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 ...

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Convex optimization for symmetric (but not positive definite) problems?

Can one employ convex optimization for symmetric but not positive definite problems? I tried using MATLAB's quadprog() function to solve this problem: ...
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37 views

What should be the number of boundary conditions of a PDE

As far as I know, for getting a unique solution to a PDE we should impose some boundary conditions to the PDE. "The number of required auxiliary conditions is determined by the highest order ...
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15 views

Dynamic reference configuration

I am solving a problem where I have a dynamic reference configuration for a solid, i.e., I am updating the reference mesh at every timestep. The problem does entail large deformations. If my solid is ...
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1answer
80 views

Quadrature order for finite elements and time dependent discontinuous Galerkin

When setting up a finite element system you have to use quadrature to calculate the integrals. I'm having trouble understanding what order rule to use. I know of some rules of thumb, for example with ...
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1answer
42 views

FENICS subdomains - restriction/ prolongation operators

I am trying to implement my own multigrid method in fenics. Is there any "smart/ fenics" way how to assemble subdomains and obtain restriction/ prolongation operators ? Thanks!
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2answers
75 views

How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices

I am wondering how Dirichlet boundary conditions in global sparse finite element matrices are actually implemented efficiently. For example lets say that our global finite element matrix was: $$K = ...
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5 views

Log-out on Computational Science [migrated]

Is it possible to log out from this website? I am trying to find the log out button - but not very successfully.
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93 views

How to compute this double integral?

Let $$T=1, K=100, S_0=100, \sigma=0.05, r=0.15. $$ Define $\nu:=\frac{2r}{\sigma^2}-1$ and $$H(y,z)=\frac{z e^{\pi^2 /4y}}{\pi \sqrt{\pi y}}\int_0^{\infty} e^{-z \cosh(u) -u^2/(4y)} \sinh(u) ...
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2answers
71 views

3D Divergence-free Stokes equations

I would like to have a divergence-free formulation of the Navier-Stokes equations for creeping flow (a.k.a. Stokes equation, N.-S. without the inertial terms) in 3D, for purposes of tracking particle ...
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94 views

Are there any possible applications of real-time Finite Element Analysis?

FEA when applied to solid mechanics / structural engineering result in the solution to the equation $$ F = K\cdot x $$ $F$ being the forces, $K$ the stiffness of the system and $x$ represents the ...
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296 views

What is the purpose of the test function in Finite Element Analysis?

In the wave equation: $$c^2 \nabla \cdot \nabla u(x,t) - \frac{\partial^2 u(x,t)}{\partial t^2} = f(x,t)$$ Why do we first multiply by a test function $v(x,t)$ before integrating?
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1answer
161 views

Finite Difference Beam Propagation Method problem

I am trying to implement the finite difference beam propagation method to study the propagation of a TE light signal through a waveguide. However, my solutions are exponentially growing, and display ...
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71 views

Dissipative time-stepping scheme for first order in time system

When solving semi-discrete equations (originating from finite element models, for example), which are second-order in time of the form \begin{equation} M\ddot d + C\dot d + Kd = F, \end{equation} ...
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2answers
113 views

Absorbing boundary conditions for acoustics in Discontinuous Galerkin

Note: I'm trying to implement a Discontinuous Galerkin method, as kind of a way to learn about these things. As of now, I've taken the acoustic wave equation $c^2 \nabla \cdot \nabla u(x,t) - ...
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1answer
46 views

Raviart-Thomas elements global definition and compact support

As per the suggestion by Christian in the comments here, as part of my continuing quest to understand the Raviart-Thomas (RT) elements I'd like to know how exactly the RT elements are defined ...
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2answers
60 views

Looking for reference on Streamline Upwind Petrov Galerkin finite elements for incompressible unsteady Navier-Stokes

I am looking for a relatively simple book/paper that explains the basic Streamline Upwind Petrov Galerkin (SUPG) method for solving the incompressible unsteady Navier-Stokes equations. Most of the ...
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1answer
108 views

Raviart-Thomas elements on reference square

I'd like to learn how the Raviart-Thomas (RT) element works. To that end I'd like to analytically describe how the basis functions look on the reference square. The goal here is not to implement it ...
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1answer
105 views

H(curl) conforming Nédélec-Elements to satisfy div(B)=0

Most authors are very clear that it's very dangerous to just use $\mathrm{H}(curl)$ conforming edge elements, which are divergence free, to satisfy $\mathrm{div}(\mathbf{B})=0$ and implement this ...
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54 views

generalized eigenvalue problem, SLEPcEigenSolver no eigenvalues with negative reals

We are working on the model of a kinematic fluid dynamo in FEniCS. It tries to explore whether a magnetic field can be excited if a conducting fluid flows with a certain velocity field through some ...
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1answer
113 views

Darcy flow finite elements

The Darcy equations for porous media flow are given by: $\frac{\mu}{\kappa}\mathbf{u} - \nabla p = \mathbf{0}$ $\nabla\cdot\mathbf{u} = 0$ where $\kappa$ is the permeability and can in general be ...
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1answer
54 views

How to calculate divergence and vorticity from a velocity field using finite elements

I am in the process of writing a finite element solver for the Navier-Stokes equations and am having trouble computing things like the divergence and vorticity correctly. Currently to compute the ...
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2answers
143 views

FEM: Symmetry of stiffness matrix

After my first course in the Finite element method I understand that elemental and global stiffness matrices are symmetric. What is the reason behind symmetry of stiffness matrices? Is it because ...
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41 views

Petrov-Galerkin enrichment method for Darcy equation

I was reading about Petrov-Galerkin Enrichment Method for Darcy equations. Here are a couple papers that discuss this in detail: A Petrov–Galerkin enriched method: A mass conservative finite element ...
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126 views

Verification in Eigenvalue problems

Let us start with a problem of the form $$(\mathcal{L} + k^2) u=0$$ with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
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1answer
92 views

Implementing the pressure correction method using finite elements

Ok so I am nearing the completion of my finite element Navier-Stokes solver that uses the $\theta$-method for time stepping and the pressure correction method for the pressure. I am following the ...
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1answer
65 views

Method of Manufactured Solutions for non-differentiable coefficients

The Method of manufactured is commonly used for verification of computational science codes. I want to use the method for verification of Navier-Cauchy (elasticity) equations with periodic and ...
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1answer
160 views

Solving Linear Systems in Julia

To give you some context, I am currently implementing a simple finite element solver in Julia. I am getting run-times that are 70% of a Matlab code. (Both codes are essential equivalent in ...
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1answer
56 views

How to update velocity to include pressure when using P2/P1 elements

I am in the process of writing a finite element code to solve the Navier-Stokes equations using the theta method for time stepping (basically Crank-Nicholson for diffusion and forward Euler for ...
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2answers
142 views

What's the difference between C0 penalty methods and Discontinuous Galerkin methods?

I am trying to understand the DG FEM methods, but I got lost in their definitions. In some papers I read that the "C0 penalty method" is one example of the DG method, but sometimes they are separated ...
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2answers
100 views

How to calculate efficiently mesh edges midpoints?

I have a 2D mesh of triangles used in Finite Element method to discretize the domain. I want to calculate the midpoints of all the edges because I want to use $\mathbb{P}^2$ elements. I am using ...
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49 views

Stability analysis for explicit time discretization in the Finite Element Method

I have been looking for stability analysis of general reaction-diffusion problems, of the form $\frac{\partial u}{\partial t}=\nabla\cdot D\nabla u-k\,u$ , to be solved using the standard Finite ...
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25 views

Converting a labelled pixel grid to an isoregion geometry file

I am working with ASCII text files that each contain a grid of integer labels. The grid can be interpreted like the pixels of an indexed raster image. e.g. $\begin{pmatrix} 1& 1& 1& ...
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130 views

What exactly causes mesh locking in thin plate bending problems?

In thin plate bending problems, it would be very nice if we could model a thin rectangular plate with a mesh of many elements in the transverse directions, but only a single element in the ...
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110 views

How to correctly normalize modulus and phase of an eigenvector?

I am solving a linear stability problem using finite element discretization. Then, I have a generalised eigenvalue problem: $$ \lambda M x = J x.$$ I obtain complex eigenvalue and eigenvectors from ...
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1answer
144 views

What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
3
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1answer
84 views

P2/P0 and P2/(P1+P0) finite elements for stokes and darcy equations

Typically I have seen P2/P1 elements used for Stokes equation, but I want to use P2/(P1+P0) and P2/P0 elements because I want to ensure local mass conservation. When I say P2/(P1+P0), I simply mean ...
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12 views

Resonant interactions, instability of nonlinear waves- which numerical method:HOSE or FNPT-based?

I am studying non linear interactions for water waves. Resonant interations type problems or instability of waveforms are the kinds of problems of interest. Resonant interactions and related ...
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2answers
108 views

Relationship between FEM solutions of PDE with different spatial resolutions

I use FEM to simulate deformations of elastic objects for animation applications in computer graphics. The governing equation is generally with the form: $$ \mathbf{M}\ddot{\mathbf{u}} ...
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1answer
50 views

How does the animation work in eigenvalue problem of FEM

I have used free vibration analysis in FEM. After analysis, we can usually use animation to see the motion of each eigenmode (In Abaqus or Comsol, I would choose either half harmonic or full ...
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1answer
94 views

openfoam - Programming customized PDEs

I am looking for a method to automatically solve custom PDEs on a custom control volume. Specifically I would like to solve equations similar (but not exactly alike) to: $$\frac{\partial y}{\partial ...
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82 views

Arpack and Matlab give different values for eigenvalues

I am solving a generalized eigenvalues problem with inversed complex shift: $$(M-\sigma J)^{-1}J \boldsymbol{x} = \boldsymbol{x} \nu \enspace .$$ My matrices are obtained from a finite element ...
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40 views

Integro-differential eigenvalue problem, COMSOL

In my research I encounter an eigenvalue integro-differential equation of the form: $$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace ...
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0answers
47 views

Selecting a Finite Element discretization for subsurface flows

Are there guidelines to follow in selecting the right FE discretization. Specifically, for subsurface flow models like Darcy's equation (both single and multi-phase), Richard's equation, etc. I know ...
4
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1answer
315 views

How to formulate lumped mass matrix in FEM

When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...
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2answers
129 views

How to deal with nonlinear term in Navier Stokes equations (finite element code)

I am trying to solve the Navier Stokes equations using the finite element method. I plan on using the pressure correction method to deal with the pressure and an implicit time stepping scheme for ...
2
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1answer
112 views

Is my matrix symmetric?

I obtained a mass matrix through Finite Elements discretization. Now, I want to check if it is symmetric. To do that I subtract to my matrix $M$ its transposed $M^T$. The result is another matrix of ...
3
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1answer
99 views

Meshless Methods and the Kronecker Delta Property

In texts I often read that meshless methods, as opposed to the FEM, do not exhibit the Kronecker Delta Property [1]: $N_I(x_J)\neq\delta_{IJ} = \begin{cases}1 &\mbox{if } I=J \\0 ...
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41 views

Gauss Integration over Zero Order Element

I'm working with the Boundary Element Method and want to integrate an expression over a triangular region. I would like to use Gauss Integration to do this, but I'm having trouble since the triangular ...
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65 views

Proof of convergence for Newton method in finite element analysis

The Newton method in FEA is to solve for a non-linear equation where stiffness matrix is a function of displacement: And to assemble the stiffness matrix from constitutive laws, we need the tangent ...
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2answers
168 views

Is “tangent stiffness matrix” the same as “stiffness matrix”?

I'm trying to implement nonzero Displacement Boundary Conditions in VegaFEM on a non-linear model, using the method outlined in §3.6.2 of University of Colorado's intro to FEM (modify $f = Ku$: set ...