Non linear finite element theory

[chandran]

I wanted to get some thought in non linear fea.

Let me say there is a spring of initial stiffness k.

Let me say in a non linear analysis the load is divided into 20 steps of 1N each(Total load is 20N).
1.1st iteration.1N is applied and the displacement is arrived
2.2nd iteration 2N is applied. The stiffness matrix(tangent stiffness) is derived from iteration 1. The displacement(u) is calculated. F=ku is the calculated value of resisting force of the spring.

Now i want to understand how any fea software is able to calculate the actual resisting force the spring produces.Because after this step only the software is able to calculate the residual force(residual force=Calculated resisting force-actual resisting force).

can anyone explain this for a rod
(Take the example of a axially loaded rod.

The non linearity arrises from material non linearity).

 

[PerennialII]

Hi Chandran,

A decent example is to consider the behavior of the modified Newton-Raphson algorithm in a 1D case when material nonlinearities appear. In the modified routine the tangent stiffness, the stiffness matrix more usually, is not updated or is updated only at times so it applies well to your question.

Consider first that you derive a solution for an initial loadstate, call that for example state "1" (this could be an initial linear step in a nonlinear analysis).

We now seek a solution in state "2", which can contain a nonlinear response. We can write the force in state "2" as

$$P_{2}=P_{1}+k_{1}\Delta u_{1}$$

where $k_{1}$ is the stiffness in state "1" and $\Delta u_{1}$ is an increment of displacement. Now in nonlinear FE analysis, we can solve the $\Delta u_{1}$ since we know the applied loads and the stiffness in state "1". From this solution we obtain a new displacement estimate for state "2", which is

$$u^{i+1}=u_{1}+\Delta u_{1}$$

Now we compute a new residual force between the state corresponding to $u^{i+1}$ and "2", and redo the iteration. When the force residual converges towards 0, $u^{i+1}$ converges to $u_{2}$. Notice that I used "i" in there rather than state "2", being at the crux of it all, since several iterative steps, "i"s, may be required before reaching state "2".

So the keys are understanding the force residual driving the solution and updates to the displacement done on the basis of it, the stiffness acting as a 'projection' direction in the nonlinear P-u curve (you can think of it totally analogous to classic Newton's method).