Shelling Hexahedral Complexes for Mesh Generation

We present a new approach for the generation of hexahedral finite element meshes for solid bodies in computer-aided design. The key idea is to use a purely combinatorial method, namely a shelling process, to decompose a topological ball with a prescribed surface mesh into combinatorial cubes, so-called hexahedra. The shelling corresponds to a series of graph transformations on the surface mesh which is guided by the cycle structure of the combinatorial dual. Our method transforms the graph of the surface mesh iteratively by changing the dual cycle structure until we get the surface mesh of a single hexahedron. Starting with a single hexahedron and reversing the order of the graph transformations, each transformation step can be interpreted as adding one or more hexahedra to the so far created hex complex. Given an arbitrary solid body, we first decompose it into simpler subdomains equivalent to topological balls by adding virtual 2-manifolds. Second, we determine a compatible quadrilateral surface mesh for all created subdomains. Then, in the main part we can use the shelling of topological balls to build up a hex complex for each subdomain independently. Finally, the combinatorial mesh(es) are embedded into the given solids and smoothed to improve quality. Communicated by T. Nishizeki, R. Tamassia and D. Wagner: submitted January 1999; revised August 2000 and July 2001. The author was partially supported by the special program “Efficient Algorithms for Discrete Problems and Their Applications” of the Deutsche Forschungsgemeinschaft (DFG) under grant Mo 446/2-3. M. Müller-Hannemann, Shelling Hexahedral Complexes, JGAA, 5(5) 59–91 (2001)60