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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Volume change of a deformable cylinder with a uniform spinning angular velocity

Consider a deformable cylinder without gravity with a uniform spinning angular velocity and the cylinder is not in contact with anything. In theory this cylinder shouldn't change its cross sectional ...
feynman's user avatar
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How to derive the deformation matrix $F$ of axisymmetric problem?

I am working on a 2D axisymmetric problem and am wondering how to derive the gradient operation matrix for an element and the deformation matrix, I have no clue how to do it and after searching on the ...
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How to express the inner product of a tensor function using matrix Frobenius inner product

Let $\{ \phi_i(x)\}_{i=1}^{M}$ and $\{ \psi_j(y)\}_{i=1}^{N}$ be bases for the finite elements spaces for $x$- ($V_{x,h}$)and $y$- direction($V_{y,h}$), respectively. Given a function $w_h \in V_h = ...
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Galerkin projection in AMG

In the context of Classical AMG for elliptic problems discretised with finite elements (DG or CG), one has the (fine) matrix of the problem, say $A_0$, and the coarser operators of the hierarchy $\{...
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Best way to find the meshing with lowest number of elements allowing a max error of 1%?

So, I have a reference simulation meshing that's 100% reliable but it's costly in terms of time so I'm looking for optimizing it by finding the minimum mesh allowing at most 1% of error with respect ...
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Iterative solvers for problems in solid and structural mechanics

I am looking for comprehensive literature (papers, books, reports etc..) on iterative solvers for solid and structural mechanics problems to understand the best iterative solvers and preconditioners ...
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How to handle non bilinear weak form?

I solved the 2D heat equation using the finite element method. It all went well first with the adiabatic case, however problems occured when I introduced cooling with the enviroment. I modeled the ...
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Iterative solver for high order DG methods (3D Laplace problem)

I have a 3D Laplace problem on quite a complicated geometry where I am using Discontinuous Galerkin method. My mesh is composed by hexas, hence I am employing classical tensor product basis functions $...
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Jacobian for 6-noded triangle in 3D to calculate the area

I would like to calculate the surface area of a 6-noded triangle element, i.e., the face of a 10-noded tetrahedral element in 3D space. A typical solution is to calculate the surface integral of the ...
reox's user avatar
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FEM textbooks recommendation

I would really appreciate if you could recommend me some textbooks that explain FEM process well and have many exercises to solidify the knowledge. So far, Brenner and Zienkiewicz seem to be a bit ...
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maxwell and kelvin models via prony series in abaqus

abaqus documentation only shows generalized models and their formulae. What are the prony series in abaqus for the simplest maxwell and kelvin models?
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How do we implement the balance of stress on interface in ALE FSI method?

""we aim for a consistent variational-monolithic coupling scheme in which we need all equations defined in the same domain; therefore, $\mathrm{ALE}_{f x}$ was introduced. In variational ...
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How to compute overall inertia properties from FE mass matrix?

How should I evaluate the overall mass and moments of inertia (polar and transverse) of a finite elements model, having its mass matrix? Given that the global mass matrix is composed of symmetric mass ...
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mesh size restriction of spatial discretization in FEMs and FDMs

Is there any mesh size restriction of spatial discretization in FEMs and FDMs (finite difference)? If a mesh is very coarse there is still perhaps nothing to stop the program from running and cause ...
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how Petrov-Galerkin FEM different from others and advantages

How is the Petrov-Galerkin FEM different from more common FEMs? Could anyone give a layman description? What's the advantages of using the Petrov-Galerkin FEM?
feynman's user avatar
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Nodal functionals in finite element analysis

I have a quintic Hermite basis functions [-1,1] for FEM applications, I wanted to check if it's nodal functionals are proper. Could someone explain nodal functionals in details and give an example of ...
Nomad's user avatar
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Simple, easy to install and use Python FEM solver (and example) for 2D cylindrical Laplace equation

In my earlier question Interpolating the gradient of a cylindrically symmetric potential field expected obey the Laplace equation, especially near/across r=0 I give a simple example of a Jacobi ...
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How to implement the interface extension of fluid "displacement" in ALE?

In ALE, we first set a referenced space for fluid, then we extend the boundary fluid displacement to the whole fluid region, take harmonic extension as an example, we need $$\Delta \left ( \hat{u} \...
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Keeping the surface flat for finite element analysis

I ran the FEM analysis for a simple supported beam problem using solid elements in plane strain condition, with linear kinematics and linear elastic material. I'm not happy with the solution since ...
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Any code on ALE method for N-S equations or advection-diffusion equations?

I'm currently reading the book computational fluid-structure interaction, but the book doesn't provide any code, and I can't understand the ALE method, could you please provide some codes written in ...
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Reference request: graph Laplacian approximation for domains/manifolds

Is there any reference on solving the heat equation on irregular geometries via creating a mesh and using the graph Laplacian instead of FEM techniques? (Convergence to real solution). That is to say, ...
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Continuous Finite Element vs Space Frames

Finite Element allows one to divide a continuous medium into a mesh and compute structural properties of it. Instead of discretizing the continuous, why not start with a discrete method to begin with? ...
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How to generate mesh for space-time FEM method in FEniCS?

suppose that I have a region called $\Omega $,suppose that this is a 2D or 3D region, how to generate the corresponding space-time mesh by using FEniCS? and how to extract the boundary and implement ...
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Rolling friction settings in COMMERCIAL (Ansys) finite element software

Is there a standard way of implementing rolling friction in COMMERCIAL finite element software? There's no book about such a subject in COMMERCIAL FEM software though there are books on algorithms ...
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Quintic Hermite shape functions

I am trying to use quintic Hermite basis functions for FEM applications, could someone please direct me to the general formula that would help me generate quintic Hermite shape functions? In natural ...
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FEA with order reduced for a large system with a very localized force application

I have a large structure with many DOFs. But the application of force is very localized say over one or two nodes. If I try to run FEA on this with Guyan Reduction, I think the efficiency of reduction ...
s6292_1997's user avatar
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Why can the weak forms of the Stokes and continuity equations be combined into a single equation?

Consider this Stokes equations, $$ \left\{ \begin{array}{r} - \mu \Delta \vec{u} + \nabla P = \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{array} \right. $$ Weak form I is: $$ \...
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Solving of KU=F leads to numpy.linalg.LinAlgError: Singular matrix

Solver.py: ...
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How to calculte the contributions of a lift operator in FEM?

This question is largely motivated by the famous paper "UNIFIED ANALYSIS OF DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS" The Bassi and Rebay biliniar DG form is given in page 11 in ...
CuteCompute's user avatar
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Query about FE approximation of a Poisson equation with non-constant coefficients

Consider the standard weak form of the Poisson equation with coefficient $\alpha$: \begin{equation} \int_\Omega \dfrac{\partial v}{\partial x} \dfrac{ \partial \left( \alpha u \right) }{\partial x} = ...
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How to embed linear elasticity/constrain solver in non-linear soft-body dynamics

I often do simulation of dynamics of mass points connected by strings (e.g. molecular dynamics, soft body dynamics etc.). Typically I do it simply by integration of equations of motion by e.g. verlet ...
Prokop Hapala's user avatar
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How to get Gauss points for nodal force vector (surface integral) in tetrahedral elements in the isoparametric coordinate system $({\xi \eta \zeta})$?

I am aware of the following question Evaluating the surface integral in an FEM (Finite Elements Method) procedure. But they use the volumetric coordinates while I want to use the cartesian ...
Mr Thomas Anderson's user avatar
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Why multigrid is inefficient?

I am trying to solve the Stokes equation containing viscosity nonlinearity using the open source finite element software underworld2 with nested PETSc. The resolution is 2000*200. The solution results ...
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Seeking open-source PDE Solver for inhomogeneous material properties

I'm currently in search of an open-source PDE solver (Finite Element Method is preferred) that can effectively handle the challenge of material properties coefficients associated with each element in ...
Sadjad Abedi's user avatar
3 votes
2 answers
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what preconditioner for incompressible hyperelasticity in 3d (similar to stokes equation?)?

I am working on modeling incompressible elasticity at finite strains. $$ \mathrm{Div} \boldsymbol P = \boldsymbol 0, \quad \boldsymbol X \in \Omega_0 \subset \mathbb R^3, \\ J = 1, \qquad \boldsymbol ...
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uniform refinement is not working in gmsh

I get 0 node and elements when I use -refine command line flag in gmsh. Explicitly writing RefineMesh; in the script produces the original mesh and not a refined one. Am I missing something while ...
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Recommendations for some new books about computational contact mechanics in solid mechanics

I want to simulate some frictional contact problems, but I'm not familiar with this field, could you please recommend some new books as introductions?thank you
吴yuer's user avatar
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3D Quadrature schemes with points on boundary

In one dimension there are two types of quadrature schemes. asymmetric rules like Newton-Cotes like formulas (Trapezodi, Simpson), and Clenshaw-Curtis place sampling points on boundary of the ...
Prokop Hapala's user avatar
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eigenvalues of inhomogeneous Helmholtz equation violate superposition using FEM

I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM) with FEniCSx. The equation is; $c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 ...
Ekrem Ekici's user avatar
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1 answer
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Solving PDEs using FEM using cubic Hermite polynomials

everyone. I am a beginner in Numerical mathematics, I have some idea of how to use Galerkin method to solve PDEs numerically, but so far I had no luck finding an example of how to solve a simple PDE ...
Nomad's user avatar
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2 votes
1 answer
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references for optimization in the context of parameter identification with finite elements

i am performing parameter identification for a non-linear partial differential equation (elasticity) that I solve with finite elements. My optimization problem is a non-linear least squares data-...
Simon's user avatar
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A confusion about the bubble function in lumped mass FEM

I am studying knowledge related to lumped mass finite elements. As is well known, lumped mass finite element methods higher than 2nd order on simplex mesh require the construction of new function ...
Owen Jun's user avatar
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How to set Neumann BC for coupled transport problem in weak form?

Consider $$\begin{aligned} \partial_t v + b\cdot \nabla \phi &=0 \\ \partial_t \phi + b\cdot \nabla v &= 0 \end{aligned}$$ for $v:(x,y)\mapsto \mathbb{R}$, unknown and time-dependent ($...
l'étudiant's user avatar
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Confusion between standard finite element and mass-lumping finite element methods

Consider the following equation, subject to homogeneous Neumann boundary condition. $$ u_t = \Delta u + f(u). $$ The weak formulation is as follows: $$ (u_t,w) = (\Delta u, w) + (f(u),w), \quad \...
Owen Jun's user avatar
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Most promising reduced order modeling method

Many players in the field of engineering simulation software are investing on digital twinning and reduced order modeling techniques, meaning that the field bears potential. I was wondering if among ...
Lilla's user avatar
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Is the weak form on this book about a level set FSI problem wrong?

I'm trying to repeat a level set FSI problem on the book <Level Set Methods for Fluid-Structure Interaction>, on the page 89, the provided freefem code define a weak form of the discretized ...
吴yuer's user avatar
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FEM for nonlinear first-order ODE

Currently I am trying to solve nonlinear Ricatti equation using FEM (Matlab language): $$\frac{d r(z)}{dz} = i k(z) r(z) + \frac{i k(z)}{2}(\epsilon(z) - 1)(1 + r(z))^2$$ $z = [-h/2, h/2]$ and $r(-h/2)...
Andrew's user avatar
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1 answer
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Does the weighted residual method not use energy minimization in any form?

I've come across several texts/papers utilizing the concept of a minimum potential energy state corresponding to an equilibrium state, and I know that it is used in FEM formulations that are based on ...
SNIreaPER's user avatar
2 votes
1 answer
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FEM for a biphasic drying problem

I have the following PDE originating from a biphasic drying model, where $\xi \in [0,\Xi]$ is a radial coordinate attached to the dry skeleton of a wet cylindrical body: $$ \frac{\partial u}{\partial ...
IPribec's user avatar
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How to determine whether the symmetric stiffness matrix is positive definite or not? Is it related to the problem?

For two-dimensional or three-dimensional elliptic equations, when will the stiffness matrix be asymmetric and positive definite? This affected the solution efficiency so much that I had to choose an ...
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