Three-dimensional Semi-generalized Point Placement Method for Delaunay Mesh Refinement

A number of approaches have been suggested for the selection of the positions of Steiner points in Delaunay mesh refinement. In particular, one can define an entire region (called picking region or selection disk) inside the circumscribed sphere of a poor quality element such that any point can be chosen for insertion from this region. The two main results which accompany most of the point selection schemes, including those based on regions, are the proof of termination of the algorithm and the proof of good gradation of the elements in the final mesh. In this paper we show that in order to satisfy only the termination requirement, one can use larger selection disks and benefit from the additional flexibility in choosing the Steiner points. However, if one needs to keep the theoretical guarantees on good grading then the size of the selection disk needs to be smaller. We introduce two types of selection disks to satisfy each of these two goals and prove the corresponding results on termination and good grading first in two dimensions and then in three dimensions using the radius-edge ratio as a measure of element quality. We call the point placement method semi-generalized because the selection disks are defined only for mesh entities of the highest dimension (triangles in two dimensions and tetrahedra in three dimensions); we plan to extend these ideas to lower-dimensional entities in the future work. We implemented the use of both two- and three-dimensional selection disks into the available Delaunay refinement libraries and present one example (out of many choices) of a point placement method; to the best of our knowledge, this is the first implementation of Delaunay refinement with point insertion at any point of the selection disks (picking regions).