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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Find intersections between mesh and curve inside it
I have a simple square mesh, and a curve (discretised by another mesh) inside it. Here a picture worths thousand words. What I want to achieve is to find, for every cell $K$ of the circular (...
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1answer
89 views
Mesh refinement in the Finite Element Method
I need some good references on how to implement programmatically the hp-refinement of meshes in the Finite Element Method in two/three-dimension. I've searched the web a lot and read many articles and ...
2
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1answer
84 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 ...
2
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1answer
108 views
Total stored potential energy of finite element mesh from nodal point displacements and strain energy density function only
I am interested in calculating the total potential energy stored in a finite element mesh given its nodal point displacements alone. The forces that created the displacements are irrelevant because ...
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37 views
P1 Finite element discretisation of laplace-neumann eigenproblem
I am looking for help in the FE discretisation of the Laplace eigenkproblem with Neumann boundary conditions; that is,
$$-\int_{\Omega} \nabla u \cdot \nabla v= \lambda \int_{\Omega} uv,$$
or
$$Ax=\...
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66 views
How to implement large rotations in total lagrangian formulation (nonlinear FEM)?
I have developed an Octave script to solve the nonlinear Euler-Bernoulli beam equations with linearized von Karman-strains, i.e. higher-order terms are dropped. The simulation results agree with ...
0
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1answer
35 views
How to define a 3D surface from a set of points
Sorry in advance if this question has already been asked, I found nothing to help.
I want to buid a 3D box whose top surface is topography. This topography is defined by a DEM i.e. a set of points (x,...
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1answer
176 views
Solving Poisson equations as mixed Laplace using $RT_0-P_0$ pair
I'm trying to solve \begin{cases} - \Delta p=f \text{ in } \Omega\\ p=0 \text{ on } \partial \Omega \end{cases} with in $\Omega = [-1,1]^2$ by writing it as
\begin{cases} u + \nabla p=0 \\ -\...
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1answer
60 views
Building blocks for solving a vector valued problem
This question is a follow-up of this previous one. I decided to solve the linear elasticity \begin{cases}- \nabla \sigma(u)=f \\ u=0 \text{ on } \partial \Omega\end{cases}
with P1 Lagrangian finite ...
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2answers
443 views
FEM for vector valued problems: reference request
I'm a PhD working in a computational mechanics lab. I come from a Math department, and I have a good background for what concerns the basics of finite elements, like inf-sup conditions, DG, non-...
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29 views
Move just some points of the finite-element mesh (triangle/tetrahedron) and interpolate the solution in the new mesh
I have a finite element mesh, and I want that the mesh has some specific points and edges, as I show in the picture. I think that that is possible in a mesh software.
I'm solving a evolutive (time-...
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1answer
51 views
The effect of grid size on the total flux when solving Darcy flow with mixed finite element method
I am solving Darcy flow now with mixed finite element method. The Dary flow is
$$\begin{equation}\begin{aligned}k^{-1}\mathbf{q} + \nabla h=0, \text{ in } \Omega\\ %
\nabla\cdot \mathbf{q} = 0, \text{ ...
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2answers
237 views
Second Piola-Kirchoff Stress Tensor of Neo-Hookean solid at "zero deformation"
The strain energy of an incompressible Neo-Hookean solid is given as:
$$
W = C_{10}(I_1 - 3)
$$
Implying that at zero deformation $W = 0$, because $F = I \implies C = F^TF = I \implies I_1 = 3$
...
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0answers
89 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 ...
3
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1answer
219 views
Inverse of the Jacobian in the Finite Element Method
In Bathe's Finite Element Procedures 2014 P346, the Jacobian is defined as follows:
\begin{equation}
\mathbf{J} = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} \\ \...
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0answers
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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
$$C_T(t)=-\left&...
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1answer
76 views
Implementation of mixed hybrid finite element method
The mixed hybrid finite element method (MHFEM) is based on the mixed finite element method (MFEM). So, I'd recall the implementation of MFEM.
The mixed formulation of Poisson equation reads
$$\begin{...
2
votes
1answer
292 views
Solving Schrodinger Equation with finite element and Crank-Nicolson?
I have asked this in Mathematic section, but received no reply.
Please let me ask here to see if threr is any difference.
The Schrodinger equation without potential has the following form:
$$\...
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0answers
65 views
How to write a simple finite element solver in python in order to solve Poisson equation in 2D
I would like to write a simple finite element solver in python in order to solve 2D Poisson equation and then visualize it.
$$
-\nabla^{2} u(x,y)=f(x,y), \quad x,y \quad in \quad \Omega\\
u(x,y) = u_D ...
3
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1answer
94 views
Comparison on adaptive mesh refinement on finite elements and finite differences
My current work requires using (Adaptive Mesh Refinement) AMR to resolve multi scale physics. I have a general question whether finite element is better than finite difference in this aspect or not.
I ...
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2answers
385 views
Modelling question: example of a physical phenomenon with this jump condition at an interface?
in our finite element class we were talking about interface problems our teacher came up with the following, where $K_i$ are two given functions and $u_i$ is the restriction of the solution $u$ to $\...
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1answer
100 views
Elementary matrix of Raviart-Thomas elements
We can use the $RT0$ to solve the Darcy equation, i.e.
$$k^{-1}\mathbf{u}+\nabla p = 0, \text{ in } \Omega,$$
$$-\nabla \cdot \mathbf{u} = 0, \text{ in } \Omega,$$
$$p = p_D \text{ on } \partial\Omega,...
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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 ...
3
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2answers
159 views
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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2answers
145 views
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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2answers
90 views
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 ...
1
vote
2answers
142 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 ...
2
votes
1answer
110 views
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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0answers
49 views
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 ...
3
votes
1answer
70 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,$$
but I am having trouble finding a discrete analog of this equation which is numerically ...
0
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0answers
71 views
Fourier integral over elements
Suppose I have a triangular element with vertices ${\vec{r_1},...,\vec{r_3}}$ and a function $f(\vec{r})$. I want to calculate the fourier integral over this triangle such that:
$$F(k_x,k_y)=\int \int ...
3
votes
1answer
109 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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111 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
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1answer
84 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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2answers
197 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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2answers
228 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
119 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 ...
6
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1answer
185 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
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1answer
120 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
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1answer
124 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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1answer
125 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
66 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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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
90 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
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2answers
135 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
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0answers
90 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
119 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
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1answer
116 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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0answers
74 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 ...
