Characterization of the quadrilateral meshes of a surface which admit a compatible hexahedral mesh of the enclosed volume

摘要

A popular three-dimensional mesh generation scheme is to start with a quadrilateral of the surface of a volume, and then attempt to fill the interior of volume with hexahedra, so that the hexahedra touch the surface in exactly the given quadrilaterals. Folklore has maintained that there are many quadrilateral meshes for which no such compatible hexahedral mesh exists. In this paper we give an existence proof which contradicts this folklore: A quadrilateral mesh need only satisfy some very weak conditions for there to exist a compatible hexahedral mesh. For a volume that is topologically a ball, any quadrilateral mesh composed of an even number of quadrilaterals admits a compatible hexahedral mesh. We extend this to volumes of higher genus: There is a construction to reduce to the ball case if and only if certain cycles of edges are even.