All Questions
1,333 questions
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72
views
Developing a poisson equation solver for arbitrary input geometry and boundary conditions
I need to develop a Poisson equation solver that can take a input geometry and boundary conditions, produces a FEM mesh, generates the matrices and solve it. Problem is I want a solver that can use ...
- 101
0
votes
0
answers
35
views
Help Understanding the Weak Equation for Fully Developed Flow (Inlet) in COMSOL
I'm working on a simulation in COMSOL Multiphysics and trying to understand the Fully Developed Flow (Inlet) boundary condition. The documentation mentions:
The Fully Developed Flow boundary condition ...
0
votes
0
answers
78
views
Discontinuous Galerkin method: Rusanov flux implementation
I'm trying to implement the Rusanov flux as a global matrix for solving a PDE. The Rusanov flux for an element (e) and its neighbouring element (k) reads: $$f^{(*, e, k)}=\frac{1}{2}\left[f^{(e)}+f^{(...
- 53
1
vote
1
answer
94
views
How to apply a slope limiter (minmod) to the Discontinuous Galerkin (DG) method?
I've been trying to solve a PDE using the DG method for a while now. The DG method is implemented correctly but I still need to implement a slope limiter and would like to use the minmod limiter. The ...
- 53
2
votes
1
answer
81
views
$H^2-$ conforming finite element on cube
I read in a book about an $H^2$-conforming element on a rectangle, the Bogner-Fox-Schmit Rectangle element, and I was wondering if it has a three-dimensional extension to a cube. The degree of freedom ...
- 121
2
votes
1
answer
198
views
FEM for Poisson equation using C1 continuous element
When utilizing the standard Bubnov-Galerkin method and $C^1$ continuous element (such as Argyris, Bell, and HCT) on the Poisson equation \begin{align} \nabla^2u=-f \text{ in } \Omega \\ u=g \text{ on }...
- 21
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0
answers
53
views
Which time integrator should I pair with an interior penalty DG
I am currently using a discontinuous Galerkin method to discretize the following wave equation $$\begin{cases}u_{tt}=\Delta u,&\text{on }\Omega \\ u=0,& \text{on }\partial\Omega\end{cases},$$
...
- 176
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votes
1
answer
65
views
Sampling pattern for arbitrary regular polygons?
Assume you are given an integer $n$ and want to produce a sampling pattern with that many points on each side.
The square patterns is trivial, you just do n rows of n equidistant points at regular ...
- 379
2
votes
1
answer
65
views
How to derive the $\tau$ in the SUPG method for compressible flow?
I want to understand the formulation and derivation of $\tau$ matrix in the SUPG stablization term, and we sometimes get(there are many variation of SUPG stablization)$$
\begin{aligned}
& \tau_c=\...
- 211
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votes
0
answers
28
views
Combining Study Steps with Multiple Geometries in COMSOL
Currently, I am using COMSOL to first solve an eigenmode problem in a simple element and subsequently extracting the displacement from the surface of said element and the results from the eigenmode-...
- 1
4
votes
0
answers
67
views
Preconditioner for "Generalized stokes" problem
Let $\mu>0$. For the Stokes problem,
\begin{align}
-\mu \Delta u + \nabla p &= f
\\
\nabla \cdot u & = 0
\\
u&= 0 \text{ on } \partial \Omega,
\end{align}
after discretization (say, ...
- 181
2
votes
1
answer
86
views
Why would finite deformation theory be necessary in an updated Lagrangian formulation?
I am recently informed about the large deformation theory, and its concepts like curvilinear coordinates. But so far I understand in an updated Lagrangian formulation, the reference configuration is ...
- 151
2
votes
1
answer
506
views
Finite EIement Method: Why are the matrices called "mass matrix" and "stiffness matrix"?
I'm studying the discontinuous Galerkin method at the moment, but there is one point I do not understand. It's the naming of the matrices. If we for example have a simple advection PDE $$\partial_t \...
- 53
1
vote
1
answer
77
views
How is the test function of Raviart-Thomas elements derived?
I know that the test function of RT0 elements (for 2D triangles) is
$$\mathbf\psi_j(\mathbf x) = \sigma_j \frac{E_j}{2|T|}(\mathbf x - \mathbf P _j),$$
which is an interpolation for the normal ...
- 335
1
vote
2
answers
69
views
Counting the total number of Degrees of Freedom in a Finite Element Model containing different types of elements
I was wondering what is, computationally speaking, the best way of counting the total number of DoFs in a model when dealing with elements of different kinds, especially SOLID and SHELLS that could ...
- 13
2
votes
0
answers
71
views
Can I change the coordinates of the problem to avoid dynamic mesh?
I am looking to simulate the velocity field of a fluid around a fish. Fortunately, some analytical functions have been derived for specific cases (e.g., A Generalized Slender-Body Theory for Fish-Like ...
- 21
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votes
1
answer
61
views
Bad numerical solution in a Finite Element code implementing Taylor--Hood method to solve the stationary Stokes problem
I'm implementing the stationary Stokes problem using the Taylor--Hood finite element method:
$-2\nu\Delta u+\nabla p=f$ in $\Omega$
$\nabla\cdot u=0$ in $\Omega$
$u=0$ on $\partial\Omega$
The exact ...
- 527
0
votes
0
answers
55
views
Is there a clever way to define the lift for the Dirichlet BC? [duplicate]
Consider the Poisson problem with Dirichlet boundary conditions:
\begin{equation}
\nabla^2 u(x) = f(x), \quad x \in \Omega \\
u(x) = g(x), \quad x \in \partial\Omega
\end{equation}
To lift the ...
- 506
0
votes
0
answers
46
views
How to estimate a finite element function with exponential matrix
I would like to know if there are any similar theorems related to the mass lumping finite element method:
For any function $\boldsymbol{u} \in H^m_{per}(\Omega)$
$$
\|I_h e^{\tau \Delta} \boldsymbol{u}...
- 141
2
votes
1
answer
125
views
Solving advection-diffusion equation on non-rectangular domain
I am trying to solve a PDE similar to the advection-diffusion equation:
$$
\frac{\partial T}{\partial t} + (\vec{u} \cdot \nabla)T = D \Delta T
$$
(where $\vec{u}$ is a known advecting vector field) ...
- 121
1
vote
1
answer
155
views
Numerical Gauss-Lobatto quadrature in DG and instability
I am trying to implement the DG method for the solution of non-linear hyperbolic problems. The aim is mainly educational. I want to make sure I understand exactly what is happening and not just using ...
6
votes
1
answer
769
views
Electromagnetic Eigenvalue problem in FEM yielding spurious solutions
I have written an Electromagnetic FEM solver for waveguide Eigenvalue problems. I already wrote it in two different libraries, sparselizard and dolfinx. The relevant scripts are available here for ...
- 61
2
votes
1
answer
81
views
L2 bounds for fem local basis functions
In a paper on Discontinuous Galerkin, the authors state that the following estimate can be proved using a simple scaling argument:
$$||\nabla\phi||_{L^2(K)}^2 \le C h_K^{d-2}$$
where $\phi$ is any ...
- 47
2
votes
2
answers
83
views
Jacobian of 2D element in 3D domain [duplicate]
I know how to calculate the Jacobian matrix for 2D elements in 2D and for 3D elements in 3D.
But for any 2D element(a quadrilateral element) which is embedded in 3D space, how do I calculate this ...
- 41
1
vote
0
answers
43
views
How to use Transfinite option in Gmsh to mesh an stl file
I'm working on the creation of a structured mesh for the FEM analysis of the ascending aorta. I start from an STL file and I apply algorithm 8, recombination with Blossom and then the algorithm 11 for ...
2
votes
1
answer
233
views
Shape functions on the triangle using vertex values and derivatives
Is it possible to approximate a function over a 2-d triangle $\mathcal{T}$ with vertexes $\mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3$, assuming that we know the values $f(\mathbf{p}_i)$, gradient $\...
- 645
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answers
128
views
Heat Equation for fast source with FiPy
I'm trying to solve the following differential equation with FiPy, basically laser irradiation on a surface
$$
\rho_{s}C_{p,s}\frac{\partial T}{\partial t} = k_{s}\frac{\partial^{2}T}{\partial x^{2}} +...
- 11
5
votes
2
answers
219
views
Finite element accuracy on non-affine quadrilateral meshes
I have a simple finite element code (continuous Galerkin method, $Q_1$ tensor product spaces) for $-\Delta \psi = f$ on the unit square with essential boundary conditions. I want to claim that for any ...
- 123
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votes
0
answers
42
views
Any leads on how the choice of shape functions influence finite element stability?
I have seen something similar in the publication "The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations" where the arguments for stability rely on the use of Legendre-Gauss-...
- 506
2
votes
2
answers
226
views
(Isoparametric) Mapping of physical coordinates to their equivalent parametric coordinates on a reference element
I have some experiece with finite element methods (FEM), in general. However, I mainly worked with Cartesian grids -- i.e. using orthogonal (non-curved) elements.
Recently, I became interested in a ...
- 125
2
votes
1
answer
60
views
Good references for the P3/P1dc element
I am struggling to find some good references for the P3/P1dc element (cubic element for velocity and linear piecewise discontinuous for pressure) for the Stokes/Navier-Stokes equations.
Is there a ...
- 964
0
votes
2
answers
94
views
Source for scalability challenge for number of finite element nodes per process
Context
In distributed simulation of a finite element mesh with $N$ nodes and $P$ processes, a professor stated to me that "achieving good scaling for more than 25,000 finite element nodes per ...
- 265
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
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0
answers
66
views
How to correctly discretize volume elements in different geometries?
I am solving a reaction-diffusion problem in one dimension for a catalyst particle to get the internal effectiveness factor ($\eta$),as given below:
$$ \eta = \frac{\int_0^{V_p}{R_i\ dV}}{R_i^{surf}\...
- 23
1
vote
1
answer
256
views
Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
- 11
1
vote
0
answers
50
views
How can I apply a mixed boundary condition to a multi-material heat transfer problem using Crank-Nicolson?
I am working on a mixed material model for a melting material and need to enforce both a Dirichlet and Neumann type condition at the interface. Subject to an external surface heat flux at the top of ...
- 11
3
votes
1
answer
191
views
how to compute the rate of deformation gradient in finite-element context?
I am implementing hyper visco-elastic material models similar to those from Pioletti et al. see here
There, a viscous potential, e.g
$W_v = \eta [I_1-3]J_2 \quad \text{with} \quad J_2 = \mathrm{tr}(\...
0
votes
0
answers
38
views
Thermo Hydraulic Mechanical modeling of energy wall slab in Comsol multiphysics
I am currently working on a complex simulation project involving an energy wall slab, and I need assistance in accurately modeling and validating it using COMSOL Multiphysics. Here are the details of ...
3
votes
1
answer
168
views
Any FEM book recommendations that focus on stability and proofs on error bounds?
Everything from descrete stability proofs for steady state and time dependent problems. energy stability, stability of mixed methods, nonlinear problems, vector valued problems in fluid/structural/EM, ...
- 506
2
votes
2
answers
107
views
Getting singular matrices for lid driven cavity problem
I was trying to solve the lid driven cavity problem using the galerkin method with SUPG stabilization. I was using GMRES method as my solver and I am also getting a solution. And the solution looks ...
- 41
0
votes
0
answers
74
views
Lumped (diagonal) vs. consistent (non-diagonal, symmetric) mass matrix in Nastran
I've been tinkering with DMAP to explore the procedure followed by Nastran when solving a complex modes analysis.
I've reached a passage I cannot understand: at some point Nastran formulated what it ...
1
vote
0
answers
63
views
Immersed Boundary FEM reference recommendation
I want to do some Fluid-Structure Interaction using the Immersed Boundary FEM.
Could you please recommend some books or lecture notes on it?
- 211
0
votes
1
answer
124
views
What do diagonal (DOF-to-self) terms of stiffness matrix physically mean?
I am used to interpreting each entry of a solid mechanic system's stiffness matrix as a 1D (linear or angular) spring joining one DOF (column index) to another (row index).
But this interpretation ...
3
votes
0
answers
49
views
Can finite element exterior calculus be used for the proof of discrete stability?
I've heard about Finite Element Exterior Calculus (FEEC) and its applications in numerical simulations, but can FEEC be utilised to prove discrete stability in computational methods? If so, can the ...
- 506
-1
votes
1
answer
84
views
2
votes
1
answer
93
views
How to constraint the tangential gradient on a boundary in FEniCS?
The problem I'm considering is a 2D scalar PDE.
The domain $\Omega$ is a disk with two holes $\partial\Omega_1$ and $\partial\Omega_2$ and an external boundary $\partial\Omega_0$.
The PDE and boundary ...
0
votes
1
answer
72
views
1
vote
0
answers
84
views
Gradient of field computed using FEM (L2 Projection)
From my understanding, in the finite element context, an L2 projection can be formulated as the following linear system:
$$
M\ \phi = \beta
$$
where $M$ is the mass matrix, $\phi$ the resulting ...
- 31
4
votes
0
answers
95
views
Computational efficiency of Galerkin projection in AMG
I have been using recently AMG as preconditioner for CG with several meshes for simple elliptic problems discretised with linear elements on "complicated" three dimensional geometries and I ...
- 435
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votes
1
answer
88
views