Exploring Non-Linear FEA: Applying a Force and Beyond

[chandran]

How non linear fea works. I am studying that and i can't find any material in the web which has some basics.

How the Newton raphson method is used in non linear fea

suppose i apply a force f on a structure after that how non linear fea works?

 

[PerennialII]

About the Newton-Raphson method ... you can view it as a generalization of Newton's method, where in this case the system response is sequentially linearized until convergence (the modified version classically doesn't update the coefficient matrix). Below a link about the method and FE solvers. So what you do in structural applications when your force - displacement response in nonlinear you linearize it sequentially until you've reached a convergent solution (typically e.g. using force residual/norm criteria).

http://www.aaue.dk/bm/ansys/b8km-fe05.pdf
http://uranus.ee.auth.gr/lessons/1/10.html

 

[chandran]

What is meant by a residual? How non linear works?

In each iteration what is checked against what? and why it is checked?

 

[PerennialII]

The nonlinearities were touched in general here :

https://www.physicsforums.com/showthread.php?t=74375

... to elaborate, when we're considering a typical structural mechanics problem, nonlinearity e.g. in material behavior (such as elastoplasticity) reflects upon the load - displacement response of the structure. The curve is no longer linear, and thus no longer follows from e.g. Hookean material behavior, but becomes nonlinear. In the FEA equations this will appear similarly as nonlinearities usually do, the stiffness matrix becoming nonlinear and dependent on the displacement field (i.e. the solution itself).

Because of this coupling, within the numerical process the solution needs to be established iteratively, meaning that you solve the problem by successive approximations forming a (hopefully) convergent series. The difference between two iterations is referred to as the 'residual', to which there are generally different measures and criteria. If we stay in the context of the Newton-Raphson method, the residual of force is usually defined by comparing the external and internal force vectors at a specific node (typically different implementations have different convergence criteria, commonly the norm of the residual force vector is compared to the norm of the external force vector and required to be a small fraction of it).