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.

Filter by
Sorted by
Tagged with
5
votes
1answer
269 views

Are serendipity elements still polynomially complete when the quadrilateral is skewed?

I have not implemented these elements before, but I like their reduced cardinality compared to (e.g.) a tensor product of Lagrange interpolants, which is very "overcomplete" (especially for orders>2) ...
0
votes
2answers
3k views

Local and global coordinates in FEM

Am I right that in FEM we can associate a local coordinate system with every node, not with every element?
4
votes
1answer
3k views

Meaning of CFL condition on parabolic problems

I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method. I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
9
votes
2answers
5k views

How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?

In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
2
votes
1answer
152 views

Diffusion-Transport problem FEM

I was looking at a book of FEM on problems of Diffusion-Transport. $$-div(\mu \nabla u) + b \cdot \nabla u + \gamma u = f \qquad in~\Omega$$ $$u = 0 \qquad in~\partial\Omega\text{ (in the boundaries)...
3
votes
2answers
361 views

Finite element stabilization schemes for incompressible flow

I am looking for an easy to implement stabilization scheme that can be used with equal order ($P_1-P_1$ or $Q_1-Q_1$) finite elements for fluid flow. Is there something like this or should I stick to ...
4
votes
2answers
2k views

Mesh simple 2d CAD boundry drawing

I sincerely apologise if this question is a duplicate. Though it is clearly a question that must have been asked and answered a 1000 times I can't find any reasonable solution. How do I take a simple ...
3
votes
1answer
354 views

Limitations of Domain Decomposition Method (DDM) in Finite Element Analysis (FEA)?

The use of DDM in FEA makes parallel solution of the whole analysis e.g. assembly, solver etc possible. DDM splits the model in domains and runs them in parallel. Since there are interconnected nodes ...
3
votes
1answer
154 views

Approximation of a linear function with polynomials of degree 1

If I have the following problem $$-\mu u'' + u' = 1$$ with boundary conditions $u(0) = u'(1) = 1$ in the interval $\Omega = (0,1)$. The exact solution is $$u(x) = x + 1$$ Will the FEM approximation ...
8
votes
1answer
528 views

Nonlinear wave equation - Finite element or finite difference

I would like to know the which is more advantageous when it comes to solving nonlinear hyperbolic equations, Finite Element or Finite difference methods? Which method will be better in capturing ...
1
vote
1answer
792 views

How to include parameters in plot title in comsol? [closed]

I have a comsol model. I want to draw a few plots for different values of a global parameter called plot_z, e.g. if plot_z can ...
1
vote
1answer
142 views

FEM oscillations for polynomials of degree 1

I have the following eliptic 1-D problem $$-\mu u'' + \beta u' = 1$$ $$u(0) = u'(1) = 1$$ where $\mu = 10e^{-5}$ and $\beta = 1$. For this specific problem I am using the following space steps $h=[0.1,...
2
votes
1answer
135 views

Is it necessary to project the initial condition onto the variational space in a fully discrete galerkin method?

I'm solving a simple 1D heat diffusion problem $$u_t=u_{xx},\quad \Omega\times[0,T]$$ $$u=0\quad, \partial\Omega\times [0,T]$$ $$u(x,0)=f$$using a fully discrete galerkin finite element method. This ...
6
votes
1answer
210 views

Neumann BCs in cylindrical geometry (FEM)

I was wondering where I could get a detailed account (either in print or online) on applying a Neumann/mixed Boundary condition along the $r=0$ axis in an axially symmetric geometry. Though this is a ...
9
votes
4answers
1k views

When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...
4
votes
2answers
643 views

Orthonormalized Bernstein polynomials using Gram-Schmidt

I was wondering, before trying to do that myself, has anyone attempted to do orthonormalization of Bernstein polynomials using Gram-Schmidt? I discussed this with several people and have been told ...
1
vote
1answer
487 views

How to apply a Galerkin finite element method to a linear, one-dimensional boundary value problem

I have the following boundary value problem: $$-(\alpha u')' + \gamma u = f $$ in $\Omega = (a,b)$ with b.c. $u(a) = u(b) = 0$ and $\alpha > 0, \gamma ≥ 0$ and $f:(a,b) \to \Re $ The weak ...
1
vote
2answers
120 views

how to approach time zero when the equation is not defined at that point

Some background: define a process $Y_t=\frac{1}{t}\int_0^tX_sds$, where $X_t$ is a standard Brownian motion. Then I define a function $u(t,X_t,Y_t)$ and require it to be a martingale. Thus, by ...
5
votes
1answer
1k views

Mixed boundary conditions Finite Element Method

I have the following problem in Finite Element Method $$ -(\alpha u')' + \beta u' + \gamma u = f$$ with $ \Omega = (0, 1)$, $ u(0) = 0 $ and $ u'(1) = 3 $ to be able to write the weak formulation ...
2
votes
1answer
388 views

rate of convergence for the second order accurate method on two dimensional grid

I am solving $u_t=u_{xx}+u_{yy}$ with a solver which is locally $O(dx^2+dy^2+dt^2)$. I use the following norm for the error between the numerical vector solution and the analytical solution $$||e||_2=\...
2
votes
1answer
295 views

Partitioning of mesh with 20 noded hexahedral elements [closed]

Is there a way to partition a mesh consisting of 20 noded hexahedral elements for parallel processing? I used METIS for partitioning mesh with 8 noded hexahedron elmements, which works fine but i don'...
2
votes
3answers
2k views

Scalar vs. vector potential for magnetostatics

When trying to solve a magneto-static boundary-value problem (BVP) ($\nabla \times \mathbf{H} = \mathbf{J}$ and $\nabla \cdot \mathbf{B} = 0$), one can use either the magnetic vector potential $\...
6
votes
2answers
4k views

Deriving the element stiffness matrix for 2D linear elasticity

I'm following the derivation from Finite Element Method using Matlab 2nd Edition, pg 311-315, which derives of the local stiffness matrix for planar isotropic linear elasticity as follows: Force ...
4
votes
2answers
1k views

Should the discrete $L^{\infty}$ norm error increase as the mesh refines?

I'm working on an finite element code to solve the boundary value problem: $$-\frac{d}{dx}\left[ k \frac{du}{dx} \right] = f $$ $$u(0)=u(1)=0$$ The matlab code is available here. I'm testing this ...
8
votes
1answer
586 views

Using algebraic multigrid for preconditioning convection-diffusion operators

I implemented a Navier Stokes based on FEM discretization and PETSc for solving the linear system of equations. To create an efficient solution procedure, I follow the paper "Efficient ...
9
votes
2answers
4k views

How to remove Rigid Body Motions in Linear Elasticity?

I want to solve $K u = b$ where $K$ is my stiffness matrix. However some constraints may be missing an therefore some rigid body motion may be still present in the system (due to eigenvalue zero). ...
8
votes
2answers
571 views

Helmholtz and Biharmonic equation examples with exact solution

I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare my numerical solutions with it. I was able to find quite a few ...
6
votes
2answers
420 views

FEM toolbox for discretization of higher order PDEs

Is there any (open source) FEM toolbox that allows the direct discretization of higher order PDEs without the need to split them up into systems of second order?
2
votes
2answers
529 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 $$...
5
votes
4answers
520 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 ...
4
votes
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
votes
1answer
174 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
votes
1answer
339 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
votes
1answer
235 views

Looking for parMetis visualizer?

Is there any visualizer for parMetis (mpmetis), which can visualize FEM mesh grids after partitioning?
8
votes
2answers
427 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
votes
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
votes
1answer
160 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
votes
2answers
946 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} \...
3
votes
2answers
606 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
votes
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-...
8
votes
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 ...
16
votes
3answers
5k 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
votes
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 ...
2
votes
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 ...
2
votes
0answers
62 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.)...
29
votes
9answers
2k 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
votes
5answers
587 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
votes
3answers
938 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
votes
1answer
267 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''(...