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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Setting up diffusion with integral B.C. in Fenics
I'm trying to model diffusion through a cylindrical domain $D = \{ (x,y,z) : x^2 + y^2 \leq 1, \;\; 0 \leq z \leq 1\}$.
The is an initial concentration of the diffusant at the upper flat surface, ...
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24 views
Divergence issues when using intrinsic cohesive elements approach
When I model the strain localisation of a microscopic sample (or say RVE ) with cohesive elements approach, the convergence performance looks very terrible. I have to use extremely time increments (...
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1answer
69 views
Fenics: solving the same PDE multiple times
I am new to Fenics and just started reading the tutorial Solving PDEs in Python. For simplicity, we can refer to simplest example, page 17 (the linear poisson equation), despite not necessary.
My ...
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1answer
72 views
Lattice Boltzmann methods vs Navier stokes/ other eulerian methods for *water* simulation
Note, there is already a question here, however the answers don't answer the original question, let alone specific considerations when dealing with nearly in-compressible fluids (water). Another ...
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39 views
Finite element lemma proof
I was curious if anyone could help or provide a reference for the proof to the following lemma
Lemma:
Let $P_{1}$ be the set of polynomials of the first degree and let
$W = w(x) : w \in C([0,1]), ...
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2answers
40 views
Evaluation of slope at iteration ith - Newton-Raphson method
I'd like to know how Ansys computes the slope (=stiffness matrix) at point x1 in figure. I'm studying the way in which Ansys uses the Newton-Raphson method when there are nonlinearities.
In the slide ...
3
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1answer
50 views
Weighted QR Implementation
Say I want a QR decomposition of matrix $A$, where orthogonality of $Q$ is with respect to a generic non-degenerate positive-definite bilinear form $\phi$ (in my case, $\phi$ is "defined" by a finite-...
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25 views
Finite difference/element method : time step and spatial resolution close to a finite singularity
I'm using the finite element method (FEM), but my question is quite a global question. It's related to this question but it is not the same.
Let's assume we have this equation : $$\partial_t c - u\...
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77 views
numerical instabilities in Fluid Dynamics, Finite Element Method
I'm looking for references to understand where the numerical instabilities come from in hydrodynamics in general, and notably when the Péclet number: $Pe>1$. I'm using the finite element method.
...
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49 views
Derivatives over a Finite Element mesh
I have a data extracted from Comsol on some node points and I know the coordinates of each node.
Does anyone know how Comsol calculate the partial derivative from the values at each node and also ...
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36 views
FEniCS implementation of Maxwell equations for a dipole antenna
someone knows where I can find a FEniCS implementation of Maxwell equations for a dipole or other type of antenna? I mean a dipole antenna with an arbitrary geometry of every 'leg' in the dipole.
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1answer
116 views
Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?
For numerical methods of the Stokes equations, with appropriate boundary:
$$-\nabla^{2} \vec{u}+\nabla p=\overrightarrow{0}$$
$$\nabla \cdot \vec{u}=0$$
one may use FDM (finite difference method) ...
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17 views
Inconsistent potential over a cylindrical surface in COMSOL
I made the following construction in COMSOL (This is a cut):
Two cylinders, the inner one in the middle is a solid cylindrical conductor. The thick outer cylindrical shell, along with the two small ...
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63 views
How to implement the following Finite Element method for Burgers' equation?
I am trying to replicate this result.
It involves using the Galerkin finite element approach onto the viscous Burgers' equation. However, my implementation (in R) seems to be giving me wrong results....
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1answer
53 views
Determination of Young's Modulus for a Finite Element Code
I am writing a finite element code for my final year project of BS Mechanical Engineering. The geometry is an integration of several parts composed of different materials. I don't have exact values of ...
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1answer
40 views
How to define $P0-$ Piecewise constant basis function in finite element method?
Suppose if we take $X_h(G)$ as finite element space then this space (space of piecewise constant basis function)is defined as
$$X_h=\{v: v|_{T}=c_{T}, T \in \mathbb{T}\},$$
where $\mathbb{T}$ is a ...
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39 views
Cubature rule in unit Sphere in $\mathbb{R}^{3}$
I need to find the cubature rule for the following integration
$$\int_{S^{2}} f(s,\tilde{s})d\tilde{s} ds,$$
where $S^2$ is the unit sphere in $\mathbb{R}^{3}$.
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1answer
82 views
Reference request: Riks method (Nonlinear FEM)
I'm struggling to find a good detailed reference explaining the Arc-length method or, more generally, Riks method and its derivations. I looked for the classical books in nonlinear mechanics (the ones ...
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1answer
83 views
Iterative solution of ill-conditioned matrix systems
I want to solve a matrix system of the form $Ax=b$ where $A$ is ill-conditioned. The matrix system comes from a structural simulation problem which was discretized using finite elements. I do not have ...
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1answer
101 views
Approximate $\|\Delta f\|^2_{L^2(\Omega)}$ in finite element context
I have minimization problem of the form
$$
G(f) + \|\Delta f\|^2_{L^2(\Omega)} \to \min
$$
over all $f\in C^2(\Omega)$, $\Omega$ being closed and bounded.
Let us forgot about $G$; I'm interested in ...
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15 views
Equi-order in pressure correction schme of Navier-Stokes equation
I am wondering if there is an stabilized equi-order scheme in pressure correction scheme in solving Navier-Stokes equation? Usually P2-P1 element combination is used to solve NS equation, and a PSPG ...
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1answer
72 views
Good reference on the implementation and limitations of SDIRK methods
For the solution of many PDE, implicit high-order time integration schemes are required. I am specifically interested in schemes that do not require a constant time step.
I am well acquainted with ...
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22 views
Automatic single point constraint
A lot of modern FE codes have an option called AUTOSPC. Examples are Nastran or Marc. I know that this option removes degrees of freedom to avoid a singular matrix system. But how to determine ...
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1answer
118 views
How to use Newton-Raphson method to handle nonlinear terms in coupled system of PDEs?
I'm trying to solve the Nonlinear Schrodinger's Equation (NLSE) in 2D using Finite Elements, but I don't know how to handle the nonlinear term. I suppose I have to apply the Newton-Raphson algortihm ...
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60 views
Shape functions in Euler Bernoulli Beam Equation
Does anyone have a intuitive explanation of why Hermite polynomials have to be utilized as the shape functions in the FEM solution of the Euler Bernoulli Beam 4th order ODE?
I have been learning FEM ...
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40 views
References to solve system of differential equations which describe the evolution of sandpile surface using the finite element method
I want to solve the following nonlinear system in 1D
\begin{cases}
\dot{R} + v \frac{\partial R }{\partial x} - \frac{\partial }{\partial x}\left( D \frac{\partial R }{\partial x} \right) -\Gamma =...
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0answers
51 views
Strain propagation from surface to bulk in COMSOL
I am trying to simulate strain propagation from the surface into the bulk. I have a rectangular semiconductor block (~2 μm thick) on top of which metal gates (~25 nm thick) are deposited as seen in ...
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1answer
172 views
Should we always expect FEM error plots to be straight lines?
The error estimates in FEM are usually of the form
$$||u^h-u||\leq Ch.$$
Taking logarithm on both sides, we obtain
$$\log ||u^h-u||\leq \log C + \log h.$$
This estimate implies that the error lies ...
4
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1answer
89 views
Original paper on the augmented Lagrangian method in FEM
I am writing a paper in which I want to cite the earliest reference to the augmented Lagrangian method in FEM. For the pure Lagrangian method in FEM, the classical work of Babuška [1] is the original ...
3
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1answer
148 views
calculation of the right hand side of DG FEM (with code)
I got stuck with Hestaven/Warburton's dG-FEM Matlab code.
Starting with the file AdvecRHS1D.m, we see in line 11
...
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1answer
222 views
Stability of hyperbolic PDE and DG-FEM
In the book of Hesthaven and Warburton on discontinuous Galerlkin methods in example 2.3 (regarding solutions of the wave equation), the authors regard the following PDE:
$$\frac{\partial u }{\...
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1answer
193 views
DG-FEM integration by parts
I am going through the book of Hesthaven and Warburton on discontinuous Galerkin methods. I have difficulties understanding some basic steps in the calculations.
Consider the PDE:
$$\frac{\partial u}...
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1answer
64 views
Abaqus, ANSYS, and FVM solver for thermal expansion problem converges to different values
Is it reasonable for a FEM and FVM code to converge to slightly different solutions for the same physical problem (identical BCs, geometry, properties, etc...), provided stability constraints are ...
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2answers
324 views
(FEM) 1D time-dependent heat equation convergence problem
I'm simulating a simple 3-node bar with convection BCs at the edges to validate my FEM code. The following data was used:
Initial temperature = 25 ºC
Temperature surrounding the rod = 10 ºC
Thermal ...
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72 views
Algebraic multigrid for coupled equations
As far as I understand is algebraic multigrid(AMG) a method that was intentionally developed to solve linear systems where every grid point or node has a single DOF. When AMG should now be used for ...
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31 views
How to solve the Poisson equation with KINK aligned with mesh facet
I have a problem that solving the Poisson equation with kink ( discontinuous gradient but solution is continuous ) in the analytical solution, I want to solve this problem with FEM. To approximate ...
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1answer
100 views
References for the nonlinear reaction-diffusion equation using Finite Element Methods
I want to study how to solve the following PDE
\begin{cases}
-\nabla \cdot(\ k(x,y) \ \nabla u \ ) + \beta(x,y)\ u^2 = f(x,y), \ (x,y) \in \Omega \subset \mathbb{R^2} \\
\hspace{0.5cm} u = ...
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2answers
155 views
Order of element vs Degrees of freedom of the element
I have read that the order of the element is the order of the polynomial used to approximate/represent the field variable in that element. If we consider a one-dimensional, 2 degrees of freedom ...
3
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1answer
72 views
How to show the stability of $L^2$ projection?
If $\mathcal{T}_h$ is a regular and quasi-uniform triangulation of $\Omega$, and $V_h$ is the $H^1$-conforming linear finite element space. Moreover, let $P_h$ be the $L^2$ projection to $V_h\subset H^...
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2answers
67 views
Good references for dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)
Can anyone help me with good references (books or papers) where I can learn about dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)?
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1answer
212 views
Spectral Element vs Finite Element
I am trying to understand the difference between SEM and FEM. If I go by this paper, spectral element methods are a subset of FEM methods and the only difference lies in the choice of basis functions. ...
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2answers
142 views
Electromagnetism FEM (FEniCS) interpolation - leakage effect
As for the background of what is going on:
I'm using FEniCS that is dedicated FEM solver
The problem I'm solving is magnetostatic problem where the governing PDE is $$ \bf{\nabla} \times \frac{1}{\mu}...
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1answer
62 views
Split solution of FEM problem depending on number of DOF
Assume we have a 3D finite element structural problem discretized with hexahedral elements with 8 nodes and 3 degrees of freedom per node. Instead of solving the global stiffness matrix system for all ...
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69 views
Trouble with creating correct element matrices in Finite Element Analysis for a cantilever beam
I'm trying to solve for displacements of a cantilever beam numerically with FEA. The beam is modeled as a 3D-solid made up of a set of 8-noded hexahedral elements, which are in their undeformed state, ...
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1answer
72 views
Is there any fundamental difference between meshing for FEM, FVM and FDM?
I am a novice to the field of computational science and have just started studying the FDM and FEM (haven't started on FVM yet). While trying the understand the subject I got this question and trying ...
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Testing the SUPG method and other methods for hyperbolic equations
I am interesting in integrating the simple equation
$$
\frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla \phi = 0
$$
with a Dirichlet boundary condition at the influx boundary ($\mathbf{u} \...
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3answers
2k views
Mathematically, why does mass matrix / load vector lumping work?
I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented a code where the load vector is assembled in a lumped fashion rather than ...
3
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1answer
185 views
What are all these functions in FEM? Shape function vs Basis Function vs Trial Function vs Test Function vs Interpolation Function
I am a novice in FEM. I have some experience with FDM which was pretty straight forward. Since I have a confusion with a number of concepts, I will try to break them down by writing down what I have ...
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49 views
Strange spectral (finite) element results of a solid plate
I tried to implement the spectral element method1 as proposed in [1]. I simulated an aluminum plate and a vertical concentrated force was applied at the middle of the top left quarter of the plate2. I ...
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30 views
Metis: how to use and tutorial recommendation
I am new to METIS and trying to use it in my fortran code. I read the manual online. But still, I am not sure about how to implement it my code. I tried the test cases in the graphs directory. For ...
