In February, Gary Miller presented our results on static mesh refinement: the problem was to, given an input geometry, produce an output tetrahedral mesh where every tetrahedron has good aspect ratio. This requires adding some additional points to the input; he showed how to minimize (to within a constant factor) the number of points in the output, and how to run in optimal sequential time.
When the mesh domain changes after the first timestep, we now need to solve the dynamic mesh refinement problem: how to maintain a small but good quality mesh as we change the input. I will show that after addding or removing points from the input, the mesh refinement algorithm of Bern, Eppstein, and Gilbert can recompute the output quality mesh in O(log L/s) time. This uses the automatic dynamization techniques of Acar, which ensures that the dynamic algorithm is no harder to implement than the static algorithm.
In two dimensions, we can extend this to do even better: by dynamizing a post-processing algorithm of Har-Peled and Ungor, we can maintain a mesh as small as is known how to produce in practice, at no additional asymptotic cost.
I expect this joint work with Umut Acar (TTI-C) to be of particular interest to geometers, dynamic algorithmaticians, and scientific computers.