MULTI-MODALDIAGNOSTICANDPROGNOSTICTECHNIQUESFOR NDEAPPLICATIONS By PortiaBanerjee ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof ElectricalandComputerctorofPhilosophy 2018 ABSTRACT MULTI-MODALDIAGNOSTICANDPROGNOSTICTECHNIQUESFORNDE APPLICATIONS By PortiaBanerjee Withrapidtechnologicalbreakthroughs,roleofnon-destructiveevaluation(NDE)has shiftedfromassessingstructuralintegritytobuildingcomplexsystemswithreliabledefect classi˝cationanddecisionmakingcapabilities.WidespreaduseofNDEinindustriessuchas aviation,nuclear,constructionandautomotive,haveresultedinincreasedamountofNDE datawhichisbeyondcapacityforhumananalystsanddemandsautomatedsignalclassi˝ca- tion(ASC)systemsforaccurateandconsistentsignalinterpretation.AtypicalASCsystem processesNDEsignalsandclassi˝essignalcategoriesbasedonappropriatefeatures.Despite strikingbene˝tsofASCsystems,classi˝cationresultsareoftena˙ectedduetoinherentam- biguityofnon-discriminativefeatures,inadequatetrainingsamplesornoisymeasurements. Asaresult,uncertaintyquanti˝cationindefectclassi˝cationiscriticalinNDEapplications wheretheperformanceofastructuredependsonthereliabilityoftheASCresults.Arelia- bilitymeasurethataccountsforsystemuncertaintiescanhelpinmonitoringitsperformance andautomatically˛aggingindicationswhereoperatorinterventionisrequired.Inaddition todiagnosis,i.e.,reliablecharacterizationofcurrenthealthstatus,damageprognosisorpre- dictionofsystem'sremaining-useful-life(RUL)isanotheressentialaspectofNDE.Accurate healthprognosisensuressystemreliabilityandaidsinestimatingresidualserviceabilityof acomponentwhichinturnreducesrepairorreplacementcosts.Moreover,combininginfor- mationfrommultiplesensorsinmulti-modalNDEsystemscane˙ectivelyimprovedamage growthmodelingandpredictionofsystem'sRUL.Thisdissertationpresentsthreemajor contributionstothe˝eldofNDEdiagnosisandprognosis: 1. UncertaintyinASCsystemsisquanti˝edinastatisticalframeworktodevelopacon- ˝dencemetric(CM)associatedwithASCresults.Bybootstrappingandweighting Bayesposteriorprobabilitywithestimatednoisedistribution,e˙ectofmeasurement noiseisembeddedintotheproposedCM.E˙ectivenessoftheCMisdemonstrated onexperimentaldatafromeddycurrentinspectionofsteamgeneratortubes.Fur- ther,thebene˝tofCMinimprovingclassi˝cationperformanceisexploredusinga con˝dence-rated-classi˝cationtechnique. 2. Particle˝ltering(PF)frameworkisdevelopedforpredictionofimpactdamageprop- agationincompositematerialswhichutilizesbothphysicalmodelbasedonmodi˝ed Paris'lawandinspectiondataobtainedfromNDEsystem. 3. JointlikelihoodupdationisproposedinexistingPFalgorithmwhichenablesopti- mizationofdamagemodelparametersateverytimestepbydiscardingnoisyorbiased measurementsfrommultiplesources.Prognosisresultsonacompositespecimensub- jectedtofatiguetestingandinspectedusingtwoNDEmodalities,validatethebene˝t ofmulti-sensorprognosisapproachoversingle-sensormethods.Additionaladvantage ofmulti-sensorpredictioninreductionofparticlecountwithinthePFalgorithmis demonstrated,therebyreducingthetotalcomputationtimeandresources. Overall,areliabilitymetricandprognosismethodologyisdiscussedforamulti-sensor systemthatcanbeextendedtomultipleapplications. Tomyfamily,fortheirunconditionalloveandsupport. iv ACKNOWLEDGMENTS ThisPhDjourneyisaculminationofperseverance,blessings,supportandguidancefrom severalwell-wishersinmylife,towhomIamandwillremaingratefulforever. Firstly,averyspecialgratitudegoesouttomyadvisorsDr.LalitaUdpaandDr.Yiming Deng,wholaidthefoundationofresearchethicsinme.IamthankfultoDr.Udpafor trustingmewithopportunitiesduringmyearlierdaysofPhD,forprovidingtimelyadvice oncareerplansandaidingmeintechnicalwriting.IamgratefultoDr.Dengforestablishing thebasicknowledgeaboutprognosticmethods,providingtechnicalsuggestionswhenIwas stuckwithfailedexperimentsandintroducingmetothereliabilitycommunity.Hisconstant wordsofencouragementhasbeenamajorstress-deterrentandmotivatedmetoadopta careerinpost-doctoralresearch. Iowemyacknowledgmenttootheradvisorsinmycommittee,especiallyDr.Mahmoodul HaqandDr.SatishUdpa.ToDr.Haq,Iamthankfulforhiscriticalinsightsandtechnical know-howinthe˝eldofcompositesandstructuralengineering.Despitehavingabackground ofElectricalEngineering,Iwasabletoconductamajorpartofmyresearchondamage progressionincompositesowingtohisinvaluableguidance.IwouldliketothankDr.Satish Udpaforhisimportantandthought-provokingquestionsregardingpracticalapplicabilityof thisresearchatitsmultiplestageswhichshapedthe˝naloutcome.Therichlegacycreated byhimandDr.LalitaUdpaintheNDEcommunity,willcontinuetoinspiremeandmy colleaguesthroughoutourcareers. IamimmenselygratefultomypastandcurrentcolleaguesofNDElabwhohavebeen friendsaswellasasourceofgoodadviceandcollaboration,particularlyMorteza,Pavel,Ra- jendra,SuhailandZhiyiforassistingmewithseveraltime-intensiveexperimentsandcoding. v Thankyouforcreatingapositive,motivatingandcollaborativeworkenvironment,idealfor inter-disciplinaryresearch.IextendmygratitudetofellowstudentsattheCompositeVehicle ResearchCenterwhomanufacturedcompositespecimensrequiredforthisresearch. Thisjourneywouldnothavebeencompletewithouttheunconditionalloveandblessings frommyparentsandsister.Theirsacri˝ce,enthusiasmandconstantsupporthasbeenand willalwaysbeakeyreasonbehindmyachievements.Iwouldalsotakethisopportunityto acknowledgemygrandfather,Dr.AsitKr.Mukherjee,whohimselfbeingadoctoratein thefamilyhasbeenoneofmyearliestinspirations.Hiscommitment,zealandremarkable professionalexperiences,instilledinmethepassiontowards"knowingtheunknown". Iowemyheartfeltgratitudetofriendsandfamilywhosupportedmeduringthetough yearsinthePh.D.pursuit. Finally,thankyouSaptarshi,forbeingaconstantwitness,colleague,friendandcritic throughouttheseyears.Yourcommitmenttowardsworkiscontagiousandencouragesme toreachoutformygoals. vi TABLEOFCONTENTS LISTOFTABLES .................................... x LISTOFFIGURES ................................... xi Chapter1Introduction ............................... 1 1.1Motivation&Objective.............................1 1.2ScopeandOrganizationofthedissertation...................4 Chapter2DiagnosticsinNDE ........................... 5 2.1Introduction....................................5 2.2AutomatedSignalClassi˝cationinNDE....................7 2.3StatisticalmeasuresinNDE...........................9 2.3.1Hit/Missresponse.............................10 2.3.2ProbabilityofDetection(POD)andProbabilityofFalseAlarm(PFA)11 2.3.3avs ^ a Model...............................13 2.3.4a90/95Con˝denceBoundsonPODcurve...............14 2.3.5ReceiverOperatingCharacteristics(ROC)...............16 2.3.6Con˝denceMetric.............................17 2.4Existingcon˝dencemetrics............................19 2.4.1Con˝denceinbinaryform........................19 2.4.2Con˝denceintermsofprobability....................20 2.4.3SimilarityRatioinClustering......................20 2.4.4MembershipFunctionsinNeuralNetworks...............21 2.4.5PosteriorProbabilityinDensityEstimationtechniques........23 2.5BayesCon˝dence.................................23 Chapter3ComprehensiveCon˝denceMetricinNDE ............ 27 3.1Introduction....................................27 3.2FactorsA˙ectingReliabilityinNDESignalClassi˝cation..........29 3.2.1Quantityandrepresentativenessoftrainingdata............30 3.2.2Qualityoffeatures............................30 3.2.3Noisestatisticsoftestdata........................31 3.3Comprehensive(Boosted)BayesCon˝dence..................32 3.3.1Bootstrapping...............................33 3.3.2Incorporationofnoisefactor.......................37 3.4SimulationResults................................39 Chapter4Con˝denceMetricEvaluation:EddyCurrentinspectionof SteamGeneratortubes. ........................ 41 4.1Introduction....................................41 4.2PrincipleofEddyCurrentTesting........................43 vii 4.3AutomatedAnalysisofSGTubeInspectiondata................44 4.3.1SignalPre-processing...........................45 4.3.2ROIDetection...............................46 4.3.3FeatureExtractionandClassi˝cation..................47 4.4NoiseAnalysisinFieldDatafromEddyCurrentInspection.........48 4.5Con˝denceofClassi˝cationwithNoiseConsideration..............52 Chapter5Con˝dence-RatedClassi˝cationinNDE .............. 56 5.1Introduction....................................56 5.2Background....................................58 5.2.1ADABOOST...............................58 5.3Con˝denceratedclassi˝cation:proposedmethod................59 5.4Results.......................................62 5.4.1SimulationResults............................62 5.4.2ExperimentalResults...........................64 Chapter6PrognosisinNDE ............................ 68 6.1Introduction....................................68 6.2TheoryofReliability...............................68 6.2.1RemainingUsefulLife(RUL)......................71 6.3LiteratureReviewonRULPrognosis......................74 6.3.1Model-basedmethods...........................74 6.3.2Data-basedmethods...........................74 6.3.3Integratedmethods............................75 6.3.3.1Regressionbasedmodels....................76 6.3.3.2Markovianbasedmodels....................77 6.3.3.3StochasticFiltering.......................79 6.4TheoryofBayesianUpdating..........................80 6.4.1Bayesupdateofmodelparametersusingsyntheticdata........83 6.5BayesianUpdatingbasedonParticleFiltering(PF)Approach.........85 6.5.1PFestimateofmodelparametersusingsyntheticdata........87 Chapter7SingleSensorPrognosisinCompositesbyDirectCondition Monitoring ................................ 92 7.1Introduction....................................92 7.2ConditionbasedMaintenanceofGFRP.....................94 7.2.1ImpactDamageinGFRP........................94 7.2.2OpticalTransmissionScanning(OTS).................95 7.3ProposedPrognosticframeworkfordelaminationgrowthmodel.......97 7.3.1DamagePropagationModel.......................97 7.3.2ParticleFilteringbasedPrognosisofDelaminationAreainGFRP..100 7.4ExperimentalSetupandResults.........................104 7.4.1GFRPSpecimenandExperimentalSetup...............104 7.4.2PrognosisResults.............................106 viii Chapter8SingleSensorPrognosisinCompositesbyIndirectCondition Monitoring ................................ 113 8.1Introduction....................................113 8.2ExperimentalSetup................................115 8.2.1SpecimenGeometryandMaterial....................115 8.2.2FatiguetestingofGFRPunderModeIfailure.............116 8.3NDEofFatigueDamageinComposites.....................118 8.3.1DelaminationdetectionusingOTS...................118 8.3.2DelaminationdetectionusingGW....................122 8.3.3OverallFrameworkofDamagePrognosis................125 8.4DamagePrognosisResults............................126 8.4.1PredictionofDelaminationAreabyLogarithmicRegression.....127 8.4.2PredictionofDelaminationAreabyKalman˝ltering.........128 8.4.3PredictionofDelaminationAreabyParticle˝ltering.........131 Chapter9Multi-sensorPrognosisinComposites ............... 134 9.1Introduction....................................134 9.2LiteratureReviewofDataFusionTechniques..................139 9.3JointLikelihoodComputationinParticleFiltering...............141 9.4Experimentalset-up...............................145 9.4.1SpecimenGeometryandMaterial....................145 9.4.2FatiguetestingofGFRPundertensileloading.............146 9.5NDEdataacquisition...............................147 9.5.1FatiguedamagedetectionbyOTS)...................147 9.5.2FatiguedamagedetectionbyGW....................151 9.6PrognosisResults.................................154 9.6.1PFprognosisonOTSdata........................155 9.6.2PFprognosisonGWdata........................158 9.6.3PFprognosisonAverageofTwoSensorsdata.............160 9.6.4PFPrognosisonTwoSensorDatabyJointLikelihoodComputation.161 Chapter10Conclusion ................................ 166 APPENDICES ...................................... 169 AppendixABayesianNetworksformulti-sensorfusion...............170 AppendixBSti˙nessofCompositeMaterials.....................172 BIBLIOGRAPHY .................................... 176 ix LISTOFTABLES Table4.1:Con˝denceofclassi˝cationofdefectsinsteamgeneratortubesusingRPC probe.......................................54 Table6.1:BayesUpdatinghistoryofparameter m forsyntheticcrackgrowthpath..85 Table7.1:CumulativeenergiesofconsequtivelowvelocityimapactsonGFRPsample 1and2......................................104 Table7.2:CumulativeenergiesofconsequtivelowvelocityimapactsonGFRPsample3104 Table7.3:CumulativeenergiesofmorenumberofimpactsonGFRPsample S 1 from 150Jto450J..................................110 Table8.1:OTSandGWmeasurementsfromModeIfatiguetestingofGFRPatin- termediateloadcycles..............................126 Table9.1:Militaryapplicationsofdatafusion,from[1].................139 Table9.2:Non-militaryapplicationsofdatafusion,from[1]..............140 Table9.3:LoadingcyclesforintermediateOTSandGWinspectionsontestGFRP specimen.....................................155 x LISTOFFIGURES Figure2.1:SchematicofforwardandinverseprobleminNDE.............6 Figure2.2:Agenericautomateddataanalysissystem..................8 Figure2.3:Schematicoffeaturespace(fromtrainingdatabase)withclassi˝cation thresholdandtestdata x ...........................9 Figure2.4:(a)ExampleNDEsignal(b)Classi˝cationbetweendefectandnon-defect (noise)indicationsfromNDEsignal.....................10 Figure2.5:MatrixoffourpossibleoutcomesfromanNDEprocedurefor˛awdetection12 Figure2.6:Signal/noisedistributionfor(a)large˛aw(b)medium˛awand(c)small ˛aw.......................................12 Figure2.7: a vs ^ a modelwithPODcurvegeneration([2])................13 Figure2.8:PODcurvesforexampledatasetcalculatedatdi˙erentthresholdparam- eters(adoptedfrom[3])............................15 Figure2.9:(a)Loglikelihoodratiospaceforregressionparameters(b)PODwith95% likelyparameters[4]..............................16 Figure2.10:ROCcurvesforexampledatasethavingdi˙erentsized˛aws[3]......17 Figure2.11:(a)Self-evaluationinASCsystemwithcon˝dencemetrics(b)Single-pass systemsinNDE................................19 Figure2.12:2-meansclusteringonasyntheticdataset..................21 Figure2.13:Theprobabilisticneuralnetwork.......................22 Figure2.14:Asamplefeaturespacewithdecisionboundary(dashedline)separating twoclasses...................................24 Figure2.15:Bayesiancon˝denceforone-dimensionalcaseinatwo-classclassi˝cation. (Example:testdataisx=1)..........................25 Figure3.1:E˙ectofdiscriminativequalityoffeaturesoncon˝dencemeasure.....32 xi Figure3.2:(a)BootstrapppingBayescon˝dence,(b)Con˝dencehistogramwith C 0 : 95 valueastheredline..............................34 Figure3.3:E˙ectofnumberoftrainingdataoncon˝dencehistogramand C 0 : 95 value fortestdata X (a) C 0 : 95 =0.4033forn=10(b) C 0 : 95 =0.6776forn=15(c) C 0 : 95 =0.7277forn=20............................36 Figure3.4:Demonstrationofcon˝dencecomputationbyweightingwithnoise....38 Figure3.5:(a)2Dscatterplotoftrainingandtestdata x withnoisedistributions.(b) Classi˝cationcon˝denceof x in'Red'classwithrespecttovaryingnoise levelsoftestdata...............................40 Figure4.1:(a)Eddycurrentgenerationand˛owinaconductingspecimen(b)Change inimpedanceofcoilinadefectanddefectfreeregion(Xaxis:resistance, Yaxis:inductance)[5]............................44 Figure4.2:(a)RPCcon˝guration(b)Postprocessededdycurrentsignal(RPCprobe at300KHz)ofadefectiveSGtube[5]....................44 Figure4.3:AutomtatedECdataanalysissystemwithcon˝dencemetriccomputation module.....................................45 Figure4.4:Variousstagesofautomatedsignalprocessing(a)Rawdata,(b)Cali- brateddata,(c)TSPsuppression,(d)thresholding(e)ROIdetection[6].46 Figure4.5:Scatterplotforsteam-generatortubedatashowingfeaturesfromtraining datafrombothclassesandtestdata x ....................47 Figure4.6:(a)Imaginarychannelimageofasampleeddycurrentresponsesignalat 300KHzwithrectangularROIboxindicatingcircumferential˛aw.(b) Signalwithmaskedtubesupportand˛awregion(c)Noisehistogram. (d-f)Repeatedforrealchanneldata.....................49 Figure4.7:ExperimentalnoisemodelledasabivariateGaussiandistribution.....50 Figure4.8:(a)Noise-onlysignalfromECTofTube1usingPancakeandPlus-point probesat200Hzand300kHz,(b)NoisehistogramofTube1signals, (c)Noise-onlysignalfromECTofTube2usingPancakeandPlus-point probesat200Hzand300kHz,(d)NoisehistogramofTube2signals....51 Figure4.9:Scatterplotforsteam-generatortubedatashowingexperimentalnoise distributionforatestfeature x .......................52 xii Figure4.10:EddycurrentresponsesignalofthreeSGtubeswithROIsconsistingofde- fectsanda˙ectedbydi˙erentnoiselevels..................53 Figure5.1:Automatedanalysissystemwithcon˝dencefeedback............56 Figure5.2:(a)Trainingdata(b)testdatawithtrueclasslabelsoftwoclasses:red andblue....................................62 Figure5.3:Trainingerrorrateversusnumberofweakclassi˝ersin(a) traditional ADABOOST(b) Bayescon˝dence-rated ADABOOST.Testdataclassi- ˝edwithADABOOSTmodel(c) traditional ADABOOST:Errorrateon testdata= 35% (d) Bayescon˝dence-rated ADABOOST:Errorrateon testdata= 35% ................................63 Figure5.4:Comparisonofclassi˝cationperformanceoftraditionalADABOOSTand con˝dence-ratedADABOOST........................63 Figure5.5:Asamplepost-processededdycurrentsignalofdefectiveSGtube.Red rectangularboxes:ROIscontainingfalseindications(classi˝edas non- defect );Greenrectangularboxes:ROIscontainingtruedefects......64 Figure5.6:(a)Trainingdata(b)testdatawithtrueclasslabelsoftwoclasses:red (non-defect)andblue(defect).........................65 Figure5.7:Trainingerrorrateversusnumberofweakclassi˝ersin(a) traditional ADABOOST(b) Bayescon˝dence-rated ADABOOST.Testdataclassi- ˝edwithADABOOSTmodel(c) traditional ADABOOST:Errorrateon testdata= 19 : 40% (d) Bayescon˝dence-rated ADABOOST:Errorrate ontestdata= 14 : 93% ............................66 Figure5.8:Eddycurrentresponsesignalaftercalibrationof3defectiveSGtubes (Imaginarychannel,pluspointprobe,at300KHz).Greenboxes:ROIs (truedefects)classi˝edasdefectsbybothtraditionalandCon˝dence- basedADABOOST.Redboxes:ROIs(truedefects)classi˝edasnon- defectsbytraditionalADABOOSTbutcorrectlyclassi˝edasdefectsby Con˝dence-basedADABOOST........................66 Figure6.1:Statevariable X ( t ) andTimetoFailure t ofasystem.[7]........69 Figure6.2:Distributionfunction F ( t ) andprobabilitydensityfunction f ( t ) .[7]...70 Figure6.3:Reliabilityorsurvivorfunction R ( t ) .[7]..................71 Figure6.4:IllustrationofdamagepathprognosisandRULprediction[8]......72 xiii Figure6.5:(a)1degreepolynomial˝ttingonmeasurementdataleadingtounder estimationofRUL(b)2degreepolynomial˝ttingonmeasurementdata leadingtooverestimationofRUL......................73 Figure6.6:BayesUpdatingProcess...........................82 Figure6.7:(a)BayesUpdatingofparameter m (b)Crackgrowthexampleforsynthetic datasetwithestimatedcrackgrowthpath..................84 Figure6.8:IllustrationofBayesestimationusingparticle˝lteringtechnique.....86 Figure6.9:IllustrationofresamplingbyinverseCDFmethod[8]............89 Figure6.10:(a)CrackgrowthpredictionusingPFalgorithm(b)PredictedRULhis- togram(c)Traceofupdatingofparameter C (d)Traceofupdatingof parameter m .................................91 Figure7.1:(a)HealthyGFRPsample(nodamage),(b)DelaminatedGFRPsample afterE=20Jimpact..............................95 Figure7.2:Experimentalsetupofopticaltransmissionscanningsystemwithimpacted sampleundertest...............................96 Figure7.3:(a)( top )Healthy(noimpact)GFRPsample;( bottom )GFRPsampleafter E=30Jimpact(b)OTScanof( top )healthy;( bottom )impactedsample(c) Segmentationofdelaminationsin( top )healthy;( bottom )impactedsample.97 Figure7.4:Delaminationareapropagationwithrespecttoincreaseincumulativeim- pactenergy...................................99 Figure7.5:(a)Measurementnoisecollectedbyphotodetectorwithoutspecimeninab- senceoflasersource(b)Measurementnoisehistogram : N ( =0 : 2535 ;˙ = 0 : 0091) .....................................101 Figure7.6:OTSscansofGFRPsample(a)healthy(b)-(p)aftereachconsequtive impactfrom1to15asmentionedinTable7.1...............105 Figure7.7:Growthofdelaminationareaforthreesampleswithincreasedcumulative impactenergies(Solidcurve-S1,Dashedcurve-S2,Dottedcurve-S3)..106 Figure7.8:Predictionofdelaminationareacurvesbasedondi˙erentnumberofavail- ablemeasurements(a)n=9,(b)n=10,(c)n=11,(d)n=12.Thetrue measureddelaminationareacurveisplottedindashedlines........107 xiv Figure7.9:ComparisonofdamageareaprognosisbyParismodel(dashedline)and Paris-Parismodel(solidline)for(a)1delamination(b)2delaminations (c)3delaminations(d)4+delaminations..................108 Figure7.10:Predictionofdelaminationareacurvesbasedondi˙erentnumberofavail- ablemeasurements(a)n=8,Parismodel(b)n=7,Parismodel(c)n=8, Paris-Parismodel(d)n=7,Paris-Parismodel.Thetruemeasuredde- laminationareacurveisplottedindashedlines...............109 Figure7.11:OTSscansofGFRPsample(a)-(f)aftereachconsequtiveimpactfrom 16to21asmentionedinTable7.3,(g)Cameraimageofsample1after 450 Jimpact(H)Enlargedimageofdelaminationreachingtheedgesand breakageof˝bersdenotingitsend-of-life...................111 Figure7.12:Growthofdelaminationareaforsample1withincreasedcumulativeim- pactenergiesuptoend-of-life.........................111 Figure7.13:RULpredictionfor(a)1delamination(b)2delaminations(c)2delami- nationsand(d)4delaminations.......................112 Figure8.1:Double-cantileverbeam(DCB)specimenforModeIfatiguetests,accord- ingto ASTMD 5528 ..............................116 Figure8.2:(a)ExperimentalsetupforModeIGFRPsamplesubjectedtocyclicload- inginMTSmachine,(b)EnlargedimageofGFRPsampleunderModeI test.......................................117 Figure8.3:(a)GFRPsampleunderModeIfatiguetestsafter160Kloadcycles(b) OTSimageofGFRPsamplewithdelaminationindications........119 Figure8.4:OTSimagesofaGFRPsample(a)Healthysampleandonbeingsubjected toMode1cyclicloadingafter(b)20Kcycles(c)40Kcycles(d)60Kcycles (e)80Kcycles(f)100Kcycles(g)120Kcycles(h)140Kcycles(i)160cycles.120 Figure8.5:(a)FatiguedGFRPsampleafter160Kloadcycles(b)OTSimageofde- laminatedsample(c)Binaryimagedenotingdelaminationareaidenti˝ed afterimageprocessing.............................121 Figure8.6:PlotofnumberofloadcyclesversusdelaminationareafromOTSmea- surements...................................121 Figure8.7:(a)SchematicofGWexperimentalsetup(b)Excitedandreceivedsignals inhealthysample...............................123 xv Figure8.8:(a)Receivedsignalforahealthysampleandsampleafter20K-160Kcycle (b) TOF betweenreceivedandexcitedsignalwithincreaseinnumberof fatiguecycles..................................124 Figure8.9:CorrelationbetweenTOFfromguidedwavesignalsanddelaminationarea fromOTSimages...............................125 Figure8.10:Damageprognosis˛owchartusingguidedwaveandopticaltransmission data.......................................126 Figure8.11:(a)PredicteddelaminationareafrompredcitedGWmeasurementsusing correlationcurve(b)Updationoflogarithmicrate ` m 0 ateveryestimation step.......................................127 Figure8.12:(a)PredicteddelaminationareafrompredcitedGWmeasurementsusing correlationcurve(b)Updationoflogarithmicrate ` m 0 ateveryestimation step.......................................131 Figure8.13:(a)PredicteddelaminationareafrompredcitedGWmeasurementsusing correlationcurve(b)Updationoflogarithmicrate ` m 0 ateveryestimation step......................................132 Figure8.14:Errorcomparisonofprognosismethodsforpredictionofdelaminationarea fromguidedwavemeasurements.......................133 Figure9.1:(a)ReferenceGFRPspecimenfailingafter1386cycleswhilesubjected tension-tensionfatiguetestunderconstantload(Maxload= 70% offail- ureload,Stressratio=0.1,Frequency=3Hz),(b)TestGFRPspecimen (identicalmanufacturingconditions)failingafter2250cyclessubjectedto identicalfatiguetestingconditions(c)Normalizedsti˙nessdegradationof referenceandtestspecimenfromMTSmeasurements...........135 Figure9.2:(a)DigitalcameraimageofGFRPsamplewithdelamination,underMode Ifracturetest(b)Low-frequencyeddycurrentinspectionusngTRcoilat 10MHz(c)Near-˝eldmicrowavescanat7.5GHz(d)Opticaltransmission scanat2.5mW.................................136 Figure9.3:Bayesiannetworkinmulti-sensorparticle˝lteringframework.......144 Figure9.4:(a)Experimentalsetupfortensileopen-holeGFRPcouponsubjectedto cyclicloadinginMTSmachine,(b)HealthyandbrokenGFRPcoupons subjectetofatigue...............................146 xvi Figure9.5:(a)Stress-strainhysteresisloopofGFRPspecimenatdi˙erentintervalsof fatiguecycles(b)Stress-strainslopeorsti˙nessmodulus( S )asafunction ofnumberofloadcycles...........................148 Figure9.6:(a)Healthyopen-holeGFRPcoupon(b)OTSimageofhealthyGFRP coupon(c)OTSimageofGFRPcouponafter 900 fatiguecyclesat 70% offailureloadandstressratioof 0 : 1 .....................149 Figure9.7:OTSimagesofanopen-holedGFRPcouponunderfatigueloading:(a) Healthy( 0 cycles)(b) 200 cycles(c) 400 cycles(d) 600 cycles(e) 800 cycles (f) 900 cycles(g) 1000 cycles(h) 1100 cycles(i) 1200 cycles(j)Total failureat 1386 cycles..............................150 Figure9.8:(a)Increaseindelaminationareainopen-holedGFRPcouponunderfa- tigueloading,fromOTSmeasurements(b)Correlationbetweennormal- izedsti˙nessfromMTSmeasurementsanddelaminationareafromOTS images.....................................150 Figure9.9:GuidedWaveinspectionofGFRPspecimen(a)Experimentalsetup(b) Schematicofpressuresensitiveskin.....................152 Figure9.10:(a)TimeshiftinGWsignalsinprogressivelydamagedGFRPspecimen underfatiguecycles(b)Enlargedregionin S 0 modeofreceivedGWsignals.153 Figure9.11:(a)Increasein TOF inopen-holedGFRPcouponunderfatigueloading, fromGWmeasurements(b)Correlationbetweennormalizedsti˙nessfrom MTSmeasurementsanddelaminationareafromGWimages.......154 Figure9.12:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberof availableOTSmeasurementsinParis-Parismodel(a)n=4,(b)n=8(c) n=12(d)n=16.................................157 Figure9.13:RULpredictionforvaryingnumberofavailableOTSmeasurements(NRMSE= 0 : 1761 ).157 Figure9.14:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberof availableGWmeasurementsinParis-Parismodel(a)n=4,(b)n=8(c) n=712(d)n=16................................159 Figure9.15:RULpredictionforvaryingnumberofavailableGWmeasurements(NRMSE= 0 : 1441 ).159 Figure9.16:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberof availableAVGmeasurementsinParis-Parismodel(a)n=4,(b)n=8(c) n=12(d)n=16.................................161 Figure9.17:RULpredictionforvaryingnumberofavailableAVGmeasurements(NRMSE= 0 : 1507 ).161 xvii Figure9.18:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberof availablemeasurementsusingjointlikelihoodcomputationinParis-Paris model(a)n=4,(b)n=8(c)n=12(d)n=16.................162 Figure9.19:RULpredictionforvaryingnumberofavailableOTSandGWmeasure- mentsusingjointlikelihoodcomputation(NRMSE= 0 : 065 ).........164 Figure9.20:(a)ErrorcomparisonforvaryingnumberofavailableOTSandGWmea- surements,(b)ErrorcomparisonforvaryingnumberofparticlesinPF algorithm....................................164 FigureA.1:ExampleofBayesianNetworkwithchildrennodes( X 1 ;X 2 ;:::;X M )and theirparentnode(P).............................170 FigureB.1:Materialdirectionsinaspecimen x 1; y 2; z 3 ............173 xviii Chapter1 Introduction 1.1Motivation&Objective Withadvancementoftechnologyinrecentyears,the˝eldofnondestructiveevaluation (NDE)andtestinghaveshifteditsgearsfromclassicalapproachestomorecomplexinter- disciplinaryoperations.TraditionalNDEsystemshavefocusedmostlyonevaluationof mechanicalcomponentsforthedetectionandcharacterizationofdefectsinmaterialsor structures.Howeverwithdiscoveryofnewscienti˝cmethodsandimagingsystems,the˝eld hasgrownbothinscopeandacrossdisciplines.NDEengineeringisnolongerjustrestricted todetectingandcharacterizingdefectsbutextendstoanalyzing'riskversusrewards'and 'remainingusefullife'ofsystemsandcomponents.Overall,industrieshavestarteddemand- ingdesigners,systemintegratorsandoperatorstocollaborateanddevelopv evaluationsolutions. IntegratedNDEinspectionprimarilycomprisestwoequallyimportantprocesses(i)diag- nosticsofsystems'healthand(ii)prognosticsorpredictionofremainingoperationallifetime. NDEdiagnosticscomprisesidenti˝cationofdistinguishingfeatureswhichareindicatorsof anyanomalyordeteriorationofgeneralhealthofindustrialcomponents.Existingandemerg- ingNDEmonitoringtechniquesincludemechanical,electrical,electromagneticoroptical methodsthatcansuccessfullyimageorindicatepresenceof˛awswithoutcompromising 1 theirusefulness.OneofthecrucialaspectsofNDEdiagnosticsismaintainingreliabilityand accuracyofitsevaluationperformance.Withincreaseinamountofinformationfromnu- merousNDEapplicationssuchasdefectcharacterizationinsteamgeneratortubes,natural gastransmissionpipelines,aircraftenginesandcomponents,arti˝cialheartvalvesandmany more,automateddataanalysissystemshavebecomenecessary.Indealingwithsuchlarge volumesofdata,manualanalysisbesidesbeingtime-consumingisofteninconsistentwhich demandstheneedforautomatedsignalclassi˝cation(ASC)systemstoidentifyanomalies withreducederrorbyapplyingsuitablesignalprocessingalgorithmsontheacquiredNDE responsesignal.Often,expensiveremedialoperationsareinvolvedbasedontheclassi˝cation resultsrequiringmoreaccuracyandconsistencyinASCsystems.Afteradefectisdetected inatube,itiseitherreplacedorrepairedwhichisbothtime-consumingandexpensive.On theotherhand,detectionofpotentiallyharmfulanomalieswhichmaybefatalandshould notbemissedatanycost.Suchdefectsshouldbeidenti˝edwithgreatercertaintythanthose generatedfrombenigndiscontinuities.Hencedesignofself-evaluatingautomateddataanal- ysissystemshavebecometheneedofthehourwheresafetyandserviceabilityofstructural componentscanbemetwhilenecessarylevelofoperatorinterventionisminimized. ThesecondcrucialprocessinmodernNDEsystemsistheprognosisofstructuralaging overtime.Prognosisdealswithpredictingfuturehealthofasystem,speci˝callytopredict thetimeuntilwhichthesystemisdeemedtobesafe.Thediagnosticstepfeedsvital informationtotheprognosticarmwhereinpastandpresenthealthindicatorsareusedto predictfuturehealthofastructure.Bycalculatingthelong-termreliabilityorprognosis ofremainingusefullife,failurescanbeavoidedenablingthemaximumserviceabilityofthe component.Thisisextremelybene˝cialtoindustriessinceitensuresmaximumusagefrom thecomponent. 2 AnotherimportantadvancementinmodernNDEisthepracticeofmulti-modalsensing andinspectiontechniquesforcharacterizingmaterialsorstructures.Rapiddevelopmentin sensingandcomputingtechnologieshasenabledtheuseofmorethanonesensorforsimulta- neousconditionbasedmaintenance(CBM)ofacomponent.Manytimesasinglemeasure- menttechniquehaslimitedcapabilitiesforcharacterizingstructuralhealthofacomponent duetotheirresolutionconstraints.Di˙erentsensorsaresensitivetodi˙erentstagesofdegra- dationandcanportraymultipleperspectivesoftheunderlyingdamagegrowthpath,thus providingmoreinformationaboutsystemhealth.Asaresult,fusionofmeasurementsfrom multiplesensorshelpsreducetheuncertaintyofindividualsensorsignalandenhancesthe reliabilityofprognosis.Datafusiontechniquesareapromisingenhancementinthe˝eldof NDEwhereincurrentmeasurementsystemscombinedwithadvancedstatisticalprocessing canprovidemorereliableresults. TheprincipleobjectiveofthisthesisistoprovideadetailedinvestigationofNDEdiag- nosticsandprognostictoolsthataimatenhancingreliability,accuracyandconsistencyof damagedetectionandcharacterizationsystems.Speci˝cally,sourcesofuncertaintiestypi- callyencounteredinNDEmeasurementsystemsandtheire˙ectsonthe˝naldiagnosisof defectsarestudied.Acon˝dencemetricbasedonBayesposteriorprobabilityhasbeenpro- posedwhichcanincorporateseveralfactorsofuncertaintytoprovideacomprehensivemetric tothe˝nalinspectionresults.Further,useofstatisticalestimationandoptimizationtools suchasparticle˝lteringmethodareemployedforpredictionofdamagegrowthincompos- itematerials.Resultsfromprognosisofdelaminationinglass˝berreinforcedpolymersin associationwithdatafusionfrommultipleNDEmodalitiesarepresentedinthisthesis. 3 1.2ScopeandOrganizationofthedissertation Therearetenchaptersinthisdissertation.Chapter1introducesthemotivationand objectivesofthisstudy.Theremainingofthereportcanbebroadlycategorizedintwoparts. Chapters2-5discussestheproblemofdiagnosticsinNDEinwhichtheoverallbackgroundof existingstatisticalaspectsinNDEdiagnosticsisdiscussedinchapter2.Chapter3focuseson theimportanceofcon˝dencemetricinNDEsignalclassi˝cationanddescribestheproposed methodofcomputingacomprehensiveBayescon˝dence.Resultsonapplyingcon˝dence assessmentonrealdatafromeddycurrentinspectionofheatexchangetubesarepresented inchapter4andimprovingexistingclassi˝cationalgorithmsbyincorporationofcon˝dence metricisdemonstratedinchapter5. Chapters6-9aredevotedtoprognosticsinNDE.Thebackgroundreviewandtheory ofprognosisisdiscussedinchapter6.Chapter7describesparticle˝lteringtechniquefor predictingdamagepropagationmodelandresiduallifebasedonNDEdataacquiredby directconditionmonitoring.Resultsobtainedbyapplyingtheproposedmethodonstudyof impact-damagegrowthincompositesarepresentedinthesamechapter.Prognosisresultson indirectconditionmonitoringofcompositejointssubjectedtoModeIfatiguemechanismare presentedinchapter8.Chapter9discussesthebene˝tofmulti-modalNDEmeasurementson theprognosisofend-of-lifeofacomponent.Ajointlikelihoodupdatemethodisproposedto particle˝lteringframeworkwhichenablesoptimizationofdamagegrowthmodelparameters ateverytimestepbydiscardingnoisyorbiasedmeasurements.Predictionresultsofmatrix sti˙nessdegradationintensilecompositecouponssubjectedtorun-to-failurefatiguetests arepresented.Theoverallcontributionofthisresearchinthe˝eldofNDEandfuturescope ofworkaresummarizedinChapter10. 4 Chapter2 DiagnosticsinNDE 2.1Introduction Nondestructiveevaluation(NDE)encompassesthestudyandinspectionofobjectswith- outcompromisingtheirstructuralintegrity.InatypicalNDEinspection,atestobjectis stimulatedbyanexternalenergysourceandtheresponseoftheenergyinteractionwith thetestmaterialisrecorded.AschematicofofatypicalNDEsystemwiththeassociated forwardandinverseproblems,isdepictedinFigure2.1.Forwardprobleminvolvespredic- tionothedefectsignalgiventhematerial,defectparametersandexcitationenergy.This canbedoneviaexperimentalmethodsusingappropriateenergysourcesorviamathemati- calmodelswhichcansimulateunderlyinggoverningequations(eg:FiniteElementModel). Ontheotherhand,detectionandcharacterizationofdefectsbasedonNDEmeasurements formstheinverseproblem.Thisincludesprocessestorealizepropertiesofthestructurefrom theNDEresponseimage/signal.Inversiontechniquesinindustriesincludedevelopmentof dataanalysisandimageprocessingmethodologiestointerpretNDEmeasurementsforvi- sualization,fullpro˝lereconstructionorclassi˝cationofdefectsinstructures.Fullpro˝le reconstructionisrequiredfordeterminingsizeandshapeofdefects,whereasclassi˝cation isappliedtodistinguishdefectindicationsfrommeasurementnoiseanddecideifa˛awis seriousenoughtorenderacomponentunacceptableorunusable. 5 Figure2.1:SchematicofforwardandinverseprobleminNDE. InNDE,automatedclassi˝cationsystemsareusedtoanalyzelargevolumeofmeasure- mentdata.Forexample,defectsatrivetsitesinaircraftwingsisiscommonlyinspected usingelectromagneticNDEmethods.Eachaircraftwingcontainsthousandsofrivets,with defectsinonlyafewofthem.NDEinspectionofsuchstructuresgeneratehugeamountof informationthatneedstobeprocessedandclassi˝edintodefectandnon-defectcategories. OtherNDEapplicationssuchasinspectionofgaspipelinesextendinguptohundredsofmiles orinspectionofthousandsoftubesinheatexchangeunitsbymultipleprobesproducelarge volumesofdata.Insuchcases,manualanalysisofindividualmeasurementstakeexcessive time.Besides,errorsduetohumanfatigueoftenleadtoinconsistentandinaccurateclas- si˝cationresults.Performanceofmanualanalysisdependsonleveloftrainingacquiredby theNDEoperatorwhichmayvaryfrompersontoperson.Thereforeindustriesaremoving towardsthuseofautomatedsystemsthatcananalyzelargevolumeofNDEmeasurements fasterandwithhigheraccuracy,consistencyandreliability[9][10].Innuclearindustries, 6 single-passsystemsorsingle-party-analysisispreferredovertwo-party-analysiswherebyNDE signalsareanalyzedbyautomateddataanalysissystemsandonlyafewselectedsignalsare reviewedbyreviewanalysts.Thisreducescostofhumanresourcesaswellasdowntimeof thepowerplant.Moreover,shorterandmoreaccurateinspectionsbyautomatedsystems haveasigni˝canteconomicimpactontheoverallstation'soperationalcost,sinceeachdayof stationshutdowncanresultinmillionsofdollarsinlostrevenue.Thus,shorterinspections andpreventionofunplannedshutdownscanhelpthestationssavemillionsofdollars[11]. 2.2AutomatedSignalClassi˝cationinNDE AschematicofatypicalAutomatedSignalClassi˝cation(ASC)systemisshownin˝gure 2.2.Itcomprisesthreemajorcomponents-(1)Signalenhancement,(2)FeatureExtraction (3)Classi˝cation.Signalenhancementtechniquesimprovesthesignal-to-noiseratioofinput rawsignalusingmethodsrangingfromsimpleaveragingandlow-pass˝lteringmethods[12] tomoresophisticatedtechniquessuchaswaveletshrinkagede-noising[13]andadaptivenoise cancellation.Noisecontainedinasignalcanbeattributedtoseveralsourcesincludingin- strumentation,probewobbleandvariationsinlift-o˙orfromunwantedre˛ectionscausedby thespecimen'ssurfaceroughness.Dependingonthecharacteristicsofnoise,di˙erent˝lter- ingtechniquesareimplemented.Oncenoiseisremovedfrominputsignal,regions-of-interest (ROI)orpotentialdefectlocationsareidenti˝edbyimplementingadaptivethresholds. Afterdatareductionstep,meaningfulfeaturesareextractedfromtheROIswhichare abletodiscriminatedefectsfromnoiseindications.Featureextractionservestwomajor functions,namelydatacompressionandinvariance.Ajudiciouslyselectedfeaturevector containsmostofthediscriminatoryinformationandyetbesubstantiallysmallerindimension 7 Figure2.2:Agenericautomateddataanalysissystem. relativetotheoriginalsignalvector.This,inturn,improvestheclassi˝cationaccuracy andreducestheoverallcomputationale˙ort.Moreover,NDEsignalsareoftenacquired undervaryingtestconditionsandtheirresultsaresensitivetofactorssuchasvariationsin probecharacteristics,scanningspeed,operatingfrequencies,testobjectconductivityand permeabilityvalues,instrumentdrift,gainsettings,etc.Featureextractionservesasan importantstepinASCofNDEsignalswherefeaturesarechosensothattheyareinvariant tochangesintestconditionsortestspecimenproperties. Afterfeatureextraction,thefeaturevectorissenttotheclassi˝cationmodule.Signal classi˝cationtechniques,basedonpatternrecognitionprinciples,areusedtoclassifysignals intooneofaknownsetofclasses.Suchmethodsmaybeemployedtodiscriminatebe- tweenmultipletypesofdefectsorbetweendefectsandbenignsources.Severalclassi˝cation algorithmshavebeenusedinNDEsuchasK-meansclustering[14],neuralnetworks[15], support-vectormachineanddensityestimationtechniques.Theparametersoftheclassi˝er aredeterminedo˜ineusingadatabankofsignalsfromknowndefecttypes,referredtoas thetrainingdatabase.SimilarfeaturesareextractedfromthetestROIandsentasinput totheclassi˝cationalgorithmtoobtaintheoutputclassofthetestsignal.Aschematicof featurespace,obtainedfromtrainingdatabase,withclassi˝cationthresholdandtestdatais showninFigure2.3,indicatedby" x ".Basedonthelocationofthetestfeaturepoint,the testdataisclassi˝edintoeitheroftheclasses. 8 Figure2.3:Schematicoffeaturespace(fromtrainingdatabase)withclassi˝cationthreshold andtestdata x . 2.3StatisticalmeasuresinNDE NDEmeasurementsprovideindirectindicationofdefectlocations.Forexample,eddycur- renttestinggeneratesacomplexvoltagesignalfromwhichrelevantfeaturesareextracted andclassi˝edintopositive(defect)andnegative(non-defect)indicationsbyhumanorauto- mateddiscriminators.Positivesignalsmaybegeneratedfromnon-defectsourcessuchas surfaceroughness,grainsructure,variationsingeometryandmaterialproperties.Itisim- portanttonotethatsuchsignalsconstitutetheapplicationnoiseinherenttoaspeci˝cNDE procedureandisdi˙erentfromelectronicormeasurementnoisewhichcanbeeliminated by˝lteringoraveragingtechniques.DiscriminationthresholdofNDEsignalsmustbeset suchthatthedefectindicationsexceedthelevelofapplicationnoise.InFigure.2.4(a), anexamplesignal/imageobtainedfromeddycurrenttechniqueisshown.Ahistogramof thepixelsfromthe'defect'and'nondefectornoise'indicationstypicallyformsabimodal distributionandathresholdcanbeselectedtoclearlydistinguishthedefectpixelsfromthe noiseindications,asdemonstratedinFigure2.4(b). Althoughcontrolmeasuresareappliedtoensureaconsistentoutput,measurementsfrom 9 (a)(b) Figure2.4:(a)ExampleNDEsignal(b)Classi˝cationbetweendefectandnon-defect(noise) indicationsfromNDEsignal. anNDEsystemvarieswithincontrolparameters.Speci˝cally,ifthesameNDEexperiment isrepeatedmultipletimes,itisunlikelytoobtainthesameresulteverytimebecauseofslight variationsinhardware,materialproperties,geometryorsurfacecondition.Asaresult,a probabilitydistributionofsignalisgeneratedattheoutputinsteadofadeterministicresult. DuetotheinherentstochasticnatureofanyNDEprocess,severalstatisticalmeasuressuch asprobabilityofdetection(POD),probabilityoffalsealarms(PFA),Receiver-Operating Characteristic(ROC)curveandcon˝denceboundsarede˝nedtocharacterizedetectionca- pabilityofanNDEprocedure.Thesemeasuresareobtainedbyusingdatafromexperiments. TheobjectiveofthesemeasuresissolelytocharacterizeinspectioncapabilityoftheNDE methodbyprovidingestimatesandcon˝denceboundsforimportantquantitiesasdescribed inthefollowingsections. 2.3.1Hit/Missresponse ThenameisderivedfromtheabilityofsomeNDEprocedurestodetectonly thepresenceorabsenceofa˛aw,providingnoquantitativeinformationabout˛awcharac- teristics.Binaryresponsesofthistypearemostcommonformethodssuchastheliquid- 10 penetrantimagingandradiography,forwhichtheremaybelimitedabilitytomeasurethe ˛awsize.Forahit/missdata,theresponseistypicallyde˝nedas: Y = 8 > > > < > > > : 1 ; ifdefectisdetected 0 ; ifdefectisnotdetected (2.1) 2.3.2ProbabilityofDetection(POD)andProbabilityofFalseAlarm (PFA) WhenNDEassessmentforcrackdetectionisperformed,theinspectioncapabilityofthe procedurecannotbefullycharacterizedbyasimpleHit/missresponse.Asshowninmatrix inFigure2.5thepossibleoutcomesfromatypicalinspectionsystemare: (a) Truepositive(TP):Acrackexistsandisdetected,whereM(A,a)isthetotalnumber oftruepositivesandP(A,a)istheprobabilityoftruepositive. (b) Falsepositive(FP):Nocrackexistsbutisidenti˝edbytheNDEsystem,whereM(A,n) isthetotalnumberoffalsepositivesandP(A,n)istheprobabilityoffalsepositive. (c) Falsenegative(FN):Acrackexistsbutisnotdetected,whereM(N,a)isthetotal numberoffalsenegativesandP(N,a)istheprobabilityoffalsenegative. (d) Truenegative(TN):Nocrackexistsandisnotdetected,whereM(N,n)isthetotal numberoftruenegativesandP(N,n)istheprobabilityoftruenegative. TocompletelycharacterizedetectioncapbilityofaNDEsystem,twomeasuresarede˝ned. Theprobabilityofdetection(POD)orprobabilityforatruepositiveP(A,a)canbeexpressed as: P ( A;a )= M ( A;a ) M ( A;a )+ M ( N;a ) or Totaltruepositivecalls Totalnumberofdefects . 11 Figure2.5:MatrixoffourpossibleoutcomesfromanNDEprocedurefor˛awdetection (a)(b)(c) Figure2.6:Signal/noisedistributionfor(a)large˛aw(b)medium˛awand(c)small˛aw. Similarly,theprobabilityoffalsealarm(PFA)orprobabilityforafalsepositiveP(A,n) canbeexpressedas: P ( A;n )= M ( A;n ) M ( A;n )+ M ( N;n ) or Totalfalsealarms Totalnumberofnon-defects . Foragiven˛awsize,distributionsforapplicationnoiseanddefectsignalaredepicted inFigure2.6.Theregiontotherightofthechosendecisionthresholdcorrespondstothe PODwhereastheregiontotheleftofthethresholdrepresentsthePFA.Itisobviousthat theshadedregionsrepresentingPODandPFAdependslargelyonthedistributionofnoise anddefectsignalaswellasonthechoiceofdecisionthreshold. Underidealconditions,suchasresponsefromalarge˛aw,thesignalandnoisedistribu- tionsarewellseparatedandcanbeclearlydiscriminatedbythechosenthreshold,asshown 12 Figure2.7: a vs ^ a modelwithPODcurvegeneration([2]). inFigure2.6(a).ThiscorrespondstothemostdesirableoutputwithhighPODandlow PFA.Formedium˛awsasshowninFigure2.6(b),thereissomeoverlapbetweenthetwo distributions.Ifthresholdisthesameasthepreviouscase,thisNDEinspectionwillbe characterizedwithlowerPODandhigherPFAthancase(a).Similarly,detectionofthe smallest˛awsismostchallengingsincethenoiseanddefectsignalscannotbeseparated resultingintolowestPODandhighestPFA. 2.3.3avs ^ a Model CalculationofPODcanbeextendedto˛awsofmultiplesizestogenerateaPODcurve. Suppose a isthetrue˛awsize,thesignalresponseestimatedfromtheoutputofNDE inspectioncorrespondingtoa˛awsize a istermedas ^ a .Underidealconditions,measurement ^ a issupposedtobeexactlyequaltotruesize a andcorrespondtotheblacksolidlinein a versus ^ a plotinFigure2.7.However,inNDEinspectionsthetruesizeisunknownand relationshipbetween a and ^ a isinferredonlyfromthemeasurementdata.Accordingto empiricalstudiesin[16],itwasfoundthatanormal-theoryregressionmodel,withstandard 13 devaition ˙ ,canbeappliedtologarithmictransformationon a and ^ a suchthat: Y = N ( = 0 + 1 x i ;˙ ) (2.2) where, Y =log^ a , X =log a , 0 and 1 aretheregressionparameters.Fora a versus ^ a modelinNDE,athreshold a th isset;whenever ^ a exceedsthethreshold,theROIisclassi˝ed asa˛awandthecorrespondingPODiscalculated.ThePODiscalculatedforvarying˛aw sizesandaPODcurveisgeneratedasshowninFigure2.7.TheProbabilityofDetection (POD)curveisfurtherde˝nedas Pr (^ a>a th j a )=1 ˚ ( a th ( 0 + 1 log a ) ˙ ) (2.3) where ˚ ( Z ) isthestandardnormalcdf.Figure2.8illustratestheestimatedPODcurvesfora datasetwithvaryingthresholdparameters.Thesecurvesareusefultoexaminethetrade-o˙ betweennumberofhitsversusmisses.PODfunctionscanbede˝nedformoregeneralNDE modelsbyincludingtheinspectionfactorsuniquetotheNDEprocedure.DetailsofPOD studiesinNDEareavailablein[17,18,19]. 2.3.4a90/95Con˝denceBoundsonPODcurve PODcurvesarecriticalinassessingthedetectioncapabilityofanyNDEmeasuring system.However,accuracyofaPODcurveisitselfdependentontheestimationofthe regressionparameters 0 and 1 .Slightchangeintheseparameterscana˙ectthePOD curvegreatlyandthereforeitisnecessarytoposecon˝denceboundsonthemtoallowfor discrepanciesintheestimatedPODvalues. 14 Figure2.8:PODcurvesforexampledatasetcalculatedatdi˙erentthresholdparame- ters(adoptedfrom[3]). Consideraplotoftheloglikelihoodratiofordi˙erentvaluesof and ˙ ,asshownin Figure2.9(a).AccordingtoKnoppetal.[4],ifthepairofparametersismovedfromtheir maximumlikelihoodestimate(MLE)positiondenotedby + ,theloglikelihoodchanges,as illustratedbythecontourlines.Oneofthecontours,shownbythedottedline,isthe95% con˝denceboundfortheparameterestimatesbasedonthesedata.Inotherwords,the true and ˙ pairisexpectedtobecontainedwithinthecon˝denceellipsein95%offuture experimentssimlartothisone.PODcurvesarethenconstructedforallthepointsalongthe 95%con˝denceellipseasshowninFigure2.9(b).TheenvelopeofallthesePOD(a)curves representsthecon˝denceboundsonthePOD(a)curve.ThePODcurvecorrespondingto theMLEofparametersisshownastheblacksolidlinein˝gure2.9(b).Thepointwherethe estimatedPODcurveintersectsPOD=0.9,isknownasthea90/95valuewhichrepresents thatin95outof100similarexperiments,theoutput˛awsizehavingPODof0.9willlie withintheestimatedcon˝dencebounds. 15 Figure2.9:(a)Loglikelihoodratiospaceforregressionparameters(b)PODwith95%likely parameters[4]. 2.3.5ReceiverOperatingCharacteristics(ROC) TheROCcurveisaplotofProbabilityofFalseAlarmonthehorizontalaxisandProbabil- ityofDetectionontheverticalaxis,asshowninFigure2.10.TheROCfunctionisgenerated byvaryingdetectionthresholdoverallpossiblevalues.ROCfunctionswereoriginallyde- velopedtoillustratethee˙ectofchoiceofthresholdontheprobabilityofmisclassi˝cation inradarapplications[20]. Ifasetofmeasurements,containingagroupof˛awsofsimilarsize,isrepeatedlyassessed, thePODandPFAcanbecalculatedwhichformsasinglepointontheROCcurve.This processisrepeatedbyseveraloperatorsofvaryinglevelsofpro˝ciency(denotingvarying thresholds)andtheROCcurveisgenerated.Asuperiordiscriminatingperformanceofthe NDEinspectionwillresultinhighPODandlowPFA,orthetop-leftregionoftheROC curveisconsideredasthepreferredthreshold. 16 Figure2.10:ROCcurvesforexampledatasethavingdi˙erentsized˛aws[3]. 2.3.6Con˝denceMetric StatisticalmeasuressuchasPODandROCcurveswiththeircon˝denceboundsassesses theinherentdetectionperformanceofanyNDEmeasuringsystem.WhileROCcurveaidsin selectingtheoptimumthresholdfordetectingdefectofaparticularsize,PODcurveshows thee˙ectof˛awsizeondetectioncapabilityfora˝xedthreshold.Boththesecurvesare criticalforassessingtheminimum˛awsizethatcanbeaccuratelydetectedusingtheNDE procedure. However,thesemeasuresdonotdealwiththecompletepictureofsystemreliability inNDE.Apartfrominherentuncertaintiesofthemeasuringsystem,classi˝cationbyau- tomatedsystemsarea˙ectedbyotherfactorswhicharenottakenintoaccountineither ofthesemeasures.Aninspectionsystemwithhighdetectioncapabilitycanstillproduce inaccurateresultsiftheASCsystemisunder-trainedorsub-optimalsignalfeaturesarese- lected.Further,whilecomputingPODandPFA,onlyapplicationnoiseisconsideredwhich isinherenttotheNDEtechnology.Randomnoiseinmeasurementswhichmayoccurdueto 17 probelift-o˙variations,unexpectedchangeinstructuralgeometryormachinefatiguea˙ects classi˝cationresultswhichisnotcapturedbyeitherPODorROCmeasure. Mostimportantly,bothPODandROCcurvesaregeneratedusingexperimentaldata withknown˛awsizes,forcharacterizingthemeasurementsystembeforeapplyingto˝eld data.Oncontrary,when˝elddataisinspectedbyNDEprocedure,theASCsystemisun- awareofdefectsizesandthe˝elddatacanbesigni˝cantlydi˙erentfromexperimentaldata usedtocomputePODorROCcurves.Thetestdataofunknowndefectpro˝leisprocessed andthe˝nalclassi˝cationresultsarebasedsolelyontrainingandselectedfeatures.Asa result,existingPODandROCcurvescannotquantifyreliabilityofASCsystemwhichis a˙ectedbynumberanddistributionoftrainingsignals,qualityoffeaturesandmeasurement noiseintestdata.AreliabilitymeasureoftheASCsystemisde˝nedintermsofcon˝dence metric(CM)toquantifyuncertaintiesassociatedwithclassi˝cationofeveryROI.Assess- mentofCMtoindividual˝elddataobservationsisanecessarytoolinNDEdiagnostics sincepotentiallyharmfulanomaliesareexpectedtobedetectedwithgreaterprobability thanbenigndiscontinuitiesandanASCsystemwithsuchcapabilitycanautomatically˛ag indicationsforwhichoperatorinterventionisrequired.Asdepicted,inFigure2.11(a),NDE data3and4identi˝edwithlowcon˝dencecanbefurtheranalyzedbeforedirectlyreplacing orrepairingthe'defective'component.Byreviewingonlyselectedsignals(havinglowCM), single-passsystemscanbereliablyusedinindustriestherebysavingbothtimeandcostof humanresources. 18 Figure2.11:(a)Self-evaluationinASCsystemwithcon˝dencemetrics(b)Single-passsys- temsinNDE. 2.4Existingcon˝dencemetrics Thefollowingsectionisdedicatedtoareviewofsomeexisingcon˝dencemetricsto understandtheircapabilitiesandshortcomingsindi˙erentclassi˝ers. 2.4.1Con˝denceinbinaryform Initialpartofliteratureonclassi˝cationalgorithmshasrestrictedcon˝dencemeasure tohaveabinaryform.Grunwaldetal.[21]useshighcon˝dence(sure)andlowcon˝dence (unsure)asthetwolabelstodenotewhetheragivenindicationiscorrect(C)orincorrect(I). Baileyetal.[22]buildontheconceptofarejectionregionimplementinguncertaintyenvelopes (UE)thatareassociatedwithunsureclassi˝cations.Whentestedwithalargenumberof classi˝ers,thepercentageofclassi˝ersthatcorrectlyclassi˝esthesameexemplaristhe levelofcon˝denceassociatedwiththatexemplar.Insuchcases,auser-de˝nedthreshold isappliedtoformtheUE.Anydatafallingintotheuncertaintyenvelopeisunsure,and anydatafallingoutsidetheuncertaintyenvelopeissure.Thus,thecon˝dencemeasureis essentiallyabinaryindicator,eithersureorunsure.Similarconceptshavebeenusedby Krzanowski[23]andJacobsen[24].Although,thesecon˝dencemeasuresgiveagoodestimate 19 ofsuccess-failureanalysisofclassi˝cation,theusageofauser-de˝nedthresholdmakesthese approachesheuristicinnatureanddonotparticularlyaddressthee˙ectofuncertaintiesin aclassi˝cation. 2.4.2Con˝denceintermsofprobability Themostpopularmethodofquantifyinguncertaintyinclassi˝cationhasbeenintheform ofprobability.DuetoinherentcharacteriticsofaNDEsystem,noiseandsignalconditional distributionsoverlapandatestdatafallsunderboththeclasseswithdi˙erentprobabilities. Di˙erentclass˝cationalgortihmsinliteraturesuchasK-nearestneighborSimilarityRatio[25] provideprobabilityscoreswhichcanbede˝nedascon˝dencemetrics.Afewotherapproaches arediscussedfurther. 2.4.3SimilarityRatioinClustering Clustering[14]isanintuitivemeansofclassi˝cationthatusesthefactthatpatternsfrom thesameclasstendtobesimilartoeachother.Membersofaclasstendtoclusteraround apointinfeaturespace.Itisasimplealgorithmwhichminimizestheobjectivefunction: J = k X j =1 n X i =1 x ( j ) i c j 2 (2.4) where x ( j ) i c j 2 isachosendistancemeasurebetweenadatapoint x ( j ) i andthecluster centre c j . n isthetotalnumberofdatapointsand k isthenumberofclassesthedata isgroupedinto.ThesimplestformofclusteringistheK-meansalgorithmwhichassigns datapointstothegroupthathastheclosestcentroid.Figure2.12showstheresultof applicationof2-meansclusteringonasyntheticdataset.Onepossiblecon˝dencemetric 20 Figure2.12:2-meansclusteringonasyntheticdataset. inclusteranalysisisformulatedbasedonclosenessofadatapointtotheclusterwiththe closestcenter.Apointwhichisclosertothecentreofitsassignedclusterwillbeassociated withhighercon˝denceofclassi˝cationcomparedtoapointwhichisfarfromthecluster. Con˝denceassociatedwithclassi˝cationofadatapointcanbecomputedas: C =1 d m P k i =1 d i (2.5) where d i isthedistanceofdatapoint i from k clustercentresand d m isitsdistancefromthe clustertowhichitisclassi˝edto. 2.4.4MembershipFunctionsinNeuralNetworks Neuralnetworkshavebeenusedsuccessfullyinpatternrecognitionlargelyduetotheir simplelearningalgorithmsandabilitytogeneratecomplexdecisionboundaries.Theycon- sistofweightedinterconnectionsofsimpleprocessingunitscalledneurons.Eachweight representstheinterconnectionstrengthbetweentwocells.Learningoccursbyaprocessof adaptingtheweightstore˛ectmappingofaninputtoadesiredoutput.Pradeep[26]uses 21 Figure2.13:Theprobabilisticneuralnetwork. probabilisticneuralnetwork(PNN)astheclassi˝cationschemeappliedonadatabaseof ultrasonicsignalsobtainedfrominspectionoftubesinnuclearpowerplants.Hefurtheruses membershipfunctiontorepresentthecon˝denceassociatedwitheverysignalclassi˝cation. Figure2.13showsthearchitectureofaPNN.Theinputpatternismultipliedbythein- terconnectionweightsandsenttothesecondlayerpatternnodesrepresentingthetraining dataset.Eachpatternunitimplementsmemebershipfunction MF de˝nedas: MF = exp 1 2 x c ˙ 2 ! (2.6) where x istheinputand c , ˙ arethecenterandspreadoftheGaussianmembershipfunction ofthatpatternnode.Theoutputofeachpatternunit j isthedegree t j towhichtherule ˝res. t j = MF j y j (2.7) 22 Theoutputsofpatternnodesbelongingtothesameclassaresentasinputtothesummation layerandtheircorrespondingmembershipvaluesareaggregated.Theinputpatternis assignedtotheclasshavingthemaximumoutputfromsummationlayer.Classi˝cation con˝dencecanbeinterpretedastheaggregatedmembershipvalues i : i ( x )= X j 2 C i t j (2.8) 2.4.5PosteriorProbabilityinDensityEstimationtechniques Inclassi˝cationviadensityestimationtechniques,conceptofposteriorprobabilitycon- tainsrelevantinformationtoassesstheaccuracyofclassi˝cationresult[27,28,29].The datapointstobeclassi˝edareassumedtobegeneratedbyaunderlyingprobabilitydensity functionofrespectiveclasses.Classi˝cationisperformedbyestimatingthedensityfunctions for'defect'and'non-defect'classandassigningadatatotheclasshavingmaximumdensity value.DensityestimationtechniquesincludeparametricapproachessuchasMaximumLike- lihoodEstimateornon-parametricmethodssuchasParzenwindowclassi˝erandK-nearest neighbors.Atypicalcon˝dencemetricindensityestimationtechniqueistheBayesposterior probabilityortheBayesCon˝dence. 2.5BayesCon˝dence Duringtrainingofthesystem,featuresfromtrainingdataareplottedinahyper-dimensional featurespaceandadecisionboundaryisobtainedsuchthattheclassi˝cationerrorismini- mizedasshowninFigure2.14.Thisdecisionboundarypartitionsthefeaturespaceintwo categories,defect( class 1 )andnon-defect( class 2 ).Featuresfromtestsignalareextracted 23 Figure2.14:Asamplefeaturespacewithdecisionboundary(dashedline)separatingtwo classes. andthesignalisclassi˝edintotheclassdependingonthelocationofthetestdatarelative tothedecisionboundary. AccordingtoBayestheorem,if x isapatternvectorfromaclass A i ,theposteriorprob- abilityofclass A i given x iswrittenastheconditionalprobability P ( A i j x ) .Thus,the probabilitydistributionofaclassisconditionedonevidenceobtainedfromtrainingdata and x isassignedtotheclasshavingmaximumposteriorprobabilitydensityfunction.[30] Con˝denceinclassi˝cationcanbede˝nedastheprobabilityofmakingacorrectdecision. Consequently,thecon˝denceofclassifyingatestdatainclass A i istheposteriorprobability function,givenbythewell-knownBayesrule.[31] P ( A i j x )= p ( x j A i ) P ( A i ) P c i =1 p ( x j A i ) P ( A i ) (2.9) where, P ( A i j x ) istheclass-conditionaldensityforaclass A i P ( A i ) ispriorprobabilityofclass A i P ( A i j x ) isposteriorprobabilityofclass A i giventhepatternvector x . 24 Figure2.15:Bayesiancon˝denceforone-dimensionalcaseinatwo-classclassi˝cation.(Ex- ample:testdataisx=1). Ifnootherinformationabouttheclassesisavailable,thepriorprobabilitiescanbe assumedtobeequal.Fora2-classproblem( Defect and Non defect ),theequation(2) reducesto Con˝denceof x classi˝edas Defect is: P ( Defect j x )= p ( x j Defect ) p ( x j Defect )+ p ( x j Non Defect ) (2.10) Hence,foragiventestdata x ,theclassi˝cationcon˝denceof x asdefectcanbeobtained usingtheaboveequation.Theconditionalprobabilitydensityfunctionsforthetwoclasses p ( x j Defect ) and p ( x j Non defect ) areestimatedfromthetrainingset.Figure2.15shows therepresentationofcon˝denceina1-dimensionalcase(wherethetrainingsetisrepresented byonefeature).AssumingGaussiandistributionforclass-conditionaldensityfunctionsof thetwoclasses,thetwoGaussianplotsrepresenttheestimateddistributionofthefeatures fromtrainingsampleslabeledasclass A 1 andclass A 2 .Thecon˝denceof x classi˝edas class A 1 iscalculatedas C A 1 = P ( A 1 j x ) =0.176/(0.176+0.0829)=0.6798or67.98%. Oneofthemajorchallengesofusingsimplytheposteriorprobabilitydirectlyisthat priorsareoftenunknown,aspointedoutbyRichardetal[32].Moreover,suchameasure 25 isheavilydependentoncorrectestimationofdensityfunctions.Throughoutliterature, althoughcon˝dencemeasurehasbeende˝nedindi˙erentways,allapproachesdealwitha singleobjectiveoftryingtocomeupwiththebestpossiblewaytoincludethee˙ectsofall potentialuncertaintiesencounteredinexistingNDEclassi˝cation. 26 Chapter3 ComprehensiveCon˝denceMetricin NDE 3.1Introduction Althoughawiderangeofbothrule-basedandpatternrecognition-basedclassi˝cation algorithmshavebeenstudiedforvariousNDEapplications[33,34],theestimationofa con˝dencemeasurehasremainedunder-emphasizedinNDEliterature.Therearenumerous uncertaintiesinvolvedinNDEsystems.Goebeletal.claimthatNDEsensordataisoften highlynoisyandnumberoftrainingsamplesavailableislimited[35].Althoughutilizing classi˝erensemblesimprovesclassi˝cationperformancefornoisyNDEdata,thereliability ofclassi˝cationresultshavenotbeenevaluated.Besides,accuracyofaclassi˝erdepends onthediscriminativequalityofthefeaturesused.Therehasbeeninvestigationofnoise- invariantfeaturestoimproveclassi˝cationperformance,forinstancein[36],buttheire˙ect onreliabilityisnotyetveri˝ed.Afewquantitativestudiesonreliabilityofclassi˝cation systemshavebeenconductedoverthepastyears[37,23],butnomethodofcon˝dence estimationseemstobewidelyacceptedtilldate.Inpractice,thesourcesa˙ectingreliability ofsignalclassi˝cationinNDEsystemsoccursimultaneously.Whiletheyhavebeendiscussed beforebyNDEspecialists,ajointquanti˝cationandincorporationoftheirimpactinthe 27 formofasinglereliabilitymeasureforeverydecisionmaderemainsunsolved. Muchofthecurrentstudies[27,28,29]useposteriorprobabilityorsimilarmeasuresof con˝dence.Asmentionedinchapter2,posteriorprobabilityofoccurrenceofaneventis representativeofinter-classsimilaritiesandintra-classdistanceandthus,maybeusedasa measureofinherentambiguityofclassesanddiscriminativequalityoffeatures.Howeverthe majorconcerninsuchapproachesliesintheestimationoftheparametersofdensityfunctions fromtrainingset.Bayescon˝dencetakesintoaccountthee˙ectofqualityoffeaturesex- tractedfromsignalassumingthattheclass-conditionaldensityfunctionsareknowna-priori. Onthecontrary,inpractice,lackofadequatetrainingdatacausesestimatedparametersto besigni˝cantlydi˙erentfromtheirtruevalueswhicha˙ectsthecalculationofcon˝dence ofclassi˝er.Tothebestofourknowledge,thisfactorhasremainedunder-emphasizedin existingliteratureoncon˝dencemeasures.Inourstudy,thedi˙erencebetweentrueparam- etersandestimatedvaluesisreducedandthee˙ectofsizeoftrainingdataisincorporated inBayescon˝dencebyapplyingbootstrapmethod[38]. Theothercauseofunreliabilityinclassi˝cationdecision,irrespectiveoftheclassi˝ca- tiontechnique,isthemeasurementnoise.ParticularlyinNDE,theabsolutenoiselevel andabsolutestrengthofadefectsignaldependsonanumberoffactors.Forexamplein ultrasounddetection,measurementnoisedependsonprobesizeandfocalproperties,probe frequency,inspectionpath,couplingbetweentransducerandsample,inherentnoisinessof themetalmicrostructure,etc.Similarlyineddycurrenttesting,themajornoisesources aretemperaturevariations,probelift-o˙,changesintheelectromagneticpropertiesofthe materialsuchaselectricalconductivityormagneticpermeabilityandchangesintestspeed [39].ResearchersinNDEhaveexploredadvancedsignalprocessingtechniquesfordetecting di˙erentsourcesofnoiseanddistinguishingsignalsarisingfromtruedefectsinpresenceof 28 noise[40,41].Howevertothebestofauthor'sknowledge,existingcon˝denceassessmentsdo notincorporatenoiseinformation.Incaseofuncertaintyanalysis,estimationtheorysuggest thatthevarianceoftheestimatordependslargelyonthevarianceofnoiseinobservation whichthereforea˙ectsthecon˝denceofclassi˝cationresultandmotivatedourstudyon proposinganupdatedcon˝dencemetric. Inthischapter,theprimarysourcesofuncertaintiesencounteredinatypicalASCsystem inNDEhavebeenidenti˝ed.Aframeworkhasbeendevelopedtoincorporatetheire˙ect onclassi˝cationperformanceintoasinglequantity.Inlieuofthecommonlyusedsimplistic assumptionof˝xeddistributions[42],weassumethatparametersofthedistributionofa classarerandomvariables.Weutilizebootstrapmethodto˝ndempiricaldistributionof parametersoftheclassconditionaldensitiesbasedonwhichadistributionofcon˝denceis obtained[38].Fromthisdistribution,di˙erentinterpretationsofthecon˝dencemeasuremay beprovided.Analyticalresultsshowhowstatisticalpropertiesofthecon˝dencedistribution arerepresentativeoftheunderlyingsourcesofuncertaintiesinASCsystems. 3.2FactorsA˙ectingReliabilityinNDESignalClassi˝- cation Thereliabilityofclassifyingasignalasdefectislargelya˙ectedbytheaccuracyin estimationofthedensityfunctionsoftheclasses.Uncertaintiesinparametricestimation oftheclass-conditionaldensitiesleadtoerrorsinclassi˝cationresultsintermsofmissing truedefectsorcausingfalsealarms.Ideally,acomprehensivecon˝dencemeasureinASC systemsshouldbeabletoquantifythee˙ectofthefactorsa˙ectingreliabilityofNDEsignal classi˝cationandprovideself-evaluationofitsresults.Thefollowingfactorswereidenti˝ed 29 andtheirindividuale˙ectswerestudied. 3.2.1Quantityandrepresentativenessoftrainingdata SinceBayescon˝dencereliesonparametricstatistics,accuracyofestimatedcon˝dence heavilydependsonthenumberoftrainingsamplesused. AccordingtoChebyshevsInequality, P ( j x n j " ) ˙ 2 n" 2 (3.1) where, istruemean, x n istheexpectedmean, ˙ isthevarianceofthedistributionand n isthenumberofsamples.Thisstatesthattheestimatedstatisticalparametersofclass distributionstendtoconvergetothetruedistributionasthenumberofsamplesincreases [43].Subsequently,thecon˝denceassociatedwithdecisionofatestsignalbyanASCsystem whichistrainedwithmoretrainingsampleswouldbehigher,consideringthatthetraining dataisrepresentativeoftheclassirrespectiveofitssize. Anotherdesiredpropertyoftrainingdatasetisthatitshouldberepresentativeofits classes.Forinstance,anidealtrainingdatasetofdefectsignalsshouldcontainsignalsob- tainedfromdefectsofallpossibledepth,widthoranyotherparameterthata˙ectssignal features.Ifsomeregionofthefeaturespaceisunder-representedduetolackofenough samples,computationofclassi˝cationcon˝denceofatestdatawillbeinaccurate. 3.2.2Qualityoffeatures Featuresselectedtodescribethetrainingdatashouldpossessdiscriminativeproperty. Con˝denceofasignalbeingadefectismorewhenitsfeatureliesclosertothemeanof 30 distributionofclassdefectandfartherfromthemeanoftheotherclass.Discriminative propertyoffeatureensuresthatinter-classdistanceishighandintra-classvarianceislow whichenablesseparationoftwoclassesinthefeaturespace.Atestdatawhichliesfarther fromthemeanofdistributionofanotherclassandclosertothemeanofitsownclassislikely tohaveahighervalueofcon˝denceassociatedwithit[44].Thisconceptcanbeexpressed quantitativelyas: Confidence / d 1 d 2 (3.2) d1:Inter-classdistance(distancebetweenmeansofbothdistributions) d2:Intra-classdistance(varianceofeachclassdistribution) Figure3.1showsthecasewherethesametestdataisassociatedwiththesameclas- si˝cationresultbutintuitivelyhasdi˙erentcon˝denceduetodi˙erenceindiscriminative propertyoffeaturesetchosen.Fig.3.1a)indicateshigherinter-classdistanceleadingto distinctclustersresultinginhighercon˝denceassignedtoatestdata.Fig.3.1b)uses feature3andfeature4todescribeanoverlappingfeaturespaceandthereforecon˝denceof thesametestdata(tobeinclass1)islow. 3.2.3Noisestatisticsoftestdata Thesignal-to-noiseratio(SNR)ofthetestdataa˙ectscon˝denceofitsclassi˝cation. NoiseisgeneratedduringmeasurementinNDEsystemswhichmaybedi˙erentfordi˙erent testsamples.Anoisytestsignalwillhaveinaccuracyincomputedfeatureswhichinherently a˙ectsitsclassi˝cationcon˝dence.Hence,togenerateamorecomprehensivecon˝dence metric,itisimportanttoincorporatenoisecharacteristicsintotheposteriorprobabilty measure.Thee˙ectofnoisestatisticsonNDEclassi˝cationcon˝denceandthemethodof 31 (a)(b) Figure3.1:E˙ectofdiscriminativequalityoffeaturesoncon˝dencemeasure. incorporatingitintocalculationofcon˝dencearediscussedingreaterdetailsinlatersections. 3.3Comprehensive(Boosted)BayesCon˝dence Assumingknownparameterscharacterizingtheclass-conditionaldensityfunctions,ex- istingBayesposteriorprobabilitysu˚cientlydenotesapossiblemeasureofreliabilityin classi˝cationresults.HoweverinNDEapplications,densityfunctionsareunknownandcon- ˝denceofsignalanalysisdependsstronglyontheaccuracyofparametricestimationaswell asthenoisemodel.Inthisthesis,thesetwoissuesareaddressed.Anewmetricofreliability isproposedbasedontraditionalBayescon˝dencewhichsuccessfullyincorporatese˙ectsof uncertaintiesduetolimitednumberoftrainingdataandnoiseinmeasurements.Apopular sub-samplingtechniqueknownasbootstrappingisappliedforcalculationofposteriorprob- abilitysuchthattheestimationerrorisreduced,followedbyincorporationofnoisestatistics fromNDEsignalintothecon˝denceassessment.Thedetailsoftheproposedmethodare describedinthefollowingsubsectionsandsummarizedinAlgorithm1and2. 32 3.3.1Bootstrapping Bootstrappingisatechniqueofsub-samplingwithreplacement[38].Ateveryiterationof thealgorithm,asubsetofthetotaltrainingdataset( D t )isselectedandmaximumlikelihood estimatorsofparametersoftheclass-conditionaldensityfunctionsareobtained. AGaussianmixturemodel(GMM)isimplementedon D t tomodelclass-conditionalpdf forallclasses [ ! 1 ;::::;! c ] as: p t ( x ) ! i =( k X i =1 ˚ i N ( i ; i )) t (3.3) wherethe i th vectorcomponentischaracterizedbynormaldistributionswithweights ˚ i , means ~ i andcovariancematrices i of k componentsinGMM[45]. Usingestimatedvaluesofmeanvectorandcovariancematrix,theBayesposteriorproba- bilityiscalculatedaccordingtoEquation2.9.Repeatingtheprocessonothersetofsamples fora˝xednumberofiterationsprovidesadeeperinsightintothebehavioroftheentire statisticalpopulation.Inlieuofdeterministicapproach,parametersofthedistributionof aclassareassumedtoberandomvariablesunderthisframeworkandhenceadistribu- tionofcon˝denceisobtained,insteadofone-shotcon˝dencecalculation.Theprocedureis illustratedinFigure3.2. Fromthecon˝dencehistogram, C 0 : 95 measureiscalculatedtotherightofwhich95%of thetotalareaunderthehistogramlies.Classi˝cationcon˝denceof C 0 : 95 associatedwitha NDEsignalsigni˝esthat95timesoutof100,theASCensuresthatthesignalwillbelong tothereportedclasswithacon˝denceof C 0 : 95 .Ifthehistogramofcon˝dencevaluesis denotedas h =[ h 1 ;::::::h n ] 33 (a) (b) Figure3.2:(a)BootstrapppingBayescon˝dence,(b)Con˝dencehistogramwith C 0 : 95 value astheredline. 34 h isde˝nedas: h n = X x 2 [0 ; 1] n ( x ) (3.4) Where, n :1 ; 2 ;::::;N ; N isthetotalnumberofbinsofhistogramand n isanindicator functionsuchthat, n ( x )= 8 > < > : 1 if n 1 N x n N 0 otherwise (3.5) Themetric C 0 : 95 isfurtherde˝nedas: C 0 : 95 = index 1 N + index N 2 (3.6) where, index =max j 0 @ N X k = j h 0 k 0 : 95 1 A (3.7) h 0 n = h n P n 2 N h n (3.8) Asaruleofthumb,atleast 75% ofthetrainingsamplesareselectedineachiterationtogen- erateunbiasedestimatesoftheparameters.Additionally,bootstrappingreinforcesthee˙ect ofnumberoftrainingsamplesoncon˝denceevaluation,depictedusingsimulateddataset inFigure3.3withtrainingdataofsize(a)10,(b)15and(c)20respectively.Presence ofmorenumberofrepresentativetrainingdatareducesthevarianceofthecon˝dencehis- togramwhichre˛ectshighercertaintyinclassi˝cationresults[46].Itisimportanttonote thattrainingdatadistributionisunchangedforthethreecases,onlythenumberofdata varies. 35 (a)(b) (c) Figure3.3:E˙ectofnumberoftrainingdataoncon˝dencehistogramand C 0 : 95 valuefor testdata X (a) C 0 : 95 =0.4033forn=10(b) C 0 : 95 =0.6776forn=15(c) C 0 : 95 =0.7277for n=20. 36 3.3.2Incorporationofnoisefactor AsstatedbyMannetal.[47],thelevelofcon˝dencethatcanbeattributedtoclassi˝- cationishighlysensitivetopriorassumptionsregardingthenatureofthebackgroundnoise, aswellastheappropriatenessofthestatisticalmodelfornoise.Typicalmeasurementsfrom anNDEexperimentcontains˛awresponsea˙ectedbydi˙erentsourcesofnoise.Formost casesespeciallywhenthe˛awsizeisrelativelysmall,itisverydi˚culttodeterminewhether anobservationarisesduetoa˛awormeasurementnoiseleadingtolowcon˝denceinclas- si˝cation.Inthiswork,assumingadditivenoisewerecognizethatfeaturesfromanoisy signalarenotdeterministicinnature;insteadtheyarerandomvariableswhosedistribution isa˙ectedbythenoisepdf.NoisestatisticsisextractedfromthemeasuredNDEsignaland characterizedbyitsdensityfunction.Duringtrainingofalgorithm,posteriorprobability functionofthefeaturespaceiscalculated.Subsequently,theBayesposteriorprobability functionisweightedwiththenoisedensityfunctionaccordingtoequation3.3,inorderto evaluatecorrespondingclassi˝cationcon˝dence[48]. ( P W ( x )) ! = P x 2 A p n ( x x ) : P p ( x ) ! dx P x 2 A p n ( x x ) dx (3.9) where, A isthefeaturespace, P p ( x ) ! isthecalculatedposteriorprobabilityfunction ofthesignalclassi˝edintoclass ! , p n ( x x ) istheestimatednoisedensityfunctionofthe signaland ( P W ( x )) ! isthenoise-weightedposteriorprobabilityfunctionoftheclassi˝ed signal. Theprocessofweightingposteriorprobabilitywithnoisedistributionisdemonstratedin ˝gure3.4.Bayescon˝denceoftestdata x withoutnoiseiscomputedas C 1 (valueof C ( x ) at x ).Withe˙ectofnoise,theclassi˝cationcon˝denceiscalculatedas: 37 C n = C 1 n 1 + C 2 n 2 + C 3 n 3 + C 4 n 4 + ::: + C M n M n 1 + n 2 + n 3 + n 4 + ::: + n M 8 n 2 N (3.10) Figure3.4:Demonstrationofcon˝dencecomputationbyweightingwithnoise. Theproposedmethodofcomputingclassi˝cationcon˝denceofanoisytestdataimple- mentedinbootstrapframeworkisdescribedinAlgorithm1and2. Algorithm1 Trainingalgorithm Input: Trainingdata D = f x i ;y i g ; i 2f 1 ; 2 ;::::;N g withtruelabels y i 2f ! j g ; j 2 f 1 ; 2 ;::::;c g ; Output: Posteriorpdfoftheclassforwhichcon˝denceiscalculated= P t p ( x ) ! j 1: Initialize t =1 ; 2: for t =1 to T (Numberofiterations) do 3: Selectatrainingsubset D t containing M samplesdrawnfrom D ; 4: FitGMMto D t andmodelclass-conditionalpdfforallclasses [ ! 1 ;::::;! c ] as: p t ( x ) ! i = ( P k i =1 ˚ i N ( i ; i )) t wherethe i th vectorcomponentischaracterizedbynormaldistributionswithweights ˚ i ,means ~ i andcovariancematrices i of k componentsinGMM 5: Estimate ( ~ t ; t ) ! i forallclasses [ ! 1 ;::::;! c ] ; 6: Calculatetheposteriorprobabilityfortheclassforwhichcon˝denceisobtained(e.g. ! j ): P t p ( x ) ! j = p t ( x ) ! j P c i =1 ( p t ( x ) ) ! i ; 7: endfor 38 Algorithm2 Generatingcon˝denceofclassi˝cationofatestdata Input: Testdata whosecon˝denceistobeevaluated: x ; Classi˝edlabelof x = y ; PDFofnoiseoftestdata= p n ( x x ) ; Output: Con˝denceoftestdataclassi˝edintoclass ! j = C ! j ; 1: for t =1 to T do 2: Calculateposteriorpdfweightedbynoisepdf: P t W ( x ) ! j = P x 2 A p n ( x x ) : P t p ( x ) ! j P x 2 A p n ( x x ) where A :{featurespace}; 3: Calculateweightedposteriorprobabilityfortestdata: P t W ( x ) y ; 4: endfor 5: Formulateahistogramof ( P W ( x )) ! j with M binss.t. h =[ h 1 ;::::::h M ] ; 6: De˝ne h m as: h m = X x 2 [0 ; 1] m ( x ) (3.11) where, m :1 ; 2 ;::::;M ; M isthetotalnumberofbinsofhistogramand m isanindicator functionsuchthat, m ( x )= ˆ 1 if m 1 M x m M 0 otherwise (3.12) 7: Calculateclassi˝cationcon˝denceof x = C ! j = C 0 : 95 ; C 0 : 95 = index 1 M + index M 2 (3.13) where, index =max j 0 @ M X k = j h 0 k 0 : 95 1 A (3.14) h 0 m = h m P m 2 M h m (3.15) 3.4SimulationResults Aparametricstudyshowinge˙ectofnoisevarianceonclassi˝cationcon˝denceisdemon- stratedinthissection.Theproposedalgorithmisappliedonasyntheticexampleof2- dimensionalfeaturespaceclassi˝edintotwoclasses-'red'and'blue'.The2Dfeatureplot showninFig.2(a)isobtainedbyrandomsamplingofdatapointsfromabivariatedistribu- 39 tion.Thesepointsinthefeaturespacesignifythefeaturesfromtrainingdatasetand "X" marksthetestdata ( x ) whoseclassi˝cationcon˝denceiscalculated.Simulatednoiseis modelledbywhiteGaussianuncorrelatednoisemodel: p n ( x ) ˘ N (0 ;˙ 2 n ) andisdenotedby theellipticalcontoursaroundthetestdata x inFig.2(a).Thenoisevarianceisvariedfrom 0to0.05andcorrespondingcon˝denceofthetestdataiscomputedaccordingtoalgorithms 1and2.Itshouldbenotedthatthemagnitudeof ˙ 2 n determinesthepoweroftheinjected noise. (a)(b) Figure3.5:(a)2Dscatterplotoftrainingandtestdata x withnoisedistributions.(b) Classi˝cationcon˝denceof x in'Red'classwithrespecttovaryingnoiselevelsoftestdata. Fig.2(b)showsthatasvarianceofnoisedensityfunctionincreases,thecon˝denceof classifyingthetestdatareduces,indicatedbybluecurve C n .Thereddashedlinedenotes con˝dencecalculatedwithouttakingnoiseintoconsideration C .Itisareasonableargument thatiftheoutputmeasurementfromNDEinspectionisa˙ectedbyahighlevelofnoise, correspondingfeatureswillbeincorrect,leadingtolessreliableclassi˝cationdecisionwhich isre˛ectedbyitslowclassi˝cationcon˝dence.Ontheotherhand,ifa˛aw-signalisa˙ected bylownoise,thecontributionofnoiseweightsontheBayesposteriorprobabilityislowand hencethecon˝denceofclassi˝cationishigher. 40 Chapter4 Con˝denceMetricEvaluation:Eddy CurrentinspectionofSteamGenerator tubes. 4.1Introduction Oneofthe˝eldswhereNDEisusedextensivelysince1950sisthenuclearindustry. Steamgenerators(SG)areheatexchangetubesusedinnuclearindustriesfortransferring heatfromtheprimarylooptothehotpressurizedwatercirculatingontheoutsidetopro- ducesteam,whichisusedtoruntheturbines.TheseSGtubesarecontinuouslyexposedto hightemperature,vibrationsandcorrosiveenvironmentoftenresultinginvarioustypesof degradationssuchasmechanicalwearbetweentubeandtubesupportplates,outerdiameter stresscorrosioncracking(ODSCC),pitting,volumetricchanges,primarywaterstresscorro- sioncracking(PWSCC),andintergranularattack(IGA).Tubewallthinningorformation ofcrackscausesharmfulradioactivegasesleakfromtheprimarysidetothesecondaryside whichmaybecatastrophictoenvironmentorleadtounscheduledplantshutdowns.Hence thereisastrongeconomicincentivetobuildNDEsystemsinordertoperiodicallymonitor thegeneralhealthofSGtubes. 41 Multi-frequencyandmulti-coilprobeeddycurrenttesting(ECT)hasbeenane˙ective NDEtechniquesusedforin-serviceSGtubeinspectionastheyareverywellsuitedfor detectingdefectsinconductingmaterials[49,50,51].Duetoharshenvironmentfaced bytheheatexchangeunitsandtheircomplexgeometries,oftenothermethodssuchas ultrasonics,radiography,liquid-penetrantsoropticalscanningareincapableofproducing strongindicationofanomalieswhichmakesECTanobviouschoice.Eddycurrentinspection hasproventobebothfastande˙ectiveindetectingandsizingmostofthedegradation mechanismsthatoccurredintheearlygenerators.ByusingECTitispossibletodetectand sizedefectseveninthepresenceofartifactsthatusuallycomplicatetheanalysisprocedure. Moreover,collectionofdataatseveraltestfrequenciessimultaneouslydecreasesin-service inspectiontimeandhumanexposuretimetoradiation.Threemajortypesofmultifrequency eddycurrentprobesareusedinpracticethebobbincoil,therotatingprobecoil(RPC) andthearraysensors. StructuralhealthmanagementofSGtubesandrelateduncertaintyquanti˝cationtech- niqueshavebeenanimportantNDEproblem[52].Withrapidincreaseintheamountofdata obtainedfromheatexchangertubebyECinspections,thereisahighdemandofautomated signalanalysissystemsthatcanprovideaccurateandconsistentsignalinterpretationand avoiderrorsbyhumananalysts.ThedataacquiredfromSGtubeinspectionmustbeana- lyzedaccuratelyandinnearreal-time.Generally,theanalysisrequirementisaclassi˝cation ofthesignalinto˛awandnon-˛awcategories.Insuchscenarios,computationofreliability ofeachclassi˝cationbecomescriticalsothatspeci˝cclassi˝cationswithlowcon˝dencecan bereportedtoNDEanalystforfurtherinvestigation.Inthischapter,con˝denceofsignal classi˝cationhasbeenstudiedforeddycurrentdatacollectedusingRPCprobefromin- spectionofSGtubesconsistingofvolumetric˛aws.Thisstudywasconductedasapart 42 ofaprojectfundedbytheElectricPowerResearchInstitute(EPRI),USA.ECdatafrom varioustubegeometrieswerecollectedbyEPRIandanalysedbyanautomatedanalysis softwaredevelopedbyNonDestructiveEvaluationLaboratoryofMichiganStateUniversity [5,6].Con˝dencemetricswereassignedtotheclassi˝cationresultsfromthisdataanalysis software. Thischapterbrie˛yreviewstheprincipleofECtechniqueandexistingsignalprocess- ingmethodsperformedonECdatatoenhance˛awindicationbyincreasingsignal-to-noise (SNR)ratio.Further,noisefromtypicaleddycurrentsignalsobtainedfromSGtubein- spectionbyRPCprobeisstudiedtoextractstatisticalparametersofthenoisedistributions. Finally,boostedBayesCon˝dence,proposedinchapter3,wascalculatedforevery˛aw indicationsbytakingnoisestatisticsintoconsideration. 4.2PrincipleofEddyCurrentTesting Eddycurrent(EC)techniqueworksontheprincipleofelectromagneticinduction.When analternatingcurrentsourceisbroughtclosetoanelectricallyconductingmaterial,an alternatingmagnetic˝eldisinducedinthematerialwhichcausescurrentto˛owinside thematerialintheformofclosedpathlikeeddies;theirdirectionbeingoppositetothe inducedcurrent˛owaccordingtoLenz'slaw.Opposingsecondary˝eldgeneratedbythe inducedcurrentinthesampleinteractswiththeprimary˝eldandreducesthecharacteristic impedanceoftheexcitationcoil,asdemonstratedinFigure.4.1.Moreover,presenceofa defectinthesampleamendsthepathoftheinducededdieswhichinturnchangesthecoil impedancesubstantially.ThischangeincoilimpedanceformstheNDEsignal(containing informationaboutsampledefects)recordedbytheECprobes[39]. 43 Figure4.1:(a)Eddycurrentgenerationand˛owinaconductingspecimen(b)Changein impedanceofcoilinadefectanddefectfreeregion(Xaxis:resistance,Yaxis:inductance) [5]. (a)(b) Figure4.2:(a)RPCcon˝guration(b)Postprocessededdycurrentsignal(RPCprobeat 300KHz)ofadefectiveSGtube[5]. 4.3AutomatedAnalysisofSGTubeInspectiondata State-of-thearteddycurrenttestingofSGtubesbyrotatingprobecoil(RPC)isdemon- stratedinFigure4.2.Theseprobesacquiresatwo-dimensionaldatadepictingimpedance changeintheformofacomplexvoltagewhichisafunctionofaxialandcircumferentialpo- sitionoftheprobeinthetubestructure.Imaginarycomponentofatypicalpostprocessed signalfromeddycurrenttubeinspectionisshowninFigure.4.2.Theabscissaandordinate denotethecircumferentialandaxialdirectionoftheSGtuberespectively.Similartoany NDEdataanalysissystem,eddycurrentdataare˝rstsubjectedtosignalprocessingfollowed byidentifcationofregionsofinterests(ROIs).FromtheROIsorthepossiblelocationof 44 Figure4.3:AutomtatedECdataanalysissystemwithcon˝dencemetriccomputationmod- ule. ˛aws,discriminativefeaturesarechosentoeventuallyclassifyaROIintodefectornon-defect category. The˛owchartofexistingautomatededdycurrentanalysissystemsalongwithaddedcon- ˝dencemetriccomputationmoduleapplicabletoinspectiondatafromSGtubesispresented inFigure8.10.AbriefoverviewofexistingmethodsinECdataanalysisisdiscussedinthe followingsubsections. 4.3.1SignalPre-processing PotentialROIsshownbyrectangularboxesinFigure.4.2(b)areidenti˝edafterrigorous signalprocessingalgorithmsontherawsignalwhichincludescalibration,tubesupportsignal (TSP)suppressionandnoiseremoval.DetailsofthesealgorithmsaredescribedbyUdpa etal.in[6].SeveralothersignalprocessingmethodologiesforanalysingECsignalshave 45 Figure4.4:Variousstagesofautomatedsignalprocessing(a)Rawdata,(b)Calibrateddata, (c)TSPsuppression,(d)thresholding(e)ROIdetection[6]. beendiscussedextensivelyinliteraturesuchaslinearandnonlinearmixing[53]andwavelet transforms[54,40].Often˛awindicationsarecorruptedbynoiseand/ornondefectsignals duetotheprobelift-o˙andsurroundingtubestructureswhichadverselya˙ectdetectionand characterizationofdefects.Hence,itbecomesnecessarytoenhancetheSNRoftheECT signalsbyusingsignalprocessingmethods[55,56]beforeimplementingtherecognition techniques. 4.3.2ROIDetection OncetherawsignaliscleanedanditsSNRisimproved,possible˛awlocationsareidenti- ˝edbyselectingpeaksignalsorsignalsaboveapre-de˝nedthresholdadaptively[57].Signals atdi˙erentfrequenciesareintegratedappropriatelytodeterminethepotentialdefectindi- cations.However,the˝nalresultoftencontainsignalsfromnon-˛awsaswell(forinstance, signalsfromexternaldeposits).TheenitreprocessofadaptivethresoldingfollowedbyROI selectioninatypicalSGtubeinspectionsignalisdemonstratedinFigure.4.4. 46 4.3.3FeatureExtractionandClassi˝cation SignalcharacteristicsorsalientfeaturesareextractedfromtheROIswhichcandistin- guishadefectROIfromanon-defectone.Featurescanbecomputedusingthesignalintime domainsuchaspeak-to-peakvalueoftherealandimaginarycomponentsofthecomplex eddycurrentsignal,itsphaseangleorenergy[58].Transformedfeatures(eg:Fourierde- scriptors[59])orstatisticalfeatures(eg:principalcomponents[60])havebeenusedaswellin existingECdataanalysis.ThesefeaturesarecalculatedfromeachpotentialROIfromdata obtainedatseveralexcitationfrequencies.Figure.4.5showsthetwo-dimensionalfeature spacespannedbytherealandimaginarycomponentsofthecomplexECsignalacquiredat 300kHzbyRPC.EachfeaturepointrepresentsanROIcollectedfromallthetrainingdata whoserealcategories(orgroundtruth)areknown.Theredlabelleddatapointsaredefect ROIswherasthebluelabelledonesaretheROIswhichwereselectedafterthresholdingstep buteventuallydidnotrepresenta˛aw. Figure4.5:Scatterplotforsteam-generatortubedatashowingfeaturesfromtrainingdata frombothclassesandtestdata x . Featuresarefedintoarule-baseorneuralnetworksorothermachinelearningalgorithms toclassifythemintodefectornon-defectclass.Severalclassi˝ersforanalysiseddycurrent 47 signals[9]includingimpedanceplanediagrams[61],inverseanalysis[62]andarti˝cialneural networks[15]havebeenstudiedinliterature.IntheexampletestdatashowninFigure.4.2, redandgreenboxesdenoteROIsinthetubewhichhavebeenclassi˝edasnon-defectsand defectrespectivelybyrule-basedclassi˝erintheautomatedanalysisalgorithm. Thesubsequentstepofanautomatedsignalclassi˝erinNDEistoassessthereliability ofitsclassi˝cationresultsbyassigningcon˝dencemetrics.Ourproposedcon˝dencemeasure incorporatesfeatures,classi˝cationresultsandnoisestatisticsoftheacquiredECsignalas shownin˝gure8.10.Hence,thenextsectionfocussesonthestudyofnoiseinECsignal obtainedfromSGtubeinspection. 4.4NoiseAnalysisinFieldDatafromEddyCurrentIn- spection Inourstudy,noiseextractedfromECinspectionsignalsismodelledasamixtureofone ormoreGaussiandensityfunctions(GMM).Therationalebehindthisassumptionisbased onapreviousworkbySafdarnejadetal.[63]wheretheauthorshavenotonlyshownthat theexperimentalnoisepresentincomplexECTsignalsfromSGtubeinspectionadheresto GaussiandistributionbutfurtherreportedthatGMMalongwithnoise˝lteringalgorithms enhancesperformanceofnoisyECsignalanalysis.Anotherimportantcharacteristicofthe noiseisitsadditivenature.Itisknownthatsteamgeneratorsconsistofseveraltubes˝xed withsupportsfromtheoutside.Ideallyinnoise-freescenario,theECTsignalfromSG tubeinspectioncontainindicationsonlyfromthetubesupportsandfromananomalyifit exists.Signalatthefree-spanregion(inbetweenthetubesupports)isassumedtobezero inabsenceofanynoise.Howeverinrealexperiments,theECTsignalatthefree-spanisnot 48 zeroeverywherebutcontainssomelowvoltagemeasurementswhichoriginatesduetoprobe wobble,mechanicalmotionofprobeandmeasurementnoise.Therefore,thesignalfrom free-spanregionisconsideredasthenoise-onlymeasurementwhichismodeledasadditive tothenoise-freemeasurement.ThisassumptionisbackedbyapreviousstudywhereOlin etal.[3]usedasequenceofNDEsignalsatdi˙erentpositionsonaunitcontainingno˛aws toprovideinformationaboutthedistribution. (a)(b)(c) (d)(e)(f) Figure4.6:(a)Imaginarychannelimageofasampleeddycurrentresponsesignalat300KHz withrectangularROIboxindicatingcircumferential˛aw.(b)Signalwithmaskedtube supportand˛awregion(c)Noisehistogram.(d-f)Repeatedforrealchanneldata. Fig.4.6explainstheprocessofestimatingnoisedistributionofatypicaleddycurrentre- sponseimage.Imaginaryandrealcomponentsoftheeddycurrentinspectionimageobtained fromadefectiveSGtubeat300KHzareshowninFig.4(a)andFig.4(d);ROIscontaining ˛awindicationsandsignalfromthetubesupportaremasked,asshowninFig.4(b)and Fig.4(e),andtherestofthesignalisusedtoextractnoise-onlyinformationrepresentedas 49 Figure4.7:ExperimentalnoisemodelledasabivariateGaussiandistribution. thenoisehistograms(Fig.4(c)andFig.4(f)).Itisimportanttonotethat˛awsignalsin thevicinityofexternalsupportstructures(suchastubesupportplates(TSP))aredistorted bythepresenceofthesupportstructuresandhencenotincludedasthenoise-onlyregion. Since,experimentalnoiseiscomplexwithrealandimaginaryvalues,theresultantnoise ismodelledas2DdistributionasshowninFig.4.7.Inthisexample,noisedistribution hasbeenmodelledasabivariateGaussiandensityfunctionwhosestatisticalparameters areestimatedbymaximizingthelikelihoodfunction.Theestimatednoisedistributionis describedbytheellipticalcontoursinFig.4.7. Itshouldbenotedthatexperimentalnoiseisspeci˝ctotubeinspected,probeandthe inspectionfrequency.Fig.4.8showstwodi˙erenttubesthatareinspectedusingtwodi˙erent kindsofRPCprobe(pancakeandplus-pointprobe)at200KHzand300KHz.Fig.6(a) andFig.6(c)showthenoise-onlysignalobtainedfromTube1andTube2inspections respectivelywhereasFig.6(b)andFig.6(d)showthecorrespondingnoisehistogramsof thetwotubesinspectedbytwoprobesattwofrequencies-200KHzand300KHz.Dueto suchuniquenatureofexperimentalnoise,itisabsolutelyimperativetostudythenatureof inspectionnoisebeforecomputingclassi˝cationcon˝denceofaROIpresentinaparticular SGtube. 50 (a) (b) (c) (d) Figure4.8:(a)Noise-onlysignalfromECTofTube1usingPancakeandPlus-pointprobes at200Hzand300kHz,(b)NoisehistogramofTube1signals,(c)Noise-onlysignalfromECT ofTube2usingPancakeandPlus-pointprobesat200Hzand300kHz,(d)Noisehistogram ofTube2signals. AttimeswhenGaussianfunctiondoesnotserveasthebest˝tteddistribution,other pdfssuchaslog-normal,exponential,gammaorbetafunctionscanbechosenwhichhasa highergoodnessof˝tonthenoisedata.However,theproposedcon˝dencemetricevaluation approachcanstillbeappliedtothosecaseswithnochangeappliedtothealgorithm.As describedbefore,theposteriorpdfwillbeweightedbythenoisedensityfunctionirrespective ofitsform. 51 Figure4.9:Scatterplotforsteam-generatortubedatashowingexperimentalnoisedistribu- tionforatestfeature x . 4.5Con˝denceofClassi˝cationwithNoiseConsidera- tion. Ineddycurrentinspection,magnitudeandphasebasedinformationformcrucialfeatures todiscriminatesignalsofadefectivesamplefromahealthysample[6].Inthisthesis,the peak-to-peakvalueofrealandimaginarycomponentsofthecomplexeddycurrentsignal arechosenassuitablefeaturesforcon˝denceanalysis.Fig.4.9denotesthefeatureplotusing featuresextractedfrom10tubesinthetrainingdatabase,eachcontainingoneormore˛aws. ExperimentalnoiseisextractedfromoneofthetestROIsandmodelledasbivariatedensity functionshownbyellipticalcontoursatthetestdatalocation.Classi˝cationcon˝dence ofthetestdatapointiscalculatedaccordingtoalgorithms1and2bymultiplyingBayes posteriorpdfwithnoise-weightsinabootstrapframework. InFig.4.9,con˝denceofclassifyingtestROIas'defect'withouttakingnoiseintoaccount iscalculatedas90.41%.Afterincorporatingthee˙ectofnoise,thecon˝dencereduced to80.15%.Similarly,allROIsclassi˝edas'defects'wereassessed,noiseintheirresponse 52 signalswereextractedandtheirclassi˝cationcon˝dencemetricswereobtained.Theresults arerecordedinTable4.1.Animportantthingtonoteisthatmodelingtrainingdatawith Gaussianpdfmaynotalwaysbeavalidassumptiondependingonthedataandshould bechosencarefullybeforecalculatingBayesposteriorprobability.Statisticaltoolssuchas quantile-quantileplotsshouldbecomputedonthetrainingdatatoverifythevalidityofthe Gaussianassumptiononthetrainingdataset.Ifdatadoesnot˝tanormaldistribution, otherpdfsshouldbeconsideredornon-parametric(kernelbased)approachesmaybeavailed. (a)(b)(c) Figure4.10:EddycurrentresponsesignalofthreeSGtubeswithROIsconsistingofdefects anda˙ectedbydi˙erentnoiselevels. InFig.4.10,eddycurrentsignalsfromthreeSGtubesareshown.TheROIswere identi˝edbyautomateddataanalysissoftwareandcon˝denceofclassifyingeachofthemas defectswascalculated.Itcanbeseenthatastheeddycurrentresponseimagegetsnoisier, classi˝cationofthedefectROIsbecomesmoredi˚cult.Hencecertaintyoftheautoanalysis resultsdecreaseswithhighernoiselevelwhichcorrespondstolowerclassi˝cationcon˝dence. 53 Table4.1:Con˝denceofclassi˝cationofdefectsinsteamgeneratortubesusingRPCprobe. Tube-ID No.of˛aws Noise-mean ( n ) 10 4 Noise-cov ( n ) 10 3 Noise-free Con˝dence C 0 Con˝dence withnoise C n 1 4 1 : 6 4 : 2 2 : 90 : 5 0 : 52 : 7 2 6 6 4 0 : 99 1 0 : 95 0 : 59 3 7 7 5 2 6 6 4 0 : 98 0 : 99 0 : 65 0 : 55 3 7 7 5 2 1 0 : 18 1 : 2 5 : 80 : 6 0 : 60 : 8 0 : 98 0 : 82 3 2 0 : 043 0 : 025 1 : 70 : 09 0 : 092 : 7 1 1 0 : 99 0 : 99 4 2 0 : 022 0 : 026 0 : 90 : 5 0 : 50 : 9 0 : 99 1 0 : 98 0 : 99 5 1 0 : 19 0 : 23 1 : 5 0 : 75 0 : 750 : 8 0 : 99 0 : 94 6 3 4 : 5 0 : 5 2 : 2 1 : 1 1 : 11 : 2 2 4 0 : 99 1 0 : 90 3 5 2 4 0 : 86 0 : 99 0 : 61 3 5 7 3 0 : 16 0 : 34 2 : 3 1 : 9 1 : 92 : 6 2 4 0 : 98 0 : 99 0 : 96 3 5 2 4 0 : 58 0 : 86 0 : 67 3 5 8 1 0 : 12 0 : 002 3 : 4 0 : 17 0 : 172 : 8 0 : 90 0 : 80 9 3 4 : 6 2 : 9 1 : 8 0 : 59 0 : 591 : 3 2 4 0 : 63 1 0 : 99 3 5 2 4 0 : 61 0 : 99 0 : 95 3 5 10 1 0 : 96 0 : 38 1 : 54 : 7 4 : 73 : 04 0 : 99 0 : 85 SGtubesdepictedinFig.8(a),(b)and(c)correspondtotubeID4,5and7inTable4.1and area˙ectedbylow,mediumandhighnoiselevelrespectively.Theclassi˝cationcon˝dence oftheseROIsarerecordedasa)99.8%,98.03%(b)94.23%and(c)58.36%,86.86%,67.82%. Thisitcanbeconcludedthattheproposedcon˝dencemetriciswellrepresentativeofnoise 54 inNDEresponsesignal. Anotherinterestingobservationtodeducefromtheseresultsisthatalowcon˝dence metriccanserveasanindicatorofsmaller˛awswhicharemoredi˚culttobediagnosed.For example,thetwo˛awsinTubeID1and9weretwoofthesmallest˛awsinthedatabaseused forthisstudyandtheywereassociatedwithcon˝dencevaluesof0.59and0.63respectively evenbeforetheirexperimentalnoisewasconsidered.Insuchcases,itmightbeusefulto segregatethestudyintoclassesof'larger˛aw'and'smaller˛aw'beforeevaluatingthe con˝dencemetrics.Thisshallbelookedintoinmoredetailasafutureextensionofthis research. 55 Chapter5 Con˝dence-RatedClassi˝cationinNDE 5.1Introduction AfteridentifyingtheunderlyingfactorsofuncertaintiesinatypicalNDEdataclassi˝ca- tionsystemandgeneratingasuitablecon˝dencemetricforclassi˝cationofNDEsignal,the nextideawastostudythepossibilityofimprovingperformanceofASCsystemsbyincor- poratingknowledgeofitsclassi˝cationcon˝dence.Thebasicideaistousethecon˝dence metricasafeedbacktotheclassi˝erasshownintheschematicallyin˝gure5.1.Thebene˝t ofsuchasystemisthatitnotonlygeneratesaself-evaluatingmetricofreliability,butalso utilizesitasafeedbackandretrainsthesystemtoachievealowererrorrateonblindtesting. Figure5.1:Automatedanalysissystemwithcon˝dencefeedback. Thischapterdescribesthedevelopmentofacon˝dence-rated-classi˝erensembleapproach isdevelopedtoclassifyeddycurrentdatainto'defect'and'nondefect'classwhichincor- 56 poratesunderlyingstatisticalcharacteristicsofdata.Reliabilitymeasurede˝nedbyBayes con˝dence,inthepreviouschapters,isfedintotheautomateddataanalysissystemsuch that˝nalclassi˝cationofNDEsignalsisenhanced.OurworkisinspiredfromShapire's ADABOOST(traditionalboosting)[64]algorithmwiththemodi˝cationofmaximizingcon- ˝denceofclassi˝cationateveryiterationoftheclassi˝erensembleinadditiontominimizing themisclassi˝cationerror.Suchanapproachhelpstodetectdefectswithweakerindications whicharemissedotherwise. Oneofthelatestcontributionsinthe˝eldofmachinelearningisthedevelopmentofen- sembleclassi˝ersknownasboostingormixtureofexperts.Inabroadersense,anensemble systemisacombinationofoutputsfrommanyindividualclassi˝erssuchthatthecombined classi˝erperformsbetterthanalltheindividualones.Thereareseveraladvantagesofusing ensemblesystems,oneofthembeingitsabilityofstatisticallearningfromlimitedamountof trainingdatawhichisparticularlyrelevantforanalysingNDEdata.In1990,Schapire[64] provedthatastrongclassi˝ercanbegeneratedthroughaweightedcombinationofseveral weakclassi˝ersanddevelopedADABOOSTalgorithmwhichwasfollowedbyextensiveem- piricalandtheoreticalstudy[65][66][67].Otherversionsofensemblebaseddecisionsystems includemixtureofexperts[68],classi˝erfusion[69]andcommitteesofneuralnetworks[70]. Bene˝tsofensemblebasedsystemsinautomateddecisionmakingapplicationshavealsobeen recentlydiscoveredbycomputationalintelligencecommunityandNDEresearchers.Polikar etal.[71]developedLearn++algorithm,basedonensembleclassi˝ers,whichachievesincre- mentallearningondatafromultrasonicweldinspectionwhereintheclassi˝erisabletolearn newinformationwithoutforgettingpreviouslyacquiredknowledge. 57 5.2Background 5.2.1ADABOOST Boostingisatechniqueofcombiningagroupofweaklearnersintoastrongclassi˝er withmuchlowererrorrate.Aweaklearnerisasimpleclassi˝erwhichproducesprediction resultsofaninstancejustbetterthanrandom-guessing.Boostingcreatesanensembleof classi˝ersbystrategicallyresamplingthedata.The˝nalclassi˝cationisthenobtainedby combiningpredictionresultsofweakclassi˝ersusingmajorityvoting.In 1997 ,Freundand SchapireintroducedADABOOSTalgorithmwhichgenerateshypothesesbytrainingweak learnersoninstancesdrawnfromaniterativelyupdateddistributionofthetrainingdata [72].Thisupdateensuresthatinstanceswhicharemisclassi˝edinpreviousclassi˝erare morelikelytobeincludedintrainingdataofthenextclassi˝er.Thepseudocodeforthe originalADABOOSTalgorithmisprovidedinAlgorithm3. Let S = f ( x 1 ;y 1 ) ;::::; ( x m ;y m ) g beasequenceof m trainingsampleswhereeachinstance x i 2 X representsafeaturevectorandeachlabel y i 2 Y representsthetrueclassof x i . AlthoughADABOOSTcanbeexendedtomulticlassproblems,inthispaperwelimitour discussiontoabinaryclassi˝cationschemesuchthat Y = 1 ; +1 g . Forade˝nedsetof T iterations,aweakclassi˝eristrainedonthetrainingsequence S . Thedistribution D 1 ( i ) isinitializedtobeuniformwhichsigni˝esthatat t =1 ,allinstances ( x i )areequallylikelytobeselectedfortrainingthe˝rstweakclassi˝er. D 1 ( i )=1 =m ; (5.1) Witheveryiteration,theweightdistributionisupdatedaccordingtotheequation5.2and 58 aweakhypothesis h t isgeneratedforeveryinstance h t ( x i ) 8 i suchthat h t ( x i ) 2 H t ;H t = [ 1 ; +1] . D t +1 ( i )= D t ( i ) e ( t y i h t ( x i ) ) Z t (5.2) t = 1 2 ln 1+ r t 1 r t (5.3) where r t = P i D t ( i ) y i h t ( x i ) Wheneverthereismismatchinsignof h t and y t ,itsigni˝esmisclassi˝cationofthat instanceanditsweightisincreased.Correctlyclassi˝edinstanceshavetheirweightsun- changed.Theparameter controlsthein˛uenceofeachoftheweakhypothesisandis de˝nedbyequation5.3.The˝naloutputoftheclassi˝erensemble H ( x ) isthesignedsum- mationofalltheweakhypothesesgivenbyequation5.4.Duringtestingofblinddata x ,the ˝nalhypothesis H ( x ) iscalculatedanditsclassispredictedbasedonitssign 1 ; +1 g . H ( x )= sign ( T X t =1 t h t ( x )) (5.4) 5.3Con˝denceratedclassi˝cation:proposedmethod Incon˝dence-ratedboostingproposedbySchapireandSinger[73],thechosencon˝dence measureisheuristicinnatureanddoesnotquantifythesourcesofuncertainties.Inthis thesis,typicaluncertaintiespresentinNDEdataanalysisarequanti˝edintermsofcon˝dence measurewhichincludese˙ectofquantityoftrainingdata,qualityoffeaturesandnoisein testdata.Thereforeitisamorecomprehensivemeasureofreliabilitywhichcanbeusedas feedbacktoclassi˝cationalgorithmtoincreasetheclassi˝cationaccuracy. 59 Algorithm3 ADABOOST Input: Trainingdata S = f ( x i ;y i ) g ;i =1 ; 2 ;::::;m withcorrectlabels y i 2 Y , Y =[ 1 ; +1] ; 1: Initialize D 1 ( i )=1 =m ; (5.5) and t =1; (5.6) 2: T = Totalno.ofiterationsinclassi˝erensemble; Output: FinalhypothesisH(x) 3: for t T do 4: TrainWeaklearnerusingdistribution D t ; 5: Getweakhypothesis h t ( x i ) 8 i suchthat h t ( x i ) 2 H t ;H t =[ 1 ; +1] ; 6: Calculate t = 1 2 ln 1+ r t 1 r t (5.7) where r t = P i D t ( i ) y i h t ( x i ) 7: Update D t +1 ( i )= D t ( i ) e ( t y i h t ( x i ) ) Z t (5.8) where Z t isanormalizationfactorchosentohave D t +1 asadistributionfunction; 8: endfor 9: Finalhypothesis H ( x )= sign ( T X t =1 t h t ( x )) (5.9) Thepseudocodeofproposedcon˝dence-ratedADABOOSTisshowninAlgorithm4.The primarydi˙erencefromtraditionalADABOOSTisthatthepredictionofeveryinstanceby eachweakhypothesisismultipliedwithitsassociatedBayescon˝dence. h t ( x i )= C t;i ( x i ) H t ;H t = 1 ; +1 g (5.10) where, C t;i = p t pos ( x ) y j = p t ( x ) y j P 2 i =1 ( p t ( x )) y i (5.11) Byweightingthehypothesisofeveryweaklearnerwiththecon˝dencemetric,thesamples 60 whichareclassi˝edwithahighercon˝dencebuttothewrongclassareassociatedwithlower weights.Hence,theobjectivefunctionismodi˝edsuchthatforeveryiterationclassi˝cation con˝denceismaximizedalongwithminimizingtheerrorrate. Algorithm4 Bayescon˝dence-ratedADABOOST Input: Trainingdata S = f ( x i ;y i g ;i =1 ; 2 ;::::;m withtruelabels y i 2 Y , Y = 1 ; +1 g ; Output: FinalhypothesisH(x) 1: Initialize D 1 ( i )=1 =m (5.12) and t =1 (5.13) 2: T = Totalno.ofiterationsinclassi˝erensemble; 3: for t T do 4: Estimate ( t ;˙ t ) y i forallclasses y i 5: Calculateclass-conditionalpdf p t ( x ) y i forallclasses y i 2 Y , Y = 1 ; +1 g using estimated ( t ;˙ t ) y i 6: Calculatetheposteriorprobabilityfortheclassforwhichcon˝denceiscalculated(say y j ): C t;i = p t pos ( x ) y j = p t ( x ) y j P 2 i =1 ( p t ( x )) y i (5.14) 7: TrainWeaklearnerusingdistribution D t ; 8: Getweakhypothesis h t ( x i ) 8 i suchthat h t ( x i )= C t;i ( x i ) H t ;H t = 1 ; +1 g (5.15) 9: Obtain t = 1 2 ln 1+ r t 1 r t (5.16) where r t = P i D t ( i ) y i h t ( x i ) ; 10: Update: D t +1 ( i )= D t ( i ) e ( t y i h t ( x i ) ) Z t (5.17) where Z t isanormalizationfactorchosentohave D t +1 asadistributionfunction; 11: endfor 12: Finalhypothesis H ( x )= sign ( T X t =1 t h t ( x )) (5.18) 61 5.4Results 5.4.1SimulationResults Theproposedmethodofcon˝dence-ratedADABOOSTisappliedonclassi˝cationof syntheticdataintotwoclasses-redandblueanditsperformanceiscomparedwithtradi- tionalADABOOSTperformance.Datasetusedfortrainingandvalidationtestingofthetwo methodsareshownin˝gure5.2. Figure5.2:(a)Trainingdata(b)testdatawithtrueclasslabelsoftwoclasses:redandblue. Figure5.3illustratestheclassi˝cationresultsofthetwomethods.Itisfoundthatafter 16iterations,theerrorrateontrainingdatahasreachedtozerointraditionalADABOOST classi˝er,butisat 5% forcon˝dence-ratedADABOOST.However,theerrorrateonthetest dataclassi˝cationiscalculatedas 35% and 25% fortraditionalADABOOSTandcon˝dence- ratedADABOOSTrespectively.Thisisduetothefactthattheprincipalobjectiveof ADABOOSTistominimizetrainingerrorwhichoftenleadstoover˝ttingofthemodel. Itresultsinmisclassi˝cationsonablindtestsetwhichisre˛ectedfromtheerrorinthe testdataset.Whencon˝dence-ratedADABOOSTisapplied,itincreasesthegeneralization propertyoftheclassi˝erbyclassifyingtestdatawithlowererror. Figure5.4showsthatinboththeapproachesofclassi˝erensemble,theerrorrateon 62 Figure5.3:Trainingerrorrateversusnumberofweakclassi˝ersin(a) traditional AD- ABOOST(b) Bayescon˝dence-rated ADABOOST.Testdataclassi˝edwithADABOOST model(c) traditional ADABOOST:Errorrateontestdata= 35% (d) Bayescon˝dence-rated ADABOOST:Errorrateontestdata= 35% testdatadecreaseswithincreaseinnumberoftrainingsamples,althoughcon˝dence-rated ADABOOSTposesapproximately 5% lowererrorratethantheother. Figure5.4:Comparisonofclassi˝cationperformanceoftraditionalADABOOSTand con˝dence-ratedADABOOST. 63 5.4.2ExperimentalResults AsdescribedinNDEapplicationpresentedinchapter4,con˝dence-ratedclassi˝eren- sembleisutilizedtoclassifyexperimentaldataobtainedfromeddycurrentinspectionof steamgenerator(SG)tubes.SGtubeswhicharecontinuouslyexposedtoharshenvironmen- talconditionsarea˙ectedbyvarioustypesofdegradations.Thereisdemandfromindustry forthedevelopmentofautomatedsignalclassi˝cationsystemsthatcanprovideaccurate andconsistentsignalinterpretationwithcapabilityofcomputingitsreliability.Atypical post-processedsignalfromeddycurrenttubeinspectionisshownin˝gure6.TheregionsOf interest(ROIs)denotingthepossiblelocationsofthe˛awsareidenti˝edbytheASCsystem asshownbytherectangularboxes. Figure5.5:Asamplepost-processededdycurrentsignalofdefectiveSGtube.Redrectan- gularboxes:ROIscontainingfalseindications(classi˝edas non-defect );Greenrectangular boxes:ROIscontainingtruedefects. DiscriminatoryfeaturesareextractedfromtheseROIsoftrainingdataandusedto developtheclassi˝ermodel.Inthisexperiment,peakvaluesofrealandimaginarysignal fromeachROIareusedasfeaturesforclassifcation.Asinthecaseofsyntheticdataset,the totalavailabletrainingdatafromexperimentswasdividedintotwosets:onetotrainthe classi˝ermodelandtheothertovalidatetheperformanceoftheclassi˝er.Bothmethods ofclassi˝erensemblewerecompared.Thefeatureplotsofthetrainingandtestdatasetare 64 shownin˝gure5.6. Figure5.6:(a)Trainingdata(b)testdatawithtrueclasslabelsoftwoclasses:red(non- defect)andblue(defect). Con˝denceofclassi˝cationofeverytrainingsampleiscalculatedbythecomprehen- siveBayesposteriorprobabilityasstatedinalgorithm2andthenimplementedintothe con˝dence-rated-classi˝cationframework.Classi˝cationresultsbythetwoADABOOST methodsareshownin˝gure5.7.Asinsyntheticdataset,con˝dence-ratedADABOOSTis abletocorrectlyclassifymoretestinstancesrelativetothetraditionalADABOOST.Afew ˛aws,asindicatedinFigure5.8,aredetectedcorrectlybyapplyingcon˝dence-feedbackto classi˝erensemblethatweremissedbytraditionalADABOOST.Threeeddycurrentimages ofdefectivesteamgeneratortubesaredepictedhavingROIsideniti˝edbytheASCsystem. Thegreenboxesindicatedefectswhicharecorrectlyidenti˝edbyboththeclassi˝cation methods.TheredrectangularROIboxesindicatemoresubtle˛awsandaretheoneswhich areclassi˝edasnon-defectsbytraditonalADABOOST,butcorrectlyidenti˝edasdefects bythecon˝dence-ratedADABOOST.Asaresult,errorratereducedfrom 19 : 40% to 14 : 93% intheproposedASCsystemhavingcon˝dencefeedback. Inthisthesis,con˝dence-basedADABOOSThasbeenvalidatedonatwo-classclassi- ˝cationproblem.Thisapproachnotonlyemphasizestheimportanceofaselfevaluation 65 Figure5.7:Trainingerrorrateversusnumberofweakclassi˝ersin(a) traditional AD- ABOOST(b) Bayescon˝dence-rated ADABOOST.Testdataclassi˝edwithADABOOST model(c) traditional ADABOOST:Errorrateontestdata= 19 : 40% (d) Bayescon˝dence- rated ADABOOST:Errorrateontestdata= 14 : 93% (a)(b)(c) Figure5.8:Eddycurrentresponsesignalaftercalibrationof3defectiveSGtubes(Imaginary channel,pluspointprobe,at300KHz).Greenboxes:ROIs(truedefects)classi˝edasdefects bybothtraditionalandCon˝dence-basedADABOOST.Redboxes:ROIs(truedefects) classi˝edasnon-defectsbytraditionalADABOOSTbutcorrectlyclassi˝edasdefectsby Con˝dence-basedADABOOST. 66 measureinASCsystems,butfurtherutilizesitforimprovingclassi˝cationofNDEsignals. Theproposedclassi˝erexploitstheadvantagesofaboostingalgorithmwhileavoidingthe problemofover-˝tting.Weakerindicationsoftubedefectsfromaneddycurrentresponse signalwhicharemisclassi˝edbytraditionalADABOOST,arecorrectlyclassi˝edwitha con˝dence-basedensemblesystem. 67 Chapter6 PrognosisinNDE 6.1Introduction Inadditiontoassessingclassi˝cationcon˝denceofNDEdataobtainedfromperiodic monitoringofstructuresandindustrialcomponents,studyofstructuralreliabilitybasedon theacquireddataisanequallycriticaltasktoachieve.Theprimaryobjectiveoflong-term reliabilityanalysisinNDEisdefectgrowthpredictionordamageprognosis.AscitedbyFar- raretal.[74],damageprognosis(DP)isde˝nedasestimationoftheremaininguseful life(RUL)ofequipmentbytakingintoconsiderationtheinformationgatheredfrommon- itoringsystems,designinformation,pastoperationexperienceandoperatingenvironment ofthesystemorequipmenAccurateanddynamicRULpredictionenablesindustriesto maximizeusageofacomponentbeforeitencountersacatastrophicfailure.Integratedstruc- turalhealthmonitoringanddamageprognosis(SHM-DP)strategiescoupledwith nondestructiveevaluation(NDE)techniquesarebecomingfundamentalengineering toolsfornear-real-timestructuralintegrityassessmentandpredictivemaintenance. 6.2TheoryofReliability AccordingtoInternationalOrganizationforStandardization(ISO),reliabilityisde˝ned as"theabilityofanitemtoperformarequiredfunction,undergivenenvironmentalandop- 68 Figure6.1:Statevariable X ( t ) andTimetoFailure t ofasystem.[7] erationalconditionsandforastatedperiodoftime(ISO8402)".Insimplerterms,reliability iscalculatedastheprobabailitythatagivencomponentorentitycanoperatesatisfacto- rilyforaspeci˝edtimeperiodintheactualapplicationforwhichitisintendedwithout experiencingafailure. Thestateofansystemattime t maybedescribedbythestatevariable X ( t ) . X ( t ) is de˝nedas: X t = 8 > > > < > > > : 1 ; ifsystemisfunctioningattimet 0 ; ifsystemisnotfunctioningattimet (6.1) Supposethesystemstartsoperatingattime t =0 .Thetimeelapsingfromitsstart timetotheinstantitencountersafailureistermedasthetimetofailure( T ).Therelation between X ( t ) and( T )isdemonstratedinFigure6.1. Itisquiteobviousthatduetopresenceofseveraluncertaintiesduringtheoperation ofthesystem,timetofailure( T )cannotbeinterpretedasa˝xedvaluebutasarandom variablewithaprobabilitydensityfunction f ( t ) anddistributionfunction: F ( t )= Pr ( T t )= Z t 0 f ( u ) du for t> 0 (6.2) where F ( t ) denotestheprobabilitythattheitemfailswithinthetimeinterval (0 ;t ] .The 69 Figure6.2:Distributionfunction F ( t ) andprobabilitydensityfunction f ( t ) .[7] pdf f ( t ) andCDF F ( t ) areillustratedinFigure6.2. Itshouldbenotedthattheoperationtime t doesnotindicatetheclocktime.Itcan includeanyothermetricswhichcountstheageorusageofthesystemsuchasnumberof loadingcyclesofamechanicalpart,numberofkilometersacarhasbeendriven,numberof rotationsofabearingetc. Thereliabilityfunctionofasystemcanbede˝nedas R ( t ) where: R ( t )=1 F ( t )= Pr ( T>t ) fort> 0 (6.3) orequivalently R ( t )=1 Z t 0 f ( u ) du = Z inf t f ( u ) du (6.4) Hence R ( t ) istheprobabilitythattheitemdoesnotfailinthetimeinterval (0 ;t ] ,or,inother words,theprobabilitythattheitemsurvivesthetimeinterval (0 ;t ] andisstillfunctioning attimet.Thereliabilityfunction R ( t ) isalsocalledthesurvivorfunctionandisillustrated inFigure6.3.Thereareseveralotherstatisticalmeasuresandfunctionswhichareuseful instudyofreliabilitytheorysuchasFailureRatefunctionorMeanTimetoFailureetc. whichareoutsidethescopeofthisresearchandhencenotdiscussedinthisthesis.Readers 70 Figure6.3:Reliabilityorsurvivorfunction R ( t ) .[7] interestedinsuchtopicscan˝ndthede˝nitionsandexplanationsinthebookbyRausand andArnljot[7]. 6.2.1RemainingUsefulLife(RUL) Amongthebroaderde˝nitionofreliabilitymeasure R ( t ) ,themetricmostcommonlyused indamageprognosisisremainingusefullife(RUL)ofasystemunderoperation.RULof anysystemcanbede˝nedbyarandomvariablewhichdependsonsystem'scurrentage,its operationenvironmentandhealthinformationacquiredfromperiodicNDEofthesystem. Ifthehistoryofinspectiondataacquireduptotime t isdenotedby Y ( t ) ,theprimarygoal ofprognosisistoestimateexpectationoftheRULpdf: E ( R t j Y t ) . TheprocessofdamageprognosisfollowedbyRULcalculationisdemonstratedin˝gure 6.4.Firstlyahealthindex(HI)isde˝nedwhichcharacterizesthedamagelevelofasystem orstructureatagiventimeinstant t .Afterregulartimeintervals(orloadingcycles), measurementsarerecordedandHIiscalculatedateverytimestepuptothecurrenttime (say k ).Theseconstitutethemeasurementdatashownbytheblackdotsin˝gure6.4.The objectiveofdamageprognosisistoconstructthedamagepropagationpathuptothecurrent timeusingthemeasurementHIvaluesaswellaspredictHIforfuturetime( ^ ˝ )tillthesystem 71 Figure6.4:IllustrationofdamagepathprognosisandRULprediction[8] . reachesaprede˝nedfailurethreshold.FailurethresholdisusuallydecidedasthevalueofHI whenthesystemisexpectedtocrashorfailandisgenerallyobtainedfromdomainexperts inthespeci˝capplication˝eld/industry. RUL =^ ˝ k (6.5) Ata˝rstglance,damageprognosismayseemlikeatrivialproblemofpolynomial˝ttingon themeasurementdataandthenextendingthe˝ttedcurveuptothethresholdtocalculate theRUL.However,thismaynotbeafeasibleapproachinmostpracticalcases.Figure6.5 illustratesthereasonwhyRULpredictionusingasimplecurve˝ttingsolutionmaynotalways leadtothecorrectsolution.Inmanycasesthedamagepropagationisacomplexanddynamic phenomenonwhichnotonlydependsonthematerialanddimensionsofthestructurebutalso onexternalcharacteristicssuchaspressure,temperatureorotherenvironmentalconditions etc.Insuchcases,adamagepropagationcurvegeneratedfrom˝rstfewmeasurementsisvery di˙erentfromthetruedamagegrowthpath.WrongestimationofRULcanbedangerous especiallyifitisover-estimated(illustratedinFigure6.5(a)).A˝rstorderpolynomial 72 ˝tisappliedonthemeasurementdataacquiredupto750hoursandthedamagegrowth curveisestimated.Ifdamagesizeof0.03mmisconsideredasthefailurethreshold,then theoptimumtimeatwhichthesystemshouldbestoppedisat2200hours.However,the estimateddamagegrowthlinereachesthethresholdof0.03muchbeyond2200hoursand thereforetheequipmentwillcontinueoperationbeyonditssafetylimitwhichmaybefatal. Figure6.5:(a)1degreepolynomial˝ttingonmeasurementdataleadingtounderestimation ofRUL(b)2degreepolynomial˝ttingonmeasurementdataleadingtooverestimationof RUL Ontheotherhand,under-estimationofRULleadstosuboptimalperformanceofthe equipmentasthesystemisstoppedtooearly,asshowninFigure6.5(b).A 2 nd order polynomialcurveis˝ttedonthesamemeasurementdatawhichreachesthefailurethreshold of0.03mmat1630hours(muchearlierthan2200hours).Asaresult,thesystemisstopped earlierthanitssafetylimit(2200-1630=570hoursbeforeexpectedfailure).Boththesecases shouldbeavoidedandthereforestatisticalmethodsareadoptedforaccurateRULestimation. Prognosticapproachesshouldideallybeabletoincorporateunderlyinguncertaintiesinvolved inthedamagepropagationprocessinordertoprovideaccuratepredictionresults.Areview ofcurrentstate-oftheartinthistopicisprovidedinthefollowingsection. 73 6.3LiteratureReviewonRULPrognosis 6.3.1Model-basedmethods Existingprognosismethodscanberoughlyclassi˝edintomodel-based(orphysics-based) anddata-drivenmethods.Model-basedmethodspredicttheequipmenthealthcondition usingcomponentphysicalmodels,suchas˝niteelement(FE)models,anddamageprop- agationmodelsbasedondamagemechanics.Suchmethodsuse˝xedmodelparameters dependingonmaterialpropertiesandgenerallydonotuseconditionmonitoringdatafor predictionofdamageevolution.Severalmodel-basedsystemshasbeenstudiedoverthepast years.Kacprzynskietal.[75]presentedaprognosistoolusing3DgearFEmodelingto studydamageinititationandpropagationinhelicoptergears.LiandLee[76]proposeda gearprognosisapproachbasedonFEmodelingwhereanembeddedmodelwasproposed toestimateFouriercoe˚cientsofthemeshingsti˙nessexpansion.Thestrip-yieldmodel includedintheNASGROsoftwaredevelopedin[77]iswidelyusedtosimulatecrackgrowth undervariableamplitudeloading.Ifaccuratemodelscanbedevelopedforeverymechani- calstructureanddamagetype,model-basedprognosiscanprovidepredictionresultswith highprecision.However,buildingauthenticphysicalmodelsfordescribingtheequipment dynamicresponseanddamagepropagationisachallengingtaskinitselfwhichrequiresa thoroughunderstandingofthesystem.Ifanyimportantphysicalphenomenonismissed,the predictionofdegradationwillbeerroneousresultingincatastrophicconsequences.. 6.3.2Data-basedmethods Ontheotherhand,data-drivenprognosticmethodsmodelstherelationshipbetween equipmentageandconditionmonitoringdatabytrainingtheprognosticsystemonhistor- 74 icaldata.Gebraeeletal.[78]usedArti˝cialNeuralNetwork(ANN)formonitoringrolling bearingelementsandpredictingfatiguecrackpropagationfromvibration-baseddegradation signals.Bayesianupdatingmethodshavebeeninvestigatedinequipmentprognosticsfor utilizingthereal-timeconditionmonitoringdata[79].Data-drivenmethodsdonotrelyon physicalmodels,andonlyutilizethecollectedconditionmonitoringdataforhealthpredic- tion.Accuracyofthesemethodsstronglyrelyuponthetrainingdatacharacteristics.As aresult,theymayfailtoproduceaccuratepredictionifinsu˚cientorunder-representative trainingdataisused.Resultsfromthesemethodsmaysometimesbecounter-intuitiveas theydonotconsiderunderlyingphysicsofthesystemandthereforemaybeerroneousat times. 6.3.3Integratedmethods Byincorporatingbene˝tsofbothmodel-basedanddata-basedprognosticapproaches, integratedorhybridmethodshavegainedalotofpopularityinrecentyears[80].Underthese methods,physicsbaseddegradationmodelsareconsideredbuttheparametersofthephysical modelunderlyingthedamagegrowthprocessarenot˝xed.Insteadtheyareestimated utilizingthedatafromCBMofthestructure.Bayesianinference[81]isacommontechnique implementedinseveralstudieswhereinthemodelparametersareupdatedateveryinstanta newinspectiondataisreported,therebyincreasingtheaccuracyofestimatedphysicalmodel. Anothercrucialbene˝tofintegratedmethodsistheirabilitytoincorporateuncertainity duetomodelaswellasmeasurementsintotheiralgorithmwhichmakesthemabetter representativeofpracticalsystems.BayesianinferencehasbeenusedbyShankaretal.[82] toestimateparametersof˝niteelementmodel,surrogatemodelandcrackgrowthmodel incylindricalstructuressubjectedtofatigue.Anotherhybridapproachistheparticle˝lter 75 basedframeworkdevelopedbyOrchardandVachtsevanos[83]forthefailureprognosisof planetarycarrierplates. Aliteraturereviewandmathematicaltheoriesofafewbroadcategoriesofprognostic approachesarediscussedbelowforunderstandingthestate-of-artmethodologiesinthisarea. 6.3.3.1Regressionbasedmodels Regression-basedmethodsarehavegainedpopularityinindustriesandacademic˝elds forestimationofequipmentlifeduetothesimplicityofthesemodels.[84,85].Theyfall intothecategoryofdata-basedprognosis.Thefundamentalprincipleofthesemethodsis thatthehealthofthesystemsunderstudycanbemappedbysomekeyfeaturesobtained fromconditionmonitoring(CM)ofsystemsandRULcanbeestimatedbytrending,and predictingtheseCMfeaturesuptoaprede˝nedthreshold.LuandMeeker[84]werethe˝rst authorstopresentageneralnonlinearregressionmodeltocharacterizethedegradationpath ofapopulationofunits.Accordingtothegeneraldegradationmodel,theobservedsample degradation Y ( t ) attime t canberepresentedas Y ( t )= D ( t ; ; )+ ( t ) ,where D ( t ; ; ) istheactualpathattime t , isthe˝xede˙ectregressioncoe˚cientscommonforallunits, istherandome˙ectrepresentingindividualunitcharacteristics,and ( t ) istherandom errortermdescribedby N (0 ;˙ ) .Here, and ( t ) areassumedtobeindependentofeach other.Usingthismodel,theRULatsamplingtime t i canbede˝nedas: X t i = x t i : D ( t i + x t i ; ; ) w j D ( t i ; ; ) ( t k ) (6.9) 79 and y t = g ( x t ; ) (6.10) where g isafunctiontobedetermined, isanoiseterm, k isthetimeofthelast monitoringcheckpointand ( t k ) istheintervalbetweenthecurrentandthelastcheckpoint. Inaddition,Luoetal.[98]usedamultiple-model˝ltertoestimatethemeanandvariance oftheRULwithoutconsideringthedistributionofRULexplicitly. Amongallstochastic˝lteringtechniques,themostcommonlyusedprocessistheBayesian updatingbasedonthephilosophythatonecanincorporatepriorknowledgeaboutthedegra- dationphenomenoninthemodelandupdatethemodelasmoremeasurementsarecollected. OneofthewaysofachievingBayesianupdatingprocessistheparticle˝lteringapproachin- vestigatedbyOrchardandVachtsevanos[83].Theyusedanon-linearstate-spacemodel (withunknowntime-varyingparameters)andaparticle˝lteringalgorithmthatcanupdate thecurrentstateestimate.Inthisthesis,thisapproachwasusedtocomputethedamage growthcurveincompositematerialsandthereforethisalgorithmwillbedescribedinmore detailinthefollowingsections. 6.4TheoryofBayesianUpdating AccordingtotheBayesianpointofview,observationdata X isconsideredarandom variablegeneratedfromanunderlyingpdf f ( x; ) ; 2 . isalsode˝nedbyarandom variablewithdensity f ( ) whichdescribestheprobabilityofoccurrenceofaparameter valuefrom ,beforeanyobservationismade.Hence f ( ) iscalledthepriordensityof . TheobjectiveofBayesupdatingistoobtain ^ ,theestimatedvalueof thatcharacterizes theunderlyingpdfgeneratingtheobservationdata. 80 Withthisinterpretation,jointdensityof X and , f X; ( x; ) ,isgivenby: f X; ( x; )= f X j ( x j ) :f ( ) (6.11) Themarginaldensityof X is, f X ( x )= Z f X; ( x; ) d = Z f X j x j ) :f ( ) d (6.12) Hence,theconditionaldensityof given X = x becomes, f j X ( j x )= f X; ( x; ) f X ( x ) = f X j ( x j ) f ( ) f X ( x ) (6.13) f j X ( j x ) expressestheprobabilitydistributionof afterhavingobserved X = x , and f j X ( j x ) isthereforecalledtheposteriordensityof .Itshouldbenotedthatwhen Xisobserved, f X ( x ) occursinEquation6.13asaconstant.Hence f j X ( j x ) isalways proportionaltof f X j ( x j ) f ( ) : f j X ( j x ) / f X j ( x j ) f ( ) (6.14) TheBayesianapproachisusedforupdatinginformationabouttheparameter .First, aninitialprobabilitydensityfor isassignedbeforeobservationsof X = x iscollected. Whenthe˝rstmesurementin X isavailable,thepriordistributionof isupdatedtothe posterioridistributionof ,given X = x .Thisprocessisrepeatedandinthenextiteration theposteriordistributionof ,given X = x ,ischosenasthenewpriordistribution.When anothermeasurementin X isobserved,itleadstoasecondposteriordistribution,andso 81 Figure6.6:BayesUpdatingProcess on.ThisupdatingprocessisillustratedinFigure6.6. The˝naltaskoftheBayesupdatingprocessistoestimatethevalue of thatgenerates anobservedvalue x of X .Wedenotethisestimateby b ( x ) .Theoptimumestimateisthe onethatminimizestheexpectationofmeansquarederror(MSE): E [( b ( x ) 2 ] . E [( b ( x ) 2 ]= Z 1 Z ( b ( x ) 2 f X; ( x; ) (6.15) Byusingequation6.14, E [( b ( x ) 2 ]= Z 1 f X ( x )[ Z ( b ( x ) 2 f j X ( j x ) ] dx (6.16) E [( b ( x ) 2 ] isminimizedwhenforeach x , ( x ) ischosentominimize [ R ( b ( x ) 2 f j X ( j x ) ] whosesolutionis E j X ) . Hence,accordingtoBayesinferenceorupdatingprocess,theestimateof isthemean oftheposteriordistributionof . b = E j X ) (6.17) 82 6.4.1Bayesupdateofmodelparametersusingsyntheticdata Forprognosticsapplication,theprimarygoalistoestimatethedamagegrowthpathbased onmeasurementdata,asillustratedinFigure6.4.Ifdatabasedmethodisadoptedinsuch applications,nodamagemodelneedstobede˝ned;butdatadrivenmodelsmaynotalways yieldaccurateresultsasdiscussedearlier.Henceintegratedmethodsareabetteralternative whereadamagepropagationmodelisde˝nedwhoseparametersareestimatedusingthe measurementdatabyBayesupdatingprocess.Itshouldbenotedthatinthisthesis,damage growthmodelfordegradationincompositesisde˝nedbasedonexperimentalmeasurements acquiredfromperiodicNDEinspections.Althoughmeasurementswereassumedtoimply theunderlyingmodel,physicsbasedrelationbetweendamagelevelandloadingcycleshave notbeenstudiedinthisresearch.Physics-basedrelationshipsforaspeci˝cgeometryand loadingconditionsdependentirelyonthestructuralmechanicsofacomponentandwillbe incorporatedinfutureextensionsofthisresearch. Asyntheticmeasurementdatasetisconsideredtodemonstratetheimplementationof Bayesupdatingprocedureusingasimpleexponentialdamagemodelde˝nedas: k = Ct m (6.18) where, k representsthecracklengthpropagatingovertime t . C and m arethemodel parameters.Inthisexample,thevalueof C iskept˝xedat 9 : 12 10 3 and m isestimated usingtheBayesupdating. 6 syntheticmeasurementsareselectedbyaddingrandomnoise tothetruecracklengthvaluesattime 1 ; 10 ; 20 ; 30 ; 40 and 50 seconds.Thesereplicatethe observationsthatareobtainedfromexperimentswherethecracklengthismeasuredafter ˝xedintervalsoftimeorloadingcycles.Figure6.7(b)showsthetruecrackgrowthcurve 83 (byusingtruevalueof m =1 : 48 )alongwiththemeasurementsselected.Onlythese6 measurementdatawereusedtoestimatetheunknownparameter m ofthedamagemodel toeventuallypredictthecrackpropagationpath. (a)(b) Figure6.7:(a)BayesUpdatingofparameter m (b)Crackgrowthexampleforsynthetic datasetwithestimatedcrackgrowthpath. Thepriordistributionof m ischosenas: f prior ( m ) ˘N (2 : 6 ; 0 : 5 2 ) (6.19) Ateachinspectiontimeinstant,theposteriordistributionofthecurrentiterationbecomes thepriordistributionforthenextupdatingtime.Theupdatinghistoryforthecrackgrowth parametersisshowninTable6.1.Thusitcanbeconcludedthatbyrepeatingtherecursive processasnewmeasureddatabecomesavailable,theestimatedparametersconvergesto theirtruevalues[99]. 84 Table6.1:BayesUpdatinghistoryofparameter m forsyntheticcrackgrowthpath. 6.5BayesianUpdatingbasedonParticleFiltering(PF) Approach Inrecentyears,recursiveBayesianframeworkhasbeenusedextensivelyinfaultdiagnos- ticsandprognosticsapplications[100,101].Inthisapproach,observeddataisincorporated intothea-prioristateestimationbyconsideringthelikelihoodofmeasuredvalues.Particu- larly,sequentialMonteCarlo(SMC)technique,alsoreferredtoasparticle˝ltering(PF)has gainedpopularityinengineeringdomainowingtotheirconsistenttheoreticalfoundationto handlemodelnon-linearitiesornon-Gaussianobservationnoise[83,102].Inthisapproach, theconditionalprobabilityisapproximatedbya'swarm'ofpoints,knownas'particles'. Theparticlesconstitutediscretesampleswithassociatedweightsrepresentingthediscrete probabilitymasses.Particlescanbegeneratedandrecursivelyupdatedgivenanon-linear processmodel,ameasurementmodel,asetofavailablemeasurements Z = f z k ;k 2 N g and aninitialestimationforthestateprobabilitydensityfunction(pdf) p ( x 0 ) .Usingthisidea OrchardandVachtsevanos[83]presentedafailureprognosticmodeltopredicttheevolution intimeofthefaultindicatorandcomputetheRULpdfofthefaultysubsystem. UnderPFframework,theBayesianupdateisprocessedinasequentialwaywithparticles havingprobabilityinformationofunknownparameters.Itisbasedonastate-transition 85 functionorthedamagepropagationmodel f andmeasurementfunction h [103]. a k =( f; k ; k ) (6.20) z k = h ( a k ;! k ) (6.21) where k isthetimestepindexorindexofloadingcycleatwhichsampleisscanned, a k is damagestate, k isparametervectorand z k isthemeasurementdata. k and ! k arethe modelandmeasurementnoiserespectively.Inprognosticapplications,themeasurementis assumedtobea˙ectedbywhiteGaussiannoise ! k ˘ N (0 ;˙ ) .Therefore,theunknown parametersare = f a;;˙ g ,includingthedamagestate a whichisobtainedbasedonthe modelparameters . Figure6.8illustratestheprocessofBayesupdationusingparticle˝lteringtechnique. Figure6.8:IllustrationofBayesestimationusingparticle˝lteringtechnique. Next,asyntheticdatasetisusedasanexampletodemonstratethePFalgorithmfor 86 estimationofdamagegrowthpathandremainingusefullife(RUL). 6.5.1PFestimateofmodelparametersusingsyntheticdata Asimpleexponentialdamagepropagationmodelisde˝nedwherethecrackgrowsexpo- nentiallywithtime(8.4).Thetruevaluesoftheparametersareselectedas C = 22 : 62 and m =3 : 8 andthetruecrackgrowthpathisshowninFigure6.10. a = Ct m (6.22) ForapplyingPFalgorithm,bothparameters C and m areconsideredunknownand representedbyasetof n =5000 particles.Forthesakeofsimplicity,measurementnoiseis modelledaswhiteGaussiandensitywithstandarddeviation( ˙ )of0.01.Asyntheticdataset of25pointsarechosen,alsodenotedinFigure6.10,whichformthemeasurementsusedin Bayesupdatingbyparticle˝ltering.Theparticle˝lteringapproachcanbedescribedinthe followingsteps. (a) Initialization :At k =1 step, n samplesofallparametersaredrawnfrominitial(prior) distribution.Thepriordistributionparametersareslectedeitherbasedondomain knowledgeorintelligentdataprocessingfromavailablemeasurements.Mostoften, experimentsareconductedmultipletimesandthe˝rstinspectionobservationsare usedastheprior.Inthisexample,thepriordistributionsfortheunknownparameters aresetas: a 0 ˘N (0 : 01 ; (5 10 4 ) 2 ) m 0 ˘N (4 ; (0 : 02) 2 ) , log C 0 ˘N ( 22 : 33 ; (1 : 12) 2 ) 87 Inthisexample,theparameter C and m followlog-normalandnormaldistribution respectively. (b) Prediction :Posteriordistributionsofthemodelparametersevaluatedattheprevious ( k 1) th stepareusedaspriordistributionsatthecurrentstep ( k th ) intheformof particles. Also,damagestateatthecurrenttimestepistransmittedfromthesamplesatthe previousstepaccordingtothedamagepropagationmodel(8.4). a k = C k t k ) m k + a k 1 (6.23) where t k isthetimegapbetween ( k 1) th and k th inspectionstep. (c) Updating :Inthisstep,thelikelihoodiscalculatedsuchthatBayesinferencecan beevaluatedaccordingtoEquation6.11.Giventhatmeasurementnoise ! k follows normaldistribution,thelikelihoodcanbecomputedas: L ( z k j a i k ;m i k ;C i k )= 1 z k p 2 ˇ˘ i k exp 2 4 1 2 ln z k i k ˘ i k ! 2 3 5 ;i =1 ;:::n (6.24) where, ˘ i k = v u u u t ln 2 4 1+ ˙ a i k ( m i k ;C i k ) ! 2 3 5 (6.25) and i k =ln h a i k ( m i k ;C i k ) i 1 2 ( ˘ i k ) 2 (6.26) (d) Resampling :Resamplingisthestepinwhichanexistingsetofparticlesisreplaced 88 byanewset.ItisparticularlyessentialinPFinordertoavoiddegeneracyofweights [104]inwhichafewparticlesdominatetherest,afterthe˝rstfewiterations.This oftenleadstoinaccurateestimateswithlargevariances. Severalresamplingtechniquesarediscussedinliteraturesuchassingledistribution sampling[105]thresholds/grouping-basedresampling[106]orvariablesizeresampling [107].Inourapplication,asequentialimportanceresamplingtechnique,speci˝cally theinverseCDFmethodisappliedtoachievetheresamplingprocess[102]wherebya particleoftheparameterhavingtheCDFvaluegeneratedrandomlyischosenandthe processisrepeated n timesinordertoobtain n resampledparticlesattheendof k th iteration. Figure6.9:IllustrationofresamplingbyinverseCDFmethod[8]. Figure6.9illustratestheaboveprocess.Herearandomvalueisgeneratedfrom U (0 ; 1) whichbecomesaparticularCDFvalue(e.g., 0 : 45 inthe˝gure).Finally,asampleof theparameterhavingtheCDFvalueisfound,whichismarkedbyarectangleinthe ˝gure.Byrepeatingthisprocess n times,nsamplesareobtained.Notethatsince samplesexistinadiscreteform,thesamplehavingtheclosestvaluetotheCDFvalue 89 isselected. ThePDFconstitutedfromtheseresampledparticlesformstheposteriordistribution ofthecurrentiterationorthepriordistributionofthenextiteration. (e) Prognosis :Oncethemodelparametersareestimated,thedamagestateispropagated fromthecurrentstateuptothethresholdvalue.ThePDFoftheparticlesincurrent stateissubtractedfromthoseatthethresholdvalueinordertoevaluatethePDFof theRUL.ThemedianRULalongwithitscon˝denceintervalsarecalculatedfromthe PDFoftheRUL. Figure6.10demonstratestheresultsofapplyingparticle˝lteringonthesyntheticdataset. Thetruevaluesofparameterswereknowna-prioriandservedasareferencetocomparethe accuracyofestimatedvaluesusingthesyntheticdata.Theestimatedcrackgrowthpath withitscon˝denceboundsmatchescloselytothetruepathasseeninFigure6.10(a).Also, theestimatedRULhistogramisplottedinFigure6.10(b)usingafailurethresoldatcrack length =0 : 03 units.ThemedianRULat 1200 cyclesiscomputedas 950 cycleswithits 95% con˝denceboundsat 750 and 1200 cycles.Thismeansthatifthecurrentinspectionisdone at 1200 cycles,after 950 cyclesthecracklengthispredictedtoreachitsfailurethresoldof 0 : 03 mm.TheresultsshowclosealignmentofpredictedRULwithitstruevaluewhichis 1000cycles. Itisimportanttonotethatinstatisticalprognosis,observedconditionmonitoring(CM) datafromperiodicNDEcanbeclassi˝edintodirectandindirectCM[108,109].Datafrom directCMdescribestheunderlyingdamagestatedirectlysuchascracklengthordamage areaextractedfromNDEimagingtechniquesorsti˙nessdataobtainedfromstraingauge. FordirectCM,predictionofRULisequivalenttopredictionoftheCMdatatoreacha 90 (a)(b) (c)(d) Figure6.10:(a)CrackgrowthpredictionusingPFalgorithm(b)PredictedRULhistogram (c)Traceofupdatingofparameter C (d)Traceofupdatingofparameter m prede˝nedfailurethresholdlevel.Ontheotherhand,indirectCMprovidesdatawhichcan indirectlyorpartiallyindicatethehealthstatusofastructure.Inthesecases,failureevent datamaybeneededinadditiontoCMdataforRULestimation.ExamplesofindirectCM aretime-of-˛ightdataobtainedfromultrasonicwaves,amplitudedatafromeddycurrent signalsorfeaturesfromotherNDEtechniquesfromwhichstructuralhealthcanbededuced indirectly.Chapter7focusesonprognosisusingdirectCMwhereasapplicationsonindirect CMarediscussedinchapters8and9. 91 Chapter7 SingleSensorPrognosisinComposites byDirectConditionMonitoring 7.1Introduction Inlastfewdecades,compositematerialshavegainedimmensepopularityandreplaced metalsoralloysinseveralindustriesnamelyaviation,automotive,spaceandconstruction owingtotheirsalientpropertiesoflight-weightness,highspeci˝csti˙nessandstrength. Despitetheirhighenvironmentalandfatigueresistance,laminated˝bre-reinforcedpolymers (FRP)areoftenvulnerableto˛awsduringfabricationandservicesuchasfatiguecracks ordisbondsinadhesivemetal-compositejoints.HencethereisaneedforNDEexperts todevelopmethodologiesforinspectingcompositematerials.Also,industrialcomponents madeofcompositematerialsaresubjectedtoawiderangeofstressesduringtheirservice life.Dynamicloadingiscommonespeciallyinaircraftcomponentssuchasdroppingoftools duringmaintenanceorhailstormswhileinservicewhichposeseriousthreattotheremaining usabilityandreliabilityofsuchcomponents.Ifacompositelaminateissubjectedtorepeated low-velocityimpactofsu˚cientenergy,itmaycreatedamageinternallyintheformof delaminationswhichmayremaininvisiblebutcansigni˝cantlycompromisethestructure's integrity.Hence,severalanalyticalandexperimentalinvestigations[110,111,112,113, 92 114]havebeenconductedontheinitiationandevolutionofimpactdamagesincomposite materialswhichdemonstratedthattheextentandrateofgrowthofsuchdamagedepend onthematerial,manufacturingprocess,hybridization,energylevelsandgeometryofthe impactor.Vulnerabilityofcompositematerialspropelstheneedforrobustprognosticsand healthmonitoringtechniques.Inthischapterwefocusonprognosisofdamageaccumulation inGFRPsamplesduetorepeatedlowvelocityimpacts. Accuratehealthprognosticsiscriticalforcondition-based-maintenanace(CBM)andfor reducingoveralllife-cyclecosts.UnderCBM,dataiscollectedfromvariousnon-destructive evaluation(NDE)techniquessuchasvibration,acousticemission,X-rayimagingetc.are utilizedforstructuralhealthinspectionandpredictionofRUL.SeveralNDEtechniques arediscussedinliteratureforinspectingimpactdamageincompositelaminates.Meola etal.[115]demonstratedtheuseofinfraredthermographytoimagedelaminationsinthe sample.X-raycomputedtomographyhasbeenpopularaswelltoinspectdelaminationsin GFRP[116,117].Inthiswork,opticaltransmissionscanning(OTS)wasusedtodetectand locatedamageintroducedinaGFRPcompositeplatebysuccessivelowvelocityimpacts. OTShasbeenrecentlyproposedbyKhomenkoetal.[118]asanovelopticalmethodfor quantitativeNDEofGFRPstructures.Thetechniquecanbeusedwhenaccesstoboththe topandbottomsurfacesofthetestsampleareavailable.Inadditiontobeingnon-contact, rapid,cost-e˚cientandsafe,itprovideshigh-resolutionopticaltransmittance(OT)scansof aGFRPsample.Detailsofthismethodhasbeendescribedinsection2. Thischapterpresentstwocrucialcontributionstoresearchinprognosisofcomposite materialsbydirectCMutilizingexperimentaldatafromOTSofGFRPsamples.Firstly,an optimizeddamagepropagationmodelisdescribedusingimprovedParis'lawfordelamination growthinthesample.Lackofrobustmodelscapableofdescribingthecriticaltransition 93 fromahealthytoaprogressivelydamagedsampleuptothecompletecollapseofthematerial makesestimationoffailurethresholdsmorechallengingforcompositestructures.Although crackpropagationmodelsinmetalshasbeenstudiedextensively,damagepropagationin GFRPspeci˝callythegrowthofdelaminationsintroducedbylow-velocityimpactshavenot beenaddressedyet.Secondly,anintegratedprognosismethodisimplementedtoestimate damageareagrowthinGFRPwhereindatafromOTSformstheCBMdatatobeused forestimationofthefuturedamageareaundertheframeworkofparticle-˝ltering.With growingdemandofGFRPinindustries,prognosticstudiesonsuchmaterialshavebecome imperativeandareaddressedinthisstudy. 7.2ConditionbasedMaintenanceofGFRP 7.2.1ImpactDamageinGFRP Oneofthemostcommondegradationmechanismsencounteredincompositesamples isdelaminationformedbylowvelocityimpacts.Impactdamageincompositesoccurin thetransversedirectionwheretheylackthrough-thicknessreinforcementandthetransverse damageresistanceispoor.Asaresult,theimpactforcetendstobreakthe˝bresinthe polymerandeventuallyleadstoformationofairgapsordelaminationsinsidethematerial whichmayormaynotbevisiblydetected[119].Delaminationinacompositeplateis causedduetointerlaminarstresseswhicharedependentonspecimengeometryandloading parameterssuchasdimensionsofspecimen,typeofboundaryconditions,shapeofimpactor, impactenergy,etc.Interlaminarstrengthisstronglyrelatedtothematerialproperties, i.e.,fracturetoughnessofmatrixandbondingstrengthbetween˝berandmatrix.E˙ect ofimpactdamageonresidualcompressivestrengthofGFRPlaminateshavebeenstudied 94 previouslybyseveralresearchers[113,114]. Figure7.1showsahealthysampleofGFRPlaminateandafterbeingsubjectedtoan impactof20J.Asdelaminationareaincreases,thestrengthofthestructurereduceswhich eventuallyleadstofailure. Figure7.1:(a)HealthyGFRPsample(nodamage),(b)DelaminatedGFRPsampleafter E=20Jimpact. 7.2.2OpticalTransmissionScanning(OTS) Inthispaper,experimentaldatafromOpticalTransmissionScanning(OTS)system [120]isusedforassessingandpredictingdelaminationgrowthinGFRPduetorepeated impacts.Opticaltransmissionscanning(OTS)hasemergedasaviabletechniqueforrapid andnon-contactnondestructiveevaluation(NDE)ofglass˝berreinforcedpolymer(GFRP) composites[118].Earlierworks[121,120]highlightedthecapabilitiesofOTSinquantifying lowvelocityimpactdamageinmultilayerGFRPsamples,which,incombinationwithad- vancedimageprocessing,allowedforaccuratecharacterizationofmultipledelaminationsand theircontours.Theresultsobtaineddemonstratedexcellentagreementwithwell-established NDEtechniques. Figure7.2showstheimageoftheOTSsetup.Itcomprisesatranslationstage,alaser 95 sourcethatilluminatestheGFRPsamples,andadownstreamphotodetectorplacedunder- neaththesample.Thephotodetectorrecordsthepowertransmittedthroughthesample afteritisilluminatedbythelasersource.Consequentlytheoutputpowerdependsonthe transmissionpropertiesofthesamplebeingtested.Hence,presenceofdelamination(airgap) insidethesamplealtersthetransmittedradiationreceivedbythephotodetectorswhichis capturedbytheOTSsystem.Imagesfrominspectionofahealthyandimpactedsampleby theOTSispresentedinFigure7.3. Figure7.2:Experimentalsetupofopticaltransmissionscanningsystemwithimpactedsam- pleundertest. OTShasbeendemonstratedasasuccessfulfastandnon-contacttechniquetodetect delaminationsinGFRPandvalidatedusingadigitalcameraimageofthecross-sectionof theGFRPsample[118].TheauthorsfurtherappliedadvancedsignalprocessingontheOT imagesinordertodeterminethedelaminationcontoursasafunctionofthenumberof˝ber layersthathavebeena˙ectedbyimpact.Asdenotedin˝gure7.3(c),theOTSimage ofimpactedGFRPissegmentedintofoursetsofdelaminations,rangingfrom1to4+, quantifyingtheextentandseverityofdamage.AdetaileddescriptionoftheOTSoperat- ingprinciple,imageprocessingprocedurefordeterminingthesegmentsofdelaminationsin GFRPandcomputationoftheareaofeachdelmainationsegmentisillustratedin[118].The 96 Figure7.3:(a)( top )Healthy(noimpact)GFRPsample;( bottom )GFRPsampleafterE=30J impact(b)OTScanof( top )healthy;( bottom )impactedsample(c)Segmentationofdelam- inationsin( top )healthy;( bottom )impactedsample. resultsobtaineddemonstratedexcellentagreementwithotherwell-establishedNDEtech- niques.OngoingworkoftheauthorsisfocusedonextendingthecapabilitiesofOTSto3D imagingsuchthattheycanbetailoredtowardsscanningtheGFRPstructureunderloading conditionsinindustrialapplications. 7.3ProposedPrognosticframeworkfordelaminationgrowth model 7.3.1DamagePropagationModel Inthisapplication,damagepropagationmodelusedfordescribingpropagationofde- laminationareainsideaGFRPsampleduetorepeatedlow-velocityimpactsisbasedon ParisLawwhichde˝nestherelationshipbetweencrackgrowthrateandstressstateofthe 97 structure,asgivenbyequation1. da dN = C K ) m (7.1) where a isthecracklength, da dN isthecrackgrowthratepercycle, N isthetotalnumber ofload/impactcycles, K istherangeofthestressintensityfactorand C and m arethe Parislawparameters. K canbefurtherinterpretedas: K = Y p ˇa (7.2) where, Y isadimensionlessconstantdependingonthecrackshapeandgeometryofthe specimenforagivenstressrangeinfatiguecrackgrowthmodels. Frommaterialstructurestheory,itcanbeinferredthatmostofexistingcrackgrowth modelsarebasedontheempiricalParis'law[122]tode˝netherelationshipbetweencrack growthrateandstressstateofthestructure.Thereareseveralprognosticstudiesinliterature whichadherestoParislawtopredictcrackgrowthinmetallicstructuressuchasanalysisof axialcrackgrowthinUH-60planetarycarrierplate[83],aluminiumalloyspecimens[123]or fatiguecracksinSAE1045steel[124].However,unlikecrackgrowthinmetals,delamination inGFRPsamplesduetorepeatedimpactsbehavedi˙erently.Inparticular,the˝bre/matrix interphaseproperties,whicharea˙ectedby˝bresurfacetreatment,playanimportantrole indeterminingthefailuremechanisms,theextentofdamageandthethresholdenergyof thecomposite[125].Manyresearchershavestudiedimpactdamagesincompositematerials [112,119]andmonitoredtherelationshipbetweendelaminationareaandimpactenergy.A typicaldamagepropagationcurveispresentedinFigure7.4.Thedamageareaisfound 98 toincreaserapidlyinthe˝rstfewimpactsandthenslowdownbeyondathreshold.Such behaviourofdelaminationareagrowthhasbeenstudiedbeforebyWuetal.[112]wherehe reportedthattheareaextendsataslowerrateafterthe˝rstfewimpactsduetoconstraints fromthefour-˝xedendboundariesin (0 ; 90 ) cross-plylaminates. Figure7.4:Delaminationareapropagationwithrespecttoincreaseincumulativeimpact energy. ApplyingParisLawdirectlywasnotsuitableformodelingandpredictingdegradation processesinGFRPsamplesandhenceamodi˝edversionreferredastheParis-Parismodel basedonPiecewise-deterministicMarkovprocesses(PDMPS)isproposed.Themathemat- icaldetailsaredescribedin[123]wheretheauthorspresentedfatiguecrackgrowth(FCG) predictionapproachusing"Parismodelwithonejump".Inthispaper,thedamageprop- agationplotisdividedintotworegions-RegionIandRegionII.Insteadofconsideringa singleexponentialmodel,twodi˙erentexponentialmodelsareconsideredbeforeandafter thethresholdor'jump'indamagepropagationcurve.Parislawisusedinboththeregions butde˝nedbydi˙erentsetofparameters.Overall,the˝veparametersintheParis-Paris modeltobeestimatedare: f m 1 ;C 1 ;m 2 ;C 2 ;E g where m 1 ;C 1 and m 2 ;C 2 aretheparame- tersofParismodelbeforeandaftertheloadingcycle E .ItshouldbenotedthatthePDMPs 99 mayuseothercrackpropagationlawssuchasParis-Formanlaw[126]orParis-Erdoganlaw [122]forotherapplicationsandthenumberof'jumps'maynotberestrictedtoone.More 'jumps'willleadtomoreregionsandmoreparameterstobeestimated,withoutchangingthe underlyingtheory.Inourstudy,theParis-Parismodelwasusedformodellingthegrowthof delaminationareainGFRPsampleswithone'jump'inthedamagegrowthcurve. 7.3.2ParticleFilteringbasedPrognosisofDelaminationAreain GFRP Theparticle˝lteringapproach,describedinchapter6,hasbeenimplementedinthis studyfortheprognosisofdelaminationareainaGFRPsample.Theoverallalgorithm ismodi˝edtoestimateunknownparametervector oftheParis-Parismodelwhere = f m 1 ;C 1 ;m 2 ;C 2 ;E (Note: T isreplacedby E sincewemeasuredelaminationareaafter ˝xedintervalsofimpactenergyinsteadoftimeorloadingcycle). DamageareaobtainedfromOTSmeasurement( z k )at k th observationisassumedtobe equivalenttothetruedamagearea( a k )withadditivenoise,asdescribedinequation7.3, where ! k ˘N (0 ;˙ 2 ) . z k = a k + ! k (7.3) Theassumptionofadditivenormalmeasurementnoisesisbackedbyexperimentalevi- dence.MeasurementnoiseintheOTSsystemisessentialygeneratedduetovibrationsinthe equipmentgantry,noiseinphotodetectorandexternallights(eg:fromcomputerscreensin theopticallaboratory).Noisefromexternallightscontributetomajorityoftheexperimental noisewhereastheotherfactorscanbeneglected.Inordertoquantifythenoisedistribu- tion,OTSsystemwasusedtoscana 40 mm 50 mm areawithoutspecimeninabsenceof 100 lasersource.TheoutputimagecapturedbyOTSphotodetectorsindepictedinFigure7.5 (a).Itcanbefurtherconcludedthatthemeasurementnoisefollowsanormaldistribution withmeanat 0 : 2535 andstandarddeviationof 0 : 0091 .Similarobservationswerefoundfor repeatedOTSscanswithoutspecimen. (a)(b) Figure7.5:(a)Measurementnoisecollectedbyphotodetectorwithoutspecimeninabsence oflasersource(b)Measurementnoisehistogram : N ( =0 : 2535 ;˙ =0 : 0091) . Therefore,theunknownparameterstobeestimatedare = f a;;˙ g ,includingthe damagestate a thatdependsonthemodelparameters .Theparticle˝lteringapproach tailoredtowardsthisapplicationissummarizedinthefollowingsteps. (a) Initialization :At k =1 step, n samplesofallparametersaredrawnfrominitial (prior)distribution. (b) Prediction :Posteriordistributionsofthemodelparametersevaluatedattheprevious ( k 1) th stepareusedaspriordistributionsatthecurrentstep ( k ) . Usingequation7.1,damagestateatthecurrenttimestepispredictedfromtheparam- etersestimatedatthepreviousstepaccordingtoequation7.4. da = C Y p ˇa m dN (7.4) 101 TheParismodelisre-writtenintheformofstate-transitionfunctioninequation7.5. Itshouldbenoted,loadingcycleinterval dN ofParislawforcrackpropagationin metalsisreplacedby E ortheintervalofimpactenergybetweentwoconsecutive impactsonGFRP. a k a k 1 = C Y p ˇa k 1 m dN (7.5) Whenthecumulativeenergyoftheimpactsislowerthantheunknown'jump'energy E ,thedamageupdatefollowstheParislawwithparameters f C 1 ;m 1 g .Beyond E , themodelshiftstoRegionII(inFig.2)whereinthedamagepropagatesaccordingto ParisLawwithparameters f C 2 ;m 2 g . a k = 8 > > > < > > > : C k 1 Y p ˇa k 1 m k 1 E + a k 1 ; if E k E C k 2 Y p ˇa k 1 m k 2 E + a k 1 ; if E k E (7.6) (c) Updating :Inthisstep,thelikelihoodiscalculatedaccordingtoBayesinferencede- notedinEquation3.Assumingthatmeasurementnoise ! k followsnormaldistribution, thelikelihoodiscomputedas: L ( z k j a i k ;m i k ;C i k )= 1 z k p 2 ˇ˘ i k exp 2 4 1 2 ln z k i k ˘ i k ! 2 3 5 ;i =1 ;:::n (7.7) where, ˘ i k = v u u u t ln 2 4 1+ ˙ a i k ( m i k ;C i k ) ! 2 3 5 (7.8) 102 and i k =ln h a i k ( m i k ;C i k ) i 1 2 ( ˘ i k ) 2 (7.9) The n particlesforeachparameterin at k th iterationareassociatedwithweights thatcorrespondtothePDFvalueofthe i th particleinmeasurement z k ascalculated byEquation8.Parislawparameters f m;C g arechosendi˙erentlybeforeandafter thecumulativeimpactenergy E k crossesthe'jump'energy E . (d) Resampling :Sampleswithhigherlikelihoodareduplicatedwhereastheoneswith lowerlikelihoodareeliminated.Thisstepcapturestheessenceofoptimizationby particle˝lteringsuchthatthe'good'particlesaretransmittedtothenextiteration, therebyre˝ningtheestimationofmodelparameters. (e) Remaining-Useful-Life(RUL)computation :Oncethemodelparametersarees- timated,thedamagestateispropagatedfromthecurrentstateuptothethreshold valueorend-of-life EOL .Aftereverymeasurementstate,everyparticlewhichcrosses thefailurethresholdareidenti˝edanditsRULiscomputedas RUL n = EOL n , n beingthecurrentobservationtimeinstant.PDFofRULisgeneratedbycomputing theRULofalltheparticles.ThemedianandmeanvalueoftheRULalongwithits con˝denceintervalsarecalculatedfromtheRULPDF. 103 7.4ExperimentalSetupandResults 7.4.1GFRPSpecimenandExperimentalSetup Threeeight-layeredS2-glassreinforcedlaminates S 1 ;S 2 and S 3 ofdimensions 100 100 4 : 7 mm weresubjectedtoasequenceoflow-velocityimpactsbydrop-weighttestswith˝xed massof 17 Kg withdi˙erentenergies(ordi˙erentvelocities)andscannedateveryinterval usingtheOTStechnique. S 1 and S 2 weresubjectedto15impactswithenergiesstated inTable7.1,whereas S 3 wasimpactedwith14impactsofdi˙erentenergiesupto 89 J ,as denotedinTable7.1.Thevelocityoftheimpactvarieddependingontheimpactenergy suchthatfora10Jimpact,thevelocityofdrop-weighttestswasrecordedas1m/s,whereas for50Jand100J,themeasuredvelocitywasaround2.41m/sand3.39m/srespectively. Table7.1:CumulativeenergiesofconsequtivelowvelocityimapactsonGFRPsample1and 2. Table7.2:CumulativeenergiesofconsequtivelowvelocityimapactsonGFRPsample3 TheOTSsetupusedinthisexperimentconsistedofaniBeam-smart- 640 s laserdiodewith 640 nm fundamentalwavelengthusedasthelightsource.Ithad 1 : 5 mm beamdiameterand upto 150 mW outputpower.ThetransmittedradiationwasregisteredusingaDET36ASi detectorwith 350 1100 nm wavelengthrange, 14 ns risetimeand13 mm 2 activearea.The voltageontheoutputofthephotodetectorwasdirectlyproportionaltoregisteredradiation power.TheXY-coordinatestagewithsteppermotorsallowedforrapidinspectionofthe 104 GFRPsampleswithalateralresolutionof 0 : 25 mm . Figure7.6:OTSscansofGFRPsample(a)healthy(b)-(p)aftereachconsequtiveimpact from1to15asmentionedinTable7.1. Attheendofeachimpact,theGFRPsamplewasinspectedbyOTStherebyproducing 15scansforS1andS2and14scansforS3.TheOTSimagesfor S 1 after15repeatedimpacts arepresentedinFigure7.6.BasedontheimagesegmentationasshowninFigure7.3(c), delaminationareafor 1 ; 2 ; 3 ; 4+ delminationswerecalculatedfromeachOTSscanforthe threesamples.Thedamageareagrowthcurvewithrespecttocumulativeimpactenergiesis plottedinFigure7.7whichveri˝esthedamagegrowthbehaviourcausedbyrepeatedimpacts inthethreesamples.Thesecurvesareconsideredasthegroundtruthforourapplication andestimationofdamagegrowthparametersusingourproposedmethodisvalidatedagainst them.Itisobservedthatthe`knee'ofthehealthindexcurveorthe`jump'energycorrespond toanapproximatevalueof20Jforallthreesamples,evenwhenthesamplewereimpacted withdi˙erentintervalsofimpactenergy(orvelocity)duetothesamegeometryofspecimens andthelocationofimpact. 105 Figure7.7:Growthofdelaminationareaforthreesampleswithincreasedcumulativeimpact energies(Solidcurve-S1,Dashedcurve-S2,Dottedcurve-S3). 7.4.2PrognosisResults Inordertoimplementparticle˝lteringalgorithmtopredictdamagegrowthcurvefrom initialmeasurements,initialdistributionfortheparametersweresetas: a 0 ˘N (20 ; (0 : 01) 2 ) m 10 ˘N (4 ; (0 : 02) 2 ) , log C 10 ˘N ( 22 : 33 ; (1 : 2) 2 ) m 20 ˘N (2 : 87 ; (0 : 1) 2 ) , log C 20 ˘N ( 22 : 2 ; (0 : 1) 2 ) E 0 ˘N (20 : 02 ; (0 : 45) 2 ) Thepriordistibutionof E wheretheParislawparameters'jump'fromRegionIto ReguionIIishighlysensitivetotheOTSmeasurementsforindividualGFRPspecimens. Slopedi˙erenceateverymeasurementcyclewithrespecttoitslasttwopredicteddelami- nationareaswascalculatedaccordingtoequation(7.10)and E 0 ( i )= E k 1 if S diff ( k )= max ( S diff ) .ThisprocesswasrepeatedfordamagegrowthcurvesinthethreeGFRPsam- 106 ples( i =1 to 3 )andthemeanandsamplevariancewereusedtode˝nethepriorPDFof E 0 . S diff ( k )= a k a k 1 E k E k 1 a k 1 a k 2 E k 1 E k 2 (7.10) Toverifytheparticle˝lteringprognosticsapproach,parametersoftheParis-Parismodel wereestimatedwithvaryingnumberofavailablemeasurementsfrom9to12andtheesti- matedcurvesarepresentedinFigure7.8. Figure7.8:Predictionofdelaminationareacurvesbasedondi˙erentnumberofavailable measurements(a)n=9,(b)n=10,(c)n=11,(d)n=12.Thetruemeasureddelamination areacurveisplottedindashedlines. Itisobservedthatthepredictionbecamemoreaccuratewithnumberofobservations Figure7.8.TheRMSEofestimatedvauescomparedtotheOTSmeasurementswascom- 107 putedaccordingtoequation7.11andplottedinFigure7.9whichshowsadecreasingtrend withincreasingnumberofobservations. RMSE = v u u t 1 n n X i =1 ( a i ^ a i ) 2 (7.11) Tofurtherdemonstratethebene˝tofanoptimizedParis-ParismodeloveraregularParis model,RMSEwascomputedfornumberofavailablemeasurementsincreasingfrom10to13 andplottedinFigure7.9.AlthoughRMSEislessthan 0 : 15 forboththemodelswhenmore than 10 measurementswereconsidered,predictionismoreaccuratewithParis-Parismodel sinceithaslowerRMSEingeneralthantheregularParismodel. (a)(b) (c)(d) Figure7.9:ComparisonofdamageareaprognosisbyParismodel(dashedline)andParis- Parismodel(solidline)for(a)1delamination(b)2delaminations(c)3delaminations(d) 4+delaminations. Aninterestingthingtonoteisthatwhennumberofavailablemeasurementsofdelamina- 108 tionareaweremorethan8orwhenthecumulativeimpactenergywashigherthanthe'jump' energy( E =20 J ),thepredicteddamagegrowthcurvesmatchedthetruemeasuredgrowth accurately,withmaximumRMSEof 0 : 07 .Howeverwhenthenumberofmeasurementswas either8or7i.ebeforethedamagegrowthcurvechangeditsgrowthrate,estimationbecomes morechallengingastheimpending'jump'energyneedstobepredictedaccuratelyevenbe- foreitisreached.ThisiswhereanoptimizedParis-ParismodeloutperformsaregularParis model. Figure7.10:Predictionofdelaminationareacurvesbasedondi˙erentnumberofavailable measurements(a)n=8,Parismodel(b)n=7,Parismodel(c)n=8,Paris-Parismodel(d) n=7,Paris-Parismodel.Thetruemeasureddelaminationareacurveisplottedindashed lines. ResultsarepresentedinFigure7.10(a)and(b)forestimationofdamagegrowthcurve 109 wth8and7measurementsusingParis-Parismodel.ThemaximumRMSEiscalculated tobe 0 : 1 .Ontheotherhand,whensamemeasurementswereusedtopredictthedamage growthcurveusingregularParismodelwithonesetofparameters f m 1 ;C 1 g ,theestimation failed,asshownin˝gure7.10(a)and(b).Clearlyforthesecases,theParismodelcouldnot capturethe'jump'inthedamagegrowthcurve,thereforeresultinginwrongpredictionofits futurevalues.If6orlessmeasurementswerechosen,boththemodelsfailedtoaccurately predictthe'jump'inthedamagegrowthcurveduetolackofsu˚cientinformation. Inordertocomputetheremaining-useful-lifeoftheGFRPsamplefromtheinitialOTS measurements,sample S 1 wassubjectedtomorenumberofimpactswithhigherenergy intervals,asstatedinTable7.3.Attheendof 450 J,OTSimageoftheGFRPspecimenin Figure7.11(g-h)showsthatdelaminationhadreachedtooneofitsedgesleadingtobreakage of˝bresatthatend,hencedenotingitsend-of-life(EOL)asfurtherusageofthespecimen couldnotbecontinued.Thenetdelaminationareaafter 450 Jwascalculatedas7803.8 mm 2 or 78 : 03% oftotalareaofthesample.Figure7.12presentsthedamagegrowthcurveupto EOLforsample1. Table7.3:CumulativeenergiesofmorenumberofimpactsonGFRPsample S 1 from150J to450J Asetofdamagethresholdswasset ˝ d ; d =1 ; 2 ; 3 ; 4 foreverydelaminationscorresponding tothe 21 st impactortotalenergyof450J: ˝ 1 =955 sqmm, ˝ 2 =1362 sqmm, ˝ 3 =2065 sqmm, ˝ 4 =3422 sqmm.RULwascalculatedontheParis-Parismodelfordi˙erent numberofobservationsrangingfrom14to21andthecorrespondingresultisillustratedin 110 Figure7.11:OTSscansofGFRPsample(a)-(f)aftereachconsequtiveimpactfrom16to 21asmentionedinTable7.3,(g)Cameraimageofsample1after 450 Jimpact(H)Enlarged imageofdelaminationreachingtheedgesandbreakageof˝bersdenotingitsend-of-life. Figure7.12:Growthofdelaminationareaforsample1withincreasedcumulativeimpact energiesuptoend-of-life. Figure7.13.At 14 th observation,thecumulativeimpactenergywas 90 J ,hencetrueRUL is 450 J 90 J =360 J whereasat 21 st observation,thecumulativeimpactenergywas 450 J , hencetruevalueofRULis 0 . ThemeanandmedianofestimatedRULvaluesalongwiththeir 90% con˝denceboundsis showninFigure7.13.Thetwoshadedconesofaccuracyat20%and30%oftrueRULenable 111 (a)(b) (c)(d) Figure7.13:RULpredictionfor(a)1delamination(b)2delaminations(c)2delaminations and(d)4delaminations. comparisonofpredictionaccuracyandprecision.Predictionprecisionclearlyimproveswith timeasthe90%con˝denceintervalofestimatedRULdecreaseswithadditionofmore measurements.ThetrueRULlieswithinthecon˝denceintervalsformostofthecases. Infactfordelamination2,3and4,themeanofestimatedRULexactlymatchesthetrue RULwhen18ormoremeasurementsareusedforprediction.However,itcanbeseenthat RULestimationerrorishigh( 50% )when16orfewermeasurementsareused(truevalue lieswithin 90% CI)whichindicatesthatthemodelanditsvariancestructuredonotfully capturethedamagedynamicsattheearlierstagesofdelaminationgrowth.Inorderto improveRULpredictionwithlessermeasurements,amoreaccuratedamagegrowthmodel shouldbeinvestigated. 112 Chapter8 SingleSensorPrognosisinComposites byIndirectConditionMonitoring 8.1Introduction PredictionoffuturedamagestateusingNDEdatafromdirectconditionmonitoring(CM) ofacompositespecimenisdiscussedinchapter7.Accurateestimationofdelaminationarea couldbeachievedwithhigh-resolutionopticaltransmissionscanning(OTS)system,particu- larlysuitablefortransparentGFRPs.However,oftenindustriesdemandin-situmonitoring ofslow-growingdefectsinstructuressuchasfatigue-induceddelaminationincomposites. Airplanewingsorautomobilepartsmadeofcompositesarefrequentlysubjectedtoawide spectrumofloadingpatternsduringtheirserviceresultinginslowprogressionofcracks causedbyfatigue.Fatigue-induceddelaminationincompositejointsposesseriousthreat totheirremainingusability[127,128]propelingseveralanalyticalandexperimentalinves- tigationsontheinitiationandevolutionoffatiguecracksincomposites[129,130].Fatigue behavioroftenresultsinformationofair-gapsinbetweenthematrixlayersknownasdelam- inationwhichmaybehiddenininternallayersandnotvisibleonoutersurfaces.Therefore, complexdamagemechanismsincompositesdemandtheuseofNDEandSHMtechniques notonlytodetectdamagesattheinitialstagesoffatiguebutalsotoprovideindirectCM 113 dataforfuturehealthprognosis. AlthoughsomeNDEtechniquessuchaspulse-echoultrasonics[131],far-˝eldmicrowave imaging[132]andsonicinfraredimagingtechnique[133]arecapableofdiagnosingdelami- nationincomposites,accurateprognosisoffatiguedamageincompositesusingNDEdata remainsachallengingtask.FirstlyinthecaseofmostNDEtechniques,noknownphysics- basedmodelsareavailablefordescribingfatigue-damageprogressionincompositejoints [134].Asdiscussedbefore,unlikemetals,compositesareheterogenousinnaturewherea slightchangeinthematerialorgeometrycanresultintoanentirelydi˙erentandcomplex damagemechanismresultinginuncertainNDEinspectionresults.Crackordelamination growthbehaviorincompositesstronglydependonthemanufacturingprocess,mechanical propertiesofmaterial(s),presenceofimpuritiesorinclusionsinresinandothercomplex micro-levelphenomenonwhicharedi˚culttobeincorporatedintoknownelectromagnetic, acousticoropticalmeasurements,particularlyforcompositescuredfrommultipleandnewer materials.Asaresultinmostpracticalapplications,prognosisissolelydependentonin- directCMdatafromperiodicNDE/SHMofthecompositestructures.Secondly,mostof in-situmonitoringsystemscannotprovideaccurateestimationoftheslow-growingdefectin adhesivejointsespeciallyintheearlystagesoffatigue.Prognosisbasedonnoisyestimates collectedunderuncertainenvironmentinherentlyleadstoover˝ttingonthetrainingdata andwrongpredictionoffuturedamagestates. ThischapterpresentstheprognosticcapabilitiesoftwomethodsusingindirectCMdata: regressionbasedprediction[85]andstochastic˝lteringbasedonBayesinference[81]ina sequentialMonteCarloframework,suchasKalman˝lterandparticle˝lter.Delamination areaispredictedforaGFRPspecimensubjectedtomodeIfracturemechanismundercyclic loading.IndirectCMdataisprovidedbyguidedwave(GW)[135]signalswhicharegenerated 114 throughsurface-mountedpiezoelectrictransducers,thereforefacilitatingon-linemonitoring ofcompositestructureswhiletheyareinservice.Delaminationareacomputedfromperiodic OTSmeasurementsareconsideredasthegroundtruth.Predictionresultsofbothapproaches arecomparedtodemonstratethebene˝tofdynamicparameterupdateinNDEprognosis applications. 8.2ExperimentalSetup 8.2.1SpecimenGeometryandMaterial GFRPcompositesamples,usedinthemodeIfatigueexperiment,weremanufactured usingvacuumassistedliquidmoldingprocess.ThereinforcementwasS2-glassplainweave fabricwitharealweightof 818 g=m 2 ,namelyShield-StrandS,providedbyOwensCorning. TheGFRPsamplescomprisedsixlayersofsuchfabricsstackedatthesameangle.The distributionmediumwasResin˛ow 60 LDPE/HDPEblendfabricfromAirtechAdvanced MaterialsGroupandtheresin, SC 15 ,wasatwoparttoughenedepoxyobtainedfrom AppliedPoleramic.GFRPplateofsize 300 150 mm 2 wasmanufacturedina 914 : 4 609 : 6 mm 2 aluminummoldwithpointinjectionandpointventing.Twote˛onsheetsof dimensions 50 150 mm 2 withdensity 2 : 16 g=cm 3 andtensilestrengthof 3900 psi wereinserted inbetweenthirdandfourthlayerofGFRPfabricsatthetwoedgesoftheplate.Afterthe materialswereplaced,themoldwassealedusingavacuumbagandsealanttape,anditwas theninfusedundervacuumat 29 inHg.Theresin-infusedpanelwascuredinaconvection ovenat 60 o C fortwohoursandpost-curedat 94 o C forfourhours.Finally,double-cantilever beam(DCB)sampleswithdimensionsof L =150 mm , b =25 mm and h =2 mm werecut fromthemanufacturedGFRPplateusingadiamondsawandpianohingeswereattached 115 Figure8.1:Double-cantileverbeam(DCB)specimenforModeIfatiguetests,accordingto ASTMD 5528 . usinghigh-strengthcyanoacrylateglue.Thedesignofthesampleadheresto ASTMD 5528 standardformode1fatiguetesting,asshownin˝gure8.1.Figure8.3(a)showsaDCB sampleusedinourexperimentswhichismadeof6layerswithate˛onsheetoflength 50 mm insertedfromtheedgeinbetweensecondandthirdlayeroftheplate.Asthete˛on insertsareultra-thin,theyhavenomechanicalcontributiontothesamplebutareusedsolely tocreateinitialdelaminationinthespecimen.EachDCBspecimenischaracterizedwith Young'smodulusof 26 GPa ,density 1907 Kg=cm 3 andthepoison'sratioof 0 : 17 (materialis assumedtobequasi-isotropic). 8.2.2FatiguetestingofGFRPunderModeIfailure AccordingtoASTMstandardE1823,fatigueinmechanicalsystemsisde˝nedas: processofprogressivelocalizedpermanentstructuralchangesoccurringinamaterialsub- jectedtoconditionsthatproduce˛uctuatingstressesatsomepointorpointsandthatmay culminateincracksorcompletefractureafterasu˚cientnumberof˛uctuations".When astructureissubjectedtocyclicloading,theappliedstressisnotconstantbutchanges withtimeleadingtofatiguefailure.Strikingcharacteristicoffatigueisthatduetorepeated 116 variableloading,localizedstressconcentrationpointsarecreatedatwhichcrackisinitiated andthesystemfailsatstressvaluesbelowtheyieldstrengthofthematerial.Hence,fatigue posesseriousthreatintheoverallreliabilityofmaterialsandrequiresaccurateprognosis. Practicalmechanicalsystemsundergovariableloadinginseveralscenariossuchas; 1. ChangeinthemagnitudeofappliedloadExample:punchingorshearingoperations. 2. ChangeindirectionofloadapplicationExample:aconnectingrod. 3. ChangeinpointofloadapplicationExample:arotatingshaft. (a)(b) Figure8.2:(a)ExperimentalsetupforModeIGFRPsamplesubjectedtocyclicloadingin MTSmachine,(b)EnlargedimageofGFRPsampleunderModeItest. Susceptibilitytodelaminationisoneofthemajorweaknessesofmanyadvancedlaminated compositestructures.Althoughprogressingatalowerrate,fatiguecaninducelocalmatrix crackingincompositesleadingtodelaminationsinadhesivejointsormatrixlaminates,which signi˝cantlycompromisesstructure'shealthandcanbecatastrophic.Owingtoitsindustrial importance,fatiguemechanismshavebeenstudiedextensivelywithregardstocomposite materials[136,137,138].Inthispaper,e˙ectoffatigueloadingisstudiedonreliabilityof aDCBGFRPsampleunderModeIcyclicloading.TheGFRPspecimenissubjectedto 117 tension-tensionfatiguetestingin810MaterialTestSystem(MTS)machinewith50kNload cell.At˝rst,criticaldisplacementwherethespecimencracksisrecordedbyintroducing monotonicloading.Theprocessisrepeatedon 5 similarspecimensandtheaveragecritical displacementiscomputed.Fatigueloadingisthenconductedonanewsampleunderconstant displacementat 5 Hzwithdisplacementratioof0.1andmaximumstressequalto 70% of criticaldisplacement.TheexperimentalsetupforDCBGFRPsamplesubjectedtocyclic loadinginMTSmachinefollowsASTMStandardD6115andisillustratedinFigure9.4. 8.3NDEofFatigueDamageinComposites Forreliabilityanalysis,interruptedfatiguetestsareperformedontheDCBsample.Start- ingfromitspristinecondition,cyclicloadingispausedafterevery20,000cyclesandthe specimenisinspectedusingtwoNDEmethods.Thisprocesswascontinuedupto120,000 cycles.NDEmeasurementsalongwithfeaturesindicatingthestructuraldamagegrowthis describedinthissection. 8.3.1DelaminationdetectionusingOTS AdetaileddescriptionoftheOTSoperatingprinciplefordetectionofimpactdamages inGFRPisdiscussedinchapter7.Similarexperimentalsetupisusedforinspectionofthe DCBGFRPspecimensubjectedtoModeIfailureundercyclicloading.AGFRPsample withdelaminationshowninFigure8.3(a)isinspectedbyOTSimagingsystemandthe resultingimagedataispresentedinFigure8.3(b).Lightisobstructedbythete˛onsheet insertedwithintheDCBspecimenresultinginnopowertransmittedthroughthatregion. Detailedpro˝leofdelaminationstartingfromtheedgeofte˛onsheetisvisblefromtheOT 118 scan,showninFigure8.3(b). (a) (b) Figure8.3:(a)GFRPsampleunderModeIfatiguetestsafter160Kloadcycles(b)OTS imageofGFRPsamplewithdelaminationindications. OTSimagesoftheDCBGFRPsampleobtainedafterevery 20 K cyclesoffatigueloading upto160KloadcyclesispresentedinFigure8.4.AniBeam-smart-640slaserdiodewith640 nmfundamentalwavelength,1.5mmbeamdiameterand3.1mWoutputpowerwasused asthelightsource.TheOTSsystemwasplacedonanactivevibrationisolationtableand opticalscanswereacquiredindarkambiencewitha1mmstepsize. FromtheOTSimages,extentofdelaminationcanbeobservedastheregionbetween endofte˛onandthebeginningofhealthypartofthesample.Asexpected,delamination growsinsidethesamplewithincreaseinnumberofloadcycles.Areaofdelaminationfrom thescannedimageiscomputedusingimageprocessingalgorithmimplementedinMATLAB, asdepictedinFigure8.5.Thedelaminatedareaisidenti˝edusingsegmentationviafast marchingmethod[139]togeneratethegrayscaleimageshowninFigure8.5(b).Thetotal numberofpixelsthatare`turnedon'providestheareaofdelaminationintermsofpixels ( d pix ). ThepiezoelectricsensorsattachedtotheGFRPsamplemarkasreferencepointsandare 119 Figure8.4:OTSimagesofaGFRPsample(a)Healthysampleandonbeingsubjectedto Mode1cyclicloadingafter(b)20Kcycles(c)40Kcycles(d)60Kcycles(e)80Kcycles(f) 100Kcycles(g)120Kcycles(h)140Kcycles(i)160cycles. usedtocalculatethephysicalareaofdelaminationfrom d pix .Speci˝cuseofthesensorsare describedinthefollowingsection.Usingcluster-based-segmentationfollowedbyconnected components[19],locationofthetwopztsensorsareidenti˝edandthepixeldistancebetween theirinneredgesisrecordedas l pix .Additionally,edgedetectionalgorithmisimplemented todeterminetheupperandloweredgesofthesampleanditspixelwidthisrecordedas w pix . MeasuringthephysicaldistancebetweentwoPZTsensors ( L phy ) andwidthofthesample ( W phy ) ,thedelaminationarea ( D phy ) iscalculatedaccordingtoequation8.1.Inthispaper, L phy =10 cm and ( W phy )=2 : 5 cm . D phy = ( d pix ) ( l pix w pix ) ( L phy W phy ) cm 2 (8.1) AreaofdelaminationiscomputedforeachoftheOTSimagesdepictedinFigure8.4,after everyinterval20Kloadcycles.Plotofdelaminationareaagainstnumberofloadcyclesis showninFigure8.6.Theinitialdamageareacomputedfromthehealthysampleisdeducted fromallsuccessiveareameasurements.Khomenkoetal.[120]successfullydemonstrated 120 (a) (b) Figure8.5:(a)FatiguedGFRPsampleafter160Kloadcycles(b)OTSimageofdelaminated sample(c)Binaryimagedenotingdelaminationareaidenti˝edafterimageprocessing. OTSasavalidtechniquetodetectdelaminationinGFRPinducedbyrepeatedlow-velocity impactsandvalidatedscannedresultsbyobservingdamageinacross-sectionoftheimpacted samplesafterbeingcutbydiamond-saw.Similartocracklengthinfatigue-crack-growth (FCG)prediction,delaminationareaservedasasuitablehealthindicatoroftheDCBGFRP samplesubjectedtoModeIfatiguetesting. Figure8.6:PlotofnumberofloadcyclesversusdelaminationareafromOTSmeasurements. 121 8.3.2DelaminationdetectionusingGW Oneofthein-situNDE/SHMtechniqueswhichhasbeenusedforreal-timemonitoringof aerospaceandautomobilecomponentsisguidedwave(GW)sensing[140,141].Delamination detectionusingGWtechniquecanbeachievedusingPZTsensorsmountedonthesurface ofcompositelaminateswhichcantransmitandreceiveguidedwavesignalsinpitch-catch con˝guration[135].Theexcitationfrequencyisidenti˝edbystudyingdispersioncurvesfor selectedmaterialssuchthatcomplexwavemodesareavoidedandtheanti-symmetric A 0 and symmetric S 0 modesareexcited.PZTswithresonantfrequencyclosetoexcitationfrequency aremountedonbothendsofthespecimen(seeFig8.3(a)).Waveformgeneratorexcites thetransducerandgeneratestheguidedwave,whichpropagatesthroughthespecimenand pickedupbyasecondtransducer.Thereceivedsignalscanbeobservedviaanoscilloscope. SchematicoftheexperimentalsetupforGWinspectionofGFRPspecimenisdepitedin ˝gure8.7(a). AccordingtoGWtheory[142],geometricalpropertiesofthewaveguide,especiallyspeci- menthickness,determinethemodecontentoftheGWsignalatthereceiverPZTsensor.In ModeIfracturetests,growthofdelaminationresultsinchangeofthicknessofthewaveguide atthecracktipwhichmodi˝esthedispersioncurvesorleadstomodeconversion[143].Ear- lierworks[144]con˝rmthatthegroupvelocityoftheGWsignalisreducedasdelamination growsandtherefore,analyzingthetimeof˛ight(TOF)fromthereceivedGWsignal,asde- pictedinFigure8.7(b),providesinformationaboutpresenceofinternalair-gaps(ordamage) inthecompositelaminate. Thesame6-layeredGFRPsampleismonitoredusingGWsetupinadditiontoOTS,after every 20 K fatiguecycles.Asdelaminationareaincreases,timeof˛ightbetweenreceivedand 122 (a)(b) Figure8.7:(a)SchematicofGWexperimentalsetup(b)Excitedandreceivedsignalsin healthysample. transmittedsignalincreases.TheincrementalchangeinTOFofreceivedGWsignalis computedfor9roundsoftension-tensionloadingofthesample.Figure8.8(a)showsthe phaseshiftinreceivedGWsignalasthesampleprogressesfromhealthytodelaminated layersafterevery20Kcycles.Figure8.8(b)illustratesthe TOF fromhealthyto160K fatiguecyclesatanintervalof20Kcycles.Asteadygrowthin TOF isnoticedwhichcan becorrelatedtotheincreaseindelaminationinsidethespecimen. Inordertoquantifye˙ectofdelaminationgrowthintheGWmeasurements,di˙erenceof TOF ofreceivedGWsignalsbetweenthedelaminatedandhealthyspecimeniscomputed. TOF k = TOF k TOF 1 8 k =1 ; 2 ;:::; 9 (8.2) TOF ofGWsignalswerecomparedwiththedelaminationareaextractedfromOTS imagesoffatigue-inducedsamples.Apositivecorrelationbetweenthetwoparameters,as showninFigure8.9,demonstratesthatmonitoringTOFofreceivedGWsignalscanbeused toestimatetheareaofdelaminaioninGFRPspecimens.A2nddegreepolynomialcurve, 123 (a) (b) Figure8.8:(a)Receivedsignalforahealthysampleandsampleafter20K-160Kcycle(b) TOF betweenreceivedandexcitedsignalwithincreaseinnumberoffatiguecycles. accordingtoequation8.3,wasestimatedbasedonthemeasurementsandthenusedto predictthedelaminationareafromguidedwavesignalsrecordedatthereceiverPZTsensor. Fromtheexperimentaldataset,thecoe˚cientswerecomputedas p 1 = 9 : 1005 10 9 , p 2 =0 : 4 10 6 and p 3 =0 : 297 . ^ Area = p 1 TOF ) 2 + p 2 TOF )+ p 3 (8.3) 124 Figure8.9:CorrelationbetweenTOFfromguidedwavesignalsanddelaminationareafrom OTSimages. 8.3.3OverallFrameworkofDamagePrognosis Inourapplication,damageareaintheadhesivejointwasderivedfromsensormeasure- mentsobtainedatregularintervalsoffatigueprogression.Fatiguetestswereintermediately stoppedonthetrainingspecimenstoextracttheguidedwave(GW)datafromattachedPZT sensorsandimagedusingOTStechnique.Featuresdeterminingdegradationofstructural healthwereextractedfromtheGWsignalsandcomparedwiththedelaminationareacom- putedfromOTSimages.Finally,featuresfromtestspecimen,extractedafterintermediate fatiguecycles,wereimplementedviaregressionandstochastic˝lteringapproachestopredict futurefeaturevaluesfromwhichthefuturedelaminationareawascomputed.Predictedarea wasthencomparedwithOTSimagedatatoassesstheperformanceofthedamageprognosis algorithm.Theentireapproachisdescribedinthe˛owchartof˝gure8.10. MeasurementsfromOTSandGWsensorsonaGFRPspecimensubjectedtointerrupted fatigueloadingisrecordedintable8.1. 125 Figure8.10:Damageprognosis˛owchartusingguidedwaveandopticaltransmissiondata. Table8.1:OTSandGWmeasurementsfromModeIfatiguetestingofGFRPatintermediate loadcycles. 8.4DamagePrognosisResults PredictionoffuturedelaminationareainaGFRPspecimenbasedoninitialGWmeasure- mentsisperformedusingtwodynamicdata-drivenpredictionapproaches,namelykalman ˝lterandparticle˝lter.Thepredictionaccuracyforeachofthesemethodsarecompared withregressionbasedstaticestimationapproach.Startingwiththe˝rst3GWmeasure- ments( T 1:3 =0 ; 20 K; 40 K cycles), TOF iscomputedforthenextmeasurementtime-point whichisat 60 K loadcyces.Allmeasurementsupto k th observationareutilizedtopredictthe 126 TOF for ( k +1) th observation.Delaminationarea Ar k +1 isthencomputedfrompredicted TOF usingthecorrelationexpressiongivenbyequation8.3.Thisprocessisrepeatedupto 160 K cycles. 8.4.1PredictionofDelaminationAreabyLogarithmicRegression Basedondamage-propagationcurvedepictedinFigure8.8(b),asimplelogarithmicfunc- tion,asdescribedinequation8.4,isimplementedtomodelchangeinTOFmeasurements inDCBcompositesampleovertime(numberofloadingcycles).Logarithmicregressionis achievedby˝ttingafunctionoftheform8.4onthemeasurements TOF 1: k toestimatemodel parameter m andhence, TOF k +1 . TOF = mlog ( T ) (8.4) Resultsofdelaminationareapredictionunderstaticapproachusinglogarithmicregres- sionispresentedinFigure8.11(a).Updatedvaluesofparameter m atevery k th observation timeisplottedin˝gure8.11(b). (a)(b) Figure8.11:(a)PredicteddelaminationareafrompredcitedGWmeasurementsusingcor- relationcurve(b)Updationoflogarithmicrate ` m 0 ateveryestimationstep. 127 Itcanbeobservedthatestimatesoffuturedamageareafrompredicted TOF valuesdo notmatchthetruedamageareaobtainedfromOTSmeasurements.Besides,thepredicted valueshavehighvariancesthattranslatetolargecon˝denceintervalswhichmakesthese resultsunacceptable.Primaryreasonforthehighpredictionerrorinregressionbasedprog- nosisisthelackoflargenumberofNDEmeasurements.Regressioncanachieveaccurate estimationonlywhenalargeamountofdataisavailablewhichisseldomthecaseinindus- trialapplications.Hence,otherprognosistechniquessuchasstochastic˝lteringisexplored forpredictionofdamageareafromfewerGWmeasurements. Itshouldbenotedthatinthisthesis,logarithmicfunctionisselectedtomodelpropa- gationofGWmeasurementswithincreasingfatiguecycles,duetolackofknownphysics- based-modelsthatcande˝ne TOF ofGWsignalsinDCBwovencompositesundercyclic load.IfunderlyingphysicsofguidedwavepropagationinGFRPplatescanbemodeled accurately,improvedmodel-based-predictionofdamagegrowthmaybeachieved. 8.4.2PredictionofDelaminationAreabyKalman˝ltering Asdiscussedinchapter6,Bayesinference[145]isawidelyusedapproachforparameter estimation ^ .Thisapproachderivestheposteriordistributionofparametersbyupdating aninitialpriorestimatemultipliedwithlikelihoodfunctionobtainedfrommeasurements, accordingtoequation6.14.Particularlyinfatiguedamageprognosis,Bayesianinferencehas beenimplementedbyPengetal.[146]forprobabilisticprognosisinfatiguetestoflapjoints, Enricoetal.[102]forfaultprognosisinnon-linearcomponentsandAnetal.[81]forcrack growthmodelingunderModeIfracturetests. Apartfromparticle˝ltering,approximatesolutionofBayesinferencecanbeachievedby anotherstochastic˝lteringapproachknownasKalman˝ltering[147],speci˝callysuitable 128 forlinearsystemswithGaussiannoise.Since,thelogarithmicmodelfor TOF adheresto alinearsystem,Kalman˝lterwasexploredforthisprognosisapplication. Christeretal.[148]appliedKalman˝lterforestimatingrefractorythicknessinanin- ductorfurnacefromaseriesofmeasurements,containingmeasurementnoiseandmodel uncertainties.Kalman˝lterisatypicaltoolusedforoptimalestimationofunknownpa- rametersinlinearsystems,withGaussianmeasurementandprocessnoise.Inthispaper, anempiricalrelationshipisestablishedbetweenfatiguecycleandchangeinTOFofreceived GWsignalsfromdamagedGFRPspecimens,asstatedinequation(8.4).Itisimportantto notethatthisempiricalmodelisvalidonlyforthegivenspecimengeometry,materialand ModeIloadingconditions. Thislogarithmicrelationshipisrepresentedinastatespacemodel,whichisderivedin equations(8.5)-(8.9), x k +1 = A x k (8.5) y k +1 = C x k +1 (8.6) where, x isthestatevector, A isthestate-transitionmatrixand C istheobservationmatrix. x k = 2 6 4 TOF k m k 3 7 5 (8.7) A = 2 6 4 1 log T ) 01 3 7 5 (8.8) 129 C = 2 6 4 1 0 3 7 5 (8.9) ThedevelopeddamageevolutionstatespacemodelcanbeusedinKalman˝lter(KF) algorithmforprognosis.Ingeneral,KFalgorithmfollowstwostepsi.e.predictionand measurementupdate.Inpredictionstep,thestates x ,errorcovariance P _ ( k j k 1) and output y _ ( k j k 1) forthe k th fatigueintervalispredictedwiththeinformationavailable from k 1 th fatigueintervalasshowninequation(8.5).ThepredictionstepofKalman˝lter computeschangeinTOF x k fornextiterationfromtheexperimentalGWdata z k according toequations(8.11)-(8.13). x ( k j k 1) = A x ( k 1) (8.10) P ( k j k 1) = A P ( k 1) A T (8.11) y ( k j k 1) = C x ( k j k 1) (8.12) (8.13) Whenanewmeasurementisobtained,estimatedparameters( x k )areupdatedaccording toequations(8.15)-(8.16)where K k isthekalmangain, P k istheerrorcovarianceand R =0 : 025 isthemeasurementnoise.Thefuturedamageareaishencepredictedfrom estimated x k whichisupdatedonceanewGWmeasurementisavailable. K k = P ( k j k 1) C T ( C P ( k j k 1)+ R ) 1 (8.14) x k = x ( k j k 1) + K k ( z k C x ( k j k 1) ) (8.15) P k =(1 K k C ) P ( k j k 1) (8.16) 130 ImplementingKalman˝lteronthesamedatasetgeneratedpredictionresultsdepicted inFigure8.12.Theinitialdistributionofparametersarecomputedusingthe˝rsttwoGW observations,asdenotedinequation8.18. a = TOF 1 (8.17) m = TOF 2 TOF 1 ) log T ) (8.18) (a)(b) Figure8.12:(a)PredicteddelaminationareafrompredcitedGWmeasurementsusingcor- relationcurve(b)Updationoflogarithmicrate ` m 0 ateveryestimationstep. 8.4.3PredictionofDelaminationAreabyParticle˝ltering Asexplainedinchapters6and7,underparticle˝lteringframework,Bayesinferenceis processedinsequentialmannerwithparticlesassociatedwithprobabilityweights[103,149]. PredictionofdelaminationareaincompositelaminatesunderModeIfatiguetestingisbased onthedamagepropagationmodelgivenbythelogarithmicfunctioninequation(8.19)where T k isthetimegapbetween ( k 1) th and k th inspectionstep. a k = m k log T k )+ a k 1 (8.19) 131 UnlikethecaseofKalman˝ltering,inthiscasenoisevarianceistreatedasanunknown parameterwhichisestimatedbytheparticle˝lteringalgorithm.Assumingzeromodel noise,theconditionalprobabilityoftheNDEmeasurementscanbededucedas, L ( z k j a i k )= 1 z k p 2 ˇ˙ i k exp 2 4 1 2 z k a i k ˙ i k ! 2 3 5 (8.20) Startingwithuniforminitialdistributionsforalltheparametersinequation8.21and n =5000 particles,theestimateddamageareacurvealongwithupdatingpathof'm'are denotedinFigure8.13. a ˘ Uniform (0 ; 1) m ˘ Uniform (0 ; 1) ˙ ˘ Uniform (0 : 01 ; 0 : 05) (8.21) (a)(b) Figure8.13:(a)PredicteddelaminationareafrompredcitedGWmeasurementsusingcor- relationcurve(b)Updationoflogarithmicrate ` m 0 ateveryestimationstep. Inordertocomparepredictionperformanceofthethreemethods,thenormalizedroot meansquarederror(RMSE)iscomputedaccordingtoequation8.22,forvariablenumberof observationsandplottedinFigure8.14.Thepredictionerrorislowerinthedynamicdata- drivenapproachesbyapproximately 10 15% ,especiallyattheearlierstagesofdamage 132 progressionwhenfewermeasurementsareconsidered.Therefore,itcanbeconcludedthat boththedynamicdata-driventechniquesviaKalmanandparticle˝lteringoutperformsthe staticregressionbasedapproachowingtocapabilityofsequentialupdateoffunctionparam- etersbyincorporatinguncertaintiesofnon-linearmodelandmeasurementnoise.Moreover, resamplingbasedonlikelihoodcomputationwithinparticle˝lteringtechniquedrivesthees- timationtowardstheoptimumparametervalueevenwhenfewermeasurementsareavailable. HencethepredictionerrorislowerthanKalman˝lteringattheearlierstagesofdamagearea growth.Withadditionalmeasurementsafter 120 K loadcycles,predictionresultsfromboth the˝ltersbecomecomparable. NRMSE = q 1 n P n i =1 ( a i ^ a i ) 2 mean ( a ) (8.22) Figure8.14:Errorcomparisonofprognosismethodsforpredictionofdelaminationareafrom guidedwavemeasurements. 133 Chapter9 Multi-sensorPrognosisinComposites 9.1Introduction OneoftherecentextensionsofNDEistheuseofmorethanonesensingtechniquefor inspectionofstructures.TraditionalNDEprognosticsfocusesonanalyzingasinglesensor signalwhenaunitrunsunderasingleoperationalcondition[150,151].Inmostpractical situations,astochasticmodelis˝rstdevelopedatthetrainingstagebasedonhistorical results.Inspectiondatafromatestunitisthenappliedontothemodeltopredictitsfuture healthstate.Theseapproachesaree˙ectiveundertheassumptionthatsinglesensordata isabletocapturetheentirestochasticnatureofthedegradationprocess.Unfortunately,as systembecomesmorecomplex,severaluncertaintyfactorscomeintoplayduringdamage propagationwhereinmeasurementsfromonesensormaysu˙erfromnoise,outliersorbiases [152,153].Insuchcases,relyingonsinglesensordatabecomesinsu˚cienttoaccuratelypre- dictthegrowthofunderlyingdegradationmechanism,leadingtoinaccurateandunreliable remaining-useful-life(RUL)prediction. Assessingfatiguebehaviourofanystructureisanimportantaspectofitsreliability analysis.Fatigueinmechanicalsystemsoccurswhenastructureissubjectedtocontinuous cyclicloadingresultinginprogressive,localizedandpermanentstructuralchanges.Repeated variableloadingcreateslocalizedstressconcentrationpointsinaspecimenatwhichcrack 134 isinitiatedandthesystemfailsatstressvaluesbelowtheyieldstrengthofthematerial. Hence,fatigueposesseriousthreatintheoverallreliabilityofmaterialsanddemandsaccu- rateprognosis,especiallyatitsinitialstages.Althoughprogressingatalowerrate,fatigue induceslocalmatrixcrackingincompositesleadingtoglobaldamages,whichsigni˝cantly compromisesstructure'soverallhealth.Owingtoitsindustrialimportance,fatiguemech- anismshavebeenstudiedextensivelywithregardstocompositematerials.In[154,138], Bayesianmodelisdiscussedforparameterestimationoffatiguedamagepropagationbased onmodi˝edParislaw.Owen[155]presentedanexponentialcumulativedamagemodelfor estimationofstrengthofcarbon˝berpolymers.Krugeretal.[156]studiedanenergybased approachforfatiguedamagemodelinFRPunderplaneloading. (a)(b)(c) Figure9.1:(a)ReferenceGFRPspecimenfailingafter1386cycleswhilesubjectedtension- tensionfatiguetestunderconstantload(Maxload= 70% offailureload,Stressra- tio=0.1,Frequency=3Hz),(b)TestGFRPspecimen(identicalmanufacturingconditions) failingafter2250cyclessubjectedtoidenticalfatiguetestingconditions(c)Normalizedsti˙- nessdegradationofreferenceandtestspecimenfromMTSmeasurements. Reliabilityassessmentoffatiguebehaviorismorechallengingincompositematerials, comparedtometals,owingtopoorlyunderstoodnatureofdamagepropagation.Unlike metals,cyclicloadingincompositesresultsinsimultaenousformationofcomplexdamages 135 consistingofmatrixcracking,˝berbreakageanddelamination,whichdonotfollowknown crackpropagationmodelssuchasParis-Forman[157]orParis-Erdoganlaw[122].Mostim- portantly,variationsincompositemanufacturingsuchasimproperresinmixingproportions orpresenceofimpuritiesresultsinlargedi˙erencesoftensilesti˙nessfromonespecimento another,evenwhensubjectedtoexactsameloadingconditions.Asshownin˝gure9.1(a) and(b),twoglass˝ber-reinforcedpolymers(GFRP)specimensmanufacturedunderiden- ticalconditionsandsubjectedtosamefatigueloadmayfailatsigni˝cantlydi˙erenttime instants.Normalizedsti˙nessdegradationcurvesforthereferenceandthetestspecimen underidenticalloadingconditions,computedfrommechanicaltestingsystem(MTS),are plottedin˝gure9.1(c).AlthoughbelongingtothesameGFRPplate,minorvariations innumberof˝bersor˝berorientationresponsibleforspecimen'stensilestrengthleadto signi˝cantdi˙erenceinfailuretimeofthetwospecimens.Therefore,life-cyclestudiesin- ferredfrommechanicaltestingonareferencespecimenmaynolongerremainvalidforatest samplewhichposesseriousissueonRULpredictionincompositestructures. Figure9.2:(a)DigitalcameraimageofGFRPsamplewithdelamination,underModeI fracturetest(b)Low-frequencyeddycurrentinspectionusngTRcoilat10MHz(c)Near- ˝eldmicrowavescanat7.5GHz(d)Opticaltransmissionscanat2.5mW. OnepossiblewaytoovercometheseissuesistousemultipleNDEsensorsfortracking 136 defectsinacompositestructure[158].Di˙erentsensitivityofindividualNDEtechniquescan providedistinctinferencesofthedamagemechanism,whichifjudiciouslycombinedcanpre- ciselydescribetheoverallsti˙nessdegradationofthestructure.However,individualsensor informationmaybeincoherent,uncertain,fuzzyorevenincon˛ictwhichdemandsdevel- opmentofrobustdatafusionmethodstoestimatethetruedamagestatusofthespecimen. Figure9.2showsaGFRPdelaminatedsampleinspectedusingthreeNDEtechniques:(a) low-frequencyeddycurrentinspectionusngTRcoilat10MHz,(b)near-˝eldmicrowavescan at7.5GHzand(c)opticaltransmissionscanat2.5mW.Thelengthofdelaminationinferred fromeachofthesetechniquesarenotequaltothetruedelaminationlength( l 0 =7 cm )and evenvariesfromeachother.Therefore,usingonlyonetechniqueisnotidealforaccurate prognosissinceincorrectevaluationofdamagelengthatanobservationtimeleadstoin- correctpredictionoflengthatafuturetimeinstant.Insuchcases,fusionofinformation gatheredfrommultipleNDEsensorsisapossiblesolutionforreducingpredictionerrors. Despiteseveralmulti-sensorfusionprocessesbeenreportedinliterature[159,160,161], implementationofe˙ectivedatafusionsystemsforpredictionofcompositesti˙nessisnon- trivial.Inpractice,ifindividualNDEdataarebiasedandtheirunderlyinguncertaintyor varianceisnottakenintoaccount,prognosisbasedonfuseddatamayproduceworseresults thanwhatcouldbeobtainedfromthe'best'sensor[1].Moreover,someoftheexistingdata fusiontechniquessuchastheclusterbasedfusion[162]assumesmeasurementsatconsecutive timeinstantstobestatisticallyindependentwhichisspeci˝callynotapplicableindamage prognostics.Inthecaseofcompositeswherethestructuredeterioratesfromitspristinestate tototalfailure,correlationexistsbetweenNDEobservationsatconsecutivetimeinstants whichneedstobeincorporatedintothefusionmethodology.Besides,clusterbasedfusion approaches[162,163]areabletoprovideaccuratepredictionresultsonlywhendatafroma 137 largenumberofsensorsareavailablesincenumberofclustersisusuallyselectedas N s = 3 , N s beingthenumberofsensors.SinceNDEofcompositesinautomotiveoraviationstructures isusuallyanexpensiveandtime-consumingprocess,mostindustrialapplicationsrelyon inspectionfrom 1 or 2 NDEsystemsandthereforedemandadatafusiontechniqueforfewer sensordata. UsingmultiplesensorsforNDEinspectionraisestwomainchallenges.Firstly,sensors mayhavedi˙erentsensitivityatdi˙erentstagesofthedegradation.Forexample,thermal cameraisoftenincapableofimagingsmallcracksinmetalsattheirinitialstagebutcan sensethemoncethecrackisofasubstantialsize[164].Thus,contributionofmeasurements fromdi˙erentsensorstothefusedpathshouldchangewithtime.Thisbringsinthenotionof associatingdynamicandnon-uniformweightstoindividualsensorswhilegeneratingthefused path.Secondly,signalscollectedfrommultiplesensorsareoftencorrelatedandeachsignal onlycontainspartialinformationofthedegradedunit.Agoodexampleforsuchscenarios istheonewhereasampleisinspectedusingopticalandacoustictechnique.Regularoptical methodsdonotprovideinformationregardingthedepthofvolumetricdefectsinsamples whichcanbeobtainedfromtheacousticmethods.Insuchcases,datafusionmethodsshould bedesignedfore˙ectivecombinationofinformationfrommultiplesensorstoachievebetter characterizationofsystemhealth.Besides,sinceallsensorsmeasurethesamedegradation process,theirmeasurementsarehighlycorrelatedandhenceshouldbetreatedjointly. Inthischapter,amulti-sensorprognosismethodologyisproposedbasedonjointlike- lihoodcomputationinparticle˝lteringframeworktopredictresidualsti˙nessofaGFRP specimensubjectedtofatigue.Threemajorcontributionsinreliabilityassessmentofcom- positematerialsaredemonstratedthroughthisstudy-1)aparis-parismodelisdiscussedfor potentialmodelingofnormalizedsti˙nessdegradationofGFRPtensilecouponundercyclic 138 loadingconditions,whichhavenotbeenreportedbefore,2)improvementinpredictionre- sultsusingtwoindependentNDEsensorsoversingle-sensorprognosisisestablished,both inRULcomputationaswellaspredictionerrordomainand3)possiblereductioninnum- berofparticlesusedinparticle˝ltersisachievedbyimplementingmulti-sensorprognosis basedonjointlikelihoodcomputationwhichmayresultinsigni˝cantbene˝tinloweringthe computationtimeandcost. 9.2LiteratureReviewofDataFusionTechniques Historically,datafusiontechniqueswereprimarilydevelopedformilitaryapplications (statedinTable9.1)suchasautomatedtargetrecognition,remotesensing,battle˝eld surveillance,andautomatedthreatrecognitionsystems.Laterthetechniqueswereadopted inseveralcivilianapplicationsassummarizedinTable9.2.Forourstudy,wefocusonthe applicationofdatafusionforconditionbasedmaintenanceofstructuresasanextensionto existingNDEtechnology. Table9.1:Militaryapplicationsofdatafusion,from[1]. Measurementdatacanbecombinedorfusedatmultiplestagesresultingindatalevel 139 Table9.2:Non-militaryapplicationsofdatafusion,from[1]. fusion,featurelevelfusionordecisionlevelfusion[165,166].Datalevelfusioncombinesthe rawdatameasuredbyindividualsensorstoformanuni˝edindicator[159,160].Datalevel fusioncanbeimplementedwhenthesensorsarecommensurate,i.etheyhavesimilaroutput measurementssuchascombiningdatafromtwoacousticsensorsortwoeddycurrentsensors acquiringdataatdi˙erentfrequencies.Featurelevelfusioniscombinationofrepresentative featuresfromsensordataandconcatenatingthemtoformanewfeaturevectorwhichis thenfedtopatternrecognitionapproachessuchasneuralnetwork,clusteringetc[161,167]. Themostcommonexampleoffeaturelevelfusionisthehumancognitivesystem.Finally, decisionlevelfusionisobtainedbycombininginferencesfromindividualsensorsaftereach sensorhasmadeapreliminarydecisioninordertoextractmorecomprehensiveinformation [168].InthecaseofconditionbasedmaintenancebyNDEtechniques,decisionlevelfusion combinesdamagepropagationpathpredictedbymultipleNDEsensorsandthencomputes the˝nalresiduallifeusingthefusedpath. Severalstatisticaltoolsandsignalprocessingtechniqueshavebeenincorporatedinthe pastfortheobjectiveofdatafusion.Typicaldecisionlevelfusionincludeevidentialreason- 140 ing[169],Bayesianinference[170],andDemethod[171].Besides,pattern recognitionapproacheshavebeenincorporatedindecisionlevelfusionsuchasarti˝cialneu- ralnetworkbasedfusion[172]andclusterbasedfusion[162].Adetailedreviewofpopular datafusiontechniquescanbefoundin[1]. Despitethesequalitativenotionsandquantitativecalculationsofimprovedsystemoper- ationbyusingmultiplesensorsandfusionprocesses,actualimplementationofe˙ectivedata fusionsystemsisnottrivialatall.Inpractice,fusionofsensordatamayproduceworse resultsthanwhatcouldbeobtainedfromthe'best'sensor.Thiscanhappenespecially whenindividualsensordataarebiasedandtheirunderlyinguncertaintyorvarianceisnot takenintoaccountwhilefusingtheirdecisions.Moreover,someoftheexistingdatafusion techniquessuchastheclusterbasedfusiononlyconsidersthemeasurementsataparticular timeinstantwhichisspeci˝callynotapplicableinprognostics.Correlationsexistbetween observationsfromsensorsatconsecutivetimeinstantswhichneedstobeincorporatedinto thefusionmethodology.Inthischapter,alltheabovechallengesareaddressedbydevelop- ingadatafusionframeworkbasedonweightedcombinationofsensordatadependingonits consistencyandqualityofinspectionsignal.Themethodologywillbeimplementedforprog- nosisandreliabilityanalysisofdelaminationgrowthinglass˝berreinforcedpolymer(GFRP) compositessubjectedtofatiguetesting. 9.3JointLikelihoodComputationinParticleFiltering Inthisstudy,integratedprognosticsunderparticle˝lteringframeworkisimplemented forpredictionofsti˙nessdegradationincomposites.Similartopredictionofimpactdamage areainGFRPdescribedinchapter7[149],sti˙ness( s )degradationinGFRPtensilecoupons 141 causedbyfatigueismodeledaccordingtotheParis-Parismodel[123].Whencompositesare subjectedtotensileloading,di˙erentdamagesoccurinsequentialphases;thematrixbegins tocrackattheinitialstagesoffatiguefollowedbydelaminationgrowthduringmid-lifeand ˝berbreakagetowardstheend-of-life(EOL)[138].Sincematrixsti˙nessisrelativelylower than˝berstrengthincomposites,overallstructuralsti˙nessdropsrapidlyinthe˝rstfew loadcyclesandthendecreasesatalowerrateuntilfailure.Suchasti˙nessdegradationcurve canthereforebedescribedbytheParis-ParismodelbasedonPiecewise-deterministicMarkov processes(PDMPS)whereParislawisdescribedbytwosetsofparameters( m 1 ;C 1 ;m 2 ;C 2 ) beforeandafteratransitiontime N ,denotedbyequation9.1. ds dN = 8 > > > < > > > : C 1 ( Y p ˇs ) m 1 ; if N N C 2 ( Y p ˇs ) m 2 ; if N N (9.1) Periodicsti˙nessvaluesobtainedfromNDEmeasurements,denotedby z k ,areincor- poratedforupdationofmodelparameterswhere z k isconsideredasnoisyestimateoftrue sti˙nessvalue s k ofthecompositespecimenattimeinstant T k . z k = s k + ! k (9.2) ! k ˘N (0 ;˙ 2 ) (9.3) InexistingPFalgorithm,distributionof i th particleisupdatedbasedonitslikelihoodgiven theevidenceorthemeasurementdata z k ,asdenotedinequation9.4.Itisimportanttonote thatdi˙erentParislawparameters f m 1 ;C 1 g and f m 2 ;C 2 g areselectedbeforeandafterthe 142 loadingcycle N k crossesthe'jump'cycle N . L ( z k j s i k )= 1 z k p 2 ˇ˘ i k exp 2 4 1 2 ln z k i k ˘ i k ! 2 3 5 (9.4) where, ˘ i k = v u u u t ln 2 4 1+ ˙ s i k ! 2 3 5 (9.5) i k =ln h s i k i 1 2 ( ˘ i k ) 2 (9.6) InordertoincorporatedatafrommultipleNDEsources,likelihoodofparticlesarecom- putedaccordingtotheprincipleofBayesiannetwork(seeAppendixA.),asdepictedin ˝gure9.3.Formulti-sensorNDEsystems,iftruesti˙nessparameter s i k ofastructureis known,evidencefromindividualNDEtechniques{ z 1 k ;z 2 k ;:::;z M k }canbeconsideredtobe statisticallyindependent.Forexample,astructurewithaparticularsti˙nesscanbeimaged usingNDEsensor1aswellasNDEsensor2.Owingtodi˙erenceinphysicsoftheNDE methods,featuresextractedfromindividualNDEsignalscanbedi˙erent,yetbothcanbe usedtocharacterizethesamestructuralsti˙ness.Anychangeinthesti˙nessvalueextracted fromonesensorimagedoesnota˙ectsti˙nessmeasurementfromsecondsensor.Therefore, accordingtothetheoryofconditionalindependence,thejointlikelihoodfor i th particlecan becomputedfrom M measurementsusingequation9.7,whereindividuallikelihoodsare obtainedusingequation9.4-9.6.Additionaladvantageofthisapproachliesinthefactthat singlesensorlikelihoodiscomputedincorporatingthemodelandmeasurementnoiseofthe correspondingNDEsensorwhichfacilitatesdynamicupdatingofweightsfromindividual 143 sensorsontheresultantsti˙nessestimation. L ( z 1 k ;z 2 k ;:::;z M k j s i k )= M Y j =1 L ( z j k j s i k ) (9.7) Figure9.3:Bayesiannetworkinmulti-sensorparticle˝lteringframework. Itshouldbenotedthattheassumptionofconditionalindependenceremainsvalidonly whendi˙erentNDEsensorsareusedforinspectionofsamestructuralsti˙ness.Ifmultiple featuresareextractedfromthesameNDEresult(eg:eddycurrentmeasurementsobtained atmorethanonefrequencies)andimplementedintothemulti-sensorframework,conditional independencebetweenmeasurementswillnotbeapplicable.Insuchcases,correlationbe- tweeneachmeasurementhastobeconsideredwhilecomputingthejointlikelihoodofeach particle. Asdescribedinchapter6,resamplinginPFalgorithmisachievedthroughinverseCDF methodsuchthatparticleswithlikelihoodgreaterthanarandomnumbergeneratedfrom U (0 ; 1) areduplicatedandothersarediscarded[102].Inthisstudy,itisassumedthatthe end-of-life(EOL)ofthecompositestructureisknowna-priorifrompreviousexperiments foraspeci˝cgeometryandmaterial.Underfatiguetests,GFRPspecimensfailedat 30% ofinitialsti˙nessobtainedatpristinecondition.Thesti˙nessmodelparameters k are updatedupto k = L iterations,where L isthetotalnumberofobservedmeasurements. 144 AfterLiterations,futuresti˙nessispredictedusingequations7.6untilitreaches 30% of initialsti˙ness.RULafterLiterationsishencecomputedas RUL L =( L EOL L ) cycles where L EOL istheloadcycleatEOL.PDFofRULisgeneratedbycomputingtheRUL ofalltheparticlesandtheRULmedianandmeanalongwithitscon˝denceintervalsare calculatedfromtheRUL'sPDF. 9.4Experimentalset-up 9.4.1SpecimenGeometryandMaterial Forourexperiment,four-layered (0 = 90) GFRPspecimenswerefabricatedusingVac- uumAssistedResinTransferMolding(VARTM)technique.Thereinforcementconsistedof S2-glassplainweavefabricwitharealweightof 818 g=m 2 providedbyOwensCorninganddis- tributionmediumcomprisingResin˛ow 60 LDPE/HDPEblendfabricobtainedfromAirtech AdvancedMaterialsGroup.Atwoparttoughenedepoxyresin, SC 15 ,wasusedfromAp- pliedPoleramic.TheGFRPplate (150 300 mm 2 ) wasmanufacturedina 609 : 6 914 : 4 mm 2 aluminummoldwithpointinjectionandpointventing.Aftertheglassfabricwithresin transfermediumwereplacedonthemoldandsealedusingavacuumbagandsealanttape, thereinforcementwasinfusedundervacuumat29in-Hgfollowingbycuringinaconvection ovenat 60 o Cfortwohoursandpost-curingat 94 o Cforfourhours.Finally,open-holetensile couponswithdimensionsof 250 mm 25 mm 2 mm andcenterholediameterof 6 mm were cutfromthemanufacturedGFRPplateusingawater-cooleddiamondsaw,accordingto ASTMD 7615 =D 7615 M standard,asdepictedinFigure9.6(a). 145 9.4.2FatiguetestingofGFRPundertensileloading Inthischapter,e˙ectoffatigueloadingonsti˙nessdegradationisstudiedonanopen- holeGFRPsampleundertension-tensioncyclicloadona810MaterialTestSystem(MTS) machinewith50kNloadcell,accordingto ASTMD 3479 =D 3479 Mstandard .Theexper- imentalsetupforopen-holeGFRPsamplesubjectedtocyclicloadinginMTSmachineis illustratedinFigure9.4withalaserextensometertotracktheaxialdisplacementwhilethe sampleisunderload.At˝rst,averagefailureload( F L )wherethespecimenbreakswas recordedbyintroducingmonotonicloadingto˝vesimilarsamples.Thenatestsamplewas subjectedtocyclicloadingatconstantloadequalto 70% of F L ,frequencyof 3 Hzandstress ratioof 0 : 1 .Axialload(F)andaxialdisplacement( L )wascontinuouslyrecordedbythe MTSmeasurementandlaserextensometerrespectively. (a)(b) Figure9.4:(a)Experimentalsetupfortensileopen-holeGFRPcouponsubjectedtocyclic loadinginMTSmachine,(b)HealthyandbrokenGFRPcouponssubjectetofatigue. Tensilesti˙nessofanymaterialisgivenbyitsYoung'smodulus( E ),asde˝nedby equation9.8,where istheaxialstrainundergonebythespecimensubjectedtoaxialstress ˙ , F istheconstantloadappliedtothespecimeninaxialdirection, A isthecross-sectional 146 areaofsampleperpendiculartothedirectionofappliedforce, L istheoriginallengthofthe sampleand L denotesthechangeinspecimenlengthcausedbyloading.Cross-sectional area( A )andoriginallength( L )ofthespecimenareconstant,thereforethesti˙nessis directlyproportionaltotheratiooftheaxialloadandchangeinlengthofthespecimen undercyclicloading. E = ˙ = F=A L=L / F L (9.8) Foracompositespecimenundercyclicloading,sti˙nessmodulus S iscomputedastheslope oftheloaddisplacement(orstressstrain)hysteresisloop,i.e.,theslopeoftheline connectingthemaximumstressandminimumstresspoint[173,174].Asdepictedin˝gure 9.5(a),theslopeofhysteresisloopreduceswithincreasingloadcycles.Sti˙nesscomputed fromthisslope,versusnumberofloadingcyclesforatrainingGFRPspecimenisplottedin Figure9.5(b).Thesti˙nessmoduluswasnormalizedwithrespecttothemaximumsti˙ness modulus( S 0 )computedinitspristinecondition.Detailsofcomputingtensilestrengthof compositematerialisderivedinAppendixB. 9.5NDEdataacquisition 9.5.1FatiguedamagedetectionbyOTS) OneoftheNDEsensorsusedinthisstudyisbasedonanopticaltransmissionscanning (OTS)systemdevelopedbyKhomenkoetal.[118].FormationofairgapsinsideGFRPma- terialintroduceschangesinitsopticalpropertiessuchasradiationabsorptionandscattering, whicharecapturedbytheOTSsystem.Earlierworkshavedemonstratedthecapabilityof OTStoimageimpactdamagesinGFRPspecimensandallowedforaccuratecharacterization 147 ofmultipledelaminationsandtheircontours[120,149].Theresultsobtaineddemonstrated excellentagreementwithcameraimagesusingdyepenetrant.Besides,OTSshowedgreat potentialforqualitycontrol(QC)andothercrucialNDEapplicationssuchascharacteriza- tionofthicknessvariations,improperresinproportionsandmixingandinclusionsofforeign objects. (a)(b) Figure9.5:(a)Stress-strainhysteresisloopofGFRPspecimenatdi˙erentintervalsoffatigue cycles(b)Stress-strainslopeorsti˙nessmodulus( S )asafunctionofnumberofloadcycles. Inthisstudy,theexperimentalsetupusedforNDEdataacquisitionconsistedofan iBeam-smart-640slaserdiodesourceemittinglightofwavelength 640 nm , 1 : 5 mm beam diameterandmaximumoutputpowerof 150 mW .AphotodetectorunderneaththeGFRP specimenrecordedthethrough-transmissionpowerandmappedto 0 10 V valuesuchthat directtransmissioninairwithoutspecimencorrespondedto 10 V .Thelaserpowerwas˝xed at 1 : 9 mW inordertoobtainhighestsignal-to-noiseratioandto˝xtransmissionvoltage closeto 9 : 8 V atthehealthysectionsofthesample.Thesespeci˝cationsprovidedhigh contrastimagesofdamagedordelaminatedregionsinthespecimen,asshowninFigure9.6. ThegoalofNDEprognosisistoinfersti˙nessofthestructurefrommulti-modalNDE techniquesincludingOTSandGWcollectedatperiodicintervalsoffatigueloading,starting 148 Figure9.6:(a)Healthyopen-holeGFRPcoupon(b)OTSimageofhealthyGFRPcoupon (c)OTSimageofGFRPcouponafter 900 fatiguecyclesat 70% offailureloadandstress ratioof 0 : 1 . fromitspristineconditionuptoend-of-life.OTSimagesforanopen-holeGFRPcoupon (trainingspecimen)subjectedtofatiguetestintheMTSmachine,arepresentedin˝gure 9.7.Atcyclicloadingof 70% offailureloadandloadratioof0.1,thespecimenfailedafter 1386 cycles. StrongindicationsontheOTSimagesre˛ectthepresenceofairgaphiddeninsidethe compositelayerscausedbycontinuoscyclicloading,whicheventallyleadstolossofsti˙ness inthecompositematrix.PixelsassociatedwithdamagewereextractedfromtheOTSimages viahistogramthresholding[175].Itisknownfromstructuralmechanicstheory,theopen holeinatensilecouponresultsinstressconcentrationzonearoundtheholeandthematerial startstocrack(ordelaminateincaseofcomposites)surroundingthehole.TheOTSimages supportstheabovetheoryandthereforea 100 mm lengthofthesampleisconsideredfor damageareacomputation,keepingtheholeatthecenter.Damageareacomputedforeach oftheOTSimagesin˝gure9.7isplottedinFigure9.8(a). 149 Figure9.7:OTSimagesofanopen-holedGFRPcouponunderfatigueloading:(a)Healthy ( 0 cycles)(b) 200 cycles(c) 400 cycles(d) 600 cycles(e) 800 cycles(f) 900 cycles(g) 1000 cycles(h) 1100 cycles(i) 1200 cycles(j)Totalfailureat 1386 cycles. (a)(b) Figure9.8:(a)Increaseindelaminationareainopen-holedGFRPcouponunderfatigue loading,fromOTSmeasurements(b)Correlationbetweennormalizedsti˙nessfromMTS measurementsanddelaminationareafromOTSimages. Normalizedsti˙nesscanbeinterpretedfromthedamageareainanOTSimageusing calibrationcurveobtainedfromtrainingspecimen,depictedinFigure9.8(b).Itisimportant 150 torepeattheexperimentsonmultiplespecimensinordertoassessthereproducibilityofthe NDEmethodaswellascalculateunderlyingmodeluncertaintyandmeasurementnoise variance.Asecondorderpolynomialcurve,givenbyequation9.9,isimplementedtode˝ne therelationshipbetweennormalizedsti˙ness( ^ S )anddamagearea( Ar OTS )fromOTSimage. ForthesetofGFRPspecimensusedinourstudy,theparametersofthepolynomialcurve werecomputedas p 1 = 1 : 12 10 7 , p 2 = 3 : 66 10 4 and p 3 =1 : 014 . ^ S = p 1 ( Ar OTS ) 2 + p 2 ( Ar OTS )+ p 3 (9.9) 9.5.2FatiguedamagedetectionbyGW Guidedwave(GW)sensingtechniqueisanin-situNDEmethodwhichcapturesthe changeinacousticwavespropagatingthroughstructuresinpresenceofananomaly[140, 141,121].CapabilityofGWsensingfordetectionoffatiguedamageinGFRPadhesive jointsviasurfacemountedPZTsensorshavebeensuccessfullyestablishedinchapter8.In thisstudy,GWsensingisimplementedviaasensingskinwithpressuresensitiveadhesive. InsteadofmountingPZTsensorsonthespecimen,thetransducersareembeddedona sensingskinwithpressuresensitiveadhesiveforrepeatedbondinganddebondingasshown inFig.9.9.TheexperimentalsetupusedforGWinspectionofGFRPspecimenisshownin ˝gure9.9(a).Gaussianpulsewith50KHzcentralfrequencywasgeneratedusingfunction generatortoexcitethetransmitterPZT. Incomparisontopermanentlybondedtransducers,thesesensingskinsarereusable.Be- sides,distancebetweenthetwotransducersareheldconstantirrespectiveofanyplastic (permanent)straininthespecimen.Neglectingplasticstraininthematerial,theobserved timeof˛ight(TOF)changeinguidedwavesignalscanbesolelyaccountedtothespeci- 151 (a) (b) Figure9.9:GuidedWaveinspectionofGFRPspecimen(a)Experimentalsetup(b) Schematicofpressuresensitiveskin. mensti˙nessdegradationwhicharisesfromvarioussourceofdamagesuchas˝berbreakage, matrixcracking,delaminationetc.Groupvelocityofacousticwave( c )traversingthrough specimendependsonitssti˙nessalonglongitudinaldirectiongivenbyitsYoung'smodulus E 1 accordingtoequation9.10. isPoisson'sratio, ˆ isdensity, ! isangularfrequencyand d isthethicknessoftheplate.AssumingthechangeinPoisson'sratiotobenominal,fora ˝xedfrequencyandspecimengeometry,thevelocityofacousticwavesdecreaseswithreduc- tionofitssti˙nessmodulus.GiventhedistancebetweenPZTtransducersdonotchangein thesensingskin,timetakenbytheGWsignaltoreachthereceiverPZTsensorismoreina 152 damagedGFRPspecimen.Hence,changeinTOFofGWsignalsserveassuitableindicator ofsti˙nessdegradation. c =2 s E 1 3 ˆ (1 2 ) p !d (9.10) GWdatawascollectedfromtheopen-holeGFRPcouponinintermediateloadcyclesasit graduallyprogressedfromhealthytototalfailure,withthehelpoftheGWsensingskin.Raw ultrasonicsignalswereaveraged64timespriortoplottingandwere˝lteredwithabandpass ˝lterwithcuto˙frequenciesof5kHzand400kHz.Asthecompositespecimenunderwent matrixcrackingfollowedbyformationofdelaminaionand˝berbreakage,structuralsti˙ness reducedwhichcausedaphase-shiftinthegroup S 0 modeoftheGWsignals.Atime-shift wasobservedintheGWsignalatthereceiverPZT,asdepictedin˝gure9.10. TOF was hencecomputedateveryloadcycleintervalandplottedin˝gure9.11(a). (a)(b) Figure9.10:(a)TimeshiftinGWsignalsinprogressivelydamagedGFRPspecimenunder fatiguecycles(b)Enlargedregionin S 0 modeofreceivedGWsignals. SimilartoOTSsensing,normalizedsti˙nessobtainedfromMTSmeasurementscould becorrelatedwiththeGWsignalfeatures.OncontrarytoOTS,GWsignalpropagatign throughthedamagedregionofcompositeplateprovidsamoreglobalassessmentofdamage incompositesincludinge˙ectofmatrixcracking,delaminationand˝brebreakageonthe overallsti˙nessreduction.A2ndorderpolynomialcurveis˝ttedonthecorrelationcurves, 153 (a)(b) Figure9.11:(a)Increasein TOF inopen-holedGFRPcouponunderfatigueloading,from GWmeasurements(b)Correlationbetweennormalizedsti˙nessfromMTSmeasurements anddelaminationareafromGWimages. asdepictedin˝gure9.11(b)andthecalibrationcoe˙cientsareobtainedfromthetraining specimensas p 1 = 1 : 5 10 9 , p 2 = 6 : 5 10 3 and p 3 =1 : 009 . ^ S = p 1 TOF ) 2 + p 2 TOF )+ p 3 (9.11) 9.6PrognosisResults Particle˝lteringbasedprognosiswasappliedtotheOTSandGWdatacollectedfrom GFRPspecimenssubjectedtofatiguetestingandthepredictionresultsarereportedinthis section.Initialdistributionofunknownparameters( )andcorrelationcoe˚cients( p 1 ;p 2 ;p 3 ) ofNDEdataandsti˙nessmeasuredfromMTSsystemwereobtainedfromtrainingsample. PFalgorithmwiththeestimatedparameterswasthenimplementedinanidenticaltestwhere GFRPspecimenwassubjectedtofatigueloadingwithconditionsasrecordedintable9.3. Startingfromitspristinecondition,thetestspecimenwassubjectedtoprogressivefa- tiguedegradationuntilitfailedafter 2250 cycles.Sti˙nesscomputedfrommeasurements 154 Table9.3:LoadingcyclesforintermediateOTSandGWinspectionsontestGFRPspecimen. fromtheMTSandlaserextensometerwereconsideredasthegroundtruthinthisstudy. Bene˝tofusingtwoNDEsensordataoversinglesensorprognosisisassessedandresults fromimplementingtheproposedjointlikelihoodcomputationapproachiscomparedwith predictiononaverageofsensormeasurements. 9.6.1PFprognosisonOTSdata Sti˙nesscomputedfromOTSmeasurements f z OTS g usingequation9.9wereusedto predictunknownparametersinParis-Parismodeldescribingthesti˙nessdegradationin fatigue-inducedGFRPtestspecimen.Initialdistributionofparameterswereobtainedfrom trainingdatasetandsetas: s 0 ˘N (0 : 01 ; (0 : 001) 2 ) m 10 ˘N (4 ; (0 : 6) 0 : 01 ) ; log C 10 ˘N ( 10 ; (0 : 1) 2 ) m 20 ˘N (0 : 3 ; (0 : 01) 2 ) ; log C 20 ˘N ( 10 ; (0 : 1) 2 ) (9.12) T 0 ˘N (750 ; (10) 2 ) ! ˘N (0 : 09 ; (0 : 001)) 155 InitialdistributionofnoisewascharacterizedbasedonexperimentalevidenceofNDE measurementsontrainingspecimens.Predictionresultswithdi˙erentnumberofOTSob- servationsarepresentedin˝gure9.12.ThelikelihoodofeachparticleinthePFalgorithmis updatedaccordingtothesingle-sensorprognosisframework,asgiveninequation9.13.With increasingnumberofavailableOTSmeasurements,thepredictedsti˙nesscurveconvergesto thetruesti˙nesscalculatedfromMTSmeasurementsalongwithdecreasing 95% con˝dence intervals. L ( z OTS k j s i k )= 1 z OTS k p 2 ˇ˘ i k exp 2 4 1 2 ln z OTS k i k ˘ i k ! 2 3 5 (9.13) Similartotrainingspecimen,sti˙nesspredictionofthetestspecimenwascontinuedup tillthecompositesti˙nessreducedtolessthan 30% ofitsinitialsti˙nessinpristinecondition. TheestimatedRULvaluesatallfatiguestagesalongwiththeir 95% con˝denceintervalsare illustratedin˝gure9.13.When 2 OTSobservationswereavailable,thespecimenhadalready beensubjectedto100cycles,thereforethetrueRULwascomputedas 2250 1000=2150 cycleswhereastrueRULattheendof 2250 cycleswas 0 sinceitreacheditsEOL.Prediction accuracyofRULintermsofnormalizedmeansquarederror(NRMSE),accordingtoequation 9.14,was 0 : 1761 where O isthenumberofobservations.SinceRULpredictedfromOTS measurementsislowerthanitstruevalueformostofthecases,itdoesnotleadtousage ofGFRPstrucutrebeyonditssafetylimit.However,portionofitsresiduallifemayremain unexploitedduetounderestimationofRULbysinglesensorNDE. NRMSE = q 1 O 1 P O i =2 ( RUL i ^ RUL i ) mean ( RUL i ) (9.14) 156 (a)(b) (c)(d) Figure9.12:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberofavailable OTSmeasurementsinParis-Parismodel(a)n=4,(b)n=8(c)n=12(d)n=16. Figure9.13:RULpredictionforvaryingnumberofavailableOTSmeasurements (NRMSE= 0 : 1761 ). 157 9.6.2PFprognosisonGWdata Predictionresultsofsti˙nessreductioninGFRPspecimenviaimplementationofPF basedprognosisonGWsensingdataispresentedinthissection.Normalizedsti˙nessis computedfromthe TOF ofGWreceivedsignalsusingthecalibrationcoe˚cientsinequa- tion9.11.ParticleswereupdatedinthePFapproachbyresamplingaccordingtotheir likelihoodvaluescomputedbyequation9.13,with f z OTS g beingreplacedby f z GW g orthe sti˙nessvaluesofthespecimenatdi˙erentstagesoffatigueinferredfromGWmeasurements. Predictionresultsoffuturesti˙nessvaluesusingtheParis-Parismodelaredenotedin˝gure 9.14.SimilartoOTSdata,initialnoisedistributionwascharacterizedfromGWexperiments ontrainingspecimens.InitialdistributionofotherparametersinPFalgorithmwerekept unchanged,inordertocomparethepredictioncapabilityofthetwoNDEtechniques. Itisobviousfrom˝gure9.14thatasnumberofavailablemeasurementsincreases,the predictedsti˙nesscurvebecomesmorerepresentativeofthetruesti˙nessvalues.thecon- ˝denceintervalreduces.SimilartoOTSmeasurementresults,theRULiscomputedfor di˙erentnumberofavailableGWmeasurementsassumingthatthespecimen'sEOLoccurs at 2250 cycles.Figure9.15presentstheaccuracyofRULestimationcomparedtothetrue valuesateveryintermediatestageoffatiguetesting.NRMSEforRULpredictionusingGW measurementswasobtainedas 0 : 1441 .Comparing˝gures9.13and9.15,itcanbeconcluded thatGWmeasurementscandescribethedamagegrowthprogressionmoreaccuratelythan OTSdata,thereasonbeingthatGWdataprovideglobalassessmentofdamagestatusin- cludingmatrixcracks,˝berbreakageanddelaminationwhereas,OTSsystemcapturese˙ect ofdelaminationonsti˙nessdegradation.Diagnosisofsti˙nessfromOTSmeasurements lacksthecontributionfrommatrixcrackingand˝berbreakage,therebyleadingtohigher 158 (a)(b) (c)(d) Figure9.14:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberofavailable GWmeasurementsinParis-Parismodel(a)n=4,(b)n=8(c)n=712(d)n=16. errorinitsprognosisresults. Figure9.15:RULpredictionforvaryingnumberofavailableGWmeasurements (NRMSE= 0 : 1441 ). 159 9.6.3PFprognosisonAverageofTwoSensorsdata Withanaimtoexploitthebene˝tsofbothNDEsensors,PFbasedprognosiswasimple- mentedonadatasetobtainedbyaveragingOTSandGWmeasurementsateverytime-step. ThelikelihoodforeachparticleinthePFframeworkwascomputedaccordingtoequation 9.13with f z OTS g beingreplacedby f z AVG g ,where f z AVG g is: f z AVG g = f z OTS g + f z GW g 2 (9.15) KeepingallotherparametersoftheParis-Parismodelunchanged,futuresti˙nessval- ueswerepredicted,givenvaryingnumberofknownmeasurementsandthecorresponding estimatedsti˙nesscurvesaredenotedinFigure9.16.Asexpected,thepredictedcurvewas closertothetruesti˙nesscomputedfromMTSmeasurementswithincreasingnumberof observations.ThecorrespondingRULpredictionfordi˙erentobservedmeasurementsusing averagedataispresentedinFigure9.17.NRMSEofpredictedRULwascomputedas 0 : 1507 whichshowsthatasimpleaveragingoftwosensordatadoesnotprovidehigheraccuracyin itsprognosisresults.Sti˙nessdegradationinacompositematerialisinherentlyadynamic processwhichcannotbecapturedbystaticweightedcombinationofthetwosensordata. AccuracyofOTSandGWmeasurementsvariesatdi˙erentloadcycleswhichrequiredy- namicupdatingofweightsonthe˝nalpredictionresult.Onthecontrary,averagingleadto higherNRMSEofRULpredictioncomparedtothatofGWmeasurements. 160 (a)(b) (c)(d) Figure9.16:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberofavailable AVGmeasurementsinParis-Parismodel(a)n=4,(b)n=8(c)n=12(d)n=16. Figure9.17:RULpredictionforvaryingnumberofavailableAVGmeasurements (NRMSE= 0 : 1507 ). 9.6.4PFPrognosisonTwoSensorDatabyJointLikelihoodCom- putation InaGFRPspecimenwithaopen-holeatthecenterundergoingfatiguetest,overall sti˙nessreductioncanbeaccreditedtothedamagegrowtharoundthehole.Particularly 161 forcomposites,damageincludesmultiplestructuralphenomenonoccuringsimultaneously. HoweveranalysingOTSandGWdata,itwasobservedthattheindividualsensorsonly providedpartialrepresentationofdamagestatusinsideaGFRPspecimensubjectedto fatigue.Ononehand,sti˙nessinferredfromOTSmeasurementsaccountedfortheincrease indelaminationareaaroundthehole,whereasontheotherhand,sti˙nesscomputedfrom GWmeasurementscapturedoveralldamagemechanismthroughoutthespecimenlengthand notlimitedtotheregionaroundhole.Besides,measurementsfromindividualNDEsensors werea˙ectedbyvariablenoiseateveryinspection.Sincedi˙erentsensordataprovides di˙erentcontributiontothesti˙nessreduction,itiscrucialtoimplementjointlikelihoodin BayesiannetworkwithinthePFalgorithm,forthisapplication. (a)(b) (c)(d) Figure9.18:Predictionofsti˙nessdegradationcurvebasedondi˙erentnumberofavailable measurementsusingjointlikelihoodcomputationinParis-Parismodel(a)n=4,(b)n=8(c) n=12(d)n=16. 162 Usingsameparametersasbefore,jointlikelihoodparticlesateveryiterationiscalculated byequation9.13,replacing f z OTS g with f z JL g where L f z JL g iscomputedaccordingto conditionalindependenceofOTSandGWmeasurementsinBayesiannetworks. L f z JL g = L f z OTS g L f z GW g (9.16) Asshownin˝gure9.18,predictedsti˙nessbyjointlikelihoodconvergesclosertotheground truthwithincreasingnumberofobservedmeasurements.Moreover,comparedtotheprevi- ousresults,themostaccurateRULprognosisisachieved,withNRMSEof 0 : 065 whenjoint likelihoodistakenintoaccount,asdenotedin˝gure9.19.ThemeanofRULdistribution liedwithin 20% errorboundfromtruevalueswithexactmatchingofmedianRULatmost observationcycles.TheprimaryreasonforhigheraccuracyofRULprognosisbyjointlikeli- hoodcomputationoftwosensordataisduetothefactthatthisapproachalloweddynamic updateofweightscontributingtothetruesti˙nessvalueunlikesimpleaveragingoftwo data.Especiallyatearlierstagesoffatiguewhenfewermeasurementswereavailable,deci- sionfusionfrombothsensorswithunequalweightsbasedontheirsti˙nessmodeluncertainty andmeasurementnoiseleadtomoreaccuratepredictionofsti˙nessdegradationinGFRP specimen. Foradditionalcomparisonbetweentheprognosisapproaches,NRMSEiscalculatedfor everypredictedsti˙nesscurveusingdi˙erentnumberofobservedmeasurementsandplotted in˝gure9.20.Bene˝tofproposedjointlikelihoodbasedPFalgorithmoverotherapproaches isevidentfromFig.9.20(a)especiallyintheearlierstagesoffatigue.Further,the˝nal errorafter 16 measurementsreaches 3% ,therebyreinforcingtheproposedmethodasavalid predictiontechnique. 163 Figure9.19:RULpredictionforvaryingnumberofavailableOTSandGWmeasurements usingjointlikelihoodcomputation(NRMSE= 0 : 065 ). (a)(b) Figure9.20:(a)ErrorcomparisonforvaryingnumberofavailableOTSandGWmeasure- ments,(b)ErrorcomparisonforvaryingnumberofparticlesinPFalgorithm. ItisimportanttonotethatincreasingnumberofparticlesinPFalgorithmreducesthe estimationerror,butleadstohighercomputationtime[176].Advantageofjointlikelihood inPFalgorithminproducingaccuratepredictionresultsatlowerparticlecountcompared totheothersingle-sensorprognosisisdepictedin˝gure9.20(b).Implementingthejoint likelihoodapproachon 16 observations,NRMSEreachesto 0 : 04 using 50 particleswhereas ittakesalmost 500 particlesforsinglesensororaveragedataprognosis.Computationtime isdoubledwhen500particlesareusedcomparedto50particles.Besides,averagingof 164 twomeasurementsdoesnotguaranteehigherestimationaccuracycomparedtothesingle sensordata.However,jointlikelihoodcomputationensuresthelowesterrorforallparticle counts.Therefore,theproposedmethodofcomputingjointlikelihoodofmeasurementsfrom multi-modalNDEsystemdemonstratesanaddedadvantageofreducingparticlecountinPF algorithm.Reductionofparticleshavesigni˝cantimpactinreducingoverallcomputation timeandresources,therebyachievingreal-timeprognosisofindustrialstructures. 165 Chapter10 Conclusion Inthisstudy,theimportanceofselfevaluationinexistingautomatedNDEsignalanalysis systemhasbeendiscussed.SourcesofuncertaintiesinatypicalNDEsignalclassi˝cation systemandtheire˙ectsonclassi˝cationcon˝dencehavebeenidenti˝ed.Bene˝tsofBayes posteriorprobabilityasastrongmeasureofreliabilityhasbeenimplementedwhichcaptures thee˙ectofinterclassdistanceandintra-classvarianceinthefeaturespace.Inaddition tothat,e˙ectofinspectionnoisehasbeenincorporatedintocon˝dencecalculation.Ithas beenshownthatbootstrappingandweightingBayesposteriorprobabilitywiththenoise statisticsofthetestdataachievesamorecomprehensivecon˝dencemetricassociatedwith classi˝cationofnoisyNDEdata.Further,implementationoftheproposedapproachon steamgeneratortubeinspectiondatashowspossibleapplicationofthemethod. Infuture,otherfactorsofreliabilityinNDEanalysissuchase˙ectofa-prioriinformation aboutthemechanicalstructureandhistoricalinspectionresultscanbestudied.Another highlyimportantproblemtobeaddressedisthee˙ectofill-˝ttingofstatisticalmodelon thedata.Ifdatadoesnotfollownormaldistribution,theproposedcon˝dencemetricwillfail tocapturethereliabilityofclassi˝cationresultsaccurately.Insuchcases,amoreadaptive reliabilitymeasurebasedonnon-parametricstatisticalmodelisnecessary.Thechallenging taskofevaluatingclassi˝cationcon˝dencewithlimiteddata,missingdataorpresenceof outliersshouldbeinvestigated. 166 ThesecondpartofthethesispresentsanNDEapproachforconditionbasedmaintenance andreliabilityanalysisofstructuresunderoperation.Predictionofdelaminationgrowthin GFRPsamplessubjectedtolowvelocityimpactsisdiscussed.Imagesfromopticaltrans- missionscanningsystemwereusedforextractingdelaminationareafromimpactedsamples whichisarapidandnoncontactscanningtechniqueinadditiontobeingcoste˙ectiveand easytobeimplementedinindustries.Resultsfromimplementationofparticle˝lteringap- proachtoestimatedelaminationpropagationpathandremainingusefullifetimeofaGFRP samplearepromising.DuetouniquecharacteristicofGFRPresin,thedelaminationarea growthhadasudden'jump'atthetransitionimpactenergywhichmadethepredictionall themorechallenging.ApplyingtwoParismodelswithdi˙erentparametersforcapturingthe 'jump'insteadofasingleParismodelenhancestheprognosisperformanceoftheapproach andre˝nedestimationofthedelaminationpropagationpathandRUL. Despitestrikingbene˝tsoftheParis-Parismodel,oneofthelimitationsofthismodelis thatitstronglydependsontheinitialdistributionofthe'jump'energy.Ifthe'jump'energy ishighlydi˙erentfromthetruevalue,themodelfailstocorrectlyestimatethetransitionand yieldsasub-optimalresultandhencepredicteddelaminationcurveisinaccurate.Moreover thedelaminationpathcouldnotbeestimatedwhenfewermeasurementswereavailable duetolackofenoughinformationtopredictthe'jump'energy.Infuture,theproposed algorithmshouldbeinvestigatedonothercompositesamplesbyincorporatingadditional factorsa˙ectinginter-laminardelaminationsuchascomplexdamagegrowthduetovibration followingimpactsorcomplicatedspecimengeometry.Insuchcases,thedamagegrowth modelhastobemodi˝edwithoutchangingtheoverallframeworkofthepredictionapproach. AnobviousextensiontotheNDEprognosticsistheuseofmulti-sensorinformationto re˝nethepredictionofresiduallifeofasystemunderoperation.Itisevidentfromprognosis 167 resultshatOTSandGWmeasurementscomplementeachotherforestimatingcomposite's sti˙nessfromNDEmethods.OTScanimagedelaminationaccuratelybutcannotdetect matrixcrackingwhichoccursattheinitialstagesoffatigue.Ontheotherhand,overall e˙ectofmatrixcrackanddelaminationiscapturedwithinchangeinTOFofGWsignals. Judicioususageofbothmeasurementsenableshigherpredictionaccuracy,evenforearlier stagesoffatigue.OTS-sti˙nessmodelerrorishigherthanGW-sti˙nessmodelerror,which canbefedintothePFmodel,therebyautomaticallyadjustingfusionweightsduringjoint likelihoodcomputation.PFprognosisbyjointlikelihoodachieveshighestRULprediction andlowestpredictionerror(NRMSE),therebyvalidatingtheproposedprognosisapproach basedonjointlikelihoodcomputation. Resultsareencouragingandcanbeimplementedusingmorethan2sensors,without changingtheBayesianNetworkframeworksincetheassumptionofconditionalindependence staysvalidinmultiplesensorframework.Infuture,predictionresultscanbefurtherre˝ned byreplacingempiricalcorrelationcurvesbetweenNDEfeaturesandstructuralsti˙nesswith physics-basedmodels.Besides,Paris-Parismodelshouldbeinvestigatedformorespecimens undervaryingloadconditions.Overall,theproposedprognosismethodcanbeusedfor reliabilityassessmentofanymulti-sensorynetworkacrossvariousapplication˝elds. 168 APPENDICES 169 AppendixA BayesianNetworksformulti-sensor fusion Bayesiannetworks,alsoknownasbeliefnetworksisapopularmethodformodeling uncertainandcomplexdomainssuchasenvironmentalmodelling[177],faultdiagnosis[178] andforensicscience[179].Bayesiannetworksareatypeofprobabilisticgraphicalmodel thatrepresentsasetofvariables(nodes),andtheirconditionaldependencies(arrows)viaa directedacyclicgraph(DAG),asshownin˝gureA.1.Inthisexample,thereare M children nodes( X 1 ;X 2 ;:::;X M )fromtheparentvariable P . FigureA.1:ExampleofBayesianNetworkwithchildrennodes( X 1 ;X 2 ;:::;X M )andtheir parentnode(P). TheprimaryadvantageofBayesnetworkistodecomposethejointdistributionsofall variablesbyexploitinglocalMarkovpropertyofvariables,therebyreducingdimensionalityof themodeltomakeitcomputationallyfeasible.LocalMarkovpropertyofvariablesdictates thatthethejointprobabilitydensityfunctioncanbewrittenasaproductoftheindividual 170 densityfunctions,conditionalontheirparentvariables[180]. p ( x )= Y v 2 V p ( x v j x pa ( v ) ) (A.1) Now,foranysetofrandomvariables,theprobabilityofanymemberofajointdistribution canbecalculatedfromconditionalprobabilitiesusingthechainrule(givenatopological orderingofX)asfollows: P ( X 1 = x 1 ;:::;X M = x m )= M Y v 2 V P ( X v = x v j X v +1 = x v +1 ;:::;X M = x m ) (A.2) Byconditionalindependenceofvariables,foreach X j whichisaparentof X v thejoint likelihoodcanthereforebecomputedas: P ( X 1 = x 1 ;:::;X M = x m )= M Y v 2 V P ( X v = x v j X j = x j ) (A.3) Bayesiannetworksareparticularlysuitablefordecisionfusioninpracticalapplications owingtotheirfavorablefeaturessuchas: Theyfacilitatelearningaboutcausalrelationshipsbetweenvariables[180]. Theyprovideamethodforavoidingover˝ttingofdata[181] Theycanshowgoodpredictionaccuracyevenwithrathersmallsamplesizes[182] 171 AppendixB Sti˙nessofCompositeMaterials Tensileloadappliedtoanyspecimenstretchesitsmaterial.Thechangeinlengthof thespecimenwithrespecttoitsoriginallengthistermedasthestrain.Now,forisotropic materialssuchasmetals,therelationshipbetweenstress( ˙ )andstrain( " )isindependent ofthedirectionofappliedforce.Hence,sti˙nessinisotropicmaterialscanbede˝nedbya singleparametercalledYoung'smodulus( E )whichrelatesthestressandstrainaccording toequationB.1. ˙ = E" (B.1) Materialsinwhichtheirmechanicalpropertiesdi˙erindi˙erentdirectionsareknownto beanisotropic.Compositematerialsbelongtothiscategoryandthereforesti˙nesscompu- tationismorecomplicatedinpolymerscomparedtometals.Foranisotropicmaterials,the stress-strainbehaviorisgivenbythegeneralizedHooke'slaw,givenbyequationB.2.Apart fromtheYoungsmodulii,materialpropertiesarealsogivenbythePoisson'sratio( )which istheratioofthestrainperpendiculartoagivenloadingdirection,tothestrainparallelto thegivenloadingdirection.Eg: 12 = " 2 " 1 forunixialloadindirection1. 172 FigureB.1:Materialdirectionsinaspecimen x 1; y 2; z 3 . (B.2) The C matrixconsistingof36constantsisknownasthegeneralized sti˙nessmatrix inwhichthesubscripts1to6denotethesixpossibledirectionsofsti˙nesschangeinthe matrixsubjectedtoexternalload.1,2and3refertothelongitudinal( x )andtransverse directions( y;z )asshownin˝gureB.1,whereas " 4 ;" 5 and " 6 denotesthestrainalong xz;yz and xy directions. Acompositewithunidirectional˝berorientationcanbeconsideredasanorthotropic materialisonewhichhasthreeorthogonalplanesofmicrostructuralsymmetry.Asex- plainedin[183],materialsymmetry(equalnormalstresses ˙ 1 = ˙ ; 1 ;˙ 2 = ˙ ; 2 ,oppositeshear stresses ˙ 6 = ˙ ; 6 )inherentintheorthotropicmaterialreducesthenumberofindependent elasticconstants.Asaresult,thesti˙nessmatrixisreducedtonineindependentelastic constants,accordingtoequationB.3. 173 (B.3) ExpandingtheelasticconstantsintermsofYoungsmodulus( E ),Poisson'sratio( )and shearmodulus( G ),equationB.4isobtained. (B.4) Inourapplicationoftensileloading,onlyaxialstressalongthedirectionof˝bers( ˙ 1 ) ispresent.Further,intensilecoupons,thewidthofthespecimenbeingverysmall,strains in z directioncanbeneglected.Besides,inorthotropicmaterialthereisnoshearcoupling withrespecttothematerialaxes,i.e.,normalstressesresultinnormalstrainsonlyand shearstressesresultinshearstrainsonly.Hence,byretainingonlythe x;y componentsof normalstrainsandinvertingthecompliancematrixofequationB.4,thesti˙nessmatrixcan 174 begeneratedasequationB.5. 2 6 6 6 6 6 4 ˙ 1 ˙ 2 ˙ 6 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 S 11 S 12 0 S 21 S 22 0 00 S 66 3 7 7 7 7 7 5 2 6 6 6 6 6 4 " 1 " 2 " 6 3 7 7 7 7 7 5 (B.5) where, S 11 = E 1 1 12 21 , S 22 = E 2 1 12 21 , S 12 = 12 E 2 1 12 21 = 21 E 1 1 12 21 and S 66 = G 12 . 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