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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2answers
123 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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90 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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1answer
98 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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81 views

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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1answer
64 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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1answer
111 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
112 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 ...
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1answer
53 views

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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67 views

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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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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2answers
128 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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81 views

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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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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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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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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92 views

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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1answer
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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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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1answer
86 views

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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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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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
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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56 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 ...
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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
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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1answer
133 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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52 views

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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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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1answer
133 views

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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1answer
62 views

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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1answer
106 views

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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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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1answer
78 views

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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74 views

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 ...
3
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1answer
130 views

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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1answer
90 views

Classical global estimate for $H^1$ error

I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$. $$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
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2answers
165 views

Sum of norms over elements is not equal to norm over the whole $\Omega$

In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$: $$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
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1answer
175 views

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
73 views

MPI vs OPENMP usage for boundary element method

I have an implementation question with MPI and OPENMP. I have a large three-dimensional surface mesh divided into elements. I want to loop over each element (outer loop) and compute an integral over ...
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1answer
61 views

Displacement/Pressure Finite Element formulation for Large Deformations (from Bathe)

On page 563 of Finite Element Procedures by Klaus-Jürgen Bathe the author states that governing equations of the displacement/pressure finite element formulation for large deformations is given as $$ \...
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1answer
159 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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2answers
166 views

What is the difference between $u_h$ and $I_h(u)$ in finite element literature?

In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
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0answers
119 views

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 ...
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1answer
23 views

How to decrease error in (FTCS) forward time centered space method?

I am using the FTCS method for solving differential equations. I know that the condition for stable output is $$ \frac{\alpha \Delta t}{\Delta x ^2} < \frac{1}{2} $$ But when I use the distance ...
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0answers
92 views

Confusion about preconditioner for incompressible Navier-Stokes equation with implicit-explicit method

Consider the time-dependent Navier-Stokes equation $$u_t + (u \cdot \nabla) u - \Delta u + \nabla p = f$$ $$\operatorname{div}(u)=0$$ Looking at deal.ii tutorials, I've notice that there are ...
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1answer
67 views

Finite Volume on Cubed Sphere

The US weather model uses an uncommon (?) discretization called 'Finite Volume on Cubed Sphere'. To avoid the singularities that occur at the poles when using lat/lon discretization, they instead ...
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1answer
79 views

deal.ii - ParaView "warp by scalar" of my output is not continuous

During our finite element course, we've solved the linear elasticity problem in 2D on a square (GridGenerator::hyper_cube) with $Q_1$ bilinear finite elements in ...

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