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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Schur complement formulation of linear system
Consider a system of the following form:
$$(A+K)x=b$$
where $A$ is symmetric, positive definite and block diagonal (in fact, a block diagonal matrix made of stiffness matrices arising from FEM ...
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Estimating forces on a model from the displacements of nodes
In any FEM problem involving mechanics, we try to solve the differential equation for the displacement field, $u$ given the force vector in the nodes, $F$. In industry, we often see our automobiles ...
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Are my boundary conditions in my stiffness matrix correct?
I am trying to find a P1 Lagrange finite element solution to the following ODE:
$\begin{cases}-u''+u'+u=f~~~~~~~~\text{in}~~(0,1)\\ u(0)=1, u'(1)=0\end{cases}$
Where $f(x)=-2e^{x}+2\left(1-x\right)e^{...
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C^1 continuous element for a triangle?
I am looking for an element for FEM that is piecewise $C^1$ continuous across triangles (i.e. $C^1$ continuous on the edge separating 2 triangles of the mesh).
I have heard about the Bell element:
...
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It is possible to solve integro-differential equations using in Fenics?
I am interested in solve the following integro-differential equation:
\begin{align}
\frac{\partial{\rho(\theta, t)}}{\partial{t}} = D \frac{\partial{\rho(\theta, t)}}{\partial{\theta^2}} - \beta \...
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FreeFEM++ converting equation into code
I am trying to solve the following problem nuemrically:
$$u_t = \Delta u + \sin t$$
To that effect I scanned the documentation of FreeFEM, the closest example I can find to my problem is the Thermal ...
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Solving a boundary value problem with variable number of coupled equations
Let's assume the equation
$$
\nabla^2u_n(\vec{r})+a_n(\vec{r})u_n(\vec{r})=\sum_{m=1}^{N}b_{nm}u_m(\vec{r}),\quad n=1,2,\dots,N,\quad \vec{r}\in\varOmega\tag{1}\label{eq1},
$$
is to be solved for $u_n$...
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2D FEM element with 6 DOF per node (including rotation)
I'm writing finite element code in 3D, and I wish to connect vertical (Euler-Bernoulli) beams to a horizontal plate. The deformations of a beam are well described, but for the plate it is more ...
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Interface condition for 1D Helmholtz equation using finite element method
I want to implement a 1D Helmholtz equation with jump condition. The domain is $x=[0,1]$ and both ends have Dirichlet boundaries($p$=0). The 1D strong formulation is;
$$c^2\nabla^2p + w^2p=0 \qquad \...
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Pyton get fem solving simple 2D differential equation?
I need to solve the differential equation:
$$u_t = \Delta u + \sin t$$
On a 2D domain with homogeneous Dirichlet boundary conditions. Tod o this I am trying to use the python package getfem.
I am ...
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Is there any book about fundamental FEM theory using similar terminology as Comsol MultiPhysics?
I am considering to solve complexed PDE systems, like in this post, using Comsol MultiPhysics. The PDEs are different from the General Form provided by Comsol. The Weak Form module may be worth a ...
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How to compute numerically the $H^{1/2}$ norm of a function
I'm, in the context of FEM. Let's say I have a discrete function $g$ living on the boundary of my domain $D$. I need to compute numerically $||g||_{1/2,\partial D}$.
The definition I know is the ...
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Explicit polynomial for quadratic elements? (FEM)
In this resource the linear (barycentric) elements are explicitly given:
The geometry placement of higher order elements is also given but not expression for the polynomial of $P_2$ is given. I am ...
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Could you recommend some books on FEM that explain various data-structures in FEM?
I want to understand the data structure of elements, elements around elements, and so on, and various other data structures in FEM, could you please recommend some books?
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Elements on a triangle (FEM)
I am trying to learn about 2D FEM methods. I am trying to understand the generalization of Lagrange polynomial basis from 1D into 2 variable polynomials over triangle domains.
The most basic element ...
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Is there an obvious n-dimensional elasticity complex?
Finite element exterior calculus helps to furnish nice stable elements for many common elliptic PDE by looking at the de Rham complex
$$\Lambda^0(\Omega) \stackrel{d_1}{\to} \Lambda^1(\Omega) \...
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Continuous vs discontinuous space-time FEM
What are some reasons for approximating a (e.g. parabolic) PDE using the space-time method, with continuous finite elements in time, vs discontinuous finite elements in time?
Are there e.g. ...
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How to get an "optimal point" for refinement in FEM adaptive mesh refinement?
Consider the following 1D problem
\begin{align*}
\begin{cases}
\displaystyle
-\frac{d^2u}{dx^2} = f(x), \hspace{0.5cm} x\in (a,b) \\[4mm]
u(a) = u_{a}, \ \ u(b) = u_{b}
\end{cases}
\end{align*}
I ...
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Discrete Laplacian operator with finite element discretization
Let a function $u \in H^1_0(\Omega)$ defined by its values at the mesh nodes. Can we compute its Laplacian using the matrix resulting from the finite elements discretization of Laplace's equation? I ...
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Translating poisson equation to sfepy code
As the title suggests, I am trying to convert the Poisson equation:
Into sfepy code that can solve the differential euqation problem numerically.
I am trying to modify the tutorial on lienar ...
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1
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Cell-based vs face-based finite element methods
Notation:
Denote $T_{h} = \left\{K\right\}$ to be a face-conforming triangulation of a domain $\Omega$ such that $K_{i} \cap K_{j} = \emptyset$ for $i \neq j.$ Additionally, denote $\mathcal{V}_{h} = ...
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Conceptual doubt regarding 2D conjugate heat transfer modelling (COMSOL and Mathemtica)
I have been dealing with some conceptual flaws in my understanding of modelling, which I will elaborate herein. I am modelling conjugate heat transfer of a reciprocating fluid, which flows with ...
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books/paper recommendation on computational thermal-turbulence by using FEM
I have just learned basic FEM for 2D N-S euqation, now my teacher let me to do the following problem, the document of this problem is in large fluid problem, the system of equations is listed in that ...
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Is it really necessary to solve a system of linear equations in the Finite Element Method?
When we solve some boundary value problem by Finite Element Method, the appropriate system of linear equations is built, $$Ax=b.$$
Usually we use the solution x just for plugging it into some ...
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solve a coupled PDE system with some discontinuity by a mixed FEM
$$
\begin{cases}
u_t=f(u,v,\nabla{v}),\\
\Delta{[g(u)v]}=0.
\end{cases}$$
I want to solve the above PDE system by a mixed FEM, that is, $u_t=f(u,v,\nabla{v})$ by discontinuous Galerkin (DG), where $f$...
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How can I solve this PDE system by discontinuous Galerkin method?
As is known to all, the discontinuous Galerkin method (DG) was first used to solve the equation $u_t+u_x=0$. Now I have the following system of PDEs:
$$
\begin{cases}
u_t=f(u,v,\nabla{v}),\\
\Delta{v}...
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Matrix-free FEM references
I've seen that many people are using matrix-free fem codes in my community (mechanical engineering). I have to admit that I googled a bit and I didn't manage to find a good reference for the subject. ...
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Partition of unity in FEM with bubble functions
In FEM with bubble functions, the field ($\boldsymbol{u}$) is approximated as a linear combination of the standard one ($\tilde{\boldsymbol{u}}$) plus the bubble field ($\boldsymbol{u}^b$). That is,
\...
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Errors imposing boundary conditions weakly with DG
I am using interior penalty discontinuous Galerkin to solve a simple Laplace problem:
\begin{align*}
\nabla u=0
\end{align*}
with prescribed 0 and 1 Dirichlet boundary conditions on opposite edges of ...
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computing higher order derivatives with linear elements
Consider the following equation on $(0,1)$, with Dirichlet boundary conditions on both ends.
$$
\frac{d}{dx}\left(k(x)\frac{du}{dx}\right) = 0
$$
Let us solve this using simple linear finite elements. ...
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$u_t+a u_x=0$ solved by Discontinuous Galerkin
If $u_t+a \cdot u_x=0$ under a periodic boundary condition (to mimic an infinite domain) is solved by Discontinuous Galerkin (DG), how to implement periodic boundary condition and the other details in ...
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guiding principles of integration by parts in FEMs
Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is also a first ...
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(Dynamic FE analysis) Proportionality between time step and displacement
I am currently adopting a Newmark algorithm, for which the system of equation is
$$(M+{\beta}{\Delta}t^2{K}){d_i}=\beta{\Delta}t^2{F_i}+Md_{i-1}+{\Delta}tM\dot{d}_{i-1}+{\Delta}t^2M\left(\frac{1}{2}-\...
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FEM Basis for functions of two variables in $\mathbb{R}^2$ (Applied to Linear Full Stokes Equations)
I'm trying to do FEM for a very basic version of the linear full Stokes equations in two dimensions. Say we are working in the grid $[0,1]\times[0,1]$ in the $xy-$plane.
To solve an FEM problem for a ...
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time discretization via FEM to solve ODEs
It seems that FEMs are never seen to implement time discretization. Is time discretization via FEMs to solve ODEs ever possible?
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How to implement pressure stabilization in matlab while solving steady stokes equation
the equation is $$
\left\{\begin{array}{l}
-\nabla \cdot \mathbb{T}(\mathbf{u}, p)=\mathbf{f} \text { in } \Omega, \\
\nabla \cdot \mathbf{u}=0 \text { in } \Omega, \\
\mathbf{u}=\mathbf{g} \text { on ...
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answer
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Meaning of Degree of Freedom in FEM
Assume we want to solve the Poisson eq. with the FEM on some Domain $\Omega$, i.e.
$$\begin{cases} -\Delta u = f, \; \Omega\\
u = 0, \; \partial \Omega \end{cases}$$
For the sake of the discussion let ...
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How do the current FEM opensource libraries compare?
Almost all FEM libraries are good enough, but I want to start with a FEM package and stick to it for some time. Instead of trying all of them, or going with what everyone else is using, I want to ...
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Once a method is implemented in a computer program, then does learning the theory become less useful use of time?
Once a method is implemented in a computer program, then does learning the theory become less useful use of time?
This has confused me. I used to view that all things should be studied from first ...
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Geometrically nonlinear finite element problem and mesh distortion
In my research on topology optimization of fluid-structure interaction problems (2D), I am using a geometrically nonlinear model to represent the structure. Body/surface forces are extracted from the ...
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Finite Element Modelling of Hyperelastic Material under 2D Plane Strain Conditions
I am currently working on writing a MATLAB code for running a finite element simulation of a hyperelastic material in 2D. Since I am building this simulation as a part of a fluid-structure interaction ...
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Problem with my Octave code (unsteady heat equation with FEM)
I want help with my Octave code regarding the unsteady heat equation.
My geometry and mesh are generated with FreeFEM++, so there is no problem with that (I tried it with the steady problem with no ...
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VTK arbitrary-order Lagrange elements - node positions/ordering on reference tri/quad/hex/tet elements
VTK has had support for visualizing arbitrary-order Lagrange elements for a while now, but there aren't many resources out there (as far as I know) facilitating their use. Notably, these elements rely ...
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GMSH: How can I extrude a surface in a non-linear way?
I am developing a model to represent a dam, shown in the picture below. The geometry in the white circle is the curtain, which is required to be curved. I extruded this 2D face (including the ...
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Solving second order coupled differential equations in python
as I have to design a reactor and therefore have to get its length x, I have to solve the following differential equations:
$$D_{eg}\tfrac{d^2A_g}{dx^2}-u_g\tfrac{dA_g}{dx} = k_la_b\left(\tfrac{A_g}{...
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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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Gmsh: what is the right syntax to iterate over a list of numbers with a For loop on the native language?
I'm new to Gmsh and after browsing many sites I haven't found the answer for this silly error.
Using the native language of the software, I'm assigning a value to the Transfinite property of a bunch ...
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0
answers
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How to incorporate a continuity of 1st derivatives at an internal interface in a 2nd order Poisson equation in finite elements method?
How should I modify the following matrix formulation of a finite element (FE) approximation of Poisson equation at the (internal) interface denoted by the nodes $i-1$ to $i+1$ having a continuous 1st ...
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votes
1
answer
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Decoupling Stokes problem into two problems: velocity and pressure, using FEM
I have seen finite difference methods for fluid equations (Stokes and Navier--Stokes) that solve a pressure problem first and then a fluid problem. That is, although they solve two different problems, ...
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ON the Kronecker product form of the laplacian matrix
It's well known that if we use 2nd order, centered, finite differences for the Laplace operator, we have that the matrix can be written as $$K=I \otimes A + A \otimes I $$ where $I$ is the identity ...
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