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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18 views

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 ${\vec{r_1},...,\vec{r_4}}$ 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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56 views

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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80 views

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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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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284 views

Global to local transformation matrix in 2D frame structures

In section 3.2 of this paper [1], where 2D planar frame structures are being analyzed, the authors mentioned a transformation matrix to be used in extracting the element displacement vector from the ...
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1answer
91 views

Determining the voxels between two boundary surfaces

Issue description I am working on human brain tACS simulations where I have the models of the skin, skull, csf, brain and ventricles in STL format. The shape does not matter and there are no ...
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119 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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187 views

Implementation of Z^2 error estimator in Abaqus for adaptive mesh refinement

Currently, I am working on a remeshing routine for my simulations (Abaqus 6.14-1) using python scripts. The simulation deals with the Brinell indentation test and as the remeshing software I use Gmsh ...
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2answers
146 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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105 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 ...
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1answer
106 views

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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66 views

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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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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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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113 views

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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Is this estimated time accurate with the type of work I'm wanting to deliver in LS-Dyna?

I'm a 3d model designer and I wanted to use the 3dsplot to export it to blender and then make use of the simulation for rendering a beautiful video in BLENDER, in order to achieve that I learned a ...
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1answer
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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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518 views

How to simulate thermal expansion in a 2D plane using FEA?

I am trying to model 2D thermal expansion of a square area inside another square using FEATool. I have simulated plane strain by incorporating forces pointing along the $[1 \,\,\, -1]^T$ direction ...
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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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Evaluating the surface integral in an FEM (Finite Elements Method) procedure

I want to evaluate the Surface force integral in an FEM procedure. The basic reference tet is shown in the figure. The faces are numbered corresponding to the node opposite to them. For example the ...
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106 views

Adaptive Lagrangian-Eulerian methods and practical benchmark results

Does anyone know of any published study that talks about the practical aspects of running Adaptive Lagrangian-Eulerian techniques for solid and/or fluid mechanics problems? I'm looking for things ...
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1answer
114 views

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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2answers
87 views

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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1answer
104 views

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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130 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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2answers
168 views

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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3answers
582 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 (...
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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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1answer
318 views

Lower bound for bilinear form in FEM

I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries. I did some research and found: $$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(...
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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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568 views

Difference between MoM and FEM

Method of Moments and Finite Element Methods are two of the most used methods in computational electromagnetics to solve electromagnetic equations. As it is known, in FEM sparse matrixes are used ...
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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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81 views

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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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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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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148 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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1answer
483 views

DG local equation, how to interpret mean-averaged test function

In the paper http://www.sciencedirect.com/science/article/pii/S0045782509003521, an HDG element-local equation is described on page 584 equation (4), with one of the equations taking the following ...
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Discrete divergence free functions

I'm studying the weak formulation of NS equations. During the analysis, the book I'm using (Quarteroni-Valli, page 301-302), defined $$Z_h=\{v_h \in V_h: (\operatorname{div}(v_h),q_h)=0 \quad \forall ...
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1answer
65 views

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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1answer
134 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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1answer
96 views

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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3answers
143 views

preconditioner for Laplace "without" boundary values

I'm looking at solving systems with the FEM discretization $$ -\int_\Omega (\Delta u) v = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot\nabla u) v. $$ without applying Dirichlet- or ...
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1answer
162 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^...
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343 views

Proving solution existence and uniqueness of the Helmholtz equation with Robin boundary conditions with complex coefficients

I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation $$ \iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)...
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1answer
160 views

Time discretization Navier Stokes equation

This question is a follow-up of this one. The weak form of Navier Stokes equation is (assuming $v,q$ test functions for the velocity and the pressure, respectively) $$(\frac{du}{dt},v)_{\Omega} + (\...
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1answer
368 views

Global stiffness matrix from element stiffness matrices for a thin rectangular plate (Kirchhoff plate)

I have the element stiffness matrix for a thin "kirchhoff" plate. The plate is 3 [m] x 5 [m] and is simply supported on all edges. It's thickness is 0,2 [m]. On the plate there acts a ...

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