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.
1,214
questions
0
votes
0
answers
16
views
Recommendations for some new books about computational contact mechanics in solid mechanics
I want to simulate some frictional contact problems, but I'm not familiar with this field, could you please recommend some new books as introductions?thank you
- 151
0
votes
0
answers
36
views
3D Quadrature schemes with points on boundary
In one dimension there are two types of quadrature schemes.
asymmetric rules like Newton-Cotes like formulas (Trapezodi, Simpson), and Clenshaw-Curtis place sampling points on boundary of the ...
- 907
0
votes
0
answers
49
views
eigenvalues of inhomogeneous Helmholtz equation violate superposition using FEM
I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM) with FEniCSx. The equation is;
$c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 ...
- 11
1
vote
1
answer
136
views
Solving PDEs using FEM using cubic Hermite polynomials
everyone.
I am a beginner in Numerical mathematics, I have some idea of how to use Galerkin method to solve PDEs numerically, but so far I had no luck finding an example of how to solve a simple PDE ...
2
votes
1
answer
88
views
references for optimization in the context of parameter identification with finite elements
i am performing parameter identification for a non-linear partial differential equation (elasticity) that I solve with finite elements.
My optimization problem is a non-linear least squares data-...
- 123
0
votes
0
answers
81
views
A confusion about the bubble function in lumped mass FEM
I am studying knowledge related to lumped mass finite elements. As is well known, lumped mass finite element methods higher than 2nd order on simplex mesh require the construction of new function ...
- 81
0
votes
0
answers
55
views
How to set Neumann BC for coupled transport problem in weak form?
Consider
$$\begin{aligned}
\partial_t v + b\cdot \nabla \phi &=0 \\
\partial_t \phi + b\cdot \nabla v &= 0
\end{aligned}$$
for $v:(x,y)\mapsto \mathbb{R}$, unknown and time-dependent ($...
3
votes
0
answers
62
views
Confusion between standard finite element and mass-lumping finite element methods
Consider the following equation, subject to homogeneous Neumann boundary condition.
$$
u_t = \Delta u + f(u).
$$
The weak formulation is as follows:
$$
(u_t,w) = (\Delta u, w) + (f(u),w), \quad \...
- 81
3
votes
0
answers
182
views
Most promising reduced order modeling method
Many players in the field of engineering simulation software are investing on digital twinning and reduced order modeling techniques, meaning that the field bears potential.
I was wondering if among ...
- 157
1
vote
1
answer
77
views
Is the weak form on this book about a level set FSI problem wrong?
I'm trying to repeat a level set FSI problem on the book <Level Set Methods for Fluid-Structure Interaction>, on the page 89, the provided freefem code define a weak form of the discretized ...
- 151
1
vote
0
answers
136
views
FEM for nonlinear first-order ODE
Currently I am trying to solve nonlinear Ricatti equation using FEM (Matlab language):
$$\frac{d r(z)}{dz} = i k(z) r(z) + \frac{i k(z)}{2}(\epsilon(z) - 1)(1 + r(z))^2$$
$z = [-h/2, h/2]$ and $r(-h/2)...
- 31
5
votes
1
answer
267
views
Does the weighted residual method not use energy minimization in any form?
I've come across several texts/papers utilizing the concept of a minimum potential energy state corresponding to an equilibrium state, and I know that it is used in FEM formulations that are based on ...
- 73
2
votes
1
answer
176
views
FEM for a biphasic drying problem
I have the following PDE originating from a biphasic drying model, where $\xi \in [0,\Xi]$ is a radial coordinate attached to the dry skeleton of a wet cylindrical body:
$$
\frac{\partial u}{\partial ...
- 544
0
votes
0
answers
65
views
How to determine whether the symmetric stiffness matrix is positive definite or not? Is it related to the problem?
For two-dimensional or three-dimensional elliptic equations, when will the stiffness matrix be asymmetric and positive definite? This affected the solution efficiency so much that I had to choose an ...
- 1
2
votes
2
answers
111
views
Defining the restriction operator in nested multigrid FEM
I'm new in the multigrid approach and I'm trying to implement an algorithm from this paper: https://www.researchgate.net/publication/242913687_A_Multigrid_Algorithm_for_the_p-Laplacian
I'm stuck with ...
- 21
0
votes
0
answers
85
views
Why aren't mortar domain decomposition techniques used as much as schwarts type DD?
Schwartz type domain decomposition techniques require a transmission condition which can be hard to come by. Mortar type techniques enforce continuity with a Lagrange multiplier across domains. Are ...
- 394
1
vote
0
answers
37
views
How does the time needed for force propagation increase with the discretization (using symplectic Euler schemes)?
I am trying to model a physical system by (lets say the system is a long deformable object, on which I can apply forces). It can be described by Cosserat Rod theory and discretized, by modelling it ...
- 11
2
votes
1
answer
155
views
Does this second-order implicit Runge-Kutta method have a name?
I am studying the time-integration of the following paper,
Young, L. C. (1981). A finite-element method for reservoir simulation. Society of Petroleum Engineers Journal, 21(01), 115-128.
A copy (PDF)...
- 544
0
votes
0
answers
45
views
recommendation on some papers/books about frontal solver used in FEM
I'm reading a program about computational plasticity, this program use frontal solver to solve the program, but I'm not familiar with frontal solver even after reading some papaers, so could you ...
- 151
1
vote
0
answers
31
views
FEM 3D Solid with multiple materials
From Springer, Engineering Computation of Structures a mass matrix is defined from shape functions N based from the density tho of the material.
I expect to have to analyze a solid of multiple ...
1
vote
1
answer
93
views
0
votes
1
answer
75
views
In level set fluid structure interaction method why can we rewrite the elastic force in this form while there is no shear force?
when we consider a Immersed Membrane Without Shear, we can define a regularized elastic energy as $$\mathcal{E}(\varphi)=\int_{\Omega} E(|\nabla \varphi|) \frac{1}{\varepsilon} \zeta\left(\frac{\...
- 151
0
votes
1
answer
119
views
FE simulation of traction separation relations
I am trying to move completely open source. One of the things my research group does is cohesive zone modelling using traction separation laws, which I currently implement in ABAQUS. How easy would it ...
- 33
3
votes
2
answers
103
views
Can a mixed boundary conditions in 1D Linear FE lead to non positive definite stiffness?
Consider the finite element discretisation of a general second order linear differential equation, with mixed boundary conditions at the end. The boundary conditions are given as $\alpha_0u'(0)+\...
- 33
0
votes
3
answers
170
views
Dirichlet condition in finite element method
I'm trying to understand the approach described in these questions:
How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices
How to apply non zero ...
- 145
1
vote
1
answer
106
views
in Finite Element, which approximation requires less number of unknowns: B-splines vs Shape functions vs Spectral Elements
In Finite Element Analysis, one requires to approximate the solution in terms of basis functions. My questions is, which method in general is better? by 'better', I mean which involves less number of ...
2
votes
0
answers
71
views
How can I validate my time integration scheme for my dynamic linear elasticity FEM code with a manufactured solution or similar?
I am solving for a dynamic linear elasticity problem:
\begin{equation}
\dfrac{\partial^2 u}{\partial^2 t} - \nabla \cdot \sigma = f
\end{equation}
where $ u \in R^2 $ with sufficient BCs and ...
- 394
2
votes
1
answer
135
views
Quadrature over a triangle. Computing the dot products of the gradients
From the book "Understanding the Finite Element Method" by Mark S. Gockenbach (Chapter 7), there is stated an integral:
$$ \int_{T_k} \kappa \nabla \phi_i \cdot \nabla\phi_j. $$
for a ...
- 23
3
votes
0
answers
98
views
Implementation of the roller constraint
What could be the best way to implement the roller constraint in finite element code, i.e. constraint of the type
$$\mathbf{u} \cdot \mathbf{n} = 0$$
I plan to enforce it in the weak sense by ...
- 229
2
votes
1
answer
122
views
error estimate inequalities in finite elements
In finite element textbooks there are error estimate inequalities like
$e<~h^p$ or $e<~N^{-p/d}$, where h is the FE length scale, d the # of dimensions, p the FE order, N the # of DOFs. Are ...
- 225
0
votes
0
answers
75
views
Appropriate values for fluid-structure interaction model using finite element method in MATLAB
I'm having a program to compute the solution for a fluid-structure interaction model (blood and blood vessel walls) using FEM in MATLAB. The program is running fine but the function that I'm ...
3
votes
2
answers
167
views
Solving systems of advection-diffusion-reaction equations with finite element methods
I have been doing a lot of self-study on numerically solving PDEs so that I can solve system of linear and nonlinear Advection-Diffusion-Reaction (ADR) systems on complex meshes.
I have been watching ...
- 287
1
vote
2
answers
491
views
PhD in scientific computing to be a scientific programmer
Intro and disclaimer: this question concerns developing a career in Scientific Computing in industry, starting from an (applied) mathematics background, say an MSc. It definitely arises from my ...
- 243
1
vote
0
answers
66
views
shape functions of interpolating a piecewise polynomial with continuous 0-th and 1-st derivatives
shape functions are the basis functions that interpolate a function in a subdomain using polynomials. linear interpolation is probably the most convenient approach which results in so-called "...
1
vote
0
answers
65
views
No reasonable solution for cylindrical waveguide using FreeFEM++
I am using FreeFEM++ to simulate the cylindrical waveguide, but no reasonable solution is returned. Here I'm using the 94 GHz light ($\lambda$ = 3.17mm) as input, and radius of the waveguide is 1.27mm....
- 11
0
votes
0
answers
49
views
PETSC: Solving a simpler PDE results in error while solving the original equation works in snes/tutorials/ex13.c
In snes/tutorials/ex13.c,
there is a function SetupPrimalProblem(),
which sets up the $f_0$ and $f_1$ in ...
- 101
0
votes
0
answers
69
views
How to get damping matrix for structural model in FE analysis
I need to implement in C a method of obtaining transient solution of damped FE models based on modal results for a structural model (imported CAD geometry) defined with hysteretic (structural) damping....
- 1
0
votes
0
answers
56
views
non zero dirichlet boundary condition entered in weak form
i am trying to write a julia code for linear elasticity in my case i dont have body force and traction but there is a nonzero drichlet bc(ubc) i want to engage the bc in weak form in linear part. is ...
- 1
0
votes
0
answers
38
views
Equilibrium position finding with DSM
I've coded a framework that can be used to simulate the dynamic behavior of a system discretized by particles (nodes) that are connected by spring-damper elements. However, I want to compare it to a ...
5
votes
0
answers
111
views
References on the theory of Petrov-Galerkin methods for more "basic" problems
In my reading on various aspects of FEM, Petrov-Galerkin methods often arise in the study of solutions of convection-dominated systems, such as Hughes' work on Navier-Stokes, or systems where optimal ...
- 2,054
5
votes
2
answers
351
views
Why does the choice of basis functions change the approximate solution in FEM even when the same space is spanned?
Let's say we have the the weak form laplace problem
Find $u$ $\in$ $H^1$ what satisfies
\begin{equation}
\int_\Omega \nabla u \cdot \nabla v \, d\Omega = \int_\Omega f v \, d\Omega
\end{equation}
...
- 394
2
votes
1
answer
66
views
Quads with some nodes shifted
I'm new to gmsh. I would like to produce a mesh of quads where some nodes are shifted.
I managed to have construct a structured mesh of quads. However, I would like now to shift some of the points to ...
- 23
0
votes
1
answer
90
views
Pretension a truss mechanism (using the direct stiffness method)
Currently, I'm working on a mechanical mechanism where nodes are connected via beams. This is very comparable to planar truss mechanism analysis, but in my case, the deformations are large (relative ...
- 1
0
votes
0
answers
58
views
Plot 2D piecewise function in MATLAB
I'm trying to use this method Plot 2D piecewise constant in matlab in a finite elements mesh for my data. However, I have problems with the values of n1, n2 and n3. I guess this is why it doesn't show ...
0
votes
1
answer
113
views
Mass Matrix of Tetra10 Element
How or where can I find the mass matrix of a 10 nodes tetraedron element for FEM computation? Every nodes has 3 dof (x,y,z) so that the element has 30 dof. The Stiffness Matrix is a 30x30 matrix and ...
- 23
0
votes
2
answers
163
views
Determination of the domain of nonlinearity in a Neo-Hook solid model (Finite elements)
For a FEM simulation of a Neo-Hook solid model, how do we know we are in the "regime" of nonlinearity of the solid? In other words, how do I know the hyperelastic material law is really used,...
- 23
1
vote
1
answer
114
views
how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)
So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following:
$\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$
I am now familiar with ...
- 35
1
vote
0
answers
66
views
How can I optimize the solution of linear systems in my Finite Element Method code for solving nonlinear computational mechanics problems?
Currently, I use Eigen3 for linear algebra operations on sparse matrices and either UMFPACK or CHOLMOD from SuiteSparse to solve sparse linear systems. However, as my model grows larger and the need ...
1
vote
1
answer
132
views
2D Heat equation solved with finite element method converges in skewed way
I tried to solve the 2D heat equation with the finite element method, using triangles as elements. Currently generated by a Delaunay triangulation. The base function I'm currently using is basically ...
0
votes
1
answer
100
views
finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
- 35