## General Mechanical

#### Update of the stiffness matrix by the Newton Raphson method

• Emperor
Subscriber

Hello to all,

I would like to ask for your expertise on finite elements in order to know how to make the right choice on the different methods of solving the finite element calculation.

In Ansys, the resolution of the equation K*U=F is done by the Newton-Raphson method. Several resolution techniques exist, among them: the full, asymmetric and modified Newton-Raphson method (respectively NROPT,full NROPT,unsym NROPT,modi).

The difference in its resolutions is that for the first two methods, the stiffness matrix is updated at each iteration contrary to the last one which updates the stiffness matrix only on the first iterations (so there is less inversion of the matrix and less formulation). The final choice of these two methods could be done by what means? (we can always proceed by scanning the cases and see the difference of the results, which can be long I think)

Are there some structures that require an update of the stiffness matrix at each iteration (I guess yes as for hyperelastic materials) and others not? If the stiffness matrix is not updated at each iteration the accuracy is not the same I think, so the last method of solving should not exist?

thank,

• 1shan
Ansys Employee
Check out the 2 images below. -

The first one is a regular newton raphson and the second one is modified newton raphson. In the regular NR method you can see that at each incremental displacement the tangent slope is calculated(slope is decreasing), while in the modified NR method the slope is calculated just once, at start and the same slope is used (all lines are parallel)to progress ahead till convergence. For regular NR since you calculate tangent at each point you know that you are moving in the "right direction" and hence only a few iterations are needed. But the "right direction" comes at a cost - you need to evaluate stiffness matrix at each point which is a computationally expensive. On the other hand for modified NR since the stiffness is not updated at each point, you don't know if you are moving in the "right direction" or not and hence more iterations are usually needed than regular NR to achieve convergence. But since you are computing stiffness matrix only once, each iteration is faster. In short, Regular NR - less iterations but each iteration is time consuming, Modified NR- more iterations but each iteration is faster.
Regards Ishan.
• Emperor
Subscriber
Thank you for the answer because it is much clearer.
In a simulation how to choose the most efficient one? or is it arbitrary?
For a behavior with more curvature it is clear that the update of the stiffness matrix is absolute.