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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Solution of thermal analysis using finite element

I want to solve a thermal analysis using finite elements. The governing equation is $$C \frac{dT}{dt}+K T = Q$$. When using backward differencing for time, the resulting equation is quite straight ...
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Immersed boundary method in FEniCS?

I have looked at the FEniCS tutorials and documentation but I cannot find any mention to the possibility of implementing an immersed boundary method (IBM) for fluid dynamics. In particular, I want ...
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Necessary information that a toplogical optimisation solver needs to collecte from a pre-processed CAD model

I am developing a solver that gets a CAD model as entry and does the topological optimisation calculation on it. My solver is inspired by the open source codes presented in literature. Since it is ...
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Fast algorithm for computing lower mode shapes and natural frequencies in MATLAB using sparse stiffness and mass matrices

I am looking for a fast algorithm for computing eigenvalues and eigenvectors from sparse stiffness and mass matrices in MATLAB. The eig(K, M) doesn't work with ...
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Produce vertex displacements from volumetric shrinkage data on unstructured meshes

I was wondering what would be an efficient way to produce compatible displacements for mesh nodes/vertices if the computed data is volume shrinkage of each element/cell in the unstructured mesh? ...
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67 views

Multi-domain 3D Geometries for MATLAB PDE Toolbox

In principle the PDE Toolbox in MATLAB can handle multi-domain 3D geometries as noted here. This feature and the associated function geometryfromMesh were introduced in MATLAB R2018a. The associated ...
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How to use natural logarithm inside Expression on FENICS

I'm trying to evaluate the exact solution of heat diffusion in circular plate. I'm not able to use the natural logarithm inside expression. ...
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What kind of problem or matrices are suitable for multigrid method?

For Poisson or Convection-diffusion equation as follows: $$ -\Delta u=f,\qquad u|_\Omega = g. $$ or $$ -\Delta u +\vec{w}.\nabla u=f,\qquad u|_\Omega = g. $$ using FDM or FEM discretization, we can ...
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How to check if my stiffness matrix is correct

I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "...
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67 views

Lumped matrices in thermal analysis using finite elements

The governing equation of the transient heat transfer problem is $$C \frac{dT}{dt}+K T = Q$$ $C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $...
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70 views

(FEM) Efficient CRS vectors evaluation using elements connectivities

What is an efficient way of evaluating the column (col_ind) and the row (row_ptr) vectors for the CRS (Compressed Row Storage) storage format using the Connectivity Array? The Connectivity Array is a ...
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37 views

Four-noded rectangular element shape matrices

I works on a project where I need to compute a modal analysis of an acoustic cavity. The cavity is rigid which translates the problem to the following equation $$\frac{\partial^2p}{\partial{}x^2}+\...
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Four-noded rectangular element shape functions

I works on a project where I need to compute a modal analysis of an acoustic cavity. The cavity is rigid which translates the problem to the following equation $$\frac{\partial^2p}{\partial{}x^2}+\...
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52 views

Defining Current Density in a FEM model (MATLAB)

I'm attempting to solve the Poisson equation in 3D for a magnetic vector potential in the presence of a current source. To validate my code, I'm initially looking to reproduce the model described in ...
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Understanding the Eisenstat-Walker method for choosing the tolerance of a linear solver when solving a non-linear PDE

We are working on the solution of large non-linear PDE (say the Navier-Stokes equation) which we solve using Newton's method with an analytical formulation of the jacobian. For very large systems, we ...
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84 views

Calculating the jacobian of norm and square root terms in the Finite Element Method

In the code that my group is writing (Lethe) we use a stabilized approach to solve the Navier-Stokes equation. The GLS stabilized method we use has a stabilisation term which contains a stabilization ...
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What is appropriate boundary condition for Poisson pressure equation?

I'm doing CFD simulations in unstructured grids. Well, it's a bit different from conventional unstructured grids that are used mainly in FEM or FVM as tetrahedral meshes. Mine is a voxelized mesh of ...
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Can we use interpolation function of different order to represent different degrees of freedom in a FEM element?

Consider a line element in FEM. Let each node have 3 DOF. They are x and y translation DOF and temperature. Can we use interpolation functions of different orders for the translation DOFs and ...
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Efficient Alternatives to Operator Splitting in NLSE

Lately i've been trying to decide my thesis theme and i've become interested in adaptive finite elements and finite volumes algorithms. However, I need my thesis to fit into a physics related theme. ...
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Extracting FEM matrices in matlab pde toolbox

I am trying to follow the dynamic linear elasticity in Matlab, link here. My goal is to extract the FE Matrices using the function assembleFEMatrices in matlab and solve the resulting system of second-...
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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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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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88 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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161 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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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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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 ...
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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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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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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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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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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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132 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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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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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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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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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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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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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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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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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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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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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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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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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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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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158 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 ...