Generation of tetrahedral mesh of variable element size by sphere packing over an unbounded 3D domain

Abstract

This paper describes an algorithm for the generation of tetrahedral mesh of specified element size over an unbounded three-dimensional domain. Starting from an arbitrary point in space (defined as the origin) and guided by the concept of advancing front, spheres of size compatible with the specified element size are packed tightly together one by one to form a cluster of spheres of different sizes. The compactness of the cluster of spheres is achieved by packing spheres at a site closest to the origin in a densest manner with tangent but no overlapping with as many other spheres as possible. In view of these criteria, a rotational mechanism between spheres is innovated, which allows the newly inserted sphere to follow the path by rotation between existing spheres until the lowest point is reached. The centres of the packing spheres provide ideal locations for Delaunay point insertion to form a triangulation of tetrahedral elements of size compatible with the specified value. Spheres of random size distribution or of size specified by a node spacing function are packed by the proposed algorithm. In all test examples, high-quality tetrahedral elements of size consistent with the specified node spacing are generated. The process is fast and robust, and the time complexity for mesh generation is expected to be almost linear; however, in the present implementation, a quasi-linear time relationship is observed as a search for the nearest node on the front is proposed in the sphere packing process. The memory requirement has been kept to a minimum as no additional data structure other than the adjacency relationship of the tetrahedral elements as required by the Delaunay triangulation is stored. The structure of the advancing frontal surface needs not be explicitly constructed nor updated as it is simply not required in the sphere packing process.