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The main contribution of this paper is a new mesh generation technique for producing 3D tetrahedral meshes. Like many existing techniques, this one is based on the Delaunay triangulation (DT). Unlike existing techniques, thk is the first Delaunay-based method that is mathematically guaranteed to avoid slivers. A sliver is a tetrahedral mesh-element that is almost completely flat. For example, imagine the tetrahedron created as the (3D) convex hull of the four corners of a square; th~ tetrahedron has nicely shaped faces — all faces are 45 degree right-triangles — but the tetrahedron has zero volume. Slivers in the mesh generally lead to poor numerical accuracy in a finite element analysis. The Delaunay triangulation (DT) has been widely used for mesh generation. In 21), the DT maximizes the minimum angle for a given point set; thus, small angles are avoided. There is no analogous property involving angles in 3D. We make use of the Empty Circle Property for the DT of a set of point sites: the circumcircle of each triangle is empty of all other sites. In 3D, the analogous property holds: the circumsphere of each tetrahedron is empty of all other sites. The Empty Circle Property can be used as the definition of the DT. There is a vsst literature on mesh generation with most of the material emanating from the various applications communities. We refer the reader to the excellent survey by Bern and Eppstein [BE92]. We consider here only work related to the topic of mesh generation with mathematical quality guarantees. Chew [Che89] showed how to use the DT to triangulate any 2D region with smooth boundaries and no sharp corners to attain a mesh of uniform density in which all angles are greater than 30 degrees. An optimality theorem for meshes of nonuniform density was developed by Bern, Eppstein and Gilbert [BEG94] using a quadtree-based approach. Ruppert [Ru93] later showed that a modification of Chew’s algorithm could also attain the same strong results

Guaranteed-quality Delaunay meshing in 3D (short version)