Shell-closure is critical to the repair of CAD-models described in the .STL file-format,
the de facto solid freeform fabrication industry-standard. Polyhedral CAD-models that
do not exhibit shell-closure, i.e. have cracks, holes, or gaps, do not constitute valid solids
and frequently cause problems during fabrication. This paper describes a solution for
achieving shell-closure of polyhedral CAD-models. The solution accommodates nonmanifold
conditions, and guarantees orientable shells. There are several topologically
ambiguous situations that might arise during the shell-closure process, and the solution
applies intuitively pleasing heuristics in these cases.Mechanical Engineerin

Bohn, John Helge

Wozny, Michael J.

English

Texas ScholarWorks

Automatic CAD-model Repair: Shell-ClosureJan Helge B~hn and Michael J. WoznyRensselaer Design Research CenterRensselaer. Polytechnic InstituteTroy, New York 12180-3590(518) 276-6751bahn@rdrc.rpLeduAbstractShell-closure is critical to the repair of CAD-models described in the .STL file-format,the de facto solid freeform fabrication industry-standard. Polyhedral CAD-models thatdo not exhibit shell-closure, i.e. have cracks, holes, or gaps, do not constitute valid solidsand frequently cause problems during fabrication. This paper describes a solution forachieving shell-closure of polyhedral CAD-models. The solution accommodates non-manifold conditions, and guarantees orientable shells. There are several topologicallyambiguous situations that might arise during the shell-closure process, and the solutionapplies intuitively pleasing heuristics in these cases.1 IntroductionIt is critical that the CAD-models used in solid freeform fabrication (SFF) are robustto avoid failure during manufacturing. Unfortunately, CAD-models described in the .STLfile-format [1], the de facto industry standard, frequently lack this robustness [2, 3, 4].Typical problems include punctured shells, inconsistent facet-orientations, and internalwalls and structures.With the .STL file-format, the CAD-model solid is defined by its surface, Le. thesolid is defined by a closed shell enclosing its material. If this shell is punctured (i.e.open), then it can no longer properly contain the material, and this consequently causesthe failures commonly associated with cracks, gaps, and holes (Figure 1.1). Similarproblems arise when the facet-orientations across a shell are inconsistent. This too causesuncertainty regarding what side of the shell contains material. Finally, structural problemsmight arise from internal walls and structures within the solid. These internal featurescan create irregularities in the solidified material during the processing. For instance,with stereolithography [5] and selective laser sintering (SLS) [6], excess amounts ofunsolidified material might get trapped inside the solidified matrix, and consequentlyresult in uncontrolled shrinkage and surface warping (Figure 1.2).The .STL file-format is capable of representing a robust, well-formed solid. Unfor-tunately, contemporary CAD-systems frequently fail to achieve this goal. There are twoCopyright © 1992 Rensselaer Design Research Center. All rights reserved.86Slice Plane..Stray Scan-VectorsFigure 1.1 Shell-punctures causes the laser to producestray scan-vectors.Figure 1.2 Internal walls cause structuraldiscontinuities that can lead to excessiveshrinkage.design environments currently available for CAD; solid- and surface-modelers. Solid-modelers generally avoid punctured shells, but frequently generate needless internal wallsand structures, and inconsistent orientations. In addition to these problems, surface-modelers frequently fail to properly mate adjacent surfaces; this results in puncturedshells.These problems are commonly resolved by the SFF-equipment operators prior tofabrication. The operator might attempt to edit the .STL file directly, or, if available,edit the original CAD-model to eliminate the problems. However, manual model repairbecomes excessively expensive when the number of facets reaches 10,000 to 100,000+,and the number of errors run into the hundreds. This has lead to the development ofautomatic CAD-model repair.Brock Rooney & Associates [7] perform limited CAD-model repair in their .STLtranslators, ensuring that their .STL output does not contain punctured shells. Their pro-prietary shell-closure solution is not complete. It does not handle orientability-problemssatisfactorily, and it has limited success with certain simple structures. However, theyreport few problems in general with their .STL output when run on the software-robuststereolithography process of 3D Systems, Inc. In the rare cases where their shell-closurealgorithms fails, the CAD-model must either be regenerated at the design-level, or berepaired manually as described above.3D Systems, Inc. [8] provides its customers with a limited, proprietary model-repairfacility. It too is intended to ensure that the model does not fail during fabrication, orif the repair fails, to provide indications to where it must be repaired manually. Theprogram attempts to repair simple orientation problems, minor punctures, and facet-intersections, but leaves internal walls, and fails for large punctures and non-orientableshells. It performs the repair under the restriction of several user-specified parameters,and if these are set too conservatively, an intermediate result is generated. This outputcan be processed iteratively until a valid, though not necessarily correct result occurs.The basic problem of CAD-model repair is that of inferring information that has beenlost somewhere in the design process. While it might be easy in some cases to detect thepresence of a problem, it is frequently impossible to determine with certainty what thecorrect solution is. This is especially true if the designer's intent is lost. Without it, aneducated guess must be made whenever an ambiguity is encountered. The goal of ourwork is to match the quality of manual repair that is performed without the knowledgeof the designer's intent.87\Figure 2.1 A non-manifold condition can be viewed locally as a set of two-manifold conditions.2 Shell-ClosureThis paper presents a complete solution for shell-closure of polyhedral shells. Indoing so, it also solves the orientability problems, and it simplifies the complexity of themodel-topology for more efficient processing during fabrication. In particular, it ensures apseudo two-manifold topological organization of the facets to facilitate high-speed slicing[4, 9]. This topological organization also simplifies the subsequent resolution of shell-intersection and shell-nesting problems, which are necessary for the removal of needlessinternal walls.A (two-) manifold structure is homeomorphic to a two-dimensional disk at everypoint across its surface, i.e., each such point has a neighborhood that is topologicallyequivalent to a two-dimensional disk. A non-manifold structure violates this constraint;two cubes that touch along a common edge is one such example. However, if the facetsat a non-manifold location are paired up properly, then this situation can be described asa set of (pseudo) two-manifold conditions [10] (Figure 2.1).Our solution for shell-closure follows a three-stage strategy: First, we identify thelamina edges. Second, we combined these edges into closed, non-self-intersecting loopsof boundary-curves; commonly known as directed Jordan curves. And finally, we createlids that fit these Jordan curves and that close the open shells.3 Lamina EdgesA lamina edge is generally considered a special case of a non-manifold edge; ithas only a single adjacent face [11]. Lamina edges are of interest because they makeup the boundaries of the shell-punctures, and of the corresponding lids which we willcreate to close the shells. However, if the definition of a lamina edge is generalized,in particular generalizing the definition of a puncture, then we can expand the use ofcreating lids for shell-closure to also solve erroneous facet-orientations and complicatednon-manifold conditions.Expanded definition: A lamina edge exists along each face-edge that does not mateand pair properly this particular face with any other face along this edge. These laminaedges can be found as follows: Let all face-edges be candidates for lamina edges. Then,for each face-edge, attempt to find an adjacent face of compatible orientation such that88Surface Normal} ={... }3.1 Two laminaof two faces aboutcandidates (a,b) are removed from the set of candidates (LEe) after a successful pairing3.3 Two unpaired facets along a non-manifoldyield two coincident lamina edges.adjacent about the common edge.are removed from the set of laminaafter this process is exhausted,4.... WOIlA.'Vl"LLLj;. ULL.v'VIL,VU. '"''VA.''''' curves from the set of lamina edges,.UAl..'V "'v"'"',,...u.F-, boundary curve-segments, orto a single shell, and is terminatedFigure 4.1 Simple, self-connectingloop (unambiguous).Figure 4.3 Cyclic with single direct returns.Figure 4.2 Linear combination (unambiguous).Figure 4.4 Cyclic with multiple direct returns.Figure 4.5 Trivial and nontrivial cases of cyclic with non-direct returns.by two endpoints that might coincide. This can be achieved as follows: Initially, considereach lamina edge as an arc. Then, iteratively, for each vertex with only two adjacentarcs, merge those two arcs into a single arc, causing this vertex to no longer have anadjacent arc. The final set of arcs form a set of directed graphs (as defined in graphtheory [12]) of various complexities that describe the shell-puncture boundaries, andthese puncture-boundaries must be separated prior to lid-creation.We have identified five classes of puncture-boundaries of increasing identification-complexity, which when combined, form the above mentioned sets of directed graphs.Among these classes, three occur in our industrial .STL test-suite, while the remainingtwo, the most complex classes, persist as unobserved possibilities: The most trivial classof all is the simple combination; a single self-connecting arc. It is unambiguous andtrivial to detect, and it forms a complete puncture-boundary by itself (Figure 4.1).The linear combination is also unambiguous; two arcs that are adjacent to the sameshell and that form a closed loop. These arcs form a link between two separate subgraphs,each of which is less complex that their combination (Figure 4.2).The remaining three classes are ambiguous because they offer multiple closed-loopalternatives, and because the Jordan curves are located in three-dimensional space ratherthan in a simplifying two-dimensional plane. Fortunately, one can make some reasonable,90alternatives: For instance,separate shells. This is basedsolids; consequently, all facetsshell. This dictates that, fornot be adjacent to same shell....V ...'VA-A-LJV'Y by the formation of cyclici"""Tl'~i"n1nn arc for each of itshas been eliminated asrare, but present, our...AA'L,' .... "'.AA we not yet en-returns is an ambiguous problemarc, multiple options remain. ItU ........ILLLd'AAE:, not only topological-,'l"nQTf't"I1nn problem where the.......u.J--,v.u ...u of connectingcase, and it has a gracefulmore ambiguous. Fur-with minimum weight problemarc-pairs, we end up withJl"'-V..........JlAA..., back to the example inarcs are not shown), yieldingarc), the above shell-mergerdownward arc) as an ac-V"'''''.l.AAJUA..., arcs as possible matcheswill cause the merger of twowhere further loop-extractiontwo possible combinations.these are combinations..... .I.'V''-'''''' ..... loop (Figure 4.5). Some.I.......,.l.V,....."" .. A ........I..I. 7 ........'.I....,JL......., arcs into longer parts, butare more and ambiguous. One possiblearcs onto a suitable two-manifold structure,aV./l........ ""'.., ...... ""'.... from the more complexthese problems appear toour strategy is to identify the lessoriginal graph. This simplifies......... ' ........... 'VAA ....... unambiguous constructions. Assub-problems.o 1 2rr([r 3 4r5 6(f'r 7 8 9r1'Figure 5.1 Lid-creation by iterative removal of the vertex with the smallest angle.•Figure 5.2 Iterative post-processing to improve the quality of the lid-surface.5 Creating LidsThe task of creating lids is greatly simplified once the corresponding directed Jordancurves have been identified: The puncture is closed by an iterative algorithm thatdecrements the Jordan curve by eliminating one vertex at a time; the vertex is eliminatedfrom further consideration by adding a facet that spans it and its two neighboring verticesalong the shrinking Jordan curve. At each iteration, we remove the vertex along theJordan curve that spans the smallest angle with its two neighboring vertices along theremaining, decremented Jordan curve (Figure 5.1). This avoids the clearly undesirablesplitting of angles, and it will in general produce excellent results, especially for longnarrow punctures which abound in industrial .STL files [7, 8].However, this heuristic algorithm is not without drawbacks. It is a time-localalgorithm in that it finds the best selection at any given time, without keeping theoverall optimal solution in mind. An example of this is the selection of the thirdfacet in Figure 5.1; it forces the subsequent undesirable creation of facet number four.Fortunately, such problems are easily rectified with a simple, inexpensive iterative post-processing step as illustrated in Figure 5.2: For each adjacent pair of lid-facets, flip theircommon diagonal edge if this increases their minimum facet-angle.A more serious problem is the difficulty of finding the angles spanned about thevertices. While angles between vectors in 3D-space are in the range 0-180 degrees, itis clear, from the 2D-plane, that the angles between facet-edges are in the range 0-360degrees. It turns out, however, that a conservative decision to assign the range 180-360degrees to a vertex whenever there is a doubt to which half of the total range it belongsto, simply postpones its elimination. Eventually, one of the vertex's neighboring vertices92Orientation DiscontinuityFigure 5.3 Cutting a non-orientable Mobius-strip along its orientation-discontinuity, results in a orientable panel.will be eliminated, and its angles must be recomputed. Usually, this clarifies the situation,and the vertex might be assigned the range 0-180 degrees instead.It is interesting to note that the above lid-creation algorithm is capable of creating anon-orientable shell if the open shell is non-orientable. An example of this would be theclosure of a Mobius-strip. However, with our definition of lamina edges, non-orientableshells are cut such that they become orientable prior to lid-creation. In particular, theMobius-strip becomes a simple panel (Figure 5.3). As a consequence, since the lid-creation algorithm maintains the directionality of the diminishing Jordan curves, we areguaranteed to maintain orientability throughout the shell-closure; and more importantly,the final closed shells are therefore always orientable.6 ConclusionsThis paper has presented a complete solution for shell-closure of polyhedral shells,the first step towards well-formed and manufacturable solids. The solution is based ontopological principals, and consequently resolves orientability and non-manifold problemsin addition to eliminating shell-punctures (i.e. cracks, holes, and gaps). As a result, thefacet-model output consists entirely of closed, orientable shells, and the resulting facet-models can therefore be viewed as a set of two-manifold shells.This topological organization simplifies the subsequent resolution of shell-intersectionand shell-nesting problems, and it is a prerequisite for the detection and removal of zero-volume parts and other needless internal walls that create structural discontinuities in thefabricated parts.7 AcknowledgmentsThis paper is taken in part from a thesis to be submitted in partial fulfillment for thedegree of Doctor of Philosophy in the Department of Electrical, Computer, and SystemsEngineering at Rensselaer Polytechnic Institute.This research was supported by NSF Grant DDM-8914212 as a subcontract throughthe University of Texas Solid Freeform Fabrication program, and other grants of theRensselaer Design Research (RDRC) Industrial Associates Program. Any opinions,findings, conclusions, or recommendations expressed in this publication are those of93the authors and do not necessarily reflect the views of the National Science Foundationor any of the industrial sponsors.We would like to thank Dick Aubin, Pratt & Whitney (a division of United Tech-nologies), for providing a number of industrial .STL models. Also, special thanks to themembers of the RDRC: Stephen Rock for his early work on this project and his subse-quent collaboration; James Miller and Robert O'Bara for their development of facet-modelvisualization tools; and Peter Wilson for his helpful review of this paper.References[1] "Stereolithography Interface Specification," 3D Systems, Inc., June 1988.[2] S. 1. Rock and M. J. Wozny, "A Flexible File Format for Solid FreeformFabrication," in: Solid Freeform Fabrication Symposium Proceedings, L. Marcus,1. J. Beaman, J. W. Barlow, D. L. Bourell, and R. H. Crawford (eds.), TheUniversity of Texas at Austin (Austin, TX), August 12-14 1991, pp. 1-12.[3] 1. Arline, "A user survey of problems with rapid prototyping systems," PrototypingReport, vol. 1, no. 6, November 1991, pp. 7-8.[4] S. J. Rock, "Solid Freeform Fabrication and CAD System Interfacing," Master'sthesis, Rensselaer Polytechnic Institute, Troy, NY, December 1991.[5] R. A. Schubert, "StereoLithography," in: NCGA'89 Conference Proceedings, vol. 2,(Philadelphia, PA), April 17-20, 1989, pp. 182-186.[6] C. R. Deckard, Selective Laser Sintering (CAD/CAM). Ph.D. thesis, The Universityof Texas at Austin, 1989.[7] B. Rooney. Private communication, Brock Rooney & Associates, Birmingham,MI, May 27, 1992.[8] C. Alexander. Private communication, 3D Systems, Inc., Valencia, CA, May 29,1992.[9] S. J. Rock and M. J. Wozny, "Utilizing Topological Information to IncreaseScan Vector Generation Efficiency," in: Solid Freeform Fabrication SymposiumProceedings, H. L. Marcus, 1. 1. Beaman, 1. W. Barlow, D. L. Bourell, and R.Crawford (eds.), The University of Texas at Austin (Austin, TX), August 12-141991, pp. 28-36.[10] M. MantyHi, "Boolean Operations of 2-Manifolds through Vertex NeighborhoodClassification," ACM Transactions on Graphics, vol. 5, no. 1, January 1986, pp. 1-29.[11] K. J. Weiler, Topological Structures for Geometric Modeling. Ph.D. thesis,Rensselaer Polytechnic Institute, Troy, NY, August 1986. (RDRC-TR 86032).[12] F. S. Roberts, Applied Combinatorics. Prentice-Hall, 1984.94

Automatic CAD-model Repair: Shell-Closure

Abstract Shell-closure is critical to the repair of CAD-models described in the .STL file-format, the de facto solid freeform fabrication industry-standard. Polyhedral CAD-models that do not exhibit shell-closure, i.e. have cracks, holes, or gaps, do not constitute valid solids and frequently cause problems during fabrication. This paper describes a solution for achieving shell-closure of polyhedral CAD-models. The solution accommodates nonmanifold conditions, and guarantees orientable shells. There are several topologically ambiguous situations that might arise during the shell-closure process, and the solution applies intuitively pleasing heuristics in these cases

Jan Helge B~hn

Michael J Wozny

CiteSeerX

Automatic CAD-model Repair: Shell-Closure
Jan Helge B~hn and Michael J. Wozny
Rensselaer Design Research Center
Rensselaer. Polytechnic Institute
Troy, New York 12180-3590
(518) 276-6751
bahn@rdrc.rpLedu
Abstract
Shell-closure is critical to the repair of CAD-models described in the .STL file-format,
the de facto solid freeform fabrication industry-standard. Polyhedral CAD-models that
do not exhibit shell-closure, i.e. have cracks, holes, or gaps, do not constitute valid solids
and frequently cause problems during fabrication. This paper describes a solution for
achieving shell-closure of polyhedral CAD-models. The solution accommodates non-
manifold conditions, and guarantees orientable shells. There are several topologically
ambiguous situations that might arise during the shell-closure process, and the solution
applies intuitively pleasing heuristics in these cases.
1 Introduction
It is critical that the CAD-models used in solid freeform fabrication (SFF) are robust
to avoid failure during manufacturing. Unfortunately, CAD-models described in the .STL
file-format [1], the de facto industry standard, frequently lack this robustness [2, 3, 4].
Typical problems include punctured shells, inconsistent facet-orientations, and internal
walls and structures.
With the .STL file-format, the CAD-model solid is defined by its surface, Le. the
solid is defined by a closed shell enclosing its material. If this shell is punctured (i.e.
open), then it can no longer properly contain the material, and this consequently causes
the failures commonly associated with cracks, gaps, and holes (Figure 1.1). Similar
problems arise when the facet-orientations across a shell are inconsistent. This too causes
uncertainty regarding what side of the shell contains material. Finally, structural problems
might arise from internal walls and structures within the solid. These internal features
can create irregularities in the solidified material during the processing. For instance,
with stereolithography [5] and selective laser sintering (SLS) [6], excess amounts of
unsolidified material might get trapped inside the solidified matrix, and consequently
result in uncontrolled shrinkage and surface warping (Figure 1.2).
The .STL file-format is capable of representing a robust, well-formed solid. Unfor-
tunately, contemporary CAD-systems frequently fail to achieve this goal. There are two
Copyright © 1992 Rensselaer Design Research Center. All rights reserved.
86
Slice Plane
..
Stray Scan-Vectors
Figure 1.1 Shell-punctures causes the laser to produce
stray scan-vectors.
Figure 1.2 Internal walls cause structural
discontinuities that can lead to excessive
shrinkage.
design environments currently available for CAD; solid- and surface-modelers. Solid-
modelers generally avoid punctured shells, but frequently generate needless internal walls
and structures, and inconsistent orientations. In addition to these problems, surface-
modelers frequently fail to properly mate adjacent surfaces; this results in punctured
shells.
These problems are commonly resolved by the SFF-equipment operators prior to
fabrication. The operator might attempt to edit the .STL file directly, or, if available,
edit the original CAD-model to eliminate the problems. However, manual model repair
becomes excessively expensive when the number of facets reaches 10,000 to 100,000+,
and the number of errors run into the hundreds. This has lead to the development of
automatic CAD-model repair.
Brock Rooney & Associates [7] perform limited CAD-model repair in their .STL
translators, ensuring that their .STL output does not contain punctured shells. Their pro-
prietary shell-closure solution is not complete. It does not handle orientability-problems
satisfactorily, and it has limited success with certain simple structures. However, they
report few problems in general with their .STL output when run on the software-robust
stereolithography process of 3D Systems, Inc. In the rare cases where their shell-closure
algorithms fails, the CAD-model must either be regenerated at the design-level, or be
repaired manually as described above.
3D Systems, Inc. [8] provides its customers with a limited, proprietary model-repair
facility. It too is intended to ensure that the model does not fail during fabrication, or
if the repair fails, to provide indications to where it must be repaired manually. The
program attempts to repair simple orientation problems, minor punctures, and facet-
intersections, but leaves internal walls, and fails for large punctures and non-orientable
shells. It performs the repair under the restriction of several user-specified parameters,
and if these are set too conservatively, an intermediate result is generated. This output
can be processed iteratively until a valid, though not necessarily correct result occurs.
The basic problem of CAD-model repair is that of inferring information that has been
lost somewhere in the design process. While it might be easy in some cases to detect the
presence of a problem, it is frequently impossible to determine with certainty what the
correct solution is. This is especially true if the designer's intent is lost. Without it, an
educated guess must be made whenever an ambiguity is encountered. The goal of our
work is to match the quality of manual repair that is performed without the knowledge
of the designer's intent.
87
\
Figure 2.1 A non-manifold condition can be viewed locally as a set of two-manifold conditions.
2 Shell-Closure
This paper presents a complete solution for shell-closure of polyhedral shells. In
doing so, it also solves the orientability problems, and it simplifies the complexity of the
model-topology for more efficient processing during fabrication. In particular, it ensures a
pseudo two-manifold topological organization of the facets to facilitate high-speed slicing
[4, 9]. This topological organization also simplifies the subsequent resolution of shell-
intersection and shell-nesting problems, which are necessary for the removal of needless
internal walls.
A (two-) manifold structure is homeomorphic to a two-dimensional disk at every
point across its surface, i.e., each such point has a neighborhood that is topologically
equivalent to a two-dimensional disk. A non-manifold structure violates this constraint;
two cubes that touch along a common edge is one such example. However, if the facets
at a non-manifold location are paired up properly, then this situation can be described as
a set of (pseudo) two-manifold conditions [10] (Figure 2.1).
Our solution for shell-closure follows a three-stage strategy: First, we identify the
lamina edges. Second, we combined these edges into closed, non-self-intersecting loops
of boundary-curves; commonly known as directed Jordan curves. And finally, we create
lids that fit these Jordan curves and that close the open shells.
3 Lamina Edges
A lamina edge is generally considered a special case of a non-manifold edge; it
has only a single adjacent face [11]. Lamina edges are of interest because they make
up the boundaries of the shell-punctures, and of the corresponding lids which we will
create to close the shells. However, if the definition of a lamina edge is generalized,
in particular generalizing the definition of a puncture, then we can expand the use of
creating lids for shell-closure to also solve erroneous facet-orientations and complicated
non-manifold conditions.
Expanded definition: A lamina edge exists along each face-edge that does not mate
and pair properly this particular face with any other face along this edge. These lamina
edges can be found as follows: Let all face-edges be candidates for lamina edges. Then,
for each face-edge, attempt to find an adjacent face of compatible orientation such that
88
Surface Normal
} ={... }
3.1 Two lamina
of two faces about
candidates (a,b) are removed from the set of candidates (LEe) after a successful pairing
3.3 Two unpaired facets along a non-manifold
yield two coincident lamina edges.
adjacent about the common edge.
are removed from the set of lamina
after this process is exhausted,
4
.... WOIlA.'Vl"LLLj;. ULL.v'VIL,VU. '"''VA.''''' curves from the set of lamina edges,
.UAl..'V "'v"'"',,...u.F-, boundary curve-segments, or
to a single shell, and is terminated
Figure 4.1 Simple, self-connecting
loop (unambiguous).
Figure 4.3 Cyclic with single direct returns.
Figure 4.2 Linear combination (unambiguous).
Figure 4.4 Cyclic with multiple direct returns.
Figure 4.5 Trivial and nontrivial cases of cyclic with non-direct returns.
by two endpoints that might coincide. This can be achieved as follows: Initially, consider
each lamina edge as an arc. Then, iteratively, for each vertex with only two adjacent
arcs, merge those two arcs into a single arc, causing this vertex to no longer have an
adjacent arc. The final set of arcs form a set of directed graphs (as defined in graph
theory [12]) of various complexities that describe the shell-puncture boundaries, and
these puncture-boundaries must be separated prior to lid-creation.
We have identified five classes of puncture-boundaries of increasing identification-
complexity, which when combined, form the above mentioned sets of directed graphs.
Among these classes, three occur in our industrial .STL test-suite, while the remaining
two, the most complex classes, persist as unobserved possibilities: The most trivial class
of all is the simple combination; a single self-connecting arc. It is unambiguous and
trivial to detect, and it forms a complete puncture-boundary by itself (Figure 4.1).
The linear combination is also unambiguous; two arcs that are adjacent to the same
shell and that form a closed loop. These arcs form a link between two separate subgraphs,
each of which is less complex that their combination (Figure 4.2).
The remaining three classes are ambiguous because they offer multiple closed-loop
alternatives, and because the Jordan curves are located in three-dimensional space rather
than in a simplifying two-dimensional plane. Fortunately, one can make some reasonable,
90
alternatives: For instance,
separate shells. This is based
solids; consequently, all facets
shell. This dictates that, for
not be adjacent to same shell
....V ...'VA-A-LJV'Y by the formation of cyclic
i"""Tl'~i"n1nn arc for each of its
has been eliminated as
rare, but present, our
...AA'L,' .... "'.AA we not yet en-
returns is an ambiguous problem
arc, multiple options remain. It
U ........ILLLd'AAE:, not only topological-,
'l"nQTf't"I1nn problem where the
.......u.J--,v.u ...u of connecting
case, and it has a graceful
more ambiguous. Fur-
with minimum weight problem
arc-pairs, we end up with
Jl"'-V..........JlAA..., back to the example in
arcs are not shown), yielding
arc), the above shell-merger
downward arc) as an ac-
V"'''''.l.AAJUA..., arcs as possible matches
will cause the merger of two
where further loop-extraction
two possible combinations.
these are combinations
..... .I.'V''-'''''' ..... loop (Figure 4.5). Some
.I.......,.l.V,....."" .. A ........I..I. 7 ........'.I....,JL......., arcs into longer parts, but
are more and ambiguous. One possible
arcs onto a suitable two-manifold structure,
a
V./l........ ""'.., ...... ""'.... from the more complex
these problems appear to
our strategy is to identify the less
original graph. This simplifies
......... ' ........... 'VAA ....... unambiguous constructions. As
sub-problems.
o 1 2
rr([r
3 4
r
5 6
(f'r
7 8 9
r1'
Figure 5.1 Lid-creation by iterative removal of the vertex with the smallest angle.
•
Figure 5.2 Iterative post-processing to improve the quality of the lid-surface.
5 Creating Lids
The task of creating lids is greatly simplified once the corresponding directed Jordan
curves have been identified: The puncture is closed by an iterative algorithm that
decrements the Jordan curve by eliminating one vertex at a time; the vertex is eliminated
from further consideration by adding a facet that spans it and its two neighboring vertices
along the shrinking Jordan curve. At each iteration, we remove the vertex along the
Jordan curve that spans the smallest angle with its two neighboring vertices along the
remaining, decremented Jordan curve (Figure 5.1). This avoids the clearly undesirable
splitting of angles, and it will in general produce excellent results, especially for long
narrow punctures which abound in industrial .STL files [7, 8].
However, this heuristic algorithm is not without drawbacks. It is a time-local
algorithm in that it finds the best selection at any given time, without keeping the
overall optimal solution in mind. An example of this is the selection of the third
facet in Figure 5.1; it forces the subsequent undesirable creation of facet number four.
Fortunately, such problems are easily rectified with a simple, inexpensive iterative post-
processing step as illustrated in Figure 5.2: For each adjacent pair of lid-facets, flip their
common diagonal edge if this increases their minimum facet-angle.
A more serious problem is the difficulty of finding the angles spanned about the
vertices. While angles between vectors in 3D-space are in the range 0-180 degrees, it
is clear, from the 2D-plane, that the angles between facet-edges are in the range 0-360
degrees. It turns out, however, that a conservative decision to assign the range 180-360
degrees to a vertex whenever there is a doubt to which half of the total range it belongs
to, simply postpones its elimination. Eventually, one of the vertex's neighboring vertices
92
Orientation Discontinuity
Figure 5.3 Cutting a non-orientable Mobius-strip along its orientation-discontinuity, results in a orientable panel.
will be eliminated, and its angles must be recomputed. Usually, this clarifies the situation,
and the vertex might be assigned the range 0-180 degrees instead.
It is interesting to note that the above lid-creation algorithm is capable of creating a
non-orientable shell if the open shell is non-orientable. An example of this would be the
closure of a Mobius-strip. However, with our definition of lamina edges, non-orientable
shells are cut such that they become orientable prior to lid-creation. In particular, the
Mobius-strip becomes a simple panel (Figure 5.3). As a consequence, since the lid-
creation algorithm maintains the directionality of the diminishing Jordan curves, we are
guaranteed to maintain orientability throughout the shell-closure; and more importantly,
the final closed shells are therefore always orientable.
6 Conclusions
This paper has presented a complete solution for shell-closure of polyhedral shells,
the first step towards well-formed and manufacturable solids. The solution is based on
topological principals, and consequently resolves orientability and non-manifold problems
in addition to eliminating shell-punctures (i.e. cracks, holes, and gaps). As a result, the
facet-model output consists entirely of closed, orientable shells, and the resulting facet-
models can therefore be viewed as a set of two-manifold shells.
This topological organization simplifies the subsequent resolution of shell-intersection
and shell-nesting problems, and it is a prerequisite for the detection and removal of zero-
volume parts and other needless internal walls that create structural discontinuities in the
fabricated parts.
7 Acknowledgments
This paper is taken in part from a thesis to be submitted in partial fulfillment for the
degree of Doctor of Philosophy in the Department of Electrical, Computer, and Systems
Engineering at Rensselaer Polytechnic Institute.
This research was supported by NSF Grant DDM-8914212 as a subcontract through
the University of Texas Solid Freeform Fabrication program, and other grants of the
Rensselaer Design Research (RDRC) Industrial Associates Program. Any opinions,
findings, conclusions, or recommendations expressed in this publication are those of
93
the authors and do not necessarily reflect the views of the National Science Foundation
or any of the industrial sponsors.
We would like to thank Dick Aubin, Pratt & Whitney (a division of United Tech-
nologies), for providing a number of industrial .STL models. Also, special thanks to the
members of the RDRC: Stephen Rock for his early work on this project and his subse-
quent collaboration; James Miller and Robert O'Bara for their development of facet-model
visualization tools; and Peter Wilson for his helpful review of this paper.
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Automatic CAD model repair: Shell-closure

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Automatic CAD-model Repair: Shell-Closure

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