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第1页共72页Aformaltheoryforestimatingdefeaturing-inducedengineeringanalysiserrorsSankaraHariGopalakrishnan,KrishnanSureshDepartmentofMechanicalEngineering,UniversityofWisconsin,Madison,WI53706,UnitedStatesReceived13January2006;accepted30September2006AbstractDefeaturingisapopularCAD/CAEsimplificationtechniquethatsuppressessmallorirrelevantfeatureswithinaCADmodeltospeed-updownstreamprocessessuchasfiniteelementanalysis.Unfortunately,defeaturinginevitablyleadstoanalysiserrorsthatarenoteasilyquantifiablewithinthecurrenttheoreticalframework.Inthispaper,weprovidearigoroustheoryforswiftlycomputingsuchdefeaturing-inducedengineeringanalysiserrors.Inparticular,wefocusonproblemswherethefeaturesbeingsuppressedarecutoutsofarbitraryshapeandsizewithinthebody.Theproposedtheoryexploitstheadjointformulationofboundaryvalueproblemstoarriveatstrictboundsondefeaturinginducedanalysiserrors.Thetheoryisillustratedthroughnumericalexamples.Keywords:Defeaturing;Engineeringanalysis;Errorestimation;CAD/CAE1.IntroductionMechanicalartifactstypicallycontainnumerousgeometricfeatures.However,notallfeaturesarecriticalduringengineeringanalysis.Irrelevantfeaturesareoftensuppressedordefeatured,priortoanalysis,leadingtoincreasedautomationandcomputationalspeed-up.Forexample,considerabrakerotorillustratedinFig.1(a).Therotorcontainsover50distinctfeatures,butnotallofthesearerelevantduring,say,athermalanalysis.AdefeaturedbrakerotorisillustratedinFig.1(b).Whilethefiniteelementanalysisofthefull-featuredmodelinFig.1(a)requiredover150,000degreesoffreedom,thedefeaturedmodelinFig.1(b)required<25,000DOF,leadingtoasignificantcomputationalspeed-up.第2页共72页Fig.1.(a)Abrakerotorand(b)itsdefeaturedversion.Besidesanimprovementinspeed,thereisusuallyanincreasedlevelofautomationinthatitiseasiertoautomatefiniteelementmeshgenerationofadefeaturedcomponent1,2.Memoryrequirementsalsodecrease,whileconditionnumberofthediscretizedsystemimproves;thelatterplaysanimportantroleiniterativelinearsystemsolvers3.Defeaturing,however,invariablyresultsinanunknownperturbationoftheunderlyingfield.Theperturbationmaybesmallandlocalizedorlargeandspread-out,dependingonvariousfactors.Forexample,inathermalproblem,supposeonedeletesafeature;theperturbationislocalizedprovided:(1)thenetheatfluxontheboundaryofthefeatureiszero,and(2)nonewheatsourcesarecreatedwhenthefeatureissuppressed;see4forexceptionstotheserules.Physicalfeaturesthatexhibitthispropertyarecalledself-equilibrating5.Similarlyresultsexistforstructuralproblems.Fromadefeaturingperspective,suchself-equilibratingfeaturesarenotofconcernifthefeaturesarefarfromtheregionofinterest.However,onemustbecautiousifthefeaturesareclosetotheregionsofinterest.Ontheotherhand,non-self-equilibratingfeaturesareofevenhigherconcern.Theirsuppressioncantheoreticallybefelteverywherewithinthesystem,andcanthusposeamajorchallengeduringanalysis.Currently,therearenosystematicproceduresforestimatingthepotentialimpactofdefeaturingineitheroftheabovetwocases.Onemustrelyonengineeringjudgmentandexperience.Inthispaper,wedevelopatheorytoestimatetheimpactofdefeaturingonengineeringanalysisinanautomatedfashion.Inparticular,wefocusonproblemswherethefeaturesbeingsuppressedarecutoutsofarbitraryshapeandsizewithinthebody.Twomathematicalconcepts,namelyadjointformulationandmonotonicityanalysis,arecombinedintoaunifyingtheorytoaddressbothself-equilibratingandnon-self-equilibratingfeatures.Numericalexamplesinvolving2ndorderscalarpartialdifferentialequationsareprovidedtosubstantiatethetheory.Theremainderofthepaperisorganizedasfollows.InSection2,wesummarizepriorworkondefeaturing.InSection3,weaddressdefeaturinginducedanalysiserrors,anddiscusstheproposedmethodology.ResultsfromnumericalexperimentsareprovidedinSection4.Aby-productoftheproposedworkonrapiddesignexplorationisdiscussedinSection5.Finally,conclusionsandopenissuesarediscussedinSection6.第3页共72页2.PriorworkThedefeaturingprocesscanbecategorizedintothreephases:Identification:whatfeaturesshouldonesuppress?Suppression:howdoesonesuppressthefeatureinanautomatedandgeometricallyconsistentmanner?Analysis:whatistheconsequenceofthesuppression?Thefirstphasehasreceivedextensiveattentionintheliterature.Forexample,thesizeandrelativelocationofafeatureisoftenusedasametricinidentification2,6.Inaddition,physicallymeaningfulmechanicalcriterion/heuristicshavealsobeenproposedforidentifyingsuchfeatures1,7.Toautomatethegeometricprocessofdefeaturing,theauthorsin8developasetofgeometricrules,whiletheauthorsin9usefaceclusteringstrategyandtheauthorsin10useplanesplittingtechniques.Indeed,automatedgeometricdefeaturinghasmaturedtoapointwherecommercialdefeaturing/healingpackagesarenowavailable11,12.Butnotethatthesecommercialpackagesprovideapurelygeometricsolutiontotheproblem.theymustbeusedwithcaresincetherearenoguaranteesontheensuinganalysiserrors.Inaddition,opengeometricissuesremainandarebeingaddressed13.Thefocusofthispaperisonthethirdphase,namely,postdefeaturinganalysis,i.e.,todevelopasystematicmethodologythroughwhichdefeaturing-inducederrorscanbecomputed.Weshouldmentionheretherelatedworkonreanalysis.Theobjectiveofreanalysisistoswiftlycomputetheresponseofamodifiedsystembyusingprevioussimulations.OneofthekeydevelopmentsinreanalysisisthefamousShermanMorrisonandWoodburyformula14thatallowstheswiftcomputationoftheinverseofaperturbedstiffnessmatrix;othervariationsofthisbasedonKrylovsubspacetechniqueshavebeenproposed1517.Suchreanalysistechniquesareparticularlyeffectivewhentheobjectiveistoanalyzetwodesignsthatsharesimilarmeshstructure,andstiffnessmatrices.Unfortunately,theprocessof几何分析canresultinadramaticchangeinthemeshstructureandstiffnessmatrices,makingreanalysistechniqueslessrelevant.Arelatedproblemthatisnotaddressedinthispaperisthatoflocalglobalanalysis13,wheretheobjectiveistosolvethelocalfieldaroundthedefeaturedregionaftertheglobaldefeaturedproblemhasbeensolved.Animplicitassumptioninlocalglobalanalysisisthatthefeaturebeingsuppressedisself-equilibrating.3.Proposedmethodology3.1.ProblemstatementWerestrictourattentioninthispapertoengineeringproblemsinvolvingascalarfieldugovernedbyageneric2ndorderpartialdifferentialequation(PDE):.).(fauucAlargeclassofengineeringproblems,suchasthermal,fluidandmagneto-staticproblems,maybereducedtotheaboveform.Asanillustrativeexample,considerathermalproblemoverthe2-Dheat-blockassemblyillustratedinFig.2.TheassemblyreceivesheatQfromacoilplacedbeneaththeregionidentifiedascoil.Asemiconductordeviceisseatedatdevice.Thetworegionsbelongtoandhavethesame第4页共72页materialpropertiesastherestof.Intheensuingdiscussion,aquantityofparticularinterestwillbetheweightedtemperatureTdevicewithindevice(seeEq.(2)below).Aslot,identifiedasslotinFig.2,willbesuppressed,anditseffectonTdevicewillbestudied.Theboundaryoftheslotwillbedenotedbyslotwhiletherestoftheboundarywillbedenotedby.Theboundarytemperatureonisassumedtobezero.Twopossibleboundaryconditionsonslotareconsidered:(a)fixedheatsource,i.e.,(-krT).n=q,or(b)fixedtemperature,i.e.,T=Tslot.Thetwocaseswillleadtotwodifferentresultsfordefeaturinginducederrorestimation.Fig.2.A2-Dheatblockassembly.Formally,letT(x,y)betheunknowntemperaturefieldandkthethermalconductivity.Then,thethermalproblemmaybestatedthroughthePoissonequation18:)1()().)(00).(slctslctslctcoilcoilTTboronqhkaonTinininQTkBCPDEGiventhefieldT(x,y),thequantityofinterestis:)2(),(),(devicedycTyxHTComputedevicewhereH(x,y)issomeweightingkernel.Nowconsiderthedefeaturedproblemwheretheslotissuppressedpriortoanalysis,resultinginthesimplifiedgeometryillustratedinFig.3.Fig.3.Adefeatured2-Dheatblockassembly.Wenowhaveadifferentboundaryvalueproblem,governingadifferentscalarfieldt(x,y):