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.

Filter by
Sorted by
Tagged with
1
vote
0answers
274 views

Alternatives to Newton-Raphson for nonlinear elasticity via finite element

As far as I have seen, solving problems of nonlinear elasticity using the finite element method proceeds by linearizing, either around the initial configuration (total Lagrangian approach) or around ...
1
vote
1answer
137 views

How would one solve the wave equation using the finite element method?

I would like to solve $$\alpha u_{tt} = -\nabla^2u$$ with $\frac{\partial u}{\partial n} = 1$. On using a Galerkin approximation I obtain $$M\ddot{c}=\frac{1}{\alpha}(Dc+b)$$ where $M$ is the mass ...
1
vote
0answers
61 views

Radially symmetric system of PDEs in deal.II

I am trying to solve the radially symmetric polar form of the PDE with homogeneous Neumann BC in deal.II on a unit circle: $$ u_t = \Delta u - \nabla \cdot (u \nabla h) $$ $$ h_t = \Delta h $$ I am ...
2
votes
1answer
88 views

Taylor expansion of error - Finite elements approximation

In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the ...
0
votes
1answer
73 views

Deriving weak form of a set of scalar equations

I have the equilibrium equation in elasticity for a static case.i.e Div T=0. For certain implementation, I have to get the x and y component equations and then derive the weak form separately. How is ...
0
votes
1answer
105 views

Non-linearities in modal analysis of flexible beam

I'm trying to analyse the behaviour of a wind turbine blade rigidly connected to a wall during fatigue testing, being excited in it's first mode in two orthogaonal directions simultaneously (the first ...
0
votes
2answers
138 views

Nédélec Elements and Newton-Methods

If you want to develop numerical algorithms for variational inequalities, you often choose a Semismooth Newton Method. In many cases, this approach involves derivatives of $\max$ or $\min$ functions ...
0
votes
1answer
148 views

Finite difference method for the electric field of the electron gun

Can anybody help me to find books or MATLAB code examples for solving electric field of the electron gun(fig.1)with finite difference method? Python code examples are also perfect. The electron gun ...
3
votes
1answer
270 views

Periodic boundary condition in solid

I want to solve a small deformation solid structure problem applying periodic boundary conditions in FEM. The geometry is a square and the equations are: $$ \text{div} \, \sigma = 0 \\ \sigma = f(\...
4
votes
1answer
94 views

Interpolation estimates for $H^1$ into $P_1$

As far as I can tell, nodal interpolation estimates proven with Bramble / Hilbert require higher regularity of the functions being interpolated. E.g. with linear elements in 2 dimensions, one needs $v ...
1
vote
0answers
59 views

Why do higher order finite elements (Q2) do not perform well for large Peclet number flows, as compared to Q1 finite elments?

I a solving the 2d steady state convection-diffusion problem on the famous flow around a cylinder in rectangular domain benchmark. My numerical results show that with Q1 finite elements, the solver is ...
3
votes
1answer
78 views

FEM problem: how to get a feeling for size of problem

The following problem is given: $$ - \bigtriangleup u = f, \quad \partial_n u = g \quad \text{on } \Gamma_N, \quad u = 0 \quad \text{on } \Gamma_D $$ with $\Gamma_N$ or $\Gamma_D$ denoting the ...
3
votes
1answer
343 views

Radiation boundary condition (heat transfer)

I am looking for reference on how to implement nonlinear boundary conditions. Specifically, I am interested in implementing a radiation boundary condition for heat transfer with the FEM: $-k \frac {\...
7
votes
2answers
300 views

How to calculate/derive analytic FEM Newton Jacobian

I trying to wrap my head of derivation of the analytic FEM Jacobian for the Newton method. Say we have a nonlinear Poisson problem of the (weak) form $$ \int a(u)\nabla\ u\cdot \nabla v = \int f v $$ ...
1
vote
0answers
210 views

Is reduced stiffness matrix positive definite too?

The stiffness matrix $K$ in a finite element analysis is a symmetric positive definite matrix. When we introduce essential boundary conditions, we remove rows and columns associated with prescribed ...
4
votes
0answers
157 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
1
vote
0answers
34 views

How do 'virtual kinematics/functions' play a role during deriving weak form formulations for physical problems?

I wanna ask a question that confuses me quite a long time. I saw many guys, in the context of computational mechanics, they seemed to choose the virtual functions or kinematics in a way that some ...
0
votes
1answer
146 views

Introducing EigenModes from 2D FEM into 3D FEM

This particular FEM question concerns waveguides and FEM 3D simulation. To excite a waveguide with waveport (TE10 and so on), we typically have to solve for eigenvalues ($k$) of helmholtz equation ...
4
votes
3answers
344 views

How can I make sure the flow is divergence-free when I use moving mesh?

I am using projection method and P2/P1 finite element method to solve the incompressible Navier-Stokes equations while the mesh is constantly adapted as the body moves (edge swapping, splitting and ...
6
votes
0answers
363 views

Understanding Boundary Condition in FEM

I am trying to understand Dirichlet and Neumann boundary conditions in FEM and I wanted to know if my inference is correct. To articulate my understanding, lets consider a simple case of TE and TM ...
0
votes
1answer
184 views

stiffness matrix for 3D regular grid in FEM

I have understood the stiffness matrix for 3D truss, and programmed Ku=f from scratch (in Java) to find the displacements. Then I moved to 3D solid but lost in too many concepts and equations, such ...
0
votes
1answer
464 views

Finite Element Analysis for Laminated Plates with Holes or Patches

As the title says, I am trying to code in FEM a plate structure that either has a hole in one of the layers or one of the layers is made of patches of plates, rather than one whole plate. However, ...
1
vote
1answer
141 views

Parallel calculation in finite elements

I am trying to solve a 1 Dimensional eigenvalue of poisson problem: $$\nabla \phi ^2 +\nabla \phi = k\phi$$ with the boundary condition: $\phi (0)=0 , \nabla \phi(1) = 0 $. I could solve this ...
0
votes
1answer
67 views

Principle of virtual work - extra term needed for deformation dependent loading?

I'm working on a problem in nonlinear elasticity, for which the external forces (loadings) depend on the displacements. Following Klaus-Jürgen Bathe's book "Finite Element Procedures", the virtual ...
3
votes
1answer
227 views

FEM current toy problem

I am solving the Dirichlet problem $$ \begin{cases} \Delta u = 0, \\ u|_{\partial D} = f, \end{cases} $$ in a $2d$ domain $D$ using the finite element method. What I want to get is the ...
8
votes
1answer
239 views

Resources on mesh generation for finite element methods

I know that this is not really apart of the rules as this is a recommendation question and these don't really have an answer per say. But, like this forum posting: https://stackoverflow.com/questions/...
4
votes
1answer
870 views

How to apply non zero Dirichlet boundary condition in finite elements?

I am writing a code for steady state heat transfer on a rectangular domain. I am specifying temperature on the edges - nonzero Dirichlet boundary condition. The equations can be written in form of $$...
0
votes
1answer
93 views

What is $H^{\frac{1}{2}}(\Gamma_g)$

In the second answer here one considers a function $w\in H^{\frac{1}{2}}(\Gamma_g)$. What is this space and what is the finite element functions that one should use that belongs to this space? I have ...
3
votes
2answers
315 views

Finite Difference and Finite Volume as special cases of Finite Element

I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE". However I have never come across a brief document that substantiates that claim, ...
2
votes
2answers
176 views

Question about motivation of mixed FE method

I'm trying to understand what a mixed FE method is, and I was recommended this pdf: http://www.ima.umn.edu/~arnold/papers/mixed.pdf (Arnold, Mixed Finite Element Methods for Elliptic Problems). On ...
1
vote
1answer
188 views

Raviart Thomas Mixed Finite Element with Mixed boundary conditions reference request

This is perhaps a more focused version of this question. Using standard notation, I have a code that works and solves the following PDE using a Raviart Thomas mixed method. $$\begin{align} 0 &= ...
6
votes
0answers
126 views

Do practice and theory differ substantially when implementing Neumann Boundary Conditions using a Mixed Method?

I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both ...
8
votes
2answers
302 views

Lagrange multipliers space is too rich in a mathematical view

Background: Lagrange multiplier method has been employed in numerous fields, such as contact problems, material interfaces, phase transformation, stiff constraints or sliding along interfaces. It is ...
1
vote
0answers
188 views

the augmented global stiffness matrix is not positive semi-definite using Lagrange Multipliers method within FEM

The augmented global stiffness matrix is not positive semi-definite when using Lagrange Multipliers method to enforce boundary constraints on a simple square domain of integral form: I am considering ...
0
votes
1answer
771 views

Stiffness matrix computation for 4 node quadrilateral element

I am writing a finite element code for heat transfer (scalar field problem) and starting from simple 4 node quadrilateral element. I tried computing conductance (stiffness) matrix in the physical ...
2
votes
1answer
168 views

Galerkin method for a system of nonlinear PDEs

Suppose I have a nonlinear system of PDEs. I am actually interested in Navier-Stokes, but, for the sake of simplicity and example, suppose I had $$ \frac{\partial f}{\partial t} - f \frac{\partial g}{...
3
votes
0answers
93 views

Piecewise Constant Enrichment of the continuous galerkin method

I am interested to study crack propagation in a hyperelastic material in a variational setting. The crack surface exhibits a jump discontinuity. The function space for displacement field should ...
3
votes
1answer
121 views

Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes

I am thinking to use discontinuous galerkin FEM (DGFEM) method to estimate discontinuous displacement field $u: \Omega \rightarrow \mathbb{R}^2$ at the crack surface of a material. The domain is ...
6
votes
1answer
169 views

Pressure definition/convergence issues for the Incompressible Navier-Stokes when using a stabilized P1-P1 finite element formulation

I believe this might be a recurring topic, but i have not found a post that directly related to this issue. I come from a finite volume background and my experience is more with predictor-corrector ...
1
vote
0answers
57 views

Plotting yield surface for geotechnical constitutive model

I have found the following example which is produced as documentation for RS3 geotechnical FE software. This is a validation document that goes through an analytical verification of triaxial testing ...
1
vote
1answer
131 views

Comparison between FEM libraries & languages

Is there any modern online resources which compares the most popular finite element method libraries/packages/languages?
1
vote
1answer
116 views

Correct way to model an embedded reinforcement (non linear FEM)?

I need to add to an existing FEM solver some embedded reinforcement element. This would give me the possibility to model/solve concrete structure (reinforced with steel rebar) taking into account the ...
2
votes
1answer
120 views

Some questions on Trace (operators) on the boundary in the context of PDEs

Background: The solution space of original problem (which requires a fine enough mesh to resolve the microstructure) can be split into a macroscale solution space and microscale solution space. This ...
4
votes
2answers
790 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? ...
2
votes
2answers
243 views

error estimator VS error indicator in the context of FEM error estimation

Is an error indicator an index for size measurement of something? And how is the relationship between the error estimator and error indicator? Is the error indicator just used for the (e.g. derivative)...
3
votes
1answer
143 views

Bubnov-Galerkin method in 1D: how to handle convective-type nonlinearity?

Consider the BVP: find $u = u(x)$, for $x \in (0,1)$ that satisfies \begin{align} u'' + u u' = f, \\ u'(0) = g_n, u(1) = g_d. \end{align} To derive the weak form for this BVP, we multiply the first ...
1
vote
1answer
75 views

What does Consistent Variation mean?

Consistent variation occurs in papers multiple times when reading . So what does it really mean from a maths view? I searched online and found an answer for it. Someone explained it as ‘Solution is ...
0
votes
1answer
86 views

Is FEM discretization error equivalent to quadrature error of bi-linear term in the weak form?

In FEM, to discrete the weak form, elements are needed to build the FE approximation basis space. FE method involves integration of the weak form. Due to discretization, the solution is an ...
1
vote
2answers
443 views

Create mesh for complicated 3D object for finite element analysis

I see images of steel connections, concrete dams, and other complicated 3D objects in papers which finite element analysis has been performed on them. My questions are: How these objects are created ...
3
votes
1answer
464 views

p-refinement in adaptive methods

Some reference in adaptive techniques say that when the solution $u$ is smooth enough we can use p-refinement instead of h-refinement. And when we have for example singularity, we should use h-...