VariableAmplitudeFatigueCrackGrowthUsingSurrogateModelsandExactXFEMReanalysisMatthewPaisMarch31,20112/36AcknowledgementsDr.KimforyourcommentsandhelpthroughoutmytimeatUF,youhavetaughtmehowtoresearchanddealwithobstaclesencounteredalongthewayAlexCoppeforourcollaborationontheidentificationofequivalentdamageparametersfromXFEMdataDr.DavisandNuriYeralanforourcollaborationontheformulationandimplementationoftheexactXFEMreanalysisalgorithmDr.Petersforourcollaborationontheuseofanelement-basedenrichmentschemeformodelingweakdiscontinuitiesindependentoftheFEmeshRichardPippywhooriginallycreatedthefiniteelementwingboxmodelwhichwasmodifiedaspartoftheanalysisofAFRLflightdataBenSmarslokandEricTeugelatARFLforprovidingtheflightdataFelipeVianaforourcollaborationontheuseofkrigingforenablinghigher-orderintegrationoffatiguecrackgrowthmodelsandDr.HaftkaforanidealeadingtothecreationofthevariablestepsizealgorithmCommitteemembersforyourwillingnesstoserveandcontributionstotheimprovementofmydissertationMDOgroupforallthefeedbackwhichcontributedtotheimprovementofthiswork3/361Black,Structuralhealthmonitoring:Compositesgetsmart,2008.2Coppe,Simplifieddamagegrowthmodelscanstillyieldaccurateprognosis,submitted.3An,Experimentalstudyonidentifyingcracksofincreasingsizeusingultrasonicexcitation,2011.AcousticEmissionSensorsDataEstimateofDamageSize3EstimateofMaterialProperties/RemainingLife2AddWeightConditionalMaintenanceUncertaintyinLocation/SizeTraditionalSensorsDataConvertDatatoStressHistoryDamageGrowthonDigitalTwinLifeEstimateofPanel/AirplaneInDevelopment4/36xxxyyyNNNR=minmaxBiaxialVariableAmplitudeLoadingVariableAmplitudeLoadingConstantAmplitudeLoadingNOverloadUnderloadProportionalloading xx

:xy

:yy–ratioisconstantasafunctionofN

Non-proportionalloading

xx

:xy

:yy–ratiochangesasafunctionofN5/36xxxyyyNN?NaorN6/36OverviewFatiguecrackgrowthStressintensityfactorevaluationExtendedFiniteElementMethod(XFEM)ExactXFEMreanalysisalgorithmIntegrationoffatiguecrackgrowthmodelsSurrogatemodelsforhigh-orderintegrationSurrogatemodelsforvariableintegrationstepsizeVariableamplitudefatiguecrackgrowthfromflightdataCracktipplasticityConversionofflightdatatobiaxialstresshistoriesExampleproblemanddiscussionConclusions7/36ComputationalFatigueHigh-cyclefatigue(104-108cyclestofailure)FatiguegrowthmodelisordinarydifferentialequationSomeanalyticalequationsforKforsimplegeometriesFiniteelementmethodtocalculateKforcomplexgeometryConstantamplitudefatigueCrackgrowthincrementaorNreducesnumberofsimulationsCreateserrorinintegrationoffatiguegrowthmodel/crackpathBiaxialvariableamplitudefatigueCrackhaspreferredgrowthdirectionforhistoryorforeachcycle(non-proportional)Nopriorknowledgeofcrackpath,howtohavea-Krelationship?Plasticityaccelerates/slowscrackgrowth8/36ParisModel1Paris,Arationalanalytictheoryforfatigue,TheTrendinEng.,1961.C1Kthm9/36DirectandIndirectSolutiontoODEDirectSolutiontoODEIndirectsolutiontoODECalculateKI,KIICalculatecrackgrowthdirectionCalculateafromfatiguemodelGrowcrackbycalculateddirection/magnitudeGivenacrackConvertKI,KIIintoKeqUsegivenaNumberofelapsedcyclesforiteration,N.Assumedcrackgrowthforiteration,a.10/36ExtendedFiniteElementMethodContourintegralstocalculateKI,KIITheextendedfiniteelementmethod1

(XFEM)eliminatesneedtorecreatemesh2locallyaroundcracktipascrackgrowthsEnrichmentfunctions/DOFsaddedtodisplacementapproximationandareactiveinelementswithdiscontinuityEnrichedelementTraditionalelementCracktipenrichednodeHeavisideenrichednode1Belytschko,Elasticcrackgrowthinfiniteelementswithminimalremeshing,Int.J.Num.Meth.Eng.,1999.2Maligno,Athree-dimensionalnumericalstudyoffatiguecrackgrowthusingremeshing,Eng.Frac.Mech.,2010.3Osher,Frontspropagatingwithcurvaturedependentspeed,J.Comp.Phys.,1988.11/36ExactXFEMReanalysisReanalysisalgorithms1modifyfactorizationfromsmallchangetoKAscrackgrows,smallchangesinKCracktipterms,discardandreplaceHeavisideterms,appendnewFirstiterationCalculateandfactorKintoLandDSubsequentiterationsModifyfactorizationofLandDDetailsforfindingfill-reducingorderingpermutationavailableindissertationCracktipenrichednodeHeavisideenrichednodeHeavisideenrichednodefrompreviousiteration1AbuKassim,Staticreanalysis:Areview,J.Struct.Eng.,1987.2Amestoy,Anapproximateminimumdegreeorderingalgorithm,SIAMJ.MatrixAnal.App.,1996.3Davis,DynamicsupernodesinsparseCholeskyupdate/downdateandtriangularsolves,ACMTran.Math.Soft.,2009.12/36ExampleProblemH=W=2m,a=0.25m,a=0.05Computationaltimenormalizedbytimeforfullassemblyorfactorization/solvingfromscratchSavingsinassembly(~80%)Savingsinfactorizationandsolving(~70%)Moresimulationswithinacomputationalbudget!13/36DirectandIndirectSolutiontoODEDirectSolutiontoODEIndirectsolutiontoODECalculateKI,KIICalculatecrackgrowthdirectionCalculateafromfatiguemodelGrowcrackbycalculateddirection/magnitudeGivenacrackConvertKI,KIIintoKeqUsegivenaNumberofelapsedcyclesforiteration,N.Assumedcrackgrowthforiteration,a.14/36IntegrationChallengeODEEulerMidpointSlope=f(xi,yi)Slope=f(xi+1/2,yi+1/2)EulerMidpointy(x)DataEuler15/36SurrogateforDirectIntegrationFindK(N)atdiscretepointsusingfiniteelementsimulations(expensive)FitsurrogatetoK-Nhistory(cheap)UsesurrogatetoextrapolatetoNi+1/2Higher-orderintegrationwithoutadditionalexpensivesimulationsSameprinciplecanbeextendedtocrackgrowthdirectioninmixed-modeloadingForwardEulerMidpointai+1EulerMidpointKi+1/2(KRG)Surrogate(KRG)16/36VariableStepSizeAlgorithmTargeterror:et(0.001)Rateofchange:α

(0.1)et,αcalibratedfromseveralmaterials,initialcracksizes,crackgeometriesForerror<et,increaseNi+1Forerror>et,decreaseNi+1Removesneedtoassumecrackgrowthincrementapriorierrorerror17/36InclinedCrackinFinitePlateMat.R=0.1R=0.5Aust.Mart.xfailEuler1.541.541.541.54Var.1.541.541.551.54yfailEuler1.091.091.091.09Var.1.091.091.091.09NfailEuler29,50015,00068,30080,600Var.29,60015,00068,60080,700IterEuler29,50015,00068,30080,600Var.3561575224518/36EulerVersusVariableKrigingEulerapproximationwithstepsfromvariablekrigingtoaluminumwithR=0.5CrackgrowthEulerapproximation:0.34mVariablekriging:0.41m21%different!CracktipcoordinatesEuler(1.47,1.09)Variablekriging:(1.54,1.09)19/36xxxyyyNNNR=minmaxMulti-axialVariableAmplitudeLoadingVariableAmplitudeLoadingConstantAmplitudeLoadingNOverloadUnderloadProportionalloading xx

:xy

:yy–ratioisindependentofN

Non-proportionalloading

xx

:xy

:yy–ratioisdependentuponN20/36BiaxialNon-ProportionalVariableAmplitudeFatigueFornon-proportionalloadingthecrackgrowthdirectioniscycledependentCannotchooseaorN-assumesaconstantcrackpathovermanycyclesFiniteelementsimulationforeacha,NtofindKandθChangingcracktipplasticitymustbeconsideredAmountofplasticityisinverselyproportionaltoda/dNNOverloadUnderloadConstantOverloadUnderloadConstant21/36ModifiedParisModelMR,normalizesRratiotoR=0Constantsand1,scaletoR=0SinglevalueofC,mforallRMP,loadinteractionsasaresultofvariableamplitudeinducedplasticityConstantn,modifiescrackgrowthratefrompast,currentplasticityKth,thresholdstressintensityfactorVerifiedforrangeofvariableamplitudefatigueproblemsR=0.5R=0.0R=-0.5R=0.5R=0.0R=-0.51Xiaoping,Anengineeringmodeloffatiguecrackgrowthundervariableamplitudeloading,Int.J.Fatigue,2008.22/36WhynotmodelplasticityinXFEM?Elgeudj1,2hasintroducedenrichmentfunctionsforpower-lawhardeningmaterialswithconfinedplasticitymodelObservationsresultinlittledifference(~3%)betweenplasticandelasticstressintensitysolutionsforconfinedplasticitymodel,agreeingwiththeworkofAnderson3inthetraditionalFEMPlasticXFEMsolutionCracktipelement:≈214additionalgausspointsHeavisideelement:≈244additionalgausspointsPlasticitynotsodifferentthatasimplermodelcannotreproducebehavior1Elguedj,AppropriateextendedfunctionforX-FEMsimulationofplasticfracturemechanics,Comp.Meth.App.Mech.Eng.,2006.2Elguedj,Mixed-augmentedLagrangian-XFEMforelastic-plasticfatiguecrackgrowthandunilateralcontact,Int.J.Num.Meth.Eng.,2007.3Anderson,FatigueCrackInitiationandGrowthinShipStructures,Ph.D.Thesis,TechnicalUniversityofDenmark,1998.23/36NUnderloadNOverloadConstantOverloadConstantUnderloadKopenversusMPOverloadKopen↑,da/dN↓-Mp↓,da/dN↓UnderloadKopen↓,da/dN↑-Mp↑,da/dN↑24/36SummaryofAFRLDataAirForceResearchLaboratory(AFRL)providednormalizeddatafor19flightsatcentroid180,588datapointsintotalDatacollection:PeaksandvalleysofaccelerationsFlightevents(e.g.landinggearup/down)EveryminuteotherwiseNormalizeddata:Normal,lateral,longitudinal,roll,pitch,andyawaccelerations;roll,pitch,yawrates;airspeed;altitude;angleofattack(α);flapangle;fuelquantity;MachnumberScaleddataforselectedvariableslinearlytoexpectedvaluesforacommercialaircraftflightEx:α

norm=0-1,α=0-10,α=10αnormrollyawpitchz,normaly,lateralx,longitudinal25/36ScaledDatatoStressHistoryInertiaeffectmodeledfromAFRLaccelerationdata,stiffnesseffectfromlinearAbaqusfiniteelementsimulationsCalculatebiaxialstressesforassumedpressuredistributionsforlift(xx,L,xy,L,yy,L)anddrag(xx,D,xy,D,yy,D)4analysisforlift(α=-5,0,5,10),1analysisfordragFitsurrogatemodelforliftstressasafunctionofangleofattackMagnitudeoflift(wo)anddrag(qo)fromsimplemodelusingAFRLdataLocationofinterest26/36StressHistoryforFlightID2Loadingisnon-proportional,eveninregionswhereitappearstobeconstantEffectmaybelargeorsmall,impossibletoguess27/36StressHistorytoCyclicStressHistoryNeedtoconvertstresshistoryintocyclicstresshistoryRainflowcounting1oneachcomponent(e.g.xx,xy,yy)givesuniquecyclichistoryNeedsinglecyclicbiaxialstresshistoryRainflowcountingonequivalentstressSuperimposecyclestostresscomponentsforuseinXFEManalysis1Nieslony,Determinationoffragmentsofmulitaxialserviceloadingstronglyinfluencingfatigue,Mech.Sys.Sign.Proc.,2009.28/36VariableAmplitudeAnalysisRainflowcountingidentified37,007cyclesDuetonon-proportionality,74,014XFEMsimulationsneededSimplifiedgeometryusedinanalysisStressesweremagnifiedby10toencouragecrackgrowthWithreanalysisSimulationtime:about2daysWithoutreanalysisApproximatetime:about17daysBasedupon100iterationswithoutreanalysisandextrapolationBestcasescenario29/36CrackGrowthDiscussionCrackgrowthoccursinabout2,500/37,000cyclesKrarelyexceedKth(2.2MPam1/2)Largerstressesearlycreatesincreasedplasticity,slowinggrowthlaterPeaksandvalleysinKtendtoberelatedtoincreasesincracklength30/36CrackPathDiscussionMaximumstressisgenerallyyy,whichagreeswithhorizontalgrowthtrendInitialgrowthdirectionissouthwest-yyandxyaredominantFutureiterationsofgrowtharemorechallengingtodiscernConstantlychangingcracktipcoordinatesystemConstantlychangingratiobetweenbiaxialcomponentsN=1N=5,000N=10,000N=20,000N=30,00031/361Black,Structuralhealthmonitoring:Compositesgetsmart,2008.2Coppe,Simplifieddamagegrowthmodelscanstillyieldaccurateprognosis,submitted.3An,Experimentalstudyonidentifyingcracksofincreasingsizeusingultrasonicexcitation,2011.AcousticEmissionSensorsDataEstimateofDamageSize3EstimateofMaterialProperties/RemainingLife2AddWeightConditionalMaintenanceUncertaintyinLocation/SizeTraditionalSensorsDataConvertDatatoStressHistoryDamageGrowthonDigitalTwinLifeEstimateofPanel/AirplaneInDevelopment32/36StructuralHealthMonitoringUsecrackgrowthhistorytoidentifymaterialpropertiesCandmEstimateremaininglifeInsteadofhavingaccurateK,canequivalentCandmbeidentified?GeneratedcrackgrowthhistoryusingexactXFEMreanalysisCurrentlyconsideringvariableamplitudeloadinganddifferentfatiguemodels1Coppe,Simplifieddamagegrowthmodelscanstillyieldaccurateprognosis,submitted.33/36ConclusionsFatiguecrackgrowthisexpensiveduetonumberofcyclestofailureChoiceofaorNreducescost,butpoorchoiceaffectsaccuracyofcrackgrowthprediction;maynotbevalidforallfatigueproblemsSurrogatemodelsenablehigh-orderintegrationoffatiguecrackgrowthmodelandprovidemeanstodynamicallyadjustintegrationstepsizewithoutneedtochooseaorNExactXFEMreanalysismakesrepeatedcrackgrowthsimulationsmoreaffordablethroughthedirectmodificationofanexistingCholeskyfactorizationTheexactXFEMreanalysiswasusedinthefatiguecrackgrowthanalysisofapanelsubjectedtonon-proportionalbiaxialstresshistoryInitialintegrationofSHMprognosisandXFEMhavebegun34/36PossibleAreasforFutureWorkApplicationofexactXFEMreanalysisOptimizationproblemforGwhencrackreachesinterfaceConsiderbranching,interfacewithotherenrichmentfunctionsApplicationofsurrogateintegrationtowearanalysish:weardepthk:wearconstants:slidingdistancep:contactpressure35/36OutcomesofWork1Dand2DMATLABXFEMcodesWebsite()AbaqusXFEMtutorials(7)1Dand2DMATLABXFEMcodesfordownloadKrigingintegrationexamplefilesfordownload1750+visitorsamonthfromabout65countriesGooglesearchrankings:AbaqusXFEM-1stAbaqusCrackTutorial-1stMATLABXFEM-1st(2Dcodes)and4th(1Dcodes)XFEM-5thXFEMReanalysis-1stthrough7thXFEMSurrogate-1stthrough3rdInteractionwithabout160individualsaroundtheworldInternshipatIdahoNationalLaboratorywithNationalandHomelandSecurityInterviews(SIMULIA/Abaqus)andpost-docs(CardiffUniversity,UCLA)36/36PublicationsJournalPapers(4)PaisM,KimNH.Predictingvariableamplitudefatiguelifethroughuseofadigitaltwin,inpreparation.PaisM,YeralanS,DavisT,KimNH.CrackgrowthandoptimizationintheXFEMframeworkthroughanexactreanalysisalgorithm,InternationalJournalofNumericalMethodsinEngineering,submitted.PaisM,VianaFAC,KimNM.Enablinghigh-orderintegrationoffatiguecrackgrowthwithsurrogatemodel,Intern