6

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 strains in the same way, so their rotations are done via the same $6\times 6$ matrices. A reference that I consider good for Voigt's notation is Auld's book, and ...


5

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 rotation is to set the y-displacement at node 2 to zero. You could also constrain the x-displacement at either node 3 or node 4 to prevent the rotation. One way ...


4

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 applied forces on the nodes of triangles, so distributed loads must be approximated). You can see the effect of this by looking at only a single triangle ...


4

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 cubes in the model, the two element edges that cross that face don't align with each other but instead cross each other. A typical face in a non-conforming mesh ...


4

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 can not be checked in an efficient way for large problems. The question you specifically ask is how you can detect whether constraints have been applied, and ...


3

TL;DR How can I determine which constrain I need to apply to the system to make problem solved? Or how can I determine which rigid body constrain I should apply to the system? The constraints are given by the boundary conditions of your problem, so you should know them before you have a numerical method as the FEM. In that sense, that is more a physics ...


3

There is absolutely nothing wrong with converting the second-order system to first-order form and then using appropriate numerical methods to solve it. Both implicit and explicit Euler methods can be used. However, both have only first-order accuracy, i.e. the error is proportional to the time step size. And, of course, explicit Euler is not stable unless ...


3

What you are looking for is a Discrete Kirchhoff Quadrilateral plate or DKQ plate. Seems you are looking for a very straight forward formulation that simply give you the global stiffness matrix. But i'm afraid that most codes I've seen are dealing with integration and transformation. You can search for DKQ source code. There are documents for java which ...


2

Here is how you want to test this and you need only two elements in the mesh. You want to define your left BC so it will reproduce a constant stress state as follows: assuming $u$ is the displacement in the x-direction and $v$ the displacement in the y-direction, set $u=0$ at the two nodes on this edge and $v=0$ at the bottom node on this edge. The two nodes ...


2

If I understand your question correctly you're solving a linear elasticity problem using conjugate gradient and it's preconditioned with a preconditioned AMG solver? It seems to me that this may be overkill for a pretty well behaved problem, and that could be why you don't see much of a speed-up. Just to elaborate a bit. I think it makes more sense to just ...


2

As you said, "If displacement not be constrained, equation above can not be solved, because the system can have rigid body motion" So you should try to apply constraints that will not allow the body to move i.e. translate or rotate. In 2D there are 2 translations (along x and y axis) and one rotation (along z axis) to be killed. In 3D there are 3 ...


1

I think that the answer by @helloworld922 is misleading. The first image shown in the answer seems to be an effect of the Poisson effect, a contraction in one direction due to loads applied in the other direction. If you want to obtain a state of constant stress in your simulation you need to change the boundary conditions that you are applying, namely: all ...


1

This seems to be more of a Physics question than a Computational Science one. Due to the symmetry of your problem, you can conclude that the solution is of the form $$\mathbf{u} = u_r \hat{\mathbf{e}}_r(r)\, ,$$ since the selection of the zenithal and azimuthal angles is arbitrary. This turns the PDE system \begin{equation} (\lambda + \mu) \...


Only top voted, non community-wiki answers of a minimum length are eligible