Project file on GitHub
## Equations

## Problem specification and analytical solution

## Input files

## Results and evaluation

We consider homogeneous parabolic equation here: \[ \begin{equation} \rho C_\textrm{p}\frac{\partial T}{\partial t} + \nabla\cdot(\lambda \nabla T) = 0 \quad \text{in }\Omega \end{equation}\] w.r.t boundary conditions \[ \eqalign{ T(x, t=0) = T_0,\cr T(x) = g_D(x) &\quad \text{on }\Gamma_D,\cr k{\partial T(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N, } \] where \(T\) could be temperature, the subscripts \(D\) and \(N\) denote the Dirichlet- and Neumann-type boundary conditions, \(n\) is the normal vector pointing outside of \(\Omega\), and \(\Gamma = \Gamma_D \cup \Gamma_N\) and \(\Gamma_D \cap \Gamma_N = \emptyset\).

We solve the Parabolic equation on a line domain \([60\times 1]\) with \(k = 3.2\) and \(\rho C_\textrm{p} = 2.5e10^6\) w.r.t. the specific boundary conditions:

\[ \eqalign{ k {\partial T(x) \over \partial n} = 2 &\quad \text{on } (x=0) \subset \Gamma_N.\cr k {\partial T(x) \over \partial n} = 0 &\quad \text{on } (x=60) \subset \Gamma_N. } \]

The solution of this problem is

TODO: is not rendered correct

\[ \begin{equation} T(x,t) = \frac{2q}{\lambda}\left((\frac{\alpha t}{\pi})^{\frac{1}{2}} e^{-x^2/4\alpha t} - \frac{x}{2}\textrm{erfc}( \frac{x}{2\sqrt{\alpha t}})\right), \end{equation} \]

where \(T_\textrm{b}\) is the boundary temperature, \(\textrm{erfc}\) is the complementary error function and \(\alpha = \lambda/(C_p \rho)\) is the thermal diffusivity.

The main project file is `line_60_heat.prj`

. It describes the processes to be solved and the related process variables together with their initial and boundary conditions. It also references the mesh and geometrical objects defined on the mesh.

As of now a small portion of possible inputs is implemented; one can change: - the mesh file - the geometry file - introduce more/different Dirichlet boundary conditions (different geometry or other constant values) - introduce more/different Neumann boundary conditions (different geometry or other constant values)

The geometries used to specify the boundary conditions are given in the `line_60_heat.gml`

file.

The input mesh `line_60_heat.vtu`

is stored in the VTK file format and can be directly visualized in Paraview for example.

The result, written in the `.vtu`

file, can be visualized with Paraview, for example.

Loading the `line_60_heat_pcs_0_ts_65_t_5078125.000000.vtu`

file in Paraview and Plotting over line. Compared to the analytical solution ‘temperature_analytical.vtu’, the results are very good:

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This article was written by Tianyuan Zheng. If you are missing something or you find an error please let us know.
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