Several promising approaches for hexahedral mesh
generation work as follows: Given a prescribed
quadrilateral surface mesh they first build the
combinatorial dual of the hexahedral mesh. This
dual mesh is converted into the primal hexahedral
mesh, and finally embedded and smoothed into the given
domain. Two such approaches, the modified Whisker Weaving
algorithm by Fowell and Mitchell, as well as a method
proposed by the author, rely on an iterative elimination
of certain dual cycles in the surface mesh.
An intuitive interpretation of the latter method is that
cycle eliminations correspond to complete sheets of
hexahedra in the volume mesh.
Although these methods can be shown to work in principle,
the quality of the generated meshes heavily relies on the
dual cycle structure of the given surface mesh.
In particular, it seems that difficulties in the
hexahedral meshing process and pure mesh qualities
are often due to self-intersecting dual cycles.
Unfortunately, all previous work on quadrilateral surface
mesh generation has focused on quality issues of the
surface mesh alone but has disregarded its suitability
for a high-quality extension to a three-dimensional mesh.
In this paper, we develop a new method to generate
quadrilateral surface meshes without self-intersecting
dual cycles. This method reuses previous b-matching
problem formulations of the quadrilateral mesh refinement
problem. The key insight is that the b-matching
solution can be decomposed into a collection of simple
cycles and paths of multiplicity two, and that
these cycles and paths can be consistently embedded into
the dual surface mesh.
A second tool uses recursive splitting of components into
simpler subcomponents by insertion of internal
two-manifolds. We show that such a two-manifold can be
meshed with quadrilaterals such that the induced dual
cycle structure of each subcomponent is free of
self-intersections if the original component satisfies
this property. First experiments indicate that the
hexahedral mesh quality can
greatly benefit from both methods proposed in this
paper.
