Automatic 2D Abstraction and Hexahedral Meshing by Sorting a Delaunay Mesh

A framework is presented for the automatic 2D abstraction, decomposition and block-structured hexahedral meshing of a volume defined by a closed triangular mesh. No proofs, necessary or sufficient conditions are provided. Instead, constructive definitions and instructions describe a procedure for automatically building a 2D abstraction and an all-hexahedral mesh. First, a constrained Delaunay tetrahedralization of the interior of the object is built and its tetras are partitioned into groups based on their internal edge and face counts. Through labeling, organized structures are observed where 64-tetras (6 internal edges and 4 internal faces) bookend stacks of 54-tetras (5 internal edges and 4 internal faces). Stacks of 54-tetras, connected four-to-each 64-tetra, form a network of primary prism that populate the entire object near and along its edges. The exposed edges of the primary prism form rails that partition the input mesh into two categories of patches. Source and target patch pairs that define extrudable sub-volumes, and edge patches that define secondary prism. Degeneracies are remedied through affine transformations and an efficient local mesh manipulation process. The mid-mesh of the extrusion pairs is then computed and extended in order to obtain a 2D manifold that is a far simpler, albeit incomplete, 2D abstraction of the object than a customary mid-surface/medial axis representation. The result is a partition of the volume into extrudable, prismatic and tetrahedral blocks that trivially leads to a single block-structured hexahedral mesh.