9
votes
Accepted
Why is the FVM traditionally used in CFD, and FEM in computational structures?
The finite element method is actually quite widely used in fluid flow problems, for example for the Stokes and Navier-Stokes equations.
The delineation between the methods is more along the following ...
- 52.4k
9
votes
Accepted
4th order tensor rotation - sources to refer
There are two main ways to write stress/strain tensors as 6 components vectors:
Voigt notation, that is the most common; and
Mandel-Kelvin notation, that has the advantage of writing stress and ...
- 8,209
7
votes
Accepted
Derivative of the inverse of the Right Cauchy-Green Deformation Tensor wrt itself
Nick Alger gives a nice explanation. Here is another one, possibly slightly simpler because it avoids the "should stay roughly the same" part.
Let's say you want to compute the derivative of any ...
- 52.4k
7
votes
Accepted
Why are the linear elasticity equations referred to as the "equilibrium" equations?
The general idea comes from this example.
Imagine a rigid pendulum of length $l_0$ hanging a mass $m_0$. The equation of motion can be derived from a minimum principle (Lagrangian).
Denoting with $\...
- 1,618
7
votes
Is it really necessary to solve a system of linear equations in the Finite Element Method?
I think your question is actually pretty fundamental and deserves a thoughtful answer.
Paraphrasing a bit, your question is perhaps motivated by the observation that engineering design is often ...
- 4,604
6
votes
Accepted
Traction -> stress; stress->displacement gradient
In short, no. Neumann boundary conditions should be specified in terms of traction. This is clear when you move from a strong-form statement of the boundary value problem to a weak-form.
Strong form:
...
- 1,522
6
votes
How does one calculate reaction force in FEA?
To calculate the reaction forces at a node, Abaqus (or any structural FE code) simply sums the internal forces for all elements attached to that node. The reaction forces are the negative of that sum.
...
- 5,844
6
votes
Accepted
Stability criterion for waves in anisotropic solids
Wave equations like this can be rewritten as a hyperbolic system of first-order conservation laws:
$$q_t + \nabla \cdot F(q) = 0.$$
The stable time step for any explicit numerical discretization ...
- 16.4k
6
votes
Accepted
Euler-Bernoulli beam element versus continuum beam element
The standard, displacement formulation quadrilateral is notoriously bad at
representing bending behavior, especially with only one element through the
thickness of the beam. This is often referred to ...
- 5,844
6
votes
Extracting system matrices from FEM software
All of the major finite element libraries (such as libMesh; FEniCS; or the project I run, deal.II) provide you with ready access to the system matrix and or any other matrices you need. They typically ...
- 52.4k
6
votes
Accepted
Is steady linear elasticity inherently ill-conditioned?
The condition number for the stiffness matrix of any method (finite elements, finite volumes, finite differences) applied to a second order differential operator always grows as ${\cal O}(h^{-2})$ ...
- 52.4k
6
votes
Modelling question: example of a physical phenomenon with this jump condition at an interface?
$\vec\Phi = K_i \nabla u_i$ is the flux across the interface. For example, if $u$ is the thermal energy density and $K$ the thermal conductivity, then $\vec\Phi$ is the thermal energy flux. Energy ...
- 52.4k
5
votes
Time step relationship with number of elements or material properties
It sounds as if you're running a time-dependent linear elasticity simulation, right? Most likely, you're running an "Explicit" time-stepping scheme, which means that all of your information at time $...
- 1,522
5
votes
Accepted
Why the solid FEM problem can not be solved after constraining 3 degrees of freedom?
The particular set of constraints you have chosen does not prevent a rigid body rotation about node 1. Thus the stiffness matrix is singular, as you have noted.
One way to prevent this rigid body ...
- 5,844
5
votes
Accepted
Uniaxial stretching solution not uniform in FEM code
The reason that this particular mesh does not give the correct, uniform
displacement solution to this problem is that it is "non-conforming."
Specifically, at the intersection of the two ...
- 5,844
5
votes
Accepted
Transition from 2D to 3D finite element code, what are the inevitable modifications to be implemented?
So so many places you have to rewrite. The whole mesh handling (accessing faces and edges from cells, neighbors from cells, ...). Shape functions. Dealing with the question of how the normal vector of ...
- 52.4k
5
votes
Accepted
Why FEM for incompressible materials is ill-posed?
For incompressible materials, as the extent of incompressibility increases, the bulk modulus approaches infinity (for the isotropic case). This is what causes ill-conditioning.
Consider the case of ...
- 810
5
votes
If FEM is exact at the nodes, why do first and second-order elements give very different results?
They should both converge to the same limit solution, but are expected to do so at different rates of convergence.
The best way to verify the convergence rates is to use a test-setting with known ...
- 2,411
5
votes
Accepted
Finite Element Modelling of Hyperelastic Material under 2D Plane Strain Conditions
The plane-stress model is physically more meaningful for modelling thin structures. This is the reason for its popularity. It's rare to encounter problems with hyperelastic material models where the ...
- 810
4
votes
Developing a C++ solid mechanics program
Specific answers to this question are probably time-limited. However, the following general approach (from the great Eric S. Raymond) works very well:
Rule of Modularity: Write simple parts ...
4
votes
Time step relationship with number of elements or material properties
The most commonly used explicit ODE solver in structural analysis is the central difference method.
Because it is explicit, the solution becomes unstable if the time step is larger than a so-called ...
- 5,844
4
votes
Accepted
1D analytical solution vs FEM solution for a bar under compression
It is a matter of boundary conditions on the longitudinal faces.
As you noted, the axial stress and strain for a linear isotropic material will satisfy the following relation:
\begin{equation}
\...
- 472
4
votes
Simulation of a complicated projectile motion
You need to know which equations you need to solve inc initial conditions.
IMO, a disadvantage of click&result software is that it's not transparent to what you are actually solving.
Why don't ...
- 277
4
votes
Accepted
Structural mechanics - traction free = Zero displacement gradient?
The traction is defined as
$$
\mathbf{t} = \mathbf{n} \cdot \boldsymbol{\sigma}
$$
In terms of components, the zero-traction condition is
$$
t_j = \sum_i n_i \sigma_{ij} = 0
$$
From the above ...
4
votes
How does one calculate reaction force in FEA?
Once the solution of the problem is known, i.e. you know displacement vector, to calculate reaction/internal forces an integral is evaluated
\begin{equation}
\mathbf{f}^\textrm{int} = \sum_{e=1}^{...
- 906
4
votes
Accepted
I wrote a 2D Finite Element program for Axial Loaded Plates, but the results are unexpected
The primary problem is that the CST approximation has a different displacement response depending on the orientation of mesh elements relative to the applied element loading (you're only allowed to ...
- 1,751
4
votes
How to determine global stiffness matrix is constrained or not
You already know that at least theoretically, unconstrained matrices have a null space and consequently eigenvalues that are equal to zero. But, in practice, this is a meaningless condition because it ...
- 52.4k
4
votes
Global stiffness matrix from element stiffness matrices for a thin rectangular plate (Kirchhoff plate)
What you are looking for is a Discrete Kirchhoff Quadrilateral plate or DKQ plate. Seems you are looking for a very straight ...
- 141
4
votes
How are the classical set of equilibrium equations for linear elasticity derived?
You take an arbitrary volume $V$ and use the translational and rotational equilibrium equations over it. They read
\begin{align*}
\int\limits_A \mathbf{t}\mathrm{d}A + \int\limits_V \mathbf{f} \mathrm{...
- 8,209
4
votes
Accepted
How to solve a linear problem A x = b in PETSC when matrix A has zero diagonal enteries?
Use of Lagrange multipliers produces a saddle-point problem,
$$ \begin{pmatrix} A & B^T \\ B & 0 \end{pmatrix} \begin{pmatrix} u \\ \lambda \end{pmatrix} = \begin{pmatrix} b \\ 0 \end{pmatrix} ...
- 25.4k
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