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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Classical global estimate for $H^1$ error

I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$. $$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
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Sum of norms over elements is not equal to norm over the whole $\Omega$

In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$: $$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
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Discrete divergence free functions

I'm studying the weak formulation of NS equations. During the analysis, the book I'm using (Quarteroni-Valli, page 301-302), defined $$Z_h=\{v_h \in V_h: (\operatorname{div}(v_h),q_h)=0 \quad \forall ...
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MPI vs OPENMP usage for boundary element method

I have an implementation question with MPI and OPENMP. I have a large three-dimensional surface mesh divided into elements. I want to loop over each element (outer loop) and compute an integral over ...
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Displacement/Pressure Finite Element formulation for Large Deformations (from Bathe)

On page 563 of Finite Element Procedures by Klaus-Jürgen Bathe the author states that governing equations of the displacement/pressure finite element formulation for large deformations is given as $$ \...
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Time discretization Navier Stokes equation

This question is a follow-up of this one. The weak form of Navier Stokes equation is (assuming $v,q$ test functions for the velocity and the pressure, respectively) $$(\frac{du}{dt},v)_{\Omega} + (\...
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What is the difference between $u_h$ and $I_h(u)$ in finite element literature?

In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
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How to treat nonlinear radiation term in heat equation using Finite-element method?

I am trying to solve the time-dependant non-linear heat equation with radiation. This equation is coupled to the radiative transfer equation but for the purpose of my question, this does not matter ...
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How to decrease error in (FTCS) forward time centered space method?

I am using the FTCS method for solving differential equations. I know that the condition for stable output is $$ \frac{\alpha \Delta t}{\Delta x ^2} < \frac{1}{2} $$ But when I use the distance ...
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Confusion about preconditioner for incompressible Navier-Stokes equation with implicit-explicit method

Consider the time-dependent Navier-Stokes equation $$u_t + (u \cdot \nabla) u - \Delta u + \nabla p = f$$ $$\operatorname{div}(u)=0$$ Looking at deal.ii tutorials, I've notice that there are ...
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Finite Volume on Cubed Sphere

The US weather model uses an uncommon (?) discretization called 'Finite Volume on Cubed Sphere'. To avoid the singularities that occur at the poles when using lat/lon discretization, they instead ...
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deal.ii - ParaView "warp by scalar" of my output is not continuous

During our finite element course, we've solved the linear elasticity problem in 2D on a square (GridGenerator::hyper_cube) with $Q_1$ bilinear finite elements in ...
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How to couple the vibro-acoustic equations by Mortar method for non-matching meshes?

Assume we have two domains $\Omega_a$ a acoustic domain with boundary $\Gamma_a$ and $\Omega_s$ a domain of a solid body with boundary $\Gamma_s$. $\Omega_a$ and $\Omega_s$ have the common interface $\...
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Ritz projection error estimate for bilinear Lagrange element?

For second order elliptic problem \begin{align} -\Delta u=f,\quad in ~~\Omega\\ u=0,\quad on~~ \partial\Omega, \end{align} we have for the Ritz projection for $P_1$ conforming element \begin{align} \|...
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Applying displacement control loading using lagrange multipliers in the material non-linear finite element method

Hi I am trying to implement a simple plasticity based finite element code. I am not clear how to set up displacement control applied through Lagrange multipliers. In case of a linear problem, I did ...
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Stable finite elements for the mixed form of the elasticity equations

The mixed form of the elasticity equations is to find the unique critical point of the Hellinger-Reissner functional $$J(u, \sigma) = \int_\Omega\left(\frac{1}{2}A\sigma : \sigma - (\nabla\cdot\sigma)\...
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Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?

I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann ...
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Bubble function property: how to prove it?

I'm studying error estimators and I need a check on an estimate. In our course, we've been given the following definition of bubble function (in 2D): It's a function defined on a triangle $T$ such ...
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1 answer
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Computing the residual in a Dual Weighted Residual (DWR) method

I am in the process if computing the Dual-Weighted Residual (DWR) for a linear PDE with a linear functional but I am struggling with the residual part of the calculation. For example suppose we want ...
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Displacement field not correct?

Consider the elastic equation $$- \operatorname{div}(C \nabla \mathbf{u}) = \mathbf{f}$$ as presented in step-8. Here $\mathbf{u}$ is the displacement vector, let's consider the 2d case. As you can ...
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Confusion about bilinear form for elasticity equation in deal.ii tutorial

I'm learning how to solve vector-valued problems with deal.II library. In particular, I'm looking at the following introduction from the official website https://www.dealii.org/current/doxygen/deal.II/...
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2 answers
287 views

Dyadic operations, fourth order tensors and Tensor algebra

I am trying to understand the dyadic operation for a while since I am interested in Elasticity problems. I believe an intuitive understanding (rather than assuming) will give me good problem solving ...
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Hanging nodes in deal.ii tutorials: how is the continuity constraint imposed?

While looking at step6 of deal.ii tutorials, I decided to try to understand how the constraints coming from hanging nodes are imposed. So I started by watching video lecture 16 by prof. Bangerth As ...
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250 views

Weak Formulation of Poisson's Equation for Electrostatics with Surface Charge

I am currently attempting to use FEnICS to solve an electrostatic problem with two materials of different permittivity $\varepsilon_1$ and $\varepsilon_2$ forming an interface: Consider a domain $\...
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How to interpolate stress at unknown points from the stress values available based on geometrical position for constant load?

I am working on a combined contact, bending, and torsion problem. I have data on geometrical points and their instantaneous stress components. However, based on the available data, I have to ...
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Transient advection equation with stabilized FEM

I am interested in solving the transient advection equation $\left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-...
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discretizing surface integral using nodal DG method

I am currently learning nodal DG methods, primarily through the book by Warburton, and am a bit confused on how to handle surface integrals using straight edged elements. On page 187 (and on page 214)...
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Constrained optimization for non-linear equations in octaveGNU

I have installed Optim1.6.1 package. I would like to solve a system of equations in non linear finite element analysis using constraints as u=1 at certain nodes. u=0 at certain nodes. Typically I find ...
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Reference for mass matrix assembly

I will now explain what I understand to be the process of a finite element mass matrix assembly. I would like a reference which does something similar, or if I am mistaken about the process please let ...
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How to implement point source or volume source in finite element implementations

I'm trying to do a simple implementation to study the advection-diffusion-reaction dynamics in a straight pipe. I have points positioned along the length of the pipe (blue dots in the image above). I ...
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RCM better than Nested dissection? (For FEM discretizations in 2D and 3D)

I realize this might be a too general question but here goes nothing: I am trying different re-ordering strategies and checking the fill-in of $A=LU$. I have 2D ($p=1$, $h=1/40$ on $\Omega = [-1,1]^2$)...
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What is the weak form of a vector type Laplace equation?

For a scalar variable $u$, the weak form of the following Laplacian: $$\nabla^2 u =0 $$ with the assumption that $v$ vanishes at the boundary is $$\int \nabla v . \nabla u \, d\Omega = 0 $$ which is ...
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Validity of algorithm for assembling the finite element global stiffness matrix

This blog post describes an algorithm for constructing the finite element global stiffness matrix as two basic steps: [Step 1] Initialize a table with the elements shared by a pair of connected node ...
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2 answers
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How to create a good preconditioner for a system of linear equations that is created with FEM applied on the time harming Maxwell eqution?

I set out to solve the time harmonic Maxwell equation numerically which was discritzed using FEM and with the use of Nedelec elements as basis and test functions. The equation reads: $$ \nabla \times \...
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3 votes
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381 views

Jacobian Matrix of 2D element mapped to 3D

Note: I previously posted this question to MathStackExchange, but got no attention there. So I'm rewritting and trying over here. Problem summary Given a common¹ set of shape functions defined at ...
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Weak form of Elliptic problem with mixed Dirichlet & Neumann conditions

Let $\Omega \subset \mathbb{R}$ in a bounded polygon domain and $f:\Omega \to \mathbb{R}$ known function.We split the boundary into two parts $\partial \Omega_{1}$ and $\partial \Omega_{2}$ such that $...
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3 votes
2 answers
397 views

Is mesh orthogonality important for FEM?

While studying mesh quality metrics in literature and software documentation, I've seen discussions about mesh orthogonality in Finite Volume Method (FVM) contexts, but not for Finite Element Method (...
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Confused on how Method of Manufactured solutions works?

I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique so my starting equation ...
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3 votes
1 answer
170 views

Crouzeix-Raviart for Stokes on elasticity form

I have tried to solve the driven cavity problem with incompressible Stokes flow using the standard Crouzeix-Raviart non-conforming P1/P0 element (linear velocity, constant pressure, velocity-nodes at ...
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2 answers
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Why is FEM theory only ever written down for simplices?

I noticed that in most theoretic literature (Braess, Ciarlet, ...) for Finite Elements, the method is only described in a detailed way for simplex triangulations, while other forms of hexahedral ...
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FEM port Boundary definition for electromagnetics and wave guides

We are currently in the process of implementing ports in our EM FEM simulation SW. We have come across the definition of boundary conditions for the ports, and we do not understand the equation for ...
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A priori error estimates - finite element method - mixed boundary conditions

Consider the problem $$ \left\{\begin{array} {rcl} -\Delta u & = 0 & \text{ in } \Omega \\ u & = 0 & \text{ on } \Gamma_D \\ \frac{\partial u}{\partial n} &= g &\text{ on } \...
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1 answer
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Shape function for quarter point element from degenerate quadrilateral

I want an element as the one shown following. Nodes 6 and 8 are in the quarter position. Whether eight node quadrilateral element or six node triangle function can be used directly? If not, how to ...
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2 answers
166 views

Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous

I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too. ...
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1 vote
1 answer
285 views

Finite Element Method for 1D Poisson Equation with Inhomogeneous Boundary Conditions

Im trying to solve the Poisson equation in 1D: $$-u_{xx} = f(x), \hspace{6mm} u(a) = d1, \hspace{2mm} u(b) = d2$$Assuming a uniform partition such that $x_n = a + nh$, where $h = (b-a)/N$ and $n \in [...
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Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations

I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
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Error $L_{2}$ convergence in Finite Element for Poisson Equation

I have written a Matlab code to solve the equation $-u'' = f$ with conditions $u(0) = u'(1) = 0$ on the domain $x \in [0,1]$. I tested the code with $f(x) = -2, \forall x\in [0,1]$. I check the plot ...
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2 answers
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Efficient schemes for solving the extended Saddle point problem

I am interested in knowing some efficient techniques for solving the following extended Saddle point problem. \begin{align} \begin{bmatrix} A & B^T & C^T \\ B & 0 & 0 \\ C & ...
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1 answer
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Finite element method for high-frequency electromagnetics

I am writing a project about the Finite element method for use in high-frequency solutions of Maxwell's equations. This could be for use in antenna design and similar. I have some trouble ...
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Boundary conditions for an FEM approximation of the Laplace operator

Using FEM, I want to approximate the Laplacian $$u = \nabla \cdot \nabla h \, ,$$ where $h(x,y)$ is an FEM approximated scalar field on the same mesh, i.e. piecewise differentiable. I am using MOOSE ...
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