AUTOMATICDETAILREDUCTIONFOR MESHGENERATIONAPPLICATIONS TimothyJTautges . 1
SandiaNationalLaboratories Albuquerque, , NM, UniversityoWisconsin-Madison, f Madison,WI
[email protected] ABSTRACT Increasing resolutionincomputationalsimulationi generation. Because ovarying f resolutionrequirem of geometry detailsare classifiedbtyheir removal them withneighboring topology;blend removalremov neighbors; bridge removalrequiresmore advancedde methodsinclude defining size a measure,anddetect Resultsare givenfor automatic detailreductionap
enabling s the use ooriginal f CADdesignmodelsas ents,geometricdetailremovalis cariticalparto method:directdetailremovalseekstoremove deta esblendsby firstpartitioning blends thencomposi composition, andinot s described here. Key capabi ing geometricdetailsbased otnhose. Blenddetect plied tsoeveralrealparts.
Keywords:mesh generation,computationalgeometry,
detailreduction
1 INTRODUCTION Computer hardware speed has been increasing at a remarkablepaceoverthelastfewdecades,fulfilli nga predictionknowntodayaM s oore’sLaw [1]. Morerecently, afocusedeffortondevelopingnumericalmodelings oftware whichcantakeadvantageofthefastesthardwareha sbeen sponsoredbyDOE’sASCI [2]andSciDAC [3]programs. Theanalysiscodesdevelopedundertheseprogramsa renow capableopf redictingcomponentresponseaut nprece dented levelsofdetailandcomplexity. Inresponsetoth esenew capabilities,theresolutionofanalysesbeingperf ormedon thefastestcomputers hasalsoincreasedovertime. Although there have been manyadvances in mesh generation technologyrecently [4][5]theseadvanceshavenotkeptup withincreasesincomputerspeedandresolution. Therefore, meshgenerationtime(includingbothinteractivean dcpu time)forthesemodelsisstillincreasingrelative tothe analysistime for high-endproblems. Formanyanalyses,anincreaseinresolutioninthe modelisoftenaccompaniedbyanincreaseindetail domainmodel,orthegeometryofthecomponentbein analyzed. Onthespectrumbetweenfullydetailedg (i.e.theoriginalCADdesignmodel)andageometry almostnodetail,typicalresolutionsforlargemod
the basisof mesh this f process. Three classes ilsdirectly bcyompositing ting the pieceswith litiesfor allthese ionialso s described.
analysis inthe g eometry with elsare
approachingthefullydetailedendofthespectrum. problemslikethis,startingthemeshingprocessus original CAD design model is becoming an option. However,designmodelsoftenmustbe convertedinto suitableformeshgenerationandfiniteelementana Thisconversionprocesscanbelengthy,andifnot carefully,mustberepeatedafterevensmallchange originaldesignmodel. Thispaperdescribesasolu onepartotfhisprocess,theremovalodf esigndet CADmodels.
For ingthe models lysis. done stothe tionfor ailsfrom
Designmodelsareusuallyconstructedforthepurpo sesof manufacturing,andthereforecontainfinedetailst hatarepart oftheas-manufacturedcomponent.Examplesofthes e detailsincludefillets,whichdissipatestresssin gularities,and rounds,whichremovesharpedges,oftenforcosmeti c purposes. Thesedetailscancomplicatethemeshge neration process,especiallyforhexmeshgenerationbutals oforother typesofmeshes. Therefore,analystsoftenremove design detailsbefore meshingageometry;thatis,theycoarsenthe resolutionofthegeometricmodelsothatitmorec losely matchestheresolutionofthetargetanalysis. Thi s simplification process reduces the subsequent mesh generationtime,for little costintermsof analys isaccuracy. Removalofgeometricdetailcanbetime-consuming, requiring manual manipulation of the CAD model. Automationofthisprocesscouldsignificantlyredu
1 SANDIA ISA MULTIPROGRAMLABORATORYOPERATEDBYSA LOCKHEEDMARTINCOMPANY,FORTHEUNITEDSTATESDEPA CONTRACTDE-AC04-94AL85000.
NDIACORPORATION,A RTMENTOFENERGYUNDER
cethe
overallmeshgenerationtime.Therehavebeenvari approachestoautomaticdetailremoval;thesearer below.
ous eviewed
1.1 Previousefforts Because the appearance of integrated design-analysi s packagesisrelativelyrecent,therearenotmanyi ntegrated solutionsforgeometricdetailremovaldeliveredin the marketplace.However,thereareseveralresearche fforts whichhave addressedthisproblem. WhenstartingwithadesignmodelinaCADsystem, the mostcommonapproachtoremovingdetailsistoremo ve themdirectlyintheCADsystem. Thisios ftendon ebythe analyst,orbythedraftsmanwiththeanalystlooki ngoverhis shoulder.Obviously,thisapproachrequiresknowle dge aboutthe CADsystem (hence thedraftsman!). Howev er,this approachcanalsointroduceproblemswithconsisten cy, wheresmallchangestotheinitialdesignmodelaft erstarting themodelgenerationarenotreflectedintheanaly sismodel. Also,thisapproachisdifficulttoreproduceafter model changesresultingfromanalysis.Therefore,manual ly removingdetailcanserveaasnimpossiblebottlene ckinthe design-analysisiterationsoftenusedinindustry. Finally, changesmadeintheCADsystemcanbde ifficultto reverse, incaseswheretheanalysisrequiresmoreresolutio nina givenarea thaninitially thought. OnesolutionbeingdiscussedaSandia t NationalLab oratories andelsewhereitsousemulti-levelorlayereddesi gnmodels. Inthisapproach,acomponentisdesignedinprogre ssively higher-resolution layers. Modern CAD systems like Pro/Engineer [6]andSolidWorks [7]havebuilt-insupport forlayeringdetailinamodel,andsimplifythere movalof details(bysuppressingoneom r orelayers). Onea dvantage ofthisapproachitshatitreflectshowdesignis usuallydone, startingwithanabstractconceptandmovingtoward a detaileddesign. However,therearealsodisadvant ageswhen thesemodelsareusedinanalysis. Ifthedesired resolution doesnotcloselymatchthatof pa re-definedlayer, changesto themodelarestillrequired.Evenworse,thelaye red approachdoesnotworkwhenthedesiredanalysisre solution variesacrossthemodel;inthiscase,thefullestdetaillayer mustbeusedovertheentiremodel,againrequiring the removalofdetailatotherlocations. Also,fined etailsinthe modelareoftennotthe resultoffine featuresin feature-based modelers,butrathertheinteractionbetweenlarger features; thesedetailstypicallywillnotberemovedbysupp ressing fine-resolutionlayersintheCADsystem. Finally, weoften donothavetheluxuryofdesigningacomponentfro m scratch;rather,weareworkingwithmodelsconstru ctedand modifiedovertheyears. Thesemodelsareoftendi fficultto convertintolayeredmodels. Whilewefeelthatth elayered approachhaspromiseandshouldbiencorporatedint odesign practices,werecognizethatitwillbeawhilebefo rethis change propagates down to the draftsmen actually constructing these models. Therehavebeenseveraleffortsaimedatremovingd afteraninitialmeshhasbeengenerated,e.g.Ref. approach,detailsareremovedbyperformingwell-kn meshtransitions,e.g.collapsingthefaceoatet f
etails [8]. Inthis own rahedronto
removethetetelement.Whiletheprocessofremov ing elementsfromameshusingprimitiveoperationsis wellunderstood,theproblemremainsofgeneratingthei nitial meshonthefineresolutionofthemodel. Thiscan beboth time-andmemory-intensive.Also,incaseswheret he desiredanalysisresolutionismuchcoarserthanth eoriginal model,thisapproachseemsquitewasteful;theseca seswould probably stillrequiresomeinitialcoarseningotf hegeometric modelbeforemeshingbegan.Shefferhasinvestigate dthe clusteringosurface f facetsintogroupshavingsim ilarnormal vectorsom r atchingothergeometriccriteria [9]. Whilethis solvestheproblemofgeneratinganinitialmesh,i trelies heavilyongeometricinformation,andseemstonot account for topologicaldetails. Therehavebeenseveralgeometry-basedsolutionspr esented intheliterature [10][11][12];ourapproachim s ostsimilarto these. Armstronget.alpresent m a ethodbasedon themedial axistransformofanobject [10]. WhiletheMATisquite usefulforprovidingdetailsaboutthenon-localas pectsof geometry,it ypicallydoesnotcontaininformation about “imprints”onthegeometry,orgroupsosfurfacesw herethe geometryitshesamebutthetopologyvaries. Blac ker [12] discussesapproachesbasedondetectingsmallgeome tric features,andremovingthefeaturesusing“virtual topology” operations. Keyissuesthatariseinalltheseeff orts,andin theresearchreportedinthispaper,arethelevel ofuser interactionrequired,especiallyfortheadjusting ofmultiple freeparameters,andofcoursetheoverallrobustne sswhen appliedtocomplicatedmodels.Itisnotclearfro mthe descriptionsof [10][11][12]iftheseissuesareaddressed sufficiently toproduce usable a capability.
1.2 Contribution Whilethesolutiondescribedinthispaperdoesnot resolvethekeyissuesouf serinteractionandrobu doescontainslightlydifferentapproachestosome whichimprovesonpastwork. Adifferentsizemeas introducedwhichmoreaccuratelyreflectscharacter Thispaperalsodiscussescaseswheresimplevirtua operationsarenotsufficientforremovingdetails, providessolutions;thisdirectlyimprovesrobustne paperpresentsagroupingofeatures f intothreecl addressedbyadifferentmethodofremoval,andela onthosemethods.Inparticular,anewmethodfor automatically identifying blendsisintroduced.
fully stness,it ofthem ureis isticsize. topology l and ss. This asses,each borates
1.3 Organization Thispaperarrangedafsollows. Section2describe assumetobetheinitialconditionsfordetailremo describesourgeneralapproach.Section3discusse importantissueofcharacteristicsize,andintrodu methodformeasuringit. Sections45,and6descr approachestodirectdetailremoval,blendremoval, bridgeremoval,respectively.Section7containsa discussion othis f researchand givesconclusions.
w s hatwe val,and sthe cesour ibeour and final
2 ASSUMPTIONS&GENERALAPPROACH
assumethatchangesdescribedin1and2 completed. Thispaper describes how todostep 3.
Beforedescribingourresearcheffortsindetailre moval,itis usefultodescribethestartingconditions.Thege ometry whichservesasthe basisforfiniteelementmodelsusually comesfromoneoftwosources:theanalystconstruc tsthe geometricmodelfromthebottom-up,sometimesusing an existingCADmodelassource a oprimitives; f or,t heanalyst startswithafully-detailedCADmodelwhichhasbe en constructedforother(i.e.non-CAE)purposes,e.g. a manufacturingCADmodel. Thelattersourceisbeco ming morecommon,alteastforhigher-endproblems,and isthe basisof thisresearch.
2.1 DesignandAnalysisModelsandthe AnalysisProcess Itisrarethatmeshgenerationcanbedonedirectl fully-detailedCADmodel,whichwerefertoasthe model”. Moretypically,changesarenecessarytoa thegeometrywhichitshebasisothe f mesh,which toasthe“analysismodel”. Thesechangesfallint categories: 1.
2.
3.
yonthe “design rriveat werefer othree
Geometry Healing: that is, theremoval of geometricdetailwhichdoesnotreflectdesign intentandwhichusuallyappearsastheresultof translationbetweenCADsystems.Although geometryhealingcanbeanarduousprocess, commercialsoftwaresolutionsarebeginningtobe effective [13][14]. Thisissueisnotaddressedin thispaper.
Forthepurposesothis f research,weassumethatt pointis gaeometricmodelwherealldetailswhich designintenthavebeenremoved,andwherethemode beenidealized,suchthatthegeometrydefinesasp commontomostdownstreamapplications.Thatis,w
The stepsabove feedintoaonverallanalysisproce
ss:
4. Generatemesh 5. Analyze & interpretresults 6. Repeat3-5anseeded,makingchangesinsteps3 4toaccountfordesignchangesomesh r resolution changes,respectively.
or
Animportantgoal,onewhichiscomingwithinreach useanalysisasthebasisofproductdesignactivit However,theefficiencyoftheapproachabovedepen the detailsuppressionstep being:
is ,to ies. dson
•
Automatic, to minimize the user interaction required per analysisiteration;
•
Reversible,toallowuseradjustmentoaf utomatic simplifications;and
•
Repeatable,sothatdetailreductioncanbe performedonamodelslightlydifferentthatthe previousone(e.g.afterlocaldesignmodelchanges resulting from interpretation oanalysis f results).
Thedetailreductionapproachdescribedinthispap distinguishedfrom pasteffortsinallthree catego
eris ries.
2.2 ThreeTypesofDetails Thereare twostepstoour detailsuppressionappro weaddressseparately. First,criteriamustbecho determinethetopologicalentitiesdesignatedas“s refertotheseas“detailentities”).Theotherst removalodf etailentitiesafterbeingdesignateds identifythreedetailremovalmethods,distinguishe typesodf etailentitiestheyapplytoaswellast methoditself:
ModelIdealization: wherethevariousgeometric componentsaremadeto“fittogether”.This requiresremovingsmallgapsandoverlapsinthe model.Notethatthesedetailsarenotoutside designintent.Forexample,wehaveobserved “pressfits”,wherepinsweredesignedlargerthan theholetheyfitin,whichappearasoverlapping bodiesinamodel.Gapsinamodelaremore common,forexamplebeingusedtorepresentbolts whichareshorterthantheholestheyfitinto. We assertthatinmostofthesecases,thesedetails representtheneedsofdownstreamapplications, e.g.manufacturing,andshouldnotbepartotfhe originaldesignmodel, butadded afterward. DetailSuppression: Thisstepisspecializedfor analysismodelgeneration,wheresmalldetailsin thedesignmodelareremovedtoavoidresolving theminthemesh. Thisisdonebothtosimplify meshgenerationandtoavoidlargenumbersof elementsoabrupt r transitionsinmeshsizerequire toresolve the smallfeatures.
abovehavebeen
ach,which senwhich mall”(we episthe mall. We dbythe heremoval
Directremoval: Certaingeometricdetailscanbedirectly removed from the model using “Virtual Topology” operations. Blendsuppression: Blendsarecommonlyusedindesigns, butareoftennotresolvedinanalyses. Removalof these featuresusuallyrequiresacombinationofvirtual topology operations. Bridgefeatureremoval: Bridgefeaturesarepartsofa geometricmodelwhichlinkdistinctregionsoafge ometry witheachother. Thesefeaturescanbeeitherposi tive(a bridge)onr egative(athrough-hole);however,thei removal issimilar. d
hestarting are outside hl as ace e
Afterdiscussingthefirstissueodf efiningwhati removalof eachothese f typesof detailsisdescri
small,the bed.
3 CHARACTERISTICLENGTHAND “SMALL”DEFINITION Beforedeterminingwhether gaivenfeatureismall beabletomeasureitssize, The S. simplestsize
we , must measurefor
anobjectisitsextent;thatis,thelength,area, caurve,surface,or volume,respectively:
andvolumeof
S= Lc ength(c) S= A s rea(s)
Figure 2-4showapartresolvedtoseveralmeshsizes,wit theappropriatelevelofdetailinthegeometricmo resolved. Obviously,therearecaseswherethiste willnotwork,especiallyforhexahedralmeshing;h the generaltrend omore f detailasmeshirefined s
h del chnique owever, iscorrect.
S= V v olume(v)
S= Lc ength(c) 1/2
S= (vVolume(v))
1/3
st shape,
= 4A/P h(s)
(4)
S= D v
= 6V/A h(v)
(5)
Sqrt(area)
0.8 0.6 0.4 0.2
(2)
0 0
(3)
S= D s
HydDiam
1
(1)
Theproblemwiththesenormalizationsisthatthey over-estimatethecharacteristicsize,forexample thin surfaces. Therefore, weborrowa measure of characteristicsizefromfluidflowmodeling,that diameter.Thehydraulicdiameterforsurfacesand, extension,volumes,are [15]:
wherePisthesurfaceperimeter,andAandVarea volume,respectively.
1.2
10 Surface#
20
tendto forlong, 10
ofhydraulic by
8 sqrt(A)/Dh
S= (sArea(s))
esdirectly dvolume thatthey
Measure
Thismeasuremakessenseforcurves,sinceirelat t tomeshsize. However,themeasuresforsurfacean shouldreallybenormalizedtounitsolfength,so canbecomparedtorequestedmeshsize.Thesimple normalizationwouldbetoassumeasquareocr ubic andcompute characteristic a lengthbased otnhat:
reaand
Error!Referencesourcenotfound. showsthesize measurescomputedby(2)and(4)forvarioussurfac esina realpart. Thehydraulicdiameteriseithermuchs mallerthan oraboutthesameasthesquarerootofthearea. Forsome surfaces,thedifferenceisignificant(morethan afactorof five),andtendstobelargerforlargersurfaces. The hydraulicdiameter-basedsizemeasuretendstogive amore accurate representationocharacteristic f size.
6 4 2 0 0.0001
0.001
0.01
0.1
Figure 1: Sizemeasurescomputedbyvarious methods for various entities (top); ratio of sqrt(A)/Dhversusarea(bottom).
Thenotionofafeaturewhichis“small”onlymakes sense whenmeasuredrelativetosomethingelse. Whenapp liedto themeshgenerationprocess,thatsomethingelsesh ouldbe themeshsize,sincethatisthequantityaffected bydetail size. Thedeterminationofwhat’ssmallcomparedt othe localmeshsizeS issomewhatarbitrary;forthepurposeof m thisresearch,wechoose1/3localmeshsize,assum ingthe smallestelementinmesh a ins osmallerthan1/3t heaverage meshsize. Usingthelocalmeshsizeasarelativemeasureof whatis smallhastwoprimaryadvantages. First,itallows thesize measuretovaryacrossthemodel,ratherthanbeing constant forallbodies. Thisim s orenatural,sinceifit t m s oreclosely thewaymeshsizesvaryacrossatypicalmesh.Als o, assumingthedetailremovalprocessisreversiblea nd repeatable,definingdetailsizeusingthelocalme shsize, featurescanberemovedorrestoredsimplybychang ingthe localmeshsize.Thisprovidesamoredirectcoupl ing betweenthegeometricfeaturesizeandthelocalme shsize. Thatis,itprovidesmoregeometricdetailinareas wherethe meshismall s enoughtoresolve thatdetail, andvi ca versa.
1
Area
Figure 2:Apartresolvedtoacoarsemeshsize,0.5, includingappropriateamountofgeometricdetail.
Figure 3: Apartresolvedtoacoarsemeshsize, 0.15, includingappropriateamountofgeometric detail.
Figure 5:Simplecasesofdirectremovalofdetail curve(top),andsurface(bottom).
Figure 4: Apartresolvedtoacoarsemeshsize, 0.04, includingappropriateamountofgeometric detail.
4 DIRECTREMOVAL Inthedirectremovalmethodofdetailsuppression, entitiesareremovedusingvirtualtopology“compos operations [16]. Thesimplestcasesofdirectremovalare shownin Figure 5;here,eachdetailentityhasonlyoneother entitywithwhichtocomposite,andthesmalldetai representedbythedetailentityirsemovedbythe Inmanycases,however,thereismorethanoneadja entitywhichcouldbecompositedwiththedetailen Someofthesecombinationsyieldtopologicallyinco models,whileothersaretopologicallyvalidbutdo removethesmalldetailrepresentedbythedetaile Severalexamplesotfhesesituationsareshownin Tochoosethebestpairofentitiestocompositeto introducethenotionotopological f andgeometricc compatibility.
detail ite”
l operation. cent tity. rrect not ntity. Figure 6. gether,we omposite-
Figure 6:Examplesofcompositeoperationswhich producetopologicallyinvalidresults(top)orwhic donotremovethesmalldetail(bottom).
h
TwoentitiesN and are tobteopologically compositei N j said compatibleitfhecompositeoperation leavesthetopologyin avalidconfiguration,i.e.whereentitiesareboun dedby lower-ordertopology(exceptinthecaseoperiodi f ecntities). LetN of dhave set a ofadjacententitiesN= {N j: idimension =j1,n}ofsamedimension,whereentitiesareadja centif d-1 theyshareoneormoreentitiesn ={N i ∩ Nj}(d-1) of k dimensiond-1. ThenN and are if i N j composite-compatible d-1 d-1 andonlyithe f parentsonf =={N N n is/are k i, Proof: j}. k removedaps artofthecompositeoperation[16]. Ho wever,if d-1 parent(nkd-1)={N i,N j, Nk},removingn leavesN k k unbounded,i.e.producestopologicallyill-defined entities,as in Figure 6(left). Inmanycasesthereareseveralchoices ofneighboring entitieswhicharetopologicallycompatiblewithth edetail entityN Intihis . situation,N istjheentitywhichicslosestin geometrytoN i,suchthattheircompositewillbeas
compatibility. Forexample,compositingsurfacesA eliminatecurvec,whilepossible,isclearlynota choiceacsompositingsurfacesAandThe B. bestc compositinghigher-dimensionentitiescanbequanti measuringgeometriccompatibility,andchoosingent whichhavecompatibilityclosesttounity. Thealg usedtpoerform directcurve removalisshownin
continuousaspossible.Wedenotethisasgeometric composite-compatibility. Specifically,weindicate geometric composite-compatibilityC as closetocontinuoustwo g how entitiesare: C
g
i•tj)(nk)
abs(t =
i•n j)(nk)
abs(n =
(curves)
(6)
(surfaces) (7)
wheretangentsonormals r aremeasuredlocalto Cclose tuonity are preferred. g
theentity n
Athirdsituationwhichiqs uitecommoninpractice thereisnoentityofequaldimensionwhichistopo composite-compatiblewiththedetailentityN showsanexamplewherethedetailentity(curvea) haveaneighboringentitywhichistopologicallyco compatible. Insituationslikethis,thedetailen removedsimplybycombiningitwithaneighboringe rather,entitiesofhigherdimensionmustbecompos Thishastheneteffectofreducingthevalenceof entitiesnk,eventuallyyieldingacomposite-compat of entitiesN and i N j.
k.
iswhere logically Figure 7 i. doesnot mpositetitycannotbe ntity; ited. oneotfhe iblepair
Forexample,in Figure 7,thedetailcurve,curvea,is boundedbytwoverticesovf alence4. Curveacann otbe directly combinedwithany othe f curvesadjacentt overtex 1, sincethatwouldleavetheothercurvesconnectedt overtex1 withoutavertexononeend;thesameproblemexist swith vertex However, 2. vertex1canbereducedtovale nce2by eliminatingsomeoftheincidentcurves;thisisdo neby compositingsurfacepairsboundedbyincidentcurve For s. example,curvebisremovedbycompositingsurfaces Aand B. Curvedisremoved bycompositingsurfacesCandD. Afterthesecompositeoperations,vertex1hasvale nce2and , curve can a bceompositedwithcurve c.
b A
B c
1 2 a
–SharesanedgewithS1 –DoesnotcontainC –IscompatiblewithS1 –S(S1US2)>eps
3.MarkvertexVthatsharesS1,S2,C 4.PutS2onlistA 5.Repeat2-4untilValence(V) size(A) +1=2on ro newsurfacesinA 6.IfValence(V)>2,repeat2-4withothersurface incidentonC,buildinglistB 7.IfValence(V)>2,clearlistsA&B,repeat2-5 othervertexofC 8.IfValence(V) size(A) size(B) +2=2,composite surfacesinlistAtogether,sameforB 9.CompositeCwithcurveadjacenttoCandV
with
twosetsof atibility set(s)of nganyof lldoesnot entities, derentity
Thenotionogf eometriccompatibilityhasbeendesc ribedin otherworks,e.g.Ref. [9].Ifgeometriccompatibilityis enforcedonanequalsettingatsopologicalcompati bility,as intheseworks,theresultcaneasilybethepersis tenceof smalldetails. Forexample,thecylindricalprotru sionsand thesmallledgeonthepartin Error!Referencesourcenot found.aregeometricallyincompatiblewiththeirneighbor s, andbythatmeasurewouldnotberemovedusingdire ct removal. However,thesurfacearetopologicallyco mpositecompatiblewiththeirneighbors. Moreover,weasse rtthatif weareremovingasmalldetail,geometriccompatibi lityias mootpointandshould bgeivenlower priority.
A+B 2 c+a
C+D
Intheaboveexample,somechoicesofhigher-dimens composites are better than others, due to geometric
;
4.1 GeometricversusTopological Compatibility
C+D D
Figure 7:Exampleofadetailentity(curvea)having nocomposite-compatibleneighborsofthesame dimension(topleft). CompositesurfacesAandB andsurfacesCandD(topright). Valenceofverte 1 reduced so that curves a and c can be composited(bottom).
1.ChoosesurfaceS1withsmallestDhincidentonC addtolistA 2.FindanothersurfaceS2that:
Notethatthealgorithmin Figure 8allows compositedentities,eachsethavinggeometriccomp withothersurfacesintheset. Notealsothatthe surfacestobecompositedareidentifiedbeforedoi theactualcompositing;thisway,itfheresultsti yieldapairoftopologicallycomposite-compatible theoperationiasbandonedforthecurrentlower-or nk,andthealgorithmmovesontoanotherlower-order bounding entity nk.
d C
● For smallcurveC:
Figure 8:Algorithmusedtoperformdirectdetail removal,includingidentificationoftopologically andgeometricallycomposite-compatibleentities.
A+B
12 a
c
andCto sgooda hoicefor fiedby itypairs orithm Figure 8.
x
ion
Loweringthepriorityofgeometriccompatibilityha advantageousside-effectofremovingsomeofthefr parametersdescribedin [9]. Thisisveryimportanttothe automationofdetailremoval,andneedstobeinves further.
san ee tigated
4.2 Examples Figure 9and Figure 10show examplesof directremoval,on caontrivedpart to illustrate variouscases(Figur e 9)and a few realpartstoshow robustnessandapplicability toreal problems(Figure 10).
Figure 9:Directdetailremovalonacontrivedpart; detailvertex,twodetailcurves,andadetailsurf ace. Alldetailentitiesareremovedautomaticallyto produceabrickwith8,12and6 vertices,edges, andfaces,respectively.
Figure 10:Directdetailremovalfor2parts,showing originalpartandpartafterdetailremoval.
5 BLENDSUPPRESSION Blendsaredesignfeaturesincludedinamodelfor aesthetic purposes(e.g.roundingoffsharpedges)otroavoi dstress singularities(fillets). Astheirnameimplies,bl endsmake smoothtransitionsbetweentwoomore r surfaces. B lendsare oneotfhemostcommondesignmodeldetailsremoved for analysis. Removingblendsrequirestheautomaticd etection ofblends;onceblendsarecorrectlyidentified,th eirremoval isstraightforward. Blendremovalrequiresbothpa rtitionand compositeoperations.Blenddetectionandremoval is discussedbriefly here;amorecompletedescriptionwillbe givenifuture na paper.
5.1 Blendtopology&nomenclature Althoughblendsareusedinvirtuallyeverydesign theyareoftenreferredtousingdifferentnomencla Figure 11showsanexampleofabrickwiththreesurface pairsblendedtoformrounds;nomenclatureforvari oftheblendsareindicatedinthefigure,andbrie below.
system, ture. ousparts fly discussed
2 1
C
Throughhole
A A 1
B
2 C
C A
Sidecap
Figure 11:Blendtopology,includingblendsheets (A),blendblend(B),blendedfaces(C);andspring curves(1)andcrossingcurves(2). Onlyselected curvesarelabeled. •
Blendedface: oneotfwosurfacesbetweenwhich the blendforms samoothtransition.
•
Blendsheet(BS): thesurfacewhichtransitions between pair a of blended faces.
•
Blendblend(BB): ablendsheetwhichtransitions betweenthree omore r blendsheets.
•
Springcurve(SC): atopologicalcurveseparating ablendsheetfromablendedface. Notethatthe transitionbetweenthesurfacesaastpringcurvei C1-continuous.
•
Figure 12:Afewexamplesofmorecomplicated blends.
5.2.1 Geometriccriteria Consideroneotfheblendsshownin Figure 11. Findinga blendfirstrequiresidentifyingspringandcrossin gcurves. Figure 13showstwoviewsofablendsheet,viewingthe blend from the top(top) andfrom the side (bottom) .
s
n1+ •n1- ≥(1- ε)
Crossingcurve(XC): acurvewhichcutsacrossa blendsheetandwhichjoinstwospringcurves,one oneachsideoftheblendsheet.Notethatthe transitionbetweensurfacesacatrossingcurvemay or may notbe C1-continuous.
+
etween curve asmall
(GC1)
-
Wheren1 andn1 denotethenormalstosurfacesoneither side oP1 f from the blendsheet.
5.2 Automaticblenddetection Atitssimplest,ablendischaracterizedbyablen dsheet whichtransitionsbetweentwoblendedsurfaces,suc hthatthe surfacesareC1-continuouswheretheymeet. Ifall blends werelikethis(andnoothernon-blendsurfaceswer e), detectingthemwouldbeasimplematteroflooking for curvedsurfaceswithC1-continuoustransitionstoa djacent surfaces,withcrossingcurvesatheend. However blends , ofteninteractwitheachother,andwithotherfeat ures;for example, Figure 12showsafewexamplesofmore complicatedblends. Hence, saimpletopologicalch eckinsot sufficientfor detecting allbutthe simplestblend s. Thissectiondescribesaheuristicmethodforfindi sheetsandblendblends,aswellastheboundingsp crossingcurves.Oncethesesurfacesandcurvesar identified,decomposingthemtoremovethedetails simplematter. Theblendfindingheuristicconsist geometricandtopologicalchecks,alongwithnon-lo topologicalchecks.
SpringcurvesrepresentC1-continuoustransitionsb twosurfaces;therefore,acurvecannotbeaspring unlessthenormalsonadjacentsurfacesarewithin tolerance oeach f other:
ngblend ringand e isa os lfocal cal
BlendsheetsformaC1-continuoustransitionbetwee surfaces;thisismosteasilydoneusingasurface constantradiusofcurvatureintheplaneorthogona blendsheetandpassingthroughthecrossingcurve Assumingc0isaconstant-radiuscurve,theradius curvaturecanbceomputedfromitsendpointsande surface normalsusing basic trigonometry: α/2) = c/cos(
α /2)
cos (α /2)= 1/2(1-cos(
β))
= dtan( r 2
ntwo with tlothe c0. of ndpoint (8)
cos(β)= n1••n2 C= 1/2 | | =r
P 1P 2 | P1P2|/(8(1-n1•n2))1/2 = computed (8) radius
Howdoesthatradiuscomparewiththeactualradius curvatureotfhesurfaceinthatlocation? Thisca byprojectingtheprincipalcurvaturesontothevec the twocrossing curve end points: c1= txn1 c2= -txn2
1 2
κ1|c1= c1 (• κa + κb |)
P1
of nbefound torjoining
κ2|c2 = -c2 (• κa+ κb |) -1
κ1,κ2))=
R= (avg(
markedascrossingcurves;theremainingcurvessat GC1aremarkedaspring s curves.
P2
measured (9) radius
where κa and κb are the principalcurvatureson the surface. Weassertthatiafcurveisacrossingcurve,its radiusocurvature f willbeclosetothemeasuredr endpoints:
computed adiusathe t
r/R ≥(1- ε)
(GC2)
Inordertopreventdegeneratecases,i.e.when curvatureiisnfinite,andcaseswheresurfacecurv widelyalongacrossingcurve,weaddthefollowing criteria:
radiusof aturevaries two
abs((κ1- κ2)/avg(κ1, κ2))< ε
(GC3)
abs(κ1)> ε
(GC4a)
abs(κ2)> ε
(GC4b)
Theseconstraintsenforcecertain a amountofunifo smoothnessacrossblendsheets,butdoesnotrequir radiusocfurvaturetobeconstant. Thisflexibili for robustness.
rmityand ethe tyicsritical
n1
GeometriccriteriaGC1-4identifycurveswhich canbe springorcrossingcurves.However, thesecriteriaare necessary,butnotsufficient,fordeterminingthes ecurves. Statedanotherway,therecanbeC1-continuouscurv esor other curvessatisfyingGC1-4imodel na whichdon otbound blendsheets.Obviously,othercriteriaareneeded to correctly identify blendsheetsandblends.
5.2.2 Localtopologicalcriteria A blendsheetis distinguishedfrom saurface bound ed bayn arbitrary collectionospring f and crossing curves by the specific arrangementof the curvesonthe surface. We observe thatblendsheetstypically contain: • 2chains of spring curves,where eachchainias seriesof spring curvesarrangedend-to-end with tangentson either side ointerior f vertices approximately parallel, • 2chains of crossing curves • Noother curves Thisleadstothefollowinglocaltopologycriteria forblend sheets: #chains of SC’s== 2
P1
#chains of XC’s== 2
t1 c1 n2
c2 P2
P1
t2
β
n1
r c1
r
d1
n2 d2
P2
on nsitionis atcaurve surfaces. irends, romthe heetsare
Blendblendssometimeshave“offsets”,wherethecr curvesdonotjoindirectlybutratherareseparate curves. These offset curves are still C1-continuou transitions,buttheytransitionbetweentheblend some othe f blendedfacesassociatedwiththe blend
ossing dbyoffset s blendand sheets.
#SCchains== == 0|| #XCchains
ocal
(LTC2a) (LTC2b)
ForeachSCchain,thesurfaceoppositetheblend blendissharedbytheblendsheetscorresponding tothe XC chains oneither side othe f SCchain (LTC2c)
Figure 13:Identificationofcrossingcurves(top) andspringcurves(bottom)usinggeometriccriteria GC1-4. Notethatforcrossingcurves(GC2-GC4),thereare criteriainvolvingtheend-pointnormalsam s easure blendedsurfaces. Thismeansthatacrossingcurve satisfythespringcurvecriteria,althoughthisis thiswillbeimportantforevaluatingnon-localtop criteria.Inthegeometriccriteriastageofblend curvessatisfyingthecrossingcurveconstraintsGC
(LTC1b)
Blendblendsaredefinedassurfaceswhichtransiti between3ormoreblendsheets. Since smooth a tra typical,theblendblendwillmeeteachblendsheet representing C1-continuous a transitionbetweenthe However,blendsheetsmeetblendblendsonlyathe t sincethespringcurvesseparatetheblendsheetsf blendedfaces. Therefore,blendblendsandblends separatedbcyrossing curves.
X # Cchains>= 3 α
(LTC1a)
Basedontheseobservations,weusethefollowingl topology criteriafor blendblends:
c2 c0
isfying
no dforthe alsocan notrequired; ological finding, 2-4are
Topological criteria LTC1 and LTC2, although quite restrictiveintheallowedarrangementsofcurveso sheetsandblends,stillarenotsufficientforrob identifying blendsinarbitrary models.
nblend ustly
5.2.3 Non-localtopologicalcriteria
5.2.4 Blenddecomposition
Theproblemwiththeabovealgorithm,asisttands now,is thatcertaincurvescanbebothspringorcrossing curves, sinceasubsetotfhecrossingcurvescanalsosati sfyspring curveconstraints. Thus,asubsetocf urvescanbechanged fromcrossingcurvestospringcurves. ByLTC1and LTC2, thiscurveclassificationaffectstheclassificatio nofadjacent surfaces. Therefore,theidentificationofbothcu rvesand surfacesassociatedwithblendsicsoupled,andmus be t done concurrently. Forthisreason,weobservethatthe reinonas localcomponent to the blend detection process.
Describing the process of decomposing blendsheetsand blendsandcompositing the piecesinto adjacentsur facesis omittedfor brevity. However,since SC’sand XC’s are already identifiedfor eachblend, thisprocessis straightforward,andfollowsthe processnormally d one interactively.
Solvingthenon-localpartofthisprocesscanbed eitheraheuristicoranoptimizationtechnique. T approachisoutsidethescopeofthispaper,andwi describedidifferent na paper.
oneusing helatter llbe
Themostchallengingpartoftheblenddetectional gorithmis tochoosenon-localtopologicalconstraintsthatfo rcethe algorithmtoconvergetoareasonablesolution. We choose theseconstraintsbyobservingthefollowing:inty pical models,thesurfacesadjacenttospringandcrossin gcurves canonlybeclassifiedinafewpossibleways.The se arrangementsareshownin Figure 14forcrossingcurves (top),indicatingnon-localconstraintNTC1,andsp ring curves(bottom),indicating non-localconstraintNT C2. AfterclassifyingcurvesandsurfacesusingGC1-2a nd LTC1-2,eachcrossingandspringcurveischeckeda gainst thearrangementsin Figure 14. Acertainnumberocf urves willviolateNTC1-2;howdowedecidewhichcurves to change?CurveswhichareidentifiedasSC’sandwh ich violateNTC1areresettonoclassification,while XC’swhich violate NTC2andwhichsatisfyGC1arere-classifie daSC’s s (otherwisetheyareresettonoclassification).A fterreclassifyingcurves,thesurfacesboundingthesecur vesarereevaluatedagainstLTC1-2,andtheprocessirsepeat ed. Since theseiterationsmonotonicallydecreasetheclassif icationof curvesandsurfacestowardsnoclassification,this process willconvergeeventually(althoughiis tpossiblet oconverge on solution a consisting ono fblends). BS
BS
*
BS
BS
*
BS *
BB
BS
BS
BS *
BS
BS
BB
XC
BS
BS
XC
XC
Figure 14: Allowed arrangements of surfaces adjacenttocrossingcurves(top)andspringcurves (bottom).
The algorithm describedabove isummarized s in • • • •
• •
*
Figure 15.
For eachcurve,evaluate GC1-2, classify accordingly For eachsurface adjacenttospring/crossing curve,evaluate LTC1-2,classifyaccordingly For eachspring/crossing curve,evaluate NTC1mark invalidcurve(s) For eachinvalidcrossing curve: o IfGC1isatisfied, s mark aspring s curve,else mark ano sclassification o Mark adjacentsurfaces For eachinvalidspring curve,mark ano s classification& mark adjacentsurfaces Re-classify allmarkedsurfaces
2,
Figure 15: Overall algorithm for finding and suppressingblends.
5.4 Examples The algorithm in Figure 15has beenimplementedinthe CUBITmeshgenerationtoolkitfrom Sandia National Laboratories[17]. Implementing these algorithmsin production-quality meshing toolkitallowstesting d “real” analysisproblems. Severalexamplesof blen detectionare shownin Figure 16. These blendswere detectedfully automatically.
a irectly on d
5.5 TheFinePrint Therearemanytopologicalarrangementsoblends f f practicewhicharen’thandledbythealgorithmabov naturalquestiontheniwhether s thisalgorithm wil moregeneralgeometrycontainingthesetypesoble f discussbelowseveralofthemorecommoncasesthat andhowwepostulatetheycouldbehandledwithour algorithm.
*
XC
5.3 Algorithm
One arrangementthatisquite commonisurfaces s co twoexternalloops,where the curvesoneachothe f form saingle chainospring f curves. Thisisfoun examplewhen fillet a isaddedathe t base ocafyl boss. Currently,since blend a sheetmustcontain andcrossing curves,our algorithm willnotdetect of blends. Thiscanbceorrected inseveralways. solution ito ssplit periodic surfaces;thiswillu these typesof blendsheetsintotwoblendsheets, theneasily detectedboyur algorithm. The other w handle two-loop surfacesistojointhe twoloopsw
oundin The e. work l with nds. We appear,
ntaining loops fdor indrical both spring these types The easiest sually split whichare ay to ithtwo
virtualcurves,eachominimal f length. These curv passthe crossing curve test,andagainthe blends be detectedeasily.
eswill heetswill
Figure 17:Examplesofblendsnotyethandledby thisalgorithm; roll-onblends(left),andface-ed blends(right).
ge
Somemorecomplexblendconfigurationscontainblen d sheetswhichintersecttheboundaryofablendedsu rface (sometimesreferredtoasa“roll-on”blend),orwh erea surfaceisblendedwithanedgeinthemodel;these configurationsareshownin Figure 17.Althoughour algorithmhasnotyetbeenimplementedtohandleth ese configurations,ourgeneralapproachiasmenableto handling these. Roll-onblendscanbheandledbdyetecting theroll-on point(automaticallyorwithuserguidance)andadd ingthe roll-onedgestotheadjacentspringcurvechain. Face-edge blendsare probably difficulttohandle withoutadd itionaluser input.
6 BRIDGEFEATUREREMOVAL Figure 16: Examples of blend detection and removal. Blendsheetsareshadeddark,andblend blendsareshadedmedium Anothercommonarrangementistohaveblendsheets which arecutwiththrough-holesoblind r holes;anexamp leothis f arrangementisshownin Figure 12.Simplecases,for examplewhentheholeintersectsonechainofsprin gor crossingcurvesentirely,canbehandledifthecur ves boundingtheholearedetectedandconsideredpart ofthe adjacentchain. Preliminaryworkhasbeendoneto detect thesesituations;however,thisworkhasnotprogre ssed enoughtoreportatthistime.Anothersimplecase, wherethe holeformsitsownloopontheinterioroftheblen dsheetor blend,couldbheandledbiygnoringthesecurvesin theblend determination step (althoughtheywould haveto be accountedfor during the partitioning step).
Thethirdclassofeatures f tobhe andledbyautoma ticfeature suppressionarereferredtoasbridgefeatures.Br idge featuresarecharacterizedbyamaterialbridgewit hcrosssection smallenoughtosatisfythedefinitionof“small”. Thisbridgecanbeapositivefeature,e.g.athinwalled section, or naegative feature,e.g. tahrough-hole . Algorithmsfordetectingandsuppressingthesetype detailsareoutsidethescopeofthispaper;theyw describedinanother paper.
7
sof illbe
DISCUSSION,CONCLUSIONS
Themeshgenerationprocesscurrentlydominatesthe analysistime;mostofthistimeisspentinintera manipulationsothe f model. Thisitshetimewhich reducedtoeliminate the meshgeneration bottleneck
overall ctive mustbe .
Akeypartofreducingthemeshingbottleneckisus inga geometry-centricprocess,wheremeshingisfocused onthe geometricmodel. Inthisway,muchofthedatarel atedto meshingcanbeassignedtogeometry,andcanpersis across t multiplemeshingoperations. Thisapproachisgain ingin popularity,indicatedbythenumberofdifferentin tegrated packagesformeshing/analysis/post-processor,e.g. SDRC Ideas [17], Ansys [19],and Pro/Engineer [6]suitesof tools. Thenaturaloutcomeotfhisapproachitsostartth processfromthedesignmodelinsideaCADsystem. Assumingthestartingmodeliswell-defined,accord
emeshing ingtothe
definitioninSection3,thedetailsinthemodelw hichmust beremovedforagivenanalysisarenotgeometrypr oblems, perse,butaredetailsnotrelevanttothatanalys Whether is. a detailisimportanttoananalysiswillchangebetw een analysistypesandevenwithinananalysis.Thus, detail reductionimeshing-time as decision,ratherthana partofthe CADdesignprocess.Furthermore,sincefinedetail s complicatethemeshingprocess,itworksbestifth esedetails areremovedfromthemodelbefore thestartofmeshing. Finally,becauseresolutionrequirementsoftenvary overthe courseofagivenanalysis,detailreductionmustb e reversible,sothatdetailscanbreecoveredithe f ayre relevant after all. Thealgorithmsdescribedinthispapermeetthereq describedabove,largely through their use ovirtu f Thealgorithmsalsoarerelativelyefficient,since mostlyonlocalcriteriaandminimizetheamountof searchingrequiredtodetectsmallfeatures. Thea arerobust,asdemonstratedbytheexamplesinSect and5.4. Finally,sincethesealgorithmsdonotde informationparticulartoany one solidmodelingsy are portable acrosssolidmodelers. Thereareseveralareaswherethisworkmustbeext completedinordertoprovide fully a capableautom suppressioncapability. Removalofsmallvolumede notyetbeenimplemented,althoughdetectionothe f worksthesameasurfaces. Inblendremoval,more topologylikeroll-onblendsandface-edgeblendsh behandled;theseconfigurationsarenotobservedo realparts,sothisissueisnotascritical. Fina removalhasnotbeenaddressed;thisias nimportan anydetailsuppressiontechnique,becauseotfheab ofthrough-holesintypicaldesignmodels. Thisca willbe implementedsoon.
uirements altopology. theyrely global lgorithms ions4.2 pendon stem,they endedor aticdetail tailshas sedetails complex aveyetto ftenin lly,bridge part t of undance pability
8r .S.
[3]“Scientific Discovery throughAdvancedComputin Office oScience, f U.S.Departmentof Energy,March 24, 2000.
g”,
al
[5]StevenJOwen, . “A Survey oUnstructured f Mesh GenerationTechnology”, Proc. 7thInternational Meshing Roundtable, SAND98-2250,Sandia National Laboratories,Oct.1998. [6]Parametric Technology Corporation, http://www.ptc.com.
isiting el 7th
[9]A.Sheffer,“ModelSimplificationfor Meshing U Face Clustering”,toappear,Computer AidedDesign.
sing
[10]CG. .Armstrong,S.J.Bridgett,R.I.Donaghy, McCune,R.M. McKeag andD.J. Robinson Techniques , for Interactive andAutomatic Idealisation oCAD, f proc.NumericalGridGeneration inComputational Fi Simulations,Ed. M.Cross.,B.K. Soni,J. F.Thomp J.Hauser, P.R.Eiseman,Proceedingsof the 6th InternationalConference,heldathe t University of Greenwich,pp.643-662,July 1998.
R.W.
[11]Mobley,Anton VMichael ., P.Carroll,and Scot Canann,"AnObjectOrientedApproachtoGeometry Defeaturing for Finite ElementMeshing",7th InternationalMeshing Roundtable,Sandia National Labs,pp.547-563,October 1998.
A. t
in eld son,
[12]TedBlacker,Alla Sheffer,JanClements,Miche l Bercovier,“Using VirtualTopology toSimplify the MeshGeneration Process”,TrendsinUnstructured MeshGeneration,ASMEAppliedMechanicsDivision, VolumeAMD-Vol 220, 1997. [13]BarequetGillandSubodbKumar,"Repairing CAD Models", Proceeding IEEE Visualization, Phoenix,AZ IEEE, pp.363-370,October 1997.
[15]RB. .Bird,W.E. Stewart,E.N. Lightfoot,“T Phenomena”,Wiley & Sons(1960).
[1]GordonEMoore, . “Crammingmore components onto integratedcircuits”,Electronics, Volume 38,Numbe (1965).
[4]Timothy JTautges, . “TheGeneration oHexahedr f Meshesfor Assembly Geometry:Survey and Progress”, Int.J.Numer.Meth.Engng.,50: 2617-2642(2001).
[8]M.S. Shephard,M. W.Beall,R. M.O’Bara,“Rev the Elimination othe f Adverse Effectsof SmallMod FeaturesinAutomatically GeneratedMeshes”, Proc. InternationalMeshing Roundtable,Sandia National Laboratories,SAND98-2250,pp. 347-364(1998).
,
[14]ACISHealing Husk,SpatialTechnology Inc., http://www.spatial.com.
8 REFERENCES
[2]“AcceleratedStrategic Computing Initiative”,U Departmentof Energy, http://www.asci.doe.gov/index.htm.
[7]SolidWorks,SolidWorksCorp., http://www.solidworks.com.
[16]JasonAKraftcheck, . “VirtualGeometry:A Mech for Modification oCAD f ModelTopology for Improved Meshability”,Master’sThesis,University oWiscon f Madison,2000. [17]TD. .Blacker etal.,'CUBITmeshgeneration environment,Vol. 1:User'smanual',SAND94-1100, Sandia NationalLaboratories,Albuquerque,New Mexico,May 1994. [18]SDRCIdeasMaster Series,StructuralDynamics ResearchCorp., http://www.sdrc.com. [19]ANSYS,ANSYS,Inc., http://www.ansys.com.
ransport anism sin-
