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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50 views

Weak form of Elliptic problem with mixed Dirichlet & Neumann conditions

Let $\Omega \subset \mathbb{R}$ in a bounded polygon domain and $f:\Omega \to \mathbb{R}$ known function.We split the boundary into two parts $\partial \Omega_{1}$ and $\partial \Omega_{2}$ such that $...
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Biharmonic equation mixed formulation, mixed boundary conditions

I have the following formulation of the biharmonic equation: $$\Delta u(x) = v(x), \quad x \in \Omega\setminus K$$ $$-\Delta v = 0, \quad x \in \Omega\setminus K$$ $$u(x) = g(x), \quad x \in K$$ $$v(x)...
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Implementing fourth-order PDE into COMSOL [closed]

I would like to solve the following partial differential equation for the scalar function $\Psi(x,z)$: $$\epsilon^2\nabla^4 \Psi - \dfrac{\partial^2 \Psi}{\partial x^2}=0$$ on the domain $\Omega_\rm{f}...
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274 views

Is mesh orthogonality important for FEM?

While studying mesh quality metrics in literature and software documentation, I've seen discussions about mesh orthogonality in Finite Volume Method (FVM) contexts, but not for Finite Element Method (...
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Confused on how Method of Manufactured solutions works?

I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique so my starting equation ...
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122 views

Crouzeix-Raviart for Stokes on elasticity form

I have tried to solve the driven cavity problem with incompressible Stokes flow using the standard Crouzeix-Raviart non-conforming P1/P0 element (linear velocity, constant pressure, velocity-nodes at ...
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Why is FEM theory only ever written down for simplices?

I noticed that in most theoretic literature (Braess, Ciarlet, ...) for Finite Elements, the method is only described in a detailed way for simplex triangulations, while other forms of hexahedral ...
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Python book for Finite Element Methods [duplicate]

Does any body knows the book/reference for Finite element Method for solid mechanics in python. Could you please share it to me.
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FEM port Boundary definition for electromagnetics and wave guides

We are currently in the process of implementing ports in our EM FEM simulation SW. We have come across the definition of boundary conditions for the ports, and we do not understand the equation for ...
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A priori error estimates - finite element method - mixed boundary conditions

Consider the problem $$ \left\{\begin{array} {rcl} -\Delta u & = 0 & \text{ in } \Omega \\ u & = 0 & \text{ on } \Gamma_D \\ \frac{\partial u}{\partial n} &= g &\text{ on } \...
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70 views

Shape function for quarter point element from degenerate quadrilateral

I want an element as the one shown following. Nodes 6 and 8 are in the quarter position. Whether eight node quadrilateral element or six node triangle function can be used directly? If not, how to ...
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139 views

Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous

I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too. ...
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Finite Element Method for 1D Poisson Equation with Inhomogeneous Boundary Conditions

Im trying to solve the Poisson equation in 1D: $$-u_{xx} = f(x), \hspace{6mm} u(a) = d1, \hspace{2mm} u(b) = d2$$Assuming a uniform partition such that $x_n = a + nh$, where $h = (b-a)/N$ and $n \in [...
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Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations

I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
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Error $L_{2}$ convergence in Finite Element for Poisson Equation

I have written a Matlab code to solve the equation $-u'' = f$ with conditions $u(0) = u'(1) = 0$ on the domain $x \in [0,1]$. I tested the code with $f(x) = -2, \forall x\in [0,1]$. I check the plot ...
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Efficient schemes for solving the extended Saddle point problem

I am interested in knowing some efficient techniques for solving the following extended Saddle point problem. \begin{align} \begin{bmatrix} A & B^T & C^T \\ B & 0 & 0 \\ C & ...
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Finite element method for high-frequency electromagnetics

I am writing a project about the Finite element method for use in high-frequency solutions of Maxwell's equations. This could be for use in antenna design and similar. I have some trouble ...
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Boundary conditions for an FEM approximation of the Laplace operator

Using FEM, I want to approximate the Laplacian $$u = \nabla \cdot \nabla h \, ,$$ where $h(x,y)$ is an FEM approximated scalar field on the same mesh, i.e. piecewise differentiable. I am using MOOSE ...
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87 views

preconditioner for Laplace “without” boundary values

I'm looking at solving systems with the FEM discretization $$ -\int_\Omega (\Delta u) v = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot\nabla u) v. $$ without applying Dirichlet- or ...
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116 views

Step3 in deal.II - Convergence of the mean

I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
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Proof of kinematic relationship between updated and total Lagrangian 2nd Piola-Kirchhoff stresses

On page 587 of Finite Element Procedures by Bathe the author gives the following kinematic transformations $$ {}^t\tau_{ij} = \frac{{}^t\rho}{{}^o\rho} \; {}^t_ox_{i,r} \; {}^t_oS_{rs} \; {}^t_ox_{j,...
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Finite element method for an equation requiring switch between spectral and temporal domain

Some equations (such as the non-linear schrödinger equation for pulse propagation) are more easily solved in the spectral form, but still need a representation in the temporal domain to calculate ...
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Solving coupled PDEs with self-consistency condition

I am figuring out how to attack a problem (the Usadel equations of superconductivity) in which I need to solve a set of nonlinear PDEs for the fields $\{G_i (r)\}$ $$ U(G_i(r), \nabla G_i(r), \Delta(r)...
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Lumped mass matrices for higher-order finite elements for CFD

Given that some of the mass lumping techniques, for example, row-sum lumping does not produce practically viable lumped mass matrices for all the element shapes, what are the techniques used for mass ...
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How to apply the boundary condition when global stiffness matrix is stored in csr format? [duplicate]

I am solving the poisson equation and I constructed the global stiffness matrix in compressed row storage format. Then I wrote the preconditioned conjugate gradient solver for solving the system of ...
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Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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1D Poisson equation and quadratic basis functions assembly

I'm solving the simple Poisson problem $$-u''(x)=1$$ in the interval $[0,1]$ with $u(0)=u(1)=0$. I discretised my domain as done here, i.e. with ...
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Discretization of a nonlinear boundary value problem

I am trying to use finite element method to discretize the following problem \begin{align} \min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega, \...
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Renumbering the nodes for quadratic basis functions for a 2D domain

I have a simple triangulation for a 2D domain, described by the connectivity matrix $T$ and by the point matrix $P$. For didactic purposes, I assembled the stiffness matrix for $-\Delta u = f$ by ...
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Differences in using Clausius-Duhem inequality vs Principle of Virtual Work/Power in derriving constitutive equations?

I am a novice getting my toes wet in continuum mechanics and nonlinear elasticity. I have seen papers that use both approaches in developing constitutive connections to compliment balance equations of ...
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Who uses finite elements with higher continuity?

Lagrange elements of any polynomial describe piecewise continuous functions. Typically, those functions are differentiable. Mixed finite element methods use vector fields of even less continuity, such ...
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Spline interpolation for vector-valued data in 3D space

I have output from a 3D linear elasticity finite element simulation which uses linear tetrahedral elements, such that the displacement is continuous over the nodes but the gradient is not ($C_0$ ...
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118 views

FEM solution for Poisson is not exact at nodes

Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$ I know that the FEM solution (...
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C.S.R method in finite element matrix assembly

I have solved the 2D Poisson equation using finite element method with simplex triangular element in MATLAB. First, I generated the triangular mesh using pdetool in ...
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diffusivity matrix assembly in nonlinear finite element analysis

I want to solve a diffusion analysis using finite elements. According to fick's law, governing equation is $$\frac{\partial h}{\partial t} = D \frac{\partial^{2} h}{\partial x^{2}}$$ . h is relative ...
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Poisson equation, stiffness matrix positive definiteness, Dirichlet boundary conditions

I have a question regarding the positive definiteness of the stiffness matrix. Specifically, I believe that it should be positive definite only when at least one Dirichlet point is given, so I would ...
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Finite element interpolation

I have a finite element solver that I implemented in MATLAB. I am calculating a specific potential function that is constant in each tetrahedral element. The question is, I want to be able to ...
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velocity in CFL condition

I am studying the evolution of the density and velocity field of a core in a molecular cloud in 1 D. I defined the radial grid (let us say x between 100 and 101) and the time grid. I am using the ...
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Fourier transform in finite element

I have a finite element solver where I am using tetrahedral elements. I am solving for electric potentials and then calculate the current densities in each element, which are constant in each element. ...
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Sparse linear solver in fortran working with REAL16

I need some (direct) sparse linear solver for fortran, which works with REAL16 data type. Any suggestions? Both Pardiso and MUMPS support only REAL8. (identical question: https://math.stackexchange....
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Finite elements with CFL condition - How to obtain correct order of convergence

I have discretized a PDE with continuous finite element method in spatial variable and with implicit Euler or Crank-Nicolson in temporal variable. In both cases, I have error estimates in $L_2$ norm ...
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1answer
103 views

L2 projection with bounds

In some problems we are currently working on, we are working with discontinuous functions that are defined on a finite element mesh and are established using Lagrangian particles. To obtain them on a ...
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69 views

relation between different tangent stiffness

I need to find a relation between the tangent stiffness $L_1$ of the first Piola-Kirchhoff stress tensor with the tangents stiffness $L_2$ of the second Piola-kirchoff stress tensor. They are defined ...
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112 views

How can I implement second order derivatives of shape functions of a 3D elements?

I am developing an Abaqus UEL with 3D 8 nodes brick elements and I need second order derivatives of the shape functions, I have already mapped the first order derivatives from the element coordinates ...
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Mesh partitioning with METIS

I am trying to use METIS-5.1.0 edition in order to partion a FE mesh. For demostration purposes I created 2x2 rectangle mesh and tried to partition it. However, I notice a weird behaviour in my code. ...
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Output solution vector in Petsc

I am using petsc to solve a linear elasticity problem discretized by finite elements.The initial mesh is read by a mesh file and the distribution in each processor is done using METIS.I am using only ...
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1answer
59 views

How to solve a Stiffness matrix?

I am quite new in this field. For my university I prepared a stiffness matrix to solve for a project group. This matrix consists of 450 equations with 450 unknows (it's a Matlab script) and I have the ...
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Fortran compiles for legacy Finite Element Fortran program (1980)

The version of Fortran used come from Montreal Ecole Polytechical in 1980. I need a compiler for Fortran for Windows 7 or Windows 8.
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68 views

Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?

I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
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1answer
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Solving PDEs in parallel

I have read different approaches on how to solve pdes in parallel which are discretized using finite element method. For example: Non-overlapping domain decomposition approach as mentioned in https://...

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