Geometry in Action

From:  (Bill Thurston)
Newsgroups:     sci.math
Subject:        Re: Hexahedral decomposition of polyhedra
Date:           25 Oct 1993 04:28:19 GMT
Organization:   U.C. Berkeley Math. Department.

In article <29sb7v$>,
>In the process of generating hexahedral finite element meshes,
>the following question is important: What are the conditions
>on a polyhedron that allow a hexahedral decomposition
>of the polyhedron without modifying its boundary? One
>condition is that all the faces be quads, i.e., have four
>edges. What are the other conditions? Note, for example, that
>it is possible to mesh a tetrahedron with hexes, but not
>without modifying its faces, i.e., adding vertices. 
>Any help would be appreciated. 
>   Mac Casale

Hexahedron here means combinatorially equivalent to a cube.
This can be taken as a question in combinatorial topology,
where the faces can be curvilinear, or one can insist that
the hexahedra really have linear faces.   Let's look at
the topological question first.

One way to view this problem is to dualize, that is, take
the dual subdivision.   The dual of a cell division of the
sphere by quadrilaterals
is characterized by the property that edges meet in 4's.  You
can think of it as a subdivision of the sphere obtained by
drawing some possibly self-intersecting  closed curves that meet
themselves and meet each other transversely.  This is like
a knot or link projection.   

A hexahedral cell division of the ball is dual to a subdivision
obtained by a collection of surfaces inside the ball, intersecting

Each double curve (where a surface intersects itself) has to either
be closed, or meet the boundary in two points.  So a necessary condition
is that there be an even number of quadrilaterals.  (There do exist
convex polyhedra with an odd number of quadrilateral sides, so
this is a nonvacuous condition.)

Given a system of curves that does have an even number of self-intersections,
it is possible to find a surface with transveral self-intersections having
the system of curves as its boundary.  (not necessarily dual to
a hexahedral decomposition yet though, at this point).
First, a simple closed curve bounds a disk.  Second, if there is any
pair of self-intersections connected by an arc on the system of
curves, you can pair them up with a double line of surface, and
reduce the problem to one with two fewer double points.  The only
remaining possibility is that each component of the curve is
a figure eight.  You can connect any two such together.

This gives a way to construct the surface with boundary the given
curves.  It may not be dual to a hexahedral decomposition, because
for instance there may be no interior vertices (corresponding to
hexahedra) at all.   So, add a bunch of spheres with enough intersection
points with the existing surface to make lots of interior vertices.
This turns it into the dual of a hexahedral decomposition.

This answers the topological question.
As for the geometric question:  My first guess is it can be done
geometrically if it can be done topologically, but it
looks very tricky.  If you're using curvilinear coordinates, then
it's irrelevant.  Which question are you really trying to answer?

It is known that any
combinatorial subdivision of the sphere into quadrilaterals can
be realized by a convex polyhedron; one strategy would be first
to try to show everything can be gotten into convex form (by constructing
quadrilateral-cross-section tubes), then analyzing further obstructions.

By the way, one easy construction for certain hexahedral decompositions:
a tetrahedron divides into 4 hexahedra, so if you start with something
with triangulated boundary and divide into quadrilateals, you can
always extend it.

	Bill Thurston

Part of Geometry in Action, a collection of applications of computational geometry.
David Eppstein, Theory Group, ICS, UC Irvine.

Last update: .