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 ...
1
vote
1answer
29 views
FEniCS: how to access coordinates when writing an equation for a trial function
I need to solve the following equation in FEniCS:
$$
\boldsymbol{\nabla} \cdot \begin{pmatrix}
(1+\alpha^2y^2)\frac{\partial u}{\partial x} - xy\alpha^2\frac{\partial u}{\partial y} \\
- ...
4
votes
1answer
75 views
Fenics: time-independent Sine-Gordon equation
Is there a code for the equation
$$
\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2} = \sin(u)
$$
or for the sine gordon equation in two dimensions because I want to change some ...
2
votes
0answers
52 views
How to apply periodic boundary conditions to a Raviart-Thomas finite element space in Fenics?
I'm trying to use periodic boundary conditions within a Raviart-Thomas finite element space in Fenics (dolfin 1.2.0) in a Ubuntu 12.04 (amd64) machine (with python 2.7). If other FE space is used, ...
8
votes
2answers
71 views
Which preconditioners (and solver) in PETSc for indefinite symmetric systems should I use?
My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow):
\begin{pmatrix}A & B^T \\ B & C\end{pmatrix}
where $C = 0$ is the typical case (I have even ...
3
votes
1answer
58 views
Compute spatial second derivatives in Isogeometric analysis
Motivation:
In isogeometric analysis, state variables(e.g. displacement) are defined in the parametric domain, which can be mapped to the physical domain by $\boldsymbol{\xi}\mapsto \boldsymbol{x}$ ...
-1
votes
1answer
47 views
xml Mesh format -Fenics
I imported an xml file format and the nodal connectivty seems to be different than what FeniCS usually follow..
Is it possible to sort the nodes or element connectivity same as Fenics ???because ...
6
votes
0answers
143 views
+100
Simple turbulence model appropriate for buoyancy-driven cavity like problem
Which turbulence model is suitable for resolving incompressible buoyancy-driven flow of a fluid within an cylindrical ampoule?
I prefer turbulence model which is sufficiently simple so that fully ...
7
votes
3answers
196 views
Poisson equation: Impose full gradient as boundary condition via Lagrange multipliers
I have a physical problem governed by the Poisson equation in two dimensions
$$
-\nabla^2 u = f(x,y), \; in \; \Omega
$$
I have measurements of the two gradient components $\partial{u}/\partial{x}$ ...
1
vote
0answers
30 views
Finite Element, NASTRAN, DMAP, how to print the differential stiffness matrix in .f06 output file
We are working on an optimization problem in which we can approximate the eigenvalue calculation by assuming a constant eigenvector, using the formula:
$$
...
5
votes
1answer
95 views
Stabilization of convection-dominated flow and turbulence modeling
Are stabilization techniques for convection-dominated flows like SUPG+PSPG, interior penalty methods, etc. able to handle turbulent flows without tubulence model being employed, at least up to some ...
5
votes
1answer
169 views
Implementing Explicit formulation of 1D wave equation in Matlab
So the theory is straightforward. We have:
$$\frac{\partial^2U}{\partial t^2}=c^2 \frac{\partial^2U}{\partial x^2}$$
discretizing it gives:
$$\frac{U(i+1,j)- 2U(i,j) + U(i-1,j)}{(\Delta t)^2} = c^2 ...
2
votes
1answer
88 views
Compute heat flux from Galerkin solution
I have a Galerkin solution for a heat equation
$$
u_t = \Delta u + f
$$
with Dirichlet conditions
$$
u=0, \qquad x \in \partial\Omega
$$
The time discretization is done using a BDF scheme. How can ...
1
vote
0answers
41 views
Is there MPI_Star instead of MPI_Win (window) for MPI_Get?
In FEM applications, each processor needs only data from neighboring processors, which gives a stencil or star pattern of communication. In MPI_Win for MPI_Get, one needs to setup a group of ...
6
votes
1answer
134 views
What is the impact of C++11 move semantics in the context of scientific computing?
C++11 introduces move semantics which can, for example, improve code performance in situations where C++03 would need to perform a copy construction or copy assignment. This article reports that ...
3
votes
3answers
123 views
Online Poisson Solver
I'm wondering if anyone can point to a browser-based FEM (or other) 2D PDE solver for simple elliptic problems. It seem like there ought to be a javacript implementation that would allow for the ...
4
votes
2answers
73 views
Existence of incomplete cholesky factorization
What is the current state of research on the existence of incomplete cholesky factorizations (in the context of preconditioning) for symmetric positive definite matrices?
I wonder in particular ...
3
votes
4answers
98 views
What is the meaning of “preasymptotic” and “superconvergent”?
Precisely the title of the question.
I have encountered these terms in two areas: conjugate gradient method, and adaptive finite elements.
0
votes
0answers
35 views
Local axis follows origin node rotation
I'd like to define a local axis (unit vectors l, m and n) which once defined follow the rotation of the origin node, i.e. regardless of the deformation the local axis should be basically the same as ...
3
votes
0answers
48 views
Example of level set method
I am looking for easily to understand example of level set method used to track phases interfaces. I would like to solve it using FEM because my solution is based on the FEM solution of second Fick ...
6
votes
0answers
68 views
Energy conservation in the solution of the Helmholtz equation
This might be a silly question, but I know very little about the theoretical properties finite elements, so here goes. Suppose you were to solve the Helmholtz equation (let's say in 2D) with a ...
10
votes
4answers
169 views
Calculation of the sparsity structure for finite element matrices
Question: What methods are available to accurately and efficiently calculate the sparsity structure of a finite element matrix?
Info: I'm working on a Poisson Pressure Equation solver, using ...
3
votes
5answers
204 views
FEM library with support of simplex elements
My question can be rephrased as "FEM library like deal.II but for simplex elements".
Our scientific group works with very complicated 3D geometries, so usually we prefer tetrahedral meshes for our ...
4
votes
1answer
75 views
How to treat hexahedral element with shifted hanging node?
When using the Hexpress grid generator one gets hexahedral cells, possibly with hanging nodes. Because of a smoothing step, the hanging nodes can be shifted: they are not necessarily on the straight ...
2
votes
1answer
357 views
Finite Element Method: 2-D Poisson's Equation in Matlab, Gaussian quadrature
I'm having trouble understanding how to code 2-D Poisson's Equation with Dirichlet boundary conditions. What I have thus far is
Constructed square mesh with triangular elements
Assembled stiffness ...
2
votes
1answer
115 views
FEM for non-divergence form elliptic equation
The FEM is usually used with a weak form of PDE. But for the non-divergence form elliptic operator
$$
-a_1(x,y) \frac{\partial^2}{\partial x^2} - a_2(x,y) \frac{\partial^2}{\partial y^2}
$$
or ...
4
votes
2answers
95 views
I'm having trouble debugging multigrid. What to do?
I've spent far too much time coding and debugging multigrid. While I clearly can't post all of my code as it would be silly to ask someone to go through all that code, is there anything I should pay ...
4
votes
3answers
123 views
implicit vs. explicit domain decomposition methods
I've been working on a finite element code on unstructured methods, which I've parallelized using the Schur complement method. Here's a summary of how I did it:
Assign each triangle of the mesh to a ...
5
votes
0answers
79 views
Understanding and implementing the Heterogeneous Multiscale Finite Element Method
I'm following the explanation given by Weinan E and Bjorn Engquist (1994), pp 26-29, and I have a few questions about it. To understand my questions, I'll first try to explain what think I know, and ...
0
votes
2answers
131 views
Local and global coordinates in FEM
Am I right that in FEM we can associate a local coordinate system with every node, not with every element?
2
votes
1answer
204 views
Meaning of CFL condition on parabolic problems
I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method.
I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
5
votes
1answer
275 views
How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?
In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
2
votes
1answer
91 views
Diffusion-Transport problem FEM
I was looking at a book of FEM on problems of Diffusion-Transport.
$$-div(\mu \nabla u) + b \cdot \nabla u + \gamma u = f \qquad in~\Omega$$
$$u = 0 \qquad in~\partial\Omega\text{ (in the ...
1
vote
1answer
106 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 ...
5
votes
2answers
154 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 ...
2
votes
1answer
86 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
93 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 ...
5
votes
1answer
179 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 ...
0
votes
0answers
50 views
How to include parameters in plot title in comsol?
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
90 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 ...
2
votes
1answer
99 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 ...
4
votes
1answer
86 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 ...
7
votes
4answers
168 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
110 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 ...
2
votes
1answer
153 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
89 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
185 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 ...
1
vote
1answer
89 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 ...
2
votes
1answer
121 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 ...
2
votes
3answers
255 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 ...
4
votes
2answers
847 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 ...
