Probabilistic approaches to compute uncertainty intervals and sensitivity factors of ultrasonic simu

F.Rupina,⇑,G.Blatmana,S.Lacazea,T.Fouquetb,B.ChassignoleaabElectricitédeFrance,RechercheetDéveloppement,LesRenardières,77818MoretsurLoing,FranceElectricitédeFrance,RechercheetDéveloppement,92141Clamart,Francearticleinfoabstract

Forcomprehensionpurpose,numericalcomputationsaremoreandmoreusedtosimulatethepropaga-tionphenomenaobservedduringexperimentalinspections.However,thegoodagreementbetweenexperimentalandsimulateddatanecessitatestheuseofaccurateinputdataandthusagoodcharacter-izationoftheinspectedmaterial.Generallytheinputdataareprovidedbyexperimentalmeasurementsandareconsequentlytaintedwithuncertainties.Thus,itbecomesnecessarytoevaluatetheimpactoftheseuncertaintiesontheoutputsofthenumericalmodel.Theaimofthisstudyistoperformaproba-bilisticanalysisofanultrasonicinspectionofanausteniticweldcontainingamanufactureddefectbasedonadvancedtechniquessuchaspolynomialchaosexpansionsandcomputationofsensitivityfactors(Sobol,DGSM).ThesimulationofthisconfigurationwiththefiniteelementcodeATHENA2Dwasper-formed6000timeswithvariationsoftheinputparameters(thecolumnargrainorientationandtheelas-ticconstantsofthematerial).The6000setsofinputparameterswereobtainedfromadaptedstatisticallaws.Theoutputparameters(theamplitudeandthepositionofthedefectecho)distributionswerethenanalyzedandthe95%confidenceintervalsweredetermined.Ó2013ElsevierB.V.Allrightsreserved.Articlehistory:Received18October2012Receivedinrevisedform24October2013Accepted11December2013Availableonline21December2013Keywords:UltrasoundWeldinspectionSensitivityanalysisUncertaintyFiniteelementmodeling1.IntroductionThepropagationofultrasonicwavesinheterogeneouscomplexmediaisachallengeforpowerplantoperators.Theyarebound,forobvioussafetyreasons,tostringentrequirementsconcerningtheirabilitytoreliablydetectsizedefectsduringin-serviceinspections.Nevertheless,thepresenceofcomplexmaterialslikemultipassweldsinausteniticstainlesssteelgreatlydisturbsthewaveprop-agationinducingdeviationand/ordivisionofthebeamassociatedwithanimportantattenuationofthewaveamplitude.Asaconse-quence,interpretationofdatacomingfromaweldinspectioncanbeachallenge.Inordertoovercomethisdifficultyandtoprovideanaccurateunderstandingofthewave-to-mediuminteraction,numericaltoolsareusedmoreandmoretosimulatetheultrasonicbeampropagationinheterogeneousmedia[1–5].InparticularFi-niteElement(FE)codeshaveprovedtheirefficiencytoaccuratelyreproducecomplexphenomenasuchasbeamdeviation,divisionorattenuation[6–9].Nevertheless,theaccuracyofsimulateddatanecessitatestheuseofaccurateinputdataandthusagoodcharac-terizationoftheinspectedmaterial.Inparticular,forwavepropa-gation,itisnecessarytodetermineboththeelasticpropertiesandthemicrostructureofthematerial.⇑Correspondingauthor.Tel.:+33160737479;fax:+33160736889.E-mailaddress:fabienne.rupin@edf.fr(F.Rupin).0041-624X/$-seefrontmatterÓ2013ElsevierB.V.Allrightsreserved.http://dx.doi.org/10.1016/j.ultras.2013.12.006Theausteniticweldstructuregenerallyexhibitselongatedgrains.Theorientationofthelongaxisofthesecolumnargrainsvariesinsidethewelddependingontheweldingprocessandchamfergeometry[10].Inpreviousstudies,ithasbeenshownthattheweldcouldbedescribedinseveralhomogeneousdomainsinwhichthefrontierandthecolumnargraindirectionaredeter-minedeitherbyimageprocessing[6,7,11]orusingtheMINAmod-elwhichusesinformationextractedfromtheweldingnotebookandprocesstopredictthegrainsorientations[12,13].Theelasticconstantsassociatedwitheachdomainarethenassessedusingultrasonic[14,15]orX-raydiffraction[16]measurementsobtainedinahomogeneoussamplewhereallthecolumnargrainsarealigned.Thisscaleofdescriptioncombinedwithanaccuratenumericalmodelisabletosimulatecomplexwavepropagationphenomenasuchasbeamdistortionandghostechoes.However,theinputdataareprovidedbyexperimentalmeasurementsandareconsequentlytaintedwithuncertainty.Thus,itbecomesneces-sarytoevaluatetheimpactofthisuncertaintyontheoutputsofthenumericalmodel.Uncertaintypropagationcanbeperformedinsimulationbyusingaprobabilisticapproach.Inthiscontext,theinputparame-terscanbedescribedasrandomvariableswithappropriatestatis-ticaldistributions.ThustheFEmodelcanbeevaluatedatvariousrandomlysampledvaluesoftheinputparameters(MonteCarlomethod).Eventuallythestatisticalpropertiesoftheresultingsetofoutputparameterscanbeanalyzedthrough,forexample,the1038F.Rupinetal./Ultrasonics54(2014)1037–1046computationoftheirstatisticalmoments,suchasthemeanandthestandarddeviation.Moreover,evaluatingthesensitivityofthere-sponsestoeachinputvariableorinteractionsthereofisofparticu-larinterest.Sensitivitymeasurementsbasedontheoutputvariancehavebeenwidelyusedforthelasttwodecades.TheyaresometimesreferredtoastheSobol’sindices[17].ThelattermaybeestimatedusingacrudeMonteCarloscheme,meaningtherandomsamplingofalargenumberofinputparameters.How-everitrequiresahugenumberofmodelevaluationstogetsuffi-cientlyaccurateestimates.Toovercomethisproblem,ithasbeenshownintheliteraturethatitwaspossibletosubstituteatime-consumingmodelbyananalyticalformula,knownasmetamodel,responsesurfaceorsurrogatemodel,whichisfastertoevaluate,seeforinstance[18].Inparticular,underrelativelyweakassump-tions,therandomoutputsmayberepresentedexplicitlyinabasismadeofpolynomialswhichareorthonormalwithrespecttothejointdistributionoftheinputvariables.Sucharepresentationisknownasthepolynomialchaos(PC)expansion[19,20].Inthiscon-dition,theprobabilisticfeaturesofthemodelresponsearecom-pletelydeterminedbycomputingthePCcoefficients,i.e.theresponsecomponentsinthePCbasis.ThePCmethodologyre-vealedparticularlyefficienttocomputethesensitivityindicesofvarioustestfunctionsaswellasmorecomplexphysicalmodels[21,22].Inthisstudy,themainobjectiveistoapplyasensitivityanalysisonawelldocumentedweldinspectionconfigurationinordertobothdeterminetheimpactofinputdatauncertaintiesonnumeri-calsimulationandtoevaluatetheweightofeachinputparameterinthecomputation.Toaddressthesequestions,amethodologywasimplementedtocoupleaFiniteElement(FE)code(ATHENA2D)withaprobabilisticsoftware(OpenTURNS)inordertoperformastatisticalanalysisonanimportantnumberofsimulations.ThispaperthuspresentsanewapproachtousenumericalsimulationsofUTinspectionofausteniticweldsandinparticular,awaytoaddconfidenceinter-valstonumericalresults.2.Numericalmodel2.1.ATHENA2DThenumericalsimulationswereperformedwiththe2DversionoftheEDF(ElectricitédeFrance)FiniteElements(FE)codeATHE-NA.TheprincipalcharacteristicsofATHENAhavealreadybeenpublishedelsewhere[23,24].Inshort,thecodeisdedicatedtothesimulationofwavepropagationinallkindsofelasticmediaandinparticular,heterogeneousandanisotropicmaterialslikewelds.Itisbasedonthesolvingoftheequationsofelastodynamicinthecalculationzoneexpressedwiththestressandvelocitiesofdisplacements(Eqs.(1)and(2)):@r@t¼CeðvÞinXð1Þq@v@tÀdivr¼finXð2Þwherevisthevelocityofthedisplacement,eðvÞ¼1ðrvþrTthedeformationtensor,Cisthesymmetricelastic2vÞistensor,qisthedensity,fisthesourceandXisthecaculationdomain.Theparticularityofthecodereliesonthefactthatthediscret-isationofthecalculationdomainusesasquareandregularmeshwhilethedefectofcomplexgeometrycanbedescribedusingaseparatemeshusingthefictitiousdomainsmethod.Thisallowscombiningthespeedofregularmeshcalculationswiththepossi-bilitytomodelarbitrarilyshapeddefects.Furthermore,ATHENAgivesthepossibilitytouseperfectlymatchedabsorbinglayerstodefinetheboundariesofthecalculationdomaininordertoavoidreflectioncomingfromthedomainlimits.Finally,ATHENAintegratesthepossibilitytosimulatevariousinspectionconfigurations(pulse-echo,tandem,TOFD)withawiderangeoftransducers,materialsanddefects.2.2.InspectionconfigurationTheinspectionconfigurationwastodetectamanufacturedde-fectlocatedina40mmthickVgrooveweldmadeof316Lsteel.Theabovementioneddefectisasidedrilledhole(SDH)of1.5mmdiameterimplantedat25mmdepthinthebasemetalandat15mmfromthecentralaxisoftheweld.Thecharacteristicsoftheweldhavealreadybeenpublished[6,11,15]andithasbeenshownthatthe2DversionofATHENAwasabletoreproducetheexperimentalamplitudeanddeviationofthebeamduringitsprop-agationthroughtheweld.The2Dapproximationwasreliableinthisparticularcasebecausethepropagationplaneisaplaneofsymmetryoftheorthotropicmaterial.Theinputdataofthesepre-viousstudiesaresummarizedinFig.1andTables1and2.Theweldwasdescribedasaheterogeneousmediummadeof7homoge-neousmaterials.Eachmaterialwascharacterizedbytheorienta-tionofthemainaxisofthecolumnargrainswithregardtotheverticalaxis.Thedeterminationoftheorientationwasperformedbyimageprocessingofametallographicpictureoftheweld(Fig.1).Theelasticcoefficientofthe316Lsteelweldweredeter-minedusinganultrasonicprocessperformedonahomogeneoussampleharvestedintheweld[14,25].Onlythefourcoefficientsin-volvedintheequationsofthepropagationinthesymmetryplaneareusedbyATHENA.Theultrasonicprobewasa20mmdiametersingletransducerwitha2.25MHzcentralfrequency.Aplexiglasswedgewasusedinordertoobtaina45°incidenceangleforlongitudinalwave(L45)inthematerial.TenexperimentalBscanswereproducedatvariouspositionswithregardtotheweldingaxis.Thescanningstepandincrementwidthwere0.5and2mmrespectively.Thesimulatedinspectionconfigurationwasa18stepBscan(Fig.1).Thestepwidthwastwicetheexperimentalone(1mm)inordertoreducecalculationtime.ThisdifferencewasconsideredtohavelittleinfluenceontheBscanwithregardtotheprobecharacteristics.2.3.ParametersidentificationTheinputparametersofthecodeweredefinedpreviously.However,inthisstudy,onlytheelasticcoefficientsandtheorien-tationsofthecolumnargrainsofthewelddescriptionareconsid-eredtobeimperfectlyknownastheyarederivedfromexperimentalevaluationandconsequently,exhibitacertaindispersion.Theoutputparametersofthemodelconcernedbytheuncer-taintypropagationaretheamplitudeofthedefectechoesA,itstimepositiontandthepositionpofthetransducerwhenthemax-imumofthedefectechoisrecorded(Fig.2).TheyareobtainedfromtheanalysisofthesimulatedandexperimentalBscans.Finally,11inputparameters(4elasticcoefficientsand7orien-tations)and3outputparameterswereidentified.3.ProbabilisticapproachtoperformuncertaintyandsensitivityanalysesForthesakeofformalization,theinputparametersofthesim-ulationmodel,namelytheelasticconstantsandtheorientations,aredenotedbyxiandaregatheredinavectorx.Themodelre-sponses,namelythemaximumamplitudeandtherelatedtimeF.Rupinetal./Ultrasonics54(2014)1037–10461039Fig.1.Metallographicpicture(left)anddescriptionoftheweldin7homogeneousdomains(middle)andinspectionconfiguration(right).Table1

Elasticcoefficients(GPa)ofthe316Lstainlesssteelwelddeterminedbyultrasonicmeasurements.TheelastictensorisexpressedusingtheVoigtnotationwiththeplane13asthemainplane.C11246C33218C13148C551053.2.UncertaintyanalysisbasedonMonteCarlosampling3.2.1.PrinciplesoftheMonteCarlomethodMonteCarlosamplingisprobablythemostclassicalschemeforcarryingoutanuncertaintyanalysis.Thisapproachcanbeoutlinedasfollows:(1)Generatearandomsample{x(1),...,x(N)}oftheinputrandomvectorXforagivensamplesizeN.(2)Foreachx(i),performasimulationf(x(i))andcollecttheasso-ciatedoutputvectory(i).(3)Conductastatisticalanalysisoftheobtainedresponsesam-ple{y(1),...,y(N)}(forinstancedrawingthehistogramorcomputingthesamplemomentssuchasthemeansandthestandarddeviationsoftheresponsecomponents).3.2.2.EstimatesofthemomentsoftheresponsecomponentsLetusconsidersomecomponentYjofY.TheMonteCarloesti-mateofitsmeanvalueisgivenby:Table2

Orientationh(°)oftheprincipalaxisofthecolumnargrainsofeachhomogeneousdomainoftheweld.DomainOrientation13521831441158617-31andposition,aredenotedbyyiandaregatheredinavectory.Themodelisrepresentedbyafunctionfthattakesxasargumentsandthatreturnsaresponsey=f(x).3.1.ProbabilisticframeworkAsmentionedintheprevioussection,theinputparametersareaffectedbyuncertainty.Thisistakenintoaccountbydescribingthemasrandomvariables,denotedbyX=(X1,...,XM),withpre-scribedprobabilitydistributions.Afteruncertaintypropagationthroughthemodel,theresponsesarealsorandomvariablesde-notedbyY=(Y1,...,YQ)=f(X).ThustheobjectiveoftheanalysisistoestimatethedistributionofYandtodeterminesomerelatedquantitiesofinterestsuchasthemeanvalues,thestandarddevia-tionsand90%-uncertaintyintervalsoftheYi’s.Wearealsointer-estedinquantifyingthesensitivityofthemodelresponsestoeachinputrandomvariableandcombinationsthereof.^YjlYj󰀃E½Yj󰀄%lN1XðiÞ󰀃yjNi¼1whereE[Z]denotesthemathematicalexpectationofanyrandomvariableZofprobabilitydensityfunctionpZ(z)andsupportDZ:E½Z󰀄¼ZDZzpZðzÞdzMoreover,theestimateofthevariancereads:^2r2Yj󰀃Var½Yj󰀄%rYj󰀃N21XðiÞ^YjÞðyjÀlNÀ1i¼1whereVar[Z]denotesthevarianceofanyrandomvariableZ:Fig.2.Ontheleft:exampleofBscansimulatedwithATHENA2D.Ontheright:theAscancorrespondingtothemaximumamplitudeoftheecho.Theparametersofinterestarethemaximumofamplitudeoftheecho(maximumordinateoftheAscan),itstimeofflight(abscissaoftheAscan)andtheprobepositioncorrespondingtothemaximumofthedefectecho(abscissaoftheBscan).1040F.Rupinetal./Ultrasonics54(2014)1037–1046Var½Z󰀄¼E½ðZÀE½Z󰀄Þ2󰀄IfthesamplesizeNislargeenough,theso-calledcentrallimittheoremprovidesthefollowingconfidenceintervaloflevela(e.g.a=95%or90%)fortheactualmeanl\"Yj:Pl^r^Yjr^Y#jYjÀtð1ÀaÞ=2pffiffiffiNffi6lYj6l^Yjþtð1ÀaÞ=2pffiffiffiffiN¼awheret(1-a)/2denotesthequantileoflevel(1Àa)/2oftheStudentdistributionwith(NÀ1)degreesoffreedom,whichcanbecom-putedinmostscientificsoftwarelibrariesorfoundinstatisticalhandbooks.Asexpected,theprecisionoftheestimateincreaseswithN.Althoughaconfidenceintervalfortheresponsevariancecannotbederivedinsuchaclosedform,itcanbeestimatedusingabootstrapmethod[26].3.2.3.UncertaintyintervalsoftheresponsecomponentsLetusconsiderthesamplefyð1ÞðNÞj;...;yjgofacomponentYiofthemodelresponseresultingfromMonteCarlosimulations.Thissampleissortedinincreasingorderandthesortedsampleisde-notedby:yÃð1Þj6ÁÁÁ6yÃðNÞjTheempiricalquantileoflevelcisdefinedasthe[cN]-thelementyÃð½cN󰀄Þjinthesortedsampleinwhich[z]representstheclosestinte-gertotherealnumberz.Forinstance,ifN=100,theempiricalquantileoflevelc=5isthefifthsmallestelementinthesample.Inthiscase,theuncertaintyintervaloflevelafortheoutputYicanbeapproximatedby:PhyÃð½Nð1ÀaÞ=2󰀄ÞÃð½Nð1þaÞ=2󰀄Þij6Yj6yj%að3Þ3.3.Sensitivityanalysisbasedonpolynomialchaosexpansion3.3.1.DefinitionofglobalsensitivityindicesTheevaluationofthesensitivityofthemodelresponsesYjtoeachinputrandomvariableXiwillbeinvestigatedthroughthecomputationofparticularstatisticscommonlyreferredtoastheSobolsensitivityindicesintheliterature:SVar½E½YjjXi󰀄󰀄i½Yj󰀄¼Var½Yj󰀄whereE½ÁjÁ󰀄denotestheconditionalexpectationoperator.Broadlyspeaking,theSobolindicesrepresentthepartofthetotalvarianceofYjthatisexplainedbythevarianceofthevariableXitakensepa-rately(i.e.nointeractionwithothervariablesaretakenintoac-count).TheeffectoftheinteractionbetweentwovariablesXi1andXi2isquantifiedbythefollowingindex:S¼Var½E½Yj󰀃󰀃XiXi󰀃󰀃1;2󰀄ÀE½Yj󰀃Xi1󰀄ÀE½Yj󰀃Xi2󰀄󰀄i1;i2½Yj󰀄Var½Yj󰀄Itisalsopossibletoevaluatetheinteractioneffectbetweenanynumberofvariables.However,wewouldrathercomputeinprac-ticetheso-calledtotalSobolindicesinordertoevaluatetheimpactofavariableXitakenseparatelyaswellasininteractionwithanyothervariable.Theseindicesarecastas:STi½Yj󰀄¼Si½Yj󰀄þXSi;k½Yj󰀄þXSi;k;l½Yj󰀄ÁÁÁþS1;2;...;M½Yj󰀄k–ikC33>C13>C55Precisely,thelowerboundsaresetequalto95%andtheupperboundsto105%ofthemeanvalues:ai¼0:95li;bi¼1:05liMoreover,theshapeparametersaiandbiarechosensuchthatthemeansoftherandomvariablesareequaltoexperimentallyob-servedvaluesliandthattheircoefficientsofvariation(i.e.theirstandarddeviationsdividedbytheirmeans)areallequalto2.5%.Thecoefficientofvariationwasestimatedusingthedispersionob-servedintheliteratureandtheerrorobservedontheexperimentaldeterminationoftheelasticconstants[15].ThepropertiesofthedistributionsofthevariousinputrandomvariablesarereportedinTable3.4.2.DistributionanalysisTheprobabilisticapproachespresentedintheprevioussection,namelyMonteCarlosamplingandPCexpansion,wereappliedtothesimulationofultrasoundnondestructivetesting.ThemethodsareimplementedbycouplingthefiniteelementcodeAthena2DwiththeopensourcesoftwareOpenTURNS[33]developedbytheEADS,EDFandPhimecacompanies.OpenTURNSisdedicatedtothetreatmentofuncertaintyinnumericalsimulationandincludesmanybasicandadvancedstatisticalmethods.ItisavailableasamoduleofthePythonlanguage.OnceOpenTURNSiscoupledtoanumericalmodel,itisabletogeneratevariousinputparameterssetsaccordingtoaspecificexperimentaldesign,tolaunchthecor-respondingconfigurationcalculationsandfinally,toperformthestatisticalanalysesoftheoutputparameters.Thecalculationswerecarriedoutusingacalculationclusterandexhibiteda2mincalcu-lationdurationforonepositionoftheprobe.Furthermore,asATHENAallowsparallelcomputing,itispossibletocalculateeachofthe18probepositionsatonce(using18coresoftheclustersimultaneously).Inaddition,aPythonmoduleallowedthelaunchofseveralcalculationconfigurationssimultaneouslyonseveralnodes.Forthesakeofefficiency,thejobqueueoftheclusterwasmonitoredinsuchawaythatanewcalculationwasperformedassoonasanodewasidentifiedidle.Thisledtoasubstantialtimegainwhenperformingintensivesimulationcomparedtoapurelysequentialscheme.Finally,atotalofN=6000MonteCarlosimula-tionsofthefiniteelementmodelwereperformed.Inthepresentcase,6000Bscanimageswereobtainedafterafewdaysofcalcula-tion.Inordertoautomaticallyextracttherelevantinformationofeachimage,aroutinewasdevelopedtofindnumericallythemax-imumamplitudeofthedefectechoaswellastherelatedtimeandtransducerposition.ThehistogramsoftheresultingN–samplesofresponses(max-imumamplitude,positionandtime)weredrawnforavisualTable3Propertiesofthedistributionsoftherandominputvariables.VariablesDistributionsMomentsBoundsC11(GPa)Betal1=247;r1=6a1=235;b1=259C33(GPa)l2=218;r2=5a2=207;b2=229C13(GPa)l3=148;r3=4a3=141;b3=155C55(GPa)l4=105;r4=3a4=100;b4=110h1(°)Normall5=35;r5=5–h2(°)l5=18;r6=5–h3(°)l5=14;r7=5–h4(°)l5=11;r8=5–h5(°)l5=8;r9=5–h6(°)l5=1;r10=5–h7(°)l5=À31;r11=5–inspectionoftheoutputprobabilitydistributions(Figs.3–5).Ithastobenotedthattheoutputquantitieswerenormalizedasfollows:~yyiÀyi;mini¼y2½0;1󰀄i;maxÀyi;minwhereyi,minandyi,maxdenoterespectivelytheminimumandthemaximumofthecomputedvaluesofvariableYi.Figs.4and5revealtheexistenceofanunexpectedbimodalre-sponseofthetimeandthepositionparameters(fromastatisticalpointofviewbimodalitymeanstheexistenceoftwolocalmaximaofthedensityfunction).Precisely,alocalmaximumofthedistribu-tionofthepositionisobservedatverylowvaluesofthisparameter(i.e.closetozero)whilealocalmaximumisalsoobservedforquitehighvaluesofthetime(i.e.closeto0.85).Inordertoidentifythesourceofthebimodality,afewBscansassociatedwitheachofthetimemodeswereexamined.OneBscanrelatedtothemajortime,andanotheronerelatedtotheminortimemodearecomparedinFig.6.TheBscanrelatedtothemajormoderevealsaneasilyrecognis-ableSDHecho.Onthecontrary,theBscanassociatedwiththemin-ormodeshowsaspreadecho,whichbeginsattheveryfirstpositionoftheB-scanandexhibitsatimepositionofthemaximumamplitudeinconsistentwiththepositionofthemanufacturedde-fect.Forthesakeofunderstanding,thewavefieldsassociatedwiththetwosimulationcasesarecalculatedattheinitialpositionoftheprobe(Fig.7).Fig.7showsthatintheminormodecase,theconfigurationfa-vouredtheinteractionbetweenthewaveandthetaperoftheweld.Thishasbeenmadepossiblebythefactthatsmallchangesofthematerialpropertiesinducedmodificationsofthewavefield(skewing,splitting,divergence,etc.)sothatthewavecanintercepttheinnerweldbeadwithanalmostspecularcondition.Themod-ificationofthebeamcanbeexplainedbythefactthattheultra-sonicbeamcanbeeitherfocusedordivergedinfunctionoftheincidenceanglethatexistsbetweenthewaveandthecolumnargraindirection.Indeed,whentheincidenceanglebetweenthelon-gitudinalwaveandthegrainmainaxisisabout45°,aconvergencephenomenonoccurs.Onthecontrary,whenthisincidenceangleisaround0°and90°,itleadstoadivergenceofthebeam[34].Inaheterogeneousmedium,thepredictionoftheinfluenceofthecolumnarmicrostructureonthebeampropagationcanbedifficultandconsequentlyitsassessmentnecessitatestheuseofnumericalsimulation.Finally,inthisconfiguration,thedisturbancesoftheultrasonicbeamledtointerferencebetweentheechoescomingfromthetaperoftheweldandtheSDHinducingtheimpossibilityofuncouplingthesetwocontributions.ThisinterpretationwasconfirmedbycomputingaconfigurationusinganinputdatasetFig.3.HistogramoftheMonteCarlosampleofnormalizedamplitudes.F.Rupinetal./Ultrasonics54(2014)1037–104610434.3.UncertaintyandsensitivityanalysesofthemaximumamplitudeInthissection,wefocusedonastatisticalanalysisoftheMonteCarlosimulationsthatyieldednonproblematicinterpretations.Inotherwords,werejectedfromtheresponsesampleallthesimula-tionsrelatedtotheminormodeofthedetectiontime,leadingtounimodaldistributionsofthethreemodelresponses.Precisely,thesimulationsleadingtoadetectiontimegreaterthan0.5werediscarded.Eventually5632simulationsoutof6000wereretained.4.3.1.Uncertaintyanalysis90%-uncertaintyintervalsofthemaximumamplitude,thetimeandthepositionwereestimatedusingthequantile-basedmethodoutlinedinSection3.2.3.TheirboundsarereportedinTable4to-getherwiththemeanvaluesoftheparametersaswellastherel-^Yi,withaiativelengthsoftheintervalsdefinedbyli¼ðbiÀaiÞ=l^YidenotingandbidenotingthelowerandtheupperboundsandlthesamplemeanofvariableYi.Itappearsthattherelativelengthassociatedwiththeamplitudeandthetimewererespectivelyequalto33%(correspondingtoavariationof2.8dB)and6%.Be-sides,thecorrespondingdispersionsobservedinexperimentalmeasurementswererespectivelyequalto26%and6%(thesemi-lengthsoftheconfidenceintervalsweresetequalto1.63rwhereristhestandarddeviationoftheexperimentalvalues).Neverthe-less,thecomputedamplitudevariabilitywasfoundtobelargerthantheexperimentalone.Thisdiscrepancycouldbeexpectedforseveralreasons.Firstofall,thestatisticalstudyhasbeenper-formedusingconservativestatisticalassumptions.Inparticular,thedispersionoftheelasticpropertieswasobtainedfromabiblio-graphicstudyandthusincludestwotypesofvariability:theonethatisobservedfromoneweldtoanother(manufacturedusingthesameprocess)andtheonecomingfromtheevolutionoftheweldmicrostructureintheweldingdirection.Secondofall,theexperimentalvariabilitywasdeterminedfromonly10B-scansre-cordedatvariouspositionswithregardtotheweldingdirection.Thiscouldconducttoanunderestimationoftheexperimentaldis-persion.Finally,thevariabilitycouldbeenhancedbytheparame-terindependencehypothesis.Indeed,thisleadstoevaluatetheFEmodelwithunrealisticinputdatasets,inducingagreatervarianceoftheresponses.Concerningtheprobepositionp,therewasverylittlevariabilityobservedbothinexperimentalandcomputeddata:themaximumwasfoundatthe9th(±1)probeposition.Usingthevaluesoftheoutputparametersobtainedpreviouslyandacalibrationprocedureintheisotropicmedium,itispossibletorepositionthedefectbothvertically(Z)andhorizontally(X).ThemeanvaluesledtoarepositioningerrorofDZ=À2.9mmindepthandDX=0.9mminthelateraldirection.ThisisconsistentwithFig.4.HistogramoftheMonteCarlosampleofpositions.Fig.5.HistogramoftheMonteCarlosampleoftimes.leadingtotheminorbutwithaflatbottomsurface,i.e.theweldtapperandinnerbeadareremoved.TheresultingBscanonlyexhibitedtheSDHandasexpectednospuriousechoeswereob-served.Nevertheless,theexistenceoftheminormodeisparticu-larlytroublesomebecauseitmaypreventfromcorrectlyinterpretingtheultrasonicinspection.Inaddition,fromamethod-ologicalviewpoint,thebimodalityoftheresponsedensityfunc-tionsmayinduceaverypoorconvergencerateofthePCapproximation,whichmaythusleadtointractablecalculations.Nevertheless,only6%ofthesimulationswereconcernedbythismixingoftwodifferentechoes.Fig.6.TwoBscanstakenfromtheMonteCarlosamplingofmodeloutputs.TheBscanontheleftcorrespondstothemajormodeofthetimedistribution.Theoneontherightcorrespondstotheminormode.1044F.Rupinetal./Ultrasonics54(2014)1037–1046Fig.7.Wavefieldsattheinitialpositionofthesensorsforthetwosimulationsrelatedtotheregulardetectionpattern(left)andtheminormode(right).Table4Meanvaluesandboundsofthe90%-uncertaintyintervalsoftheoutputparameters.Max.amplitude(Â10À11a.u.)a16.1^Y1l7.2b18.5l10.33Time(ls)a215.05^Y2l15.41b215.77l20.06experimentalresultsobservedpreviously[35].Furthermore,theconfidenceintervalsdeterminedthroughtheuncertaintyanalysisallowanassessmentofthedispersionontherepositioningvalues.Inthepresentcase,thedispersionintheprobepositionhadnoim-pactonZpositionoftheechowhileitinducedanincreaseofDXof2mm.Onthecontrary,thetimedispersionplayedaroleonbothXandZ.TherangeofDXwasÀ0.1toÀ1.6mmwhiletherangeofDZwasÀ2.3toÀ3.6mm.4.3.2.SensitivityanalysisThesensitivityanalysisiscarriedoutonlyfortheoutputparam-eterexhibitingasignificantvariability,namelytheechoamplitude.Inthisrespect,aPCapproximationofthemaximumamplitudewasconstructed.ItwasmadeofnormalizedHermitepolynomialsanditscoefficientswereestimatedusing2500valuesamongthewholeMonteCarlosample.TherestofthesamplewasusedtoevaluatetheR2accuracycoefficient.AnoptimalvalueofthenumberPoftermsintheremainderofthePCserieswasdeterminedbycom-putingR2forvariousvaluesofthePCsize,andbyretainingeven-tuallythePCassociatedwiththehighestR2.TheoptimalPwasequalto364,whichcorrespondstoaPCcontainingHermitepoly-nomialswithatotaldegreenotgreaterthan3(cubicapproxima-tion).TheassociatedR2wasequalto0.8.Accordingtotheauthors’experience,thiscorrespondstoanaccurateenoughapproximationinordertocalculatereliableestimatesoftheSobolindices.The‘‘simple’’andthetotalSobolindiceswerederivedfromthePCcoefficientsaccordingtoEqs.(8)and(9).Theirvaluesareplot-tedinFig.8.Thereisasignificantimpactofvariousvariableinter-actionsasthesumofthetotalindicesisstronglygreaterthan100%.Itisobservedthatalmostallinputparametershaveanon-negligibleimpactonthemaximumamplitudewheninteractionswereaccountedfor(excepttheelasticconstantC55whosesimpleandtotalSobolindicesarelessthan5%).Itcanalsobenoticedthatparametersh1,h2andh3onlyinfluencetheresponseininteractionwithothervariables.h1andh7appeartobethemostimportantvariables.Thiscanbeexplainedbythefactthattheycorrespondrespectivelytothefirstandthelastmediathatthewavecrossesintheweld,priortointeractingwiththedefect.Inaddition,h1mostlyinfluencestheresponsewheninteractionwithotherparametersaretakenintoaccountwhereash7influencesthere-Fig.8.SimpleandtotalSobolindicesofthemaximumamplitudewithanincidenceangleequalto45°.Thevalueofeachindexisgiveninthelegend.sponseseparately.Surprisingly,onlytheelasticcoefficientC33hap-penedtohaveasignificantinfluenceontheamplitudevariability.Thepossibilitytoidentifythemostinfluentparameters,whentherelationshipbetweentheinputandtheoutputofthemodelisnotstraightforward,isofgreatinterest.Inparticular,itcanprovideusefulinformationwithregardstothequalificationprocesswhichrequiresthedeterminationoftheimpactofvariousparametersoninspectionresults.Inaddition,everyparticularconfigurationcanexhibitdifferentparametersofinfluence.Toillustratethispoint,thesamestudywasperformedonthesameconfigurationbutwitha60°incidenceangle(L60).TheresultsaresummarizedinFig.9.Itcanbeshownthattheclassificationoftheinfluenceoftheinputparametersisdifferentfromthe45°configuration(L45).Thesig-nificantelasticcoefficienthappenedtobeC11whereastheorienta-tionsofdomains2and4becomeasimportantastheorientationofdomain1.Nevertheless,theorientationofdomain7remainedthemostimportantparameter.5.DiscussionInthisstudy,theuncertaintyanalysisofaFEmodelofaweldultrasonicinspectionwasperformedusingprobabilisticap-proaches.Inparticular,adirectMonteCarlosamplingschemewasusedinordertoestimatethemeanvaluesoftheoutputvari-ablestogetherwiththeir90%-confidenceintervals(CI).Inaddition,theso-calledpolynomialchaosmethodenabledthederivationofsensitivityindices(theSobolindices).Theuncertaintyanalysisrevealedthattheamplitudewasaf-fectedbyasignificantvariabilitywhereasthetimeandtheprobeF.Rupinetal./Ultrasonics54(2014)1037–10461045positionsoftheechowerelittleaffectedbythevarianceofthein-putparameters.Thelatterresultwasinagreementwithamea-sureddispersionobtainedbyapreviousexperimentalinvestigation.Onthecontrary,thecomputedvariabilityoftheamplitudewaslargerthantheexperimentalone.Howeverthiswasexpectedsincethemeasuredvarianceprobablyunderesti-matestheactualoneandthesimulateddispersionisbasedoncon-servativeassumptions.Furthermore,thestrongprobabilistichypothesisofindependenceledtoaconservative90%-CIastheFEmodelwasevaluatedforunrealisticsetsofinputparameters,increasingthevariance.However,thecurrentmethodologysuc-ceededinreproducingtheexperimentalvariability.ThepossibilitytoestimateconfidenceintervalsontheoutputparametersofATHENAisofgreatimportancewithregardtothevalidationofthecode.Indeed,itgenerallyconsistsincomparisonsbetweenexperimentalandcomputeddata.Whendifferencesexistbetweenthetwokindsofdata,thequestionofthesignificanceofthesedis-crepancieshastobeaddressed.Thisnecessitatestheknowledgeofthevarianceofbothresults.Furthermore,theuncertaintyanalysisshowedthatthetimeandtheprobepositionsofthemaximumoftheechowerebothlittleaffectedbytheuncertaintiesontheinputdata.Asaconsequence,inthisinspectionconfiguration,theuncer-taintyanalysisshowedthattherepositioningofthedefectwaslit-tlesensitivetothevariabilityoftheinputdata.Finally,theuncertaintyanalysisprovidesveryusefulinformationwithregardtothequalificationprocedureoftheUTmethodswhichnecessi-tatestheevaluationoftheinfluenceofeachparameteroninspec-tionresults.Accordingtothesensitivityanalysis,therewasalmostnoneg-ligibleinputvariable.Nonetheless,theorientationsplayedamajorroleforexplainingtheoutputvariance,especiallyh1andh7whichcorrespondtothedomainsneartheweldchamfer.Moregenerally,theoverallorientationparametersrepresent75%and83%ofthevariancefortheL45andL60configurationsrespectively.Thiscon-firmsthatanaccuratedeterminationoftheweldmicrostructureisofcrucialimportanceinsimulationstudieswithATHENA.More-over,forthisconfiguration,theuncertaintyexistingontheelasticcoefficientsseemstohavelittleinfluenceontheresults.Thedeter-minationoftheelastictensorappearsconsequentlytobelesscrit-ical.Nevertheless,thedifferenceobservedbetweentheL45andtheL60casesshowsthenecessitytosystematicallycarryoutasensi-tivityanalysistoestimatetheinfluentialparameterswhentherelationshipbetweeninputandoutputdataiscomplex.However,ithastobenotedthattheattenuationwasnottakenintoaccountinthisworkbecauseitwouldhaverequirednontrivialadaptationsoftheOpenTURNS–ATHENAcouplingscheme.In-deed,theconsiderationoftheattenuationinATHENAisperformedbyintroducingaspecifictensorDinEq.(1)leadingtothenewFig.9.SimpleandtotalSobolindicesofthemaximumamplitudewithanincidenceangleequalto60°.equation:@rbythelocal@tþDr¼CeðvÞ(3).Inthiscase,theweldisdescribedorientationofthegrainsandthetwotensorsCandD.Thetwolatteraredeterminedbyadjustingexperimentaldataofbothvelocitiesandattenuationmeasurements.Intheelasticcase(whenD=0),theChristoffel’seigenvaluesaredirectlyrelatedtothelongitudinalandthetransversalvelocitieswhereasinthemodelwithattenuation,fromacomplextensor~theChristoffel’stensorisconstructedC¼ixðixþDÞÀ1C(4).ConsequentlytheChristoffel’seigenvaluesarecomplexandcontainboththeveloci-tiesandtheattenuationsofthetransverseandcompressionwaves[6,36,37].Finally,Eqs.(3)and(4)inducethefactthatDandCarelinkedtogethervia~C:Inthisconfiguration,anyvariationoftheattenuationwillleadtoamodificationoftheelasticity.TheOpen-TURNS–ATHENAcouplingschememustthenbeadaptedinordertobeabletodriveoneparameterindependentlyfromtheother.Thisdevelopmentisunderprogressandwillbethesubjectoffur-therpublications.Inaddition,theprobabilisticdistributionsoftheinputrandomvariablesshouldbeselectedinamorerealisticfashion.Forexam-ple,itislikelythattheerrorbarsarenotidenticalfromoneelasticcoefficienttoanother[14].Furthermore,itwouldberelevanttotakeintoaccounttherelationshipsdependencebetweentheorien-tations.Inparticular,itisnecessarytoproperlyrepresentthecor-relationsbetweentwoneighbordomains.Suchanimprovementoftheweldmicrostructuredescriptionwouldprobablyreducetheappearanceofinterferencebetweentheinnerweldbeadandthedefectechoes.Indeed,suchamixedindicationhasnotbeenob-servedexperimentallyandcanpossiblyresultfromunrealisticsetsofinputparameters.However,thiswillrequiretomodifytheprob-abilisticmodelandhencetoperformawholenewsetofsimula-tions.Moreover,theSobolindicesarenolongervalidinthepresenceofstatisticalcorrelation.Thusalternativeimportancefac-torswillhavetobecomputed,suchasindicesbasedonacovari-ancedecompositionofthemodelresponse[38],whichresultsfromageneralizationoftheusualvariancedecomposition.Othersensitivitymeasures,referredtoasmomentindependent[39],willalsobeinvestigated.Broadlyspeaking,suchmeasuresquantifytheshiftbetweentheprobabilitydensityfunctionoftheresponseandtheoneobtainedwhenfixingeachinputparameter.Asaconclusion,despitetheabovementionedlimitations,thepresentstudyledtothedefinitionofageneralmethodologytodeterminatethevariabilityoftheoutputparametersandtoiden-tifythemostinfluentialonesforanynumericalsimulationofultra-sonicweldinspection.Finally,theuncertaintyanalysisrepresentsanimprovementintherobustnessandtheaccuracyassessmentofthesimulationresults.References[1]R.Alford,K.Kelly,D.M.Boore,Accuracyoffinite-differencemodelingoftheacousticwaveequation,Geophysics39(6)(1974)834–842.[2]P.Fellingeretal.,NumericalmodelingofelasticwavepropagationandscatteringwithEFIT-ElastodynamicFiniteIntegrationTechnique,WaveMotion21(1)(1995)47–66.[3]G.Ghoshal,J.Turner,Numericalmodeloflongitudinalwavescatteringinpolycrystals,IEEET.Ultrason.Ferr.56(7)(2009)1419–1428.[4]P.Calmon,O.Roy,SimulationofUTexamination:modelingofthebeam-defectinteraction,Rev.Progr.QNDE13(1994)101–108.[5]S.Shahjahanetal.,Comparisonbetweenexperimentaland2-DnumericalstudiesofmultiplescatteringinInconel600Òbymeansofarrayprobes,Ultrasonics54(1)(2014)358–367.[6]B.Chassignoleetal.,ModellingtheattenuationintheATHENAfiniteelementscodefortheultrasonictestingofausteniticstainlesssteelwelds,Ultrasonics49(8)(2009)653–658.[7]B.Chassignoleetal.,Ultrasonicandstructuralcharacterizationofanisotropicausteniticstainlesssteelwelds:Towardsahigherreliabilityinultrasonicnon-destructivetesting,NDTE.Int.43(4)(2010)273–282.[8]B.Chassignoleetal.,Ultrasonicexaminationofanausteniticweld:illustrationofthedisturbancesoftheultrasonicbeam,Rev.Progr.QNDE28(B)(2008)1886–1893.1046F.Rupinetal./Ultrasonics54(2014)1037–1046[9]H.Hannemannetal.,Ultrasonicwavepropagationinreal-lifeausteniticV-Buttwelds:numericalmodelingandvalidation,Rev.Progr.QNDE19(A)(1999)145–152.[10]Wiley,WeldingMetallurgy,seconded.,vol.480,2002.[11]B.Chassignoleetal.,Ultrasonicpropagationinausteniticstainlesssteelwelds-Approximatemodelandnumericalmethodsresultsandcomparisonwithexperiments,Rev.Progr.QNDE19(A)(1999)153–160.[12]J.Moysanetal.,Modellingthegrainorientationofausteniticstainlesssteelmultipassweldstoimproveultrasonicassessmentofstructuralintegrity,Int.J.Press.Ves.Pip.80(2)(2003)77–85.[13]J.Yeetal.,Influenceofweldingpassesongrainorientation–Theexampleofamulti-passV-weld,Int.J.Pres.Ves.Pip.93–94(2012)17–21.[14]D.Ducretetal.,Characterisationofanisotropicelasticconstantsofcontinuousaluminafibrereinforcedaluminiummatrixcompositeprocessedbymediumpressureinfiltration,ComposPartA:Appl.Sci.Manuf.31(1)(2000)45–55.[15]B.Chassignoleetal.,CharacterizationofausteniticstainlesssteelweldsforultrasonicNDT,Rev.Progr.QNDE19(B)(1999)1325–1332.[16]Butterworths,ed.TextureAnalysisinmaterialsscience:mathematicalmethods,London,1982.[17]I.M.Sobol,Sensitivityestimatesfornonlinearmathematicalmodels,Math.Model.Comp.Exp.1(1993)407–414.[18]R.H.Myers,D.C.Montgomery(Eds.),Responsesurfacemethodology:processandproductoptimizationusingdesignedexperiments,seconded.,Wiley,&Sons,2002.[19]R.G.Ghanem,P.D.Spanos(Eds.),Stochasticfiniteelements-Aspectralapproach,Springer,Verlag,1991.[20]C.Soize,R.Ghanem,Physicalsystemswithrandomuncertainties:chaosrepresentationswitharbitraryprobabilitymeasure,SIAMJ.Sci.Comput.26(2)(2004)395–410.[21]G.Blatman,B.Sudret,Sparsepolynomialchaosexpansionsandadaptivestochasticfiniteelementsusingaregressionapproach,ComptesRendusMécanique336(6)(2008)518–523.[22]G.Blatman,B.Sudret,Efficientcomputationofglobalsensitivityindicesusingsparsepolynomialchaosexpansions,Reliab.Eng.Syst.Safety95(11)(2010)1216–1229.[23]E.Becache,P.Joly,C.Tsogka,Fictitiousdomains,mixedfiniteelementsandperfectlymatchedlayersfor2Delasticwavepropagation,2000.[24]E.Bécache,P.Joly,C.Tsogka,Ananalysisofnewmixedfiniteelementsfortheapproximationofwavepropagationproblems,SIAMJ.Numer.Anal.37(4)(2000)1053–1084.[25]M.A.Ploixetal.,AcousticalcharacterizationofausteniticstainlesssteelweldsforexperimentalandmodellingNDT,J.Adv.Sci.17(1)(2005)76–81.[26]B.Efron,R.J.Tibshirani(Eds.),AnIntroductiontotheBootstrap,Chapman&Hall,NewYork,1993.[27]B.Sudret,Probabilisticmodelsfortheextentofdamageindegradingreinforcedconcretestructures,Reliab.Eng.Syst.Safe.93(3)(2008)410–422.[28]S.K.Choietal.,Polynomialchaosexpansionwithlatinhypercubesamplingforestimatingrespo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