Finite element code for transient one-dimensional heat conduction in Python.

Introduction

This repository presents a one-dimensional transient heat conduction problems using both linear and quadratic elements. The code has been implemented in a Python package and is available for download. The derivation of the equations is documented in this pdf file.

Three validation cases are presented at the end of this document to show that the code is implemented correctly. Further, the validation cases can be used as examples and inspiration.

Minimal example

Consider a domain of length L with a uniform initial temperature. The temperature at one end is raised to a fixed value, while the other end is kept at the initial temperature. The code below shows how to calculate the evolution of temperature distribution in the domain with time. A more elaborate implementation for this case, together with a comparison with the analytical solution, can be found in step_change.py.

import numpy as np
import thermesh as tm

# Domain information
L = 0.01
k, rho, cp = 0.72, 1560, 1450
cpeek = tm.isothermal_model(k, rho, cp)  # constitutive model

# Mesh generation using linear elements
nn = 11  # number of nodes
z = np.linspace(0, L, nn)  # node locations
mesh = tm.Mesh(z, tm.LinearElement)

# Boundary conditions
bc = [{"T": 0.0},  # T on the left
      {"T": 1.0}]  # T on the right

# Domain generation and initialization
domain = tm.Domain(mesh, cpeek, bc)
domain.set_T(np.zeros(nn))

# Solve
solver = {"dt": 0.1, "t_end": 100.0, "theta": 0.5}  # settings
t, T = tm.solve_ht(domain, solver)

Install

You can simply clone the repository to your folder of choice using git:

git clone https://github.com/wjbg/thermesh.git

All functions are reasonably well-documented and the annotated examples should be sufficient to get you started.

License

Free as defined in the MIT license.