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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How to determine the orientation of convex/concave hexahedra?
I am writing a code that checks the orientation of a list of vertices (along with face connectivity) describing both convex and concave hexahedra. The face connectivity table stores the list of vertex ...
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1answer
117 views
Write incompressible Navier Stokes as ODE in $(\mathbf{u},p)$
Consider the Navier stokes equation after the discretization with conforming finite elements with source term $f=0$. We have the algebraic structure of a saddle point problem:
$$M \dot{u} = f- Au -B^...
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130 views
Material properties for a node in a 2-material FEM code
I'm trying to debug an FEM that I inherited, and I unfortunately do not have much knowledge of FEM. I only know FD and FVM.
If you're modeling a system with 2 materials, there will be an interface ...
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58 views
How to Calculate magnetic and electric field in 2D Magnetotelluric using Edge based Finite Element
I calculate 2D Model of Magnetotelluric responses which are apparent resistivity and phase. I do the calculation for Transverse Electric (TE) mode. Then I used edge based finite element with ...
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1answer
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Locking phenomena for $P1 - P0$ elements
Consider the Stokes problem and the usual divergence operator $B:V \rightarrow Q'$, $\langle Bv, q\rangle = b(v,q)=(\operatorname{div} v,q)$
and its discrete versione $B_h : V_h \rightarrow Q_h'$.
In ...
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141 views
About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation
I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$:
$$A u + B^t p = f$$
$$Bu = g$$
where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
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1answer
76 views
Finite element method for Stokes and Navier--Stokes with square elements only
I wanted to learn how to implement a code for the Stokes and Navier--Stokes equations 2D/3D. I already know how to implement it when the elements are triangles or tetrahedral.
Do exists finite element ...
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Understanding inf-sup conditions for classical saddle point problems
I'm studying the inf-sup conditions for saddle point problems. I'm referring to the usual one $$\begin{cases}Au + B^t p = f \\Bu=g \end{cases}$$ In the book I'm using (Ern - Guermond: Theory and ...
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1answer
126 views
Proof of R. Verfürth paper on adaptive mesh and bubble functions
I'm studying adaptive meshes, and my professor wrote the following property for a bubble function ( see this scicomp post for the definition I'm using)$b_T$ defined on a triangle $T$.
$$||b_T \phi ||_{...
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1answer
85 views
Classical global estimate for $H^1$ error
I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$.
$$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
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Sum of norms over elements is not equal to norm over the whole $\Omega$
In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$:
$$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
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135 views
Discrete divergence free functions
I'm studying the weak formulation of NS equations. During the analysis, the book I'm using (Quarteroni-Valli, page 301-302), defined $$Z_h=\{v_h \in V_h: (\operatorname{div}(v_h),q_h)=0 \quad \forall ...
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1answer
63 views
MPI vs OPENMP usage for boundary element method
I have an implementation question with MPI and OPENMP.
I have a large three-dimensional surface mesh divided into elements. I want to loop over each element (outer loop) and compute an integral over ...
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59 views
Displacement/Pressure Finite Element formulation for Large Deformations (from Bathe)
On page 563 of Finite Element Procedures by Klaus-Jürgen Bathe the author states that governing equations of the displacement/pressure finite element formulation for large deformations is given as
$$
\...
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135 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)
$$(\frac{du}{dt},v)_{\Omega} + (\...
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What is the difference between $u_h$ and $I_h(u)$ in finite element literature?
In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
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How to treat nonlinear radiation term in heat equation using Finite-element method?
I am trying to solve the time-dependant non-linear heat equation with radiation. This equation is coupled to the radiative transfer equation but for the purpose of my question, this does not matter ...
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How to decrease error in (FTCS) forward time centered space method?
I am using the FTCS method for solving differential equations. I know that the condition for stable output is
$$
\frac{\alpha \Delta t}{\Delta x ^2} < \frac{1}{2}
$$
But when I use the distance ...
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85 views
Confusion about preconditioner for incompressible Navier-Stokes equation with implicit-explicit method
Consider the time-dependent Navier-Stokes equation
$$u_t + (u \cdot \nabla) u - \Delta u + \nabla p = f$$
$$\operatorname{div}(u)=0$$
Looking at deal.ii tutorials, I've notice that there are ...
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1answer
64 views
Finite Volume on Cubed Sphere
The US weather model uses an uncommon (?) discretization called 'Finite Volume on Cubed Sphere'.
To avoid the singularities that occur at the poles when using lat/lon discretization, they instead ...
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1answer
77 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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How to prove the stability of the interpolation?
From the book (Vidar Thomee, Galerkin finite element methods for parabolic problems), there holds (see Lemma 13.3)
\begin{align}\label{eq}
\|\nabla I_h u\|_{L_{\infty}}\leq C\|\nabla u\|_{L_{\infty}},\...
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How to couple the vibro-acoustic equations by Mortar method for non-matching meshes?
Assume we have two domains $\Omega_a$ a acoustic domain with boundary $\Gamma_a$ and $\Omega_s$ a domain of a solid body with boundary $\Gamma_s$.
$\Omega_a$ and $\Omega_s$ have the common interface $\...
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57 views
Ritz projection error estimate for bilinear Lagrange element?
For second order elliptic problem
\begin{align}
-\Delta u=f,\quad in ~~\Omega\\
u=0,\quad on~~ \partial\Omega,
\end{align}
we have for the Ritz projection for $P_1$ conforming element
\begin{align}
\|...
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1answer
67 views
Applying displacement control loading using lagrange multipliers in the material non-linear finite element method
Hi I am trying to implement a simple plasticity based finite element code. I am not clear how to set up displacement control applied through Lagrange multipliers. In case of a linear problem, I did ...
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2answers
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what finite elements are stable for the mixed form of the elasticity equations?
The mixed form of the elasticity equations is to find the unique critical point of the Hellinger-Reissner functional
$$J(u, \sigma) = \int_\Omega\left(\frac{1}{2}A\sigma : \sigma - (\nabla\cdot\sigma)\...
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Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?
I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann ...
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1answer
101 views
Bubble function property: how to prove it?
I'm studying error estimators and I need a check on an estimate. In our course, we've been given the following definition of bubble function (in 2D):
It's a function defined on a triangle $T$ such ...
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1answer
89 views
Computing the residual in a Dual Weighted Residual (DWR) method
I am in the process if computing the Dual-Weighted Residual (DWR) for a linear PDE with a linear functional but I am struggling with the residual part of the calculation.
For example suppose we want ...
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1answer
75 views
Displacement field not correct?
Consider the elastic equation $$- \operatorname{div}(C \nabla \mathbf{u}) = \mathbf{f}$$ as presented in step-8. Here $\mathbf{u}$ is the displacement vector, let's consider the 2d case.
As you can ...
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1answer
131 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
131 views
Dyadic operations, fourth order tensors and Tensor algebra
I am trying to understand the dyadic operation for a while since I am interested in Elasticity problems. I believe an intuitive understanding (rather than assuming) will give me good problem solving ...
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Hanging nodes in deal.ii tutorials: how is the continuity constraint imposed?
While looking at step6 of deal.ii tutorials, I decided to try to understand how the constraints coming from hanging nodes are imposed. So I started by watching video lecture 16 by prof. Bangerth
As ...
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Can this volume intgral be expressed as a convex function?
This question is related to the following:
https://math.stackexchange.com/q/4151405/685910 - the context is summarized below for clarity.
In the setting of convex optimization, I am looking for a ...
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1answer
118 views
Weak Formulation of Poisson's Equation for Electrostatics with Surface Charge
I am currently attempting to use FEnICS to solve an electrostatic problem with two materials of different permittivity $\varepsilon_1$ and $\varepsilon_2$ forming an interface:
Consider a domain $\...
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How to interpolate stress at unknown points from the stress values available based on geometrical position for constant load?
I am working on a combined contact, bending, and torsion problem. I have data on geometrical points and their instantaneous stress components. However, based on the available data, I have to ...
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73 views
Transient advection equation with stabilized FEM
I am interested in solving the transient advection equation
$\left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-...
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1answer
84 views
discretizing surface integral using nodal DG method
I am currently learning nodal DG methods, primarily through the book by Warburton, and am a bit confused on how to handle surface integrals using straight edged elements. On page 187 (and on page 214)...
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Constrained optimization for non-linear equations in octaveGNU
I have installed Optim1.6.1 package. I would like to solve a system of equations in non linear finite element analysis using constraints as u=1 at certain nodes. u=0 at certain nodes. Typically I find ...
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2answers
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Reference for mass matrix assembly
I will now explain what I understand to be the process of a finite element mass matrix assembly. I would like a reference which does something similar, or if I am mistaken about the process please let ...
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2answers
257 views
How to implement point source or volume source in finite element implementations
I'm trying to do a simple implementation to study the advection-diffusion-reaction dynamics in a straight pipe.
I have points positioned along the length of the pipe (blue dots in the image above).
I ...
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1answer
84 views
RCM better than Nested dissection? (For FEM discretizations in 2D and 3D)
I realize this might be a too general question but here goes nothing:
I am trying different re-ordering strategies and checking the fill-in of $A=LU$.
I have 2D ($p=1$, $h=1/40$ on $\Omega = [-1,1]^2$)...
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1answer
173 views
What is the weak form of a vector type Laplace equation?
For a scalar variable $u$, the weak form of the following Laplacian:
$$\nabla^2 u =0 $$
with the assumption that $v$ vanishes at the boundary is
$$\int \nabla v . \nabla u \, d\Omega = 0 $$
which is ...
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1answer
66 views
Validity of algorithm for assembling the finite element global stiffness matrix
This blog post describes an algorithm for constructing the finite element global stiffness matrix as two basic steps:
[Step 1] Initialize a table with the elements shared by a pair of connected node ...
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160 views
How to create a good preconditioner for a system of linear equations that is created with FEM applied on the time harming Maxwell eqution?
I set out to solve the time harmonic Maxwell equation numerically which was discritzed using FEM and with the use of Nedelec elements as basis and test functions. The equation reads:
$$ \nabla \times \...
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Jacobian Matrix of 2D element mapped to 3D
Note: I previously posted this question to MathStackExchange, but got no attention there. So I'm rewritting and trying over here.
Problem summary
Given a common¹ set of shape functions defined at ...
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Weak form of Elliptic problem with mixed Dirichlet & Neumann conditions
Let $\Omega \subset \mathbb{R}$ in a bounded polygon domain and $f:\Omega \to \mathbb{R}$ known function.We split the boundary into two parts $\partial \Omega_{1}$ and $\partial \Omega_{2}$ such that $...
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Biharmonic equation mixed formulation, mixed boundary conditions
I have the following formulation of the biharmonic equation:
$$\Delta u(x) = v(x), \quad x \in \Omega\setminus K$$
$$-\Delta v = 0, \quad x \in \Omega\setminus K$$
$$u(x) = g(x), \quad x \in K$$
$$v(x)...
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2answers
301 views
Is mesh orthogonality important for FEM?
While studying mesh quality metrics in literature and software documentation, I've seen discussions about mesh orthogonality in Finite Volume Method (FVM) contexts, but not for Finite Element Method (...
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Confused on how Method of Manufactured solutions works?
I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique
so my starting equation ...
