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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Contact analysis does not converge due to the projection falls outside valid domain

I implemented Node-To-Surface contact algorithm (Wriggers, Peter, Computational contact mechanics., Berlin: Springer (ISBN 3-540-32608-1/hbk). xii, 518 p. (2006). ZBL1104.74002.). The code is done by ...
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Test functions of Raviart-Thomas elements?

The test functions of general finite elements are like interpolation functions (if my understanding is correct). But how about test functions of Raviart-Thomas elements? Let's raise the $RT0$ element ...
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6 votes
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Is their a book/paper about mixed finite element method for engineering students (non-math)?

There are a lot of books about FEM, which are really friendly to engineering students. Through these books, we can know how to use shape/test functions based on the variational principle. But I'd like ...
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Numerical methods for Vlasov's equation

Vlasov equation is pretty straightforward It would be easy to solve with Fem packages like firedrake, but in my case I have 6d distribution function: it depends on 3d vector of spatial coordinates ...
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161 views

Nitsche's method for imposition of Dirichlet boundary conditions: implementation standpoint

I'm trying to understand how Nitsche's method works in practice. I understood the theoretical principle behind it, but what I can't understand is its implementation. More precisely, I'd like to solve ...
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3 votes
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How to enforce fluid and solid dynamic coupling in fluid-structure interactions using the finite element method?

I apologize in advance if the question has been posted before or if it sounds a bit naive. I am writing my own code in MATLAB for a staggered finite element solver for fluid-structure interaction ...
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How to apply Neumann boundary conditions in Newton's method [closed]

Suppose that I have a very long and tedious set of differential equations. After discretization, I can get a mapping $f:\mathbb{R}^N \to \mathbb{R}^N$ such that solutions $\phi =(\phi_0,...,\phi_{N-1})...
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2D axisymmetric Magnetostatics Finite Element Method solver

I am new to the field of magnetostatics. I wish to write a 2D finite element code for obtaining the magnetic field inside a solenoid coil. I have started with a 2D code and have followed the method as ...
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3 votes
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Instability at the boundary of a finite difference simulation of a hyperbolic PDE

I want to simulate the hyperbolic partial differential equation $$W_{tt} + V W_{tx} + k_E V W_x + k W_t = 0,$$ but I am having trouble finding a discrete analog of this equation which is numerically ...
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Fourier integral over elements

Suppose I have a triangular element with vertices ${\vec{r_1},...,\vec{r_3}}$ and a function $f(\vec{r})$. I want to calculate the fourier integral over this triangle such that: $$F(k_x,k_y)=\int \int ...
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Discontinuous pressure elements for incompressible Navier-Stokes

I am looking for some LBB-stable velocity-pressure combinations for incompressible Navier-Stokes where the pressure space is element-wise discontinuous, preferably with a linear variation elementwise. ...
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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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Roundoff errors in FEM computations - generalized eigenvalues

This is a continuation of my previous question. I am trying to effectively compute a bound for the roundoff errors in some FEM computation (2d polygons, triangular meshes). Below I will write some of ...
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2 votes
2 answers
267 views

Different sources of error in Finite Element computations

Consider the problem $-\Delta u = f$ in $\Omega$, with $u=0$ on $\partial \Omega$. Suppose that $\Omega$ is a polygon and that we approximate the solutions to the previous problem using Lagrange ...
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1 vote
2 answers
236 views

Why aren't face integrals for an element calculated in FEM but they show up in FVM?

Consider the Laplace problem: \begin{align} -\nabla^2 u = f \qquad \text{in } \Omega \\ u = 0 \qquad \text{on } \Gamma \end{align} The weak problem is find $u_h \in V \subset H^1$ such that $\...
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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 ...
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PETSc-like library for Julia

I want to build an application for Material Point Method (and probably other meshfree methods too) in Julia and I am looking for library for direct and iterative solvers that can help me with it. One ...
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1 answer
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Lagrange multiplier for boundary conditions in pure Neumann problem

I'm trying to solve $-u''=\cos(2 \pi x)$ with boundary conditions $u'(0)=u'(1)=0$ and the constraint $\int_{0}^1 u = 0$ I have to use linear finite elements, so let's assume that I have $M$ degrees of ...
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3 votes
1 answer
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How to measure the error of Finite Element approximation in satisfying the PDE?

In Galerkin methods, we seldom can measure the accuracy of an approximation by tracking the value of the residual. For example, take the wave equation: $ u_{tt} = u_{xx}$, $(x,t) \in (0,L) \times (0,T)...
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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 ...
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Finite element modelling of thermal expansion of 3D solid bodies

I want to solve the thermal expansion of solid by using FEM approach. When I developed the model based on the the principle the minimum potential energy, the solutions for thermal expansion are not ...
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Varying Young modulus in FEM simulation

I'm working on a project for which I have inherited some FEM code. This implemented FEM calculates, given some force field, displacements on a discretised square using square elements and assumes a ...
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1 vote
2 answers
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What is known about C0 triangular finite elements with nonstandard mesh point placement?

I'm curious about the general case, but for ease of explaining lets just take the case of a $P^2(\Omega)$ approximation. For simplicity, let's also just consider the reference element $(0,0), (0,1), (...
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2 votes
2 answers
143 views

How is FEM used in structural engineering?

I have learned about the finite element method (FEM) as a method for solving boundary problems given by a PDE. The way I learned it is to approximate the solution by a linear combination of test ...
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2d advection-diffusion: cell Péclet number and numerical stability

I am studying the numerical resolution of 2d advection-diffusion problems with finite element methods. $$\frac{\partial u}{\partial t} + \beta\cdot\nabla u = \nabla\cdot(\nabla u) \, .$$ It is said in ...
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Area of 8-node rectangular serendipity finite element

I am trying to compute the area of an 8-node rectangular serendipity finite element from the equation $$ \sum_{i= 1}^8 det \, J(\xi,\eta) \cdot W_i $$ based on Gaussian quadrature, where $$ J(\xi,\eta)...
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1 answer
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Question about step in the proof of standard discrete trace inequality

I'm studying from Guermond lecture notes available at https://www.math.tamu.edu/~guermond/M661_FALL_2019/chap12.pdf (see Lemma 12.8( Discrete trace inequality).) Consider the simple case $p=r$, i.e. ...
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2 votes
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131 views

Variational loss of hp-Variational Physics Informed Neural Networks for 2D-Poisson Equation in Tensorflow

I am trying to reproduce the results from the hp-VPINN paper (https://arxiv.org/pdf/2003.05385.pdf) on tensorflow (v1) for Poisson's equation, particularly the two-dimensional Poisson equation. In one ...
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Weird "oscillatory" modes appearing in FEM simulations

I am using COMSOL to solve a mathematical model involving thermoelectric hydrodynamic (TEMHD) flow. I am running a very large parameter sweep and using the solutions obtained to make some plots. ...
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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 ...
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2 votes
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Bounds for the optimal bandwidth of 2D/3D FEM stiffness matrices

is anyone here aware of whether there exist bounds on the optimal bandwidths of 2D/3D FEM stiffness matrices? Edit: more specifically, I would like to have bounds on the minimum bandwidth after ...
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Basis function in a tetraedron for finite elements contex

In the finite element method we need to know a base for the fem spaces. For example, a base for the space $P_1(\hat{K})=<\{1-x-y,x,y,z\}>$ is a typical base for the polynomials of degree less ...
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How is the integral of a projection over an element $T$ computed in practice? (deal.II related)

I'm studying an error estimator for the equation $\nabla\cdot(\beta u) + cu = f$ and it contains the following term $$||f - cU_h - \Pi(f-c U_h) ||_T$$ where : $\Pi$ is the local orthogonal $L^2$ ...
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2 votes
2 answers
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Single hexahedral element stiffness matrix problem, help me find the mistake

Attached below is some code I wrote to solve a basic problem: finding the node displacements of a cube with two vertices constrained (vertices 6 and 7 with coordinates ...
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SIPG method for $-\nabla \cdot (\nu \nabla u)=f$

Consider the diffusion equation with a coefficient $\nu$: $$-\nabla \cdot (\nu \nabla u)=f$$ with Dirichlet boundary conditions $u = g_D$ in $\partial \Omega$. If the coefficient would be constant, ...
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1 vote
2 answers
162 views

Software and tutorial for FEM

i'm looking for some advice for finite element analisys. i'm a biomedical engineering student with few knowledge about the FEM. Tools like Comsol and Ansys are very powerfull but also complex and i ...
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Use of 7-node rectangular serendipity element

I am would like to use 8-node quadrilateral serendipity elements to model a problem. However it seems to me that mid-nodes are not required at the boundary, as shown in the diagram below i.e. elements ...
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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 ...
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1 vote
1 answer
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Finite element method with two different Dirichlet boundary conditions

I have the problem like this $$ -\triangle u = f \ \ on\ \Omega \\ u = g_1 \ \ on \ \partial \Omega_1 \\ u = g_2 \ \ on \ \partial \Omega_2 $$ If we choose $$ V_1 = \{ \nu_1 \in H^1 : \nu_1 = 0 \ ...
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Discontinuous Galerkin order of convergence on arbirary refined mesh: step-12 deal.ii tutorial

I'm learning DG methods and in order to practice a little bit I'm using the deal.ii library. In particular, I'm looking at step-12, where they solve $$\operatorname{div}(\beta u) = 0$$ $$u = g_D \text{...
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4 votes
1 answer
157 views

Discontinuous Galerkin: confusion about the weak formulation for linear advection equation

In an introduction to Discontinuous galerkin methods, I have some problems in checking the weak formulation. I'm looking at page 16 here The context is the advection reaction equation: $$\operatorname{...
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1 vote
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How to determine the orientation of convex/concave hexahedra?

I am writing a code that checks the orientation of a list of vertices (along with face connectivity) describing both convex and concave hexahedra. The face connectivity table stores the list of vertex ...
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1 answer
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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^...
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1 answer
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Material properties for a node in a 2-material FEM code

I'm trying to debug an FEM that I inherited, and I unfortunately do not have much knowledge of FEM. I only know FD and FVM. If you're modeling a system with 2 materials, there will be an interface ...
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1 answer
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How to Calculate magnetic and electric field in 2D Magnetotelluric using Edge based Finite Element

I calculate 2D Model of Magnetotelluric responses which are apparent resistivity and phase. I do the calculation for Transverse Electric (TE) mode. Then I used edge based finite element with ...
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1 vote
1 answer
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Locking phenomena for $P1 - P0$ elements

Consider the Stokes problem and the usual divergence operator $B:V \rightarrow Q'$, $\langle Bv, q\rangle = b(v,q)=(\operatorname{div} v,q)$ and its discrete versione $B_h : V_h \rightarrow Q_h'$. In ...
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5 votes
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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 $A: V \rightarrow V', B:V \rightarrow Q'$ ...
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1 answer
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Finite element method for Stokes and Navier--Stokes with square elements only

I wanted to learn how to implement a code for the Stokes and Navier--Stokes equations 2D/3D. I already know how to implement it when the elements are triangles or tetrahedral. Do exists finite element ...
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2 votes
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Understanding inf-sup conditions for classical saddle point problems

I'm studying the inf-sup conditions for saddle point problems. I'm referring to the usual one $$\begin{cases}Au + B^t p = f \\Bu=g \end{cases}$$ In the book I'm using (Ern - Guermond: Theory and ...
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3 votes
1 answer
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Proof of R. Verfürth paper on adaptive mesh and bubble functions

I'm studying adaptive meshes, and my professor wrote the following property for a bubble function ( see this scicomp post for the definition I'm using)$b_T$ defined on a triangle $T$. $$||b_T \phi ||_{...
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