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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29
votes
9answers
1k views

What is a good way to run parameter studies in C++

The problem I'm currently working on a Finite Element Navier Stokes simulation and I would like to investigate the effects of a variety of parameters. Some parameters are specified in an input file ...
7
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5answers
573 views

Recommended Route for Mastering Inverse PDE Problems

I would like to master Inverse PDE Problems particularly with the use of Finite Elements. My problem is I don't know where to start. Should I begin by reading a book on Inverse Problems or on PDE-...
9
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3answers
862 views

Construction of $C^1$/$H^2$-conforming finite element basis for triangular or tetrahedral mesh

In the paper Hierarchical Conforming Finite Element Methods for the Biharmonic Equation, P. Oswald claimed Clough-Tocher type elements has $C^1$-continuity while being a cubic polynomial on each ...
7
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1answer
258 views

Forced viscous damping in elastodynamics

I have an 2D elastodynamics problem, that is a problem which is driven by the Cauchy equation: $$\rho\ddot u-\mathrm{div}\sigma=\rho f$$ where $u$ is the displacement, $\sigma$ the Cauchy stress ...
11
votes
2answers
5k views

FEM: singularity of the stiffness matrix

I'm solving the differential equation $$ \left( \sigma^{2}(x) u ''(x) \right)'' = f(x), \;\;\; 0 \leqslant x \leqslant 1 $$ with initial conditions $u(0) = u(1) = 0$, $u''(...
5
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2answers
477 views

What does symmetrize mean? (imposing multifreedom constraints to stiffness matrix)

I have a small FEM implementation program. And I want to add imposing multifreedom constraints (MFC) feature to it. The theory of master-slave method is given here (page 10 for general case). ...
10
votes
2answers
299 views

What about this simple error estimate for linear PDE?

Let $\Omega$ be a convex polygonally bounded Lipschitz domain in $\mathbb R^2$, let $f \in L^2(\Omega)$. Then the solution of the Dirichlet problem $\Delta u = f$ in $\Omega$, $\operatorname{trace} u ...
8
votes
2answers
129 views

Initial guesses for perturbed linear systems

Suppose you solve a linear system $Au = f$ by an iterative method, e.g. conjugate gradients or Richardson iteration. Then you try to solve a linear system that is slightly perturbed in the matrix and ...
9
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3answers
2k views

Standard format for finite element meshes

Does there exist a standard format for finite element meshes which is widely used in the industry? Thanks!
31
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9answers
4k views

Modern resources for learning FEM

I need to get started using Finite Element Methods. I am about to start reading Numerical solutions of partial differential equations by the finite element method by Claes Johnson, but it's dated 1987....
21
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4answers
2k views

How to incorporate the boundary conditions with the Galerkin method?

I've been reading some resources on the web about Galerkin methods to solve PDEs, but I'm not clear about something. The following is my own account of what I have understood. Consider the following ...
4
votes
0answers
289 views

How does one handle the source term in the Shallow Water Equations when using the discontinuous galerkin method? [closed]

I use the discontinuous galerkin method to solve the steady flow 1D shallow water equations with a bump at the bottom. This flow is frictionless. I use the runge-kutta method to approximate the time ...
7
votes
1answer
175 views

Conjugate Gradient with Hierarchical Basis Functions: How can the hierarchical base be decomposed?

I'm trying to implement a Conjugate Gradient solver using Hierarchical Basis Functions, following this paper. In section 3 the paper says that the hierarchical basis matrix $S$ can be decomposed into ...
6
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2answers
2k views

Shape regularity in higher dimensions

In Finite Element theory, and other methods in scientific computing for PDEs, one uses meshes which fulfill several regularity criteria, many of them being equivalent. It is of interest to have ...
7
votes
2answers
2k views

Reference implementation of Nédélec-Elements

Does anybody know of an implementation of Nédélec elements that does not come along with a huge bulk of additional software? Is there a small library written in a language like Python, Matlab, or ...
6
votes
2answers
2k views

Which libraries have good implementations of Basis splines?

I'm looking to use the finite element method with B-splines as my function basis. Which C/C++ libraries have good B-spline support? Specifically, I'm looking for an implementation of a stable ...
15
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1answer
5k views

How to derive the Weak Formulation of a Partial Differential Equation for Finite Element Method?

I have taken a basic introduction to Finite Element Method, which did not emphasize a sophisticated understanding of a 'weak formulation'. I understand that with the galerkin method, we multiply both ...
11
votes
1answer
287 views

What are the possible numerical schemes for a diffusion equation with a nonlinear reaction term?

For some simple convex domain $\Omega$ in 2D, we have some $u(x)$ satisfying the following equation: $$ -\mathrm{div}(A\nabla u)+cu^n = f $$ with certain Dirichlet and/or Neumann boundary conditions....
10
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1answer
216 views

Quadrature rules, methodologies, and references

There is at least one quite comprehensive encyclopaedia of quadrature rules that doesn't seem to have been updated in quite a while and has restricted access. This source refers to several classical ...
6
votes
0answers
431 views

How to do FEM in sector elements?

Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there are a lot of ...
10
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2answers
1k views

Are 8 Gauss points required for second order hexahedral finite elements?

Is it possible to get second order accuracy for hexahedral finite elements with fewer than 8 Gauss points without introducing unphysical modes? A single central Gauss point introduces an unphysical ...
4
votes
2answers
1k views

Solution oscillations with a small timestep in backward Euler

I am using backward Euler in a FEM scheme for a convection-diffusion problem. On a given mesh, I can take arbitrarily large time steps, as expected. But if I decrease time step, at some point it will ...
3
votes
2answers
1k views

Looking for a library or algorithms to perfom clipping 3D unstructured meshes by a set of surfaces

We have a 3D (volume) unstructured, possibly hybrid, degenerative irregular mesh data structure that we are capable of generating (mostly composed of hexahedra and general polyhedra, using a mix of ...
9
votes
1answer
640 views

What numerical quadrature to choose to integrate a function with singularities?

For example, I would like to numerically compute the $L^2$-norm of $\displaystyle u = \frac{1}{(x^2+y^2+z^2)^{1/3}}$ in some domain that includes zero, I tried Gauss quadrature and it fails, it is ...
8
votes
2answers
195 views

Is there one general approach to build a projection methods for different problems?

My question is probably going to be too general to answer it with a couple words. Could you please suggest a good reading in that case. Projection methods are used to reduce size of the solution space ...
46
votes
5answers
8k views

What are criteria to choose between finite-differences and finite-elements

I am used to thinking of finite-differences as a special case of finite-elements, on a very constrained grid. So what are the conditions on how to choose between Finite Difference Method (FDM) and ...
5
votes
5answers
836 views

Introductions to hp-FEM

do you know good introductions into or surveys $hp$-adaptive finite elements? In particular I do not know how the heuristics for choosing spatial refinement or increased polynomial degree are ...
13
votes
1answer
1k views

What are possible methods to solve compressible Euler equations

I would like to write my own solver for compressible Euler equations, and most importantly I want it to work robustly in all situations. I would like it to be FE based (DG is ok). What are the ...
9
votes
2answers
620 views

Coupling FEM DG methods to Riemann solvers

Are there any good papers and or codes that couple discontinuous galerkin finite element solvers with Riemann solvers? I need to explore coupling elliptic and hyperbolic problems but most splitting ...
12
votes
1answer
407 views

What spatial discretizations work for incompressible flow with anisotropic boundary meshes?

High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this ...

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