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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Finite elements algorith for a fluid in a tube with an elastic obstacle (Fluid-Solid coupling)
I want to solve the model of a tube with an elastic obstacle, something like a simple model of an vessel with a valve. The fluid is given by an evolutionary incompressible Navier--Stokes equations, ...
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1answer
141 views
Finite element (1D) for steady state non-linear problem
I need to solve with linear finite elements the equation $$\frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ \sqrt{u} \frac{\partial u}{\partial x} \Bigr] =0$...
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40 views
FEM solution becoming wider as number of nodes increase
My FEM scheme uses a 4-node quadrilateral element with bilinear shape functions. The simple problem I'm solving is. $\nabla ^2 f = 5$ But as I increase the number of nodes, the plot of the solution ...
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1answer
119 views
FEM does not match exact solution
I am trying to solve :
$$-u''(x) + u(x) = \sin(2\pi x)\, ,\quad 0<x<1\, ,$$
$t>0$,
with $u(0) = u(1) = 0$. That has as exact solution
$$u(x) = \frac{\sin(2\pi x)}{1 + 4\pi^2}\, .$$
But the ...
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1answer
108 views
Non-Linear advection diffusion with nondifferetiable advection term
I'm looking at Murray's book: Mathematical biology: an introduction , first volume, pag. 404
In particular, I'm interested to solve the following PDE:
$$\partial_t u = \partial_x (\text{sign}(x) u) + \...
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39 views
Split of complex parts in weak form
I am working on a numerical model to simulate the acoustic and elastic wave propagation in frequency domain via the Finite Element Method. Basically, the problem is to solve the Helmholtz equation in ...
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2answers
151 views
1D FEM for nonlinear diffusion coefficient
I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$
in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$.
...
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Hi I am trying to model a 2D Lug angle using Gmsh 4.6. How can I combine transfinite quad and regular full quad meshes in the following geo file?
I need transfinite mesh a small section of the bolt hole to insert a crack. However, The transfinite mesh and regular full quad mesh seem being incompatible and throwing errors. How can I combine ...
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1answer
57 views
Imposing pressure variation instead of Dirichlet boundary conditions on Finite Element Method
I always see Finite Element codes solving PDE with Dirichlet or Neumann boundary conditions. But, I have a problem now consisting of a straight cylinder with a circular base (a simple 3D tube), with ...
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1answer
78 views
Global stiffness matrix from element stiffness matrices for a thin rectangular plate (Kirchhoff plate)
I have the element stiffness matrix for a thin "kirchhoff" plate. The plate is 3 [m] x 5 [m] and is simply supported on all edges. It's thickness is 0,2 [m]. On the plate there acts a ...
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FEM shape functions on triangular elements: transition from 2D to 3D
I'm writing a code for solving PDEs through the finite element method. In particular, I'm facing with 3D problems, in which I don't know how to calculate shape functions derivatives on the boundaries (...
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1answer
76 views
Weak formulation for advection diffusion reaction
I need a check on the following exercise about weak formulations and finite elements.
Consider the advection diffusion system
$$
\begin{cases}
-(\mu u')' + \beta u' + \gamma u = f \\
u(a)=0 \\
u(b) = ...
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64 views
How to calculate the interior value of triangular element in edge (vector) finite element?
I was using an edge (vector) finite element to solve electromagnetic diffusion (two-dimensional cases). The element that I used was a triangular element. I have got the result of the finite element in ...
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Periodic Boundary Conditions for a Unit cell subjected to Combined Loading-(Uniaxial + Shear, Hydrostatic + Shear)
I am trying to simulate the mechanical behaviour of a unit cell of a periodic structure subjected to combined loading cases:
[1] Uniaxial and shear
[2] Hydrostatic and Shear
[3] Shear and shear on ...
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3answers
316 views
Finite-difference software for solving custom equations
Are there any good, easy to use, software for simulating the evolution of systems of generic differential equations? I know there are custom programs for various specific circumstances (such as ...
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Parallelisation strategies for mixed FE formulations
Mixed FE formulations with LBB-stable elements require two different meshes for the primary and the constraint variables, for example, displacement and pressure. With continuous approximation for the ...
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1answer
57 views
Going From Blender Structure defined by triangles to full 3D mesh (Using GMSH?)
I currently have created a model airplane in Blender by drawing a closed volume with triangular planes. I want to do a FEM calculation on this object, meaning I need a fine 3D tetrahedral mesh of this ...
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58 views
What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method?
What are the practical differences between a Galerkin finite element method vs an orthogonal collocation method? The best reference I've found thus far as been a paper titled The Performance of the ...
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Condition number of finite element stiffness matrix
In the FEM, there are certain applications in which the condition number $(\kappa)$ of the overall global stiffness matrix $(A)$ is computed or reduced by some preconditioner $(P)$. Are there any ...
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5answers
268 views
FEM and High Performance Computing
Suppose we want to solve an FEM problem in terms of HPC. What is the most usual way to do it:
Using an open-source software like mfem,deal.ii etc.. or,
Assembly the system by your own(read mesh file,...
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implementation of shell elements in a topology optimization algorithm
I am working on developing a topology optimization solver based on the finite element method and I want to add a triangular shell element in it. I used the classic finite element method but I didnāt ...
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61 views
FEM Meshing artifact at nodes with fewer neighbors
I wrote a 2D-FEM solver to solve some diffusion process and wanted to verify my code with a test problem.
The input was $f(x,y) = x^2+y^2$ and I applied the stiffness matrix on it to get $\Delta f = 4$...
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3answers
199 views
Is the imaginary part needed in this problem?
Before jumping into my question, let me contextualize it.
I'm doing numerical simulations of a Helmholtz scattering problem
$$\Delta p + \kappa^2 p = 0\, .$$
The incident pressure wave $p^{inc}$ will ...
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3answers
79 views
How to determine global stiffness matrix is constrained or not
Background
In solid fem, we often solve $$\mathbf{Ku}=\mathbf{p}$$
where $\mathbf{K}$ is global stiffness matrix, $\mathbf{u}$ is displacement, $\mathbf{p}$ is global load vector.
If displacement not ...
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1answer
47 views
traction boundary conditions in elasticity
I have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My ...
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1answer
77 views
Pressure interpolation in the Q2-P1 element
The Q2-P1 element is one of the popular finite elements for incompressible flow problems in the mathematics community. For this element, the velocity field is approximated using bi-quadratic shape ...
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2answers
61 views
Uniaxial stretching solution not uniform in FEM code
I am trapped here for a long time. I wrote a toy Matlab FEM code. I want to run the follow simulation.
Mesh
Suppose we have a cube, and we divide it into subcube along $x,y,z$ axis, then each subcube ...
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1answer
93 views
Solving large sparse system
I am working on a problem with very large sparse matrices. I'd like to compute $A^{-1} B$, that is a crucial part of converting DAE to ODE (and there is no workaround). Here size of $A$ is 2E+5 x 2E+5 ...
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2answers
138 views
Weak form of the Navier-Cauchy equation
I am trying to obtain the weak form of the Navier-Cauchy equation, which is
$$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$
...
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1answer
107 views
FEM with elastic inhomogeneous properties leads to mesh-induced anisotropy
I'm solving an elastic homogenization problem and I'm having problems with mesh artifacts.
I would like to first give a brief summary of what I do: I have a system with inhomogeneous (but isotropic) ...
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149 views
$P0$ elements for $H1$
Are there $P0$ (zero degree/constant element) nonconforming methods for approximating solutions in $H1$? More specifically, I have the equation:
$$u-f - T\Delta u = 0$$
Which can be interpreted as ...
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2answers
164 views
Integration by parts with cross derivatives to obtain the weak form [duplicate]
Iām trying to write the weak form of the Navier-Cauchy equation in the component form, where $u_1$ and $u_2$ are the displacement components:
$$-(2 \mu +\lambda) \frac{\partial ^2 u_1}{\partial x_1 ^2}...
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Cell-centered DG extension to the two-point flux approximation scheme
A current problem that I am working on requires me to compute the solution from the heat diffusion evolution on a discontinuous function. More precisely - I have a Delaunay triangulation and within ...
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Some formulations of domain coupling lead to saddle point problems. Is this merely an artifact?
Background I wanted to learn how to couple FEM and BEM (for the Poisson equation), because I wanted to better understand how open boundary conditions look like. Therefore I worked through the relevant ...
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1answer
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Simple reference problems for time harmonic Maxwell equations
For Navier-Stokes problems we can often choose a relatively simple verification problem such as the lid driven cavity, flow over a cylinder, or flow over a backward facing step to verify our ...
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3answers
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I wrote a 2D Finite Element program for Axial Loaded Plates, but the results are unexpected
TLDR: I used Python to write a 2D Finite Element program using 'Constant Strain Triangles' and my beam keeps pointing slightly upwards instead of straight sideways (like the force). I'm new to FEA and ...
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At what l/d ratio will a frame element start to behave as a shell element?
I'm working in ETABS. There are few columns of dimension 300mmx1400mm. The height of building is 36.6 meter above ground level and the dimension of building is 26mx68m. I'm getting the time period of ...
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56 views
How to construct a Fortin Operator for Crouzeix-Raviart Element?
I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element.
The continuous Stokes Equation in the weak formulation is
Find $u \in H_0^1(\Omega, \mathbb{R}^...
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1answer
66 views
Why does smoothed aggregation multigrid method used as preconditioner in conjugate gradient slows down the solution time?
I'm solving a system of linear equations obtained from the FEM discretization of a simple linear elasticity problem on a cube with zero displacements at one plane and a load on the opposite one. The ...
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1answer
64 views
Fast way to build stiffness directly as CSC matrix
I have been working on finite element code in Fortran 2008, and have implemented my own sparse matrix types. I have found that mapping local stiffness matrices (real type) to a global COO sparse type ...
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1answer
78 views
Methodology Suggestion for Wave-propagation Problem using Finite Elements
I want to simulate the propagation of a sinusoidal plane wave in a rectangular domain using Finite Elements Method. First, the wave should propagate through a fluid medium, then it will encounter a ...
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2answers
85 views
Solution of symmeric/non-symmetric linear system
I would like to understand what happens in the following:
I have a really simple Poisson problem, in 1D, with $u_0 = u_N = 0$. I assembled the stiffness matrix and the right-hand side, and I applied ...
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1answer
95 views
Partial differential equation FEM application
I have a PDE which looks like Helmholtz wave equation on one dimensional domain.
$$\dfrac{d^2u(x)}{dx^2}+\pi^2u(x)=f(x)$$
where $-\infty <x<\infty $
Also, $f(x)= 1$ for $-0.25<x<0.25$, I ...
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COMSOL Circularl polarization
I'm having some problems trying to implement circularly polarized light in COMSOL Muliphysics.
For a isotropic homogenous media, I've obtained without problems the TE and TM reflectance curves. ...
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Determining the voxels between two boundary surfaces
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 intersections. I want to ...
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Solve 1D saturation equation using FEM
I want to solve the following 1D equation to model the time evolution of saturation in a porous medium using finite elements (I'm a beginner).
The permeability $K$ and the pressure $p$ depend on the ...
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Adding a boundary surface on a STL model in Gmsh
Background info
I want to run brain simulations with models from the PHM repository (not relevant to the problem but suits the narrative) which are shipped in STL format. The meshes of the STL have ...
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0answers
91 views
Stokes problem with imposed acceleration on boundaries (projection scheme)
I am trying to solve FSI problems with finite elements and using a projection scheme (I am taking as reference the review of Guermond:
Guermond, J. L.; Minev, P.; Shen, Jie, An overview of ...
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1answer
65 views
FDM on nonlinear PDEs
I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{āu}{āt} = F(u,t)$.
In order to perform time discretization with FDM (finite ...
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Add orthogonality constraint to PDE over entire domain finite element
I have a nonlinear differential equation for $u:\mathbb{R}^2\rightarrow\mathbb{R}^3$ that I can express in the form:
$ \nabla (g(u)_{ij} \nabla(u)^j) = 0 $
which is similar to the "Nonlinear Poisson" ...
