Generation of Polyhedral Delaunay Meshes

Abstract

A polyhedral mesh fulfills the Delaunay condition if the vertices of each polyhedron are co-spherical and each polyhedron circum- sphere is point-free. If Delaunay tessellations are used together with the finite volume method, it is not necessary to partition each polyhedron into tetrahedra; co-spherical elements can be used as final elements. This paper presents a mixed-element mesh gen- erator based on the modified octree approach that has been adapted to generate polyhedral Delaunay meshes. The main difference with its predecessor is to include a new algorithm to compute Delaunay tessellations for each 1-irregular cuboids (cuboids with at most one Steiner point on their edges) that minimize the number of mesh elements. In particular, we show that when Steiner points are located at edge midpoints, 24 different co-spherical elements can appear while tessellating 1-irregular cubes. By inserting internal faces and edges to these new elements, this number can be reduced to 13. When 1-irregular cuboids with aspect ratio equal to √2 are tessellated, 10 co-spherical elements are required. If 1-irregular cuboids have aspect ratio between 1 and √2, all the tessellations are adequate for the finite volume method. The proposed algorithm can be applied to any point set to compute the Delaunay tessellation inside the convex hull of the point set. Simple polyhedral Delaunay meshes generated by using the adapted mesh generator are shown.