Frame Field Singularity Correctionfor Automatic Hexahedralization

We present an automatic hexahedralization tool, based on a systematic treatment that removes some of the singularities that would lead to degenerate volumetric parameterization. Such singularities could be abundant in automatically generated frame fields guiding the interior and boundary layouts of the hexahedra in an all hexahedral mesh. We first give the mathematical definitions of the inadmissible singularities prevalent in frame fields, including newly introduced surface singularity types. We then give a practical framework for adjusting singularity graphs by automatically modifying the rotational transition of frames between charts (cells of a tetrahedral mesh for the volume) to resolve the issues detected in the internal and boundary singularity graph. After applying an additional re-smoothing of the frame field with the modified transition conditions, we cut the volume into a topologically trivial domain, with the original topology encoded by the self-intersections of the boundary of the domain, and solve a mixed integer problem on this domain for a global parameterization. Finally, a properly connected hexahedral mesh is constructed from the integer isosurfaces of $(u,v,w)$ in the parameterization. We demonstrate the applicability of the method on complex shapes, and discuss its limitations.