2D Axisymmetric Heat Transfer Problem with uniform Neumann BCs

Files

• Comprehensive test file: main.cpp
• Reference results for comparison: convergence_output_ref.csv

Statement of the Problem

This test corresponds to a 2D axisymmetric simulation of heat transfer in a solid. The domain $\Omega$ is an axisymmetric domain $[0,1]\times[0,1]$. The axis of symmetry is located at $r=0$.

$$\begin{align} \rho C_p\frac{\partial T}{\partial t}&=\nabla \cdot k \nabla T + S(r,z,t)\text{ in }\Omega \end{align}$$

In equation (1), $S(r,z,t)$ is the source term allowing the exact solution:

$$\begin{align} T(r,z,t) &= e^{-t} cos (\pi r^2) cos(\pi z) - z \end{align}$$

A convergence analysis is carried out to ensure the consistency of the results.

Initial condition

The initial condition is given by:

$$\begin{align} T(r,z) &= cos (\pi r^2) \cos(\pi z) - z \end{align}$$

Boundary Conditions

• Uniform Neumann boundary conditions are prescribed on the upper and lower surfaces.

$$\begin{align} {\bf{n}} \cdot{} k \nabla T &= 1 \text{ on }\Gamma_{bottom} \\[6pt] {\bf{n}} \cdot{} k \nabla T &= -1 \text{ on }\Gamma_{top} \end{align}$$

• Homogeneous Neumann boundary conditions are prescribed on the left and right boundaries.

$$\begin{align} {\bf{n}} \cdot{} k \nabla T &= 0 \text{ on } \Gamma_{left} \text{ and } \Gamma_{right} \end{align}$$

Parameters Used for the Test

For this test, all physical parameters are equal to one.

Numerical Scheme

• Time integration: Euler Implicit over the interval $t\in[0,0.01]$ with a time-step $\delta t=10^{-3}$.
• Spatial discretization for convergence analysis: uniform grid with $N={30, 60, 90, 120}$ nodes in each spatial direction, with $\mathcal{Q}_1$

Results

Figures 1 shows the results of convergence analysis with $\mathcal{Q}_1$.

Figure 1: convergence analysis with $\mathcal{Q}_1$ finite elements