All Questions
1,333 questions
5
votes
0
answers
372
views
Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations
I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
- 1,157
5
votes
0
answers
95
views
Numerical methods for the continuity equation with Sobolev vector field
Consider the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
- 51
5
votes
0
answers
1k
views
Gmsh for 3D volume with inclusions [closed]
In an attempt to create three-dimensional volumes with inclusions in Gmsh I stumble upon a problem which was non-existent in the two-dimensional case.
I'm using the OpenCASCADE geometry kernel ...
- 93
5
votes
0
answers
185
views
Geometric Multigrid for Conform and Non–″ Elements: Restriction Operators
First of all, let me set up some notations.
Suppose we have a hierarchy of meshes $\mathcal{T}_0 \subset \mathcal{T}_1 \subset \dots \subset \mathcal{T}_n$. For simplicity I restrict myself here to $...
- 901
5
votes
0
answers
5k
views
What should be the number of boundary conditions of a PDE [closed]
As far as I know, for getting a unique solution to a PDE we should impose some boundary conditions to the PDE. "The number of required auxiliary conditions is determined by the highest order ...
- 643
5
votes
0
answers
370
views
Is it possible to predict the null space of a structure from contributing elements null spaces?
I am trying to solve an almost incompressible problem with heterogeneous properties by domain decomposition. Solution with CG converges slowly or divergerces completely. My problem becomes ill-...
- 101
5
votes
0
answers
1k
views
Poisson equation with pure Neumann boundary conditions (using FEM)
I am trying to derive the correct variational form for the Poisson equation with pure Neumann boundary conditions, and an additional contraint $\int_{\Omega} u \, {\rm d} x = 0$, as described in this ...
- 51
5
votes
0
answers
249
views
Conjugate residual/gradient convergence checking in practice
Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
- 511
5
votes
0
answers
263
views
Negative viscosity stabilized by fourth order terms
I am trying to solve a "Navier-Stokes"-type problem where the viscosity is negative. Of course this renders the equation unstable and thus I add a fourth order term, so the entire equation becomes:
$$...
- 51
4
votes
4
answers
2k
views
What is the meaning of "preasymptotic" and "superconvergent"?
Precisely the title of the question.
I have encountered these terms in two areas: conjugate gradient method, and adaptive finite elements.
- 3,720
4
votes
5
answers
888
views
FEM library with support of simplex elements
My question can be rephrased as "FEM library like deal.II but for simplex elements".
Our scientific group works with very complicated 3D geometries, so usually we prefer tetrahedral meshes for our ...
- 291
4
votes
1
answer
400
views
Under what circumstances is parallel scaling of the finite element method not "solved"?
I see things like this deal.II example, (ctrl + f for "superMUC") which seems to show some pretty impressive scaling (nearly twice as fast for twice as many CPU cores for a wide range of ...
- 155
4
votes
2
answers
202
views
What is the difference between $u_h$ and $I_h(u)$ in finite element literature?
In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
- 435
4
votes
2
answers
348
views
Analog of perfectly matched layers for finite element methods
Is there an analog of perfectly matched layers for finite element methods? References or small examples are much appreciated.
- 2,165
4
votes
3
answers
900
views
What condition ensures the global continuity of the solution in the FEM?
I know this is trivial but I don't seem to understand it. In which step of the FE formulation do we enforce the global continuity of the solution? Or in other words, how the construction of the local ...
- 146
4
votes
2
answers
2k
views
Example of level set method
I am looking for easily to understand example of level set method used to track phases interfaces. I would like to solve it using FEM because my solution is based on the FEM solution of second Fick ...
4
votes
2
answers
178
views
Good references for dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)
Can anyone help me with good references (books or papers) where I can learn about dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)?
- 63
4
votes
1
answer
966
views
Interpolating a mathematical function using a Hermite Cubic Finite Element Space
I have a Hermite Cubic Finite Element Space on a computer in the form of Matlab m-files. More specifically, I can evaluate four "shape functions" $N_1, N_2, N_3,$ and $N_4$, for which the following ...
- 399
4
votes
1
answer
169
views
What is the origin of the preasymptotic convergence behaviour in FEM?
When you have fine-scale features (e.g. boundary layers) in the solution, its FEM approximation on coarse meshes converge at strange apparent rates. Looking at Cea's lemma, is this behaviour because ...
- 73
4
votes
4
answers
213
views
FEM textbooks recommendation
I would really appreciate if you could recommend me some textbooks that explain FEM process well and have many exercises to solidify the knowledge. So far, Brenner and Zienkiewicz seem to be a bit ...
- 65
4
votes
2
answers
417
views
computing higher order derivatives with linear elements
Consider the following equation on $(0,1)$, with Dirichlet boundary conditions on both ends.
$$
\frac{d}{dx}\left(k(x)\frac{du}{dx}\right) = 0
$$
Let us solve this using simple linear finite elements. ...
- 852
4
votes
1
answer
381
views
guiding principles of integration by parts in FEMs
Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is also a first ...
- 335
4
votes
1
answer
160
views
A priori FEM estimates without $H^2$ regularity
In basic lectures on finite elements theory, people always assume $H^2$ regularity of the solution in order to derive $O(h^k)$ a priori estimates in the norms $H^{2-k}$, $k=1,2$. For simplicity let's ...
- 259
4
votes
2
answers
327
views
Getting started with finite element modelling
I'm a high-schooler building a small vehicle for an independent study. I've had finite element modelling recommended to me as a way to save time during the design process, and I'd like to try it out. ...
- 157
4
votes
2
answers
3k
views
Efficently invert tiny matrix in Fortran
I have a piece of code in Fortran90 in which I have to solve both a non-linear (with the Newton-Raphson method, for which I have to invert the Jacobian matrix) and a linear system of equations. When I ...
- 41
4
votes
4
answers
4k
views
How to deal with nonlinear term in Navier Stokes equations (finite element code)
I am trying to solve the Navier Stokes equations using the finite element method. I plan on using the pressure correction method to deal with the pressure and an implicit time stepping scheme for ...
- 1,929
4
votes
2
answers
271
views
Why is FEM theory only ever written down for simplices?
I noticed that in most theoretic literature (Braess, Ciarlet, ...) for Finite Elements, the method is only described in a detailed way for simplex triangulations, while other forms of hexahedral ...
- 3,065
4
votes
3
answers
707
views
I wrote a 2D Finite Element program for Axial Loaded Plates, but the results are unexpected
TLDR: I used Python to write a 2D Finite Element program using 'Constant Strain Triangles' and my beam keeps pointing slightly upwards instead of straight sideways (like the force). I'm new to FEA and ...
- 43
4
votes
3
answers
4k
views
Lattice Boltzmann methods vs Navier stokes/ other eulerian methods for *water* simulation
Note, there is already a question here, however the answers don't answer the original question, let alone specific considerations when dealing with nearly in-compressible fluids (water). Another ...
- 161
4
votes
2
answers
132
views
Analytical testing/quality control for scientific software in professional setting
I am charged with maintaining a buildserver on Teamcity which is meant to test our in house FE software. Currently our test suite consists of a list of benchmarks which run every time a commit is made ...
- 251
4
votes
2
answers
303
views
Textbook/Manual on Implicit FEM Methods
I've recently been interested in learning about implicit finite element methods. I've found this post, but I'd like to learn more about them, specifically about how simulations like these from LS-DYNA ...
- 383
4
votes
1
answer
2k
views
Inverse isoparametric mappings for quadrilateral finite elements
I have an isoparametric mapping $F_{E}: \hat{E} \to E$ where $\hat{E}$ is a reference quadrilateral (square) and $E$ is some quadrilateral in the domain $\Omega$. If $E$ is a parallelogram, then the ...
- 947
4
votes
2
answers
379
views
P versus Q elements
I am currently developing a project that uses finite elements for multi-dimensional PDEs and I'm still wondering if I will use P elements (triangles in 2D and tetra in 3D) or Q elements (squares in 2D ...
- 1,266
4
votes
1
answer
144
views
Reconciling vector and scalar notation
I am trying to derive Galerkin type weak formulation for the Stokes equations. I'm having a bit of a problem reconciling the notation in the integration by parts. I know that the answer I'm looking ...
- 625
4
votes
3
answers
251
views
vector PDEs on manifolds
What are the subtleties involved in solving vector PDEs on manifolds? Can someone suggest a reference summarizing the problems involved?
Specifically I want to solve a vector Helmholtz equation with ...
- 176
4
votes
1
answer
2k
views
Is there a mesh generator that will generate zero thickness elements for interfaces?
I have a geometry and a finite element method where there are a couple internal interfaces across which I want to enforce some fairly simple jump conditions.
One recommendation I've gotten has been ...
- 30.4k
4
votes
1
answer
491
views
Multi threaded finite element assembly implementation
What is typically the best way to multi thread the assembly loop in a finite element code? Does anyone have experience with implementing this, that they can share? I can think of a couple of ways of ...
- 203
4
votes
1
answer
3k
views
How to apply non zero Dirichlet boundary condition in finite elements?
I am writing a code for steady state heat transfer on a rectangular domain. I am specifying temperature on the edges - nonzero Dirichlet boundary condition. The equations can be written in form of
$$...
- 145
4
votes
3
answers
1k
views
Pre/Post-processor for an academic finite element solver
I'm currently developing a finite element solver for academic/research purposes. Therefore I'm searching for a pre- and postprocessor in my toolchain.
For a previous project I have used gmsh as a ...
- 186
4
votes
1
answer
5k
views
what is the difference between non-conformal and conformal?
So far what I understand is that two neighbouring elements are conformal if their edges and faces match exactly, whereas with non-conformal elements this is not the case. For instance, h-refinement ...
- 311
4
votes
1
answer
395
views
Finite differences vs. elements: Accuracy and implementation
I am trying to solve the 2D Poisson equation numerically:
$ \frac{\partial ^2 \phi}{\partial x^2} + \frac{\partial ^2 \phi}{\partial y^2} = 1 $
with the Dirichlet boundary condition $\phi = 0$.
I ...
- 41
4
votes
1
answer
349
views
Localization of Sub-problems in the Multiscale Finite Element Method
Background:
I'm trying to understand the Multiscale Finite Element Method and I'm reading Effendiev & Hou (2009) Multiscale Finite Element Method. Suppose I'm working with a poisson equation of ...
- 12.2k
4
votes
1
answer
4k
views
Stress due to the mismatch of thermal expansion coefficients of two different attached materials in COMSOL
I'm simulating the thermo-electro-mechanical behavior of a copper wire which is surrounded by silicon dioxide. In other words, the wire segments is under mechanical and thermal loads and at the same ...
- 183
4
votes
1
answer
387
views
FEM for non-divergence form elliptic equation
The FEM is usually used with a weak form of PDE. But for the non-divergence form elliptic operator
$$
-a_1(x,y) \frac{\partial^2}{\partial x^2} - a_2(x,y) \frac{\partial^2}{\partial y^2}
$$
or ...
- 1,319
4
votes
2
answers
2k
views
Mesh simple 2d CAD boundry drawing
I sincerely apologise if this question is a duplicate. Though it is clearly a question that must have been asked and answered a 1000 times I can't find any reasonable solution.
How do I take a simple ...
- 328
4
votes
1
answer
113
views
Why the solid FEM problem can not be solved after constraining 3 degrees of freedom?
I write a simple MATLAB code for solving solid FEM problem.
The problem looks like that
(1) (2)
x-------x
| / |
| / |
| / |
x-------x
(3) (4)
...
- 323
4
votes
1
answer
216
views
Taylor expansion of error - Finite elements approximation
In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the ...
- 1,077
4
votes
3
answers
961
views
How can I make sure the flow is divergence-free when I use moving mesh?
I am using projection method and P2/P1 finite element method to solve the incompressible Navier-Stokes equations while the mesh is constantly adapted as the body moves (edge swapping, splitting and ...
4
votes
1
answer
425
views
Finite element mesh software
I'm looking for a program to obtain meshes to finite element codes 2D and 3D as complete as possible, preferably in fortran 90 or C/C++.
For example, softwares "Triangle" or "TetGen" generate meshes ...
- 527
4
votes
1
answer
358
views
Finite Difference Beam Propagation Method problem
I am trying to implement the finite difference beam propagation method to study the propagation of a TE light signal through a waveguide. However, my solutions are exponentially growing, and display ...
- 143