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
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 ...
0
votes
1answer
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
3
votes
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 ...
0
votes
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 ...
3
votes
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{\...
1
vote
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-...
1
vote
0answers
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 ...
4
votes
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
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, ...
5
votes
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 ...
0
votes
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 ...
0
votes
0answers
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 $...
2
votes
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 ...
0
votes
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 ...
4
votes
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. ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
4
votes
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
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-...
0
votes
0answers
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 ...
1
vote
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,...
4
votes
0answers
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 ...
5
votes
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
$$
...
1
vote
2answers
261 views
How does one calculate reaction force in FEA?
I wrote a UEL (User Element in Abaqus) for one element and compared to a reference UEL which used standard FEM, where the results agreed satisfactorily, except the reaction force. The stress, strain, ...
0
votes
1answer
73 views
Structural mechanics simulation using FLUENT compared to analytical solution
This is a continuation of my previous post, 1D analytical solution vs FEM solution for a bar under compression. For some reason, I cannot comment in it.
The analytical solution to the 1-D static ...
2
votes
1answer
78 views
Non-linear flux interface condition - variational formulation
Context: I am working on implementing this paper and I am struggling to come up with a variational formulation for the Butler-Volmer interface conditions.
To simplify my question I consider the ...
0
votes
1answer
59 views
Lagrange multipliers in minimization problem with bilinear forms
Let $V$ a Hilbert space, $a:V\times V\rightarrow \mathbb{R}$ a bounded, symmetric and positive bilinear form and $f:V\rightarrow\mathbb{R}$ bounded.
Is well known that problem
$$\left\lbrace\begin{...
0
votes
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 ...
2
votes
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
...
1
vote
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 ...
2
votes
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
...
0
votes
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 ...
0
votes
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
$$\nabla u=\begin{pmatrix}\dfrac{\partial u_1}{\partial x}&\dfrac{\...
0
votes
1answer
79 views
how i can prove the exists and unique of the solution of the Helmothz equation with a robin boundary condition with complex coeficient
I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation
$$
\iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)...
4
votes
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 ...
3
votes
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 ...
2
votes
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 ...
2
votes
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 $\vec{r}$ ...
1
vote
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 ...
1
vote
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 ...
4
votes
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 ...
1
vote
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 ...
1
vote
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 ...
