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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How is the surface Jacobian determinant calculated in FEM?

I am currently trying to evaluate surface forces on a structure. I came across P356 in Bathe's Finite Element Procedures 2014 (example 5.8) in which he related the edge derivative from the global to ...
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How can I define an equipotential surface/volume in FEniCS?

I want to solve electrostatic problem for potential. Charge density and medium permittivity are known, so is the potential of a grounded surface. I know how I can implement that. But I would like to ...
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Derivation of the second Piola-Kirchhoff tensor

I tried many formulas to find the components of the second Piola Kirchhoff. I need help to derive equations 27-29 on reference 1. [![enter image description here][1]][1] we have $$ \mathbf{C}=C_{11} \...
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A priori estimates in finite elements for inhomogeneous heat equation

Consider the problem $$\partial_t u-\Delta u = f\\ u(\Sigma_1)=f_D\\ \partial_\nu u (\Sigma_2)=f_N\\u(0)=u_0$$ where the sides of the space-time cylinder $\Sigma_i$ are disjoint (one of them could be ...
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Numerical methods for $u_t+c u_x= \frac{-c}{x}u$?

I am looking for possible numerical methods to solve the PDE $$u_t+c u_x= \frac{-c}{x}u$$ I am particularly interested in a Finite elements method, although I am also curious if you can expose some ...
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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- ...
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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 ...
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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 & = ...
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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 ...
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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, (...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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3D Euler Bernoulli Beam for Nonlinear FEA

Anybody has any experience in coding 3D beam elements? I am trying to write a C++ code for a 3D euler bernoulli beam. For 2D, I used Reddy for coding 2D for non linear FEA. How should I proceed with ...
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How to solve spatially discretised PDEs (method of lines) in solve_ivp or ODEint?

I can discretise the spatial domain of a system of PDEs using the method of lines, converting the system of PDEs to a system of ODEs (with a time derivative only). These equations (for context they ...
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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 ...
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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 ...
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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_{...
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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 ...
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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 ...
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Why is the solution in FEM bounded by zero?

Consider the following problem: $$ -\nabla^2 u = f,$$ Referencing to this post: FEM, we write the problem in variational form I'm assuming Dirichlet boundary conditions here): Find $u\in H^1_0(\Omega)$...
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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)*...
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Why does the total Lagrangian and updated Lagrangian formulations gives different results?

I'm studying Finite Element Procedures by K.J. Bathe and I have trouble understanding how the the total and updated lagrangian formulations should give same equations. Taking a simple truss element as ...
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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 ...
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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 ...
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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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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 ...
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1 answer
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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 ...
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2 votes
1 answer
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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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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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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 ...
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1 answer
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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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2 votes
1 answer
203 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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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 ...
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10 votes
2 answers
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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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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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1 answer
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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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3 votes
2 answers
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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 ...
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3 votes
1 answer
280 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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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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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{...
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2 votes
1 answer
331 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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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 ...
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3 votes
1 answer
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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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2 votes
2 answers
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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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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,...
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