Questions tagged [finite-element-method]
Tag for the usage of "FiniteElement" Method embedded in NDSolve and implementation of finite element method (fem) in mathematica.
1,360
questions
97
votes
17
answers
5k
views
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, ...
- 38.4k
78
votes
3
answers
9k
views
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 ...
- 998
71
votes
4
answers
5k
views
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 ...
user484
60
votes
1
answer
3k
views
What's inside InterpolatingFunction[{{1., 4.}}, <>]?
I'm curious what's inside the InterpolationFunction object?
For example:
...
- 27.2k
55
votes
3
answers
5k
views
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....
- 1,939
54
votes
2
answers
4k
views
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 ...
- 22.2k
52
votes
4
answers
7k
views
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 ...
- 15.6k
52
votes
2
answers
2k
views
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 ...
- 101k
41
votes
4
answers
7k
views
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 ...
- 3,366
40
votes
3
answers
5k
views
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 ...
- 2,284
36
votes
4
answers
7k
views
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 ...
- 3,912
35
votes
1
answer
3k
views
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 ...
- 15.6k
32
votes
7
answers
5k
views
How to generate a mesh with quadrilateral elements?
I have the following code that generates a finite element mesh:
...
- 2,820
31
votes
1
answer
948
views
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 ...
- 15.6k
31
votes
1
answer
683
views
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.
...
- 2,571
28
votes
1
answer
4k
views
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 ...
- 15.6k
27
votes
1
answer
4k
views
How do I use the new nonlinear finite element in Mathematica 12 for this equation?
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[0] == AiryAi[0], 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
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 ...
user52420
22
votes
2
answers
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 ...
- 37k
21
votes
4
answers
6k
views
How do I solve a PDE with a strange boundary condition?
How do I solve the PDE with boundary value like this
$$u(t,x,y)=0, \textrm{when } F(x,y)=0$$
using DSolve?
As a specific example, I want to solve heat equation
$$\frac{\partial u}{\partial t}=\...
- 313
21
votes
3
answers
7k
views
How to discretize a nonlinear PDE fast?
I wish to numerically solve the following PDE. Although there are some complete discussions for solving PDEs in tutorial/NDSolvePDE, there is no hint for the nonlinear case by discretization. Thus, I ...
- 1,795
21
votes
2
answers
2k
views
Least effort to handle a point source inside the domain of PDE(s)
By point source I mean a constrained condition at one point inside the domain of PDE(s). For example:
$$\frac{\partial ^2u(t,x,y)}{\partial t^2}=\frac{\partial ^2u(t,x,y)}{\partial x^2}+\frac{\...
- 59.5k
21
votes
2
answers
1k
views
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 ...
- 311
21
votes
2
answers
1k
views
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 ...
- 945
21
votes
1
answer
1k
views
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 ...
- 6,443
21
votes
1
answer
647
views
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:
...
- 5,001
21
votes
1
answer
1k
views
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 ...
- 867
21
votes
1
answer
439
views
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 ...
- 38.5k
20
votes
3
answers
2k
views
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:
...
- 313
20
votes
3
answers
589
views
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 ...
- 22.2k
20
votes
1
answer
413
views
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 ...
- 231k
19
votes
6
answers
2k
views
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 ...
- 305
19
votes
4
answers
1k
views
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 ...
- 3,804
19
votes
3
answers
706
views
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" ...
- 3,912
19
votes
2
answers
789
views
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: ...
- 38.4k
19
votes
1
answer
450
views
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 ...
- 38.5k
18
votes
5
answers
3k
views
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 ...
- 1,312
18
votes
3
answers
3k
views
18
votes
1
answer
764
views
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,
...
- 14k
18
votes
1
answer
2k
views
Solver for unsteady flow with the use of the Navier-Stokes and Mathematica FEM
There are many commercial and open code for solving the problems of unsteady flows. We are interested in the possibility of solving these problems using Mathematica FEM. Previously proposed solvers ...
- 38.5k