Unconstrained Paving & Plastering: A New Idea for All Hexahedral Mesh Generation

Abstract

Unconstrained Plastering is a new algorithm with the goal of generating a conformal all-hexahedral mesh on any solid geometry assembly. Paving[1] has proven reliable for quadrilateral meshing on arbitrary surfaces. However, the 3D corollary, Plastering [2][3][4][5], is unable to resolve the unmeshed center voids due to being over-constrained by a pre-existing boundary mesh. Unconstrained Plastering attempts to leverage the benefits of Paving and Plastering, without the over-constrained nature of Plastering. Unconstrained Plastering uses advancing fronts to inwardly project unconstrained hexahedral layers from an unmeshed boundary. Only when three layers cross, is a hex element formed. Resolving the final voids is easier since closely spaced, randomly oriented quadrilaterals do not over-constrain the problem. Implementation has begun on Unconstrained Plastering, however, proof of its reliability is still forthcoming.

Full text

Unconstrained Paving & Plastering: A New Idea
for All Hexahedral Mesh Generation
Matthew L. Staten, Steven J. Owen, Ted D. Blacker
Sandia National Laboratories
1
, Albuquerque, NM, U.S.A.,
mlstate@sandia.gov
, sjowen@sandia.gov, tdblack@sandia.gov
Summary: Unconstrained Plastering is a new algorithm with thegoalofgeneratinga con-
formal all-hexahedral mesh onany solid geometry assembly. Paving[1] has proven reliable
for quadrilateral meshing on arbitrary surfaces. However, the 3D corollary, Plastering
[2][3][4][5], is unable to resolve the unmeshed center voids due to being over-constrained
by a pre-existing boundary mesh. Unconstrained Plastering attempts to leveragethe bene-
fits of Paving and Plastering, without the over-constrained nature of Plastering. Uncon-
strained Plastering uses advancing fronts to inwardly project unconstrained hexahedral lay-
ers from an unmeshed boundary. Only when three layers cross, is a hex element formed.
Resolving the final voids is easier since closely spaced, randomlyoriented quadrilateralsdo
not over-constrain the problem. Implementation has begun on Unconstrained Plastering,
however, proof of its reliability is still forthcoming.
Keywords: mesh generation, hexahedra, plastering, sweeping, paving
1 Introduction
The search for a reliable all-hexahedral meshing algorithm continues. Manyresearchers
have abandoned the search, relying upon the widely availableand highly robust tetrahedral
meshing algorithms [6]. However, hex meshes are still preferable for many applications,
and depending on the solver, still required. This paper introducesa new method for hexa-
hedral mesh generation called Unconstrained Plastering.
1.1 Previous Research
For all-quadrilateral meshing, Paving [1] and its many permutations [7][8] have proven re-
liable. Paving starts with pre-meshed boundary edges which are classified intofronts and
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400 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
advanced inward. As fronts collide, they are seamed, smoothed, and transitioned until only
a small unmeshed void remains (usually 6-sided or smaller). Then atemplate is inserted
into this void resulting in quadrilaterals covering theentire surface.
Paving’s characteristic of maintaining high quality, boundary-aligned rows of elements is
what has made it a successful approach to quad meshing. In addition, because of itsability
to transition in element size, Paving is ableto match nearly any boundary edge mesh.
There have been many attempts to extend Paving to arbitrary 3D solid geometry. While
valuable cont
rib
ut
io
ns to
the
lit
er
atur
e,
th
es
e at
te
mpt
sh
av
e no
t result
ed
in
reli
ab
le
gener
al
algorithms for hexahedral meshing.Plastering [2][3][4][5] was one of the first attempts. In
Plastering, the bounding surfaces of the solid are quad meshed, fronts are determined and
then advanced inward. However, once opposing fronts collide, the algorithm frequently has
deficiencies. Unless the number,size, and orientation of the quadrilateral faces on oppos-
ing fronts match, Plastering is rarely ableto resolve the unmeshed voids.
Many creative attempts havebeen made to resolve this unmeshed void left behind by plas-
tering. Since arbitrary 3D voids can be robustly filled with tets, the idea of plastering in a
few layers, followed by tet-meshing the remaining void was attempted [9][10]. Transitions
between the tets and hexes weredone with Pyramids [11] and multi-point constraints. The
Geode-Template [12] provided a method ofgenerating an all-hex mesh by refining both the
hexes and tets. However, this required an additional refinement of the entire mesh resulting
in meshes much larger than required. In addition, the Geode-Template was unable to pro-
vide reasonableelement quality.
A
draw
-back
of
p
avi
ng i
s
th
e nee
d
fo
r ex
pen
si
ve i
nte
rs
ec
ti
on c
alcul
ati
on
s.
An
alt
ernati
ve
to
Paving called Q-Morph [7] eliminated the need for intersection calculations by first triangle
meshing the surface.This triangle mesh isthen “transformed” into a quad mesh. Using a
similar advancing front technique to paving, triangles are locally reconnected, repositioned,
and combined to form quads. Q-Morph is able to formhigh-quality quadrilateral elements
with similar characteristics to paving. Q-Morph has proven to be arobust and reliable quad
meshingalgorithmin common use in several commercial meshingpackages.
An attempt at extending Q-Morph to a hex-dominant meshing algorithm was done with H-
Morph [13]. This algorithm takes an existing tetrahedral mesh and applies local connec-
tivitytransformations to the elements. Groupsof tetrahedral are then combined to form
hi
gh
-q
uali
ty
he
xa
he
dra. The adv
anc
in
g fr
on
t approach
wa
s also
used
for
o
rd
eri
ng
an
d pri
-
oritizing tetrahedral transformations. Although H-Morph had the desirable characteristics
of regular layers near the boundaries, itwas unable to reliably resolve the interior regions to
form a completely all-hex mesh since it also attempted to honor a pre-meshed quadbound-
ary.
Recognizing the difficulty of defining the full connectivity of a hex mesh using traditional
geometry-based advancing front approaches, the Whisker-Weaving algorithm [14][15][16]
attempted to address the problem from a purely topological approach. It attempts to first
generate the complete dual of the mesh, from which the primal, or hex elements, are readily
obtainable. Although wisker-weaving can in most cases generate a successful dual topol-
ogy, resulting hex elements are often poorly shaped or inverted.
Plastering, H-Morph, Whisker Weaving and all of their permutations are classified as Out-
side-In-Methods. They start with a pre-definedboundary quad mesh and then attempt to
use that to define the hexconnectivity on the inside. Another class of Hex meshing algo-
rithms can be classified as Inside-Outmethods [17][18][19]. These algorithmsfill the in-
side of the solid with elements first, often using an octree-based grid. This grid is then

Unconstrained Paving & Plastering 401
adapted to fitthe boundary. These methods place high quality elements on the interior of
the volume, however, they typically generate extremely poor quality elementson the
boundary. In addition, traditional Inside-Out methods are unable to mesh assemblies with
conformalmeshes. These inside-out methods seem particularly popular with the metal
forming industry, but of less appeal in structural mechanics applications.
Sweeping based methods [20][21][22]are among the most widely used hexahedral based
me
sh
in
g al
go
rith
ms
in
in
du
stry
to
day.
Swe
epi
ng
,
ho
we
ve
r,
a
pp
lie
s
on
ly
to
so
lid
s
wh
ic
ha
re
2.5D, or solids which can be decomposed into 2.5D sub-regions. There has been a consid-
erable amount of research in sweeping and many successful implementations have been
published. It is typically quite simpleto decompose and sweep simple to mediumcomplex-
ity solids. However, as more complexity is added to the solid model, the task ofdecompos-
ing the solids into 2.5D sub-regions can be daunting, and insome regards, an art-form re-
quiring significant creativity and experience.
Advancing front methods haveproven ideal for triangle, quadrilateral and even tetrahedral
meshes.They have been successful in these arenas because of the smaller number of con-
strai
nt
si
mp
os
ed
by
the
con
ne
cti
vi
ty of
the
se si
mp
le
elem
ent
sha
pe
s.
H
exa
he
dr
al me
sh
es,
on the other hand, must maintain a connectivity of eight nodes, 12 edges, and sixfacesper
element,with strict constraints on warping and skewness. As a result, unlike tetrahedral
meshes,minor local changes to the connectivity of a hex mesh can have severe conse-
quences to the global mesh structure. For this reason, current hexahedral advancing front
methods where the boundary is prescribed apriori have rarely succeeded for general geo-
metric configurations.
Current advancing front methods, while having the high ideal of maintaining the integrity
of a prescribed boundary mesh, frequently fail because the very boundary mesh they are at-
tempting to maintain over-constrains the problem, creating a predicament which can be in-
tractable.
To resolve this issue, we introduce a new concept, known as Unconstrained Plastering.
With this approach, we relax the constraint of prescribing a boundary apriori quad mesh.
While still maintaining the desirable characteristics of advancing front meshes, Uncon-
strainedPlastering is free to define the topology of its boundary mesh asaconsequence of
the interior meshing process. Itis understood thatnot prescribingan apriori boundary quad
mesh can have implications on the traditional bottom-up approach to mesh generation.
These implications, however, are significantly outweighed bythe prospect ofautomating
the all-hex mesh generation process through a more top-down approach to the problem that
Unconstrained Plastering offers.
1.2 Unconstrained Plastering, an Unproven Concept
Unconstrained Plastering is a new approach that is unique from all the others. Although it
contains similarities to other existing algorithms, primarily Plastering, it should not be con-
sidered an extension of Plastering.
Finally, Unconstrained Plastering is not a finished work. Rather, Unconstrained Plastering
is an idea that holds promise for hex meshing researchers and should be studied further.
Im
pl
em
en
tati
on
has be
gu
n on
a pr
ot
ot
ype an
d there is
reas
on
to
be
opt
imi
st
ic th
at it has
a
greater potential for success than others. However, evidence of it being robust enough to
handle general purpose mesh generation for industrial applications is forthcoming.

402 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
2
Un
co
ns
tr
ain
ed
Pa
ving
To best understand the general concept behind Unconstrained Plastering, we first examine
the 2D corollary, Unconstrained Paving.
2.1 Advancing Unconstrained Rows
Fig. 1 shows a geometricsurface ready for quad meshing. If we were to pavethis surface,
we would first mesh each ofthe surfaces boundary curves, after which we would advance a
row of quads along one of the boundary curves as can be seen in Fig.2. In this case four
quads were added becausethe curve along which the row was paved was pre-meshed with
four mesh edges.
If, instead, the surface was being meshed with Unconstrained Paving, the boundary curves
would not be pre-meshed withedges.Advancing,or paving, an unconstrained row would
resultinFig. 3. Inthis case a row of quads have been inserted, however, we do not know
how many quads will be in that row. The number of quads in this row is determined as ad-
jacent rows cross it. Fig. 4 shows what the mesh looks like after a second row is advanced.
Since the second row advanced crossed the first row advanced, a single quadis formed
(shaded) in the corner where the two rows cross. However, both of the rows still have an
undetermined number of quads in them.
Fig. 1 Example Surface
Fig. 2 Example Surface with meshed
boundary and one row paved
Fig. 3 One row advanced with Un-
constrained Paving
Fig. 4 Two rows advanced withUn-
constrained Paving

Unconstrained Paving & Plastering 403
In Fig. 5, several additional rows have been advanced and 12 quadrilateral elements have
been formed where the various unconstrained rows have crossed. At this point, the un-
meshed portion of the surface has been subdivided into two sub-regions (sub-region A &
B). It is important to note that both of these unmeshed regions are completely uncon-
strained. For example, sub-region Ais bound by five edges,however, none of these edges
has been meshed. Sub-regionA is free to be meshed with as many divisions as needed
al
on
g all of th
ese
edges
.
Fig. 5 Additional rows advanced with
Un
co
ns
tr
aine
d
Pa
vin
g
Fig. 6 Transition row inserted based
on large angle between two adjacent
rows
2.2 Transitioning Unconstrained Rows
Like traditional Paving, Unconstrained Paving has the ability to insert irregular nodes
(nodes with more or less than four adjacent quads) in order to transition and fit the shape of
the surface. In traditional Paving, this is done by assigningstates to the fronts based on an-
gles
wit
h
ad
jacen
t fro
nt
s.
Un
co
nst
ra
in
ed
Pl
ast
ering
is
no
diff
erent
.
Th
e start
an
d
end
o
f
an
advanced unconstrained row likewise depends upon states and angles. Fig.6 shows the ad-
vancement of an additional row, which, because of anglesisthe advancement of two previ-
ously advanced rows.
Fig.
7 F
ro
nt
A
cann
ot
bea
dva
nc
ed
normally because Edge B is too short
Fig. 8 Tran
si
ti
on
ro
w in
se
rted
b
as
ed
on front sizes
Fig. 7 shows an additional case where rows must be advanced with care. Front A is the
next front to advance, however, Edge B is too short even though angles indicate that the

404 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
advancement of Front A should extend to Edge B . In this case, Front A can be advanced as
shows in Fig. 8.
Unconstrained rows continue to advance as previously described. Rows bend through the
mesh as required to maintain proper quadrilateral connectivity ensuring that all quadrilat-
eral elements created are of proper size. In addition, Paver-like row smoothingand seam-
ing, along withthe insertion of tucks and wedges [1] are additional operations that can be
performed on the unconstrained rows. Fig. 9 illustrates the example surface and how it may
lo
ok
after sev
eral
mor
e
ro
ws
are adv
anc
ed.
All edg
es
on
th
e un
meshed
sub
-regi
ons
A and
B are less than two times the desired element size, and so we stopadvancing fronts. At this
point, it is time to resolve the unmeshed voids.
Fig. 9 Unconstrained rows advanced
leaving only small unmeshed voids
an
dc
on
ne
ct
ing
tub
es
;
Qu
ad
are
sha
de
d, connec
ti
ng
tubes are
white.
Fig. 10
Unm
esh
ed
void
s ha
ve
bee
n
me
sh
ed
2.3 R
esolving Unmes
hed Voids
In
general
,t
he
unmeshe
d sub-regi
ons wi
ll be any general
pol
yg
on,
wi
th
any
number of
sides. It is assumed that each polygon will be convex. If it is not convex, that would sug-
gest that an additionalrow needs to be advanced before resolving the unmeshed void. It is
also assumed that the size of the polygon is roughly one-to-twotimes the desired element
size. Ifit is larger than this, then additional unconstrained rows should be advanced until
the remaining polygon is one-to-two times the desired element size.
Also, note that the unmeshed region is completely unconstrained. Each of the edges on the
unmeshed polygons areconnected to theboundary of the mesh through “connecting tubes”.
Con
ne
ct
ing
tub
es
a
re
th
ew
hi
te
re
gion
s in
F
ig
.
9 tha
t
ha
ve
be
en
cr
osse
d by
onl
y a
sing
le
row. The
edges
of
eac
h polygon ca
n be meshe
d with any number of
edges, whi
ch
w
ill be
propagated back to the boundary through the connecting tubes.
At this point, the polygon is meshed witha template quad mesh similar to the templates
used to fill the voids during Paving [1]. The template inserted is based on the relative
lengths of edges, and anglesbetween edges. In the general case, any convex polygon can
be meshed with midpoint subdivision[23]. Midpoint subdivisionmeshes convex polygons
by adding a node at the centroid ofthe polygon and connecting it to nodes added at the cen-
ter of each polygon boundary edge. The number ofnew quads formed is equal to the num-

Unconstrained Paving & Plastering 405
ber of points defining the polygon. Although midpoint subdivision can always be used to
meshthe void, simpler templatesare often possible.
In Fig. 9, since sub-region B is already four-sided and is ofproper size and shape, it can be
converted into a single quadrilateral element. However, sub-region Ais meshed withmid-
point subdivision since it has five sides. The resulting mesh is illustrated inFig. 10.
Bef
or
e Unco
ns
trai
ned Pa
vi
ng
isf
inish
ed, th
e connect
in
g tu
bes
mu
st
be exam
in
ed
fo
r si
ze
.
In Fig. 10 Connecting Tube A is much too wide. This can befixed by advancing afew
more rows until the proper size is obtained as shown in Fig. 11. Traditional quadrilateral
cleanup operations and smoothing can then be performed to finalize the mesh connectivity
and quality as shown in Fig. 12.
Fig. 11 Sizes in connectingtubes
have been resolved
Fig. 12 Final quad mesh after clean-
ing and smoothing
2.4 Unconstrained Paving with Multiple Surfaces
In real world models, rarelyis the geometry confined to a single surface. For example,
sheet metal parts inthe autoindustry representing automobile hoods often contain thou-
sands of surfaces. Each of these surfaces mustshare nodes andelement edges with its
neighboring surfaces across its boundary edges in order to ensurea conformal mesh.
Typically, algorithms thatdo notpre-mesh the curves of surface before meshinghave diffi-
culty ensuringa conformal mesh [17][18][19]. However, Unconstrained Paving can be ex-
tended to ensure conformal meshes between any number of surfaces. The penalty, how-
ever, is that all of the surfaces must be meshed at the same time. For example, Fig. 13
illustrates four adjacent surfaces which require a conformal mesh. Fig.14 shows the same
model with one unconstrained row advanced. The row was advanced in three ofthe sur-
faces. Fig. 15 shows several additional rowsinserted and the formation ofa tuck in surface
2. Notice that curves which are shared by more than one surface are double-sided fronts
advancing into both adjacentsurfaces.
After additionalrowsare advanced, Fig. 16 shows the small unmeshed voids and the con-
necting tubes. It is important to note that when meshing multiple surfaces at once, the con-
necting tubes impose additional constraints on how the unmeshed voids canbemeshed.
How
ev
er, th
ese con
st
rai
nts can al
ways
be satisfi
ed with
mi
dp
oi
nt su
bdiv
is
io
ns
in
ce th
is
would split each edge in everyconnecting tube exactly once. Fig. 17 shows what the mesh
may look like before a final pass through cleanupand smoothing.

406 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
In the current research, Unconstrained Paving is usedonly as a thought experiment to help
illustrate the concepts of Unconstrained Plastering. While implementation of Uncon-
strained Paving may be beneficial current unstructured quadrilateral meshing techniques
satisfy FEA needs.For this reasonthe current research focuses implementation and proto-
typing efforts on the 3D Unconstrained Plasteringproblem.
Fig. 13 Multipleadjacent surfaces re-
qu
iri
ng
a
c
on
fo
rm
al
me
sh
Fi
g. 14 An
unco
nst
ra
ined
row
has
been advanced extending through
mu
lt
ip
le
s
ur
fa
ces
Fi
g.
15
Ad
di
tio
na
l
ro
ws
are
a
dva
nc
ed
including a tuck
Fig. 16 Onlysmall voids and con-
nec
ti
ng tub
es
rem
ain
Fig. 17
Unm
es
he
dv
oi
ds
and
c
onnec
ti
ng
t
ub
es ar
e me
shed
3 Un
co
ns
tr
ai
ne
dP
las
teri
ng
The basic principles of Unconstrained Paving extend to 3D as Unconstrained Plastering.
The basic algorithm is as follows and is described in the following sections.

Unconstrained Paving & Plastering 407
1. Start with a solid assembly with unmeshed volume boundaries.
2. Define fronts, which initially are the surfaces ofthe volumes.
3. While the unmeshed voidsofthe solids are larger than twice the desired element
size:
a. Select afront to advance,
b.
Based
on s
izes
of
fr
on
ts
, an
d an
gl
es wit
h
ad
jac
ent
fr
on
ts
, de
term
in
e
which adjacent fronts should be advanced withthe current front.
c. Advance the fronts
d. Form unconstrained columns ofhexahedra where 2 layers cross.
e. Form actual hexahedral elements where 3 layers cross.
f. Perform layer smoothing and seaming.
g. Insert tucks and wedges as needed.
4. Identify unmeshed voids, connectingtubes, andconnecting webs.
5. Define constraints between unmeshed voids through connecting tubes.
6. Mesh the interior voids with either midpoint subdivisionor T-Hex.
7. Sweep the connecting tubes between voids and out to the boundary.
8. Split connecting webs as needed.
9. Smooth all nodes to improve element quality.
3.1
Advancing Unconstr
ained La
ye
rs
The model in Fig. 18 will be used an example of Unconstrained Plastering. Fig. 19 shows a
single unconstrained hexahedral layer advanced. The new surface displayed in Fig. 19
represents the topofthe layer ofhexes which will be adjacent to the advanced surface. The
region between the boundary and the new surface represents an unconstrained layer of
hexahedra. It is still unknown how many hexes will be in this layer, however, we do know
that it will contain a single layer of hex elements.
In Fig. 20, a second layer is advanced, which crosses the first layer advanced previously.
When two layers cross, a column of hexahedra is formed, however, the size and number of
hexahedr
a in this colu
mn wil
ln
ot
be det
er
mined until additional hex layers cros
s th
is
col
-
umn.
Fig. 18 Unconstrained Plastering ex-
ample model
Fig. 19 One unconstrained hex layer
has been advanced

408 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
Fig. 20 Whentwo layers cross, an
unconstrained column of hexes is de-
fined
Fig. 21 When three layers cross a
hexahedral element is defined
In Fig. 21, a third layer is advanced, which crosses both of the previously defined layers.
Where ever three layers cross, a hexahedral element is formed. In Fig. 21, a single hex ele-
me
nt
isd
efi
ne
d in
th
e lower left
co
rner. Not
e th
at un
til
now
, no
fi
nal
de
cisio
ns have be
en
imposed on placement of hexahedra. Itis only when three orthogonallayers intersect that
hex placement becomes finalized.
The process continues in Fig. 22, Fig. 23, and Fig. 24.Each time two layers cross a column
of hexahedra is defined.Each time a third layer cross a column, a single hexahedral ele-
ment is defined. During this process, there will be transition layers inserted with logic simi-
lar to that described in section 2.2. Layers are advanced until the unmeshed void is ap-
proximately twice the desired element size.
Fig. 22 Additionallayers are ad-
vanced
Fig. 23 Additional layers are ad-
vanced
Fig. 24 Additional layers are advanceduntil theunmeshed void is small

Unconstrained Paving & Plastering 409
3.2 Front Processing Order
A front to advance is a group of one or more adjacentsurfaces which are advanced to-
gether. The order that fronts are processed in Unconstrained Plastering is very important.
This is an area where additional research is required. However, factors to consider when
choosing the next front to advance include:
1. Number of layers away from the boundary the front is. Fronts closer to the
boundary shouldbe processed first.
2. If the front is “complete” or not. A complete front isa group of surfaces which
are completely surrounded by what are referred to as “ends” in Paving and Sub-
mapping[24][25]. For example, in Fig. 25, Surface 1is complete since its bound-
ary is adjacent to a cylindrical face which is perpendicular to Surface 1. In con-
trast, Surface 2 is incompletesince it is bounded on one of its loops by an “end”,
but is bound on its other loop with a “corner” [24]. The bestway toproceed
would be advance Surface 1 several times until it becomes even with Surface 2, at
which the front from Surface 1and Surface 2 would be combined and advanced
as a single front.
3. The size of the front. Smaller fronts should probably be processed first.
4. How much distance there is ahead of the front before a collision will occur.
Fronts with a lot of room to advance should probably be processed first.
Fig. 25 Surface 1 is a “complete” front, while Surface 2 is “incomplete”
3.2 Resolving Unmeshed Voids
Like
Un
constrained Pavi
ng,
th
ere will be unm
es
he
d voi
ds
at
th
e
cent
er
of
each volum
e
be-
ing meshed. The unmeshed voids can be easily identified because they are the regions in
space that have not been crossed by any hex layers. Fig. 26 illustrates the unmeshed void at
the center of the example model.In general, these voids define general polyhedra. It is as-
sumed thatthese polyhedra are convex.If they are not convex, that suggests that an addi-
tionallayer should be advanced before resolving the voids. Although we have no theoreti-
cal basis to prove that advancing additional rows will always ensure a convex polyhedra,
experience has shown that it does. Likewise, it is assumed that all edges and faces of the
polyhedra are approximately twice the desired element size, or less. If they are larger than
this, it suggests that an additional layer should be advanced.
In addition to identifying the unmeshed voids, we must also identify the connecting tubes
and connecting webs. Connecting tubes are those regions in space which have been crossed
by only a single hex layer as illustrated in Fig. 27. In order to define a hex, three layers
must cross, which gives the connecting tubes two degreesof freedom, allowing them to be

410 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
swept as a one-to-one sweep[22]. The direction of the sweep is perpendicular to the single
layers which already cross the connecting tube. The number of layers in the sweep is the
same as the number of hex layers the connecting tube crosses between the unmeshed poly-
hedra and the boundary surface.
Connecting webs are those regions in space which have been crossed byonly two hex lay-
ers as illustrated in Fig. 28. Connecting webs only have a single degree of freedom. Essen-
tially, connecting webs represents a layer of hexahedra which will be split thesame number
of
ti
me
s
th
at
the
adjacen
t co
nnect
in
g
tu
be
s are sp
lit.
In
th
is
exa
mpl
e
mod
el,
th
ere is
one
small connecting web section which is not attached to any connecting tubes or unmeshed
void. This willhappen in locations where seaming has been performed, since seaming will
often eliminate or split the unmeshed void. In the example model, the front that was ad-
vanced from the front-right surface was seamed with the fronts extending from the hole.
Fig. 26 The unmeshed void Fig. 27 The connecting tubes
Fi
g. 28 The co
nnec
ti
ng
web
s
After the unmeshed voids, connecting tubes, and connecting webshave been identified, the
unmeshed void is meshed using either midpoint subdivision[23], or T-Hex. Midpoint sub-
division is the preferable method since it generates higher quality elements. To determine
if
midpoint su
bdivi
si
on
is
possibl
e, a si
mpl
e count
of the num
be
r of
curves connec
te
d
to
each vertex on the unmeshed polyhedra is done. If there are any verticeswhich have four
or more connected curves, then midpoint subdivision is not possible. The unmeshed void
in Fig.26can be meshed with midpointsubdivision as illustrated in Fig.29.
Fig. 29 Midpointsubdivision of the
unmeshed void
Fig. 30 The connecting tubes are
swept

Unconstrained Paving & Plastering 411
Fig. 31 The connecting webs are split
Fig. 32 Final mesh using midpoint
subdivision of voids; connecting tubes
are shaded; connecting webs are
cross-hatched
After the unmeshed void is meshed, the connecting tubes are swept as shown in Fig. 30 us-
ing the mesh from the unmeshed void as the source. Finally, the connecting webs can be
split asillustrated inFig. 31. The final mesh on the example model, after some global
smoothing, is shown in Fig. 32. The connecting tubes exposed to the boundary ofthe mesh
are shaded dark and the exposed connecting webs are cross-hatched.
If the polyhedra cannot be meshed with midpoint subdivision,itis meshed with the T-Hex
tem
pl
ate in
st
ead. Tod
o
this
,
we fi
rs
t
tak
e each
non
-tri
angu
lar poly
go
n
on
eac
hu
nm
es
hed
polyhedra and split it into triangles. If the polygon being split is connected to other un-
meshed polyhedra through connecting tubes, we must be careful that the face is split the
same on both polyhedra so the sweeper can match them up through the connectingtubes.
To ensure that they are split the same, a node can be added at the center of the face and tri-
angles are formed using each edge on the polygon and the newly created center node. After
each face is split into triangles, the polyhedra are meshed with tets. Since we are assuming
that the unmeshed void is 1-2 times the desired element size, we would like to mesh these
polyhedra without introducing any nodes interior tothe polyhedra. Not putting any new
nod
es in
th
e po
lyhed
ra
wil
l
also hel
p
wi
th
e
le
ment qu
al
it
y since T-
Hex
m
esh
es
are wo
rst
when T-Hexing around a node surrounded completely by tet elements. After the polyhedra
are tet meshed, each tet issplit into four hexahedral elementsusing the T-Hex template
shown in Fig. 33. The T-Hex mesh for the polyhedra in the example problem is shown in
Fig. 34 and the mesh on the connecting tubes inFig. 35. Since, in this example,wewere
meshing only a single solid, there were no constraints between multiple unmeshed voids.
Therefore, the quadrilateral face on the top of the polyhedra was split into only two trian-
gles before tetrahedralization. Fig. 36 shows the final mesh after some global smoothing.
The connecting tubes exposed to the boundary of the mesh are shaded dark and the exposed
connecting webs are cross-hatched.
Fig.
33 T-He
x
tem
plate
Fig. 34 T-Hex on unmeshed void

412 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
Fig. 35 The connecting tubes are
swept with T-Hex mesh
Fig. 36 Final mesh using T-Hex on
voids; connecting tubes are shaded;
connecting webs are cross-hatched
T-Hex has long been known as a guaranteed way to get an all hexahedral mesh on nearly
any solid geometry. However, the quality ofthe elements that result is rarely sufficient for
most solver codes. Critics of Unconstrained Plastering will point to the use of T-Hex on in-
terior voids as a major downfall of Unconstrained Plastering. However, before that judg-
ment can be made, the following should be considered:
1. T-Hex is only used when interior voids have a vertex with a valence of four or
more. Inmost cases, the interior voids can be meshed with midpointsubdivision.
2. The worst quality hexahedra in T-Hex meshes are found adjacenttonodes which
were completely surrounded by tets in the initial tet mesh. This is because a tet
mesh can havenodes with a valence of 15 or more, which results in the same
nu
mb
er of
hex
ahed
ra
w
hen
th
e
T-H
ex
tem
plate
is
app
li
ed
.T
hi
s
ca
se
s
ho
ul
d
no
t
appear duringUnconstrained Plastering, since we assume that enough uncon-
strained layers have been advanced to make the interior voidssmall enough to be
tet meshed with nointeriornodes.
3. The T-Hex looking elements that are swept to the boundary through connecting
tubes are not poor in quality since a swept T-Quad mesh is much higher quality
than a traditional T-Hex mesh.
4. Any poor quality hexahedra that are formed by Unconstrained Plastering will be
in
th
e
in
te
rio
rv
oi
ds
wh
ic
h
sh
ou
ld
be
on
th
e
de
ep
in
te
rio
r
of
the
v
ol
ume
s,
wi
th
the
exception of thinparts which require only one or two layers of hexahedra through
the thickness.
3.4 Unconstrained AssemblyMeshing
Even though Unconstrained Plastering does not seem capable of honoring existing bound-
ary quad meshes, it can be used to mesh assemblies of solids and still get a conformal mesh.
Li
ke
Unc
on
st
ra
ined
P
av
in
g,
h
ow
ev
er
, al
l of
t
he
vol
um
es in
th
e ass
em
bl
ym
us
t
be me
shed
at
once. F
ig
. 37
illustrat
es
a
s
imp
le
assembl
ym
ode
l to be meshed. F
ig
.
38
and Fig
.3
9 show
the same model after 3 unconstrained layers have been advanced. Fig. 39 shows that the
unconstrained layers have been extended and advanced through both of the solids in the as-
sembly. Like Unconstrained Paving, surfaces which are shared by both volumes will be-
have as double-sided fronts advancing into bothvolumes.

Unconstrained Paving & Plastering 413
Fig. 37 Assembly model
Fig. 38 Assembly model withthree
advanced unconstrained plastering
layers
Fig. 39 Detailof Fig. 38
3.5 Element Quality with Unconstrained Plastering
Implementation of Unconstrained Plasteringhas not progressed far enough to make any
claims onelement quality. However, like other advancing front algorithms, Unconstrained
Plastering willhave the tendency to put the highest quality element near the boundary.
Subsequent publications on Unconstrained Plastering will report element quality findings as
the research matures.
One limitation that Unconstrained Plastering will have compared to Unconstrained Paving
is the lack of hexahedral cleanup operations. Unlike quadrilateral cleanup, hexahedral
cleanup operations are limited due to the highly constrained nature of hexahedra [26][27].
As a result, Unconstrained Plastering will be required to create hexahedral topology that
willpermit good element quality rather than relying on a post-processing cleanup step to fix
poor elements.
4 Implementation
Deta
il
s
As stated earlier, implementation has begun on a full3D implementation of Unconstrained
Plastering. The authors have chosen to use a faceted surface based approach. The basic al-
gorithm that is being followed is:

414 Matthew L. Staten, Steven J. Owen, Ted D. Blacker
1. Triangle mesh all of the boundary surfaces using an element size approximately
equal to the desired hexahedral element size.
2. Traverse through this triangle mesh to eliminate any unnecessary CAD artifacts
(i.e. small angles, slivers, etc.
3. Divide the triangles up into groups which form Surfaces or Fronts. They are
grouped together considering the original CAD topology, but also dihedral angles
between the original CAD surfaces.
4. For each volume in the assembly being meshed, form a “cell”. Each cell has
three associated layer ids. These initial cells get (UNDEFINED,UNDEFINED,
UNDEFINED) as their initial layer ids.
5. While any cell is larger than twice the desired element size:
a. Choose a set of Surfaces to advance to form a new layer.
b. Advance the triangle mesh on these surfaces into the volume. A fac-
et
ed su
rf
ac
e is
cr
eat
ed of
fs
et
by
the
d
esi
rede
le
me
nt
si
ze to
th
e ad
van
c-
ing surfaces.
c. Forma new cell betweeneach newly created surface its corresponding
front surface.This new Cell inherits the layer idsfrom the cell being
advanced into. It is also assigned a new layer id which represents the
layer just created.
d. Smooth and seam the newly advanced faceted surface with its
neighboring surfaces.
6. Create constraints between the unmeshed voids through the connecting tubes.
7. Mesh each unmeshed void with either midpoint subdivision orT-Hex.
8. Sweep the connecting tubes.
9. Split the connecting webs as needed.
10.Send the entire mesh to a smoother for global smoothing. Smoothing is needed
on curves and surfaces, in addition to the nodes on the interior ofthe volumes.
5 Conclusions
Th
e co
ncept
of ad
van
cin
g un
co
ns
tr
ained
rows of
qua
ds
an
d la
yers
o
f
he
xahed
ra
has
been
introduced through the algorithms of Unconstrained Paving and Unconstrained Plastering.
The concept ismost relevant with Unconstrained Plastering since it eliminates the problems
of resolving highly constrained unmeshed voids which iscommon with most other advanc-
ingfront hexahedral meshing algorithms.
The algorithmspresented are able to mesh assembly models with conformal meshes with
the penalty thatall of the volumes/surfaces in the model must be meshed at the sametime.
Meshing all of the volumes in an assembly at once increases memory requirements since
the mesh onthe entire assembly will need to bein the mesher’s internal datastructures at
once, which are typically larger than mesh storage datastructures.
ImplementationofUnconstrained Plastering has begun and the authors are optimistic that it
will be more successful than other free hex meshing algorithms. However, additional re-
search is required before any claims will be made.

Unconstrained Paving & Plastering 415
Unconstrained Paving is also presented which is a potential improvement upon traditional
advancing front quadrilateral meshing algorithms. However, since the quadrilateral mesh-
ing problem already has several solutions, the priority of researching and implementing
Unconstrained Paving is lower than that ofUnconstrained Plastering.
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fe
re
nc
es
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