第1頁共72頁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)required25,000DOF,leadingtoasignificantcomputationalspeed-up.第2頁共72頁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頁共72頁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幾何分析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頁共72頁materialpropertiesastherestof.Intheensuingdiscussion,aquantityofparticularinterestwillbetheweightedtemperatureTdevicewithindevice(seeEq.(2)below).Aslot,identifiedasslotinFig.2,willbesuppressed,anditseffectonTdevicewillbestudied.Theboundaryoftheslotwillbedenotedbyslotwhiletherestoftheboundarywillbedenotedby.Theboundarytemperatureonisassumedtobezero.Twopossiblebound