All Questions
1,333 questions
5
votes
3
answers
403
views
Why do structured and unstructured discretizations give different errors?
It is necessary for me to solve a Poisson problem with a numerical method on a square domain with two types of triangular mesh: uniform triangular mesh (using uniform distribution nodes on square) and ...
- 533
5
votes
4
answers
1k
views
Mesh generator that can do 2D & 3D elements combined?
I'm trying to analyze a circuit card assembly (CCA). The biggest problem is always trying to mesh the thin copper layers along with the thicker epoxy layers between. I'm making the approximation that ...
- 151
5
votes
3
answers
2k
views
Online Poisson Solver
I'm wondering if anyone can point to a browser-based FEM (or other) 2D PDE solver for simple elliptic problems. It seem like there ought to be a javacript implementation that would allow for the ...
- 10.9k
5
votes
1
answer
396
views
Are serendipity elements still polynomially complete when the quadrilateral is skewed?
I have not implemented these elements before, but I like their reduced cardinality compared to (e.g.) a tensor product of Lagrange interpolants, which is very "overcomplete" (especially for orders>2)
...
- 5,076
5
votes
2
answers
779
views
Orthonormalized Bernstein polynomials using Gram-Schmidt
I was wondering, before trying to do that myself, has anyone attempted to do orthonormalization of Bernstein polynomials using Gram-Schmidt?
I discussed this with several people and have been told ...
- 1,413
5
votes
2
answers
219
views
Finite element accuracy on non-affine quadrilateral meshes
I have a simple finite element code (continuous Galerkin method, $Q_1$ tensor product spaces) for $-\Delta \psi = f$ on the unit square with essential boundary conditions. I want to claim that for any ...
- 123
5
votes
2
answers
238
views
Which way is the right way to compute the integrals in finite element methods?
Finite element methods involve integrals of functions that are not polynomials, and these integrals must be computed numerically.
For example, suppose that $f$ is the right-hand side of a Poisson ...
- 3,720
5
votes
2
answers
290
views
How to create a good preconditioner for a system of linear equations that is created with FEM applied on the time harming Maxwell eqution?
I set out to solve the time harmonic Maxwell equation numerically which was discritzed using FEM and with the use of Nedelec elements as basis and test functions. The equation reads:
$$ \nabla \times \...
- 506
5
votes
1
answer
325
views
Solving PDEs in parallel
I have read different approaches on how to solve pdes in parallel which are discretized using finite element method. For example:
Non-overlapping domain decomposition approach as mentioned in https://...
- 481
5
votes
4
answers
2k
views
FEM shape functions on triangular elements: transition from 2D to 3D
I'm writing a code for solving PDEs through the finite element method. In particular, I'm facing with 3D problems, in which I don't know how to calculate shape functions derivatives on the boundaries (...
5
votes
1
answer
961
views
Advantages and disadvantages of space-time finite element methods
I have heard of space-time finite element methods. Although I was able to find some articles that describe the different possible methods from a mathematical point of view (thanks to Space-time finite ...
- 53
5
votes
2
answers
956
views
Recommended visualization tools for higher order finite element solutions?
Is there any software available which can directly render higher order finite element results? In particular, 3D finite elements would be preferable. It seems gmsh has some capability in this regard ...
- 193
5
votes
1
answer
2k
views
When should a geometric stiffness matrix for truss elements include axial terms?
Bathe's Finite Element Procedures shows the "nonlinear strain stiffness matrix" for a 2D truss element as
$$
\frac {^tP} {L_0 + \Delta L}
\left[ \begin{array}{ccc}
1 & 0 & -1 & 0 \\
0 &...
- 223
5
votes
5
answers
10k
views
scipy.sparse: Set row/column in sparse matrix to the identity without changing sparsity
I'm using the SciPy sparse.csr_matrix format for a finite element code. In applying the essential boundary conditions, I'm setting the desired value in the right ...
- 211
5
votes
1
answer
1k
views
boundary conditions for electrostatic problem
I'm solving an electrostatic problem governed by Laplace equation $$-\nabla \cdot (\rho^{-1} \nabla u) = 0$$ in the following domain: a brick ($\Omega_1$) with a cylindrical inclusion ($\Omega_2$), ...
- 291
5
votes
2
answers
498
views
Existence of incomplete cholesky factorization
What is the current state of research on the existence of incomplete cholesky factorizations (in the context of preconditioning) for symmetric positive definite matrices?
I wonder in particular ...
- 3,720
5
votes
1
answer
248
views
trilinear hex elements
Do the faces of tri-linear hex elements have to be planar? Three nodes define a plane. If the fourth node does not lie on the plane, then the nodes are not planar and the face is not plane. In general,...
- 852
5
votes
2
answers
493
views
Prescribe solution of a PDE at specific points
I am using MATLAB's PDE toolbox to solve the differential equation
$-\nabla\cdot\left(c(x)\nabla u(x)\right) + a(x)u(x) = f(x)$
The particular problem in question is an electrostatic problem, but ...
- 185
5
votes
2
answers
2k
views
How to implement adaptive mesh refinement using conformal triangles
I am trying to implement adaptive mesh refinement for a finite element code. The code uses (at least for now) linear triangles and so when I do the mesh refinement I want the triangular mesh to remain ...
- 1,929
5
votes
2
answers
655
views
Choice of spaces for mixed formulation for Poisson Equation Or Darcy equation
Consider the mixed formulation of the Poisson/Darcy system for a region $\Omega$:
$\alpha \mathbf{v} + \nabla p = f \\
\mathrm{div}[\mathbf{v}] = 0 $
with the boundary conditions
$\mathbf{v}\cdot ...
- 445
5
votes
2
answers
434
views
Finite element discretization of Reaction-diffusion problem with Dirac source term
I'm writing a code using continuous piecewise linear finite elements on triangular
grids to solve the diffusion-reaction problem. the source function f is a Dirac mass at the center. How can i compute ...
- 177
5
votes
1
answer
233
views
What Linear Equation Solver should be used for a problem with many dirichlet conditions?
I am solving a laplace equation on a finite-element mesh (tetrahedral, triagonal) and have many say 99% dirichlet conditions compared to the number of unknowns. Is there an efficient way to solve this ...
- 51
5
votes
2
answers
317
views
I'm having trouble debugging multigrid. What to do?
I've spent far too much time coding and debugging multigrid. While I clearly can't post all of my code as it would be silly to ask someone to go through all that code, is there anything I should pay ...
- 583
5
votes
3
answers
993
views
implicit vs. explicit domain decomposition methods
I've been working on a finite element code on unstructured methods, which I've parallelized using the Schur complement method. Here's a summary of how I did it:
Assign each triangle of the mesh to a ...
- 10.5k
5
votes
1
answer
2k
views
Mixed boundary conditions Finite Element Method
I have the following problem in Finite Element Method
$$ -(\alpha u')' + \beta u' + \gamma u = f$$
with $ \Omega = (0, 1)$, $ u(0) = 0 $ and $ u'(1) = 3 $
to be able to write the weak formulation ...
- 1,039
5
votes
2
answers
399
views
Why does the choice of basis functions change the approximate solution in FEM even when the same space is spanned?
Let's say we have the the weak form laplace problem
Find $u$ $\in$ $H^1$ what satisfies
\begin{equation}
\int_\Omega \nabla u \cdot \nabla v \, d\Omega = \int_\Omega f v \, d\Omega
\end{equation}
...
- 506
5
votes
1
answer
289
views
Write incompressible Navier Stokes as ODE in $(\mathbf{u},p)$
Consider the Navier stokes equation after the discretization with conforming finite elements with source term $f=0$. We have the algebraic structure of a saddle point problem:
$$M \dot{u} = f- Au -B^...
- 435
5
votes
2
answers
359
views
Preconditioner for the GMRES method in the Uzawa algorithm
I'm trying to solve
\begin{equation}\left\{
\begin{split}
\frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\frac1\rho\nabla p&=f\;\;\;\text{in }\Lambda\\
u&=0\;\;\;\text{on }\partial\...
- 283
5
votes
2
answers
169
views
Stokes Equation in "two-fold saddle point" form?
Are there papers that deal with the (nondimensionalized) Stokes equation for incompressible fluid flow in a "doubly mixed" form like the following?
\begin{align*}
0&=\underline{\epsilon} + \frac{...
- 1,000
5
votes
1
answer
3k
views
Simulate electric fields due to surface charges in simple circuits using python
I want to simulate the electric fields in simple circuits using Python and only free software. My first goal is to reproduce the images given in (1) which are made by the commercial ANSYS Maxwell ...
- 273
5
votes
1
answer
348
views
Method of Manufactured Solutions for non-differentiable coefficients
The Method of manufactured is commonly used for verification of computational science codes. I want to use the method for verification of Navier-Cauchy (elasticity) equations with periodic and Bloch-...
- 8,622
5
votes
1
answer
393
views
Are additional penalty terms necessary to solve elliptic PDE's with DG-FEM?
After a cursory glance at several references to discontinuous galerkin finite element methods for the elliptic poisson PDE, i notice that all of them emphasize using penalty methods where an ...
- 12.2k
5
votes
1
answer
1k
views
Which finite-element framework allows the easy implementation of HDG methods?
I have tinkered around with the implementation of hybridizable Discontinuous Galerkin (HDG) methods in DUNE-PDELab and DUNE-FEM for my PhD but since then I have not had the time to work on these codes ...
- 1,074
5
votes
2
answers
697
views
Direct solvers and domain decomposition for FEM
In the finite element method, in order to use MPI, we need to decompose the domain into sub-domains first. Then my question is whether we can solve each sub-domain using a direct solver?
Of course, ...
- 295
5
votes
1
answer
2k
views
Shell vs frame element model stiffness differences
I have a model of a tall, slender structure that I am investigating using both shell and $3D$ frame elements.
The shell elements are type MITC4, $4$-node membrane elements. The frame elements are ...
5
votes
1
answer
408
views
How much measure theory should I know to understand the proofs in Brenner & Scott's FEM book?
I've been reading Larson and Bengzon's recent book on finite element methods, which has been good for getting an understanding of basic theory and computational procedures. The finite element book by ...
- 30.4k
5
votes
1
answer
4k
views
FEM: which is the correct way to impose Dirichlet B.C
I know the Neumann B.C. is implicit in FEM language. However, I have seen at least two ways to impose Dirichlet B.C.
e.g. for the following problem 1D,
$$\nabla^2 u + \nabla u= 0, u_{left}= 1, u_{...
- 593
5
votes
1
answer
7k
views
Implementing Explicit formulation of 1D wave equation in Matlab
So the theory is straightforward. We have:
$$\frac{\partial^2U}{\partial t^2}=c^2 \frac{\partial^2U}{\partial x^2}$$
discretizing it gives:
$$\frac{U(i+1,j)- 2U(i,j) + U(i-1,j)}{(\Delta t)^2} = c^2 \...
- 153
5
votes
2
answers
2k
views
Solution oscillations with a small timestep in backward Euler
I am using backward Euler in a FEM scheme for a convection-diffusion problem. On a given mesh, I can take arbitrarily large time steps, as expected. But if I decrease time step, at some point it will ...
- 186
5
votes
1
answer
161
views
Analysis of nonlinear finite element methods
I have been doing a lot of reading on the development of finite element methods and their analysis using, e.g., functional analysis. I am clear on the formulation of the weak form of a PDE and ...
- 63
5
votes
1
answer
135
views
Approximate $\|\Delta f\|^2_{L^2(\Omega)}$ in finite element context
I have minimization problem of the form
$$
G(f) + \|\Delta f\|^2_{L^2(\Omega)} \to \min
$$
over all $f\in C^2(\Omega)$, $\Omega$ being closed and bounded.
Let us forgot about $G$; I'm interested in ...
- 3,146
5
votes
1
answer
910
views
Dirichlet boundary conditions in generalized eigenvalue problem
Let us consider a problem of the form
$$(\mathcal{L} + k^2) u(\mathbf{x})=0\, ,\quad \forall \mathbf{x} \in \Omega$$
with Dirichlet boundary conditions
$$u(\mathbf{x}) = 0, \quad \forall \mathbf{x} ...
- 8,622
5
votes
2
answers
670
views
Boundary condtions on nonlinear FEM time integration
I'm using the finite element method to obtain the time response of a structure harmonically excited. I'm using a linear displacement function to obtain the stiffness matrix and the consistent mass ...
- 231
5
votes
1
answer
243
views
Using finite element error estimators for adaptive mesh refinement
I am in the process of implementing adaptive mesh refinement for a finite element code that solves the Poisson equation. I have had some trouble finding good references on deciding which elements to ...
- 1,929
5
votes
2
answers
572
views
What does symmetrize mean? (imposing multifreedom constraints to stiffness matrix)
I have a small FEM implementation program. And I want to add imposing multifreedom constraints (MFC) feature to it. The theory of master-slave method is given here (page 10 for general case).
...
- 501
5
votes
1
answer
107
views
Prediction of sphere (i.e. roast) core temperature heated in an oven
The real-life problem
Assume I put a spherical roast with initially constant temperature of start_temp=25 (°C) into an oven with ...
- 161
5
votes
1
answer
147
views
Non-conforming bi-linear finite element
The four-noded bi-linear rectangle element, which sometimes goes under the name Melosh element, is non-conforming unless the element sides are aligned.
Out of curiosity I have implemented this element ...
- 188
5
votes
1
answer
168
views
Effect of subdomain topologies on overlapping additive Schwarz?
Is there a reference on the effect of subdomain topology on performance of the overlapping additive Schwarz method for (high order) finite elements? For example, taking subdomains to be vertex ...
- 3,162
5
votes
0
answers
94
views
How to get an "optimal point" for refinement in FEM adaptive mesh refinement?
Consider the following 1D problem
\begin{align*}
\begin{cases}
\displaystyle
-\frac{d^2u}{dx^2} = f(x), \hspace{0.5cm} x\in (a,b) \\[4mm]
u(a) = u_{a}, \ \ u(b) = u_{b}
\end{cases}
\end{align*}
I ...
- 51
5
votes
0
answers
161
views
About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation
I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$:
$$A u + B^t p = f$$
$$Bu = g$$
where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
- 435