Efficiently Hex-Meshing Things with Topology

Discrete & Computational Geometry, Sep 2014

A topological quadrilateral mesh $$Q$$ of a connected surface in $$\mathbb {R}^3$$ can be extended to a topological hexahedral mesh of the interior domain $$\varOmega$$ if and only if $$Q$$ has an even number of quadrilaterals and no odd cycle in $$Q$$ bounds a surface inside $$\varOmega$$. Moreover, if such a mesh exists, the required number of hexahedra is within a constant factor of the minimum number of tetrahedra in a triangulation of $$\varOmega$$ that respects $$Q$$. Finally, if $$Q$$ is given as a polyhedron in $$\mathbb {R}^3$$ with quadrilateral facets, a topological hexahedral mesh of the polyhedron can be constructed in polynomial time if such a mesh exists. All our results extend to domains with disconnected boundaries. Our results naturally generalize results of Thurston, Mitchell, and Eppstein for genus-zero and bipartite meshes, for which the odd-cycle criterion is trivial.

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Jeff Erickson. Efficiently Hex-Meshing Things with Topology, Discrete & Computational Geometry, 2014, 427-449, DOI: 10.1007/s00454-014-9624-3