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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2answers
515 views

Ill-conditioned Gram Matrix Assembly

I'm trying to find the best approximation to the function $e^x$ in the finite dimensional polynomial space $P_4$ with respect to the standard basis vectors $B=\{1,x,x^2,x^3,x^4\}$ with inner product $$...
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4answers
512 views

Where to find Leszek Demkowicz's finite element codes or alternatives?

I know that long back Dr. Leszek Demkowicz finite elements codes(1Dhp,2Dhp,3Dhp) were available in his website. I'm finding it difficult to locate it now. Is there any alternatives available to these ...
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1answer
2k views

Finite Volume Method vs. Finite Element Method for Eulerian and Lagrangian Reference Frames

As far as I can tell, finite volume methods are primarily for eulerian reference frames (such as in fluid dynamics simulations) while finite element methods are easily ameanable to lagrangian ...
4
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1answer
172 views

How to express the derivatives with respect to the initial coordinates {x,y,z} when 3D coordinate system is mapped into 2D manifold {e,n}?

This is often required in the stiffness matrix evaluation. Thanks to this thread I found out how to perform integrals over 2D surface. I've tried to evaluate the integral of the single shape function ...
12
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1answer
318 views

How to integrate polynomial expression over 3D 4-node element?

I want to integrate a polynomial expression over a 4-node element in 3D. Several books on FEA cover the case where integrating is performed over an arbitrary flat 4-noned element. The usual procedure ...
3
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1answer
231 views

Looking for parMetis visualizer?

Is there any visualizer for parMetis (mpmetis), which can visualize FEM mesh grids after partitioning?
8
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2answers
400 views

Octree cubes to tetrahedrons

I'm trying to learn more about volume meshing and have decided to try to implement a simple volume mesher. The strategy I have chosen is to subdivide my space using an octree, refined based on some ...
6
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1answer
1k views

Appropriate space for weak solutions to an elliptical pde with mixed inhomogeneous boundary conditions

I'm working with the following mixed inhomogeneous boundary value problem: $\nabla(\kappa\nabla u)=f$ in $\Omega$ with $\partial\Omega = \Omega_1 \bigcup\Omega_2$ such that $u=g$ on $\partial\...
2
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1answer
157 views

FEM simulation of a material being stretched

I'm trying to teach myself FEM. The problem I have in mind is to completely investigate a plastic (e.g rubber band) being stretched (therefore, boundary conditions move as you stretch). I imagine this ...
13
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2answers
891 views

Impose the compatibility conditions for mixed finite elements method in Stokes equation

$\newcommand{\v}[1]{\boldsymbol{#1}}$ Suppose we have following Stokes flow model equation: $$ \tag{1} \left\{ \begin{aligned} -\mathrm{div}(\nu \nabla \v{u}) + \nabla p &= \v{f} \\ \mathrm{div} \...
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2answers
597 views

Solving Poisson equation with free boundaries and adaptively refined mesh

Assume we want to solve the Poisson equation $$ \Delta u = f $$ with free (Neumann) boundary conditions. So, the right hand side function $f$ must fulfill the compatibility condition to integrate to ...
15
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1answer
2k views

Visualizing discontinuous Galerkin/finite element data

I would like to visualize simulation results, obtained using the discontinuous Galerkin (DG) approach, within ParaView. Similarly to finite volume methods, the problem domain is divided into cube-...
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1answer
2k views

PDE discretization with the method of rothe and the method of lines (Modular implementation)

The Heat equation is discretized in space with FV (or FEM), and a semi-discrete equation is obtained (system of ODEs). This approach, known as the method of lines, allows to easily switch from one ...
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3answers
4k views

Finding which triangles points are in

Suppose I have a 2D mesh consisting of nonoverlapping triangles $\{T_k\}_{k=1}^N$, and a set of points $\{p_i\}_{i=1}^M \subset \cup_{k=1}^N T_K$. What is the best way to determine which triangle each ...
3
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1answer
1k views

Rigid Body Elements

I am currently developing structural FEM solver in FORTRAN. My question is about Rigid Body Elements (Multi Point Constraints). In NASTRAN there is RBE2 element defined by one independent and one or ...
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0answers
3k views

COMSOL - Implementing Perfectly Matched Layers (3D)

If there are any COMSOL gurus here, I want to simulate a 3D RF system infinite in 1 direction. I realize I need to use PML's, but am unsure how to configure them. it seems that simply scaling them by ...
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0answers
61 views

Complexity of direct solvers? [duplicate]

Possible Duplicate: How to reorder variables to produce a banded matrix of minimum bandwidth? What is the time and space complexity of direct sparse solvers (e.g., UMFPACK, SUPERLU, PARDISO, etc.)...
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9answers
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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 ...
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5answers
565 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-...
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3answers
829 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
252 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 ...
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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''(...
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2answers
474 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
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2answers
294 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 ...
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2answers
128 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 ...
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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!
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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....
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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 ...
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0answers
286 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
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1answer
174 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 ...
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2answers
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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 ...
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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 ...
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1answer
4k 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
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1answer
282 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....
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1answer
215 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 ...
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0answers
419 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 ...
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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 ...
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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 ...
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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
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1answer
637 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 ...
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2answers
193 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 ...
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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 ...
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5answers
821 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 ...
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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
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2answers
596 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
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1answer
400 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 ...