Materials,StructuralIntegrityandReliabilitySolutionsThroughInnovativeEngineering

ComparisonofNon-StandardSimulationMethodsforPerformingExtremelyLowProbabilityAssessments

RobertE.Kurth,CédricJ.Sallaberry

EngineeringMechanicsCorporationofColumbus(Emc2)3518RiversideDrive-Suite202

Columbus,OH43221-1735Phone:(614)459-3200Fax:(614)459-6800

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•  In1985,theNRCproposedaModificationofGeneralDesignCriterion(GDC)4RequirementsforProtectionofDynamicEffectsofPostulatedPipeRupturesin10CFRPart50.▫  TheproposedamendmentswouldmodifyGDC4toallowforthedemonstrationof

pipinganalysestoserveasabasisforexcludingconsiderationofdynamiceffectsassociatedwithcertainpiperuptures.

▫  Theseanalysesconstitutewhatiscommonlyreferredtoastheleak-before-break(LBB)concept.

•  Thefinalrulerequiresadeterministicfracturemechanicsanalysis•  Thesupportingsafetyanalysismustdemonstratethatasubstantialrangeofstablepipecracksizescanexistwithdetectableleaks•  Thattheprobabilityoffluidsystemspipingruptureisextremelylowunderconditionsconsistentwiththedesignbasisforthepiping.•  Thedefinitionof“extremelylowprobability”ofpiperuptureisgivenontheorderof10-6perreactoryearwhenallpiperupturelocationsareconsidered.•  Thisprobabilityincludestheprobabilityofaninitiatingeventoccurringsuchasanearthquakeoranaccident.

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•  Presently,theStandardReviewPlan(SRP)3.6.3doesnotallowforassessmentofpipingsystemswithactivedegradationmechanisms,suchasPrimaryWaterStressCorrosionCracking(PWSCC),whichhasoccurredinsystemsthathavepreviouslybeengrantedLBBexemptions

•  Along-termgoalhasbeentodevelopamodular-based,probabilisticfracturemechanicscodecapableofdeterminingtheprobabilityofruptureforReactorCoolantSystem(RCS)components.

▫  Theneedforthismodular-basedcodeisstronglydrivenbytheneedtoquantitativelyassessanLBB-approvedpipingsystem’scompliancewithGDC-4onaninterval(time)basis.

▫  BasedontheterminologyofGDC-4,thisprogramandcodeisentitledExtremelyLowProbabilityofRupture(xLPR).

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•  xLPR2.0hasbeencompleted▫  ItemploystheGoldSim®platformfortheprobabilisticcalculations▫  ItisafullyQA’edcodeforuseinregulatoryassessments▫  Becauseofthisoverheadeachsimulation,whichcoverstheentirepipelinelife,

takesapproximately4seconds

•  BecausetheabilitytocomplywithGDC4implies,forasimplerandomsamplingmethod,5to10millionsimulationswouldbeneededalternativestrategieshavebeendeveloped▫  Simplerandomsampling▫  LatinHypercubeSampling(LHS)▫  ImportanceSampling▫  AdaptiveImportancesampling*

•  Thefocusofthispaperisontheadaptiveimportancesampling

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*DetailsofthesetopicscanbefoundinPSAM103:“Efficientsamplingstrategiestoestimateextremelylowprobabilities”,C.Sallaberry,R.Kurth,F.Brust,E.Twombly

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SRS LHS DPD

Better multi-dimensional coverage for DPD. Helps in conjoint influence

Densely cover of each input with LHS. Helps if extreme value are needed

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•  Thedetailsofthemethodology,whichislabeledGRC(GoldenRuleClusteringsincethemethodisbasedontheFibonaccisequence)•  Insteadweillustratetheapplicationtotwoproblems:▫  AtheoreticalproblemtoillustratetheGRCmethod▫  Ananalysisoftheapipingproblemwhere

!  Timetofirstcrackinitiation!  Timetofirstleakage!  Timeofpipefailure

arecalculated

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•  GRCmethodis▫  Variancereductiontechnique▫  Purposeistooversampletheregionofinterestandunder-sampletheregions

withnoevents.

▫  InthetheoreticalproblemThreeareasofinterest,morethantwostandarddeviationsfromthecentroid,needtohavetheirprobabilityofoccurrenceestimated

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•  Forthetheoreticalproblemwewishtofindthepeaks•  Adaptivesamplingfocusestheinputsontheareaswheretheresponseis“ofinterest”

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•  Inthepictureonthebottomleftthe3DsurfaceisprojectedontotheX-Yplane▫  RedpointsareSRS▫  Bluepointsareadaptive

•  Onthebottomrightthevaluesbelowalowerlimitareremoved

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•  AcomparisonofthenormaldistributiontheoreticalCDF,theDPDrepresentation,andtheGRCrepresentationareshowntotheleft

•  BoththeDPDandGRClieonthetheoreticalline.Agoodstart.

•  TheDPDpoints,shownasanopenblackcircle,areuniformlyspacedalongthexaxisandthenmovedtotheconditionalmeanvalue

•  TheGRCpointsdependonthevalueoftheinputwhenaresponseofinterestisfound

•  Forthissampletheinputwasat~15%value

•  50%oftheGRCpointsareplacedbelowthisvalue,theremainderabove

9

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•  TheCDFandCCDFareshownfortheyieldstrength

•  The“pivot”pointisnearthemean

•  ThepointsaretheDPDvaluesaftertheadaptivealgorithmhasbeenapplied

•  ClearlytherewillbemoresamplestakennearthepivotpointwhenthesamplingisbasedonthesimplerandomsamplingoftheintervalnotonthevalueoftheCDF

•  TheDPDPDFareusedtofindtheweightingvaluefortheresponse

10

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•  BecausethefocusofGRC4isonanextremelylowprobabilityofrupturethreequantitiesareexamined▫  Timetofirstcrackinitiation▫  Timetofirstleakage▫  Timeofrupture

•  BecauseoftheQArequirementsofxLPRtheGoldSim®platformcannotimplementtheGRCmethod

•  Therefore,apreprocessortoxLPRhasbeenwrittenwhichimplementstheGRCmethod

•  ThispreprocessoriscallPROMETHEUSandhasbeenreleasedasabetaversion,PROMETHEUS1.0

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•  Acomparisonoftheresultsforaprobabilisticanalysisisshown

•  2,500simulationusingSRSwithLHSxLPR2.0(dottedline)

•  100,000SRSsample(dashedline)

•  2,500GRCsample(solidline)

•  All3methodsconvergetosample60yearresult

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•  GRCresultestimatesthe100,000SRSresultthroughtheentirerange•  Theuncertaintyislarger•  Theabilitytoestimatethe1in100,000pointallowsthexLPRcodetofocus

samplingusingstandardimportancesamplingmethods

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•  Theuseofavarietyofsamplingmethodshavebeenstudiedtoallowefficientanalysisschemestobedevelopedto▫  Estimateextremelylowprobabilitiesofavarietyofevents▫  Identifytheareasoftheinputspacewhichleadtheeventsofinterest

•  Muchsmallersamplesizescanbeusedtoidentifyareasofinterestforfocusedsampling•  Comparisonofthemethodsshowsthat▫  TheSRS,LHS,andDPDmethodsallleadtostatisticallyequivalentmeanresults▫  TheGRCmethodfindsverylowprobabilityeventsmoreefficientlybutthecost

islargeruncertainty

•  TheuseoftheGRCmethodallowslessefficient(basedonthenecessarynumberofsimulations)tobefocusedontheregionsoftheinputspaceleadingtorareevents•  Thisfocusincreasestheefficiencyofthesecodes

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