Non-ManifoldModels
Thenormalobjectsthatyoumeetineverydaylifearecalled‘‘manifold’’objects.Whichmeans,puttingitglibly,thatateverypointonthesurfacetheneighbour-hoodaroundthepointishomeomorphictoadisc.Youmay,ormaynotwanttoknowthat.Figure6.1,showsanalternativewayofunderstandingthis,fromBraid,that,ateverypointontheoutsideoftheobject,asmallenoughspherewillbecutintotwopieces,oneinsidetheobjectandoneoutside.
ThefollowingpottedhistoryiswhatIbelievetobetrue,butifsomeoneeverwritesadefinitivehistoryofCADthentheremaybeotherfactorsofwhichIamnotaware.
IntheoriginalBoundaryRepresentationmodellingsystemsonlyvalidsolidswereconsidered.IntheBUILDresearchsystem,asanintermediatestep,itwaspossibletorepresentflatobjects,butthesewereusuallyonlyshapeswhichweretobeswept.Towardstheendofthe1970saninternordicproject,GPM,wassetuptodevelopmethodsfor‘‘GeometricProductModelling’’wassetupincorporatinganumberofmodellingmethods.InDenmarktheusersystemwasdeveloped.InNorwayan‘‘AssembledPlateConstruction’’(APC)module(SINTEF)andasurfacemoduleweredeveloped(SI).InSwedenandFinlandthevolumetricmodellingmodulewasdeveloped.TheAPCmodulewasaspecialised,advancedmoduleformodellingconstructionsmadefromthinplates.Aspartofthevolumemoduleitwasintendedtobeabletointerfacewithboththismoduleandthesurfacemodule,sothinplatemodelswereintroducedaspartofthevolumetricmodellingsystem[1].
OneoftheSwedishideaswasthat,bymixingdifferentrepresentationsinthesamemodellingframework,youcouldrepresentdifferentstagesandlevelsofmodels.Inthebeginningyoumighthaveasimplesketch.Thismightthenbefleshedoutintopartialmodels,idealisationsofthevolumetricshape.Forproductionneedsthesemightneedtobeexpandedintofullvolumetricmodels.Asthelifetimeofaproductdevelopsitmayproveusefultogobacktoidealisationsandmaybeevensketches.ThisisdescribedbyKjellberg[2],who,asfarasIknow,pioneeredthismethod,buthisdissertationisinSwedishsoisnotsoaccessible.OneoftheexamplesheusedisillustratedinFig.6.2,asimplifiedmodelofanexcavator.
I.StroudandH.Nagy,SolidModellingandCADSystems,
DOI:10.1007/978-0-85729-259-9_6,ÓSpringer-VerlagLondonLimited2011
315
3166Non-ManifoldModels
Fig.6.1Manifoldobjectdefinition
Thewholeexcavatorisanassemblyofmodelswithdifferentcharacteristics.Thebodyoftheexcavatorandthetracksaresolidmodels,thearmisawireframemodelandthescoopisacompoundsheetmodel.
So,fornon-manifoldmodelsinCAD,itisnecessarytodistinguishbetweenthreetypesofspecialandnon-manifoldmodel:1.Wireframemodels.2.Sheetmodels.
3.Non-manifoldsolidmodels.
Thesecanbeintegratedintothesamedatastructureforeasytransitionbetweenapplications.Forwireframemodelstheloopandfaceinformationis‘‘ignored’’,thatis,settoNULL.Sheetmodelsarelikedegeneratesolidmodels,withthelimitedgescorrespondingtothinfaces.Non-manifoldsolidshavecoincidingportions.Somesystemskeepsolidsapartfromsheetmodelsandwireframemodels,othersintegratethem.Thisisastrategicquestionwhich,ofcourse,affectstheuser
butisuptotheCADdeveloper.Neitherstrategyisparticularlybad,theybothhaveadvantagesanddisadvantages.Integratingthemmeansthattheuserhasafluentdesignenvironmentandthatcommonfunctionalityisshared.Keepingthemapartmeansthatthereislesschanceoferror,increatingsheetmodelsinsteadofsolids.Onethingthatshouldneverbedone,though,istohavesolidswithsheetand/orwireframeelementsintegratedintothesamemodel,suchastheoneshowninFig.6.3.Thisispossible,butsheetandwireframemodelsareidealisationsofsomething,whereasolidmodelisafullsolid.Integratingthemwouldmeanthatyouhaveamodelwhichhastobeinterpreteddifferentlyindifferentplaces,whichisnotparticularlyagoodidea.Nobodydoesthis,asfarasIknow,soitshouldnotbeaproblem.
6.1DatastructureNeeds
Acommonmethodforimplementingnon-manifoldmodellingistousetheso-called‘‘STAR’’representation.Thedevelopmentthatallowedthiswastointro-ducetheloop-edgelinkssothatedgescouldrefertomorethantwoloops(loopsarefaceboundaries).IntheoriginalBoundaryRepresentation(Brep)datastructure,inBUILD,therewasafixedrestrictionthatthereweretwoloopsateveryedge.Theycouldbethesameloop,buttherewerenevermore,becausesuchobjectswereunrealisable.Theadditionofloop-edgelinkstothedatastructure,whichappearedintheGPMVolumeModule,allowedringsoflinksaroundtheedgeand,hence,anynumber.Figure6.4showsthis.
Arequirementforthismethodofrepresentationisthatthelinksareorderedaroundtheedge.Figure6.5illustratesthis.Ontheleftofthefigureyouseeanormalcase.Thelargeblackdotrepresentstheedge,seenincross-section.Thelinesrepresentfacesandthesmalldotsarejusttoindicatewherethereismaterial.Turningaroundtheedgecounter-clockwise,asindicatedbythearrow,thelinksbetweentheedgeandthefacesareclassifiedas‘‘enters’’or‘‘leaves’’dependingon
3186Non-ManifoldModels
leavesenters
leaves
leaves
leavesenters
entersleaves
entersenters
leavesentersleavesenters
Fig.6.5Orderingedgelinksaroundtheedge
whetheryouenterthematerialorleaveit.Thereisasequenceofalternatingpairsinacorrectfigure.OntherightofFig.6.5isanincorrectcase.Thesequenceis‘‘enters–leaves–enters–leaves–enters–enters–leaves–leaves’’.Thedouble‘‘enters’’and‘‘leaves’’indicatethatthereismaterialwithinmaterial,i.e.thatthereisaself-intersectingobject.Notethatthisorderingprocedurecanbedoneforvolumeobjects,butforsheetobjectstheenters–leavesclassificationswithcoincide,andorderingisdifficult.
Thestarrepresentationisnotnecessarytoimplementnon-manifoldmodels,thenon-manifoldconditionisageometriccondition,notatopologicalone.IntheGPMprojectedgeswereallowedtorefertoonlytwofaces,eventhoughloop-edgelinkswerepartofthedatastructure.Itwasfeltthatitwasmorenaturaltoduplicateedges.Edgeduplicationalsohasanadvantageinthatthemeaningofthe
datastructureentityconnectionsisunambiguous.Withthestarrepresentationitisnotclearwhethertheobjectpartsarejustconnectedatthenon-manifoldedge,orwhethertheyjustmisseachother.Itisimportanttoknowthisbecauseperforminganoperationonanon-manifoldedgeshouldentailaconversionbeforetheoper-ation.Thisconversioninvolvespairinguptheloop-edgelinksandduplicatingedges.AvisualcomparisonisshowninFig.6.6.Foranon-manifoldedge,asshownatthetopofthefigure,thestarversionisshownatthebottomleftandthedegenerateversiononthebottomright.Forthestarversionanadvantageisthatthelinksareassociated,itisclearthattheedgeisaspecialcase.Unlessthereisalinkbetweentheduplicatedelementsthisconnectionisnotexplicit.
TheambiguityproblemisillustratedgraphicallyinFig.6.7.Thestararrangementisshownatthetopofthefigure.Ifyougroupedgelink1withedgelink2andlinks3and4thenyougetthedoubleedgejoinedcaseatthebottomleft.Ifyougrouplink2withlink3and4with1thenyougetthearrangementonthebottomright,wheretheobjectsareseparate,onlybeingjoinedattheverticesoftheedges.
TheproblemisthatthereisnowayfortheCADsystemtoknowwhichoneyoumean.Thismeansthatoperations,suchaschamfering,couldhavetwointerpre-tations,asillustratedinFig.6.8,asalreadymentionedinSect.4.7.Ofcourse,whatyouwouldlikeistheCADsystemtoaskyouwhichyoumean,butatthemomentthetrendistoignoretheseedges.
Thepointofexplainingthisistwo-fold.Thefirstistoexplainwhy,sometimes,yougetdiscriminationbetweenmodelelementstowhichyouwouldliketoapplyanoperation.Secondly,toexplainthenotionofedgeduplicationandedge-linkpairing,whichishowtointerpretthemulti-linkedges.
6.2WireframeModels
Wireframemodelsareanotherexampleofpartialmodelswhichcanbeusefulforspecialpurposes,suchassketchingthecentre-linesofpipework.AtonetimeCADsystemsusedwireframemodelsexclusively.
6.2.1WireframeDatastructure
Thedatastructureofwireframemodelsisverysimple,consistingof‘‘nodes’’and‘‘links’’,asshowninFig.6.9.Thenodesarerepresentedbyverticesandthelinksbyedgestousethesameelementsasforsheetobjectsandsolids.
Whiletheseareenoughforsimpleshapes,thelackofsurfaceinformationisahandicapformanyfunctions,fromdrawingtomanufacturing.AwireframegraphicsviewofanobjectisshowninFig.6.10toemphasisthis.
6.2.2ImpossibleWireframes
Itisalsopossibletomakeobjectswhicharenotrealisable,asinFig.6.11.SheetobjectsandvolumeobjectsareEulerianobjects,whichmeansthattheyfollowtheformuladescribedinSect.2.7.3.Thismeansthattherearerestrictionsonhowyoubuildmodels,whichprecludesmodelssuchasthatinthefigure.Othermodels,suchasMöbiusstrips,orKleinbottlescanalsobecreatedusingwireframetechniques,butareexcludedusingvolumetrictechniques.Moreusualthantheserecreationalobjects,though,isthatitispossibletocreateerroneousobjects.
6.2.3WireframesandModelling
Anoldresearchtopicwastheautomaticconversionofwireframemodelstosolids.Itisnotpossibletoguaranteeaconversionandsomecounter-examplesexistofobjectswhichcannotbeconverted.Afeasibleuseforwireframemodelsisasasupportforsketchingortheycanbeusedasidealisationsandthenconverted,aswiththeoperationdescribedinSect.4.13.
Onecurrentuseforwireframesisfordefiningtwo-dimensionalshapestobesetintosurfaces,ashasalreadybeendescribedinSect.3.7.Theycanalsobeusedformodellingcurvesinageometricpackageand,forexample,swepttocreatesur-faces,aswillbedescribedinSect.6.3.1.
6.2.4WireframeExperiments
6.2.4.1CreatingPipework
MakeashapelikethatontheleftofFig.6.12.Extrudeacirclealongthispathtocreateasimplepipe.Youcanfinishthepipeusingtheshellingoperationtocreatetheinterior.
Ifyouhaveashapelikethatontherightofthefigureyoucannotcreateitinonepiecewiththeextrusionalongpathoperation.Youcancreatethebasicshapeas
withthefigureonleftandthenaddtheadditionalshapewithasecondextrusionalongapath.
Thequestionsconcernhowtoperformthevariouspartsoftheoperation.If,afterthefirstextrusion,thebasicshapeisturnedintoaextrudedshapethenthesecondoperationwillcreateinteriorelementsratherthanthedesiredshape.Thesecondshapeshouldbeaddedbeforetheshellingoperationtocreatetheinterior.However,thiscreatesthefinalobjectinonepiece,butitwouldnormallybecreatedinseveralshapedpieceswhichwouldbeweldedtogether.Thiscanbedonebycreatingtheoutershapeasasolidmodel,separatingitintoelementaryparts,andthencreatingtheindividualshelledpiecestobemade.
Thepurposeofthislongexplanationistosaythat,whilethefacilitiesmayexisttocreatesimplemodels,realapplicationsneedtobebasedaroundcorrectinter-pretationsratherthanusingstandardtools.
6.3SheetModels
Sheetmodelsareausefultoolforrepresentingidealisationsofthin-platemodelsorforrepresentingsurfaces,asdescribedinSect.5.8.1.Sheetmodelsarenon-manifoldbecausetheyareinfinitelythin,butinsomeapplicationsitisquitenaturaltousethemratherthanvolumetricmodels.TheGPMAPCmodulewasmentionedatthebeginningofthischapter,andtherewasasuccessfuloilrigplatformdesignapplicationbasedonit.Otherapplications,suchaslayouts,formodellingshapestobecutfromclothorshapestobecutfromthinmetalsheets,donotneedvolumetricmodels.Itismoreefficientandmorenaturaltousesheetmodels.
Animportantpartoftheuseofsheetmodelsistheirinterpretationasideali-sationsofthin-platemodels.Inthisrespect,theduplicationofedgesismorenaturalthanusingastarrepresentationforcoincidentedges.Theduplicatededgeslieondifferentsidesofthesheetobjectsandwouldbeslightlyapartifthesheetobjectwereexpandedtoproduceavolumetricmodel.ThiswasoneofthereasonsthatthismethodwasusedintheGPMvolumemodule.
6.3.1ExtrudingWireframeModels
Acommonoperationistoextrudecurvestoproducesurfacemodels.Thismeansthatedgesgofrombeingnon-EuleriantobeingEulerianaspartoftheextrusionprocess.Theedgesgofrombeinglinkedthroughverticestobeinglinkedintochainsasbordersoffaces.ThiswasalsodiscussedbrieflyinSect.4.2.ItisimportanttoknowhowwireextrusionisintegratedintotheCADsystem,whethersheetmodelsareseparatefromvolumetricmodelsorcoexist.
BranchingwireobjectshavealreadybeenmentionedinSect.4.11.TheseareusuallyexcludedfromextrusionoperationsbyCADsystems,sowillnotbedealtwithfurtherhere.
Figure6.13showsanobjectthatwasusedforcomparisonofsolidmodellingsystemsforaseminarorganisedbytheCAM-Iorganisationin1983.Itisanobject,reportedlyofagun-platform,whichiscomposedofthin-walledparts.
AspartoftheGPMVolumemoduledemonstration,thispartwasshownbothasasheetmodelandasasetofflattenedshapestobecutfromplatematerial,showninFig.6.14.ThemethodfordoingthisisdescribedinStroud[3].TheoperationhasnotappearedincommercialCADsystems,asfarasIknow,butitshowswhatcouldbedoneaspartofaspecialapplication.
6.3.2JoiningSheetObjects
JoiningsheetobjectsrepresentingsurfaceportionshasalreadybeendescribedinSect.5.8.1.TheprocedureisshowninFig.6.15.Eachedgehastwoloop-edgelinks,linkedinachain.Theedge-linkpairsareregroupedtoformtwochains,inthefigure,oroneiftheedgesaremerged.
Figure6.16showshowtheloop-edgelinksarerearranged.Thecasewheretheedgesareinthesamedirectionisshowninthelefthandcolumnofthefigure.Thecasewheretheedgesareintheoppositedirectionisshownintheright-handcolumn.Theoriginalarrangementisshownatthetop.Edgee1hasleftlinkL1andrightlinkR1.Theedgetowhichitistobejoinedise2,withleftlinkL2andrightlinkR2.
Iftheedgesareinthesamedirectionandbotharekept,middleleft,thenlinkL1ispairedwithlinkR2andlinkL2ispairedwithlinkR1.Iftheedgesaremerged,bottomleft,thenthelinksarearrangedinthecircularsequenceL1;R2;L2;R1.Iftheedgesareinoppositedirections,andbotharekept,middleright,thenL1ispairedwithL2,whichbecomesarightlink,andlinkR1ispairedwithR2,whichbecomesaleftlink.Iftheedgesaremerged,bottomright,thenthelinksaremergedintothesequence:L1,L2(whichbecomesarightlink),R2(whichbecomesaleftlink),R1.
6.3.3VolumeModelstoSheetModels
Section4.9describesthesimplewayofconvertingvolumemodelstosheetmodelsasonestepintheshellingprocess.
AnoperationthatyoufindinCADsystemsistounfold,orflatten,volumemodels.ThiswasdescribedinSect.4.10.
Thesemethodsallowadesignertocreateapartasasolidandthenconvertittoasheetmodeltobemadefromthinmaterial.Convertingavolumemodeltoasheetmodelsisafirststepinatleastoneflatteningalgorithm.Thisconversionallowstheconcaveedgesalongwhichtheobjectistobebenttobemarkedasgroovedforfinishingoperationsaftercutting.
6.3.4SheetModelExperiments
6.3.4.1ExtrudingWires
Firstofall,createanopenshapeasasketchinthevolumemodellingpartandextrudeitinastraightlineorcirculararc,asinFig.6.17.Thisisthesameexperimentasforextrusionandisintendedtoshowwhetherornotsheetmodelsandvolumemodelsareintegratedorseparate.ThequestionisnotwhethertheCADsystemcandoitornot,thesearesimpleshapes,butwhethertheyareallowedtocoexistornot.
Anotherextrusionexperimenttotryonsimpleshapesinvolveswireswithoneedgeintheextrusiondirection,asshowninFig.6.18.Theshapeonthetopleftofthefigureshouldprobablycauseanerror,astheextrudedshape,bottomleft,
wouldhaveadanglingedge,whichisnotagoodidea.Theshapeshowntopmiddle,though,ismoredebatable.Theresultofanextrusionwouldbeavalidshape,bottommiddle,thoughtheinternaledges,showndotted,wouldhavetobehandledproperly.Thethirdshape,topright,wouldcauseproblems,becausetheextrusionwouldleaveasinglewireedgeinthemiddle,ortheshapewouldhavedegenerateparts.
TestthesethreeshapestoseeiftheCADsystemallowsthemornot.Fortheshapeontheright,makesurethatthemiddleedgeislongerthantheextrusiondistance.
6.3.4.2ExtrudingBranchingWires
Thisisanotherexperimentthathasbeensuggestedbefore,inthesectiononextrusion.Extrudingbranchingedgesshouldnotbeaproblem,exceptforcriticalcaseswhereoneormoreedgesareintheextrusiondirectionortherearecoin-cidentedges.ThereasonfornotimplementingthisisastrategicdecisionbyCADimplementersaboutthecomplexityallowed.Theproblemistoworkouttheconnectionsatthecomplexbranch-points.Ifthereareatmosttwoedgesateveryvertexthenthereisnoproblem.Iftherearemorethenageometrictestisneededtosortoutpairings.
Oneoftheedgesistakenasabaseedge,thezerodegreeedgeandtheotheredgesareorderedusingtheirtangentvectors,projectedontotheplanedefinedbythecommonvertexandtheextrusiondirection(Fig.6.19).ThismethodwasdevelopedbyMüller[4].Notethatedges3and4havethesametangentdirection,
butthisissortedoutinMüller’smethodbyusingthecurvature.Themethodwillnotwork,though,iftherearecoincidentedgesoredgesparalleltotheextrusiondirection.
Analternativeistoslicetheedgestocreateadegenerateface,whichisthenextruded.SeeFig.6.20.Thiscanthenbeextrudedusinganormalfaceextrusionandthentheedgescollapsedbackasafinishingprocess.
TheactualdetailsofhowthisisdoneinaparticularCADsystemarenotreallyimportant,thequestioniswhetherornotthesystemallowsyoutoextrudebranchingwires.
6.3.4.3JoiningThreeorMoreSheetObjects
DedicatedCADsystemsforthinplatemodellingshouldbeabletohandlethiscase,thoughaspartofageneralCADsystemthismaynotbeallowed.AsimpletestisshowninFig.6.21.CreatealineontheZ=0plane,sayfromðÀ50;0Þtoð0;0Þandextrudeit60.Createasecondline,fromð30;À50Þtoð0;0Þandextrudethis60,aswell.Finally,createathirdline,fromð30;50Þtoð0;0Þandextrudethis60,thesamedistanceforallthreesheetobjects.Nowtryjoiningthem.Inthesamewaythatbranchingwirescanbehandled,sheetmergingcanbehandlediftheCADsystemdeveloperispreparedtoinvesttheeffort.
6.3.4.4JoiningSheetModelsWithoutMatchingEdges
ThisexperimentistotesthowmuchefforthasbeenputintosheetmodellingintheCADsystem.ThisisequivalenttoaBooleanoperationonsheetmodelsandisnotimpossible,technically,justrequiressomeeffortfromtheCADsystemimplementer.OntheZ=0plane,drawasemicircle,or,ifyouwish,afullcircle,radius25,say.Extrudethis50tocreatethefirstsheet.Nowmakealinejusttouchingthemiddleofthesemicircle,from(25,0)to(50,0),say,asillustratedinFig.6.22.Trytojointhemandcheckwhetherornotthesystemallowsthis.
6.4PartialModels
Partialmodelsareaspecialtypeofsheetmodelwhichareusefulasmechanismsforapplications.Theyhavefacesandsurfacesononesidebutnothing,ormaybeoneunsurfacedface,ontheotherside,aswasdoneintheBUILDsystem.Ifthereisnothingonthebackofthepartialmodel,asisnormal,thentheboundaryedgesareonlypartiallydefined,withonlyoneloop-edgelink.
YouwouldnotexpecttoseetheseaspartoftheCADsystem,theyaretoolsforapplicationsorintermediateresults,perhaps.Partialmodelsactasthoughtheyhaveunlimitedmaterialbehindthem.Providedanyoperationdoesnotgobeyondtheboundariesofthepartialmodeltheyactasnormalvolumes.AnapplicationusingpartialmodelswillbedescribedinSect.10.3.2.
AsimpleillustrationofthedifferencesbetweensheetobjectsandpartialobjectsisshowninFig.6.23.Ifyouweretoaddacylindertoasquareshape,showninFig.6.23ayouwouldgettheresultinFig.6.23bifthesquareshapewereasheetobjectandtheresultinFig.6.23cifthesquareshapewereapartialobject.Thereasonisthatthesheetobjectwouldhavetwointersectionsandbothtopandbottomofthecylinderwouldbeclassifiedasoutsidethesheet.Ontheotherhand,withapartialobject,therewouldbeonlyoneintersection,withthedefinedface,andonlythetopofthecylinderwouldbeclassifiedasoutsidetheobject.Inaddition,theadditionofthecylindertothesheetobjectshouldreallybeclassifiedasinvalid,sincethiswouldcreateamixedobjectwithvolumetricandsheetparts.
6.5Non-ManifoldVolumeModels
Bewareofnon-manifoldportionsinvolumetricmodels.
Althoughcreatingsuchmodelpartsistechnicallycorrect,integrationinalloperationsseemspatchy,atleastatthetimeofwriting.Theproblemofhandling
3326Non-ManifoldModels
starrepresentationsandpairingloop-edgelinkshasalreadybeenmentioned.Aswithsheetmodels,thiscomesbacktohavingaclearmethodofinterpretationofwhatthenon-manifoldedgemeans:isitwherethereismaterialorisitwhereobjectstouchwithoutbeingjoined?Similarconsiderationsexistforobjectsjusttouchingatvertices.
Note,also,thattherearetwokindsofnon-manifoldnessthatyoumightcreate.Thefirstkindiswherethereareedgeswithmorethantwofacesorverticeswithmultipleedgesets,thesecondiswheretwoelementstouchwithouthavingcommontopologicalelements.AnexampleofthesecondtypeisshowninFig.6.24.Theobjectwascreatedusinganextrusionalongapath,describedinSect.4.2.8.
Thesortofobjectshowninthefigureishardforamodellingsystemtoresolvebecausethenormalpoint-setconsiderationscannotbeusedtosortouttheobject.Pointshaveneighbourhoodswhicharenothomeomorphictodiscs,ifyouprefertohaveitthatway.AnattempttosubtracttheobjectinFig.6.24fromablockfailedbecause,thesystemsaid,itcouldnotsortoutthetangentialrelationships.
Sortingoutthistypeofnon-manifoldobjectwouldrequireaspecialtypeofBooleanoperation,forevaluatingself-intersectingobjects,which,insteadoftakingtwoobjectsandcomparingtheirfaces,comparesthefacesofasingleobjectwitheachother.Itisnotimpossibletoimplement,butIhavenotyetseensuchanoperationinaCADsystem.Suchanoperationislinkedwithanothertopic,called‘‘modelchecking’’or‘‘modelhealing’’,whichwillbedescribedbrieflyinSect.14.3.
Fig.6.24Non-manifoldsolidwithtouchingelements
6.5Non-ManifoldVolumeModels333
6.5.1DatastructureImprovements
Theinherentambiguityofthestarrepresentationascommonlyimplementedisadrawback,buttherearewaysofimprovingit.ThebestworkIknowinthisareawasbyLuoandLukacs[5,6].Theyintroducedelementscalled‘‘bundles’’forhandlingnon-manifoldverticesand‘‘wedges’’fornon-manifoldedges.Bundlesandwedgescanbethoughtofasbeingequivalenttoloopsforfacesinthattheyallowmultipleassociationsbetweendatastructureelements.Afulltreatmentofthis,though,liesoutsidethescopeofthischapter,whichisintendedtodealwithpracticalaspectsoftheuseofnon-manifoldsolids.
Thereasonformentioningthisisthatthecurrentstructuremightchangetoavoidthisambiguity.Ifitdoes,inthefuture,youmaybepromptedtotellthesystemwhetheryoumeanobjectsto‘‘touch-and-join’’or‘‘touch-but-miss’’.Ifyouareasked,thenitmeansthatthesystemcandistinguishbetweenthecasesandoperationssuchaschamferorblendonnon-manifoldedgesandverticesmaywork.Ifyougetthiskindofmessage,checkwhathappensbyblendingthenon-manifoldmodelpart.
6.5.2Non-ManifoldVolumeApplications
Tosomeextent,non-manifoldmodellingisasolutioninsearchofaproblem.Theuseofsheet-andwireframe-modelsasidealisationsisaclearandusefulfacilitythathasproveditsusefulness.Theusesfornon-manifoldsolidmodelsislessclear.Onesuggestionhasbeentousethemforfinite-elementmod-elling.However,finiteelementmodelstendtohavealargenumberofele-mentsandtheoverheadsassociatedwithavolumewouldmakethesedifficulttohandle.
OnemoreappropriateapplicationareaisthatdescribedbyLuoandLukacsandinvolvestheuseofnon-manifoldvolumemodelsforprocessplanning.ThisisadevelopmentbuildingonanideaproposedbyMalcolmSabin,oneofthemostinfluentialfiguresinCAD/CAM.Ataconferencein1983SabinsuggestedbuildingbackaCADmodeltoitsstockasawayofmanufacturingplanning.LukacsandLuo’sideawastokeepthebuilt-backvolumesseparateandtolinkthemwiththeCADmodeltocreateacompound,non-manifoldmodel.Thesep-aratepartscouldthenbeeasilyidentifiedformanufacturingplanning,IknowofnoCADsystemsusingsuchmethods,though.
6.5.3VolumetricExperiments
Thefirstthreeofthesecomefrompreviousexercises.Theyaresimplewaysofcreatingnon-manifoldsolids.
6.5.3.1TouchingEdgesbyExtrusion
ThisexercisewasdescribedinSect.4.2.Makeasquareshape100Â100.Extrudeit100units.Onthetopface,createanewsquare,100Â100,whichtouchesoneoftheedgesofthetopfaceandextrudethis100unitstocreatetheobject.SeeFig.6.25.6.5.3.2TouchingVerticesbyExtrusion
AnotherexercisefromSect.4.2.Thisisasimilarexercisetothepreviousone,exceptthattheshapestouchonlyatavertex.Makeasquareshape100Â100.Extrudeit100units.Onthetopface,createanewsquare,100Â100,whichtouchesoneoftheedgesofthetopfaceandextrudethis100unitstocreatetheobject.SeeFig.6.26.
6.5.3.3ExtrudingTouchingShapes
YetanotherexercisefromSect.4.2.Makeapairoftrianglestouchingatavertex,asshowninFig.6.27.Theorderofdefinitionshownbep1-p2-p3-p1andp4-p5-p2-p5.Youshouldnottrymakingthisasp1-p5-p4-p3-p1,whichwillnotcreatetheimportantvertexinthemiddle.
Theactualsizeofthetrianglesisnotimportant,justthattheytouchatonevertex.Extrudethetouchingtrianglesaboutthelengthofthesidetogiveareasonablethickness.Thequestioniswhathappenswiththecommonvertex,isitoneedgeortwo?6.5.3.4TouchingFaces
ThiscreatestheshapeshowninFig.6.24.IntheYZplane,Y=0,createtheshapeshowninFig.6.28.OntheXYplane,Z=0,createa25Â25squareshape
centredaboutoneoftheendverticesoftheshapeshown,markedAandBinthefigure,andextrudeitalongthepathtocreatetheshape.
ThismethodofcreatingtouchingfacesmaynotworkinsomesystemswhichuseBooleanoperationstojoinsub-extrusionelements.
6.5.3.5ChamferingtoCreateTouchingElements
CreateathinshapesuchasthatshowninFig.6.29a.Thearmsmightbe75unitslongand10unitswide.Extrudetheshape75togiveavolumetricmodellikethatshowninFig.6.29c.NowchamfertheconvexedgeasshownintwodimensionsinFig6.29b,andinthreedimensionsinFig.6.29d.
pffiffiffi
Ifthedepthofthechamferiscorrect,armthicknessÂ2,thenyougetanon-manifoldedge,showndottedinFig.6.29d.IfBooleanoperationsareusedtocreatethechamferthentheedgemightbeastarnon-manifoldedge.Ifasimplelocaloperationisusedthentheedgemaytouchtheface,butnotbeassociatedwithit.Youcantestthisbyattemptingtoblendtheedge.
6.6ChapterSummary
Thischapterdealswiththesubjectofnon-manifoldandidealisedmodels.Whileidealisedmodelshaveaclearuseasdesignsketches,non-manifoldvolumemodelshavealessclearroleinmodelling.BeingabletoworkwithidealisationsaspartofthedesignprocessaddsfluencytoCADuseandmayalsohelpbyprovidingclearerdesignintent.
6.7Non-ManifoldExercises6.7.1IdealisationMatching
Topracticeidentifyingcaseswhereidealisationsareuseful,identifywhetheridealisationsmightbeusefulforthefollowingdesigntasksand,ifso,whichtype,ortypes,ofidealisationyouwoulduse.Writeshortnotestojustifyyouranswer.1.2.3.4.5.6.7.
Adrillingplatformstructure.Factorypipinglayout.Abuildingdesign.Ashipdesign.Carbodydesign.
TheGehauseRohteil,showninFig.1.73.TheEiffelTower.
6.7.2BuildingtheExcavator
CreateanassemblymodeloftheexcavatorshowninFig.6.2useyourownjudgementastowhichelementstocreateunifiedandwhichtokeepseparate.
3386Non-ManifoldModels
Writedownalistofthedifferentmodelsinyourassembly,togetherwiththetypeofmodeltheyare.Howwouldyouconverttheidealisationsintovolumetricobjects?Whichoperationswouldyouuse?
References
1.Kjellberg,T.A.,Lindholm,G.,Sorgen,A.,Haglund,G.:GPMReport10:GPM—SpecifikationVolymgeometri.DepartmentofManufacturingSystems.KTH,Stockholm,Sweden.Confi-dentialdocument,ISBN91-85212-54-7(1980)
2.Kjellberg,T.A.:IntegreradDatorstödforMänskligProblemlösningochMänskligKom-munikationinomVerkstadstekniskProduktionbegränsattillProdukt-utveckling,Produktions-beredning,Kon-struktionochTill-verknings-beredning.EnsystemansatsbaseradpaaProdukt-modeller.Ph.D.Dissertation,DepartmentofManufacturingSystems,KTH,Stock-holm,Sweden(inSwedish)(1982)
3.Stroud,I.A.:BoundaryRepresentationModellingTechniques.Springer,Heidelberg(2006)4.Müller,M.,Stroud,I.,Xirouchakis,P.:Preprocessingofdegenerateslicesinlayeredmanufacturing.In:Topping,B.H.V.(ed.)DevelopmentsinEngineeringComputationalTechnology,pp.63–68.Civil-CompPress,Scotland,ISBN0-948749-70-9(2000)
5.Luo,Y.:SolidmodellingforregularobjectsRenewedtheory,datastructureandEuleroperators.Ph.D.Dissertation,ComputerandAutomationInstitute(1991)
6.Luo,Y.,Lukacs,G.:Aboundaryrepresentationforformfeaturesandnon-manifoldsolidobjects.In:TheFirstACM/SIGGRAPHSymposiumonSolidModelingFoundationsandCAD/CAMApplications,Austin,TX,June(1991)
因篇幅问题不能全部显示，请点此查看更多更全内容