In the spirit of “breaking the ice” on these forums, I’ll relay a few related questions I’m commonly asked regarding IGA and that I perhaps struggle the most at explaining effectively. These questions crop up during 1:1 conversations with other analysts, professors, and researchers. Unfortunately I don’t always have a straightforward response to questions regarding the merits of having a smooth basis function. For instance, here are a few questions I was asked last week by a researcher from LANL:
What are the practical benefits of having a smooth basis function in a “simple” simulation (i.e. linear or non-linear without contact). For instance:
Convergence rate rules? Do these offset the computational cost?
Is there value in having smooth derived values (e.g. smooth stress/strain fields)
How often does the Bezier extraction need to be performed? Is it expensive (RAM and or FLOPS), particularly in problems with adaptive meshing? Is it scalable?
How does the stable time step in an explicit dynamics simulation vary as a function of smoothness?
EDIT : Even a succinct list of good reports on these topics would be nice – we’ve talked before about the difficulties in “separating the wheat from the chaff” regarding IGA papers.
I’m not the best to answer questions 1 and 2, but I can answer question 3 for you as I have been studying that for a while now. We haven’t published anything on it yet, but I can give a quick overview.
This is a complicated question. A huge reason why linear elements became the industry standard is that higher order Lagrange elements display inferior time steps even for the same number of DOFs. The main cause of this is the lower continuity between the elements. This is because the exact eigenfunctions are typically of high continuity and lowering the continuity adds kinks to the discrete eigenfunctions. Raising the order of the basis emphasizes the kinks even more.
Using NURBS allows us to smooth these kinks out, but another problem emerges where the boundary elements are less constrained when compared to the interior elements. This results in inferior time steps when compared to linear elements.
We have been experimenting with U-splines to fix this problem. We constrain the boundary elements more by lowering the order on the boundary without lowering the continuity. It appears that this fixes the issue and we have been able to run explicit simulations with vastly superior time steps to those using linear elements. In one problem the time step estimate was 1.24e-7. Using linear elements, the power method estimated a time step of 1.29e-7. Using a U-spline without modifying the boundary basis, the time step was 1.06e-7 which is worse than the linear case. After modifying the boundary elements, the time step jumped to 1.78e-7.
Essentially, there are many spline spaces that are higher order that can be chosen for the same linear mesh. If we choose the wrong one, we destroy the time step such as when using higher order Lagrange elements. However, if we choose the right one we get a vastly better time step when compare to linear elements.
