Hello everyone, I am solving the diffusion problem driven by stress and concentration gradient, the equation is as follows:

The governing equation is shown in the figure above, where c is the concentration, p is the stress, D(T) is the diffusion coefficient, T is the temperature, and everything else is constant. **Where the p stress term is a constant**.

I’m sorry I don’t have a minimum work example, because I don’t know where to start, but what I want to ask is: what examples should I look at for implementation?Because I didn’t find the information.

Consider the FEniCSx tutorial. Assuming you’re familiar with PDEs and their discretisation by the finite element method, you could take a look at: Solving the Poisson equation — FEniCSx tutorial

followed by The heat equation — FEniCSx tutorial.

Thank you for your answer. I am very familiar with the simulation of heat transfer, but perhaps my basic knowledge is not reliable. The current problem is, how should I obtain the variational form of the governing equation when the stress p is constant?

Does stress p have to be expanded by distribution integral like concentration c? Thank you very much!

If p is constant in space then that term vanishes. Otherwise, assuming p is known a priori, that term would comprise a component of the ‘right-hand-side’. It would just be part of your complete forcing function which I believe you’ve encapsulated in the variable S_\text{ext}.

Thank you very much!