Add BC and Overview fixes, partial fixes to Poiseuille

docs/src/bc_debug.md

@@ -27,7 +27,8 @@ end
 
 Advection = @decapode begin
   C::Form0
-  (V, ϕ)::Form1
+  ϕ::Form1
+  V::Constant
 
   ϕ == ∧₀₁(C,V)
 end
@@ -64,8 +65,7 @@ compose_diff_adv = @relation (C, V) begin
   superposition(ϕ₁, ϕ₂, ϕ, C_up, C)
 end
 
-to_graphviz(compose_diff_adv, box_labels=:name, junction_labels=:variable,
-            graph_attrs=Dict(:start => "2"))
+to_graphviz(compose_diff_adv, box_labels=:name, junction_labels=:variable, prog="circo")
 ```
 
 ```@example Debug
@@ -99,6 +99,7 @@ boundary conditions and so we will use the `plot_mesh` from the previous
 example instead of the mesh with periodic boundary conditions. Because the mesh
 is only a primal mesh, we also generate and subdivide the dual mesh.
 
+`Rectangle_30x10` is a default mesh that is downloaded via `Artifacts.jl` when a user installs Decapodes. Via CombinatorialSpaces.jl, we can instantiate any `.obj` file of triangulated faces as a simplicial set.
 
 ```@example Debug
 using CombinatorialSpaces, CombinatorialSpaces.DiscreteExteriorCalculus
@@ -117,26 +118,30 @@ fig
 ```
 
 Finally, we define our operators, generate the simulation function, and compute
-the simulation. Note that when we define the boudary condition operator, we
+the simulation. Note that when we define the boundary condition operator, we
 hardcode the boundary indices and values into the operator itself. We also move
 the initial concentration to the left, so that we are able to see a constant
 concentration on the left boundary which will act as a source in the flow. The
 modified initial condition is shown below:
 
 ```@example Debug
 using LinearAlgebra
+using MultiScaleArrays
 using MLStyle
 using CombinatorialSpaces.DiscreteExteriorCalculus: ∧
+include("../../examples/boundary_helpers.jl")
 
 function generate(sd, my_symbol; hodge=GeometricHodge())
   op = @match my_symbol begin
     :k => x -> 0.05*x
     :∂C => x -> begin
+      boundary = boundary_inds(Val{0}, sd)
       x[boundary] .= 0
       x
     end
     :∧₀₁ => (x,y) -> begin
-      ∧((0,1), x,y)
+      ∧(Tuple{0,1}, sd, x,y)
+    end
     x => error("Unmatched operator $my_symbol")
   end
   return (args...) -> op(args...)
@@ -162,12 +167,15 @@ fₘ = sim(plot_mesh_dual, generate)
 velocity(p) = [-0.5, 0.0, 0.0]
 v = ♭(plot_mesh_dual, DualVectorField(velocity.(plot_mesh_dual[triangle_center(plot_mesh_dual),:dual_point]))).data
 
-prob = ODEProblem(fₘ, c, (0.0, 100.0))
-sol = solve(prob, Tsit5(), p=v);
+u₀ = construct(PhysicsState, [VectorForm(c)], Float64[], [:C])
+params = (V = v,)
+
+prob = ODEProblem(fₘ, u₀, (0.0, 100.0), params)
+sol = solve(prob, Tsit5());
 
 # Plot the result
 times = range(0.0, 100.0, length=150)
-colors = [sol(t)[1:nv(plot_mesh)] for t in times]
+colors = [findnode(sol(t), :C) for t in times]
 
 # Initial frame
 fig, ax, ob = mesh(plot_mesh, color=colors[1], colorrange = extrema(vcat(colors...)))
@@ -177,7 +185,7 @@ framerate = 30
 
 # Animation
 record(fig, "diff_adv_right.gif", range(0.0, 100.0; length=150); framerate = 30) do t
-ob.color = sol(t)[1:nv(plot_mesh)]
+  ob.color = findnode(sol(t), :C)
 end
 ```
 

docs/src/overview.md

@@ -22,7 +22,7 @@ surfaces of the mesh. Below, we provide a very simple Decapode with just a
 single variable `C`.
 and define a convenience function for visualization in later examples.
 
-```
+```@example DEC
 using Decapodes
 using Catlab, Catlab.Graphics
 
@@ -36,7 +36,7 @@ to_graphviz(Variable)
 The resulting diagram contains a single node, showing the single variable in
 this system. We can then add a second variable:
 
-```
+```@example DEC
 TwoVariables = @decapode begin
   C::Form0
   dC::Form1
@@ -69,7 +69,7 @@ some actual PDE systems! One classic PDE example is the diffusion equation.
 This equation states that the change of concentration at each point is
 proportional to the Laplacian of the concentration.
 
-```
+```@example DEC
 Diffusion = @decapode begin
   (C, Ċ)::Form0
   ϕ::Form1
@@ -81,7 +81,7 @@ Diffusion = @decapode begin
   ∂ₜ(C) == Ċ
 end;
 
-draw_equation(Diffusion)
+to_graphviz(Diffusion)
 ```
 
 The resulting Decapode shows the relationships between the three variables with
@@ -101,7 +101,8 @@ visualization, as well as a mapping between the points on the periodic and
 non-periodic meshes.
 
 See CombinatorialSpaces.jl for mesh construction and importing utilities.
-```
+
+```@example DEC
 using Catlab.CategoricalAlgebra
 using CombinatorialSpaces, CombinatorialSpaces.DiscreteExteriorCalculus
 using CairoMakie
@@ -128,6 +129,8 @@ function.
 Note that we chose to define `k` as a function that multiplies by a value `k`. We could have alternately chosen to represent `k` as a Constant that we multiply by in the Decapode itself.
 
 ```@example DEC
+using MLStyle
+
 function generate(sd, my_symbol; hodge=GeometricHodge())
   op = @match my_symbol begin
     :k => x -> 0.05*x
@@ -156,14 +159,17 @@ fig
 Finally, we solve this PDE problem using the `Tsit5()` solver and generate an animation of the result!
 
 ```@example DEC
+using MultiScaleArrays
 using OrdinaryDiffEq
 
-prob = ODEProblem(fₘ, c, (0.0, 100.0))
+u₀ = construct(PhysicsState, [VectorForm(c)], Float64[], [:C])
+
+prob = ODEProblem(fₘ, u₀, (0.0, 100.0))
 sol = solve(prob, Tsit5());
 
 # Plot the result
 times = range(0.0, 100.0, length=150)
-colors = [sol(t)[point_map] for t in times]
+colors = [findnode(sol(t), :C)[point_map] for t in times]
 
 # Initial frame
 fig, ax, ob = mesh(plot_mesh, color=colors[1], colorrange = extrema(vcat(colors...)))
@@ -173,7 +179,7 @@ framerate = 30
 
 # Animation
 record(fig, "diffusion.gif", range(0.0, 100.0; length=150); framerate = 30) do t
-ob.color = sol(t)[point_map]
+ob.color = findnode(sol(t), :C)[point_map]
 end
 ```
 
@@ -203,7 +209,8 @@ end
 
 Advection = @decapode begin
   C::Form0
-  (V, ϕ)::Form1
+  ϕ::Form1
+  V::Form1
   ϕ == ∧₀₁(C,V)
 end
 
@@ -215,13 +222,17 @@ Superposition = @decapode begin
   Ċ == ⋆₀⁻¹(dual_d₁(⋆₁(ϕ)))
   ∂ₜ(C) == Ċ
 end
+true # hide
 ```
+
 ```@example DEC
 to_graphviz(Diffusion)
 ```
+
 ```@example DEC
 to_graphviz(Advection)
 ```
+
 ```@example DEC
 to_graphviz(Superposition)
 ```
@@ -238,8 +249,7 @@ compose_diff_adv = @relation (C, V) begin
   superposition(ϕ₁, ϕ₂, ϕ, C)
 end
 
-to_graphviz(compose_diff_adv, box_labels=:name, junction_labels=:variable,
-            graph_attrs=Dict(:start => "2"))
+to_graphviz(compose_diff_adv, box_labels=:name, junction_labels=:variable, prog="circo")
 ```
 
 After this, the physics can be composed as follows:
@@ -259,6 +269,8 @@ now requires another value to be defined, namely the velocity vector field. We d
 this using a custom operator called `flat_op`. This operator is basically the flat
 operator from CombinatorialSpaces.jl, but specialized to account for the periodic mesh.
 
+We could instead represent the domain as a the surface of a an object with equivalent boundaries in 3D.
+
 ``` @setup DEC
 function closest_point(p1, p2, dims)
     p_res = collect(p2)
@@ -292,9 +304,15 @@ end
 ```
 
 ```@example DEC
+using LinearAlgebra
+using MLStyle
+
 function generate(sd, my_symbol; hodge=GeometricHodge())
   op = @match my_symbol begin
     :k => x -> 0.05*x
+    :∧₀₁ => (x,y) -> begin
+      ∧(Tuple{0,1}, sd, x,y)
+    end
     x => error("Unmatched operator $my_symbol")
   end
   return (args...) -> op(args...)
@@ -306,12 +324,15 @@ fₘ = sim(periodic_mesh, generate)
 velocity(p) = [-0.5, -0.5, 0.0]
 v = flat_op(periodic_mesh, DualVectorField(velocity.(periodic_mesh[triangle_center(periodic_mesh),:dual_point])); dims=[30, 10, Inf])
 
-prob = ODEProblem(fₘ, c, (0.0, 100.0))
-sol = solve(prob, Tsit5(), p=v);
+u₀ = construct(PhysicsState, [VectorForm(c)], Float64[], [:C])
+params = (V = v)
+
+prob = ODEProblem(fₘ, c, (0.0, 100.0), params)
+sol = solve(prob, Tsit5());
 
 # Plot the result
 times = range(0.0, 100.0, length=150)
-colors = [sol(t)[point_map] for t in times]
+colors = [findnode(sol(t), :C)[point_map] for t in times]
 
 # Initial frame
 fig, ax, ob = mesh(plot_mesh, color=colors[1], colorrange = extrema(vcat(colors...)))
@@ -321,7 +342,7 @@ framerate = 30
 
 # Animation
 record(fig, "diff_adv.gif", range(0.0, 100.0; length=150); framerate = 30) do t
-ob.color = sol(t)[point_map]
+ob.color = findnode(sol(t), :C)[point_map]
 end
 ```