Radiation boundary condition (heat transfer)

Question

I am looking for reference on how to implement nonlinear boundary conditions. Specifically, I am interested in implementing a radiation boundary condition for heat transfer with the FEM:

$-k \frac {\partial T} {\partial n} =F (T^{4} - T_{\infty}^{4})$

with $T$ temparature, $F$ is some factor, $k$ the thermal conductivity.

An ideal reference has the theory and a simple (1D/2D) example and possibly some code to try/verify. I have looked at Finite Element Analysis for Heat Transfer (Hou-Cheng Huang) but for the price the review was not favorable. Maybe some knows of some lecture nodes, a deal ii or fenics implementation I could look at?

Answer 1

This is not particularly difficult once you realize that the nonlinear boundary condition simply yields a nonlinear term in the weak formulation. Let's assume that you want to solve the steady state problem, i.e., $$ -k \Delta T = f \qquad \text{in} \ \Omega\\ -k \frac{\partial T}{\partial n} = F(T^4 - T_\infty^4) \qquad \text{on} \ \partial\Omega. $$ You convert this into the weak form by multiplying the equation by a test function and integrating by parts, which yields $$ (\nabla \varphi,k\nabla T)_\Omega -(\varphi,k\partial_n T)_{\partial\Omega} = (\varphi,f)_\Omega $$ Now realize that you can substitute the boundary condition in the second term on the left to obtain $$ (\nabla \varphi,k\nabla T)_\Omega +(\varphi,F T^4)_{\partial\Omega} = (\varphi,f)_\Omega + (\varphi,F T_\infty^4)_{\partial\Omega} $$ You then only have to implement a nonlinear solver for this problem -- but that works the same way as for all other nonlinear problems and you can find many references for this. (Among them are my video lectures on nonlinear problems.)