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Why even bother to simulate the left half? Why not just simulate the right half with the left BC being the constant value? Seems simpler and more accurate to what you want to actually model, because currently you're just going to have a very low slope on the left half due to the low conductivity, but it won't be constant.


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From the comments, I believe you are struggling with the FEM formulations and therefore fail to see what happens if the diffusion coefficient tends to zero. I don't have access to your book, but I will post here an answer which shows how to discretize the heat equation with the standard Galerkin method. You will then see that, if there is no diffusion, there ...


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In these slide there are some comments about the energy. At pag 4 it focus on the fact that this energy is not a physical energy, but it is a mathematical tool. At pag 8 it observes that: From a physical point of view it seems reasonable that a the energy will decrease in a system without any heat source And after this the author defines another ...


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I was expecting to see my solution in the right half and then having the left half mostly constant with a value of 0 and rapidly changing close to the interface for continuity. This would make sense given that the left half is mostly an insulating material. This assumption is wrong. Small conductivity simply means that you have small heat flux, but by no ...


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The choice of the boundary conditions is dependent on the physics you want to model. If you're doing heat diffusion through a lake, and you want constant temp on top you'll have a different boundary condition than for constant heat flux out of the top (which appears to be what you want). I'm assuming that you want the bottom of the lake to have no heat flux (...


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numpy arrays and methods are incredibly helpful. They are usually optimized and much faster than looping in python. Always look for a way to use an existing numpy method for your application. np.roll() will allow you to shift and then you just add. I learned to use convolve() from comments on How to np.roll() faster?. I haven't checked if this is faster or ...


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The natural convection of the oil shouldn't be an issue and can be solved without any 'assumptions' to be made. The challenge comes with the diodes temperature control. CFD doesn't currently have a temperature controlled heat flux, so some assumptions will have to be made. The simplest assumption would be to just assign a temperature of 150C and assume it ...


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Thanks for asking your question. It really made me delve alot deeper into the topic of boundary conditions. I am still a rookie in this topic, but I'll try to answer your question anyways. Your major problem seems to be that your units are not correct. Your equation for radiative heat flux has the unit $[\frac{\text{W}}{\text{m}^2}]$, while the Neumann ...


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Since Laplacian is an elliptic operator you are looking for the Cholesky decomposition of the assumed constant diffusion matrix $D$: $$D = LL^T$$ Therefore the parabolic equation may be written as: $$\partial_t u=-\nabla^T(D\nabla u)=-(L^T\nabla)^T(L^T\nabla u)=-\tilde{\nabla}^T(\tilde{\nabla} u)=-\tilde{\nabla}^2u$$ Naming $\tilde{\nabla}=L^T\nabla$ and ...


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Combinatorial Laplacian ? It depends on what you expect from your solution. Something reasonable to expect is that your solution should be independent of the way you mesh your surface (or as independent as possible). If you want that, then clearly a combinatorial graph Laplacian is not what you need, since the result would be completely different if you ...


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If you have a cloud of points and you don't want to use mesh-based methods like FEM or FVM, a possibility is to use a mesh-free method like the Finite Point Method. For instance, you could have a look at this article: Tatari, M., Kamranian, M., & Dehghan, M. (2011). The finite point method for the p-Laplace equation. Computational Mechanics, 48(6), 689-...


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Regarding your issue of multigrid taking longer than Gauss-Seidel: It could be that you didn't code the interpolation/restriction operations very efficiently. Do you make sure to take full advantage of the sparsity of the matrices? We write the coarse grid operator as something like RAI, but you shouldn't multiply out the matrices directly. All three are ...


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Let's assume the following heat transient heat transfer equation in 1D : $$ \frac{\partial T} {\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} $$ If we take its finite difference approximation using an explicit time scheme we obtain : $$ \frac{T_i^{t+\Delta t} - T_i^{t}}{\Delta t} = \frac{\alpha}{(\Delta x)^2} (T_{i-1}^{t}-2T_i^{t}+T_{i+1}^{t}) $...


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In the comments Christian directed me towards lateral Cauchy problems and the fact that this is a textbook example of an ill-posed problem. Following this lead, I found that this is more specifically know as the sideways heat equation. As an alternative to the suggested quasireversibility method (again Christian), there is a proposed sequential solution in ...


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