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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How to apply Dirichlet and Neumann boundary condition at the corner in Finite-Element-Method?
I have a 2D rectangular domain. The governing equation on this domain is Laplace equation:
∇2f=0
In the left and right edge there is Neumann boundary conditon : ∂f∂n=a
n is the normal vector to the ...
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One dimensional $C^1$ finite elements
I tried to solve a one dimensional biharmonic equation with finite elements. I wanted to use a conforming approach (as I simply do not know a lot about other approaches) and therefore was looking for ...
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Convergence of FEM on curved boundaries, and inhomogenous boundary data
In a smooth domain in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ let's consider $-\Delta u = f$ with $u=g$ on a part of the boundary and $\partial_\nu u = w$ on another part of the boundary, which is far ...
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Integration of (d-1)-dimensional functions on finite element surfaces
I am trying to integrate a function $\hat u$ on the common surface of discontinuous finite elements. The function $\hat u$ lives in a $d-1$-dimensional space of functions defined on the element ...
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Convergence rate for not smooth solution with classical $P^1$ Lagrangian FEM
I'm using classical $P^1$ finite elements to solve $- \Delta u = f$ with Dirichlet BC in a 2D domain $\Omega$.
I know from theory that the solution is not in $H^1$ for my particular choice of $f$, so ...
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Which way is the right way to compute the integrals in finite element methods?
Finite element methods involve integrals of functions that are not polynomials, and these integrals must be computed numerically.
For example, suppose that $f$ is the right-hand side of a Poisson ...
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Advantages of using BEM for contact problem over FEM
Are there any advantages of using BEM (Boundary Element Method) for contact problem over FEM (Finite Element Method) apart from lesser computational cost?
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What is the difference between non-linear elastic simulation and linear elastic simulation with plasticity?
I'm learning how to do Finite Element calculations using Comsol Multiphysics.
In Comsol, Linear Elastic Material and Nonlinear Elastic Material are available as material models:
Using Linear Elastic ...
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stable solutions for Large-scale ODEs under boundary value problem
I'm doing FEM and have a problem about getting numerically stable solution for ODEs problems like:
$$ \frac{\mathrm{d}}{\mathrm{d}x}\mathbf{Y} = \mathbf{AY}, x\in[x_1,x_2]$$
in which $\mathbf{Y}$ ...
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deal.II and curved faces: how can I get the curved description
I'm not a deal.II expert, and while studying step-6 I was reading the documentation of the MappingQ1 class in the deal.II documentation. At some point in the description (https://www.dealii.org/...
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What are the prerequisites and resources to self-learn the Boundary Element Method for Contact Mechanics problems?
What are the prerequisites to learning BEM?
In which order is it advisable to learn BEM and FEM - either one before the other, or does it not matter?
What are some good resources to self-learn BEM?
P....
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How is the surface Jacobian determinant calculated in FEM?
I am currently trying to evaluate surface forces on a structure. I came across P356 in Bathe's Finite Element Procedures 2014 (example 5.8) in which he related the edge derivative from the global to ...
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How can I define an equipotential surface/volume in FEniCS?
I want to solve electrostatic problem for potential. Charge density and medium permittivity are known, so is the potential of a grounded surface. I know how I can implement that.
But I would like to ...
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Derivation of the second Piola-Kirchhoff tensor
I tried many formulas to find the components of the second Piola Kirchhoff. I need help to derive equations 27-29 on reference 1.
[![enter image description here][1]][1]
we have
$$
\mathbf{C}=C_{11} \...
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A priori estimates in finite elements for inhomogeneous heat equation
Consider the problem
$$\partial_t u-\Delta u = f\\
u(\Sigma_1)=f_D\\
\partial_\nu u (\Sigma_2)=f_N\\u(0)=u_0$$
where the sides of the space-time cylinder $\Sigma_i$ are disjoint (one of them could be ...
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Numerical methods for $u_t+c u_x= \frac{-c}{x}u$?
I am looking for possible numerical methods to solve the PDE
$$u_t+c u_x= \frac{-c}{x}u$$
I am particularly interested in a Finite elements method, although I am also curious if you can expose some ...
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A Finite Element Method for a first order PDE?
I want to develop a finite element method to solve for $u(x,t)$ the PDE:
$$u_t+c u_x= \frac{-c}{x}u$$
where $c$ is a constant.
so I am trying the following ( as Rothe's method? ) :
Letting $k= t_n- ...
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Applying Stress Boundary Conditions in Commercial Finite Element Analysis Codes
I am trying to replicate a finite element analysis given in a research paper titled On the Detection of Stress Singularities in Finite Element Analysis 1 by G.B.Sinclair et. al.
The geometry of the ...
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positivity preservation for mixed finite element discretization
I'm interested in mixed discretizations of the diffusion (and related) equations:
$$\begin{align}
\frac{\partial h}{\partial t} + \nabla\cdot \mathbf q & = f \\
k^{-1}\mathbf q + \nabla h & = ...
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What FEM solver should be used for matrix-valued FE spaces?
I am pretty new to using FE solvers. I am trying to solve a system of (up to) 9 complex equations. We write these as a matrix equation (here), (with the implied sum over $j$, for each component ...
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Finding the weak form of a PDE with a tensor argument
I am trying to solve for the order parameter ($A$) in the Ginzburg Landau equations. I had asked on the math SE site but was recommended to ask here.
We are trying to solve the following equation, (...
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A priori FEM estimates without $H^2$ regularity
In basic lectures on finite elements theory, people always assume $H^2$ regularity of the solution in order to derive $O(h^k)$ a priori estimates in the norms $H^{2-k}$, $k=1,2$. For simplicity let's ...
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Why "Right" and "Left" Cauchy-Green tensor?
$C=F^TF$ is called the "Right" Cauchy-Green tensor, and $b=FF^T$ is called the "Left" Cauchy-Green tensor.
I suppose in $C=F^TF$ the non-transposed $F$ stands on the right, and in $...
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Taylor-Hood elements for Darcy's equation
I would like to know if Taylor-Hood elements $P_2$-$P_1$ form a stable pair for the mixed approximation of Darcy's equation ( or Poisson's equation) with Dirichlet B.C. In the literature I only find ...
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Reference request for finite elements theory
Consider a domain $\Omega \subseteq \mathbb{R}^{2,3}$ which is non convex and with $C^2$ boundary. Could you recommend a good reference where it is explained how:
without needing isoparametric ...
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FEM applied to heat equation and incompatible conditions
Consider the problem $$u_t - \Delta u = f \text{ on } \Omega\times (0,T)\\u=0 \text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) \text{ on } \Omega$$
with $g$ NOT vanishing on the boundary. If I ...
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What condition ensures the global continuity of the solution in the FEM?
I know this is trivial but I don't seem to understand it. In which step of the FE formulation do we enforce the global continuity of the solution? Or in other words, how the construction of the local ...
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Can I use Q0 finite elements when there are gradients involved?
Could I apply a Q0-discretization to, say, the Poisson equation $\Delta \phi = f$ (where by Q0 I mean piecewise constant, and thus non-continuous, elements)?
Solving this with FEM, at least as I know ...
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How can I correctly determine velocity of a point inside a grid after using mixed finite element method to solve Poisson equation?
I am using the mixed finite element method (MFEM) to solve the Poisson equation:
$$\Delta h = 0,$$where $h$ denotes hydraulic pressure. The MFEM could determine the normal flux rate, $q_n$, through ...
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calculating the Laplacian of the field variable in estimating the local residual error in the finite element method
to perform adaptive refinement in the finite element method according to the explicit residual method, the quantity
$$\eta_K^2=h_K^2\left\lVert r\right\rVert_{L_2(K)}^2+h_K\left\lVert R\right\rVert_{...
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Applying flux limiters on vertices/faces instead of quadrature points
I have a couple of questions with regards to the usage of flux/slope limiters, which I was hoping to get some input. I apologise in advance if my questions are trivial, but flux limiters lie way ...
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Algebraic Multigrid on fine mesh vs. just starting with coarse mesh?
So if I've understood it correctly, the Algebraic Multigrid Method (AMG) basically takes a fine mesh, coarsens it, solves the coarse mesh and projects the solution back on the fine mesh.
Wouldn't it ...
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Assembly of the Isoparametric Quadratic load vector in Matlab [duplicate]
I work to solve PDE using FEM in the case P2 on Matlab. I try to correctly assemble load vector using quadratic Lagrange shape functions $$b_i =(f,\phi_i)=\sum_{q=1}^{nq}f(r_q,s_q)*\phi_{i}(r_q,s_q)*...
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Thin-plate FEM simulation
I want to simulate a vibration of a thin plate (the Kirchhoff-Love model) on triangular meshes. Can you advise me on an introductory-level review of different elements for thin-plate FEM simulation?
I ...
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numerical integration of integrals in the p-adaptive version of the finite element method
In the p-adaptive version of the finite element method, elements are allowed to have shape functions with arbitrary different polynomial orders. Therefore regarding a 2D problem with quadrilateral ...
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Find intersections between mesh and curve inside it
I have a simple square mesh, and a curve (discretised by another mesh) inside it. Here a picture worths thousand words. What I want to achieve is to find, for every cell $K$ of the circular (...
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Mesh refinement in the Finite Element Method
I need some good references on how to implement programmatically the hp-refinement of meshes in the Finite Element Method in two/three-dimension. I've searched the web a lot and read many articles and ...
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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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Total stored potential energy of finite element mesh from nodal point displacements and strain energy density function only
I am interested in calculating the total potential energy stored in a finite element mesh given its nodal point displacements alone. The forces that created the displacements are irrelevant because ...
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How to implement large rotations in total lagrangian formulation (nonlinear FEM)?
I have developed an Octave script to solve the nonlinear Euler-Bernoulli beam equations with linearized von Karman-strains, i.e. higher-order terms are dropped. The simulation results agree with ...
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How to define a 3D surface from a set of points
Sorry in advance if this question has already been asked, I found nothing to help.
I want to buid a 3D box whose top surface is topography. This topography is defined by a DEM i.e. a set of points (x,...
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Solving Poisson equations as mixed Laplace using $RT_0-P_0$ pair
I'm trying to solve \begin{cases} - \Delta p=f \text{ in } \Omega\\ p=0 \text{ on } \partial \Omega \end{cases} with in $\Omega = [-1,1]^2$ by writing it as
\begin{cases} u + \nabla p=0 \\ -\...
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Building blocks for solving a vector valued problem
This question is a follow-up of this previous one. I decided to solve the linear elasticity \begin{cases}- \nabla \sigma(u)=f \\ u=0 \text{ on } \partial \Omega\end{cases}
with P1 Lagrangian finite ...
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FEM for vector valued problems: reference request
I'm a PhD working in a computational mechanics lab. I come from a Math department, and I have a good background for what concerns the basics of finite elements, like inf-sup conditions, DG, non-...
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The effect of grid size on the total flux when solving Darcy flow with mixed finite element method
I am solving Darcy flow now with mixed finite element method. The Dary flow is
$$\begin{equation}\begin{aligned}k^{-1}\mathbf{q} + \nabla h=0, \text{ in } \Omega\\ %
\nabla\cdot \mathbf{q} = 0, \text{ ...
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Second Piola-Kirchoff Stress Tensor of Neo-Hookean solid at "zero deformation"
The strain energy of an incompressible Neo-Hookean solid is given as:
$$
W = C_{10}(I_1 - 3)
$$
Implying that at zero deformation $W = 0$, because $F = I \implies C = F^TF = I \implies I_1 = 3$
...
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Quadrature rules for non-linear finite element problems
For solving linear problems stemming from PDEs with the FEM, such as the Poisson equation or the wave equation, it is customary to use the "simplest" numerical quadrature that exactly ...
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Inverse of the Jacobian in the Finite Element Method
In Bathe's Finite Element Procedures 2014 P346, the Jacobian is defined as follows:
\begin{equation}
\mathbf{J} = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} \\ \...
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Numerical calculation of out-of-time order correlators (OTOCs)
I want to numerically calculate OTOCs for different quantum mechanical systems. Consider the following Hamiltonian
$$H=p_x^2+p_y^2+x^2y^2$$
and I want to calculate the following OTOC
$$C_T(t)=-\left&...
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Implementation of mixed hybrid finite element method
The mixed hybrid finite element method (MHFEM) is based on the mixed finite element method (MFEM). So, I'd recall the implementation of MFEM.
The mixed formulation of Poisson equation reads
$$\begin{...
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