AKMCSi一种基于krig的方法来处理小故障概率和耗时模型外文文献

AKMCSi一种基于krig的方法来处理小故障概率和耗时模型外文文献 AK-MCSi:A Kriging-based methodto deal with small failureprobabilities andtime-consuming modelsNicolasLelivre?,Pierre Beaurepaire*,Ccile Mattrand*,Nicolas Gayton*UniversitClermont Auvergne,RS,SIGMA Clermont,Institut Pascal,F-63000Clermont-Ferrand,Francea rt ic le in foArticle history:Received17MayxxReceived inrevised form15January2018Aepted20January2018Keywords:Reliability analysisAK-based methodsClusteringSmallfailure probabilitiesab st ra ctReliability analysesstill remainchallenging todayfor manyapplications.First,assessing small failureprobabilities istedious because of thevery largenumber ofcalculations required.Secondly,mechanicalsystem modelscan requireconsiderable numerical efforts.To deal with theseproblems,classical reliabil-ity analysis methods maybe binedwith thoseof meta-modeling,to enable the construction of amodellike the former numericalmodel butwith fewertime-consuming evaluations.Among theseapproaches,the ActiveLearning ReliabilityMethod,bining Kriging and Monte Carlo Simulations,has receivedattention overthe lastfew years.This methodhas several drawbacks,such as the diff i cultyto assess small failure probabilitiesand itsinability toparallelize putations.The proposed paperfocuses on the improvementof sucha methodto solveboth these issues.It introducesa sequentialMonte Carlo Simulationtechnique todeal withsmall failure probabilities.A multipointenrichment tech-nique is also proposedto allowparallelization and thus to reduce numerical efforts.Both thesenew tech-niques give rise to the proposalof a new,more conservativestopping condition for learning.Theeff i ciency of this newmethod,called AK-MCSi,is thendemonstrated usingthree examplesfor whichtheresults showa signif i cantreduction in the timerequired and/or the number of iterations neededforan aurateevaluation of the failure probability.?2018Elsevier Ltd.All rightsreserved.1.IntroductionMost elementsin realengineering systems,such asmaterialproperties,loads orgeometrical parameters,are subjectto uncer-tainties.Reliability analysisprovides anappropriate frameworktodealwith such uncertaintiesand to evaluate failure probabili-ties.This approachis increasinglybeing used in manyf ields16.To performreliability analyses,multiple techniquescan befoundin the literature.The CrudeMonte CarloSimulation(MCS)is widelyused and is consideredas thereference method.How-ever,this methodrequires signif i cantnumerical efforts,especiallywith time-consuming systemmodels,low failure probabilities orlarge-scale models.Some approximationtechniques have thusbeen proposed,such asthe FirstOrder ReliabilityMethod(FORM)or itsimprovement,the SecondOrder ReliabilityMethod(SORM)7.These techniquesaim to approximate the limit statewithrespectively alinear orquadratic functionbased on the knowledgeof the mostprobable failurepoint(MPP).However,as theyarebased on a simplifying hypothesis(the limitstate islinear orquad-ratic),they may not be applicable inmany practicalcases.Thus,FORM andSORM maycause largeerrors when evaluating failureprobability8.More advancedsimulation methodshave also beenproposed,such asImportance Sampling9or SubsetSimulations10.Even thoughthe numericaleffort decreaseswith suchmeth-ods,it canstill beprohibitively high.For severalyears now,meta-modeling hasappeared to be usefulin order todevelop more eff i cient structuralreliability methods.The mainidea isto calibratea simplif i edmodel of the originalsys-tem based on alimited set of evaluations of its response.The so-called meta-model,which is a simplif i edmathematical modelanddoes notrequire much time to be evaluated in practice,canthus be used in place of the initialtime-consuming modelusingtraditional reliability analysismethods.Several meta-models orsurrogate models have been proposed in theliterature,such aspolynomialresponse surfaces11,artif icial neuralwork12,Kriging13,support vectormachines14and polynomialchaosexpansions15,16.The readershould referto17,18for reviewsonsurrogate modeling.Surrogate models and traditionalreliabilitymethods arethus plementary,and whenbined ares:/doi/10.1016/j.strusafe.2018.01.00xx7-4730/?2018Elsevier Ltd.All rightsreserved.?Corresponding author.?Principal correspondingauthor.E-mail addresses:nicolas.lelievresigma-clermont.fr(N.Lelivre),pierre.beaurepairesigma-clermont.fr(P.Beaurepaire),cecile.mattrandsigma-clermont.fr(C.Mattrand),nicolas.gaytonsigma-clermont.fr(N.Gayton).Structural Safety73 (2018)111Contents listsavailable atScienceDirectStructural Safetyjournalhomepage:.elsevier./locate/strusafeexpected toreduce the number ofevaluations of the initialtime-consuming model.Indeed,the calculationof failure probability isperformedon thepreviously calibratedsurrogate model19.However,a biascan beintroduced since the failure probability isassociated with the surrogate modeland not the originalmodel.Advanced meta-model methodshavethus been proposed2023to trytoreducethis bias.We focushere on the Kriging surrogate modeldue to thestraightforward estimationof the so-called Krigingvariance,whichis anappropriate measureof thepossible errorsmade by the sur-rogate model.However,the maindrawback ofKriging surrogatemodelsis that the Designof Experiments(DoE)is chosenbeforethe calibrationstep19.A largeDoE,which can be veryexpensivefrom aputational pointof view,must thusbe evaluatedtoobtain asuitable approximation.Adaptive variantsof Krigingmeta-modeling,such asEff i cient GlobalReliability Analysis(EGRA)24and Active learning reliability method biningKriging andMonte Carlo Simulations(AK-MCS)25were f i rstproposed tosolvethis issue.They aimto identifyiteratively atwhich inputpointthe initialtime-consuming modelhas to be evaluated.Sincethen,several versionshave beenintroduced bydifferent authors,especially regardingthose relatedto AKstrategy.Fig.1summa-rizes the AK-based methods.The latterhavebeenapplied forreli-ability analysis,optimization under uncertainty andgeometricalconformity analysis.As theformer AK-based methodwas devel-oped for the purposeof reliabilityanalysis,this applicationgathersthe mostcontributions.AK-MCS is a classif i cationmethod,i.e.a methodwhich aims toclassify a population into the failure domain orthe safetydomain.This methodappears to be eff i cientfor thispurpose and thus toestimate failureprobabilities.However,AK-MCS islimited bysev-eraldrawbacks.Indeed,it requiresthe generationof a Monte Carlopopulation,which couldprovoke putermemory problems,especially when trying toassess verysmallfailureprobabilities,sincethe population must be verylarge.Moreover,system reliabil-ity analysesor problemswith timedependence cannotbe handledwith AK-MCS.Finally,the useof distributed puting facilitiesincreasesprocessing powerand enablesparallel putations;however,AK-MCS cannothandle simultaneousmultiple calcula-tions.Several versionsof AK-MCS havetherefore been proposedto tryto solvetheseissues.First,MCS can be replacedby moreadvancedreliability methods.Indeed,Echard et al.26use theimportance sampling strategy.Their newmethod isnamed AK-IS.Importance samplingimplies thesearch for the MPP.To per-form thissearch AK-IS usesFORM,which canonly fi nda singleMPPand thuscould involveconsiderable errorsin estimating thefailure probability.However,under theright conditionsAK-IS eff i-ciently allowssmallfailureprobabilities to be assessed.Lv etal.27develop theActive learningreliabilitymethodbiningKriging andLine Sampling(AK-LS).The linesampling techniqueiseff icient whendealing withplex failure domains,since itsalgorithmexplores theglobal designspace inmultiple directionsandfi nds theintersection between these directionsand the failuredomain.Huang etal.28propose anActive learningreliabilitymethod biningKriging andSubset simulations(AK-SS).Cadiniet al.29propose toreplace theFORM algorithmused in AK-IS tofindthe MPPby themetaIS algorithmproposed byDubourg etal.30.They calledtheir newmethod metaAK-IS2.This metaISalgo-rithm seems tobemoreeff icientthan FORM and thusimproves theability of AK-IS toestimatefailureprobabilities.Tong etal.31bine subsetsimulations andimportancesampling.This isnamedAK-SSIS.Two methodsare alsoproposed in the literaturetoconsider time-dependent problems:Hu andDu32bihe mixedEGO methodwithAK-MCS andHu andMahadevanFig.1.State of the artof AK-based methods.2N.Lelivre etal./Structural Safety73 (2018)11133propose amethod entitledSILK.Finally,Fauriat andGayton34propose threemethods tosolve systemreliability problems.The bestone identifi esthe performance functions whichimpactthe globalsystem,and itdoes notevaluate the performance func-tions havingonly amoderate inf l uence on failure probability.Theirtechnique reducesthe numericalefforts requiredandisreferred toasAK-SYS.As mentionedpreviously,AK-MCS hasalsobeenadapted tootherfields.Regarding optimizationunderuncertainty,Moustaphaet al.35present analgorithm which fi rst improvesthe surrogatemodelin theoverall domainby using theAK-MCS learning func-tion,then refi nesit locallyin theregion exploredby theoptimiza-tion.Liu etal.36use aKriging surrogate model with the AKlearning function toevaluate worst-case designsfor optimization.It dealswith structuraldesign optimizationunder reliabilitycon-straints includinginterval uncertainty.AK-MCS hasbeen adaptedtometrology problems.It can beused to checkthe conformityoflarge surfaces,as proposedby Dumasetal.37in theirmethodnamed AK-ILS.Such problemscan infact beviewed asclassif ica-tion ones,since two domains exist:the pliantand thenon-pliant domains.We proposehere threeimprovements to the initialAK-MCS tosolvesome of the drawbackspresented previously.They mightalsobe involvedin anyother AK-based method.First,even thoughasurrogate modelis usedinstead of the mechanicalmodel,its eval-uation ona large population canrequire toomuchtimeand mem-ory.Dividing thelargepopulationusedtoassess failureprobabilitiesis thesolution presented in this paper assequentialMonte Carlo Simulations.Next,distributed puting facilitiesare increasinglyusedinscience andengineering,but theoriginalAK-MCS method is onlybased ona singlesample ofenrichmentby iteration.Thus,this paperproposes a clustering techniquewhichenables severalevaluationsof the performance function tobe performed simultaneously.Finally,these twotechniques haveshownthat the initial stopping condition for learning appearsnotto be conservative enough to obtainsuff icient auracy.A newstopping criterion isthus proposed.These improvementslead ustorename AK-MCS asAK-MCSi.The proposedpaper isstructured asfollow:Section2presentsthe theoryof classicalKriging surrogate modelsand the formerAK-MCS.Section3presents the proposed AK-MCSi with the threeenhancements.Some examplesare then presentedin Section4toillustrate theeff iciency of theproposedimproved method.Someconclusions arefi nally drawnin Section5.2.Reliability analysisusing aKrigingsurrogate model2.1.Description of the reliabilityanalysis problemOneaim of a reliabilityanalysis istoevaluate the probabilityofourrence associatedwith apredef i ned failureevent.The systemischaracterized byitsresponse,def i ned bytheso-called perfor-mancefunction G?X?,which isexpressed withrespect torandominput variablesX?fX1;.;X mg T.This set of randominput vari-ables ischaracterized byits jointprobability density functionf?X?.Failure probabilityis def i ned asthe probability thata sampleliesin the failuredomainX f.Generally,this failuredomain isdef i ned by negativeperformance function values,i.e.G?X?0.The safetydomain isthus def i ned by positiveperformancefunc-tionvaluesG?X?0.The junctionbetweenthesetwodomains,i.e.G?X?0,is calledthe limitstate.Failure probabilityis thusdef i ned bytheprobability that the performance function isnega-tive,which reads:p f?Prob?G?X?0?Z1G?X?f?X?dX?1?where1?is theindicator functiondef i ned as:1G?X?1if G?X?600otherwise?2?The performancefunction Gcanbeplex,and can thus consider-ably increasethe putationalcost of the failure probability eval-uation.These plexperformance functionscan therefore bereplaced bysurrogate modelsbG19.In reliabilityanalysis,standard spaceis monlyused andisdef i nedasa spacewhere allvariables are independently andnor-mally distributedwith zero mean andunit variance.In theremain-der ofthis paper,all Xareindependentstandard normalrandomvariables.An iso-probabilistic transformationcanbeperformedto respectthis requirement.2.2.The Krigingsurrogate modelKriging,theorized byKrige38and developedby Matheron13,aims toapproximate afunction from a givendataset ofcou-ples ofinput andoutput parameters.In reliabilityanalysis,thesur-rogatemodelis usedtoapproximatethe performancefunctionG.In sucha context,the Krigingsurrogate modelis based on theassumptionthat theperformancefunction is a realization of a sta-tistical Gaussian process G?x?i?tobecharacterized,which isdef inedas:G?x?i?Xpj?1b jf j?x?i?Z?x?i?3?where x?i?is the ith sampleof the set ofrandom variablesX.Allthese x?i?are storedin the DoE.P pj?1b jf j?x?i?isadeterministic termwithbasis functionsf?x?i?ff1?x?i?;.;f p?x?i?gand regressioncoeff icients b?fb1;.;b pg.Z isa stationaryzeromeanGaussianprocess with the covariancebetween twosamples x?i?and x?j?definedby Eq. (4).C ZZ?x?i?;x?j?r2Z R?x?i?x?j?;h?4?where r2Zis thevariance of the Gaussian process,R?x?i?x?j?;h?theauto-correlation functionand hisavector ofso-called hyperparam-eters drivingthe auto-correlation function.A low-order polynomialfunctionbasis and the Gaussianauto-correlation function arewidely used25,31,35for reliabilityand optimizationapplications.The calibrationstep of the Krigingsurrogate modelis supposedtoidentify the optimal valuesof thehyperparameters h.Several tech-niques existfor thispurpose butthe MaximumLikelihood Estimationis the mostwidely-used.It leadsto thefollowing optimizationproblem:h?Arg minhr2Z?h?det R?h?1m?5?wherer2Z?1m?G?Fb?T R?h?1?G?Fb?.G gathersthe responsesof theperformancefunction assessedfor the DoE samples,i.e.the terms ofG areG?i?G?x?i?.This optimization problem canbe solvednumer-ically withthe DACEmodule ofScilab andMatlab39.The plex-ityofthis optimization problem increaseswhen theset ofvariablesX is large.Now consider the sample x?0?,which isnot includedin theDoE.Its Krigingapproximation isgiven bythe BestLinear UnbiasedPre-dictor(BLUP)which isarealizationofaGaussian variablewith ameangiven in(Eq. (6)and variancegiven in(Eq. (7).l bG?x?0?f T0b?r T0R?1G?Fb?6?N.Lelivre etal./Structural Safety73 (2018)1113r2b G?x?0?r21?r T0R?1r0?v T0FT R?1F?1v0?7?In thissetting,b?F TR?1F?1F TR?1G andv0?F TR?1r0?f0.f0denotes thevector ofbasis functionsevaluated forx?0?;F isa matrixsuchthat theterm atrow iand columnj isdefinedbyF ij?f j?x?i?andr0is the correlation vectorbetween x?0?and x.Since noassumption on thelimitstate shapeis made,contraryto e.g.FORMandSORM,the consideredspace haslittle inf l uenceonthe Krigingmeta-model24.This methodcanthusbeper-formed eitherin physicalspace orin standardspace.As mentionedin Section2.1,all techniquesare appliedin standardspace in theremainder ofthis paper.2.3.AK-MCSActive learningand Kriging-based Monte CarloSimulations(AK-MCS25)isa two-nested-step methodwhich aimsto classifyaMonteCarlo(MC)population intothe safeor failuredomains.Thetwo nested steps of the method concern theadaptive calibrationofthe Krigingsurrogatemodeland the evaluation of the failureprob-ability.The fl owchart of AK-MCS is presented below:1.Generation ofaMonteCarlo population of np samplesu?i?in thedesignspace.This population is sampledaording tothe jointprobabilitydensityfunctionof theset ofrandom variables.Itis generallyremended thatgood auracyfor the failureprobability isobtained ifnpP100=p fwhich mightbe diffi culttoevaluate withoutany priorknowledge aboutthis failureprobability.2.Def i nition ofthe initialDoE byusing severalpossible techniquessuch as,for example,random samplingor LatinHypercubeSampling40.The performancefunction is then putedforthese samples.3.Calibration ofthe Krigingsurrogate modelbGbased ontheDoE.4.Evaluation ofthe failureprobability.Once the Kriging surrogatemodelis calibrated,it servesto calculateapproximate valuesofG forthe pleteMonteCarlo population fromstep1.The probabilityof failureis thenestimated with(Eq. (8).p f1n pXpi?11bG?x?i?8?5.Activelearning.The subsequentsample in the MC population isidentifi edfortheuping iterationof AK-MCS.This is basedon the learning function U in Eq. (9),which isevaluated onthewhole MC population.U?x?i?j l bG?x?i?jr bG?x?i?9?6.Stopping conditionforlearning.The stopping condition isdefinedby Eq. (10).If thiscondition is satisf i ed,the surrogatemodel isconsideredaurate enoughandthe learning process is stopped(Step8).However,if thecondition isnot satisfi ed,the algorithmgoeson to step7.min?U?P2?10?7.Update theDoE.The DoEis enrichedwiththe sampleminimizing thefunction U(Eq. (9).The performancefunctionis thenevaluated for this newsample andthe methodgoes backto step3.8.End ofthe process.The resultoftheAK-MCS methodis thefinalestimation ofthefailureprobability.Compared toclassical meta-modeling methods,whichfirstcali-brate the surrogatemodelfromaDoE andthen performthe relia-bilityanalysis,the twonestedsteps of AK-MCS allowanimprovement ofthe surrogatemodel,originally calibratedfrom amoderate-sized DoE,in thearea ofinterest.The number of callstotheperformancefunction isthus reduced,andthe methodcon-verges faster.However,this methodstruggles toassess smallfailureprobabil-ities.Indeed,forthispurpose the size ofthe MonteCarlopopula-tion has tobehigh,which cangiveriseto timeand memoryproblems.This paperconsiders thenumericaleffortsrequired toperformstep4of AK-MCS,which canbe considerableif theMonteCarlo population islarge.3.The proposedAK-MCSiAK-MCSi isan improvementontheoriginal AK-MCS.We pro-pose threeenhancements:a sequentialMonteCarlosimulationtechnique todealwithsmallfailureprobabilities,a multipointenrichmentstrat