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 ...

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Convergence of the second derivative of the finite element solution

Let $u_h$ is the finite element solution of a fourth order equation (like biharmonic equation), using polynomial degree two. If the convergence rate of $u_h$ be $2$, what is the convergence rate of ...
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1answer
63 views

FEM: Obtaining the Weak Form

In the the Finite Element Method (FEM), we attempt to obtain the Weak Form of the described equation. I understand that this is an attempt to reduce the order regularity of the equation, but what are ...
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3answers
72 views

Angle of rotation at a point in a deformed triangle

I have a 2D triangle which deforms with each vertex moving by some small ($\sin(x) \approx \tan(x) \approx x$) displacement vector. The displacement of any point in the triangle is linearly ...
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0answers
49 views

Solving system of equations with zeros on diagonal [on hold]

I am using finite elements, Newton Raphson technique and MUMPS direct solver, to solve for $P$ in this equation: $$\frac{\delta K}{\delta x} \frac{\delta P}{\delta x}= \frac{\delta K B}{\delta x} + ...
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1answer
56 views

Continuous vs discontinuous pressure elements in fluid flow problems

When solving fluid flow problems using finite elements you typically end up requiring that $p\in L^2$. For Darcy flow problems, a popular choice of elements is the Raviart-Thomas element and ...
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47 views

Approximation properties of FEM projections operators on a boundary

We have an elliptic projection $$P: V \rightarrow V_{h}$$ which satisfies $$\Vert u - Pu \Vert_{L^{2}(\Omega_{e})} \leq Ch^{k+1} \enspace .$$ Can we say anything about $\Vert u - Pu \Vert_{L^{2}( ...
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1answer
54 views

How to deal with transition elements in adaptive fem

It is necessary for me to solve a Poisson equation by adaptive finite element method with transition elements technique to get conforming mesh. For the first local refinement everything is OK, ...
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459 views

Choice of basis in FEM

Briefly, what are the different types of basis used in FEM and why is the nodal basis so popular and advantageous in the finite element context?
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2answers
103 views

Finite element results by different meshes

There are some technique to generate mesh in a domain. My qustion is that: Is there any difference between the results using different techniques for mesh generation? If yes which one is better. For ...
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1answer
56 views

Gradients on 2D FEM Triangular Mesh

A scalar field f approximated on a triangular mesh using the FEM method can be given as $f(x,y) \approx \bar{f}(x,y) = \sum_{n=1}^3 \left(\phi_n(x,y) \cdot f_n \right)$ I am of the understanding, ...
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2answers
69 views

Simple question about find the element indicator

It is necessary for me to find the element indicator for poisson equation by linear basis in FEM. Therefore I have to find the following: $$\eta_k=h_k||f+\Delta ...
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1answer
84 views

Uniqueness of coefficients for shape functions of higher-order finite elements

In the FEM we usually use a trick to find the coefficients for shape functions: Finding $M^{-1}$ in the system $MC=I$, where for example in the linear case: $$ \begin{aligned} M&=[1 ,x_1,y_1;1 ...
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35 views

Comparing the following mesh grids for a star-shaped plate

Between these four different meshes for the same shape, compare them whilst considering the different results for displacement and stress. Moreover the difference in computational cost and accuracy of ...
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1answer
50 views

1D uniform flow test case for compressible flow

Are there any one dimensional hyperbolic test cases for a 1D uniform flow? (I need the test case for the purpose of showing the satisfaction of geometric conservation law on a moving mesh).
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1answer
102 views

a few questions on understanding geometric conservation law

I was wondering if anyone could help with understanding the geometric conservation law for moving domains. I came across Link1, and have tried to understood the paper by Farhat et al Link2. So far ...
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44 views

convergence of a method

I want to show convergence of a finite element method for a higher order equation.I have a coupled equations that solved together and gives two variables as answer $[u, v]$. $$w+\Delta u=0$$ ...
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1answer
59 views

Order of accuracy of DGFEM or FEM

I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is ...
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1answer
54 views

Integrate over a face of a bubble function

I'm working in a finite element code where I need to calculate $$\int_F b_F$$ where $F$ is a face of some tetrahedron $K$ in the mesh, and $b_F$ is the usual bubble function defined by ...
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3answers
104 views

Reconstructing a continuous function from finite element method. Is there a faster algorithm for doing so?

Lets say I've decomposed a continuous function $y(x)$ over some domain $L_x$ using known finite element method with local basis $Q_i(x)$. Suppose $L$ is divided into $M$ "elements". If I want to know ...
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4answers
135 views

How to make a good mesh in a biologically accurate model with very small domains

I have been trying to make a biologically accurate 2D spatial model of tissue layers, where different physiological processes happen. This includes mainly chemical reactions, diffusion and fluxes over ...
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67 views

Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)

Edit: the question is solved ! Thank you for your time ! This problem arises when I tried to compute the valua of Asian call otions using Inverse Laplace transform method. Let $r=\mu = 0.15; \sigma ...
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0answers
18 views

Duality/Lagrangian condition and Variational Inequality of a cost functional

Given the functional $J_A(y,u) = \frac{1}{2}||y-y_d||^2_{L^2(w,\omega)} + \frac{\lambda}{2}||u||^2_{L^2(w^{-1}, \omega)}$ where w is a function belongs to Muckenhoupt class. Given the optimization ...
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1answer
91 views

Using finite element error estimators for adaptive mesh refinement

I am in the process of implementing adaptive mesh refinement for a finite element code that solves the Poisson equation. I have had some trouble finding good references on deciding which elements to ...
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1answer
132 views

Which preconditioning for large linear elasticity problem?

The problem I want to solve is the displacement formulation of the linear elasticity : $$ \nabla \cdot \sigma = 0 \quad \text{in} \quad \Omega \\ \sigma = \lambda ( \nabla \cdot u ) I + \mu (\nabla ...
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1answer
124 views

what is the difference between non-conformal and conformal?

So far what I understand is that two neighbouring elements are conformal if their edges and faces match exactly, whereas with non-conformal elements this is not the case. For instance, h-refinement ...
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1answer
72 views

Is it possible to show global conservative properties FEM as it is done in FVM?

I know that in FVM, it is possible to show that a discretisation scheme is conservative by adding the discrete terms over a few control volumes and showing that all terms cancel apart from those ...
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2answers
347 views

In FEM, why is the stiffness matrix positive definite?

In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation? For instance, we can consider the ...
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1answer
95 views

Are FEM or DGFEM methods based on integrals or PDEs?

I know that FVM is based on the integral form of conservation laws, and FDS is based on PDEs. What I'm confused by, is whether FEM and DGFEM formulations are based on integral or pde form of ...
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50 views

difference/similarities between DGFEM and FVM and FEM methods

I have been to trying to understand the discontinous galerkin method. My understanding is that in comparison to FVM, the essential difference is in the weighting function used and by setting the ...
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1answer
148 views

What is a “hanging node” in the finite element meshing?

When reading literatures about finite element method, the term "hanging nodes" can often be encountered. Could anyone tell me what indeed is a hanging node? Thanks in advance.
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62 views

understanding interior penalty jump of basis function

If cardinal basis functions are used (i.e. $\psi_{ij}=1$ iff $i=j$, and 0 otherwise) in interior penalty methods for elliptic equations such as SIPG & NIPG, shoudn't the jump in basis functions ...
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1answer
93 views

BCs in a coupled problem

Consider a thermo-mechanical coupled problem, where coupling exists from both the sides, mechanical loading producing thermal effects and vice versa. In such a case, is it necessary to always ...
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81 views

Time discretization of wave equation

I am trying to model the seismic wave equation and have therefore been reading about discretization schemes and their stability. I recently came across an insightful paper on 'Galerkin FEM methods for ...
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2answers
133 views

Understanding Neumann BC

I understand what Neumann BC means physically and how to imply them. However, I am not able to perfectly understand it the way it is represented mathematically as $\partial u / \partial n$ where $n$ ...
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1answer
82 views

How can i get gauss-lobatto points on a quadrilateral?

How can I get Gauss-Lobatto points on a quadrilateral or a triangle in $x$-$y$ plane? I am only getting abscissa coordinates and weights by solving Lobatto polynomials using Lobatto quadrature. ...
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17 views

How to model charge density waves in anisotropic arrays of metallic nanoparticles

The system I am investigating (theoretically) is an anisotropic array of metallic nanoparticles (arranged in such a way as to mimic strain). Each nanoparticle can host a charge density oscillation ...
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2answers
105 views

Finding rate of convergence by curve fitting in Matlab

I have some data: number of nodes $N$ and error in energy norm corresponing to it. I have seen in some references that the rate of convergence is reported by $$\| u-u_h\| _E=CN^{\alpha} $$ How can ...
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1answer
77 views

Help with Variational Formulation

Am solving a system of pdes in Fenics and need to write the variational form. Everything looks good expect a term like the following $$ \nabla \xi\nabla \cdot U $$ $\mathbf{U} = (U,V) $ is a vector ...
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48 views

Solving the transient advection-diffusion equation analytically

Can the following 1D transient advection diffusion and corresponding initial and boundary conditions: $$ \frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} - D\frac{\partial^2 c}{\partial ...
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42 views

How to model softening of concrete?

I have a question about acquiring the softening part of the compressive stress-stain curve of concrete under uniaxial static loading. I have tried very different options: Concrete Damaged Plasticity ...
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2answers
99 views

Evaluate integral on Boundary - FEM

I am implementing a MatLab program to solve the equation given in this paper, which involves solving integrals coming from the variational formulation of the problem. One of them is $$\int_{\partial ...
2
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1answer
75 views

how can I find the mesh size

Let there is triangular mesh for a simple domain (domain is not important). If this mesh is not uniform (the mesh is produced by adaptive technique), what is the mesh size? Is there a mesh size for ...
3
votes
1answer
126 views

Meshing options to generate number of the sides of and element (tetgen-triangle)

I wrote a finite element code in fortran 90. This code is really fast, except the meshing process. I used triangle and tetgen for meshing in 2D and 3D, respectively, so this process is fast, of ...
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2answers
106 views

Prescribe solution of a PDE at specific points

I am using MATLAB's PDE toolbox to solve the differential equation $-\nabla\cdot\left(c(x)\nabla u(x)\right) + a(x)u(x) = f(x)$ The particular problem in question is an electrostatic problem, but ...
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0answers
59 views

Solving the Jump function in discontinuous Galerkin method

I am trying to solve the elasticity equation using Crouzeix-Raviart element. Since this can not be solved using continuous FEM I am trying to solve this by discontinuous Galerkin method. In the ...
4
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0answers
134 views

Boundary conditions in the Finite Element Method at only one side of computational domain

I want to solve a Sturm-Liouville problem in 1D, i.e., \begin{align} [p(x)\ u'(x)]'+q(x)u(x) = f(x) \end{align} with boundary conditions \begin{align} u(0)=a \hspace{1cm} u'(0)=b \end{align} How do I ...
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1answer
76 views

Solve FEM matrix from coupled system

I'm developing an FEM solver for a coupled system. I have diffusion and potential equations which result in positive definite matrices for each equation, but the coupling makes the overall system ...
2
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1answer
59 views

Finite strain FEM using an existing code that solves small strain elasticity

I have an existing FEM code that solves the linear elasticity problem. I would like to use the same code for large strain rates, still using a simple material law (Saint Venant–Kirchhoff model). [The ...
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2answers
151 views

How to implement adaptive mesh refinement using conformal triangles

I am trying to implement adaptive mesh refinement for a finite element code. The code uses (at least for now) linear triangles and so when I do the mesh refinement I want the triangular mesh to remain ...
2
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0answers
90 views

FEniCS: both normal and shear stress boundary conditions for elasticity? [closed]

I would like to have both the normal (xx) component and shear (xy) component of a 2D (stress) tensor defined on a boundary (y=const, for instance) for an equation which is of the type $$ \nabla \cdot ...