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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A Finite Element Method for a first order PDE?

I want to develop a finite element method to solve for $u(x,t)$ the PDE: $$u_t+c u_x= \frac{-c}{x}u$$ where $c$ is a constant. so I am trying the following ( as Rothe's method? ) : Letting $k= t_n- ...
NotaChoice's user avatar
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1 answer
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Applying Stress Boundary Conditions in Commercial Finite Element Analysis Codes

I am trying to replicate a finite element analysis given in a research paper titled On the Detection of Stress Singularities in Finite Element Analysis 1 by G.B.Sinclair et. al. The geometry of the ...
Ali Baig's user avatar
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positivity preservation for mixed finite element discretization

I'm interested in mixed discretizations of the diffusion (and related) equations: $$\begin{align} \frac{\partial h}{\partial t} + \nabla\cdot \mathbf q & = f \\ k^{-1}\mathbf q + \nabla h & = ...
Daniel Shapero's user avatar
1 vote
1 answer
116 views

What FEM solver should be used for matrix-valued FE spaces?

I am pretty new to using FE solvers. I am trying to solve a system of (up to) 9 complex equations. We write these as a matrix equation (here), (with the implied sum over $j$, for each component ...
Izek H's user avatar
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Finding the weak form of a PDE with a tensor argument

I am trying to solve for the order parameter ($A$) in the Ginzburg Landau equations. I had asked on the math SE site but was recommended to ask here. We are trying to solve the following equation, (...
Izek H's user avatar
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121 views

A priori FEM estimates without $H^2$ regularity

In basic lectures on finite elements theory, people always assume $H^2$ regularity of the solution in order to derive $O(h^k)$ a priori estimates in the norms $H^{2-k}$, $k=1,2$. For simplicity let's ...
Lilla's user avatar
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Why "Right" and "Left" Cauchy-Green tensor?

$C=F^TF$ is called the "Right" Cauchy-Green tensor, and $b=FF^T$ is called the "Left" Cauchy-Green tensor. I suppose in $C=F^TF$ the non-transposed $F$ stands on the right, and in $...
MaxD's user avatar
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Taylor-Hood elements for Darcy's equation

I would like to know if Taylor-Hood elements $P_2$-$P_1$ form a stable pair for the mixed approximation of Darcy's equation ( or Poisson's equation) with Dirichlet B.C. In the literature I only find ...
Cherif's user avatar
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Reference request for finite elements theory

Consider a domain $\Omega \subseteq \mathbb{R}^{2,3}$ which is non convex and with $C^2$ boundary. Could you recommend a good reference where it is explained how: without needing isoparametric ...
Lilla's user avatar
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FEM applied to heat equation and incompatible conditions

Consider the problem $$u_t - \Delta u = f \text{ on } \Omega\times (0,T)\\u=0 \text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) \text{ on } \Omega$$ with $g$ NOT vanishing on the boundary. If I ...
Lilla's user avatar
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What condition ensures the global continuity of the solution in the FEM?

I know this is trivial but I don't seem to understand it. In which step of the FE formulation do we enforce the global continuity of the solution? Or in other words, how the construction of the local ...
Davide Papapicco's user avatar
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2 answers
93 views

Can I use Q0 finite elements when there are gradients involved?

Could I apply a Q0-discretization to, say, the Poisson equation $\Delta \phi = f$ (where by Q0 I mean piecewise constant, and thus non-continuous, elements)? Solving this with FEM, at least as I know ...
MaxD's user avatar
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How can I correctly determine velocity of a point inside a grid after using mixed finite element method to solve Poisson equation?

I am using the mixed finite element method (MFEM) to solve the Poisson equation: $$\Delta h = 0,$$where $h$ denotes hydraulic pressure. The MFEM could determine the normal flux rate, $q_n$, through ...
Tingchang Yin's user avatar
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92 views

calculating the Laplacian of the field variable in estimating the local residual error in the finite element method

to perform adaptive refinement in the finite element method according to the explicit residual method, the quantity $$\eta_K^2=h_K^2\left\lVert r\right\rVert_{L_2(K)}^2+h_K\left\lVert R\right\rVert_{...
Masa's user avatar
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Applying flux limiters on vertices/faces instead of quadrature points

I have a couple of questions with regards to the usage of flux/slope limiters, which I was hoping to get some input. I apologise in advance if my questions are trivial, but flux limiters lie way ...
gnikit's user avatar
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Algebraic Multigrid on fine mesh vs. just starting with coarse mesh?

So if I've understood it correctly, the Algebraic Multigrid Method (AMG) basically takes a fine mesh, coarsens it, solves the coarse mesh and projects the solution back on the fine mesh. Wouldn't it ...
MaxD's user avatar
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Assembly of the Isoparametric Quadratic load vector in Matlab [duplicate]

I work to solve PDE using FEM in the case P2 on Matlab. I try to correctly assemble load vector using quadratic Lagrange shape functions $$b_i =(f,\phi_i)=\sum_{q=1}^{nq}f(r_q,s_q)*\phi_{i}(r_q,s_q)*...
A. AchbK's user avatar
1 vote
4 answers
394 views

Thin-plate FEM simulation

I want to simulate a vibration of a thin plate (the Kirchhoff-Love model) on triangular meshes. Can you advise me on an introductory-level review of different elements for thin-plate FEM simulation? I ...
user42439's user avatar
1 vote
1 answer
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numerical integration of integrals in the p-adaptive version of the finite element method

In the p-adaptive version of the finite element method, elements are allowed to have shape functions with arbitrary different polynomial orders. Therefore regarding a 2D problem with quadrilateral ...
Masa's user avatar
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2 answers
292 views

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 (...
FEGirl's user avatar
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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 ...
Masa's user avatar
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371 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 ...
bob_bill's user avatar
2 votes
1 answer
300 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 ...
Olumide's user avatar
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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 ...
Tepa's user avatar
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1 vote
1 answer
105 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,...
Nathalie Glinsky's user avatar
2 votes
1 answer
267 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 \\ -\...
bob_bill's user avatar
1 vote
1 answer
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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 ...
bob_bill's user avatar
10 votes
2 answers
679 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-...
bob_bill's user avatar
2 votes
1 answer
85 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{ ...
Tingchang Yin's user avatar
3 votes
2 answers
773 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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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 ...
Andreas Longva's user avatar
3 votes
1 answer
558 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} \\ \...
Mohamed Abdelhamid's user avatar
3 votes
0 answers
126 views

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&...
ghost's user avatar
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1 answer
177 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{...
Tingchang Yin's user avatar
2 votes
1 answer
463 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: $$\...
WhatsupAndThanks's user avatar
2 votes
0 answers
787 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 ...
Dude's user avatar
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3 votes
1 answer
256 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 ...
WhatsupAndThanks's user avatar
3 votes
2 answers
411 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 $\...
FEGirl's user avatar
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2 votes
1 answer
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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,...
Tingchang Yin's user avatar
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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 ...
kstn's user avatar
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3 votes
2 answers
374 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 ...
Tingchang Yin's user avatar
6 votes
2 answers
264 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 ...
Tingchang Yin's user avatar
0 votes
2 answers
178 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 ...
Moonwalker's user avatar
3 votes
2 answers
407 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 ...
FEGirl's user avatar
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3 votes
2 answers
274 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 ...
Mohamed Abdelhamid's user avatar
1 vote
0 answers
218 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})...
Andrew Yuan's user avatar
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191 views

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 ...
Shekhar's user avatar
3 votes
1 answer
86 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 ...
kevinkayaks's user avatar
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0 answers
75 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 ...
strahd's user avatar
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3 votes
1 answer
176 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. ...
Chenna K's user avatar
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