9 votes
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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 ...
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9 votes
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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 ...
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  • 8,091
7 votes
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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 ...
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7 votes
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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 $\...
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  • 1,606
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. ...
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  • 5,744
6 votes
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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: ...
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  • 1,522
6 votes
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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 ...
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6 votes
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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 ...
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  • 5,744
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 ...
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6 votes
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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 ${\cal O}(h^{-2})$ ...
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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 ...
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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 $...
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  • 1,522
5 votes
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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 ...
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  • 5,744
5 votes
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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 ...
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  • 5,744
5 votes
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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 ...
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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 ...
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4 votes
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Buckling reference using the FEM

I suggest you take a look at this site: http://shellbuckling.com/index.php Dave Bushnell has pretty much written the book on computational techniques for buckling analysis. There is a wealth of ...
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  • 5,744
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 ...
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  • 5,744
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 ...
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  • 277
4 votes
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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 ...
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4 votes
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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} \...
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  • 472
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}^{...
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  • 906
4 votes
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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 ...
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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 ...
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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 ...
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  • 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{...
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  • 8,091
4 votes
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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} ...
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  • 25.3k
4 votes

Modelling question: example of a physical phenomenon with this jump condition at an interface?

In addition to Wolfgang Bangerth's explanation of temperature and concentration, let me give an other application where such interface conditions arise: (linear) elasticity, which has a similar ...
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4 votes

Second Piola-Kirchoff Stress Tensor of Neo-Hookean solid at "zero deformation"

From an old set of notes, I think that since your material is incompressible, the stress is determined by the strain energy density function $W$ only upto the hydrostatic pressure. The Cauchy stress ...
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  • 438
3 votes
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Why does the displacement have to be small to use the infintesimal strain elasticity equations?

In general, in the linearized theory of elasticity, it is only the displacement gradients that need to be small compared to unity. The displacements don't need to be small. (Note that when we say ...
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