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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51 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
38 views
A priori error estimates - finite element method - mixed boundary conditions
Consider the problem
$$ \left\{\begin{array} {rcl}
-\Delta u & = 0 & \text{ in } \Omega \\
u & = 0 & \text{ on } \Gamma_D \\
\frac{\partial u}{\partial n} &= g &\text{ on } \...
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1answer
66 views
Shape function for quarter point element from degenerate quadrilateral
I want an element as the one shown following.
Nodes 6 and 8 are in the quarter position.
Whether eight node quadrilateral element or six node triangle function can be used directly？
If not, how to ...
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2answers
133 views
Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous
I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too.
...
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1answer
39 views
Finite Element Method for 1D Poisson Equation with Inhomogeneous Boundary Conditions
Im trying to solve the Poisson equation in 1D: $$-u_{xx} = f(x), \hspace{6mm} u(a) = d1, \hspace{2mm} u(b) = d2$$Assuming a uniform partition such that $x_n = a + nh$, where $h = (b-a)/N$ and $n \in [...
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70 views
Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations
I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
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62 views
How to prove the Lipschitz continuity of the following functions？ [migrated]
If $f(x)=\frac{\cos(x)-\cos(a)}{x-a}$, where $a$ is a fixed number, how to prove the following inequality
\begin{equation}
|f(x_1)-f(x_2)|\leq C|x_1-x_2|,\quad \forall~~ x_1,x_2\in \mathrm{R}.~~~~~~~~...
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70 views
Error $L_{2}$ convergence in Finite Element for Poisson Equation
I have written a Matlab code to solve the equation $-u'' = f$ with conditions $u(0) = u'(1) = 0$ on the domain $x \in [0,1]$. I tested the code with $f(x) = -2, \forall x\in [0,1]$. I check the plot ...
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2answers
93 views
Efficient schemes for solving the extended Saddle point problem
I am interested in knowing some efficient techniques for solving the following extended Saddle point problem.
\begin{align}
\begin{bmatrix}
A & B^T & C^T \\
B & 0 & 0 \\
C & ...
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1answer
112 views
Finite element method for high-frequency electromagnetics
I am writing a project about the Finite element method for use in high-frequency solutions of Maxwell's equations. This could be for use in antenna design and similar.
I have some trouble ...
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77 views
Boundary conditions for an FEM approximation of the Laplace operator
Using FEM, I want to approximate the Laplacian
$$u = \nabla \cdot \nabla h \, ,$$
where $h(x,y)$ is an FEM approximated scalar field on the same mesh, i.e. piecewise differentiable.
I am using MOOSE ...
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2answers
84 views
preconditioner for Laplace “without” boundary values
I'm looking at solving systems with the FEM discretization
$$
-\int_\Omega (\Delta u) v = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot\nabla u) v.
$$
without applying Dirichlet- or ...
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1answer
115 views
Step3 in deal.II - Convergence of the mean
I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
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49 views
Proof of kinematic relationship between updated and total Lagrangian 2nd Piola-Kirchhoff stresses
On page 587 of Finite Element Procedures by Bathe the author gives the following kinematic transformations
$$
{}^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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99 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 $\{G_i (r)\}$
$$ U(G_i(r), \nabla G_i(r), \Delta(r)...
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62 views
Lumped mass matrices for higher-order finite elements for CFD
Given that some of the mass lumping techniques, for example, row-sum lumping does not produce practically viable lumped mass matrices for all the element shapes, what are the techniques used for mass ...
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3answers
90 views
How to apply the boundary condition when global stiffness matrix is stored in csr format? [duplicate]
I am solving the poisson equation and I constructed the global stiffness matrix in compressed row storage format. Then I wrote the preconditioned conjugate gradient solver for solving the system of ...
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0answers
50 views
Solving Laplace equation with constraint on boundary
I have found the following PDE problem in a paper:
Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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71 views
1D Poisson equation and quadratic basis functions assembly
I'm solving the simple Poisson problem $$-u''(x)=1$$ in the interval $[0,1]$ with $u(0)=u(1)=0$.
I discretised my domain as done here, i.e. with
...
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1answer
78 views
Discretization of a nonlinear boundary value problem
I am trying to use finite element method to discretize the following problem
\begin{align}
\min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega,
\...
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1answer
76 views
Renumbering the nodes for quadratic basis functions for a 2D domain
I have a simple triangulation for a 2D domain, described by the connectivity matrix $T$ and by the point matrix $P$.
For didactic purposes, I assembled the stiffness matrix for $-\Delta u = f$ by ...
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0answers
30 views
Differences in using Clausius-Duhem inequality vs Principle of Virtual Work/Power in derriving constitutive equations?
I am a novice getting my toes wet in continuum mechanics and nonlinear elasticity. I have seen papers that use both approaches in developing constitutive connections to compliment balance equations of ...
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3answers
181 views
Who uses finite elements with higher continuity?
Lagrange elements of any polynomial describe piecewise continuous functions. Typically, those functions are differentiable.
Mixed finite element methods use vector fields of even less continuity, such ...
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0answers
33 views
Spline interpolation for vector-valued data in 3D space
I have output from a 3D linear elasticity finite element simulation which uses linear tetrahedral elements, such that the displacement is continuous over the nodes but the gradient is not ($C_0$ ...
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1answer
116 views
FEM solution for Poisson is not exact at nodes
Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$
I know that the FEM solution (...
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0answers
72 views
C.S.R method in finite element matrix assembly
I have solved the 2D Poisson equation using finite element method with simplex triangular element in MATLAB.
First, I generated the triangular mesh using pdetool in ...
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2answers
104 views
diffusivity matrix assembly in nonlinear finite element analysis
I want to solve a diffusion analysis using finite elements. According to fick's law, governing equation is
$$\frac{\partial h}{\partial t} = D \frac{\partial^{2} h}{\partial x^{2}}$$
. h is relative ...
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75 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
90 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 ...
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0answers
27 views
velocity in CFL condition
I am studying the evolution of the density and velocity field of a core in a molecular cloud in 1 D.
I defined the radial grid (let us say x between 100 and 101) and the time grid.
I am using the ...
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0answers
37 views
Fourier transform in finite element
I have a finite element solver where I am using tetrahedral elements. I am solving for electric potentials and then calculate the current densities in each element, which are constant in each element. ...
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0answers
63 views
Sparse linear solver in fortran working with REAL16
I need some (direct) sparse linear solver for fortran, which works with REAL16 data type. Any suggestions? Both Pardiso and MUMPS support only REAL8. (identical question: https://math.stackexchange....
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1answer
83 views
Finite elements with CFL condition - How to obtain correct order of convergence
I have discretized a PDE with continuous finite element method in spatial variable and with implicit Euler or Crank-Nicolson in temporal variable. In both cases, I have error estimates in $L_2$ norm ...
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1answer
97 views
L2 projection with bounds
In some problems we are currently working on, we are working with discontinuous functions that are defined on a finite element mesh and are established using Lagrangian particles. To obtain them on a ...
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1answer
67 views
relation between different tangent stiffness
I need to find a relation between the tangent stiffness $L_1$ of the first Piola-Kirchhoff stress tensor with the tangents stiffness $L_2$ of the second Piola-kirchoff stress tensor. They are defined ...
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1answer
103 views
How can I implement second order derivatives of shape functions of a 3D elements?
I am developing an Abaqus UEL with 3D 8 nodes brick elements and I need second order derivatives of the shape functions, I have already mapped the first order derivatives from the element coordinates ...
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0answers
51 views
Mesh partitioning with METIS
I am trying to use METIS-5.1.0 edition in order to partion a FE mesh. For demostration purposes I created 2x2 rectangle mesh and tried to partition it. However, I notice a weird behaviour in my code. ...
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0answers
29 views
Output solution vector in Petsc
I am using petsc to solve a linear elasticity problem discretized by finite elements.The initial mesh is read by a mesh file and the distribution in each processor is done using METIS.I am using only ...
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1answer
57 views
How to solve a Stiffness matrix?
I am quite new in this field. For my university I prepared a stiffness matrix to solve for a project group. This matrix consists of 450 equations with 450 unknows (it's a Matlab script) and I have the ...
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82 views
Fortran compiles for legacy Finite Element Fortran program (1980)
The version of Fortran used come from Montreal Ecole Polytechical in 1980.
I need a compiler for Fortran for Windows 7 or Windows 8.
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1answer
68 views
Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?
I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
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1answer
121 views
Solving PDEs in parallel
I have read different approaches on how to solve pdes in parallel which are discretized using finite element method. For example:
Non-overlapping domain decomposition approach as mentioned in https://...
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1answer
42 views
Finite element analysis software for acoustic and electrostatic
I need to do a simulation for my thesis project involving some piezoelectric nanoparticles in a fluid beamed with ultrasounds. I'm looking for a software for such simulation and for now it seems me ...
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1answer
66 views
How to compute gradient of each node in finite element method?
I have a problem with computing gradient of each node in finite element method. I can get the value of each node. But how can I get the gradient?
I know $u = \sum u_i \phi_i$ where $\phi_i$ are the ...
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1answer
86 views
Finite Element Analysis based on contractions of a subset of edges in enterior of mesh
I am modelling a problem that is "driven", not by typical boundary conditions but, by contractions in its interior.
In a finite element analysis, I can specify the new lengths (not ...
1
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1answer
71 views
Transition from 2D to 3D finite element code, what are the inevitable modifications to be implemented?
Imagine we have a simple 2D FEM solver (we are dealing with solid mechanics) and we would like to develop it to a 3D FEM solver (let's say for the same solid mechanics problem) in this case what are ...
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0answers
67 views
Finite elements convergence issue with 2D elliptic equation
I deal with a system of coupled 2D Helmholtz-like equations solved via the P1 FEM on a given geometry. Let's consider, for instance, the following simplified coupled problem:
for $i\in[-I,I]$ we have $...
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0answers
126 views
Problem with solving coupled ODE and DAE equations with mass matrix (Error using daeic12 (line 77) This DAE appears to be of index greater than 1)
I am trying to solve 6 ODE equations coupled with 1 DAE one. The ODE equations have been discritized in space domain and ode15s MATLAB solver is used to solve the equations in time domain. I have ...
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1answer
298 views
Gauss-Lobatto quadrature and nodal points for FEM
By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.)
...
