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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What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method?

What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method? The best reference I've found thus far as been a paper titled The Performance of the ...
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Condition number of finite element stiffness matrix

In the FEM, there are certain applications in which the condition number $(\kappa)$ of the overall global stiffness matrix $(A)$ is computed or reduced by some preconditioner $(P)$. Are there any ...
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FEM and High Performance Computing

Suppose we want to solve an FEM problem in terms of HPC. What is the most usual way to do it: Using an open-source software like mfem,deal.ii etc.. or, Assembly the system by your own(read mesh file,...
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implementation of shell elements in a topology optimization algorithm

I am working on developing a topology optimization solver based on the finite element method and I want to add a triangular shell element in it. I used the classic finite element method but I didn’t ...
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FEM Meshing artifact at nodes with fewer neighbors

I wrote a 2D-FEM solver to solve some diffusion process and wanted to verify my code with a test problem. The input was $f(x,y) = x^2+y^2$ and I applied the stiffness matrix on it to get $\Delta f = 4$...
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Is the imaginary part needed in this problem?

Before jumping into my question, let me contextualize it. I'm doing numerical simulations of a Helmholtz scattering problem $$\Delta p + \kappa^2 p = 0\, .$$ The incident pressure wave $p^{inc}$ will ...
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Matrix Calculation Different between Python and Matlab

I am transferring a finite element code from Matlab to Python. A problem occurs at the last step when I try to solve the displacement $U = F/K$. I have checked that the calculated $F$ and $K$ are same ...
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How to determine global stiffness matrix is constrained or not

Background In solid fem, we often solve $$\mathbf{Ku}=\mathbf{p}$$ where $\mathbf{K}$ is global stiffness matrix, $\mathbf{u}$ is displacement, $\mathbf{p}$ is global load vector. If displacement not ...
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traction boundary conditions in elasticity

I have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My ...
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Pressure interpolation in the Q2-P1 element

The Q2-P1 element is one of the popular finite elements for incompressible flow problems in the mathematics community. For this element, the velocity field is approximated using bi-quadratic shape ...
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Uniaxial stretching solution not uniform in FEM code

I am trapped here for a long time. I wrote a toy Matlab FEM code. I want to run the follow simulation. Mesh Suppose we have a cube, and we divide it into subcube along $x,y,z$ axis, then each subcube ...
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Solving large sparse system

I am working on a problem with very large sparse matrices. I'd like to compute $A^{-1} B$, that is a crucial part of converting DAE to ODE (and there is no workaround). Here size of $A$ is 2E+5 x 2E+5 ...
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Weak form of the Navier-Cauchy equation

I am trying to obtain the weak form of the Navier-Cauchy equation, which is $$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$ ...
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FEM with elastic inhomogeneous properties leads to mesh-induced anisotropy

I'm solving an elastic homogenization problem and I'm having problems with mesh artifacts. I would like to first give a brief summary of what I do: I have a system with inhomogeneous (but isotropic) ...
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$P0$ elements for $H1$

Are there $P0$ (zero degree/constant element) nonconforming methods for approximating solutions in $H1$? More specifically, I have the equation: $$u-f - T\Delta u = 0$$ Which can be interpreted as ...
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Integration by parts with cross derivatives to obtain the weak form [duplicate]

I’m trying to write the weak form of the Navier-Cauchy equation in the component form, where $u_1$ and $u_2$ are the displacement components: $$-(2 \mu +\lambda) \frac{\partial ^2 u_1}{\partial x_1 ^2}...
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Cell-centered DG extension to the two-point flux approximation scheme

A current problem that I am working on requires me to compute the solution from the heat diffusion evolution on a discontinuous function. More precisely - I have a Delaunay triangulation and within ...
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Some formulations of domain coupling lead to saddle point problems. Is this merely an artifact?

Background I wanted to learn how to couple FEM and BEM (for the Poisson equation), because I wanted to better understand how open boundary conditions look like. Therefore I worked through the relevant ...
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Simple reference problems for time harmonic Maxwell equations

For Navier-Stokes problems we can often choose a relatively simple verification problem such as the lid driven cavity, flow over a cylinder, or flow over a backward facing step to verify our ...
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I wrote a 2D Finite Element program for Axial Loaded Plates, but the results are unexpected

TLDR: I used Python to write a 2D Finite Element program using 'Constant Strain Triangles' and my beam keeps pointing slightly upwards instead of straight sideways (like the force). I'm new to FEA and ...
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At what l/d ratio will a frame element start to behave as a shell element?

I'm working in ETABS. There are few columns of dimension 300mmx1400mm. The height of building is 36.6 meter above ground level and the dimension of building is 26mx68m. I'm getting the time period of ...
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How to construct a Fortin Operator for Crouzeix-Raviart Element?

I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element. The continuous Stokes Equation in the weak formulation is Find $u \in H_0^1(\Omega, \mathbb{R}^...
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Why does smoothed aggregation multigrid method used as preconditioner in conjugate gradient slows down the solution time?

I'm solving a system of linear equations obtained from the FEM discretization of a simple linear elasticity problem on a cube with zero displacements at one plane and a load on the opposite one. The ...
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1answer
63 views

Fast way to build stiffness directly as CSC matrix

I have been working on finite element code in Fortran 2008, and have implemented my own sparse matrix types. I have found that mapping local stiffness matrices (real type) to a global COO sparse type ...
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Methodology Suggestion for Wave-propagation Problem using Finite Elements

I want to simulate the propagation of a sinusoidal plane wave in a rectangular domain using Finite Elements Method. First, the wave should propagate through a fluid medium, then it will encounter a ...
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Solution of symmeric/non-symmetric linear system

I would like to understand what happens in the following: I have a really simple Poisson problem, in 1D, with $u_0 = u_N = 0$. I assembled the stiffness matrix and the right-hand side, and I applied ...
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94 views

Partial differential equation FEM application

I have a PDE which looks like Helmholtz wave equation on one dimensional domain. $$\dfrac{d^2u(x)}{dx^2}+\pi^2u(x)=f(x)$$ where $-\infty <x<\infty $ Also, $f(x)= 1$ for $-0.25<x<0.25$, I ...
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COMSOL Circularl polarization

I'm having some problems trying to implement circularly polarized light in COMSOL Muliphysics. For a isotropic homogenous media, I've obtained without problems the TE and TM reflectance curves. ...
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Determining the voxels between two boundary surfaces

I am working on human brain tACS simulations where I have the models of the skin, skull, csf, brain and ventricles in STL format. The shape does not matter and there are no intersections. I want to ...
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Solve 1D saturation equation using FEM

I want to solve the following 1D equation to model the time evolution of saturation in a porous medium using finite elements (I'm a beginner). The permeability $K$ and the pressure $p$ depend on the ...
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Adding a boundary surface on a STL model in Gmsh

Background info I want to run brain simulations with models from the PHM repository (not relevant to the problem but suits the narrative) which are shipped in STL format. The meshes of the STL have ...
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Stokes problem with imposed acceleration on boundaries (projection scheme)

I am trying to solve FSI problems with finite elements and using a projection scheme (I am taking as reference the review of Guermond: Guermond, J. L.; Minev, P.; Shen, Jie, An overview of ...
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FDM on nonlinear PDEs

I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$. In order to perform time discretization with FDM (finite ...
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Add orthogonality constraint to PDE over entire domain finite element

I have a nonlinear differential equation for $u:\mathbb{R}^2\rightarrow\mathbb{R}^3$ that I can express in the form: $ \nabla (g(u)_{ij} \nabla(u)^j) = 0 $ which is similar to the "Nonlinear Poisson" ...
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Finding isocontour lines, populating points on said lines (in MATLAB)

I'm trying to extract data from a potential field. I have an example code of a potential field, using meshgrid in MATLAB. I've pasted the code below. What I'd like to do next is 'convert' this field ...
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Spurious eigenvalues in a finite element eigenvalue problem

This post is closely related to this one and uses the exact same setup: a mix of quadratic and cubic basis functions in a finite element approach, where variables $u_1$ and $u_2$ are quadratic and $...
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Mesh partition quality

I am working on static FE mesh partitioning and in order to achieve a good quality partitioning I want to know how to drecrease interprocessor communication by increasing the connectivity of elements ...
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Open Boundary Conditions for Solving Navier Stokes in moving ALE domain

I'm trying to solve the problem of a body freely rising in a fluid due to gravity, the density of the body is just slightly less than that of the fluid. The body is rigid, so I have to solve a ...
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Boundary conditions in a finite element eigenvalue problem

I've been reading multiple papers and related posts for a while now, but I can't seem to find a specific answer to the issues I'm having so I hope someone can clarify things here. I'll provide some ...
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Applying boundary Conditions on FEM

I have a partial differential equatons as shown below. $$\dfrac{d}{dx}((1+x)\dfrac{du(x)}{dx})=0$$ With the following boundary conditions. $$u(0)=0, u(3)=10$$ To solve it using FEM, I multiplied the ...
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Galerkin method for heat equation

I'm working out the Galerkin method for the heat equation $$\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0$$ subject to $u(0,t)=0,u_x(1,t)=v(t)$. I want to use a Fourier basis ...
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Norm of operator in finite element discretization of Heat equation

I am solving the heat equation discretized spatially via FEM and temporally via backward Euler. I get the system $$M \dot{u} = K u +f$$ where $u$ is a vector representing the solution at spatial ...
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Difference between MoM and FEM

Method of Moments and Finite Element Methods are two of the most used methods in computational electromagnetics to solve electromagnetic equations. As it is known, in FEM sparse matrixes are used ...
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Adaptive Lagrangian-Eulerian methods and practical benchmark results

Does anyone know of any published study that talks about the practical aspects of running Adaptive Lagrangian-Eulerian techniques for solid and/or fluid mechanics problems? I'm looking for things ...
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Implementation of Z^2 error estimator in Abaqus for adaptive mesh refinement

Currently, I am working on a remeshing routine for my simulations (Abaqus 6.14-1) using python scripts. The simulation deals with the Brinell indentation test and as the remeshing software I use Gmsh ...
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Practical way to build element stiffness matrix for 3D FEM simulations

The element stiffness matrix in 3D FEM problems is build as follows: $$ K = \int\limits_{[-1,1]^3} B^T C B\, |J| \mathrm{d}r\, \mathrm{d}s\, \mathrm{d}t$$ The integral can be solved using e.g. Gauss ...
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Lattice spring models vs. finite element models

I am a beginning graduate student in the field of continuum mechanics. It is my understanding that most problems in this field are numerically solved via finite element methods (FEM). However, I have ...
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Parallel mesh partitioning

When a mesh partitioning takes place and every process works on a part of the mesh is any way to rename the global numbering of nodes(on each process) into a local numbering?Is there any software that ...
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How to solve odd-order differential equations in FEM? Petrov-Galerkin?

I've recently learned about using weighted residuals with the Galerkin method to numerically approximate even-order differential equations (for linear elements, I'm still a beginner). It seems for odd-...
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Normalized legendre and quadrature basis for discontinous Galerkin method

I've successfully implemented a 1D DG code with non-normalized Legendre basis and I've now moved onto developing a 2D code using tensor products. For my 2D code I've chosen to have normalized ...

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