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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questions
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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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0answers
43 views
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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38 views
Can this volume intgral be expressed as a convex function?
This question is related to the following:
https://math.stackexchange.com/q/4151405/685910 - the context is summarized below for clarity.
In the setting of convex optimization, I am looking for a ...
4
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1answer
100 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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0answers
90 views
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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66 views
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^{-...
2
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1answer
80 views
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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41 views
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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2answers
65 views
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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2answers
126 views
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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1answer
81 views
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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1answer
142 views
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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1answer
51 views
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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2answers
154 views
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:
$$ \nabla \times \...
3
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0answers
60 views
Jacobian Matrix of 2D element mapped to 3D
Note: I previously posted this question to MathStackExchange, but got no attention there. So I'm rewritting and trying over here.
Problem summary
Given a common¹ set of shape functions defined at ...
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115 views
Weak form of Elliptic problem with mixed Dirichlet & Neumann conditions
Let $\Omega \subset \mathbb{R}$ in a bounded polygon domain and $f:\Omega \to \mathbb{R}$ known function.We split the boundary into two parts $\partial \Omega_{1}$ and $\partial \Omega_{2}$ such that $...
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126 views
Biharmonic equation mixed formulation, mixed boundary conditions
I have the following formulation of the biharmonic equation:
$$\Delta u(x) = v(x), \quad x \in \Omega\setminus K$$
$$-\Delta v = 0, \quad x \in \Omega\setminus K$$
$$u(x) = g(x), \quad x \in K$$
$$v(x)...
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2answers
294 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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0answers
57 views
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 ...
3
votes
1answer
130 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 ...
4
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2answers
185 views
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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61 views
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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0answers
56 views
A priori error estimates - finite element method - mixed boundary conditions
Consider the problem
$$ \left\{\begin{array} {rcl}
-\Delta u & = 0 & \text{ in } \Omega \\
u & = 0 & \text{ on } \Gamma_D \\
\frac{\partial u}{\partial n} &= g &\text{ on } \...
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1answer
73 views
Shape function for quarter point element from degenerate quadrilateral
I want an element as the one shown following.
Nodes 6 and 8 are in the quarter position.
Whether eight node quadrilateral element or six node triangle function can be used directly？
If not, how to ...
2
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2answers
143 views
Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous
I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too.
...
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1answer
44 views
Finite Element Method for 1D Poisson Equation with Inhomogeneous Boundary Conditions
Im trying to solve the Poisson equation in 1D: $$-u_{xx} = f(x), \hspace{6mm} u(a) = d1, \hspace{2mm} u(b) = d2$$Assuming a uniform partition such that $x_n = a + nh$, where $h = (b-a)/N$ and $n \in [...
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0answers
76 views
Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations
I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
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0answers
78 views
Error $L_{2}$ convergence in Finite Element for Poisson Equation
I have written a Matlab code to solve the equation $-u'' = f$ with conditions $u(0) = u'(1) = 0$ on the domain $x \in [0,1]$. I tested the code with $f(x) = -2, \forall x\in [0,1]$. I check the plot ...
3
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2answers
105 views
Efficient schemes for solving the extended Saddle point problem
I am interested in knowing some efficient techniques for solving the following extended Saddle point problem.
\begin{align}
\begin{bmatrix}
A & B^T & C^T \\
B & 0 & 0 \\
C & ...
3
votes
1answer
132 views
Finite element method for high-frequency electromagnetics
I am writing a project about the Finite element method for use in high-frequency solutions of Maxwell's equations. This could be for use in antenna design and similar.
I have some trouble ...
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82 views
Boundary conditions for an FEM approximation of the Laplace operator
Using FEM, I want to approximate the Laplacian
$$u = \nabla \cdot \nabla h \, ,$$
where $h(x,y)$ is an FEM approximated scalar field on the same mesh, i.e. piecewise differentiable.
I am using MOOSE ...
0
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2answers
89 views
preconditioner for Laplace “without” boundary values
I'm looking at solving systems with the FEM discretization
$$
-\int_\Omega (\Delta u) v = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot\nabla u) v.
$$
without applying Dirichlet- or ...
1
vote
1answer
118 views
Step3 in deal.II - Convergence of the mean
I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
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0answers
51 views
Proof of kinematic relationship between updated and total Lagrangian 2nd Piola-Kirchhoff stresses
On page 587 of Finite Element Procedures by Bathe the author gives the following kinematic transformations
$$
{}^t\tau_{ij} = \frac{{}^t\rho}{{}^o\rho} \; {}^t_ox_{i,r} \; {}^t_oS_{rs} \; {}^t_ox_{j,...
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56 views
Finite element method for an equation requiring switch between spectral and temporal domain
Some equations (such as the non-linear schrödinger equation for pulse propagation) are more easily solved in the spectral form, but still need a representation in the temporal domain to calculate ...
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0answers
103 views
Solving coupled PDEs with self-consistency condition
I am figuring out how to attack a problem (the Usadel equations of superconductivity) in which I need to solve a set of nonlinear PDEs for the fields $\{G_i (r)\}$
$$ U(G_i(r), \nabla G_i(r), \Delta(r)...
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0answers
79 views
Lumped mass matrices for higher-order finite elements for CFD
Given that some of the mass lumping techniques, for example, row-sum lumping does not produce practically viable lumped mass matrices for all the element shapes, what are the techniques used for mass ...
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3answers
105 views
How to apply the boundary condition when global stiffness matrix is stored in csr format? [duplicate]
I am solving the poisson equation and I constructed the global stiffness matrix in compressed row storage format. Then I wrote the preconditioned conjugate gradient solver for solving the system of ...
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0answers
53 views
Solving Laplace equation with constraint on boundary
I have found the following PDE problem in a paper:
Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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0answers
75 views
1D Poisson equation and quadratic basis functions assembly
I'm solving the simple Poisson problem $$-u''(x)=1$$ in the interval $[0,1]$ with $u(0)=u(1)=0$.
I discretised my domain as done here, i.e. with
...
1
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1answer
81 views
Discretization of a nonlinear boundary value problem
I am trying to use finite element method to discretize the following problem
\begin{align}
\min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega,
\...
1
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1answer
77 views
Renumbering the nodes for quadratic basis functions for a 2D domain
I have a simple triangulation for a 2D domain, described by the connectivity matrix $T$ and by the point matrix $P$.
For didactic purposes, I assembled the stiffness matrix for $-\Delta u = f$ by ...
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0answers
30 views
Differences in using Clausius-Duhem inequality vs Principle of Virtual Work/Power in derriving constitutive equations?
I am a novice getting my toes wet in continuum mechanics and nonlinear elasticity. I have seen papers that use both approaches in developing constitutive connections to compliment balance equations of ...
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3answers
192 views
Who uses finite elements with higher continuity?
Lagrange elements of any polynomial describe piecewise continuous functions. Typically, those functions are differentiable.
Mixed finite element methods use vector fields of even less continuity, such ...
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0answers
38 views
Spline interpolation for vector-valued data in 3D space
I have output from a 3D linear elasticity finite element simulation which uses linear tetrahedral elements, such that the displacement is continuous over the nodes but the gradient is not ($C_0$ ...
1
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1answer
123 views
FEM solution for Poisson is not exact at nodes
Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$
I know that the FEM solution (...
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0answers
74 views
C.S.R method in finite element matrix assembly
I have solved the 2D Poisson equation using finite element method with simplex triangular element in MATLAB.
First, I generated the triangular mesh using pdetool in ...
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2answers
111 views
diffusivity matrix assembly in nonlinear finite element analysis
I want to solve a diffusion analysis using finite elements. According to fick's law, governing equation is
$$\frac{\partial h}{\partial t} = D \frac{\partial^{2} h}{\partial x^{2}}$$
. h is relative ...
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0answers
100 views
Poisson equation, stiffness matrix positive definiteness, Dirichlet boundary conditions
I have a question regarding the positive definiteness of the stiffness matrix. Specifically, I believe that it should be positive definite only when at least one Dirichlet point is given, so I would ...
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1answer
98 views
Finite element interpolation
I have a finite element solver that I implemented in MATLAB. I am calculating a specific potential function that is constant in each tetrahedral element. The question is, I want to be able to ...
