Questions tagged [finite-element]
A means of solving ordinary and partial differential equations. The domain of the problem is broken up into elements, and the solution in each element is expanded in a basis of functions. The Finite Element Method lends itself well to adaptive refinement, irregular geometry, and good error estimates.
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Which preconditioners (and solver) in PETSc for indefinite symmetric systems should I use?
My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow):
\begin{pmatrix}A & B^T \\ B & C\end{pmatrix}
where $C = 0$ is the typical case (I have even ...
12
votes
1
answer
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How to integrate polynomial expression over 3D 4-node element?
I want to integrate a polynomial expression over a 4-node element in 3D. Several books on FEA cover the case where integrating is performed over an arbitrary flat 4-noned element.
The usual procedure ...
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Space-time finite element discretization for time-dependent PDEs
In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, ...
11
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1
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Raviart-Thomas elements on reference square
I'd like to learn how the Raviart-Thomas (RT) element works. To that end I'd like to analytically describe how the basis functions look on the reference square. The goal here is not to implement it ...
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$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)
I know that the piecewise linear finite element approximation $u_h$ of
$$
\Delta u(x)=f(x)\quad\text{in }U\\
u(x)=0\quad\text{on }\partial U
$$
satisfies
$$
\|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)}
$...
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Are 8 Gauss points required for second order hexahedral finite elements?
Is it possible to get second order accuracy for hexahedral finite elements with fewer than 8 Gauss points without introducing unphysical modes? A single central Gauss point introduces an unphysical ...
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What is the difference between implicit FEM and explicit FEM?
What is the difference between explicit FEM and implicit FEM exactly? According to the post here, it seems that the only difference is whether implicit or explicit time integration is used.
As I ...
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FEM for vector valued problems: reference request
I'm a PhD working in a computational mechanics lab. I come from a Math department, and I have a good background for what concerns the basics of finite elements, like inf-sup conditions, DG, non-...
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Resources on mesh generation for finite element methods
I know that this is not really apart of the rules as this is a recommendation question and these don't really have an answer per say. But, like this forum posting: https://stackoverflow.com/questions/...
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What numerical quadrature to choose to integrate a function with singularities?
For example, I would like to numerically compute the $L^2$-norm of $\displaystyle u = \frac{1}{(x^2+y^2+z^2)^{1/3}}$ in some domain that includes zero, I tried Gauss quadrature and it fails, it is ...
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Who uses finite elements with higher continuity?
Lagrange elements of any polynomial describe piecewise continuous functions. Typically, those functions are differentiable.
Mixed finite element methods use vector fields of even less continuity, such ...
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Is discontinuous Galerkin really any more parallelizable than continuous Galerkin?
I've always heard that easy parallelization was one of the advantages of DG methods, but I don't really see why any of those reasons don't also apply to continuous Galerkin.
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Stabilization of convection-dominated flow and turbulence modeling
Are stabilization techniques for convection-dominated flows like SUPG+PSPG, interior penalty methods, etc. able to handle turbulent flows without tubulence model being employed, at least up to some ...
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Finite element convergence rates for mixed problems
I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh.
...
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implementing higher order derivatives for finite element
I am implementing higher order derivatives for FEM. Example, to solve a Poisson problem, biharmonic or triharmonic PDE one needs first, second or third order derivatives respectively.
As usually ...
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3
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Mixed Finite Element Method for the Stokes System—Some Implementation Details
I am currently working on my bachelor’s diploma. The research concerns mixed finite element method for the 2D Stokes system
$$
- \Delta \boldsymbol u + \nabla p = \boldsymbol f, \quad \boldsymbol x \...
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1
answer
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Why are functional representations of systems important in numerical applications?
I tried asking a similar question in SE.Physics, and I got some information regarding the abstract side of this, but I figured I should post here to get more complete information about the numerical ...
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Which preconditioning for large linear elasticity problem?
The problem I want to solve is the displacement formulation of the linear elasticity :
$$
\nabla \cdot \sigma = 0 \quad \text{in} \quad \Omega \\
\sigma = \lambda ( \nabla \cdot u ) I + \mu (\nabla \...
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1
answer
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Appropriate space for weak solutions to an elliptical pde with mixed inhomogeneous boundary conditions
I'm working with the following mixed inhomogeneous boundary value problem:
$\nabla(\kappa\nabla u)=f$ in $\Omega$
with $\partial\Omega = \Omega_1 \bigcup\Omega_2$ such that
$u=g$ on $\partial\...
6
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2
answers
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Implementing the pressure correction method using finite elements
Ok so I am nearing the completion of my finite element Navier-Stokes solver that uses the $\theta$-method for time stepping and the pressure correction method for the pressure. I am following the ...
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1
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Find the direction of the gradient on a finite element mesh
Suppose we have a triangular mesh of a two dimensional shape $\Omega$, and on this mesh we define a P1 finite element structure. I know that given $u,v$ by their values at the vertices of the ...
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Do practice and theory differ substantially when implementing Neumann Boundary Conditions using a Mixed Method?
I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both ...
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1
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How are rigid bodies implemented in finite element codes
I am writing a finite element code for structural analysis, and I want to implement rigid bodies. How is this usually done? Say that I have a square mesh, with one half of the mesh being defined rigid ...
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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 ...
5
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2
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Solving Poisson equation with current BC using FEM
I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$
Insulator (Neumann BC)
Electrode set at some ...
5
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1
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Mixed boundary conditions Finite Element Method
I have the following problem in Finite Element Method
$$ -(\alpha u')' + \beta u' + \gamma u = f$$
with $ \Omega = (0, 1)$, $ u(0) = 0 $ and $ u'(1) = 3 $
to be able to write the weak formulation ...
5
votes
1
answer
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Dirichlet boundary conditions in generalized eigenvalue problem
Let us consider a problem of the form
$$(\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} ...
5
votes
2
answers
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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\...
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1
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What are all these functions in FEM? Shape function vs Basis Function vs Trial Function vs Test Function vs Interpolation Function
I am a novice in FEM. I have some experience with FDM which was pretty straight forward. Since I have a confusion with a number of concepts, I will try to break them down by writing down what I have ...
5
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1
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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_{...
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Do I need to impose boundary conditions in the Jacobian matrix?
In the framework of Finite Element Method, when the Newton method is used, we solve $J(x^k) \delta x = -f(x^k)$, and the increment $\delta x$ would not change some entries from $x^k$ related to ...
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Introductions to hp-FEM
do you know good introductions into or surveys $hp$-adaptive finite elements?
In particular I do not know how the heuristics for choosing spatial refinement or increased polynomial degree are ...
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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,...
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$L^2$-error in FEM: how to compute integral over reference element?
I have the following problem. The domain is $(0,1)$ and we consider a uniform triangulation on $\hat{\Omega}$ with elements $K_i = [i/N,(i+1)/N]$ and $X_h^1$ the linear finite element space. I wrote ...
4
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1
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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 ...
4
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answers
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Finite Difference and Finite Volume as special cases of Finite Element
I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE".
However I have never come across a brief document that substantiates that claim, ...
4
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2
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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
$$
Let us solve this using simple linear finite elements. ...
4
votes
0
answers
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Extrapolation after successive finite element refinement
I wish to compute a certain function $\lambda$ (in my case an eigenvalue of the Laplacian) to a certain accuracy, which I wish to guarantee, if possible. I started with a finite element method. I know ...
4
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1
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Meaning of CFL condition on parabolic problems
I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method.
I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
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2
answers
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Efficently invert tiny matrix in Fortran
I have a piece of code in Fortran90 in which I have to solve both a non-linear (with the Newton-Raphson method, for which I have to invert the Jacobian matrix) and a linear system of equations. When I ...
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1D FEM for nonlinear diffusion coefficient
I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$
in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$.
...
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FEM shape functions on triangular elements: transition from 2D to 3D
I'm writing a code for solving PDEs through the finite element method. In particular, I'm facing with 3D problems, in which I don't know how to calculate shape functions derivatives on the boundaries (...
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1
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Using Gmsh to create a mesh with zero thickness (quad) interface elements
I acknowledge the following post where a similar question is posed and a very nice answer has been provided:
Is there a mesh generator that will generate zero thickness elements for interfaces?
...
4
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1
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Finite element error for second order ODE at nodes equal to zero
I coded a finite element method with linear basis elements for the problem $$-u'' = f(x), x\in[0,1], u(0) = u(1) = 0$$ The nodes are uniformly spaced and I will denote them as $x_i$. I initially ...
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answer
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Bubnov-Galerkin method in 1D: how to handle convective-type nonlinearity?
Consider the BVP: find $u = u(x)$, for $x \in (0,1)$ that satisfies
\begin{align}
u'' + u u' = f, \\
u'(0) = g_n, u(1) = g_d.
\end{align}
To derive the weak form for this BVP, we multiply the first ...
3
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0
answers
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numerical instabilities in Fluid Dynamics, Finite Element Method
I'm looking for references to understand where the numerical instabilities come from in hydrodynamics in general, and notably when the Péclet number: $Pe>1$. I'm using the finite element method.
...
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$H^1$-convergence rate of finite element method for Poisson equation, depending on element order
I wanted to verify my FEM-program by applying the method of manufactured solutions, while solving the Poisson equation in two dimensions using the continuous Galerkin method
$$-\nabla^2u=f$$
with
$$u=...
3
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3
answers
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What numerical methods are used to model deformations in elastic physics?
What numerical methods are used to model deformations in elastic physics? For example, here's an example of a hyperelastic deformation in Ansys:
Perhaps more simply than hyperelasticity, for linear ...
3
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2
answers
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Does length unit in FEM affect numerical condition?
I try to solve a system of coupled PDEs using FEM. Unfortunately, the originating matrix has very poor condition. After days of double checking and thinking, I suspect the following reason:
Given a ...
3
votes
1
answer
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Jump condition for elliptic equation in standard finite element method
Consider the elliptic PDE
$ -\mathrm{div}(k(x)\nabla u) = f(x) \in \Omega$
and
$u = 0$ on $\partial \Omega$
If $k(x)$ is piecewise constant across an interface $\Gamma$ in the domain, then we ...