PROVABLY ROBUST DIRECTIONAL VERTEX RELAXATION FOR GEOMETRIC MESH OPTIMIZATION

We introduce an iterative algorithm called directional vertex relaxation that seeks to optimally perturb vertices in a mesh along prescribed directions without altering element connectivities. Each vertex update in the algorithm requires the solution of a max-min optimization problem that is nonlinear, nonconvex, and nonsmooth. With relatively benign restrictions on element quality metrics and on the input mesh, we show that these optimization problems are well posed and that their resolution reduces to computing roots of scalar equations regardless of the type of the mesh or the spatial dimension. We adopt a novel notion of mesh quality and prove that the qualities of mesh iterates computed by the algorithm are nondecreasing. The algorithm is straightforward to incorporate within existing mesh smoothing codes. We include numerical experiments which are representative of applications in which directional vertex relaxation will be useful and which reveal the improvement in triangle and tetrahedral mesh qualities possible with it.