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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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45 views

How to manipulate odd derivatives into the weak form?

Admittedly, my knowledge of FEM for hyperbolic equations is weak (no pun intended). First, let's consider the linear 1D advection equation: $$\frac{\partial u}{\partial t} + a\frac{\partial u}{\...
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19 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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30 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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1answer
79 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 ...
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1answer
56 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
59 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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23 views

Alternative to Fabric Mechanics DFMA

I would like to ask if there an alternative to DFMA fabric mechanics software for FEM modelling and maybe this software using MD as well, but I am not sure. For more details: DFMA stands for Design ...
7
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1answer
155 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
102 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
57 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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64 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
51 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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25 views

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 code where the load vector is assembled in a lumped fashion rather than an ...
3
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1answer
141 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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15 views

finding rotations between 2 points in COMSOL

So I have the results of an eigenfrequency analysis, and would like to find out how much an object has rotated in a certain plane. For instance, in the picture below, I'd like to find out how much the ...
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48 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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0answers
21 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 ...
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2answers
78 views

3d vs 2d finite element method

Is the theory of 3d finite element method just an assembly of 2d finite element analysis by putting planes on top of each other, or, a much more comple and different theory applies for 3d, with ...
2
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1answer
103 views

Why do we use hermite interpolation for finite element method in beams?

Why not just Lagrange polynomials basis functions
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1answer
98 views

(FEM) Reorder nodes or use sparse matrix storing techniques

Is it necessary to reorder nodes (using Reverse Cuthill-Mckee algorithm, for example) if I am already using a CSR or CSC storing technique? Because, since CSR/CSC only stores non-zero elements I guess ...
5
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0answers
77 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 $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
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33 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
65 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 ...
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1answer
136 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{\...
5
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1answer
152 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 ...
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0answers
71 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/...
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36 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(\...
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71 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
93 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
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1answer
61 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 ...
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3answers
225 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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60 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 ...
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1answer
61 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: ...
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51 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/...
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1answer
52 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) ...
5
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1answer
77 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} ...
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1answer
61 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 $...
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58 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 ...
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1answer
138 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 $$\...
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1answer
194 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 ...
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1answer
82 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 ...
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1answer
99 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?
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3answers
143 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=...
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1answer
39 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 ...
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0answers
136 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 ...
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1answer
58 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 ...
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1answer
49 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 ...
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0answers
54 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 ...