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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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100 views

Does a generic method for solving a system of PDEs exist?

There are generic methods for solving systems of ODEs numerically. Are there generic methods for PDEs? If so, what are they? If not, why not? To elaborate... Any set of ODEs can be written in ...
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47 views

How is nonlinear flux interface term assembled for Discontinuous Galerkin method for hyperbolic conservation laws?

For example, for 1D Burgers equation $$ u u_x = 0 \\ $$ equivalently, $$ \frac{dF(u)}{dx} = 0\\ F(u)= \frac{u^2}{2} $$ If I want to obtain $A_{ij},i\ne j$ for two DOFs ($U_i$ and $U_j$) of two ...
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49 views

Confusion about Rayleigh beam formulation

Sorry for disturbing you, my friends. Recently, I am reading books on Rayleigh beam theory where the rotary inertia can be taken into account without introducing rotational degrees of freedom. For ...
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33 views

How to analyze the dispersion and dissipation of a certain FEM?

In Finite Difference method or Finite volume textbook. Dispersion/Dissipation can be analyzed by set $u = u_0\exp{\omega t +\mathbf{kx}}$。 However, I cannot find something about this kind of analysis ...
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1answer
73 views

Why Coercivity is so important in FEM framework?

I know Lax-Milgram theorem is fundamental to FEM. But it did not explain what will happen if coercivity is not met. My understanding is if it is met, eigen value of the operator (or its corresponding ...
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3answers
97 views

Why is the test function space in FEM chosen with homogeneous boundary conditions?

It is so confusing, especially when I learns discontinuous galerkin method in broken Sobolev space and weak Dirichlet boundary condition. If the trial function is chosen with homogeneous boundary ...
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0answers
40 views

Should I expect computational gains using a second-order splitting method here?

I am trying to solve a three-dimensional baroclinic transport problem. The hydrodynamic (three-dimensional shallow water) equations are: \begin{align} \nabla \cdot \vec{v} = 0, \tag{1} \\ \frac{\...
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1answer
70 views

A reference request for computational plasticity

My background is in applied mathematics and I'm trying to learn plasticity. I have successfully understood the theory and finite element implementation of: linear elasticity, hyperelasticity (Neo-...
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71 views

Basic approach for numerical solution of PDE

I'm looking for some guidance on how to write a program to numerically solve a PDE. As an example for comparison in 1D: $$\frac{d^2u}{dx^2} = f\;\;\;\;u(0) = 0\;\;\;\;u(1) = 0$$ We could try 6 ...
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2answers
97 views

Analytical testing/quality control for scientific software in professional setting

I am charged with maintaining a buildserver on Teamcity which is meant to test our in house FE software. Currently our test suite consists of a list of benchmarks which run every time a commit is made ...
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1answer
48 views

Oscillation term in a posteriori error estimator

Assume that in the a (residual type) posteriori error estimator of some PDE is a term of the form $h_T\|g\|_{L^2(\Omega)}$ involved where $h_T$ is the diameter of an element and $g$ is some known data ...
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1answer
101 views

Influence of node numbering in a FEM problem?

In a FEM mesh, does the order of node numbering in an element has any importance? I'm currently trying to code my own FEM solver, which seems to work fine with quadrilateral elements, however I'm ...
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0answers
74 views

What is the mathematical and physical principle behind of RBE2 element?

I am writing a 3d linear finite element code to solve the standard linear elasticity equation on a tetrahedron mesh of a gearbox. Notice that, the two rectangular plates above the gearbox are fixed, ...
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1answer
113 views

How is rigid bodies implemented in finite element codes

I am writing a finite element code for structural analysis, and I want to implement rigid bodies. How is this usually done? Say that I have a square mesh, with one half of the mesh being defined rigid ...
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1answer
93 views

Parallelization of FEM calculations

I need to conduct some FEM calculations and I am wondering whether parallelization would be a good idea. The trouble is that my model is not especially large so it takes few seconds to solve a single ...
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36 views

Simplest meaningful PDE/FEM calculation for mechanical stress due to heat

W have a complicated structure on which we do some FEM calculations regarding electrical potentials and heat distribution. The equations have the form $\nabla\kappa\nabla u = f + g\rvert_{N}$ where $...
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1answer
91 views

Help debugging finite element solution in nonlinear elasticity

I'm writing some code to solve problems in nonlinear elasticity using finite element methods. I have been following Bathe's book but I am having trouble with some nagging details. My question is ...
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0answers
25 views

CGFEM - Adding a solvent flux to a 1D adv-diff system as a source term

I have a system consisting of a narrow pipe with porous walls where the inlet conditions are flow rate and initial concentration, and the goal is to determine the change in concentration along the ...
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2answers
127 views

Getting started with finite element modelling

I'm a high-schooler building a small vehicle for an independent study. I've had finite element modelling recommended to me as a way to save time during the design process, and I'd like to try it out. ...
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1answer
64 views

How to simulate thermal expansion in a 2D plane using FEA?

I am trying to model 2D thermal expansion of a square area inside another square using FEATool. I have simulated plane strain by incorporating forces pointing along the $[1 \,\,\, -1]^T$ direction ...
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0answers
46 views

Getting started with FEM: Ill-conditioned matrix when evaluating flux terms in conservation law?

I have a system of conservation laws of the form $$ \frac{\partial \mathbf{q}}{\partial t} + \nabla \cdot \mathbf{F}\!\left(\mathbf{q}\right) = 0 $$ I want to use finite elements to solve this ...
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20 views

Enforced (prescribed) displacements at more than 1 node of FE model [duplicate]

If I have a structural finite element model (could be continuum or frame elements), I was wondering if there is a way to enforce a prescribed displacement at more than 1 node in the model in a ...
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2answers
138 views

Unstructured mesh vs hybrid structured/unstructured for numerical simulations

While answering one of the questions on meshing process, I encountered a lack of understanding on my end for the comparison of the mesh quality. First, consider an unstructured mesh created in GMSH ...
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1answer
76 views

How to implement Galerkin Method of Lines / FEM with black box integrators in scipy

Suppose I have some time dependent PDE, which can be written in the strong form as $$ \frac{\partial u}{\partial t} + \mathcal{L}(u) = f $$ Where $\mathcal{L}$ is some differential operator. If I ...
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1answer
60 views

Reordering algorithm for minimization of ram usage of a skyline matrix

The stiffness matrix of $Ax=B$ system of linear equations, where $A$ is an $n\times n$ symmetric matrix stored in the form of symmetric skyline matrix, that is associated with a finite element model ...
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1answer
66 views

FEM-Laplace with Dirichlet in only a few points: Nonsingular operator?

Let's consider the FEM discretization of the Laplace operator without boundary conditions, i.e., $$ a(u,v) = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot \nabla u) v. $$ For one-...
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38 views

Applying base excitation to a MATLAB state-space

I have a state space model that was provided to me by exporting it from an external FEA program. The model can be described as $\dot x = Ax + Bu$ $y = Cx + Du$ This model assumes forces and ...
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1answer
57 views

FE discretisation of normal to displacement vector

Having shape functions $N_i(\xi,\eta), i = 1,...,N_n$ and, a normal vector $n = (n_x,n_y,n_z)$, a thickness function $F_\tau (\zeta), \tau = 1,...,N_\tau$ and nodal variables $\mathbf{Q}_u = (Q_u,Q_v,...
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63 views

Data transfer in the context of adaptive mesh in finite element method (FEM)

In the context of adaptive mesh in FEM, after a new mesh is created, the data on the old mesh are to be transferred to the new mesh. For the data on the integration points (IPs), it seems the usual ...
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2answers
119 views

Residual norm of PDE discretization: correspondence in the continuous problem?

Solving a linear PDE like $$ \Delta u = f \quad\text{on } \Omega,\\ n\cdot \nabla u = 0 \quad\text{on } \Gamma, $$ with Finite Elements usually goes like this: Create the discretization $Au=b$ via $$ ...
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2answers
261 views

How does one calculate reaction force in FEA?

I wrote a UEL (User Element in Abaqus) for one element and compared to a reference UEL which used standard FEM, where the results agreed satisfactorily, except the reaction force. The stress, strain, ...
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1answer
73 views

Structural mechanics simulation using FLUENT compared to analytical solution

This is a continuation of my previous post, 1D analytical solution vs FEM solution for a bar under compression. For some reason, I cannot comment in it. The analytical solution to the 1-D static ...
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1answer
78 views

Non-linear flux interface condition - variational formulation

Context: I am working on implementing this paper and I am struggling to come up with a variational formulation for the Butler-Volmer interface conditions. To simplify my question I consider the ...
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1answer
59 views

Lagrange multipliers in minimization problem with bilinear forms

Let $V$ a Hilbert space, $a:V\times V\rightarrow \mathbb{R}$ a bounded, symmetric and positive bilinear form and $f:V\rightarrow\mathbb{R}$ bounded. Is well known that problem $$\left\lbrace\begin{...
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1answer
90 views

Discontinuous Galerkin - Inhomogeneous Dirichlet B.C. for 1D Poisson Equation

I am trying to get some code working for the 1D Poisson equation using the textbook: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications. I use the following formulation (for ...
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0answers
47 views

2D reaction-diffusion system simulation

I am a complete beginner in numerical simulation and I am pretty lost about how to tackle this problem. I have been trying for some time to find the steady state (or simulate), the following system ...
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1answer
50 views

Integral transformations for isoparametric quadrilateral elements

Suppose I have a reference quadrilateral on $[-1, -1] \times [-1, 1]$ with reference coordinates $\xi, \eta$ and a mapping to an isoparametric quadrilateral in 'physical space' described by ...
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1answer
104 views

1D analytical solution vs FEM solution for a bar under compression

I simulated the compression problem in ANSYS and compared to the analytical solution and found some discrepancies. The classical solution to the 1-D compression problem is: \begin{align} u(x) = Cx ...
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1answer
49 views

Gmsh exporting wrong mesh DATA [closed]

so hopefully I'll be using gmsh to make meshes out of 2-D cross sections with o thickness. I tried to make a structured mesh with quad elements of a rectangular ...
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0answers
76 views

Implementing fem with a polynolial space such that $\nabla u=-\nabla u^t$ and its orthogonal space

I need to use the polynomial (vector) space such that $\nabla u=-\nabla u^t$, where the gradient of a vector $u=(u_1,u_2)$ is $$\nabla u=\begin{pmatrix}\dfrac{\partial u_1}{\partial x}&\dfrac{\...
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1answer
79 views

how i can prove the exists and unique of the solution of the Helmothz equation with a robin boundary condition with complex coeficient

I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation $$ \iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)...
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1answer
73 views

Computation of residual error indicator in adaptive mesh refinement (FEM)

I am attempting to implement an $h$-adaptive FEM scheme for the following simple 1D problem: $$ -u''(x) = f(x) \;\text{in}\;(0,1)\\ u(0) = u(1) = 0. $$ To this end, I begin with a coarse, uniform ...
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0answers
94 views

Finite Element Stabilization for Drift-Diffusion/Advection-Diffusion Equations

I've tried my best to look through the relevant suggested similar questions when posting this, and hopefully this contains enough new material to not be considered a duplicate. I'm currently trying ...
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0answers
50 views

Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?

I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
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1answer
75 views

Calculation of the EFIE integral

I need help computing the following integral: $$ \int_{}\frac{(1+jk|\vec{r}-\vec{r}^\prime|)e^{-jk|\vec{r}-\vec{r}^\prime|}}{|\vec{r}-\vec{r}^\prime|}d\vec{r}^\prime $$ in this integral $\vec{r}$ ...
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1answer
67 views

finite differences on a slanted grid — advection diffusion equation

I used pretty much all my expertise with finite differences to solve an advection-diffusion equation with space-dependent coefficient with a grid in the $x$,$z$ domain with regular spacings. Something ...
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1answer
112 views

Can we simulate rigid body motion using finite element analysis?

I was wondering if we could model rigid body motion of bodies using finite element models. Particularly I'm interested to know if we can model motion of objects with no constraints or with some ...
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0answers
70 views

Conservation in finite element codes [duplicate]

Typical finite volume methods are conservative, because fluxes (of e.g., mass or energy) are always between neighboring cells. Is the same generally true for finite element codes? Do I correctly ...
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0answers
117 views

Alternatives to Newton-Raphson for nonlinear elasticity via finite element

As far as I have seen, solving problems of nonlinear elasticity using the finite element method proceeds by linearizing, either around the initial configuration (total Lagrangian approach) or around ...
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1answer
130 views

How would one solve the wave equation using the finite element method?

I would like to solve $$\alpha u_{tt} = -\nabla^2u$$ with $\frac{\partial u}{\partial n} = 1$. On using a Galerkin approximation I obtain $$M\ddot{c}=\frac{1}{\alpha}(Dc+b)$$ where $M$ is the mass ...