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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questions with no upvoted or accepted answers
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Sequential approach to solving coupled PDEs
I'm dealing with a coupled system of three transient, non-linear convection-diffusion equations. Let's just say to simplify the problem that they take the following form:
$$
-\nabla\cdot(D_{1}(u_{2},...
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Simple turbulence model appropriate for buoyancy-driven cavity like problem
Which turbulence model is suitable for resolving incompressible buoyancy-driven flow of a fluid within an cylindrical ampoule?
I prefer turbulence model which is sufficiently simple so that fully ...
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What can be done with Finite Element Method and not with the Finite Volume Method, and vice versa?
What are some applications where you would absolutely go for either FEM, but not FVM, or vice versa? What are some applications where both methods are equally suited?
I worked with the FEM so far and ...
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votes
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Numerical implementation of the Dirichlet-to-Neumann map
I am solving the Dirichlet problem
$$
\begin{cases}
\Delta u = 0, \\
u|_{\partial D} = f,
\end{cases}
$$
in a $2d$ domain $D$ using the finite element method. What I want to get is the ...
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answers
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Matrix-free FEM references
I've seen that many people are using matrix-free fem codes in my community (mechanical engineering). I have to admit that I googled a bit and I didn't manage to find a good reference for the subject. ...
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How to check if my stiffness matrix is correct
I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "...
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votes
0
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FEM : energy minimization VS PDE solving
Engineering FEM
When I studied engineering, I learned the traditional approach for finite elements for elasticity. The point was to solve the PDE $-div(\sigma)=f$ as:
Multiply your PDE with a test ...
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0
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$L^2$ norm error estimates of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
- 61
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votes
0
answers
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Understanding Boundary Condition in FEM
I am trying to understand Dirichlet and Neumann boundary conditions in FEM and I wanted to know if my inference is correct. To articulate my understanding, lets consider a simple case of TE and TM ...
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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 ...
- 1,000
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Will penalty-augmented stiffness matrix cause numerical issues in eigenvalue analysis?
In the finite element method, we often construct the constraints of the system by adding penalty-function terms ( which often are many many magnitudes, up to $10^6$ order bigger than the largest ...
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Boundary conditions in the Finite Element Method at only one side of computational domain
I want to solve a Sturm-Liouville problem in 1D, i.e.,
\begin{align}
[p(x)\ u'(x)]'+q(x)u(x) = f(x)
\end{align}
with boundary conditions
\begin{align}
u(0)=a \hspace{1cm} u'(0)=b
\end{align}
How do I ...
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votes
0
answers
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views
How to do FEM in sector elements?
Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there
are a lot of ...
- 1,319
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votes
0
answers
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views
References on the theory of Petrov-Galerkin methods for more "basic" problems
In my reading on various aspects of FEM, Petrov-Galerkin methods often arise in the study of solutions of convection-dominated systems, such as Hughes' work on Navier-Stokes, or systems where optimal ...
- 1,832
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votes
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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 ...
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0
answers
153
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 $A: V \rightarrow V', B:V \rightarrow Q'$ ...
- 241
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votes
0
answers
286
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 ...
- 1,147
5
votes
0
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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 $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
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votes
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answers
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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 $...
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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
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votes
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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 ...
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5
votes
0
answers
245
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 $x_*$ ...
- 491
5
votes
0
answers
254
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:
$$...
- 51
4
votes
0
answers
220
views
Spurious oscillations in solving diffusion problems using finite elements
I've been struggling with this problem for a while so I hope someone can help me here.
I'm trying to solve the McNabb-Foster equations for hydrogen diffusion in metals using a simple 1D finite element ...
- 41
4
votes
0
answers
122
views
Quadrature rules for non-linear finite element problems
For solving linear problems stemming from PDEs with the FEM, such as the Poisson equation or the wave equation, it is customary to use the "simplest" numerical quadrature that exactly ...
4
votes
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answers
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views
Global reconstruction defined elementwise in a-posteriori error estimator
This question is a follow-up of this previous one. In "Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems" by Georgoulis et al., an error estimator is ...
- 241
4
votes
0
answers
641
views
Construction of Prolongation and Restriction Operator for Geometric Multigrid (2D-FEM): Resulting in a Decreasing Solution
Consider the following problem,
$$ -\Delta u(x) = f(x), \qquad x \in \Omega \\ u(x) = 0,\qquad x \in \partial \Omega$$
with $\Omega = [0,1]\times [0,1]$ being the domain and $\partial \Omega$ being ...
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votes
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answers
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views
How to apply an integrated constrain condition in FEM?
I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below:
$$\frac{\...
4
votes
0
answers
90
views
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 ...
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votes
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answers
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views
Once and for all: Which FEM plattform should I use for a very large multiphysics simulation?
I'm fooling around with the decision on how to build a multiphysics simulation for too long now (also several questions in this forum). First, I thought it would be possible/necessary to write most of ...
- 1,463
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votes
0
answers
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views
Galerkin FEM error when using even number of elements
Intro:
I am developing a Galerkin FEM code in matlab - starting small with a simple 1D ODE. The equation I'm trying to solve is: $$a u_x = cos(x), x \in [0, 2\pi]$$ Which has a known exact solution $$...
- 674
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votes
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answers
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views
Implementation of the roller constraint
What could be the best way to implement the roller constraint in finite element code, i.e. constraint of the type
$$\mathbf{u} \cdot \mathbf{n} = 0$$
I plan to enforce it in the weak sense by ...
- 229
3
votes
0
answers
80
views
Is there any book about fundamental FEM theory using similar terminology as Comsol MultiPhysics?
I am considering to solve complexed PDE systems, like in this post, using Comsol MultiPhysics. The PDEs are different from the General Form provided by Comsol. The Weak Form module may be worth a ...
- 31
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votes
0
answers
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views
Is there an obvious n-dimensional elasticity complex?
Finite element exterior calculus helps to furnish nice stable elements for many common elliptic PDE by looking at the de Rham complex
$$\Lambda^0(\Omega) \stackrel{d_1}{\to} \Lambda^1(\Omega) \...
- 9,813
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votes
1
answer
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views
Distributed Lagrange multiplier approach to impose constraint in Poisson equation
I'm trying to understand how Lagrange multipliers are applied in order to impose constraints in PDEs. Consider $B \subset \Omega$. For instance, a square inside another square domain $\Omega$. Let's ...
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votes
0
answers
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views
Numerical calculation of out-of-time order correlators (OTOCs)
I want to numerically calculate OTOCs for different quantum mechanical systems. Consider the following Hamiltonian
$$H=p_x^2+p_y^2+x^2y^2$$
and I want to calculate the following OTOC
$$C_T(t)=-\left&...
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3
votes
0
answers
72
views
Typo in a-priori error estimate in a Discontinuous Galerkin paper
I'm looking at this famous paper which is available in the link below:
Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
- 241
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votes
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How to treat nonlinear radiation term in heat equation using Finite-element method?
I am trying to solve the time-dependant non-linear heat equation with radiation. This equation is coupled to the radiative transfer equation but for the purpose of my question, this does not matter ...
- 31
3
votes
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answers
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Hanging nodes in deal.ii tutorials: how is the continuity constraint imposed?
While looking at step6 of deal.ii tutorials, I decided to try to understand how the constraints coming from hanging nodes are imposed. So I started by watching video lecture 16 by prof. Bangerth
As ...
- 241
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votes
0
answers
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Jacobian Matrix of 2D element mapped to 3D
Note: I previously posted this question to MathStackExchange, but got no attention there. So I'm rewritting and trying over here.
Problem summary
Given a common¹ set of shape functions defined at ...
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votes
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answers
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views
What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method?
What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method? The best reference I've found thus far as been a paper titled The Performance of the ...
- 747
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votes
0
answers
66
views
Some formulations of domain coupling lead to saddle point problems. Is this merely an artifact?
Background I wanted to learn how to couple FEM and BEM (for the Poisson equation), because I wanted to better understand how open boundary conditions look like. Therefore I worked through the relevant ...
- 2,141
3
votes
0
answers
127
views
How to construct a Fortin Operator for Crouzeix-Raviart Element?
I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element.
The continuous Stokes Equation in the weak formulation is
Find $u \in H_0^1(\Omega, \mathbb{R}^...
- 459
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votes
0
answers
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Correct approach for thermal finite element simulation of layered assembly
I would like to optimise the heat transfer on a PCB. Several dies are on the top and cooling air is going through the fins in heat sink on the bottom. The assembly consists of several layers like ...
- 560
3
votes
0
answers
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views
Solving saddle point problem having non-invertible top-left block with a PETSc nested matrix
My system is a symmetric FE problem with lagrange multipliers:
$Z=\begin{pmatrix}A & C^T \\ C & 0\end{pmatrix}$
The matrix $A$ is positive semi-definite, non-invertible. The whole matrix is ...
- 151
3
votes
0
answers
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views
What is appropriate boundary condition for Poisson pressure equation?
I'm doing CFD simulations in unstructured grids. Well, it's a bit different from conventional unstructured grids that are used mainly in FEM or FVM as tetrahedral meshes. Mine is a voxelized mesh of ...
- 2,322
3
votes
0
answers
105
views
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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3
votes
0
answers
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views
Strain propagation from surface to bulk in COMSOL
I am trying to simulate strain propagation from the surface into the bulk. I have a rectangular semiconductor block (~2 μm thick) on top of which metal gates (~25 nm thick) are deposited as seen in ...
- 31
3
votes
0
answers
110
views
Numerical solution to N-dimensional diffusion on simplex?
Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
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votes
0
answers
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Structural Analysis Library
Can anyone recommend a structural analysis library that satisfies the following requirements:
C++ API
Simulate both beam elements and shell (slab) elements
Both static and dynamic analysis
Free and/...
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