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.
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24 views
Fastest method for elliptic pde
What is the fastest method (language, solver, etc.) to numerically solve elliptic equations such as Poisson or Stokes equations in a cube in 3D with mixed boundary conditions?
Especially, a ...
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0answers
16 views
Infinity norm inverse estimate
Let $V_h$ be the space of linear finite elements on triangles. From the scaling argument, we can obtain the following inverse estimate inequality:
$$ \| v_h \|_t \leq ch^{m-t} \| v_h \|_m, $$
for ...
4
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0answers
30 views
Numerical methods for the continuity equation with Sobolev vector field
Consider the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
1
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0answers
27 views
Boundary conditions for solving the time-independent SE for the hydrogen atom
I am trying to solve the schrodinger equation for the hydrogen atom numerically, using finite elements, with matlab's solvepdeeig(). I have a hard time getting the solution to be right, and it seems ...
0
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1answer
36 views
How to efficently plot a finite element mesh solution with matplotlib
I am looking for the most efficient way to plot a mesh using matplotlib given the following information, coordinates of each node, what nodes belong to each element, and the value each node has. Below ...
4
votes
1answer
122 views
Is a symmetric bilinear form necessary to ensure a weak formulation has a solution?
Problem
I want to convert the general second order linear PDE problem
\begin{align}
\begin{cases}
a(x,y)\frac{\partial^2 u}{\partial x^2}+b(x,y) \frac{\partial^2 u}{\partial y^2} +c(x,y)\frac{\...
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0answers
32 views
Rigid FEM applicability for Viscoelastic Material
I have seen some scholars implement rigid FEM on viscoelastic materials, I would like to know whether this sound approach.
In the following reference, they have implemented RFEM on viscoelastic ...
5
votes
1answer
137 views
Advantages and disadvantages of space-time finite element methods
I have heard of space-time finite element methods. Although I was able to find some articles that describe the different possible methods from a mathematical point of view (thanks to Space-time finite ...
2
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0answers
69 views
Numerical solution to N-dimensional diffusion on simplex?
Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
2
votes
0answers
33 views
Solver for generalized eigenvalue problem with multipoint constraints
We have the following generalized eigenvalue (set of) problem(s)
$$[K_R(\kappa)]\{u_R\} = \omega^2[M_R(\kappa)]\{u_R\}\quad \forall \kappa \in [\kappa_0, \kappa_1]$$
with
\begin{align}
&K_R(\...
1
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0answers
67 views
Constraining the total volume in Finite Element Methods
I have a diffusion problem which can be broken down to be:
$-\Delta u = f(u) $ on $\Omega ~/~ \Omega_{int}$
$u = 1$ on $\Omega_{int}$
Note that this is an internal Dirichlet constraint to the ...
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1answer
85 views
Finite Element - Flux Calculation
I am solving an advection-diffusion equation using the FEM and am having trouble calculating my fluxes.
I start with the equation,
$$\frac{\partial n}{\partial t} = \frac{\partial j_{n}}{\partial x}\...
3
votes
1answer
58 views
Parallelizing FEM for elliptical PDEs with n >1
For a little personal project, I am picking up my FEM skills again. I learned a lot about the theory back in university and I am able to implement a simple FEM solver for specific problems but I was ...
5
votes
3answers
196 views
Why is the FVM traditionally used in CFD, and FEM in computational structures?
Most CFD codes use FVM. Most computational structures codes use FEM.
Why is the FEM not frequently used in CFD, and why is FVM not frequently used in FEM?
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0answers
55 views
Finite element method for Surface integrals using polar coordinates
I am trying to solve a 2D elliptic PDE (see complete electrode model for electrical impedance tomography) using the finite element method (FEM) over a circular region $\Omega$. I have discretized the ...
1
vote
1answer
58 views
Area and volume of P2 elements
Context:
For a fluid solver, I need to compute areas and volumes of curved elements. My curved elements are quadratic triangles and quadratic tetrahedra, defined by their Lagrange nodes.
Question:
...
2
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0answers
47 views
Structural Analysis Library
Can anyone recommend a structural analysis library that satisfies the following requirements:
C++ API
Simulate both beam elements and shell (slab) elements
Both static and dynamic analysis
Free and/...
1
vote
1answer
49 views
Mixed formulation in 1D
I have been working on a hybrid dimensional model using the mixed FEM formulation, in which 3D elements and 2D elements are combined by certain relationships between the degrees of freedom (DOFs) ...
6
votes
1answer
68 views
Dirichlet boundary conditions in generalized eigenvalue problem
Let us consider a problem of the form
$$(\mathcal{L} + k^2) u(\mathbf{x})=0\, ,\quad \forall \mathbf{x} \in \Omega$$
with Dirichlet boundary conditions
$$u(\mathbf{x}) = 0, \quad \forall \mathbf{x} ...
0
votes
1answer
53 views
Imposing zero mean condition in FEM
I wanted to solve a periodic elliptic equation of the form $$-\nabla\cdot(A\nabla u)=-\nabla\cdot F$$ on $Y=[0,2\pi)^d$ using FreeFem++, where $A$ and $F$ are $Y$-periodic. The space of solutions is $...
1
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0answers
51 views
Finite Element Model of Euler-Bernoulli Beam Theory with Isoparametric Element
In the formulation of Euler-Bernoulli Beam Theory, there are two degrees of freedom at a point, $w$ and $\frac{dw}{dx}$. Typically, the finite element model of this theory uses cubic polynomial for ...
1
vote
1answer
130 views
Normalization of polynomials for discontinuous Galerkin methods (DGM)
I was curious if someone could share their opinion on this matter. I have noticed that some people in literature normalize their Legendre polynomials, i.e. divide or multiply the polynomial by $$\...
1
vote
0answers
65 views
How to solve potential flow with FEM, stream function, and the Kutta condition?
I'm trying to solve two-dimensional potential flow over airfoils with
the finite element method, using the stream function formulation
($\Delta\psi = 0$, $u = -\partial\psi/\partial y$, $v =
\partial\...
8
votes
1answer
187 views
Size of jump for piecewise discontinuous approximations
If one has a sufficiently smooth function $u$ that is approximated by a piecewise constant function $u_h=\Pi^0_h u$ on a mesh of cell size $h$ (where $\Pi^0_h$ is the $L_2$ projection onto the ...
1
vote
1answer
74 views
Node renumbering in a 2D mesh
I have a 2D domain which is discretized using Q4 elements. I have the nodal positions and the element connectivity matrix. I would now like to renumber the nodes in such a way that all the interior ...
1
vote
1answer
97 views
Relation between conjugate gradient method and finite elements method
What is difference beetwen this two method? Are these methods far from each other or are these methods complement each other? Could you take an example?
3
votes
3answers
129 views
$H^1$-convergence rate of finite element method for Poisson equation, depending on element order
I wanted to verify my FEM-program by applying the method of manufactured solutions, while solving the Poisson equation in two dimensions using the continuous Galerkin method
$$-\nabla^2u=f$$
with
$$u=...
0
votes
1answer
36 views
Given co-ordinates of 8 vertices, how to calculate the outward normal and surface area for each face of a irregular hexahedron?
I am working on an FEA mesh of hexahedron elements. The elemental level calculations involve finding the surface normals and area for each surface of a hex element. I preferred the vector cross ...
3
votes
0answers
108 views
Gmsh for 3D volume with inclusions [closed]
In an attempt to create three-dimensional volumes with inclusions in Gmsh I stumble upon a problem which was non-existent in the two-dimensional case.
I'm using the OpenCASCADE geometry kernel ...
-1
votes
1answer
51 views
Finite element convergence rate and possion's ratio
I am running simulations of a cantilever beam where it is fixed on one end and negative force applied to the other end. The first simulation is with 4-node linear quadrilateral elements and the other ...
1
vote
1answer
48 views
Strain from FEM simulations to strain gauge measurements
I am looking for some intuition in making comparisons of FEM simulations to experimental measurements. In particular, I am interested in comparisons to strain gauge readings, and perhaps even LVDTs.
...
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0answers
31 views
Is it possible to obtain a 'relaxing lengths' for 8 node hexaedral element
I am using 3D FEM with 8-node brick elements to model a certain type of growth/expansion, which is not a plastic deformation. First, I apply a pressure/load and I get displacement and the new position ...
2
votes
0answers
51 views
Why do stabilized formulations for the Navier-Stokes equation maintain the convergence rate for high order polynomial interpolation?
I have a quick questions which has been troubling me lately.
When reading the FENICS Finite Element Book they assess various approaches to solver the Stokes equation. Obviously, they discuss the ...
1
vote
0answers
85 views
Lower bound for bilinear form in FEM
I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries.
I did some research and found:
$$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(...
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votes
1answer
28 views
FEM 1D poisson substitution integral issue
I'm trying to solve
$
\begin{cases}
-u''=f \\
u(0)=0 \\
u(1)= \alpha
\end{cases}
$
with FEM using reference elements and local coordinates.
So we have the global ...
1
vote
2answers
115 views
Method to find PDE equation coefficient satisfying mean solution?
What is the best approach to go about solving a PDE problem of the type
\begin{equation}
k^3\Delta u - k(\mathbf{1}\cdot\nabla u) = 0\, ,\\
u=g\; \text{on}\; \Gamma_D\, ,\\
mean(u) = u_\text{...
3
votes
0answers
68 views
Element Preconditioner
Im just working on a preconditioner for the linear equation system $Ax = b$ arising in FEM for elliptic PDE. $A$ is a s.p.d Matrix with real valued entries. I read something about the element by ...
3
votes
2answers
163 views
Recommended visualization tools for higher order finite element solutions?
Is there any software available which can directly render higher order finite element results? In particular, 3D finite elements would be preferable. It seems gmsh has some capability in this regard ...
2
votes
0answers
29 views
Solving Compressible Euler in Primitive Variables with Galerkin P1 FEM
I have implemented a small compressible Euler solver, discretizing in primitive variables (rho, u, v, p) with standard Galerkin FEM P1 triangular elements, and mixed isotropic and anisotropic/...
-1
votes
1answer
67 views
Relation of Condition of a Matrix and Convergency
Can anybody explain me the relation between the condition of a Matrix and the convergency of a problem.
For example how is the relation between the condition of the stiffness Matrix occuring in FEM ...
3
votes
0answers
89 views
$L^2$ norm error estimates of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
1
vote
1answer
208 views
What is the difference between Abaqus and Calculix contact input?
I would like to say first that am new at using Calculix.
I'm using Abaqus/CAE to create a cup deep drawing simulation and everything worked perfectly but my objective is to run the same exact ...
0
votes
1answer
90 views
Penalization parameter for DG with jump penalization
I adapted this FEniCS code for my problem and I'm wondering if there is any good resource about how to choose the penalty parameter $\alpha$? Best case would be, if I can define it through some ...
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votes
2answers
84 views
Taylor-Hood finite hexahedral elements, pressure diverging
I am developing a FEM fluid solver using the Taylor-Hood algorithm, i.e. quadratic interpolation for velocity, and linear for pressure.
I have developed the code for 2-D quadrilaterals and triangles, ...
2
votes
1answer
108 views
Prevent single node spikes in a FEM-simulation (using continuous Galerkin)
I am trying to solve a non-linear time-dependent heat equation
$$\partial_tT=\nabla \left(k_T(T)\nabla T\right) + f$$ (similar to question Solving a non-linear heat equation with the galerkin method ...
-1
votes
1answer
85 views
seminorm of solutions of Laplace equation
If $u_1$ and $u_2$ are solutions of (weak-form) Laplace equation on a connected domain $\Omega$, with Dirichlet boundary values $u_{\partial\Omega, 1}$ and $u_{\partial\Omega, 2}$, respectively. If $$...
-1
votes
1answer
88 views
Compute outward normal and surface area for 8 noded brick element in FEA
I have a cube which is divided into 8 small cubes by bisecting each edge, I am trying to find out the surface area of each of the faces and the corresponding outward normals for them. This operation ...
2
votes
0answers
98 views
Which SciPy nonlinear solver when Jacobian is analytically known and sparse?
I have a nonlinear function fun with n inputs and n outputs. I also have a function jac which calculates the Jacobian, which is ...
5
votes
1answer
66 views
Stabilization parameter for an elliptic equation
I simply want to solve the elliptic equation:
$$
-\kappa \nabla^2 u + u = f
$$
where $f\in [0,1]$. When using continuous Galerkin with Lagrange elements, I have noticed that $\kappa$ has to be greater ...
-1
votes
1answer
108 views
Connectivity matrix in Finite Element Method in Triangular elements
Imagine a simple triangular base mesh in finite element method with an unknown number of elements (varying by the user). How can connectivity matrix be coded to be generated automatically?
