Required Weak Form for Steady State Solution of 3D Heat Equation

variational formulation
1

Hi everyone,

I am very new to Fenics and to finite element analysis in general. I am currently trying for find the polar coordinate (r, \phi, z) form of a time independent heat equation (i.e. I wish to find the steady state solution of the system). The problem I am attempting to model is shown below: The top surface (d\Omega_1) is illuminated by a Guassian laser of power P_{max} and width w. A is an arbitrary constant. On all boundaries, I assume the system radiates proportional to T^4, thus creating non-linear boundary conditions.

model
model1525×1895 135 KB

I wish to find the steady state solution of this system when conduction and radiative cooling balance out so T does not chance significantly with time (say less than 0.1K per sec) at every point on the mesh. Is this possible to do given that this problem has not Dirichlet BC’s? If it is possible, what is the weak form of the heat equation I should use?

Thanks in advance.

2

Hi there,

You can find the derivation of heat equation in Cartesian coordinates with radiating boundaries in Section 2.2 (page 123) including the FEniCS code in this book:

https://link.springer.com/book/10.1007%2F978-981-10-2444-3

If your library is not allowing you to get the book, write me an e-mail.

Please have a look at the supply code in this web site for the Errata of the book, FEniCS had some changes since then

http://bilenemek.abali.org

Best, Emek