Spacing Control and Sliver-free Delaunay Mesh

|We are often required to generate a Delaunay mesh whose element size is within a constant factor of a control spacing function, i.e., well-conformed, in addition to the fact that each mesh element has small aspect ratio, i.e., well-shaped. However, generating wellshaped Delaunay meshes is an open problem for a long time. Observe that slivers have small radius-edge ratio thus the Delaunay triangulation of well-spaced point set can not guarantee a sliver-free mesh. In this paper, we present a re nement-based method that, given a PLC domain with no acute input angles, guarantees to generate a well-shaped and well-conformed Delaunay mesh. Speci cally, for any tetrahedron generated by this algorithm, its radius-edge ratio is at most a small constant %0 > 2, which can be given as an input parameter. Moreover, we show that there is a constant 0 > 0 depending on %0 such that V=L 3 0, where V is the volume of and L is the shortest edge length of . Thus, the algorithm generates a well-shaped Delaunay mesh: the aspect ratio of each tetrahedron is at most a constant depending on %0. The size of each tetrahedron element is also within a small constant factor of the given control spacing.