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第1页共72页AformaltheoryforestimatingdefeaturinginducedengineeringanalysiserrorsSankaraHariGopalakrishnan,KrishnanSureshDepartmentofMechanicalEngineering,UniversityofWisconsin,Madison,WI53706,UnitedStatesReceived13January2006accepted30September2006AbstractDefeaturingisapopularCAD/CAEsimplificationtechniquethatsuppressessmallorirrelevantfeatureswithinaCADmodeltospeedupdownstreamprocessessuchasfiniteelementanalysis.Unfortunately,defeaturinginevitablyleadstoanalysiserrorsthatarenoteasilyquantifiablewithinthecurrenttheoreticalframework.Inthispaper,weprovidearigoroustheoryforswiftlycomputingsuchdefeaturinginducedengineeringanalysiserrors.Inparticular,wefocusonproblemswherethefeaturesbeingsuppressedarecutoutsofarbitraryshapeandsizewithinthebody.Theproposedtheoryexploitstheadjointformulationofboundaryvalueproblemstoarriveatstrictboundsondefeaturinginducedanalysiserrors.Thetheoryisillustratedthroughnumericalexamples.KeywordsDefeaturingEngineeringanalysisErrorestimationCAD/CAE1.IntroductionMechanicalartifactstypicallycontainnumerousgeometricfeatures.However,notallfeaturesarecriticalduringengineeringanalysis.Irrelevantfeaturesareoftensuppressedordefeatured,priortoanalysis,leadingtoincreasedautomationandcomputationalspeedup.Forexample,considerabrakerotorillustratedinFig.1a.Therotorcontainsover50distinctfeatures,butnotallofthesearerelevantduring,say,athermalanalysis.AdefeaturedbrakerotorisillustratedinFig.1b.WhilethefiniteelementanalysisofthefullfeaturedmodelinFig.1arequiredover150,000degreesoffreedom,thedefeaturedmodelinFig.1brequired25,000DOF,leadingtoasignificantcomputationalspeedup.第2页共72页Fig.1.aAbrakerotorandbitsdefeaturedversion.Besidesanimprovementinspeed,thereisusuallyanincreasedlevelofautomationinthatitiseasiertoautomatefiniteelementmeshgenerationofadefeaturedcomponent1,2.Memoryrequirementsalsodecrease,whileconditionnumberofthediscretizedsystemimprovesthelatterplaysanimportantroleiniterativelinearsystemsolvers3.Defeaturing,however,invariablyresultsinanunknownperturbationoftheunderlyingfield.Theperturbationmaybesmallandlocalizedorlargeandspreadout,dependingonvariousfactors.Forexample,inathermalproblem,supposeonedeletesafeaturetheperturbationislocalizedprovided1thenetheatfluxontheboundaryofthefeatureiszero,and2nonewheatsourcesarecreatedwhenthefeatureissuppressedsee4forexceptionstotheserules.Physicalfeaturesthatexhibitthispropertyarecalledselfequilibrating5.Similarlyresultsexistforstructuralproblems.Fromadefeaturingperspective,suchselfequilibratingfeaturesarenotofconcernifthefeaturesarefarfromtheregionofinterest.However,onemustbecautiousifthefeaturesareclosetotheregionsofinterest.Ontheotherhand,nonselfequilibratingfeaturesareofevenhigherconcern.Theirsuppressioncantheoreticallybefelteverywherewithinthesystem,andcanthusposeamajorchallengeduringanalysis.Currently,therearenosystematicproceduresforestimatingthepotentialimpactofdefeaturingineitheroftheabovetwocases.Onemustrelyonengineeringjudgmentandexperience.Inthispaper,wedevelopatheorytoestimatetheimpactofdefeaturingonengineeringanalysisinanautomatedfashion.Inparticular,wefocusonproblemswherethefeaturesbeingsuppressedarecutoutsofarbitraryshapeandsizewithinthebody.Twomathematicalconcepts,namelyadjointformulationandmonotonicityanalysis,arecombinedintoaunifyingtheorytoaddressbothselfequilibratingandnonselfequilibratingfeatures.Numericalexamplesinvolving2ndorderscalarpartialdifferentialequationsareprovidedtosubstantiatethetheory.Theremainderofthepaperisorganizedasfollows.InSection2,wesummarizepriorworkondefeaturing.InSection3,weaddressdefeaturinginducedanalysiserrors,anddiscusstheproposedmethodology.ResultsfromnumericalexperimentsareprovidedinSection4.AbyproductoftheproposedworkonrapiddesignexplorationisdiscussedinSection5.Finally,conclusionsandopenissuesarediscussedinSection6.第3页共72页2.PriorworkThedefeaturingprocesscanbecategorizedintothreephasesIdentificationwhatfeaturesshouldonesuppressSuppressionhowdoesonesuppressthefeatureinanautomatedandgeometricallyconsistentmannerAnalysiswhatistheconsequenceofthesuppressionThefirstphasehasreceivedextensiveattentionintheliterature.Forexample,thesizeandrelativelocationofafeatureisoftenusedasametricinidentification2,6.Inaddition,physicallymeaningfulmechanicalcriterion/heuristicshavealsobeenproposedforidentifyingsuchfeatures1,7.Toautomatethegeometricprocessofdefeaturing,theauthorsin8developasetofgeometricrules,whiletheauthorsin9usefaceclusteringstrategyandtheauthorsin10useplanesplittingtechniques.Indeed,automatedgeometricdefeaturinghasmaturedtoapointwherecommercialdefeaturing/healingpackagesarenowavailable11,12.Butnotethatthesecommercialpackagesprovideapurelygeometricsolutiontotheproblem...theymustbeusedwithcaresincetherearenoguaranteesontheensuinganalysiserrors.Inaddition,opengeometricissuesremainandarebeingaddressed13.Thefocusofthispaperisonthethirdphase,namely,postdefeaturinganalysis,i.e.,todevelopasystematicmethodologythroughwhichdefeaturinginducederrorscanbecomputed.Weshouldmentionheretherelatedworkonreanalysis.Theobjectiveofreanalysisistoswiftlycomputetheresponseofamodifiedsystembyusingprevioussimulations.OneofthekeydevelopmentsinreanalysisisthefamousSherman–MorrisonandWoodburyformula14thatallowstheswiftcomputationoftheinverseofaperturbedstiffnessmatrixothervariationsofthisbasedonKrylovsubspacetechniqueshavebeenproposed15–17.Suchreanalysistechniquesareparticularlyeffectivewhentheobjectiveistoanalyzetwodesignsthatsharesimilarmeshstructure,andstiffnessmatrices.Unfortunately,theprocessof几何分析canresultinadramaticchangeinthemeshstructureandstiffnessmatrices,makingreanalysistechniqueslessrelevant.Arelatedproblemthatisnotaddressedinthispaperisthatoflocal–globalanalysis13,wheretheobjectiveistosolvethelocalfieldaroundthedefeaturedregionaftertheglobaldefeaturedproblemhasbeensolved.Animplicitassumptioninlocal–globalanalysisisthatthefeaturebeingsuppressedisselfequilibrating.3.Proposedmethodology3.1.ProblemstatementWerestrictourattentioninthispapertoengineeringproblemsinvolvingascalarfieldugovernedbyageneric2ndorderpartialdifferentialequationPDE..fauucAlargeclassofengineeringproblems,suchasthermal,fluidandmagnetostaticproblems,maybereducedtotheaboveform.Asanillustrativeexample,considerathermalproblemoverthe2DheatblockassemblyΩillustratedinFig.2.TheassemblyreceivesheatQfromacoilplacedbeneaththeregionidentifiedasΩcoil.AsemiconductordeviceisseatedatΩdevice.ThetworegionsbelongtoΩandhavethesame第4页共72页materialpropertiesastherestofΩ.Intheensuingdiscussion,aquantityofparticularinterestwillbetheweightedtemperatureTdevicewithinΩdeviceseeEq.2below.Aslot,identifiedasΩslotinFig.2,willbesuppressed,anditseffectonTdevicewillbestudied.TheboundaryoftheslotwillbedenotedbyΓslotwhiletherestoftheboundarywillbedenotedbyΓ.TheboundarytemperatureonΓisassumedtobezero.TwopossibleboundaryconditionsonΓslotareconsideredafixedheatsource,i.e.,krT.ˆnq,orbfixedtemperature,i.e.,TTslot.Thetwocaseswillleadtotwodifferentresultsfordefeaturinginducederrorestimation.Fig.2.A2Dheatblockassembly.Formally,letTx,ybetheunknowntemperaturefieldandkthethermalconductivity.Then,thethermalproblemmaybestatedthroughthePoissonequation181.00.slctslctslctcoilcoilTTboronqhkaonTinininQTkBCPDEGiventhefieldTx,y,thequantityofinterestis2,,devicedycTyxHTComputedevicewhereHx,yissomeweightingkernel.Nowconsiderthedefeaturedproblemwheretheslotissuppressedpriortoanalysis,resultinginthesimplifiedgeometryillustratedinFig.3.Fig.3.Adefeatured2Dheatblockassembly.Wenowhaveadifferentboundaryvalueproblem,governingadifferentscalarfieldtx,y
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