# Questions tagged [finite-element-method]

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

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### Future enhancements for the finite element method

How should the finite element method (FEM) framework in the language be extended to be more useful? With the release of version 12.0 all fundamental FEM solvers (linear, nonlinear, stationary, ...
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### Numerically solving Helmholtz equation in 2D for arbitrary shapes

I would like to solve the Helmholtz equation with Dirichlet boundary conditions in two dimensions for an arbitrary shape (for a qualitative comparison of the eigenstates to periodic orbits in the ...
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### Can Mathematica solve Plateau's problem (finding a minimal surface with specified boundary)?

Given a closed curve $\mathcal C$ in three dimensions, is it possible to use Mathematica's built-in functionality to compute a minimal surface whose boundary is $\mathcal C$? For simplicity, let us ... 3k views

### What's inside InterpolatingFunction[{{1., 4.}}, <>]?

I'm curious what's inside the InterpolationFunction object? For example: ...
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### Mathematica vs. Comsol for finite element analysis?

Being relatively new to finite element analysis, I was wondering how expert users assess Mathematica's capabilities in solving PDEs via the finite element method compared to other commercial tools (e....
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### Numerically solving Helmholtz equation in 3D for arbitrary shapes

Context While studying manifold Learning I got interested in finding the eigenvectors of the Laplacian. (also in connection to this problem of solving the heat equation) Following this and that ...
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### Dynamic Euler–Bernoulli beam equation

I am trying to solve for the vibration of a Euler–Bernoulli beam. The equation is $\frac{\partial ^2u(t,x)}{\partial t^2}+\frac{\partial ^4u(t,x)}{\partial x^4}=0$ For the boundary conditions I ...
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### A geometric multigrid solver for Mathematica?

Cross posted to community.wolfram.com Mathematica ships a variety of linear solvers through the interface LinearSolve[A, b, Method -> method] the most ...
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### Creating a 2D meshing algorithm in Mathematica

As what is proving to be a difficult, but entertaining task, I am attempting to adapt a 2D meshing algorithm created for MATLAB and port it to Mathematica. I understand meshing functions already exist ...
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### Nonlinear FEM Solver for Navier-Stokes equations in 2D

it's me again. I'm trying to obtain a numerical solution to Navier-Stokes equations in 2D in a non-rectangular region. So far, this guide was very helpful, but he is using finite differences, which is ...
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### Why is raytracing so slow?

I'm trying to get some rays to bounce in a circle. But I want to be able to control the reflections, i.e. the direction the rays bounce in the circle. I have a MWE below, and it is severely limited by ...
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### Calculating a potential function using the finite element method

This is my first attempt to use the Finite Element method available in version 10. There are questions and I am very open to suggestions. My example is flow around a cylinder which is a well known ...
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### How to generate a mesh with quadrilateral elements?

I have the following code that generates a finite element mesh: ...
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### Finite element simulation of Airy waves

I am attempting to solve for waves on a water surface starting with a two dimensional solution. The equations are that the water must satisfy Laplace's equation everywhere with a time dependent ...
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### Detecting collisions in FEM

Say I want to study the deformation of a pitchfork when you have it fixed on the bottom and push one side. ...
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### Stress calculations using finite elements

A standard engineering problem is to calculate stresses in a structure due to applied forces. With the inclusion of the finite element method in version 10 this question attempts to investigate how ...
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With Mathematica 12 we get new technology for nonlinear finite elements. Out of curiosity, I just wanted to solve the following equation $$\frac{d}{dx} \left( c(x) \left[\frac{d}{dx} u(x)\right]^p \... • 5,323 26 votes 5 answers 2k views ### Plot a partition of the sphere given vertices of polygons I saw in this question that Mathematica can draw spherical triangles. I guess something similar can be done to plot a spherical polygon. I am interested in something similar: I have a set of points ... • 645 26 votes 3 answers 14k views ### How to solve ODE with boundary at infinity y''[x] - x y[x] == 0 y == AiryAi, y[∞] == 0 The analytic solution to this ODE is the Airy function y[x] == AiryAi[x] if ... • 669 26 votes 1 answer 6k views ### How to numerically solve a 1-d time-independent Schrödinger equation? The point is to solve the eigensystem of the given Hamiltonian. I tried ParametricNDSolve combined with FindRoot to search for ... • 1,040 25 votes 1 answer 893 views ### How to model wooden joints with mathematica's FEM? This is a dovetail joint: and I'd like to see the stresses and deformation on the joint.I haven't seen any modeling of disconnected regions with FEM, only connected regions, so I'm curious if you can ... • 2,571 24 votes 2 answers 3k views ### Optimizing a Numerical Laplace Equation Solver Laplace's Equation is an equation on a scalar in which, given the value of the scalar on the boundaries (the boundary conditions), one can determine the value of the scalar at any point in the region ... • 2,103 24 votes 2 answers 2k views ### ListContourPlot interpolation fails if x and y axes have different scales To summarize what is below: ListContourPlot doesn't work when the domain has different x and y scales (fails when x/y~10^4, which seems surprisingly small). Is it possible to call ListContourPlot on ... • 529 24 votes 2 answers 901 views ### 3D Elastic waves in a glass Take an empty glass, hit the side, the glass will make a sound that can be recorded using ... • 38.5k 24 votes 1 answer 885 views ### Mismatch between Mathematica and COMSOL in 3D FEM problem I would like to solve an advection-diffusion problem on a torus domain. There are three Dirichlet conditions: One at the inlet (concentration c=0), one at the outlet (c=0.5) and one at the wall (... • 1,939 23 votes 7 answers 670 views ### How to make the boundary of a 3D region smooth? I want to draw this region,but the surface is rough.I tried to find options to improve the surface but failed. ... • 7,597 23 votes 4 answers 776 views ### Making holes from maze generated Graphics3D ... • 487 23 votes 1 answer 1k views ### Implement fractional Laplacian What is a way to implement the Fractional Laplacian with Mathematica? How can we apply such implementation to numerically solve the problem$$(-\Delta)^su = 1 \text{ in } B_1(0), \\ u = 0 \text{ in ... 853 views

### Getting rid of spikes in the PDE solution

Bug introduced in 10.0 and fixed in 10.3 Note: In 10.0, Rationalize[fd, 0] was needed or mesh generation would fail. Preamble: I am solving a PDE in a domain ...
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### Partial Differential Equation in Parallel

is there any native way to implement multi-core parallel solving of PDE in Wolfram Mathematica? WM 10 now supports Finite Elements Method, but it is actually useless without parallelization. Usually ...
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### How to access FEM shape functions

A couple of days ago I asked here about surface meshes and plotting on surfaces. Now I have another question: How can I access the surface or boundary element shape functions? I would like to ...
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### Couple a PDE and ODE in NDSolve

I would like to solve an example of non-stationary heat transfer with a coupled PDE and ODE. Let's assume that we have 1 dimensional bar of length $L$ with uniform initial temperature. The right end ...
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### How to fix broken InterpolatingFunction?

Bug introduced in 8.0 and fixed in 9.0.0 I have an InterpolatingFunction based on irregularly-gridded data, like this: ...
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### Eigenfunctions of the Laplacian on an arbitrary mesh

So, I've constructed a mesh over which I'd like to find eigenfunctions of Laplace's equation with a free boundary (a zero Neumann boundary condition along the edge). Mostly because I figured an ...
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### Violin with f-holes and FEM simulation of Helmholtz resonance in 3D

This question is about FEM simulation of violin vibration modes in 3D. There are several problem around. One of them is Helmholtz resonance. Air inside the violin body has own frequencies dependent ...
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### Numerically solve the initial value problem for the 1-D wave equation

I want to solve the standard 1-dimensional wave equation $y_{xx}=y_{tt}$ using NDSolve (for $y(x,t)$) with the following conditions: ...
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### Inverting differential equation using finite element methods

tl;dr; How to use FEM tools to extract models needed to invert PDEs. Context In astrophysics, one is interested in so-called 'cosmic archeology' which involves recovering the origin of a given ...
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### What are pattern sparse arrays and how do I use them?

I found a few references to something called a "pattern sparse array" that does not contain any values, only positions. What is this data structure? How and when do I use it? From the documentation ...
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### Poisson equation with pure Neumann boundary conditions

Dear Mathematica users, I would like to numerically solve a, as the title says, Poisson equation with pure Neumann boundary conditions $-\nabla^2(\psi)=f$ $\nabla(\psi)\cdot \text{n}=g$ Is it ...
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### How to apply different equations to different parts of a geometry in PDE?

I want to solve two coupled partial differential equations on two dimension. There are two variables v and m. The geometry is a ...
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### Using Regions, can I model a reflecting wavefront?

I was wondering if I could get Regions and Mathematica's shapes to do all the hard work for me in making a "droplet in a pond" simulation. I do not want the "waves" ...
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### Large deformation of solids

Link to notebook with this question and code I'd like to understand how large deformations of solid mechanics work and how they are implemented. For this am looking at the following reference problem: ...
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### 3D stable fluids algorithm to simulate transition from laminar to turbulent flow

This algorithm is 3D extension of our 2D algorithm published on this page and here. We suppose that with this code we can simulate transition from laminar to turbulent flow. In this example we compute ...
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### periodic boundary conditions and NDEigensystem

Mathematica 10 has a splendid new function, NDEigensystem, that makes it possible to solve Sturm-Liouville problems numerically in a single step. I have not however been able to find a way to get it ...
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### What is NDSolveFEM*?

I stumbled on this: ?"NDSolveFEM*" ...
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### Plotting the eigenmodes of a cylindrical shell

There are many examples of eigenmodes computations for surfaces with Mathematica, such as: https://www.wolfram.com/mathematica/new-in-10/pdes-and-finite-elements/solve-a-wave-equation-in-2d.html, ...
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