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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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Could you please recommend some books or lectures about immersed boundary FEM methods?

I want to learn IB for FEM and do some FSI problems, could you please recommend some great books or lecture notes on it?thank you
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What do diagonal (DOF-to-self) terms of stiffness matrix physically mean?

I am used to interpreting each entry of a solid mechanic system's stiffness matrix as a 1D (linear or angular) spring joining one DOF (column index) to another (row index). But this interpretation ...
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How to constraint the tangential gradient on a boundary in FEniCS?

The problem I'm considering is a 2D scalar PDE. The domain $\Omega$ is a disk with two holes $\partial\Omega_1$ and $\partial\Omega_2$ and an external boundary $\partial\Omega_0$. The PDE and boundary ...
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How to assemble K local matrix for a hexahedron cell in a piecwise linear fashion?

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Can finite element exterior calculus be used for the proof of discrete stability?

I've heard about Finite Element Exterior Calculus (FEEC) and its applications in numerical simulations, but can FEEC be utilised to prove discrete stability in computational methods? If so, can the ...
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4 answers
656 views

If FEM is exact at the nodes, why do first and second-order elements give very different results?

I'm looking at the solution to a structural mechanics problem that is modeled with first-order elements and then as a comparison with second-order elements. It is clear that the first-order elements ...
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1 answer
132 views

Can domain decomposition methods also be applied to linear systems resulting from finite difference discretizations?

Question In general, do domain decomposition methods (DDMs) require a linear system of equations $Au = f$ to be formed by finite element discretization/method (FEM) of a PDE? Or could one simply use a ...
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1 answer
66 views

How to expand the C matrix for three aditional degrees of freedom for rotational forces?

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Gradient of field computed using FEM (L2 Projection)

From my understanding, in the finite element context, an L2 projection can be formulated as the following linear system: $$ M\ \phi = \beta $$ where $M$ is the mass matrix, $\phi$ the resulting ...
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Computational efficiency of Galerkin projection in AMG

I have been using recently AMG as preconditioner for CG with several meshes for simple elliptic problems discretised with linear elements on "complicated" three dimensional geometries and I ...
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Formulation of K for tetrahedral elements for the Finite Element Method

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Computing Tangential Derivative using the Dirichlet value

Let $\Gamma$ be a smooth boundary of a domain $\Omega$. Let $u = g$ on $\Gamma$. How can I compute the tangential derivative of the function $u$ using the information that $u = g$ on $\Gamma$? Please ...
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1 answer
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"Cleanest, most professional" approach to implement a finite differences scheme in dimension greater than two

Coding a finite difference algorithm in 1D does not require a complex mesh. In higher dimensions, you would need a mesh and its connectivity graph to compute the differential operators. Finite ...
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Interpolation constant on triangles

There are quite a few references regarding the estimation for the interpolation error for the piece-wise affine finite elements. I find one particular estimate interesting (and useful in my case), ...
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Lowest order Raviart Thomas elements

I have questions regarding the implementation of Lowest order Raviart Thomas elements on quadrilaterals, some papers use the basis functions in this form: for example the right edge: $N = \left[\dfrac{...
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How to ensure solution evolves forward in Modified Riks Method by Crisfield?

I am writing a Matlab based script for solving nonlinear FEA problems using Modified Riks method by Crisfield. As a starting point, I am solving $y = x^3 - 2 x^2 - x + 2$. Around the function minima ...
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How to derive excitation equation used in Finite-Element-Method waveport-driven simulation?

I am working my way through getdp's linear waveguide example in 3D. It appears to me that the electric fields are excited by the waveports, as defined in the formulations file on lines 161-164: ...
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Galerkin Method - Why does integration-by-parts eliminate need to enforce Neumann boundaries?

I've posted this to MathStackexchange but I figured I'd also post here as well as I have yet to receive an answer on my original post, and that I would be more likely to encounter users of the ...
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How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE

Setting I am solving for $u(x,y,t)$ the wave equation $\partial_{tt} u - \partial_{xx} u = f$ on $(x,y)\in\Omega=[0,1]\times \mathbb{R}$ by splitting it into an equivalent first order system: $$\...
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Transformation matrix for global displacements derivates to local ones

The derivatives of the displacements in the coordinate system $\bar{x} \bar{y} \bar{z}$ is given by \begin{equation} \begin{aligned} \{\bar{L}\} & = \begin{Bmatrix} ...
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Constructing metric terms for high order elements

Given 27 $(x,y,z)$ coordinates in 3D space which describe a generally curved quadratic hexahedron, which correspond to the HEXA_27 reference element figure with planar faces in $(\xi, \eta, \zeta)$ ...
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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 ...
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1 answer
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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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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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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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2 answers
112 views

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 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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166 views

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 ...
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4 answers
206 views

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 ...
3 votes
1 answer
211 views

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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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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2 answers
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Help with basic Triangular Finite Elements, constructing stiffness/mass matrices

I am going through this example problem (picture below) and working it out myself. The problem is that I do not get the same mass matrix (M) as the example problem. I am able to get the same ...
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How to simulate thermal expansion in a 2D plane using FEA?

I am trying to model 2D thermal expansion of a square area inside another square using FEATool. I have simulated plane strain by incorporating forces pointing along the $[1 \,\,\, -1]^T$ direction ...
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2 answers
649 views

Boundary condtions on nonlinear FEM time integration

I'm using the finite element method to obtain the time response of a structure harmonically excited. I'm using a linear displacement function to obtain the stiffness matrix and the consistent mass ...
1 vote
1 answer
142 views

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 ...
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1 answer
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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?
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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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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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1 answer
103 views

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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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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Determining the voxels between two boundary surfaces

Issue description I am working on human brain tACS simulations where I have the models of the skin, skull, csf, brain and ventricles in STL format. The shape does not matter and there are no ...
3 votes
1 answer
387 views

Distributed Lagrange multiplier approach to impose constraint in Poisson equation

I'm trying to understand how Lagrange multipliers are applied in order to impose constraints in PDEs. Consider $B \subset \Omega$. For instance, a square inside another square domain $\Omega$. Let's ...
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1 answer
146 views

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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1 answer
78 views

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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1 answer
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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, ...
1 vote
1 answer
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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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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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