Small polyhedron reconnection: A new way to eliminate poorly-shaped tetrahedra

Local transformation, or topological reconnection, is one of effective procedures of mesh improvement method, especially for three-dimensional tetrahedral mesh. Although the existing local transformations such as 2-3/3-2 flip are effective in removing poorly-shaped tetrahedra, it is still possible to improve the quality of mesh further by expanding the space of transformation region. The authors recently proposed a new local transformation operation, small polyhedron reconnection (or SPR for abbreviating), which seeks the optimal tetrahedralization of a polyhedron with a certain number of vertexes and faces (typically composed of 20 to 40 tetrahedral elements). In this paper, the framework of SPR approach for mesh quality improvement based on the SPR operation is presented. The main idea is to take a poorly-shaped or "worst" element as the core and construct a small polyhedron by adding 20-40 elements surrounding it, then find the optimal tetrahedralization of this small polyhedron through SPR operation. By replacing the original tetrahedra with the optimal tetrahedralization, the quality of the mesh is improved. Experimental investigations with tetrahedral finite element meshes show that the SPR approach is quite effective in improvement of mesh quality with acceptable time cost, and works well in combining with a 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 of further study.Mathematics, AppliedMathematicsEICPCI-S(ISTP)