Questions tagged [finite-element-method]
Tag for the usage of "FiniteElement" Method embedded in NDSolve and implementation of finite element method (fem) in mathematica.
1,356
questions
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How to unite the region of ParametricPlot and set up the mesh for FEM?
I have used the ParametricPlot to plot the sector of the disk and use Show to combine them in one picture, however it cannot be ...
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votes
1
answer
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How to extract gauss coordinates in AceGen/ AceFEM?
I am trying to implement a 2D turbulence problem with the model of Spalart-Allmaras. Therefore, I need to extract the global gauss coordinates to define parameter 'd' which stands for the distance to ...
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How to transform the default coordinate when set up the mesh with FEM method?
I want to establish the electromagnetic field of the motor based on the finite element method, because the governing equations are different in different regions, so we need to first build the shape ...
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votes
1
answer
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Inaccuracy for FEM for 3D Heat Equation
I'm simulating the heat transfer within a cylindrical rod, with external heating from its sides.
...
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Mesh `Cylinder[{{0,0,0},{0,0,1}},1]` with inner boundary `x=y=0`
In a meshed cylinder Cylinder[{{0,0,0},{0,0,1}},1] I need to specify DirichletConditions along ...
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NDSolve stops solving past a certain time
This is a continuation of the post I've made Unable to solve Delay PDEs Error in Boussinesq Approximation. I apologise if I shouldn't have posted a seperate question for this but I think that the ...
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Steady state fluid flow for downward flow past an obstacle
I was attempting to simulate fluid flow past a circular obstacle. The following is the code which I used
...
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Unable to solve Delay PDEs Error in Boussinesq Approximation
I'm trying to solve the set of equations below describing the flow of a pot of water being heated slightly. The equations are 2D axisymmetric in nature.
...
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AceGen/ AceFEM: Wrong number of nodes?
I am trying to create an element T1 (linear triangle) to solve the steady, incompressible Navier-Stokes equations. Therefore, my SMSTemplate command reads as follows
...
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Periodic boundary condition conflicts with DirichletCondition?
I want to solve a simple diffuse equation in a cylinder.
$\nabla\cdot(\hat{c}\nabla\rho_P）=0$
$\rho_P(x,y,\theta)==0,\sqrt{x^2+y^2}=R$
$\rho_P(0,0,\theta)==1,\sqrt{x^2+y^2}=0$
$\rho_P(x,y,2\pi)=\rho_P(...
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Solving PDE with Dirichlet, Neumann and Boundary conditions
I am trying to solve the following PDE:
$$
u_{xx} + u_{yy}
=
\begin{cases}
- \cos(x) \quad -\pi/2 \leq x \leq \pi/2, \\
0 \quad \text{otherwise}
\end{cases}
$$
The domain is $\Omega = [-\pi,\pi] \...
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Message Text not Found in FEM Model
I'm trying to solve for the heat transfer in 2D within a fluid. The geometry I'm using is a rectangle of water heated from the bottom. The code I'm using is as follows:
...
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FEM code working well in v.12.1, has warning in v.13, and not working in v.13.2.1
Let us consider sound in a glass. Numerical model has been described here. FEM code is given by
...
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PDE - The minimal damping factor of 1/10000 has been reached, NDSolveValue: PDESolve could not find a solution
I am trying to find a solution to a system of partial differential equations using Mathematica.
$$
\begin{align}
D_{e}\frac{\partial^{2}C_{A}}{\partial r^{2}}+\frac{D_{e}}{r}\frac{\...
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How to solve 2 coupled PDEs with inner boundary condition?
I want to solve 2 coupled PDEs with inner region boundary conditions.
Diffuse equation
There are 2 kinds of particles in my system, $p$ and $a$. Their diffusion equations are as follows:
$\hat{D}=\...
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How can I solve a three-dimensional Gross-Pitaevskii equation?
I want to solve a three-dimensional Gross-Pitaevskii equation with an uniform potential. The related papers are Emergence of a Turbulent Cascade in a Quantum Gas and Synthetic dissipation and cascade ...
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How to plot solution and approximate solution in one graph?
I want to solve a BVP $y^{\prime\prime}(t)+y^{2}(t)-t^{4}-2=0$, with BCs $y(0)=0, y(1)=1$. Solving $x^{\prime\prime}=0$ with the BCs, we get the initial approximation $x_{0}=t$. The exact solution is ...
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CPU time in iteration process
Let $Tx=\frac{x}{2}$ for all $x\in[0,1]$. Let $x_{0}\in[0,1]$ and set an iterative sequence $\{x_{n}\}$ by the method $x_{n+1}=Tx_{n}$. Now if $x_{0}=0.8$, then I get a convergent sequence towards the ...
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Mesh region from mesh edges only
I have a collection of lines ordered by pairs of points, as below, and wish to get a MeshRegion from them, with the mesh edges being the given lines. (That is, the ...
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How to make gaussian pulse move using Boundary Conditions (one spatial dimension)?
I'm trying to solve a system of normalized coupled PDE's to model Raman scattering. Essentially, a gaussian light pulse of 300ps duration enters the raman active medium at ζ = 0 which then drives the ...
2
votes
2
answers
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Symbolic solution for Laplace equation with pure Neumann b.c. and constraint at a point
Starting problem:
$$
\begin{cases}
\nabla^2 f = 0 & \text{on} \; \Omega \\
\frac{\partial}{\partial\mathbf{n}} f = y\,n_x - x\,n_y & \text{on} \; \partial\Omega
\end{cases}
$$
with $\Omega \...
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Post-process NDSolveValue result
For example, after solving the Poisson equation $-\nabla^2 T = 1$ on the unit circle using NDSolveValue and the following code
...
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65
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Wrong Eigenfunctions with NDEigensystem because of discontinuities
I'm trying to solve a 1D Photonic Crystal with PeriodicBoundaryConditions and a step function permittivitty but even if I get the proper band diagram my eigenfunctions don't correspond to the ones ...
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votes
1
answer
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How to access implementations of MaterialModel?
In the documentation for Hyperelasticity there is an example of the MaterialModel for NeoHookean and the St Venant Kirchoff models. These are very useful in helping develop ones own materials models. ...
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Decomposing an eigenfunction of helmholtz equation into plane waves
I have defined a region and found the eigenmodes of the 2D Helmholtz equation in it:
...
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votes
1
answer
287
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FEM - Linear grid covering a curved shape
I have a disk (or any other smooth curve) and wish to define a grid of squared cells everywhere, except for the boundary where they should be right triangles or pentagons with three right angles. In ...
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votes
2
answers
298
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How to use MMA to solve the minimal surface?
There is a great old post, but since MMA greatly improves the ability of solving differential equations, especially the Region can be used to define the range of variables. So I ask it again. As the ...
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votes
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mesh / region creation in unit meters or mm:
How can we know the unit of mesh or create a mesh in specified length units:
such as, for example I have:
...
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How do I pose Neumann boundary condition to suppress particles flux into zero point?
This is continuation from my previous post How to ensure for a solution of NDSolve to be positive? [https://mathematica.stackexchange.com/questions/278777/how-to-...
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Complex equations: NDSolveValue / FEM,
So, I am trying to solve a simple problem using the FEM method.
The distribution of Voltage over a region,
The region is,
...
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5
votes
2
answers
230
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Meshing an irregular domain using quads to solve conjugate heat transfer problem
I am trying to mesh the following domain to solve a heat transfer + fluid flow problem:
The continuity+momentum equations are to be solved in $ABGH$, while the energy equation is to be solved across ...
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Resolving of internal boundaries when using DistMesh generator
Mesh generator ToElementMesh allows to resolve internal boundaries of the domain to be tesselated:
...
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Solving an ODE and a PDE
I am trying to solve coupled sets of (partial) differential equations. For my ODE, I use
...
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answer
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How to specify derivative boundary conditions for the gradient to be normal to the boundary?
I'm currently modeling an electric field with 2 charges. To do so, I use NDSolveValue to solve a Laplacian with 2 Dirichlet conditions on the voltages of the ...
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Running of mesh generator DistMesh under version Mathematica 13.2
I am trying to run the element mesh generator DistMeshfrom FEMAddOns package by using version Mathematica 13.2 under Ubuntu 20.04.
...
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Boundary of mesh gives non uniform result
I wish to extract a boundary mesh but for some reason the Frontier option only works on a portion on it:
...
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Incorrect eigenvalues obtained from NDEigensystem
I have a matrix H1 given by
H1[kx_, ky_] := {{0, kx + ky}, {kx - ky, 0}};
and H2 which ...
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0
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Finding eigensystem using NDEigensystem
I am trying to calculate eigensystems for a matrix Ap, which I have provided an example of which in this thread. The matrix in general has a dependency on a ...
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Stitch 2 diffusion equations together at the boundary
I wanted to stitch 2 diffusion equations together. Here, n1 and n2 are solved in the domain {x,-1,1}, and I wanted to impose: n1[1,t] = n2[1,t] and D[n1(x,t),x] = -D[n2(x,t),x] at x=1. At x=-1, I am ...
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Obtain the coodinates of boundary mesh with particular mesh marker
This is a very simple question, however, I haven't been able to find straight forward documentation for this. I have a region that I have converted to an element mesh (which we can visualize with '...
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votes
2
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NDSolve very slow on 2D heat equation
I am trying to solve the 2D heat equation
$$
\begin{cases}
u_{t}-u_{x x}-u_{y y}=f \\
u(0, x, y)=\sin (2 \pi x) \sin (2 \pi y) \\
u(t, 0, y)=0 \\
u(t, x, 1)=0 \\
u_{x}(t, 1, y)=2 \pi e^{-t} \sin (2 \...
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How can we enhance the performance of `NDEigensystem`?
I am trying to reproduce eigenvalues in Tabl2 II in this paper (also available on the arxiv) for B=-12 but the results are slightly different from those in the ...
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votes
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Implementation of FEM to steady-state coupled fluid flow and heat transfer
This is a steady-state conjugate heat transfer problem (the time-independent version of this problem). The problem is conjugate as the energy equation is being solved in thermally connected solid and ...
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Numerical instability when using NDSolve to simulate phase separation
I'm using NDSolve to simulate the following model of phase separation:
where rho_T=rho_1+rho_2. The term with chi causes rho_1 and rho_2 to repel each other. However, there is a numerical instability ...
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0
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93
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Implementation of Neumann-like boundary condition
If we wish to solve an elliptic PDE, say $\nabla^2\phi=\text{given}$, on a domain $\Omega$ with Neumann boundary conditions, $\hat{\mathbf{n}}\cdot \nabla \phi\Big|_{\partial \Omega}=\text{given}$, ...
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votes
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Finite Element analysis: distribution of sine wave over a distance
I am trying to solve the following one-dimensional problem:
(to better understand and extend the FEM for a more complex problem),
...
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answer
64
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Neumann b.c are not satisfied in NDSolve PDE
I am solving a PDE (p[x,t]) with 1 spatial dimension {x,-10,10} and 1 time dimension {t,0,100}. There is an external function, U. The code reads
...
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Modelling heat transfer in periodically reversing flow
This is a heat transfer problem, which involves reciprocating (fully-reversing) fluid flow over a heated block of solid. The objective is to determine the temperature field in the solid and the fluid ...
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