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Questions tagged [finite-element-method]

Tag for the usage of "FiniteElement" Method embedded in NDSolve and implementation of finite element method (fem) in mathematica.

38 questions with no upvoted or accepted answers
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From where to learn finite element method?

Can anyone reccomend me a book or site for learning finite element method with mathematica, besides wolfram language official site, https://reference.wolfram.com/language/FEMDocumentation/tutorial/...
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297 views

Create vtk file of ParaView from Mathematica [FEM post-processing]

I'm doing FEM in Mathematica and want to visualize the results in Paraview (e.g., create a video of the displacement field). Now I have the element mesh, the displacement and stress fields. How can I ...
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496 views

benchmark of Mathematica's FEM?

Does anyone have any numbers on how mathematica compares to other commercial (eg ANSYS) or free FEM softwares (eg. FreeFEM++, FEniCS, elmer)? If this is too vague say solving the diffusion equation in ...
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407 views

FEM Mesh for pipes and tubes

I work with pipe systems and would like to use finite elements within NDSolve for regions inside or exterior to a pipe. How to do I make a mesh of the region ...
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316 views

Stress Operators for Finite Element Analysis

For stress analysis in the finite element method you need a stress operator. For two dimensional plane stress it can be found here. For two dimensional plain strain it can be found here. I need the ...
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292 views

How to use periodic boundary conditions on the derivative of u[x,y,z]?

I have a structural mechanical problem, where i try to calculate the strain field in the modelled material. I simplified the problem in the following example: There is a material with two phases, ...
7
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0answers
420 views

Symbolic Weak Form

Usually I write the weak form by hand for my FEM code, but it's a little annoying and mechanic sometimes. So, I wonder, is there any way to generate the symbolic weak form in Mathematica? For ...
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79 views

Direction of NeumannValue on boundary

When solving partial differential equations NeumannValue is used to specify the flux across the boundary. For normal fluxes the detailed notes state They appear on the boundary ∂Ω of the region Ω ...
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315 views

How to increase boundary mesh quality?

Update: I tracked down the current issue to errors generated by numerically integrating over the boundary mesh. What's the proper way to integrate a function over the boundary of an implicit region? ...
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145 views

NDSolve post-processing: Calculate the flow over a FEM-boundary

I'm trying to calculate the heat loss from buried pipes with NDSolve`FEM. For this I set Dirichlet conditions at the model top and bottom and temperature dependant Neumann conditions at the contact ...
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52 views

Interface points of NDEigensystem

When solving an eigenvalue problem with "NDEigensystem", e.g. a 1D Eigenvalue problem with the interval composed of different materials, which should be solved by the pure numerical method such as FEM,...
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143 views

Postprocessing of FEM: extremely slow speed of NIntegrate

I've gone through the documentation for NDSolve FEM. There are quite a lot tips and techniques for accelerating NDSolve, and only quite few tips for post processing. Somehow I found out that the ...
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183 views

Reducing a FEM 3D Interpolation Function to a 2D Interpolation Function

I'm trying to solve the heat equation in 3D using FEM and back out a slice in the xz plane. My problem is when I define a function that is a slice in the xz plane it uses the full 3D interpolation ...
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124 views

Stress analysis in a disk with circular holes

Writing the following code: ...
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128 views

Coupled parabolic differential equations with time delay

Is it possible for NDSolve to solve delay partial differential equations with simple Neumann boundary conditions? An example I have is as below: ...
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953 views

FDTD Electromagnetic Simulations with Mathematica

Mathematica has some Finite Element capabilities (as explained here). Do you think mathematica is a realistic/sensible option for electromagnetic simulations? (aka making a movie of electric fields ...
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268 views

How to Implement a “Function Object” suitable for NDSolve and Finite Element Method?

I have a bunch of methods that shoud return an expression that behave like a function, i.e. an expression that, when supplied with suitable arguments, evaluate to something. A simple ...
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258 views

How can I improve the solution of my PDE?

I want to improve the last code to find a better solution for umi. For now I'm using only MaxStepFraction. Also, when I put no ...
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503 views

Nonlinear FEM and FindRoot

I'm trying to develop a kind of nonlinear FEM application using mathematica to solve a bvp like the following: $$ \gamma(u') ~u^{iv} + 2 \gamma'(u') u''' u''+ u''^3 = f(x) $$ where $u = \tilde{u}(x)...
3
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1answer
317 views

1st-order linear ODE system gives inaccurate/biased solutions

Consider an ODE eigensystem $$ t(y+\frac{1}{s})a(y)+[(q+\frac{1}{2}+\frac{s}{2}y)+s(y\partial_y+\frac{1}{2})]b(y)=\lambda a(y)\\ t(y+\frac{1}{s})b(y)+[(q+\frac{1}{2}+\frac{s}{2}y)-s(y\partial_y+\frac{...
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80 views

Non constant coefficient in heat equation

I have to solve the following heat equation over a cylindrical domain. In cylindrical coordinates the PDE Equation reads: ...
2
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1answer
65 views

Variation of the biharmonic equation with Neumann conditions

I am currently writing a script to plot the solution of a variant of the biharmonic equation. In this case the equation I want to solve is ...
2
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0answers
63 views

Formulating equations for 3D stress in the finite element method

I would like to know how you formulate equations for the finite element method for stress calculations. We know the answer because user21 has put it here. It involves usage of ...
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0answers
116 views

Using of FindMinimum in constrained problem

There is a constrained optimization problem based on finite element method. We should find the optimal distribution of density inside the our region for minimazing the energy of deformation. The ...
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65 views

AceFEM: Matrix condensation and elimination of local unknowns

Lately I have been working with non-linear mixed hybrid elements. The basic article that I took for my research is one by T.H.H. Pian dn K. Sumihara: Rational approach for assumed stress finite ...
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242 views

Use Mathematica FEM-functionality for further calculations

I would like to use MMA FEM abilities for example to apply galerkin's method to solve intergal equations or nonstandard numerical applications in a given mesh m. Is it possible to extract the "node-...
2
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245 views

Solving a coupled PDEs using FEM with Robin conditions

The same type of question has been answered here but for single PDE. The main issue I am facing is how to write the weak form for two PDEs. $$\epsilon^2\frac{\partial ^2u}{\partial x^2}+\frac{\...
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82 views

AceFEM: Tie - Option Issue

I try to use the "Tie" Option of the SMTAnalysis[] function. In general, I want to have some additional Nodes (two kinds of them) at certain (not all) coordinates tied and at some not. What I have is ...
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0answers
126 views

AceGen: Solving the dynamic equation in the middle of the time step

Note: This question is about AceFEM and AceGen package for automatic code generation for finite element analysis. If the question is not specific enough, I will try to refrase it, but I am a fairly ...
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0answers
166 views

ElementMesh (rendering?) issue

The appearance of the following ElementMesh is bad as you can see for example on the inside surface of the shell. The same problem happens converting to a ...
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0answers
109 views

Eliminating instabilities in a transient finite element solution at a discontinuity near t = 0

I have found a transient solution to the 1D heat equation where the initial condition is discontinuous. The results are accurate except for when time is small. The initial condition looks like this: $...
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221 views

Puzzling NDSolve[] behavior for PDE (smooth solution, inconsistent with boundary conditions)

Consider the following: NDSolve[{D[z[x, y], x, x] + D[z[x, y], y, y] == 0, z[x, 0] == Sin[x], z[0, y] == Cos[y]}, z[x, y], x, y] {{z[x, y] -> ...
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85 views

Problems with the eigenvalues calculated using NDEigenvalue

I'm trying to solve a Sturm-Liouville problem like $\qquad -\psi''(z)+(\frac{1}{z}+2\,z)\psi'(z)=\lambda\,\psi(z)$ using NDEigensystem in order to learn how to ...
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48 views

AceFEM: Manipulation of SMTData[“MatrixGlobal”] after building but before solving

I want to manipulate the global stiffness matrix before I solve the system of equations. Unfortunately, the command SMTNewtonIteration[] builds up the global matrix and the global vector and solve the ...
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110 views

PeriodicBoundaryConditions an a square

I hope the answer is simple, but I can't see the solution: I want to solve a PDE for u[x,y] under Periodic boundary conditions on say a square size 10 x 10. I.e. <...
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81 views

How to draw a triangle shape function plot with shape function area being (s, t,1 - s - t)

I want to draw a dynamic graph showing the varying shape of a triangular element with the change in the point P[x, y] as shown below. I want to find the shape ...
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235 views

Incorrect solution of 2D unsteady heat equation with Neumann condition

I want to resolve a PDE model, which is 2D heat diffusion equation with Neumann boundary conditions. The key problem is that I have some trouble in solving the equation numerically. Consider the ...
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129 views

System of 2 stationary coupled non-linear PDEs on a rectangular domain

I need to solve the following system of 2 coupled non-linear PDEs, desirably on a domain of arbitrary shape, but it would be a nice start if it was possible on a simple rectangular domain with ...