Optimal Tetrahedralization for Small Polyhedron: A New Local Transformation Strategy for 3-D Mesh Generation and Mesh Improvement

摘要

Local transformation, or topological reconnection, is one of effective procedures of mesh improvement method, especially in three-dimensional situation. The commonly used local transformations for tetrahedral mesh involve changing in mesh topology (i.e. node-element connectivity relationship) within a relatively small region composed of several tetrahedra, such as 2-3 flip, 3-2 flip, 2-2 flip, 4-4 flip, etc. Although these local transformations are easy to implement and effective in removing poorly-shaped tetrahedra, it is still possible to improve the quality of mesh further by expanding the space of transformation region. In this paper, the concept of optimal tetrahedralization for small polyhedron and corresponding small polyhedron re-connection (or SPR for abbreviating) approach are presented. As a new local transformation scheme and a potential substitute for the existing ones, the presented method seeks for the optimal tetrahedralization of a polyhedron with a certain number of vertexes and faces (typically composed of 20 to 40 tetrahedral elements) rather than simply making a selection from several possible configurations within a small region that consists of several tetrahedra, and therefore will give better results than existing ones. Despite of quite high time complexity of the optimal searching algorithm, the presented approach can be significantly speeded up by some deliberate strategies. Experimental investigation and results on tetrahedral finite element mesh show that the SPR approach is quite effective in improvement of mesh quality with acceptable time cost, and more suitable for combining with smoothing approach. Although further researches are required for a more definite conclusion, the presented approach can be utilized as a powerful and effective tool for tetrahedral mesh generation and mesh improvement. We believe that the superior performance of the SPR approach makes it worthy to be further studied.