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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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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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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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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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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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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. ...