4 votes
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Flux sign and face normal confusion in finite volume method

When dealing with conservation laws like your case, you can often make use of the divergence theorem (as you did). You can then express the fact that the total mass within your integration region is ...
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4 votes
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Solving geodesics on triangular meshes gives negative distances

The solution to the final Poisson equation is defined only up to an additive constant. So you just need to shift the solution vector so the smallest value is zero.
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  • 156
3 votes

Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?

Expanding (sort of) on @MPIchael's answer, you can pick any smooth function you like and plug it into the heat equation to give a problem to then work the other way. In numerical methods, we call this ...
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3 votes
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Lumped matrices in thermal analysis using finite elements

No, you can't lump the $K$ matrix: that would not be a consistent approximation to the second-order differential operator it is supposed to represent. But if you're trying to be a bit more formal, ...
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2 votes

(FEM) 1D time-dependent heat equation convergence problem

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 ...
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  • 172
2 votes

Finite Difference Grid Spacing and Scaling

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 ...
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  • 1,127
2 votes

Simulating the heat equation with insulating material

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 ...
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  • 1,902
2 votes

Flux sign and face normal confusion in finite volume method

Normal direction depends on the cell that you are writing equation for. the word outward is relative to the cell under study. In order to write equation for each of cells, i.e. $\Sigma \nabla T.n S_f=...
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  • 235
2 votes
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Solution method of nonlinear heat transfer analysis

The Newton-Raphson method can be used to solve non-linear systems of equations. The first step is to write your system as a root finding problem: $$ f(T_n) = \left( \frac{C}{\Delta t} + K \right) ...
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2 votes

Flux sign and face normal confusion in finite volume method

An extended answer. For more arbitrary meshes you have to consider that generally CFD/FEM solvers rely on generic data-structures with element and side lists: Element list Side list Consider the ...
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2 votes
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Physical interpretation of L2 norm of heat equation solution

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 ...
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2 votes

Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?

TL DR: $$u_1(x_1) = \cos(2\pi~(\frac{x_1}{L_1}) - \pi) + 1$$ $$u_2(x_2) = \cos(2\pi~(\frac{x_2}{L_2}) - \pi) + 1$$ $$u(x,t) = \exp(-a t) u_1(x_1) u_2(x_1)$$ How to construct it: Sines and cosines are ...
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2 votes
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Incorporating heat flux into Laplace Equation

Because a heat flux has a direction, and from what you are describing, you are adding a heat flux in the normal direction -- out of the domain. So if you started with a constant temperature plane and ...
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1 vote

How to solve heat equation in spherical coordinates with finite differences?

Deriving the CFL condition was handled by @nicoguaro in the comments. The coordinate singularity problem at $r = 0$ was addressed in this paper: "Numerical Treatment of Polar Coordinate ...
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1 vote
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Heat equation in non-dimensional form behaving differently than in usual format

The applied division is fine, what went wrong here, is the application of Stoke's theorem. If you multiply with the test function you get following term: $$\int \frac{1}{c_p}\nabla\left(-k\nabla u\...
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  • 1,285
1 vote

How to use the Thomas-Algorithm to the Heat-diffusion-equation correctly

For a linear PDE, like the Laplace equation, when you discretize it you should get a linear system. Since you're 1D, the Thomas algorithm should be able to solve the system, and it's executed by ...
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1 vote
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Simulating the heat equation with insulating material

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 ...
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1 vote

Produce vertex displacements from volumetric shrinkage data on unstructured meshes

Based on our discussion in the comments, it seems vtkCellDataToPointData is what you want to convert volume shrinkage, which is stored as cell data, to nodal values ...
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1 vote
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(FEM) 1D time-dependent heat equation convergence problem

You don't need MFnx answer to find out why if $k = 0$, $h$ also MUST be zero ($h = 0$). Look at your initial PDE for heat transfer: $$\rho C_{p}\frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{...
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1 vote
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Heat equation with Neumann and Dirichlet conditions on same boundary

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 ...
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1 vote
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Heat diffusion - Is this the correct approach to include Newmann boundary conditions?

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 ...
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1 vote
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Python Finite Difference Schemes for 1D Heat Equation: How to express for loop using numpy expression

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. ...
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1 vote

Modeling Diodes in Autodesk CFD

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 ...
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1 vote

Solve 3-D Heat equation with Neumann boundaries

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 ...
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  • 255
1 vote

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

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: $$...
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  • 1,606
1 vote

Computing geodesic distances with diffusion

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 ...
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1 vote

Solving the diffusion/heat equation for a randomly distributed set of points in 3D

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 ...
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1 vote

V-cycle Multigrid for 2D transient heat transfer on a square plate using finite difference

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 ...
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