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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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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73 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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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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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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68 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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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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110 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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55 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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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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48 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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168 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 ...
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87 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 ...
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127 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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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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185 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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60 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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284 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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68 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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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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96 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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151 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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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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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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187 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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118 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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61 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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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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64 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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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 ...
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159 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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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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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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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 ...
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116 views

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

Why not just Lagrange polynomials basis functions
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(FEM) Nodes reordering for 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 ...
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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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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 ...
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122 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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147 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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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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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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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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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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116 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}\...
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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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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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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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63 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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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/...