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.
1,049
questions
2
votes
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37 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 ...
0
votes
0answers
37 views
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 ...
3
votes
1answer
69 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. ...
6
votes
0answers
91 views
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 ...
4
votes
1answer
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 ...
2
votes
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 ...
1
vote
2answers
124 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 $\...
3
votes
1answer
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 ...
3
votes
0answers
85 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 ...
0
votes
1answer
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 ...
3
votes
1answer
115 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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0answers
30 views
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 ...
5
votes
1answer
115 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 ...
2
votes
1answer
55 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 ...
1
vote
0answers
68 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 ...
1
vote
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), (...
2
votes
2answers
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 ...
2
votes
0answers
84 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) \, .$$
It is said in ...
1
vote
1answer
104 views
Area of 8-node rectangular serendipity finite element
I am trying to compute the area of an 8-node rectangular serendipity finite element from the equation
$$
\sum_{i= 1}^8 det \, J(\xi,\eta) \cdot W_i
$$
based on Gaussian quadrature, where
$$
J(\xi,\eta)...
1
vote
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. ...
2
votes
0answers
62 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 ...
1
vote
0answers
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. ...
4
votes
0answers
70 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 ...
2
votes
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 ...
0
votes
2answers
56 views
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 ...
0
votes
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 $L^2$ ...
2
votes
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 ...
1
vote
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, ...
1
vote
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 ...
0
votes
0answers
40 views
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 ...
3
votes
0answers
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 ...
1
vote
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
$$
V_1 = \{ \nu_1 \in H^1 : \nu_1 = 0 \ ...
2
votes
1answer
96 views
Discontinuous Galerkin order of convergence on arbirary refined mesh: step-12 deal.ii tutorial
I'm learning DG methods and in order to practice a little bit I'm using the deal.ii library. In particular, I'm looking at step-12, where they solve
$$\operatorname{div}(\beta u) = 0$$
$$u = g_D \text{...
4
votes
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: $$\operatorname{...
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vote
0answers
52 views
How to determine the orientation of convex/concave hexahedra?
I am writing a code that checks the orientation of a list of vertices (along with face connectivity) describing both convex and concave hexahedra. The face connectivity table stores the list of vertex ...
5
votes
1answer
162 views
Write incompressible Navier Stokes as ODE in $(\mathbf{u},p)$
Consider the Navier stokes equation after the discretization with conforming finite elements with source term $f=0$. We have the algebraic structure of a saddle point problem:
$$M \dot{u} = f- Au -B^...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
5
votes
0answers
144 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 $A: V \rightarrow V', B:V \rightarrow Q'$ ...
0
votes
1answer
79 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 ...
2
votes
0answers
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
votes
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 $T$.
$$||b_T \phi ||_{...
1
vote
1answer
90 views
Classical global estimate for $H^1$ error
I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$.
$$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
1
vote
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 $T_m$:
$$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
5
votes
1answer
175 views
Discrete divergence free functions
I'm studying the weak formulation of NS equations. During the analysis, the book I'm using (Quarteroni-Valli, page 301-302), defined $$Z_h=\{v_h \in V_h: (\operatorname{div}(v_h),q_h)=0 \quad \forall ...
1
vote
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 ...
0
votes
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
$$
\...
3
votes
1answer
160 views
Time discretization Navier Stokes equation
This question is a follow-up of this one.
The weak form of Navier Stokes equation is (assuming $v,q$ test functions for the velocity and the pressure, respectively)
$$(\frac{du}{dt},v)_{\Omega} + (\...
4
votes
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 ...