Sequential Optimization and Reliability Assessment

1DETC’02:ASME2002DESIGNENGINEERINGTECHNICALCONFERENCESANDCOMPUTERSANDINFORMATIONINENGINEERINGCONFERENCESeptember29–October2,2002,Montreal,CanadaDETC2002/DAC-34127SEQUENTIALOPTIMIZATIONANDRELIABILITYASSESSMENTMETHODFOREFFICIENTPROBABILISTICDESIGNXiaopingDuIntegratedDesignAutomationLaboratory(IDEAL)DepartmentofMechanical&IndustrialEngineeringUniversityofIllinoisatChicagoWeiChenIntegratedDesignAutomationLaboratory(IDEAL)DepartmentofMechanical&[e-mail]optimizationdesignofferstoolsformakingreliabledecisionswiththeconsiderationofuncertaintyassociatedwithdesignvariables/parametersandsimulationmodels.Inaprobabilisticdesign,suchasreliability-baseddesignandrobustdesign,thedesignfeasibilityisformulatedprobabilisticallysuchthattheprobabilityoftheconstraintsatisfaction(reliability)exceedsthedesiredlimit.Thereliabilityassessmentforprobabilisticconstraintsofteninvolvesaniterativeprocedure;therefore,twoloopsareinvolvedinaprobabilisticoptimization.Duetothedouble-loopprocedure,thecomputationaldemandisextremelyhigh.Toimprovetheefficiencyofaprobabilisticdesign,anovelmethod–sequentialoptimizationandreliabilityassessment(SORA)isdevelopedinthispaper.TheSORAmethodemploysasingle-loopstrategywhereaserialofcyclesofoptimizationandreliabilityassessmentisemployed.Ineachcycleoptimizationandreliabilityassessmentaredecoupledfromeachother;noreliabilityassessmentisrequiredwithinoptimizationandthereliabilityassessmentisonlyconductedaftertheoptimization.Thekeyconceptoftheproposedmethodistoshifttheboundariesofviolateddeterministicconstraints(withlowreliability)tothefeasibledirectionbasedonthereliabilityinformationobtainedinthepreviouscycle.Hencethedesignisquicklyimprovedfromcycletocycleandthecomputationalefficiencyisimprovedsignificantly.Twoengineeringapplications,thereliability-baseddesignforvehiclecrashworthinessofsideimpactandtheintegratedreliabilityandrobustdesignofaspeedreducer,arepresentedtodemonstratetheeffectivenessoftheSORAmethod.1.INTRODUCTIONTraditionaloptimizationdesignsarepushedtothelimitsofsystemfailureboundaries,leavingverylittleornoroomforaccommodatinguncertaintiesinengineeringdesign.Consequently,deterministicoptimizationdesignsobtainedwithoutanyconsiderationofuncertaintiesmaybesensitivetothevariationofsystem(leadingtoqualityloss),risky(highlikelihoodofundesiredeventsorlowconstraintsatisfaction),orconservativeandthereforeuneconomicifdeterministicsafetyfactorsarelargerthanrequired.Itisthereforeimportanttoincorporateuncertaintyinengineeringdesignoptimizationanddevelopcomputationaltechniquesthatenableengineerstomakeefficientandreliabledecisions.Probabilisticdesignmethodshavebeendevelopedandhavebeenappliedinengineeringdesign.Thetypicalprobabilisticdesignmethodsincludereliability-baseddesign(WuandWang,1996;Carter,1997;GrandhiandWang,1998)androbustdesign(Chen,etal,1996;DuandChen,2000a).Reliability-baseddesignemphasizeshighreliabilityofadesignbyensuringtheprobabilisticconstraintsatisfactionatdesiredlevels,whilerobustdesignfocusesonmakingthedesigninerttothevariationsofsysteminputthroughoptimizingmeanperformanceofthesystemandminimizingitsvariancesimultaneously.Oneimportanttaskofaprobabilisticdesignisuncertaintyanalysis,throughwhichweunderstandhowmuchtheimpactoftheuncertaintyassociatedwiththesysteminputisonthesystemoutputbyidentifyingtheprobabilisticcharacteristicsofsystemoutput.Wethenperformsynthesis(optimization)underuncertaintytoachieveourdesignobjectivebymanagingandmitigatingtheeffectsofuncertaintyonsystemoutput(systemperformance)(DuandChen,2000b).Inspiteofthebenefitsofprobabilisticdesign,oneofthemostchallengingissuesforimplementingprobabilisticdesignisassociatedwiththeintensivecomputationaldemandofuncertaintyanalysis.Tocapturetheprobabilisticcharacteristicsofsystemperformanceatadesignpoint,weneedtoperformanumberofdeterministicanalysesinthevicinityofthenominalpoint,eitherusingsimulationapproach(forinstance,MonteCarlosimulation)orotherprobabilisticanalysismethods(such2asreliabilityanalysis).Manyresearcheshavebeenconcentratingondevelopingpracticalmeanstomakeprobabilisticdesigncomputationallyfeasibleforcomplexengineeringproblems.Ourfocusinthisstudyistodevelopanefficientprobabilisticdesignapproachtofacilitatedesignoptimizationsthatinvolveprobabilisticconstraints.Reliability-baseddesignissuchtypeofprobabilisticoptimizationproblems(Reddy,etal.,1993;Wang,etal.,1995;ChenandHasselman,1997;Tu;etal.,1999)inwhichdesignfeasibilityisformulatedasreliabilityconstraints(ortheprobabilityofconstraintsatisfaction).Theconventionalapproachforsolvingaprobabilisticoptimizationproblemistoemployadouble-loopstrategyinwhichtheanalysisandthesynthesisarenestedinsuchawaythatthesynthesisloop(outerloop)performstheuncertaintyanalysis(innerloopforreliabilityassessment)iterativelyformeetingtheprobabilisticobjectiveandconstraints.Asthedouble-loopstrategymaybecomputationallyinfeasible,varioustechniqueshavebeendevelopedtoimproveitsefficiency.Thesetechniquescanbeclassifiedintotwocategories:oneist