Array-Based Half-Facet Data Structure for Mixed-Dimensional and Non-manifold Meshes

20 Pages·2013·1.13 MB·English

AHF: Array-Based Half-Facet Data Structure
for Mixed-Dimensional and Non-manifold
Meshes
VladimirDyedov1,(cid:2),NavamitaRay1,DanielEinstein2,
XiangminJiao1,(cid:2)(cid:2),andTimothyJ.Tautges3
1 Dept.ofAppliedMath.&Statistics,StonyBrookUniversity,StonyBrook,NY11744
[e-mail]
2 BiologicalSciencesDivision,PacificNorthwestNationalLaboratory,
Richland,WA99352
3 Math.&ComputerScienceDivision,ArgonneNationalLaboratory,Argonne,IL60439
Summary. We present an Array-based Half-Facet mesh data structure, or AHF,
for efficient mesh query and modification operations. The AHF extends the com-
pact array-based half-edge and half-face data structures (T.J. Alumbaugh and X.
Jiao, Compact array-based mesh data structures, IMR, 2005) to support mixed-
dimensionaland non-manifoldmeshes. The design goals of our data structure in-
clude generality to support such meshes, efficiency of neighborhood queries and
mesh modification, compactness of memory footprint, and facilitation of interop-
erability of mesh-based application codes. To accomplish these goals, our data
structure uses sibling half-facetsas a core abstraction, coupledwith other explicit
and implicit representations of entities. A unique feature of our data structure is
a comprehensive implementation in MATLAB, which allows rapid prototyping,
debugging, testing, and deploymentof meshing algorithms and other mesh-based
numerical methods. We have also developed C++ implementation built on top
of MOAB (T.J. Tautges, R. Meyers, and K. Merkley, MOAB: A Mesh-Oriented
Database, Sandia National Laboratories, 2004). We present some comparisons of
thememoryrequirementsandcomputationalcosts,andalsodemonstrateitseffec-
tivenesswithafewsampleapplications.
Keywords: mesh generation, data structure, non-manifold, mixed-dimensional
meshes,siblinghalf-facets,MATLAB.
1 Introduction
Mesh data structures are an important technology underlyingmeshing algorithms
(such as mesh generation and modification) and mesh-based numerical methods
(cid:2) Currentaddress:CD-adapco,Austin,TX78750.
(cid:2)(cid:2) Correspondingauthor.
J.Sarrate&M.Staten(eds.),Proceedingsofthe22nd 445
InternationalMeshingRoundtable,
DOI:10.1007/978-3-319-02335-9_25,(cid:2)c SpringerInternationalPublishingSwitzerland2013
446 V.Dyedovetal.
(suchasfinite elementandfinitevolumemethods).While meshdatastructurehas
been investigated and used since the inception of mesh generation and computa-
tional geometry [1, 2, 8, 10, 11, 13, 16, 17], the increasing complexities and the
ever-changing demands of the application codes often pose new requirements to
mesh data structures. Examplesof such new demandingapplicationsincludecou-
pled multiphysicssimulations and multi-componentsystems, which may pose di-
verse requirements within each code as well as requirements on interoperability
acrossdifferentcodes.Therefore,meshdatastructurehascontinuedtobeanactive
researchtopicinrecentyears.
In light of the new challenges of meshing applications, we summarize a few
requirementsofmeshdatastructures,whichwillserveasourdesigngoals:
Generality: Supportmixed-dimensional,non-manifold(orientedornon-oriented)
meshesin1-D,2-D,and3-D,andbeeasilygeneralizabletohigherdimensions.
Efficiency: Supportalllocaladjacencyqueriesinconstanttime,assumingtheva-
lenceofeachmeshentityisboundedbyasmallconstant.
Simplicity: Besimpleandintuitive,andbeeasytoimplement.
Extensibility: Allowextensibilityforperformanceandparallelization.
Interoperability: Facilitateinteroperabilitywithapplicationcodes,suchassimu-
lationcodes,multigridsolvers,etc.
Compactnessofmemoryfootprint: Require minimal storage in addition to ele-
mentconnectivity.
In this paper, we strive to develop a mesh data structure that meets all the above
requirements. We refer to our data structure as the Array-based Half-Facet data
structure, or AHF. The AHF unifies the array-based half-edge and half-face data
structuresforsurfaceandvolumemeshes[1],andfurthergeneralizesthemtosup-
port mixed-dimensional and non-manifold meshes by introducing the concept of
sibling half-facets. It can be used to perform efficient mesh queries and
modification.
Thekeycontributionsofthispaperaretwofold:First,weintroduceasimpledata
model for mixed-dimensional and non-manifold meshes. Our data model is easy
toimplementandisefficientin bothmemoryandcomputationalcost. Inaddition,
someofitsdatafieldscanbecreateddynamicallyforfine-grainoperationsandbe
removedafterwards.Second,asanarray-baseddatastructure,AHFfacilitatesbetter
interoperabilityacrossdifferentapplicationcodes,differentprogramminglanguages
(such as MATLAB, C/C++, FORTRAN, etc.), and different hardware platforms.
OurC++implementationontopofMOAB[17]furtherimprovesitsinteroperabil-
ity through the iMesh interface (http://www.itaps.org/). In addition, our
data structure is uniquein its comprehensiveimplementationin MATLAB, which
allows rapid prototyping and deploymentof meshing algorithms and other mesh-
basednumericalmethodsinaproductivefashion.
Theremainderofthepaperisorganizedasfollows.Section2reviewssomeback-
ground knowledge and related mesh data structures. Section 3 describes our data
model for non-manifold and mixed-dimensional meshes. Section 4 describes the
AHF:Array-BasedHalf-FacetDataStructure 447
algorithmsfortheconstruction,query,meshmodificationoperations,aswellastheir
implementationsinMATLAB.Section5describestheC++implementationontop
ofMOAB.Section6presentssomecomparisonsofAHFagainstothersintermsof
storageandcomputationalcost.Section7concludesthepaperwithadiscussion.
2 Background and RelatedWork
In this section, we review some terminology and other existing data structures,
whichwillestablishthefoundationofourproposeddatastructure.
2.1 Terminology
Wedevelopdatastructuresforrepresentingdiscretegeometricandtopologicalob-
jectsin1-D,2-D,or3-D,arisingfromnumericalcomputationsinengineeringand
scientific applications.These objectscorrespondto curves, surfaces, andvolumes,
respectively, typically embedded in two- or three-dimensional Euclidean spaces.
Topologically, a d-dimensional object is a manifold with boundary if every point
inithasaneighborhoodhomeomorphictoeitherad-dimensionalballorhalf-ball,
wherethepointswhoseneighborhoodishomeomorphictoahalf-ballareboundary
(orborder)points.Inpractice,itis quitecommonto havetopologicalobjectsthat
arenon-manifold,especiallyforcurvesandsurfaces.Thesenon-manifoldsaretyp-
ically composed of a union of a finite number of manifolds with boundaries, and
sometimes embeddedin a higher-dimensionalmanifold structure. In 3-D space, a
surface is oriented if it is possible to make a consistent choice of surface normal
vectorateverypoint;otherwiseitisnon-oriented.
Inoursetting,ameshisasimplicialcomplexrepresentingdiscretelyageometric
ortopologicalobject.Wesayameshis1-D,2-D,or3-Diftheobjectthatitrepre-
sentsis topologically1-D,2-D, or3-D,respectively.We say a mesh isa manifold
ornon-manifoldifitsgeometricrealizationisamanifoldornon-manifold,respec-
tively.Ameshiscomposedof0-D,1-D,2-D,and3-Dentities,whichwerefertoas
vertices,edges,faces,andcells,respectively.Typically,afaceiseitheratriangleor
quadrilateral,andacellisatetrahedron,prism,pyramid,orhexahedron,especially
forfiniteelementmethods,althoughgeneralpolygonsandpolyhedraarealsooften
usedin finitevolumemeshes.We focusonfinite-elementmeshes,andwillbriefly
describethegeneralizationtopolyhedralmeshesinSection7.
Inad-dimensionalmesh,werefertothed-dimensionalentitiesaselements,and
refer to the (d − 1)-dimensional sub-entities as its facets. More specifically, the
facetsofacellareitsfaces,thefacetsofafaceareitsedges,andafacetofanedge
are its vertices. Each facet has an orientation with respect to the containing ele-
ment.Forexample,eachedgeofatrianglehasadirection,andalltheedgesforman
orientedloop.Thusitmakessensetocallthefacetsashalf-facets.Eachfacetmay
havemultipleincidentelements,especiallyfornon-manifoldentities.Werefertoall
448 V.Dyedovetal.
such half-facetsas sibling half-facets.A half-facetwithoutanysibling is aborder
half-facet,andrefertoverticesincidentonaborderhalf-facetasabordervertex.A
meshissaidtobeconformalifthepairwiseintersectionofanytwoentitiesiseither
anotherentityorisempty.Inthispaper,weconsideronlyconformalmeshes,which
maybemanifoldornon-manifold.Inthecaseofsurfacemeshes,themeshmaybe
orientedornon-oriented.
Insomeengineeringapplications,especiallyincoupledormulti-componentsys-
tems, the domain of interest may be composed of a union of topologically 1-D,
2-D,and3-Dobjects,suchasamixtureofcables,thin-shells,andsolids.Werefer
to such a domainand its mesh as mixed-dimensional.1 We refer to a subsetof the
mesh corresponding to a 1-D, 2-D, and 3-D object in the domain as a sub-mesh.
Itiscommon,althoughnotrequired,forthesub-meshestosharesomemesh enti-
ties, especiallysharedvertices.Our goalisto designdata structuresforconsistent
representationsformixed-dimensionalmesheswithsharedentities.
2.2 Half-EdgeDataStructure
The half-edge data structure, a.k.a. the doubly-connected edge list (DCEL), is a
popular data structure for 2-D and surface meshes (see e.g. [5]), especially ori-
ented, manifold, polygonalsurface meshes with or without boundary.The DCEL
uses edgesas the core object. The edge within each face is called a directed edge
orhalf-edge.Inanorientedmanifoldsurfacemesh,supposetheedgeswithineach
facecanbeorderedincounter-clockwiseorderwithrespecttooutwardnormal(or
upwardnormalfor2-Dmeshes).Eachedgehastwoincidentfaces,andthetwohalf-
edgeshaveoppositeorientationsandhencearesaidtobeoppositeortwinofeach
other.Anedgeontheboundarydoesnothaveatwinhalf-edge.
TherearevariousimplementationsofDCEL.Atypicalimplementation,suchas
thoseinCGAL[10,7],OpenMesh[3]andSurface_Mesh[15],storesthemappings
fromeachhalf-edgetoitsoppositehalf-edge,itspreviousandnexthalf-edgewithin
itsface,itsvertices,itsincidentface,aswellasthemappingfromeachvertexand
eachfacetoanincidenthalf-edge.Morecompactrepresentations,suchas[1],can
beobtainedbystoringonlythemappingbetweenoppositehalf-edge,optionallythe
mapping from each vertex to an incident half-edge, along with the element con-
nectivity.Theaboveimplementationsdonotsupportnon-manifoldornon-oriented
meshes,whichweovercomeintheproposeddatastructure.
2.3 Half-FaceDataStructure
A generalization of the concept of DCEL to volume meshes is the so-called the
half-face data structure [1, 13]. Within each cell, suppose the edges of each face
are orientedin counter-clockwiseorder with respectto the outward normalof the
cell.Werefertotheorientedfacesashalf-faces.Fortypicalmeshesinengineering
1Thesemeshesaresometimesreferredtoasmixedorhybridmeshes,whichweavoidhere
sincetheymayalsorefertomesheswithmixed-typesofelementsofthesamedimension.
AHF:Array-BasedHalf-FacetDataStructure 449
applications,eachfaceintheinteriorofavolumemeshhastwocorrespondinghalf-
faceswithoppositeorientations,whicharesaidtobeoppositeortwinofeachother.
Thedatastructurein[1]wasdesignedfor3-manifoldwithboundarycomposed
of the standard elements. Although this is the typical case in applications, a vol-
umemeshmay alsobe non-manifold.Forexample,thiscanhappenif the domain
contains two objects that intersect at a single vertex or along an edge. For gener-
ality, we consider volume meshes that may be manifold or non-manifold,and al-
lowthemtobeorientedornon-oriented.OpenVolumeMesh[13]wasdevelopedfor
general polytopal meshes, which may be non-manifold.However, the data model
in OpenVolumeMeshis quite complicated, because unlike the edges in a cell, the
faces within a cell do not have a natural topologicalorder. A mesh data structure
for non-manifold meshes in arbitrary dimensions was proposed in [4]. This data
structurestoresnotonlytheadjacencyinformationofthehighestdimensionalele-
mentsbutalso some adjacencyinformationof intermediateelements, whichincur
extrastorageoverhead.Inaddition,initsproposedform,itcouldnothandlemixed-
dimensionalmeshesthatcontainsubmeshesofdifferentdimensions.Weseekasim-
plerandamoreunifieddatamodel.
2.4 Pointer-BasedVersus Array-Based Implementations
A mesh data structure may be implemented using either pointers or arrays . The
pointer-basedimplementationsaremorecommon,sincetheyarerelativelyeasyto
manipulate. For example, the DCEL implementation in CGAL is pointer-based.
Other examples, which are not based on half-edges or half-faces, include FMDB
[16],MSTK[8],libMesh[11],etc.Insuchanimplementation,theentitiesarerep-
resentedas“objects”explicitly,andpointers(orhandles)areusedtorefertothese
explicitobjects.
In contrast, in an array-based implementation, we do not represent the entities
in the meshas objects.Instead,anattribute ofallthe entitiesof thesame typeare
storedinoneorafewarrays,andtheattributesforasingleentitymaybedistributed
indifferentarrays.AnentitymaybereferencedthroughanIDor“handle”,which
can bemappedeasily to arrayindices.The half-edgeand half-facedata structures
in[1],MOAB[17],andOpenVolumeMesh[13]arearray-based.
In our work, we choose to use array-based, pointer-free implementations for a
numberof reasons.First, in an array-basedimplementation,we can treatinterme-
diate dimensional entities (such as half-facets) as implicit entities, and reference
themwithoutformingexplicitobjects.Thiscanleadto significantsavingsinstor-
age, especially on computers with 64-bit pointers. Second, using arrays can also
lead to faster memoryaccess and hence better efficiency.In addition, array-based
implementationsalsoofferbetterinteroperabilityacrossapplicationcodes,different
programminglanguages,anddifferenthardwareplatforms(suchasbetweenGPUs
andCPUs).
450 V.Dyedovetal.
3 Data Model forNon-manifoldand Mixed-Dimensional
Meshes
Thebasichalf-edgeandhalf-facedatastructuresaresimple,buttheywererestricted
to oriented, manifoldmeshes (with or withoutboundary)in 2-D and 3-D, respec-
tively. In this section, we present a simple and unified generalization to mixed-
dimensionalmeshes,whichmaybenon-manifoldand/ornon-oriented.
3.1 UnificationofHalf-Edgesand Half-Faces
Weunifythedatamodelsofhalf-edgeandhalf-facedatastructures.Inthisunified
datamodel,thekeyabstractionsforad-dimensionalmeshare:
vertices: 0-dimensionalentities(a.k.a.nodes);
elements: d-dimensionalentities(facesandcellsin2-Dand3-D,respectively);
half-facets: (d−1)-dimensionalsub-entitiesofad-dimensionalentity(half-edges
andhalf-facesofa2-Dand3-Delement).
Inthisdatamodel,weassumethattheelementhasastandardnumberingconvention
foritsverticesanditsfacets.Forstandardelements,wefollowtheconventionofthe
CGNS(CFDGeneralNotationSystem)[12,18],asillustratedinFig.1.Wedonot
3 4 3 4 5 6 7 6
3 4 3 4 3 4 5 5 8 56
1 2 1 2 1 2 1 3 4 34 5 12 3 1 4 33 2 43 15 23 2
1 21 2 2 1 2 1 24 1
Fig.1CGNSnumberingconventionsfor2-Dand3-Delements.Underscorednumberscor-
respondtolocaledgeIDs,andcircledonescorrespondtolocalfaceIDs.
require explicit representationof intermediate dimensionalentities between 1 and
d−1.Instead,wetreatthehalf-facetasanimplicitentity,andrefertoahalf-facet
usingtheelementIDanditslocalIDwithintheelement.Werefertothisdatamodel
asthehalf-facetdatastructure.
Theaboveunifiedmodelisnotlimitedtod = 2and3.Infact,itcanbeapplied
to any d, as long as the numbering convention is predefined for the vertices and
itsfacets. Inparticular,ford = 1,werefertothisrepresentationforcurvesasthe
half-vertexdatastructure,whichisa specializationofthehalf-facetdatastructure
to curves.For d ≥ 2, it providesa compactrepresentation,since the intermediate
dimensionalentitiesarenotstoredbutreferencedimplicitlyinstead.Inaddition,this
datamodelcanalsobeusedformesheswithhigh-orderelements(suchassix-node
trianglesor10-nodetetrahedra),wherethemid-edge,mid-faceormid-cellnodesdo
notaffectthedefinitionandidentificationofthehalf-facets.
AHF:Array-BasedHalf-FacetDataStructure 451
We store the data structure similar to that in [1]. The element connectivity is
storedinarraysinamannersimilartoatypicalfiniteelementcode.Themappings
betweensiblinghalf-facetsare storedina 2-D arrayorin separatearrays,oneper
uniqueelementtype.Let |E| and f denote the numberof elementsand the maxi-
mumnumberoffacetsinanelement,respectively.We denoteeachhalf-facetbya
two tuple (cid:3)eid,lfid(cid:4),whereeiddenotesthe elementID, whichstarts from1, and
lfid denotes the local facet ID, which starts from 0. For typical meshes, the two
tuple can be encodedinto a single 32-bitunsigned integer,by using first d bits to
storingthelocalfacetIDofad-dimensionalelement,andusingtheremainingbits
tostoringtheelementID.Thisallowsuptoabout500millionelementsforvolume
meshes.Forverylargemeshes,weassigna32-bitunsignedintegertotheelement
IDandanunsigned8-bitintegertothelocalID,andstoretheminseparatearrays.
Thisallowsupto4billionelementswithminimalextrastorageoverhead.Depend-
ingonwhetherthehalf-facetIDsareencodedinasingleintegerorintwointegers,
wecanstorethemappingsineitherasinglearrayortwoarrays,respectively,where
eacharrayisofsize|E|×f.Inaddition,thevertextohalf-facetmappingisstoredin
asingle1-Darray.Formostmeshes,weneedtostoreonlyoneincidenthalf-facet.
Somecomplicationsmayarisefornon-manifoldvertices,whichweaddressnext.
3.2 GeneralizationtoNon-manifoldMeshes
In a non-manifold mesh, there can be more than two elements abutting the same
facet, unlikea manifoldmesh wherethere can beonlyup to two. We still referto
an oriented facet within an element as a half-facet, but it no longer has a “twin”
or“opposite”half-facetingeneral.Werefertothehalf-facetscorrespondingtothe
samefacetassiblinghalf-facets.Theorientationsoftwosiblinghalf-facetsarenot
requiredtobe oppositetoeach other,andthereforethisgeneralizationalso allows
representingnon-orientedmeshes,suchastheMöbiusstrip.
An important issue is the storage for mapping between the sibling half-facets.
Insteadofdoubly-connectedlinkedlist fortwin half-facets,weuse acycliclinked
list, which allows us to preserve the storage structure and also to traverse all the
elements incident on a half-facet. Figure 2 shows an example of a non-manifold
edge(edgejoiningvertex2and3)presentinthegiventriangulatedmesh,aswellas
the the element connectivity,sibling half-edgemap and vertexto half-edgemaps.
Notethatwedonotnecessarilyneedtosortthehalf-facetsinanyparticularorder.In
fact,anorderingmaynotbewell-definedinsomecases.However,thedatastructure
doesnotexcludetheuserfromorderingthehalf-facets.
In a d-dimensional non-manifoldmesh, when d > 1, there may exist a vertex
whoseneighborhoodisnotad-dimensionalballorhalf-ball,butinsteadtheunion
of two or mored-dimensionalballor half-ballsthatintersectatan entity of lower
than(d−1)dimension(suchasatthevertexinasurfacemesh,oratavertexoredge
inavolumemesh).Werefertosuchavertexasanon-manifoldvertex.Someminor
modificationtothedatastructureisnecessaryinthissetting.Inparticular,atanon-
manifoldvertex,weneedtostorea1-to-nmappinginsteadofa1-to-1mappingto
452 V.Dyedovetal.
(cid:6)
elementconnectivity
(cid:5)
element vertices
(cid:5)
1 1 2 3
(cid:2) (cid:3) (cid:4)
2 2 4 3
(cid:2)
(cid:3) 3 2 5 3
(cid:4)
4 2 3 6
(cid:7)
vertextohalf-edges
siblinghalf-edges vertices v2he
element sibhes 1 (cid:2)1,0(cid:3)
1 nil (cid:2)2,2(cid:3) nil 2 (cid:2)3,0(cid:3)
2 nil nil (cid:2)3,2(cid:3) 3 (cid:2)4,1(cid:3)
3 nil nil (cid:2)4,0(cid:3) 4 (cid:2)2,1(cid:3)
4 (cid:2)1,1(cid:3) nil nil 5 (cid:2)3,1(cid:3)
6 (cid:2)4,2(cid:3)
Fig. 2Example of anon-manifold mesh, along withitselement connectivity, sibling half-
edges,andmappingfromeachvertextoanincidenthalf-edge
itsincidenthalf-facets.Notethatthisalsocoversthecasewherethereexistanedge
inavolumemeshwhoseneighborhoodistheunionoftwoormored-dimensional
ballorhalf-ballsthatintersectonlyatthevertex.
3.3 GeneralizationtoMixed-DimensionalMeshes
The concept of sibling half-facets unifies the half-vertex,half-edge, and half-face
data structures for 1-D, 2-D, and 3-D meshes, which may be manifold or non-
manifoldwithboundary.Inamixed-dimensionalmesh,thesub-meshesofdifferent
dimensions can share entities. In particular, it is most common for the meshes to
sharevertices.However,theseentitiesmayhavedifferentrepresentations.
Thisunificationallowsaneasyextensiontosupportmixed-dimensionalmeshes,
which may be composed of sub-meshes of 1-D, 2-D, and 3-D. Figure 3 shows a
diagramofatypicalhalf-facetdatastructure,wherethehalf-verticesandhalf-edges
areonlyrequiredforexplicitedgesandfacesinthemesh,respectively.Wereferto
this data structure for mixed-dimensionalmeshes as Array-based Half-Facetdata
structure,orAHF.Thisdatastructureisverysimpleandmodular,astheindividual
sub-meshesofdifferentdimensionsareself-contained,andtheycanbemaintained
separately.Theyalsoallowusto traversebetweenmultipledimensionsefficiently.
The interactions of the different dimensions are all performedthrough the shared
vertices.
AHF:Array-BasedHalf-FacetDataStructure 453
sibhf sibhe
Cells Exp-Faces
v2hf
v2he
faces
cells
edges
Exp-Edges Vertices
v2hv
sibhv
Fig.3TypicalAHFformixed-dimensional meshesiscomposed half-vertex(black, forex-
plicitedgesonly),half-edge(blue,forexplicitfacesonly),andhalf-face(red)datastructures.
4 Construction, Query, andModificationofAHF
Inthissection,wedescribesomedetailedalgorithmsfortheconstructionofAHF,as
wellassomequeryandmodificationoperations.SinceAHFisarray-based,theseal-
gorithmscanbeimplementedinanyprogramminglanguages,includingMATLAB,
C/C++,FORTRAN,etc.WewillalsodescribeourimplementationinMATLAB.
4.1 ConstructionofAHF
Inthehalf-facetdatastructure,therearetwocomponents:sibhfs(siblinghalf-facets
) and v2hf (vertexto half-facet).The formeris centralto AHF, as nearly all adja-
cencyqueriesrequireit.Thesesiblinghalf-facetsshouldmaptoeachotherandform
acycle.Thelatterarray,v2hf,isoptionalformanyoperations,itcanbecreatedonly
whenneeded.
Ingeneral,weconstructtheAHFintwosteps,whichconstructthesetwomap-
pings,respectively.Inamixed-dimensionalmesh,theAHFforthesubmeshofeach
each is constructedindependentlyof each other.In the following,we describethe
twostepsinamannerindependentlyofthedimensionofthemesh.
IdentificationofSiblingHalf-Facets
During the first step, we determine the sibling half-facets and construct a cyclic
mapping between them. The key components of this step are two intermediate
mappings:
v2hfs: amappingfromeachvertextoitsincidenthalf-facetsinwhichthevertex
haslargestID;
v2adj: a mapping from each vertex to its adjacent vertices in each of the above
incidenthalf-facets.
Algorithm1 outlinesthe procedurefor the first stage, which is applicableto half-
facetsinarbitrarydimensions,anditisparticularlyefficientin1to3dimensions.
454 V.Dyedovetal.
Algorithm1.Determinationofsiblinghalf-facets.
Input: elems:elementconnectivity
Output: sibhfs:cyclicmappingsofsiblinghalf-facets
1: foreachelementseinelemsdo
2: foreachfacetf inedo
3: v←vertexwithlargestIDwithinf;
4: us←setofadjacentverticesofvinf;
5: Appendf intov2hfs(v),andappendusintov2adj(v,f);
6: endfor
7: endfor
8: foreachelementseinelemsdo
9: foreachfacetf inedo
10: ifsibhfs(f)isnotsetthen
11: v←vertexwithlargestIDwithinf,andus=v2adj(v,f);
12: Findhalf-facetsinv2hfs(v)s.t.v2adj(v,·)=us;
13: Formacyclicmappingforthesehalf-facetsinsibhfs;
14: endif
15: endfor
16: endfor
The computational cost of Algorithm 1 is linear, assuming that the number of
facets incident on a vertex is bounded by a small constant, say c. To analyze the
storagerequirement,let|H|denotethenumberofhalf-facetsinthemesh.Theout-
put requires approximately|H| integers. The algorithm requires two intermediate
mapsv2hfandv2adj,which requirec|H| integerstotal. Thisintermediatestorage
requirementcanbefurtherreducedbyequallydistributingtheverticesintobbuck-
etsandprocesseachbucketinaseparatepass,whichthenwouldrequireonly|H|/b
integers.
ConstructionofIncidentHalf-FacetofVertex
During the second step, we construct a mapping from each vertex to an incident
half-facet. This is done by utilizing the sibling half-facets obtained from the first
step,asoutlinedinAlgorithm2.
When determining the incident half-facets, we give higher-priorities to border
half-facets,so thatfromits outputv2hf,we candeterminewhethera vertexv is a
bordervertexbysimplycheckingwhetherv2hf(v)isaborderhalf-facet.Inaddition,
a minor variant of Algorithm 2 can be used to construct a bitmap of vertices to
determinewhetheralltheverticesareborderverticeswithoutformingv2hf.These
functions can be useful, for example when extracting the boundary of a mesh or
whenimposingboundaryconditionsinnumericalcomputations.Thecomputational
costofAlgorithm2isalsolinearinthenumberofverticesplusthenumberofhalf-
facets. It does not require any intermediate storage. In addition, the AHF can be
usedtoextracttheinternalboundariesbetweendifferentmaterialsinamesh.