force of the [25] surface tension model.\n\nKeywords\n\nsurface_tension_coefficient=1.0: Modifies the intensity of the surface tension-induced force, enabling the tuning of the fluid's surface tension properties within the simulation.\n\n\n\n\n\n","category":"type"},{"location":"systems/weakly_compressible_sph/#TrixiParticles.SurfaceTensionAkinci","page":"Weakly Compressible SPH (Fluid)","title":"TrixiParticles.SurfaceTensionAkinci","text":"SurfaceTensionAkinci(surface_tension_coefficient=1.0)\n\nImplements a model for surface tension and adhesion effects drawing upon the principles outlined by [25]. This model is instrumental in capturing the nuanced behaviors of fluid surfaces, such as droplet formation and the dynamics of merging or separation, by utilizing intra-particle forces.\n\nKeywords\n\nsurface_tension_coefficient=1.0: A parameter to adjust the magnitude of surface tension forces, facilitating the fine-tuning of how surface tension phenomena are represented in the simulation.\n\n\n\n\n\n","category":"type"},{"location":"tutorials/tut_dam_break/","page":"Example file","title":"Example file","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials_template/tut_dam_break.md\"","category":"page"},{"location":"tutorials/tut_dam_break/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_dam_break/","page":"Example file","title":"Example file","text":"# 2D dam break simulation based on\n#\n# S. Marrone, M. Antuono, A. Colagrossi, G. Colicchio, D. le Touzé, G. Graziani.\n# \"δ-SPH model for simulating violent impact flows\".\n# In: Computer Methods in Applied Mechanics and Engineering, Volume 200, Issues 13–16 (2011), pages 1526–1542.\n# https://doi.org/10.1016/J.CMA.2010.12.016\n\nusing TrixiParticles\nusing OrdinaryDiffEq\n\n# Size parameters\nH = 0.6\nW = 2 * H\n\n# ==========================================================================================\n# ==== Resolution\nfluid_particle_spacing = H / 40\n\n# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model\nboundary_layers = 4\nspacing_ratio = 1\n\nboundary_particle_spacing = fluid_particle_spacing / spacing_ratio\n\n# ==========================================================================================\n# ==== Experiment Setup\ngravity = 9.81\n\ntspan = (0.0, 5.7 / sqrt(gravity))\n\n# Boundary geometry and initial fluid particle positions\ninitial_fluid_size = (W, H)\ntank_size = (floor(5.366 * H / boundary_particle_spacing) * boundary_particle_spacing, 4.0)\n\nfluid_density = 1000.0\nsound_speed = 20 * sqrt(gravity * H)\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=1, clip_negative_pressure=false)\n\ntank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,\n n_layers=boundary_layers, spacing_ratio=spacing_ratio,\n acceleration=(0.0, -gravity), state_equation=state_equation)\n\n# ==========================================================================================\n# ==== Fluid\nsmoothing_length = 3.5 * fluid_particle_spacing\nsmoothing_kernel = WendlandC2Kernel{2}()\n\nfluid_density_calculator = ContinuityDensity()\nviscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)\n# nu = 0.02 * smoothing_length * sound_speed/8\n# viscosity = ViscosityMorris(nu=nu)\n# viscosity = ViscosityAdami(nu=nu)\n# Alternatively the density diffusion model by Molteni & Colagrossi can be used,\n# which will run faster.\n# density_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)\ndensity_diffusion = DensityDiffusionAntuono(tank.fluid, delta=0.1)\n\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, smoothing_kernel,\n smoothing_length, viscosity=viscosity,\n density_diffusion=density_diffusion,\n acceleration=(0.0, -gravity), correction=nothing,\n surface_tension=nothing)\n\n# ==========================================================================================\n# ==== Boundary\nboundary_density_calculator = AdamiPressureExtrapolation()\nboundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n boundary_density_calculator,\n smoothing_kernel, smoothing_length,\n correction=nothing)\n\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model, adhesion_coefficient=0.0)\n\n# ==========================================================================================\n# ==== Simulation\n# `nothing` will automatically choose the best update strategy. This is only to be able\n# to change this with `trixi_include`.\nsemi = Semidiscretization(fluid_system, boundary_system,\n neighborhood_search=GridNeighborhoodSearch{2}(update_strategy=nothing))\node = semidiscretize(semi, tspan, data_type=nothing)\n\ninfo_callback = InfoCallback(interval=100)\n\nsolution_prefix = \"\"\nsaving_callback = SolutionSavingCallback(dt=0.02, prefix=solution_prefix)\n\n# Save at certain timepoints which allows comparison to the results of Marrone et al.,\n# i.e. (1.5, 2.36, 3.0, 5.7, 6.45).\n# Please note that the images in Marrone et al. are obtained at a particle_spacing = H/320,\n# which takes between 2 and 4 hours.\nsaving_paper = SolutionSavingCallback(save_times=[0.0, 0.371, 0.584, 0.743, 1.411, 1.597],\n prefix=\"marrone_times\")\n\n# This can be overwritten with `trixi_include`\nextra_callback = nothing\n\nuse_reinit = false\ndensity_reinit_cb = use_reinit ?\n DensityReinitializationCallback(semi.systems[1], interval=10) :\n nothing\nstepsize_callback = StepsizeCallback(cfl=0.9)\n\ncallbacks = CallbackSet(info_callback, saving_callback, stepsize_callback, extra_callback,\n density_reinit_cb, saving_paper)\n\nsol = solve(ode, CarpenterKennedy2N54(williamson_condition=false),\n dt=1.0, # This is overwritten by the stepsize callback\n save_everystep=false, callback=callbacks);\n\n","category":"page"},{"location":"general/neighborhood_search/#Neighborhood-Search","page":"Neighborhood Search","title":"Neighborhood Search","text":"","category":"section"},{"location":"general/neighborhood_search/","page":"Neighborhood Search","title":"Neighborhood Search","text":"The neighborhood search is the most essential component for performance. We provide several implementations in the package PointNeighbors.jl. See the docs of this package for an overview and a comparison of different implementations.","category":"page"},{"location":"general/neighborhood_search/","page":"Neighborhood Search","title":"Neighborhood Search","text":"note: Usage\nTo run a simulation with a neighborhood search implementation, pass a template of the neighborhood search to the constructor of the Semidiscretization. A template is just an empty neighborhood search with search radius 0.0. See copy_neighborhood_search and the examples below for more details.semi = Semidiscretization(system1, system2,\n neighborhood_search=PrecomputedNeighborhoodSearch{2}())The keyword argument periodic_box in the neighborhood search constructors can be used to define a periodic domain. See the PointNeighbors.jl docs for more details.periodic_box = PeriodicBox(min_corner=[0.0, -0.25], max_corner=[1.0, 0.75])\nsemi = Semidiscretization(system1, system2,\n neighborhood_search=GridNeighborhoodSearch{2}(; periodic_box))","category":"page"},{"location":"tutorials/tut_falling/","page":"Example file","title":"Example file","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials_template/tut_falling.md\"","category":"page"},{"location":"tutorials/tut_falling/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_falling/","page":"Example file","title":"Example file","text":"using TrixiParticles\nusing OrdinaryDiffEq\n\n# ==========================================================================================\n# ==== Resolution\nfluid_particle_spacing = 0.02\nsolid_particle_spacing = fluid_particle_spacing\n\n# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model\nboundary_layers = 3\nspacing_ratio = 1\n\n# ==========================================================================================\n# ==== Experiment Setup\ngravity = 9.81\ntspan = (0.0, 1.0)\n\n# Boundary geometry and initial fluid particle positions\ninitial_fluid_size = (2.0, 0.9)\ntank_size = (2.0, 1.0)\n\nfluid_density = 1000.0\nsound_speed = 10 * sqrt(gravity * initial_fluid_size[2])\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=1)\n\ntank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,\n n_layers=boundary_layers, spacing_ratio=spacing_ratio,\n faces=(true, true, true, false),\n acceleration=(0.0, -gravity), state_equation=state_equation)\n\nsphere1_radius = 0.3\nsphere2_radius = 0.2\nsphere1_density = 500.0\nsphere2_density = 1100.0\n\n# Young's modulus and Poisson ratio\nsphere1_E = 7e4\nsphere2_E = 1e5\nnu = 0.0\n\nsphere1_center = (0.5, 1.6)\nsphere2_center = (1.5, 1.6)\nsphere1 = SphereShape(solid_particle_spacing, sphere1_radius, sphere1_center,\n sphere1_density, sphere_type=VoxelSphere())\nsphere2 = SphereShape(solid_particle_spacing, sphere2_radius, sphere2_center,\n sphere2_density, sphere_type=VoxelSphere())\n\n# ==========================================================================================\n# ==== Fluid\nfluid_smoothing_length = 3.0 * fluid_particle_spacing\nfluid_smoothing_kernel = WendlandC2Kernel{2}()\n\nfluid_density_calculator = ContinuityDensity()\nviscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)\ndensity_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)\n\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, fluid_smoothing_kernel,\n fluid_smoothing_length, viscosity=viscosity,\n density_diffusion=density_diffusion,\n acceleration=(0.0, -gravity))\n\n# ==========================================================================================\n# ==== Boundary\nboundary_density_calculator = BernoulliPressureExtrapolation()\nboundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n boundary_density_calculator,\n fluid_smoothing_kernel, fluid_smoothing_length)\n\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model)\n\n# ==========================================================================================\n# ==== Solid\nsolid_smoothing_length = 2 * sqrt(2) * solid_particle_spacing\nsolid_smoothing_kernel = WendlandC2Kernel{2}()\n\n# For the FSI we need the hydrodynamic masses and densities in the solid boundary model\nhydrodynamic_densites_1 = fluid_density * ones(size(sphere1.density))\nhydrodynamic_masses_1 = hydrodynamic_densites_1 * solid_particle_spacing^ndims(fluid_system)\n\nsolid_boundary_model_1 = BoundaryModelDummyParticles(hydrodynamic_densites_1,\n hydrodynamic_masses_1,\n state_equation=state_equation,\n boundary_density_calculator,\n fluid_smoothing_kernel,\n fluid_smoothing_length)\n\nhydrodynamic_densites_2 = fluid_density * ones(size(sphere2.density))\nhydrodynamic_masses_2 = hydrodynamic_densites_2 * solid_particle_spacing^ndims(fluid_system)\n\nsolid_boundary_model_2 = BoundaryModelDummyParticles(hydrodynamic_densites_2,\n hydrodynamic_masses_2,\n state_equation=state_equation,\n boundary_density_calculator,\n fluid_smoothing_kernel,\n fluid_smoothing_length)\n\nsolid_system_1 = TotalLagrangianSPHSystem(sphere1,\n solid_smoothing_kernel, solid_smoothing_length,\n sphere1_E, nu,\n acceleration=(0.0, -gravity),\n boundary_model=solid_boundary_model_1,\n penalty_force=PenaltyForceGanzenmueller(alpha=0.3))\n\nsolid_system_2 = TotalLagrangianSPHSystem(sphere2,\n solid_smoothing_kernel, solid_smoothing_length,\n sphere2_E, nu,\n acceleration=(0.0, -gravity),\n boundary_model=solid_boundary_model_2,\n penalty_force=PenaltyForceGanzenmueller(alpha=0.3))\n\n# ==========================================================================================\n# ==== Simulation\nsemi = Semidiscretization(fluid_system, boundary_system, solid_system_1, solid_system_2)\node = semidiscretize(semi, tspan)\n\ninfo_callback = InfoCallback(interval=10)\nsaving_callback = SolutionSavingCallback(dt=0.02, output_directory=\"out\", prefix=\"\",\n write_meta_data=true)\n\ncallbacks = CallbackSet(info_callback, saving_callback)\n\n# Use a Runge-Kutta method with automatic (error based) time step size control.\nsol = solve(ode, RDPK3SpFSAL49(),\n abstol=1e-6, # Default abstol is 1e-6\n reltol=1e-3, # Default reltol is 1e-3\n save_everystep=false, callback=callbacks);\n\n","category":"page"},{"location":"tutorials/tut_beam/","page":"Example file","title":"Example file","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials_template/tut_beam.md\"","category":"page"},{"location":"tutorials/tut_beam/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials/tut_beam/","page":"Example file","title":"Example file","text":"using TrixiParticles\nusing OrdinaryDiffEq\n\n# ==========================================================================================\n# ==== Resolution\nn_particles_y = 5\n\n# ==========================================================================================\n# ==== Experiment Setup\ngravity = 2.0\ntspan = (0.0, 5.0)\n\nelastic_beam = (length=0.35, thickness=0.02)\nmaterial = (density=1000.0, E=1.4e6, nu=0.4)\nclamp_radius = 0.05\n\n# The structure starts at the position of the first particle and ends\n# at the position of the last particle.\nparticle_spacing = elastic_beam.thickness / (n_particles_y - 1)\n\n# Add particle_spacing/2 to the clamp_radius to ensure that particles are also placed on the radius\nfixed_particles = SphereShape(particle_spacing, clamp_radius + particle_spacing / 2,\n (0.0, elastic_beam.thickness / 2), material.density,\n cutout_min=(0.0, 0.0),\n cutout_max=(clamp_radius, elastic_beam.thickness),\n tlsph=true)\n\nn_particles_clamp_x = round(Int, clamp_radius / particle_spacing)\n\n# Beam and clamped particles\nn_particles_per_dimension = (round(Int, elastic_beam.length / particle_spacing) +\n n_particles_clamp_x + 1, n_particles_y)\n\n# Note that the `RectangularShape` puts the first particle half a particle spacing away\n# from the boundary, which is correct for fluids, but not for solids.\n# We therefore need to pass `tlsph=true`.\nbeam = RectangularShape(particle_spacing, n_particles_per_dimension,\n (0.0, 0.0), density=material.density, tlsph=true)\n\nsolid = union(beam, fixed_particles)\n\n# ==========================================================================================\n# ==== Solid\n# The kernel in the reference uses a differently scaled smoothing length,\n# so this is equivalent to the smoothing length of `sqrt(2) * particle_spacing` used in the paper.\nsmoothing_length = 2 * sqrt(2) * particle_spacing\nsmoothing_kernel = WendlandC2Kernel{2}()\n\nsolid_system = TotalLagrangianSPHSystem(solid, smoothing_kernel, smoothing_length,\n material.E, material.nu,\n n_fixed_particles=nparticles(fixed_particles),\n acceleration=(0.0, -gravity),\n penalty_force=nothing)\n\n# ==========================================================================================\n# ==== Simulation\nsemi = Semidiscretization(solid_system,\n neighborhood_search=PrecomputedNeighborhoodSearch{2}())\node = semidiscretize(semi, tspan)\n\ninfo_callback = InfoCallback(interval=100)\n\n# Track the position of the particle in the middle of the tip of the beam.\nmiddle_particle_id = Int(n_particles_per_dimension[1] * (n_particles_per_dimension[2] + 1) /\n 2)\nstartposition_x = beam.coordinates[1, middle_particle_id]\nstartposition_y = beam.coordinates[2, middle_particle_id]\n\nfunction deflection_x(v, u, t, system)\n return system.current_coordinates[1, middle_particle_id] - startposition_x\nend\n\nfunction deflection_y(v, u, t, system)\n return system.current_coordinates[2, middle_particle_id] - startposition_y\nend\n\nsaving_callback = SolutionSavingCallback(dt=0.02, prefix=\"\",\n deflection_x=deflection_x,\n deflection_y=deflection_y)\n\ncallbacks = CallbackSet(info_callback, saving_callback)\n\n# Use a Runge-Kutta method with automatic (error based) time step size control\nsol = solve(ode, RDPK3SpFSAL49(), save_everystep=false, callback=callbacks);\n\n","category":"page"},{"location":"general/density_calculators/#density_calculator","page":"Density Calculators","title":"Density Calculators","text":"","category":"section"},{"location":"general/density_calculators/","page":"Density Calculators","title":"Density Calculators","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"density_calculators.jl\")]","category":"page"},{"location":"general/density_calculators/#TrixiParticles.ContinuityDensity","page":"Density Calculators","title":"TrixiParticles.ContinuityDensity","text":"ContinuityDensity()\n\nDensity calculator to integrate the density from the continuity equation\n\nfracmathrmdrho_amathrmdt = sum_b m_b v_ab cdot nabla_r_a W(Vert r_a - r_b Vert h)\n\nwhere rho_a denotes the density of particle a and r_ab = r_a - r_b is the difference of the coordinates, v_ab = v_a - v_b of the velocities of particles a and b.\n\n\n\n\n\n","category":"type"},{"location":"general/density_calculators/#TrixiParticles.SummationDensity","page":"Density Calculators","title":"TrixiParticles.SummationDensity","text":"SummationDensity()\n\nDensity calculator to use the summation formula\n\nrho(r) = sum_b m_b W(Vert r - r_b Vert h)\n\nfor the density estimation, where r_b denotes the coordinates and m_b the mass of particle b.\n\n\n\n\n\n","category":"type"},{"location":"general/semidiscretization/#Semidiscretization","page":"Semidiscretization","title":"Semidiscretization","text":"","category":"section"},{"location":"general/semidiscretization/","page":"Semidiscretization","title":"Semidiscretization","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"semidiscretization.jl\")]","category":"page"},{"location":"general/semidiscretization/#TrixiParticles.Semidiscretization","page":"Semidiscretization","title":"TrixiParticles.Semidiscretization","text":"Semidiscretization(systems...; neighborhood_search=GridNeighborhoodSearch{NDIMS}())\n\nThe semidiscretization couples the passed systems to one simulation.\n\nArguments\n\nsystems: Systems to be coupled in this semidiscretization\n\nKeywords\n\nneighborhood_search: The neighborhood search to be used in the simulation. By default, the GridNeighborhoodSearch is used. Use nothing to loop over all particles (no neighborhood search). To use other neighborhood search implementations, pass a template of a neighborhood search. See copy_neighborhood_search and the examples below for more details. To use a periodic domain, pass a PeriodicBox to the neighborhood search.\nthreaded_nhs_update=true: Can be used to deactivate thread parallelization in the neighborhood search update. This can be one of the largest sources of variations between simulations with different thread numbers due to particle ordering changes.\n\nExamples\n\nsemi = Semidiscretization(fluid_system, boundary_system)\n\nsemi = Semidiscretization(fluid_system, boundary_system,\n neighborhood_search=GridNeighborhoodSearch{2}(update_strategy=SerialUpdate()))\n\nperiodic_box = PeriodicBox(min_corner = [0.0, 0.0], max_corner = [1.0, 1.0])\nsemi = Semidiscretization(fluid_system, boundary_system,\n neighborhood_search=GridNeighborhoodSearch{2}(; periodic_box))\n\nsemi = Semidiscretization(fluid_system, boundary_system,\n neighborhood_search=PrecomputedNeighborhoodSearch{2}())\n\nsemi = Semidiscretization(fluid_system, boundary_system,\n neighborhood_search=nothing)\n\n\n\n\n\n","category":"type"},{"location":"general/semidiscretization/#TrixiParticles.SourceTermDamping","page":"Semidiscretization","title":"TrixiParticles.SourceTermDamping","text":"SourceTermDamping(; damping_coefficient)\n\nA source term to be used when a damping step is required before running a full simulation. The term -c cdot v_a is added to the acceleration fracmathrmdv_amathrmdt of particle a, where c is the damping coefficient and v_a is the velocity of particle a.\n\nKeywords\n\ndamping_coefficient: The coefficient d above. A higher coefficient means more damping. A coefficient of 1e-4 is a good starting point for damping a fluid at rest.\n\nExamples\n\nsource_terms = SourceTermDamping(; damping_coefficient=1e-4)\n\n\n\n\n\n","category":"type"},{"location":"general/semidiscretization/#TrixiParticles.restart_with!-Tuple{Any, Any}","page":"Semidiscretization","title":"TrixiParticles.restart_with!","text":"restart_with!(semi, sol)\n\nSet the initial coordinates and velocities of all systems in semi to the final values in the solution sol. semidiscretize has to be called again afterwards, or another Semidiscretization can be created with the updated systems.\n\nArguments\n\nsemi: The semidiscretization\nsol: The ODESolution returned by solve of OrdinaryDiffEq\n\n\n\n\n\n","category":"method"},{"location":"general/semidiscretization/#TrixiParticles.semidiscretize-Tuple{Any, Any}","page":"Semidiscretization","title":"TrixiParticles.semidiscretize","text":"semidiscretize(semi, tspan; reset_threads=true)\n\nCreate an ODEProblem from the semidiscretization with the specified tspan.\n\nArguments\n\nsemi: A Semidiscretization holding the systems involved in the simulation.\ntspan: The time span over which the simulation will be run.\n\nKeywords\n\nreset_threads: A boolean flag to reset Polyester.jl threads before the simulation (default: true). After an error within a threaded loop, threading might be disabled. Resetting the threads before the simulation ensures that threading is enabled again for the simulation. See also trixi-framework/Trixi.jl#1583.\n\nReturns\n\nA DynamicalODEProblem (see the OrdinaryDiffEq.jl docs) to be integrated with OrdinaryDiffEq.jl. Note that this is not a true DynamicalODEProblem where the acceleration does not depend on the velocity. Therefore, not all integrators designed for DynamicalODEProblems will work properly. However, all integrators designed for ODEProblems can be used.\n\nExamples\n\nsemi = Semidiscretization(fluid_system, boundary_system)\ntspan = (0.0, 1.0)\node_problem = semidiscretize(semi, tspan)\n\n\n\n\n\n","category":"method"},{"location":"systems/boundary/#Boundary-System","page":"Boundary","title":"Boundary System","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundarySPHSystem","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundarySPHSystem","page":"Boundary","title":"TrixiParticles.BoundarySPHSystem","text":"BoundarySPHSystem(initial_condition, boundary_model; movement=nothing, adhesion_coefficient=0.0)\n\nSystem for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.\n\nArguments\n\ninitial_condition: Initial condition (see InitialCondition)\nboundary_model: Boundary model (see Boundary Models)\n\nKeyword Arguments\n\nmovement: For moving boundaries, a BoundaryMovement can be passed.\nadhesion_coefficient: Coefficient specifying the adhesion of a fluid to the surface. Note: currently it is assumed that all fluids have the same adhesion coefficient.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundaryDEMSystem","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryDEMSystem","page":"Boundary","title":"TrixiParticles.BoundaryDEMSystem","text":"BoundaryDEMSystem(initial_condition, normal_stiffness)\n\nSystem for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in a future releases.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundaryMovement","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryMovement","page":"Boundary","title":"TrixiParticles.BoundaryMovement","text":"BoundaryMovement(movement_function, is_moving; moving_particles=nothing)\n\nArguments\n\nmovement_function: Time-dependent function returning an SVector of d dimensions for a d-dimensional problem.\nis_moving: Function to determine in each timestep if the particles are moving or not. Its boolean return value is mandatory to determine if the neighborhood search will be updated.\n\nKeyword Arguments\n\nmoving_particles: Indices of moving particles. Default is each particle in BoundarySPHSystem.\n\nIn the example below, movement describes particles moving in a circle as long as the time is lower than 1.5.\n\nExamples\n\nmovement_function(t) = SVector(cos(2pi*t), sin(2pi*t))\nis_moving(t) = t < 1.5\n\nmovement = BoundaryMovement(movement_function, is_moving)\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#boundary_models","page":"Boundary","title":"Boundary Models","text":"","category":"section"},{"location":"systems/boundary/#Dummy-Particles","page":"Boundary","title":"Dummy Particles","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Boundaries modeled as dummy particles, which are treated like fluid particles, but their positions and velocities are not evolved in time. Since the force towards the fluid should not change with the material density when used with a TotalLagrangianSPHSystem, the dummy particles need to have a mass corresponding to the fluid's rest density, which we call \"hydrodynamic mass\", as opposed to mass corresponding to the material density of a TotalLagrangianSPHSystem.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Here, initial_density and hydrodynamic_mass are vectors that contains the initial density and the hydrodynamic mass respectively for each boundary particle. Note that when used with SummationDensity (see below), this is only used to determine the element type and the number of boundary particles.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"To establish a relationship between density and pressure, a state_equation has to be passed, which should be the same as for the adjacent fluid systems. To sum over neighboring particles, a smoothing_kernel and smoothing_length needs to be passed. This should be the same as for the adjacent fluid system with the largest smoothing length.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"In the literature, this kind of boundary particles is referred to as \"dummy particles\" (Adami et al., 2012 and Valizadeh & Monaghan, 2015), \"frozen fluid particles\" (Akinci et al., 2012) or \"dynamic boundaries Crespo et al., 2007. The key detail of this boundary condition and the only difference between the boundary models in these references is the way the density and pressure of boundary particles is computed.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Since boundary particles are treated like fluid particles, the force on fluid particle a due to boundary particle b is given by","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"f_ab = m_a m_b left( fracp_arho_a^2 + fracp_brho_b^2 right) nabla_r_a W(Vert r_a - r_b Vert h)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The quantities to be defined here are the density rho_b and pressure p_b of the boundary particle b.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BoundaryModelDummyParticles","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryModelDummyParticles","page":"Boundary","title":"TrixiParticles.BoundaryModelDummyParticles","text":"BoundaryModelDummyParticles(initial_density, hydrodynamic_mass,\n density_calculator, smoothing_kernel,\n smoothing_length; viscosity=nothing,\n state_equation=nothing, correction=nothing)\n\nBoundary model for BoundarySPHSystem.\n\nArguments\n\ninitial_density: Vector holding the initial density of each boundary particle.\nhydrodynamic_mass: Vector holding the \"hydrodynamic mass\" of each boundary particle. See description above for more information.\ndensity_calculator: Strategy to compute the hydrodynamic density of the boundary particles. See description below for more information.\nsmoothing_kernel: Smoothing kernel should be the same as for the adjacent fluid system.\nsmoothing_length: Smoothing length should be the same as for the adjacent fluid system.\n\nKeywords\n\nstate_equation: This should be the same as for the adjacent fluid system (see e.g. StateEquationCole).\ncorrection: Correction method of the adjacent fluid system (see Corrections).\nviscosity: Slip (default) or no-slip condition. See description below for further information.\n\nExamples\n\n# Free-slip condition\nboundary_model = BoundaryModelDummyParticles(densities, masses, AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length)\n\n# No-slip condition\nboundary_model = BoundaryModelDummyParticles(densities, masses, AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length,\n viscosity=ViscosityAdami(nu=1e-6))\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#Hydrodynamic-density-of-dummy-particles","page":"Boundary","title":"Hydrodynamic density of dummy particles","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"We provide six options to compute the boundary density and pressure, determined by the density_calculator:","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"(Recommended) With AdamiPressureExtrapolation, the pressure is extrapolated from the pressure of the fluid according to Adami et al., 2012, and the density is obtained by applying the inverse of the state equation. This option usually yields the best results of the options listed here.\n(Only relevant for FSI) With BernoulliPressureExtrapolation, the pressure is extrapolated from the pressure similar to the AdamiPressureExtrapolation, but a relative velocity-dependent pressure part is calculated between moving solids and fluids, which increases the boundary pressure in areas prone to penetrations.\nWith SummationDensity, the density is calculated by summation over the neighboring particles, and the pressure is computed from the density with the state equation.\nWith ContinuityDensity, the density is integrated from the continuity equation, and the pressure is computed from the density with the state equation. Note that this causes a gap between fluid and boundary where the boundary is initialized without any contact to the fluid. This is due to overestimation of the boundary density as soon as the fluid comes in contact with boundary particles that initially did not have contact to the fluid. Therefore, in dam break simulations, there is a visible \"step\", even though the boundary is supposed to be flat. See also dual.sphysics.org/faq/#Q_13.\nWith PressureZeroing, the density is set to the reference density and the pressure is computed from the density with the state equation. This option is not recommended. The other options yield significantly better results.\nWith PressureMirroring, the density is set to the reference density. The pressure is not used. Instead, the fluid pressure is mirrored as boundary pressure in the momentum equation. This option is not recommended due to stability issues. See PressureMirroring for more details.","category":"page"},{"location":"systems/boundary/#1.-[AdamiPressureExtrapolation](@ref)","page":"Boundary","title":"1. AdamiPressureExtrapolation","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The pressure of the boundary particles is obtained by extrapolating the pressure of the fluid according to Adami et al., 2012. The pressure of a boundary particle b is given by","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"p_b = fracsum_f (p_f + rho_f (bmg - bma_b) cdot bmr_bf) W(Vert r_bf Vert h)sum_f W(Vert r_bf Vert h)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the sum is over all fluid particles, rho_f and p_f denote the density and pressure of fluid particle f, respectively, r_bf = r_b - r_f denotes the difference of the coordinates of particles b and f, bmg denotes the gravitational acceleration acting on the fluid, and bma_b denotes the acceleration of the boundary particle b.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" AdamiPressureExtrapolation","category":"page"},{"location":"systems/boundary/#TrixiParticles.AdamiPressureExtrapolation","page":"Boundary","title":"TrixiParticles.AdamiPressureExtrapolation","text":"AdamiPressureExtrapolation(; pressure_offset=0.0)\n\ndensity_calculator for BoundaryModelDummyParticles.\n\nKeywords\n\npressure_offset=0.0: Sometimes it is necessary to artificially increase the boundary pressure to prevent penetration, which is possible by increasing this value.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#2.-[BernoulliPressureExtrapolation](@ref)","page":"Boundary","title":"2. BernoulliPressureExtrapolation","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Identical to the pressure p_b calculated via AdamiPressureExtrapolation, but it adds the dynamic pressure component of the Bernoulli equation:","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"p_b = fracsum_f (p_f + frac12 rho_textneighbor left( frac (mathbfv_f - mathbfv_textbody) cdot (mathbfx_f - mathbfx_textneighbor) left mathbfx_f - mathbfx_textneighbor right right)^2 times textfactor +rho_f (bmg - bma_b) cdot bmr_bf) W(Vert r_bf Vert h)sum_f W(Vert r_bf Vert h) ","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where mathbfv_f is the velocity of the fluid and mathbfv_textbody is the velocity of the body. This adjustment provides a higher boundary pressure for solid bodies moving with a relative velocity to the fluid to prevent penetration. This modification is original and not derived from any literature source.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" BernoulliPressureExtrapolation","category":"page"},{"location":"systems/boundary/#TrixiParticles.BernoulliPressureExtrapolation","page":"Boundary","title":"TrixiParticles.BernoulliPressureExtrapolation","text":"BernoulliPressureExtrapolation(; pressure_offset=0.0, factor=1.0)\n\ndensity_calculator for BoundaryModelDummyParticles.\n\nKeywords\n\npressure_offset=0.0: Sometimes it is necessary to artificially increase the boundary pressure to prevent penetration, which is possible by increasing this value.\nfactor=1.0 : Setting factor allows to just increase the strength of the dynamic pressure part.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#5.-[PressureZeroing](@ref)","page":"Boundary","title":"5. PressureZeroing","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"This is the simplest way to implement dummy boundary particles. The density of each particle is set to the reference density and the pressure to the reference pressure (the corresponding pressure to the reference density by the state equation).","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" PressureZeroing","category":"page"},{"location":"systems/boundary/#TrixiParticles.PressureZeroing","page":"Boundary","title":"TrixiParticles.PressureZeroing","text":"PressureZeroing()\n\ndensity_calculator for BoundaryModelDummyParticles.\n\nnote: Note\nThis boundary model produces significantly worse results than all other models and is only included for research purposes.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#6.-[PressureMirroring](@ref)","page":"Boundary","title":"6. PressureMirroring","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Instead of calculating density and pressure for each boundary particle, we modify the momentum equation,","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"fracmathrmdv_amathrmdt = -sum_b m_b left( fracp_arho_a^2 + fracp_brho_b^2 right) nabla_a W_ab","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"to replace the unknown density rho_b if b is a boundary particle by the reference density and the unknown pressure p_b if b is a boundary particle by the pressure p_a of the interacting fluid particle. The momentum equation therefore becomes","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"fracmathrmdv_amathrmdt = -sum_f m_f left( fracp_arho_a^2 + fracp_frho_f^2 right) nabla_a W_af\n-sum_b m_b left( fracp_arho_a^2 + fracp_arho_0^2 right) nabla_a W_ab","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the first sum is over all fluid particles and the second over all boundary particles.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"This approach was first mentioned by Akinci et al. (2012) and written down in this form by Band et al. (2018).","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" PressureMirroring","category":"page"},{"location":"systems/boundary/#TrixiParticles.PressureMirroring","page":"Boundary","title":"TrixiParticles.PressureMirroring","text":"PressureMirroring()\n\ndensity_calculator for BoundaryModelDummyParticles.\n\nnote: Note\nThis boundary model requires high viscosity for stability with WCSPH. It also produces significantly worse results than AdamiPressureExtrapolation and is not more efficient because smaller time steps are required due to more noise in the pressure. We added this model only for research purposes and for comparison with SPlisHSPlasH.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#No-slip-conditions","page":"Boundary","title":"No-slip conditions","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"For the interaction of dummy particles and fluid particles, Adami et al. (2012) impose a no-slip boundary condition by assigning a wall velocity v_w to the dummy particle.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The wall velocity of particle a is calculated from the prescribed boundary particle velocity v_a and the smoothed velocity field","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"v_w = 2 v_a - fracsum_b v_b W_absum_b W_ab","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the sum is over all fluid particles.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"By choosing the viscosity model ViscosityAdami for viscosity, a no-slip condition is imposed. It is recommended to choose nu in the order of either the kinematic viscosity parameter of the adjacent fluid or the equivalent from the artificial parameter alpha of the adjacent fluid (nu = fracalpha h c 2d + 4). When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"warning: Warning\nThe viscosity model ArtificialViscosityMonaghan for BoundaryModelDummyParticles has not been verified yet.","category":"page"},{"location":"systems/boundary/#Repulsive-Particles","page":"Boundary","title":"Repulsive Particles","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Boundaries modeled as boundary particles which exert forces on the fluid particles (Monaghan, Kajtar, 2009). The force on fluid particle a due to boundary particle b is given by","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"f_ab = m_a left(tildef_ab - m_b Pi_ab nabla_r_a W(Vert r_a - r_b Vert h)right)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"with","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"tildef_ab = fracKbeta^n-1 fracr_abVert r_ab Vert (Vert r_ab Vert - d) Phi(Vert r_ab Vert h)\nfrac2 m_bm_a + m_b","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where m_a and m_b are the masses of fluid particle a and boundary particle b respectively, r_ab = r_a - r_b is the difference of the coordinates of particles a and b, d denotes the boundary particle spacing and n denotes the number of dimensions (see Monaghan & Kajtar, 2009, Equation (3.1) and Valizadeh & Monaghan, 2015). Note that the repulsive acceleration tildef_ab does not depend on the masses of the boundary particles. Here, Phi denotes the 1D Wendland C4 kernel, normalized to 177 for q=0 (Monaghan & Kajtar, 2009, Section 4), with Phi(r h) = w(rh) and","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"w(q) =\nbegincases\n (17732) (1 + (52)q + 2q^2)(2 - q)^5 textif 0 leq q 2 \n 0 textif q geq 2\nendcases","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The boundary particles are assumed to have uniform spacing by the factor beta smaller than the expected fluid particle spacing. For example, if the fluid particles have an expected spacing of 03 and the boundary particles have a uniform spacing of 01, then this parameter should be set to beta = 3. According to Monaghan & Kajtar (2009), a value of beta = 3 for the Wendland C4 that we use here is reasonable for most computing purposes.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The parameter K is used to scale the force exerted by the boundary particles. In Monaghan & Kajtar (2009), a value of gD is used for static tank simulations, where g is the gravitational acceleration and D is the depth of the fluid.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The viscosity Pi_ab is calculated according to the viscosity used in the simulation, where the density of the boundary particle if needed is assumed to be identical to the density of the fluid particle.","category":"page"},{"location":"systems/boundary/#No-slip-condition","page":"Boundary","title":"No-slip condition","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"By choosing the viscosity model ArtificialViscosityMonaghan for viscosity, a no-slip condition is imposed. When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"warning: Warning\nThe no-slip conditions for BoundaryModelMonaghanKajtar have not been verified yet.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"boundary\", \"monaghan_kajtar\", \"monaghan_kajtar.jl\")]","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryModelMonaghanKajtar","page":"Boundary","title":"TrixiParticles.BoundaryModelMonaghanKajtar","text":"BoundaryModelMonaghanKajtar(K, beta, boundary_particle_spacing, mass;\n viscosity=nothing)\n\nBoundary model for BoundarySPHSystem.\n\nArguments\n\nK: Scaling factor for repulsive force.\nbeta: Ratio of fluid particle spacing to boundary particle spacing.\nboundary_particle_spacing: Boundary particle spacing.\nmass: Vector holding the mass of each boundary particle.\n\nKeywords\n\nviscosity: Free-slip (default) or no-slip condition. See description above for further information.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#open_boundary","page":"Boundary","title":"Open Boundaries","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"boundary\", \"open_boundary\", \"system.jl\")]","category":"page"},{"location":"systems/boundary/#TrixiParticles.OpenBoundarySPHSystem","page":"Boundary","title":"TrixiParticles.OpenBoundarySPHSystem","text":"OpenBoundarySPHSystem(boundary_zone::Union{InFlow, OutFlow};\n fluid_system::FluidSystem, buffer_size::Integer,\n boundary_model,\n reference_velocity=nothing,\n reference_pressure=nothing,\n reference_density=nothing)\n\nOpen boundary system for in- and outflow particles.\n\nArguments\n\nboundary_zone: Use InFlow for an inflow and OutFlow for an outflow boundary.\n\nKeywords\n\nfluid_system: The corresponding fluid system\nboundary_model: Boundary model (see Open Boundary Models)\nbuffer_size: Number of buffer particles.\nreference_velocity: Reference velocity is either a function mapping each particle's coordinates and time to its velocity, an array where the i-th column holds the velocity of particle i or, for a constant fluid velocity, a vector holding this velocity.\nreference_pressure: Reference pressure is either a function mapping each particle's coordinates and time to its pressure, a vector holding the pressure of each particle, or a scalar for a constant pressure over all particles.\nreference_density: Reference density is either a function mapping each particle's coordinates and time to its density, a vector holding the density of each particle, or a scalar for a constant density over all particles.\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"boundary\", \"open_boundary\", \"boundary_zones.jl\")]","category":"page"},{"location":"systems/boundary/#TrixiParticles.InFlow","page":"Boundary","title":"TrixiParticles.InFlow","text":"InFlow(; plane, flow_direction, density, particle_spacing,\n initial_condition=nothing, extrude_geometry=nothing,\n open_boundary_layers::Integer)\n\nInflow boundary zone for OpenBoundarySPHSystem.\n\nThe specified plane (line in 2D or rectangle in 3D) will be extruded in upstream direction (the direction opposite to flow_direction) to create a box for the boundary zone. There are three ways to specify the actual shape of the inflow:\n\nDon't pass initial_condition or extrude_geometry. The boundary zone box will then be filled with inflow particles (default).\nSpecify extrude_geometry by passing a 1D shape in 2D or a 2D shape in 3D, which is then extruded in upstream direction to create the inflow particles.\nIn 2D, the shape must be either an initial condition with 2D coordinates, which lies on the line specified by plane, or an initial condition with 1D coordinates, which lies on the line specified by plane when a y-coordinate of 0 is added.\nIn 3D, the shape must be either an initial condition with 3D coordinates, which lies in the rectangle specified by plane, or an initial condition with 2D coordinates, which lies in the rectangle specified by plane when a z-coordinate of 0 is added.\nSpecify initial_condition by passing a 2D initial condition in 2D or a 3D initial condition in 3D, which will be used for the inflow particles.\n\nnote: Note\nParticles outside the boundary zone box will be removed.\n\nKeywords\n\nplane: Tuple of points defining a part of the surface of the domain. The points must either span a line in 2D or a rectangle in 3D. This line or rectangle is then extruded in upstream direction to obtain the boundary zone. In 2D, pass two points (A B), so that the interval A B is the inflow surface. In 3D, pass three points (A B C), so that the rectangular inflow surface is spanned by the vectors widehatAB and widehatAC. These two vectors must be orthogonal.\nflow_direction: Vector defining the flow direction.\nopen_boundary_layers: Number of particle layers in upstream direction.\nparticle_spacing: The spacing between the particles (see InitialCondition).\ndensity: Particle density (see InitialCondition).\ninitial_condition=nothing: InitialCondition for the inflow particles. Particles outside the boundary zone will be removed. Do not use together with extrude_geometry.\nextrude_geometry=nothing: 1D shape in 2D or 2D shape in 3D, which lies on the plane and is extruded upstream to obtain the inflow particles. See point 2 above for more details.\n\nExamples\n\n# 2D\nplane_points = ([0.0, 0.0], [0.0, 1.0])\nflow_direction=[1.0, 0.0]\n\ninflow = InFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,\n open_boundary_layers=4)\n\n# 3D\nplane_points = ([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0])\nflow_direction=[0.0, 0.0, 1.0]\n\ninflow = InFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,\n open_boundary_layers=4)\n\n# 3D particles sampled as cylinder\ncircle = SphereShape(0.1, 0.5, (0.5, 0.5), 1.0, sphere_type=RoundSphere())\n\ninflow = InFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,\n extrude_geometry=circle, open_boundary_layers=4)\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#TrixiParticles.OutFlow","page":"Boundary","title":"TrixiParticles.OutFlow","text":"OutFlow(; plane, flow_direction, density, particle_spacing,\n initial_condition=nothing, extrude_geometry=nothing,\n open_boundary_layers::Integer)\n\nOutflow boundary zone for OpenBoundarySPHSystem.\n\nThe specified plane (line in 2D or rectangle in 3D) will be extruded in downstream direction (the direction in flow_direction) to create a box for the boundary zone. There are three ways to specify the actual shape of the outflow:\n\nDon't pass initial_condition or extrude_geometry. The boundary zone box will then be filled with outflow particles (default).\nSpecify extrude_geometry by passing a 1D shape in 2D or a 2D shape in 3D, which is then extruded in downstream direction to create the outflow particles.\nIn 2D, the shape must be either an initial condition with 2D coordinates, which lies on the line specified by plane, or an initial condition with 1D coordinates, which lies on the line specified by plane when a y-coordinate of 0 is added.\nIn 3D, the shape must be either an initial condition with 3D coordinates, which lies in the rectangle specified by plane, or an initial condition with 2D coordinates, which lies in the rectangle specified by plane when a z-coordinate of 0 is added.\nSpecify initial_condition by passing a 2D initial condition in 2D or a 3D initial condition in 3D, which will be used for the outflow particles.\n\nnote: Note\nParticles outside the boundary zone box will be removed.\n\nKeywords\n\nplane: Tuple of points defining a part of the surface of the domain. The points must either span a line in 2D or a rectangle in 3D. This line or rectangle is then extruded in downstream direction to obtain the boundary zone. In 2D, pass two points (A B), so that the interval A B is the outflow surface. In 3D, pass three points (A B C), so that the rectangular outflow surface is spanned by the vectors widehatAB and widehatAC. These two vectors must be orthogonal.\nflow_direction: Vector defining the flow direction.\nopen_boundary_layers: Number of particle layers in downstream direction.\nparticle_spacing: The spacing between the particles (see InitialCondition).\ndensity: Particle density (see InitialCondition).\ninitial_condition=nothing: InitialCondition for the outflow particles. Particles outside the boundary zone will be removed. Do not use together with extrude_geometry.\nextrude_geometry=nothing: 1D shape in 2D or 2D shape in 3D, which lies on the plane and is extruded downstream to obtain the outflow particles. See point 2 above for more details.\n\nExamples\n\n# 2D\nplane_points = ([0.0, 0.0], [0.0, 1.0])\nflow_direction = [1.0, 0.0]\n\noutflow = OutFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,\n open_boundary_layers=4)\n\n# 3D\nplane_points = ([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0])\nflow_direction = [0.0, 0.0, 1.0]\n\noutflow = OutFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,\n open_boundary_layers=4)\n\n# 3D particles sampled as cylinder\ncircle = SphereShape(0.1, 0.5, (0.5, 0.5), 1.0, sphere_type=RoundSphere())\n\noutflow = OutFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,\n extrude_geometry=circle, open_boundary_layers=4)\n\nwarning: Experimental Implementation\nThis is an experimental feature and may change in any future releases.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/#open_boundary_models","page":"Boundary","title":"Open Boundary Models","text":"","category":"section"},{"location":"systems/boundary/#method_of_characteristics","page":"Boundary","title":"Method of characteristics","text":"","category":"section"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"boundary\", \"open_boundary\", \"method_of_characteristics.jl\")]","category":"page"},{"location":"systems/boundary/#TrixiParticles.BoundaryModelLastiwka","page":"Boundary","title":"TrixiParticles.BoundaryModelLastiwka","text":"BoundaryModelLastiwka()\n\nBoundary model for OpenBoundarySPHSystem. This model uses the characteristic variables to propagate the appropriate values to the outlet or inlet and have been proposed by Lastiwka et al. (2009). For more information about the method see description below.\n\n\n\n\n\n","category":"type"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"The difficulty in non-reflecting boundary conditions, also called open boundaries, is to determine the appropriate boundary values of the exact characteristics of the Euler equations. Assuming the flow near the boundaries is normal to the boundary and free of shock waves and significant viscous effects, it can be shown that three characteristic variables exist:","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"J_1, associated with convection of entropy and propagates at flow velocity,\nJ_2, downstream-running characteristics,\nJ_3, upstream-running characteristics.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Giles (1990) derived those variables based on a linearized set of governing equations:","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"J_1 = -c_s^2 (rho - rho_textref) + (p - p_textref)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"J_2 = rho c_s (v - v_textref) + (p - p_textref)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"J_3 = - rho c_s (v - v_textref) + (p - p_textref)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the subscript \"ref\" denotes the reference flow near the boundaries, which can be prescribed.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Specifying the reference variables is not equivalent to prescription of rho, v and p directly, since the perturbation from the reference flow is allowed.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Lastiwka et al. (2009) applied the method of characteristic to SPH and determined the number of variables that should be prescribed at the boundary and the number which should be propagated from the fluid domain to the boundary:","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"For an inflow boundary:\nPrescribe downstream-running characteristics J_1 and J_2\nTransmit J_3 from the fluid domain (allow J_3 to propagate upstream to the boundary).\nFor an outflow boundary:\nPrescribe upstream-running characteristic J_3\nTransmit J_1 and J_2 from the fluid domain.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"Prescribing is done by simply setting the characteristics to zero. To transmit the characteristics from the fluid domain, or in other words, to carry the information of the fluid to the boundaries, Negi et al. (2020) use a Shepard Interpolation","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"f_i = fracsum_j^N f_j W_ijsum_j^N W_ij","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"where the i-th particle is a boundary particle, f is either J_1, J_2 or J_3 and N is the set of neighboring fluid particles.","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"To express pressure p, density rho and velocity v as functions of the characteristic variables, the system of equations from the characteristic variables is inverted and gives","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":" rho - rho_textref = frac1c_s^2 left( -J_1 + frac12 J_2 + frac12 J_3 right)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"u - u_textref= frac12rho c_s left( J_2 - J_3 right)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"p - p_textref = frac12 left( J_2 + J_3 right)","category":"page"},{"location":"systems/boundary/","page":"Boundary","title":"Boundary","text":"With J_1, J_2 and J_3 determined, we can easily solve for the actual variables for each particle.","category":"page"},{"location":"tutorials_template/tut_setup/#Setting-up-your-simulation-from-scratch","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For different setups and physics, have a look at our other example files.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"using TrixiParticles\nusing OrdinaryDiffEq","category":"page"},{"location":"tutorials_template/tut_setup/#Resolution","page":"Setting up your simulation from scratch","title":"Resolution","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Now, we define the particle spacing, which is our numerical resolution. We usually call the variable fluid_particle_spacing, so that we can easily change the resolution of an example file by overwriting this variable with trixi_include. In 2D, the number of particles will grow quadratically, in 3D cubically with the spacing.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary. In particular, we require boundary_layers >= compact_support. The value for the compact support for each kernel can be found in the smoothing kernel overview.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_particle_spacing = 0.05\nboundary_layers = 3\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/#Experiment-setup","page":"Setting up your simulation from scratch","title":"Experiment setup","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. (Image: Experiment Setup) First, we define the physical parameters gravitational acceleration, simulation time, initial fluid size, tank size and fluid density.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"gravity = 9.81\ntspan = (0.0, 1.0)\ninitial_fluid_size = (1.0, 0.9)\ntank_size = (1.0, 1.0)\nfluid_density = 1000.0\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sound_speed = 10.0\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=7)\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The speed of sound here is numerical and not physical. We artificially lower the speed of sound, since the physical speed of sound in water would lead to prohibitively small time steps. The speed of sound in Weakly Compressible SPH should be chosen as small as possible for numerical efficiency, but large enough to limit density fluctuations to about 1%.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"TrixiParticles.jl requires the initial particle positions and quantities in form of an InitialCondition. Instead of manually defining particle positions, you can work with our pre-defined setups. Among others, we provide setups for rectangular shapes, circles, and spheres. Initial conditions can also be combined with common set operations. See this page for a list of pre-defined setups and details on set operations on initial conditions.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Here, we use the RectangularTank setup, which generates a rectangular fluid inside a rectangular tank, and supports a hydrostatic pressure gradient by passing a gravitational acceleration and a state equation (see above).","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size,\n fluid_density, n_layers=boundary_layers,\n acceleration=(0.0, -gravity), state_equation=state_equation)\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/#Fluid-system","page":"Setting up your simulation from scratch","title":"Fluid system","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_length = 1.2 * fluid_particle_spacing\nsmoothing_kernel = SchoenbergCubicSplineKernel{2}()\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"You can find an overview over smoothing kernels and corresponding smoothing lengths here.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For stability, we need numerical dissipation in form of an artificial viscosity term. Other viscosity models offer a physical approach based on the kinematic viscosity of the fluid.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We choose the parameters as small as possible to avoid non-physical behavior, but as large as possible to stabilize the simulation.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The WCSPH method can either compute the particle density directly with a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating its change over time. We choose the latter approach here by using the density calculator ContinuityDensity, which is more efficient and handles free surfaces without the need for additional correction terms.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_density_calculator = ContinuityDensity()\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, smoothing_kernel,\n smoothing_length, viscosity=viscosity,\n acceleration=(0.0, -gravity))\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/#Boundary-system","page":"Setting up your simulation from scratch","title":"Boundary system","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To model the boundary, we use particle-based boundary conditions, in which particles are sampled in the boundary that interact with the fluid particles to avoid penetration. In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation. See here for a comprehensive overview over boundary models.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length)\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model)\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/#Semidiscretization","page":"Setting up your simulation from scratch","title":"Semidiscretization","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The key component of every simulation is the Semidiscretization, which couples all systems of the simulation. All methods in TrixiParticles.jl are semidiscretizations, which discretize the equations in time to provide an ordinary differential equation that still has to be solved in time. By providing a simulation time span, we can call semidiscretize, which returns an ODEProblem that can be solved with a time integration method.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"semi = Semidiscretization(fluid_system, boundary_system)\node = semidiscretize(semi, tspan)\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/#Time-integration","page":"Setting up your simulation from scratch","title":"Time integration","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"info_callback = InfoCallback(interval=50)\nsaving_callback = SolutionSavingCallback(dt=0.02)\n\ncallbacks = CallbackSet(info_callback, saving_callback)\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"See Visualization for how to visualize the solution. For the simplest visualization, we can use Plots.jl:","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"using Plots\nplot(sol)\nsavefig(\"tut_setup_plot.png\"); nothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"(Image: plot)","category":"page"},{"location":"tutorials_template/tut_setup/#Replacing-components-with-custom-implementations","page":"Setting up your simulation from scratch","title":"Replacing components with custom implementations","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.","category":"page"},{"location":"tutorials_template/tut_setup/#Custom-smoothing-kernel","page":"Setting up your simulation from scratch","title":"Custom smoothing kernel","text":"","category":"section"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)\n q = r / h\n\n if q < 2\n return 1 / (pi * h^2) * exp(-q^2)\n end\n\n return 0.0\nend\n\nfunction TrixiParticles.kernel_deriv(kernel::MyGaussianKernel, r, h)\n q = r * h\n\n if q < 2\n return 1 / (pi * h^2) * (-2 * q) * exp(-q^2) / h\n end\n\n return 0.0\nend\n\nTrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_kernel = MyGaussianKernel()\nnothing # hide","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"and run the simulation file again.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.","category":"page"},{"location":"tutorials_template/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"),\n smoothing_kernel=MyGaussianKernel());\nnothing # hide","category":"page"},{"location":"news/","page":"News","title":"News","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/NEWS.md\"","category":"page"},{"location":"news/#Changelog","page":"News","title":"Changelog","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"TrixiParticles.jl follows the interpretation of semantic versioning (semver) used in the Julia ecosystem. Notable changes will be documented in this file for human readability.","category":"page"},{"location":"news/#Version-0.2.3","page":"News","title":"Version 0.2.3","text":"","category":"section"},{"location":"news/#Highlights","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"Transport Velocity Formulation (TVF) based on the work of Ramachandran et al. \"Entropically damped artificial compressibility for SPH\" (2019) was added.","category":"page"},{"location":"news/#Version-0.2.2","page":"News","title":"Version 0.2.2","text":"","category":"section"},{"location":"news/#Highlights-2","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"Hotfix for threaded sampling of complex geometries.","category":"page"},{"location":"news/#Version-0.2.1","page":"News","title":"Version 0.2.1","text":"","category":"section"},{"location":"news/#Highlights-3","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"Particle sampling of complex geometries from .stl and .asc files.","category":"page"},{"location":"news/#Version-0.2.0","page":"News","title":"Version 0.2.0","text":"","category":"section"},{"location":"news/#Removed","page":"News","title":"Removed","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"Use of the internal neighborhood search has been removed and replaced with PointNeighbors.jl.","category":"page"},{"location":"news/#Development-Cycle-0.1","page":"News","title":"Development Cycle 0.1","text":"","category":"section"},{"location":"news/#Highlights-4","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/#Discrete-Element-Method","page":"News","title":"Discrete Element Method","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"A basic implementation of the discrete element method was added.","category":"page"},{"location":"news/#Surface-Tension-and-Adhesion-Model","page":"News","title":"Surface Tension and Adhesion Model","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"A surface tension and adhesion model based on the work by Akinci et al., \"Versatile Surface Tension and Adhesion for SPH Fluids\" (2013) was added to WCSPH.","category":"page"},{"location":"news/#Support-for-Open-Boundaries","page":"News","title":"Support for Open Boundaries","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"Open boundaries using the method of characteristics based on the work of Lastiwka et al., \"Permeable and non-reflecting boundary conditions in SPH\" (2009) were added for WCSPH and EDAC.","category":"page"},{"location":"news/#Pre-Initial-Release-(v0.1.0)","page":"News","title":"Pre Initial Release (v0.1.0)","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"This section summarizes the initial features that TrixiParticles.jl was released with.","category":"page"},{"location":"news/#Highlights-5","page":"News","title":"Highlights","text":"","category":"section"},{"location":"news/#EDAC","page":"News","title":"EDAC","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"An implementation of EDAC (Entropically Damped Artificial Compressibility) was added, which allows for more stable simulations compared to basic WCSPH and reduces spurious pressure oscillations.","category":"page"},{"location":"news/#WCSPH","page":"News","title":"WCSPH","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"An implementation of WCSPH (Weakly Compressible Smoothed Particle Hydrodynamics), which is the classical SPH approach.","category":"page"},{"location":"news/","page":"News","title":"News","text":"Features:","category":"page"},{"location":"news/","page":"News","title":"News","text":"Correction schemes (Shepard (0. Order) ... MixedKernelGradient (1. Order))\nDensity reinitialization\nKernel summation and Continuity equation density formulations\nFlexible boundary conditions e.g. dummy particles with Adami pressure extrapolation, pressure zeroing, pressure mirroring...\nMoving boundaries\nDensity diffusion based on the models by Molteni & Colagrossi (2009), Ferrari et al. (2009) and Antuono et al. (2010).","category":"page"},{"location":"news/#TLSPH","page":"News","title":"TLSPH","text":"","category":"section"},{"location":"news/","page":"News","title":"News","text":"An implementation of TLSPH (Total Lagrangian Smoothed Particle Hydrodynamics) for solid bodies enabling FSI (Fluid Structure Interactions).","category":"page"},{"location":"general/util/#Util","page":"Util","title":"Util","text":"","category":"section"},{"location":"general/util/","page":"Util","title":"Util","text":"Modules = [TrixiParticles]\nPages = [\"util.jl\"]","category":"page"},{"location":"general/util/#TrixiParticles.examples_dir-Tuple{}","page":"Util","title":"TrixiParticles.examples_dir","text":"examples_dir()\n\nReturn the directory where the example files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.\n\nCopied from Trixi.jl.\n\nExamples\n\nreaddir(examples_dir())\n\n\n\n\n\n","category":"method"},{"location":"general/util/#TrixiParticles.validation_dir-Tuple{}","page":"Util","title":"TrixiParticles.validation_dir","text":"validation_dir()\n\nReturn the directory where the validation files provided with TrixiParticles.jl are located. If TrixiParticles is installed as a regular package (with ]add TrixiParticles), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.\n\nCopied from Trixi.jl.\n\nExamples\n\nreaddir(validation_dir())\n\n\n\n\n\n","category":"method"},{"location":"general/util/#TrixiParticles.@autoinfiltrate","page":"Util","title":"TrixiParticles.@autoinfiltrate","text":"@autoinfiltrate\n@autoinfiltrate condition::Bool\n\nInvoke the @infiltrate macro of the package Infiltrator.jl to create a breakpoint for ad-hoc interactive debugging in the REPL. If the optional argument condition is given, the breakpoint is only enabled if condition evaluates to true.\n\nAs opposed to using Infiltrator.@infiltrate directly, this macro does not require Infiltrator.jl to be added as a dependency to TrixiParticles.jl. As a bonus, the macro will also attempt to load the Infiltrator module if it has not yet been loaded manually.\n\nNote: For this macro to work, the Infiltrator.jl package needs to be installed in your current Julia environment stack.\n\nSee also: Infiltrator.jl\n\nwarning: Internal use only\nPlease note that this macro is intended for internal use only. It is not part of the public API of TrixiParticles.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.\n\n\n\n\n\n","category":"macro"},{"location":"tutorials_template/tut_falling/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials_template/tut_falling/","page":"Example file","title":"Example file","text":"!!include:examples/fsi/falling_spheres_2d.jl!!\n","category":"page"},{"location":"code_of_conduct/","page":"Code of Conduct","title":"Code of Conduct","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/CODE_OF_CONDUCT.md\"","category":"page"},{"location":"code_of_conduct/#Code-of-Conduct","page":"Code of Conduct","title":"Code of Conduct","text":"","category":"section"},{"location":"code_of_conduct/","page":"Code of Conduct","title":"Code of Conduct","text":"Contributor Covenant Code of ConductOur PledgeWe as members, contributors, and leaders pledge to make participation in our community a harassment-free experience for everyone, regardless of age, body size, visible or invisible disability, ethnicity, sex characteristics, gender identity and expression, level of experience, education, socio-economic status, nationality, personal appearance, race, religion, or sexual identity and orientation.We pledge to act and interact in ways that contribute to an open, welcoming, diverse, inclusive, and healthy community.Our StandardsExamples of behavior that contributes to a positive environment for our community include:Demonstrating empathy and kindness toward other people\nBeing respectful of differing opinions, viewpoints, and experiences\nGiving and gracefully accepting constructive feedback\nAccepting responsibility and apologizing to those affected by our mistakes, and learning from the experience\nFocusing on what is best not just for us as individuals, but for the overall communityExamples of unacceptable behavior include:The use of sexualized language or imagery, and sexual attention or advances of any kind\nTrolling, insulting or derogatory comments, and personal or political attacks\nPublic or private harassment\nPublishing others' private information, such as a physical or email address, without their explicit permission\nOther conduct which could reasonably be considered inappropriate in a professional settingEnforcement ResponsibilitiesCommunity leaders are responsible for clarifying and enforcing our standards of acceptable behavior and will take appropriate and fair corrective action in response to any behavior that they deem inappropriate, threatening, offensive, or harmful.Community leaders have the right and responsibility to remove, edit, or reject comments, commits, code, wiki edits, issues, and other contributions that are not aligned to this Code of Conduct, and will communicate reasons for moderation decisions when appropriate.ScopeThis Code of Conduct applies within all community spaces, and also applies when an individual is officially representing the community in public spaces. Examples of representing our community include using an official e-mail address, posting via an official social media account, or acting as an appointed representative at an online or offline event.EnforcementInstances of abusive, harassing, or otherwise unacceptable behavior may be reported to Michael Schlottke-Lakemper, Sven Berger, or any other of the principal developers responsible for enforcement listed in Authors. All complaints will be reviewed and investigated promptly and fairly.All community leaders are obligated to respect the privacy and security of the reporter of any incident.Enforcement GuidelinesCommunity leaders will follow these Community Impact Guidelines in determining the consequences for any action they deem in violation of this Code of Conduct:1. CorrectionCommunity Impact: Use of inappropriate language or other behavior deemed unprofessional or unwelcome in the community.Consequence: A private, written warning from community leaders, providing clarity around the nature of the violation and an explanation of why the behavior was inappropriate. A public apology may be requested.2. WarningCommunity Impact: A violation through a single incident or series of actions.Consequence: A warning with consequences for continued behavior. No interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, for a specified period of time. This includes avoiding interactions in community spaces as well as external channels like social media. Violating these terms may lead to a temporary or permanent ban.3. Temporary BanCommunity Impact: A serious violation of community standards, including sustained inappropriate behavior.Consequence: A temporary ban from any sort of interaction or public communication with the community for a specified period of time. No public or private interaction with the people involved, including unsolicited interaction with those enforcing the Code of Conduct, is allowed during this period. Violating these terms may lead to a permanent ban.4. Permanent BanCommunity Impact: Demonstrating a pattern of violation of community standards, including sustained inappropriate behavior, harassment of an individual, or aggression toward or disparagement of classes of individuals.Consequence: A permanent ban from any sort of public interaction within the community.AttributionThis Code of Conduct is adapted from the [Contributor Covenant][homepage], version 2.0, available at https://www.contributor-covenant.org/version/2/0/codeofconduct.html.Community Impact Guidelines were inspired by Mozilla's code of conduct enforcement ladder.[homepage]: https://www.contributor-covenant.orgFor answers to common questions about this code of conduct, see the FAQ at https://www.contributor-covenant.org/faq. Translations are available at https://www.contributor-covenant.org/translations.","category":"page"},{"location":"systems/entropically_damped_sph/#edac","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility (EDAC) for SPH","text":"","category":"section"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"fracmathrmd p_amathrmdt = - rho c_s^2 nabla cdot v + nu nabla^2 p","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"which is derived by Clausen (2013). This equation is similar to the continuity equation (first term, see ContinuityDensity), but also contains a pressure damping term (second term, similar to density diffusion see DensityDiffusion), which reduces acoustic pressure waves through an entropy-generation mechanism.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The pressure evolution is discretized with the SPH method by Ramachandran (2019) as following:","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The first term is equivalent to the classical artificial compressible methods, which are commonly motivated by assuming the artificial equation of state (StateEquationCole with exponent=1) and is discretized as","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"- rho c_s^2 nabla cdot v = sum_b m_b fracrho_arho_b c_s^2 v_ab cdot nabla_r_a W(Vert r_a - r_b Vert h)","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where rho_a, rho_b, r_a, r_b, denote the density and coordinates of particles a and b respectively, c_s is the speed of sound and v_ab = v_a - v_b is the difference in the velocity.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The second term smooths the pressure through the introduction of entropy and is discretized as","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"nu nabla^2 p = fracV_a^2 + V_b^2m_a tildeeta_ab fracp_abVert r_ab^2 Vert + eta h_ab^2 nabla_r_a\nW(Vert r_a - r_b Vert h) cdot r_ab","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where V_a, V_b denote the volume of particles a and b respectively and p_ab= p_a -p_b is the difference in the pressure.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The viscosity parameter eta_a for a particle a is given as","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"eta_a = rho_a fracalpha h c_s8","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where it is found in the numerical experiments of Ramachandran (2019) that alpha = 05 is a good choice for a wide range of Reynolds numbers (0.0125 to 10000).","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"note: Note\nThe EDAC formulation keeps the density constant and this eliminates the need for the continuity equation or the use of a summation density to find the pressure. However, in SPH discretizations, mrho is typically used as a proxy for the particle volume. The density of the fluids can therefore be computed using the summation density approach. [19]","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"entropically_damped_sph\", \"system.jl\")]","category":"page"},{"location":"systems/entropically_damped_sph/#TrixiParticles.EntropicallyDampedSPHSystem","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"TrixiParticles.EntropicallyDampedSPHSystem","text":"EntropicallyDampedSPHSystem(initial_condition, smoothing_kernel,\n smoothing_length, sound_speed;\n pressure_acceleration=inter_particle_averaged_pressure,\n density_calculator=SummationDensity(),\n transport_velocity=nothing,\n alpha=0.5, viscosity=nothing,\n acceleration=ntuple(_ -> 0.0, NDIMS), buffer_size=nothing,\n source_terms=nothing)\n\nSystem for particles of a fluid. As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure. See Entropically Damped Artificial Compressibility for SPH for more details on the method.\n\nArguments\n\ninitial_condition: Initial condition representing the system's particles.\nsound_speed: Speed of sound.\nsmoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.\nsmoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.\n\nKeyword Arguments\n\nviscosity: Viscosity model for this system (default: no viscosity). Recommended: ViscosityAdami.\nacceleration: Acceleration vector for the system. (default: zero vector)\npressure_acceleration: Pressure acceleration formulation (default: inter-particle averaged pressure). When set to nothing, the pressure acceleration formulation for the corresponding density calculator is chosen.\ndensity_calculator: Density calculator (default: SummationDensity)\ntransport_velocity: Transport Velocity Formulation (TVF). Default is no TVF.\nbuffer_size: Number of buffer particles. This is needed when simulating with OpenBoundarySPHSystem.\nsource_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure, t) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping. Note that these source terms will not be used in the calculation of the boundary pressure when using a boundary with BoundaryModelDummyParticles and AdamiPressureExtrapolation. The keyword argument acceleration should be used instead for gravity-like source terms.\n\n\n\n\n\n","category":"type"},{"location":"systems/entropically_damped_sph/#transport_velocity_formulation","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Transport Velocity Formulation (TVF)","text":"","category":"section"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"Standard SPH suffers from problems like tensile instability or the creation of void regions in the flow. To address these problems, Adami (2013) modified the advection velocity and added an extra term to the momentum equation. The authors introduced the so-called Transport Velocity Formulation (TVF) for WCSPH. Ramachandran (2019) applied the TVF also for the EDAC scheme.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The transport velocity tildev_a of particle a is used to evolve the position of the particle r_a from one time step to the next by","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"fracmathrmd r_amathrmdt = tildev_a","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"and is obtained at every time-step Delta t from","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"tildev_a (t + Delta t) = v_a (t) + Delta t left(fractildemathrmd v_amathrmdt - frac1rho_a nabla p_textbackground right)","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where rho_a is the density of particle a and p_textbackground is a constant background pressure field. The tilde in the second term of the right hand side indicates that the material derivative has an advection part.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The discretized form of the last term is","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":" -frac1rho_a nabla p_textbackground approx -fracp_textbackgroundm_a sum_b left(V_a^2 + V_b^2 right) nabla_a W_ab","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where V_a, V_b denote the volume of particles a and b respectively. Note that although in the continuous case nabla p_textbackground = 0, the discretization is not 0th-order consistent for non-uniform particle distribution, which means that there is a non-vanishing contribution only when particles are disordered. That also means that p_textbackground occurs as prefactor to correct the trajectory of a particle resulting in uniform pressure distributions. Suggested is a background pressure which is in the order of the reference pressure but can be chosen arbitrarily large when the time-step criterion is adjusted.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The inviscid momentum equation with an additional convection term for a particle moving with tildev is","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"fractildemathrmd left( rho v right)mathrmdt = -nabla p + nabla cdot bmA","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"where the tensor bmA = rho vleft(tildev-vright)^T is a consequence of the modified advection velocity and can be interpreted as the convection of momentum with the relative velocity tildev-v.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"The discretized form of the momentum equation for a particle a reads as","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"fractildemathrmd v_amathrmdt = frac1m_a sum_b left(V_a^2 + V_b^2 right) left -tildep_ab nabla_a W_ab + frac12 left(bmA_a + bmA_b right) cdot nabla_a W_ab right","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"Here, tildep_ab is the density-weighted pressure","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"tildep_ab = fracrho_b p_a + rho_a p_brho_a + rho_b","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"with the density rho_a, rho_b and the pressure p_a, p_b of particles a and b respectively. bmA_a and bmA_b are the convection tensors for particle a and b respectively and are given, e.g. for particle a, as bmA_a = rho v_aleft(tildev_a-v_aright)^T.","category":"page"},{"location":"systems/entropically_damped_sph/","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"Entropically Damped Artificial Compressibility for SPH (Fluid)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"fluid\", \"transport_velocity.jl\")]","category":"page"},{"location":"systems/entropically_damped_sph/#TrixiParticles.TransportVelocityAdami","page":"Entropically Damped Artificial Compressibility for SPH (Fluid)","title":"TrixiParticles.TransportVelocityAdami","text":"TransportVelocityAdami(background_pressure::Real)\n\nTransport Velocity Formulation (TVF) to suppress pairing and tensile instability. See TVF for more details of the method.\n\nArguments\n\nbackground_pressure: Background pressure. Suggested is a background pressure which is on the order of the reference pressure.\n\nnote: Note\nThere is no need for an artificial viscosity to suppress tensile instability when using TransportVelocityAdami. Thus, it is highly recommended to use ViscosityAdami as viscosity model, since ArtificialViscosityMonaghan leads to bad results.\n\n\n\n\n\n","category":"type"},{"location":"authors/","page":"Authors","title":"Authors","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/AUTHORS.md\"","category":"page"},{"location":"authors/#Authors","page":"Authors","title":"Authors","text":"","category":"section"},{"location":"authors/","page":"Authors","title":"Authors","text":"TrixiParticles.jl's development is coordinated by a group of principal developers, who are also its main contributors and who can be contacted in case of questions about TrixiParticles.jl. In addition, there are contributors who have provided substantial additions or modifications. Together, these two groups form \"The TrixiParticles.jl Authors\" as mentioned under License.","category":"page"},{"location":"authors/#Principal-Developers","page":"Authors","title":"Principal Developers","text":"","category":"section"},{"location":"authors/","page":"Authors","title":"Authors","text":"Erik Faulhaber, University of Cologne, Germany\nNiklas Neher, High-Performance Computing Center Stuttgart (HLRS), Germany\nSven Berger, Helmholtz Center Hereon, Germany","category":"page"},{"location":"authors/#Contributors","page":"Authors","title":"Contributors","text":"","category":"section"},{"location":"authors/","page":"Authors","title":"Authors","text":"The following people contributed major additions or modifications to TrixiParticles.jl and are listed in alphabetical order:","category":"page"},{"location":"authors/","page":"Authors","title":"Authors","text":"Sven Berger\nErik Faulhaber\nGregor Gassner\nNiklas Neher\nHendrik Ranocha\nMichael Schlottke-Lakemper","category":"page"},{"location":"tutorials_template/tut_dam_break/#Example-file","page":"Example file","title":"Example file","text":"","category":"section"},{"location":"tutorials_template/tut_dam_break/","page":"Example file","title":"Example file","text":"!!include:examples/fluid/dam_break_2d.jl!!\n","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/docs/src/tutorials_template/tut_setup.md\"","category":"page"},{"location":"tutorials/tut_setup/#Setting-up-your-simulation-from-scratch","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For different setups and physics, have a look at our other example files.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"using TrixiParticles\nusing OrdinaryDiffEq","category":"page"},{"location":"tutorials/tut_setup/#Resolution","page":"Setting up your simulation from scratch","title":"Resolution","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Now, we define the particle spacing, which is our numerical resolution. We usually call the variable fluid_particle_spacing, so that we can easily change the resolution of an example file by overwriting this variable with trixi_include. In 2D, the number of particles will grow quadratically, in 3D cubically with the spacing.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary. In particular, we require boundary_layers >= compact_support. The value for the compact support for each kernel can be found in the smoothing kernel overview.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_particle_spacing = 0.05\nboundary_layers = 3\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/#Experiment-setup","page":"Setting up your simulation from scratch","title":"Experiment setup","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. (Image: Experiment Setup) First, we define the physical parameters gravitational acceleration, simulation time, initial fluid size, tank size and fluid density.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"gravity = 9.81\ntspan = (0.0, 1.0)\ninitial_fluid_size = (1.0, 0.9)\ntank_size = (1.0, 1.0)\nfluid_density = 1000.0\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sound_speed = 10.0\nstate_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,\n exponent=7)\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The speed of sound here is numerical and not physical. We artificially lower the speed of sound, since the physical speed of sound in water would lead to prohibitively small time steps. The speed of sound in Weakly Compressible SPH should be chosen as small as possible for numerical efficiency, but large enough to limit density fluctuations to about 1%.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"TrixiParticles.jl requires the initial particle positions and quantities in form of an InitialCondition. Instead of manually defining particle positions, you can work with our pre-defined setups. Among others, we provide setups for rectangular shapes, circles, and spheres. Initial conditions can also be combined with common set operations. See this page for a list of pre-defined setups and details on set operations on initial conditions.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Here, we use the RectangularTank setup, which generates a rectangular fluid inside a rectangular tank, and supports a hydrostatic pressure gradient by passing a gravitational acceleration and a state equation (see above).","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size,\n fluid_density, n_layers=boundary_layers,\n acceleration=(0.0, -gravity), state_equation=state_equation)\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/#Fluid-system","page":"Setting up your simulation from scratch","title":"Fluid system","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_length = 1.2 * fluid_particle_spacing\nsmoothing_kernel = SchoenbergCubicSplineKernel{2}()\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"You can find an overview over smoothing kernels and corresponding smoothing lengths here.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"For stability, we need numerical dissipation in form of an artificial viscosity term. Other viscosity models offer a physical approach based on the kinematic viscosity of the fluid.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We choose the parameters as small as possible to avoid non-physical behavior, but as large as possible to stabilize the simulation.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The WCSPH method can either compute the particle density directly with a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating its change over time. We choose the latter approach here by using the density calculator ContinuityDensity, which is more efficient and handles free surfaces without the need for additional correction terms.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"fluid_density_calculator = ContinuityDensity()\nfluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,\n state_equation, smoothing_kernel,\n smoothing_length, viscosity=viscosity,\n acceleration=(0.0, -gravity))\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/#Boundary-system","page":"Setting up your simulation from scratch","title":"Boundary system","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To model the boundary, we use particle-based boundary conditions, in which particles are sampled in the boundary that interact with the fluid particles to avoid penetration. In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation. See here for a comprehensive overview over boundary models.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,\n state_equation=state_equation,\n AdamiPressureExtrapolation(),\n smoothing_kernel, smoothing_length)\nboundary_system = BoundarySPHSystem(tank.boundary, boundary_model)\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/#Semidiscretization","page":"Setting up your simulation from scratch","title":"Semidiscretization","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"The key component of every simulation is the Semidiscretization, which couples all systems of the simulation. All methods in TrixiParticles.jl are semidiscretizations, which discretize the equations in time to provide an ordinary differential equation that still has to be solved in time. By providing a simulation time span, we can call semidiscretize, which returns an ODEProblem that can be solved with a time integration method.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"semi = Semidiscretization(fluid_system, boundary_system)\node = semidiscretize(semi, tspan)\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/#Time-integration","page":"Setting up your simulation from scratch","title":"Time integration","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"info_callback = InfoCallback(interval=50)\nsaving_callback = SolutionSavingCallback(dt=0.02)\n\ncallbacks = CallbackSet(info_callback, saving_callback)\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"See Visualization for how to visualize the solution. For the simplest visualization, we can use Plots.jl:","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"using Plots\nplot(sol)\nsavefig(\"tut_setup_plot.png\"); nothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"(Image: plot)","category":"page"},{"location":"tutorials/tut_setup/#Replacing-components-with-custom-implementations","page":"Setting up your simulation from scratch","title":"Replacing components with custom implementations","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.","category":"page"},{"location":"tutorials/tut_setup/#Custom-smoothing-kernel","page":"Setting up your simulation from scratch","title":"Custom smoothing kernel","text":"","category":"section"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)\n q = r / h\n\n if q < 2\n return 1 / (pi * h^2) * exp(-q^2)\n end\n\n return 0.0\nend\n\nfunction TrixiParticles.kernel_deriv(kernel::MyGaussianKernel, r, h)\n q = r * h\n\n if q < 2\n return 1 / (pi * h^2) * (-2 * q) * exp(-q^2) / h\n end\n\n return 0.0\nend\n\nTrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"smoothing_kernel = MyGaussianKernel()\nnothing # hide","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"and run the simulation file again.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.","category":"page"},{"location":"tutorials/tut_setup/","page":"Setting up your simulation from scratch","title":"Setting up your simulation from scratch","text":"trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"),\n smoothing_kernel=MyGaussianKernel());\nnothing # hide","category":"page"},{"location":"systems/total_lagrangian_sph/#tlsph","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH","text":"","category":"section"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"A Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers, 2021, Belytschko et al., 2000).","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The governing equations with respect to the initial configuration are given by:","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"fracmathrmDbmvmathrmDt = frac1rho_0 nabla_0 cdot bmP + bmg","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"where the zero subscript denotes a derivative with respect to the initial configuration and bmP is the first Piola-Kirchhoff (PK1) stress tensor.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The discretized version of this equation is given by O’Connor & Rogers (2021):","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"fracmathrmdbmv_amathrmdt = sum_b m_0b\n left( fracbmP_a bmL_0arho_0a^2 + fracbmP_b bmL_0brho_0b^2 right)\n nabla_0a W(bmX_ab) + fracbmf_a^PFm_0a + bmg","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"with the correction matrix (see also GradientCorrection)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmL_0a = left( -sum_b fracm_0brho_0b nabla_0a W(bmX_ab) bmX_ab^T right)^-1 in R^d times d","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The subscripts a and b denote quantities of particle a and b, respectively. The zero subscript on quantities denotes that the quantity is to be measured in the initial configuration. The difference in the initial coordinates is denoted by bmX_ab = bmX_a - bmX_b, the difference in the current coordinates is denoted by bmx_ab = bmx_a - bmx_b.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"For the computation of the PK1 stress tensor, the deformation gradient bmF is computed per particle as","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmF_a = sum_b fracm_0brho_0b bmx_ba (bmL_0anabla_0a W(bmX_ab))^T \n qquad = -left(sum_b fracm_0brho_0b bmx_ab (nabla_0a W(bmX_ab))^T right) bmL_0a^T","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"with 1 leq ij leq d. From the deformation gradient, the Green-Lagrange strain","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmE = frac12(bmF^TbmF - bmI)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"and the second Piola-Kirchhoff stress tensor","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmS = lambda operatornametr(bmE) bmI + 2mu bmE","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"are computed to obtain the PK1 stress tensor as","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmP = bmFbmS","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Here,","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"mu = fracE2(1 + nu)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"and","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"lambda = fracEnu(1 + nu)(1 - 2nu)","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"are the Lamé coefficients, where E is the Young's modulus and nu is the Poisson ratio.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The term bmf_a^PF is an optional penalty force. See e.g. PenaltyForceGanzenmueller.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"solid\", \"total_lagrangian_sph\", \"system.jl\")]","category":"page"},{"location":"systems/total_lagrangian_sph/#TrixiParticles.TotalLagrangianSPHSystem","page":"Total Lagrangian SPH (Elastic Structure)","title":"TrixiParticles.TotalLagrangianSPHSystem","text":"TotalLagrangianSPHSystem(initial_condition,\n smoothing_kernel, smoothing_length,\n young_modulus, poisson_ratio;\n n_fixed_particles=0, boundary_model=nothing,\n acceleration=ntuple(_ -> 0.0, NDIMS),\n penalty_force=nothing, source_terms=nothing)\n\nSystem for particles of an elastic structure.\n\nA Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers 2021, Belytschko et al. 2000). See Total Lagrangian SPH for more details on the method.\n\nArguments\n\ninitial_condition: Initial condition representing the system's particles.\nyoung_modulus: Young's modulus.\npoisson_ratio: Poisson ratio.\nsmoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.\nsmoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.\n\nKeyword Arguments\n\nn_fixed_particles: Number of fixed particles which are used to clamp the structure particles. Note that the fixed particles must be the last particles in the InitialCondition. See the info box below.\nboundary_model: Boundary model to compute the hydrodynamic density and pressure for fluid-structure interaction (see Boundary Models).\npenalty_force: Penalty force to ensure regular particle position under large deformations (see PenaltyForceGanzenmueller).\nacceleration: Acceleration vector for the system. (default: zero vector)\nsource_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping.\n\nnote: Note\nThe fixed particles must be the last particles in the InitialCondition. To do so, e.g. use the union function:solid = union(beam, fixed_particles)where beam and fixed_particles are of type InitialCondition.\n\n\n\n\n\n","category":"type"},{"location":"systems/total_lagrangian_sph/#Penalty-Force","page":"Total Lagrangian SPH (Elastic Structure)","title":"Penalty Force","text":"","category":"section"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"In FEM, underintegrated elements can deform without an associated increase of energy. This is caused by the stiffness matrix having zero eigenvalues (so-called hourglass modes). The name \"hourglass modes\" comes from the fact that elements can deform into an hourglass shape.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Similar effects can occur in SPH as well. Particles can change positions without changing the SPH approximation of the deformation gradient bmF, thus, without causing an increase of energy. To ensure regular particle positions, we can apply similar correction forces as are used in FEM.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Ganzenmüller (2015) introduced a so-called hourglass correction force or penalty force f^PF, which is given by","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"bmf_a^PF = frac12 alpha sum_b fracm_0a m_0b W_0abrho_0arho_0b bmX_ab^2\n left( E delta_ab^a + E delta_ba^b right) fracbmx_abbmx_ab","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"The subscripts a and b denote quantities of particle a and b, respectively. The zero subscript on quantities denotes that the quantity is to be measured in the initial configuration. The difference in the initial coordinates is denoted by bmX_ab = bmX_a - bmX_b, the difference in the current coordinates is denoted by bmx_ab = bmx_a - bmx_b. Note that Ganzenmüller (2015) has a flipped sign here because they define bmx_ab the other way around.","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"This correction force is based on the potential energy density of a Hookean material. Thus, E is the Young's modulus and alpha is a dimensionless coefficient that controls the amplitude of hourglass correction. The separation vector delta_ab^a indicates the change of distance which the particle separation should attain in order to minimize the error and is given by","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":" delta_ab^a = fracbmepsilon_ab^a cdot bmx_abbmx_ab","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"where the error vector is defined as","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":" bmepsilon_ab^a = bmF_a bmX_ab - bmx_ab","category":"page"},{"location":"systems/total_lagrangian_sph/","page":"Total Lagrangian SPH (Elastic Structure)","title":"Total Lagrangian SPH (Elastic Structure)","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"schemes\", \"solid\", \"total_lagrangian_sph\", \"penalty_force.jl\")]","category":"page"},{"location":"systems/total_lagrangian_sph/#TrixiParticles.PenaltyForceGanzenmueller","page":"Total Lagrangian SPH (Elastic Structure)","title":"TrixiParticles.PenaltyForceGanzenmueller","text":"PenaltyForceGanzenmueller(; alpha=0.1)\n\nPenalty force to ensure regular particle positions under large deformations.\n\nKeywords\n\nalpha: Coefficient to control the amplitude of hourglass correction.\n\n\n\n\n\n","category":"type"},{"location":"license/","page":"License","title":"License","text":"EditURL = \"https://github.com/trixi-framework/TrixiParticles.jl/blob/main/LICENSE.md\"","category":"page"},{"location":"license/#License","page":"License","title":"License","text":"","category":"section"},{"location":"license/","page":"License","title":"License","text":"MIT LicenseCopyright (c) 2023-present The TrixiParticles.jl Authors (see Authors) \nCopyright (c) 2023-present Helmholtz-Zentrum hereon GmbH, Institute of Surface Science \n \nPermission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the \"Software\"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.","category":"page"},{"location":"getting_started/#getting_started","page":"Getting started","title":"Getting started","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"If you have not installed TrixiParticles.jl, please follow the instructions given here.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"In the following sections, we will give a short introduction. For a more thorough discussion, take a look at our Tutorials.","category":"page"},{"location":"getting_started/#Running-an-Example","page":"Getting started","title":"Running an Example","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"The easiest way to run a simulation is to run one of our predefined example files. We will run the file examples/fluid/hydrostatic_water_column_2d.jl, which simulates a fluid resting in a rectangular tank. Since TrixiParticles.jl uses multithreading, you should start Julia with the flag --threads auto (or, e.g. --threads 4 for 4 threads).","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"In the Julia REPL, first load the package TrixiParticles.jl.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> using TrixiParticles","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"Then start the simulation by executing","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"))","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"The easiest way to quickly visualize the result is to use Plots.jl:","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> using Plots; plot(sol)","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"This will open a new window with a 2D visualization of the final solution: (Image: plot_hydrostatic_water_column)","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"For more information about visualization, see Visualization.","category":"page"},{"location":"getting_started/#Running-other-Examples","page":"Getting started","title":"Running other Examples","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"You can find a list of our other predefined examples under Examples. Execute them as follows from the Julia REPL by replacing subfolder and example_name","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> trixi_include(joinpath(examples_dir(), \"subfolder\", \"example_name.jl\"))","category":"page"},{"location":"getting_started/#Modifying-an-example","page":"Getting started","title":"Modifying an example","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"You can pass keyword arguments to the function trixi_include to overwrite assignments in the file.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"With trixi_include, we can overwrite variables defined in the example file to run a different simulation without modifying the example file.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"julia> trixi_include(joinpath(examples_dir(), \"fluid\", \"hydrostatic_water_column_2d.jl\"), initial_fluid_size=(1.0, 0.5))","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"This for example, will change the fluid size from (09 10) to (10 05).","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"To understand why, take a look into the file hydrostatic_water_column_2d.jl in the subfolder fluid inside the examples directory, which is the file that we executed earlier. You can see that the initial size of the fluid is defined in the variable initial_fluid_size, which we could overwrite with the trixi_include call above. Another variable that is worth experimenting with is fluid_particle_spacing, which controls the resolution of the simulation in this case. A lower value will increase the resolution and the runtime.","category":"page"},{"location":"getting_started/#Set-up-you-first-simulation-from-scratch","page":"Getting started","title":"Set up you first simulation from scratch","text":"","category":"section"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"See Set up your first simulation.","category":"page"},{"location":"getting_started/","page":"Getting started","title":"Getting started","text":"Find an overview over the available tutorials under Tutorials.","category":"page"},{"location":"examples/#examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/#Fluid","page":"Examples","title":"Fluid","text":"","category":"section"},{"location":"examples/#Accelerated-Tank-2D-(fluid/accelerated_tank_2d.jl)","page":"Examples","title":"Accelerated Tank 2D (fluid/accelerated_tank_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Dam-Break-2D-(fluid/dam_break_2d.jl)","page":"Examples","title":"Dam Break 2D (fluid/dam_break_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Dam-Break-3D-(fluid/dam_break_3d.jl)","page":"Examples","title":"Dam Break 3D (fluid/dam_break_3d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Falling-Water-Column-(fluid/falling_water_column_2d.jl)","page":"Examples","title":"Falling Water Column (fluid/falling_water_column_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Hydrostatic-Water-Column-(fluid/hydrostatic_water_column_*.jl)","page":"Examples","title":"Hydrostatic Water Column (fluid/hydrostatic_water_column_*.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Moving-Wall-(fluid/moving_wall_2d.jl)","page":"Examples","title":"Moving Wall (fluid/moving_wall_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Oscillating-Drop-(fluid/oscillating_drop_2d.jl)","page":"Examples","title":"Oscillating Drop (fluid/oscillating_drop_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Periodic-Channel-(fluid/periodic_channel_2d.jl)","page":"Examples","title":"Periodic Channel (fluid/periodic_channel_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Fluid-Structure-Interaction","page":"Examples","title":"Fluid Structure Interaction","text":"","category":"section"},{"location":"examples/#Dam-Break-with-Elastic-Plate-(fsi/dam_break_plate_2d.jl)","page":"Examples","title":"Dam Break with Elastic Plate (fsi/dam_break_plate_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Falling-Sphere-2D-(fsi/falling_sphere_2d.jl)","page":"Examples","title":"Falling Sphere 2D (fsi/falling_sphere_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Falling-Spheres-2D-(fsi/falling_spheres_2d.jl)","page":"Examples","title":"Falling Spheres 2D (fsi/falling_spheres_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/#Structure-Mechanics","page":"Examples","title":"Structure Mechanics","text":"","category":"section"},{"location":"examples/#Oscillating-Beam-(solid/oscillating_beam_2d.jl)","page":"Examples","title":"Oscillating Beam (solid/oscillating_beam_2d.jl)","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"overview/#Overview","page":"Overview","title":"Overview","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"The actual API reference is not listed on a single page, like in most Julia packages, but instead is split into multiple sections that follow a similar structure as the code files themselves. In these sections, API docs are combined with explanations of the theoretical background of these methods.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"The following page gives a rough overview of important parts of the code.","category":"page"},{"location":"overview/#Program-flow","page":"Overview","title":"Program flow","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"To initiate a simulation, the goal is to solve an ordinary differential equation, for example, by employing the time integration schemes provided by OrdinaryDiffEq.jl. These schemes are then utilized to integrate mathrmdumathrmdt and mathrmdvmathrmdt, where u represents the particles' positions and v their properties such as velocity and density. During a single time step or an intermediate step of the time integration scheme, the functions drift! and kick! are invoked, followed by the functions depicted in this diagram (with key parts highlighted in orange/yellow).","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: Main Program Flow)","category":"page"},{"location":"overview/#Structure","page":"Overview","title":"Structure","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"What we refer to as schemes are various models such as Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) or Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH). These schemes are categorized based on the applicable physical regimes, namely fluid, solid, gas, and others. Each scheme comprises at least two files: a system.jl file and an rhs.jl file. The system.jl file provides the data structure holding the particles of this scheme and some routines, particularly those for allocation and the main update routines, excluding system interactions. The interactions between particles of this scheme (and with particles of other schemes) are handled in the rhs.jl file.","category":"page"},{"location":"callbacks/#Callbacks","page":"Callbacks","title":"Callbacks","text":"","category":"section"},{"location":"callbacks/","page":"Callbacks","title":"Callbacks","text":"Modules = [TrixiParticles]\nPages = map(file -> joinpath(\"callbacks\", file), readdir(joinpath(\"..\", \"src\", \"callbacks\")))","category":"page"},{"location":"callbacks/#TrixiParticles.DensityReinitializationCallback","page":"Callbacks","title":"TrixiParticles.DensityReinitializationCallback","text":"DensityReinitializationCallback(; interval::Integer=0, dt=0.0)\n\nCallback to reinitialize the density field when using ContinuityDensity [42].\n\nKeywords\n\ninterval=0: Reinitialize the density every interval time steps.\ndt: Reinitialize the density in regular intervals of dt in terms of integration time.\nreinit_initial_solution: Reinitialize the initial solution (default=false)\n\n\n\n\n\n","category":"type"},{"location":"callbacks/#TrixiParticles.InfoCallback-Tuple{}","page":"Callbacks","title":"TrixiParticles.InfoCallback","text":"InfoCallback()\n\nCreate and return a callback that prints a human-readable summary of the simulation setup at the beginning of a simulation and then resets the timer. When the returned callback is executed directly, the current timer values are shown.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.PostprocessCallback","page":"Callbacks","title":"TrixiParticles.PostprocessCallback","text":"PostprocessCallback(; interval::Integer=0, dt=0.0, exclude_boundary=true, filename=\"values\",\n output_directory=\"out\", append_timestamp=false, write_csv=true,\n write_json=true, write_file_interval=1, funcs...)\n\nCreate a callback to post-process simulation data at regular intervals. This callback allows for the execution of a user-defined function func at specified intervals during the simulation. The function is applied to the current state of the simulation, and its results can be saved or used for further analysis. The provided function cannot be anonymous as the function name will be used as part of the name of the value.\n\nThe callback can be triggered either by a fixed number of time steps (interval) or by a fixed interval of simulation time (dt).\n\nKeywords\n\nfuncs...: Functions to be executed at specified intervals during the simulation. Each function must have the arguments (v, u, t, system), and will be called for every system, where v and u are the wrapped solution arrays for the corresponding system and t is the current simulation time. Note that working with these v and u arrays requires undocumented internal functions of TrixiParticles. See Custom Quantities for a list of pre-defined functions that can be used here.\ninterval=0: Specifies the number of time steps between each invocation of the callback. If set to 0, the callback will not be triggered based on time steps. Either interval or dt must be set to something larger than 0.\ndt=0.0: Specifies the simulation time interval between each invocation of the callback. If set to 0.0, the callback will not be triggered based on simulation time. Either interval or dt must be set to something larger than 0.\nexclude_boundary=true: If set to true, boundary particles will be excluded from the post-processing.\nfilename=\"values\": The filename of the postprocessing files to be saved.\noutput_directory=\"out\": The path where the results of the post-processing will be saved.\nwrite_csv=true: If set to true, write a csv file.\nwrite_json=true: If set to true, write a json file.\nappend_timestep=false: If set to true, the current timestamp will be added to the filename.\nwrite_file_interval=1: Files will be written after every write_file_interval number of postprocessing execution steps. A value of 0 indicates that files are only written at the end of the simulation, eliminating I/O overhead.\n\nExamples\n\n# Create a callback that is triggered every 100 time steps\npostprocess_callback = PostprocessCallback(interval=100, example_quantity=kinetic_energy)\n\n# Create a callback that is triggered every 0.1 simulation time units\npostprocess_callback = PostprocessCallback(dt=0.1, example_quantity=kinetic_energy)\n\n\n\n\n\n","category":"type"},{"location":"callbacks/#TrixiParticles.SolutionSavingCallback","page":"Callbacks","title":"TrixiParticles.SolutionSavingCallback","text":"SolutionSavingCallback(; interval::Integer=0, dt=0.0, save_times=Array{Float64, 1}([]),\n save_initial_solution=true, save_final_solution=true,\n output_directory=\"out\", append_timestamp=false, prefix=\"\",\n verbose=false, write_meta_data=true, max_coordinates=2^15,\n custom_quantities...)\n\nCallback to save the current numerical solution in VTK format in regular intervals. Either pass interval to save every interval time steps, or pass dt to save in intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).\n\nAdditional user-defined quantities can be saved by passing functions as keyword arguments, which map (v, u, t, system) to an Array where the columns represent the particles in the same order as in u. To ignore a custom quantity for a specific system, return nothing.\n\nKeywords\n\ninterval=0: Save the solution every interval time steps.\ndt: Save the solution in regular intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).\nsave_times=[] List of times at which to save a solution.\nsave_initial_solution=true: Save the initial solution.\nsave_final_solution=true: Save the final solution.\noutput_directory=\"out\": Directory to save the VTK files.\nappend_timestamp=false: Append current timestamp to the output directory.\n'prefix=\"\"': Prefix added to the filename.\ncustom_quantities...: Additional user-defined quantities.\nwrite_meta_data=true: Write meta data.\nverbose=false: Print to standard IO when a file is written.\nmax_coordinates=2^15: The coordinates of particles will be clipped if their absolute values exceed this threshold.\ncustom_quantities...: Additional custom quantities to include in the VTK output. Each custom quantity must be a function of (v, u, t, system), which will be called for every system, where v and u are the wrapped solution arrays for the corresponding system and t is the current simulation time. Note that working with these v and u arrays requires undocumented internal functions of TrixiParticles. See Custom Quantities for a list of pre-defined custom quantities that can be used here.\n\nExamples\n\n# Save every 100 time steps\nsaving_callback = SolutionSavingCallback(interval=100)\n\n# Save in intervals of 0.1 in terms of simulation time\nsaving_callback = SolutionSavingCallback(dt=0.1)\n\n# Additionally store the kinetic energy of each system as \"my_custom_quantity\"\nsaving_callback = SolutionSavingCallback(dt=0.1, my_custom_quantity=kinetic_energy)\n\n\n\n\n\n","category":"type"},{"location":"callbacks/#TrixiParticles.SteadyStateReachedCallback","page":"Callbacks","title":"TrixiParticles.SteadyStateReachedCallback","text":"SteadyStateReachedCallback(; interval::Integer=0, dt=0.0,\n interval_size::Integer=10, abstol=1.0e-8, reltol=1.0e-6)\n\nTerminates the integration when the change of kinetic energy between time steps falls below the threshold specified by abstol + reltol * ekin, where ekin is the total kinetic energy of the simulation.\n\nKeywords\n\ninterval=0: Check steady state condition every interval time steps.\ndt=0.0: Check steady state condition in regular intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).\ninterval_size: The interval in which the change of the kinetic energy is considered. interval_size is a (integer) multiple of interval or dt.\nabstol: Absolute tolerance.\nreltol: Relative tolerance.\n\n\n\n\n\n","category":"type"},{"location":"callbacks/#TrixiParticles.StepsizeCallback-Tuple{}","page":"Callbacks","title":"TrixiParticles.StepsizeCallback","text":"StepsizeCallback(; cfl::Real)\n\nSet the time step size according to a CFL condition if the time integration method isn't adaptive itself.\n\nThe current implementation is using the simplest form of CFL condition, which chooses a time step size that is constant during the simulation. The step size is therefore only applied once at the beginning of the simulation.\n\nThe step size Delta t is chosen as the minimum\n\n Delta t = min(Delta t_eta Delta t_a Delta t_c)\n\nwhere\n\n Delta t_eta = 0125 h^2 eta quad Delta t_a = 025 sqrth lVert g rVert\n quad Delta t_c = textCFL h c\n\nwith nu = alpha h c (2n + 4), where alpha is the parameter of the viscosity and n is the number of dimensions.\n\nwarning: Experimental implementation\nThis is an experimental feature and may change in future releases.\n\nReferences\n\n[21], [14], [43], [44]\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.UpdateCallback-Tuple{}","page":"Callbacks","title":"TrixiParticles.UpdateCallback","text":"UpdateCallback(; interval::Integer, dt=0.0)\n\nCallback to update quantities either at the end of every interval time steps or in intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).\n\nKeywords\n\ninterval=1: Update quantities at the end of every interval time steps.\ndt: Update quantities in regular intervals of dt in terms of integration time by adding additional tstops (note that this may change the solution).\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#custom_quantities","page":"Callbacks","title":"Custom Quantities","text":"","category":"section"},{"location":"callbacks/","page":"Callbacks","title":"Callbacks","text":"The following pre-defined custom quantities can be used with the SolutionSavingCallback and PostprocessCallback.","category":"page"},{"location":"callbacks/","page":"Callbacks","title":"Callbacks","text":"Modules = [TrixiParticles]\nPages = [\"general/custom_quantities.jl\"]","category":"page"},{"location":"callbacks/#TrixiParticles.avg_density-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.avg_density","text":"avg_density\n\nReturns the average_density over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.avg_pressure-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.avg_pressure","text":"avg_pressure\n\nReturns the average pressure over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.kinetic_energy-NTuple{4, Any}","page":"Callbacks","title":"TrixiParticles.kinetic_energy","text":"kinetic_energy\n\nReturns the total kinetic energy of all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.max_density-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.max_density","text":"max_density\n\nReturns the maximum density over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.max_pressure-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.max_pressure","text":"max_pressure\n\nReturns the maximum pressure over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.min_density-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.min_density","text":"min_density\n\nReturns the minimum density over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.min_pressure-Tuple{Any, Any, Any, TrixiParticles.FluidSystem}","page":"Callbacks","title":"TrixiParticles.min_pressure","text":"min_pressure\n\nReturns the minimum pressure over all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"callbacks/#TrixiParticles.total_mass-NTuple{4, Any}","page":"Callbacks","title":"TrixiParticles.total_mass","text":"total_mass\n\nReturns the total mass of all particles in a system.\n\n\n\n\n\n","category":"method"},{"location":"gpu/#GPU-Support","page":"GPU Support","title":"GPU Support","text":"","category":"section"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"GPU support is still an experimental feature that is actively being worked on. As of now, the WeaklyCompressibleSPHSystem and the BoundarySPHSystem are supported on GPUs. We have tested this on GPUs by Nvidia and AMD.","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"To run a simulation on a GPU, we need to use the FullGridCellList as cell list for the GridNeighborhoodSearch. This cell list requires a bounding box for the domain, unlike the default cell list, which uses an unbounded domain. For simulations that are bounded by a closed tank, we can use the boundary of the tank to obtain the bounding box as follows.","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"search_radius = TrixiParticles.compact_support(smoothing_kernel, smoothing_length)\nmin_corner = minimum(tank.boundary.coordinates, dims=2) .- search_radius\nmax_corner = maximum(tank.boundary.coordinates, dims=2) .+ search_radius\ncell_list = TrixiParticles.PointNeighbors.FullGridCellList(; min_corner, max_corner)","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"We then need to pass this cell list to the neighborhood search and the neighborhood search to the Semidiscretization.","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"semi = Semidiscretization(fluid_system, boundary_system,\n neighborhood_search=GridNeighborhoodSearch{2}(; cell_list))","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"At this point, we should run the simulation and make sure that it still works and that the bounding box is large enough. For some simulations where particles move outside the initial tank coordinates, for example when the tank is not closed or when the tank is moving, an appropriate bounding box has to be specified.","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"Then, we only need to specify the data type that is used for the simulation. On an Nvidia GPU, we specify:","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"using CUDA\node = semidiscretize(semi, tspan, data_type=CuArray)","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"On an AMD GPU, we use:","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"using AMDGPU\node = semidiscretize(semi, tspan, data_type=ROCArray)","category":"page"},{"location":"gpu/","page":"GPU Support","title":"GPU Support","text":"Then, we can run the simulation as usual. All data is transferred to the GPU during initialization and all loops over particles and their neighbors will be executed on the GPU as kernels generated by KernelAbstractions.jl. Data is only copied to the CPU for saving VTK files via the SolutionSavingCallback.","category":"page"},{"location":"general/interpolation/#Interpolation","page":"Interpolation","title":"Interpolation","text":"","category":"section"},{"location":"general/interpolation/","page":"Interpolation","title":"Interpolation","text":"Modules = [TrixiParticles]\nPages = [joinpath(\"general\", \"interpolation.jl\")]","category":"page"},{"location":"general/interpolation/#TrixiParticles.interpolate_line-Tuple{Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_line","text":"interpolate_line(start, end_, n_points, semi, ref_system, sol; endpoint=true,\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nInterpolates properties along a line in a TrixiParticles simulation. The line interpolation is accomplished by generating a series of evenly spaced points between start and end_. If endpoint is false, the line is interpolated between the start and end points, but does not include these points.\n\nSee also: interpolate_point, interpolate_plane_2d, interpolate_plane_2d_vtk, interpolate_plane_3d.\n\nArguments\n\nstart: The starting point of the line.\nend_: The ending point of the line.\nn_points: The number of points to interpolate along the line.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nendpoint=true: A boolean to include (true) or exclude (false) the end point in the interpolation.\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nA NamedTuple of arrays containing interpolated properties at each point along the line.\n\nnote: Note\nThis function is particularly useful for analyzing gradients or creating visualizations along a specified line in the SPH simulation domain.\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating along a line from [1.0, 0.0] to [1.0, 1.0] with 5 points\nresults = interpolate_line([1.0, 0.0], [1.0, 1.0], 5, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_plane_2d-Tuple{Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_plane_2d","text":"interpolate_plane_2d(min_corner, max_corner, resolution, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nInterpolates properties along a plane in a TrixiParticles simulation. The region for interpolation is defined by its lower left and top right corners, with a specified resolution determining the density of the interpolation points.\n\nThe function generates a grid of points within the defined region, spaced uniformly according to the given resolution.\n\nSee also: interpolate_plane_2d_vtk, interpolate_plane_3d, interpolate_line, interpolate_point.\n\nArguments\n\nmin_corner: The lower left corner of the interpolation region.\nmax_corner: The top right corner of the interpolation region.\nresolution: The distance between adjacent interpolation points in the grid.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nA NamedTuple of arrays containing interpolated properties at each point within the plane.\n\nnote: Note\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used, which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating across a plane from [0.0, 0.0] to [1.0, 1.0] with a resolution of 0.2\nresults = interpolate_plane_2d([0.0, 0.0], [1.0, 1.0], 0.2, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_plane_2d_vtk-Tuple{Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_plane_2d_vtk","text":"interpolate_plane_2d_vtk(min_corner, max_corner, resolution, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false, output_directory=\"out\", filename=\"plane\")\n\nInterpolates properties along a plane in a TrixiParticles simulation and exports the result as a VTI file. The region for interpolation is defined by its lower left and top right corners, with a specified resolution determining the density of the interpolation points.\n\nThe function generates a grid of points within the defined region, spaced uniformly according to the given resolution.\n\nSee also: interpolate_plane_2d, interpolate_plane_3d, interpolate_line, interpolate_point.\n\nArguments\n\nmin_corner: The lower left corner of the interpolation region.\nmax_corner: The top right corner of the interpolation region.\nresolution: The distance between adjacent interpolation points in the grid.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\noutput_directory=\"out\": Directory to save the VTI file.\nfilename=\"plane\": Name of the VTI file.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nnote: Note\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used, which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating across a plane from [0.0, 0.0] to [1.0, 1.0] with a resolution of 0.2\nresults = interpolate_plane_2d([0.0, 0.0], [1.0, 1.0], 0.2, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_plane_3d-Tuple{Any, Any, Any, Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_plane_3d","text":"interpolate_plane_3d(point1, point2, point3, resolution, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nInterpolates properties along a plane in a 3D space in a TrixiParticles simulation. The plane for interpolation is defined by three points in 3D space, with a specified resolution determining the density of the interpolation points.\n\nThe function generates a grid of points on a parallelogram within the plane defined by the three points, spaced uniformly according to the given resolution.\n\nSee also: interpolate_plane_2d, interpolate_plane_2d_vtk, interpolate_line, interpolate_point.\n\nArguments\n\npoint1: The first point defining the plane.\npoint2: The second point defining the plane.\npoint3: The third point defining the plane. The points must not be collinear.\nresolution: The distance between adjacent interpolation points in the grid.\nsemi: The semidiscretization used for the simulation.\nref_system: The reference system for the interpolation.\nsol: The solution state from which the properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nA NamedTuple of arrays containing interpolated properties at each point within the plane.\n\nnote: Note\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used which is not as accurate as a real surface reconstruction.\n\nExamples\n\n# Interpolating across a plane defined by points [0.0, 0.0, 0.0], [1.0, 0.0, 0.0], and [0.0, 1.0, 0.0]\n# with a resolution of 0.1\nresults = interpolate_plane_3d([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 0.1, semi, ref_system, sol)\n\n\n\n\n\n","category":"method"},{"location":"general/interpolation/#TrixiParticles.interpolate_point-Tuple{Any, Any, Any, ODESolution}","page":"Interpolation","title":"TrixiParticles.interpolate_point","text":"interpolate_point(points_coords::Array{Array{Float64,1},1}, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\ninterpolate_point(point_coords, semi, ref_system, sol;\n smoothing_length=ref_system.smoothing_length, cut_off_bnd=true,\n clip_negative_pressure=false)\n\nPerforms interpolation of properties at specified points or an array of points in a TrixiParticles simulation.\n\nWhen given an array of points (points_coords), it iterates over each point and applies interpolation individually. For a single point (point_coords), it performs the interpolation at that specific location. The interpolation utilizes the same kernel function of the SPH simulation to weigh contributions from nearby particles.\n\nSee also: interpolate_line, interpolate_plane_2d, interpolate_plane_2d_vtk, interpolate_plane_3d, .\n\nArguments\n\npoints_coords: An array of point coordinates, for which to interpolate properties.\npoint_coords: The coordinates of a single point for interpolation.\nsemi: The semidiscretization used in the SPH simulation.\nref_system: The reference system defining the properties of the SPH particles.\nsol: The current solution state from which properties are interpolated.\n\nKeywords\n\nsmoothing_length=ref_system.smoothing_length: The smoothing length used in the interpolation.\ncut_off_bnd=true: Boolean to indicate if quantities should be set to NaN when the point is \"closer\" to the boundary than to the fluid in a kernel-weighted sense. Or, in more detail, when the boundary has more influence than the fluid on the density summation in this point, i.e., when the boundary particles add more kernel-weighted mass than the fluid particles.\nclip_negative_pressure=false: One common approach in SPH models is to clip negative pressure values, but this is unphysical. Instead we clip here during interpolation thus only impacting the local interpolated value.\n\nReturns\n\nFor multiple points: A NamedTuple of arrays containing interpolated properties at each point.\nFor a single point: A NamedTuple of interpolated properties at the point.\n\nExamples\n\n# For a single point\nresult = interpolate_point([1.0, 0.5], semi, ref_system, sol)\n\n# For multiple points\npoints = [[1.0, 0.5], [1.0, 0.6], [1.0, 0.7]]\nresults = interpolate_point(points, semi, ref_system, sol)\n\nnote: Note\nThis function is particularly useful for analyzing gradients or creating visualizations along a specified line in the SPH simulation domain.\nThe interpolation accuracy is subject to the density of particles and the chosen smoothing length.\nWith cut_off_bnd, a density-based estimation of the surface is used which is not asaccurate as a real surface reconstruction.\n\n\n\n\n\n","category":"method"},{"location":"#TrixiParticles.jl","page":"Home","title":"TrixiParticles.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TrixiParticles.jl is a high-performance particle simulation framework designed to overcome challenges of particle-based numerical methods in multiphysics applications. Existing frameworks often lack user-friendliness, involve complex configuration, and are not easily extensible for development of new methods. In the future we also want to provide seamless scalability from CPU to Exascale-level computing with GPU support. TrixiParticles.jl addresses these limitations with an intuitive interface, straightforward configuration, and an extensible design, facilitating efficient simulation setup and execution.","category":"page"},{"location":"","page":"Home","title":"Home","text":"TrixiParticles.jl focuses on the following use cases:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Development of new particle-based methods and models. By providing an extensible architecture to incorporate additional particle methods easily and not focusing on a single model or numerical method.\nAccurate, reliable and efficient physics-based modelling of complex multiphysics problems by providing a flexible configuration system, tools, high performance and a wide range of validation and test cases.\nEasy setup of accessible simulations for educational purposes, including student projects, coursework, and thesis work through extensive documentation, community engagement and readable configuration files.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Its features include:","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Incompressible Navier-Stokes\nMethods: Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH), Entropically Damped Artificial Compressibility (EDAC)\nModels: Surface Tension\nSolid-body mechanics\nMethods: Total Lagrangian SPH (TLSPH)\nFluid-Structure Interaction\nOutput formats:\nVTK","category":"page"},{"location":"#Examples","page":"Home","title":"Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"\n \n \n \n \n \n \n \n \n
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2D Dam Break
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Moving Wall
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Oscillating Beam
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Dam Break with Elastic Plate
\n
","category":"page"},{"location":"#Quickstart","page":"Home","title":"Quickstart","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Installation\nGetting started","category":"page"},{"location":"","page":"Home","title":"Home","text":"If you have any questions concerning TrixiParticles.jl you can join our community on Slack or open an issue with your question.","category":"page"},{"location":"#Start-with-development","page":"Home","title":"Start with development","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To get started with development have a look at these pages:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Installation\nDevelopment\nContributing","category":"page"},{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/#Fluid","page":"Tutorial","title":"Fluid","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Setting up your simulation from scratch\nSetting up a dam break simulation","category":"page"},{"location":"tutorial/#Mechanics","page":"Tutorial","title":"Mechanics","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Deforming a beam","category":"page"},{"location":"tutorial/#Fluid-Structure-Interaction","page":"Tutorial","title":"Fluid-Structure Interaction","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Setting up a falling structure","category":"page"},{"location":"reference-pointneighbors/#pointneighbors","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.jl API","text":"","category":"section"},{"location":"reference-pointneighbors/","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.jl API Reference","text":"CurrentModule = PointNeighbors","category":"page"},{"location":"reference-pointneighbors/","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.jl API Reference","text":"Modules = [PointNeighbors]","category":"page"},{"location":"reference-pointneighbors/#PointNeighbors.DictionaryCellList","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.DictionaryCellList","text":"DictionaryCellList{NDIMS}()\n\nA simple cell list implementation where a cell index (i, j) or (i, j, k) is mapped to a Vector{Int} by a Dict. By using a dictionary, which only stores non-empty cells, the domain is potentially infinite.\n\nThis implementation is very simple, but it neither uses an optimized hash function for integer tuples, nor does it use a contiguous memory layout. Consequently, this cell list is not GPU-compatible.\n\nArguments\n\nNDIMS: Number of dimensions.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.FullGridCellList","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.FullGridCellList","text":"FullGridCellList(; min_corner, max_corner, search_radius = 0.0,\n periodicity = false, backend = DynamicVectorOfVectors{Int32},\n max_points_per_cell = 100)\n\nA simple cell list implementation where each (empty or non-empty) cell of a rectangular (axis-aligned) domain is assigned a list of points. This cell list only works when all points are inside the specified domain at all times.\n\nOnly set min_corner and max_corner and use the default values for the other arguments to create an empty \"template\" cell list that can be used to create an empty \"template\" neighborhood search. See copy_neighborhood_search for more details.\n\nKeywords\n\nmin_corner: Coordinates of the domain corner in negative coordinate directions.\nmax_corner: Coordinates of the domain corner in positive coordinate directions.\nsearch_radius = 0.0: Search radius of the neighborhood search, which will determine the cell size. Use the default of 0.0 to create a template (see above).\nperiodicity = false: Set to true when using a PeriodicBox with the neighborhood search. When using copy_neighborhood_search, this option can be ignored an will be set automatically depending on the periodicity of the neighborhood search.\nbackend = DynamicVectorOfVectors{Int32}: Type of the data structure to store the actual cell lists. Can be\nVector{Vector{Int32}}: Scattered memory, but very memory-efficient.\nDynamicVectorOfVectors{Int32}: Contiguous memory, optimizing cache-hits.\nmax_points_per_cell = 100: Maximum number of points per cell. This will be used to allocate the DynamicVectorOfVectors. It is not used with the Vector{Vector{Int32}} backend.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.GridNeighborhoodSearch","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.GridNeighborhoodSearch","text":"GridNeighborhoodSearch{NDIMS}(; search_radius = 0.0, n_points = 0,\n periodic_box = nothing,\n cell_list = DictionaryCellList{NDIMS}(),\n update_strategy = nothing)\n\nSimple grid-based neighborhood search with uniform search radius. The domain is divided into a regular grid. For each (non-empty) grid cell, a list of points in this cell is stored. Instead of representing a finite domain by an array of cells, a potentially infinite domain is represented by storing cell lists in a hash table (using Julia's Dict data structure), indexed by the cell index tuple\n\nleft( leftlfloor fracxd rightrfloor leftlfloor fracyd rightrfloor right) quad textor quad\nleft( leftlfloor fracxd rightrfloor leftlfloor fracyd rightrfloor leftlfloor fraczd rightrfloor right)\n\nwhere x y z are the space coordinates and d is the search radius.\n\nTo find points within the search radius around a position, only points in the neighboring cells are considered.\n\nSee also (Chalela et al., 2021), (Ihmsen et al. 2011, Section 4.4).\n\nAs opposed to (Ihmsen et al. 2011), we do not sort the points in any way, since not sorting makes our implementation a lot faster (although less parallelizable).\n\nArguments\n\nNDIMS: Number of dimensions.\n\nKeywords\n\nsearch_radius = 0.0: The fixed search radius. The default of 0.0 is useful together with copy_neighborhood_search.\nn_points = 0: Total number of points. The default of 0 is useful together with copy_neighborhood_search.\nperiodic_box = nothing: In order to use a (rectangular) periodic domain, pass a PeriodicBox.\ncell_list: The cell list that maps a cell index to a list of points inside the cell. By default, a DictionaryCellList is used.\nupdate_strategy = nothing: Strategy to parallelize update!. Available options are:\nnothing: Automatically choose the best available option.\nParallelUpdate(): This is not available for all cell list implementations, but is the default when available.\nSemiParallelUpdate(): This is available for all cell list implementations and is the default when ParallelUpdate is not available.\nSerialUpdate()\n\nReferences\n\nM. Chalela, E. Sillero, L. Pereyra, M.A. Garcia, J.B. Cabral, M. Lares, M. Merchán. \"GriSPy: A Python package for fixed-radius nearest neighbors search\". In: Astronomy and Computing 34 (2021). doi: 10.1016/j.ascom.2020.100443\nMarkus Ihmsen, Nadir Akinci, Markus Becker, Matthias Teschner. \"A Parallel SPH Implementation on Multi-Core CPUs\". In: Computer Graphics Forum 30.1 (2011), pages 99–112. doi: 10.1111/J.1467-8659.2010.01832.X\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.ParallelUpdate","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.ParallelUpdate","text":"ParallelUpdate()\n\nFully parallel update by using atomic operations to avoid race conditions when adding points into the same cell. This is not available for all cell list implementations, but is the default when available.\n\nSee GridNeighborhoodSearch for usage information.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.PeriodicBox","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.PeriodicBox","text":"PeriodicBox(; min_corner, max_corner)\n\nDefine a rectangular (axis-aligned) periodic domain.\n\nKeywords\n\nmin_corner: Coordinates of the domain corner in negative coordinate directions.\nmax_corner: Coordinates of the domain corner in positive coordinate directions.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.PolyesterBackend","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.PolyesterBackend","text":"PolyesterBackend()\n\nPass as first argument to the @threaded macro to make the loop multithreaded with Polyester.@batch.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.PrecomputedNeighborhoodSearch","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.PrecomputedNeighborhoodSearch","text":"PrecomputedNeighborhoodSearch{NDIMS}(; search_radius = 0.0, n_points = 0,\n periodic_box = nothing, update_strategy = nothing)\n\nNeighborhood search with precomputed neighbor lists. A list of all neighbors is computed for each point during initialization and update. This neighborhood search maximizes the performance of neighbor loops at the cost of a much slower update!.\n\nA GridNeighborhoodSearch is used internally to compute the neighbor lists during initialization and update.\n\nArguments\n\nNDIMS: Number of dimensions.\n\nKeywords\n\nsearch_radius = 0.0: The fixed search radius. The default of 0.0 is useful together with copy_neighborhood_search.\nn_points = 0: Total number of points. The default of 0 is useful together with copy_neighborhood_search.\nperiodic_box = nothing: In order to use a (rectangular) periodic domain, pass a PeriodicBox.\nupdate_strategy: Strategy to parallelize update! of the internally used GridNeighborhoodSearch. See GridNeighborhoodSearch for available options.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.SemiParallelUpdate","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.SemiParallelUpdate","text":"SemiParallelUpdate()\n\nLoop over all cells in parallel to mark cells with points that now belong to a different cell. Then, move points of affected cells serially to avoid race conditions. This is available for all cell list implementations and is the default when ParallelUpdate is not available.\n\nSee GridNeighborhoodSearch for usage information.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.SerialUpdate","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.SerialUpdate","text":"SerialUpdate()\n\nDeactivate parallelization in the neighborhood search update. Parallel neighborhood search update can be one of the largest sources of error variations between simulations with different thread numbers due to neighbor ordering changes.\n\nSee GridNeighborhoodSearch for usage information.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.ThreadsDynamicBackend","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.ThreadsDynamicBackend","text":"ThreadsDynamicBackend()\n\nPass as first argument to the @threaded macro to make the loop multithreaded with Threads.@threads :dynamic.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.ThreadsStaticBackend","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.ThreadsStaticBackend","text":"ThreadsStaticBackend()\n\nPass as first argument to the @threaded macro to make the loop multithreaded with Threads.@threads :static.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.TrivialNeighborhoodSearch","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.TrivialNeighborhoodSearch","text":"TrivialNeighborhoodSearch{NDIMS}(; search_radius = 0.0, eachpoint = 1:0,\n periodic_box = nothing)\n\nTrivial neighborhood search that simply loops over all points.\n\nArguments\n\nNDIMS: Number of dimensions.\n\nKeywords\n\nsearch_radius = 0.0: The fixed search radius. The default of 0.0 is useful together with copy_neighborhood_search.\neachpoint = 1:0: Iterator for all point indices. Usually just 1:n_points. The default of 1:0 is useful together with copy_neighborhood_search.\nperiodic_box = nothing: In order to use a (rectangular) periodic domain, pass a PeriodicBox.\n\n\n\n\n\n","category":"type"},{"location":"reference-pointneighbors/#PointNeighbors.copy_neighborhood_search-Tuple{PointNeighbors.AbstractNeighborhoodSearch, Any, Any}","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.copy_neighborhood_search","text":"copy_neighborhood_search(search::AbstractNeighborhoodSearch, search_radius, n_points;\n eachpoint = 1:n_points)\n\nCreate a new uninitialized neighborhood search of the same type and with the same configuration options as search, but with a different search radius and number of points.\n\nThe TrivialNeighborhoodSearch also requires an iterator eachpoint, which most of the time will be 1:n_points. If the TrivialNeighborhoodSearch is never going to be used, the keyword argument eachpoint can be ignored.\n\nThis is useful when a simulation code requires multiple neighborhood searches of the same kind. One can then just pass an empty neighborhood search as a template and use this function inside the simulation code to generate similar neighborhood searches with different search radii and different numbers of points.\n\n# Template\nnhs = GridNeighborhoodSearch{2}()\n\n# Inside the simulation code, generate similar neighborhood searches\nnhs1 = copy_neighborhood_search(nhs, 1.0, 100)\n\n# output\nGridNeighborhoodSearch{2, Float64, ...}(...)\n\n\n\n\n\n","category":"method"},{"location":"reference-pointneighbors/#PointNeighbors.foreach_point_neighbor-Union{Tuple{T}, Tuple{T, Any, Any, Any}} where T","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.foreach_point_neighbor","text":"foreach_point_neighbor(f, system_coords, neighbor_coords, neighborhood_search;\n points = axes(system_coords, 2), parallel = true)\n\nLoop for each point in system_coords over all points in neighbor_coords whose distances to that point are smaller than the search radius and execute the function f(i, j, x, y, d), where\n\ni is the column index of the point in system_coords,\nj the column index of the neighbor in neighbor_coords,\nx an SVector of the coordinates of the point (system_coords[:, i]),\ny an SVector of the coordinates of the neighbor (neighbor_coords[:, j]),\nd the distance between x and y.\n\nThe neighborhood_search must have been initialized or updated with system_coords as first coordinate array and neighbor_coords as second coordinate array.\n\nNote that system_coords and neighbor_coords can be identical.\n\nArguments\n\nf: The function explained above.\nsystem_coords: A matrix where the i-th column contains the coordinates of point i.\nneighbor_coords: A matrix where the j-th column contains the coordinates of point j.\nneighborhood_search: A neighborhood search initialized or updated with system_coords as first coordinate array and neighbor_coords as second coordinate array.\n\nKeywords\n\npoints: Loop over these point indices. By default all columns of system_coords.\nparallel=true: Run the outer loop over points thread-parallel.\n\nSee also initialize!, update!.\n\n\n\n\n\n","category":"method"},{"location":"reference-pointneighbors/#PointNeighbors.initialize!-Tuple{PointNeighbors.AbstractNeighborhoodSearch, Any, Any}","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.initialize!","text":"initialize!(search::AbstractNeighborhoodSearch, x, y)\n\nInitialize a neighborhood search with the two coordinate arrays x and y.\n\nIn general, the purpose of a neighborhood search is to find for one point in x all points in y whose distances to that point are smaller than the search radius. x and y are expected to be matrices, where the i-th column contains the coordinates of point i. Note that x and y can be identical.\n\nSee also update!.\n\n\n\n\n\n","category":"method"},{"location":"reference-pointneighbors/#PointNeighbors.update!-Tuple{PointNeighbors.AbstractNeighborhoodSearch, Any, Any}","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.update!","text":"update!(search::AbstractNeighborhoodSearch, x, y; points_moving = (true, true))\n\nUpdate an already initialized neighborhood search with the two coordinate arrays x and y.\n\nLike initialize!, but reusing the existing data structures of the already initialized neighborhood search. When the points only moved a small distance since the last update! or initialize!, this is significantly faster than initialize!.\n\nNot all implementations support incremental updates. If incremental updates are not possible for an implementation, update! will fall back to a regular initialize!.\n\nSome neighborhood searches might not need to update when only x changed since the last update! or initialize! and y did not change. Pass points_moving = (true, false) in this case to avoid unnecessary updates. The first flag in points_moving indicates if points in x are moving. The second flag indicates if points in y are moving.\n\nwarning: Experimental Feature: Backend Specification\nThe keyword argument parallelization_backend allows users to specify the multithreading backend. This feature is currently considered experimental!Possible parallelization backends are:ThreadsDynamicBackend to use Threads.@threads :dynamic\nThreadsStaticBackend to use Threads.@threads :static\nPolyesterBackend to use Polyester.@batch\nKernelAbstractions.Backend to launch a GPU kernel\n\nSee also initialize!.\n\n\n\n\n\n","category":"method"},{"location":"reference-pointneighbors/#PointNeighbors.@threaded-Tuple{Any, Any}","page":"PointNeighbors.jl API Reference","title":"PointNeighbors.@threaded","text":"@threaded x for ... end\n\nRun either a threaded CPU loop or launch a kernel on the GPU, depending on the type of x. Semantically the same as Threads.@threads when iterating over a AbstractUnitRange but without guarantee that the underlying implementation uses Threads.@threads or works for more general for loops.\n\nThe first argument must either be a parallelization backend (see below) or an array from which the backend can be derived to determine if the loop must be run threaded on the CPU or launched as a kernel on the GPU. Passing KernelAbstractions.CPU() will run the GPU kernel on the CPU.\n\nPossible parallelization backends are:\n\nPolyesterBackend to use Polyester.@batch\nThreadsDynamicBackend to use Threads.@threads :dynamic\nThreadsStaticBackend to use Threads.@threads :static\nKernelAbstractions.Backend to execute the loop as a GPU kernel\n\nIn particular, the underlying threading capabilities might be provided by other packages such as Polyester.jl.\n\nwarning: Warning\nThis macro does not necessarily work for general for loops. For example, it does not necessarily support general iterables such as eachline(filename).\n\n\n\n\n\n","category":"macro"}] } diff --git a/previews/PR514/systems/boundary/index.html b/previews/PR514/systems/boundary/index.html index 059521cd7..4f1e448f3 100644 --- a/previews/PR514/systems/boundary/index.html +++ b/previews/PR514/systems/boundary/index.html @@ -1,22 +1,67 @@ -Boundary · TrixiParticles.jl
GitHub

Boundary System

TrixiParticles.BoundarySPHSystem — Type
BoundarySPHSystem(initial_condition, boundary_model; movement=nothing)

System for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.

Arguments

Keyword Arguments

source
TrixiParticles.BoundaryDEMSystem — Type
BoundaryDEMSystem(initial_condition, normal_stiffness)

System for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.

Experimental Implementation

This is an experimental feature and may change in a future releases.

source
TrixiParticles.BoundaryMovement — Type
BoundaryMovement(movement_function, is_moving; moving_particles=nothing)

Arguments

  • movement_function: Time-dependent function returning an SVector of $d$ dimensions for a $d$-dimensional problem.
  • is_moving: Function to determine in each timestep if the particles are moving or not. Its boolean return value is mandatory to determine if the neighborhood search will be updated.

Keyword Arguments

  • moving_particles: Indices of moving particles. Default is each particle in BoundarySPHSystem.

In the example below, movement describes particles moving in a circle as long as the time is lower than 1.5.

Examples

movement_function(t) = SVector(cos(2pi*t), sin(2pi*t))
+Boundary · TrixiParticles.jl
GitHub
Boundary System
TrixiParticles.BoundarySPHSystem — Type
BoundarySPHSystem(initial_condition, boundary_model; movement=nothing, adhesion_coefficient=0.0)
System for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.
Arguments
Keyword Arguments
  • movement: For moving boundaries, a BoundaryMovement can be passed.
  • adhesion_coefficient: Coefficient specifying the adhesion of a fluid to the surface.  Note: currently it is assumed that all fluids have the same adhesion coefficient.
source
TrixiParticles.BoundaryDEMSystem — Type
BoundaryDEMSystem(initial_condition, normal_stiffness)
System for boundaries modeled by boundary particles. The interaction between fluid and boundary particles is specified by the boundary model.
Experimental Implementation
This is an experimental feature and may change in a future releases.
source
TrixiParticles.BoundaryMovement — Type
BoundaryMovement(movement_function, is_moving; moving_particles=nothing)
Arguments
  • movement_function: Time-dependent function returning an SVector of $d$ dimensions                      for a $d$-dimensional problem.
  • is_moving: Function to determine in each timestep if the particles are moving or not. Its   boolean return value is mandatory to determine if the neighborhood search will be updated.
Keyword Arguments
  • moving_particles: Indices of moving particles. Default is each particle in BoundarySPHSystem.
In the example below, movement describes particles moving in a circle as long as the time is lower than 1.5.
Examples
movement_function(t) = SVector(cos(2pi*t), sin(2pi*t))
 is_moving(t) = t < 1.5
 
-movement = BoundaryMovement(movement_function, is_moving)
source
Boundary Models
Dummy Particles
Boundaries modeled as dummy particles, which are treated like fluid particles, but their positions and velocities are not evolved in time. Since the force towards the fluid should not change with the material density when used with a TotalLagrangianSPHSystem, the dummy particles need to have a mass corresponding to the fluid's rest density, which we call "hydrodynamic mass", as opposed to mass corresponding to the material density of a TotalLagrangianSPHSystem.
Here, initial_density and hydrodynamic_mass are vectors that contains the initial density and the hydrodynamic mass respectively for each boundary particle. Note that when used with SummationDensity (see below), this is only used to determine the element type and the number of boundary particles.
To establish a relationship between density and pressure, a state_equation has to be passed, which should be the same as for the adjacent fluid systems. To sum over neighboring particles, a smoothing_kernel and smoothing_length needs to be passed. This should be the same as for the adjacent fluid system with the largest smoothing length.
In the literature, this kind of boundary particles is referred to as "dummy particles" (Adami et al., 2012 and Valizadeh & Monaghan, 2015), "frozen fluid particles" (Akinci et al., 2012) or "dynamic boundaries (Crespo et al., 2007). The key detail of this boundary condition and the only difference between the boundary models in these references is the way the density and pressure of boundary particles is computed.
Since boundary particles are treated like fluid particles, the force on fluid particle $a$ due to boundary particle $b$ is given by
\[f_{ab} = m_a m_b \left( \frac{p_a}{\rho_a^2} + \frac{p_b}{\rho_b^2} \right) \nabla_{r_a} W(\Vert r_a - r_b \Vert, h).\]
The quantities to be defined here are the density $\rho_b$ and pressure $p_b$ of the boundary particle $b$.
Boundary Models
Dummy Particles
Boundaries modeled as dummy particles, which are treated like fluid particles, but their positions and velocities are not evolved in time. Since the force towards the fluid should not change with the material density when used with a TotalLagrangianSPHSystem, the dummy particles need to have a mass corresponding to the fluid's rest density, which we call "hydrodynamic mass", as opposed to mass corresponding to the material density of a TotalLagrangianSPHSystem.
Here, initial_density and hydrodynamic_mass are vectors that contains the initial density and the hydrodynamic mass respectively for each boundary particle. Note that when used with SummationDensity (see below), this is only used to determine the element type and the number of boundary particles.
To establish a relationship between density and pressure, a state_equation has to be passed, which should be the same as for the adjacent fluid systems. To sum over neighboring particles, a smoothing_kernel and smoothing_length needs to be passed. This should be the same as for the adjacent fluid system with the largest smoothing length.
In the literature, this kind of boundary particles is referred to as "dummy particles" (Adami et al., 2012 and Valizadeh & Monaghan, 2015), "frozen fluid particles" (Akinci et al., 2012) or "dynamic boundaries Crespo et al., 2007. The key detail of this boundary condition and the only difference between the boundary models in these references is the way the density and pressure of boundary particles is computed.
Since boundary particles are treated like fluid particles, the force on fluid particle $a$ due to boundary particle $b$ is given by
\[f_{ab} = m_a m_b \left( \frac{p_a}{\rho_a^2} + \frac{p_b}{\rho_b^2} \right) \nabla_{r_a} W(\Vert r_a - r_b \Vert, h).\]
The quantities to be defined here are the density $\rho_b$ and pressure $p_b$ of the boundary particle $b$.
TrixiParticles.BoundaryModelDummyParticles — Type
BoundaryModelDummyParticles(initial_density, hydrodynamic_mass,
                             density_calculator, smoothing_kernel,
                             smoothing_length; viscosity=nothing,
-                            state_equation=nothing, correction=nothing)
boundary_model for BoundarySPHSystem.
Arguments
  • initial_density: Vector holding the initial density of each boundary particle.
  • hydrodynamic_mass: Vector holding the "hydrodynamic mass" of each boundary particle.                      See description above for more information.
  • density_calculator: Strategy to compute the hydrodynamic density of the boundary particles.                       See description below for more information.
  • smoothing_kernel: Smoothing kernel should be the same as for the adjacent fluid system.
  • smoothing_length: Smoothing length should be the same as for the adjacent fluid system.
Keywords
  • state_equation: This should be the same as for the adjacent fluid system                   (see e.g. StateEquationCole).
  • correction:     Correction method of the adjacent fluid system (see Corrections).
  • viscosity:      Slip (default) or no-slip condition. See description below for further                   information.
Examples
# Free-slip condition
+                            state_equation=nothing, correction=nothing)
Boundary model for BoundarySPHSystem.
Arguments
  • initial_density: Vector holding the initial density of each boundary particle.
  • hydrodynamic_mass: Vector holding the "hydrodynamic mass" of each boundary particle.                      See description above for more information.
  • density_calculator: Strategy to compute the hydrodynamic density of the boundary particles.                       See description below for more information.
  • smoothing_kernel: Smoothing kernel should be the same as for the adjacent fluid system.
  • smoothing_length: Smoothing length should be the same as for the adjacent fluid system.
Keywords
  • state_equation: This should be the same as for the adjacent fluid system                   (see e.g. StateEquationCole).
  • correction:     Correction method of the adjacent fluid system (see Corrections).
  • viscosity:      Slip (default) or no-slip condition. See description below for further                   information.
Examples
# Free-slip condition
 boundary_model = BoundaryModelDummyParticles(densities, masses, AdamiPressureExtrapolation(),
                                              smoothing_kernel, smoothing_length)
 
 # No-slip condition
 boundary_model = BoundaryModelDummyParticles(densities, masses, AdamiPressureExtrapolation(),
                                              smoothing_kernel, smoothing_length,
-                                             viscosity=ViscosityAdami(nu=1e-6))
source
Hydrodynamic density of dummy particles
We provide five options to compute the boundary density and pressure, determined by the density_calculator:
  1. (Recommended) With AdamiPressureExtrapolation, the pressure is extrapolated from the pressure of the fluid according to (Adami et al., 2012), and the density is obtained by applying the inverse of the state equation. This option usually yields the best results of the options listed here.
  2. With SummationDensity, the density is calculated by summation over the neighboring particles, and the pressure is computed from the density with the state equation.
  3. With ContinuityDensity, the density is integrated from the continuity equation, and the pressure is computed from the density with the state equation. Note that this causes a gap between fluid and boundary where the boundary is initialized without any contact to the fluid. This is due to overestimation of the boundary density as soon as the fluid comes in contact with boundary particles that initially did not have contact to the fluid. Therefore, in dam break simulations, there is a visible "step", even though the boundary is supposed to be flat. See also dual.sphysics.org/faq/#Q_13.
  4. With PressureZeroing, the density is set to the reference density and the pressure is computed from the density with the state equation. This option is not recommended. The other options yield significantly better results.
  5. With PressureMirroring, the density is set to the reference density. The pressure is not used. Instead, the fluid pressure is mirrored as boundary pressure in the momentum equation. This option is not recommended due to stability issues. See PressureMirroring for more details.
1. AdamiPressureExtrapolation
The pressure of the boundary particles is obtained by extrapolating the pressure of the fluid according to (Adami et al., 2012). The pressure of a boundary particle $b$ is given by
\[p_b = \frac{\sum_f (p_f + \rho_f (\bm{g} - \bm{a}_b) \cdot \bm{r}_{bf}) W(\Vert r_{bf} \Vert, h)}{\sum_f W(\Vert r_{bf} \Vert, h)},\]
where the sum is over all fluid particles, $\rho_f$ and $p_f$ denote the density and pressure of fluid particle $f$, respectively, $r_{bf} = r_b - r_f$ denotes the difference of the coordinates of particles $b$ and $f$, $\bm{g}$ denotes the gravitational acceleration acting on the fluid, and $\bm{a}_b$ denotes the acceleration of the boundary particle $b$.
4. PressureZeroing
This is the simplest way to implement dummy boundary particles. The density of each particle is set to the reference density and the pressure to the reference pressure (the corresponding pressure to the reference density by the state equation).
TrixiParticles.PressureZeroing — Type
PressureZeroing()
density_calculator for BoundaryModelDummyParticles.
Note
This boundary model produces significantly worse results than all other models and is only included for research purposes.
source
5. PressureMirroring
Instead of calculating density and pressure for each boundary particle, we modify the momentum equation,
\[\frac{\mathrm{d}v_a}{\mathrm{d}t} = -\sum_b m_b \left( \frac{p_a}{\rho_a^2} + \frac{p_b}{\rho_b^2} \right) \nabla_a W_{ab}\]
to replace the unknown density $\rho_b$ if $b$ is a boundary particle by the reference density and the unknown pressure $p_b$ if $b$ is a boundary particle by the pressure $p_a$ of the interacting fluid particle. The momentum equation therefore becomes
\[\frac{\mathrm{d}v_a}{\mathrm{d}t} = -\sum_f m_f \left( \frac{p_a}{\rho_a^2} + \frac{p_f}{\rho_f^2} \right) \nabla_a W_{af}
--\sum_b m_b \left( \frac{p_a}{\rho_a^2} + \frac{p_a}{\rho_0^2} \right) \nabla_a W_{ab},\]
where the first sum is over all fluid particles and the second over all boundary particles.
This approach was first mentioned by Akinci et al. (2012) and written down in this form by Band et al. (2018).
TrixiParticles.PressureMirroring — Type
PressureMirroring()
density_calculator for BoundaryModelDummyParticles.
Note
This boundary model requires high viscosity for stability with WCSPH. It also produces significantly worse results than AdamiPressureExtrapolation and is not more efficient because smaller time steps are required due to more noise in the pressure. We added this model only for research purposes and for comparison with SPlisHSPlasH.
source
No-slip conditions
For the interaction of dummy particles and fluid particles, Adami et al. (2012) impose a no-slip boundary condition by assigning a wall velocity $v_w$ to the dummy particle.
The wall velocity of particle $a$ is calculated from the prescribed boundary particle velocity $v_a$ and the smoothed velocity field
\[v_w = 2 v_a - \frac{\sum_b v_b W_{ab}}{\sum_b W_{ab}},\]
where the sum is over all fluid particles.
By choosing the viscosity model ViscosityAdami for viscosity, a no-slip condition is imposed. It is recommended to choose nu in the order of either the kinematic viscosity parameter of the adjacent fluid or the equivalent from the artificial parameter alpha of the adjacent fluid ($\nu = \frac{\alpha h c }{2d + 4}$). When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.
Warning
The viscosity model ArtificialViscosityMonaghan for BoundaryModelDummyParticles has not been verified yet.
References
  • S. Adami, X. Y. Hu, N. A. Adams. "A generalized wall boundary condition for smoothed particle hydrodynamics". In: Journal of Computational Physics 231, 21 (2012), pages 7057–7075. doi: 10.1016/J.JCP.2012.05.005
  • Alireza Valizadeh, Joseph J. Monaghan. "A study of solid wall models for weakly compressible SPH". In: Journal of Computational Physics 300 (2015), pages 5–19. doi: 10.1016/J.JCP.2015.07.033
  • Nadir Akinci, Markus Ihmsen, Gizem Akinci, Barbara Solenthaler, Matthias Teschner. "Versatile rigid-fluid coupling for incompressible SPH". ACM Transactions on Graphics 31, 4 (2012), pages 1–8. doi: 10.1145/2185520.2185558
  • A. J. C. Crespo, M. Gómez-Gesteira, R. A. Dalrymple. "Boundary conditions generated by dynamic particles in SPH methods" In: Computers, Materials and Continua 5 (2007), pages 173-184. doi: 10.3970/cmc.2007.005.173
  • Stefan Band, Christoph Gissler, Andreas Peer, and Matthias Teschner. "MLS Pressure Boundaries for Divergence-Free and Viscous SPH Fluids." In: Computers & Graphics 76 (2018), pages 37–46. doi: 10.1016/j.cag.2018.08.001
Repulsive Particles
Boundaries modeled as boundary particles which exert forces on the fluid particles (Monaghan, Kajtar, 2009). The force on fluid particle $a$ due to boundary particle $b$ is given by
\[f_{ab} = m_a \left(\tilde{f}_{ab} - m_b \Pi_{ab} \nabla_{r_a} W(\Vert r_a - r_b \Vert, h)\right)\]
with
\[\tilde{f}_{ab} = \frac{K}{\beta^{n-1}} \frac{r_{ab}}{\Vert r_{ab} \Vert (\Vert r_{ab} \Vert - d)} \Phi(\Vert r_{ab} \Vert, h)
-\frac{2 m_b}{m_a + m_b},\]
where $m_a$ and $m_b$ are the masses of fluid particle $a$ and boundary particle $b$ respectively, $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$, $d$ denotes the boundary particle spacing and $n$ denotes the number of dimensions (see (Monaghan, Kajtar, 2009, Equation (3.1)) and (Valizadeh, Monaghan, 2015)). Note that the repulsive acceleration $\tilde{f}_{ab}$ does not depend on the masses of the boundary particles. Here, $\Phi$ denotes the 1D Wendland C4 kernel, normalized to $1.77$ for $q=0$ (Monaghan, Kajtar, 2009, Section 4), with $\Phi(r, h) = w(r/h)$ and
\[w(q) =
+                                             viscosity=ViscosityAdami(nu=1e-6))
source

Hydrodynamic density of dummy particles

We provide six options to compute the boundary density and pressure, determined by the density_calculator:

  1. (Recommended) With AdamiPressureExtrapolation, the pressure is extrapolated from the pressure of the fluid according to Adami et al., 2012, and the density is obtained by applying the inverse of the state equation. This option usually yields the best results of the options listed here.
  2. (Only relevant for FSI) With BernoulliPressureExtrapolation, the pressure is extrapolated from the pressure similar to the AdamiPressureExtrapolation, but a relative velocity-dependent pressure part is calculated between moving solids and fluids, which increases the boundary pressure in areas prone to penetrations.
  3. With SummationDensity, the density is calculated by summation over the neighboring particles, and the pressure is computed from the density with the state equation.
  4. With ContinuityDensity, the density is integrated from the continuity equation, and the pressure is computed from the density with the state equation. Note that this causes a gap between fluid and boundary where the boundary is initialized without any contact to the fluid. This is due to overestimation of the boundary density as soon as the fluid comes in contact with boundary particles that initially did not have contact to the fluid. Therefore, in dam break simulations, there is a visible "step", even though the boundary is supposed to be flat. See also dual.sphysics.org/faq/#Q_13.
  5. With PressureZeroing, the density is set to the reference density and the pressure is computed from the density with the state equation. This option is not recommended. The other options yield significantly better results.
  6. With PressureMirroring, the density is set to the reference density. The pressure is not used. Instead, the fluid pressure is mirrored as boundary pressure in the momentum equation. This option is not recommended due to stability issues. See PressureMirroring for more details.

1. AdamiPressureExtrapolation

The pressure of the boundary particles is obtained by extrapolating the pressure of the fluid according to Adami et al., 2012. The pressure of a boundary particle $b$ is given by

\[p_b = \frac{\sum_f (p_f + \rho_f (\bm{g} - \bm{a}_b) \cdot \bm{r}_{bf}) W(\Vert r_{bf} \Vert, h)}{\sum_f W(\Vert r_{bf} \Vert, h)},\]

where the sum is over all fluid particles, $\rho_f$ and $p_f$ denote the density and pressure of fluid particle $f$, respectively, $r_{bf} = r_b - r_f$ denotes the difference of the coordinates of particles $b$ and $f$, $\bm{g}$ denotes the gravitational acceleration acting on the fluid, and $\bm{a}_b$ denotes the acceleration of the boundary particle $b$.

TrixiParticles.AdamiPressureExtrapolation — Type
AdamiPressureExtrapolation(; pressure_offset=0.0)

density_calculator for BoundaryModelDummyParticles.

Keywords

  • pressure_offset=0.0: Sometimes it is necessary to artificially increase the boundary pressure to prevent penetration, which is possible by increasing this value.
source

2. BernoulliPressureExtrapolation

Identical to the pressure $p_b$ calculated via AdamiPressureExtrapolation, but it adds the dynamic pressure component of the Bernoulli equation:

\[p_b = \frac{\sum_f (p_f + \frac{1}{2} \, \rho_{\text{neighbor}} \left( \frac{ (\mathbf{v}_f - \mathbf{v}_{\text{body}}) \cdot (\mathbf{x}_f - \mathbf{x}_{\text{neighbor}}) }{ \left\| \mathbf{x}_f - \mathbf{x}_{\text{neighbor}} \right\| } \right)^2 \times \text{factor} +\rho_f (\bm{g} - \bm{a}_b) \cdot \bm{r}_{bf}) W(\Vert r_{bf} \Vert, h)}{\sum_f W(\Vert r_{bf} \Vert, h)} \]

where $\mathbf{v}_f$ is the velocity of the fluid and $\mathbf{v}_{\text{body}}$ is the velocity of the body. This adjustment provides a higher boundary pressure for solid bodies moving with a relative velocity to the fluid to prevent penetration. This modification is original and not derived from any literature source.

TrixiParticles.BernoulliPressureExtrapolation — Type
BernoulliPressureExtrapolation(; pressure_offset=0.0, factor=1.0)

density_calculator for BoundaryModelDummyParticles.

Keywords

  • pressure_offset=0.0: Sometimes it is necessary to artificially increase the boundary pressure to prevent penetration, which is possible by increasing this value.
  • factor=1.0 : Setting factor allows to just increase the strength of the dynamic pressure part.
source

5. PressureZeroing

This is the simplest way to implement dummy boundary particles. The density of each particle is set to the reference density and the pressure to the reference pressure (the corresponding pressure to the reference density by the state equation).

TrixiParticles.PressureZeroing — Type
PressureZeroing()

density_calculator for BoundaryModelDummyParticles.

Note

This boundary model produces significantly worse results than all other models and is only included for research purposes.

source

6. PressureMirroring

Instead of calculating density and pressure for each boundary particle, we modify the momentum equation,

\[\frac{\mathrm{d}v_a}{\mathrm{d}t} = -\sum_b m_b \left( \frac{p_a}{\rho_a^2} + \frac{p_b}{\rho_b^2} \right) \nabla_a W_{ab}\]

to replace the unknown density $\rho_b$ if $b$ is a boundary particle by the reference density and the unknown pressure $p_b$ if $b$ is a boundary particle by the pressure $p_a$ of the interacting fluid particle. The momentum equation therefore becomes

\[\frac{\mathrm{d}v_a}{\mathrm{d}t} = -\sum_f m_f \left( \frac{p_a}{\rho_a^2} + \frac{p_f}{\rho_f^2} \right) \nabla_a W_{af} +-\sum_b m_b \left( \frac{p_a}{\rho_a^2} + \frac{p_a}{\rho_0^2} \right) \nabla_a W_{ab},\]

where the first sum is over all fluid particles and the second over all boundary particles.

This approach was first mentioned by Akinci et al. (2012) and written down in this form by Band et al. (2018).

TrixiParticles.PressureMirroring — Type
PressureMirroring()

density_calculator for BoundaryModelDummyParticles.

Note

This boundary model requires high viscosity for stability with WCSPH. It also produces significantly worse results than AdamiPressureExtrapolation and is not more efficient because smaller time steps are required due to more noise in the pressure. We added this model only for research purposes and for comparison with SPlisHSPlasH.

source

No-slip conditions

For the interaction of dummy particles and fluid particles, Adami et al. (2012) impose a no-slip boundary condition by assigning a wall velocity $v_w$ to the dummy particle.

The wall velocity of particle $a$ is calculated from the prescribed boundary particle velocity $v_a$ and the smoothed velocity field

\[v_w = 2 v_a - \frac{\sum_b v_b W_{ab}}{\sum_b W_{ab}},\]

where the sum is over all fluid particles.

By choosing the viscosity model ViscosityAdami for viscosity, a no-slip condition is imposed. It is recommended to choose nu in the order of either the kinematic viscosity parameter of the adjacent fluid or the equivalent from the artificial parameter alpha of the adjacent fluid ($\nu = \frac{\alpha h c }{2d + 4}$). When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.

Warning

The viscosity model ArtificialViscosityMonaghan for BoundaryModelDummyParticles has not been verified yet.

Repulsive Particles

Boundaries modeled as boundary particles which exert forces on the fluid particles (Monaghan, Kajtar, 2009). The force on fluid particle $a$ due to boundary particle $b$ is given by

\[f_{ab} = m_a \left(\tilde{f}_{ab} - m_b \Pi_{ab} \nabla_{r_a} W(\Vert r_a - r_b \Vert, h)\right)\]

with

\[\tilde{f}_{ab} = \frac{K}{\beta^{n-1}} \frac{r_{ab}}{\Vert r_{ab} \Vert (\Vert r_{ab} \Vert - d)} \Phi(\Vert r_{ab} \Vert, h) +\frac{2 m_b}{m_a + m_b},\]

where $m_a$ and $m_b$ are the masses of fluid particle $a$ and boundary particle $b$ respectively, $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$, $d$ denotes the boundary particle spacing and $n$ denotes the number of dimensions (see Monaghan & Kajtar, 2009, Equation (3.1) and Valizadeh & Monaghan, 2015). Note that the repulsive acceleration $\tilde{f}_{ab}$ does not depend on the masses of the boundary particles. Here, $\Phi$ denotes the 1D Wendland C4 kernel, normalized to $1.77$ for $q=0$ (Monaghan & Kajtar, 2009, Section 4), with $\Phi(r, h) = w(r/h)$ and

\[w(q) = \begin{cases} (1.77/32) (1 + (5/2)q + 2q^2)(2 - q)^5 & \text{if } 0 \leq q < 2 \\ 0 & \text{if } q \geq 2. -\end{cases}\]

The boundary particles are assumed to have uniform spacing by the factor $\beta$ smaller than the expected fluid particle spacing. For example, if the fluid particles have an expected spacing of $0.3$ and the boundary particles have a uniform spacing of $0.1$, then this parameter should be set to $\beta = 3$. According to (Monaghan, Kajtar, 2009), a value of $\beta = 3$ for the Wendland C4 that we use here is reasonable for most computing purposes.

The parameter $K$ is used to scale the force exerted by the boundary particles. In (Monaghan, Kajtar, 2009), a value of $gD$ is used for static tank simulations, where $g$ is the gravitational acceleration and $D$ is the depth of the fluid.

The viscosity $\Pi_{ab}$ is calculated according to the viscosity used in the simulation, where the density of the boundary particle if needed is assumed to be identical to the density of the fluid particle.

No-slip condition

By choosing the viscosity model ArtificialViscosityMonaghan for viscosity, a no-slip condition is imposed. When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.

Warning

The no-slip conditions for BoundaryModelMonaghanKajtar have not been verified yet.

TrixiParticles.BoundaryModelMonaghanKajtar — Type
BoundaryModelMonaghanKajtar(K, beta, boundary_particle_spacing, mass;
-                            viscosity=nothing)

boundary_model for BoundarySPHSystem.

Arguments

  • K: Scaling factor for repulsive force.
  • beta: Ratio of fluid particle spacing to boundary particle spacing.
  • boundary_particle_spacing: Boundary particle spacing.
  • mass: Vector holding the mass of each boundary particle.

Keywords

  • viscosity: Free-slip (default) or no-slip condition. See description above for further information.
source

References

  • Joseph J. Monaghan, Jules B. Kajtar. "SPH particle boundary forces for arbitrary boundaries". In: Computer Physics Communications 180.10 (2009), pages 1811–1820. doi: 10.1016/j.cpc.2009.05.008
  • Alireza Valizadeh, Joseph J. Monaghan. "A study of solid wall models for weakly compressible SPH." In: Journal of Computational Physics 300 (2015), pages 5–19. doi: 10.1016/J.JCP.2015.07.033
+\end{cases}\]

The boundary particles are assumed to have uniform spacing by the factor $\beta$ smaller than the expected fluid particle spacing. For example, if the fluid particles have an expected spacing of $0.3$ and the boundary particles have a uniform spacing of $0.1$, then this parameter should be set to $\beta = 3$. According to Monaghan & Kajtar (2009), a value of $\beta = 3$ for the Wendland C4 that we use here is reasonable for most computing purposes.

The parameter $K$ is used to scale the force exerted by the boundary particles. In Monaghan & Kajtar (2009), a value of $gD$ is used for static tank simulations, where $g$ is the gravitational acceleration and $D$ is the depth of the fluid.

The viscosity $\Pi_{ab}$ is calculated according to the viscosity used in the simulation, where the density of the boundary particle if needed is assumed to be identical to the density of the fluid particle.

No-slip condition

By choosing the viscosity model ArtificialViscosityMonaghan for viscosity, a no-slip condition is imposed. When omitting the viscous interaction (default viscosity=nothing), a free-slip wall boundary condition is applied.

Warning

The no-slip conditions for BoundaryModelMonaghanKajtar have not been verified yet.

TrixiParticles.BoundaryModelMonaghanKajtar — Type
BoundaryModelMonaghanKajtar(K, beta, boundary_particle_spacing, mass;
+                            viscosity=nothing)

Boundary model for BoundarySPHSystem.

Arguments

  • K: Scaling factor for repulsive force.
  • beta: Ratio of fluid particle spacing to boundary particle spacing.
  • boundary_particle_spacing: Boundary particle spacing.
  • mass: Vector holding the mass of each boundary particle.

Keywords

  • viscosity: Free-slip (default) or no-slip condition. See description above for further information.
source

Open Boundaries

TrixiParticles.OpenBoundarySPHSystem — Type
OpenBoundarySPHSystem(boundary_zone::Union{InFlow, OutFlow};
+                      fluid_system::FluidSystem, buffer_size::Integer,
+                      boundary_model,
+                      reference_velocity=nothing,
+                      reference_pressure=nothing,
+                      reference_density=nothing)

Open boundary system for in- and outflow particles.

Arguments

  • boundary_zone: Use InFlow for an inflow and OutFlow for an outflow boundary.

Keywords

  • fluid_system: The corresponding fluid system
  • boundary_model: Boundary model (see Open Boundary Models)
  • buffer_size: Number of buffer particles.
  • reference_velocity: Reference velocity is either a function mapping each particle's coordinates and time to its velocity, an array where the $i$-th column holds the velocity of particle $i$ or, for a constant fluid velocity, a vector holding this velocity.
  • reference_pressure: Reference pressure is either a function mapping each particle's coordinates and time to its pressure, a vector holding the pressure of each particle, or a scalar for a constant pressure over all particles.
  • reference_density: Reference density is either a function mapping each particle's coordinates and time to its density, a vector holding the density of each particle, or a scalar for a constant density over all particles.
Experimental Implementation

This is an experimental feature and may change in any future releases.

source
TrixiParticles.InFlow — Type
InFlow(; plane, flow_direction, density, particle_spacing,
+       initial_condition=nothing, extrude_geometry=nothing,
+       open_boundary_layers::Integer)

Inflow boundary zone for OpenBoundarySPHSystem.

The specified plane (line in 2D or rectangle in 3D) will be extruded in upstream direction (the direction opposite to flow_direction) to create a box for the boundary zone. There are three ways to specify the actual shape of the inflow:

  1. Don't pass initial_condition or extrude_geometry. The boundary zone box will then be filled with inflow particles (default).
  2. Specify extrude_geometry by passing a 1D shape in 2D or a 2D shape in 3D, which is then extruded in upstream direction to create the inflow particles.
    • In 2D, the shape must be either an initial condition with 2D coordinates, which lies on the line specified by plane, or an initial condition with 1D coordinates, which lies on the line specified by plane when a y-coordinate of 0 is added.
    • In 3D, the shape must be either an initial condition with 3D coordinates, which lies in the rectangle specified by plane, or an initial condition with 2D coordinates, which lies in the rectangle specified by plane when a z-coordinate of 0 is added.
  3. Specify initial_condition by passing a 2D initial condition in 2D or a 3D initial condition in 3D, which will be used for the inflow particles.
Note

Particles outside the boundary zone box will be removed.

Keywords

  • plane: Tuple of points defining a part of the surface of the domain. The points must either span a line in 2D or a rectangle in 3D. This line or rectangle is then extruded in upstream direction to obtain the boundary zone. In 2D, pass two points $(A, B)$, so that the interval $[A, B]$ is the inflow surface. In 3D, pass three points $(A, B, C)$, so that the rectangular inflow surface is spanned by the vectors $\widehat{AB}$ and $\widehat{AC}$. These two vectors must be orthogonal.
  • flow_direction: Vector defining the flow direction.
  • open_boundary_layers: Number of particle layers in upstream direction.
  • particle_spacing: The spacing between the particles (see InitialCondition).
  • density: Particle density (see InitialCondition).
  • initial_condition=nothing: InitialCondition for the inflow particles. Particles outside the boundary zone will be removed. Do not use together with extrude_geometry.
  • extrude_geometry=nothing: 1D shape in 2D or 2D shape in 3D, which lies on the plane and is extruded upstream to obtain the inflow particles. See point 2 above for more details.

Examples

# 2D
+plane_points = ([0.0, 0.0], [0.0, 1.0])
+flow_direction=[1.0, 0.0]
+
+inflow = InFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,
+                open_boundary_layers=4)
+
+# 3D
+plane_points = ([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0])
+flow_direction=[0.0, 0.0, 1.0]
+
+inflow = InFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,
+                open_boundary_layers=4)
+
+# 3D particles sampled as cylinder
+circle = SphereShape(0.1, 0.5, (0.5, 0.5), 1.0, sphere_type=RoundSphere())
+
+inflow = InFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,
+                extrude_geometry=circle, open_boundary_layers=4)
Experimental Implementation

This is an experimental feature and may change in any future releases.

source
TrixiParticles.OutFlow — Type
OutFlow(; plane, flow_direction, density, particle_spacing,
+        initial_condition=nothing, extrude_geometry=nothing,
+        open_boundary_layers::Integer)

Outflow boundary zone for OpenBoundarySPHSystem.

The specified plane (line in 2D or rectangle in 3D) will be extruded in downstream direction (the direction in flow_direction) to create a box for the boundary zone. There are three ways to specify the actual shape of the outflow:

  1. Don't pass initial_condition or extrude_geometry. The boundary zone box will then be filled with outflow particles (default).
  2. Specify extrude_geometry by passing a 1D shape in 2D or a 2D shape in 3D, which is then extruded in downstream direction to create the outflow particles.
    • In 2D, the shape must be either an initial condition with 2D coordinates, which lies on the line specified by plane, or an initial condition with 1D coordinates, which lies on the line specified by plane when a y-coordinate of 0 is added.
    • In 3D, the shape must be either an initial condition with 3D coordinates, which lies in the rectangle specified by plane, or an initial condition with 2D coordinates, which lies in the rectangle specified by plane when a z-coordinate of 0 is added.
  3. Specify initial_condition by passing a 2D initial condition in 2D or a 3D initial condition in 3D, which will be used for the outflow particles.
Note

Particles outside the boundary zone box will be removed.

Keywords

  • plane: Tuple of points defining a part of the surface of the domain. The points must either span a line in 2D or a rectangle in 3D. This line or rectangle is then extruded in downstream direction to obtain the boundary zone. In 2D, pass two points $(A, B)$, so that the interval $[A, B]$ is the outflow surface. In 3D, pass three points $(A, B, C)$, so that the rectangular outflow surface is spanned by the vectors $\widehat{AB}$ and $\widehat{AC}$. These two vectors must be orthogonal.
  • flow_direction: Vector defining the flow direction.
  • open_boundary_layers: Number of particle layers in downstream direction.
  • particle_spacing: The spacing between the particles (see InitialCondition).
  • density: Particle density (see InitialCondition).
  • initial_condition=nothing: InitialCondition for the outflow particles. Particles outside the boundary zone will be removed. Do not use together with extrude_geometry.
  • extrude_geometry=nothing: 1D shape in 2D or 2D shape in 3D, which lies on the plane and is extruded downstream to obtain the outflow particles. See point 2 above for more details.

Examples

# 2D
+plane_points = ([0.0, 0.0], [0.0, 1.0])
+flow_direction = [1.0, 0.0]
+
+outflow = OutFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,
+                  open_boundary_layers=4)
+
+# 3D
+plane_points = ([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0])
+flow_direction = [0.0, 0.0, 1.0]
+
+outflow = OutFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,
+                  open_boundary_layers=4)
+
+# 3D particles sampled as cylinder
+circle = SphereShape(0.1, 0.5, (0.5, 0.5), 1.0, sphere_type=RoundSphere())
+
+outflow = OutFlow(; plane=plane_points, particle_spacing=0.1, flow_direction, density=1.0,
+                  extrude_geometry=circle, open_boundary_layers=4)
Experimental Implementation

This is an experimental feature and may change in any future releases.

source

Open Boundary Models

Method of characteristics

TrixiParticles.BoundaryModelLastiwka — Type
BoundaryModelLastiwka()

Boundary model for OpenBoundarySPHSystem. This model uses the characteristic variables to propagate the appropriate values to the outlet or inlet and have been proposed by Lastiwka et al. (2009). For more information about the method see description below.

source

The difficulty in non-reflecting boundary conditions, also called open boundaries, is to determine the appropriate boundary values of the exact characteristics of the Euler equations. Assuming the flow near the boundaries is normal to the boundary and free of shock waves and significant viscous effects, it can be shown that three characteristic variables exist:

  • $J_1$, associated with convection of entropy and propagates at flow velocity,
  • $J_2$, downstream-running characteristics,
  • $J_3$, upstream-running characteristics.

Giles (1990) derived those variables based on a linearized set of governing equations:

\[J_1 = -c_s^2 (\rho - \rho_{\text{ref}}) + (p - p_{\text{ref}})\]

\[J_2 = \rho c_s (v - v_{\text{ref}}) + (p - p_{\text{ref}})\]

\[J_3 = - \rho c_s (v - v_{\text{ref}}) + (p - p_{\text{ref}})\]

where the subscript "ref" denotes the reference flow near the boundaries, which can be prescribed.

Specifying the reference variables is not equivalent to prescription of $\rho$, $v$ and $p$ directly, since the perturbation from the reference flow is allowed.

Lastiwka et al. (2009) applied the method of characteristic to SPH and determined the number of variables that should be prescribed at the boundary and the number which should be propagated from the fluid domain to the boundary:

  • For an inflow boundary:

    • Prescribe downstream-running characteristics $J_1$ and $J_2$
    • Transmit $J_3$ from the fluid domain (allow $J_3$ to propagate upstream to the boundary).

  • For an outflow boundary:

    • Prescribe upstream-running characteristic $J_3$
    • Transmit $J_1$ and $J_2$ from the fluid domain.

Prescribing is done by simply setting the characteristics to zero. To transmit the characteristics from the fluid domain, or in other words, to carry the information of the fluid to the boundaries, Negi et al. (2020) use a Shepard Interpolation

\[f_i = \frac{\sum_j^N f_j W_{ij}}{\sum_j^N W_{ij}},\]

where the $i$-th particle is a boundary particle, $f$ is either $J_1$, $J_2$ or $J_3$ and $N$ is the set of neighboring fluid particles.

To express pressure $p$, density $\rho$ and velocity $v$ as functions of the characteristic variables, the system of equations from the characteristic variables is inverted and gives

\[ \rho - \rho_{\text{ref}} = \frac{1}{c_s^2} \left( -J_1 + \frac{1}{2} J_2 + \frac{1}{2} J_3 \right),\]

\[u - u_{\text{ref}}= \frac{1}{2\rho c_s} \left( J_2 - J_3 \right),\]

\[p - p_{\text{ref}} = \frac{1}{2} \left( J_2 + J_3 \right).\]

With $J_1$, $J_2$ and $J_3$ determined, we can easily solve for the actual variables for each particle.

diff --git a/previews/PR514/systems/dem/index.html b/previews/PR514/systems/dem/index.html index 7147771f0..cb1a232c3 100644 --- a/previews/PR514/systems/dem/index.html +++ b/previews/PR514/systems/dem/index.html @@ -1,3 +1,3 @@ -Discrete Element Method (Solid) · TrixiParticles.jl
GitHub

Discrete Element Method

The Discrete Element Method (DEM) is a computational technique widely used in physics, engineering, and applied mathematics for simulating the mechanical behavior of granular materials, such as powders, sand, soil, or rock, as well as other discontinua. Unlike continuum mechanics that treats materials as continuous, DEM considers individual particles or elements and their interactions. This approach provides detailed insights into the micro-mechanical behavior of materials, making it particularly valuable in fields such as geomechanics, material science, and mechanical engineering.

Fundamental Principles

The core idea behind DEM is the discretization of a material system into a finite set of distinct, interacting mass elements (particles). These elements (particles) can vary in shape, size, and properties, and they interact with each other and possibly with their boundaries through contact forces and potential fields. The motion and behavior of each mass element are governed by Newton's laws of motion, accounting for the forces and moments acting upon them.

TrixiParticles.DEMSystem — Type
DEMSystem(initial_condition, normal_stiffness, elastic_modulus, poissons_ratio;
- damping_coefficient=0.0001, acceleration=ntuple(_ -> 0.0, NDIMS), source_terms=nothing)

Constructs a Discrete Element Method (DEM) system for numerically simulating the dynamics of granular and particulate matter. DEM is employed to simulate and analyze the motion, interactions, and collective behavior of assemblies of discrete, solid particles, typically under mechanical loading. The model accounts for individual particle characteristics and implements interaction laws that govern contact forces (normal and tangential), based on specified material properties and contact mechanics.

Arguments

  • initial_condition: Initial condition of the system, encapsulating the initial positions, velocities, masses, and radii of particles.
  • normal_stiffness: Normal stiffness coefficient for particle-particle and particle-wall contacts.
  • elastic_modulus: Elastic modulus for this particle system.
  • poissons_ratio: Poisson ratio for this particle system.

Keywords

  • acceleration: Global acceleration vector applied to the system, such as gravity. Specified as an SVector of length NDIMS, with a default of zero in each dimension.
  • source_terms: Optional; additional forces or modifications to particle dynamics not captured by standard DEM interactions, such as electromagnetic forces or user-defined perturbations.
  • damping_coefficient=0.0001: Set a damping coefficient for the collision interactions.

!!! warning "Experimental Implementation" This is an experimental feature and may change in a future releases.

source

References

  • N. Bićanić. "Discrete element methods". In: Encyclopedia of Computational Mechanics (2007). doi: 10.1002/0470091355.ecm006.pub2

  • P. Cundall and O. Strack. "A discrete numerical model for granular assemblies". In: Géotechnique 29.1 (1979), pages 47–65. doi: 10.1680/geot.1979.29.1.47

  • A. Renzo and F. Maio. "Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes" In: Chemical Engineering Science 59.3 (2004), pages 525–541. doi: 10.1016/j.ces.2003.09.037

+Discrete Element Method (Solid) · TrixiParticles.jl
GitHub

Discrete Element Method

The Discrete Element Method (DEM) is a computational technique widely used in physics, engineering, and applied mathematics for simulating the mechanical behavior of granular materials, such as powders, sand, soil, or rock, as well as other discontinua. Unlike continuum mechanics that treats materials as continuous, DEM considers individual particles or elements and their interactions. This approach provides detailed insights into the micro-mechanical behavior of materials, making it particularly valuable in fields such as geomechanics, material science, and mechanical engineering.

Fundamental Principles

The core idea behind DEM is the discretization of a material system into a finite set of distinct, interacting mass elements (particles). These elements (particles) can vary in shape, size, and properties, and they interact with each other and possibly with their boundaries through contact forces and potential fields. The motion and behavior of each mass element are governed by Newton's laws of motion, accounting for the forces and moments acting upon them.

TrixiParticles.DEMSystem — Type
DEMSystem(initial_condition, normal_stiffness, elastic_modulus, poissons_ratio;
+ damping_coefficient=0.0001, acceleration=ntuple(_ -> 0.0, NDIMS), source_terms=nothing)

Constructs a Discrete Element Method (DEM) system for numerically simulating the dynamics of granular and particulate matter. DEM is employed to simulate and analyze the motion, interactions, and collective behavior of assemblies of discrete, solid particles, typically under mechanical loading. The model accounts for individual particle characteristics and implements interaction laws that govern contact forces (normal and tangential), based on specified material properties and contact mechanics.

Arguments

  • initial_condition: Initial condition of the system, encapsulating the initial positions, velocities, masses, and radii of particles.
  • normal_stiffness: Normal stiffness coefficient for particle-particle and particle-wall contacts.
  • elastic_modulus: Elastic modulus for this particle system.
  • poissons_ratio: Poisson ratio for this particle system.

Keywords

  • acceleration: Global acceleration vector applied to the system, such as gravity. Specified as an SVector of length NDIMS, with a default of zero in each dimension.
  • source_terms: Optional; additional forces or modifications to particle dynamics not captured by standard DEM interactions, such as electromagnetic forces or user-defined perturbations.
  • damping_coefficient=0.0001: Set a damping coefficient for the collision interactions.
Experimental Implementation

This is an experimental feature and may change in a future releases.

References

[9], [10], [11]

source
diff --git a/previews/PR514/systems/entropically_damped_sph/index.html b/previews/PR514/systems/entropically_damped_sph/index.html index f2ec5f057..a1fed8c24 100644 --- a/previews/PR514/systems/entropically_damped_sph/index.html +++ b/previews/PR514/systems/entropically_damped_sph/index.html @@ -1,9 +1,10 @@ -Entropically Damped Artificial Compressibility for SPH (Fluid) · TrixiParticles.jl
GitHub

Entropically Damped Artificial Compressibility (EDAC) for SPH

As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure

\[\frac{\mathrm{d} p_a}{\mathrm{d}t} = - \rho c_s^2 \nabla \cdot v + \nu \nabla^2 p,\]

which is derived by Clausen (2013). This equation is similar to the continuity equation (first term, see ContinuityDensity), but also contains a pressure damping term (second term, similar to density diffusion see DensityDiffusion), which reduces acoustic pressure waves through an entropy-generation mechanism.

The pressure evolution is discretized with the SPH method by Ramachandran (2019) as following:

The first term is equivalent to the classical artificial compressible methods, which are commonly motivated by assuming the artificial equation of state (StateEquationCole with exponent=1) and is discretized as

\[- \rho c_s^2 \nabla \cdot v = \sum_{b} m_b \frac{\rho_a}{\rho_b} c_s^2 v_{ab} \cdot \nabla_{r_a} W(\Vert r_a - r_b \Vert, h),\]

where $\rho_a$, $\rho_b$, $r_a$, $r_b$, denote the density and coordinates of particles $a$ and $b$ respectively, $c_s$ is the speed of sound and $v_{ab} = v_a - v_b$ is the difference in the velocity.

The second term smooths the pressure through the introduction of entropy and is discretized as

\[\nu \nabla^2 p = \frac{V_a^2 + V_b^2}{m_a} \tilde{\eta}_{ab} \frac{p_{ab}}{\Vert r_{ab}^2 \Vert + \eta h_{ab}^2} \nabla_{r_a} -W(\Vert r_a - r_b \Vert, h) \cdot r_{ab},\]

where $V_a$, $V_b$ denote the volume of particles $a$ and $b$ respectively and $p_{ab}= p_a -p_b$ is the difference in the pressure.

The viscosity parameter $\eta_a$ for a particle $a$ is given as

\[\eta_a = \rho_a \frac{\alpha h c_s}{8},\]

where it is found in the numerical experiments of Ramachandran (2019) that $\alpha = 0.5$ is a good choice for a wide range of Reynolds numbers (0.0125 to 10000).

Note

The EDAC formulation keeps the density constant and this eliminates the need for the continuity equation or the use of a summation density to find the pressure. However, in SPH discretizations, $m/\rho$ is typically used as a proxy for the particle volume. The density of the fluids can therefore be computed using the summation density approach.

Ramachandran (2019)

TrixiParticles.EntropicallyDampedSPHSystem — Type
EntropicallyDampedSPHSystem(initial_condition, smoothing_kernel,
+Entropically Damped Artificial Compressibility for SPH (Fluid) · TrixiParticles.jl
GitHub
Entropically Damped Artificial Compressibility (EDAC) for SPH
As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure
\[\frac{\mathrm{d} p_a}{\mathrm{d}t} =  - \rho c_s^2 \nabla \cdot v + \nu \nabla^2 p,\]
which is derived by Clausen (2013). This equation is similar to the continuity equation (first term, see ContinuityDensity), but also contains a pressure damping term (second term, similar to density diffusion see DensityDiffusion), which reduces acoustic pressure waves through an entropy-generation mechanism.
The pressure evolution is discretized with the SPH method by Ramachandran (2019) as following:
The first term is equivalent to the classical artificial compressible methods, which are commonly motivated by assuming the artificial equation of state (StateEquationCole with exponent=1) and is discretized as
\[- \rho c_s^2 \nabla \cdot v = \sum_{b} m_b \frac{\rho_a}{\rho_b} c_s^2 v_{ab} \cdot \nabla_{r_a} W(\Vert r_a - r_b \Vert, h),\]
where $\rho_a$, $\rho_b$,  $r_a$, $r_b$, denote the density and coordinates of particles $a$ and $b$ respectively, $c_s$ is the speed of sound and $v_{ab} = v_a - v_b$ is the difference in the velocity.
The second term smooths the pressure through the introduction of entropy and is discretized as
\[\nu \nabla^2 p = \frac{V_a^2 + V_b^2}{m_a} \tilde{\eta}_{ab} \frac{p_{ab}}{\Vert r_{ab}^2 \Vert + \eta h_{ab}^2} \nabla_{r_a}
+W(\Vert r_a - r_b \Vert, h) \cdot r_{ab},\]
where $V_a$, $V_b$ denote the volume of particles $a$ and $b$ respectively and $p_{ab}= p_a -p_b$  is the difference in the pressure.
The viscosity parameter $\eta_a$ for a particle $a$ is given as
\[\eta_a = \rho_a \frac{\alpha h c_s}{8},\]
where it is found in the numerical experiments of Ramachandran (2019) that $\alpha = 0.5$ is a good choice for a wide range of Reynolds numbers (0.0125 to 10000).
Note
The EDAC formulation keeps the density constant and this eliminates the need for the continuity equation or the use of a summation density to find the pressure. However, in SPH discretizations, $m/\rho$ is typically used as a proxy for the particle volume. The density of the fluids can therefore be computed using the summation density approach. [19]
TrixiParticles.EntropicallyDampedSPHSystem — Type
EntropicallyDampedSPHSystem(initial_condition, smoothing_kernel,
                             smoothing_length, sound_speed;
                             pressure_acceleration=inter_particle_averaged_pressure,
                             density_calculator=SummationDensity(),
+                            transport_velocity=nothing,
                             alpha=0.5, viscosity=nothing,
-                            acceleration=ntuple(_ -> 0.0, NDIMS),
-                            source_terms=nothing)
System for particles of a fluid. As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure. See Entropically Damped Artificial Compressibility for SPH for more details on the method.
Arguments
  • initial_condition:  Initial condition representing the system's particles.
  • sound_speed:        Speed of sound.
  • smoothing_kernel:   Smoothing kernel to be used for this system.                       See Smoothing Kernels.
  • smoothing_length:   Smoothing length to be used for this system.                       See Smoothing Kernels.
Keyword Arguments
  • viscosity:      Viscosity model for this system (default: no viscosity).                   Recommended: ViscosityAdami.
  • acceleration:   Acceleration vector for the system. (default: zero vector)
  • pressure_acceleration: Pressure acceleration formulation (default: inter-particle averaged pressure).                       When set to nothing, the pressure acceleration formulation for the                       corresponding density calculator is chosen.
  • density_calculator: Density calculator (default: SummationDensity)
  • source_terms:   Additional source terms for this system. Has to be either nothing                   (by default), or a function of (coords, velocity, density, pressure)                   (which are the quantities of a single particle), returning a Tuple                   or SVector that is to be added to the acceleration of that particle.                   See, for example, SourceTermDamping.                   Note that these source terms will not be used in the calculation of the                   boundary pressure when using a boundary with                   BoundaryModelDummyParticles and AdamiPressureExtrapolation.                   The keyword argument acceleration should be used instead for                   gravity-like source terms.
source
References
  • Prabhu Ramachandran. "Entropically damped artificial compressibility for SPH". In: Computers and Fluids 179 (2019), pages 579–594. doi: 10.1016/j.compfluid.2018.11.023
  • Jonathan R. Clausen. "Entropically damped form of artificial compressibility for explicit simulation of incompressible flow". In: American Physical Society 87 (2013), page 13309. doi: 10.1103/PhysRevE.87.013309

+                            acceleration=ntuple(_ -> 0.0, NDIMS), buffer_size=nothing,
+                            source_terms=nothing)

System for particles of a fluid. As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure. See Entropically Damped Artificial Compressibility for SPH for more details on the method.

Arguments

  • initial_condition: Initial condition representing the system's particles.
  • sound_speed: Speed of sound.
  • smoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.
  • smoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.

Keyword Arguments

  • viscosity: Viscosity model for this system (default: no viscosity). Recommended: ViscosityAdami.
  • acceleration: Acceleration vector for the system. (default: zero vector)
  • pressure_acceleration: Pressure acceleration formulation (default: inter-particle averaged pressure). When set to nothing, the pressure acceleration formulation for the corresponding density calculator is chosen.
  • density_calculator: Density calculator (default: SummationDensity)
  • transport_velocity: Transport Velocity Formulation (TVF). Default is no TVF.
  • buffer_size: Number of buffer particles. This is needed when simulating with OpenBoundarySPHSystem.
  • source_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure, t) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping. Note that these source terms will not be used in the calculation of the boundary pressure when using a boundary with BoundaryModelDummyParticles and AdamiPressureExtrapolation. The keyword argument acceleration should be used instead for gravity-like source terms.
source

Transport Velocity Formulation (TVF)

Standard SPH suffers from problems like tensile instability or the creation of void regions in the flow. To address these problems, Adami (2013) modified the advection velocity and added an extra term to the momentum equation. The authors introduced the so-called Transport Velocity Formulation (TVF) for WCSPH. Ramachandran (2019) applied the TVF also for the EDAC scheme.

The transport velocity $\tilde{v}_a$ of particle $a$ is used to evolve the position of the particle $r_a$ from one time step to the next by

\[\frac{\mathrm{d} r_a}{\mathrm{d}t} = \tilde{v}_a\]

and is obtained at every time-step $\Delta t$ from

\[\tilde{v}_a (t + \Delta t) = v_a (t) + \Delta t \left(\frac{\tilde{\mathrm{d}} v_a}{\mathrm{d}t} - \frac{1}{\rho_a} \nabla p_{\text{background}} \right),\]

where $\rho_a$ is the density of particle $a$ and $p_{\text{background}}$ is a constant background pressure field. The tilde in the second term of the right hand side indicates that the material derivative has an advection part.

The discretized form of the last term is

\[ -\frac{1}{\rho_a} \nabla p_{\text{background}} \approx -\frac{p_{\text{background}}}{m_a} \sum_b \left(V_a^2 + V_b^2 \right) \nabla_a W_{ab},\]

where $V_a$, $V_b$ denote the volume of particles $a$ and $b$ respectively. Note that although in the continuous case $\nabla p_{\text{background}} = 0$, the discretization is not 0th-order consistent for non-uniform particle distribution, which means that there is a non-vanishing contribution only when particles are disordered. That also means that $p_{\text{background}}$ occurs as prefactor to correct the trajectory of a particle resulting in uniform pressure distributions. Suggested is a background pressure which is in the order of the reference pressure but can be chosen arbitrarily large when the time-step criterion is adjusted.

The inviscid momentum equation with an additional convection term for a particle moving with $\tilde{v}$ is

\[\frac{\tilde{\mathrm{d}} \left( \rho v \right)}{\mathrm{d}t} = -\nabla p + \nabla \cdot \bm{A},\]

where the tensor $\bm{A} = \rho v\left(\tilde{v}-v\right)^T$ is a consequence of the modified advection velocity and can be interpreted as the convection of momentum with the relative velocity $\tilde{v}-v$.

The discretized form of the momentum equation for a particle $a$ reads as

\[\frac{\tilde{\mathrm{d}} v_a}{\mathrm{d}t} = \frac{1}{m_a} \sum_b \left(V_a^2 + V_b^2 \right) \left[ -\tilde{p}_{ab} \nabla_a W_{ab} + \frac{1}{2} \left(\bm{A}_a + \bm{A}_b \right) \cdot \nabla_a W_{ab} \right].\]

Here, $\tilde{p}_{ab}$ is the density-weighted pressure

\[\tilde{p}_{ab} = \frac{\rho_b p_a + \rho_a p_b}{\rho_a + \rho_b},\]

with the density $\rho_a$, $\rho_b$ and the pressure $p_a$, $p_b$ of particles $a$ and $b$ respectively. $\bm{A}_a$ and $\bm{A}_b$ are the convection tensors for particle $a$ and $b$ respectively and are given, e.g. for particle $a$, as $\bm{A}_a = \rho v_a\left(\tilde{v}_a-v_a\right)^T$.

TrixiParticles.TransportVelocityAdami — Type
TransportVelocityAdami(background_pressure::Real)

Transport Velocity Formulation (TVF) to suppress pairing and tensile instability. See TVF for more details of the method.

Arguments

  • background_pressure: Background pressure. Suggested is a background pressure which is on the order of the reference pressure.
Note

There is no need for an artificial viscosity to suppress tensile instability when using TransportVelocityAdami. Thus, it is highly recommended to use ViscosityAdami as viscosity model, since ArtificialViscosityMonaghan leads to bad results.

source
diff --git a/previews/PR514/systems/total_lagrangian_sph/index.html b/previews/PR514/systems/total_lagrangian_sph/index.html index 7507aaeae..b49c24ed5 100644 --- a/previews/PR514/systems/total_lagrangian_sph/index.html +++ b/previews/PR514/systems/total_lagrangian_sph/index.html @@ -1,11 +1,11 @@ -Total Lagrangian SPH (Elastic Structure) · TrixiParticles.jl
GitHub

Total Lagrangian SPH

A Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers 2021, Belytschko et al. 2000).

The governing equations with respect to the initial configuration are given by:

\[\frac{\mathrm{D}\bm{v}}{\mathrm{D}t} = \frac{1}{\rho_0} \nabla_0 \cdot \bm{P} + \bm{g},\]

where the zero subscript denotes a derivative with respect to the initial configuration and $\bm{P}$ is the first Piola-Kirchhoff (PK1) stress tensor.

The discretized version of this equation is given by O’Connor & Rogers (2021):

\[\frac{\mathrm{d}\bm{v}_a}{\mathrm{d}t} = \sum_b m_{0b} +Total Lagrangian SPH (Elastic Structure) · TrixiParticles.jl

GitHub

Total Lagrangian SPH

A Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers, 2021, Belytschko et al., 2000).

The governing equations with respect to the initial configuration are given by:

\[\frac{\mathrm{D}\bm{v}}{\mathrm{D}t} = \frac{1}{\rho_0} \nabla_0 \cdot \bm{P} + \bm{g},\]

where the zero subscript denotes a derivative with respect to the initial configuration and $\bm{P}$ is the first Piola-Kirchhoff (PK1) stress tensor.

The discretized version of this equation is given by O’Connor & Rogers (2021):

\[\frac{\mathrm{d}\bm{v}_a}{\mathrm{d}t} = \sum_b m_{0b} \left( \frac{\bm{P}_a \bm{L}_{0a}}{\rho_{0a}^2} + \frac{\bm{P}_b \bm{L}_{0b}}{\rho_{0b}^2} \right) \nabla_{0a} W(\bm{X}_{ab}) + \frac{\bm{f}_a^{PF}}{m_{0a}} + \bm{g},\]

with the correction matrix (see also GradientCorrection)

\[\bm{L}_{0a} := \left( -\sum_{b} \frac{m_{0b}}{\rho_{0b}} \nabla_{0a} W(\bm{X}_{ab}) \bm{X}_{ab}^T \right)^{-1} \in \R^{d \times d}.\]

The subscripts $a$ and $b$ denote quantities of particle $a$ and $b$, respectively. The zero subscript on quantities denotes that the quantity is to be measured in the initial configuration. The difference in the initial coordinates is denoted by $\bm{X}_{ab} = \bm{X}_a - \bm{X}_b$, the difference in the current coordinates is denoted by $\bm{x}_{ab} = \bm{x}_a - \bm{x}_b$.

For the computation of the PK1 stress tensor, the deformation gradient $\bm{F}$ is computed per particle as

\[\bm{F}_a = \sum_b \frac{m_{0b}}{\rho_{0b}} \bm{x}_{ba} (\bm{L}_{0a}\nabla_{0a} W(\bm{X}_{ab}))^T \\ - \qquad = -\left(\sum_b \frac{m_{0b}}{\rho_{0b}} \bm{x}_{ab} (\nabla_{0a} W(\bm{X}_{ab}))^T \right) \bm{L}_{0a}^T\]

with $1 \leq i,j \leq d$. From the deformation gradient, the Green-Lagrange strain

\[\bm{E} = \frac{1}{2}(\bm{F}^T\bm{F} - \bm{I})\]

and the second Piola-Kirchhoff stress tensor

\[\bm{S} = \lambda \operatorname{tr}(\bm{E}) \bm{I} + 2\mu \bm{E}\]

are computed to obtain the PK1 stress tensor as

\[\bm{P} = \bm{F}\bm{S}.\]

Here,

\[\mu = \frac{E}{2(1 + \nu)}\]

and

\[\lambda = \frac{E\nu}{(1 + \nu)(1 - 2\nu)}\]

are the Lamé coefficients, where $E$ is the Young's modulus and $\nu$ is the Poisson ratio.

The term $\bm{f}_a^{PF}$ is an optional penalty force. See e.g. PenaltyForceGanzenmueller.

TrixiParticles.TotalLagrangianSPHSystem — Type
TotalLagrangianSPHSystem(initial_condition,
+    \qquad  = -\left(\sum_b \frac{m_{0b}}{\rho_{0b}} \bm{x}_{ab} (\nabla_{0a} W(\bm{X}_{ab}))^T \right) \bm{L}_{0a}^T\]
with $1 \leq i,j \leq d$. From the deformation gradient, the Green-Lagrange strain
\[\bm{E} = \frac{1}{2}(\bm{F}^T\bm{F} - \bm{I})\]
and the second Piola-Kirchhoff stress tensor
\[\bm{S} = \lambda \operatorname{tr}(\bm{E}) \bm{I} + 2\mu \bm{E}\]
are computed to obtain the PK1 stress tensor as
\[\bm{P} = \bm{F}\bm{S}.\]
Here,
\[\mu = \frac{E}{2(1 + \nu)}\]
and
\[\lambda = \frac{E\nu}{(1 + \nu)(1 - 2\nu)}\]
are the Lamé coefficients, where $E$ is the Young's modulus and $\nu$ is the Poisson ratio.
The term $\bm{f}_a^{PF}$ is an optional penalty force. See e.g. PenaltyForceGanzenmueller.
TrixiParticles.TotalLagrangianSPHSystem — Type
TotalLagrangianSPHSystem(initial_condition,
                          smoothing_kernel, smoothing_length,
                          young_modulus, poisson_ratio;
                          n_fixed_particles=0, boundary_model=nothing,
                          acceleration=ntuple(_ -> 0.0, NDIMS),
-                         penalty_force=nothing, source_terms=nothing)
System for particles of an elastic structure.
A Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers 2021, Belytschko et al. 2000). See Total Lagrangian SPH for more details on the method.
Arguments
  • initial_condition:  Initial condition representing the system's particles.
  • young_modulus:      Young's modulus.
  • poisson_ratio:      Poisson ratio.
  • smoothing_kernel:   Smoothing kernel to be used for this system.                       See Smoothing Kernels.
  • smoothing_length:   Smoothing length to be used for this system.                       See Smoothing Kernels.
Keyword Arguments
  • n_fixed_particles:  Number of fixed particles which are used to clamp the structure                       particles. Note that the fixed particles must be the last                       particles in the InitialCondition. See the info box below.
  • boundary_model: Boundary model to compute the hydrodynamic density and pressure for                   fluid-structure interaction (see Boundary Models).
  • penalty_force:  Penalty force to ensure regular particle position under large deformations                   (see PenaltyForceGanzenmueller).
  • acceleration:   Acceleration vector for the system. (default: zero vector)
  • source_terms:   Additional source terms for this system. Has to be either nothing                   (by default), or a function of (coords, velocity, density, pressure)                   (which are the quantities of a single particle), returning a Tuple                   or SVector that is to be added to the acceleration of that particle.                   See, for example, SourceTermDamping.
Note
The fixed particles must be the last particles in the InitialCondition. To do so, e.g. use the union function:
solid = union(beam, fixed_particles)
where beam and fixed_particles are of type InitialCondition.
source
References
  • Joseph O’Connor, Benedict D. Rogers. "A fluid-structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU". In: Journal of Fluids and Structures 104 (2021). doi: 10.1016/J.JFLUIDSTRUCTS.2021.103312
  • Ted Belytschko, Yong Guo, Wing Kam Liu, Shao Ping Xiao. "A unified stability analysis of meshless particle methods". In: International Journal for Numerical Methods in Engineering 48 (2000), pages 1359–1400. doi: 10.1002/1097-0207
Penalty Force
In FEM, underintegrated elements can deform without an associated increase of energy. This is caused by the stiffness matrix having zero eigenvalues (so-called hourglass modes). The name "hourglass modes" comes from the fact that elements can deform into an hourglass shape.
Similar effects can occur in SPH as well. Particles can change positions without changing the SPH approximation of the deformation gradient $\bm{F}$, thus, without causing an increase of energy. To ensure regular particle positions, we can apply similar correction forces as are used in FEM.
Ganzenmüller (2015) introduced a so-called hourglass correction force or penalty force $f^{PF}$, which is given by
\[\bm{f}_a^{PF} = \frac{1}{2} \alpha \sum_b \frac{m_{0a} m_{0b} W_{0ab}}{\rho_{0a}\rho_{0b} |\bm{X}_{ab}|^2}
-                \left( E \delta_{ab}^a + E \delta_{ba}^b \right) \frac{\bm{x}_{ab}}{|\bm{x}_{ab}|}\]
The subscripts $a$ and $b$ denote quantities of particle $a$ and $b$, respectively. The zero subscript on quantities denotes that the quantity is to be measured in the initial configuration. The difference in the initial coordinates is denoted by $\bm{X}_{ab} = \bm{X}_a - \bm{X}_b$, the difference in the current coordinates is denoted by $\bm{x}_{ab} = \bm{x}_a - \bm{x}_b$. Note that Ganzenmüller (2015) has a flipped sign here because they define $\bm{x}_{ab}$ the other way around.
This correction force is based on the potential energy density of a Hookean material. Thus, $E$ is the Young's modulus and $\alpha$ is a dimensionless coefficient that controls the amplitude of hourglass correction. The separation vector $\delta_{ab}^a$ indicates the change of distance which the particle separation should attain in order to minimize the error and is given by
\[    \delta_{ab}^a = \frac{\bm{\epsilon}_{ab}^a \cdot \bm{x_{ab}}}{|\bm{x}_{ab}|},\]
where the error vector is defined as
\[    \bm{\epsilon}_{ab}^a = \bm{F}_a \bm{X}_{ab} - \bm{x}_{ab}.\]
TrixiParticles.PenaltyForceGanzenmueller — Type
PenaltyForceGanzenmueller(; alpha=0.1)
Penalty force to ensure regular particle positions under large deformations.
Keywords
  • alpha: Coefficient to control the amplitude of hourglass correction.
source
References
  • Georg C. Ganzenmüller. "An hourglass control algorithm for Lagrangian Smooth Particle Hydrodynamics". In: Computer Methods in Applied Mechanics and Engineering 286 (2015). doi: 10.1016/j.cma.2014.12.005
+ penalty_force=nothing, source_terms=nothing)

System for particles of an elastic structure.

A Total Lagrangian framework is used wherein the governing equations are formulated such that all relevant quantities and operators are measured with respect to the initial configuration (O’Connor & Rogers 2021, Belytschko et al. 2000). See Total Lagrangian SPH for more details on the method.

Arguments

  • initial_condition: Initial condition representing the system's particles.
  • young_modulus: Young's modulus.
  • poisson_ratio: Poisson ratio.
  • smoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.
  • smoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.

Keyword Arguments

  • n_fixed_particles: Number of fixed particles which are used to clamp the structure particles. Note that the fixed particles must be the last particles in the InitialCondition. See the info box below.
  • boundary_model: Boundary model to compute the hydrodynamic density and pressure for fluid-structure interaction (see Boundary Models).
  • penalty_force: Penalty force to ensure regular particle position under large deformations (see PenaltyForceGanzenmueller).
  • acceleration: Acceleration vector for the system. (default: zero vector)
  • source_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping.
Note

The fixed particles must be the last particles in the InitialCondition. To do so, e.g. use the union function:

solid = union(beam, fixed_particles)

where beam and fixed_particles are of type InitialCondition.

source

Penalty Force

In FEM, underintegrated elements can deform without an associated increase of energy. This is caused by the stiffness matrix having zero eigenvalues (so-called hourglass modes). The name "hourglass modes" comes from the fact that elements can deform into an hourglass shape.

Similar effects can occur in SPH as well. Particles can change positions without changing the SPH approximation of the deformation gradient $\bm{F}$, thus, without causing an increase of energy. To ensure regular particle positions, we can apply similar correction forces as are used in FEM.

Ganzenmüller (2015) introduced a so-called hourglass correction force or penalty force $f^{PF}$, which is given by

\[\bm{f}_a^{PF} = \frac{1}{2} \alpha \sum_b \frac{m_{0a} m_{0b} W_{0ab}}{\rho_{0a}\rho_{0b} |\bm{X}_{ab}|^2} + \left( E \delta_{ab}^a + E \delta_{ba}^b \right) \frac{\bm{x}_{ab}}{|\bm{x}_{ab}|}\]

The subscripts $a$ and $b$ denote quantities of particle $a$ and $b$, respectively. The zero subscript on quantities denotes that the quantity is to be measured in the initial configuration. The difference in the initial coordinates is denoted by $\bm{X}_{ab} = \bm{X}_a - \bm{X}_b$, the difference in the current coordinates is denoted by $\bm{x}_{ab} = \bm{x}_a - \bm{x}_b$. Note that Ganzenmüller (2015) has a flipped sign here because they define $\bm{x}_{ab}$ the other way around.

This correction force is based on the potential energy density of a Hookean material. Thus, $E$ is the Young's modulus and $\alpha$ is a dimensionless coefficient that controls the amplitude of hourglass correction. The separation vector $\delta_{ab}^a$ indicates the change of distance which the particle separation should attain in order to minimize the error and is given by

\[ \delta_{ab}^a = \frac{\bm{\epsilon}_{ab}^a \cdot \bm{x_{ab}}}{|\bm{x}_{ab}|},\]

where the error vector is defined as

\[ \bm{\epsilon}_{ab}^a = \bm{F}_a \bm{X}_{ab} - \bm{x}_{ab}.\]

TrixiParticles.PenaltyForceGanzenmueller — Type
PenaltyForceGanzenmueller(; alpha=0.1)

Penalty force to ensure regular particle positions under large deformations.

Keywords

  • alpha: Coefficient to control the amplitude of hourglass correction.
source
diff --git a/previews/PR514/systems/weakly_compressible_sph/index.html b/previews/PR514/systems/weakly_compressible_sph/index.html index 115c5aa3c..385df1aea 100644 --- a/previews/PR514/systems/weakly_compressible_sph/index.html +++ b/previews/PR514/systems/weakly_compressible_sph/index.html @@ -1,24 +1,34 @@ -Weakly Compressible SPH (Fluid) · TrixiParticles.jl
GitHub

Weakly Compressible SPH

Weakly compressible SPH as introduced by Monaghan (1994). This formulation relies on a stiff equation of state that generates large pressure changes for small density variations.

TrixiParticles.WeaklyCompressibleSPHSystem — Type
WeaklyCompressibleSPHSystem(initial_condition,
+Weakly Compressible SPH (Fluid) · TrixiParticles.jl
GitHub
Weakly Compressible SPH
Weakly compressible SPH as introduced by Monaghan (1994). This formulation relies on a stiff equation of state that generates large pressure changes for small density variations.
TrixiParticles.WeaklyCompressibleSPHSystem — Type
WeaklyCompressibleSPHSystem(initial_condition,
                             density_calculator, state_equation,
                             smoothing_kernel, smoothing_length;
                             viscosity=nothing, density_diffusion=nothing,
                             acceleration=ntuple(_ -> 0.0, NDIMS),
-                            correction=nothing, source_terms=nothing)
System for particles of a fluid. The weakly compressible SPH (WCSPH) scheme is used, wherein a stiff equation of state generates large pressure changes for small density variations. See Weakly Compressible SPH for more details on the method.
Arguments
Keyword Arguments
  • viscosity:      Viscosity model for this system (default: no viscosity).                   See ArtificialViscosityMonaghan or ViscosityAdami.
  • density_diffusion: Density diffusion terms for this system. See DensityDiffusion.
  • acceleration:   Acceleration vector for the system. (default: zero vector)
  • correction:     Correction method used for this system. (default: no correction, see Corrections)
  • source_terms:   Additional source terms for this system. Has to be either nothing                   (by default), or a function of (coords, velocity, density, pressure)                   (which are the quantities of a single particle), returning a Tuple                   or SVector that is to be added to the acceleration of that particle.                   See, for example, SourceTermDamping.                   Note that these source terms will not be used in the calculation of the                   boundary pressure when using a boundary with                   BoundaryModelDummyParticles and AdamiPressureExtrapolation.                   The keyword argument acceleration should be used instead for                   gravity-like source terms.
source
References
  • Joseph J. Monaghan. "Simulating Free Surface Flows in SPH". In: Journal of Computational Physics 110 (1994), pages 399–406. doi: 10.1006/jcph.1994.1034
Equation of State
The equation of state is used to relate fluid density to pressure and thus allow an explicit simulation of the WCSPH system. The equation in the following formulation was introduced by Cole (Cole 1948, pp. 39 and 43). The pressure $p$ is calculated as
\[    p = B \left(\left(\frac{\rho}{\rho_0}\right)^\gamma - 1\right) + p_{\text{background}},\]
where $\rho$ denotes the density, $\rho_0$ the reference density, and $p_{\text{background}}$ the background pressure, which is set to zero when applied to free-surface flows (Adami et al., 2012).
The bulk modulus, $B =  \frac{\rho_0 c^2}{\gamma}$, is calculated from the artificial speed of sound $c$ and the isentropic exponent $\gamma$.
An ideal gas equation of state with a linear relationship between pressure and density can be obtained by choosing exponent=1, i.e.
\[    p = B \left( \frac{\rho}{\rho_0} -1 \right) = c^2(\rho - \rho_0).\]
For higher Reynolds numbers, exponent=7 is recommended, whereas at lower Reynolds numbers exponent=1 yields more accurate pressure estimates since pressure and density are proportional.
When using SummationDensity (or DensityReinitializationCallback) and free surfaces, initializing particles with equal spacing will cause underestimated density and therefore strong attractive forces between particles at the free surface. Setting clip_negative_pressure=true can avoid this.
TrixiParticles.StateEquationCole — Type
StateEquationCole(; sound_speed, reference_density, exponent,
-                  background_pressure=0.0, clip_negative_pressure=false)
Equation of state to describe the relationship between pressure and density of water up to high pressures.
Keywords
  • sound_speed: Artificial speed of sound.
  • reference_density: Reference density of the fluid.
  • exponent: A value of 7 is usually used for most simulations.
  • background_pressure=0.0: Background pressure.
source
References
  • Robert H. Cole. "Underwater Explosions". Princeton University Press, 1948.
  • J. P. Morris, P. J. Fox, Y. Zhu "Modeling Low Reynolds Number Incompressible Flows Using SPH ". In: Journal of Computational Physics , Vol. 136, No. 1, pages 214–226. doi: 10.1006/jcph.1997.5776
  • S. Adami, X. Y. Hu, N. A. Adams. "A generalized wall boundary condition for smoothed particle hydrodynamics". In: Journal of Computational Physics 231, 21 (2012), pages 7057–7075. doi: 10.1016/J.JCP.2012.05.005
Viscosity
TODO: Explain viscosity.
TrixiParticles.ArtificialViscosityMonaghan — Type
ArtificialViscosityMonaghan(; alpha, beta, epsilon=0.01)
Keywords
  • alpha: A value of 0.02 is usually used for most simulations. For a relation with the          kinematic viscosity, see description below.
  • beta: A value of 0.0 works well for simulations with shocks of moderate strength.         In simulations where the Mach number can be very high, eg. astrophysical calculation,         good results can be obtained by choosing a value of beta=2 and alpha=1.
  • epsilon=0.01: Parameter to prevent singularities.
Artificial viscosity by Monaghan (Monaghan 1992, Monaghan 1989), given by
\[\Pi_{ab} =
+                            buffer_size=nothing,
+                            correction=nothing, source_terms=nothing)

System for particles of a fluid. The weakly compressible SPH (WCSPH) scheme is used, wherein a stiff equation of state generates large pressure changes for small density variations. See Weakly Compressible SPH for more details on the method.

Arguments

Keyword Arguments

  • viscosity: Viscosity model for this system (default: no viscosity). See ArtificialViscosityMonaghan or ViscosityAdami.
  • density_diffusion: Density diffusion terms for this system. See DensityDiffusion.
  • acceleration: Acceleration vector for the system. (default: zero vector)
  • buffer_size: Number of buffer particles. This is needed when simulating with OpenBoundarySPHSystem.
  • correction: Correction method used for this system. (default: no correction, see Corrections)
  • source_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure, t) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping. Note that these source terms will not be used in the calculation of the boundary pressure when using a boundary with BoundaryModelDummyParticles and AdamiPressureExtrapolation. The keyword argument acceleration should be used instead for gravity-like source terms.
  • surface_tension: Surface tension model used for this SPH system. (default: no surface tension)
source

Equation of State

The equation of state is used to relate fluid density to pressure and thus allow an explicit simulation of the WCSPH system. The equation in the following formulation was introduced by Cole (1948) (pp. 39 and 43). The pressure $p$ is calculated as

\[ p = B \left(\left(\frac{\rho}{\rho_0}\right)^\gamma - 1\right) + p_{\text{background}},\]

where $\rho$ denotes the density, $\rho_0$ the reference density, and $p_{\text{background}}$ the background pressure, which is set to zero when applied to free-surface flows (Adami et al., 2012).

The bulk modulus, $B = \frac{\rho_0 c^2}{\gamma}$, is calculated from the artificial speed of sound $c$ and the isentropic exponent $\gamma$.

An ideal gas equation of state with a linear relationship between pressure and density can be obtained by choosing exponent=1, i.e.

\[ p = B \left( \frac{\rho}{\rho_0} -1 \right) = c^2(\rho - \rho_0).\]

For higher Reynolds numbers, exponent=7 is recommended, whereas at lower Reynolds numbers exponent=1 yields more accurate pressure estimates since pressure and density are proportional (see Morris, 1997).

When using SummationDensity (or DensityReinitializationCallback) and free surfaces, initializing particles with equal spacing will cause underestimated density and therefore strong attractive forces between particles at the free surface. Setting clip_negative_pressure=true can avoid this.

TrixiParticles.StateEquationCole — Type
StateEquationCole(; sound_speed, reference_density, exponent,
+                  background_pressure=0.0, clip_negative_pressure=false)

Equation of state to describe the relationship between pressure and density of water up to high pressures.

Keywords

  • sound_speed: Artificial speed of sound.
  • reference_density: Reference density of the fluid.
  • exponent: A value of 7 is usually used for most simulations.
  • background_pressure=0.0: Background pressure.
source

Viscosity

TODO: Explain viscosity.

TrixiParticles.ArtificialViscosityMonaghan — Type
ArtificialViscosityMonaghan(; alpha, beta=0.0, epsilon=0.01)

Artificial viscosity by Monaghan ([16], [17]), given by

\[\Pi_{ab} = \begin{cases} -(\alpha c \mu_{ab} + \beta \mu_{ab}^2) / \bar{\rho}_{ab} & \text{if } v_{ab} \cdot r_{ab} < 0, \\ 0 & \text{otherwise} -\end{cases}\]

with

\[\mu_{ab} = \frac{h v_{ab} \cdot r_{ab}}{\Vert r_{ab} \Vert^2 + \epsilon h^2},\]

where $\alpha, \beta, \epsilon$ are parameters, $c$ is the speed of sound, $h$ is the smoothing length, $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$, $v_{ab} = v_a - v_b$ is the difference of their velocities, and $\bar{\rho}_{ab}$ is the arithmetic mean of their densities.

Note that $\alpha$ needs to adjusted for different resolutions to maintain a specific Reynolds Number. To do so, Monaghan (Monaghan 2005) defined an equivalent effective physical kinematic viscosity $\nu$ by

\[ \nu = \frac{\alpha h c }{2d + 4},\]

where $d$ is the dimension.

References

source
TrixiParticles.ViscosityAdami — Type
ViscosityAdami(; nu, epsilon=0.01)

Viscosity by Adami (Adami et al. 2012). The viscous interaction is calculated with the shear force for incompressible flows given by

\[f_{ab} = \sum_w \bar{\eta}_{ab} \left( V_a^2 + V_b^2 \right) \frac{v_{ab}}{||r_{ab}||^2+\epsilon h_{ab}^2} \nabla W_{ab} \cdot r_{ab},\]

where $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$, $v_{ab} = v_a - v_b$ is the difference of their velocities, $h$ is the smoothing length and $V$ is the particle volume. The parameter $\epsilon$ prevents singularities (see Ramachandran et al. 2019). The inter-particle-averaged shear stress is

\[ \bar{\eta}_{ab} =\frac{2 \eta_a \eta_b}{\eta_a + \eta_b},\]

where $\eta_a = \rho_a \nu_a$ with $\nu$ as the kinematic viscosity.

Keywords

  • nu: Kinematic viscosity
  • epsilon=0.01: Parameter to prevent singularities

References

  • S. Adami et al. "A generalized wall boundary condition for smoothed particle hydrodynamics". In: Journal of Computational Physics 231 (2012), pages 7057-7075. doi: 10.1016/j.jcp.2012.05.005
  • P. Ramachandran et al. "Entropically damped artificial compressibility for SPH". In: Journal of Computers and Fluids 179 (2019), pages 579-594. doi: 10.1016/j.compfluid.2018.11.023
source

Density Diffusion

Density diffusion can be used with ContinuityDensity to remove the noise in the pressure field. It is highly recommended to use density diffusion when using WCSPH.

Formulation

All density diffusion terms extend the continuity equation (see ContinuityDensity) by an additional term

\[\frac{\mathrm{d}\rho_a}{\mathrm{d}t} = \sum_{b} m_b v_{ab} \cdot \nabla_{r_a} W(\Vert r_{ab} \Vert, h) +\end{cases}\]

with

\[\mu_{ab} = \frac{h v_{ab} \cdot r_{ab}}{\Vert r_{ab} \Vert^2 + \epsilon h^2},\]

where $\alpha, \beta, \epsilon$ are parameters, $c$ is the speed of sound, $h$ is the smoothing length, $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$, $v_{ab} = v_a - v_b$ is the difference of their velocities, and $\bar{\rho}_{ab}$ is the arithmetic mean of their densities.

Note that $\alpha$ needs to adjusted for different resolutions to maintain a specific Reynolds Number. To do so, Monaghan (2005) defined an equivalent effective physical kinematic viscosity $\nu$ by

\[ \nu = \frac{\alpha h c }{2d + 4},\]

where $d$ is the dimension.

Keywords

  • alpha: A value of 0.02 is usually used for most simulations. For a relation with the kinematic viscosity, see description above.
  • beta=0.0: A value of 0.0 works well for most fluid simulations and simulations with shocks of moderate strength. In simulations where the Mach number can be very high, eg. astrophysical calculation, good results can be obtained by choosing a value of beta=2.0 and alpha=1.0.
  • epsilon=0.01: Parameter to prevent singularities.
source
TrixiParticles.ViscosityAdami — Type
ViscosityAdami(; nu, epsilon=0.01)

Viscosity by Adami (2012). The viscous interaction is calculated with the shear force for incompressible flows given by

\[f_{ab} = \sum_w \bar{\eta}_{ab} \left( V_a^2 + V_b^2 \right) \frac{v_{ab}}{||r_{ab}||^2+\epsilon h_{ab}^2} \nabla W_{ab} \cdot r_{ab},\]

where $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$, $v_{ab} = v_a - v_b$ is the difference of their velocities, $h$ is the smoothing length and $V$ is the particle volume. The parameter $\epsilon$ prevents singularities (see Ramachandran (2019)). The inter-particle-averaged shear stress is

\[ \bar{\eta}_{ab} =\frac{2 \eta_a \eta_b}{\eta_a + \eta_b},\]

where $\eta_a = \rho_a \nu_a$ with $\nu$ as the kinematic viscosity.

Keywords

  • nu: Kinematic viscosity
  • epsilon=0.01: Parameter to prevent singularities
source
TrixiParticles.ViscosityMorris — Type
ViscosityMorris(; nu, epsilon=0.01)

Viscosity by Morris (1997) also used by Fourtakas (2019).

To the force $f_{ab}$ between two particles $a$ and $b$ due to pressure gradients, an additional force term $\tilde{f}_{ab}$ is added with

\[\tilde{f}_{ab} = m_a m_b \frac{(\mu_a + \mu_b) r_{ab} \cdot \nabla W_{ab}}{\rho_a \rho_b (\Vert r_{ab} \Vert^2 + \epsilon h^2)} v_{ab},\]

where $\mu_a = \rho_a \nu$ and $\mu_b = \rho_b \nu$ denote the dynamic viscosity of particle $a$ and $b$ respectively, and $\nu$ is the kinematic viscosity.

Keywords

  • nu: Kinematic viscosity
  • epsilon=0.01: Parameter to prevent singularities
source

Density Diffusion

Density diffusion can be used with ContinuityDensity to remove the noise in the pressure field. It is highly recommended to use density diffusion when using WCSPH.

Formulation

All density diffusion terms extend the continuity equation (see ContinuityDensity) by an additional term

\[\frac{\mathrm{d}\rho_a}{\mathrm{d}t} = \sum_{b} m_b v_{ab} \cdot \nabla_{r_a} W(\Vert r_{ab} \Vert, h) + \delta h c \sum_{b} V_b \psi_{ab} \cdot \nabla_{r_a} W(\Vert r_{ab} \Vert, h),\]

where $V_b = m_b / \rho_b$ is the volume of particle $b$ and $\psi_{ab}$ depends on the density diffusion method (see DensityDiffusion for available terms). Also, $\rho_a$ denotes the density of particle $a$ and $r_{ab} = r_a - r_b$ is the difference of the coordinates, $v_{ab} = v_a - v_b$ of the velocities of particles $a$ and $b$.

Numerical Results

All density diffusion terms remove numerical noise in the pressure field and produce more accurate results than weakly commpressible SPH without density diffusion. This can be demonstrated with dam break examples in 2D and 3D. Here, $δ = 0.1$ has been used for all terms. Note that, due to added stability, the adaptive time integration method that was used here can choose higher time steps in the simulations with density diffusion. For the cheap DensityDiffusionMolteniColagrossi, this results in reduced runtime.

density_diffusion_2d
Dam break in 2D with different density diffusion terms
density_diffusion_3d
Dam break in 3D with different density diffusion terms
-

The simpler terms DensityDiffusionMolteniColagrossi and DensityDiffusionFerrari do not solve the hydrostatic problem and lead to incorrect solutions in long-running steady-state hydrostatic simulations with free surfaces (Antuono et al., 2012). This can be seen when running the simple rectangular tank example until $t = 40$ (again using $δ = 0.1$):

+

The simpler terms DensityDiffusionMolteniColagrossi and DensityDiffusionFerrari do not solve the hydrostatic problem and lead to incorrect solutions in long-running steady-state hydrostatic simulations with free surfaces (Antuono et al., 2012). This can be seen when running the simple rectangular tank example until $t = 40$ (again using $δ = 0.1$):

density_diffusion_tank
Tank in rest under gravity in 3D with different density diffusion terms
-

DensityDiffusionAntuono adds a correction term to solve this problem, but this term is very expensive and adds about 40–50% of computational cost.

References

  • M. Antuono, A. Colagrossi, S. Marrone. "Numerical Diffusive Terms in Weakly-Compressible SPH Schemes." In: Computer Physics Communications 183.12 (2012), pages 2570–2580. doi: 10.1016/j.cpc.2012.07.006

API

TrixiParticles.DensityDiffusionAntuono — Type
DensityDiffusionAntuono(initial_condition; delta)

The commonly used density diffusion terms by Antuono et al. (2010), also referred to as δ-SPH. The density diffusion term by Molteni & Colagrossi (2009) is extended by a second term, which is nicely written down by Antuono et al. (2012).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = 2\left(\rho_a - \rho_b - \frac{1}{2}\big(\nabla\rho^L_a + \nabla\rho^L_b\big) \cdot r_{ab}\right) - \frac{r_{ab}}{\Vert r_{ab} \Vert^2},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively and $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$. The symbol $\nabla\rho^L_a$ denotes the renormalized density gradient defined as

\[\nabla\rho^L_a = -\sum_b (\rho_a - \rho_b) V_b L_a \nabla_{r_a} W(\Vert r_{ab} \Vert, h)\]

with

\[L_a := \left( -\sum_{b} V_b r_{ab} \otimes \nabla_{r_a} W(\Vert r_{ab} \Vert, h) \right)^{-1} \in \R^{d \times d},\]

where $d$ is the number of dimensions.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

References

  • M. Antuono, A. Colagrossi, S. Marrone, D. Molteni. "Free-Surface Flows Solved by Means of SPH Schemes with Numerical Diffusive Terms." In: Computer Physics Communications 181.3 (2010), pages 532–549. doi: 10.1016/j.cpc.2009.11.002
  • M. Antuono, A. Colagrossi, S. Marrone. "Numerical Diffusive Terms in Weakly-Compressible SPH Schemes." In: Computer Physics Communications 183.12 (2012), pages 2570–2580. doi: 10.1016/j.cpc.2012.07.006
  • Diego Molteni, Andrea Colagrossi. "A Simple Procedure to Improve the Pressure Evaluation in Hydrodynamic Context Using the SPH." In: Computer Physics Communications 180.6 (2009), pages 861–872. doi: 10.1016/j.cpc.2008.12.004
source
TrixiParticles.DensityDiffusionFerrari — Type
DensityDiffusionFerrari()

A density diffusion term by Ferrari et al. (2009).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = \frac{\rho_a - \rho_b}{2h} \frac{r_{ab}}{\Vert r_{ab} \Vert},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively, $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$ and $h$ is the smoothing length.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

References

  • Angela Ferrari, Michael Dumbser, Eleuterio F. Toro, Aronne Armanini. "A New 3D Parallel SPH Scheme for Free Surface Flows." In: Computers & Fluids 38.6 (2009), pages 1203–1217. doi: 10.1016/j.compfluid.2008.11.012.
source
TrixiParticles.DensityDiffusionMolteniColagrossi — Type
DensityDiffusionMolteniColagrossi(; delta)

The commonly used density diffusion term by Molteni & Colagrossi (2009).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = 2(\rho_a - \rho_b) \frac{r_{ab}}{\Vert r_{ab} \Vert^2},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively and $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

References

  • Diego Molteni, Andrea Colagrossi. "A Simple Procedure to Improve the Pressure Evaluation in Hydrodynamic Context Using the SPH." In: Computer Physics Communications 180.6 (2009), pages 861–872. doi: 10.1016/j.cpc.2008.12.004
source

Corrections

TrixiParticles.AkinciFreeSurfaceCorrection — Type
AkinciFreeSurfaceCorrection(rho0)

Free surface correction according to Akinci et al. (2013). At a free surface, the mean density is typically lower than the reference density, resulting in reduced surface tension and viscosity forces. The free surface correction adjusts the viscosity, pressure, and surface tension forces near free surfaces to counter this effect. It's important to note that this correlation is unphysical and serves as an approximation. The computation time added by this method is about 2–3%.

Mathematically the idea is quite simple. If we have an SPH particle in the middle of a volume at rest, its density will be identical to the rest density $\rho_0$. If we now consider an SPH particle at a free surface at rest, it will have neighbors missing in the direction normal to the surface, which will result in a lower density. If we calculate the correction factor

\[k = \rho_0/\rho_\text{mean},\]

this value will be about ~1.5 for particles at the free surface and can then be used to increase the pressure and viscosity accordingly.

Arguments

  • rho0: Rest density.

References

  • Akinci, N., Akinci, G., & Teschner, M. (2013). "Versatile Surface Tension and Adhesion for SPH Fluids". ACM Transactions on Graphics (TOG), 32(6), 182. doi: 10.1145/2508363.2508405
source
TrixiParticles.BlendedGradientCorrection — Type
BlendedGradientCorrection()

Calculate a blended gradient to reduce the stability issues of the GradientCorrection.

This calculates the following,

\[\tilde\nabla A_i = (1-\lambda) \nabla A_i + \lambda L_i \nabla A_i\]

with $0 \leq \lambda \leq 1$ being the blending factor.

Arguments

  • blending_factor: Blending factor between corrected and regular SPH gradient.
source
TrixiParticles.GradientCorrection — Type
GradientCorrection()

Compute the corrected gradient of particle interactions based on their relative positions.

Mathematical Details

Given the standard SPH representation, the gradient of a field $A$ at particle $a$ is given by

\[\nabla A_a = \sum_b m_b \frac{A_b - A_a}{\rho_b} \nabla_{r_a} W(\Vert r_a - r_b \Vert, h),\]

where $m_b$ is the mass of particle $b$ and $\rho_b$ is the density of particle $b$.

The gradient correction, as commonly proposed, involves multiplying this gradient with a correction matrix $L$:

\[\tilde{\nabla} A_a = \bm{L}_a \nabla A_a\]

The correction matrix $\bm{L}_a$ is computed based on the provided particle configuration, aiming to make the corrected gradient more accurate, especially near domain boundaries.

To satisfy

\[\sum_b V_b r_{ba} \otimes \tilde{\nabla}W_b(r_a) = \left( \sum_b V_b r_{ba} \otimes \nabla W_b(r_a) \right) \bm{L}_a^T = \bm{I}\]

the correction matrix $\bm{L}_a$ is evaluated explicitly as

\[\bm{L}_a = \left( \sum_b V_b \nabla W_b(r_{a}) \otimes r_{ba} \right)^{-1}.\]

Note
  • Stability issues arise, especially when particles separate into small clusters.
  • Doubles the computational effort.

References

  • J. Bonet, T.-S.L. Lok. "Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97–115. doi: 10.1016/S0045-7825(99)00051-1
  • Mihai Basa, Nathan Quinlan, Martin Lastiwka. "Robustness and accuracy of SPH formulations for viscous flow". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927
source
TrixiParticles.KernelCorrection — Type
KernelCorrection()

Kernel correction uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al. This can be further extended to obtain a kernel corrected gradient as shown by Basa et al.

The kernel correction coefficient is determined by

\[c(x) = \sum_{b=1} V_b W_b(x)\]

The gradient of corrected kernel is determined by

\[\nabla \tilde{W}_{b}(r) =\frac{\nabla W_{b}(r) - W_b(r) \gamma(r)}{\sum_{b=1} V_b W_b(r)} , \quad \text{where} \quad -\gamma(r) = \frac{\sum_{b=1} V_b \nabla W_b(r)}{\sum_{b=1} V_b W_b(r)}.\]

This correction can be applied with SummationDensity and ContinuityDensity, which leads to an improvement, especially at free surfaces.

Note
  • This only works when the boundary model uses SummationDensity (yet).
  • It is also referred to as "0th order correction".
  • In 2D, we can expect an increase of about 10–15% in computation time.

References

  • J. Bonet, T.-S.L. Lok. "Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97-115. doi: 10.1016/S0045-7825(99)00051-1
  • Mihai Basa, Nathan Quinlan, Martin Lastiwka. "Robustness and accuracy of SPH formulations for viscous flow". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927
  • Shaofan Li, Wing Kam Liu. "Moving least-square reproducing kernel method Part II: Fourier analysis". In: Computer Methods in Applied Mechanics and Engineering 139 (1996), pages 159-193. doi:10.1016/S0045-7825(96)01082-1
source
TrixiParticles.MixedKernelGradientCorrection — Type
MixedKernelGradientCorrection()

Combines GradientCorrection and KernelCorrection, which results in a 1st-order-accurate SPH method.

Notes:

  • Stability issues, especially when particles separate into small clusters.
  • Doubles the computational effort.

References

  • J. Bonet, T.-S.L. Lok. "Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97–115. doi: 10.1016/S0045-7825(99)00051-1
  • Mihai Basa, Nathan Quinlan, Martin Lastiwka. "Robustness and accuracy of SPH formulations for viscous flow". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927
source
TrixiParticles.ShepardKernelCorrection — Type
ShepardKernelCorrection()

Kernel correction uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al.

The kernel correction coefficient is determined by

\[c(x) = \sum_{b=1} V_b W_b(x),\]

where $V_b = m_b / \rho_b$ is the volume of particle $b$.

This correction is applied with SummationDensity to correct the density and leads to an improvement, especially at free surfaces.

Note
  • It is also referred to as "0th order correction".
  • In 2D, we can expect an increase of about 5–6% in computation time.

References

  • J. Bonet, T.-S.L. Lok. "Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations". In: Computer Methods in Applied Mechanics and Engineering 180 (1999), pages 97-115. doi: 10.1016/S0045-7825(99)00051-1
  • Mihai Basa, Nathan Quinlan, Martin Lastiwka. "Robustness and accuracy of SPH formulations for viscous flow". In: International Journal for Numerical Methods in Fluids 60 (2009), pages 1127–1148. doi: 10.1002/fld.1927
  • Shaofan Li, Wing Kam Liu. "Moving least-square reproducing kernel method Part II: Fourier analysis". In: Computer Methods in Applied Mechanics and Engineering 139 (1996), pages 159–193. doi:10.1016/S0045-7825(96)01082-1
source
+

DensityDiffusionAntuono adds a correction term to solve this problem, but this term is very expensive and adds about 40–50% of computational cost.

API

TrixiParticles.DensityDiffusionAntuono — Type
DensityDiffusionAntuono(initial_condition; delta)

The commonly used density diffusion terms by Antuono (2010), also referred to as δ-SPH. The density diffusion term by Molteni (2009) is extended by a second term, which is nicely written down by Antuono (2012).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = 2\left(\rho_a - \rho_b - \frac{1}{2}\big(\nabla\rho^L_a + \nabla\rho^L_b\big) \cdot r_{ab}\right) + \frac{r_{ab}}{\Vert r_{ab} \Vert^2},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively and $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$. The symbol $\nabla\rho^L_a$ denotes the renormalized density gradient defined as

\[\nabla\rho^L_a = -\sum_b (\rho_a - \rho_b) V_b L_a \nabla_{r_a} W(\Vert r_{ab} \Vert, h)\]

with

\[L_a := \left( -\sum_{b} V_b r_{ab} \otimes \nabla_{r_a} W(\Vert r_{ab} \Vert, h) \right)^{-1} \in \R^{d \times d},\]

where $d$ is the number of dimensions.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

source
TrixiParticles.DensityDiffusionFerrari — Type
DensityDiffusionFerrari()

A density diffusion term by Ferrari (2009).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = \frac{\rho_a - \rho_b}{2h} \frac{r_{ab}}{\Vert r_{ab} \Vert},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively, $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$ and $h$ is the smoothing length.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

source
TrixiParticles.DensityDiffusionMolteniColagrossi — Type
DensityDiffusionMolteniColagrossi(; delta)

The commonly used density diffusion term by Molteni (2009).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = 2(\rho_a - \rho_b) \frac{r_{ab}}{\Vert r_{ab} \Vert^2},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively and $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

source

Corrections

TrixiParticles.AkinciFreeSurfaceCorrection — Type
AkinciFreeSurfaceCorrection(rho0)

Free surface correction according to Akinci et al. (2013). At a free surface, the mean density is typically lower than the reference density, resulting in reduced surface tension and viscosity forces. The free surface correction adjusts the viscosity, pressure, and surface tension forces near free surfaces to counter this effect. It's important to note that this correlation is unphysical and serves as an approximation. The computation time added by this method is about 2–3%.

Mathematically the idea is quite simple. If we have an SPH particle in the middle of a volume at rest, its density will be identical to the rest density $\rho_0$. If we now consider an SPH particle at a free surface at rest, it will have neighbors missing in the direction normal to the surface, which will result in a lower density. If we calculate the correction factor

\[k = \rho_0/\rho_\text{mean},\]

this value will be about ~1.5 for particles at the free surface and can then be used to increase the pressure and viscosity accordingly.

Arguments

  • rho0: Rest density.
source
TrixiParticles.BlendedGradientCorrection — Type
BlendedGradientCorrection()

Calculate a blended gradient to reduce the stability issues of the GradientCorrection as explained by Bonet (1999).

This calculates the following,

\[\tilde\nabla A_i = (1-\lambda) \nabla A_i + \lambda L_i \nabla A_i\]

with $0 \leq \lambda \leq 1$ being the blending factor.

Arguments

  • blending_factor: Blending factor between corrected and regular SPH gradient.
source
TrixiParticles.GradientCorrection — Type
GradientCorrection()

Compute the corrected gradient of particle interactions based on their relative positions (see Bonet, 1999).

Mathematical Details

Given the standard SPH representation, the gradient of a field $A$ at particle $a$ is given by

\[\nabla A_a = \sum_b m_b \frac{A_b - A_a}{\rho_b} \nabla_{r_a} W(\Vert r_a - r_b \Vert, h),\]

where $m_b$ is the mass of particle $b$ and $\rho_b$ is the density of particle $b$.

The gradient correction, as commonly proposed, involves multiplying this gradient with a correction matrix $L$:

\[\tilde{\nabla} A_a = \bm{L}_a \nabla A_a\]

The correction matrix $\bm{L}_a$ is computed based on the provided particle configuration, aiming to make the corrected gradient more accurate, especially near domain boundaries.

To satisfy

\[\sum_b V_b r_{ba} \otimes \tilde{\nabla}W_b(r_a) = \left( \sum_b V_b r_{ba} \otimes \nabla W_b(r_a) \right) \bm{L}_a^T = \bm{I}\]

the correction matrix $\bm{L}_a$ is evaluated explicitly as

\[\bm{L}_a = \left( \sum_b V_b \nabla W_b(r_{a}) \otimes r_{ba} \right)^{-1}.\]

Note
  • Stability issues arise, especially when particles separate into small clusters.
  • Doubles the computational effort.
  • Better stability with smoother smoothing Kernels with larger support, e.g. SchoenbergQuinticSplineKernel or WendlandC6Kernel.
  • Set dt_max =< 1e-3 for stability.
source
TrixiParticles.KernelCorrection — Type
KernelCorrection()

Kernel correction, as explained by Bonet (1999), uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al. This can be further extended to obtain a kernel corrected gradient as shown by Basa et al. (2008).

The kernel correction coefficient is determined by

\[c(x) = \sum_{b=1} V_b W_b(x)\]

The gradient of corrected kernel is determined by

\[\nabla \tilde{W}_{b}(r) =\frac{\nabla W_{b}(r) - W_b(r) \gamma(r)}{\sum_{b=1} V_b W_b(r)} , \quad \text{where} \quad +\gamma(r) = \frac{\sum_{b=1} V_b \nabla W_b(r)}{\sum_{b=1} V_b W_b(r)}.\]

This correction can be applied with SummationDensity and ContinuityDensity, which leads to an improvement, especially at free surfaces.

Note
  • This only works when the boundary model uses SummationDensity (yet).
  • It is also referred to as "0th order correction".
  • In 2D, we can expect an increase of about 10–15% in computation time.
source
TrixiParticles.ShepardKernelCorrection — Type
ShepardKernelCorrection()

Kernel correction, as explained by Bonet (1999), uses Shepard interpolation to obtain a 0-th order accurate result, which was first proposed by Li et al. (1996).

The kernel correction coefficient is determined by

\[c(x) = \sum_{b=1} V_b W_b(x),\]

where $V_b = m_b / \rho_b$ is the volume of particle $b$.

This correction is applied with SummationDensity to correct the density and leads to an improvement, especially at free surfaces.

Note
  • It is also referred to as "0th order correction".
  • In 2D, we can expect an increase of about 5–6% in computation time.
source

Surface Tension

Akinci-based intra-particle force surface tension and wall adhesion model

The work by Akinci proposes three forces:

  • a cohesion force
  • a surface area minimization force
  • a wall adhesion force

The classical model is composed of the curvature minimization and cohesion force.

Cohesion force

The model calculates the cohesion force based on the distance between particles and the support radius $h_c$. This force is determined using two distinct regimes within the support radius:

  • For particles closer than half the support radius, a repulsive force is calculated to prevent particle clustering too tightly, enhancing the simulation's stability and realism.
  • Beyond half the support radius and within the full support radius, an attractive force is computed, simulating the effects of surface tension that draw particles together.

The cohesion force, $F_{\text{cohesion}}$, for a pair of particles is given by:

\[F_{\text{cohesion}} = -\sigma m_b C(r) \frac{r}{\Vert r \Vert},\]

where:

  • $\sigma$ represents the surface tension coefficient, adjusting the overall strength of the cohesion effect.
  • $C$ is a scalar function of the distance between particles.

The cohesion kernel $C$ is defined as

\[C(r)=\frac{32}{\pi h_c^9} +\begin{cases} +(h_c-r)^3 r^3, & \text{if } 2r > h_c \\ +2(h_c-r)^3 r^3 - \frac{h^6}{64}, & \text{if } r > 0 \text{ and } 2r \leq h_c \\ +0, & \text{otherwise} +\end{cases}\]

Surface area minimization force

To model the minimization of the surface area and curvature of the fluid, a curvature force is used, which is calculated as

\[F_{\text{curvature}} = -\sigma (n_a - n_b)\]

Wall adhesion force

The wall adhesion model proposed by Akinci et al. is based on a kernel function which is 0 from 0.0 to 0.5 support radiia with a maximum at 0.75. With the force calculated with an adhesion coefficient $\beta$ as

\[F_{\text{adhesion}} = -\beta m_b A(r) \frac{r}{\Vert r \Vert},\]

with $A$ being the adhesion kernel defined as

\[A(r)= \frac{0.007}{h_c^{3.25}} +\begin{cases} +\sqrt[4]{- \frac{4r^2}{h_c} + 6r - 2h_c}, & \text{if } 2r > h_c \text{ and } r \leq h_c \\ +0, & \text{otherwise.} +\end{cases}\]

TrixiParticles.CohesionForceAkinci — Type
CohesionForceAkinci(surface_tension_coefficient=1.0)

This model only implements the cohesion force of the [25] surface tension model.

Keywords

  • surface_tension_coefficient=1.0: Modifies the intensity of the surface tension-induced force, enabling the tuning of the fluid's surface tension properties within the simulation.
source
TrixiParticles.SurfaceTensionAkinci — Type
SurfaceTensionAkinci(surface_tension_coefficient=1.0)

Implements a model for surface tension and adhesion effects drawing upon the principles outlined by [25]. This model is instrumental in capturing the nuanced behaviors of fluid surfaces, such as droplet formation and the dynamics of merging or separation, by utilizing intra-particle forces.

Keywords

  • surface_tension_coefficient=1.0: A parameter to adjust the magnitude of surface tension forces, facilitating the fine-tuning of how surface tension phenomena are represented in the simulation.
source
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z7+8g?B$-LRrX9N`oHEcnJM6VjH$7}-IhS<%T4F}EhU}Tw9jw$CO3@L;G#%#r1I;RU z3u)Kc`J8WT%NTG8HdQ58P>@Cx&{ucSeXvSUANG;BCbd_6JY8C~KO9@pVOr0Er;Zm4 zgXp&TuMY@-=$``a3;evCErLn~zH>ZJ8wUjRd?E~+%WjI_jR{drrwk(*+qRS?m&;yN zx9^smUfNTxH%}y0ZPa@hrl$Hl_@XhRRqs`O-09gP(*BTc9M5j?(YJIa5h4CR{GO*( z>Xg0zF60V>rD|{LCOJtZIDPgE{OYXBj_W$R#ZQ)V{bI}D=eDs1JEYKZ`_0c-YsR4g zgO1~AN#TJvRzwcw%d6mBazF{7SI5?wl3 z_ThBHQ9Zab0$h*c)D}RhY3~E7!(yp>=^lYlwVGxvZib&!xx`;&2^Dev^!S$I$%g*k z?1$p)=jYQVV&e6oS{@%CzteARkY`sV0)>hHSQBNJe))?6<66&9Ht*&Iau$G<(_UT83Wi*!Cd;U~P#OofE> z@ZEjVyFTQ<>D)>B-}``AS$dWGm4}oEF#1_lhd72#$ma`EDpClb;FqcTytZlcQwb z>Hl^s1L;nZ{5~Cck&7BD8~g2a;HqqWN`odyNWvp%W_KO$xZlY?fM#eWE^c4`&%KZc zUf&nF)Bo+;o=A6+_;(=C5cFB^z_UTtuPzFM&(Jimk!0f0iSGK{jejTqz=8L3^!5xW zSt23fTp>^1>EDfiCrN+D!D3fnRRFWGs<8GBKyN`6fJjJE#c2I^J?{|iB<}A$&5V(d zgMHAEke}Ye|10hPz$bsYH)D)+@5$|2G!hB;SL6PpGRdv-TO@-!WwhTY-pM~GW4Ynt z?i(W^p*_17bf}+6NaU5erka+x?HUmzJW>?NyJ)?O z -Example file · TrixiParticles.jl

     +Example file · TrixiParticles.jl
    GitHub

    Example file

    using TrixiParticles
+using OrdinaryDiffEq
+
+# ==========================================================================================
+# ==== Resolution
+n_particles_y = 5
+
+# ==========================================================================================
+# ==== Experiment Setup
+gravity = 2.0
+tspan = (0.0, 5.0)
+
+elastic_beam = (length=0.35, thickness=0.02)
+material = (density=1000.0, E=1.4e6, nu=0.4)
+clamp_radius = 0.05
+
+# The structure starts at the position of the first particle and ends
+# at the position of the last particle.
+particle_spacing = elastic_beam.thickness / (n_particles_y - 1)
+
+# Add particle_spacing/2 to the clamp_radius to ensure that particles are also placed on the radius
+fixed_particles = SphereShape(particle_spacing, clamp_radius + particle_spacing / 2,
+                              (0.0, elastic_beam.thickness / 2), material.density,
+                              cutout_min=(0.0, 0.0),
+                              cutout_max=(clamp_radius, elastic_beam.thickness),
+                              tlsph=true)
+
+n_particles_clamp_x = round(Int, clamp_radius / particle_spacing)
+
+# Beam and clamped particles
+n_particles_per_dimension = (round(Int, elastic_beam.length / particle_spacing) +
+                             n_particles_clamp_x + 1, n_particles_y)
+
+# Note that the `RectangularShape` puts the first particle half a particle spacing away
+# from the boundary, which is correct for fluids, but not for solids.
+# We therefore need to pass `tlsph=true`.
+beam = RectangularShape(particle_spacing, n_particles_per_dimension,
+                        (0.0, 0.0), density=material.density, tlsph=true)
+
+solid = union(beam, fixed_particles)
+
+# ==========================================================================================
+# ==== Solid
+# The kernel in the reference uses a differently scaled smoothing length,
+# so this is equivalent to the smoothing length of `sqrt(2) * particle_spacing` used in the paper.
+smoothing_length = 2 * sqrt(2) * particle_spacing
+smoothing_kernel = WendlandC2Kernel{2}()
+
+solid_system = TotalLagrangianSPHSystem(solid, smoothing_kernel, smoothing_length,
+                                        material.E, material.nu,
+                                        n_fixed_particles=nparticles(fixed_particles),
+                                        acceleration=(0.0, -gravity),
+                                        penalty_force=nothing)
+
+# ==========================================================================================
+# ==== Simulation
+semi = Semidiscretization(solid_system,
+                          neighborhood_search=PrecomputedNeighborhoodSearch{2}())
+ode = semidiscretize(semi, tspan)
+
+info_callback = InfoCallback(interval=100)
+
+# Track the position of the particle in the middle of the tip of the beam.
+middle_particle_id = Int(n_particles_per_dimension[1] * (n_particles_per_dimension[2] + 1) /
+                         2)
+startposition_x = beam.coordinates[1, middle_particle_id]
+startposition_y = beam.coordinates[2, middle_particle_id]
+
+function deflection_x(v, u, t, system)
+    return system.current_coordinates[1, middle_particle_id] - startposition_x
+end
+
+function deflection_y(v, u, t, system)
+    return system.current_coordinates[2, middle_particle_id] - startposition_y
+end
+
+saving_callback = SolutionSavingCallback(dt=0.02, prefix="",
+                                         deflection_x=deflection_x,
+                                         deflection_y=deflection_y)
+
+callbacks = CallbackSet(info_callback, saving_callback)
+
+# Use a Runge-Kutta method with automatic (error based) time step size control
+sol = solve(ode, RDPK3SpFSAL49(), save_everystep=false, callback=callbacks);
+
+
    diff --git a/previews/PR514/tutorials/tut_beam_replaced/index.html b/previews/PR514/tutorials/tut_beam_replaced/index.html deleted file mode 100644 index d669fe1dd..000000000 --- a/previews/PR514/tutorials/tut_beam_replaced/index.html +++ /dev/null @@ -1,86 +0,0 @@ - -Example file · TrixiParticles.jl
    GitHub

    Example file

    using TrixiParticles
-using OrdinaryDiffEq
-
-# ==========================================================================================
-# ==== Resolution
-n_particles_y = 5
-
-# ==========================================================================================
-# ==== Experiment Setup
-gravity = 2.0
-tspan = (0.0, 5.0)
-
-elastic_beam = (length=0.35, thickness=0.02)
-material = (density=1000.0, E=1.4e6, nu=0.4)
-clamp_radius = 0.05
-
-# The structure starts at the position of the first particle and ends
-# at the position of the last particle.
-particle_spacing = elastic_beam.thickness / (n_particles_y - 1)
-
-# Add particle_spacing/2 to the clamp_radius to ensure that particles are also placed on the radius
-fixed_particles = SphereShape(particle_spacing, clamp_radius + particle_spacing / 2,
-                              (0.0, elastic_beam.thickness / 2), material.density,
-                              cutout_min=(0.0, 0.0),
-                              cutout_max=(clamp_radius, elastic_beam.thickness),
-                              tlsph=true)
-
-n_particles_clamp_x = round(Int, clamp_radius / particle_spacing)
-
-# Beam and clamped particles
-n_particles_per_dimension = (round(Int, elastic_beam.length / particle_spacing) +
-                             n_particles_clamp_x + 1, n_particles_y)
-
-# Note that the `RectangularShape` puts the first particle half a particle spacing away
-# from the boundary, which is correct for fluids, but not for solids.
-# We therefore need to pass `tlsph=true`.
-beam = RectangularShape(particle_spacing, n_particles_per_dimension,
-                        (0.0, 0.0), density=material.density, tlsph=true)
-
-solid = union(beam, fixed_particles)
-
-# ==========================================================================================
-# ==== Solid
-# The kernel in the reference uses a differently scaled smoothing length,
-# so this is equivalent to the smoothing length of `sqrt(2) * particle_spacing` used in the paper.
-smoothing_length = 2 * sqrt(2) * particle_spacing
-smoothing_kernel = WendlandC2Kernel{2}()
-
-solid_system = TotalLagrangianSPHSystem(solid, smoothing_kernel, smoothing_length,
-                                        material.E, material.nu,
-                                        n_fixed_particles=nparticles(fixed_particles),
-                                        acceleration=(0.0, -gravity),
-                                        penalty_force=nothing)
-
-# ==========================================================================================
-# ==== Simulation
-semi = Semidiscretization(solid_system)
-ode = semidiscretize(semi, tspan)
-
-info_callback = InfoCallback(interval=100)
-
-# Track the position of the particle in the middle of the tip of the beam.
-middle_particle_id = Int(n_particles_per_dimension[1] * (n_particles_per_dimension[2] + 1) /
-                         2)
-startposition_x = beam.coordinates[1, middle_particle_id]
-startposition_y = beam.coordinates[2, middle_particle_id]
-
-function deflection_x(v, u, t, system)
-    return system.current_coordinates[1, middle_particle_id] - startposition_x
-end
-
-function deflection_y(v, u, t, system)
-    return system.current_coordinates[2, middle_particle_id] - startposition_y
-end
-
-saving_callback = SolutionSavingCallback(dt=0.02, prefix="",
-                                         deflection_x=deflection_x,
-                                         deflection_y=deflection_y)
-
-callbacks = CallbackSet(info_callback, saving_callback)
-
-# Use a Runge-Kutta method with automatic (error based) time step size control
-sol = solve(ode, RDPK3SpFSAL49(), save_everystep=false, callback=callbacks);
-
-
    diff --git a/previews/PR514/tutorials/tut_dam_break/index.html b/previews/PR514/tutorials/tut_dam_break/index.html index 486bfe2cc..e963bd8cf 100644 --- a/previews/PR514/tutorials/tut_dam_break/index.html +++ b/previews/PR514/tutorials/tut_dam_break/index.html @@ -1,3 +1,114 @@ -Example file · TrixiParticles.jl +Example file · TrixiParticles.jl
    GitHub

    Example file

    # 2D dam break simulation based on
+#
+# S. Marrone, M. Antuono, A. Colagrossi, G. Colicchio, D. le Touzé, G. Graziani.
+# "δ-SPH model for simulating violent impact flows".
+# In: Computer Methods in Applied Mechanics and Engineering, Volume 200, Issues 13–16 (2011), pages 1526–1542.
+# https://doi.org/10.1016/J.CMA.2010.12.016
+
+using TrixiParticles
+using OrdinaryDiffEq
+
+# Size parameters
+H = 0.6
+W = 2 * H
+
+# ==========================================================================================
+# ==== Resolution
+fluid_particle_spacing = H / 40
+
+# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model
+boundary_layers = 4
+spacing_ratio = 1
+
+boundary_particle_spacing = fluid_particle_spacing / spacing_ratio
+
+# ==========================================================================================
+# ==== Experiment Setup
+gravity = 9.81
+
+tspan = (0.0, 5.7 / sqrt(gravity))
+
+# Boundary geometry and initial fluid particle positions
+initial_fluid_size = (W, H)
+tank_size = (floor(5.366 * H / boundary_particle_spacing) * boundary_particle_spacing, 4.0)
+
+fluid_density = 1000.0
+sound_speed = 20 * sqrt(gravity * H)
+state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
+                                   exponent=1, clip_negative_pressure=false)
+
+tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,
+                       n_layers=boundary_layers, spacing_ratio=spacing_ratio,
+                       acceleration=(0.0, -gravity), state_equation=state_equation)
+
+# ==========================================================================================
+# ==== Fluid
+smoothing_length = 3.5 * fluid_particle_spacing
+smoothing_kernel = WendlandC2Kernel{2}()
+
+fluid_density_calculator = ContinuityDensity()
+viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)
+# nu = 0.02 * smoothing_length * sound_speed/8
+# viscosity = ViscosityMorris(nu=nu)
+# viscosity = ViscosityAdami(nu=nu)
+# Alternatively the density diffusion model by Molteni & Colagrossi can be used,
+# which will run faster.
+# density_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)
+density_diffusion = DensityDiffusionAntuono(tank.fluid, delta=0.1)
+
+fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
+                                           state_equation, smoothing_kernel,
+                                           smoothing_length, viscosity=viscosity,
+                                           density_diffusion=density_diffusion,
+                                           acceleration=(0.0, -gravity), correction=nothing,
+                                           surface_tension=nothing)
+
+# ==========================================================================================
+# ==== Boundary
+boundary_density_calculator = AdamiPressureExtrapolation()
+boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
+                                             state_equation=state_equation,
+                                             boundary_density_calculator,
+                                             smoothing_kernel, smoothing_length,
+                                             correction=nothing)
+
+boundary_system = BoundarySPHSystem(tank.boundary, boundary_model, adhesion_coefficient=0.0)
+
+# ==========================================================================================
+# ==== Simulation
+# `nothing` will automatically choose the best update strategy. This is only to be able
+# to change this with `trixi_include`.
+semi = Semidiscretization(fluid_system, boundary_system,
+                          neighborhood_search=GridNeighborhoodSearch{2}(update_strategy=nothing))
+ode = semidiscretize(semi, tspan, data_type=nothing)
+
+info_callback = InfoCallback(interval=100)
+
+solution_prefix = ""
+saving_callback = SolutionSavingCallback(dt=0.02, prefix=solution_prefix)
+
+# Save at certain timepoints which allows comparison to the results of Marrone et al.,
+# i.e. (1.5, 2.36, 3.0, 5.7, 6.45).
+# Please note that the images in Marrone et al. are obtained at a particle_spacing = H/320,
+# which takes between 2 and 4 hours.
+saving_paper = SolutionSavingCallback(save_times=[0.0, 0.371, 0.584, 0.743, 1.411, 1.597],
+                                      prefix="marrone_times")
+
+# This can be overwritten with `trixi_include`
+extra_callback = nothing
+
+use_reinit = false
+density_reinit_cb = use_reinit ?
+                    DensityReinitializationCallback(semi.systems[1], interval=10) :
+                    nothing
+stepsize_callback = StepsizeCallback(cfl=0.9)
+
+callbacks = CallbackSet(info_callback, saving_callback, stepsize_callback, extra_callback,
+                        density_reinit_cb, saving_paper)
+
+sol = solve(ode, CarpenterKennedy2N54(williamson_condition=false),
+            dt=1.0, # This is overwritten by the stepsize callback
+            save_everystep=false, callback=callbacks);
+
+
    diff --git a/previews/PR514/tutorials/tut_dam_break_replaced/index.html b/previews/PR514/tutorials/tut_dam_break_replaced/index.html deleted file mode 100644 index 5762a1dab..000000000 --- a/previews/PR514/tutorials/tut_dam_break_replaced/index.html +++ /dev/null @@ -1,108 +0,0 @@ - -Example file · TrixiParticles.jl
    GitHub

    Example file

    # 2D dam break simulation based on
-#
-# S. Marrone, M. Antuono, A. Colagrossi, G. Colicchio, D. le Touzé, G. Graziani.
-# "δ-SPH model for simulating violent impact flows".
-# In: Computer Methods in Applied Mechanics and Engineering, Volume 200, Issues 13–16 (2011), pages 1526–1542.
-# https://doi.org/10.1016/J.CMA.2010.12.016
-
-using TrixiParticles
-using OrdinaryDiffEq
-
-# Size parameters
-H = 0.6
-W = 2 * H
-
-# ==========================================================================================
-# ==== Resolution
-fluid_particle_spacing = H / 40
-
-# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model
-boundary_layers = 4
-spacing_ratio = 1
-
-boundary_particle_spacing = fluid_particle_spacing / spacing_ratio
-
-# ==========================================================================================
-# ==== Experiment Setup
-gravity = 9.81
-
-tspan = (0.0, 5.7 / sqrt(gravity))
-
-# Boundary geometry and initial fluid particle positions
-initial_fluid_size = (W, H)
-tank_size = (floor(5.366 * H / boundary_particle_spacing) * boundary_particle_spacing, 4.0)
-
-fluid_density = 1000.0
-sound_speed = 20 * sqrt(gravity * H)
-state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
-                                   exponent=1, clip_negative_pressure=false)
-
-tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,
-                       n_layers=boundary_layers, spacing_ratio=spacing_ratio,
-                       acceleration=(0.0, -gravity), state_equation=state_equation)
-
-# ==========================================================================================
-# ==== Fluid
-smoothing_length = 3.5 * fluid_particle_spacing
-smoothing_kernel = WendlandC2Kernel{2}()
-
-fluid_density_calculator = ContinuityDensity()
-viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)
-# Alternatively the density diffusion model by Molteni & Colagrossi can be used,
-# which will run faster.
-# density_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)
-density_diffusion = DensityDiffusionAntuono(tank.fluid, delta=0.1)
-
-fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
-                                           state_equation, smoothing_kernel,
-                                           smoothing_length, viscosity=viscosity,
-                                           density_diffusion=density_diffusion,
-                                           acceleration=(0.0, -gravity),
-                                           correction=nothing)
-
-# ==========================================================================================
-# ==== Boundary
-boundary_density_calculator = AdamiPressureExtrapolation()
-boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
-                                             state_equation=state_equation,
-                                             boundary_density_calculator,
-                                             smoothing_kernel, smoothing_length,
-                                             correction=nothing)
-
-boundary_system = BoundarySPHSystem(tank.boundary, boundary_model)
-
-# ==========================================================================================
-# ==== Simulation
-semi = Semidiscretization(fluid_system, boundary_system, threaded_nhs_update=true)
-ode = semidiscretize(semi, tspan)
-
-info_callback = InfoCallback(interval=100)
-
-solution_prefix = ""
-saving_callback = SolutionSavingCallback(dt=0.02, prefix=solution_prefix)
-
-# Save at certain timepoints which allows comparison to the results of Marrone et al.,
-# i.e. (1.5, 2.36, 3.0, 5.7, 6.45).
-# Please note that the images in Marrone et al. are obtained at a particle_spacing = H/320,
-# which takes between 2 and 4 hours.
-saving_paper = SolutionSavingCallback(save_times=[0.0, 0.371, 0.584, 0.743, 1.411, 1.597],
-                                      prefix="marrone_times")
-
-# This can be overwritten with `trixi_include`
-extra_callback = nothing
-
-use_reinit = false
-density_reinit_cb = use_reinit ?
-                    DensityReinitializationCallback(semi.systems[1], interval=10) :
-                    nothing
-stepsize_callback = StepsizeCallback(cfl=0.9)
-
-callbacks = CallbackSet(info_callback, saving_callback, stepsize_callback, extra_callback,
-                        density_reinit_cb, saving_paper)
-
-sol = solve(ode, CarpenterKennedy2N54(williamson_condition=false),
-            dt=1.0, # This is overwritten by the stepsize callback
-            save_everystep=false, callback=callbacks);
-
-
    diff --git a/previews/PR514/tutorials/tut_falling/index.html b/previews/PR514/tutorials/tut_falling/index.html index 086d7ae68..772bd2bb3 100644 --- a/previews/PR514/tutorials/tut_falling/index.html +++ b/previews/PR514/tutorials/tut_falling/index.html @@ -1,3 +1,132 @@ -Example file · TrixiParticles.jl +Example file · TrixiParticles.jl
    GitHub

    Example file

    using TrixiParticles
+using OrdinaryDiffEq
+
+# ==========================================================================================
+# ==== Resolution
+fluid_particle_spacing = 0.02
+solid_particle_spacing = fluid_particle_spacing
+
+# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model
+boundary_layers = 3
+spacing_ratio = 1
+
+# ==========================================================================================
+# ==== Experiment Setup
+gravity = 9.81
+tspan = (0.0, 1.0)
+
+# Boundary geometry and initial fluid particle positions
+initial_fluid_size = (2.0, 0.9)
+tank_size = (2.0, 1.0)
+
+fluid_density = 1000.0
+sound_speed = 10 * sqrt(gravity * initial_fluid_size[2])
+state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
+                                   exponent=1)
+
+tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,
+                       n_layers=boundary_layers, spacing_ratio=spacing_ratio,
+                       faces=(true, true, true, false),
+                       acceleration=(0.0, -gravity), state_equation=state_equation)
+
+sphere1_radius = 0.3
+sphere2_radius = 0.2
+sphere1_density = 500.0
+sphere2_density = 1100.0
+
+# Young's modulus and Poisson ratio
+sphere1_E = 7e4
+sphere2_E = 1e5
+nu = 0.0
+
+sphere1_center = (0.5, 1.6)
+sphere2_center = (1.5, 1.6)
+sphere1 = SphereShape(solid_particle_spacing, sphere1_radius, sphere1_center,
+                      sphere1_density, sphere_type=VoxelSphere())
+sphere2 = SphereShape(solid_particle_spacing, sphere2_radius, sphere2_center,
+                      sphere2_density, sphere_type=VoxelSphere())
+
+# ==========================================================================================
+# ==== Fluid
+fluid_smoothing_length = 3.0 * fluid_particle_spacing
+fluid_smoothing_kernel = WendlandC2Kernel{2}()
+
+fluid_density_calculator = ContinuityDensity()
+viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)
+density_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)
+
+fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
+                                           state_equation, fluid_smoothing_kernel,
+                                           fluid_smoothing_length, viscosity=viscosity,
+                                           density_diffusion=density_diffusion,
+                                           acceleration=(0.0, -gravity))
+
+# ==========================================================================================
+# ==== Boundary
+boundary_density_calculator = BernoulliPressureExtrapolation()
+boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
+                                             state_equation=state_equation,
+                                             boundary_density_calculator,
+                                             fluid_smoothing_kernel, fluid_smoothing_length)
+
+boundary_system = BoundarySPHSystem(tank.boundary, boundary_model)
+
+# ==========================================================================================
+# ==== Solid
+solid_smoothing_length = 2 * sqrt(2) * solid_particle_spacing
+solid_smoothing_kernel = WendlandC2Kernel{2}()
+
+# For the FSI we need the hydrodynamic masses and densities in the solid boundary model
+hydrodynamic_densites_1 = fluid_density * ones(size(sphere1.density))
+hydrodynamic_masses_1 = hydrodynamic_densites_1 * solid_particle_spacing^ndims(fluid_system)
+
+solid_boundary_model_1 = BoundaryModelDummyParticles(hydrodynamic_densites_1,
+                                                     hydrodynamic_masses_1,
+                                                     state_equation=state_equation,
+                                                     boundary_density_calculator,
+                                                     fluid_smoothing_kernel,
+                                                     fluid_smoothing_length)
+
+hydrodynamic_densites_2 = fluid_density * ones(size(sphere2.density))
+hydrodynamic_masses_2 = hydrodynamic_densites_2 * solid_particle_spacing^ndims(fluid_system)
+
+solid_boundary_model_2 = BoundaryModelDummyParticles(hydrodynamic_densites_2,
+                                                     hydrodynamic_masses_2,
+                                                     state_equation=state_equation,
+                                                     boundary_density_calculator,
+                                                     fluid_smoothing_kernel,
+                                                     fluid_smoothing_length)
+
+solid_system_1 = TotalLagrangianSPHSystem(sphere1,
+                                          solid_smoothing_kernel, solid_smoothing_length,
+                                          sphere1_E, nu,
+                                          acceleration=(0.0, -gravity),
+                                          boundary_model=solid_boundary_model_1,
+                                          penalty_force=PenaltyForceGanzenmueller(alpha=0.3))
+
+solid_system_2 = TotalLagrangianSPHSystem(sphere2,
+                                          solid_smoothing_kernel, solid_smoothing_length,
+                                          sphere2_E, nu,
+                                          acceleration=(0.0, -gravity),
+                                          boundary_model=solid_boundary_model_2,
+                                          penalty_force=PenaltyForceGanzenmueller(alpha=0.3))
+
+# ==========================================================================================
+# ==== Simulation
+semi = Semidiscretization(fluid_system, boundary_system, solid_system_1, solid_system_2)
+ode = semidiscretize(semi, tspan)
+
+info_callback = InfoCallback(interval=10)
+saving_callback = SolutionSavingCallback(dt=0.02, output_directory="out", prefix="",
+                                         write_meta_data=true)
+
+callbacks = CallbackSet(info_callback, saving_callback)
+
+# Use a Runge-Kutta method with automatic (error based) time step size control.
+sol = solve(ode, RDPK3SpFSAL49(),
+            abstol=1e-6, # Default abstol is 1e-6
+            reltol=1e-3, # Default reltol is 1e-3
+            save_everystep=false, callback=callbacks);
+
+
    diff --git a/previews/PR514/tutorials/tut_falling_replaced/index.html b/previews/PR514/tutorials/tut_falling_replaced/index.html deleted file mode 100644 index 360a8f855..000000000 --- a/previews/PR514/tutorials/tut_falling_replaced/index.html +++ /dev/null @@ -1,132 +0,0 @@ - -Example file · TrixiParticles.jl
    GitHub

    Example file

    using TrixiParticles
-using OrdinaryDiffEq
-
-# ==========================================================================================
-# ==== Resolution
-fluid_particle_spacing = 0.02
-solid_particle_spacing = fluid_particle_spacing
-
-# Change spacing ratio to 3 and boundary layers to 1 when using Monaghan-Kajtar boundary model
-boundary_layers = 3
-spacing_ratio = 1
-
-# ==========================================================================================
-# ==== Experiment Setup
-gravity = 9.81
-tspan = (0.0, 1.0)
-
-# Boundary geometry and initial fluid particle positions
-initial_fluid_size = (2.0, 0.9)
-tank_size = (2.0, 1.0)
-
-fluid_density = 1000.0
-sound_speed = 10 * sqrt(gravity * initial_fluid_size[2])
-state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
-                                   exponent=1)
-
-tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,
-                       n_layers=boundary_layers, spacing_ratio=spacing_ratio,
-                       faces=(true, true, true, false),
-                       acceleration=(0.0, -gravity), state_equation=state_equation)
-
-sphere1_radius = 0.3
-sphere2_radius = 0.2
-sphere1_density = 500.0
-sphere2_density = 1100.0
-
-# Young's modulus and Poisson ratio
-sphere1_E = 7e4
-sphere2_E = 1e5
-nu = 0.0
-
-sphere1_center = (0.5, 1.6)
-sphere2_center = (1.5, 1.6)
-sphere1 = SphereShape(solid_particle_spacing, sphere1_radius, sphere1_center,
-                      sphere1_density, sphere_type=VoxelSphere())
-sphere2 = SphereShape(solid_particle_spacing, sphere2_radius, sphere2_center,
-                      sphere2_density, sphere_type=VoxelSphere())
-
-# ==========================================================================================
-# ==== Fluid
-fluid_smoothing_length = 3.0 * fluid_particle_spacing
-fluid_smoothing_kernel = WendlandC2Kernel{2}()
-
-fluid_density_calculator = ContinuityDensity()
-viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)
-density_diffusion = DensityDiffusionMolteniColagrossi(delta=0.1)
-
-fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
-                                           state_equation, fluid_smoothing_kernel,
-                                           fluid_smoothing_length, viscosity=viscosity,
-                                           density_diffusion=density_diffusion,
-                                           acceleration=(0.0, -gravity))
-
-# ==========================================================================================
-# ==== Boundary
-boundary_density_calculator = AdamiPressureExtrapolation()
-boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
-                                             state_equation=state_equation,
-                                             boundary_density_calculator,
-                                             fluid_smoothing_kernel, fluid_smoothing_length)
-
-boundary_system = BoundarySPHSystem(tank.boundary, boundary_model)
-
-# ==========================================================================================
-# ==== Solid
-solid_smoothing_length = 2 * sqrt(2) * solid_particle_spacing
-solid_smoothing_kernel = WendlandC2Kernel{2}()
-
-# For the FSI we need the hydrodynamic masses and densities in the solid boundary model
-hydrodynamic_densites_1 = fluid_density * ones(size(sphere1.density))
-hydrodynamic_masses_1 = hydrodynamic_densites_1 * solid_particle_spacing^ndims(fluid_system)
-
-solid_boundary_model_1 = BoundaryModelDummyParticles(hydrodynamic_densites_1,
-                                                     hydrodynamic_masses_1,
-                                                     state_equation=state_equation,
-                                                     boundary_density_calculator,
-                                                     fluid_smoothing_kernel,
-                                                     fluid_smoothing_length)
-
-hydrodynamic_densites_2 = fluid_density * ones(size(sphere2.density))
-hydrodynamic_masses_2 = hydrodynamic_densites_2 * solid_particle_spacing^ndims(fluid_system)
-
-solid_boundary_model_2 = BoundaryModelDummyParticles(hydrodynamic_densites_2,
-                                                     hydrodynamic_masses_2,
-                                                     state_equation=state_equation,
-                                                     boundary_density_calculator,
-                                                     fluid_smoothing_kernel,
-                                                     fluid_smoothing_length)
-
-solid_system_1 = TotalLagrangianSPHSystem(sphere1,
-                                          solid_smoothing_kernel, solid_smoothing_length,
-                                          sphere1_E, nu,
-                                          acceleration=(0.0, -gravity),
-                                          boundary_model=solid_boundary_model_1,
-                                          penalty_force=PenaltyForceGanzenmueller(alpha=0.3))
-
-solid_system_2 = TotalLagrangianSPHSystem(sphere2,
-                                          solid_smoothing_kernel, solid_smoothing_length,
-                                          sphere2_E, nu,
-                                          acceleration=(0.0, -gravity),
-                                          boundary_model=solid_boundary_model_2,
-                                          penalty_force=PenaltyForceGanzenmueller(alpha=0.3))
-
-# ==========================================================================================
-# ==== Simulation
-semi = Semidiscretization(fluid_system, boundary_system, solid_system_1, solid_system_2)
-ode = semidiscretize(semi, tspan)
-
-info_callback = InfoCallback(interval=10)
-saving_callback = SolutionSavingCallback(dt=0.02, output_directory="out", prefix="",
-                                         write_meta_data=true)
-
-callbacks = CallbackSet(info_callback, saving_callback)
-
-# Use a Runge-Kutta method with automatic (error based) time step size control.
-sol = solve(ode, RDPK3SpFSAL49(),
-            abstol=1e-6, # Default abstol is 1e-6
-            reltol=1e-3, # Default reltol is 1e-3
-            save_everystep=false, callback=callbacks);
-
-
    diff --git a/previews/PR514/tutorials/tut_setup/index.html b/previews/PR514/tutorials/tut_setup/index.html index da5a422bd..d96ea1030 100644 --- a/previews/PR514/tutorials/tut_setup/index.html +++ b/previews/PR514/tutorials/tut_setup/index.html @@ -1,28 +1,147 @@ -Setting up your simulation from scratch · TrixiParticles.jl
    GitHub

    Setting up your simulation from scratch

    In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.

    For different setups and physics, have a look at our other example files.

    Resolution

    At the beginning of most simulation files, we define the numerical resolution, so that it can easily be found and changed. First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.

    using TrixiParticles
-using OrdinaryDiffEq

    Now, we define the particle spacing, which is our numerical resolution. We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary.

    fluid_particle_spacing = 0.05
-boundary_layers = 3

    Experiment setup

    We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. First, we define the physical parameters gravitational acceleration, initial fluid size, tank size, fluid density, and simulation time.

    gravity = 9.81
+Setting up your simulation from scratch · TrixiParticles.jl
    GitHub
    Setting up your simulation from scratch
    In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.
    For different setups and physics, have a look at our other example files.
    First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.
    using TrixiParticles
+using OrdinaryDiffEq
    Resolution
    Now, we define the particle spacing, which is our numerical resolution. We usually call the variable fluid_particle_spacing, so that we can easily change the resolution of an example file by overwriting this variable with trixi_include. In 2D, the number of particles will grow quadratically, in 3D cubically with the spacing.
    We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary. In particular, we require boundary_layers >= compact_support. The value for the compact support for each kernel can be found in the smoothing kernel overview.
    fluid_particle_spacing = 0.05
+boundary_layers = 3
    Experiment setup
    We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. Experiment Setup First, we define the physical parameters gravitational acceleration, simulation time, initial fluid size, tank size and fluid density.
    gravity = 9.81
 tspan = (0.0, 1.0)
 initial_fluid_size = (1.0, 0.9)
 tank_size = (1.0, 1.0)
-fluid_density = 1000.0
    In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.
    sound_speed = 10.0
+fluid_density = 1000.0
    In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.
    sound_speed = 10.0
 state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
                                    exponent=7)
    The speed of sound here is numerical and not physical. We artificially lower the speed of sound, since the physical speed of sound in water would lead to prohibitively small time steps. The speed of sound in Weakly Compressible SPH should be chosen as small as possible for numerical efficiency, but large enough to limit density fluctuations to about 1%.
    TrixiParticles.jl requires the initial particle positions and quantities in form of an InitialCondition. Instead of manually defining particle positions, you can work with our pre-defined setups. Among others, we provide setups for rectangular shapes, circles, and spheres. Initial conditions can also be combined with common set operations. See this page for a list of pre-defined setups and details on set operations on initial conditions.
    Here, we use the RectangularTank setup, which generates a rectangular fluid inside a rectangular tank, and supports a hydrostatic pressure gradient by passing a gravitational acceleration and a state equation (see above).
    tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size,
                        fluid_density, n_layers=boundary_layers,
-                       acceleration=(0.0, -gravity), state_equation=state_equation)
    Fluid system
    To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.
    smoothing_length = 1.2 * fluid_particle_spacing
-smoothing_kernel = SchoenbergCubicSplineKernel{2}()
    You can find an overview over smoothing kernels and corresponding smoothing lengths here.
    For stability, we need numerical dissipation in form of an artificial viscosity term.
    viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)
    We choose the parameters as small as possible to avoid visible viscosity, but as large as possible to stabilize the simulation.
    The WCSPH method can either compute the particle density by a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating it over time. We choose the latter approach here by using the density calculator ContinuityDensity.
    fluid_density_calculator = ContinuityDensity()
+                       acceleration=(0.0, -gravity), state_equation=state_equation)
    Fluid system
    To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.
    smoothing_length = 1.2 * fluid_particle_spacing
+smoothing_kernel = SchoenbergCubicSplineKernel{2}()
    You can find an overview over smoothing kernels and corresponding smoothing lengths here.
    For stability, we need numerical dissipation in form of an artificial viscosity term. Other viscosity models offer a physical approach based on the kinematic viscosity of the fluid.
    viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)
    We choose the parameters as small as possible to avoid non-physical behavior, but as large as possible to stabilize the simulation.
    The WCSPH method can either compute the particle density directly with a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating its change over time. We choose the latter approach here by using the density calculator ContinuityDensity, which is more efficient and handles free surfaces without the need for additional correction terms.
    fluid_density_calculator = ContinuityDensity()
 fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
                                            state_equation, smoothing_kernel,
                                            smoothing_length, viscosity=viscosity,
-                                           acceleration=(0.0, -gravity))
    Boundary system
    In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation, which generally produces the best results of the implemented methods. See here for a comprehensive overview over boundary models.
    boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
+                                           acceleration=(0.0, -gravity))
    Boundary system
    To model the boundary, we use particle-based boundary conditions, in which particles are sampled in the boundary that interact with the fluid particles to avoid penetration. In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation. See here for a comprehensive overview over boundary models.
    boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
                                              state_equation=state_equation,
                                              AdamiPressureExtrapolation(),
                                              smoothing_kernel, smoothing_length)
 boundary_system = BoundarySPHSystem(tank.boundary, boundary_model)
    Semidiscretization
    The key component of every simulation is the Semidiscretization, which couples all systems of the simulation. All methods in TrixiParticles.jl are semidiscretizations, which discretize the equations in time to provide an ordinary differential equation that still has to be solved in time. By providing a simulation time span, we can call semidiscretize, which returns an ODEProblem that can be solved with a time integration method.
    semi = Semidiscretization(fluid_system, boundary_system)
-ode = semidiscretize(semi, tspan)
    Time integration
    We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.
    OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.
    info_callback = InfoCallback(interval=50)
+ode = semidiscretize(semi, tspan)
    Time integration
    We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.
    OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.
    info_callback = InfoCallback(interval=50)
 saving_callback = SolutionSavingCallback(dt=0.02)
 
-callbacks = CallbackSet(info_callback, saving_callback)
    Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.
    sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);
    See Visualization for how to visualize the solution.
    Replacing components with custom implementations
    If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.
    Custom smoothing kernel
    To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.
    struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end
    This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.
    In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.
    function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)
+callbacks = CallbackSet(info_callback, saving_callback)
    Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.
    sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);
    
+████████╗██████╗ ██╗██╗  ██╗██╗██████╗  █████╗ ██████╗ ████████╗██╗ ██████╗██╗     ███████╗███████╗
+╚══██╔══╝██╔══██╗██║╚██╗██╔╝██║██╔══██╗██╔══██╗██╔══██╗╚══██╔══╝██║██╔════╝██║     ██╔════╝██╔════╝
+   ██║   ██████╔╝██║ ╚███╔╝ ██║██████╔╝███████║██████╔╝   ██║   ██║██║     ██║     █████╗  ███████╗
+   ██║   ██╔══██╗██║ ██╔██╗ ██║██╔═══╝ ██╔══██║██╔══██╗   ██║   ██║██║     ██║     ██╔══╝  ╚════██║
+   ██║   ██║  ██║██║██╔╝ ██╗██║██║     ██║  ██║██║  ██║   ██║   ██║╚██████╗███████╗███████╗███████║
+   ╚═╝   ╚═╝  ╚═╝╚═╝╚═╝  ╚═╝╚═╝╚═╝     ╚═╝  ╚═╝╚═╝  ╚═╝   ╚═╝   ╚═╝ ╚═════╝╚══════╝╚══════╝╚══════╝
+
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Semidiscretization                                                                               │
+│ ══════════════════                                                                               │
+│ #spatial dimensions: ………………………… 2                                                                │
+│ #systems: ……………………………………………………… 2                                                                │
+│ neighborhood search: ………………………… GridNeighborhoodSearch                                           │
+│ total #particles: ………………………………… 636                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ WeaklyCompressibleSPHSystem{2}                                                                   │
+│ ══════════════════════════════                                                                   │
+│ #particles: ………………………………………………… 360                                                              │
+│ density calculator: …………………………… ContinuityDensity                                                │
+│ correction method: ……………………………… Nothing                                                          │
+│ state equation: ……………………………………… StateEquationCole                                                │
+│ smoothing kernel: ………………………………… SchoenbergCubicSplineKernel                                      │
+│ viscosity: …………………………………………………… ArtificialViscosityMonaghan{Float64}(0.02, 0.0, 0.01)            │
+│ density diffusion: ……………………………… nothing                                                          │
+│ surface tension: …………………………………… nothing                                                          │
+│ acceleration: …………………………………………… [0.0, -9.81]                                                     │
+│ source terms: …………………………………………… Nothing                                                          │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ BoundarySPHSystem{2}                                                                             │
+│ ════════════════════                                                                             │
+│ #particles: ………………………………………………… 276                                                              │
+│ boundary model: ……………………………………… BoundaryModelDummyParticles(AdamiPressureExtrapolation, Nothing) │
+│ movement function: ……………………………… nothing                                                          │
+│ adhesion coefficient: ……………………… 0.0                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ SolutionSavingCallback                                                                           │
+│ ══════════════════════                                                                           │
+│ dt: ……………………………………………………………………… 0.02                                                             │
+│ custom quantities: ……………………………… nothing                                                          │
+│ save initial solution: …………………… yes                                                              │
+│ save final solution: ………………………… yes                                                              │
+│ output directory: ………………………………… /home/runner/work/TrixiParticles…les.jl/docs/build/tutorials/out │
+│ prefix: ……………………………………………………………                                                                  │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Time integration                                                                                 │
+│ ════════════════                                                                                 │
+│ Start time: ………………………………………………… 0.0                                                              │
+│ Final time: ………………………………………………… 1.0                                                              │
+│ time integrator: …………………………………… RDPK3SpFSAL35                                                    │
+│ adaptive: ……………………………………………………… true                                                             │
+│ abstol: …………………………………………………………… 1.0e-6                                                           │
+│ reltol: …………………………………………………………… 0.001                                                            │
+│ controller: ………………………………………………… PIDController(beta=[0.7, -0.23, …iter=default_dt_factor_limiter) │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Environment information                                                                          │
+│ ═══════════════════════                                                                          │
+│ #threads: ……………………………………………………… 1                                                                │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+#timesteps:     50 │ Δt: 1.4965e-03 │ sim. time: 1.8236e-01 (18.236%)  │ run time: 1.2462e-01 s
+#timesteps:    100 │ Δt: 5.3979e-03 │ sim. time: 4.2540e-01 (42.540%)  │ run time: 2.6655e-01 s
+#timesteps:    150 │ Δt: 5.7273e-03 │ sim. time: 6.7662e-01 (67.662%)  │ run time: 3.9337e-01 s
+#timesteps:    200 │ Δt: 4.8634e-03 │ sim. time: 9.1219e-01 (91.219%)  │ run time: 5.2495e-01 s
+────────────────────────────────────────────────────────────────────────────────────────────────────
+Trixi simulation finished.  Final time: 1.0  Time steps: 222 (accepted), 222 (total)
+────────────────────────────────────────────────────────────────────────────────────────────────────
+
+────────────────────────────────────────────────────────────────────────────────
+      TrixiParticles.jl                Time                    Allocations
+                              ───────────────────────   ────────────────────────
+      Tot / % measured:            576ms /  98.0%           95.4MiB /  98.4%
+
+Section               ncalls     time    %tot     avg     alloc    %tot      avg
+────────────────────────────────────────────────────────────────────────────────
+kick!                  1.11k    358ms   63.5%   322μs   3.77MiB    4.0%  3.47KiB
+  system interaction   1.11k    286ms   50.7%   257μs   1.05MiB    1.1%     985B
+    fluid1-fluid1      1.11k    226ms   40.0%   203μs     0.00B    0.0%    0.00B
+    fluid1-boundary2   1.11k   57.0ms   10.1%  51.2μs     0.00B    0.0%    0.00B
+    ~system intera...  1.11k   3.07ms    0.5%  2.76μs   1.05MiB    1.1%     985B
+    boundary2-fluid1   1.11k   32.7μs    0.0%  29.4ns     0.00B    0.0%    0.00B
+    boundary2-boun...  1.11k   32.6μs    0.0%  29.2ns     0.00B    0.0%    0.00B
+  update systems a...  1.11k   71.4ms   12.7%  64.1μs   2.72MiB    2.9%  2.50KiB
+    compute bounda...  1.11k   48.8ms    8.6%  43.8μs     0.00B    0.0%    0.00B
+    update nhs         1.11k   11.4ms    2.0%  10.2μs   2.72MiB    2.9%  2.50KiB
+    inverse state ...  1.11k   6.14ms    1.1%  5.52μs     0.00B    0.0%    0.00B
+    ~update system...  1.11k   5.06ms    0.9%  4.55μs   1.55KiB    0.0%    1.42B
+    update density...  1.11k   32.9μs    0.0%  29.6ns     0.00B    0.0%    0.00B
+  source terms         1.11k    425μs    0.1%   381ns     0.00B    0.0%    0.00B
+  ~kick!~              1.11k    407μs    0.1%   366ns   1.55KiB    0.0%    1.42B
+  reset ∂v/∂t          1.11k    151μs    0.0%   135ns     0.00B    0.0%    0.00B
+save solution             50    205ms   36.3%  4.09ms   90.0MiB   96.0%  1.80MiB
+  write to vtk           100    176ms   31.3%  1.76ms   86.7MiB   92.5%   888KiB
+  ~save solution~         50   25.2ms    4.5%   504μs   3.15MiB    3.4%  64.6KiB
+  update systems          50   3.09ms    0.5%  61.8μs    127KiB    0.1%  2.53KiB
+    compute bounda...     50   2.10ms    0.4%  41.9μs     0.00B    0.0%    0.00B
+    update nhs            50    494μs    0.1%  9.89μs    125KiB    0.1%  2.50KiB
+    inverse state ...     50    269μs    0.0%  5.38μs     0.00B    0.0%    0.00B
+    ~update systems~      50    229μs    0.0%  4.58μs   1.55KiB    0.0%    31.7B
+    update density...     50   1.50μs    0.0%  30.0ns     0.00B    0.0%    0.00B
+drift!                 1.11k    873μs    0.2%   784ns      976B    0.0%    0.88B
+  velocity             1.11k    526μs    0.1%   473ns     0.00B    0.0%    0.00B
+  ~drift!~             1.11k    221μs    0.0%   198ns      976B    0.0%    0.88B
+  reset ∂u/∂t          1.11k    126μs    0.0%   113ns     0.00B    0.0%    0.00B
+compute boundary p...      1   59.3μs    0.0%  59.3μs     0.00B    0.0%    0.00B
+update nhs                 1   54.9μs    0.0%  54.9μs   2.50KiB    0.0%  2.50KiB
+inverse state equa...      1   7.96μs    0.0%  7.96μs     0.00B    0.0%    0.00B
+update density dif...      1   29.0ns    0.0%  29.0ns     0.00B    0.0%    0.00B
+────────────────────────────────────────────────────────────────────────────────
    See Visualization for how to visualize the solution. For the simplest visualization, we can use Plots.jl:
    using Plots
+plot(sol)
    GKS: cannot open display - headless operation mode active
    plot
    Replacing components with custom implementations
    If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.
    Custom smoothing kernel
    To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.
    struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end
    This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.
    In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.
    function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)
     q = r / h
 
     if q < 2
@@ -42,5 +161,140 @@
     return 0.0
 end
 
-TrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h
    This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by
    smoothing_kernel = MyGaussianKernel()
    and run the simulation file again.
    In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.
    julia> trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"),
-                     smoothing_kernel=MyGaussianKernel());
    
+TrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h

    This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by

    smoothing_kernel = MyGaussianKernel()

    and run the simulation file again.

    In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.

    trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"),
+              smoothing_kernel=MyGaussianKernel());
    [ Info: You just called `trixi_include`. Julia may now compile the code, please be patient.
+
+████████╗██████╗ ██╗██╗  ██╗██╗██████╗  █████╗ ██████╗ ████████╗██╗ ██████╗██╗     ███████╗███████╗
+╚══██╔══╝██╔══██╗██║╚██╗██╔╝██║██╔══██╗██╔══██╗██╔══██╗╚══██╔══╝██║██╔════╝██║     ██╔════╝██╔════╝
+   ██║   ██████╔╝██║ ╚███╔╝ ██║██████╔╝███████║██████╔╝   ██║   ██║██║     ██║     █████╗  ███████╗
+   ██║   ██╔══██╗██║ ██╔██╗ ██║██╔═══╝ ██╔══██║██╔══██╗   ██║   ██║██║     ██║     ██╔══╝  ╚════██║
+   ██║   ██║  ██║██║██╔╝ ██╗██║██║     ██║  ██║██║  ██║   ██║   ██║╚██████╗███████╗███████╗███████║
+   ╚═╝   ╚═╝  ╚═╝╚═╝╚═╝  ╚═╝╚═╝╚═╝     ╚═╝  ╚═╝╚═╝  ╚═╝   ╚═╝   ╚═╝ ╚═════╝╚══════╝╚══════╝╚══════╝
+
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Semidiscretization                                                                               │
+│ ══════════════════                                                                               │
+│ #spatial dimensions: ………………………… 2                                                                │
+│ #systems: ……………………………………………………… 2                                                                │
+│ neighborhood search: ………………………… GridNeighborhoodSearch                                           │
+│ total #particles: ………………………………… 636                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ WeaklyCompressibleSPHSystem{2}                                                                   │
+│ ══════════════════════════════                                                                   │
+│ #particles: ………………………………………………… 360                                                              │
+│ density calculator: …………………………… ContinuityDensity                                                │
+│ correction method: ……………………………… Nothing                                                          │
+│ state equation: ……………………………………… StateEquationCole                                                │
+│ smoothing kernel: ………………………………… MyGaussianKernel                                                 │
+│ viscosity: …………………………………………………… ArtificialViscosityMonaghan{Float64}(0.02, 0.0, 0.01)            │
+│ density diffusion: ……………………………… nothing                                                          │
+│ surface tension: …………………………………… nothing                                                          │
+│ acceleration: …………………………………………… [0.0, -9.81]                                                     │
+│ source terms: …………………………………………… Nothing                                                          │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ BoundarySPHSystem{2}                                                                             │
+│ ════════════════════                                                                             │
+│ #particles: ………………………………………………… 276                                                              │
+│ boundary model: ……………………………………… BoundaryModelDummyParticles(AdamiPressureExtrapolation, Nothing) │
+│ movement function: ……………………………… nothing                                                          │
+│ adhesion coefficient: ……………………… 0.0                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ SolutionSavingCallback                                                                           │
+│ ══════════════════════                                                                           │
+│ dt: ……………………………………………………………………… 0.02                                                             │
+│ custom quantities: ……………………………… nothing                                                          │
+│ save initial solution: …………………… yes                                                              │
+│ save final solution: ………………………… yes                                                              │
+│ output directory: ………………………………… /home/runner/work/TrixiParticles…les.jl/docs/build/tutorials/out │
+│ prefix: ……………………………………………………………                                                                  │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Time integration                                                                                 │
+│ ════════════════                                                                                 │
+│ Start time: ………………………………………………… 0.0                                                              │
+│ Final time: ………………………………………………… 1.0                                                              │
+│ time integrator: …………………………………… RDPK3SpFSAL35                                                    │
+│ adaptive: ……………………………………………………… true                                                             │
+│ abstol: …………………………………………………………… 1.0e-6                                                           │
+│ reltol: …………………………………………………………… 0.001                                                            │
+│ controller: ………………………………………………… PIDController(beta=[0.7, -0.23, …iter=default_dt_factor_limiter) │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Environment information                                                                          │
+│ ═══════════════════════                                                                          │
+│ #threads: ……………………………………………………… 1                                                                │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+#timesteps:     50 │ Δt: 2.1194e-06 │ sim. time: 2.6343e-02 (2.634%)   │ run time: 2.9606e+00 s
+#timesteps:    100 │ Δt: 3.6867e-07 │ sim. time: 2.6368e-02 (2.637%)   │ run time: 3.3357e+00 s
+#timesteps:    150 │ Δt: 3.8641e-06 │ sim. time: 2.7157e-02 (2.716%)   │ run time: 3.7054e+00 s
+#timesteps:    200 │ Δt: 2.7500e-05 │ sim. time: 2.9021e-02 (2.902%)   │ run time: 4.0679e+00 s
+#timesteps:    250 │ Δt: 3.0345e-08 │ sim. time: 2.9289e-02 (2.929%)   │ run time: 4.4684e+00 s
+#timesteps:    300 │ Δt: 5.4341e-07 │ sim. time: 2.9308e-02 (2.931%)   │ run time: 4.8313e+00 s
+#timesteps:    350 │ Δt: 1.1375e-08 │ sim. time: 2.9930e-02 (2.993%)   │ run time: 5.2392e+00 s
+#timesteps:    400 │ Δt: 3.7655e-08 │ sim. time: 3.0011e-02 (3.001%)   │ run time: 5.5881e+00 s
+#timesteps:    450 │ Δt: 2.5171e-06 │ sim. time: 3.0049e-02 (3.005%)   │ run time: 5.9311e+00 s
+#timesteps:    500 │ Δt: 4.6396e-05 │ sim. time: 3.1465e-02 (3.146%)   │ run time: 6.3004e+00 s
+#timesteps:    550 │ Δt: 2.8950e-04 │ sim. time: 3.4385e-02 (3.438%)   │ run time: 6.6800e+00 s
+#timesteps:    600 │ Δt: 4.7243e-04 │ sim. time: 4.0000e-02 (4.000%)   │ run time: 7.0084e+00 s
+#timesteps:    650 │ Δt: 9.2229e-04 │ sim. time: 7.3763e-02 (7.376%)   │ run time: 7.2405e+00 s
+#timesteps:    700 │ Δt: 1.5410e-03 │ sim. time: 1.2753e-01 (12.753%)  │ run time: 7.4190e+00 s
+#timesteps:    750 │ Δt: 7.9337e-04 │ sim. time: 2.1356e-01 (21.356%)  │ run time: 7.6317e+00 s
+#timesteps:    800 │ Δt: 1.5629e-03 │ sim. time: 2.9162e-01 (29.162%)  │ run time: 7.8360e+00 s
+#timesteps:    850 │ Δt: 2.4390e-03 │ sim. time: 3.6859e-01 (36.859%)  │ run time: 8.0382e+00 s
+#timesteps:    900 │ Δt: 2.6538e-03 │ sim. time: 4.5664e-01 (45.664%)  │ run time: 8.2348e+00 s
+#timesteps:    950 │ Δt: 7.2062e-03 │ sim. time: 6.5246e-01 (65.246%)  │ run time: 8.3913e+00 s
+#timesteps:   1000 │ Δt: 6.7704e-03 │ sim. time: 9.7138e-01 (97.138%)  │ run time: 8.5305e+00 s
+────────────────────────────────────────────────────────────────────────────────────────────────────
+Trixi simulation finished.  Final time: 1.0  Time steps: 1003 (accepted), 1891 (total)
+────────────────────────────────────────────────────────────────────────────────────────────────────
+
+────────────────────────────────────────────────────────────────────────────────
+      TrixiParticles.jl                Time                    Allocations
+                              ───────────────────────   ────────────────────────
+      Tot / % measured:            8.54s /  73.0%            228MiB /  53.4%
+
+Section               ncalls     time    %tot     avg     alloc    %tot      avg
+────────────────────────────────────────────────────────────────────────────────
+kick!                  9.46k    5.64s   90.4%   596μs   22.6MiB   18.6%  2.45KiB
+  system interaction   9.46k    4.77s   76.5%   504μs   8.88MiB    7.3%     984B
+    fluid1-fluid1      9.46k    3.76s   60.2%   397μs     0.00B    0.0%    0.00B
+    fluid1-boundary2   9.46k    988ms   15.8%   104μs     0.00B    0.0%    0.00B
+    ~system intera...  9.46k   24.2ms    0.4%  2.55μs   8.88MiB    7.3%     984B
+    boundary2-boun...  9.46k    281μs    0.0%  29.7ns     0.00B    0.0%    0.00B
+    boundary2-fluid1   9.46k    277μs    0.0%  29.3ns     0.00B    0.0%    0.00B
+  update systems a...  9.46k    862ms   13.8%  91.2μs   13.7MiB   11.3%  1.49KiB
+    compute bounda...  9.46k    696ms   11.2%  73.6μs     0.00B    0.0%    0.00B
+    update nhs         9.46k   74.6ms    1.2%  7.89μs   13.7MiB   11.3%  1.49KiB
+    inverse state ...  9.46k   50.6ms    0.8%  5.35μs     0.00B    0.0%    0.00B
+    ~update system...  9.46k   41.2ms    0.7%  4.35μs   1.55KiB    0.0%    0.17B
+    update density...  9.46k    276μs    0.0%  29.1ns     0.00B    0.0%    0.00B
+  source terms         9.46k   3.65ms    0.1%   386ns     0.00B    0.0%    0.00B
+  ~kick!~              9.46k   3.11ms    0.0%   329ns   1.55KiB    0.0%    0.17B
+  reset ∂v/∂t          9.46k   1.28ms    0.0%   135ns     0.00B    0.0%    0.00B
+save solution             50    587ms    9.4%  11.7ms   99.2MiB   81.4%  1.98MiB
+  write to vtk           100    335ms    5.4%  3.35ms   86.9MiB   71.4%   890KiB
+  ~save solution~         50    248ms    4.0%  4.95ms   12.1MiB   10.0%   248KiB
+  update systems          50   4.54ms    0.1%  90.9μs    161KiB    0.1%  3.23KiB
+    compute bounda...     50   3.40ms    0.1%  67.9μs     0.00B    0.0%    0.00B
+    update nhs            50    671μs    0.0%  13.4μs    160KiB    0.1%  3.19KiB
+    ~update systems~      50    247μs    0.0%  4.93μs   1.55KiB    0.0%    31.7B
+    inverse state ...     50    228μs    0.0%  4.57μs     0.00B    0.0%    0.00B
+    update density...     50   1.61μs    0.0%  32.1ns     0.00B    0.0%    0.00B
+drift!                 9.46k   7.92ms    0.1%   837ns      976B    0.0%    0.10B
+  velocity             9.46k   5.06ms    0.1%   535ns     0.00B    0.0%    0.00B
+  ~drift!~             9.46k   1.69ms    0.0%   178ns      976B    0.0%    0.10B
+  reset ∂u/∂t          9.46k   1.17ms    0.0%   124ns     0.00B    0.0%    0.00B
+compute boundary p...      1    114μs    0.0%   114μs     0.00B    0.0%    0.00B
+update nhs                 1   39.2μs    0.0%  39.2μs   1.12KiB    0.0%  1.12KiB
+inverse state equa...      1   9.56μs    0.0%  9.56μs     0.00B    0.0%    0.00B
+update density dif...      1   30.0ns    0.0%  30.0ns     0.00B    0.0%    0.00B
+────────────────────────────────────────────────────────────────────────────────
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    GitHub

    Setting up your simulation from scratch

    In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.

    For different setups and physics, have a look at our other example files.

    Resolution

    At the beginning of most simulation files, we define the numerical resolution, so that it can easily be found and changed. First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.

    using TrixiParticles
-using OrdinaryDiffEq

    Now, we define the particle spacing, which is our numerical resolution. We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary.

    fluid_particle_spacing = 0.05
-boundary_layers = 3

    Experiment setup

    We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. First, we define the physical parameters gravitational acceleration, initial fluid size, tank size, fluid density, and simulation time.

    gravity = 9.81
-tspan = (0.0, 1.0)
-initial_fluid_size = (1.0, 0.9)
-tank_size = (1.0, 1.0)
-fluid_density = 1000.0

    In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.

    sound_speed = 10.0
-state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
-                                   exponent=7)

    The speed of sound here is numerical and not physical. We artificially lower the speed of sound, since the physical speed of sound in water would lead to prohibitively small time steps. The speed of sound in Weakly Compressible SPH should be chosen as small as possible for numerical efficiency, but large enough to limit density fluctuations to about 1%.

    TrixiParticles.jl requires the initial particle positions and quantities in form of an InitialCondition. Instead of manually defining particle positions, you can work with our pre-defined setups. Among others, we provide setups for rectangular shapes, circles, and spheres. Initial conditions can also be combined with common set operations. See this page for a list of pre-defined setups and details on set operations on initial conditions.

    Here, we use the RectangularTank setup, which generates a rectangular fluid inside a rectangular tank, and supports a hydrostatic pressure gradient by passing a gravitational acceleration and a state equation (see above).

    tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size,
-                       fluid_density, n_layers=boundary_layers,
-                       acceleration=(0.0, -gravity), state_equation=state_equation)

    Fluid system

    To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.

    smoothing_length = 1.2 * fluid_particle_spacing
-smoothing_kernel = SchoenbergCubicSplineKernel{2}()

    You can find an overview over smoothing kernels and corresponding smoothing lengths here.

    For stability, we need numerical dissipation in form of an artificial viscosity term.

    viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)

    We choose the parameters as small as possible to avoid visible viscosity, but as large as possible to stabilize the simulation.

    The WCSPH method can either compute the particle density by a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating it over time. We choose the latter approach here by using the density calculator ContinuityDensity.

    fluid_density_calculator = ContinuityDensity()
-fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
-                                           state_equation, smoothing_kernel,
-                                           smoothing_length, viscosity=viscosity,
-                                           acceleration=(0.0, -gravity))

    Boundary system

    In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation, which generally produces the best results of the implemented methods. See here for a comprehensive overview over boundary models.

    boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
-                                             state_equation=state_equation,
-                                             AdamiPressureExtrapolation(),
-                                             smoothing_kernel, smoothing_length)
-boundary_system = BoundarySPHSystem(tank.boundary, boundary_model)

    Semidiscretization

    The key component of every simulation is the Semidiscretization, which couples all systems of the simulation. All methods in TrixiParticles.jl are semidiscretizations, which discretize the equations in time to provide an ordinary differential equation that still has to be solved in time. By providing a simulation time span, we can call semidiscretize, which returns an ODEProblem that can be solved with a time integration method.

    semi = Semidiscretization(fluid_system, boundary_system)
-ode = semidiscretize(semi, tspan)

    Time integration

    We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.

    OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.

    info_callback = InfoCallback(interval=50)
-saving_callback = SolutionSavingCallback(dt=0.02)
-
-callbacks = CallbackSet(info_callback, saving_callback)

    Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.

    sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);

    See Visualization for how to visualize the solution.

    Replacing components with custom implementations

    If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.

    Custom smoothing kernel

    To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.

    struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end

    This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.

    In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.

    function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)
-    q = r / h
-
-    if q < 2
-        return 1 / (pi * h^2) * exp(-q^2)
-    end
-
-    return 0.0
-end
-
-function TrixiParticles.kernel_deriv(kernel::MyGaussianKernel, r, h)
-    q = r * h
-
-    if q < 2
-        return 1 / (pi * h^2)  * (-2 * q) * exp(-q^2) / h
-    end
-
-    return 0.0
-end
-
-TrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h

    This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by

    smoothing_kernel = MyGaussianKernel()

    and run the simulation file again.

    In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.

    julia> trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"),
-                     smoothing_kernel=MyGaussianKernel());
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       diff --git a/previews/PR514/tutorials_template/tut_dam_break/index.html b/previews/PR514/tutorials_template/tut_dam_break/index.html new file mode 100644 index 000000000..f161971bf --- /dev/null +++ b/previews/PR514/tutorials_template/tut_dam_break/index.html @@ -0,0 +1,3 @@ + +Example file · TrixiParticles.jl diff --git a/previews/PR514/tutorials_template/tut_falling/index.html b/previews/PR514/tutorials_template/tut_falling/index.html new file mode 100644 index 000000000..866b2f102 --- /dev/null +++ b/previews/PR514/tutorials_template/tut_falling/index.html @@ -0,0 +1,3 @@ + +Example file · TrixiParticles.jl diff --git a/previews/PR514/tutorials_template/tut_setup/index.html b/previews/PR514/tutorials_template/tut_setup/index.html new file mode 100644 index 000000000..5c3882c74 --- /dev/null +++ b/previews/PR514/tutorials_template/tut_setup/index.html @@ -0,0 +1,300 @@ + +Setting up your simulation from scratch · TrixiParticles.jl
      GitHub

      Setting up your simulation from scratch

      In this tutorial, we will guide you through the general structure of simulation files. We will set up a simulation similar to the example simulation examples/fluid/hydrostatic_water_column_2d.jl, which is one of our simplest example simulations. In the second part of this tutorial, we will show how to replace components of TrixiParticles.jl by custom implementations from within a simulation file, without ever cloning the repository.

      For different setups and physics, have a look at our other example files.

      First, we import TrixiParticles.jl and OrdinaryDiffEq.jl, which we will use at the very end for the time integration.

      using TrixiParticles
+using OrdinaryDiffEq

      Resolution

      Now, we define the particle spacing, which is our numerical resolution. We usually call the variable fluid_particle_spacing, so that we can easily change the resolution of an example file by overwriting this variable with trixi_include. In 2D, the number of particles will grow quadratically, in 3D cubically with the spacing.

      We also set the number of boundary layers, which need to be sufficiently large, depending on the smoothing kernel and smoothing length, so that the compact support of the smoothing kernel is fully sampled with particles for a fluid particle close to a boundary. In particular, we require boundary_layers >= compact_support. The value for the compact support for each kernel can be found in the smoothing kernel overview.

      fluid_particle_spacing = 0.05
+boundary_layers = 3

      Experiment setup

      We want to simulate a water column resting under hydrostatic pressure inside a rectangular tank. Experiment Setup First, we define the physical parameters gravitational acceleration, simulation time, initial fluid size, tank size and fluid density.

      gravity = 9.81
+tspan = (0.0, 1.0)
+initial_fluid_size = (1.0, 0.9)
+tank_size = (1.0, 1.0)
+fluid_density = 1000.0

      In order to have the initial particle mass and density correspond to the hydrostatic pressure gradient, we need to define a state equation, which relates the fluid density to pressure. Note that we could also skip this part here and define the state equation later when we define the fluid system, but then the fluid would be initialized with constant density, which would cause it to oscillate under gravity.

      sound_speed = 10.0
+state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
+                                   exponent=7)

      The speed of sound here is numerical and not physical. We artificially lower the speed of sound, since the physical speed of sound in water would lead to prohibitively small time steps. The speed of sound in Weakly Compressible SPH should be chosen as small as possible for numerical efficiency, but large enough to limit density fluctuations to about 1%.

      TrixiParticles.jl requires the initial particle positions and quantities in form of an InitialCondition. Instead of manually defining particle positions, you can work with our pre-defined setups. Among others, we provide setups for rectangular shapes, circles, and spheres. Initial conditions can also be combined with common set operations. See this page for a list of pre-defined setups and details on set operations on initial conditions.

      Here, we use the RectangularTank setup, which generates a rectangular fluid inside a rectangular tank, and supports a hydrostatic pressure gradient by passing a gravitational acceleration and a state equation (see above).

      tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size,
+                       fluid_density, n_layers=boundary_layers,
+                       acceleration=(0.0, -gravity), state_equation=state_equation)

      Fluid system

      To model the water column, we use the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This method requires a smoothing kernel and a corresponding smoothing length, which should be chosen in relation to the particle spacing.

      smoothing_length = 1.2 * fluid_particle_spacing
+smoothing_kernel = SchoenbergCubicSplineKernel{2}()

      You can find an overview over smoothing kernels and corresponding smoothing lengths here.

      For stability, we need numerical dissipation in form of an artificial viscosity term. Other viscosity models offer a physical approach based on the kinematic viscosity of the fluid.

      viscosity = ArtificialViscosityMonaghan(alpha=0.02, beta=0.0)

      We choose the parameters as small as possible to avoid non-physical behavior, but as large as possible to stabilize the simulation.

      The WCSPH method can either compute the particle density directly with a kernel summation over all neighboring particles (see SummationDensity) or by making the particle density a variable in the ODE system and integrating its change over time. We choose the latter approach here by using the density calculator ContinuityDensity, which is more efficient and handles free surfaces without the need for additional correction terms.

      fluid_density_calculator = ContinuityDensity()
+fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
+                                           state_equation, smoothing_kernel,
+                                           smoothing_length, viscosity=viscosity,
+                                           acceleration=(0.0, -gravity))

      Boundary system

      To model the boundary, we use particle-based boundary conditions, in which particles are sampled in the boundary that interact with the fluid particles to avoid penetration. In order to define a boundary system, we first have to choose a boundary model, which defines how the fluid interacts with boundary particles. We will use the BoundaryModelDummyParticles with AdamiPressureExtrapolation. See here for a comprehensive overview over boundary models.

      boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
+                                             state_equation=state_equation,
+                                             AdamiPressureExtrapolation(),
+                                             smoothing_kernel, smoothing_length)
+boundary_system = BoundarySPHSystem(tank.boundary, boundary_model)

      Semidiscretization

      The key component of every simulation is the Semidiscretization, which couples all systems of the simulation. All methods in TrixiParticles.jl are semidiscretizations, which discretize the equations in time to provide an ordinary differential equation that still has to be solved in time. By providing a simulation time span, we can call semidiscretize, which returns an ODEProblem that can be solved with a time integration method.

      semi = Semidiscretization(fluid_system, boundary_system)
+ode = semidiscretize(semi, tspan)

      Time integration

      We use the methods provided by OrdinaryDiffEq.jl, but note that other packages or custom implementations can also be used.

      OrdinaryDiffEq.jl supports callbacks, which are executed during the simulation. For this simulation, we use the InfoCallback, which prints information about the simulation setup at the beginning of the simulation, information about the current simulation time and runtime during the simulation, and a performance summary at the end of the simulation. We also want to save the current solution in regular intervals in terms of simulation time as VTK, so that we can look at the solution in ParaView. The SolutionSavingCallback provides this functionality. To pass the callbacks to OrdinaryDiffEq.jl, we have to bundle them into a CallbackSet.

      info_callback = InfoCallback(interval=50)
+saving_callback = SolutionSavingCallback(dt=0.02)
+
+callbacks = CallbackSet(info_callback, saving_callback)

      Finally, we can start the simulation by solving the ODEProblem. We use the method RDPK3SpFSAL35 of OrdinaryDiffEq.jl, which is a Runge-Kutta method with automatic (error based) time step size control. This method is usually a good choice for prototyping, since we do not have to worry about choosing a stable step size and can just run the simulation. For better performance, it might be beneficial to tweak the tolerances of this method or choose a different method that is more efficient for the respective simulation. You can find both approaches in our example files. Here, we just use the method with the default parameters, and only disable save_everystep to avoid expensive saving of the solution in every time step.

      sol = solve(ode, RDPK3SpFSAL35(), save_everystep=false, callback=callbacks);
      
+████████╗██████╗ ██╗██╗  ██╗██╗██████╗  █████╗ ██████╗ ████████╗██╗ ██████╗██╗     ███████╗███████╗
+╚══██╔══╝██╔══██╗██║╚██╗██╔╝██║██╔══██╗██╔══██╗██╔══██╗╚══██╔══╝██║██╔════╝██║     ██╔════╝██╔════╝
+   ██║   ██████╔╝██║ ╚███╔╝ ██║██████╔╝███████║██████╔╝   ██║   ██║██║     ██║     █████╗  ███████╗
+   ██║   ██╔══██╗██║ ██╔██╗ ██║██╔═══╝ ██╔══██║██╔══██╗   ██║   ██║██║     ██║     ██╔══╝  ╚════██║
+   ██║   ██║  ██║██║██╔╝ ██╗██║██║     ██║  ██║██║  ██║   ██║   ██║╚██████╗███████╗███████╗███████║
+   ╚═╝   ╚═╝  ╚═╝╚═╝╚═╝  ╚═╝╚═╝╚═╝     ╚═╝  ╚═╝╚═╝  ╚═╝   ╚═╝   ╚═╝ ╚═════╝╚══════╝╚══════╝╚══════╝
+
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Semidiscretization                                                                               │
+│ ══════════════════                                                                               │
+│ #spatial dimensions: ………………………… 2                                                                │
+│ #systems: ……………………………………………………… 2                                                                │
+│ neighborhood search: ………………………… GridNeighborhoodSearch                                           │
+│ total #particles: ………………………………… 636                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ WeaklyCompressibleSPHSystem{2}                                                                   │
+│ ══════════════════════════════                                                                   │
+│ #particles: ………………………………………………… 360                                                              │
+│ density calculator: …………………………… ContinuityDensity                                                │
+│ correction method: ……………………………… Nothing                                                          │
+│ state equation: ……………………………………… StateEquationCole                                                │
+│ smoothing kernel: ………………………………… SchoenbergCubicSplineKernel                                      │
+│ viscosity: …………………………………………………… ArtificialViscosityMonaghan{Float64}(0.02, 0.0, 0.01)            │
+│ density diffusion: ……………………………… nothing                                                          │
+│ surface tension: …………………………………… nothing                                                          │
+│ acceleration: …………………………………………… [0.0, -9.81]                                                     │
+│ source terms: …………………………………………… Nothing                                                          │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ BoundarySPHSystem{2}                                                                             │
+│ ════════════════════                                                                             │
+│ #particles: ………………………………………………… 276                                                              │
+│ boundary model: ……………………………………… BoundaryModelDummyParticles(AdamiPressureExtrapolation, Nothing) │
+│ movement function: ……………………………… nothing                                                          │
+│ adhesion coefficient: ……………………… 0.0                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ SolutionSavingCallback                                                                           │
+│ ══════════════════════                                                                           │
+│ dt: ……………………………………………………………………… 0.02                                                             │
+│ custom quantities: ……………………………… nothing                                                          │
+│ save initial solution: …………………… yes                                                              │
+│ save final solution: ………………………… yes                                                              │
+│ output directory: ………………………………… /home/runner/work/TrixiParticles…cs/build/tutorials_template/out │
+│ prefix: ……………………………………………………………                                                                  │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Time integration                                                                                 │
+│ ════════════════                                                                                 │
+│ Start time: ………………………………………………… 0.0                                                              │
+│ Final time: ………………………………………………… 1.0                                                              │
+│ time integrator: …………………………………… RDPK3SpFSAL35                                                    │
+│ adaptive: ……………………………………………………… true                                                             │
+│ abstol: …………………………………………………………… 1.0e-6                                                           │
+│ reltol: …………………………………………………………… 0.001                                                            │
+│ controller: ………………………………………………… PIDController(beta=[0.7, -0.23, …iter=default_dt_factor_limiter) │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Environment information                                                                          │
+│ ═══════════════════════                                                                          │
+│ #threads: ……………………………………………………… 1                                                                │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+#timesteps:     50 │ Δt: 1.4965e-03 │ sim. time: 1.8236e-01 (18.236%)  │ run time: 1.1514e-01 s
+#timesteps:    100 │ Δt: 5.3979e-03 │ sim. time: 4.2540e-01 (42.540%)  │ run time: 2.4311e-01 s
+#timesteps:    150 │ Δt: 5.7273e-03 │ sim. time: 6.7662e-01 (67.662%)  │ run time: 4.1094e-01 s
+#timesteps:    200 │ Δt: 4.8634e-03 │ sim. time: 9.1219e-01 (91.219%)  │ run time: 8.8401e-01 s
+────────────────────────────────────────────────────────────────────────────────────────────────────
+Trixi simulation finished.  Final time: 1.0  Time steps: 222 (accepted), 222 (total)
+────────────────────────────────────────────────────────────────────────────────────────────────────
+
+────────────────────────────────────────────────────────────────────────────────
+      TrixiParticles.jl                Time                    Allocations
+                              ───────────────────────   ────────────────────────
+      Tot / % measured:            935ms /  98.7%           95.4MiB /  98.4%
+
+Section               ncalls     time    %tot     avg     alloc    %tot      avg
+────────────────────────────────────────────────────────────────────────────────
+save solution             50    569ms   61.6%  11.4ms   90.0MiB   96.0%  1.80MiB
+  write to vtk           100    546ms   59.1%  5.46ms   86.8MiB   92.5%   888KiB
+  ~save solution~         50   20.5ms    2.2%   411μs   3.15MiB    3.4%  64.6KiB
+  update systems          50   3.08ms    0.3%  61.6μs    127KiB    0.1%  2.53KiB
+    compute bounda...     50   2.05ms    0.2%  41.0μs     0.00B    0.0%    0.00B
+    update nhs            50    522μs    0.1%  10.4μs    125KiB    0.1%  2.50KiB
+    inverse state ...     50    269μs    0.0%  5.37μs     0.00B    0.0%    0.00B
+    ~update systems~      50    238μs    0.0%  4.76μs   1.55KiB    0.0%    31.7B
+    update density...     50   1.45μs    0.0%  29.1ns     0.00B    0.0%    0.00B
+kick!                  1.11k    353ms   38.3%   317μs   3.77MiB    4.0%  3.47KiB
+  system interaction   1.11k    282ms   30.6%   254μs   1.05MiB    1.1%     985B
+    fluid1-fluid1      1.11k    223ms   24.2%   201μs     0.00B    0.0%    0.00B
+    fluid1-boundary2   1.11k   55.8ms    6.0%  50.2μs     0.00B    0.0%    0.00B
+    ~system intera...  1.11k   3.01ms    0.3%  2.70μs   1.05MiB    1.1%     985B
+    boundary2-fluid1   1.11k   32.7μs    0.0%  29.4ns     0.00B    0.0%    0.00B
+    boundary2-boun...  1.11k   32.5μs    0.0%  29.2ns     0.00B    0.0%    0.00B
+  update systems a...  1.11k   69.7ms    7.6%  62.7μs   2.72MiB    2.9%  2.50KiB
+    compute bounda...  1.11k   48.0ms    5.2%  43.1μs     0.00B    0.0%    0.00B
+    update nhs         1.11k   10.9ms    1.2%  9.78μs   2.72MiB    2.9%  2.50KiB
+    inverse state ...  1.11k   6.02ms    0.7%  5.40μs     0.00B    0.0%    0.00B
+    ~update system...  1.11k   4.86ms    0.5%  4.36μs   1.55KiB    0.0%    1.42B
+    update density...  1.11k   32.3μs    0.0%  29.1ns     0.00B    0.0%    0.00B
+  ~kick!~              1.11k    525μs    0.1%   472ns   1.55KiB    0.0%    1.42B
+  source terms         1.11k    434μs    0.0%   390ns     0.00B    0.0%    0.00B
+  reset ∂v/∂t          1.11k    164μs    0.0%   148ns     0.00B    0.0%    0.00B
+drift!                 1.11k    897μs    0.1%   806ns      976B    0.0%    0.88B
+  velocity             1.11k    534μs    0.1%   480ns     0.00B    0.0%    0.00B
+  ~drift!~             1.11k    216μs    0.0%   194ns      976B    0.0%    0.88B
+  reset ∂u/∂t          1.11k    147μs    0.0%   132ns     0.00B    0.0%    0.00B
+compute boundary p...      1   67.1μs    0.0%  67.1μs     0.00B    0.0%    0.00B
+update nhs                 1   32.8μs    0.0%  32.8μs   2.50KiB    0.0%  2.50KiB
+inverse state equa...      1   8.47μs    0.0%  8.47μs     0.00B    0.0%    0.00B
+update density dif...      1   40.0ns    0.0%  40.0ns     0.00B    0.0%    0.00B
+────────────────────────────────────────────────────────────────────────────────

      See Visualization for how to visualize the solution. For the simplest visualization, we can use Plots.jl:

      using Plots
+plot(sol)

      plot

      Replacing components with custom implementations

      If we would like to use an implementation of a component that is not available in TrixiParticles.jl, we can implement it ourselves within the simulation file, without ever cloning the TrixiParticles.jl repository. A good starting point is to check out the available implementations in TrixiParticles.jl, then copy the relevant functions to the simulation file and modify them as needed.

      Custom smoothing kernel

      To implement a custom smoothing kernel, we define a struct extending TrixiParticles.SmoothingKernel. This abstract struct has a type parameter for the number of dimensions, which we set to 2 in this case.

      struct MyGaussianKernel <: TrixiParticles.SmoothingKernel{2} end

      This kernel is going to be an implementation of the Gaussian kernel with a cutoff for compact support. Note that the same kernel in a more optimized version is already implemented in TrixiParticles.jl as GaussianKernel.

      In order to use our new kernel, we have to define three functions. TrixiParticles.kernel, which is the kernel function itself, TrixiParticles.kernel_deriv, which is the derivative of the kernel function, and TrixiParticles.compact_support, which defines the compact support of the kernel in relation to the smoothing length. The latter is relevant for determining the search radius of the neighborhood search.

      function TrixiParticles.kernel(kernel::MyGaussianKernel, r, h)
+    q = r / h
+
+    if q < 2
+        return 1 / (pi * h^2) * exp(-q^2)
+    end
+
+    return 0.0
+end
+
+function TrixiParticles.kernel_deriv(kernel::MyGaussianKernel, r, h)
+    q = r * h
+
+    if q < 2
+        return 1 / (pi * h^2)  * (-2 * q) * exp(-q^2) / h
+    end
+
+    return 0.0
+end
+
+TrixiParticles.compact_support(::MyGaussianKernel, h) = 3 * h

      This is all we need to use our custom kernel implementation in a simulation. We only need to replace the definition above by

      smoothing_kernel = MyGaussianKernel()

      and run the simulation file again.

      In order to use our kernel in a pre-defined example file, we can use the function trixi_include to replace the definition of the variable smoothing_kernel. The following will run the example simulation examples/fluid/hydrostatic_water_column_2d.jl with our custom kernel.

      trixi_include(joinpath(examples_dir(), "fluid", "hydrostatic_water_column_2d.jl"),
+              smoothing_kernel=MyGaussianKernel());
      [ Info: You just called `trixi_include`. Julia may now compile the code, please be patient.
+
+████████╗██████╗ ██╗██╗  ██╗██╗██████╗  █████╗ ██████╗ ████████╗██╗ ██████╗██╗     ███████╗███████╗
+╚══██╔══╝██╔══██╗██║╚██╗██╔╝██║██╔══██╗██╔══██╗██╔══██╗╚══██╔══╝██║██╔════╝██║     ██╔════╝██╔════╝
+   ██║   ██████╔╝██║ ╚███╔╝ ██║██████╔╝███████║██████╔╝   ██║   ██║██║     ██║     █████╗  ███████╗
+   ██║   ██╔══██╗██║ ██╔██╗ ██║██╔═══╝ ██╔══██║██╔══██╗   ██║   ██║██║     ██║     ██╔══╝  ╚════██║
+   ██║   ██║  ██║██║██╔╝ ██╗██║██║     ██║  ██║██║  ██║   ██║   ██║╚██████╗███████╗███████╗███████║
+   ╚═╝   ╚═╝  ╚═╝╚═╝╚═╝  ╚═╝╚═╝╚═╝     ╚═╝  ╚═╝╚═╝  ╚═╝   ╚═╝   ╚═╝ ╚═════╝╚══════╝╚══════╝╚══════╝
+
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Semidiscretization                                                                               │
+│ ══════════════════                                                                               │
+│ #spatial dimensions: ………………………… 2                                                                │
+│ #systems: ……………………………………………………… 2                                                                │
+│ neighborhood search: ………………………… GridNeighborhoodSearch                                           │
+│ total #particles: ………………………………… 636                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ WeaklyCompressibleSPHSystem{2}                                                                   │
+│ ══════════════════════════════                                                                   │
+│ #particles: ………………………………………………… 360                                                              │
+│ density calculator: …………………………… ContinuityDensity                                                │
+│ correction method: ……………………………… Nothing                                                          │
+│ state equation: ……………………………………… StateEquationCole                                                │
+│ smoothing kernel: ………………………………… MyGaussianKernel                                                 │
+│ viscosity: …………………………………………………… ArtificialViscosityMonaghan{Float64}(0.02, 0.0, 0.01)            │
+│ density diffusion: ……………………………… nothing                                                          │
+│ surface tension: …………………………………… nothing                                                          │
+│ acceleration: …………………………………………… [0.0, -9.81]                                                     │
+│ source terms: …………………………………………… Nothing                                                          │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ BoundarySPHSystem{2}                                                                             │
+│ ════════════════════                                                                             │
+│ #particles: ………………………………………………… 276                                                              │
+│ boundary model: ……………………………………… BoundaryModelDummyParticles(AdamiPressureExtrapolation, Nothing) │
+│ movement function: ……………………………… nothing                                                          │
+│ adhesion coefficient: ……………………… 0.0                                                              │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ SolutionSavingCallback                                                                           │
+│ ══════════════════════                                                                           │
+│ dt: ……………………………………………………………………… 0.02                                                             │
+│ custom quantities: ……………………………… nothing                                                          │
+│ save initial solution: …………………… yes                                                              │
+│ save final solution: ………………………… yes                                                              │
+│ output directory: ………………………………… /home/runner/work/TrixiParticles…cs/build/tutorials_template/out │
+│ prefix: ……………………………………………………………                                                                  │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Time integration                                                                                 │
+│ ════════════════                                                                                 │
+│ Start time: ………………………………………………… 0.0                                                              │
+│ Final time: ………………………………………………… 1.0                                                              │
+│ time integrator: …………………………………… RDPK3SpFSAL35                                                    │
+│ adaptive: ……………………………………………………… true                                                             │
+│ abstol: …………………………………………………………… 1.0e-6                                                           │
+│ reltol: …………………………………………………………… 0.001                                                            │
+│ controller: ………………………………………………… PIDController(beta=[0.7, -0.23, …iter=default_dt_factor_limiter) │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
+│ Environment information                                                                          │
+│ ═══════════════════════                                                                          │
+│ #threads: ……………………………………………………… 1                                                                │
+└──────────────────────────────────────────────────────────────────────────────────────────────────┘
+
+#timesteps:     50 │ Δt: 2.1194e-06 │ sim. time: 2.6343e-02 (2.634%)   │ run time: 2.9688e+00 s
+#timesteps:    100 │ Δt: 3.6867e-07 │ sim. time: 2.6368e-02 (2.637%)   │ run time: 3.3522e+00 s
+#timesteps:    150 │ Δt: 3.8641e-06 │ sim. time: 2.7157e-02 (2.716%)   │ run time: 3.7222e+00 s
+#timesteps:    200 │ Δt: 2.7500e-05 │ sim. time: 2.9021e-02 (2.902%)   │ run time: 4.0836e+00 s
+#timesteps:    250 │ Δt: 3.0345e-08 │ sim. time: 2.9289e-02 (2.929%)   │ run time: 4.4826e+00 s
+#timesteps:    300 │ Δt: 5.4341e-07 │ sim. time: 2.9308e-02 (2.931%)   │ run time: 4.8459e+00 s
+#timesteps:    350 │ Δt: 1.1375e-08 │ sim. time: 2.9930e-02 (2.993%)   │ run time: 5.2551e+00 s
+#timesteps:    400 │ Δt: 3.7655e-08 │ sim. time: 3.0011e-02 (3.001%)   │ run time: 5.6046e+00 s
+#timesteps:    450 │ Δt: 2.5171e-06 │ sim. time: 3.0049e-02 (3.005%)   │ run time: 5.9477e+00 s
+#timesteps:    500 │ Δt: 4.6396e-05 │ sim. time: 3.1465e-02 (3.146%)   │ run time: 6.3092e+00 s
+#timesteps:    550 │ Δt: 2.8950e-04 │ sim. time: 3.4385e-02 (3.438%)   │ run time: 6.6897e+00 s
+#timesteps:    600 │ Δt: 4.7243e-04 │ sim. time: 4.0000e-02 (4.000%)   │ run time: 7.0189e+00 s
+#timesteps:    650 │ Δt: 9.2229e-04 │ sim. time: 7.3763e-02 (7.376%)   │ run time: 7.2372e+00 s
+#timesteps:    700 │ Δt: 1.5410e-03 │ sim. time: 1.2753e-01 (12.753%)  │ run time: 7.4166e+00 s
+#timesteps:    750 │ Δt: 7.9337e-04 │ sim. time: 2.1356e-01 (21.356%)  │ run time: 7.6292e+00 s
+#timesteps:    800 │ Δt: 1.5629e-03 │ sim. time: 2.9162e-01 (29.162%)  │ run time: 7.8350e+00 s
+#timesteps:    850 │ Δt: 2.4390e-03 │ sim. time: 3.6859e-01 (36.859%)  │ run time: 8.0497e+00 s
+#timesteps:    900 │ Δt: 2.6538e-03 │ sim. time: 4.5664e-01 (45.664%)  │ run time: 8.2476e+00 s
+#timesteps:    950 │ Δt: 7.2062e-03 │ sim. time: 6.5246e-01 (65.246%)  │ run time: 8.3980e+00 s
+#timesteps:   1000 │ Δt: 6.7704e-03 │ sim. time: 9.7138e-01 (97.138%)  │ run time: 8.5449e+00 s
+────────────────────────────────────────────────────────────────────────────────────────────────────
+Trixi simulation finished.  Final time: 1.0  Time steps: 1003 (accepted), 1891 (total)
+────────────────────────────────────────────────────────────────────────────────────────────────────
+
+────────────────────────────────────────────────────────────────────────────────
+      TrixiParticles.jl                Time                    Allocations
+                              ───────────────────────   ────────────────────────
+      Tot / % measured:            8.55s /  72.7%            228MiB /  53.4%
+
+Section               ncalls     time    %tot     avg     alloc    %tot      avg
+────────────────────────────────────────────────────────────────────────────────
+kick!                  9.46k    5.65s   90.8%   597μs   22.6MiB   18.6%  2.45KiB
+  system interaction   9.46k    4.78s   76.8%   505μs   8.88MiB    7.3%     984B
+    fluid1-fluid1      9.46k    3.77s   60.6%   398μs     0.00B    0.0%    0.00B
+    fluid1-boundary2   9.46k    982ms   15.8%   104μs     0.00B    0.0%    0.00B
+    ~system intera...  9.46k   24.1ms    0.4%  2.54μs   8.88MiB    7.3%     984B
+    boundary2-fluid1   9.46k    276μs    0.0%  29.1ns     0.00B    0.0%    0.00B
+    boundary2-boun...  9.46k    275μs    0.0%  29.1ns     0.00B    0.0%    0.00B
+  update systems a...  9.46k    863ms   13.9%  91.2μs   13.7MiB   11.3%  1.49KiB
+    compute bounda...  9.46k    697ms   11.2%  73.7μs     0.00B    0.0%    0.00B
+    update nhs         9.46k   74.6ms    1.2%  7.88μs   13.7MiB   11.3%  1.49KiB
+    inverse state ...  9.46k   50.7ms    0.8%  5.36μs     0.00B    0.0%    0.00B
+    ~update system...  9.46k   40.0ms    0.6%  4.23μs   1.55KiB    0.0%    0.17B
+    update density...  9.46k    277μs    0.0%  29.3ns     0.00B    0.0%    0.00B
+  source terms         9.46k   4.05ms    0.1%   428ns     0.00B    0.0%    0.00B
+  ~kick!~              9.46k   3.03ms    0.0%   320ns   1.55KiB    0.0%    0.17B
+  reset ∂v/∂t          9.46k   1.29ms    0.0%   136ns     0.00B    0.0%    0.00B
+save solution             50    568ms    9.1%  11.4ms   99.2MiB   81.4%  1.98MiB
+  write to vtk           100    317ms    5.1%  3.17ms   86.9MiB   71.4%   890KiB
+  ~save solution~         50    246ms    4.0%  4.92ms   12.1MiB   10.0%   249KiB
+  update systems          50   4.50ms    0.1%  90.0μs    161KiB    0.1%  3.23KiB
+    compute bounda...     50   3.39ms    0.1%  67.8μs     0.00B    0.0%    0.00B
+    update nhs            50    636μs    0.0%  12.7μs    160KiB    0.1%  3.19KiB
+    ~update systems~      50    251μs    0.0%  5.01μs   1.55KiB    0.0%    31.7B
+    inverse state ...     50    223μs    0.0%  4.45μs     0.00B    0.0%    0.00B
+    update density...     50   2.82μs    0.0%  56.3ns     0.00B    0.0%    0.00B
+drift!                 9.46k   7.15ms    0.1%   756ns      976B    0.0%    0.10B
+  velocity             9.46k   4.29ms    0.1%   454ns     0.00B    0.0%    0.00B
+  ~drift!~             9.46k   1.67ms    0.0%   177ns      976B    0.0%    0.10B
+  reset ∂u/∂t          9.46k   1.19ms    0.0%   125ns     0.00B    0.0%    0.00B
+compute boundary p...      1   84.4μs    0.0%  84.4μs     0.00B    0.0%    0.00B
+update nhs                 1   30.0μs    0.0%  30.0μs   1.12KiB    0.0%  1.12KiB
+inverse state equa...      1   7.41μs    0.0%  7.41μs     0.00B    0.0%    0.00B
+update density dif...      1   30.0ns    0.0%  30.0ns     0.00B    0.0%    0.00B
+────────────────────────────────────────────────────────────────────────────────
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      GitHub

      Visualization

      Export VTK files

      You can export particle data as VTK files by using the SolutionSavingCallback. All our predefined examples are already using this callback to export VTK files to the out directory (relative to the directory that you are running Julia from). VTK files can be read by visualization tools like ParaView and VisIt.

      ParaView

      Follow these steps to view the exported VTK files in ParaView:

      1. Click File -> Open.
      2. Navigate to the out directory (relative to the directory that you are running Julia from).
      3. Open both boundary_1.pvd and fluid_1.pvd.
      4. Click "Apply", which by default is on the left pane below the "Pipeline Browser".
      5. Hold the left mouse button to move the solution around.

      You will now see the following: image

      To now view the result variables first make sure you have "fluid_1.pvd" highlighted in the "Pipeline Browser" then select them in the variable selection combo box (see picture below). Let's, for example, pick "density". To now view the time progression of the result hit the "play button" (see picture below). image

      API

      TrixiParticles.trixi2vtk — Method
      trixi2vtk(vu_ode, semi, t; iter=nothing, output_directory="out", prefix="",
+Visualization · TrixiParticles.jl
      GitHub
      Visualization
      Export VTK files
      You can export particle data as VTK files by using the SolutionSavingCallback. All our predefined examples are already using this callback to export VTK files to the out directory (relative to the directory that you are running Julia from). VTK files can be read by visualization tools like ParaView and VisIt.
      ParaView
      Follow these steps to view the exported VTK files in ParaView:
      1. Click File -> Open.
      2. Navigate to the out directory (relative to the directory that you are running Julia from).
      3. Open both boundary_1.pvd and fluid_1.pvd.
      4. Click "Apply", which by default is on the left pane below the "Pipeline Browser".
      5. Hold the left mouse button to move the solution around.
      You will now see the following: image
      It is useful to make the particles larger. For this, first make sure you have "fluid_1.pvd" highlighted in the "Pipeline Browser" then in the "Properties" window in the bottom left change "Point Size" to a larger value. image
      To now view the result variables first make sure you have "fluid_1.pvd" highlighted in the "Pipeline Browser" then select them in the variable selection combo box (see picture below). Let's, for example, pick "density". To now view the time progression of the result hit the "play button" (see picture below). image
      API
      TrixiParticles.trixi2vtk — Method
      trixi2vtk(vu_ode, semi, t; iter=nothing, output_directory="out", prefix="",
           write_meta_data=true, max_coordinates=Inf, custom_quantities...)
      Convert Trixi simulation data to VTK format.
      Arguments
      • vu_ode: Solution of the TrixiParticles ODE system at one time step.           This expects an ArrayPartition as returned in the examples as sol.u[end].
      • semi:   Semidiscretization of the TrixiParticles simulation.
      • t:      Current time of the simulation.
      Keywords
      • iter=nothing:           Iteration number when multiple iterations are to be stored in                           separate files. This number is just appended to the filename.
      • output_directory="out": Output directory path.
      • prefix="":              Prefix for output files.
      • write_meta_data=true:   Write meta data.
      • max_coordinates=Inf     The coordinates of particles will be clipped if their absolute                           values exceed this threshold.
      • custom_quantities...:   Additional custom quantities to include in the VTK output.                           Each custom quantity must be a function of (v, u, t, system),                           which will be called for every system, where v and u are the                           wrapped solution arrays for the corresponding system and t is                           the current simulation time. Note that working with these v                           and u arrays requires undocumented internal functions of                           TrixiParticles. See Custom Quantities                           for a list of pre-defined custom quantities that can be used here.
      Example
      trixi2vtk(sol.u[end], semi, 0.0, iter=1, output_directory="output", prefix="solution")
 
 # Additionally store the kinetic energy of each system as "my_custom_quantity"
-trixi2vtk(sol.u[end], semi, 0.0, iter=1, my_custom_quantity=kinetic_energy)
      source
      TrixiParticles.trixi2vtk — Method
      trixi2vtk(coordinates; output_directory="out", prefix="", filename="coordinates")
      Convert coordinate data to VTK format.
      Arguments
      • coordinates: Coordinates to be saved.
      Keywords
      • output_directory="out": Output directory path.
      • prefix="":              Prefix for the output file.
      • filename="coordinates": Name of the output file.
      Returns
      • file::AbstractString: Path to the generated VTK file.
      source
      
+trixi2vtk(sol.u[end], semi, 0.0, iter=1, my_custom_quantity=kinetic_energy)
      source
      TrixiParticles.trixi2vtk — Method
      trixi2vtk(coordinates; output_directory="out", prefix="", filename="coordinates",
+          custom_quantities...)

      Convert coordinate data to VTK format.

      Arguments

      • coordinates: Coordinates to be saved.

      Keywords

      • output_directory="out": Output directory path.
      • prefix="": Prefix for the output file.
      • filename="coordinates": Name of the output file.
      • custom_quantities...: Additional custom quantities to include in the VTK output.

      Returns

      • file::AbstractString: Path to the generated VTK file.
      source
      TrixiParticles.trixi2vtk — Method
      trixi2vtk(initial_condition::InitialCondition; output_directory="out",
+          prefix="", filename="initial_condition", custom_quantities...)

      Convert InitialCondition data to VTK format.

      Arguments

      Keywords

      • output_directory="out": Output directory path.
      • prefix="": Prefix for the output file.
      • filename="coordinates": Name of the output file.
      • custom_quantities...: Additional custom quantities to include in the VTK output.

      Returns

      • file::AbstractString: Path to the generated VTK file.
      source

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