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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4
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1answer
671 views

Is a symmetric bilinear form necessary to ensure a weak formulation has a solution?

Problem I want to convert the general second order linear PDE problem \begin{align} \begin{cases} a(x,y)\frac{\partial^2 u}{\partial x^2}+b(x,y) \frac{\partial^2 u}{\partial y^2} +c(x,y)\frac{\...
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1answer
77 views

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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1answer
156 views

SIPG method for $-\nabla \cdot (\nu \nabla u)=f$

Consider the diffusion equation with a coefficient $\nu$: $$-\nabla \cdot (\nu \nabla u)=f$$ with Dirichlet boundary conditions $u = g_D$ in $\partial \Omega$. If the coefficient would be constant, ...
2
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1answer
98 views

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 ...
2
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1answer
110 views

How to enforce fluid and solid dynamic coupling in fluid-structure interactions using the finite element method?

I apologize in advance if the question has been posted before or if it sounds a bit naive. I am writing my own code in MATLAB for a staggered finite element solver for fluid-structure interaction ...
3
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1answer
92 views

Comparison on adaptive mesh refinement on finite elements and finite differences

My current work requires using (Adaptive Mesh Refinement) AMR to resolve multi scale physics. I have a general question whether finite element is better than finite difference in this aspect or not. I ...
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4answers
450 views

Finite-difference software for solving custom equations

Are there any good, easy to use, software for simulating the evolution of systems of generic differential equations? I know there are custom programs for various specific circumstances (such as ...
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0answers
36 views

P1 Finite element discretisation of laplace-neumann eigenproblem

I am looking for help in the FE discretisation of the Laplace eigenkproblem with Neumann boundary conditions; that is, $$-\int_{\Omega} \nabla u \cdot \nabla v= \lambda \int_{\Omega} uv,$$ or $$Ax=\...
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2answers
2k views

Solution oscillations with a small timestep in backward Euler

I am using backward Euler in a FEM scheme for a convection-diffusion problem. On a given mesh, I can take arbitrarily large time steps, as expected. But if I decrease time step, at some point it will ...
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0answers
64 views

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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1answer
35 views

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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1answer
175 views

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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3answers
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Lattice Boltzmann methods vs Navier stokes/ other eulerian methods for *water* simulation

Note, there is already a question here, however the answers don't answer the original question, let alone specific considerations when dealing with nearly in-compressible fluids (water). Another ...
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2answers
2k views

What does optimal convergence rate mean? How to prove whether a numerical method, e.g. FEM, has an optimal convergence rate or not?

I searched online about the way optimal is defined mathematically,but without any information acquired? How to prove whether a numerical method, e.g. FEM, has an optimal convergence rate or not? ...
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1answer
837 views

What is the difference between Abaqus and Calculix contact input?

I'm using Abaqus/CAE to create a cup deep drawing simulation and everything worked perfectly, but my objective is to run the same exact simulation on Calculix (I am new to using this program) and ...
7
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1answer
4k views

What is numerical damping in the context of time-dependent FEM solvers?

Comsol Multiphysics (a popular FEM package) includes two time-stepping algorithms (IDA aka BDF, and Generalized-alpha), described in their documentation as follows (quoted here under Fair Use; ...
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1answer
59 views

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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2answers
756 views

abaqus solver question

I want to solve a pair of ODEs using the FEM solver of Abaqus. Is it possible for the user to supply the equation and ask Abaqus solver to solve the equations?
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2answers
512 views

Direct solvers and domain decomposition for FEM

In the finite element method, in order to use MPI, we need to decompose the domain into sub-domains first. Then my question is whether we can solve each sub-domain using a direct solver? Of course, ...
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2answers
439 views

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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2answers
236 views

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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1answer
50 views

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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0answers
29 views

Move just some points of the finite-element mesh (triangle/tetrahedron) and interpolate the solution in the new mesh

I have a finite element mesh, and I want that the mesh has some specific points and edges, as I show in the picture. I think that that is possible in a mesh software. I'm solving a evolutive (time-...
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0answers
48 views

Necessary information a topological optimisation solver needs to collect from a pre-processed CAD model

I am developing a solver that gets a CAD model as entry and does the topological optimisation calculation on it. My solver is inspired by the open-source codes presented in the literature. Since it is ...
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1answer
338 views

Lower bound for bilinear form in FEM

I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries. I did some research and found: $$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(...
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1answer
108 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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3answers
379 views

How to write integration tests for numeric simulation software?

Just to be more precise, I'll put a worthy example of my typical use case. Let's say I'm developing a FEM software that produces several temporal solutions and inserts them in an HDF5 file, along with ...
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0answers
87 views

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 ...
3
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1answer
213 views

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} \\ \...
6
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1answer
180 views

PETSc-like library for Julia

I want to build an application for Material Point Method (and probably other meshfree methods too) in Julia and I am looking for library for direct and iterative solvers that can help me with it. One ...
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1answer
75 views

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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0answers
21 views

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&...
12
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1answer
497 views

DG local equation, how to interpret mean-averaged test function

In the paper http://www.sciencedirect.com/science/article/pii/S0045782509003521, an HDG element-local equation is described on page 584 equation (4), with one of the equations taking the following ...
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1answer
291 views

Solving Schrodinger Equation with finite element and Crank-Nicolson?

I have asked this in Mathematic section, but received no reply. Please let me ask here to see if threr is any difference. The Schrodinger equation without potential has the following form: $$\...
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0answers
62 views

How to write a simple finite element solver in python in order to solve Poisson equation in 2D

I would like to write a simple finite element solver in python in order to solve 2D Poisson equation and then visualize it. $$ -\nabla^{2} u(x,y)=f(x,y), \quad x,y \quad in \quad \Omega\\ u(x,y) = u_D ...
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1answer
372 views

Proving solution existence and uniqueness of the Helmholtz equation with Robin boundary conditions with complex coefficients

I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation $$ \iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)...
3
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1answer
195 views

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} + (\...
2
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2answers
384 views

Modelling question: example of a physical phenomenon with this jump condition at an interface?

in our finite element class we were talking about interface problems our teacher came up with the following, where $K_i$ are two given functions and $u_i$ is the restriction of the solution $u$ to $\...
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1answer
127 views

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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1answer
100 views

Elementary matrix of Raviart-Thomas elements

We can use the $RT0$ to solve the Darcy equation, i.e. $$k^{-1}\mathbf{u}+\nabla p = 0, \text{ in } \Omega,$$ $$-\nabla \cdot \mathbf{u} = 0, \text{ in } \Omega,$$ $$p = p_D \text{ on } \partial\Omega,...
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0answers
57 views

Contact analysis does not converge due to the projection falls outside valid domain

I implemented Node-To-Surface contact algorithm (Wriggers, Peter, Computational contact mechanics., Berlin: Springer (ISBN 3-540-32608-1/hbk). xii, 518 p. (2006). ZBL1104.74002.). The code is done by ...
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2answers
157 views

Test functions of Raviart-Thomas elements?

The test functions of general finite elements are like interpolation functions (if my understanding is correct). But how about test functions of Raviart-Thomas elements? Let's raise the $RT0$ element ...
5
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2answers
144 views

Is their a book/paper about mixed finite element method for engineering students (non-math)?

There are a lot of books about FEM, which are really friendly to engineering students. Through these books, we can know how to use shape/test functions based on the variational principle. But I'd like ...
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1answer
155 views

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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2answers
90 views

Numerical methods for Vlasov's equation

Vlasov equation is pretty straightforward It would be easy to solve with Fem packages like firedrake, but in my case I have 6d distribution function: it depends on 3d vector of spatial coordinates ...
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2answers
131 views

Nitsche's method for imposition of Dirichlet boundary conditions: implementation standpoint

I'm trying to understand how Nitsche's method works in practice. I understood the theoretical principle behind it, but what I can't understand is its implementation. More precisely, I'd like to solve ...
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0answers
81 views

2D axisymmetric Magnetostatics Finite Element Method solver

I am new to the field of magnetostatics. I wish to write a 2D finite element code for obtaining the magnetic field inside a solenoid coil. I have started with a 2D code and have followed the method as ...
0
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1answer
119 views

Lagrange multiplier for boundary conditions in pure Neumann problem

I'm trying to solve $-u''=\cos(2 \pi x)$ with boundary conditions $u'(0)=u'(1)=0$ and the constraint $\int_{0}^1 u = 0$ I have to use linear finite elements, so let's assume that I have $M$ degrees of ...
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0answers
49 views

How to apply Neumann boundary conditions in Newton's method [closed]

Suppose that I have a very long and tedious set of differential equations. After discretization, I can get a mapping $f:\mathbb{R}^N \to \mathbb{R}^N$ such that solutions $\phi =(\phi_0,...,\phi_{N-1})...
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0answers
71 views

Fourier integral over elements

Suppose I have a triangular element with vertices ${\vec{r_1},...,\vec{r_3}}$ and a function $f(\vec{r})$. I want to calculate the fourier integral over this triangle such that: $$F(k_x,k_y)=\int \int ...

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