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I am solving the steady state heat equation with linear finite element and siplex triangular element. The domain is divided into two subdomains and they have different conductivity, let say conductivity is k=0.2 for the first subdomain and 0.1 for the second one. my problem is that I don't know how to deal with interface,what kind of boundary condition should I impose there? I appreciate your help in advance. Bests,

Evg
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  • I guess you can get the boundary condition from integrating an equation analytically in the vicinity of an interface (a boundary between subdomains). That's how we typically derive such conditions. Internal boundary conditions follow from an equation itself. – Evg Apr 04 '21 at 10:50
  • Thanks for your answer,could please help me with more details, already the only equation I have for this points is Flux equation which should be equal for k1 and k2. But this gives me nothing for my code. –  Apr 04 '21 at 12:55
  • You give no details in your question, no equations, no domains, no boundary conditions. So I can't be more specific even if I wanted to. – Evg Apr 04 '21 at 14:57
  • I want to solve k∆T=q in 2D. I have a rectangular domain divided in two subdomains. for the first subdomain, I consider Dirichlet boundary condition for two edges and Neumann boundary condition for the third one. I did the same for the other subdomain. I construct all the matrices required for the finite element method, let N be the shape function, So I built the matrices of integral NN, NNx, NxN, NNy, NyN and NyNy. I stored all this matrices in CSR format. eventually I'm solving (-NxNx(i)-NyNy(i)) T(i)=RHS(i); Now the only problem is the interface, I don't know how to treat with the interface –  Apr 04 '21 at 16:02
  • And why do you need special treatment of the interface here to begin with? Matrices `N?N?` that you compute have `k` under integrals, you can just plug appropriate values of `k` there. – Evg Apr 04 '21 at 16:24
  • I need special treatment on the interface because there k is changing. I just remember from my heat transfer courses that If there is different zones with different k, k_total can be computed by k1*k2/(k1+k2) . Is that right?you think what is the appropriate value of k on the interface? –  Apr 04 '21 at 19:20
  • The changing `k` is not a problem for FE method, it can be applied to such problems out of the box. You just need to include it into integrals. No special approximations for `k` are needed. – Evg Apr 04 '21 at 19:27
  • I know exactly what you are talking about, but in my problem there are some points on the interface that can take two values for k, how I should solve this issue? –  Apr 04 '21 at 20:15
  • This question can't be answered without looking into details of the problem. I would recommend asking at math stackexchange. SO is mostly about programming, and your quedtion is not about programming at all. – Evg Apr 04 '21 at 21:45

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