I am new to fipy, so I am just getting my feet wet. I'm quite experienced with Comsol, which I only mention because it has capabilities to approach the problem I am going to mention. I am interested in modeling thermal (or electrical) conduction in a 3D region, but that region has, on a part of one of its surfaces a thin, highly conducting region. To connect to the real world, one can envisage this being a 70 micron thick copper pad at the top surfaces of a 1mm thick printed circuit board (PCB), where the PCB material is FR4 with a much lower thermal (or electrical) conductivity than the copper. There would be a heat source on the pad. The general approach to handle the conduction in the thin region is to assume a uniform temperature across the thin dimension, integrate across that dimension, and end up with a 2D PDE in the transverse direction. What started out as a Laplace equation then becomes more like a Helmholtz. The non-derivative term is simply the sum of the fluxes out the top and bottom surfaces. The bottom flux should be equal to the flux into the 3D region below. Thus, in an analytical sense, it's quite straightforward to think about. The question is, how to handle in Fipy? I can think of a number of different ways of approaching this, but not sure which might be the best, not being that familiar with the nuances of Fipy. Ideally, one could make the 2d region's equation a boundary condition for the 3d (in the correct cell faces of the 3d region of course), but I don't know if one could specify a constraint that has a 2D transvers Laplacian of the dependent variable. Alternatively, I thought that maybe I could have two coupled equations, one for the 2d region and one for the 3d region, but then one would have a 3d mesh and the other a 2d mesh. Is that possible? Finally, perhaps one could make the thin region be just one cell thick in the thin dimension, and simply treat it as a 3d region. Doesn't seem like it would add a great deal of numerical burden with the one extra layer. Anyway, I am curious about how the "gurus" would approach this. As I mentioned, I do this in Comsol often, but it's not quite clear what they are doing "under the hood" in this instance. That is one reason why I like Fipy and am starting to get into it.
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