Linear
Complexity
Hexahedral David
Department
Eppstein
of Information
University
Mesh Generation
and
of California,
Computer
Irvine,
Abstract We show
that
any polyhedron
cal ball
with
can
partitioned
be
meeting
forming
an even number
face
into
to
O(n) The
face.
result
ditional
condition.
can
version lem
also
of the
be used
to
hexahedral
to a finite
generalizes
reduce mesh
case analysis
cubes,
satisfying The
to
amenable
hexahedra
prob-
part
to machine
these
cal work nite
and
complexes
[3].
where
are often erties
preferred
[1].
For
this
are
on the
ing. )
There
working
meshes
also
numerical been
their
actually
Steiner The
(At
Sandia, related been
the
1995,
4th
[8, 9, 10] but
t heretical
much
more
Although for this its
various version
whether
of convex
hexahedra
boundary
is unknown.
such
the
is must
as the
have
studied
open
mesh
gen-
well
un-
problems.
In
complexity
of de-
admits
a mesh
the
polyhedron’s
some
eight-sided
be
unsub-
are not
a polyhedron
1, remain
do-
to
remain
interesting
Even
we
of the
authors
respecting
hex-
location
assumed
properties
termining
or as
which
interior
computatational
prob-
of the in
of hexahedral
pose many
the
in Figure
the
mesh)
t heretical
and
variant
(which
particular
cases,
construc-
very
simple
polyhedron
shown
[12].
Annual
13 of the
to hexahedral some
prop-
generation
for
divided. heuristics
the
process).
restricting
to
quadrilateral
derstood
used,
points
subdivi-
domains,
problem,
boundary
as
adjoining
because
generation
by
But
several
of multiple
a common
of into
7(c).
of boundary
(either
problems
of new
Figure
to mesh
generation
main.
eration,
(cuboids)
mesh
on systems
on
commu-
numerical
many
fi-
simplicial
hexahedra
meshes.
agenda has
to
reason
Roundtable,
titles
on such
have or
due
of hexahedral
Meshing
in the
meshes
for
concentrating
dimensional
quadrilaterals
researchers tion
higher
of theoreti-
generation
largely
However
these
of
deal
mesh
met hods,
triangulations nity,
a great
on unstructured
element
meshes
been
in
type
mesh here
mesh
a planar has recently
this
in terms
We consider avoid
There
each tetrahedron
simultaneously
ahedral
Introduction
a tetrahedralization
it difficult
of a parallel
gener-
additional
find
as shown
is defined
for
if one allows
subdivide
[9] notes,
domains lem
solution.
1
simply then
sion can make
tech-
method
meshes,
points:
Mitchell
geometric
generation
is a straightforward
domain,
four
an ad-
same
the
the
92717
hexahedral
Steiner
sides
topological
polyhedra
bipartiteness
a topologi-
of quadrilateral
non-simply-connected
niques
CA
There ating
Science
For the planar
28
is easy:
mesh-
quadrilaterals,
work
the
to be done.
difficult Permission to make digitallltard copies of all or part of tbk material for personal or classroom use is gmnted without fee provided that the copies are not made or dktributed for pmtit or commercial advantage, the copyright notice, the title of the publication and ita date appear, and notice is given that copyright is by permission of the ACM, Inc. To copy otherwise, to mpubliah, to peat on acrvera or to dktributc to Ms, requires specific petission and/or fee. Computational Geometry’96, Philadelphia PA, USA 01996 ACM 0-89791.804-5/96/05. .$3.50
points
polygon
to find needed
timize
the
efficiently suffice
58
the
for
find
number
task
of Steiner
problem
problem into
convex
without
extra
if and
only
of sides.
number (or
a set of O(n)
[13] and
showed
boundary,
smallest
this
number
for this
Thurston dently
face to face,
on the
has an even
erals
corresponding
can be subdivided
meeting
subdivision
remains
case, the
a polygon
of quadrilat-
equivalently points)
if
It may to op-
but
Steiner
one can
points
that
[10]. Mitchell
a similar
[9] recently
characterization
indepenfor the ex-
Figure
istence
of hexahedral
First,
the
polyhedron
topological
ball
to
certain
polyhedra
the
mesh
and
ever they and
must
and
dral
if and ary
faces
even
ball
if there
are an even
parity
of the
dimension,
regardless
put,
each
faces
which
are paired and
either
contribute
complexity
tures
of the original
In this
paper
refinement
in the
near
theory. one first
cubes
before
the
mesh
to
regions
between
these
similar
idea
“buffering”
of
cubes
dled
with
some
local
to mesh
the
the
or require
There
generators
in
A
theory improve
59
sides
0(n2)
than
cells.
the
class of
those
han-
method.
practical
in
themselves:
and we have not
solution
hexahedral boundary
to
the
geo-
mesh generation
heuristic
and
will
often
to be modified.
here of a two way interaction
and
practice:
they number
soon be good
of cases remaining
generation
prove
all even-quadrilateral In the other
as heuristic
may
mesh
that
a topologi-
with
is too high
geometric meshed.
of the
not
the input
to solve the finite
remaining
are
is a possibility
between
genwith
boundary.
boundary
fail
Thurston’s
largely
of a cube) be geometri-
to a different
polyhedra
Practical
of polyhedra
forming
and
are still
generation
of quadrilateral
meshed,
completed
duality,
practically
mesh
can all
generalizes
of elements
case.
methods
on planar
of the domain and the
metric
for
combines
based
patch
results
the number
avoids
boundary
polyhedra
on the
hexahedral
collection
the
connected
satisfactorily method
of the
and
bound
it to the more
any polyhedron
by Mitchell
These
of fea-
of
has
Mitchell
our method
a finite
the method
technique of
an even number
non-simply
with
interior
algorithm
an O(n)
version
if these
with
Third,
or
[11] and others,
the interior
attempting
number
in spirit
of Schneider fills
meshes
because
can also be geometrically
many
Thurston
method
boundary
It is similar
heuristics
which
meshed,
cal ball
boundary
the
for a topological
by subdividing
cally
of the in-
that
to extend
we exhibit
(formed
domain).
Our
of a tetrahedral
problem:
in any
over we prove
geometric
meshes
an alternate
generation.
relevant
is a necessary
produce
polyhedral
we discuss
grid
manipulation eration
(nonlinear
Second,
it seems easier
that
However
may
mesh.
of bound-
has evenly
in
meshing
of cells needed
such
to the
interior.)
method
high
hexahedral
cube
First,
(Indeed,
of cubical
occurring
hexahedral
advantages
number
hexahe-
of the connectivity
up in the
Mitchell’s
graph
of faces
individual
to
subdivision,
are quadrilaterals.
number
three
Thurston
number
new
Thurston.
polyhedron
boundary
for the existance
since
How-
subdivisions
[2].
Our
convex.
has a topological
further
Bern
curved
any
from
mesh?
was also used in a tetrahedralization
second,
equivalent
that
without
be a
have
face to face.
showed
all of which
condition
elements
necessarily
meet
a topological only
And
have a hexahedral
input
generalizes
holes).
the
caveats.
has to
method
be combinatially
still
both
mesh,
some
be meshed
with
still
with
the
are not
must
Mitchell
forming
to
(although
is topologica~
boundaries cubes,
meshes,
1. Does this octahedron
met hod,
direction,
and
polyhedra even
mesh enough in our thereby can be
an impracti-
Figure
cal proof ful,
of the
2. Topological
existence
by guaranteeing
method
such as Mitchell’s get stuck
not
of meshes
that
will
mesh:
(a) cell complex;
configurate
idea
However
non-convex
[8]
Statement
of the
us define
formally
the
generation
problem
solved
here.
are given
a domain
ball
sider
other
The
more
boundary
finitely
many
of various ors,
that
by lower
the
task
boundary
covering
assume
the
dimensional
in
cells)
(two-dimensional its boundary. We wish
cell)
but
must
be a hexahedron:
lateral
wit h the
the faces, Any partition
edges,
same
of that
by
every
of four
face
every it
should
have
meeting
combinatorial
polyhedron
cell
input into
convex
and then
60
connect meeting
hexahedron
they
wit h
to pair
surfaces
hexahedra), can
be
an
and form part
of topo-
arrangement.
the
first
boundary even
part curve
arrangement) number
of
independently with
an odd
a surface that,
given
a collection
(transforming note
the
surface
surface
up curves
ions
of performing
an interior
solve the
curves
ment
cuboids
each
one
to
(extending
extending
to dual
on
and
to this surface
to an interior
eral posit ion,
Any
curves
extending
Mitchell
For the second quadri-
curves hex-
of each
finding
dual
and
problem
self-intersect
with
by
surfaces,
becomes
rangement
structure
conditions.
then
hexahedra
other;
as
center
arrangement
int erections
six quadri-
midpoints of
dual
to form
curve
a mesh
hexahedron,
transformation:
curve
in edges and
of a cube.
polyhedron the
this
com-
these
at the
problem
opposite
Thurston
the same conditions
find
the
as Given
this
quadrilateral given
of each
points
arrangement,
by our algo-
three-dimensional
can
can find
by connecting
Similarly,
such
polyhedron
of curves,
one
quadrilaterals
surface
edges as
to an arrangemesh
a
3(b)).
The the
dual of the
of each
dis-
[9]. is to treat
to an arrangement
facet
(Figure
a
(zero-
edge and face of the
and vertices
satisfies
and
produced
mesh:
simply-connected faces
one
logical
faces on its boundary,
vertices
lateral
in addition
ahedra, them
edge
vertices
has a cycle
to every
plex,
is
every
two
3(a)).
we briefly
Mitchell
authors
the
boundary
simply
(Figure
in triple
complex
as being
sides
boundary
2.)
the cell complex
apply
in-
these
of quadrilaterals,
of opposite
domain.
as its endpoints,
(Figure
Mitchell
[8], and a quadrilateral
the dual
arrangement
balls interi-
to a finite
particular,
mesh
one on the
a mesh
of
nontrivial
partition cell
has
to be a hezahedral
as above
this
cell)
into
cells.
[13] and
of both
of surfaces
being
to be cov-
cell is covered
any
the overall
quadrilaterali,zation;
topologies).
relative
method
as the
lower-dimensional
boundary
(one-dimensional
rithm
and
ment
con-
to closed
of any
cells,
condi-
partitions
our own methods,
of Thurston
hexahedral
to the
is, a collection
disjoint
is to extend
cell complex We
with
The
we
we will
is assumed that
of two cells is another
Our
assume
domain
sets equivalent
dimensional
tersection cell.
cells:
dimensions,
such
We (later
complicated cell comple~
and
describing
cuss those
mesh
equivalent
dimensions
of the domain
ered by a finite
topological
topologically
in three
output
problem
more
closed
our allows
and non-polyhedral
Thurston
Before Let
satisfies
our definition
ion.
3 2
faces, edges, and vertices.
face-to-face
tions.
heuristic
whisker-weaving
in a bad
meeting
can be help-
an incremental
(b) individual
of
of arby selfeach
number
of
for each pair. these
for surfaces
the structure
of the surface
represented
as a collection
surfaces in genarrangeof ver-
Figure
tices
(for
3. Dual of a mesh:
each
triple
edges
(segments
faces).
Each
and
pair
of hexahedra
ever
not
ing
may
should
set of surfaces
in this
way,
be connected
Thurston
surrounding
and
Mitchell
problematic
rangement
with
of surfaces our mesh Theorem
solve
of
partitioned
into
curves
●
are subdivided
surface
ar-
O(n)
a hezahedral
and
the
method
these
without Thurston
to a plane; triple
will
in-
produce
fl(n312).
an even
faces
can
be
respecting
curves 4(b)).
one If
squares
one
each side of the cube, other,
curves from
the
have
fl(n)
to form
incautiously
from
the
curves,
self-intersection
one-intersection crossing
forming
in a pattern
these
to
boundary.
two
with
squares
rectangles, boundary
arranged
to surfaces
the
of the boundary Q(n)
remaining
(Figure
siqdy
four into
quadrilaterals
solving
Any
mesh
of
have fl(n3i2)
non-self-intersecting
fl(n)
boundary
will
complexity
A cube in which
by
with
method
each of these
so this
of total
to
curves
result-
mesh
domain
points,
a mesh
problem.
quadrilateral
these Q(W)
dual
Jordan
The extends
tersection
is a collection
Thurston)
arrangement
of O(W)
and Mitchell
problem
of the
result
three-dimensional
number
a dual
in a single
this
hexahedral
1 (Mitchell,
connected
How-
edges,
intersect
The
a dual
generation
face.
curve
consists
and dual surfaces.
self-intersections.
to a
some of the ver-
regions
spheres.
with
boundary grids
a dual
determines
because
do not
and
correspond
by multiple
that
to
a common
and dual curves; (b) hexahedron
of sur-
correspond
sharing
in hexahedra
face.
of surfaces)
intersections
each edge should
every
cell complex tices
intersection
of pairwise
vertex
hexahedron
(a) quadrilaterals
each matches
into
pairs
of one
and extends
them
one side of the
overall
complexity
cube
can
be
Q(n2). However guarantee
this
on the
that
is,
This
complexity
affects
of the
the
using
method
time
the mesh;
does not provide
complexity number
of the resulting
of hexahedral
is very spent
any
as it
numerical
even small
constant
to provide
examples
of a
mesh,
cells
important, by
much in
4
it.
directly
dual
hard
surface
method
than
linearly
many
the
complexity
provide ●
constructs
factors
elements
of the
(measured
gen-
eration
can be
in which
meshes
mesh
method As we saw above,
critical. It is not
Linear-complexity
with
this more
in terms
polyhedron
boundary),
each square
is subdivided
Thurston
and
f2(n3i2)
or fl(n2
new
of
will
We
Mitchell
topological always
Our
the mesh
can produce
) hexahedra. mesh
give meshes
method
generation
meshes
We now
generation with
of with
describe
method,
O(n)
has the following
method
our
which
complexity. main
steps:
two:
A cube in which an O(@)
by O(@
grid
(Figure
4(a)).
1. We
into The
dron
61
separate from
the its
boundary
interior
by
B of the a “buffer
polyhelayer”
of
Figure
4. Bad examples
Figure
cubes. side
We do this
the
hedron’s orient ing
pairs
edges,
and
We
sitting
then
two
corresponding
pairs
of
with
pairs
of connecting
pairs
of faces
quadrilateral edges,
on the
two
and
S in-
suit,
each quadrilateral
in
same
the
edges
additional
on
corresponding
known
hexa-
vertex e the inner
triangulated found
by
common
hexahedra trahedral divided
A O(n)
connecting interior
split
should
with
each
complexity
each
this tetra-
can
triangle
be
interior
(Figure
7(c)).
This
meet
consistently
in a facet with
into
fact
assumption
so becomes
that
degree but
even faces dual
implies
Each
vertex
be proved
of B
layer
corresponds
directly.) [UI to
these
any two te-
eral to become
subdivisions
edges
corresponding quadrilaterals
subdivided cause combinat
IV I).
of the
B to S has one such edge,
subdivision