  Hexahedral meshes are an ubiquitous domain for the numerical resolution of
partial differential equations. Computing a pure hexahedral mesh from an
adaptively refined grid is a prominent approach to automatic hexmeshing, and
requires the ability to restore the all hex property around the hanging nodes
that arise at the interface between cells having different size. The most
advanced tools to accomplish this task are based on mesh dualization. These
approaches use topological schemes to regularize the valence of inner vertices
and edges, such that dualizing the grid yields a pure hexahedral mesh. In this
paper we study in detail the dual approach, and propose four main contributions
to it: (i) we enumerate all the possible transitions that dual methods must be
able to handle, showing that prior schemes do not natively cover all of them;
(ii) we show that schemes are internally asymmetric, therefore not only their
implementation is ambiguous, but different implementation choices lead to
hexahedral meshes with different singular structure; (iii) we explore the
combinatorial space of dual schemes, selecting the minimum set that covers all
the possible configurations and also yields the simplest singular structure in
the output hexmesh; (iv) we enlarge the class of adaptive grids that can be
transformed into pure hexahedral meshes, relaxing one of the tight requirements
imposed by previous approaches, and ultimately permitting to obtain much
coarser meshes for same geometric accuracy. Last but not least, for the first
time we make grid-based hexmeshing truly reproducible, releasing our code and
also revealing a conspicuous amount of technical details that were always
overlooked in previous literature, creating an entry barrier that was hard to
overcome for practitioners in the field.
