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Optimizing the Safety Margins Governing a Deterministic Design Process while Considering the Effects of a Future Test and Redesign on Epistemic Model Uncertainty

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Title:
Optimizing the Safety Margins Governing a Deterministic Design Process while Considering the Effects of a Future Test and Redesign on Epistemic Model Uncertainty
Creator:
Price, Nathaniel B
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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Language:
english
Physical Description:
1 online resource (147 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
KIM,NAM HO
Committee Co-Chair:
HAFTKA,RAPHAEL TUVIA
Committee Members:
KUMAR,ASHOK V
URYASEV,STANISLAV
Graduation Date:
8/6/2016

Subjects

Subjects / Keywords:
Aleatory contracts ( jstor )
Design engineering ( jstor )
Design evaluation ( jstor )
Design optimization ( jstor )
Engines ( jstor )
Kriging ( jstor )
Modeling ( jstor )
Random variables ( jstor )
Safety factors ( jstor )
Uncertainty ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
optimization -- redesign -- reliability -- safety-margins -- uncertainty
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

Notes

Abstract:
At the initial design stage, engineers often rely on low-fidelity models that have high uncertainty. Model uncertainty is reducible and is classified as epistemic uncertainty; uncertainty due to variability is irreducible and classified as aleatory uncertainty. In a deterministic safety-margin-based design approach, uncertainty is implicitly compensated for by using fixed conservative values in place of aleatory variables and ensuring the design satisfies a safety-margin with respect to design constraints. After an initial design is selected, testing (e.g. physical experiment or high-fidelity simulation) is performed to reduce epistemic uncertainty and ensure the design achieves the targeted levels of safety. Testing is used to calibrate low-fidelity models and prescribe redesign when tests are not passed. After calibration, reduced epistemic model uncertainty can be leveraged through redesign to restore safety or improve design performance; however, redesign may be associated with substantial costs or delays. In this work, the possible effects of a future test and redesign are considered while the initial design is optimized using only a low-fidelity model. The goal is to develop a general method for the integrated optimization of the design, testing, and redesign process that allows for the tradeoff between the risk of future redesign and the associated performance and reliability benefits. This is accomplished by formulating the design, testing, and redesign process in terms of safety-margins and optimizing these margins based on expected performance, expected probability of failure, and probability of redesign. The first objective of this study is to determine how the degree of conservativeness in the initial design relates to the expected design performance after a test and possible redesign. The second objective is to develop a general method for modeling epistemic model uncertainty and calibration when simulating a possible future test and redesign. The third objective is to apply the method of simulating a future test and redesign to a sounding rocket design example. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2016.
Local:
Adviser: KIM,NAM HO.
Local:
Co-adviser: HAFTKA,RAPHAEL TUVIA.
Statement of Responsibility:
by Nathaniel B Price.

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Source Institution:
UFRGP
Rights Management:
Copyright Price, Nathaniel B. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Classification:
LD1780 2016 ( lcc )

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OPTIMIZINGTHESAFETYMARGINSGOVERNINGADETERMINISTICDESIGNPROCESSWHILECONSIDERINGTHEEFFECTSOFAFUTURETESTANDREDESIGNONEPISTEMICMODELUNCERTAINTYByNATHANIELB.PRICEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2016

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c2016NathanielB.Price

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ACKNOWLEDGMENTSFirstandforemost,IwouldliketothankmyadvisersDr.Nam-HoKimandDr.RaphaelHaftkafromtheUniversityofFlorida,Dr.MathieuBalesdentandSebastienDefoortfromONERA-TheFrenchAerospaceLab,andDr.RodolpheLeRichefromEcoledesMinesdeSaint-Etienne.Iamgratefulfortheopportunitytohavestudiedunderyoursupervisionanddeeplyappreciateyourinsight,guidance,patience,andsupport.IwishtothankmycommitteemembersDr.AshokKumarandDr.StanislavUryasevfortheiradviceandthoughtfulquestions.SpecialthankstoDr.AshokKumarandDr.JoromeMoriofortakingthetimetoserveasreviewersofthisdissertation.ThankyoutothepastandpresentmembersoftheStructuralandMultdisciplinaryOptimizationGroupandallthestudentresearchersatONERAforyourfriendship,collaborations,andenlighteningdiscussions.Iamgratefulfortheencouragementandsupportofmyfriendsandfamilythatmadethisdissertationpossible.ThisresearchwassupportedbyAirForceOfceofScienticResearch(Contract84796)andONERA-TheFrenchAerospaceLab.Thissupportisgratefullyacknowledged. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1Motivation .................................... 13 1.2Objectives .................................... 17 1.3Outline ...................................... 18 2BACKGROUNDANDLITERATUREREVIEW ................... 19 2.1UncertaintyClassication ........................... 19 2.2Multi-delityModeling ............................. 21 2.2.1Sensitivity-BasedScalingMethods .................. 21 2.2.2GaussianProcess(GP)ModelBasedMethods ........... 23 2.3DesignUnderUncertainty ........................... 26 2.3.1DeterministicSafety-FactorBasedDesign .............. 26 2.3.2Reliability-BasedDesignOptimization(RBDO) ........... 29 2.3.2.1Reliabilityindexapproach(RIA) .............. 30 2.3.2.2Performancemeasureapproach(PMA) .......... 30 2.3.2.3Sequentialoptimizationandreliabilityassessment(SORA) 31 2.3.3RBDOwithEpistemicModelUncertainty ............... 32 2.3.4UncertaintyReductionMeasures ................... 34 2.4GlobalOptimization .............................. 34 3DECIDINGDEGREEOFCONSERVATIVENESSININITIALDESIGNCONSIDERINGAFUTURETESTANDPOSSIBLEREDESIGN .................. 37 3.1Introduction ................................... 38 3.2Methods ..................................... 41 3.2.1OptimizationofSafetyMargins .................... 41 3.2.2Monte-CarloSimulationofEpistemicModelError .......... 43 3.2.3DeterministicDesign/RedesignProcess .............. 44 3.2.3.1Initialdesign ......................... 44 3.2.3.2Testinginitialdesignandredesigndecision ........ 45 3.2.3.3Modelcalibration ....................... 46 3.2.3.4Redesign ........................... 46 3.2.4ProbabilisticEvaluation ........................ 47 4

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3.3TestCases ................................... 49 3.3.1UniaxialTensionTest .......................... 49 3.3.1.1Problemdescription ..................... 49 3.3.1.2Expectedperformanceversusprobabilityofredesign .. 50 3.3.1.3Expectedperformanceversuslevelofhigh-delitymodelerror .............................. 52 3.3.2SupersonicBusinessJetEngineDesign ............... 53 3.3.2.1Problemdescription ..................... 53 3.3.2.2Expectedperformanceversusprobabilityofredesign .. 60 3.3.2.3Expectedperformanceversuslevelofhigh-delitymodelerror .............................. 60 3.4DiscussionandConclusion .......................... 61 3.5LimitationsandFuturework .......................... 66 4CONSIDERINGSPATIALCORRELATIONSINTHEEPISTEMICMODELERRORWHENSIMULATINGAFUTURETESTANDREDESIGN ............ 68 4.1ResearchContextinRelationtoScopeofDissertation ........... 69 4.2Introduction ................................... 69 4.3Methods ..................................... 72 4.3.1OptimizationofSafetyMargins .................... 73 4.3.2Monte-CarloSimulationofEpistemicModelError .......... 75 4.3.3DeterministicDesignProcess ..................... 77 4.3.3.1Initialdesign ......................... 78 4.3.3.2Testinginitialdesignandredesigndecision ........ 78 4.3.3.3Calibrationandredesign .................. 79 4.3.4ProbabilisticEvaluation ........................ 80 4.4DemonstrationExample ............................ 82 4.4.1Overview ................................ 82 4.4.2ErrorModel ............................... 83 4.4.3Results ................................. 84 4.5DiscussionandConclusions .......................... 87 5SOUNDINGROCKETDESIGNUNDERMIXEDEPISTEMICMODELUNCERTAINTYANDALEATORYPARAMETERUNCERTAINTY ................. 93 5.1ResearchContextinRelationtoScopeofDissertation ........... 93 5.2Introduction ................................... 94 5.3Methods ..................................... 97 5.3.1PreliminaryReliability-BasedDesignOptimization(RBDO) ..... 97 5.3.2OptimizationofStandardDeviationOffsets ............. 98 5.3.3Monte-CarloSimulationofEpistemicModelError .......... 100 5.3.4DeterministicDesignProcess ..................... 101 5.3.4.1Initialdesign ......................... 102 5.3.4.2Testinginitialdesignandredesigndecision ........ 102 5.3.4.3Calibrationandredesign .................. 103 5

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5.3.5ProbabilisticEvaluation ........................ 103 5.4TestCases ................................... 106 5.4.1CantileverBeamBendingExample .................. 106 5.4.1.1Problemdescription ..................... 106 5.4.1.2Applicationoftheproposedmethod ............ 107 5.4.2MultidisciplinarySoundingRocketDesignExample ......... 112 5.4.2.1Problemdescription ..................... 112 5.4.2.2Standardatmospheremodels ............... 114 5.4.2.3Disciplinemodels ...................... 117 5.4.2.4Low-delitymodel ...................... 124 5.4.2.5Applicationoftheproposedmethod ............ 125 5.5DiscussionConclusions ............................ 129 6CONCLUSIONS ................................... 133 REFERENCES ....................................... 137 BIOGRAPHICALSKETCH ................................ 147 6

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LISTOFTABLES Table page 3-1Problemdenitionforuniaxialtensiontestexample ................ 50 3-2Uncertainparametersforuniaxialtensiontestexample .............. 50 3-3Resultsforuniaxialtensionexamplefor20%probabilityofredesign ...... 52 3-4ProblemdenitionforSSBJExample ........................ 57 3-5UncertainParametersforSSBJExample ..................... 58 3-6Coefcientsforcalculatingthrottleupperbound(Equation3) ........ 59 3-7ResultsforSSBJexamplefor20%probabilityofredesign ............ 62 4-195%condenceintervalforrelativeerrorofsurrogatemodelsbasedonLOOCV 75 4-2Parametersforcantileverbeamexample ...................... 83 5-1Parametersforcantileverbeamexample ...................... 107 5-2Coefcientsforcalculatingspeedofsoundasafunctionofaltitude(5) ... 116 5-3Inputsandoutputsofpropulsiondiscipline ..................... 118 5-4Inputsandoutputsofstructuresdiscipline ..................... 119 5-5Notationusedinweightsestimation ........................ 119 5-6CoefcientsfortankmassWER's .......................... 120 5-7Inputsandoutputsofaerodynamicsdiscipline ................... 122 5-8Inputsandoutputsoftrajectorydiscipline ..................... 124 5-9Datareadfromdesigncurve ............................ 125 7

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LISTOFFIGURES Figure page 3-1ThesafetymarginsareoptimizedbasedonaMCSofthedeterministicdesign/redesignprocess .................................. 42 3-2Flowchartshowingstepsintwo-stagedeterministicdesign/redesignprocess.Safetymarginsn=fnini,nlb,nub,nregareshownasinputsatrelevantsteps. .. 45 3-3Uniaxialtensiontest-Comparisonofexpectedcrosssectionalareaafterpossibleredesignasafunctionofprobabilityofredesignforredesignforperformance(conservativeinitialdesign)versusredesignforsafety(ambitiousinitialdesign). 53 3-4Uniaxialtensiontest-Epistemicuncertaintyincrosssectionalareafor20%probabilityofredesign. ................................ 54 3-5Uniaxialtensiontest-Epistemicuncertaintyinsafetymarginwithrespecttohigh-delitymodelfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. ............................... 54 3-6Uniaxialtensiontest-Epistemicuncertaintyinsafetymarginwithrespecttotruemodelfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. ...................................... 55 3-7Uniaxialtensiontest-Epistemicuncertaintyinreliabilityindexfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. .......... 55 3-8Uniaxialtensiontest-Epistemicuncertaintyinfailurefor20%probabilityofredesign.Theguresareplottedwithdifferentscalestoshowthechangeinthetailofthedistribution.Plotsshowoverlappingtransparenthistograms. ... 56 3-9Uniaxialtensiontest-Redesignforsafetyispreferredwhenhigh-delitymodelerrorislow,butredesignforperformanceispreferredwhenhigh-delitymodelerrorishigh.Plotisforxedprobabilityofredesignof20%. ........... 57 3-10AresponsesurfaceoftheengineperformancemapcalculatesmaximumavailablethrustatagivenMachnumber,M,andaltitude,h.Thethrottlesettingisnormalizedtooneatanaltitudeofapproximately32000ftandMach1.9. .......... 59 3-11SSBJEngine-Comparisonofexpectedengineweightafterpossibleredesignasafunctionofprobabilityofredesignforredesignforperformance(conservativeinitialdesign)versusredesignforsafety(ambitiousinitialdesign). ........ 61 3-12SSBJEngine-Epistemicuncertaintyinthrottlesettingfor20%probabilityofredesign. ....................................... 62 3-13SSBJEngine-Epistemicuncertaintyinsafetymarginwithrespecttohigh-delitymodelfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. ...................................... 63 8

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3-14SSBJEngine-Epistemicuncertaintyinengineweightfor20%probabilityofredesign. ....................................... 63 3-15SSBJEngine-Epistemicuncertaintyinprobabilityoffailurefor20%probabilityofredesign.Theguresareplottedwithdifferentscalestoshowthechangeinthetailofthedistribution.Plotsshowoverlappingtransparenthistograms. . 64 3-16SSBJEngine-Redesignforsafetyispreferredwhenhigh-delitymodelerrorislow,butredesignforperformanceispreferredwhenhigh-delitymodelerrorishigh.Plotisforxedprobabilityofredesignof20%. .............. 65 4-1TheoptimizationofthesafetymarginsisbasedonaMCSofthedeterministicdesignprocess .................................... 74 4-2Thegureontheleftshowsthedesignoptimizationwhenusingasafetymarginnini=0andxedconservativevaluesudetinplaceofaleatoryvariablesU.Thegureontherightshowsthereliabilityoftheoptimumdesignfoundontheleftbyplottingthelimit-statefunctioninstandardnormalspace. ...... 85 4-3Ontheleft,themeanandvarianceoftheerrorareplottedinanormalizeddesignspacewithxedconservativevaluesudetinplaceofaleatoryvariablesU.Ontheright,themeanandvarianceoftheerrorareplottedinstandardnormalaleatoryspaceforoptimumdesignfoundusingnini=0.Theerrorisininches. ....................................... 86 4-4Tradeoffcurvesforexpectedcost(crosssectionalareainsquareinches)asafunctionofprobabilityofredesign.Thecurvelabeled“mixed”correspondstosimultaneousoptimizationofn=fnini,nlb,nub,nreg.Thecurvelabeled“safety”correspondstooptimizingfnini,nlb,nregwithnub=+1.Thecurvelabeled“performance”correspondstooptimizingfnini,nub,nregwithnlb=.Errorbarsarebasedonsurrogatemodelsusedduringoptimization. ......... 88 4-5Histogramsofpossiblesafetymargindistributionsfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. .................. 89 4-6Jointdistributionofdesignvariablesforpossiblenaldesignsfor20%probabilityofredesign.Peakislocatedatinitialdesign. .................... 89 4-7Histogramsofcross-sectionalareadistributionsfor20%probabilityofredesign. 90 4-8Histogramsofreliabilityindexdistributionsfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. .................. 90 4-9Jointdistributionofpossiblemostprobablepoints(MPP's)for20%probabilityofredesign. ...................................... 91 5-1Thebeamissubjecttohorizontalandverticaltiploads .............. 108 9

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5-2Thegureontheleftshowsthedesignoptimizationwithstandarddeviationoffsetk=0andxedconservativevaluesudetinplaceofaleatoryvariables.Thegureontherightshowsthelimit-statefunctioninstandardnormalspacefortheoptimumdesignfoundontheleft.Thereliabilityindexisthedistanceinstandardnormalspacefromtheorigintothelimit-state. ............ 109 5-3Tradeoffcurveforexpectedcrosssectionalareaversusprobabilityofredesign 110 5-4Distributionofsafetymarginandreliabilityindexfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. .................. 112 5-5Distributionofmostprobablepoint(MPP)for20%probabilityofredesign. ... 113 5-6Distributionofoptimumdesignvariablesanddesignperformancefor20%probabilityofredesign.Peakislocatedatinitialdesign. .................... 114 5-7Designstructurematrixforsoundingrocketdesignexample.Therearecouplingsbetweenpropulsion/structures,aerodynamics/structures,andtrajectory/aerodynamics. 115 5-8Speedofsoundasafunctionofaltitude(5) .................. 116 5-9DragcoefcientasafunctionofMachnumberbasedonMissileDATCOM.PCHIPinterpolationisusedbetweendatapoints. ................. 123 5-10Asecondorderpolynomialwasttotheinertmassfractionasafunctionofthelogofthepropellantmass.Themodelisextrapolatedtotheregionofinterestforsoundingrocketdesign. ............................. 125 5-11Acloudof10,000designsin6-dimensionsisprojectedontoaonedimensionalplaneandcomparedtothelow-delitymodelprediction ............. 126 5-12TradeoffcurveforexpectedGLOWversusprobabilityofredesign ........ 127 5-13Distributionsofsafetymarginandprobabilityoffailurefor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. ............ 130 5-14Distributionofoptimumdesignvariablesfor20%probabilityofredesign.Plotsshowmarginaldistributionsof5-dimensionaljointdistribution. .......... 131 5-15DistributionsofGLOWanddrymassfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. ..................... 132 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZINGTHESAFETYMARGINSGOVERNINGADETERMINISTICDESIGNPROCESSWHILECONSIDERINGTHEEFFECTSOFAFUTURETESTANDREDESIGNONEPISTEMICMODELUNCERTAINTYByNathanielB.PriceAugust2016Chair:Nam-HoKimMajor:MechanicalEngineeringAttheinitialdesignstage,engineersoftenrelyonlow-delitymodelsthathavehighuncertainty.Modeluncertaintyisreducibleandisclassiedasepistemicuncertainty;uncertaintyduetovariabilityisirreducibleandclassiedasaleatoryuncertainty.Inadeterministicsafety-margin-baseddesignapproach,uncertaintyisimplicitlycompensatedforbyusingxedconservativevaluesinplaceofaleatoryvariablesandensuringthedesignsatisesasafety-marginwithrespecttodesignconstraints.Afteraninitialdesignisselected,testing(e.g.physicalexperimentorhigh-delitysimulation)isperformedtoreduceepistemicuncertaintyandensurethedesignachievesthetargetedlevelsofsafety.Testingisusedtocalibratelow-delitymodelsandprescriberedesignwhentestsarenotpassed.Aftercalibration,reducedepistemicmodeluncertaintycanbeleveragedthroughredesigntorestoresafetyorimprovedesignperformance;however,redesignmaybeassociatedwithsubstantialcostsordelays.Inthiswork,thepossibleeffectsofafuturetestandredesignareconsideredwhiletheinitialdesignisoptimizedusingonlyalow-delitymodel.Thegoalistodevelopageneralmethodfortheintegratedoptimizationofthedesign,testing,andredesignprocessthatallowsforthetradeoffbetweentheriskoffutureredesignandtheassociatedperformanceandreliabilitybenets.Thisisaccomplishedbyformulatingthedesign,testing,andredesign 11

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processintermsofsafety-marginsandoptimizingthesemarginsbasedonexpectedperformance,expectedprobabilityoffailure,andprobabilityofredesign.Therstobjectiveofthisstudyistodeterminehowthedegreeofconservativenessintheinitialdesignrelatestotheexpecteddesignperformanceafteratestandpossibleredesign.Thesecondobjectiveistodevelopageneralmethodformodelingepistemicmodeluncertaintyandcalibrationwhensimulatingapossiblefuturetestandredesign.Thethirdobjectiveistoapplythemethodofsimulatingafuturetestandredesigntoasoundingrocketdesignexample. 12

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CHAPTER1INTRODUCTION 1.1MotivationAccordingtoBoxandDraper[ 1 ],”Essentially,allmodelsarewrong,butsomeareuseful.”Attheinitialdesignstage,engineersoftenrelyonlow-delitymodelsthathavehighuncertainty.Modeluncertaintyisreducibleandisclassiedasepistemicuncertainty;uncertaintyduetovariabilityisirreducibleandclassiedasaleatoryuncertainty.Inadeterministicsafety-margin-baseddesignapproach,uncertaintyisimplicitlycompensatedforbyusingxedconservativevaluesinplaceofaleatoryvariablesandensuringthedesignsatisesasafety-marginwithrespecttodesignconstraints.Afteraninitialdesignisselected,testing(e.g.physicalexperimentorhigh-delitysimulation)isperformedtoreduceepistemicuncertaintyandensurethedesignachievesthetargetedlevelsofsafety.Testingisusedtocalibratelow-delitymodelsandprescriberedesignwhentestsarenotpassed.Aftercalibration,reducedepistemicmodeluncertaintycanbeleveragedthroughredesigntorestoresafetyorimprovedesignperformance;however,redesignmaybeassociatedwithsubstantialcostsordelays.Inthiswork,thepossibleeffectsofafuturetestandredesignareconsideredwhiletheinitialdesignisoptimizedusingonlyalow-delitymodel.Thegoalistodevelopageneralmethodfortheintegratedoptimizationofthedesign,testing,andredesignprocessthatallowsforthetradeoffbetweentheriskoffutureredesignandtheassociatedperformanceandreliabilitybenets.Thisisaccomplishedbyformulatingthedesign,testing,andredesignprocessintermsofsafety-marginsandoptimizingthesemarginsbasedonexpectedperformance,expectedprobabilityoffailure,andprobabilityofredesign.Inthisresearch,asafety-margin-baseddesignapproachisappliedwhileconsideringepistemicmodeluncertaintyandaleatoryparameteruncertainty.Inorderforthemethodtobeapplicableundercurrentsafety-margin-baseddesignregulations[ 2 ],theoptimum 13

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. . DesignOptimization . Test . Redesign? . FinalDesign . Calibration . DesignOptimization . no . yes Figure1-1. Aninitialdesignistested.Ifthetestisnotpassed,acalibrationandredesignprocessistriggered. designisfoundusingadeterministicsafety-margin-basedapproach.Thesafetyfactorsareoptimizedbasedonprobabilisticcriteria.Traditionally,safetyfactorshavebeenselectedbasedoncombinationofregulationsandpreviousexperience,however,simpleprobabilisticguidelinesforselectingsafetyfactorshavebeenproposed[ 3 ].Safety-margin-baseddesign,testing,andredesignprocessesareentrenchedintheaircraftindustrywherethesepracticeshaveevolvedovermorethan50years,oftenbytrialanderror[ 4 ].Morerecently,studieshaveshowntheparallelsbetweensafety-margin-baseddesignandreliability-baseddesignoptimization(RBDO)approacheswhiledevelopingmethodstoreducethecomputationalcostofRBDO[ 5 – 7 ].However,thesestudieshavenotconsideredepistemicmodeluncertainty.Ontheotherhand,whenthereisonlyepistemicmodeluncertaintyasafetymarginbalancestheneedforthenaldesigntobefeasiblewhileatthesametimenotbeingsoconservativethatdesignperformancesuffers[ 8 ].Fewstudieshaveconsideredtheeffectsofbothaleatoryparameteruncertaintyandepistemicmodeluncertainty.MahadevanandRebbahaveshownthatfailingtoaccountforepistemicmodeluncertaintymayleadtoanoverestimationofreliabilityandunsafedesignsorunderestimationofthereliabilityanddesignsthatareheavierthanneeded[ 9 ].StudiesthatusesurrogatemodelsinRBDO 14

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alsoencounterasituationofmixeduncertainty.However,unlikethisstudywhereweareinterestedinepistemicmodeluncertaintyasainherentpartofthelow-delitymodel,thesestudiesareusuallymotivatedbyadesiretoreducecomputationalcost.KimandChoihaveshownthatwhenusingresponsesurfacesinRBDOtheepistemicmodeluncertaintyresultsinuncertaintyinthereliabilityindexandadditionalsamplingcanbeusedtoavoidbeingoverlyconservative[ 10 ].Oneoftheimportantaspectsofthisresearchistheintegrationofthedesignandtestingprocess.Inthisresearch,theeffectsofafuturetestandpossibleredesignareconsideredwhileoptimizingtheinitialdesign.Sincethetestwillbeperformedinthefuture,thetestresultisanepistemicrandomvariable.Predictingpossibletestresultsrequiresaprobabilisticformulationoftherelationshipbetweenthelow-delitymodelprediction,thetruevalue,andthetestresult.Inthecontextofcalibratingcomputermodels,KennedyandO'Haganproposedthatthetrueprocesscanberelatedtoacomputermodelbymultiplyingbyanuncertainconstantscaleparameterandaddinganuncertaindiscrepancyfunction[ 11 ].Similarformulationshavesubsequentlybeenappliedinmanyotherstudies[ 12 – 17 ].Theseformulationsaresimilarinthattheyallrelatethetrueprocesstothelow-delitymodelbyaddinganuncertaindiscrepancyfunction.Theformulationsdifferintherepresentationofthescaleparameter.Methodsrangefromomittingthescaleparameter[ 13 , 14 ]toconsideringanuncertainscalingfunction[ 16 ].Asimpleralternativemethodistouseonlyuncertainscalingparameterstoformulatetherelationshipbetweenthetrueprocess,thelow-delitymodel,andthemeasurement.ZioandApostolakisreferredtothisapproachastheadjustmentfactorapproachwheretheuncertainadjustmentfactormayeitherbeadditiveormultiplicative[ 18 ].However,thissimplemodelingmethodisonlyapplicablewhenthereisaconstantmodelscalingorbias.Inadditiontotheintegrationofdesignandtesting,thisstudyalsoseekstointegratearedesignprocess.Redesignreferstochangingthedesignvariablesconditionalon 15

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thetestresult.Calibrationisperformedconditionalonthetestresultpriortoredesign.Sincethefuturetestresultismodeledasanepistemicrandomvariablethedesignvariableafterredesignisalsoconsideredarandomvariable.Itisimportanttonotethatthedesignafterredesignisrandombecauseitisuncertainattheinitialdesignstage,notbecausethereisanyinherentvariability.Villanuevaetal.developedamethodforsimulatingtheeffectsoffuturetestsandredesignwhenthereisaconstantbutunknownmodelbiasinthecalculationandmeasurement[ 19 ].Matsumuraetal.comparedRBDOconsideringfutureredesigntotraditionalRBDO[ 20 ].Villanuevaetal.,2014,showedthataminimummassintegratedthermalprotectionsystemisachievedbystartingwithaconservative(heavier)initialdesignandprimarilyusingredesigntoreducemassifthetestrevealsthedesignisoverlyconservative[ 21 ].Ingeneral,engineeringdesignisaniterativeprocessthatrequiresgatheringnewknowledgeandreningtheinitialdesign.Testingfollowedbypossibleredesignisanessentialpartoftheaircraftdesignindustry[ 22 ].Thesafety-margin-baseddeterministicdesignprocesshasarichhistorythatiswellintegratedintothedesign,testing,calibration,andredesignprocess.Currently,thereisapushtotransitionfromdeterministicdesignmethodstoprobabilisticapproaches[ 4 ].However,mostproposedprobabilisticdesignmethodsneglecttheiterativenatureofdesignandfailtoconsiderepistemicmodeluncertainty.Unlikealeatoryuncertainty,epistemicuncertaintyisreduciblebygainingnewknowledge.Accountingforthepossiblechangesinepistemicmodeluncertaintythatoccurduringthedesignprocessisanimportantpartofaprobabilisticdesignapproach.Consideringepistemicmodeluncertaintyisparticularlyimportantattheinitialdesignstagewhenthetheepistemicmodeluncertaintyisveryhigh.Whenthereishighepistemicmodeluncertainty,itisimportanttoconsidertheeffectsoffutureuncertaintyreductionmeasuressuchastestingandredesignwhileselectingtheinitialdesign. 16

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1.2ObjectivesTherstobjectiveofthisstudyistodeterminehowthedegreeofconservativenessintheinitialdesignrelatestotheexpecteddesignperformanceafteratestandpossibleredesign.Failingacriticalsafetytest(e.g.measuredsafetymargintoolow)typicallytriggersaredesignprocesstorestoresafety.Itisalsoworthwhiletoimplementaredesigntriggerassociatedwithbeingtooconservative(e.g.measuredsafetymarginistoohigh)inordertoredesignwhenitispossibletosignicantlyimprovedesignperformance.Ahighprobabilityofredesignforperformanceorredesignforsafetyshouldbeavoidedduetotheassociatedcostsandprogramdelaysrelatedtoperformingredesign.Toavoidredesignforsafety,designersmayaddmoreconservativenesstotheinitialdesignbyusingahighersafetymarginwhichtypicallyresultsinworseinitialdesignperformance.Conversely,toavoidredesignforperformance,alowersafetymargincanbeusedtoachievebetterinitialdesignperformance.Thisleadstoadilemmainwhethertostartwithamoreconservativeinitialdesignandpossiblyredesignforperformanceortostartwithalessconservativeinitialdesignandriskredesigningtorestoresafety.Thesecondobjectiveistodevelopageneralmethodformodelingepistemicmodeluncertaintyandcalibrationwhensimulatingapossiblefuturetestandredesign.Previousworkonsimulatingafuturetestandredesignhasshownimportantbenetsintermsofselectingtheinitialdesignwhenusingonlyalow-delitymodel[ 21 ].However,thismethodrequiredtheassumptionofaconstantbutunknownerrorinthelow-delitymodelandtestresult.Thisisastrongassumptionthatmaybedifculttosatisfyinthemajorityofengineeringdesignproblems.Inorderforthemethodtobeapplicabletomostengineeringproblemsamoresophisticatedmethodisneededformodeling,updating,andpropagatingtheepistemicmodeluncertainty.Inparticular,itisimportanttoconsiderepistemicmodeluncertaintiesthatarecorrelatedwithrespecttodesignvariableswhenpredictingthereliabilityforadesignthatisconsiderablydifferentfrom 17

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thedesignthatwastested.Inaddition,correlationswithrespecttoaleatoryvariablesbecomeimportantwhenpredictingthesafetyofadesignatconditionsthataredifferentfromthetestconditions.Thethirdobjectiveistoapplythemethodofsimulatingafuturetestandpossibleredesigntothedesignofasoundingrocketundermixedepistemicmodeluncertaintyandaleatoryparameteruncertainty. 1.3OutlineThisdissertationisorganizedintosixchapters.Themotivation,objectives,andoutlinearediscussedinChapter 1 .Chapter 2 providesaliteraturereviewofuncertaintyclassication,multi-delitymodeling,anddesignunderuncertainty.Chapter 3 discussestheresearchtodeterminehowthedegreeofconservativenessintheinitialdesignrelatestotheexpecteddesignperformanceafteratestandpossibleredesign[ 23 , 24 ].Inparticular,thischapteranalyzesthedilemmaofwhethertostartwithamoreconservativeinitialdesignandpossiblyredesignforperformanceortostartwithalessconservativeinitialdesignandriskredesigningtorestoresafety.Chapter 4 buildsontheworkintherstchaptertodevelopageneralizedmethodforsimulatingafuturetestandpossibleredesignthataccountsforspatialcorrelationsintheepistemicmodelerror[ 25 ].Chapter 5 discussestheapplicationofthemethodofsimulatingafuturetestandpossibleredesigntothedesignofasoundingrocketundermixedepistemicmodeluncertaintyandaleatoryparameteruncertainty[ 26 ].Chapter 6 summarizesconclusionsandprovidesperspectivesforfuturework.Thechaptersarewrittensotheycanbereadseparately,butthereisanaturalprogressioninthemethodandincreasingcomplexityoftheexamplesfromchapters 3 to 5 . 18

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CHAPTER2BACKGROUNDANDLITERATUREREVIEW 2.1UncertaintyClassicationUncertaintyisoftenbroadlyclassiedintotwocategories.Aleatoryuncertaintyisduetonaturalvariabilityandisirreducible.Epistemicuncertaintyisduetolackofknowledgeandisreducible.However,sometimesitcanbechallengingtoclassifyuncertaintyasepistemic,aleatory,oramixtureofboth.Faberarguesthattheclassicationofuncertaintyhasadependenceonmodelingscaleaswellastime[ 27 ].ThequestionofhowmodelingscaleaffectsuncertaintyclassicationhasalsobeenraisedbyO'HaganandOakleyandleadstothequestionofwhetherthereisanytruerandomnessorifalluncertaintymightbeconsideredepistemic[ 28 ].O'HaganandOakleyusethetermresidualvariabilitytodescribethevariationofarealprocesswhenrepeatedunderthesameconditions.Thefundamentalquestioniswhetherthisresidualvariabilityisduetonaturalvariability(aleatoryuncertainty)orifbyspecifyingadditionalconditionsthevariabilitycouldbeeliminatedorreduced(epistemicuncertainty).Inadditiontomodelingscale,Faberalsoidentiedtimedependenceofknowledgeasanimportantfactoraffectinguncertaintyclassication.AccordingtoFaber,theuncertaintyinamodelconcerningthefuturetransformsfromamixtureofaleatoryandepistemicuncertaintytopurelyepistemicwhenthemodeledeventisobserved.KiureghianandDitlevsendescribethistimedependenceinthecontextofassessingthereliabilityofanexistingversusafuturebuilding[ 29 ].KiureghianandDitlevsenarguethatthereisadegreeofsubjectivityinthecategorizationofuncertainties,butitisnonethelessusefultodosoinengineeringdesign.Interestingly,thequestionsregardingthefundamentaldifferencesbetweenaleatoryandepistemicuncertaintymaycontributetotheproliferationofnewmethodsformodelingandpropagatingepistemicuncertainty.Thereisconsiderablediversityinthemethodsformodelingdifferenttypesofuncertainty[ 30 ].Whileprobabilitytheoryiswidelyacceptedastheappropriatechoice 19

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formodelingaleatoryuncertainty,severalalternativemethodshavebeenproposedformodelingepistemicuncertainty.Thesealternativemethodsarepartiallymotivatedbyperceiveddifcultiesassociatedwithtryingtorepresentlackofknowledgeusingclassicalprobabilitytheoryinengineeringdesign.FersonandGinzburgdescribetheinadequaciesofprobabilitytheorywhentryingtorepresentaconstantbutunknownvaluethatlieswithinagiveninterval[ 31 ].FersonandGinzburgconcludethatintervaltheoryorprobabilityboundsanalysisisbettersuitedformodelingepistemicuncertainty.OtheralternativemethodsforrepresentingepistemicuncertaintyincludeDempster-Shaferstructuresandpossibilitytheory[ 32 , 33 ].However,accordingtoO'HaganandOakley,”...toclaimthattheonlyinformationavailableaboutaparameteristhatitliesinsomeintervalistodenythepossibilityofelicitingexpertinformationeffectively”[ 28 ].Theproperelicitationofexpertopinionisanimportanttopicwhentryingtorepresentexpertopinionusingprobabilitytheory.KadaneandWolfsonofferanoverviewofgeneralandapplicationsspecicelicitationmethodsforconstructingpriordistributionsbasedonexpertopinion[ 34 ].Moreover,aspointedoutbyO'HaganandOakley,somealternativemethodsthatmayworkwellforparameteruncertaintymightnotbeeasilyapplicabletorepresentothersourcesofuncertaintysuchasmodelinadequacy.Inadditiontoclassifyinguncertaintyasaleatoryorepistemic,itisusefultoidentifydifferentuncertaintysources.Oneclassicationofuncertaintyincomputercodes,providedbyKennedyandO'Hagan[ 11 ]andsimpliedbyO'HaganandOakley[ 28 ],istoclassifyuncertaintyasparameteruncertainty,modelinadequacy,residualvariability,andcodeuncertainty.Parameteruncertaintyisuncertaintyaboutmodelinputs.Modelinadequacyreferstothediscrepancybetweenthemodelandthetrueprocess.Residualvariabilityisthevariationoftherealprocessunderthesameconditions.Codeuncertaintymayrefertoevaluatingthecodeatpreviouslyuntriedinputs.Modelinadequacyisofparticularinterestinthisstudy,however,NilsenandAvenhavearguedthatfocusonmodeluncertaintyleadstomuddlingofriskanalysis 20

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[ 35 ].AnalternativeclassicationbyOberkampfetal.issimplytoclassifyuncertaintiesasaleatoryuncertainty,epistemicuncertainty,anderror[ 36 , 37 ].Errorisdenedas”arecognizableinaccuracyinanyphaseoractivityofmodelingandsimulationthatisnotduetolackofknowledge”andmayfurtherbesubdividedintoacknowledgedorunacknowledgederrors.Identifyingdifferentsourcesandtypesofuncertaintyisimportantbecausedifferentmethodsofmodelingandpropagatinguncertaintymaybebettersuitedfordifferenttypesofuncertainty. 2.2Multi-delityModeling 2.2.1Sensitivity-BasedScalingMethodsModelapproximationscanbedividedintotwoclasses[ 38 ].Therearelocalderivative-basedapproximations(e.g.Taylor-seriesexpansions)andglobalapproximations.Consideralow-delitymodelofasinglevariablegL(x)thatisaglobalapproximationofahigh-delitymodelgH(x).Onemethodofrelatingthehigh-delitymodelgH(x)tothelow-delitymodelgL(x)istoconsiderascalingfactor()atadesignpointx0 (x0)=gH(x0) gL(x0)(2)Anapproximationofthehigh-delitymodel^gH(x)canbeobtainedusingaconstantscalingas ^gH(x)=(x0)gL(x)(2)whereahataccentisusedtodenoteaprediction^gH()thatmaybedifferentthanthetruehighdelitymodelgH().Obviously,theaccuracyoftheapproximationwilldeteriorateatpointsthatarefarawayfromx0.Animprovedapproximationistousealinearlyvaryingscalingfunction ^(x)=(x0)+(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)0(x0)(2)whereprimedenotesthederivativewithrespecttox.Notethatthelinearapproximationofthescalingfunction^()maybeconsiderablydifferentfromthetruescaling().The 21

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linearscalingfactorcanbeformulatedas ^(x)=(x0)1+(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0)g0H(x0) gH(x0))]TJ /F5 11.955 Tf 13.15 8.09 Td[(g0L(x0) gL(x0)(2)Haftkareferstothismethodastheglobal-localapproximation(GLA)methodbecauseitcombinestheglobalapproximationgL(x)withthelocalinformationcontainedinthescalingfactor^(x0)[ 38 ].Themethodiseasilyapplicabletoanynumberofvariablesbyusingarst-orderTaylorseriesexpansion.Changetal.comparedtheGLAmethodtoaconstantscalingmethodwhenmodelingawing-boxstructure[ 39 ].Inadditiontomultiplicativescaling,itisalsopossibletouseanadditivescaling (x)=gH(x0))]TJ /F5 11.955 Tf 11.96 0 Td[(gL(x0)(2)andtoconsiderasecond-orderapproximation[ 40 ].Ganoetal.proposedanadaptivehybridscalingwheretheapproximationisbasedonaweightedaverageofamultiplicativeandadditivescalingmodel[ 40 ].Ganoetal.appliedthistypeofsensitivity-basedscalingtodevelopavariabledelityreliability-baseddesignoptimization(VF-RBDO)method[ 41 ].Oneofthedrawbackstosensitivity-basedscalingmethodsisthatnoiseinthehigh-delitymodelcanresultininaccuratederivativecalculationsandpoorapproximations[ 42 ].Similaradditiveandmultiplicativescalingmethodshavealsobeenproposedforrepresentingmodeluncertainty.ZioandApostolakisdescribeanadjustmentfactorapproach ^GH(x)=gL(x)^E(2)where^GH()isarandomvariablerepresentingthepossiblehighdelitymodeland^Eisarandomvariablerepresentingpossiblemodelbias[ 18 ].ZioandApostolakisalsodiscussusinganadjustmentfactorthatisadditiveinsteadofmultiplicative.Thismethodofrepresentingthemodelerrorhasbeenappliedinseveralstudies[ 19 – 21 , 23 , 43 , 44 ].Letthesuperscript(i)denotearealizationofthemodel^GH()=^g(i)H()corresponding 22

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totheerrorrealization^E=^e(i).Themainassumptionofthemethodisthatthereexistsanerrorrealization,9^e(i)2^E,suchthatthescaledmodelcorrespondstothetruemodel,^g(i)H(x)=gH(x).Obviously,thisisonlytrueiftherelationshipbetweenthehighandlow-delitymodelscanberepresentedbysomeconstantscaling.Therefore,theadjustmentfactormethodofrepresentingmodeluncertaintyfromZioandApostolakiscorrespondstoaconstantapproximationofthemodelscaling.Duetotheassumptionofconstantmodelbias,onlyasingleevaluationofthehigh-delitymodelisneededtoremoveallthemodeluncertainty.Animprovedmethodistoconsideranuncertainscalingfunction(multiplicativeoradditive)thatdependsonthelocationx. 2.2.2GaussianProcess(GP)ModelBasedMethodsKeaneproposedamulti-delityoptimizationformulationbasedoncreatingaKrigingsurrogateforthedifferencebetweenahighandalow-delitymodel[ 45 ].Thismethodwasshowntoworkbetterthansimplybuildingasurrogateforthehighdelitymodelalonewhenappliedtotheoptimizationofawingdesign.Ganoetal.alsousedaKriging-basedscalingfunctionandappliedthemethodtothedesignoptimizationofasupercriticalhigh-liftairfoil[ 40 ].Ganoetal.notedthatthescalingcanbeeitheradditiveormultiplicative.Anapproximationofahigh-delitymodelusingamultiplicativescalingis ^gH(x)=^(x)gL(x)(2)where^(x)GP(m(x),k(x,x0))isaGaussianprocess(GP)modelwithmeanfunctionm(x)andcovariancefunctionk(x,x0).NotethattheGPmodelisconstructedforthescalingfunction(x)andnotthehigh-delitymodelgH(x).UsingtheKrigingapproximationforthescaling,ratherthanthehigh-delitymodel,mayprovideabetterapproximationwhenthelow-delitymodelincludesphysicsofthemodeledprocess[ 40 , 45 ].However,whenitisnotverycheaptoevaluatethelow-delitymodelitmaybebettertobuildtheapproximationusingonlylimitedevaluationsofthelowandhigh-delitymodels.Inthiscase,co-krigingcanbeusedasdemonstratedbyForrester 23

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etal.onamulti-delitywingoptimizationproblem[ 12 ].Co-krigingallowsforthedirectapproximationofgH(x)whileaccountingforthehigheruncertaintyintheobservationsfromgL(x).Scalingmethodsmayhaveadvantagesovermulti-delitysurrogatessuchasco-krigingbecausetheyincorporatemoreofthephysicsfromthelow-delitymodel,buttheymaybemoreexpensivewhenthecomputationalcostofthelow-delitymodelisnotinsignicant.Acompromisebetweenthetwomethodsistorstbuildasurrogateforthelow-delitymodelandthenbuildanothersurrogateforthescalingontopofthismodel.Qianetal.demonstratedthisapproachonanelectronicscoolingapplicationinvolvingcellularmaterials[ 15 ].AvariationofthemultiplicativeoradditiveKrigingmethodistoconsiderseparatetermsfor“scale”and“location”change.Forexample,therelationshipbetweenthehighandlow-delitymodelscanbeformulatedas gH(x)=(x)gL(x)+(x)(2)where()isafunctionforscalechangeand()isafunctionforlocationchange.Thisformulationissomewhatsimilartotheproposedhybridsensitivity-basedscalingmethodofGanoetal.[ 40 ]inthatitincludesbothadditiveandmultiplicativeterms.Ganoetal.foundeitheradditivescalingormultiplicativescalingmayworkbetterdependingontheproblem,butbyincludingbothtypesofscalinginasinglemodelitalleviatedtheneedtomakethisdecisionapriori.IncludingbothscaleandlocationfunctionsintheGPmodelmayhaveasimilareffect.KennedyandO'Haganconsideredaconstant(butuncertain)scalingtermandaGPmodelofthelocationfunction()inaBayesianframeworkforthecalibrationofcomputercodes[ 11 ].KennedyandO'HaganalsousedaconstantscalingandGPlocationfunctionintheformulationforpredictingatoplevelcodewhenoneormorelowerlevelcodesareavailable[ 46 ].Bayarrietal.omittedthescalefunctionfromtheproposedBayesianframeworkforthevalidationofcomputermodels[ 47 ].Inotherstudiesthescalefunctionhasbeenbasedonlinearregression[ 15 , 17 ].Qianand 24

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WuusedGPmodelsforboththescalefunctionandthelocationfunction[ 16 ].Insomesense,includingbothascalefunction()andalocationfunctionalocationfunction()issimilartoaddingadditionaltermsinaregressionmodelandmayallowforabettertofthetruerelationship.However,evenwithaconstantscaletermtheremaybeissueswithindentiabilitybecausemanydifferentmodelparameterscouldresultinthesameobservations[ 48 ].UsingaGPmodeltorelateahighandlow-delitymodelnotonlyallowsforthemodelingofcomplexrelationshipsbetweenmodels,butalsoprovidesanestimateofthemodeluncertainty.TheuncertaintyintheGPmodelagreeswithourintuitioninthatthevarianceofthemodeluncertaintyreachesaminimumatobservationsandincreasestoamaximumvalue2asthedistancefromtheobservationsincreases.TheGPmodelframeworkcanalsobeextendedtoincludesomenoiseinthehigh-delitymodel(e.g.measurementerror) gT(xi)=gH(xi)+i=(xi)gL(xi)+(xi)+i(2)wherethesubscriptiisusedbecausethemeasurementerroriatanylocationxiisindependentidenticallydistributed(i.i.d)Gaussiannoise.Huangetal.usedtheuncertaintyestimatefromtheGPmodeltodevelopasequentialsamplingalgorithmformulti-delityoptimizationbasedonanaugmentedexpectedimprovementfunctionthataccountedforthedifferenceincomputationalcostbetweenmodelevaluation[ 14 ].Xiong,Chen,andTsoiusedtheuncertaintyestimatefromtheGPmodeltodevelopanobjectiveorientedsequentialsamplingalgorithmformulti-delityoptimization[ 17 ].Chenetal.developedadesigncondencemetricbasedontheprobabilitythatanalternativedesignisbetterthanthecurrentoptimumdesign[ 13 ]. 25

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2.3DesignUnderUncertainty 2.3.1DeterministicSafety-FactorBasedDesignAbasicformulationofadeterministicsafety-factorbaseddesignoptimizationproblemwithasingleconstraintis minxf(x)s.t.g(x,udet)>0(2)wherex2Rdisavectorofdeterministicdesignvariables,udet2Rpisavectorofconservativedeterministicvaluesusedinplaceofaleatoryrandomvariables,f()isaknownobjectivefunction,andg(,)isaknownconstraintfunction.Asafetyfactormaybeincorporatedintothespecicationoftheconservativedeterministicvaluesudet.Forexample,FederalAviationRegulations(FAR)foraircraftdesignrequireasafetyfactorof1.5appliedtotheprescribedlimitloads(themaximumloadstobeexpectedinservice)[ 2 , 49 ].ForarandomloadPwecandeneaconservativedeterministicvalue pdet=nmax(P)(2)wheren=1.5isasafetyfactor.Similarly,FederalAviationRegulations25.613requireallowablefailurestressesforcriticalmembersthatarebelow99%(or90%forredundantmembers)ofthetestfailurestresswith95%condence[ 50 ].NeglectingtheuncertaintyinthedistributionofSduetothelimitedtestingsamplesize(i.e.100%condence),wecanspecifyaconservativedeterministicvalue sdet=F)]TJ /F9 7.97 Tf 6.58 0 Td[(1S(1)]TJ /F8 11.955 Tf 11.95 0 Td[()(2)whereF)]TJ /F9 7.97 Tf 6.59 0 Td[(1S()istheinversecumulativedistributionfunction(inversecdf)ofSand=0.99istheprobabilitythattheactualfailurestressislessthantheconservativevalue. 26

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Engineersoftencompensateforuncertaintybyusingconservativenesssuchasconservativematerialproperties,conservativelimitloads,safetymargins,andsafetyfactors.VariationinmaterialpropertiesisaddressedbyFederalAviationRegulations25.613whichrequiresallowablefailurestressesforcriticalmembersthatarebelow99%(or90%forredundantmembers)ofthetestfailurestresswith95%condence[ 50 ].FederalAviationRegulations(FAR)foraircraftdesignrequireasafetyfactorof1.5appliedtotheprescribedlimitloads(themaximumloadstobeexpectedinservice)[ 49 ][ 2 ].Theuseofafactorofsafetyof1.5iswidelyacceptedintheaircraftindustryandthevaluecanbetracedbacktothe1920'sand1930'swhenitwasconsideredrepresentativeoftheratiosofdesigntooperatingmaneuverloadfactors[ 51 ].Afactorofsafetyof1.4isoftenusedinspacecraftdesign[ 52 ].Theultimatefactorofsafetyisintendedtocover[ 52 ]: 1. “Inadvertentin-serviceloadsgreaterthanthedesignlimitload.” 2. “Structuraldeectionsabovelimitloadthatcouldcompromisevehiclestructuralintegrity.” 3. “As-builtpartthicknesswithintolerance,butlessthanthatassumedinthestressanalysis.”AccordingtoZipay,Modlin,andLarsen,theultimatefactorofsafetyisnotintendedtocover[ 52 ]: 1. “...errorsinthestructuralanalysisorstructuralmathmodeling” 2. “...poordesignpractice” 3. “...statisticalmaterialpropertyvariations” 4. “...processescapes”Furthermore,Zipayetal.state“itisclearthatnoportionofthefactorofsafetycanbeusedtocorrectforthenecessaryidealizationsandpotentialerrorsthatcanoccurinusingthesetools[sophisticatedcomputationmodeling]toanalyzeacomplexstructure.” 27

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Designersandengineersmayaddadditionalconservativeness,outsideofthatspeciedbyregulations,toaccountforadditionaluncertainties.Ullmandescribesthefollowingclassicalrule-of-thumbmethodforestimatingthefactorofsafety FS=FSmaterialFSstressFSgeometryFSfailureanalysisFSreliability(2)wheretherearecontributionsfromuncertaintyinmaterialproperties,uncertaintyinload,uncertaintyinmanufacturingtolerances,uncertaintyinfailuretheory,andafactorrelatedtothedesiredlevelofreliability[ 3 , 53 ].Ullmanalsoproposessomesimplestepsforestimatingafactorofsafetybasedoncoefcientsofvariation.Theselectionofasafetyfactorhasimportantimplicationsintermsofstructuralweight.Ithasbeenestimatedthatreducingthefactorofsafetyfrom1.5to1.4mayreduceaircraftstructuralweightby4%andthatreducingthefactorofsafetyfrom1.5to1.25mayreduceweight10.5%[ 52 ].Thisisasignicantreductionconsidering,forexample,thatweightscrubactivitiesforApollowerebudgetedapproximately$10,000perpoundandforShuttle$50,000perpound[ 52 ].Inordertoachievehighlevelsofreliabilitywithoutsacricingperformance,safety-factorbaseddeterministicdesigniscoupledwithavarietyofuncertaintyreductionmeasures.Oneofthemostimportantwaysofreducinguncertaintyinaircraftdesignisthroughbuilding-blocktestingwheretestsofincreasingcomplexityareperformedstartingwithcoupontestsonmaterialstodeterminepropertiesandculminatingincomponentandfull-scalevalidationtesting[ 22 ].Safetyfactorbaseddesign,testing,andredesignprocessesareentrenchedintheaircraftindustrywherethesepracticeshaveevolvedovermorethan50years,oftenbytrialanderror[ 4 ].Moremodernprobabilisticdesignapproachesofferthepromiseofreducingcostandimprovingperformance,however,itislikelythetransitionwillbedifcultduetotherichhistoryofdeterministicbaseddesignapproaches. 28

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2.3.2Reliability-BasedDesignOptimization(RBDO)Abasicformulationofareliability-baseddesignoptimization(RBDO)problemwithasingleconstraintis minxf(x)s.t.PU[g(x,U)<0]pf(2)wherex2Rdisavectorofdeterministicdesignvariables,U2Rpisavectorofaleatoryrandomvariables,f()isaknownobjectivefunction,g(,)isaknownconstraintfunction,pfisthetargetprobabilityoffailure,andPU[]isaprobabilityoperatorwithrespecttoaleatoryuncertaintyU.Thereliabilityconstraintfrom 2 canbewrittenas FG(0)()]TJ /F4 11.955 Tf 10.77 2.66 Td[()(2)whereFG()isthecumulativedistributionfunction(cdf)forg(x,U),()isthecdfforthestandardnormaldistribution,and=)]TJ /F4 11.955 Tf 9.29 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1(pf)isthetargetreliabilityindex[ 54 ].ThecdfFG()isdenedas FG(z)=PU[g(x,U)
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2.3.2.1Reliabilityindexapproach(RIA)Usingthereliabilityindexapproach,theRBDOformulationis minxf(x)s.t.()]TJ /F8 11.955 Tf 9.3 0 Td[((x))pf(2)where()isthecumulativedistributionfunction(cdf)forthestandardnormaldistributionand()isthereliabilityindex.Thereliabilityindexcanbecalculatedthroughrst-orderreliabilitymethod(FORM)bysolvinganoptimizationproblemforthemostprobablepoint(MPP) minu)]TJ /F4 11.955 Tf 5.7 -9.69 Td[(uTu1=2s.t.g(x,u)=0(2)whereuisthevectorUtransformedtostandardnormalspace.Thesolutionto 2 istheMPPuMPPandthereliabilityindexisdenedas (x)=)]TJ /F4 11.955 Tf 5.7 -9.68 Td[(uTMPPuMPP1=2(2)ThereliabilityindexapproachwithFORMisconsideredadoubleloopRBDOstrategybecausethedesignvariablesaremanipulatedintheouterloopandthereliabilityanalysisisperformedintheinnerloop.ThisformulationisanestedoptimizationproblembecausetheFORMoptimizationproblemissolvedforeveryconstraintevaluation. 2.3.2.2Performancemeasureapproach(PMA)Intheperformancemeasureapproach,theRBDOformulationis[ 7 , 54 ] minxf(x)s.t.g(x,uinvMPP)0(2)TheinverseMPPuinvMPPisfoundthroughinverseFORM[ 54 ]bysolvinganoptimizationproblem minug(x,u)s.t.)]TJ /F4 11.955 Tf 5.69 -9.68 Td[(uTu1=2=)]TJ /F4 11.955 Tf 9.3 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(1(pf)(2) 30

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Thesolutionto 2 istheinverseMPPuinvMPP.NotethattheinverseMPPuinvMPPisequaltotheMPPuMPPonlyifthetargetreliabilityindexisequaltothereliabilityindexofthedesign(x).Thisformulationisalsoreferredtoasthepercentileformulation[ 7 ]becauseg(x,uinvMPP)isthepercentileofg(x,U)suchthat PUg(x,U)
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2.3.3RBDOwithEpistemicModelUncertaintyThemajorityofRBDOmethodsonlyconsideraleatoryparameteruncertainty.Epistemicmodeluncertaintyisofparticularinterestinthepresentresearch.Specializedmethodsarerequiredforhandlingepistemicmodeluncertaintynotonlybecauseitisadifferenttypeofuncertainty(epistemicvs.aleatory)butalsobecauseitarisesfromadifferentsource(modeluncertaintyvs.inputparameteruncertainty).OneofthefewstudiesthatsoughttoincludemodelerrorinRBDOwasbyMahadevanandRebba[ 9 ].Thisstudysimplymodeledtheepistemicmodelerrorasarandomvariable(seeSection 2.2 formethodsofmodelingepistemicmodelerror),however,thendingsdidshowimportantconsequencesofepistemicmodelerror.Inoneexample,notconsideringepistemicmodelerrorresultedinadesignthatwasheavierthanrequiredbecausethemodeloverestimatedtheprobabilityoffailure.Inanotherexample,notconsideringepistemicmodelerrorresultedinadesignthatwaslighterbutdidnotmeetthereliabilityconstraintbecausethemodelunderestimatedtheprobabilityoffailure.TherearetwoareasforimprovementinthemethoddescribedbyMahadevanandRebba.First,itmaybeimportanttomodeltheepistemicmodelerrorasvaryingwiththelocationinthedesignspaceasdiscussedinSection 2.2 .Second,itisimportanttomakeadistinctionbetweenepistemicandaleatoryvariablesinthereliabilityassessment.Ifthereisepistemicmodelerrorthenthetrueprobabilityoffailureisunknownandinsteadweshouldcalculateadistributionofpossibleprobabilitiesoffailure.Thedistributionofprobabilitiesoffailurerepresentstheuncertaintyintheprobabilityoffailureofthetruesystem.Inordertoreducethecomputationalcostofrepeatedmodelevaluationsduringuncertaintypropagation,surrogatemodelshavebeenproposedascheapapproximations.StudiesthatusesurrogatemodelsinRBDOalsoencounterasituationofmixeduncertainty.Theinterestingaspectofthesemethodsinthepresentresearchishowthesemethodshandletheepistemicmodelerrorthatisintroducedbythesurrogate 32

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approximations.InSection 2.4 ,theideathatepistemicmodelerrorresultsinadistributionofpossibleoptimumsisintroduced.Thereisalsoadistributionofpossi-bleRBDOoptimumswhenperformingRBDOwithepistemicmodelerror.Inotherwords,sincethetrueprobabilityoffailureisunknownitispossiblethatmanydifferentdesignswouldsatisfythereliabilityconstraint.However,mostsurrogatebasedapproachestoRBDOdonotconsiderthisuncertaintyintheoptimumdesignandinsteadfocusonboundingandreducingtheuncertaintyinprobabilityoffailure.OneEGOinspiredalgorithm(seeEGOinSection 2.4 )forRBDOisEfcientGlobalReliabilityAnalysis(EGRA)[ 55 ].TheEGRAmethoddenesaninllsamplingcriteriacalledexpectedfeasibilitybasedonintegratingoveraregionintheimmediatevicinityoftheconstraintboundary.TheEGRAmethodsamplesthelocationofthemaximumexpectedfeasibility,updatesthesurrogatemodel,andrepeatsthisprocessuntilthemaximumexpectedfeasibilityissmall.Onlyafterthesurrogatemodelissufcientlyrenedisthesurrogateusedtocalculatetheprobabilityoffailure.Thus,EGRAavoidsconsideringepistemicmodelerrorinthereliabilityassessmentbyperformingmanyevaluationsneartheconstraintboundarytoreducetheepistemicmodelerrortoanegligiblelevel.AnothermethodbasedonsurrogatemodelsistheresponsesurfacemethodproposedbyKimandChoi[ 10 ].Inthismethodthepredictionintervalfortheresponsesurface(i.e.epistemicmodelerror)isusedtondupperandlowerboundsonthereliabilityindex.Thepredictionintervalfortheresponsesurfaceprovidesanintervalofapossiblefutureobservationforagivencondencelevelspeciedbytheengineer.Aconservativeoptimumdesignisfoundbasedonthelowerboundofthereliabilityindex.However,toavoidbeingoverlyconservativeadditionalsamplingistriggerediftheupperboundonthereliabilityindexistoohigh.AnothersurrogatebasedapproachtoRBDOwasproposedbyDubourg[ 56 ].InthismethodprobabilityoffailureboundsareestimatedbycalculatingtheprobabilityoffailurewithrespecttoaconservativeandanunconservativeKrigingprediction. 33

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2.3.4UncertaintyReductionMeasuresUncertaintyreductionmeasures,suchastestingandqualitycontrol,canbeusedtoreducetheuncertaintyintheperformanceandreliabilityofanaldesign.However,thisuncertaintyreductionisoftennotquantied.Acaretal.foundthatthatcerticationtestsinaircraftdesignreducemodelingerrorandresultinamuchlowercalculatedprobabilityoffailurethanusingsafetyfactorsalone[ 57 ].Acar,Haftka,andJohnsonshowedthatqualitycontroltotruncatethetailonthedistributionofmaterialpropertiescanbeveryeffectivewhenthealowprobabilityoffailureisrequired[ 58 ]. 2.4GlobalOptimizationAccordingtoBoxandDraper[ 1 ],”Essentially,allmodelsarewrong,butsomeareuseful.”Ingeneral,allmodelsareapproximationsofthetrueprocessandthereforecontainsomeepistemicmodeluncertainty.Inengineeringdesign,manymethodsconsideronlyuncertaintyinmodelinputs(i.e.parameteruncertainty)andnotmodeluncertainty.Incontrast,manyglobaloptimizationalgorithmsfocusexclusivelyonepistemicmodeluncertainty.Somepopularglobaloptimizationalgorithmsreducecomputationalcostbyintroducingcheapsurrogateapproximations,however,thesurrogatemodelsalsointroducesignicantepistemicmodeluncertainty.Surrogatebasedglobaloptimizationmethodsareofinterestinthepresentdiscussionbecausethemethodsacknowledgetheeffectsofhighepistemicmodelerroronoptimizationandexploremethodsfordealingwiththisuncertainty.Theintroductionofepistemicmodeluncertaintymeansthesealgorithmsmustbalancetheneedtoexploreregionsofhighuncertaintyandexploitregionswherethesurrogatepredictshighperformance.Sinceallmodelshavesomedegreeofepistemicuncertainty,thesemethodscanbeinterpretedasrepresentativeofatypicaldesignproblemwherewewishtooptimizethetrueprocessbutinsteadsettleforoptimizingacomputermodel.Insurrogatebasedglobaloptimization,atypicalapproachistotasurrogatebasedonaninitialdesignofexperiment(DoE)andthensequentiallyaddnewpointseach 34

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iterationtoreducetheepistemicmodelerrorinregionsofinterest.Thesenewsamplepointsarecalledtheinllsamplesandmanyinllsamplingcriteriahavebeenproposed.WatsonandBarnesproposedthreedifferentinllsamplingcriteriatheydescribedaslocatingthreshold-boundedextremes,locatingregionalextremes,andminimizingsurprises[ 59 ].AnimportantaspectofthisworkbyWatsonandBarneswastranslatingengineeringobjectivesintoappropriatesamplingcriteria.Therefore,theselectionofthe“best”inllsamplingcriteriadependsontheengineeringobjective.AverypopularmethodofglobaloptimizationknownasEfcientGlobalOptimization(EGO)wasproposedbyJonesetal.[ 60 ].InEGOtheinllsamplingcriteriaisthemaximizationoftheexpectedimprovement.Theexpectedimprovementisdenedbyweightingallthepossibleimprovementsbytheassociatedprobabilitydensity.TheEGOmethodwasdevelopedforunconstrainedoptimization,butSchonlaudescribesasimplemethodofadaptingthemethodtoconstrainedoptimizationproblems[ 61 ].Schonlauproposesmultiplyingtheexpectedimprovementbytheprobabilitythateachconstraintismet(i.e.probabilityoffeasibility).AnalternativemethodofhandlingconstraintswithEGOistoaddapenaltytototheinllsamplingcriteriaintheinfeasibleregion[ 62 ].Athirdalternativeistosolvetheinllsamplingcriteriaproblemasaconstrainedoptimizationproblem[ 63 ].Parretal.comparestheperformanceofdifferentinllsamplingcriteriawithconstrainthandlingonanalyticalexamplesandasappliedtoawingdesignproblem[ 64 ].TheInformationalApproachtoGlobalOptimization(IAGO)offersadifferentperspectiveoninllsampling[ 65 ].Mostinllsamplingcriterialookforlikelylocationsoftheoptimumxandthensampleatthislocation.Incontrast,theIAGOmethodseekstochooseanewsamplebasedongainingthemostinformationaboutthelikelylocationofx.ThemethodreliesonestimatingtheprobabilitydensityofthethepossibleoptimumsX.Notethattheepistemicmodelerrorintheobjectivefunctionresultsinadistributionofpossibleglobaloptimums.Inparticular,IAGOestimatesthedistribution 35

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ofpossibleoptimumdesignsconditionalonsamplingatanewlocationxc.Firstitisnecessarytosimulateapossiblerealizationofthetruefunctionatthislocationandthenconditionalsimulationsareperformedtoestimatethedistributionofpossibleoptimumsconditionalonthatrealization.Theprocessisrepeatedformanydifferentpossiblerealizationsofthefunctionatlocationxc.AlthoughIAGOhasonlybeenappliedtounconstrainedglobaloptimization,wemightreasonthatthereisalsoadistributionofpossibleoptimumsforconstrainedoptimizationproblemsunderepistemicmodelerror.Theconceptofadistributionofpossibleoptimumsmayalsobeimportantwhenperformingreliability-baseddesignoptimization(RBDO)withepistemicmodelerror. 36

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CHAPTER3DECIDINGDEGREEOFCONSERVATIVENESSININITIALDESIGNCONSIDERINGAFUTURETESTANDPOSSIBLEREDESIGN xDesignvariablevectorUAleatoryrandomvariablevectornSafetymargineEpistemicmodelerrorf(,)Objectivefunctiong(,)Limit-statefunctionqRedesignindicatorfunctionpreProbabilityofredesignpfProbabilityoffailureE[]ExpectedvalueoperatorP[]ProbabilityoperatorVar()VarianceoperatorSubscriptsLLow-delitymodelHHigh-delitymodelTTruemodeldetDeterministicvalueiniInitialdesignreDesignafterredesignnalFinaldesignafterpossibleredesignlbLowerboundubUpperboundfFailureUAleatoryuncertainty 37

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EEpistemicuncertaintySuperscripts(i)Epistemicrealization?TargetvalueinoptimizationAccentsMeanvalue 3.1IntroductionEngineeringdesignisaniterativeprocess.Earlyinthedesignprocessengineersoftenutilizelow-delitymodelswhichmaybeassociatedwithhighuncertainty.Modeluncertaintyisclassiedasepistemicuncertaintywhenitarisesduetolackofknowledge,itisreduciblebygainingmoreinformation,andithasonlyasingletrue(butunknown)value[ 31 , 66 , 67 ].Inaddition,almostallengineeringdesignsaresubjecttoaleatoryuncertainty(e.g.loading,materialproperties,etc.).Theinputparameteruncertaintyisclassiedasaleatoryifitisduetonaturalorinherentvariability,itisirreducible,anditisadistributedquantity.Inthefuturewhenprototypesaretestedorhigh-delitysimulationsareperformed,newknowledgewillbecomeavailablethatreducesepistemicuncertaintyandmayresultinadecisiontochangetheinitialdesign.Changingtheinitialdesign,referredtoasredesignorengineeringchange(EC),isanimportantissueforindustryandengineeringmanagement[ 68 , 69 ].Redesignisoftenviewednegativelybecauseitisassociatedwithcostsanddelays,however,itisalsoanopportunityfordesignimprovement[ 68 ].Researchrelatedtoredesign,orengineeringchange,hasmostlybeenperformedatthesystemlevelrequiringahigh-levelofabstraction.ThesemethodsincludetheChangePredictionMethod(CPM)[ 70 ],theRedesignITcomputerprogram[ 71 ],apattern-basedredesignmethodology[ 72 ],acombinationofafunction-behavior-structure 38

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(FBS)linkagemodelwiththeCPMmethod[ 73 ],andaMonteCarlosimulation(MCS)basedmethodofestimatingredesignrisk[ 74 ].Atalowerlevelofabstraction,redesignistypicallytriggeredwhenaninitialdesignislaterrevealedtonotmeetspecicationsorconstraintsduetomodeluncertainty.RoserandKazmerproposedtheexibledesignmethodwhichallowsadesignertominimizetotalexpectedcostswhileconsideringpossiblefuturedesignchanges[ 75 , 76 ].Roseretal.demonstratedaneconomicmethodfordecidingbetweendesignchangeswithdifferentlevelsofuncertaintyanddifferentassociatedcosts[ 77 ].Villanuevaetal.simulatedtheeffectsoffuturetestsandredesignonanintegratedthermalprotectionsystem(ITPS)consideringtheeffectofredesignontheuncertaintyintheprobabilityoffailure[ 19 ].Matsumuraetal.comparedreliability-baseddesignoptimization(RBDO)consideringfutureredesigntotraditionalRBDO[ 20 ].Villanuevaetal.demonstratedthetradeoffbetweenexpecteddesignperformanceandprobabilityofredesignfortheITPSexample[ 21 ].Priceetal.compareddesignerversuscompanyperspectivesonstartingwithahighersafetymarginandpossiblyredesigningtoimproveperformancetostartingwithalowersafetymarginandpossiblyredesigningtoimprovesafety[ 23 ].Thisstudydevelopsageneralizedformulationofthepreviouslyapplicationspecicformulations[ 19 , 21 , 23 ]andexploreshowthedegreeofconservativenessintheinitialdesignrelatestotheexpecteddesignperformanceafterpossibleredesign.Inrelatedwork,Priceetal.introducedaKrigingsurrogatetorepresentepistemicmodeluncertaintyinordertoconsiderspatialvariationsinmodeluncertaintyinthecontextofsimulatingtheeffectsoffuturetestsandredesign[ 25 ].Redesignisoftencausedbyepistemicmodeluncertainty.Ifengineershadaccesstomodelsthatwerecapableofperfectlypredictingdesignperformancethentheinitialdesignwoulddenitelysatisfydesignconstraintsandredesigncouldlargelybeavoided.Assumingaknowntruemodel,reliability-baseddesignoptimization(RBDO)hasmostlyfocusedonensuringaprescribedlevelofreliabilitygivenknownaleatoryparameter 39

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uncertainty[ 7 , 54 , 78 ].Therefore,mostRBDOformulationsareimplicitlyconditionalonthemodelofthesystemexactlymatchingthetruephysicsofthesystem.Somestudieshavesoughttospecicallyaddresstheincorporationofmodeluncertaintyintoreliability-baseddesign[ 9 , 41 , 79 ].However,tocompensateforallthelackofknowledge(i.e.epistemicmodeluncertainty)thatispresentattheinitialdesignstagethentheinitialdesignmayneedtobeveryconservative.Inreality,engineeringdesignisaniterativeprocesswhereovertimedesignsaretested,experimentsareperformed,modelsareimproved,andnewknowledgeisgainedthatreducesepistemicuncertainty.Iftherewillbeafutureopportunitytoreduceepistemicuncertaintyandpossiblychangetheinitialdesign(i.e.redesign),thenthismayaffecttheselectionoftheinitialdesign.Typically,aninitialdesignwillhavesomesafetymarginrelativetodesignconstraintsinordertoimprovesafety,butalsotoprovidesomeinsuranceagainstfutureredesign[ 80 ].Whenselectingasafetymarginfortheinitialdesign,designersfaceadilemmainwhethertostartwithalargerinitialsafetymargin(i.e.moreconservativeinitialdesign)andpossiblyperformingredesigntoimproveperformanceversusstartingwithasmallersafetymargin(i.e.lessconservativeinitialdesign)andpossiblyperformingredesigntorestoresafety.ThisdecisiontobemoreorlessconservativeinthedesignprocessissimilartothequestionofoptimisticversuspessimisticdesignpracticesasexploredbyThornton[ 81 ].Thispaperproposesageneralmethodforoptimizingthesafetymarginsgoverningatwo-stagedeterministicdesignprocessinordertocontroltheepistemicuncertaintyinthenaldesign,designperformance,andprobabilityoffailure.Themethodconsiderstheprobabilityoffutureredesignwhileselectingtheinitialdesign.Thisallowsforthetradeoffbetweenexpectednaldesignperformanceandredesignriskwhilestillensuringreliability.Themethodisdemonstratedonasimplebarproblemandthenonanenginedesignproblem.ThemethodsaredescribedinSection 3.2 .InSection 3.3 ,themethodisappliedtothedesignofaminimumweightuniaxialtensionbarandthentotheenginedesignofa 40

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supersonicbusinessjet.ThediscussionsandconclusionsarepresentedinSection 3.4 .LimitationsoftheproposedmethodandperspectivesforfutureworkarepresentedinSection 3.5 . 3.2MethodsThedeterministicdesignprocessconsistsofselectinganinitialdesign,testingtheinitialdesign,andpossiblyperformingcalibrationandredesign.Theprocessiscontrolledbyaninitialsafetymarginnini,lowerandupperboundsonacceptablesafetymarginsnlbandnub,andaredesignsafetymarginnre.Thesesafetymarginsn=fnini,nlb,nub,nregareoptimizedasdescribedinSection 3.2.1 .TheoptimizercallsafunctiontoperformacrudeMonteCarlosimulation(MCS)ofepistemicerrorrealizationsasdescribedinSection 3.2.2 .Thecompletedesign,test,andpossiblecalibrationandredesignprocessiscarriedoutforeachrealizationofepistemicerrorasdescribedinSection 3.2.3 .Probabilityofredesign,expectedprobabilityoffailure,andexpecteddesigncostarecalculatedfromtheMCSasdescribedinSection 3.2.4 . 3.2.1OptimizationofSafetyMarginsThesafetymarginsnareoptimizedtominimizetheexpectedvalueofthedesigncostfunctionsubjecttoconstraintsonexpectedprobabilityoffailureandprobabilityofredesign.Theformulationoftheoptimizationproblemis minEE[EU[f(Xnal,U)]]w.r.tn=fnini,nlb,nub,nregs.t.EE[Pf,nal]p?fprep?renlbnubnminnnmax(3)whereEE[]istheexpectationwithrespecttoepistemicuncertainty,EU[]istheexpectationwithrespecttoaleatoryuncertainty,f(,)isanobjectivefunction,Xnal 41

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isthevectorofnaldesignvariables,Uisavectorofaleatoryrandomvariables,Pf,nalisthenalprobabilityoffailure,andpreistheprobabilityofredesign.Thenaldesignandnalprobabilityoffailureareepistemicrandomvariables.Intheobjectivefunction,themeanisrstcalculatedwithrespecttoaleatoryuncertaintyforeachdesignrealizationandthentheexpectationiscalculatedoverthemeanswithrespecttoepistemicuncertainty.TheoptimizationisbasedonaMCSasseeninFigure 3-1 .Solvingtheoptimizationproblemfordifferentvaluesofp?reresultsinatradeoffbetweenexpectedcostandprobabilityofredesign.CovarianceMatrixAdaptationEvolutionStrategy(CMA-ES)withapenalizationstrategytohandletheconstraintsisusedtosolvetheoptimizationproblem[ 82 ]. . . Optimizationofsafetymargins(Section 3.2.1 ,equation 3 ) . Simulationofdeterministicdesign/redesignprocess A. InitialdesignoptimizationFori=1,...,mrealizationsofepistemicmodeluncertainty: B. Simulatedhigh-delityevaluation(i.e.simulatedtest) C. Possiblecalibration D. Possibleredesignoptimization E. Probabilityoffailurecalculation . n . Xnal,Pf,nal,pre Figure3-1. ThesafetymarginsareoptimizedbasedonaMCSofthedeterministicdesign/redesignprocess 42

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3.2.2Monte-CarloSimulationofEpistemicModelErrorTheepistemicmodeluncertaintyandaleatoryparameteruncertaintyaretreatedseparately.TorepresentepistemicmodeluncertaintyweintroducetheepistemicrandomvariablesELandEHtorepresenttheerrorinthelowandhigh-delitymodelsrespectively.Tosimplifythepropagationofmixedepistemicmodeluncertaintyandaleatoryparameteruncertainty,itisassumedthatthereisaxedbutunknownbiasbetweenthelow-delitymodel,thehigh-delitymodel,andthetruemodel.Theassumedrelationshipbetweenthedifferentdelitymodelsis gT(x,u)=gH(x,u)+eH=gL(x,u)+eL(3)wherex2Rdisavectorofdesignvariables,Uisavectorofaleatoryrandomvariableswitharealizationu2Rp,gT(,)isthetruemodel,gH(,)isthehigh-delitymodel,gL(,)isthelow-deltymodel,eH2Risthetrueerrorinthehigh-delitymodel,andeL2Risthetrueerrorinthelow-delitymodel.Itisassumedthatthepossibleerrorsareknownbasedonexpertopinionorpreviousexperience.ThepossiblemodelerrorsELandEHaremodeledastwoindependentuniformlydistributedepistemicrandomvariableswithVar(EH)
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assumptionthattheepistemicrandomvariableELincludesthetruemodelerror.ThemeanofthepossibleerrorsisdenedaseLandeH.Themeanpredictionwithrespecttoepistemicuncertaintyofthehigh-delitymodelandtruemodelaredenedasgH(,)andgT(,)respectively.AcrudeMonteCarlosimulationofi=1,...,merrorrealizationsisperformed.InSection 3.2.3 ,design/redesignprocessisdescribedconditionalononepairoferrorsamples.Thedeterministicdesign/redesignprocessisrepeatedformanydifferenterrorrealizations.BasedontheMCS,theriskofredesignisestimated.Furthermore,theMCSexploreshowfailingafuturetestisrelatedtothenaldesignperformanceandsafety. 3.2.3DeterministicDesign/RedesignProcessAowchartofthedesign/redesignprocessisshowninFigure 3-2 .Thedesignprocessconsistsofselectinganinitialdesign,asimulatedevaluationoftheinitialdesignwithahigh-delitymodel,possibleredesign,andareliabilityassessment.Insections 3.2.3.1 to 3.2.3.3 theprocessisdescribedconditionalontheerrorrealizationsEL=e(i)LandEH=e(i)H 3.2.3.1InitialdesignTheselectionoftheinitialdesignisbasedonadeterministicsafety-margin-basedoptimizationproblem minf(x,udet)w.r.t.xs.t.gT(x,udet))]TJ /F5 11.955 Tf 11.95 0 Td[(nini0xminxxmax(3)whereudetisavectorofdeterministicvaluesthataresubstitutedforaleatoryrandomvariables.Notethatifthelow-delitymodelisbelievedtobeunbiased,eL=0,thenthemeanpredictionofthetruemodelissimplythelow-delitymodelgL(,).Thefailure 44

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. . Stage1:DesignOptimization . nini . Simulatedtest . Redesign? . nlb . nub . Calibration . Stage2:DesignOptimization . nre . FinalDesign . ReliabilityAnalysis . no . yes . MCSLoop Figure3-2. Flowchartshowingstepsintwo-stagedeterministicdesign/redesignprocess.Safetymarginsn=fnini,nlb,nub,nregareshownasinputsatrelevantsteps. domainisdenedwithrespecttothetrue(butunknown)modelgT(,)as f(x)=fu2UjgT(x,u)<0g(3)whereUisthealeatorysamplingspace.LetxinidenotetheoptimumdesignfoundfromEquation 3 usinginitialsafetymarginnini.Itisassumedthattheconservativevaluesudetarebasedonregulations(e.g.FARx25.613[ 50 ],FARx25.303[ 2 ])and/orpreviousexperience. 3.2.3.2TestinginitialdesignandredesigndecisionInthefuture,theinitialdesignxiniwillbeevaluatedwiththehigh-delitymodeltomeasurethesafetymargin.IntheMonte-Carlosimulation,thetestisbasedonasimulatedhigh-delityevaluationg(i)H(xini,udet).Ifnlbg(i)H(xini,udet)nubthentheinitialdesignwillpassthetestandbeacceptedasthenaldesign.However,ifg(i)H(xini,udet)nubthenredesignisperformedtoimproveperformancebecausetheinitialdesignistooconservative.Anindicatorfunctionfortheredesign 45

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decisionisdenotedq(i)whichisoneforredesignandzerootherwise.Redesigninitiatedduetoalowsafetymargin(g(i)H(xini,udet)nub)isreferredtoasredesignforperformance. 3.2.3.3ModelcalibrationBeforeredesign,themeanpredictionofthetruemodelgT(,)iscalibratedbasedonthetestresult.Themodeliscalibrateddeterministicallybasedonthedifferencebetweenthepredictionandthehigh-delityevaluationoftheinitialdesign.Thecalibratedmodelis g(i)calib(x,u)=gT(x,u)+e(i)calib(3)wheree(i)calib=g(i)H(xini,udet))]TJ /F4 11.955 Tf 12.85 0 Td[(gT(xini,udet).ThecalibratedmodelGcalib(,)accountsforchangesinthemodelthatmightoccurduringthefuturecalibration.Thecalibrationimprovesthemodelwhenthehigh-delitymodelismoreaccuratethanthelow-delitymodel,je(i)Hj
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Letx(i)redenotethesolutiontoEquation 3 .ThenewdesignXreisanepistemicrandomvariablebecauseitisconditionalontheunknownoutcomeofthefuturehigh-delityevaluation.However,thereisnoinherentvariability(i.e.aleatoryuncertainty)inthedesignchoice.Thenewdesignisarandomvariableonlybecauseitisunknownattheinitialdesignstage.NotethatthefeasibledesignspaceoftheredesignproblemEquation 3 isdifferentthanthefeasibledesignspaceintheinitialdesignproblemEquation 3 duetothecalibrationandtheuseofasafetymarginnrethatmaybedifferentthannini.Conditionalontheoutcomeofthefuturetest,somedesignswithimprovedperformancemaybecomeaccessibleduringredesignthatwerepreviouslyconsideredinfeasibleorsomedesignsthatwerepreviouslyconsideredreasonablemayberevealedtobeunsafe. 3.2.4ProbabilisticEvaluationEachsetofsafetymarginsnresultsinaprobabilityofredesignpre,analprobabilityoffailureafterpossibleredesignPf,nal(epistemicrandomvariable),andanalcostEU[f(Xnal,U)](epistemicrandomvariable).HistogramsofrandomvariablesareobtainedbasedonacrudeMCSasdescribedinSection 3.2.2 .TheexpectedvalueswithrespecttoepistemicmodeluncertaintythatareusedinEquation 3 areobtainedusingnumericalintegration.Theprobabilityofredesignispre=EE[Q].Afterpossibleredesign,thenaldesignis x(i)nal=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q(i)xini+q(i)x(i)re(3)TheexpectedmeandesigncostafterpossibleredesignisEE[EU[f(Xnal,U)]].Theexpectedmeandesigncostcanbewrittenintermsofconditionalprobabilitiesas EE[EU[f(Xnal,U)]]=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(pre)EU[f(xini,U)]+preEE[EU[f(Xre,U)]jQ=1](3) 47

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whereEU[f(xini,U)]istheexpectedmeandesigncostconditionalonpassingthetestandEE[EU[f(Xre,U)]jQ=1]istheexpectedmeandesigncostconditionalonfailingthetest.Thenalsafetymarginwithrespecttothehigh-delitymodelafterpossibleredesignis n(i)H,nal=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q(i)g(i)H(xini,udet)+q(i)g(i)H)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(x(i)re,udet(3)wherethehigh-delitymodelisequaltothecalibratedmodelduetothecalibrationprocessasdescribedinSection 3.2.3.3 .Thenalsafetymarginwithrespecttothetruemodelafterpossibleredesignis n(i)T,nal=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q(i)g(i)T(xini,udet)+q(i)g(i)T)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(x(i)re,udet(3)Duetoepistemicmodeluncertaintythetrueprobabilityoffailureisunknown.Arealizationoftheprobabilityoffailurefortheinitialdesignis p(i)f,ini=PUhg(i)T(xini,U)<0i(3)wherePU[]denotestheprobabilitywithrespecttoaleatoryuncertainty.Intheprobabilityoffailurecalculation,epistemicmodeluncertaintyistreatedseparatelyfromthealeatoryuncertainty.Thereisepistemicuncertaintyinthetrueprobabilityoffailurewithrespecttoaleatoryuncertaintyduetoepistemicmodeluncertainty.Inreality,thetrueprobabilityoffailureofthenaldesigndoesnotdependonmodeldelity.However,ourknowledgeofthetrueprobabilityoffailuredependsontheuncertaintyinourmodels.Toaccountformodeluncertainty,theprobabilityoffailurecalculationisrepeatedconditionalondifferentrealizationsofthetruemodelg(i)T(,)asshownin 48

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Equation 3 .Afterredesigntheprobabilityoffailureis p(i)f,re=PUhg(i)T(x(i)re,U)<0i(3)Thedesignvariablex(i)reisanepistemicrandomvariablebecauseitisconditionalontheoutcomeofthefuturetest.Thenalprobabilityoffailureafterpossibleredesignis p(i)f,nal=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q(i)p(i)f,ini+q(i)p(i)f,re(3)TheexpectedprobabilityoffailureafterpossibleredesignisEE[Pf,nal].Theexpectedprobabilityoffailurecanbewrittenintermsofconditionalprobabilitiesas EE[Pf,nal]=(1)]TJ /F5 11.955 Tf 11.95 0 Td[(pre)EE[Pf,inijQ=0]+preEE[Pf,rejQ=1](3)whereEE[Pf,inijQ=0]istheexpectedprobabilityoffailureconditionalonpassingthetestandEE[Pf,rejQ=1]istheexpectedprobabilityoffailureconditionalonfailingthetest.WecanseefromEquation 3 thattheexpectednalprobabilityoffailureisaweightedaverageoftheexpectedprobabilityoffailureoftheinitialdesignandtheexpectedprobabilityoffailureofthepossibleredesigns. 3.3TestCases 3.3.1UniaxialTensionTest 3.3.1.1ProblemdescriptionInthisexampleweconsiderthedesignofaminimumweightbarsubjecttouniaxialloading.TheproblemdenitionisshowninTable 3-1 .Thedesignissubjecttoaleatoryuncertaintyinloadingandmaterialproperties.Inaddition,thereisepistemicmodeluncertaintyinthelimit-statefunctiondescribingtheyieldingofthebar.TheuncertainparametersaredenedasshowninTable 3-2 .Thebarisdesignedtominimizethemass,orequivalentlycrosssectionalarea,subjecttoastressconstraint.Thebarisdesignedusingconservativevaluesinplaceofrandomloadsandmaterialproperties.In 49

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Table3-1. ProblemdenitionforuniaxialtensiontestexampleDescriptionNotation DesignvariableCrosssectionalarea(mm2)x=aAleatoryvariablesAppliedload,materialstrengthU=fP,SgConservativevaluesLimitload,allowablestrengthudet=f1600N,15.35MPagObjectivefunctionCrosssectionalarea(mm2)f(x)=aLimit-statefunctionYieldinggL(x,U)=S)]TJ /F5 11.955 Tf 11.95 0 Td[(P=aTargetmeanreliabilityp?f=110)]TJ /F9 7.97 Tf 6.58 0 Td[(5 Table3-2. UncertainparametersforuniaxialtensiontestexampleParameterClassicationSymbolMean,C.O.VRangeDistribution AppliedloadAleatoryP(N)10000.20[,1]NormalMaterialstrengthAleatoryS(MPa)200.12[,1]NormalErrorinlow-delitymodelEpistemicEL(MPa)0–[-4.35,4.35]UniformErrorinhigh-delitymodelEpistemicEH(MPa)0–[-2.18,2.18]Uniform thefuture,thebarwillbetested(e.g.high-delitysimulationorprototypetest)anditwillberedesignedifthesafetymarginwithrespecttothestressconstraintistoohighortoolow.TheproblemfollowsthegeneralmethoddescribedinSection 3.2 .Thelimit-statefunctionisalinearfunctionofthealeatoryparametersandallaleatoryparametersareassumedtobenormallydistributed.Therefore,thecomputationalcostisreducedbycalculatingthereliabilityindexanalyticallyforeachrealizationofepistemicmodelerror.Duetothesimplicityofthedesignproblem,theoptimumdeterministicdesigncanbeobtaineddirectlybysolvingforthevalueofthedesignvariablethatsatisesthedeterministicconstraint. 3.3.1.2ExpectedperformanceversusprobabilityofredesignTradeoffcurvesforexpectedcost,EE[f(Xnal)],versusprobabilityofredesign,pre,areshowninFigure 3-3 .ThetradeoffcurveswereobtainedbysolvingEquation 50

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3 forseveralvaluesoftheconstraintonprobabilityofredesign,p?re.Thetwocurvescorrespondtothespecialcasesofperformingredesignonlyforperformanceandperformingredesignonlyforsafety.Itwasobservedthatredesignforperformancewastheglobaloptimumsolutionandtheoptimumsafetymarginswouldconvergetothissolutionwhenallowingforbothredesignforsafetyandperformance.Theexpectedmassofthebardecreaseswithincreasingriskofredesign.Whenthereiszeroprobabilityofredesign,theinitialdesignmustbeconservativeenoughthattheexpectedprobabilityoffailureislessthanorequaltothetargetvalueof110)]TJ /F9 7.97 Tf 6.58 0 Td[(5.Tomeetthetargetexpectedprobabilityoffailuretheinitialdesignmustbeheavier.Thisisthedesignwewouldobtainifweoptimizedonlyninitominimizetheweightoftheinitialdesignwithaconstraintonexpectedprobabilityoffailure.Bothcurvesstartatthisdesignbecausetheprobabilityofredesigniszeroandthereforethereisnodifferencebetweentheredesignstrategies.Astheprobabilityofredesignincreases,redesigncanbeusedtocorrecttheinitialdesignifthehigh-delitymodelrevealsthesafetymarginistoohighortoolow.Toexplorethesimulationinmoredetail,thepointsonthetradeoffcurvecorrespondingto20%probabilityofredesignwereselected.Histogramsoftheareaofthecrosssectionofthebar,safetymarginwithrespecttothehigh-delitymodel,safetymarginwithrespecttotruemodel,reliabilityindex,andprobabilityoffailurefor20%probabilityofredesignareshowningures 3-4 , 3-5 , 3-6 , 3-7 , 3-8 .StatisticsonthemassandprobabilityoffailurebeforeandafterredesignarelistedinTable 3-3 .Whenconsideringonlyredesignforperformance,theinitialdesignisheavierandredesignisperformedifthesafetymarginisrevealedtobetoohigh.Observingahighsafetymarginiscorrelatedwiththedesignbeingverysafe.Redesignforperformancehastheeffectofincreasingtheprobabilityoffailureinordertoreducethemassiftheinitialdesignisrevealedtobeverysafe.Whenconsideringonlyredesignforsafety,theinitialdesignislighterandredesignisperformedifthesafetymarginisrevealedtobetoolow. 51

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Table3-3. Resultsforuniaxialtensionexamplefor20%probabilityofredesignDescriptionNotationRedesignforSafetyRedesignforPerformance Probabilityofredesignpre0.200.20 Costofinitialdesignf(xini)155.5170.7ExpectedcostconditionalonperformingredesignEE[f(Xre)jQ=1]191.2109.4ExpectedcostafterpossiblyperformingredesignEE[f(Xnal)]162.7158.4 ExpectedprobabilityoffailureofinitialdesignEE[Pf,ini]2.910)]TJ /F9 7.97 Tf 6.59 0 Td[(50.910)]TJ /F9 7.97 Tf 6.59 0 Td[(5ExpectedprobabilityoffailureofinitialdesignconditionalonpassingtestEE[Pf,inijQ=0]0.910)]TJ /F9 7.97 Tf 6.59 0 Td[(51.210)]TJ /F9 7.97 Tf 6.59 0 Td[(5ExpectedprobabilityoffailureofnewdesignsconditionalonfailingtestEE[Pf,rejQ=1]1.310)]TJ /F9 7.97 Tf 6.59 0 Td[(50.410)]TJ /F9 7.97 Tf 6.59 0 Td[(5ExpectedprobabilityoffailureafterpossiblyperformingredesignEE[Pf,nal]1.010)]TJ /F9 7.97 Tf 6.59 0 Td[(51.010)]TJ /F9 7.97 Tf 6.59 0 Td[(5 Observingalowsafetymarginiscorrelatedwiththedesignbeingunsafe.Redesignforsafetyhastheeffectoftruncatingthetailoftheprobabilityoffailuredistributioncorrespondingtohighprobabilitiesoffailure.Iftheinitialdesignisrevealedtobeunsafe,thecrosssectionalareaisincreasedduringredesignresultinginasafer,butheavierdesign.Theredesigndecisionisbasedonthesafetymarginwithrespecttothehigh-delitymodelandthereforesuffersfromtheerrorinthehigh-delitymodel.Theerrorinthehigh-delitymodelresultsinimperfecttruncationofthetruesafetymarginandreliabilityindexdistributionsasshowningures 3-6 , 3-7 . 3.3.1.3Expectedperformanceversuslevelofhigh-delitymodelerrorToexploretheeffectoftheerrorinthehigh-delitymodel,theratioofthestandarddeviationoftheerrorinthehigh-delitymodelrelativetothestandarddeviationoftheerrorinthelow-delitymodel,p Var(EH)=Var(EL),wasvariedfromzerotoone.Thestandarddeviationoftheerrorofthelow-delitymodelwasheldxedandboth 52

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Figure3-3. Uniaxialtensiontest-Comparisonofexpectedcrosssectionalareaafterpossibleredesignasafunctionofprobabilityofredesignforredesignforperformance(conservativeinitialdesign)versusredesignforsafety(ambitiousinitialdesign). distributionshadmeansofzero.Anerrorratioofzerocorrespondstonoerrorinthehigh-delitymodelandaratioofonecorrespondstohavingthesameerrordistributionsforbothmodels.Foreachpointonthecurves,thesafetymarginswereoptimizedbysolvingEquation 3 foraxedprobabilityofredesignof20%.AsshowninFigure 3-9 ,redesignforsafetyispreferredwhentheerrorinthehigh-delitymodelislowbutredesignforperformanceispreferredwhentheerrorinthehigh-delitymodelishigh.NotethatforthetradeoffcurveshowninFigure 3-3 theratiooftheerrorsinthemodelswasp Var(EH)=Var(EL)=0.5. 3.3.2SupersonicBusinessJetEngineDesign 3.3.2.1ProblemdescriptionThisexampleisbasedonthepropulsiondisciplinedesignproblemfromtheSobieskisupersonicbusinessjet(SSBJ)problem[ 83 ].Thedesignproblemistominimizeengineweightsubjecttoaconstraintonthemaximumnormalizedthrottlesetting.Theproblemisbasedonthescalingofabaselineenginetomeetathrust 53

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A BFigure3-4. Uniaxialtensiontest-Epistemicuncertaintyincrosssectionalareafor20%probabilityofredesign. A BFigure3-5. Uniaxialtensiontest-Epistemicuncertaintyinsafetymarginwithrespecttohigh-delitymodelfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. 54

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A BFigure3-6. Uniaxialtensiontest-Epistemicuncertaintyinsafetymarginwithrespecttotruemodelfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. A BFigure3-7. Uniaxialtensiontest-Epistemicuncertaintyinreliabilityindexfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. 55

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A BFigure3-8. Uniaxialtensiontest-Epistemicuncertaintyinfailurefor20%probabilityofredesign.Theguresareplottedwithdifferentscalestoshowthechangeinthetailofthedistribution.Plotsshowoverlappingtransparenthistograms. requirement.Iftheengineisdesignedtoprovidetherequiredthrustwhenoperatingnearidlethrottlethentheresultingenginedesignisunreasonablylargeandheavy.Iftheengineisdesignedtoprovidetherequiredthrustwhenoperatingatfullthrottlethentheenginedesigncanbesmallerandlighter.However,thereisepistemicuncertaintyinthelow-delitypredictionofthethrustoutputandthereforeitisdesirabletohavesomesafetymargintoincreasetheprobabilitythattheas-builtenginecanprovidesufcientthrust.Inaddition,thethrustoutputoftheenginevarieswithMachnumberandaltitude.Inthisexample,weconsiderthattheengineisdesignedtooperateforadistributionofaltitudes(aleatoryuncertainty).ThethrottlesettingisdenedastheratiooftheengineoutputthrustrelativetothemaximumavailablethrustatagivenaltitudeandMachnumber.Athrottlesettingof1indicatesmaximumpoweratagivenaltitudeandMachnumberandathrottlesettingof0.01isidlethrust.ThenetavailablethrustoftheengineincreaseswithMachnumber 56

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Figure3-9. Uniaxialtensiontest-Redesignforsafetyispreferredwhenhigh-delitymodelerrorislow,butredesignforperformanceispreferredwhenhigh-delitymodelerrorishigh.Plotisforxedprobabilityofredesignof20%. Table3-4. ProblemdenitionforSSBJExampleDescriptionNotation DesignvariableThrottlexAleatoryvariableAltitude(ft)U=HConservativevalueMaxaltitudeudet=56,770ftObjectivefunctionEngineweight(lbs)f(x)=WE(x)Limit-statefunctionMaximumthrottlegL(x,U)=xub(H))]TJ /F5 11.955 Tf 11.96 0 Td[(xTargetmeanreliabilityp?f=110)]TJ /F9 7.97 Tf 6.59 0 Td[(3 anddecreaseswithaltitude.Anon-dimensionalthrottlesettingvariable,x,iscreatedbynormalizingthethrottlewithrespectthepointofmaximumthrustofthebaselineengine.Thenon-dimensionalthrottlesettingisdenedas x=Sout=S0(3)whereSoutistheoutputthrustandS0=16168lbfisthemaximumthrustofthebaselineengine.IftherequiredthrustSreqisdifferentthanthethrustprovidedbythebaselineengine,thebaselineenginedesignisscaledtomatchthenewrequirement.Inthis 57

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Table3-5. UncertainParametersforSSBJExampleParameterClassicationSymbolMean,C.O.VRangeDistribution AltitudeAleatoryH(ft)525000.05[45000,60000]TruncatedNormalErrorinlow-delitymodelEpistemic^EL0–[-0.0375,0.0375]UniformErrorinhigh-delitymodelEpistemic^EH0–[-0.0075,0.0075]Uniform example,weassumeaxedthrustrequirementSreq=40000lbf.TheenginescalefactorESFisdenedas ESF=Sreq 2Sout=Sreq 2xS0(3)wherethevalueof2inthedenominatorreectsthefactthattwoenginesareusedonthejet.TheweightoftheengineWEisapproximatedasfollowingapowerlawrelationshipwithenginescalefactor WE=2WBE(ESF)1.05(3)whereWBE=4360lbistheweightofthebaselineengine.AresponsesurfaceoftheengineperformancemapforthebaselineenginecalculatesmaximumavailablethrustSavailatagivenMachnumberMandaltitudeh.Theresponsesurfacesetsanupperboundonthrottle,xub,whennormalizedbyS0 xub(M,h)=Savail(M,h) S0=1 S0)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(0+1M+2h+3M2+24Mh+25h2(3)wherethecoefcientsarelistedinTable 3-6 .ThisresponsesurfacemodelshowavailablethrustdecreaseswithincreasingaltitudeasshowninFigure 3-10 .Inthisexampleweareinterestedinminimizingtheweightoftheenginesubjecttoaconstraintonmaximumthrottle.TheproblemdenitionisshowninTable 3-4 .We 58

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Table3-6. Coefcientsforcalculatingthrottleupperbound(Equation 3 )CoefcientValuehline01.148410411.08561042)]TJ /F4 11.955 Tf 9.3 0 Td[(5.080210)]TJ /F9 7.97 Tf 6.59 0 Td[(133.20021034)]TJ /F4 11.955 Tf 9.3 0 Td[(1.466310)]TJ /F9 7.97 Tf 6.59 0 Td[(156.857210)]TJ /F9 7.97 Tf 6.59 0 Td[(6 Figure3-10. AresponsesurfaceoftheengineperformancemapcalculatesmaximumavailablethrustatagivenMachnumber,M,andaltitude,h.Thethrottlesettingisnormalizedtooneatanaltitudeofapproximately32000ftandMach1.9. consideraleatoryuncertaintyinthealtitudeandepistemicmodeluncertaintyinthemaximumthrottle,xub,asdenedinTable 3-5 .TheproblemfollowsthegeneralmethoddescribedinSection 3.2 .Theengineisdesignedusingaconservativevalueinplaceofrandomaltitude.Inthefuture,theenginewillbetested(e.g.high-delitysimulationorprototypetest)anditwillberedesignedifthesafetymarginwithrespecttothethrottleconstraintistoohighortoolow.Thatis,theenginewillberedesignedifitprovidesinsufcientthrustorthethrustissolargethatitisworthredesigningtouseasmaller,lighterengine. 59

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TheprobabilityoffailureisestimatedbasedonaMonte-Carlosimulation.Thethrottleshouldbesettotheupperboundtominimizetheengineweight.Therefore,deterministicdesignoptimizationwasavoidedbysettingthethrottletotheupperboundminusthesafetymargin. 3.3.2.2ExpectedperformanceversusprobabilityofredesignTradeoffcurvesforexpectedcost,EE[f(Xnal)],versusprobabilityofredesign,pre,areshowninFigure 3-11 .ThetradeoffcurveswereobtainedbysolvingEquation 3 forseveralvaluesoftheconstraintonprobabilityofredesign,p?re.Thetwocurvescorrespondtothespecialcasesofperformingredesignonlyforperformanceandperformingredesignonlyforsafety.Itwasobservedthatredesignforsafetywastheglobaloptimumsolutionandtheoptimumsafetymarginswouldconvergetothissolutionwhenallowingforbothredesignforsafetyandperformance.ThisresultisdifferentfromtheexampleinSection 3.3.1 whereredesignforperformancewaspreferred.Toexplorethesimulationinmoredetail,thepointsonthetradeoffcurvecorrespondingto20%probabilityofredesignwereselected.Histogramsofthethrottlesetting,weight,safetymargin,andprobabilityoffailurefor20%probabilityofredesignareshowninFigure 3-12 , 3-14 , 3-13 , 3-15 .StatisticsonthemassandprobabilityoffailurebeforeandafterredesignarelistedinTable 3-7 . 3.3.2.3Expectedperformanceversuslevelofhigh-delitymodelerrorTheratioofthestandarddeviationoftheerrorinthehigh-delitymodelrelativetothestandarddeviationoftheerrorinthelow-delitymodel,p Var(EH)=Var(EL),wasvariedfromzerotoone.Foreachpointonthecurves,thesafetymarginswereoptimizedbysolvingEquation 3 foraxedprobabilityofredesignof20%.AsshowninFigure 3-16 ,redesignforsafetyispreferredwhentheerrorinthehigh-delitymodelislowbutredesignforperformanceispreferredwhentheerrorinthehigh-delitymodelishigh.TheoveralltrendsaresimilartothoseobservedfortheexampleinSection 3.3.1 . 60

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Figure3-11. SSBJEngine-Comparisonofexpectedengineweightafterpossibleredesignasafunctionofprobabilityofredesignforredesignforperformance(conservativeinitialdesign)versusredesignforsafety(ambitiousinitialdesign). NotethatforthetradeoffcurveshowninFigure 3-11 theratiooftheerrorsinthemodelswasp Var(EH)=Var(EL)=0.2. 3.4DiscussionandConclusionThisstudypresentedageneralizedformulationofatwo-stagesafety-margin-baseddesign/redesignprocessconsideringtheeffectsofafuturetestandpossibleredesign.Thesafetymarginsthatcontrolthedeterministicdesign/redesignprocessareoptimizedtominimizetheexpectedvalueofthedesigncostfunction(i.e.maximizeexpectedperformance)whilesatisfyingconstraintsonprobabilityofredesignandexpectedprobabilityoffailure.Thefuturetestresult(i.e.high-delityevaluationofinitialdesignorprototypetest)isanepistemicrandomvariablethatispredictedbasedonthedistributionsofpossibleerrorsinthelowandhighdelitymodels.Futuretestresultsaresimulatedinordertocalculatetheprobabilityofredesign,thepossibledesignsaftercalibrationandredesign,andthenaldistributionofprobabilitiesoffailure.Byconsideringthatthedesignmaychangeinthefutureconditionalontheoutcomeofthe 61

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Table3-7. ResultsforSSBJexamplefor20%probabilityofredesignDescriptionNotationRedesignforSafetyRedesignforPerformance Probabilityofredesignpre0.200.20 Costofinitialdesignf(xini)8.301049.16104ExpectedcostconditionalonperformingredesignEE[f(Xre)jQ=1]9.641046.52104ExpectedcostafterpossiblyperformingredesignEE[f(Xnal)]8.571048.63104 ExpectedprobabilityoffailureofinitialdesignEE[Pf,ini]6.9410)]TJ /F9 7.97 Tf 6.59 0 Td[(30.9610)]TJ /F9 7.97 Tf 6.58 0 Td[(3ExpectedprobabilityoffailureofinitialdesignconditionalonpassingtestEE[Pf,inijQ=0]1.0510)]TJ /F9 7.97 Tf 6.59 0 Td[(31.2010)]TJ /F9 7.97 Tf 6.58 0 Td[(3ExpectedprobabilityoffailureofnewdesignsconditionalonfailingtestEE[Pf,rejQ=1]0.8010)]TJ /F9 7.97 Tf 6.59 0 Td[(30.1810)]TJ /F9 7.97 Tf 6.58 0 Td[(3ExpectedprobabilityoffailureafterpossiblyperformingredesignEE[Pf,nal]1.0010)]TJ /F9 7.97 Tf 6.59 0 Td[(31.0010)]TJ /F9 7.97 Tf 6.58 0 Td[(3 A BFigure3-12. SSBJEngine-Epistemicuncertaintyinthrottlesettingfor20%probabilityofredesign. 62

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A BFigure3-13. SSBJEngine-Epistemicuncertaintyinsafetymarginwithrespecttohigh-delitymodelfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. A BFigure3-14. SSBJEngine-Epistemicuncertaintyinengineweightfor20%probabilityofredesign. 63

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A BFigure3-15. SSBJEngine-Epistemicuncertaintyinprobabilityoffailurefor20%probabilityofredesign.Theguresareplottedwithdifferentscalestoshowthechangeinthetailofthedistribution.Plotsshowoverlappingtransparenthistograms. futuretestitispossibletotradeoffbetweentheriskofhavingtoredesigninthefutureandtheassociatedperformanceand/orreliabilitybenets.Whenconsideringepistemicmodeluncertaintyinadesignconstraint,thedesignerfacesadilemmainwhethertostartwithalargerinitialsafetymargin(i.e.moreconservativeinitialdesign)andpossiblyredesigntoimproveperformanceversusstartingwithasmallersafetymargin(i.e.lessconservativeinitialdesign)andpossiblyredesigningtorestoresafety.Thisstudyanalyzesthisdecisionwhenthereisaxedbutunknownconstantbiasbetweenthelow-delitymodel,high-delitymodel,andtruemodel.Intheexamplesinthisstudy,itisfoundthatthedecisionofwhethertostartwithahigherinitialsafetymarginandpossiblyredesignforperformance,ortostartwithalowerinitialsafetymarginandpossiblyredesignforsafety,dependsontheratioofthestandarddeviationoftheuncertaintyinthehigh-delitymodelrelativetothestandarddeviationofuncertaintyinthelow-delitymodel. 64

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Figure3-16. SSBJEngine-Redesignforsafetyispreferredwhenhigh-delitymodelerrorislow,butredesignforperformanceispreferredwhenhigh-delitymodelerrorishigh.Plotisforxedprobabilityofredesignof20%. Itwasobservedthattheredesignforsafetystrategywasstronglyinuencedbytheamountoferrorinthehigh-delitymodel.Itishypothesizedthattheamountoferrorinthehigh-delitymodelhasastrongerinuenceontheredesignforsafetystrategybecausetheerrorinterfereswiththeprocessoftruncatingdangerousdesigns.Thebenetofredesignforsafetyisthatitpreventsadangerousinitialdesignfromsuccessfullypassingthetest.Thissubstantiallyreducestheexpectedprobabilityoffailurewhichinturnallowstheinitialdesigntobelessconservative.However,ifthereisalargeamountoferrorinthehigh-delitymodelthenadangerousinitialdesignmaypassthetestunnoticed.Evenifthisisunlikely,thepossibilityofahighprobabilityoffailurehasasignicantinuenceonthemeanprobabilityoffailure.Tocompensate,theinitialdesignmustbemoreconservative.Ontheotherhand,whenconsideringredesignforperformanceitisnotaproblemifaverysafe(i.e.overlyconservative)initialdesignpassesthetest. 65

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Itisobservedthatredesignforsafetyandredesignforperformanceresultindifferentdistributionsofperformance(e.g.weight).Redesignforperformancecapitalizesonthefactthatitmaybepossibletoobtainasubstantialimprovementinperformanceiftheinitialdesignisrevealedtobemuchtooconservative.Theperformanceimprovementislargebuttheprobabilityofobtainingthisbenetissmallwhentheprobabilityofredesignissmall.Theinitialdesignmustbemoreconservativesinceredesignisonlyusedtoimproveperformanceandnottorestoresafety.Redesignforsafetyattemptstoobtainbetterinitialdesignperformancebyallowingforthepossibilitythatredesignmaybenecessarytorestoresafety.Iftheinitialdesignisrevealedtobeunsafethenitisfoundasmalldesignchangeisusuallysufcienttorestoresafety.Whentheprobabilityofredesignissmalltheinitialdesignislikelytopassthetestandbeacceptedasthenaldesign.Redesignforsafetyallowsforabetterinitialdesignthanredesignforperformance.However,redesignforperformancehastheadvantagethatitmaybepossibletoskiptheredesignprocesswhentimeconstraintsoutweighthepossibleperformancebenetsofredesign. 3.5LimitationsandFutureworkThisstudyisbasedontheassumptionthatthereisaxedbutunknownconstantbiasbetweenthelow-delitymodel,high-delitymodel,andtruemodel.Ifthemodelerrorisconstantacrossthejointdesign/aleatoryspace,thenthereductioninepistemicmodeluncertaintydoesnotdependonthelocationwherethehigh-delitymodelisevaluated.Ifthemodelerrorisnotconstant,thenitmayincentivizestartingwithalowersafetymargininordertohaveahigh-delityevaluationclosetothelimitsurfaceg(x,u)=0.Inrelatedwork,aKrigingsurrogateisintroducedtomodelepistemicuncertaintyinordertoaccountforspatialcorrelationsinmodeluncertainty[ 25 ].TheproposedmethodmaybecomputationallyexpensivebecauseitinvolvesaMonte-Carlosimulation(MCS)ofadesign/redesignprocessnestedinsideaglobaloptimizationproblem.Toreducethecomputationalcostsurrogatemodelscanbetto 66

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themeanprobabilityoffailureandmeandesigncostasafunctionofthesafetymargins[ 25 ].IntheformulationofthedeterministicdesignoptimizationproblemsthealeatoryvariablesUarereplacedwiththeconservativedeterministicvaluesudet.Thechoiceoftheconservativedeterministicvaluestouseinplaceofaleatoryrandomvariablesmayhaveastronginuenceonthenalresults.Futureworkwillinvestigateoptimizingthevaluesudetinadditiontothesafetymargins.Inthisstudy,aconstraintwasplacedontheexpectedprobabilityoffailureduringtheoptimizationofsafetymargins.Byonlyconstrainingtheexpectedprobabilityoffailureitispossibletoarriveatanoptimumsetofsafetymarginsthatresultsinsomeverysafedesignsbutsomeunsafedesigns.Toavoidthissituationadditionalconstraintsshouldbeincludedthatconsiderthespreadoftheprobabilityoffailuredistribution(e.g.superquantile[ 84 ]). 67

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CHAPTER4CONSIDERINGSPATIALCORRELATIONSINTHEEPISTEMICMODELERRORWHENSIMULATINGAFUTURETESTANDREDESIGN Nomenclature xDesignvariablevectorUAleatoryrandomvariablevectore(,)Modelerrorf()Objectivefunctiong(,)Limit-statefunctionnSafetymarginqRedesignindicatorfunctionpreProbabilityofredesignpfProbabilityoffailureE[]ExpectedvaluewithrespecttoepistemicuncertaintyPrU[]ProbabilitywithrespecttoaleatoryuncertaintySubscriptsLLow-delitymodelHHigh-delitymodeldetDeterministicvalueiniInitialdesignreDesignafterredesignnalFinaldesignafterpossibleredesignlbLowerboundubUpperboundSuperscripts(i)Realizationofepistemicrandomvariableorfunction 68

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?TargetvalueinoptimizationAccents^EpistemicrandomvariablesorfunctionsMeanpredictionofKrigingmodel 4.1ResearchContextinRelationtoScopeofDissertationInChapter 3 ,amethodwasintroducedforpredictingthepossibleoutcomesofafuturetestfollowedbypossibleredesigninordertooptimizethesafetymarginscontrollingadeterministicdesignprocess.Themethodwasillustratedonasimplebardesignexampleandontheconceptualdesignofanengineforasupersonicbusinessjet.However,themethodreliedontherestrictiveassumptionthatthemodelbiaswasconstantacrossthedesignspace.Inpractice,itmaybedifculttosupporttheassumptionofconstantmodelbiasintheabsenceofinitialtestdata,butifinitialtestdataisavailablethenthemodelbiascouldbecorrectedbeforeperformingtheanalysis.Therefore,amoregeneralmethodofmodelingandpropagatingepistemicmodeluncertaintyisneeded.InthischapteraKrigingsurrogateisusedtoprovideaexiblerepresentationoftheepistemicmodeluncertaintythatallowsthemethodtobeapplicabletoawiderangeofengineeringproblems. 4.2IntroductionAttheinitialdesignstageengineersoftenrelyonlow-delitymodelsthathavehighuncertainty.Thismodeluncertaintyisreducibleandisclassiedasepistemicuncertainty;uncertaintyduetovariabilityisirreducibleandclassiedasaleatoryuncertainty.Bothformsofuncertaintycanbeimplicitlycompensatedforusingconservativenesssuchasconservativematerialproperties,conservativelimitloads,safetymargins,andsafetyfactors.However,ifthedesignistooconservativethentypicallyperformancewillsuffer.Traditionalsafety-factor-baseddeterministicdesign 69

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hasreliedontestinginordertoreduceepistemicuncertaintyandachievehighlevelsofsafety.Testingisusedtocalibratemodelsandprescriberedesignwhentestsarenotpassed.Aftercalibration,reducedepistemicmodeluncertaintycanbeleveragedthroughredesigntorestoresafetyorimprovedesignperformance;however,redesignmaybeassociatedwithsubstantialcostsordelays.Integratedoptimizationofthedesign,testing,andredesignprocesscanallowthedesignertotradeoffbetweentheriskoffutureredesignandthepossibleperformanceandreliabilitybenets.Previousworkhasillustratedthistradeoffwhenthereisonlyaxedconstantmodelbias[ 20 , 21 , 23 ].Thisstudybuildsonpreviousworkbyconsideringspatialcorrelationintheepistemicmodeluncertainty.AKrigingsurrogateisusedtoprovideaexiblerepresentationoftheepistemicmodeluncertaintythatallowsthemethodtobeapplicabletoawiderangeofengineeringproblems.Inthisstudy,theepistemicmodeluncertaintyistreatedseparatelyfromthealeatoryparameteruncertaintyinthemodelinputs.Thisresultsinthechallengingtaskofpropagatingaleatoryuncertaintythroughanuncertainmodel.Furthermore,inorderforthemethodtobeapplicableundercurrentsafety-factor-baseddesignregulations[ 2 ],atraditionaldeterministicsafety-margin-baseddesignapproachisconsidered.Somestudieshaveusedtheparallelsbetweensafety-factor-baseddesignandreliability-baseddesignoptimization(RBDO)approachestoreducecomputationalcostofRBDO[ 5 – 7 ].However,thesestudieshavenotconsideredepistemicmodeluncertainty.Whenthereisonlyepistemicmodeluncertaintyasafetymarginbalancestheneedforthenaldesigntobefeasiblewhileatthesametimenotbeingsoconservativethatdesignperformancessuffer[ 8 ].Fewstudieshaveconsideredtheeffectsofbothaleatoryparameteruncertaintyandepistemicmodeluncertainty.MahadevanandRebbahaveshownthatfailingtoaccountforepistemicmodeluncertaintymayleadtoanoverestimationofreliabilityandunsafedesignsorunderestimationofthereliabilityanddesignsthataremoreconservativethanneeded[ 9 ].Studiesthatusesurrogate 70

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modelsinRBDOalsoencounterasituationofmixeduncertainty.However,unlikethisstudywhereweareinterestedinepistemicmodeluncertaintyasainherentpartofthelow-delitymodel,thesestudiesareusuallymotivatedbyadesiretoreducecomputationalcost.KimandChoihaveshownthatwhenusingresponsesurfacesinRBDOtheepistemicmodeluncertaintyresultsinuncertaintyinthereliabilityindexandadditionalsamplingcanbeusedtoavoidbeingoverlyconservative[ 10 ].Oneoftheimportantaspectsofthisstudyistheintegrationofthedesignandtestingprocess:theeffectsofafuturetestandpossibleredesignareconsideredwhileoptimizingtheinitialdesign.Sincethetestwillbeperformedinthefuture,thetestresultisanepistemicrandomvariable.Predictingpossibletestresultsrequiresaprobabilisticformulationoftherelationshipbetweenthelow-delitymodelprediction,thetruevalue,andthetestresult.Inthecontextofcalibratingcomputermodels,KennedyandO'Haganproposedthatthetruemodelcanberelatedtoacomputermodelbymultiplyingbyaconstantscaleparameterandaddingadiscrepancyfunction[ 11 ].Similarformulationshavesubsequentlybeenappliedinmanyotherstudies[ 12 – 17 ].Theseformulationsaresimilarinthattheyallrelatethetruemodeltothelow-delitymodelbyaddinganuncertaindiscrepancyfunction.Theformulationsdifferintherepresentationofthescaleparameter.Methodsrangefromomittingthescaleparameter[ 13 , 14 ]toconsideringanuncertainscalingfunction[ 16 ].Inthisstudyweconsideronlyanuncertaindiscrepancyfunctiontoformulatetherelationshipbetweenthehigh-delitymodelandthelow-delitymodel.Theuncertaindiscrepancyfunctionisconstructedinthejointdesignandaleatoryinputspaceinordertohaveepistemicmodeluncertaintiesthatarecorrelatedwithrespecttodesignandaleatoryinputs.Inadditiontotheintegrationofdesignandtesting,thisstudyalsoseekstointegratearedesignprocess.Redesignreferstochangingthedesignvariablesconditionalonthetestresult.Sincethefuturetestresultisanepistemicrandomvariablethedesignvariableafterredesignisalsorandomvariable.Villanuevaetal.developedamethodfor 71

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simulatingtheeffectsoffuturetestsandredesignwhenthereisaconstantbutunknownmodelbiasinthecalculationandmeasurement[ 19 ].Inthecontextofconstantmodelbias,Matsumuraetal.comparedRBDOconsideringfutureredesigntotraditionalRBDO[ 20 ].Villanuevaetal.alsostudiedthetradeoffbetweenfutureredesignandperformanceforanintegratedthermalprotectionsystem[ 21 ].Priceetal.comparedstartingwithamoreconservativedesignandpossiblyredesigningtoimproveperformancetostartingwithalessconservativedesignandpossiblyredesigningtoimprovesafety[ 23 ].Thesestudieshavedemonstratedthatintegratedoptimizationofdesign,testing,andredesigncanbeusedtomanageredesignriskandtradeoffbetweentheprobabilityoffutureredesignanddesignperformance.However,theassumptionofconstantmodelbiasinthesestudiesseverelylimitsthetypesofproblemswherethemethodisapplicable.InordertoapplythemethodtoabroaderrangeofgeneralengineeringproblemsthisstudyusesaKrigingmodeltorepresentmodeluncertaintywhoseconditionalsimulationsallowuncertaintypropagation.InSection 4.3 thegeneralmethodofsimulatingafuturetestandpossibleredesignisdescribed.InSection 4.4 thedemonstrationexampleofacantileverbeamisdescribed.InSection 4.5 thestudyissummarizedandtheimplicationsofthemethodandresultsarediscussed. 4.3MethodsThedesign,testing,andredesignprocessisformulateddeterministicallyintermsofaninitialsafetymarginnini,lowerandupperboundsonacceptablesafetymarginnlbandnub,andaredesignsafetymarginnre.InSection 4.3.1 theformulationoftheoptimizationofthesafetymarginsispresented.Foreachsetofsafetymargins,aMonteCarlosimulation(MCS)ofepistemicerrorrealizationsisperformedasdescribedinSection 4.3.2 .AsinglesampleintheMCSconsistsofacompletedeterministicdesign/redesignprocessasdescribedinSection 4.3.3 .TheresultsoftheMCSareusedto 72

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calculatetheprobabilityofredesign,expectedprobabilityoffailure,andexpecteddesigncostasdescribedinSection 4.3.4 . 4.3.1OptimizationofSafetyMarginsThedeterministicdesignprocessiscontrolledbyavectorofsafetymarginsn=fnini,nlb,nub,nreg.Thesafetymarginsareoptimizedtominimizeexpecteddesigncostwhilesatisfyingconstraintsonexpectedprobabilityoffailureandprobabilityofredesign.Theoptimizationofthesafetymarginsisformulatedas minnEhf(^Xnal)is.t.Eh^Pf,nalip?fprep?re(4)whereE[]isusedtodenotetheexpectationwithrespecttoepistemicuncertainty,f()isacostfunction,^Xnalisadistributionofpossiblenaldesigns,^Pf,nalisadistributionofnalprobabilityoffailure,andpreistheprobabilityofredesign.Thenaldesign^Xnalisanepistemicrandomvariablebecausethedesignmaybemodiedconditionalonthefuturetestresultwhichisunknownattheinitialdesignstage.Thenalprobabilityoffailureisanepistemicrandomvariablebecausethenaldesignisuncertainandbecausethereisepistemicmodeluncertaintyinthelimit-statefunctiong(,).Thetradeoffbetweenexpectedcostandprobabilityofredesigniscapturedbysolvingthesingleobjectiveoptimizationproblemforseveralvaluesoftheconstraintp?re.TheglobaloptimizationofthesafetymarginsisperformedusingCovarianceMatrixAdaptationEvolutionStrategy(CMA-ES)[ 82 ]andapenalizationstrategytohandletheconstraints.TheoptimizercallsasubfunctiontoperformaMCSofthedeterministicdesignprocessasshowninFigure 4-1 .TheMCSofthedeterministicdesignprocessisusedtocalculatethedistributionofpossiblenaldesigns,thedistributionofnalprobabilitiesoffailure,andtheprobabilityofredesign.Toreducethecomputationalcostofoptimizingthesafetymargins,surrogatemodelscanbetfortheexpectedcostand 73

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expectedprobabilityoffailureasafunctionofthesafetymarginsasdescribedinSection 4.3.1 . . . Optimizationofsafetymargins . Simulationofdetermin-isticdesignprocessFori=1,...,mrealizationsofmodelerror: 1. Initialdesign 2. Test 3. Calibrationifnecessary 4. Redesignifnecessary 5. Reliabilityassessment . n . Ehf(^Xnal)i,Eh^Pf,nali,pre Figure4-1. TheoptimizationofthesafetymarginsisbasedonaMCSofthedeterministicdesignprocess Inthisstudy,wedenetwodifferenttriggersforredesign.Wewillrefertoredesigntriggeredbyalowsafetymargin(lessthannlb)asredesignforsafetyandredesigntriggeredbyahighsafetymargin(greaterthannub)asredesignforperformance.Toforceonlyredesignforsafetytheupperboundonacceptablesafetymarginscanberemovedfromtheoptimizationbysettingnub=+1.Toforceonlyredesignforperformance,thelowerboundonacceptablesafetymarginscanberemovedfromtheoptimizationbysettingnlb=.Consideringonlyredesignforsafetyoronlyredesignforperformancearespecialcasesofthegeneralformulationwhereallthesafetymarginsn=fnini,nlb,nub,nregareoptimizedsimultaneously. Surrogatemodels.TheoptimizationprobleminEquation 4 maybeprohibitivelyexpensiveifaMCSisperformedforeachevaluationoftheobjectiveandconstraint 74

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Table4-1. 95%condenceintervalforrelativeerrorofsurrogatemodelsbasedonLOOCVMixedPerformanceSafety Expectedprobabilityoffailure[)]TJ /F4 11.955 Tf 9.3 0 Td[(33,25]%[)]TJ /F4 11.955 Tf 9.29 0 Td[(17,13]%[)]TJ /F4 11.955 Tf 9.3 0 Td[(17,14]%Expectedcrosssectionalarea[)]TJ /F4 11.955 Tf 9.3 0 Td[(0.12,0.12]%[)]TJ /F4 11.955 Tf 9.29 0 Td[(0.05,0.07]%[)]TJ /F4 11.955 Tf 9.3 0 Td[(0.08,0.06]% equations.Surrogatemodelswereusedtoreducethecomputationalcostoftheoptimizationofthesafetymargins.KrigingmodelsofthemeannalprobabilityoffailureEh^Pf,naliandmeannaldesigncostEhf(^Xnal)iweretasafunctionofthesafetymarginsn=fnini,nlb,nub,nreg.Themeanprobabilityoffailurewastransformedtoareliabilityindexbeforettingthesurrogatemodels.TheKrigingmodelsweretbasedonaDoEconsistingof400pointsgeneratedusingLatinhypercubesampling(LHS)andthecornerpointsinthedesignspace.EachpointintheDoErequiredaMCSofepistemicmodeluncertainty.ThesamplesizeoftheMCS(i.e.numberofconditionalsimulations)wasadaptedtoreachatargetcoefcientofvariationontheexpectednalprobabilityoffailureof5%withamaximumsamplesizeofm=5000.KrigingwithnuggetwasusedinanefforttolteroutsomeofthenoiseintroducedbyMCS.AGaussiancovariancefunctionwasusedandparameterswereestimatedbasedonMLE.Threedifferentsetsofsurrogatemodelswereconstructedcorrespondingtoamixedredesignstrategy,redesignforperformance,andredesignforsafety.Theredesignforperformanceandredesignforsafetysurrogatemodelswere3-dimensionalsurrogatemodelswhilethemixedredesignstrategyrequired4-dimensionalsurrogates.Theerrorinthesurrogatemodelswasestimatedbasedonleave-one-outcrossvalidation(LOOCV).ItshouldbenotedthatLOOCVmayoverestimatetheerrorduetothenoiselteringeffectofKrigingwithnugget.ErrorestimatesforthesurrogatemodelsarelistedinTable 4-1 . 4.3.2Monte-CarloSimulationofEpistemicModelErrorTheepistemicmodeluncertaintyandaleatoryparameteruncertaintyaretreatedseparately(see[ 66 , 67 , 85 ]).Thetruerelationshipbetweenthemodelsisassumedto 75

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beoftheform gH(x,u)=gL(x,u)+e(x,u)(4)wherex2Rdisavectorofdesignvariables,Uisavectorofaleatoryrandomvariableswitharealizationu2Rp,gH(,)isthehigh-delitymodel,gL(,)isthelow-deltymodel,ande(,)istheerrorbetweenthelow-delityandhigh-delitymodels.Typically,theerrore(,)isunknown.TheuncertaintyinthemodelerrorisrepresentedasaKrigingmodel^E(,).Thehataccentontheerrorisusedtodifferentiatebetweentherandomdistributionofpossibleerror^E(,),andtheunknown,deterministicerrore(,).Basedonthepossiblemodelerrorsthehigh-delitymodelispredictedas ^GH(x,u)=gL(x,u)+^E(x,u)(4)TheKrigingmodelforthecalculationerrorisconstructedinthejointspaceofthealeatoryvariables,u,andthedesignvariables,x.Theuncertaintyin^GH(x,u)inEquation 4 isonlyduetoepistemicmodelerror^E(,).PropagationofaleatoryuncertaintyUthroughtheuncertainmodelisdiscussedinSection 4.3.4 .Forsimplicityofnotation,wewilldenethemeanoftheKrigingpredictionfortheerrorase(,)andthemeanpredictionofthehigh-delitymodelas gH(x,u)=gL(x,u)+e(x,u)(4)Theepistemicrandomfunction^E(,)isusedtorepresentthelackofknowledgeregardinghowwellthelow-delitymodelmatchesthehigh-delitymodel.Assuminginitialtestdataisavailable,maximumlikelihoodestimation(MLE)willbeusedtoestimatetheparametersoftheKrigingmodel.Theprediction^GH(,)isviewedasadistributionofpossiblefunctions.Samplesortrajectoriesdrawnfromthisdistributionthatareconditionaloninitialtestdataarereferredtoasconditionalsimulations.Intheabsenceoftestdatatheserealizationsareunconditionalsimulations.ThesesimulationsarespatiallyconsistentMonteCarlosimulations.Let^g(i)H(,)denotethe 76

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i-threalizationof^GH(,)basedonarealization^e(i)(,)oftheKrigingmodel^E(,).Avarietyofmethodsexistforgeneratingtheseconditionalsimulations[ 86 ].Inthisstudy,theconditionalsimulationsaregenerateddirectlybasedonCholeskyfactorizationofthecovariancematrixusingtheSTKMatlabtoolboxforKriging[ 87 ]andbysequentialconditioning[ 86 ].WecanconsideraMonteCarlosimulationofmconditionalsimulationsi=1,...,mcorrespondingtompossiblefutures.Inpractice,thesamplesizemisincreaseduntiltheestimatedcoefcientofvariationofthequantityofinterest,suchasexpectedprobabilityoffailure,isbelowacertainthreshold.Letdenotetheepistemicuncertaintyspaceofthemodel^GH(,).Thereisarealization,9!2,suchthatthesimulation,^g(!)H(,),isarbitrarilyclosetothetruemodel,gH(,).ThedesignprocessconditionalononeerrorrealizationisdescribedinSection 4.3.3 .Byrepeatingthedesignprocessformanydifferenterrorrealizations(i.e.fordifferentpossiblehigh-delitymodelsthroughEquation 4 )wecandeterminethedistributionofpossiblenaldesignoutcomes.FromtheMCS,itispossibletoestimatetheriskofredesignandtopredicthowfailingatestrelatestonaldesignperformanceorsafety.Thiscaninturnbeusedtooptimizethesafetymarginsthatgovernthedeterministicdesignprocess. 4.3.3DeterministicDesignProcessThedeterministicdesignprocessiscontrolledbyavectorofsafetymarginsn.Thereisaninitialsafetymarginnini,lowerandupperboundsonacceptablesafetymarginnlbandnub,andaredesignsafetymarginnre.First,aninitialdesignisfoundbasedondeterministicoptimizationusingthemeanmodelpredictionandasafetymarginnini.Then,theoptimumdesignisevaluatedusingthehigh-delitymodeltocalculatethetruesafetymarginwithrespecttogH(,).Basedonthehigh-delityevaluation,thedesignerwillconsiderthetestpassedandkeeptheinitialdesignifthesafetymarginisgreaterthannlbandlessthannub.Thelowerboundnlbisusedtoinitiateredesignwhentheinitialdesignisrevealedtobeunsafe.Theupperboundnub 77

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isusedtoinitiateredesignwhentheinitialdesignisrevealedtobesoconservativethatitisworthwhiletoredesigntoimproveperformance.Ifthetestisfailed,acalibrationprocessisperformedtoupdatethemodelbasedonthetestresult.Finally,ifredesignisperformedanewdesignisfoundbyperformingdeterministicoptimizationusingthecalibratedmodelandasafetymarginnre.However,thefuturehigh-delityevaluationoftheinitialdesign(i.e.futuretest)isunknownandthereforemodeledasanepistemicrandomvariable.Theredesigndecision,calibration,andredesignoptimumareconditionalonaparticulartestresult.InSection 4.3.3.1 to 4.3.3.3 ,theprocessisdescribedconditionalontheerrorrealization^E(,)=^e(i)(,). 4.3.3.1InitialdesignThedesignproblemisformulatedasadeterministicsafety-margin-basedoptimizationproblem minxf(x)s.t.gH(x,udet))]TJ /F5 11.955 Tf 11.96 0 Td[(nini0(4)wheregH(,)isthemeanofthepredictedhigh-delitymodel,niniistheinitialsafetymargin,udetisavectorofconservativedeterministicvaluesusedinplaceofaleatoryrandomvariables,andf(x)isaknowndeterministicobjectivefunction.Weassumethelimit-statefunctionisformulatedsuchthatfailureisdenedasg(,)<0.LetxinidenotetheoptimumdesignfoundfromEquation 4 usinginitialsafetymarginnini.Thereisnouncertaintyintheinitialdesignxinibecausetheoptimizationproblemisdenedusingthemeanofthemodelpredictionandxedconservativevalues,udet,areusedinplaceofaleatoryrandomvariables. 4.3.3.2TestinginitialdesignandredesigndecisionApossiblehigh-delityevaluation,^g(i)H(xini,udet),oftheinitialdesignxiniissimulated.Thetestwillbepassedifnlb^g(i)H(xini,udet)nub.Ifthemeasuredsafetymarginistoolow(^g(i)H(xini,udet)
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restoresafety.Ifthesafetymarginistoohigh(^g(i)H(xini,udet)>nub)thenthedesignistooconservativeanditmaybeworthredesigningtoimproveperformance.Let^q(i)denoteanindicatorfunctionfortheredesigndecisionthatis1forredesignand0otherwise.Wewillrefertoredesigntriggeredbyalowsafetymarginasredesignforsafetyandredesigntriggeredbyahighsafetymarginasredesignforperformance.Ifthetestisnotpassedthenredesignshouldbeperformedtoselectanewdesign. 4.3.3.3CalibrationandredesignToobtainthecalibratedmodel,thetestrealization^g(i)H(xini,udet)correspondingtotheerrorinstance^e(i)(xini,udet)istreatedasanewdatapointandtheerrorinstanceisaddedtothedesignofexperimentfortheerrormodel.Theupdatedmeanofthepredictedhigh-delitymodelis g(i)H,calib(x,u)=Eh^GH(x,u)j^GH(xini,udet)=^g(i)H(xini,udet)i(4)Theredesignproblemisformulatedasadeterministicsafety-margin-basedoptimizationproblem minxf(x)s.t.g(i)H,calib(x,udet))]TJ /F5 11.955 Tf 11.95 0 Td[(nre0(4)wherethemeanofthepredictedhigh-delitymodelg(i)H,calib(,)iscalibratedconditionalonthetestresult^g(i)H(xini,udet)andnreisanewsafetymarginthatmaybedifferentthannini.Let^x(i)redenotetheoptimumdesignafterredesignfoundfromEquation 4 usingthecalibratedmodelandsafetymarginnre.ComparingtheinitialdesignprobleminEquation 4 totheredesignprobleminEquation 4 ,weseethatthereisachangeinthefeasibledesignspace.Onechangeiscontrolledbythesafetymarginnre,butthereisalsoachangebasedonthecalibratedmodelusedtocalculatethesafetymargin.Forexample,ifwechoosenini=nrethenitisstillpossibleforthefeasibledesignspacetoincreaseordecreasebasedonthecalibration.Ifthefeasibledesignspaceincreasesthensomehighperformancedesigns 79

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thatwereconsideredinfeasiblebeforethetestmaybecomefeasible.Alternatively,thefeasibledesignspacemaybereducedleadingtoworsedesignperformance.Thisrelationshipbetweenthepossiblechangeinfeasibledesignspaceandtheperformanceispreciselythechangeweareinterestedinmodelinginordertoselectthesafetymargins. 4.3.4ProbabilisticEvaluationAvectorofsafetymarginsnisassociatedwithaprobabilityofredesignpreandadistributionofnaldesigns^Xnalthattranslatesintoadistributionofprobabilityoffailureafterpossibleredesign^Pf,nal,andadistributionofdesigncostf(^Xnal).ThedistributionsareapproximatedbasedonaMonteCarlosimulationofmerrorrealizationsi=1,...,masdescribedinSection 4.3.2 .Theprobabilityofredesignispre=Eh^Qiwhere^Qistheindicatorfunctionfortheredesigndecision.Thenaldesignafterpossibleredesignis ^x(i)nal=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F4 11.955 Tf 12.2 0 Td[(^q(i)xini+^q(i)^x(i)re(4)Recall,that^q(i)=1correspondstofailingthetestandperformingredesign.TheexpecteddesigncostafterpossibleredesignisEhf(^Xnal)i.Sincetheredesigndecisiondenesapartitioningoftheepistemicoutcomespace,thelawoftotalexpectationallowstheexpectationtobewrittenas Ehf(^Xnal)i=(1)]TJ /F5 11.955 Tf 11.95 0 Td[(pre)f(xini)+preEhf(^Xre)i(4)wheref(xini)istheexpecteddesigncostconditionalonthetestbeingpassedandthedesignerkeepingtheinitialdesignandEhf(^Xre)iistheexpecteddesigncostconditionalonthetestbeingfailedandthedesignerperformingredesign.Thetrueprobabilityoffailureofthenaldesignisunknownsincethereisepistemicuncertaintyinthemodel^GH(,).Arealizationoftheprobabilityoffailureiscalculatedconditionalonanerrorrealization^E(,)=^e(i)(,).Arealizationoftheprobabilityof 80

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failurefortheinitialdesignis ^p(i)f,ini=PrUh^g(i)H(xini,U)<0i(4)wherePrU[]denotestheprobabilitywithrespecttoaleatoryuncertainty.Notethattheepistemicmodeluncertaintyistreatedseparatelyfromthealeatoryuncertaintytodistinguishbetweenthequantityofinterest,theprobabilityoffailurewithrespecttothehigh-delitymodelandaleatoryuncertainty,andthelackofknowledgeregardingthisquantity.Theerrorinthelow-delitymodel^E(,)hasnoimpactonthereliabilitywithrespecttothehigh-delitymodelgH(,).However,sincethehigh-delitymodelisunknown,theprobabilityoffailurecalculationisrepeatedmanytimesconditionalonallpossiblerealizationsofthehigh-delitymodel^g(i)H(,)asshowninEquation 4 .Arealizationofthenalprobabilityoffailureafterpossibleredesignis ^p(i)f,re=PrUh^g(i)H(^x(i)re,U)0i(4)Afterredesign,thedesignvariable^x(i)reisalsoanepistemicrandomvariableinadditiontothelimitstatefunction^g(i)H(,).Manydifferentmethodsareavailableforcalculatingtheprobabilityoffailure.Inthisstudy,rstorderreliabilitymethod(FORM)isusedtocalculatetheprobabilityoffailureforeachepistemicrealization.Thenalprobabilityoffailureafterpossibleredesignis ^p(i)f,nal=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 11.955 Tf 12.2 0 Td[(^q(i)^p(i)f,ini+^q(i)^p(i)f,re(4)Notethattheredesigndecision^q(i)shapesthenalprobabilityoffailuredistributionbecausewewillhavetheopportunityinthefuturetocorrecttheinitialdesignifitfailsthedeterministictest.TheexpectedprobabilityoffailureafterpossibleredesignisEh^Pf,nali.Asabove,theexpectationcanbewrittenas Eh^Pf,nali=(1)]TJ /F5 11.955 Tf 11.95 0 Td[(pre)Eh^Pf,inij^Q=0i+preEh^Pf,rej^Q=1i(4) 81

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wheretheEh^Pf,inij^Q=0iistheexpectedprobabilityoffailureconditionalonthetestbeingpassedandthedesignerkeepingtheinitialdesignandEh^Pf,rej^Q=1iistheexpectedprobabilityoffailureconditionalonthetestbeingfailedandthedesignerperformingredesign. 4.4DemonstrationExample 4.4.1OverviewThedemonstrationproblemisadaptedfromanexamplebyWuetal.[ 6 ].Theexampleisthedesignofacantileverbeamtominimizemasssubjecttoaconstraintontipdisplacement.TheoriginalprobleminvolvedthedesignofalongslenderbeamandthereforeusedEuler-Bernoullibeamtheory.Inthisexample,thelengthofthebeamisreducedsuchthatshearstresseffectsbecomeimportantandTimoshenkobeamtheoryismoreaccurate.Thelow-delitymodelofthetipdisplacementisformulatedbasedonEuler-Bernoullibeamtheoryandthehigh-delitymodelisformulatedbasedonTimoshenkobeamtheory.Thedesignoptimization(equations 4 and??)isperformedusingsequentialquadraticprogramming(SQP).Thelow-delitymodelofthelimitstatefunctionis gL(x,U)=d?)]TJ /F4 11.955 Tf 15.28 8.09 Td[(4l3 ewts FY t22+FX w22(4)wherex=fw,tgarethedesignvariablesandU=fFX,FYgarethealeatoryvariables.Thehigh-delitymodelofthelimitstatefunctionis gH(x,U)=d?)]TJ /F11 11.955 Tf 11.96 10.74 Td[(p (dx(x,U))2+(dy(x,U))2(4)wheredxanddyaregivenbyequations 4 and 4 .TheproblemparametersaredescribedinTable 4-2 . dx(x,U)=3lFX 2gwt+4l3FX ewt3(4) dy(x,U)=3lFY 2gwt+4l3FY ew3t(4) 82

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Table4-2. ParametersforcantileverbeamexampleParameterNotationValue Designvariables,xWidthofcrosssectionw2.5w5.5inThicknessofcrosssectiont1.5t4.5inAleatoryvariables,UHorizontalloadFXFXN(500,1002)lbsVerticalloadFYFYN(1000,1002)lbsConstantsElasticmoduluse29106psiShearmodulusg11.2106psiLengthofbeaml10inAllowabletipdisplacementd?2.2510)]TJ /F9 7.97 Tf 6.59 0 Td[(3inConservativealeatoryvaluesudetf664.5,1164.5glbsTargetmeanprobabilityoffailurep?f1.3510)]TJ /F9 7.97 Tf 6.59 0 Td[(3 Theobjectivefunctionisthecross-sectionalareaofthebeam f(x)=wt(4) 4.4.2ErrorModelItisassumedthatsomepreliminarytestdataisavailableforconstructingthesurrogatemodel^E(x,U).Inthisexample,thepreliminarytestdatacorrespondstoevaluationsofgH(,)atthe16cornerpointsofthejointdesign-aleatoryspace.Thecornerpointswereselectedforillustrationpurposesinordertoensurethereisreasonablyhighepistemicmodeluncertaintyforpointsinsidethedesigndomain.Inpractice,otherdesignsofexperiments(DoE)couldbeusedoranyavailabletestdatacouldbeusedtoconstructtheerrormodel.ThedesignspaceisdenedaccordingtotheboundsonxinTable 4-2 andboundsonUcorrespondingto)]TJ /F4 11.955 Tf 9.29 0 Td[(2to+7.BasedonthisDoEtheparametersfortheKrigingerrormodelareestimatedusingmaximumlikelihoodestimation(MLE).AGaussiancovariancefunctionwasselectedfortheKrigingmodel.Theerrormodelisconstructedinthejointspaceofdesignvariables,x,and 83

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aleatoryvariables,U.Recall,thedesignoptimizationproblemisformulatedusingxedconservativevalues,U=udet.InFigure 4-2 ,thedesignoptimizationproblemisshownalongwiththe95%condenceintervalofmodeluncertainty.InFigure 4-2 ,thereliabilityanalysisisshownfortheoptimumdesignfoundusingnini=0alongwiththe95%condenceintervalofmodeluncertainty.SelectingadifferentdesignbyusingadifferentsafetymarginwillaltertheplotshowninFigure 4-2 .However,theplotisprovidedasanexampletoshowhowthemodeluncertaintyresultsinawidecondenceintervalinthealeatoryspace.Thewidecondenceintervalsinaleatoryspacewillresultinhighuncertaintyintheprobabilityoffailure.Themeanandvarianceofthemodelerrorvarywithdesignvariables,x,andaleatoryvariables,UasshowninFigure 4-3 .ThevarianceiszeroatthecornersofthedesignspacesincethesepointscorrespondtosamplelocationsintheDoE.Althoughtheabsolutevaluesoftheerrorappearsmall,theerrorissignicantrelativetothemodelpredictionsoftipdisplacement.Forexample,theallowabletipdisplacementinthisexampleisd?=2.2510)]TJ /F9 7.97 Tf 6.59 0 Td[(3inches. 4.4.3ResultsTradeoffcurvesforexpectedcostversusprobabilityofredesignareshowninFigure 4-4 .Forzeroprobabilityofredesign,theproblemreducestondinganinitialsafetymargin,nini,thatminimizesthemassoftheinitialdesign,f(xini),whileensuringthatthemeanprobabilityoffailurefortheinitialdesign,Eh^Pf,inii,satisesthereliabilityconstraint.Withincreasingprobabilityofredesign,redesigncanbeusedtoimprovesafetyiftheinitialdesignisrevealedtobedangerousorimproveperformanceiftheinitialdesignisrevealedtobetooconservative.Redesignforsafetyallowsforalighterinitialdesignbecausetheinitialdesignwillbecorrectedifthetipdisplacementislaterrevealedtobetoohigh(i.e.unsafe).Ifredesignisrequired,thenalbeamwillbecomeheavierduringredesignbecausemakingthebeamstiffer(i.e.safer)resultsinamassincrease.Redesignforperformancestartswithaheavierinitialdesignandredesignwillbeperformedifthetipdisplacementoftheinitialdesignislaterrevealedtobetoolow 84

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A BFigure4-2. Thegureontheleftshowsthedesignoptimizationwhenusingasafetymarginnini=0andxedconservativevaluesudetinplaceofaleatoryvariablesU.Thegureontherightshowsthereliabilityoftheoptimumdesignfoundontheleftbyplottingthelimit-statefunctioninstandardnormalspace. (i.e.veryconservativeorsafe).Ifredesignisrequired,thenaldesigncanbemadelighterduringredesignbecausethebeamcanbemadelessstiff.Itisobservedthatredesignforsafetyresultsinalowermeanmassthanredesignforperformance.Itisalsoobservedthatamixedredesignstrategyoffersaslightimprovementoverredesignforsafety.However,asindicatedbytheerrorbarsitisnotclearifthisdifferenceisaresultofbiasinthesurrogatemodelsusedwhenoptimizingthesafetymargins.Thesimulationresultscanbeexploredinmoredetailbylookingatasinglepointonthetradeoffcurve.Thesafetymarginscorrespondingto20%probabilityofredesignwereselectedformoredetailedinvestigation.Figure 4-5 showsthedistributionofpossiblehigh-delitysafetymarginsfortheinitialdesignthatarepredictedbasedonthemodelerror.Bothdistributionscapturethetruesafetymarginifweweretoevaluatetheinitialdesignusingthehigh-delitymodel.Inthecaseofredesignfor 85

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A B C DFigure4-3. Ontheleft,themeanandvarianceoftheerrorareplottedinanormalizeddesignspacewithxedconservativevaluesudetinplaceofaleatoryvariablesU.Ontheright,themeanandvarianceoftheerrorareplottedinstandardnormalaleatoryspaceforoptimumdesignfoundusingnini=0.Theerrorisininches. 86

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performance,weseethatthetruesafetymarginisgreaterthannubandthereforeredesignwouldberequired.Ifwecalibrateusingthetruehigh-delityevaluationandperformredesignthetruesafetymarginisnowveryclosetonrewhichagreeswiththepredictedchangeinthesafetymargin.Figure 4-6 showsthejointdistributionsofthedesignvariablescorrespondingtothewidthandthicknessofthebeamcrosssection.Thepeakinthedistributionscorrespondstotheinitialdesign,xini.Thesafetymarginshavebeenoptimizedsuchthatthereisan80%probabilitythatthedesignwillnotrequireanychangesafterthefuturetest.Theotherdesignsinthegurecorrespondtofailingthefuturetestandperformingredesign.Figure 4-7 showsthedistributionsofcross-sectionalareacorrespondingtothedesignsinFigure 4-6 .Themassisreducedifredesignforperformanceisrequiredandthemassisincreasedifredesignforsafetyisrequired.Wecanseeinthedistributionofcrosssectionalareaforredesignforperformancethatthepredictedmassreductionafterredesignisclosetothetruevalue.ComparingthesafetymargindistributionsinFigure 4-5 tothereliabilityindexdistributionsinFigure 4-8 weobservesimilardistributionshapes.Bothdistributionscapturethetruereliabilityindexoftheinitialdesignascalculatedwithrespectthehigh-delitymodel.Afterredesignforperformancethetruereliabilityindexisreducedinordertoreducethemassofthebeam.Thetruereliabilityindexafterredesignfallswithinthepredicteddistributionofpossiblenalreliabilityindexes.Histogramsofthemost-probablepoint(MPP)areshowninFigure 4-9 .ThexeddeterministicvaluesweselectedudetareslightlyoutsidethedistributionofpossibleMPP's.However,thevaluesarenottotallyunreasonablesincetheyaremuchclosertothecenterofthedistributionthan,forexample,themeanofthedistributionswhichislocatedat[0,0]. 4.5DiscussionandConclusionsInthisstudywedescribedamethodfortheoptimizationofasafety-margin-baseddesignprocessthatallowsthedesignertotradeoffbetweentheexpecteddesignperformanceandprobabilityofredesign.Previousstudiesontheoptimizationofsafety 87

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A BFigure4-4. Tradeoffcurvesforexpectedcost(crosssectionalareainsquareinches)asafunctionofprobabilityofredesign.Thecurvelabeled“mixed”correspondstosimultaneousoptimizationofn=fnini,nlb,nub,nreg.Thecurvelabeled“safety”correspondstooptimizingfnini,nlb,nregwithnub=+1.Thecurvelabeled“performance”correspondstooptimizingfnini,nub,nregwithnlb=.Errorbarsarebasedonsurrogatemodelsusedduringoptimization. marginswhenconsideringfutureredesignrequiredanassumptionofconstantmodelbias[ 20 , 21 , 23 ].However,inengineeringdesignproblemsthemodelbiasmayvarywiththedesignvariablesaswellasthealeatoryvariables,suchasinthecaseofthecantileverbeamexample.ThisstudyimprovesonpreviousworkbyintroducingaKrigingmodelasamoregeneralmodeloftheepistemicuncertaintyinthelow-delitymodel.TheKrigingmodeloffersseveralpracticalbenetsoverthepreviousmethod.Inparticular,theKrigingmodeleasilyallowsfortheincorporationofpreliminaryhigh-delitydataandasimplecalibrationofthemodelwhennewhigh-delitydatabecomesavailable.TheKrigingerrormodelcapturestheintuitiveideathatthevarianceofthemodelerrorisgreatestinunexploredregionsandaminimumatexistingdatapoints.TheKrigingmodelalsoprovidesseveraltheoreticalimprovementsofthemethod.One 88

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A BFigure4-5. Histogramsofpossiblesafetymargindistributionsfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. A BFigure4-6. Jointdistributionofdesignvariablesforpossiblenaldesignsfor20%probabilityofredesign.Peakislocatedatinitialdesign. 89

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A BFigure4-7. Histogramsofcross-sectionalareadistributionsfor20%probabilityofredesign. A BFigure4-8. Histogramsofreliabilityindexdistributionsfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. 90

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A BFigure4-9. Jointdistributionofpossiblemostprobablepoints(MPP's)for20%probabilityofredesign. benetisthatitislikelythatthereexistsarealizationtakenfromtheKrigingmodelthatisarbitrarilyclosetotheactualerrorbetweenthelowandhigh-delitymodels.Therefore,itislikelythattherealsoexistsarealizationoftheprobabilityoffailurethatisclosetothetrueprobabilityoffailurewithrespecttothehigh-delitymodel.Inaddition,itislikelytheKrigingmodelwillconvergetothetrueerrorasmorehigh-delityevaluations(ortests)areperformed.IftheKrigingmodelconvergestothetrueerror,thenthedistributionofprobabilityoffailurewillalsoconvergetothetrueprobabilityoffailurewithrespecttothehigh-delitymodel.Previousworkwasnotcapableofmodelingtheconvergenceofthemodelerrorbecauseundertheassumptionofconstantmodelbiasonlyasinglehigh-delityevaluationwasnecessarytoremoveallepistemicuncertainty.Themethodwasappliedtoasimplecantileverbeamdesignproblemofminimizingthemass,orequivalentlycross-sectionalarea,subjecttoaconstraintontip-displacement.Onlyafewhigh-delityevaluationswereneededtoconstructtheKrigingmodelthatwasusedtoprovidethedistributionofmodeluncertainty.Adistributionofprobabilityof 91

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failurewasobtainedthroughthecombinationofFORMandaMCSoferrorrealizations(i.e.conditionalsimulations).Itwasshownthatthedistributionofpossiblereliabilityindexescapturedthetruereliabilityindexoftheinitialdesignwithrespecttothehigh-delitymodel.Furthermore,itwasshownthatthepredictedchangeinreliabilityafterredesignagreedwiththeactualredesignoutcomewhenthehigh-delitymodelwasevaluatedandredesignwasperformed.Thesafetymarginsgoverningadeterministicdesignprocesswereoptimizedtotradeoffbetweentheprobabilityofredesignandtheexpectedmassofthenaldesign.Itwasshownthatthepredictedmassreduction(i.e.performanceimprovement)agreedwiththeactualchangeinperformanceafterevaluatingthehigh-delitymodelandperformingredesign.Forthisexample,itwasfoundthatitwasbettertostartwithalessconservative,lighterdesignandimplementatestandredesignprocessthatwouldrestoresafetyiftheinitialdesignwaslaterrevealedbythehigh-delitymodeltobeunsafe.Thisprocesswascontrastedwithstartingwithamoreconservative,heavierdesignandimplementingatestandredesignprocessthatwouldimprovedesignperformanceiftheinitialdesignwaslaterrevealedbythehigh-delitymodelbetooconservative.Amixeddesignstrategywhereredesignwouldrestoresafetyorimproveperformanceconditionalontheresultsofthehigh-delityevaluationwasfoundtobecomparabletotheredesignforsafetyapproach.Itishypothesizedthatthebestredesignstrategyisproblemdependent.Ingeneral,thereisnoneedtospecifyaredesignforsafetyorredesignforperformanceaprioribecausewhenallowedtocontrolallthesafetymarginstheoptimizerwillconvergetothebestredesignstrategy. 92

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CHAPTER5SOUNDINGROCKETDESIGNUNDERMIXEDEPISTEMICMODELUNCERTAINTYANDALEATORYPARAMETERUNCERTAINTY 5.1ResearchContextinRelationtoScopeofDissertationInChapter 3 ,amethodwasintroducedforpredictingthepossibleoutcomesofafuturetestfollowedbypossibleredesigninordertooptimizethesafetymarginscontrollingadeterministicdesignprocess.InChapter 4 ,themethodwasmodiedtoremovetherestrictiveassumptionofconstantmodelbiasandthemethodwasdemonstratedonasimplecantileverbeamdesignexample.Thecantileverbeamexamplewasusefulforillustratingthemethod,butitismuchsimplerthantypicalengineeringdesignproblems.Inthischapter,asoundingrocketdesignexampleisusedtoillustratethemethodonacomplexdesignproblem.Thesoundingrocketdesignexampleissignicantlymorecomplexthanpreviousexamplesduetothe: 1. Increasedcomputationalcostofthemodels 2. Increasednumberofdesignvariables 3. Multi-disciplinarydesignconsiderations 4. Designvariablesthatareuniquetothehigh-delitymodel 5. Epistemicuncertaintyintheobjectivefunction 6. Additionaldeterministicdesignconstraints.ThemethodfromChapter 4 ismodiedtoreducethecomputationalcostsothemethodismorereadilyapplicabletorealisticdesignproblems.Theconstraintonmeanprobabilityoffailurefromchapters 3 and 4 isreplacedwithamoreconservativequantileconstrainttoensurethatsomeverysafedesignsdonotcancelouttheriskofobtainingadangerousnaldesign.Thecantileverbeamexampleisrevisitedtoillustratethechangestothemethodbeforeapplyingthemethodtothesoundingrocketdesignexample. 93

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5.2IntroductionAttheinitialdesignstageengineersusuallyrelyonlow-delitymodelsthathavehighepistemicuncertainty.Uncertaintyistypicallyclassiedasaleatoryorepistemic[ 31 , 37 , 67 ].Epistemicuncertaintyisduetolackofknowledge,isreduciblebygainingmoreinformation,andhasaxedbutunknownvalue.Aleatoryuncertaintyisduetovariability,isirreducible,andisadistributedquantity.Inengineeringdesign,asystemistypicallydesignedtoberobustwithrespecttoaleatoryvariablessuchasenviromentalconditionsormaterialvariability.Therobustnessofthesystemtoaleatoryuncertaintymaybecontrolledimplicitlythroughsafetymargins,safetyfactors[ 2 ],andconservativedesignvalues[ 50 ]orexplicitlythroughreliability-baseddesignmethods.However,therearerelativelyfewdesignmethodsthatconsiderepistemicmodeluncertainty[ 9 , 10 , 79 ].Ifthereishighepistemicmodeluncertainty,thentheremaybesignicantepistemicuncertainty(i.e.lackofknowledge)regardingthereliabilityoftheas-builtsystem.Errorsinlow-delitymodels,whichmaybeconsideredindicativeoferrorsinreliabilityestimates,areoftenrevealedinthefuturewhenhigherdelitysimulationsareperformedorprototypesaretested.Ifimprovedmodelingrevealssignicantdiscrepanciesbetweenlowandhigh-delitysimulationsorbetweensimulationsandprototypes,redesignmayberequiredtocorrecttheinitialdesign.Redesign,alsoknownasengineeringchange,istheprocessofrevisinganinitialdesignconditionalonnewknowledge[ 69 ].Typically,redesignisperformedifalow-delitymodelisrevealedtohaveunconservativebiasthatmayindicateanunsafeinitialdesign.Redesignisalsobenecialwhenaninitialdesignisrevealedtobeoverlyconservativesuchthatthedesignperformancecanbesignicantlyimproved.Redesignprovidesanopportunityfordesignimprovement,however,itisoftenviewedasaprobleminindustrybecauseredesignmaybeassociatedwithsubstantialcostsanddelays[ 68 ].Designerscouldbenetfromcontrollingtheprobabilityoffutureredesignandtradingoffbetweentheprobabilityofredesignanddesignperformance[ 21 ]. 94

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However,predictinghowthereliabilityandperformancemaychangeconditionalonfutureredesignisacomplexandcomputationallyexpensivetask.Evenwithoutconsideringredesign,thereissignicantcomputationalcostinvolvedinmixedepistemicandaleatoryuncertaintypropagation.ForexampleinatwolevelMonte-Carlosimulation(MCS),foreachepistemicrealizationsampledintheouterloopmanyaleatoryrealizationsaresampledandpropagatedthroughdesignmodelsintheinnerloopinordertocalculateadistribution,orfamily,ofdistributions[ 66 ].Two-leveluncertaintypropagationiscomputationallycostly,butprovidesthecompletedistributionofprobabilityoffailurewhichcanbeusedtocalculateavarietyofusefulstatistics,suchascondenceintervals[ 67 ].Alternatively,amodelwithepistemicmodeluncertaintycouldbereplacedwithaconservativeprediction,suchasmeanpluskstandarddeviationoffset,inordertoavoidtheexpensivetwo-leveluncertaintypropagation[ 10 , 56 ].However,theformerapproachallowsforprecisereliabilitystatementssuchas“webelievewith1-condencethattheprobabilityoffailureislessthanpf”whereastheinterpretationofthelatterapproachislessstraightforwardandmayonlyyield“pseudo-condencebounds[ 56 ]”.Thereliabilityassessmentbecomesmorecomplexwhenweconsiderthatthedesignvariablesareepistemicrandomvariables.Thatis,ifthereissomeprobabilityoffutureredesignthenthenaldesignisanepistemicrandomvariablebecauseitisunknown(e.g.incomplete,imprecise,oruncertainspecication)attheinitialdesignstage.Inthisstudy,weproposeadesignmethodthatconsidersmixedepistemicmodeluncertaintyandaleatoryparameteruncertaintyandincludesthepossibilityoffutureredesign.Itwillbeshownthatredesignactsasatypeofqualitycontrolmeasureforepistemicuncertaintybyimplementingdesignchangesinresponsetoextremeepistemicrealizations.Intheproposedmethod,aleatoryandepistemicuncertaintiesinthereliabilityassessmentarehandledsequentiallyratherthaninanestedfashion.Inapreliminarystep,traditionalRBDOisperformedwithrespecttoaleatoryparameter 95

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uncertaintyusingthemeanlow-delitymodelinordertondthemostprobablepoint(MPP)ofthealeatoryrandomvariableswithrespecttothemeanlow-delitymodel.Insubsequentsteps,aleatoryrandomvariablesarexedatthisMPPandakstandarddeviationoffsetisusedasasafetymarginwithrespecttoepistemicmodeluncertainty.Aninitialdesignisfoundbasedondeterministicoptimizationusingastandarddeviationoffsetkini.Inthefuture,theinitialdesignwillbetested(i.e.thehigh-delitymodelwillbeevaluatedattheinitialdesign)andtheredesigndecisionwillbebasedontheobserveddiscrepancybetweenthelowandhigh-delitymodels.Iftheobserveddiscrepancyislessthanklborabovekubthenredesignwillbeperformed.Duringredesignapossiblydifferentstandarddeviationoffsetkreisused.Theoutcomeofthefuturehigh-delityevaluation(i.e.futuretest)isunknownattheinitialdesignstageandthereforethedesignprocessisrepeatedinaMCS.TheMCSallowsforthecalculationoftheprobabilityofredesignandapredictionofhowfutureredesignisrelatedtonaldesignperformanceandreliability.Thestandarddeviationoffsetsk=fkini,klb,kub,kreggoverningthedesignprocessareoptimizedtominimizetheexpectedvalueoftheobjectivefunctionwhilesatisfyingconstraintsonreliabilityandprobabilityofredesign.Incontrasttopreviousworkonsimulatingtheeffectsofafuturetestandredesign[ 19 – 21 ],thisstudyaccountsforspatialcorrelationsinepistemicmodeluncertaintybyusingaKrigingmodeltorepresentmodeluncertaintyandsignicantlyreducesthecomputationalcostbyproposingacomputationallycheapapproximationofthereliabilityconstraint.Aftertheoptimizationofthestandarddeviationoffsets,thecompleteprobabilityoffailuredistributionisrecoveredthroughtwo-leveluncertaintypropagation.InSection 5.3 thegeneralmethodofsimulatingafuturetestandpossibleredesignisdescribed.InSection 5.4 themethodisdemonstratedonacantileverbeambendingexampleandthenamultidisciplinarysoundingrocketdesignproblem.InSection 5.5 thestudyissummarizedandtheimplicationsofthemethodandresultsarediscussed. 96

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5.3MethodsInSection 5.3.1 ,theconservativevaluesthatwillbeusedinplaceofaleatoryrandomvariablesarefoundbasedonpreliminaryRBDO.InSection 5.3.2 ,theformulationoftheoptimizationofthestandarddeviationoffsetsispresented.TheMonteCarlosimulation(MCS)ofepistemicerrorrealizationsisdescribedinSection 5.3.3 .AsinglesampleintheMCSconsistsofacompletedeterministicdesign/redesignprocessasdescribedinSection 5.3.4 .InSection 5.3.5 ,thecalculationoftheexpectedobjectivefunctionvalue,probabilityofredesign,andprobabilityoftheprobabilityoffailureexceedingatargetvaluearedescribed. 5.3.1PreliminaryReliability-BasedDesignOptimization(RBDO)Preliminaryreliability-baseddesignoptimization(RBDO)isperformedusingthemeanlow-delitymodelofthelimit-statefunctionandconsideringonlyaleatoryuncertainty.Insubsequentsteps,aleatoryrandomvariablesarexedattheMPPasthedesignisoptimizeddeterministically.ThepreliminaryRBDOproblemisformulatedas minEU[f(x,U)]w.r.txs.t.PU[g(x,U)0]p?f(5)whereEU[]isanexpectationoperatorwithrespecttoaleatoryuncertainty,PU[]isaprobabilityoperatorwithrespecttoaleatoryuncertainty,f(,)istheobjectivefunction,x2Rdisavectorofdesignvariables,Uisavectorofaleatoryrandomvariableswitharealizationu2Rp,gH(,)isthemeanlimit-statefunction,andp?fisthetargetprobabilityoffailure.TheformulationofthesearchfortheMPPoftheRBDOoptimumxRBDOis minjjujjw.r.tus.t.g(xRBDO,u)0(5) 97

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SincetheRBDOproblemdoesnotconsiderepistemicmodeluncertaintyinthelimit-statefunctionthereisahighprobabilitythattheresultingoptimumcouldbeveryunsafeorveryconservative.However,thecomputationalcostoftheoptimizationproblemismuchlowerthanformulatinganoptimizationwithfulltwo-levelmixedepistemic/aleatoryuncertaintypropagation.Thetaskoflocatingadesignthatisconservativewithrespecttoepistemicmodeluncertainty,butnotoverlyso,willbeaddressedintheremainderoftheproposedmethod. 5.3.2OptimizationofStandardDeviationOffsetsTheoptimizationofthestandarddeviationoffsets(i.e.safetymargins)isformulatedas minEE[EU[f(Xnal,U)]]w.r.tk=fkini,)]TJ /F5 11.955 Tf 9.3 0 Td[(klb,kub,kregs.t.PE[Pf(Xnal)p?f]prep?re0k4(5)whereEE[]anexpectationoperatorwithrespecttoepistemicuncertainty,Xnalisavectorofnaloptimumdesignvariables,PE[]isaprobabilityoperatorwithrespecttoepistemicuncertainty,Pf()istheprobabilityoffailurewithrespecttoaleatoryuncertainty,p?fisthetargetprobabilityoffailure,1)]TJ /F8 11.955 Tf 12.46 0 Td[(isthedesiredcondencelevel,andpreistheprobabilityofredesign.Thenaldesign,Xnal,isuncertainbecauseweconsiderthepossibilitythatthedesignmayneedtoberedesignedinthefutureconditionalontheoutcomeofahigh-delityevaluationoftheinitialdesign.Theprobabilityoffailure,Pf(),isuncertainbecausethereisepistemicmodeluncertaintyinthelimit-statefunctionandbecausethedesignisuncertain.Thetradeoffbetweentheexpectedobjectivefunctionvalueandprobabilityofredesigniscapturedbysolvingthesingleobjectiveoptimizationproblemforseveralvaluesoftheconstraintp?re.The 98

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globaloptimizationisperformedusingCovarianceMatrixAdaptationEvolutionStrategy(CMA-ES)[ 82 ]withapenalizationstrategytohandletheconstraints.Thecomputationalcostofthestandarddeviationoffsetsoptimizationproblemishighduetothemixedepistemicandaleatoryuncertaintyinthereliabilityconstraint.Toreducethecomputationalcost,thereliabilityconstraintisapproximatedas PE[Pf(Xnal)p?f]PE[GH(Xnal,udet)0](5)whereudetisavectorofxedconservativevaluesusedinplaceofaleatoryvariablescorrespondingtotheMPPasfoundin 5 , 5 andGH(,)isanuncertainlimit-statefunction.Thetrueprobabilityontheleft-handsideof 5 requirestwo-leveluncertaintypropagation,buttheapproximationontherightonlyconsidersepistemicuncertaintyandisthereforeonlyrequiressingleleveluncertaintypropagation.Theapproximationisinspiredbystudiesonreliability-baseddesignconsideringonlyaleatoryuncertaintywherethereliabilityconstraintisconvertedtoanequivalentdeterministicconstraint[ 5 – 7 ].Therearetwoelementsthatcontributethetheerrorintheproposedapproximation.First,theMPPisanepistemicrandomvariableduetomodeluncertaintysoanysinglepointestimatewillincursomedegreeoferror.Second,thenaldesignisanepistemicrandomvariableandwilldifferfromxRBDOwheretheMPPsearchwasperformed.ItisassumedthattheMPPwithrespecttothemeanlimit-statefunctiong(,)isareasonableapproximationofthemeanMPPwithrespecttotherealizationsoftheuncertainlimit-statefunctionG(,).Thatis,itisassumedtheMPPofthemeanisclosetothemeanoftheMPP's.Furthermore,itisassumedthatthedistributionofnaldesignsXnalwillbecenterednearxRBDO.Theapproximationisintroducedtoreducedthecostoftheoptimizationofthestandarddeviationoffsets.Thefulltwo-leveluncertaintypropagationisperformedfortheoptimumstandarddeviationoffsetsinordertorecoverthefullprobabilityoffailuredistributionandassesstheaccuracyoftheapproximation. 99

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5.3.3Monte-CarloSimulationofEpistemicModelErrorTheepistemicmodeluncertaintyandaleatoryparameteruncertaintyaretreatedseparately(see[ 66 , 67 , 85 ]).Thetruerelationshipbetweenthedifferentdelitymodelsisassumedtobeoftheform gH(x,u)=gL(x,u)+e(x,u)(5)wheregH(,)isthehigh-delitymodel,gL(,)isthelow-deltymodel,ande(,)istheerrorbetweenthelow-delityandhigh-delitymodels.Typically,theerrore(,)isunknown.TheuncertaintyinthemodelerrorisrepresentedasaKrigingmodelE(,).Basedonthepossiblemodelerrorsthehigh-delitymodelispredictedas GH(x,u)=gL(x,u)+E(x,u)(5)TheKrigingmodelfortheerrorisconstructedinthejointspaceofthealeatoryvariables,u,andthedesignvariables,x.TheuncertaintyinGH(x,u)in 5 isonlyduetoepistemicmodelerrorE(,).PropagationofaleatoryuncertaintyUthroughtheuncertainmodelisdiscussedinSection 5.3.5 .Forsimplicityofnotation,wewilldenethemeanoftheKrigingpredictionfortheerrorase(,)andthestandarddeviationasE(,).Themeanpredictionofthehigh-delitymodelis gH(x,u)=gL(x,u)+e(x,u)(5)withstandarddeviationG(,)=E(,).TheepistemicrandomfunctionE(,)isusedtorepresentthelackofknowledgeregardinghowwellthelow-delitymodelmatchesthehigh-delitymodel.Assuminginitialtestdataisavailable,maximumlikelihoodestimation(MLE)willbeusedtoestimatetheparametersoftheKrigingmodel.ThepredictionGH(,)isviewedasadistributionofpossiblefunctions.Samplesortrajectoriesdrawnfromthisdistributionthatareconditionaloninitialtestdataarereferredtoasconditionalsimulations.In 100

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theabsenceoftestdatatheserealizationsareunconditionalsimulations.ThesesimulationsarespatiallyconsistentMonteCarlosimulations.Letg(i)H(,)denotethei-threalizationofGH(,)basedonarealizatione(i)(,)oftheKrigingmodelE(,).Avarietyofmethodsexistforgeneratingtheseconditionalsimulations[ 86 ].Inthisstudy,theconditionalsimulationsaregenerateddirectlybasedonCholeskyfactorizationofthecovariancematrixusingtheSTKMatlabtoolboxforKriging[ 87 ]andbysequentialconditioning[ 86 ].WecanconsideraMonteCarlosimulationofmconditionalsimulationsi=1,...,mcorrespondingtompossiblefutures.Inpractice,thesamplesizemisincreaseduntiltheestimatedcoefcientofvariationofthequantityofinterestisbelowacertainthreshold.LetdenotetheepistemicuncertaintyspaceofthemodelGH(,).Thereisarealization,9!2,suchthatthesimulation,g(!)H(,),isarbitrarilyclosetothehigh-delitymodel,gH(,).ThedesignprocessconditionalononeerrorrealizationisdescribedinSection 5.3.4 .Byrepeatingthedesignprocessformanydifferenterrorrealizations(i.e.fordifferentpossiblehigh-delitymodelsthrough 5 )wecandeterminethedistributionofpossiblenaldesignoutcomes. 5.3.4DeterministicDesignProcessThedeterministicdesignprocessiscontrolledbyavectorofstandarddeviationoffsetsk.Thedesignprocessconsistsofndinganinitialdesign,testingtheinitialdesignbyevaluatingitwiththehigh-delitymodel,andpossiblecalibrationandredesign.Thefuturehigh-delityevaluationoftheinitialdesign(i.e.futuretest)isunknownandthereforemodeledasanepistemicrandomvariable.Theredesigndecision,calibration,andredesignoptimumareconditionalonaparticulartestresult.InSection 5.3.4.1 ,Section 5.3.4.2 ,Section 5.3.4.3 theprocessisdescribedconditionalontheerrorrealizationE(,)=e(i)(,). 101

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5.3.4.1InitialdesignThedesignproblemisformulatedasadeterministicoptimizationproblem minf(x,udet)w.r.txs.t.gH(x,udet))]TJ /F5 11.955 Tf 11.95 0 Td[(kiniG(x,udet)0(5)wheregH(,)isthemeanofthepredictedhigh-delitymodel,kiniistheinitialstandarddeviationoffset,udetisavectorofconservativedeterministicvaluesusedinplaceofaleatoryrandomvariables,andG(,)isthestandarddeviationofthelimit-statefunctionwithrespecttoepistemicmodeluncertainty.Weassumethelimit-statefunctionisformulatedsuchthatfailureisdenedasgH(,)<0.Letxinidenotetheoptimumdesignfoundfrom 5 .Thereisnouncertaintyintheinitialdesignxinibecausetheoptimizationproblemisdenedusingthemeanofthemodelpredictionandxedconservativevalues,udet,areusedinplaceofaleatoryrandomvariables. 5.3.4.2TestinginitialdesignandredesigndecisionApossiblehigh-delityevaluation,g(i)H(xini,udet),oftheinitialdesignxiniissimulated.Thetestwillbepassedifnlbg(i)H(xini,udet)nubwherenlbandnubcorrespondtolowerandupperboundsonacceptablesafetymargins.Theredesigndecisioncanbeformulatedintermsofstandarddeviationoffsetsasklbz(i)inikubwhere Zini=GH(xini,udet))]TJ /F4 11.955 Tf 12.2 0 Td[(g(xini,udet) G(xini,udet)(5)Iftheobservedsafetymarginistoolow(g(i)H(xini,udet)nub)thenthedesignistooconservativeanditmaybeworthredesigningtoimproveperformance.Letq(i)denoteanindicatorfunctionfortheredesigndecisionthatis1forredesignand0otherwise.Wewillrefertoredesigntriggeredbyalowsafetymarginasredesignforsafetyandredesigntriggeredbya 102

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highsafetymarginasredesignforperformance.Ifthetestisnotpassedthenredesignshouldbeperformedtoselectanewdesign. 5.3.4.3CalibrationandredesignToobtainthecalibratedmodel,thetestrealizationg(i)H(xini,udet)correspondingtotheerrorinstancee(i)(xini,udet)istreatedasanewdatapointandtheerrorinstanceisaddedtothedesignofexperimentfortheerrormodel.Theredesignproblemisformulatedasadeterministicoptimizationproblem minf(x,udet)w.r.txs.t.g(i)H,calib(x,udet))]TJ /F5 11.955 Tf 11.95 0 Td[(kre(i)G,calib(x,udet)0(5)wherethemeanofthepredictedhigh-delitymodelg(i)H,calib(,)andthestandarddeviation(i)G,calib(,)arecalibratedconditionalonthetestresultg(i)H(xini,udet)andkreisanewstandarddeviationoffset.Letx(i)redenotetheoptimumdesignafterredesignfoundfrom 5 .Comparingtheinitialdesignproblemin 5 totheredesignproblemin 5 ,weseethatthereisachangeinthefeasibledesignspaceduetothechangeinthestandarddeviationoffsetandcalibration.Notethatthecalibrationisconditionalonobtainingthehigh-delityevaluationg(i)H(xini,udet)inthefuture.Thatis,ifweobtaintheevaluationg(i)H(xini,udet),wecanobtainthecalibratedmodelg(i)H,calib(,),andwewillselectanimproveddesignx(i)re. 5.3.5ProbabilisticEvaluationThenaldesignafterpossibleredesignis x(i)nal=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q(i)xini+q(i)x(i)re(5)whereq(i)=1correspondstofailingthetestandperformingredesign.TheexpectedobjectivefunctionvalueafterpossibleredesignisEE[EU[f(Xnal,U)]].Theprobabilityof 103

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redesigniscalculatedanalyticallyas pre=(klb)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[((kub))(5)where()isthestandardnormalcumulativedistributionfunction(cdf).TheoptimizationofthestandarddeviationoffsetsisbasedonacomputationallycheapapproximationofthereliabilityconstraintasdescribedinSection 5.3.2 .Thekeybenetoftheproposedapproximationisthattheprobabilitycanbecalculatedanalytically.Theprobabilityofanegativesafetymarginconditionalonpassingthetestandkeepingtheinitialdesignis PE[G(xini,udet)0jQ=0]=T()]TJ /F5 11.955 Tf 9.3 0 Td[(kini)(5)whereT()isthenormalcdftruncatedtotheinterval[)]TJ /F5 11.955 Tf 9.29 0 Td[(klb,kub].Theprobabilityconditionalonperformingredesignis PE[G(Xre,udet)0jQ=1]=()]TJ /F5 11.955 Tf 9.3 0 Td[(kre)(5)Thenalprobabilityofanegativesafetymarginafterpossibleredesignis PE[G(Xnal,udet)0]=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(pre)T()]TJ /F5 11.955 Tf 9.3 0 Td[(kini)+pre()]TJ /F5 11.955 Tf 9.3 0 Td[(kre)(5)Aftersolvingtheoptimizationproblemin 5 ,thefulltwo-levelmixedaleatory/epistemicuncertaintypropagationisperformedtorecovertheprobabilityoffailuredistributionandchecktheaccuracyoftheproposedapproximation.TheprobabilityoffailureofthenaldesignisunknownsincethereisepistemicuncertaintyinthemodelGH(,).ArealizationoftheprobabilityoffailureiscalculatedconditionalonanerrorrealizationE(,)=e(i)(,).Arealizationoftheprobabilityoffailureoftheinitialdesignis p(i)f(xini)=PUhg(i)H(xini,U)<0i(5) 104

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wherePU[]denotestheprobabilitywithrespecttoaleatoryuncertainty.Notethattheepistemicmodeluncertaintyistreatedseparatelyfromthealeatoryuncertaintytodistinguishbetweenthequantityofinterest,theprobabilityoffailurewithrespecttothehigh-delitymodelandaleatoryuncertainty,andthelackofknowledgeregardingthisquantity.Theerrorinthelow-delitymodelE(,)hasnoimpactonthereliabilitywithrespecttothehigh-delitymodelgH(,).However,sincethehigh-delitymodelisunknown,theprobabilityoffailurecalculationisrepeatedmanytimesconditionalonmanydifferentrealizationsofthehigh-delitymodelg(i)H(,)through 5 .Arealizationofthenalprobabilityoffailureafterpossibleredesignis p(i)f(x(i)re)=PUhg(i)H(x(i)re,U)0i(5)Afterredesign,thedesignvariablex(i)reisalsoanepistemicrandomvariableinadditiontothelimitstatefunctiong(i)H(,).Manydifferentmethodsareavailableforcalculatingtheprobabilityoffailure.Inthisstudy,rstorderreliabilitymethod(FORM)isusedtocalculatetheprobabilityoffailureforeachepistemicrealization.Thenalprobabilityoffailureafterpossibleredesignis p(i)f(x(i)nal)=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q(i)p(i)f(xini)+q(i)p(i)f(x(i)re)(5)Notethattheredesigndecisionq(i)shapesthenalprobabilityoffailuredistributionbecausewewillhavetheopportunityinthefuturetocorrecttheinitialdesignifitfailsthedeterministictest.TheprobabilityoftheprobabilityoffailureofthenaldesignexceedingthetargetprobabilityoffailureisestimatedbyMCSas PE[Pf(Xnal)p?f]1 mmXi=1Ihp(i)f(x(i)nal)p?fi(5)whereI[]isanindicatorfunction.Thecomputationalcostofthefulltwo-levelmixedaleatory/epistemicuncertaintypropagationishighandthereforeonlyperformedaftertheoptimizationofthestandarddeviationoffsets.Forexample,morethanm=1900 105

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probabilityoffailurecalculationsarenecessarytoestimateaprobabilityoftheorder=0.05witha10%coefcientofvariation. 5.4TestCases 5.4.1CantileverBeamBendingExample 5.4.1.1ProblemdescriptionTherstexampleisthedesignofacantileverbeamtominimizemasssubjecttoaconstraintontipdisplacementadaptedfromanexamplebyWuetal[ 6 ].Thebeamissubjecttoindependentaleatoryrandomloadsinthehorizontalandverticaldirections.TheoriginalprobleminvolvedthedesignofalongslenderbeamandthereforeusedEuler-Bernoullibeamtheory.Inthisexample,thelengthofthebeamisreducedsuchthatshearstresseffectsbecomeimportantandTimoshenkobeamtheoryismoreaccurate.TheTimoshenkobeammodelplaystheroleofacomputationallyexpensivehigh-delitymodel(e.g.niteelementanalysis)andtheEuler-Bernoullibeammodelplaystheroleofaninexpensivelow-delitymodel.Thebeamisoptimizedtoensurewith95%condencethatthereliabilityindexofthenaldesignafterpossibleredesignisgreaterthan3.Thelow-delitymodelofthelimitstatefunctionis gL(x,U)=d?)]TJ /F4 11.955 Tf 15.28 8.09 Td[(4l3 ewts FY t22+FX w22(5)wherex=fw,tgarethedesignvariablesandU=fFX,FYgarethealeatoryvariables.Thehigh-delitymodelofthelimitstatefunctionis gH(x,U)=d?)]TJ /F11 11.955 Tf 11.96 10.74 Td[(p (dx(x,U))2+(dy(x,U))2(5)wheredxanddyaregivenby 5 and 5 .TheproblemparametersaredescribedinTable 5-1 . dx(x,U)=3lFX 2gwt+4l3FX ewt3(5) 106

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Table5-1. ParametersforcantileverbeamexampleParameterNotationValue Designvariables,xWidthofcrosssectionw2.5w5.5inThicknessofcrosssectiont1.5t4.5inAleatoryvariables,UHorizontalloadFXN(500,1002)lbsVerticalloadFYN(1000,1002)lbsConstantsElasticmoduluse29106psiShearmodulusg11.2106psiLengthofbeaml10inAllowabletipdisplacementd?2.2510)]TJ /F9 7.97 Tf 6.59 0 Td[(3inConservativealeatoryvaluesudetf744.7,1173.5glbsTargetprobabilityoffailurep?f=()]TJ /F8 11.955 Tf 9.3 0 Td[(?)1.3510)]TJ /F9 7.97 Tf 6.59 0 Td[(3=()]TJ /F4 11.955 Tf 9.29 0 Td[(3)Targetcondencelevel1)]TJ /F8 11.955 Tf 11.95 0 Td[(0.95 dy(x,U)=3lFY 2gwt+4l3FY ew3t(5)Theobjectivefunctionisthecross-sectionalareaofthebeam f(x)=wt(5)whichisproportionaltothemassofthebeam. 5.4.1.2Applicationoftheproposedmethod Step1:Quantifyingthemodeluncertainty.Therststepistoquantifytheuncertaintyinthelow-delitymodel.AKrigingmodelisconstructedforthediscrepancybetweenthelowandhigh-delitymodelsbasedonevaluationsatthecornerpointsinthejointdesign-aleatoryspace(4beamdesignseachwith4loadingconditions).Todemonstratethemethod,thecornerpointswerechoseninordertoensurehighmodeluncertainty.Inpractice,themodelcouldalsobeconstructedbasedondatafrompreviousdesigns.TheKrigingmodelimprovesthepredictionfromthelow-delitymodel,butmoreimportantlyitprovidescondenceintervalsforthemodeluncertainty.InFigure 5-2 ,thecondenceintervalsarisingduetomodeluncertaintyareshowninthedesignspaceandaleatoryspace. 107

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Figure5-1. Thebeamissubjecttohorizontalandverticaltiploads Step2:Selectingxedconservativevaluesforaleatoryvariables.Next,aleatoryrandomvariablesUarereplacedwithxedconservativevaluesudet.TheconservativevaluesarefoundbysolvingtheRBDOproblemproblemin 5 .TheRBDOisperformedwithrespecttoaleatoryuncertaintyconditionalonthemeanlow-delitymodel.Bysolvingtheoptimizationproblemin 5 ,weselectconservativevaluesudet=f744.7,1173.5glbs.Thesevaluescorrespondtoapproximatelythe99thand96thpercentilesoftheloads.TheRBDOproblemonlyrequiressingleleveluncertaintypropagationsinceepistemicmodeluncertaintyisxedatthemeanprediction. Step3:Optimizationofsafetymargins(i.e.standarddeviationoffsets).Inthethirdstep,theoptimumstandarddeviationoffsetsarefoundbysolving 5 108

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A BFigure5-2. Thegureontheleftshowsthedesignoptimizationwithstandarddeviationoffsetk=0andxedconservativevaluesudetinplaceofaleatoryvariables.Thegureontherightshowsthelimit-statefunctioninstandardnormalspacefortheoptimumdesignfoundontheleft.Thereliabilityindexisthedistanceinstandardnormalspacefromtheorigintothelimit-state. usingCMA-ESwithapenalizedobjectivefunction.Recallthatstandarddeviationoffsetsofmodeluncertaintyareusedduringthedesign/redesignprocessassafetymarginsagainstmodeluncertainty.InsidetheMCS,thedesignoptimization( 5 , 5 )isperformedusingsequentialquadraticprogramming(SQP).Byvaryingtheconstraintontheprobabilityofredesignp?reweobtainacurvefortheexpectedcrosssectionalareaversusprobabilityofredesignasshowninFigure 5-3 .Thetradeoffcurveisusedtodeterminehowmuchriskofredesignisacceptablegiventheexpectedperformanceimprovement.Forillustration,wewillselecttheoptimumsafetymarginsk=f0.71,0.89,2.25,3.00gcorrespondingto20%probabilityofredesignformoredetailedstudy. Step4:Fulltwo-levelmixeduncertaintypropagation.Inthefourthstep,thefulltwo-levelmixeduncertaintypropagationisperformedfortheselectedoptimum 109

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Figure5-3. Tradeoffcurveforexpectedcrosssectionalareaversusprobabilityofredesign safetymargins.Thefulltwo-levelmixeduncertaintypropagationisusedtorecovertheprobabilityoffailuredistributionandobtaindetailedresultsfortheMCSofthedesign/redesignprocess.Inthepreviousstepinvolvingtheoptimizationofthesafetymargins,aleatoryvariableswerexedandonlyepistemicmodeluncertaintywasconsidered.Inthefulltwo-levelmixeduncertaintypropagation,theprobabilityoffailureiscalculatedusingrstorderreliabilitymethod(FORM)foreachrealizationofepistemicmodeluncertainty(i.e.Krigingconditionalsimulation) Step5:Post-processingofsimulationresults.Finally,post-processingisperformedforthedatagatheredintheMCS.First,weexaminethesafetymargindistributionandthereliabilityindexdistributionshowninFigure 5-4 .ThesafetymargindistributioninFigure 5-4 showsthepossibleconstraintviolationswithrespecttoepistemicmodeluncertaintyconditionalonthexedconservativevaluesudet.Thebeamwillberedesignedifthesafetymarginislessthan)]TJ /F4 11.955 Tf 9.3 0 Td[(0.1610)]TJ /F9 7.97 Tf 6.59 0 Td[(4inchesorgreaterthan2.810)]TJ /F9 7.97 Tf 6.59 0 Td[(4inches.Itcanbeobservedthatifredesignisrequired,weexpecttohavemuchmoreprecisecontroloverthetipdisplacementof 110

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thebeamduetotheknowledgegainedfromthefuturetest.Redesignactsasatypeofqualitycontrolmeasurebyinitiatingdesignchangesinresponsetoobservinganextremesafetymargin.WecancomparethesafetymargindistributionandreliabilityindexdistributionsinFigure 5-4 .Thereisastrongcorrelationbetweentheobservedsafetymarginandthereliabilityindex(correlationcoefcient0.999).Asaresult,thesafetymarginbasedredesigncriteriaisveryusefulforidentifyingoverlyconservativeorunsafedesigns.ThesafetymarginisstronglycorrelatedwiththereliabilityindexbecausethesafetymarginiscalculatedwithrespecttotheMPPofthemeanlow-delitymodel.AsshowninFigure 5-5 ,theconservativevaluesudetprovideareasonablepointestimateoftheMPPdistribution.Thestandarddeviationoffsetshavebeenoptimizedbasedonthecomputationallycheapapproximationofthereliabilityconstraintin 5 suchthattheprobabilityofanegativesafetymarginafterpossibleredesignis5%.Afterperformingthefulltwo-levelmixeduncertaintypropagation,theprobabilityoftheprobabilityoffailureexceedingthetargetvalueof1.3510)]TJ /F9 7.97 Tf 6.59 0 Td[(3isestimatedtobeintherangeof5%to7%(95%condenceintervalwithm=2500).Inotherwords,wehavebetween93%and95%condencethattheprobabilityoffailureofthenaldesignafterpossibleredesignwillbelessthanp?f=1.3510)]TJ /F9 7.97 Tf 6.58 0 Td[(3.Second,weexaminetheoptimumdesignvariabledistributionandthecrosssectionalareadistributionshowninFigure 5-6 .ThedesignvariabledistributioninFigure 5-6 showshowthedesignvariableswillchangeifredesignisrequiredinthefuture.Thepeakcorrespondstotheinitialdesignsincethereisan80%probabilitytheinitialdesignwillbeacceptedasthenaldesign.Thedistributionofdesignvariablescanbeusedtoplanforfuturedesignchanges.ThecrosssectionalareadistributioncorrespondingtothedesignsisshowninFigure 5-6 .Althoughthechangeinthemeanareaduetoallpossibledesignchangesisrelativelysmall,therealizationsoftheareacorrespondingtoredesignmaybesignicantlydifferentthantheinitialarea.Forexample,ifredesignforperformanceisrequiredtheareaisreducedbyabout6.4%,however,thereisonly 111

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A BFigure5-4. Distributionofsafetymarginandreliabilityindexfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. abouta1%chanceofredesignforperformance.Ontheotherhand,thereisabouta19%chanceofredesignforsafetywhichisassociatedwithanincreaseinareaofapproximately2%. 5.4.2MultidisciplinarySoundingRocketDesignExample 5.4.2.1ProblemdescriptionThesoundingrocketdesignexampleisbasedonamultidisciplinarydesignoptimization(MDO)problem.Thesoundingrockethasasinglecryogenicliquidhydrogenfueledgasgeneratorengine.Theintertankandthrustframearemadefromacompositematerial.Thethrustvectorcontrol(TVC)systemiselectromechanical.Theavionicsandelectricalpowersystemhavenoredundancies.Therocketisdesignedforverticalintegration.ThedesignstructurematrixforthesoundingrocketexampleisshowninFigure 5-7 .Therearefourdisciplinescorrespondingtopropulsion,structures(sizingandweightsestimation),aerodynamics,andtrajectorysimulation.TherearevedesignvariablescorrespondingtothemassofpropellantMP,initialthrusttoweight 112

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A B C DFigure5-5. Distributionofmostprobablepoint(MPP)for20%probabilityofredesign. ratioT=W,enginechamberpressurepcc,mixtureratioP,anddiameterD.Theengineefciencyfactorisconsideredtobeanaleatoryrandomvariable.TheoutputsarethetotalmassMtot,nalaltitudeattheendofthepropulsionphasernal,andlengthtodiameterratioL=D.Thedesignproblemistominimizethetotalmasswhilesatisfyingconstraintsonthenalaltitudeandthelengthtodiameterratio.Theconstraintonthelengthtodiameterratioispurelydeterministicandisthereforesimplyincludedasanadditionaldesignconstraintinthedesignoptimizationproblemsin 5 , 5 .Thereisaleatoryuncertaintyinthenalaltitudeandtotalmass(GLOW)duetothealeatoryuncertaintyintheengineefciencyfactor. 113

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A BFigure5-6. Distributionofoptimumdesignvariablesanddesignperformancefor20%probabilityofredesign.Peakislocatedatinitialdesign. Thereisacouplingbetweenthestructuresandaerodynamicdisciplinesinthatthemaximumaxialaccelerationandmaximumdynamicpressurearerelatedtothetotalmass.Thestructuremustbesizedtowithstandtheloads,butchangesinthetotalmassarerelatedtotheloadsthroughtrajectoryandaerodynamics.Thereisacouplingbetweenstructuresandpropulsioninthattheinertmassfractionisrelatedtothethrustthroughthethrusttoweightratio.Theenginemassandthrustframemassmustbedesignedforagiventhrust,butbecausethethrusttoweightratioisspeciedbeforehandchangesinmassalterthethrust.Axedpointiterationisperformedtosatisfythecouplingconstraintswithrespecttothemaximumaxialload,maximumdynamicpressure,andinertmassfraction.ThereisaloopbetweenaerodynamicsandtrajectorybecausethedragcoefcientvarieswithMachnumber. 5.4.2.2StandardatmospheremodelsTheatmospheremodelincludesvariationsofthespeedofsound,atmosphericpressure,andairdensityasafunctionofaltitude.Thespeedofsoundvariesasa 114

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. . DesignVariables: Mp:propel-lantmass T=W:thrusttoweightratio pcc:chamberpressure P:mixtureratio D:diameter :engineefciencyfactor . CouplingVariables: T:thrustIsp:specicimpulseq:massowrateMinert:inertmassCD:dragcoefcientM:Machnumbernmaxax:maximumaxialaccelerationPmaxdyn:maximumdynamicpressure:inertmassfractionAt:throatareaAe:exhaustarea . Outputs: Mtot:totalmass rnal:maxi-mumaltitude L=D:lengthtodiameterratio . Propulsion . Structures . Aerodynamics . Trajectory . T,Isp,q,Ae . T,pcc,At . T . Mtot . . CD . nmaxax,Pmaxdyn . M Figure5-7. Designstructurematrixforsoundingrocketdesignexample.Therearecouplingsbetweenpropulsion/structures,aerodynamics/structures,andtrajectory/aerodynamics. functionofaltitude c(r)=8><>:c1(r)r)]TJ /F5 11.955 Tf 11.95 0 Td[(RE<122103m303.1416m/sr)]TJ /F5 11.955 Tf 11.95 0 Td[(RE122103m(5)where c1(r)=6x6+5x5+4x4+3x3+2x2+1x+0(5)andx=r)]TJ /F5 11.955 Tf 12.9 0 Td[(REisthealtitudeinmetersrelativetotheradiusoftheearthRE.ThecoefcientsarelistedinTable 5-2 andthevariationinthespeedofsoundisplottedin 115

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Table5-2. Coefcientsforcalculatingspeedofsoundasafunctionofaltitude( 5 )CoefcientValue 03.43941021)]TJ /F4 11.955 Tf 9.3 0 Td[(6.399010)]TJ /F9 7.97 Tf 6.59 0 Td[(322.296410)]TJ /F9 7.97 Tf 6.59 0 Td[(73)]TJ /F4 11.955 Tf 9.3 0 Td[(8.912610)]TJ /F9 7.97 Tf 6.59 0 Td[(134)]TJ /F4 11.955 Tf 9.3 0 Td[(5.739110)]TJ /F9 7.97 Tf 6.59 0 Td[(1757.628310)]TJ /F9 7.97 Tf 6.59 0 Td[(226)]TJ /F4 11.955 Tf 9.3 0 Td[(2.718210)]TJ /F9 7.97 Tf 6.59 0 Td[(27 Figure5-8. Speedofsoundasafunctionofaltitude( 5 ) Figure 5-8 .Theatmosphericpressure(Pa)decreaseswithaltitude Pa(r)=p0exp()]TJ /F5 11.955 Tf 9.29 0 Td[(pref(r)]TJ /F5 11.955 Tf 11.96 0 Td[(RE))(5)wherep0=1.0437105andpref=1.458910)]TJ /F9 7.97 Tf 6.58 0 Td[(4.Thedensityoftheair(kg/m3)decreaseswithaltitude (r)=0exp)]TJ /F5 11.955 Tf 10.5 8.08 Td[(r)]TJ /F5 11.955 Tf 11.96 0 Td[(RE href(5)where0=1.22557andhref=7254.24. 116

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5.4.2.3DisciplinemodelsThedisciplinemodelsarebasedonthedissertationofCastellini,“Multidisciplinarydesignoptimizationforexpendablelaunchvehicles”[ 88 ].Fulldetailsofthemodelscanbefoundinthedissertation.Thedisciplinemodelsarebrieysummarizedhere. Propulsion.ThepropulsiondisciplinecalculatestheperformancecharacteristicsoftheenginebasedonNASAcomputerprogramCEA(ChemicalEquilibriumwithApplications)forcalculatingchemicalequilibriumcompositionsandpropertiesofcomplexmixtures[ 89 , 90 ].Inordertoreducecomputationalcost,Krigingsurrogatemodelswerettothecharacteristicvelocity(C)andthrustcoefcient(CT)asafunctionofmixtureratio,chamberpressure,andnozzleexpansionratio.Thesurrogatemodelswereconstructedbasedonadesignofexperimentconsistingof500pointsgeneratedusingLatin-hypercubesampling.TheKrigingmodelsusedaGaussiancovariancefunctionandzeroordertrendfunctions.KrigingmodelswereconstructedinMatlabusingDACE(DesignandAnalysisofComputerExperiments)Matlabtoolbox[ 91 ].AnyepistemicmodeluncertaintyintroducedbytheKrigingsurrogatesinthepropulsiondisciplineisnotincludedintheanalysis.Thespecicimpulseiscalculatedas Isp=CCT g0(5)whereCistheKrigingpredictionofthecharacteristicvelocity,CTisthetheKrigingpredictionofthethrustcoefcient,isanefciencyfactor,andg0isthestandardaccelerationduetogravity.Thesingleefciencyfactorrepresentsthecombineddegradingeffectsofchamberandnozzlelossesaswellasmassowlosses.Thethroatareaiscalculatedas At=T CTpcc(5)whereTisthethrustandpccisthechamberpressure.Theexhaustareaiscalculatedas Ae="At(5) 117

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Table5-3. InputsandoutputsofpropulsiondisciplineInputsOutputs Chamberpressurepcc MassowrateqMixtureratioP SpecicimpulseIspNozzleexpansionratio" ThroatareaAtThrustT ExhaustareaAe where"isthenozzleexpansionratio.Themassowrateiscalculatedas q=T CCT=T Ispg0(5) Structures.Thestructuresdisciplinecalculatesthetotalinertmassoftherocketandthetotallengthoftherocket.Forthisexample,thestructuresdisciplineisdenedasthecombinationofsizingandweightsestimation.Theweightsestimationincludesenginemass,thrustframemass,tankmassincludingthermalprotectionsystem,thrustvectorcontrol(TVC),andavionicsandelectricalpowersystem.ThethrustframeandtanksaredesignedusingstructuralsafetymarginsofSSM=1.1.Allweightestimationrelationships(WER's)arebasedonthedissertationofCastellini[ 88 ].Thetotalmassoftherocketiscalculatedas Mtot=Minert+MP+MPL(5)whereMinertisthetotalinertmass,MPisthepropellantmass,andMPListhepayloadmass.Thetotalinertmassiscalculatedas Minert=Meng+MTF+MFT+MOxT+MTPS,OxT+MTPS,FT+Mavio+MEPS+Mintertank+MPLF(5)whereMengistheengine,MTFisthethrustframe,MFTisthefueltank,MOxTistheoxidizertank,MTPS,OxTisthethermalprotectionfortheoxidizertank,MTPS,FTisthethermalprotectionforfueltank,Mavioistheavionics,MEPSistheelectricalpowersystem,Mintertankistheintertank,andMPLFisthepayloadfairing.MassoftanksandintertankThemassofthefuelandoxidizertanksareusuallythelargestpartofthestructuralmassofliquidpropulsionrockets[ 88 ].TheWER'sfor 118

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Table5-4. InputsandoutputsofstructuresdisciplineInputsOutputs ThrustT TotalinertmassMinertChamberpressurepcc TotalLengthLMixtureratioP Nozzleexpansionratio" ThroatareaAt Maximumaxialaccelerationnmaxax MaximumdynamicpressurePmaxdyn DiameterD MassofpropellantMP MassofpayloadfairingMPLF Table5-5. NotationusedinweightsestimationMtotTotalMinertTotalinertMPPropellantMPLPayloadMPLFPayloadfairingMengEngine(includingTVCandnozzle)MTVCThrustvectorcontrol(TVC)MnozzleNozzleMTFThrustframeMFTFueltankMOxTOxidizertankMTPS,OxTThermalprotectionsystem-OxidizertankMTPS,FTThermalprotectionsystem-FueltankMavioAvionicsMEPSElectricalpowersystemMintertankIntertank thetankmassesarealinearregressionforfuelsandaslightlynon-linearpowerlawregressionofthetankvolumeforoxidizers.Themassofthefueltanksis MFT=6Yj=1kj((VF35.315)0.4856+800)0.4536(5)whereVFisthevolumeofthefueltankandkjarecoefcientstoaccountforloadparametersanddiscretedesignvariables.Themassoftheoxidizertankis MOxT=6Yj=1kj)]TJ /F4 11.955 Tf 5.48 -9.69 Td[((VOx35.315)1.041.0850+7000.4536(5) 119

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Table5-6. CoefcientsfortankmassWER'sDescription k1Structuralmaterialk1=1Al-Lialloy0.9Compositek2Commonbulkheadorintertankk2=Stot)]TJ /F4 11.955 Tf 11.96 0 Td[(1.5Sdome=StotCommonbulkhead1Intertankk3Horizontalorverticalintegrationk3=(1=30)(L=D)+(29=30)Horizontal1Verticalk4Maxdynamicpressurek4=)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Pmaxdyn0.16=5.76404k5Axialaccelerationk5=(SSMnmaxax)0.15=1.29134k6Tankpressurek6=1.3012+1.435910)]TJ /F9 7.97 Tf 6.59 0 Td[(6ptanks=2.7862 whereVOxisthevolumeoftheoxidizertank.ThecoefcientsusedinthemassWER'saredescribedinTable 5-6 .Themassofthethermalprotectionsystemisapproximatedasalinearfunctionofthetankssurfacearea MTPS,OxT=FT=kinsSOxT=FT(5)withkins=0.9765forliquidoxygentanksandkins=1.2695forliquidhydrogentanks.Themassoftheintertankisapproximatedasatwodimensionallinearfunctionofthelateralsurfaceandthediameter MIT=kSMk1SIT(DIT3.2808)k2(5)wherekSM=1foraluminumalloysorkSM=0.7forcomposites,k1=5.4015,andk2=0.5169.PropulsionsystemandthrustframeTheenginemassforcryogenicpropulsionandgasgeneratorfeedisapproximatedasafunctionofthethrustas Meng=aTb+Mnozzle+MTVC(5)wherea=7.5435410)]TJ /F9 7.97 Tf 6.59 0 Td[(3,b=0.88563510)]TJ /F9 7.97 Tf 6.58 0 Td[(1,c=20.2881,andMnozzleisthemassofthenozzle,andMTVCisthemassofthethrustvectorcontrolsystem.Themassofanelectromechanicalthrustvectorconrolsystemisapproximatedasafunctionofthethrust 120

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as MTVC=0.1078(T10)]TJ /F9 7.97 Tf 6.58 0 Td[(3)+43.702(5)Thethrustframemass1isapproximatedasafunctionofthethrustandmaximumaxialaccelerationas MTF=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(0.013N0.795engT0.579+0.01Neng(Meng=0.45)0.7170.45(1.5SSMnmaxaxg0)kSM(5)wherekSM=1foraluminumalloysorkSM=0.62forcompositesandNeng=1forthisexample.AvionicsandelectricalpowersystemThemassoftheavionicssystemisapproximatedas Mavio=kRL(246.76+1.3183Stot)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(0.75)(5)wherekRL=0.7fornoredundancy,kRL=1forcriticalcomponentsredundancy,orkRL=1.3forfullredundancy.Themassoftheelectricalpowersystemisapproximatedas MEPS=kRL0.405Mavio(1)]TJ /F4 11.955 Tf 11.95 0 Td[(0.18)(5) Aerodynamics.Giventheinstantaneousvelocity,altitude,andtotalmassoftherockettheaerodynamicsdisciplinecalculatesthedragforce,dynamicpressure,andaxialacceleration.TheaerodynamicsdisciplineanalysisisbasedonMissileDATCOM[ 93 ].Inordertoreducecomputationalcost,thedragcoefcientiscalculatedasafunctionoftheMachnumberbasedonPCHIP(piecewisecubichermiteinterpolatingpolynomial)interpolationbetweenvaluesinatableofMissileDATCOMevaluations.TheinterpolationbetweendatapointsforthedragcoefcientasafunctionofMachnumberisshowninFigure 5-9 . 1WERfrom[ 88 ]iscorrectedtomatchoriginalsource[ 92 ]andAriane5Vulcainenginedatapoint 121

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Table5-7. InputsandoutputsofaerodynamicsdisciplineInputsOutputs VelocityV(t) DragforceFD(t)Altituder(t) DynamicpressurePdyn(t)Totalmassm(t) Axialaccelerationnax(t)DiameterD ExhaustareaAe TheMachnumberiscalculatedas M=V c(r)(5)wherethespeedofsoundc(r)variesasafunctionofaltitudeaccordingto 5 .Theaxialaccelerationsing'siscalculatedas nax=1 mg0(T)]TJ /F5 11.955 Tf 11.96 0 Td[(FD)(5)whereFD=0.5(r)V2CDAisthedragforceandtheairdensity(r)decreaseswithaltitudeaccordingto 5 .Thethrustiscalculatedas T=Ispg0q)]TJ /F5 11.955 Tf 11.96 0 Td[(AePa(r)(5)whereAeistheexhaustareaandtheairpressurePa(r)decreaseswithaltitudeaccordingto 5 .Thedynamicpressureiscalculatedas Pdyn=0.5(r)V2(5) Trajectory.Thetrajectorydisciplinecalculatesthealtitude,velocity,andtotalmassasafunctionoftime.Thetrajectorydisciplineanalysisisbasedonatwodimensionalmodel.Theequationsofmotionare _r=V_V=1 m)]TJ /F2 11.955 Tf 5.47 -9.69 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(FD+T)]TJ /F6 7.97 Tf 13.15 5.12 Td[(GMEm r2_m=)]TJ /F5 11.955 Tf 9.3 0 Td[(q(5) 122

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Figure5-9. DragcoefcientasafunctionofMachnumberbasedonMissileDATCOM.PCHIPinterpolationisusedbetweendatapoints. whereristheradius,Visthenormofthevelocityvector,FDisthedragforce,Tisthethrust,Gisthegravitationalconstant,MEisthemassoftheearth,andmisthemassoftherocket.Equationsofmotionarederivedassumingtheightpathangle()andpitchangle()areboth90degrees.Thetrajectorydisciplineiscoupledwiththeaerodynamicsdiscipline.DuringODEintegration,thetrajectorydisciplinecallstheaerodynamicsdisciplinetoupdatetheinstantaneousvaluesofthethrustanddragforce.Thetimeatwhichmaximumdynamicpressureoccursisobtainedbyndingthepointatwhichtherateofchangeofthedynamicpressurecrosseszeroaxisfrompositivetonegative.Thederivativeofthedynamicpressureis dPdyn dt=VdV dt+0.5d dtV2(5)wherethederivativeoftheairdensityin 5 is d dt=)]TJ /F8 11.955 Tf 14.03 8.09 Td[(0 hrefdr dtexp)]TJ /F5 11.955 Tf 10.49 8.09 Td[(r)]TJ /F5 11.955 Tf 11.95 0 Td[(RE href(5) 123

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Table5-8. InputsandoutputsoftrajectorydisciplineInputsOutputs TotalmassMtot FinalaltitudernalThrustT(t) Velocityv(t)DragforceFD(t) Altituder(t)Thrustdurationtburn Totalmassm(t) 5.4.2.4Low-delitymodelAlow-delityapproximationisintroducedfortheinertmassfractionasafunctionofthemassofpropellant.Thelow-delitymodelisbasedonacurvetofthemodelprovidedinthe“HandbookofCostEngineeringandDesignofSpaceTransportation”[ 94 ].Table 5-9 liststhedatathatwasreadfromthegure(approximatedvisually).Asecondorderpolynomialwasttotheinertmassfractionasafunctionofthelogofpropellant L=(1.5879log(MP)2)]TJ /F4 11.955 Tf 11.96 0 Td[(36.1554log(MP)+217.8084)=100(5)Thedesigncurveisforrocketsthataremuchlargerthanthesoundingrocketweareinvestigatinginthisdesignexample.Therefore,wewillextrapolateoutsideoftherangeofthedesigncurveusingthepolynomialcurvet.Theextrapolationmayintroducesignicanterrorontopofthealreadyquestionableaccuracyofthelow-delitymodel.Thelow-delitymassmodelisa1-dimensionalfunction.However,inthefullycoupledsystemthemassdependsonall6design-aleatoryvariables.Tovisualizetheaccuracyofthelow-delitymodel,acloudof10,000differentdesignswasgeneratedinthe6-dimensionaldesign-aleatoryspaceusingLatin-hypercubesampling.Fixedpointiterationswereperformedforeachofthedesignstoenforcecouplingconstraintsbetweendisciplines.InFigure 5-11 ,the10,000designsareprojectedontoa1-dimensionalplaneinordertocomparewiththe1-dimensionallow-delitymodel.Itisobservedthatthelow-delitymodelcapturestheoveralltrend,butthereissignicanterror.Furthermore,thereappearstobesignicantscatterinthedesignpointsaroundthemeantrendline.Thisisbecausedifferentdesignsarebeingprojectedontothe 124

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Table5-9. DatareadfromdesigncurveMassofpropellant(kg)InertMassFraction 10,0000.19520,0000.15530,0000.13840,0000.13050,0000.125 Figure5-10. Asecondorderpolynomialwasttotheinertmassfractionasafunctionofthelogofthepropellantmass.Themodelisextrapolatedtotheregionofinterestforsoundingrocketdesign. 1-dimensionalplane.Thelow-delitymodelisincapableofrepresentingthisvariationwithrespecttodesignvariablesotherthanthemassofpropellant.Forthelow-delitymodel,r2=0.81indicatingthethemodelexplainsabout81%ofthevariation. 5.4.2.5Applicationoftheproposedmethod Step1:Quantifyingthemodeluncertainty.Therststepistoquantifytheuncertaintyinthelow-delitymodel.Thelow-delitymodeloftheinertmassfractionisrelatedtothehigh-delitymodel(i.e.coupledsystem)as H(x,u)=L(MP)+E(x,u)(5) 125

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Figure5-11. Acloudof10,000designsin6-dimensionsisprojectedontoaonedimensionalplaneandcomparedtothelow-delitymodelprediction wherex=fMP,T=W,pcc,P,Dgisthevectorofdesignvariables,u=isarealizationofthealeatoryrandomvariableU,H(,)istheinertmassfractionwhencouplingconstraintsaresatised,L()isthelow-delitymodelgivenbyEquation 5 ,andE(,)istheKrigingmodelofthediscrepancybetweenthetwomodels.Byintroducingthelow-delitymodelthepropulsion/structuresandtheaerodynamics/structurescouplingsareremoved.Ineffect,thecouplingconstraintsareincorporatedintotheconstructionoftheerrormodelE(,).Removingthecouplingseliminatestheneedforxedpointiterationsandallowsthesoundingrocketdesigntoberepresentedasasimplefeedforwardsystem.Thismaysubstantiallyreducethecomputationalcostofuncertaintypropagationrelativetoperformingxedpointiterationsforeveryrealizationofaleatoryuncertainty.However,thelow-delitymodelmayintroducesignicantepistemicmodeluncertainty,particularlywhentheKrigingmodelisconstructedbasedononlyasmallsetofinitialdata(i.e.smalldesignofexperiment).TheepistemicmodeluncertaintyresultsinadditionaluncertaintyinthenalaltitudeandGLOW. 126

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Figure5-12. TradeoffcurveforexpectedGLOWversusprobabilityofredesign Step2:Selectingxedconservativevaluesforaleatoryvariables.Next,thealeatoryrandomvariableUisreplacedwithaxedconservativevalueudet.InsteadofsolvingtheRBDOproblemin 5 ,the5thpercentileoftheengineefciencyisusedfortheconservativevalue.The5thpercentilewasselectedbecausethealtitudeisnearlyalinearfunctionoftheengineefciencyandthetargetprobabilityoffailureisp?f=0.05. Step3:Optimizationofsafetymargins(i.e.standarddeviationoffsets).Inthethirdstep,theoptimumstandarddeviationoffsetsarefoundbysolving 5 usingCMA-ESwithapenalizedobjectivefunction.InsidetheMCS,thedesignoptimization( 5 , 5 )isperformedusingsequentialquadraticprogramming(SQP).Byvaryingtheconstraintontheprobabilityofredesignp?reweobtainacurvefortheexpectedGLOWversusprobabilityofredesignasshowninFigure 5-12 .Thetradeoffcurveisusedtodeterminehowmuchriskofredesignisacceptablegiventheexpectedperformanceimprovement.Forillustration,wewillselecttheoptimumsafetymarginsk=f0.78,0.96,1.87,2.29gcorrespondingto20%probabilityofredesignformoredetailedstudy. 127

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Step4:Fulltwo-levelmixeduncertaintypropagation.Inthefourthstep,thefulltwo-levelmixeduncertaintypropagationisperformedfortheselectedoptimumsafetymargins.Thefulltwo-levelmixeduncertaintypropagationisusedtorecovertheprobabilityoffailuredistributionandobtaindetailedresultsfortheMCSofthedesign/redesignprocess.Foreachrealizationofepistemicmodeluncertainty(i.e.Krigingconditionalsimulation)theprobabilityoffailureiscalculatedusingrstorderreliabilitymethod(FORM). Step5:Post-processingofsimulationresults.Finally,post-processingisperformedforthedatagatheredintheMCS.First,weexaminethesafetymargindistributionandtheprobabilityoffailuredistributionshowninFigure 5-13 .ThesafetymargindistributioninFigure 5-13 showsthepossibleconstraintviolationswithrespecttoepistemicmodeluncertaintyconditionalonthexedconservativevaluesudet.Therocketwillberedesignedifthesafetymarginislessthan)]TJ /F4 11.955 Tf 9.3 0 Td[(0.6kilometersorgreaterthan9.5kilometers(relativetotargetof150kmassumingconservativeengineefciency).Redesignactsasatypeofqualitycontrolmeasurebyinitiatingdesignchangesinresponsetoobservinganextremesafetymargin.WecancomparethesafetymargindistributiontotheprobabilityoffailuredistributioninFigure 5-13 .Thereisastrongcorrelationbetweentheobservedsafetymarginandtheprobabilityoffailure(correlationcoefcient-0.65).Asaresult,thesafetymarginbasedredesigncriteriaisveryusefulforidentifyingoverlyconservativeorunsafedesigns.Thecorrelationcoefcientisnotasstrongasinthebeamexamplebecausethealeatoryuncertaintyintheintheengineefciencyisbounded.Duetotheboundedaleatoryuncertaintythecorrelationbetweensafetymarginandprobabilityoffailurebreaksdownwhenthesafetymarginislessthanthepointcorrespondingto100%probabilityoffailureorthesafetymarginisgreaterthanthepointcorrespondingto0%probabilityoffailure.Thestandarddeviationoffsetshavebeenoptimizedbasedonthecomputationallycheapapproximationofthereliabilityconstraintin 5 suchthatthe 128

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probabilityofanegativesafetymarginafterpossibleredesignis5%.Afterperformingthefulltwo-levelmixeduncertaintypropagation,theprobabilityoftheprobabilityoffailureexceedingthetargetvalueofp?f=0.05isfoundtobeinagreementwiththetargetvalueof=0.05.Second,weexaminetheoptimumdesignvariabledistributionshowninFigure 5-14 andtheGLOWanddrymassdistributionsshowninFigure 5-15 .Thedesignvariabledistributionis5-dimensionalsothemarginaldistributionsareshown.Thepeakcorrespondstotheinitialdesignsincethereisan80%probabilitytheinitialdesignwillbeacceptedasthenaldesign.Thedistributionofdesignvariablesisusefulforplanningforfuturedesignchanges.Itisobservedthatthechamberpressuredoesnotchangeduringredesign.Theoptimumchamberpressureisalwaystheupperboundof120barsregardlessoftheoutcomeofthefuturehigh-delityevaluation.Thechangeindiameterisrelativelysmallwithachangeontheorderof1%ifredesignisrequired.However,thepropellantmassmaychangesubstantially.Themassofpropellantmaydecreaseapproximately12%ifredesignforperformanceisrequiredorincreaseby4%ifredesignforsafetyisrequired.TherelativechangeinGLOWduetoredesignissimilartotherelativechangeinpropellantmassasseeninFigure 5-15 .ThedrymassdistributionisshowninFigure 5-15 .Ifredesignforsafetyisrequired,thedrymasswillincreasebyabout2%.Ifredesignforperformanceisrequired,thedrymasswilldecreasebyabout7%.Sinceunderestimatingthemasscorrespondstooverestimatingthealtitude,andviceversa,redesigntendstoincreasethemassofheaviermassrealizationsordecreasethemassoflightermassrealizationsbyadjustingthepropellantmassaccordingly. 5.5DiscussionConclusionsAttheinitialdesignstage,engineersoftenmustrelyonlow-delitymodelswithhighepistemicmodeluncertainty.Oneapproachtohighepistemicmodeluncertaintyistoaddasafetymargin,suchasakstandarddeviationoffset,todesignconstraints 129

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A BFigure5-13. Distributionsofsafetymarginandprobabilityoffailurefor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. toensuretheoptimumdesigniswellwithinthesafedesignspace.Ifthesafetymarginislargethenthedesignerhasmorecondencethatthedesignissafe,butdesignperformancesuffers.Ifthesafetymarginissmallthenthedesignspaceislargeranddesignswithbetterperformancebecomeaccessible,butthedesignerhaslesscondenceinthesafetyofthedesign.Iftherewillbeanopportunityinthefuturetoevaluatethedesignusinghigherdelitymodeling(ortoperformatestonaprototype),thenthisprovidesanopportunitytoredesign(i.e.correctormodify)adesignthatisrevealedtobetooconservativeorunsafe.Inthisstudyweproposeasafety-margin-basedmethodfordesignundermixedepistemicmodeluncertaintyandaleatoryparameteruncertainty.Themethodisbasedonatwostagedesignprocesswhereaninitialdesignisselectedbasedonlow-delitymodeling,buttherewillbeanopportunityinthefuturetoevaluatethedesignwithahigh-delitymodelandifnecessarycalibratethelow-delitymodelandperformredesign.Thedesignoptimizationisperformeddeterministicallybasedonxing 130

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A B C D EFigure5-14. Distributionofoptimumdesignvariablesfor20%probabilityofredesign.Plotsshowmarginaldistributionsof5-dimensionaljointdistribution. thealeatoryvariablesattheMPPofthemeanlow-delitymodelandapplyingakstandarddeviationoffsettoconstraintfunctionstocompensateformodeluncertainty.AMCSisperformedwithrespecttoepistemicmodeluncertaintybasedonconditionalsimulationsofaKrigingmodel.Byrepeatingthedeterminsticdesignprocessformanydifferentrealizationofmodeluncertaintyitispossibletopredicthowfutureredesignmaychangethedesignperformanceandreliability.Itisshownthatfutureredesignactssimilartoqualitycontrolmeasuresintruncatingextremevaluesofepistemicmodeluncertainty.Thesimulationallowsthedesignertotradeoffbetweentheexpecteddesignperformanceandtheriskoffutureredesignwhilestillachievingaspeciedcondencelevelinthereliabilityofthenaldesign.Itisfoundthatredesignforsafetyisparticularlyeffectiveattruncatinghighprobabilitiesoffailureandthereforeallowsfor 131

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A BFigure5-15. DistributionsofGLOWanddrymassfor20%probabilityofredesign.Plotsshowoverlappingtransparenthistograms. improveddesignperformanceoftheinitialdesignbybeinglessconservative.Ontheotherhand,redesignforperformanceallowsadesignertoimprovetheperformanceoftheinitialdesignifitislaterrevealedtobetooconservative.Itisfoundthattheoptimumdesignstrategyincludessomeprobabilityofbothredesignforsafetyandredesignforperformance.Themethodisdemonstratedonacantileverbeambendingexampleandthenonamultidisciplinarysoundingrocketdesignexample.Inbothexamplesitisshownthatthereisastrongcorrelationbetweenthesafetymarginandtheprobabilityoffailure.Therefore,thesimplesafetymarginbasedredesigncriteriaisusefulforidentifyinganunsafeoroverlyconservativedesign.Thistypeofqualitycontrolmeasureisalreadyincorporatedintomanyengineeringdesignapplications.Theproposedmethodallowsformoredetailedstudyoftheeffectsofredesignandallowsthedesignertoplanforfuturedesignchangesandexploretheinteractionsbetweentheprobabilityofredesign,safetymargins,designperformance,andprobabilityoffailure. 132

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CHAPTER6CONCLUSIONSEarlyinthedesignprocess,engineersmustoftenrelyoncomputationallycheap,low-delitymodelstoselectaninitialdesign.Laterinthedesignprocess,high-delitymodelsmaybeusedtoevaluatetheperformanceandsafetyoftheinitialdesign.Ifhigh-delitymodelsrevealunsatisfactorydesignperformanceorsafetyconcernsthenthisusuallytriggersaredesignprocesstondanimprovednaldesign.Redesigntypicallyresultsinundesirabledelaysandincreasedcosts,however,itisalsoanopportunityfordesignimprovement.Duetotheknowledgegainedfromthehigh-delityevaluationitispossibletocalibratelow-delitymodels,reduceuncertainty,andarriveatasaferand/orbetterperformingnaldesign.Traditionally,engineershaveusedsafetymarginstoprovideinsuranceagainstdesignfailureandreducetheprobabilityofredesignforsafety.However,ifthemarginsaretoohighthendesignperformancesuffersandifthemarginsaretoolowthenthedesignmaybeunsafe.Inthisresearch,weproposeamethodforoptimizingthesafetymarginsgoverningadesign/redesignprocess.Theresearchseekstoimproveunderstandingofthecomplexrelationshipbetweensafetymargins,designperformance,probabilityoffailure,andprobabilityofredesign.Thekeycontributionsofthisresearchareasfollows: Thedevelopmentofageneralizedmethodforsimulatingtheeffectsofafuturetestandpossibleredesignwhenmodelbiasisconstant.Thegeneralizedformulationfacilitatestheunderstandingofthemethodandallowsittobemorereadilyappliedtonewdesignexamples.Themethodalsointroducedglobaloptimizationforndingtheoptimalsafetymargins.Theuseofglobaloptimizationreplacedpointcloudbasedmethods[ 21 ]inordertoreducethenoiseintheparetofrontofoptimalexpectedperformanceandprobabilityofredesign. Adetailedinvestigationwasconductedtodeterminewhenitwasbettertoredesignforsafetyandwhenitisbettertoredesignforperformance.Itwasfoundthatthedecisiondependsinpartontheratioofthevarianceofepistemicuncertaintyinthehigh-delitymodeltotheratioofthevarianceinthelow-delitymodel.Ingeneralterms,itdependsontheamountofepistemicuncertaintyinthehigh-delitymodelrelativetotheamountofuncertaintyinthelow-delitymodel.Itwasfoundthat 133

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whentheratioislowitisbettertoredesignforsafetyandwhentheratioishighitisbettertoredesignforperformance.Ifthereisalargeamountoferrorinthehigh-delitymodel(i.e.ratioishigh)thenadangerousinitialdesignmaypassthetestunnoticedandthereforeredesignforsafetyislesseffective. Thedevelopmentofamethodforsimulatingtheeffectsofafuturetestandpossibleredesignwhenmodelbiasmaybenon-linearandhigh-delityevaluation(i.e.futuretest)providesincompleteinformation. – Non-linearmodeldiscrepancy:Ingeneral,thediscrepancybetweenalowandhigh-delitymodelmaybenon-linear.Forexample,themodelsmayagreeforsomedesignsundersomeconditionsbutexhibitlargediscrepanciesforotherdesignsorotherconditions.Therefore,aKrigingmodelwasproposedasarobustmethodforrepresentingtheunknownmodeldiscrepancy.TheKrigingmodelallowsfortheconvenientsimulationofnon-lineardiscrepancyfunctionsthroughconditionalsimulations. – Calibrationanduncertaintyreduction:Ifthediscrepancyfunctionisnon-linearthenthemethodsusedformodelcalibrationanduncertaintyreductionwithconstantmodelbiasarenolongerapplicable.Ourintuitiontellsusthatthehigh-delityevaluationofonedesignunderxedconditionsonlyreducestheuncertaintyforsimilardesignsundersimilarconditions.Forexample,ifwecompareastructuralFEmodel(low-delity)ofawingdesigntoaphysicaltestofawingprototype(high-delity)underthesameloadingconditionsthenwecanquantifytheerrorinourFEmodelforthatwingdesignunderthespeciedloads.However,wemaywishtousetheFEmodeltodesignotherwingsortopredictthebehaviorofthesamewingbutundermanydifferentrandomloadrealizations.Therefore,wemustaccountforthespatialcorrelationsinthemodeldiscrepancywhenwecalibrateourmodelandreduceepistemicmodeluncertainty.SpatialcorrelationsareeasilyhandledthroughtheuseoftheKrigingmodel. Thedevelopmentofamethodforreducingthecomputationalcostofthesafetymarginoptimizationbyexploitingthecorrelationbetweenthesafetymarginandtheprobabilityoffailure.IfthesafetymarginiscalculatedwithrespecttotheMPPofthealeatoryrandomvariables,thenobservinganegativesafetymarginiscorrelatedwithaviolationoftheprobabilityoffailureconstraint.Therefore,itwasproposedthatitmaybepossibletoapproximateaquantileconstraintontheprobabilityoffailureastheprobabilityofanegativesafetymargin.ToavoidthehighcomputationalcostofrepeatedlysearchingfortheMPPforeachrealizationofepistemicmodeluncertainty,itwasproposedthattheMPPwithrespecttothemeanmodelbeusedasapointapproximationoftheMPPdistribution.Thismethodwasshowntoproducereasonableresultsforthecantileverbeamandsoundingrocketdemonstrationexamples. 134

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Theextensionofthemethodtomulti-disciplinarysoundingrocketdesignoptimizationexample.Themethodwasextendedtoconsiderincreasedcomputationalcostofmodels,increasednumberofdesignvariables,multi-disciplinarydesignconsiderations,designvariablesthatwereuniquetothehigh-delitymodel,epistemicuncertaintyintheobjectivefunction,andadditionaldeterministicdesignconstraints. Perspectives.Basedontheworkpresentedinthisdissertationthereareseveralareasthatmaybeworthyoffurtherinvestigation.Someoftheinterestingareasforfutureworkinclude: Thedevelopmentofamethodforsimulatingmultiplefuturetests.BasedontheKrigingframeworkdevelopedinthisdissertationitistheoreticallypossibletosimulatemultiplefuturetestssuchastestreplicationstoreducemeasurementuncertaintyortestsofdifferentinitialdesignconceptstoreduceuncertaintyoveralargerareaofthedesignspace. Theconsiderationofmeasurementerrorinadditiontonon-linearmodeldiscrepancy.Themethodbasedontheassumptionofconstantmodelbiasincludedtheconsiderationofmeasurementerrorinthehigh-delitymodel,butthiswasnotincludedinlaterworkwhennon-linearmodeldiscrepancywasintroduced.BasedontheKrigingframeworkitistheoreticallypossibletoincludetheeffectofmeasurementerrorintheanalysisbyusingKrigingwithnugget. Detailedstudyoftheelicitationofepistemicmodeluncertaintyparametersfromexperts.Themethodbasedontheassumptionofconstantmodelbiashadarathercursorydiscussionofhowmodeluncertaintycanberepresentedasuniformrandomvariablesbasedonexpertopinion.Inthemethodwithnon-linearmodeldiscrepancytheerrormodelwasbasedonthepreliminarytestdataratherthanexpertopinion.Thefoundationoftheproposedmethodwouldbenetfrommoredetailedliteraturereviewregardingtheelicitationofexpertopinion(e.g.[ 28 , 34 ])andhowitrelatestodevelopingamodelforepistemicmodeluncertaintyinthecontextofthepresentwork.Inparticular,futureresearchcouldaddressincorporatingexpertopinionintothedenitionoftheKrigingcovariancefunction. Thedevelopmentofamethodtocombineexpectedperformanceandprobabilityofredesignintoasinglecostfunction.Intheory,thereisanoptimalprobabilityofredesignintermsofeconomiccostforagivendesignproblem.Thatis,theexpectedperformancebenetsofredesignlikelyoutweightheexpectedcostofredesignupuntilacertainpoint.Afterthispoint,thecostofredesignmaybegreaterthantheexpectedperformancebenets.Thedenitionofasinglecombinedcostfunctionwouldreplacethetradeoffcurvebetweenexpectedperformanceandprobabilityofredesignwithasinglecurveshowingtheprobabilityofredesignthatminimizestheexpectedcost.Relatedworksuchastheeconomic 135

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changemethodofRoseretal.[ 77 ]andtheexibledesignmethodologyofDeNeufville[ 95 ]mayprovideinsightintomodelingtheeconomicsofredesign. PreliminaryworkinthisdissertationshowedpromisingresultsforapproximatingaquantileprobabilityoffailureconstraintwithanMPPbasedsafetymarginconstraint.Thismethodwasshowntosignicantlyreducethecostofthesafetymarginoptimizationwithoutintroducingexcessiveerrorinthereliabilityconstraint.Moreresearchisneededtoidentifythelimitationsoftheproposedapproximationandtoexploredevelopingastrictlyconservativeapproximation. 136

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BIOGRAPHICALSKETCHNathanielPricewasborninSt.Augustine,Floridain1988.HegraduatedmagnacumlaudewithaBachelorofScienceinmechanicalengineeringfromtheUniversityofFlorida(UF)in2012.AsanundergraduateheheldinternshipswithE&SConsulting,Inc.inSt.Augustine,FloridaandSpaceExplorationTechnologies(SpaceX)inCapeCanaveral,Florida;heperformedundergraduateresearchintheUFMaterialsDesign&PrototypingLaboratoryundertheadvisementofDr.MicheleManuel.UndertheadvisementofDr.Nam-HoKim,hewasawardedthe2012BiomedicalEngineeringSociety(BMES)DesignandResearchAwardandthe2013KnoxT.MillsapsOutstandingUndergraduatePaperAwardforhishonorsthesisresearchonthetheeffectsofcorticalthickness,bonestrength,andscrewlengthonrigidsternalxationstability.Aftergraduation,hejoinedtheStructuralandMultidisciplinaryOptimizationGroupattheUniversityofFloridaasaPh.D.studentwherehewasawardedtheGraduateSchoolFellowshipAward.In2014,heearnedanon-thesisMasterofScienceinmechanicalengineeringwithacerticateinscienticcomputing.ThesameyearhebeganresearchatONERA-TheFrenchAerospaceLabinPalaiseau,FranceandenteredadualPhDprogramwithEcoledesMinesdeSaint-Etienne. 147