<%BANNER%>

Reliability Based Design Including Future Tests and Multi-Agent Approaches

MISSING IMAGE

Material Information

Title:
Reliability Based Design Including Future Tests and Multi-Agent Approaches
Physical Description:
1 online resource (153 p.)
Language:
english
Creator:
Villanueva, Diane C
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Haftka, Raphael Tuvia
Committee Co-Chair:
Sankar, Bhavani V
Committee Members:
Ifju, Peter G
Uryasev, Stanislav

Subjects

Subjects / Keywords:
agents -- optimization -- reliability -- surrogates -- uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
The initial stages of reliability-based design optimization involve the formulation of objective functions and constraints, and building a model to estimate the reliability of the design with quantified uncertainties.However, even experienced hands often overlook important objective functions and constraints that affect the design. In addition, uncertainty reduction measures, such as tests and redesign, are often not considered in reliability calculations during the initial stages. This research considers two areas that concern the design of engineering systems: 1) the trade-off of the effect of a test and post-test redesign on reliability and cost and 2) the search for multiple candidate designs as insurance against unforeseen faults in some designs. In this research, a methodology was developed to estimate the effect of a single future test and post-test redesign on reliability and cost.The methodology uses assumed distributions of computational and experimental errors with re-design rules to simulate alternative future test and redesign outcomes to form a probabilistic estimate of the reliability and cost for a given design. Further, it was explored how modeling a future test and redesign provides a company an opportunity to balance development costs versus performance by simultaneously designing the design and the post-test redesign rules during the initial design stage.  The second area of this research considers the use of dynamic local surrogates, or surrogate-based agents, to locate multiple candidate designs.Surrogate-based global optimization algorithms often require search in multiple candidate regions of design space, expending most of the computation needed to define multiple alternate designs. Thus, focusing on solely locating the best design may be wasteful. We extended adaptive sampling surrogate techniques to locate multiple optima by building local surrogates in sub-regions of the design space to identify optima. The efficiency of this method was studied, and the method was compared to other surrogate-based optimization methods that aim to locate the global optimum using two two-dimensional test functions, a six-dimensional test function, and a five-dimensional engineering example.
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.
Statement of Responsibility:
by Diane C Villanueva.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Haftka, Raphael Tuvia.
Local:
Co-adviser: Sankar, Bhavani V.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045819:00001

MISSING IMAGE

Material Information

Title:
Reliability Based Design Including Future Tests and Multi-Agent Approaches
Physical Description:
1 online resource (153 p.)
Language:
english
Creator:
Villanueva, Diane C
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Haftka, Raphael Tuvia
Committee Co-Chair:
Sankar, Bhavani V
Committee Members:
Ifju, Peter G
Uryasev, Stanislav

Subjects

Subjects / Keywords:
agents -- optimization -- reliability -- surrogates -- uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
The initial stages of reliability-based design optimization involve the formulation of objective functions and constraints, and building a model to estimate the reliability of the design with quantified uncertainties.However, even experienced hands often overlook important objective functions and constraints that affect the design. In addition, uncertainty reduction measures, such as tests and redesign, are often not considered in reliability calculations during the initial stages. This research considers two areas that concern the design of engineering systems: 1) the trade-off of the effect of a test and post-test redesign on reliability and cost and 2) the search for multiple candidate designs as insurance against unforeseen faults in some designs. In this research, a methodology was developed to estimate the effect of a single future test and post-test redesign on reliability and cost.The methodology uses assumed distributions of computational and experimental errors with re-design rules to simulate alternative future test and redesign outcomes to form a probabilistic estimate of the reliability and cost for a given design. Further, it was explored how modeling a future test and redesign provides a company an opportunity to balance development costs versus performance by simultaneously designing the design and the post-test redesign rules during the initial design stage.  The second area of this research considers the use of dynamic local surrogates, or surrogate-based agents, to locate multiple candidate designs.Surrogate-based global optimization algorithms often require search in multiple candidate regions of design space, expending most of the computation needed to define multiple alternate designs. Thus, focusing on solely locating the best design may be wasteful. We extended adaptive sampling surrogate techniques to locate multiple optima by building local surrogates in sub-regions of the design space to identify optima. The efficiency of this method was studied, and the method was compared to other surrogate-based optimization methods that aim to locate the global optimum using two two-dimensional test functions, a six-dimensional test function, and a five-dimensional engineering example.
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.
Statement of Responsibility:
by Diane C Villanueva.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Haftka, Raphael Tuvia.
Local:
Co-adviser: Sankar, Bhavani V.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045819:00001


This item has the following downloads:


Full Text

PAGE 1

RELIABILITYBASEDDESIGNINCLUDINGFUTURETESTSANDMULTI-AGENTAPPROACHESByDIANEVILLANUEVAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

PAGE 2

c2013DianeVillanueva 2

PAGE 3

Tomyparents,LambertoandDiwata,andmysister,Tisha 3

PAGE 4

ACKNOWLEDGMENTS FirstandforemostIwouldliketothankmyadvisorsDr.RaphaelHaftkaandDr.BhavaniSankarattheUniversityofFloridaandDr.RodolpheLeRicheandDr.GauthierPicardattheEcoledesMinesdeSaint-Etienne.Allofyourinsights,guidance,andpatiencehasbeengreatlyappreciated.Iwouldalsoliketothankthemembersofmyadvisorycommittee,Dr.PeterIfju,Dr.StanislavUryasev,andDr.Jean-PierreGeorge.Also,thankyoutothereviewersofthisdissertation,Dr.NataliaAlexandrovandDr.EduardoSouzadeCursi.Fortheirhospitalityandguidanceduringmysummervisit,IthankChristapherLangandKimBeyatNASALangleyResearchCenter.ThankyoutothemanymembersoftheStructuralandMultidiscplinaryGroupforyourfriendship,support,andthoughtfulquestionsandcommentsduringgroupmeetings.Also,thankyoutothemembersoftheInstitutHenriFayolandotherdepartmentsforyourfrienshipandhospitality,especiallywithanythingthatinvolvesspeaking,reading,orwritinginFrench.Tomyfamilyandfriends,Icannotexpresshowmuchyoursupporthasmeanttome.Noneofthiswouldbepossiblewithoutyou.Finally,IwouldliketoexpressmygratitudetoNASA(AwardNo.NNX08AB40A),theAgenceNationaledelaRecherche(FrenchNationalResearchAgency)(Ref.ANR-09-COSI-005),andAirForceOfceofScienticResearch(AwardFA9550-11-1-0066)forfundingthiswork,andtheadministrationsinFloridaandSaint-EtiennethathelpedmakethejointPhDprogrampossible. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................... 4 LISTOFTABLES ....................................... 8 LISTOFFIGURES ...................................... 9 ABSTRACT .......................................... 11 CHAPTER 1INTRODUCTION .................................... 13 1.1OutlineofDissertation .............................. 16 1.1.1Objectives ................................. 16 1.1.2OutlineofText ............................... 16 2BACKGROUNDANDLITERATUREREVIEW .................... 18 2.1Reliability-BasedDesignOptimization ...................... 18 2.1.1OptimizationMethods ........................... 19 2.1.1.1Double-loop(nested)methods ................. 19 2.1.1.2Single-loopmethods ...................... 19 2.1.2MethodstoEvaluateReliability ...................... 20 2.1.2.1Momentbasedmethods .................... 20 2.1.2.2Samplingbasedmethods .................... 21 2.2DenitionofTypesofUncertainty ........................ 22 2.3UncertaintyReductionMethodsinReliabilityBasedDesign .......... 23 2.4TheRoleofSurrogates .............................. 23 2.5IntegratedThermalProtectionSystemTestCaseDescription ......... 25 2.5.1IntegratedThermalProtectionSystem .................. 25 2.5.2ThermalandStructuralAnalysis ..................... 27 3INCLUDINGTHEEFFECTOFFUTURETESTSANDREDESIGNINRELIABIL-ITYCALCULATIONS .................................. 29 3.1MotivationforExaminingFutureTestsandRedesign .............. 29 3.2UncertaintyModeling ............................... 31 3.2.1ClassicationofUncertainties ...................... 31 3.2.2TrueProbabilityofFailureCalculation .................. 34 3.2.3Analyst-EstimatedProbabilityofFailureCalculation .......... 35 3.3IncludingtheEffectofaCalibrationTestandRedesign ............. 36 3.3.1CorrectionFactorApproach ....................... 37 3.3.2BayesianUpdatingApproach ....................... 37 3.3.2.1IllustrativeexampleofcalibrationbytheBayesianapproach 38 3.3.2.2Extrapolationerrorincalibration ................ 39 5

PAGE 6

3.3.3Test-CorrectedProbabilityofFailureEstimate .............. 41 3.3.4RedesignBasedontheTest ....................... 42 3.3.4.1Deterministicredesign ...................... 42 3.3.4.2Probabilisticredesign ...................... 42 3.4MonteCarloSimulationsofaFutureTestandRedesign ............ 43 3.5IllustrativeExample ................................ 43 3.5.1FutureTestwithoutRedesign ...................... 46 3.5.2RedesignBasedonTest ......................... 48 3.5.2.1Deterministicredesign ...................... 48 3.5.2.2Probabilisticredesign ...................... 49 3.6SummaryandConcludingRemarks ....................... 50 4ACCOUNTINGFORFUTUREREDESIGNTOBALANCEPERFORMANCEANDDEVELOPMENTCOSTS ............................... 53 4.1MotivationforAccountingforFutureRedesign ................. 53 4.2IntegratedThermalProtectionShieldDescription ................ 54 4.3AnalysisandPost-DesignTestwithRedesign .................. 56 4.4UncertaintyDenition ............................... 58 4.5DistributionoftheProbabilityofFailure ..................... 59 4.6SimulatingFutureProcessesattheDesignStage ............... 61 4.7OptimizationoftheSafetyMarginsandRedesignCriterion .......... 63 4.7.1ProblemDescription ........................... 63 4.7.2Results .................................. 66 4.7.3UnconservativeInitialDesignApproach ................. 70 4.7.4Discussion ................................. 74 4.8SummaryandDiscussiononPossibleFutureResearchDirections ...... 74 5DYNAMICDESIGNSPACEPARTITIONINGFORLOCATINGMULTIPLEOPTIMA:ANAGENT-INSPIREDAPPROACH .......................... 76 5.1MotivationandBackgroundonLocatingMultipleOptima ............ 77 5.2MotivationforMultipleCandidateDesigns .................... 79 5.3Surrogate-BasedOptimization .......................... 82 5.4AgentOptimizationBehavior ........................... 83 5.5DynamicDesignSpacePartitioning ....................... 85 5.5.1MovingtheSub-regions'Centers .................... 86 5.5.2Merge,SplitandCreateSub-regions .................. 86 5.5.2.1Mergeconvergingagents .................... 87 5.5.2.2Splitclusteredsub-regions ................... 88 5.5.2.3Createnewagents ....................... 89 5.6Six-DimensionalAnalyticalExample ....................... 90 5.6.1ExperimentalSetup ............................ 91 5.6.2SuccessestoLocateOptima ....................... 93 5.6.3AgentEfciencyandDynamics ..................... 93 5.7EngineeringExample:IntegratedThermalProtectionSystem ......... 97 6

PAGE 7

5.7.1ExperimentalSetup ............................ 99 5.7.2SuccessestoLocateOptima ....................... 99 5.7.3AgentsEfciencyandDynamics ..................... 99 5.8Discussion .................................... 103 5.9SummaryandDiscussiononPossibleFutureResearchDirections ...... 105 6FURTHERINVESTIGATIONONTHEUSEOFSURROGATE-BASEDOPTIMIZA-TIONTOLOCATEMULTIPLECANDIDATEDESIGNS ................ 107 6.1MotivationforInvestigatingSurrogate-BasedTechniques ........... 107 6.2Surrogate-BasedOptimization .......................... 109 6.2.1Multiple-StartingPoints .......................... 109 6.2.2EfcientGlobalOptimization ....................... 110 6.3NumericalExamples ............................... 111 6.3.1ExperimentalSetup ............................ 113 6.3.2Branin-HooTestFunction ......................... 114 6.3.3SasenaTestFunction ........................... 121 6.4DiscussionandSummary ............................ 126 7CONCLUSIONS .................................... 127 7.1Perspectives ................................... 129 7.1.1EfcientIdenticationofIndividualLocalOptima ............ 129 7.1.1.1Isolatingbasinsofattraction .................. 129 7.1.1.2Suspendingorallocatingfewresourcestounpromisingsub-regions .............................. 130 7.1.2VulnerabilityAnalysisandRangeofAcceptableObjectiveFunctions 131 APPENDIX ACOMPARISONOFBAYESIANFORMULATIONS ................... 133 BEXTRAPOLATIONERROR .............................. 135 CSIMULATINGATESTRESULTANDCORRECTIONFACTOR .......... 138 DEFFECTOFADDITIONALUNCERTAINTIES ..................... 140 EGLOBALVSLOCALSURROGATES ......................... 142 REFERENCES ........................................ 145 BIOGRAPHICALSKETCH .................................. 153 7

PAGE 8

LISTOFTABLES Table page 2-1ITPSmaterialproperties ................................ 26 3-1ITPSvariables ..................................... 45 3-2Distributionoferrors .................................. 45 3-3Comparingabsolutetrueerror ............................. 46 3-4Comparingtruetemperatureestimates ........................ 47 3-5ProbabilitiesofFailurewithoutRedesign(usingBayesianCorrection) ........ 47 3-6Summaryofthepercentilesofthetrueprobabilityoffailurewithoutredesign .... 48 3-7Calibrationbythecorrectionfactorapproachwithdeterministicredesign. ...... 48 3-8Calibrationbycorrectionfactorwithdeterministicredesignwithboundsonds .... 49 3-9CalibrationbytheBayesianupdatingapproachwithpfbasedredesign ....... 52 4-1Correlatedrandomvariables .............................. 55 4-2DescriptionofErrors .................................. 59 4-3Boundsofcomputationalandexperimentalerrors .................. 66 4-4Breakdownofalternativefutures ............................ 73 5-1TrueoptimaofmodiedHartman6 ........................... 91 5-2Surrogatesconsideredinthisstudy .......................... 92 5-3Multi-AgentParametersformodiedHartman6 ................... 92 5-4Trueoptimaof5-DITPSexample ........................... 99 6-1Descriptionofmethods ................................. 112 6-2Multi-agentparametersforexampleproblems ..................... 113 A-1Comparisonoffupdtest;Ptruewithdifferentformulationsofthelikelihoodfunction ..... 134 B-1CalibrationbytheBayesianupdatingapproach .................... 137 C-1Summaryofsurrogates ................................. 139 D-1Valuesoftheuncertainvariablesinthelimitstates. .................. 141 D-2Standarddeviationofthelimitstatesbeforeandafterredesign ........... 141 8

PAGE 9

LISTOFFIGURES Figure page 2-1CorrugatedcoresandwichpanelITPSconcept .................... 26 3-1Illustrationofuncertaintiesleadingtothepossibletruetemperaturedistribution .. 33 3-2Illustrationwithunconservativecalculationoftemperature .............. 34 3-3IllustrativeexampleofBayesianupdating ....................... 40 3-4IllustrationofthecalibrationusingBayesianupdating ................. 40 4-1CorrugatedcoresandwichpanelITPSconcept .................... 55 4-2Examplesoftemperaturedistributions ......................... 60 4-3Beforeandafterredesignexample .......................... 63 4-4Obtainingstatisticsforfuturealternatives ....................... 64 4-5PDFsofthesafetymargins ............................... 67 4-6Paretofrontforminimumprobabilityofredesignandmeanmass .......... 68 4-7DetailsofParetofrontforminimumpreandm .................... 69 4-8Percentageofconservativeandunconservativeredesigns .............. 69 4-9Paretofrontfortheunconservativeapproach ..................... 71 4-10DetailsofParetofrontfortheunconservativeapproach ............... 72 4-11Descriptionofredesignsforunconservative-rstapproach .............. 72 4-12Histogramsofmassafterredesign ........................... 73 5-1Integratedthermalprotectionsystem ......................... 80 5-2Feasibleandinfeasibleregionsfor3-DITPSexample ................ 81 5-3Multi-agentSystemoverview .............................. 84 5-4Illustrationtosplitanagent'ssub-region ........................ 88 5-5SuccessrateformodiedHartman6example .................... 94 5-6FormodiedHartman6,themedianf ........................ 95 5-7PercentageofaddedpointsexploitorexploreinmodiedHartman6example ... 95 5-8Foramulti-agentsystem,themediannumberofagents ............... 96 9

PAGE 10

5-9AccuracyofsurrogatesformodiedHartman6 .................... 97 5-10CorrugatedcoresandwichpanelITPSconcept .................... 98 5-11Successratefor5-DITPSexample .......................... 100 5-12Medianffor5-DITPSexample ............................ 101 5-13Numberofagentsanditerationsfor5-DITPSexample ................ 102 5-14Percentageofpointsthatexploitorexplorein5-DITPSexample .......... 102 5-15PRESSRMSforITPS5-Dexample ........................... 103 5-16eRMSforITPS5-Dexample ............................... 104 6-1ContourplotofBranin-Hoofunction .......................... 114 6-2SuccesspercentageforBranin-Hooexample ..................... 115 6-3MedianfforBranin-Hooexample ........................... 117 6-4ComparisonofpointsforBranin-Hooexample .................... 118 6-5eRMScomparisonforBranin-Hooexample ....................... 119 6-6Comparisontoconstant3agentsforBranin-Hooexample .............. 120 6-7ContourplotofSasenafunctionshowingfouroptima ................. 121 6-8SuccesspercentageforSasenaexample ....................... 122 6-9MedianfforSasenaexample ............................. 124 6-10eRMScomparisonforSasenaexample ......................... 124 6-11Comparisontoconstant3agentsforSasenaexample ................ 125 A-1IllustrativeexampleofBayesianupdating ....................... 133 B-1Comparisonoftheeextrap ................................ 136 E-1For4regions,comparisonofglobalandlocalsurrogates ............... 144 10

PAGE 11

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyRELIABILITYBASEDDESIGNINCLUDINGFUTURETESTSANDMULTI-AGENTAPPROACHESByDianeVillanuevaAugust2013Chair:RaphaelT.HaftkaCochair:BhavaniV.SankarMajor:AerospaceEngineeringTheinitialstagesofreliability-baseddesignoptimizationinvolvetheformulationofobjectivefunctionsandconstraints,andbuildingamodeltoestimatethereliabilityofthedesignwithquantieduncertainties.However,evenexperiencedhandsoftenoverlookimportantobjectivefunctionsandconstraintsthataffectthedesign.Inaddition,uncertaintyreductionmeasures,suchastestsandredesign,areoftennotconsideredinreliabilitycalculationsduringtheinitialstages.Thisresearchconsiderstwoareasthatconcernthedesignofengineeringsystems:1)thetrade-offoftheeffectofatestandpost-testredesignonreliabilityandcostand2)thesearchformultiplecandidatedesignsasinsuranceagainstunforeseenfaultsinsomedesigns.Inthisresearch,amethodologywasdevelopedtoestimatetheeffectofasinglefuturetestandpost-testredesignonreliabilityandcost.Themethodologyusesassumeddistributionsofcomputationalandexperimentalerrorswithre-designrulestosimulatealternativefuturetestandredesignoutcomestoformaprobabilisticestimateofthereliabilityandcostforagivendesign.Further,itwasexploredhowmodelingafuturetestandredesignprovidesacompanyanopportunitytobalancedevelopmentcostsversusperformancebysimultaneouslydesigningthedesignandthepost-testredesignrulesduringtheinitialdesignstage. 11

PAGE 12

Thesecondareaofthisresearchconsiderstheuseofdynamiclocalsurrogates,orsurrogate-basedagents,tolocatemultiplecandidatedesigns.Surrogate-basedglobaloptimizationalgorithmsoftenrequiresearchinmultiplecandidateregionsofdesignspace,expendingmostofthecomputationneededtodenemultiplealternatedesigns.Thus,focusingonsolelylocatingthebestdesignmaybewasteful.Weextendedadaptivesamplingsurrogatetechniquestolocatemultipleoptimabybuildinglocalsurrogatesinsub-regionsofthedesignspacetoidentifyoptima.Theefciencyofthismethodwasstudied,andthemethodwascomparedtoothersurrogate-basedoptimizationmethodsthataimtolocatetheglobaloptimumusingtwotwo-dimensionaltestfunctions,asix-dimensionaltestfunction,andave-dimensionalengineeringexample. 12

PAGE 13

CHAPTER1INTRODUCTIONDesignersintheaerospaceindustryhavetypicallyusedasafetyfactorapproachinordertocompensateforuncertainties.Thispracticeofusingsafetyfactorsindesign,alongwithsafetymarginsandknockdownfactors,isknownasdeterministicdesign.Itiscommontousesafetyfactorsthatarebasedontraditionandexperiencewithoutconsiderationofuncertainties.Therefore,thedeterministicallyoptimizeddesignmaynotleadtoaminimumcostdesign.Forexample,afailuremodewithtoohighofasafetyfactorwillbeover-designedandunnecessarilycostly.Inreliabilitybaseddesignoptimization(RBDO),thedesignisoptimizedinconsiderationoftheuncertaintiesandtheireffectontheprobabilityoffailureofthedesigntakingintoaccounteachfailuremodeandthesystemasawhole.Inprobabilisticdesign,thedesignercanoptimallyallocateriskamongstthefailuremodessuchthatmostriskisallocatedtothemostdifcultfailuremodetoprotectagainst.Lessriskisthenallocatedtothecheaper,easier-to-protect-againstmodes.Animportantstepinprobabilisticdesignistheidenticationandquanticationofun-certaintiesinthedesignorthetoolsusedinthedesignprocess.Abroadandoftenusedclassicationofuncertaintiescategorizesuncertaintyaseitheraleatory(orintrinsic)orepistemic[ 1 2 ].Thetermsaleatoryandepistemicareoftenusedinterchangeablywithvariabilityanderror,respectively.Aleatoryuncertaintygenerallyreferstoinherentuncer-tainties,suchasthoseassociatedwithphysicalpropertiesofmaterialsortheenvironment[ 3 ].Someexamplesincludethevariationsintheyieldstrengthofamaterial,appliedloads,orgeometricdimensionsofastructure.Epistemicuncertainty,orerror,arisesduetolackofknowledge.Itisoftenassociatedwiththeinabilitytoadequatelycharacterizeaphenomenonbyuseofmodels,suchasniteelementmodels,orthroughexperiments.Theseuncertaintiesareconsideredwhencalculatingthereliabilityofthestructure,and,inRBDO,thestructureisoptimizedwithconstraintsonthereliability.However,afterdesign,itiscustomaryforthecomponenttoundergovariousuncertaintyreductionmeasures 13

PAGE 14

(URMs)followedbypossibleremediation,suchasredesignorrepair,ifnecessary.ExamplesofURMsintheaerospaceeldincludethermalandstructuraltesting,inspection,healthmonitoring,maintenance,andimprovedanalysisandfailuremodeling.TheseURMsaregenerallynotconsideredattheinitialdesignstage,andtheeffectoffutureremediationarenotreectedinthereliabilitycalculationsordesignoptimization.Inrecentyears,therehasbeenamovementtoquantifytheeffectofURMsandas-sociatedremediationonthesafetyoftheproductoveritslifecycle.Muchworkhasbeencompletedintheareasofinspectionandmaintenanceforstructuresunderfatigueloading[ 4 7 ].StudiesbyAcaretal.[ 2 ]investigatedtheeffectsoffuturetestsandredesignonthenaldistributionoffailurestressandstructuraldesignwithvaryingnumbersoftestsatthecoupon,element,andcerticationlevels.Sankararamanetal.[ 8 ]proposedanoptimizationalgorithmoftestresourceallocationformulti-levelandcoupledsystems.RBDOcanbecomequitecostly,partlyduetotheneedfornumerousreliabilityassess-ments.Thoughcheaperanalyticalapproachesexist(e.g.,rstorderreliabilitymethod),computationallyexpensivesimulationmethods,suchasMonteCarlosimulation(MCS),areattractivebecausetheycanconsidertheinteractionbetweenfailuremodes,whereastheanalyticalapproachescannot.Theuseofexpensivemodelsisanothersourceofcostofanoptimizationproblem.Inmanyengineeringapplications,itisnotuncommonforcomplexsimulationstotakeuptodaysorweekstocomplete.Forinstance,considerthecostofusingaMonteCarlosimulationincombinationwithamoderatelyexpensiveniteelementmodeltoevaluatetheprobabilityoffailure.Eveniftheamountoftimerequiredtocompleteonesimulationoftheniteelementmodelisontheorderofoneminute,aMonteCarlosimulationwithasamplesizeof1000wouldtakenearlyaday!Consequently,muchresearchhasbeendevotedtotheformulationofproblemsanddevelopmentofmethodologiesthatreducethecostofRBDO.Surrogates,ormetamodels,arefrequentlyusedtoreducethecomputationalcostinoptimizationproblems.Thepurposeofasurrogateistoreplaceanexpensivemodelbya 14

PAGE 15

simplemathematicalmodel-thesurrogate-ttedtoasetofdatapointsevaluatedusingtheexpensivemodel.Thesurrogatecanthenprovidepredictionsoftheexpensivemodelatalowercost.Oneofthemostwellknownandcheapesttotsurrogatesisthepolynomialresponsesurface,butotherssuchaskriging,radialbasisneuralnetworks,andsupportvectorregressionarebecomingincreasinglypopularthoughtheycanbemorecostlytot.Surrogate-basedoptimizationgenerallyproceedsincycles,whereinonecycleanewpointisfoundthroughoptimization,thepointisaddedtothesurrogate,andthesurrogateisupdated(retusingthenewpoint).Thisupdatedsurrogateisusedinthenextcycletondanewpoint.Therecentadvancesincomputerthroughputhavebeenfollowedbyanincreasedinterestinparallelanddistributedcomputing.Parallelcomputationisnowregularlyusedtoreducethetimeandcostofexpensivesimulations,suchasniteelementmodels.Intheareaofsurrogates,thereisagrowinginterestincombiningthepredictionsobtainedwiththesimultaneoususeofmultiplesurrogatesduringoptimization,ratherthanasingleone[ 9 12 ].Theaimistoprotectagainstpoorsurrogates,possiblywhilereducingthenumberofcyclesrequiredtondtheoptimum.VianaandHaftkahavedevelopedanalgorithmtoaddseveralpointsperoptimizationcycle,whicharefoundthroughparallelsimulations[ 13 ].Theyhaveshownthatbetterresultscanbefoundinafractionofthecyclescomparedtoatraditionalimplementation.Theincreasedinterestindistributedcomputingisclearlyevidentintheareaofmulti-agentsystemsforoptimization.Withitsrootsincomputerscience,multi-agentsystemshaveanaturalconnectionwithconstrainedoptimization.Multi-agentsystemssolvecomplexproblemsbydecomposingthemintoautonomoussub-tasks.Ageneraldenitionpositsamulti-agentsystemtobecomprisedofseveralautonomousagentswithpossibledifferentobjectivesthatworktowardacommontask.Throughtheirownobjectives,theagentsasasystemreachaglobalsolutionforthewholeconstraint-basedproblem.Amulti-agentsystemcansolvedecomposedproblemssuchthattheagentsonlyknowsubproblems. 15

PAGE 16

Generally,theoptimizationframeworkconsistsindistributingvariablesandconstraintsamongseveralagentsthatcooperatetosetvaluestovariablesthatoptimizeagivencostfunction,likeinDistributedConstraintOptimizationProblem(DCOP)model[ 14 ].Anotherapproachistodecomposeproblemsortotransformproblemsindualproblemsthatcanbesolvedbyseparateagents[ 15 ](forproblemswithspecicproperties,aswithlinearproblems).Thiscooperativeapproachasbeenappliedtonumerousdistributedconstraint-basedproblems,suchaspreliminaryaircraftdesign[ 16 ]anduniversitytime-tabling[ 17 ]. 1.1OutlineofDissertation 1.1.1ObjectivesTheobjectiveofthisresearchistoaddressthefollowingtopics: 1. Futuretestsandredesigninreliabilityassessment:Developamethodologytoincorpo-ratetheeffectofatestthatwilltakeplaceinthefuture(possiblefollowedbyredesign)intothereliabilityassessmentatthedesignstage.Inaddition,considertheeffectofredesignduetoanunacceptabletestresult.AmethodologybasedonMonteCarlosamplingofuncertainties,particularlytheerrors,simulatespossibleresultsofthefuturetest,andweproposetwomethodsofmodelcalibrationandredesignbasedonthetestresult.Theaimistoexplorethereductioninuncertainties,theprobabilityoffailure,theuncertaintyintheprobabilityoffailure,andmassthatcanoccur. 2. Tradeoffoftestsvsweight:Comparethecostofperformingatestandredesigntobuildingaconservativedesignatthedesignstage.Theaimofthisresearchistoexplorewhatchangesoccurintheinitialdesignknowingthatatestwilloccur,whilealsobeingabletodesignthetestwithredesign. 3. Dynamicdesignspacepartitioningusesurrogate-basedagentstolocatemultiplecandidatedesigns:Theaimistoexploitmulti-agentsystemtechniquestoreducethecostofsolvingproblemsthatrequireexpensivefunctionevaluations.Weproposetodeneagentsbasedonsurrogates,withinspirationdrawnfrommultiplesurrogatetechniques.Thegoalistolocatemultiplelocaloptimaasameansofobtainingmultiplecandidatedesignsforinsuranceinthedesignprocess. 1.1.2OutlineofTextTheorganizationofthisworkisasfollows.Chapter 2 presentsanoverviewofreliability-baseddesignoptimizationandsometechniques,suchasMonteCarlosimulationandsurrogates,thatareusedinthiswork.Italsodescribesatestproblem,thedesignofanintegratedthermalprotectionsystem.Chapter 3 presentsamethodologytoincludethe 16

PAGE 17

effectofafuturetestandredesignonreliabilityassessments.Itshowshowperformingredesignfollowingasinglefuturetestcanpotentiallyleadtobothareductioninprobabilityoffailureandweightreductionthroughanexamplethatusestheintegratedthermalprotectionsystem.Chapter 4 usesthemodelingoffutureredesigntoprovideawayofbalancingdevelopmentcosts(testandredesigncosts)andperformance(mass)bydesigningthedesignandredesignrules.Bysimultaneouslydesigningsafetymarginsandredesigncriterionbasedonprobabilitiesandcosts,weshowthatacompanycanbalanceprobabilisticdesignandthemoretraditionaldeterministicapproach.Chapter 5 describesamethodtodynamicallypartitionthedesignspaceandlocatemultiplecandidatedesignsbysurrogate-basedoptimization.Chapter 6 furtherinvestigatestheuseofsurrogatestolocatemultiplecandidatedesigns.Thenalchapterconcludeswithasummaryofthemajoraspectsoftheworkpresentedanddescribessomeresearchdirectionsthatcouldbepursuedbasedonthiswork. 17

PAGE 18

CHAPTER2BACKGROUNDANDLITERATUREREVIEWReliability-baseddesigninvolvesevaluatingthesafetyofthedesignintermsoftheprobabilityoffailure,anddesigningtomeetaspeciedlevelofreliability.Thetermsreliabilityandprobabilityoffailurearecomplementary,inthatthemorereliablethedesign,thelowertheprobabilityoffailure.Thissectiondiscussestheformulationofreliability-baseddesignoptimizationproblems,themethodsusedtoevaluatethereliability,themethodsusedinoptimization,andvariousmethodsthatreduceuncertaintyandconsequentlyaffectthereliability.Sincesurrogate-basedmethodsarewidelyusedintheoptimizationmethodsdiscussed,thissectionconcludeswithareviewofsurrogatesandsurrogate-basedoptimization. 2.1Reliability-BasedDesignOptimizationReliability-baseddesignoptimizationisaprobabilisticapproachtooptimizationthatisattractiveinitsabilitytoallowthedesignertoprescribetherequiredlevelofreliability.RBDOproblemsareprimarilyformulatedtominimizeacostfunctionf,suchasthemass,whilesatisfyingconstraintsonthereliability.Theoptimizationoccursoverthedesignvariablesx,consideringtheuncertainrandomvariablesr.Theuncertaintypresentintherandomvariablesisdiscussedinthenextsection,Sec. 2.2 .Abasicoptimizationproblem1canbeformulatedoverthefailuremodestoformacomponentleveloptimizationproblem.Foraproblemwithnfailuremodes,theproblemcanbeformulatedas minimizexE[f(x;r)]subjecttoPf;i(x)Pallowf;ii=1:::n(2)wherePf;allowistheallowableprobabilityoffailure. 1Here,theobjectivefunctionisshownastheexpectationf.Thisisonlyanexample;theobjectivecanalsobeapercentileoff. 18

PAGE 19

System-levelfailureoccursinparallel,series,oracombinationofboth.Forparallelfailure,failuremustoccurinallmodesforsystemfailuretooccur.Forseriesseries,systemfailureresultsfromthefailureinanymode.Ifthesystem-levelfailureisconsidered,theproblemabasicformulationis minimizexE[f(x;r)]subjecttoPf;sys(x)Pallowf;sys(2)ThoughbothEqs.( 2 )and( 2 )showconstraintsonprobabilityoffailure,theseconstraintscanalsobeformulatedintermsofthereliabilityindex.ThereliabilityindexisrelatedtotheprobabilityoffailurebyPf=()]TJ /F4 11.955 Tf 7.6 0 Td[(),whereisthestandardnormalcumulativedensityfunction(CDF).Theconstraintswouldthenbeformulatedsuchthat(x;r)target,wheretargetistheminimumallowablereliabilityindex. 2.1.1OptimizationMethods 2.1.1.1Double-loop(nested)methodsInthedouble-loopapproachtoRBDO,thedesignoptimizationiscarriedoutintheouterloopandtheprobabilityoffailureisestimatedintheinnerloop.Thiscanbequitecostlyduetothemethodsusedtoevaluatethereliability(seeSec. 2.1.2 ),and,inaddition,therecanbeproblemswithconvergence(asseenintheReliabilityIndexApproach[ 18 ])sotechniqueshavebeenproposedtoreducethecomputationalcosts.Twocategorieshavebeenidentied:(i)techniquestoimprovetheefciency(e.g.fastprobabilityintegration[ 19 ],two-pointadaptivenon-linearapproximations[ 20 ],(ii)techniquesthatmodifytheformulationofprobabilisticconstraints(e.g.inversereliabilitymeasuressuchastheperformancemeasureapproach[ 18 ]ortheprobabilisticsufciencyfactor[ 21 ]). 2.1.1.2Single-loopmethodsThebasicideaofasingleloopmethodistoformulatetheprobabilisticconstraintsasdeterministicconstraintsbytwoways:(i)theapproximationoftheKarush-Kuhn-Tucker 19

PAGE 20

optimalityconditionsatthemostprobablepoint[ 22 ],(ii)ndingapproximationsofproba-bilisticdesignthroughdeterministicdesign[ 23 25 ].DuandChendevelopedthesequentialoptimizationandreliabilityassessmentmethod(SORA),whichusestheinformationfromthereliabilityassessmenttoshifttheboundariesofviolatedconstraintstothefeasibleregion[ 26 ]. 2.1.2MethodstoEvaluateReliabilityFailureisdenedbyalimitstatefunctiong,whichisafunctionofthedesignvariablesxandtherandomvariablesr.ItisoftendenedasthedifferencebetweentheresponseRandcapacityCdenedforafailuremode.Thelimitstatefunctioncanbeexpressedas g(x;r)=C(x;r))]TJ /F5 11.955 Tf 10.26 0 Td[(R(x;r)(2)whereRandCarebothfunctionsofboththedesignandrandomvariables.Failureoccurswhentheresponseexceedsthecapacity(g<0).Theprobabilityoffailurecanbeexpressedas Pf=P(g(R)<0)=Z:::Zg(R)<0fR(R)dX(2)wherefR(R)isthejointprobabilitydensityfunctionforthevectorRthatcontainstherandomvariablesr.AsMelchersexplains,theanalyticalcalculationofthisexpressionischallengingbecausethejointprobabilitydensityfunctionfR(R)isnotusuallyeasilyobtained,and,forthecaseswhenitisobtained,theintegrationoverthefailuredomainisnoteasy.Momentandsimulationbasedmethodsweredevelopedtocalculatetheprobabilityoffailure. 2.1.2.1MomentbasedmethodsInmomentbasedmethods,thevectorofvariablesismappedtoanindependentstan-dardnormalspace(knownasu-space)byatransformation.Differenttransformationsexist(e.g.,Nataftransformation),butacommontransformationistheRosenblatttransformation[ 3 27 ].Momentbasedmethodsofcalculatingthereliabilityhavetheadvantageofbeinggenerallycheaperthanothermethods.However,theycanonlyevaluatetheprobabilityoffailureofasinglemode. 20

PAGE 21

Oneofthemostcommonmomentbasedmethodsistherst-orderreliabilitymethod(FORM).Thevariablesaremappedtoanindependentstandardnormalspace(u-space)byatransformation.Thelimitstatefunctionisapproximatedaslinear,andFORMisfairlyaccuratewhenthecurvatureofthelimitstatefunctionisnottoosevere.Inthestandardnormalspace,thepointonthelimitstatefunctionwhereg(u)=0attheminimumdistancefromtheoriginisthemostprobablepoint(MPP)offailure.ThereliabilityindexisthedistancefromtheorigintotheMPP.TheMPPisexpressedas minimizeu=p uTusubjecttog(u)=0(2)whereuisthevectorofvariablesinstandardnormalspace.Secondordermethodscanbeusedwhenthecurvatureofthelimitstatefunctionishigh.Thesecond-orderreliabilitymethod(SORM)approximatesthelimitstateasaquadratic,andprovidesamoreaccurateapproximationinsuchcases. 2.1.2.2SamplingbasedmethodsMonteCarlosampling(MCS)isatechniquetonumericallyintegratetheprobabilityoffailureasexpressedinEq.( 2 ).Itrequiresrandomsamplingoftherandomvariablesrfordesignx.Thelimitstateischeckedforeachrealization.Formally,forNtrials,thisisexpressedas Pf=1 NNXi=1I[g(Ci;Ri)<0](2)whereIistheindicatorfunction,whichequals1wheng<0and0otherwise.ThemainadvantageofMCSisthatitallowstheevaluationoftheprobabilityoffailureconsideringjointfailuresbetweentwoormoremodes.TheaccuracyoftheprobabilityoffailuregivenbyEq.( 2 )isestimatedbythecoef-cientofvariationoftheprobabilityoffailuregivenbyEq.( 2 ),andapproximatedasshownwhenPfissmall[ 28 ]. CV(Pf)=s (1)]TJ /F5 11.955 Tf 10.56 0 Td[(Pf) PfNs 1 PfN(2) 21

PAGE 22

Bythisapproximation,itisseenthat,foraprobabilityoffailureof1e-6,100millionsimula-tionsareneededtoachieve10%accuracyforone-sigmalevelofcondence.Clearly,whenthecalculationofthelimitstateinvolvescomplexanalyses,suchasniteelementmodels,theaccuratecalculationofsmallprobabilitiesoffailurebecomesexpensive.Smarsloketal.developedtheseparableMonteCarlomethod(SMC)toreformulatethelimitstatewhenthetypesofuncertaintyinthelimitstate(i.e.responseandcapacity)areindependent[ 28 ].InseparableMonteCarlo,thenumberofsimulationsoftheresponseandcapacitycanbedifferent,suchthatanexpensiveresponsecanbeevaluatedafewernumberoftimes. Pf=1 MNNXi=1MXj=1I[g(Cj;Ri)<0](2) 2.2DenitionofTypesofUncertaintyManyhaveattemptedtoidentifyandclassifydifferenttypesofuncertaintythatshouldbeconsideredinareliabilityassessment.Abroadandoftenusedclassicationofuncertaintiescategorizesuncertaintyaseitheraleatory(orintrinsic)orepistemic[ 1 2 ].Thetermsaleatoryandepistemicareoftenusedinterchangeablywithvariabilityanderror,respectively.Aleatoryuncertaintygenerallyreferstoinherentuncertainties,suchasthoseassociatedwithphysicalpropertiesofmaterialsortheenvironment[ 3 ].Someexamplesincludethevariationsintheyieldstrengthofamaterial,appliedloads,orgeometricdimensionsofacomponent.Variabilitycanbereducedwithmoredata(e.g.moreteststoreducethevariationoftheyieldstrengthofamaterial),orqualitycontrol(e.g.improvedqualitycontroltoreducevariationsindimensions).Epistemicuncertainty,orerror,arisesduetolackofknowledge.Itisoftenassociatedwiththeinabilitytoadequatelycharacterizeaphenomenonbyuseofmodels,suchasniteelementmodels,orthroughexperiments.Epistemicuncertaintycanoftenbyreducedby 22

PAGE 23

simplyaddingmoreknowledgebymoreresearch,expertconsultation,andteststocalibrateanalyticalmodels,forexample. 2.3UncertaintyReductionMethodsinReliabilityBasedDesignAfterdesign,itiscustomaryforthecomponenttoundergovariousuncertaintyreductionmeasures(URMs)followedbyremediation,suchasredesignorrepair,ifnecessary.Exam-plesofURMsintheaerospaceeldincludethermalandstructuraltesting,inspection,healthmonitoring,maintenance,andimprovedanalysisandfailuremodeling.Inrecentyears,therehasbeenamovementtoquantifytheeffectofURMsonthesafetyoftheproductoveritslifecycle.Muchworkhasbeencompletedintheareasofinspectionandmaintenanceforstructuresunderfatigueloading.Fujimotoetal.[ 4 ],Toyoda-Makino[ 5 ],andGarbatovetal.[ 6 ]developedmethodstooptimizeinspectionschedulesforagivenstructuraldesigntomaintainaspeciclevelofreliability.Evenfurther,Kaleetal.[ 7 29 ]exploredhowsimultaneousdesignofthestructureandinspectionscheduleallowsthetradingofcostofadditionalstructuralweightagainstinspectioncostofstiffenedpanelsaffectedbyfatiguecrackgrowth.Therehavebeenfewstudiesthathaveincorporatedtheeffectsoffuturetestsfollowedbypossibleredesignonthedesignofastructure.StudiesbyAcaretal.[ 2 30 ]investigatedtheeffectsoffuturetestsandredesignonthenaldistributionoffailurestressandstructuraldesignwithvaryingnumbersoftestsatthecoupon,element,andcerticationlevels.Suchstudiesshowedthatthesetestswithpossibleredesigncangreatlyreducetheprobabilityoffailureofastructure,andestimatedtherequiredstructuralweighttoachievethesamereductionwithouttests.Sankararamanetal.[ 8 ]proposedanoptimizationalgorithmoftestresourceallocationformulti-levelandcoupledsystems. 2.4TheRoleofSurrogatesSurrogatemodels,ormeta-models,areoftenusedtoreducethecostassociatedwithexpensivefunctionevaluations,suchasthosefromniteelementanalysisorcomputationaluiddynamics.Someexamplesofsurrogatesincludepolynomialresponsesurfaces 23

PAGE 24

[ 31 32 ],kriging[ 33 35 ],supportvectorregression[ 36 ],andneuralnetworks[ 37 38 ].Inoptimization,surrogatesareoftenusedtoprovideapproximationsoftheobjectivefunctionand/orconstraintsinoptimization.Theuseofsurrogateshasbeenwelldocumented,andforacompletereviewsurrogatesandsurrogate-basedoptimizationtechniques,thereaderisreferredtoreferences[ 39 43 ].Traditionalsurrogate-basedoptimizationprogressesiniterations,orcycles,untilanoptimumorsuitablesolutionisfound.Inonecycle,datafromexpensivesimulationsisttoasurrogate,thesurrogateisusedtondacandidateoptimum,andtheoptimumisevaluatedbytheexpensivesimulator.Theoptimumisgenerallyaddedtothesurrogateinthenextiteration.Inrecentyears,manyhaveproposedstrategiesforusingmultiplesurrogatesforopti-mization[ 9 12 ].Vianaexplainsthattheuseofmultiplesurrogatesoverasinglesurrogatemakessensebecause(i)nosinglesurrogateworkswellforallproblems,(ii)thecostofconstructingmultiplesurrogatesisoftensmallcomparedtothecostofsimulations,and(iii)useofmultiplesurrogatesreducestheriskassociatedwithpoorlyttedmodels[ 39 ].Inparticular,theabilityofmultiplesurrogatestogivedifferentinterpretations(i.e.,predictionsanduncertaintyestimates)ofthesamedesignspaceisattractive.Multiplesurrogateshavebeenusedtosimplycomparethemultiplesolutionsgivenbyeachsurrogate.Forexample,Samadetal.[ 10 ]comparedpolynomialresponsesurface,kriging,radialbasisneuralnetwork,andaweightedaveragesurrogateintheshapeoptimiza-tionofacompressorblade,andfoundthatthemostaccuratesurrogatedidnotleadtothebestsolution.Zerpaetal.[ 44 ]showedthattheuseofmultiplesurrogateshelpedtoidentifyalternativeoptimalsolutionscorrespondingtodifferentregionsinthedesignspace.Multiplesurrogatetechniquesincludeusinganensembleofsurrogates,wherethepredictionisaweightedresultofthesurrogatepredictions[ 45 47 ].Theweightsplacedoneachsurrogatepredictionaregenerallybasedonlocalorglobalerrormetrics.Proposed 24

PAGE 25

methodsforchoosingtheweightfactorsincludeerrorcorrelation,cross-validationerror,predictionvariance,anderrorminimization.Theadditionofmultiplepointsperoptimizationcyclehasalsobeenexplored[ 48 50 ].VianaandHaftkaandlaterChaudhurietal.developedanalgorithmforaddingseveralpointsperoptimizationcyclebasedonapproximatedcomputationoftheprobabilityofimprovement(theprobabilityofbeingbelowatargetvalue)[ 13 51 ].Comparingtheirresultswithtraditionalsequentialbasedoptimizationwithkriging,theywereabletodeliverbetterresultsinafractionoftheoptimizationcyclesusingthisalgorithm. 2.5IntegratedThermalProtectionSystemTestCaseDescriptionThissectiondescribesthemaintestcasethatisusedtoillustratethemethodologiespresentedinthisproposal.TheintegratedthermalprotectionsystemwasusedasanillustrativeexampleinthearticlesinRefs.[ 52 57 ]. 2.5.1IntegratedThermalProtectionSystemLargeportionsoftheexteriorsurfaceofmanyspacevehiclesaredevotedtoprovidingprotectionfromthesevereaerodynamicheatingexperiencedduringascentandatmosphericreentry.Traditionally,thermalprotectionsystems(TPS)donotprovidestructuralsupportfunctions,andareaddedtoonlytoprotecttheunderlyingstructure,therebyaddingtothelaunchweight.ThisisthecasewiththeTPSoftheApollo,SpaceShuttleOrbiter,andX-33VentureStar.Aproposedintegratedthermalprotectionsystem(ITPS)providesstructuralloadbearingfunctioninadditiontoitsinsulationfunction,andinsodoingprovidesachancetoreducelaunchweight.OneproposedITPSdesignisthecorrugatedcoresandwichstructure,whichisillus-tratedinFig. 2-1 .Thisdesignevolvedfromstudiesforreusablelaunchvehicles(RLV)andevolvedtowardsrobustmetallicTPSconcepts[ 58 60 ],whichbeenthesubjectofseveralstudies[ 61 63 ].Thesestudieshaveshownthatthisdesignshouldbeanadequatelyrobust,weight-efcient,load-bearingstructure. 25

PAGE 26

Figure2-1.CorrugatedcoresandwichpanelITPSconcept Thedesignconsistsofatopfacesheetandwebsmadeoftitaniumalloy(Ti-6Al-4V),andabottomfacesheetmadeofberyllium(gradeS200-F,vacuumhotpressed).SaflRfoamisusedasinsulationbetweenthewebs.Thematerialpropertiesareassumedtobenormallydistributed(withtheexceptionofthedensityoftheinsulationfoam),withthenominalvaluesandcoefcientofvariationsgiveninTable 2-1 Table2-1.ITPSmaterialproperties PropertySymbolNominalCV(%) densityoftitanium1Ti4429kg m32.89densityofberyllium2Be1850kg m32.89densityoffoamS24kg m30thermalconductivityoftitaniumkTi7.6W m=K2.89thermalconductivityofberylliumkBe203W m=K3.66thermalconductivityoffoamkS0.105W m=K2.89specicheatoftitaniumcTi564J kg=K2.89specicheatofberylliumcBe1875J kg=K2.89specicheatoffoamcS1120J kg=K2.89 1Topfacesheetandwebmaterial 2Bottomfacesheetmaterial TherelevantgeometricvariablesoftheITPSdesignarealsoshownontheunitcellinFig. 2-1 .Thevariablesconsideredarethetopfacethickness(tT),bottomfacethickness(tB),thicknessofthefoam(dS),webthickness(tw),corrugationangle(),andlengthofunitcell(2p). 26

PAGE 27

Thethermalandstructuralrequirementsoftenconictduetothenatureofthemecha-nismsthatprotectagainstthefailureinthedifferentmodes.Examplesofconictsbetweenthermalandstructuralrequirementsinclude: Thinwebsallowlessheattoowtothebottomfacesheet,butaremoresusceptibletobucklingfailure. AsthedepthoftheITPSisreduced,thedesignresistsbucklingbetterbutisapoorerinsulator. Athickbottomfacesheetincreasesstressesintheweb,butdecreasesthebottomfacesheettemperature. 2.5.2ThermalandStructuralAnalysisThermalanalysisoftheITPSwasperformedusing1-Dheattransferequationsonamodeloftheunitcell.Theheatuxincidentonthetopfacesheetofthepanelishighlydependentonthevehicleshapeaswellasthevehicle'strajectory.AsinpreviousstudiesbyBapanapalli[ 61 ],incidentheatuxonaSpaceShuttle-likevehiclewasused.Alargeportionoftheheatisradiatedouttotheambientbythetopfacesheet,andtheremainingportionisconductedintotheITPS.Weconsidertheworst-casescenariowherethebottomfacesheetcannotdissipateheatbyassumingthebottomfacesheetisperfectlyinsulated.Also,thereisnolateralheatowoutoftheunitcell,sothatheatuxontheunitcellisabsorbedbythatunitcellonly.Foranin-depthdescriptionofthemodelandboundaryconditions,thereaderisreferredtotheBapanapalliandSharmareferences[ 61 63 ].ThemaximumtemperatureofthebottomfacesheetoftheITPSpaneliscalculatedusingthequadraticresponsesurfacedevelopedbyVillanuevaetal.[ 53 ]byaprocesssimilartothatofGoguetal.[ 62 ],usingtheMATLABtoolboxdevelopedbyViana[ 64 ].Itisafunctionofthepreviouslydescribedgeometricvariablesandthedensity,thermalconductivity,andspecicheatoftitaniumalloy,beryllium,andSaflRfoam.ThemaximumvonMisesstressinthewebwasalsofoundusingananalysisinAbaqus,atthetimewhenthetemperaturedifferencebetweenthetopandbottomfacesheetswasmaximum.AquadraticresponsesurfaceofthemaximumvonMisesstressintheweb 27

PAGE 28

wasdevelopedasafunctionofthegeometry,Young'smodulus,Poisson'sratio,andthecoefcientofthermalexpansion.TheoverallbucklingofthewebisassumedtobeEulerbuckling.Itismodeledasafunctionofthewebthicknessandwidthofthefoam,alongwiththecoefcientofthermalexpansionandYoung'smodulusofthewebmaterialtorepresentaloadduetothetemperaturedifferencebetweenthetopandbottom.ThemassperunitareamoftheITPSiscalculatedusingEq.( 2 )whereT,B,andwarethedensitiesofthematerialsthatmakeupthetopfacesheet,bottomfacesheet,andweb,respectively. m=TtT+BtB+wtwdS psin(2) 28

PAGE 29

CHAPTER3INCLUDINGTHEEFFECTOFFUTURETESTSANDREDESIGNINRELIABILITYCALCULATIONSItiscommontotestcomponentsaftertheyaredesignedandredesignifnecessary.Thereductionoftheuncertaintyintheprobabilityoffailurethatcanoccurafteratestisusuallynotincorporatedinreliabilitycalculationsatthedesignstage.Thisreductioninuncertaintyisaccomplishedbyadditionalknowledgeprovidedbythetestandbyredesignwhenthetestrevealsthatthecomponentisunsafeoroverlyconservative.Inthischapter,wedevelopamethodologytoestimatetheeffectofasinglefuturethermaltestfollowedbyredesign,andmodeltheeffectoftheresultingreductionoftheuncertaintyintheprobabilityoffailure.Usingassumeddistributionsofcomputationandexperimentalerrorsandgivenre-designrules,weobtainpossibleoutcomesofthefuturetestandredesignthroughMonteCarlosamplingtodeterminewhatchangesinprobabilityoffailure,design,andweightwilloccur.Inaddition,Bayesianupdatingisusedtogainaccurateestimatesoftheprobabilityoffailureafteratest.Thesemethodsaredemonstratedthroughafuturethermaltestonanintegratedthermalprotectionsystem.Weobservethatperformingredesignfollowingasinglefuturetestcanreducetheprobabilityoffailurebyordersofmagnitude,onaverage,whentheobjectiveoftheredesignistorestoreoriginalsafetymargins.Redesignforagivenreducedprobabilityoffailureallowsadditionalweightreduction. 3.1MotivationforExaminingFutureTestsandRedesignTraditionally,aerospacestructureshavebeendesigneddeterministically,employingsafetymarginsandsafetyfactorstoprotectagainstfailure.Afterthedesignstage,mostcomponentsundergotests,whosepurposeistovalidatethemodelandcatchunacceptabledesignsandredesignthem.Afterproduction,inspectionandmanufacturingaredonetoen-suresafetythroughoutthelifecycle.Incontrast,probabilisticdesignconsidersuncertaintiestocalculatethereliability,whichallowsthetrade-offofcostandperformance.Inrecentyears,therehasbeenamovementtoquantifytheeffectofuncertaintyreductionmeasures,suchastests,inspection,maintenance,andhealthmonitoring,on 29

PAGE 30

thesafetyofaproductoveritslifecycle.Muchworkhasbeencompletedintheareasofinspectionandmaintenanceforstructuresunderfatigue[ 4 7 ].StudiesbyAcaretal.[ 2 ]investigatedtheeffectsoffuturetestsandredesignonthenaldistributionoffailurestressandstructuraldesignwithvaryingnumbersoftestsatthecoupon,element,andcerticationlevels.Goldenetal.[ 65 ]proposedamethodtodeterminetheoptimalnumberofexperimentsrequiredtoreducethevarianceofuncertainvariables.Sankararamanetal.[ 8 ]proposedanoptimizationalgorithmoftestresourceallocationformulti-levelandcoupledsystems.AmethodtosimultaneouslydesignastructuralcomponentandthecorrespondingprooftestconsideringtheprobabilityoffailureandtheprobabilityoffailingtheprooftestwasintroducedbyVenterandScotti[ 66 ].Mostaerospacecomponentsaredesignedusingacomputationalmodelingtechnique,suchasniteelementanalysis.Weexpectsomeerror,oftenlabeledasepistemicuncer-tainty(associatedwithlackofknowledge),inthemodeledbehavior.Thetruevalueofthiserrorisunknown,andthusweconsiderthislackofknowledgetoleadtoanuncertainfuture.Testsareperformedtoreducetheerror,thusnarrowingtherangeofpossiblefuturesthroughtheknowledgegainedandthecorrectionofunacceptablefuturesbyredesign.Inthisstudy,weexaminetheeffectofasinglefuturethermaltestfollowedbypossibleredesignonthereliabilityandweightofanintegratedthermalprotectionsystem(ITPS).AdescriptionoftheintegratedthermalprotectionsystemispresentedinSec. 2.5.1 .AnexperimentthatndsthebottomfacesheettemperatureofasmallITPSpanelisusuallyconductedinavacuumchamberwithheatappliedtothetopfacesheetbyheatlamps.Thesidesofthepanelaretypicallysurroundedbysomekindofinsulationtopreventlateralheatloss.Thetemperatureofthebottomfacesheetisfoundwiththermocouplesembeddedintoorincontactwiththelowersurfaceofthebottomfacesheet.Thethermaltestconsideredinthisstudymeasuresthemaximumtemperatureofthebottomfacesheet,whichiscriticalduetoitsproximitytotheunderlyingvehiclestructure.Adesignisconsideredtohavefailedthermallyifitexceedsthemaximumallowabletemperature. 30

PAGE 31

InpreviousworkontheoptimizationoftheITPS,Villanuevaetal.[ 53 ]usedprobabilityoffailurecalculationsthatconsideredonlythevariabilityingeometricandmaterialparametersanderrorduetoshortcomingsintheanalyticalmodel.Expandingonthosestudies,weincludetheinformationgainedfromatestinatemperatureestimate,thereductioninuncertaintyresultingfromthetest,andtheabilityofthetesttoguideredesignfordangerousoroverlyconservativedesigns.Thereby,theobjectiveofthischapteristo: 1. Presentamethodologytobothpredictandincludetheeffectofafutureredesignfollowingatestduringthedesignstage 2. Illustratetheabilityofatestincombinationwithredesigntoreducetheprobabilityoffailureevenwhenatestshowsthatthedesigniscomputationallyunconservative 3. ExaminetheoverallchangesinmassresultingfromredesignbasedonthefuturetestTheuncertaintymodelandprobabilityoffailurecalculationsaredescribedinSection 3.2 .Section 3.3 continueswiththemethodologytocalibratethecomputationalmodelbasedonatestandincludesredesignbasedonthetest.ThemethodtosimulatefuturetestsissummarizedinSection 3.4 .Section 3.5 presentsanillustrativeexamplethatdetailstheeffectofincludingthetestandredesigninprobabilityoffailurecalculations. 3.2UncertaintyModeling 3.2.1ClassicationofUncertaintiesOberkampfetal.[ 1 ]providedananalysisofdifferentsourcesofuncertaintyinengineer-ingmodelingandsimulation,whichwassimpliedbyAcaretal.[ 2 ].WeuseclassicationsimilartoAcar'stocategorizetypesofuncertaintyaserrors(uncertaintiesthatapplyequallytoeveryITPS)orvariability(uncertaintiesthatvaryineachindividualITPS).Wefurtherdescribeerrorsasepistemicandvariabilityasaleatory.AsdescribedbyRaoetal.[ 67 ],theseparationoftheuncertaintyintoaleatoryandepistemicuncertaintiesallowsmoreunder-standingofwhatisneededtoreducetheuncertainty.Testsreduceerrorsbyallowingustocalibrateanalyticalmodels.Forexample,testingcanbedonetoreducetheuncertaintyinfailurepredictionsduetohighstresses.Variabilitycanbereducedbyloweringtolerancesin 31

PAGE 32

manufacturing.Variabilityismodeledasrandomuncertaintiesthatcanbemodeledproba-bilistically.Incontrast,errorsarexedforagivenITPSandarelargelyunknown,butheretheyaremodeledprobabilisticallyaswell.VariabilityinmaterialpropertiesandconstructionoftheITPSleadstovariabilityintheITPSthermalresponse.Morespecically,wewillhavevariabilityinthecalculatedtemperatureduetotheinputvariabilities.WesimulatethisprocesswithaMonteCarlosimulation(MCS)thatgeneratesvaluesoftherandomvariablesrbasedonanestimateddistributionandcalculatesthebottomfacesheettemperatureTcalcforeach,generatingtheprobabilitydistributionfunction.Thecalculatedtemperaturedistributionthatreectstherandomvariabilityisdenotedfcalc(T).Inestimatingtheprobabilityoffailure,wealsoneedtoaccountforthemodelingorcomputationalerror.Wedenotethiscomputationalerrorbyec,whereecismodeledasauniformlydistributedrandomvariablewithincondencelimitstheinthecomputationalmodelasdenedbytheanalyst.Unlikethevariability,theerrorhasasinglevalue,andtheuncertaintyisduetoourlackofknowledge.Foragivendesigngivenbydandr,thepossibletruetemperatureTPtruecanbefoundbyEq.( 3 )intermsofpossiblecomputationalerrorsec.Thesigninfrontofecisnegativesoapositiveerrorimpliesaconservativecalculation,meaningitoverestimatesthetemperature. TPtrue(d;r;ec)=Tcalc(d;r)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ec)(3)Sincetheanalystdoesnotknowecanditismodeledasarandomvariable,wecanformadistributionofthepossibletruetemperature,denotedasfPtrue(T).Toillustratethedifferencebetweenthetruedistributionofthetemperatureftrue(T)andpossibletruedistributionfPtrue(T),letusconsiderasimpleexamplewherethecalculatedtemperatureofthenominaldesignis1,thetruetemperatureis1.05,andthecomputationalerrorisuniformlydistributedintherange[-0.1,0.1].Thepossibletruetemperaturewithoutvariabilityareuniformlydistributedin[0.9,1.1]byEq.( 3 ).Now,letusconsideranadditionalvariabilityinthetemperatureduetomanufacturingtolerancesintherange[-0.02,0.01],suchthat 32

PAGE 33

Tcalc(d;r)isuniformlydistributedintherange[0.98,1.01].Finally,thetruetemperaturewillvaryfrom[1.03,1.06]asftrue(T),andthepossibletruetemperaturefrom[0.882,1.111]asfPtrue(T).Figure 3-1 illustrateshowwearriveatthedistributionfPtrue(T).Theinputrandomvariableshaveinitialdistributions,denotedasfinp(r),andtheserandomvariables,incombinationwiththedesignvariables,leadtothedistributionofthecalculatedtemperaturefcalc(T).Therandomcomputationalerrorisapplied,leadingtothedistributionofthepossibletruetemperaturefPtrue(T),whichhasawiderdistributionthanfcalc(T). Figure3-1.Illustrationofthevariabilityoftheinputrandomvariables,calculatedvalue,computationalerror,andresultingdistributionofpossibletruetemperature Aspreviouslynoted,ecismodeledasarandomvariablenotbecauseitisrandom,butbecauseitsvalueisunknowntotheanalyst.Toemphasizethispoint,theactualtruetemperatureisknownonlywhenweknowtheactualvalueofecasec;trueasillustratedinEq.( 3 ). Ttrue(d;r)=Tcalc(d;r)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ec;true)(3)Again,thesetruevaluesareunknowntotheanalyst.Thisdistinctionbetweentruevaluesandanalyst-estimated,possibletruevaluesisimportantandwillbeapointofcomparisonthroughoutthischapter.Figure 3-2 showsanexampleoftheprobabilitydistributionofthetruetemperatureftrue(T),aswellastheprobabilitydensityfunctions(pdf)offcalc(T)andfPtrue(T).Forthisexample,wemodeledthevariabilityinthematerialpropertiesandvariabilityingeometry 33

PAGE 34

withnormaldistributions,andthecomputationalerrorwithauniformdistribution.TheplotsofeachpdfshowtheprobabilityofexceedingtheallowabletemperatureTallow,representedbytheareawherethetemperatureexceedstheallowable. Figure3-2.Illustrationwithunconservativecalculationoftemperature.Whenincludingtheerrorintheestimate,theestimateoftheprobabilityoffailureisimproved. Wechoseanillustrationwherethecomputationalerrorisunconservativesothefcalc(T)providesanunderestimateoftheprobabilityoffailuregivenbyftrue(T).Thiscomputationalerrorbetweenthemeanoffcalc(T)andthemeanofftrue(T)isec;true.However,sinceweincludeecasarandomvariable,wewidenedthedistributionfcalc(T),resultinginfPtrue(T).Thisprovidesamoreconservativeestimateoftheprobabilitythatcancompensatefortheunconservativecalculation.Ofcourse,whentheerrorinthecalculationisconservative,thiswidedistributionwillgrosslyoverestimatetheprobabilityoffailure. 3.2.2TrueProbabilityofFailureCalculationThetrueprobabilityoffailureofadesigndwithrandomvariablesrcanbefoundwhenthetruecomputationalerrorisknown.Thisisclearlyahypotheticalsituationbecauseinrealitythetruecomputationalerrorisnotknownbytheanalyst.Here,MonteCarlosimulation(MCS)isusedtocalculatethetrueprobabilityoffailure.Thelimitstateequationgis 34

PAGE 35

formulatedasthedifferencebetweenacapacityCandresponseRasshowninEq.( 3 ). g=Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(Ttrue(d;r)C)]TJ /F5 11.955 Tf 10.26 0 Td[(R(3)SinceweconsiderfailuretooccurwhenthemaximumbottomfacesheettemperatureexceedstheallowabletemperatureTallow,theresponseisTtrueandthecapacityistheallowabletemperature.Thetrueprobabilityoffailurepf;trueisestimatedwithEq.( 3 ). pf;true=1 NNXi=1I[g(Ci;Ri)0](3)TheindicatorfunctionIequals1iftheresponseexceedsthecapacity,andequals0fortheoppositecase.ThenumberofsamplesisN. 3.2.3Analyst-EstimatedProbabilityofFailureCalculationSincethetruecomputationalerrorisunknown,thetrueprobabilityoffailureisunknownaswell.Becauseofthis,thebestestimatetheanalystcanobtainusesthecalculatedtemperatureTcalcandthecomputationalerrorthroughthepossibletruetemperatureofEq.( 3 )todeterminetheestimatedprobabilityoffailurewiththelimitstateequationformulatedasinEq.( 3 ). g=Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(TPtrue(d;r;ec)C)]TJ /F5 11.955 Tf 10.26 0 Td[(R(3)Sincethetwotypesofuncertainty(computationalerrorsandvariabilityinmaterialpropertiesandgeometry)intheresponseareindependent,SeparableMonteCarlo(SMC)sampling[ 28 ]canbeusedwhenevaluatingtheprobabilityoffailure.Thelimitstateequationcanbereformulatedsothatthecomputationalerrorisonthecapacityside,andallrandomvariablesassociatedwithmaterialpropertiesandgeometrylieontheresponseside. g=Tallow 1)]TJ /F5 11.955 Tf 10.26 0 Td[(ec)]TJ /F5 11.955 Tf 10.26 0 Td[(Tcalc(d;r)C0)]TJ /F5 11.955 Tf 10.26 0 Td[(R0(3)Thisanalyst-estimatedprobabilityoffailurepf;analystcanthenbecalculatedwithEq.( 3 ),whereMandNarethenumberofcapacityandresponsesamples,respectively. 35

PAGE 36

pf;analyst=1 MNNXi=1MXj=1I[g(Cj;Ri)0](3) 3.3IncludingtheEffectofaCalibrationTestandRedesignWeconsideratest,performedforthepurposeofvalidatingandcalibratingamodel,foraselecteddesigndtesttodeterminethetemperatureofthetestarticleTtest.Wefurtherassumethatthetestarticleiscarefullymeasuredforbothdtestandrtestsothatbothareaccuratelyknown,andthattheerrorsinthecomputedtemperaturesduetouncertaintyinthevaluesofdtestandrtestaresmallcomparedtothemeasurementerrorsandcanbeneglected.Ifnoerrorsaremadeinthemeasurementsofdtest,rtest,andTtest,thentheexperimentalresultisactuallythetruetemperatureofthetestarticle.Wedenotethiserror-freetesttemperatureTtest;true. Ttest;true=Ttrue(dtest;rtest)(3)However,thereisunknownmeasurementerrorex,whichwemodelasarandomvariablebasedonourestimateoftheaccuracyofthetest.ThemeasuredtemperatureTmeasthenincludestheexperimentalerrorex;true.Theexperimentalerrorcouldalsoincludeacomponentduetothefactthatrtestisnotperfectlyknown. Tmeas=Ttest;true 1)]TJ /F5 11.955 Tf 10.26 0 Td[(ex;true(3)Usingthecomputationalandexperimentalresults,alongwiththecorrespondingerrorestimatesforthetestarticle,weareabletorenethecalculatedvalueanditserrorforanydesigndescribedbythedesignvariablesdandrandomvariablesr.Inthisway,theresultofthesingletestcanbeusedtocalibratecalculationsforotherdesigns.Weexaminetwomethods,whichtakedifferentapproachesinusingthetestascalibration.Therstapproachintroducesasimplecorrectionfactorbasedonthetestresult.ThesecondusestheBayesianmethodtoupdatetheuncertaintyofthecalculatedvaluefordtestbasedonthetestresultandthentransfersthisupdateduncertaintytoothercalculationsasthemeansofcalibration. 36

PAGE 37

3.3.1CorrectionFactorApproachThecorrectionfactorapproachisafairlystraightforwardmethodofcalibration.Assum-ingthatthetestresultismoreaccuratethanthecalculatedresultforthetestarticle,wescaleTcalcforanyvalueofdandrbytheratioofthetestresulttothecalculatedresulttoobtainthecorrectedcalculationTcalc;corr. Tcalc;corr=Tcalc(d;r) Tmeas Tcalc(dtest;rtest)!(3) 3.3.2BayesianUpdatingApproachBeforethetest,wehaveanexpectationofthetestresultsbasedonthecomputationalresultofdtestandrtest.Wedenotethisdistributionbyfinitest;Ptrue,whichcanbeviewedasthedistributionoffPtrue(T)ofthetestarticlewithxedrandomvariablesrtest.Furthermore,itmaybeviewedasthepossibletruetemperaturedistributionofthetestarticlejustbeforethetest.Inthetest,wemeasureatemperatureTmeas.Becauseofexperimentalerrorex,thetruetestresultTtest;trueisnotequaltoTmeas(asseeninEq.( 3 )).Thepossibletruevalueofthetestresultisinsteadgivenas Tmeastest;Ptrue=Tmeas(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ex)(3)whereTmeastest;Ptrueformsthedistributionofpossibletruetestresultsavailablefromthemeasure-mentsonly.Wethushavetwodistributionsofpossibletruetestresults.Oneisbasedonthecalculatedvalueandthedistributionofthecalculationerror,andtheotherisbasedonthemeasurementandthedistributionofthemeasurementerror.TheBayesianapproachcombinesthesetwodistributionstoobtainanarrowerandmoreinformativedistribution.Inthisformulation,theprobabilitydistributionofthepossibletruetemperatureofthetestarticleftest;Ptrue(T)isupdatedas fupdtest;Ptrue(T)=ltest(T)finitest;Ptrue(T) R+1ltest(T)finitest;Ptrue(T)dT(3) 37

PAGE 38

wherethelikelihoodfunctionltest(T)istheconditionalprobabilitydensityofobtainingthetestresultTmeaswhenthetruetemperatureofthetestarticleisT.Thatis,ltestistheprobabilitydensityofT 1)]TJ /F7 7.97 Tf 5.07 0 Td[(exevaluatedatT=Tmeas.Theupdatedestimatefupdtest;Ptrue(T)isthedistributionoftheupdatedtruepossibletestresultTupdtest;Ptrue.ThisisusedtondthedistributionoftheBayesianestimateofthecomputationalerroreBayeswithEq.( 3 ). eBayes=1)]TJ /F5 11.955 Tf 26.14 10.21 Td[(Tupdtest;Ptrue Tcalc(dtest;rtest)(3)WecanthenreplacethepossibletruetemperaturegivenbyEq.( 3 )withatruetemperaturethatusestheBayesianestimateoftheerror. TPtrue(d;r;eBayes)=Tcalc(d;r)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(eBayes)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(eextrap)(3)TheadditionalerroreextrapisincludedtoaccountfortheerrorthatoccurswhenapplyingthisBayesianestimateoftheerrortosomedesignotherthanthetestdesign.ThisextrapolationerrorisfurtherdescribedinSec. 3.3.2.2 .NotethatitisalsopossibletoperformtheBayesianupdatingbyreversingtherolesofthetwopossibletruetesttemperatures.Thatis,wecouldtakethedistributionbasedonthemeasurementerrorastheinitialdistribution,andtakethecomputedresultastheadditionalinformation.However,inthiscasethelikelihoodfunctionwouldrequirerepeatedsimulationsfordifferentpossibletruetemperatures,greatlyincreasingthecomputationalcost. 3.3.2.1IllustrativeexampleofcalibrationbytheBayesianapproachToillustratehowBayesianupdatingisusedtocalibratecalculationsbasedonasinglefuturetest,weconsiderasimplecasewhereboththecomputationalandexperimentalerrorsareuniformlydistributed.Tosimplifytheproblem,wenormalizealltemperaturesbythecalculatedtemperaturesothatTcalc(dtest;rtest)=1.Theerrorboundofthecalculationis10%andtheerrorboundofthetestis7%.ThenormalizedtestresultisTmeas=1:05. 38

PAGE 39

Inthiswork,wemakethesimplifyingassumptionthatthelikelihoodfunctionisaboutTmeasratherthanT.Thatis,weuseconditionalprobabilityofobtainingthetemperatureTgiventhemeasuredtemperature.Thisallowsforauniformvalueofthelikelihoodfunctionwhereitisnonzero,whichtherebyresultsinauniformdistributionoftheupdatedBayesianestimateofthecomputationalerrorsincethedistributionoffupdtest;Ptruewillalsobeuniform.TheeffectofthisapproximationofthelikelihoodfunctionisexaminedinAppendix A .Theinitialprobabilitydistributionfinitest;Ptrue(T)andthelikelihoodfunctionltestaredescribedbyEqs.( 3 )and( 3 ),respectively. finitest;Ptrue(T)=8>>>><>>>>:1 0:2Tcalc(dtest;rtest)ifT Tcalc(dtest;rtest))]TJ /F12 11.955 Tf 10.26 0 Td[(10:1;0otherwise:(3) ltest(T)=8>>>><>>>>:1 0:14TmeasifT)]TJ /F7 7.97 Tf 5.07 0 Td[(Tmeas Tmeas0:07;0otherwise:(3)SinceTcalc(dtest)=1andthecomputationerrorboundsare10%,theinitialdistributionofthetruetemperatureisfinitest;Ptrue(T)=5ontheinterval(0.9,1.1)andzeroelsewhere.ThisisshowninFig. 3-3 .ThetestresultofTmeas=1:05resultsinalikelihoodofltest=6:803ontheinterval(0.9765,1.1235)andzeroelsewhere.Equation( 3 )isusedtondtheupdatedTtruedistributionsothatfupdtest;Ptrue(T)=8:1ontheinterval(0.9765,1.1)andzeroelsewhere.Theupdateddistributionshowsthatthetruetemperatureissomewhereontheinterval(0.9765,1.1).UsingthistemperaturedistributionalongwiththecalculatedvalueTcalc(dtest),theupdatederrordistributioneBayescanbefound.ThroughEq.( 3 ),wedeterminethateBayesisuniformlydistributedfrom-10%to2.35%. 3.3.2.2ExtrapolationerrorincalibrationFigure 3-4 illustrateshowtheBayesianapproachisusedtocalibratethecalculationsforotherdesignsdescribedbyd.Here,weconsiderthecasewhenthecalculatedtemperature 39

PAGE 40

Figure3-3.IllustrativeexampleofBayesianupdatingshowingtheinitialdistribution(top),initialdistributionandtest(middle),andupdateddistribution(bottom). islinearinthedesignvariabled,andthereisnovariability(randomvariablesxedatnominalvalues). Figure3-4.IllustrationofthecalibrationusingBayesianupdating 40

PAGE 41

Atdesigndtest,wehavethesameerrorscenariosimilartothatillustratedinFig. 3-3 .Thatis,werepresentthecalculatedtemperatureatdtestasapointonthesolidblackline,andtheerrorboundsaboutthiscalculationbythedottedblacklines.Thestarrepresentstheexperimentallymeasuredtemperature,andtheerrorbarsshowtheuncertaintyinthistemperature.BytheBayesianapproach,weobtainacorrectedtesttemperatureasrepresentedbythepointonthegreyline,aswellasupdatederrorboundsrepresentedbythegreydash-dotline.However,thiscorrectionandupdatederrorismostaccurateatthetestdesign.There-fore,weapplyanadditionalerror,theextrapolationerroreextrap,whencalibratingdesignsotherthandtest.NotethatatdtesttheupdatederrorboundsinFig. 3-4 coincidewiththeerrorboundsofthetest.Asthedesignbecomesincreasinglydifferentfromdtest,theupdatederrorboundsbecomewider.Themagnitudeofeextrapisassumedtobeproportionalthedistancebetweendanddtest,suchthat eextrap=(eextrap)maxkd)]TJ /F5 11.955 Tf 10.26 0 Td[(dtestk dlim(3)Thisdenestheextrapolationerrorsothatitismaximumwhenthedistancebetweendanddtestisatlimitofthisdistancedlimandzeroatthetestdesign.Theextrapolationerrorisameasureofthevariationoftheerrorsinthemodelawayfromthetestdesign.Inthiswork,weassumethatthemagnitudeofeextrapislinearwiththedistancebetweendanddtest,whichwouldbereasonableforsmallchangesinthedesign.However,weexaminetheeffectofthisassumptioninAppendix B whereweuseaquadraticvariation. 3.3.3Test-CorrectedProbabilityofFailureEstimateThecorrectedprobabilityoffailurepf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corrafterthetestcanbeestimatedbytheanalystusingtheupdatederrorobtainedfromtheBayesianapproach.SeparableMonteCarloisusedtocalculatepf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr. g=Tallow 1)]TJ /F5 11.955 Tf 10.26 0 Td[(eBayes)]TJ /F5 11.955 Tf 10.26 0 Td[(Tcalc(d;r)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(eextrap)C)]TJ /F5 11.955 Tf 10.26 0 Td[(R(3) 41

PAGE 42

pf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr=1 MNNXi=1MXj=1I[g(Cj;Ri)0](3) 3.3.4RedesignBasedontheTestTwocriteriaforredesignareconsidered,eachwithdifferentperspectivesonthepurposeoftheredesign.Therstcriterionisbasedontheagreementbetweenthemeasuredandcalculatedvaluesforthetestarticle.Thesecondcriterionconsiderstheprobabilityoffailureestimatedbytheanalyst. 3.3.4.1DeterministicredesignIndeterministicredesign,redesignoccurswhenthereisasignicantdifferencebetweentheexperimentallymeasuredtemperatureTmeasandtheexpectedtemperaturegivenbythecomputationalmodel.Itisassumedthatthetemperaturegivenbythecomputationalmodel(Tcalc)isthedesiredvalue.Therefore,thecomponentisredesignedtorestorethisoriginaltemperature.Thedeterministicredesigncriterionisimplementedbyimposinglimitsontheacceptableratioofthemeasuredtemperaturetothecalculatedtemperature.RedesignoccurswhenTmeas Tcalc(dtest;rtest)islessthanthelowerlimitDL(conservativecomputationalmodel)orexceedstheupperlimitDU(unconservativecomputationalmodel). 3.3.4.2ProbabilisticredesignInprobabilisticredesign,theoriginalstructureisdesignedforaspeciedprobabilityoffailure,andredesignisalsodonetoachieveaspeciedprobabilityoffailure.Itisreasonabletoselectthetargetredesignprobabilitypf;targettobethesameasthatobtainedwithproba-bilisticdesign.Thetargetredesignprobabilityoffailurecanalsobesettomakethedesignsaferafterthetest.Therefore,redesignoccurswhenthetest-correctedprobabilityoffailureestimate,givenbyEq.( 3 )isoutsidethelimitsoftheacceptablerange.ThelowerlimitofthisrangeisdenotedPL,andtheupperlimitPU. 42

PAGE 43

3.4MonteCarloSimulationsofaFutureTestandRedesignMonteCarlosimulationsareusedtosimulatetheeffectofafuturetestforadesigndescribedbydesignvariablesdandrandomvariablesrwiththegoalofsimulatingmultiplepossibleoutcomesofthistest.Tosimulateasingleoutcomeofthefuturetest,werstobtainasinglesampleofthetruecomputationalandexperimentalerrors.Usingthecalculatedvalueforthetestdesignandthetruecomputationalerror,wecanobtainthetruetemperaturebyEq.( 3 ).Next,theexperimentallymeasuredtemperatureisfoundusingEq.( 3 ).ThechoicecanbemadetocalibratebythecorrectionfactorapproachortheBayesianupdatingapproach,and,further,thechoiceofdeterministicorprobabilisticredesigncanbemade.Thetrueandcorrectedanalyst-estimatedprobabilitiesoffailureafterthetestcanthenbedetermined.Atthispoint,theeffectofonlyonepossibleoutcomeofthetesthasbeenexamined.ThemajorstepsandequationsinvolvedinthesimulationofasingleoutcomeofthetestaresummarizedinthepseudocodegiveninAlgorithm 1 1.Todetermineanotherpossibleoutcome,thetruecomputationalandexperimentalerrorsarere-sampledandtheprocessisrepeated.Therefore,fornpossibleoutcomesofafuturetest,wesamplenpairsoftheerrorsandtrueprobabilitiesoffailure,nanalyst-estimatedprobabilitiesoffailureafterthetest,anduptonupdateddesigns.Notethatthereisasingleinitialdesign,butifkofthencasesarere-designedwewillendupwithuptok+1differentdesigns. 3.5IllustrativeExampleInthisexample,wecomparetheprobabilitiesoffailureofanITPSwiththedimensionsandmaterialpropertiesofprobabilisticoptimumfoundin[ 53 ].Inthatstudy,theoptimum 1Intheimplementationofthisalgorithm,itisassumedthatallanalystsperformingthetesthavethesamevalueofrtest.Sinceeachanalystaccuratelymeasuredrtest,theeffectofthisassumptionislikelytobenegligible. 43

PAGE 44

Algorithm1Proceduretosimulatenpossibletruecomputationalerrorstocalculateproba-bilityoffailurefordesigndandr,andnpossibleoutcomesofthefuturetestwithredesignforadesigndescribedbydtestandrtest.Therandomvariablesrtestarexedforthetest.Thesetofsamplesofrarexedovertheoriginaldesignandtheredesigns. 1: Samplesetofvaluesofrandomvariablesr(thissetisxedoverthenpossibleout-comes,andfortheoriginaldesigndandanyredesignedd) 2: Samplenvaluesofecandex 3: CalculateTcalc(d;r)usingcomputationalmodelandTcalc;test(dtest;rtest) 4: Calculatepf;analystbynsamplesofecandEqs.( 3 )and( 3 ) 5: fori=1!ndo 6: Setec;true=ec(i)andex;true=ex(i) 7: Calculatepf;truebyEqs.( 3 )and( 3 ) 8: CalculateTtrue;test(dtest;rtest)byEq.( 3 )andTmeasbyEq.( 3 ) 9: ifCorrection-FactorCalibrationthen 10: CalculateTcalc;corr(d;r)byEq.( 3 ) 11: elseifBayesianCalibrationthen 12: CalculateeBayesbyEqs.( 3 )and( 3 ),andeextrapbyEq.( 3 ) 13: Updatepf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corrbynsamplesofecandEqs.( 3 )and( 3 ) 14: endif 15: ifDeterministicRedesignthen 16: ifTmeas Tcalc(dtest;rtest)DUthen 17: Redesignfor(Tcalc;corr(d;r))redesign=(Tcalc(d;r))originaldesign 18: Updatepf;truebyEqs.( 3 )and( 3 ) 19: endif 20: elseifProbabilisticRedesignthen 21: ifpf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr>pf;target+PUjjpf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr
PAGE 45

Table3-1.ITPSvariables VariableSymbolNominalCV(%) webthicknesstw1.77mm2.89bottomfacesheettB7.06mm2.89foamthicknessds71.3mm2.89topfacesheetthicknesstT1.2mm2.89halfunitcelllengthp34.1mm2.89angleofcorrugation802.89densityoftitanium1Ti4429kg m32.89densityofberyllium2Be1850kg m32.89densityoffoamS24kg m30thermalconductivityoftitaniumkTi7.6W m=K2.89thermalconductivityofberylliumkBe203W m=K3.66thermalconductivityoffoamkS0.105W m=K2.89specicheatoftitaniumcTi564J kg=K2.89specicheatofberylliumcBe1875J kg=K2.89specicheatoffoamcS1120J kg=K2.89 1 Topfacesheetandwebmaterial 2 Bottomfacesheetmaterial 3-2 .Theoriginalestimatedprobabilityoffailureis0:12%andthenominalmassperunitareais35.1kg=m2.Sincethedistributionsoftheerrorsarebounded,weremovethepossibilityofextremedifferencesbetweenthecalculatedandexperimentallymeasuredvaluesinthesimulatedthefuturetest.Withthesevaluesoftheerrors,inthemostextremecase,thetemperaturesdifferbyapproximately13%,whichoccurswhentheerrorsaresampledatopposingboundsofthedistribution(e.g.ec;true=0:1andex;true=)]TJ /F12 11.955 Tf 7.6 0 Td[(0:03).Ifnormaldistributionsoftheerrorswereused,thisdifferencecanbecomeinnite. Table3-2.Distributionoferrors ErrorDistributionBounds ecUniform10%exUniform3% Theextrapolationerroreextrapisestimatedtobe2%whendischangedby10%fromdtest,andvarieslinearlywithchangeind. eextrap=0:02kd)]TJ /F5 11.955 Tf 10.26 0 Td[(dtestk 0:1kdtestk(3) 45

PAGE 46

Itispossibletoassumeotherrelationshipsbetweenoftheextrapolationerrorandthedistanceofdfromdtest.InAppendix B ,weexaminetheeffectofassumingthemagnitudeofeextrapisquadraticwiththechangeind.Inthisexample,weexaminethebenetsofincludingafuturetestbyexaminingseveralcasesthatincludefuturetests,onewithoutredesignandonewithredesignbasedonthefuturetestbytheprocessdescribedinSection 3.4 .Wewillexamine10,000possiblefuturetestoutcomes(10,000samplesoftheerrors),anduse10,000samplesoftherandomvariables.Therefore,thetrueprobabilityoffailureiscalculatedwith10,000sampleseachoftheresponseandcapacity,whereastheanalyst-estimatedprobabilityoffailureiscalculatedwith10,000samplesofthecapacityand10,000oftheresponsebyseparableMonteCarlo.Toreducetheeffectofnoise,thesetof10,000randomvariableswasheldconstantthrougheachofthe10,000possiblefuturetestoutcomes. 3.5.1FutureTestwithoutRedesignUsingthe10,000possibleoutcomesofthesinglefuturetest,wecanestimatetheeffectivenessoftheBayesianapproachbycomparingthreecases.Intherstcase,theanalystacceptsTcalcasthebestestimateofthetestarticletemperature.Inthesecond,theanalystacceptsTmeas.Inthethird,theanalystacceptsTBayeswhereTBayesisthetemperaturewiththemaximumlikelihoodintheupdateddistribution.Sincethisexamplesimpliesthelikelihoodfunction(seeSection 3.3.2.1 )sothattheupdateddistributionisuniform,wetakethemeanthedistributionasTBayes.WecomparetheabsoluteerrorofeachfromthetruetemperatureinTable 3-3 Table3-3.ComparingabsolutetrueerrorwhenusingTcalc,TmeasandTBayesasthetestarticletemperature TcomparedMeanerror(%)Standarddeviationoferror(%) Tcalc5.02.9Tmeas1.50.8TBayes1.30.8 TheseresultsshowthattheBayesianapproachprovidestheanalystwiththemostaccurateestimateofTtrueforthetestarticle.AcceptingTmeasresultsinaslightlyincreased 46

PAGE 47

errorandacceptingonlytheoriginalTcalchastheworsterrorwithameanvalueof5%.Table 3-4 showstheinwhichnumberofoccurrencesthecomparisontemperatureisclosertothetruetemperaturecomparedtoanothertemperatureoutofthe10,000possibleoutcomesIt Table3-4.Numberofoccurrencesinwhichthecomparisontemperatureisclosertothetruetemperaturecomparedtoanothertemperatureoutof10,000possibleoutcomes ComparisonTemperatureTBayesTmeasTcalc(#ofoccurrences) BetterthanTcalc84908490BetterthanTmeas19641507BetterthanTBayes9981507Betterthanalltemperatures19629971507EqualtoTcalc22EqualtoTmeas70382EqualtoTBayes70382 wasobservedthatacceptingeitherTBayesorTmeasinsteadofTcalcwasbetterin8490cases.Ofthesecases,TBayesandTmeaswereequalin5531.Intheremainderofthe7038timesTBayesandTmeaswereequal(1507cases),acceptingTcalcwasbetter.Inaddition,wecancomparetheanalyst-estimateprobabilityoffailuretothetrueprobabilityoffailure.TheseresultsaregiveninTable 3-5 .Weobservethatthemeantrueprobabilityoffailureisequaltothatoftheoriginalestimatedprobabilityoffailurebeforethetest.Thisresultisnotunexpectedaswedidnotallowredesign,thuspreventinganychangesindesignandthustheprobabilityoffailure. Table3-5.ProbabilitiesofFailurewithoutRedesign(usingBayesianCorrection) ParameterMeanStandardDeviationMinimumMaximum pf;true(%)0.120.3902.00pf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr(%)0.120.2701.93 Itisimportanttonotethat8884outofthe10000possibleoutcomesshowthatthetrueprobabilityoffailureislessthantheoriginalestimateoftheprobabilityoffailure.Infact,themediantrueprobabilityoffailureiszero,andiszerouptothe85thpercentile.AsummaryofthepercentilesisshowninTable 3-6 47

PAGE 48

Table3-6.Summaryofthepercentilesofthetrueprobabilityoffailurewithoutredesign Percentiles25%50%75%88.8%90%95%97.5% pf;true(%)0000.150.201.101.80 Basedonthelargenumberoftrueprobabilityoffailuresthatarezero,itwouldbeexpectedthatifredesignwereimplementedtorestoretheoriginalestimatedprobabilityoffailure,mostredesignswouldincreasetheprobabilityoffailure. 3.5.2RedesignBasedonTestInthissection,weexaminetheeffectofdeterministicandprobabilisticredesignfortheexample.ThesetworedesignmethodologiesaredescribedinSec. 3.3.4 3.5.2.1DeterministicredesignWechosedeterministicredesigntooccurwhentheratioTmeas Tcalc(dtest;rtest)isgreaterthan1.05(unconservativecomputationalmodel)orlessthan0.95(conservativecomputationalmodel).Weconsideronedesignvariable,thefoamthicknessds.Thisvariablewaschosensinceithasalargeimpactonthebottomfacesheettemperature.TheresultsincludingdeterministicredesignaregiveninTable 3-7 Table3-7.Calibrationbythecorrectionfactorapproachwithdeterministicredesign. ParameterOriginalMeanStandardDeviationMinimumMaximum dS(mm)71.371.51.244.899.4mass(kg=m2)35.135.12.828.941.6pf;true(%)0.120.0007120.00900.20 1 Ofthe10000possibleoutcomesofthefuturetest,4964requiredredesign.Conservativecasesaccountfor2425oftheredesigns,andunconservativecasesaccountfor2539. 2 99.3%oftrueprobabilityfailuresarebelowthemean. Theseresultsshowthatthetrueprobabilityoffailureisgreatlyreducedwhenredesignisallowed.Inaddition,thestandarddeviationisalsoreduced.Sincetheredesignissym-metric,itdoesnotcausemuchchangeintheaveragemass.Thereasonforthisdrasticreductioninprobabilityoffailureisthesubstantialreductioninerrorthatallowedustore-designallthedesignsthathadaprobabilityoffailureabove0.12%.Sowhilethesystemwas 48

PAGE 49

designedforaprobabilityoffailureof0.12%,itendedupwithameanprobabilityoffailureof0.0007%.However,wenotealargestandarddeviationinds,withtheminimumandmaximumvaluesquitedifferentfromthedesignvalueof71.3mm.Inpractice,theredesignmaynotbeallowedtobethisdrastic.Therefore,wealsoexaminethecasewheretheboundsoftheredesigneddsarerestrictedto10%oftheoriginalnominaldS.TheseresultsaregiveninTable 3-8 Table3-8.Calibrationbycorrectionfactorwithdeterministicredesign,boundsofredesigneddsrestrictedto10%oforiginalds ParameterOriginalMeanStandardDeviationMinimumMaximum dS(mm)71.371.40.564.178.4mass(kg=m2)35.135.11.233.436.7pf;true(%)0.120.00070.00900.20 WeobservethatrestrictingtheboundsofdSdoesnotchangethetrueprobabilityoffailure,anddoesnotcauseasignicantchangeintheaveragemass. 3.5.2.2ProbabilisticredesignTheinitialdesigndoesnotnecessarilymeetthereliabilityrequirementsofthedesigner.Itcanbe,forexample,acandidatedesigninaprocessofdesignoptimization.Whenitcomestoprobabilisticredesign,onemayexaminere-designtothemeanprobabilitywithoutredesignortoatargetprobability.Hereweassumethelatter,andweexaminecaseswherethetargetredesignprobabilityispf;target=0:01%withandwithoutboundsonds.Here,werequireredesigntooccurwhentheestimatedprobabilityoffailureisnotwithin50%ofthetarget.Werequirethatallunconservative(dangerous)designsabovethe50%thresholdberedesignbutrejecttheredesignofoverlyconservativecasesifitsmassdoesnotdecreasebyatleast4.5%.Sinceonlyonedesignvariable,thefoamthickness,isconsidered,adecreaseinmasscanonlyresultfromadecreaseinfoamthickness,whichcausesanincreaseintemperature.TheresultsareshowninTable 3-9 49

PAGE 50

Withoutboundsontheredesignedds,weobservethattheanalyst-estimatedtargetprobabilityoffailureisclosetothetargetof0.01%.Itisalsoobservedthatthereisasignif-icantreductioninmass(4%reduction)andareductionintheoriginalmeantrueprobabilityoffailurefrom0.12%to0.003%.Theanalystisabletoestimatethistrueprobabilityoffailurewithreasonableaccuracy.Whenweincludetheboundsonds,thetrueprobabilityoffailureisunabletoconvergetothetargetprobabilityoffailure,butthereisbetteragreementbetweentheanalyst-estimatedprobabilitiesoffailureandthetruevalue.Thisisduetotheinclusionoftheextrapolationerrorintheprobabilityoffailureintheredesignprocess.Wealsoobservea1.7%reductioninmeanmassfromtheoriginalvalue.Onanalnote,werecognizethatthelargepercentageofredesignsisundesirable.Thispercentagecanbegreatlyreducedbylessstringentredesignruleswhilestillhavingverylowprobabilitiesoffailure. 3.6SummaryandConcludingRemarksThisstudypresentedamethodologytoincludetheeffectofasinglefuturetestfollowedbyredesignontheprobabilityoffailureofanintegratedthermalprotectionsystem.Twomethodsofcalibrationandredesignbasedonthetestwerepresented.Weobservedthatthedeterministicapproach,whichrepresentscurrentdesign/redesignpractices,leadstoagreatlyreducedprobabilityoffailureafterthetestandredesign,areductionthatusuallyisnotquantied.TheprobabilisticapproachincludestheBayesiantechniqueforcalibratingthetemper-aturecalculationandre-designtoatargetprobabilityoffailure.Itprovidesawaytomoreaccuratelyestimatethetrueprobabilityoffailureafterthetest.Inaddition,itallowsustotradeweightagainstperformingadditionaltests.ThoughthemethodologyispresentedinthecontextofafuturethermaltestandredesignontheITPS,themethodologyisapplicableforestimatingthereliabilityofalmostanycomponentthatwillundergoatestfollowedbypossibleredesign.Givenacomputational 50

PAGE 51

model,uncertainties,errors,andredesignprocedures,alongwiththestatisticaldistributions,theprocedureofsimulatingthefuturetestresultbyMonteCarlosampling,calibration,andredesigncanbereadilyapplied.Thisstudybroughttolightmanytunableparametersinthetest,suchastheboundsonthedesignvariables,thetargetprobabilityoffailureforredesign,andtheredesigncriterionitself.Byincludingtheseparametersintotheoptimization,wewillnotonlyoptimizethedesignbutoptimizethetestaswell.ThisworkisthefocusofCh. 4 51

PAGE 52

Table3-9.CalibrationbytheBayesianupdatingapproachwithprobabilityoffailurebasedredesign(pf;target=0:01%) RestrictiononredesigneddsParameterOriginalMeanStandardDeviationMinimumMaximum NoboundsdS(mm)71.365.38.947.577.7mass(kg=m2)35.133.72.129.536.5pf;true(%)0.120.003120.01600.100pf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr(%)0.120.0070.00400.015 Within10%ofdtestdS(mm)71.368.85.164.177.7mass(kg=m2)35.134.51.233.436.5pf;true(%)0.120.0030.01600.100pf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr(%)0.120.0030.00500.015 1 Ofthe10,000possibleoutcomesofthefuturetest,7835areredesigned.Withtherequirementofa4.5%decreaseinmass,5126ofthe7001conservativemodels(pf;analyst
PAGE 53

CHAPTER4ACCOUNTINGFORFUTUREREDESIGNTOBALANCEPERFORMANCEANDDEVELOPMENTCOSTSAsseeninthepreviouschapter,mostcomponentsundergotestsaftertheyarede-signedandareredesignedifnecessary.Testshelpdesignersndunsafeandoverlycon-servativedesigns,andredesigncanrestoresafetyorincreaseperformance.Ingeneral,theexpectedchangestotheperformanceandreliabilityofthedesignafterthetestandredesignarenotconsidered.Inthischapter,weexplorehowmodelingafuturetestandredesignprovidesacompanyanopportunitytobalancedevelopmentcostsversusperformancebysimultaneouslydesigningthedesignandthepost-testredesignrulesduringtheinitialdesignstage.Duetoregulationsandtradition,safetymarginandsafetyfactorbaseddesigniscommonpracticeinindustryasopposedtoprobabilisticdesign.Inthischapter,weshowthatitispossibletocontinuetousesafetymarginbaseddesign,andemployprobabilitysolelytoselectsafetymarginsandredesigncriteria.Inthisstudy,wendtheoptimumsafetymarginsandredesigncriterionforanintegratedthermalprotectionsystem.Theseareoptimizedinordertondaminimummassdesignwithminimalredesigncosts.Weobservedthattheoptimumsafetymarginandredesigncriterioncallforaninitiallyconservativedesignandusetheredesignprocesstotrimexcessweightratherthanrestoresafety.Thiswouldtwellwithregulatoryconstraints,sinceregulationsusuallyimposeminimumsafetymargins. 4.1MotivationforAccountingforFutureRedesignThepreviouschapterdescribedamethodtosimulatethesepossiblefuturesincludingtestandredesign,andstudiedtheeffectofasinglefuturethermaltestfollowedbyredesignontheinitialreliabilityestimatesofanintegratedthermalprotectionsystem(ITPS).MonteCarlosamplingoftheassumedcomputationalandexperimentalerrorswasusedtosamplefuturetestalternatives,orthepossibleoutcomesofthefuturetest.Usingthefuturealterna-tives,themethodologyincludedtwomethodsofcalibrationandredesign.Itwasobservedthatthedeterministicapproachtocalibrationandredesign,whichactedtorestoretheorig-inal(designed)safetymargin,ledtoagreatlyreducedprobabilityoffailureafterthetest 53

PAGE 54

andredesign,areductionthatusuallyisnotquantied.Aprobabilisticapproachwasalsopresented,whichprovidedawaytomoreaccuratelyestimatetheprobabilityoffailureafterthetest,whiletradingoffweightagainstperformingadditionaltests.Matsumuraetal.[ 68 ]extendedthemethodologytoincludeadditionalfailuremodesoftheITPS.Inthischapter,weshowthatmodelingfutureredesignprovidesacompanywiththeopportunitytotradeoffdevelopmentcosts(testandredesign)andperformance(mass)bydesigningtheinitialdesigncriteriaandtheredesignrules.Asregulationsandtraditiondrivecompaniestousetraditionaldeterministicdesignwithsafetymarginsandsafetyfactors,welimitourselvestodeterministicdesignandredesignprocesses.Theprobabilisticapproachcanbelimitedtoselectsafetymarginsandredesigncriteria.Thisisatwo-stagestochasticoptimizationproblem[ 69 ],atypeofproblemwhichhasbeenstudiedextensivelyintheareaofprocessplanningunderuncertainty[ 70 71 ].Here,intherststage,adecisionismadeabouttheinitialdesignbeforethetest(i.e.,aninitialoptimumdesignisfound)andthendecisionsaretakenbasedontheupdatedinformationfromthetestresult(i.e.,toredesignornot)inthesecondstage.Thefollowingsectionofthechapterwillprovideabriefdescriptionofthetestproblem,theintegratedthermalprotectionsystem.InSec. 4.3 ,theprocessoftestandredesignisdescribedindetail.Section 4.4 providesadetaileddescriptionoftheuncertaintiesconsideredinthisstudy,andSec. 4.5 describeshowtheseuncertaintiesareusedtoobtainadistributionoftheprobabilityoffailure.InSec. 4.6 ,theprocessofsimulatingthefuturetestandredesignforasinglecandidatedesignisdescribed.AnillustrativeexampleisprovidedinSec. 4.7 4.2IntegratedThermalProtectionShieldDescriptionFigure 4-1 showstheITPSpanelthatisstudied,whichisacorrugatedcoresandwichpanelconcept.Thedesignconsistsofatopfacesheetandwebsmadeoftitaniumalloy(Ti-6Al-4V),andabottomfacesheetmadeofberyllium.SaflRfoamisusedasinsulationbetweenthewebs.TherelevantgeometricvariablesoftheITPSdesignarealsoshown 54

PAGE 55

Figure4-1.CorrugatedcoresandwichpanelITPSconcept ontheunitcellinFigure 4-1 .Thesevariablesarethetopfacethickness(tT),bottomfacethickness(tB),thicknessofthefoam(dS),webthickness(tw),corrugationangle(),andlengthofunitcell(2p).ThemassperunitareaiscalculatedusingEq.( 2 ). m=TtT+BtB+wtwdS psin( 2 )whereT,B,andwarethedensitiesofthematerialsthatmakeupthetopfacesheet,bottomfacesheet,andweb,respectively.Inthischapter,thematerialpropertiesusedarethesameaslistedinTable 3-1 ,butthedensityandthethermalconductivityvaluesofthematerialsarecorrelatedasshowninTable 4-1 Table4-1.Correlatedrandomvariables VariableCorrelationCoefcient densityoftitanium0.95thermalconductivityoftitaniumdensityofberyllium0.95thermalconductivityofberylliumdensityoffoam0.95thermalconductivityoffoam Inthisstudy,weagainconsiderthermalfailuretooccurwhenthetemperatureofthebottomfacesheetexceedsanallowabletemperature,andassumethattestsofthestructurewillbeconductedtoverifythedesign.Observeddatafromthetestisusedtocalibrateerrorsinanalyticalcalculations. 55

PAGE 56

4.3AnalysisandPost-DesignTestwithRedesignItisassumedthatananalysthasacomputationalmodelbywhichtocalculatethechangeinthetemperatureofthebottomfacesheetoftheITPS,Tcalc,foradesignde-scribedbydesignvariablesdandrandomvariablesr.Therandomnessisduetovariabilitiesinmaterialproperties,manufacturing,andenvironmentaleffects.UsingTcalc,thecalculatedtemperatureisdenedas Tcalc(d;r;v0)=T0(1)]TJ /F5 11.955 Tf 10.26 0 Td[(v0)+Tcalc(d;r)(4)whereT0istheinitialtemperatureofthebottomfacesheet,whichalsohasvariabilityrepresentedbyv0.ThedesignisobtainedviaadeterministicoptimizationproblemwhichrequiresthatthecalculatedtemperaturebelessthanorequaltosomedeterministicallowabletemperatureTdetallowbysomeasafetymarginSiniasshowninEq.( 4 ).Traditionally,thevalueofthissafetymarginisdeterminedbyregulationsandpastexperience. mind=ftw;tB;dSgm(d)subjecttoT0+Tcalc(d;rnom)+SiniTdetallowtw;Ltwtw;UtB;LtBtB;UdS;LdSdS;U(4)Notethatforthedeterministicdesign,therandomvariablesareheldatthenominalvaluernomandthevariabilityintheinitialtemperatureiszero.ThesubscriptsLandUonthedesignvariablesrepresentthelowerandupperbounds,respectively.Thesolutionoftheoptimizationproblemisdenotedasdini. 56

PAGE 57

Afterthedesignstage,atestisconductedtoverifythechosendesign.Thetestisperformedonatestarticledescribedbydtest(possiblyslightlydifferentthandiniduetoman-ufacturingtolerances)andrtest1,andanexperimentallymeasuredchangeintemperature,Tmeas,isfound.Forthistestdesign,Tcalc(dtest;rtest)andTcalc(dtest;rtest)arealsocalculated.Asameansofcalibration,theexperimentallymeasuredandcalculatedtemperaturescanbeusedintheformofacorrectionfactorforthecomputationalmodel.Thatis,thecorrectedcalculatedtemperatureisgivenas Tcalc;corr(d;r;v0)=T0(1)]TJ /F5 11.955 Tf 10.26 0 Td[(v0)+Tcalc(d;r)where=Tmeas Tcalc(dtest;rtest)(4)Notethatthisresultsinanupdateddistributionofthecorrected-calculatedtemperature.Shouldthetestresultshowthatadesignisunacceptable,redesignoccurs.Thecriterionforredesignisbasedonthesafetymarginofthecorrectedcalculatedtemperatureoftheoriginaldesign.ThelowerandupperlimitsofthesafetymarginofthecorrectedtemperaturearerepresentedwithSLandSU,respectively.Thisisexpressedas Redesignif:Scorr=Tdetallow)]TJ /F14 11.955 Tf 10.26 8.2 Td[()]TJ /F5 11.955 Tf 4.1 -8.2 Td[(T0+Tcalc(dini;rnom)SU(4)Deterministicredesignisperformedsothatthecorrectedcalculatedtemperatureoftheredesign(withthecorrectionfactor)islessthanorequaltotheallowabletemperaturebyasafetymarginSre.ThissafetymarginSredoesnotnecessarilyneedtobeequaltotheinitialsafetymarginSini.Sincemoreinformationisgainedfromthetest,thedesignermaychoosetodesigntosaveweightbyreducingthesafetymargin.Thiscanbeformulatedintoanoptimizationproblemtominimizethemassgivenaconstraintonthecorrectedcalculated 1Itisassumedthatthetestarticledesignisaccuratelymeasuredsuchthatbothdtestandrtestareknown,andthereisnovariabilityintheinitialtemperature. 57

PAGE 58

temperatureofthenewredesign,wherethedesignvariablesarethegeometry. mind=ftw;tB;dSgm(d)subjecttoT0+Tcalc(d;rnom)re+SreTdetallowtw;Ltwtw;UtB;LtBtB;UdS;LdSdS;U(4)Theoptimumupdateddesignisdenoteddupd. 4.4UncertaintyDenitionAsdescribedinCh. 3 andsummarizedagainhere,thisstudyrequirestheclassicationofuncertainties.Oberkampfetal.[ 1 ]providedananalysisofdifferentsourcesofuncertaintyinengineeringmodelingandsimulation,whichwassimpliedbyAcaretal.[ 2 ].WeuseclassicationsimilartoAcar'stocategorizetypesofuncertaintyaserrors(uncertaintiesthatapplyequallytoeveryITPS)orvariability(uncertaintiesthatvaryineachindividualITPS).Wefurtherdescribeerrorsasepistemicandvariabilityasaleatory.AsdescribedbyRaoetal.[ 67 ],theseparationoftheuncertaintyintoaleatoryandepistemicuncertaintiesallowsmoreunderstandingofwhatisneededtoreducetheuncertainty(i.e.,usingteststogainmoreknowledgetherebyreducingtheerror),andtradeoffthevalueoftheinformationneededtoreducetheuncertaintyagainstthecostofthereductionoftheuncertainty.Variabilityismodeledasrandomuncertaintiesthatcanbemodeledprobabilistically.WesimulatethevariabilitythroughaMonteCarlosimulation(MCS)thatgeneratesvaluesoftherandomvariablesrbasedonanestimateddistributionandcalculatesthechangeinbottomfacesheettemperatureTcalc.Inaddition,wesamplethevariabilityv0intheinitialtemperature.ThisformsthetemperatureTcalcforeachsample,generatingtheprobabilitydistributionfunction.Thecalculatedtemperaturedistributionthatreectstherandomvariabilityisdenotedfcalc(T).Additionally,wehavevariabilityintheallowabletemperatureTallow.NotethatinCh. 3 ,thevariabilityinTallowandv0werenotincluded,andweconsidered 58

PAGE 59

thebottomfacetemperatureratherthatthatchangeintemperature.TheeffectofthisdifferentformulationandtheadditionaluncertaintiesisdiscussedinAppendix D .Incontrasttovariability,errorsarexedforagivenITPSandthetruevaluesarelargelyunknown,sotheycanbemodeledprobabilisticallyaswell.Wehaveclassiedtwosourcesoferror,whicharedescribedinTable 4-2 Table4-2.DescriptionofErrors SymbolDescription eccomputationalerrorduetomodelingofthetemperaturechangeTcalcexexperimentalerrorinmeasuringTmeas Inestimatingthetemperatureofadesign,theerrormustalsobeconsidered.Aspreviouslydescribed,thecalculatedtemperaturedistributionfcalc(T)ofthedesignreectsrandomvariability.Ifthetruevalueofthecomputationalerrorisknown,thenthetruetemperaturedistribution,ftrue(T),associatedwithfcalc(T)isknown,asshowninFig. 4-2(a) .Thetruetemperaturestillhasrandomnessduetothevariabilities.Sincetheerrorisunknownandmodeledprobabilistically,weinsteadsamplethecomputationalerrortocreateseveralpossibledistributionsofthetruetemperaturedistributions,fiPtrue(T)correspondingtotheithsampleofec.ThissamplingisillustratedinFig. 4-2(b) forfoursamplesofec.Usingtheallowabletemperaturedistribution,theproba-bilityoffailurecanbecalculatedforeachsampleofthecomputationalerror.Thisformsadistributionoftheprobabilityoffailure,whichisfurtherdescribedinthenextsection. 4.5DistributionoftheProbabilityofFailureThetruetemperatureforadesigndescribedbygeometricdesignvariablesdandrandomvariablesrcanbedenedas Ttrue(d;r;v0)=T0(1)]TJ /F5 11.955 Tf 10.26 0 Td[(v0)+(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ec;true)Tcalc(d;r)(4) 59

PAGE 60

(a)Calculated,true,andallowabletemperaturedistri-butions (b)Calculated,allowable,andsampledpossibletruetemperaturedistributions Figure4-2.Exampleillustrating (a) knowncalculatedandallowabletemperaturedistributionsandunknowntruedistribution, (b) 4possibletruetemperaturedistributionsobtainedbysamplingof4valuesofec Thelimitstatefortheprobabilityoffailuretakesintoaccountthevariabilityintheallowabletemperature2alongwiththedistributionofthetruetemperature.ThelimitstateequationgisformulatedasthedifferencebetweenacapacityCandresponseRasshowninEq.( 4 ). 2TheabsenceofthesuperscriptdetforTallowdenotestheallowabletemperaturewithvariabilitytodistinguishitfromthedeterministicallowabletemperatureTdetallow. 60

PAGE 61

gtrue=Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(Ttrue(d;r;v0)=C)]TJ /F5 11.955 Tf 10.26 0 Td[(R(4)Usingthelimitstateequation,theprobabilityoffailureiscalculatedusingSeparableMonteCarlo[ 28 ].TheprobabilityoffailurepfiscalculatedwithEq.( 4 ),whereMandNarethenumberofcapacityandresponsesamples,respectively.TheindicatorfunctionIis1ifthegislessthanzeroand0otherwise. pf=1 MNNXi=1MXj=1I[gtrue(Cj;Ri)<0](4)Asdescribedintheprevioussection,adistributionoftheprobabilityoffailurecanbeformedbysamplingthecomputationalerrorforec;trueandcalculatingtheprobabilityoffailureforeachsample.Therefore,fornsamplesofec;truetherearenprobabilitydistributionsftrue(T)fromwhichwecancalculatenpfvalues.Recallthateachsamplerepresentsapossiblefutureforthedesign.Fromthesenvalues,wecancalculatethemeanand95thpercentileoftheprobabilityoffailure.Thefollowingsectionwilldescribethisprocessofsamplingtheerrorstosimulatethefuturealternatives. 4.6SimulatingFutureProcessesattheDesignStageMonteCarlosamplingofthetruevaluesofthecomputationalandexperimentalerrorsfromtheassumederrordistributionsisusedtosimulatethefuturetestandredesignalterna-tivesfortheinitialoptimumdesigndini.Thestepstosimulateasinglealternativeofthefuturetestwithpossibleredesign,whichhasbeensimpliedandadaptedfromAlgorithm 1 tottheworkinthischapter,arelistedbelow: 1. Samplesetoferrorsecandexfromassumeddistributions(fromthis,thebeforeredesignprobabilityoffailureusingtheecsamplecanbecalculated) 2. Usethetrueecandexsamplestosimulateatestresultandcorrectionfactor(Eq.( 43 )withfurtherdetailsinAppendix C ) 3. ApplythecorrectionfactorbasedonthetestresulttoTcalc(Eq.( 4 )) 61

PAGE 62

4. CalculatethesafetymarginwiththecorrectedtemperatureandevaluateifredesignisnecessarybasedonSLandSU(Eq.( 4 )),thenredesign,ifnecessary(Eq.( 4 )) 5. Ifredesigntookplace,calculatethemassandprobabilityoffailureforthisalternativeTosimulateanotheralternativefuture,thetrueerrorsarere-sampledandtheprocessisrepeated.Fornpossiblefuturealternatives,wesamplensetsoftheerrors,andobtainntrueprobabilitiesoffailureanduptonupdateddesigns(withnmassvalues).Withthesenvalues,wecancalculatethemeanand95thpercentileoftheprobabilityoffailureandmass.InhowmanyfutureswewillneedtoredesignisdeterminedbythethewindowdenedbySL,SU.Ifaredesignisneeded,theupdateddesignwillbedeterminedbythechoiceofsafetymarginSrerequiredinredesign.Figure 4-3 illustrateshowthedistributionofTcalc;corr,probabilityoffailure,andmasschangeswithredesignforagivenSini,Sre,SL,andSUfornalternativefutures.Ifthechoiceofthesafetymarginandredesignwindowleadtokredesigns,theprobabil-ityofredesignpreis pre=k n100%(4)Figure 4-4 displaystheaboveprocess,andthecalculationofthemeanmass,meanprobabilityoffailure,and95thpercentileoftheprobabilityoffailureofacandidatedesign,fornalternativefuturesInthisgure,atestisperformedfromwhichthecorrectionfactorisobtained.Thecorrectedsafetymarginisthenusedtodetermineifredesignshouldbeperformedbasedontheredesigncriterion.Ifredesignisrequired,thenthedesigngiventheredesignsafetymarginisfound,andthemassandprobabilityoffailurearecalculated.Otherwise,theoriginalmassofthedesignandprobabilityoffailureiscalculated.Afterthisisrepeatedforthenalternatives(i.e.,nvalues),themeanmass,meanprobabilityoffailure,and95thpercentileoftheprobabilityoffailurecanbecalculated. 62

PAGE 63

(a)Tcalc;corrbeforeredesign (b)Tcalc;corrafterredesign (c)pfbeforeredesign (d)pfafterredesign (e)mbeforeredesign (f)mafterredesign Figure4-3.Illustrativeexampleofbeforeandafterredesigndistributionsof (a) (b) Tcalc;corr, (c) (d) probabilityoffailure,and (e) (f) massfornalternativefuturesforagivenSini,Sre,SL,andSU 4.7OptimizationoftheSafetyMarginsandRedesignCriterion 4.7.1ProblemDescriptionTheprocessshowninFig. 4-4 canbethoughtofastheprocessthatisusedbyadesignerinthedesignofanITPSwithagivensetofsafetymargins(SiniandSre)andredesigncriterion(SLandSU),leadingtoadistributionofthefuturemassandprobabilityoffailure.Inthissection,weexplorehowacompanymayusetheprobabilityoffailurewith 63

PAGE 64

Figure4-4.Flowchartoftheprocesstocalculatethemeanmass,meanprobabilityoffailure,and95thpercentileoftheprobabilityoffailureforacandidatedesignthatsatisestheprobleminEq.( 4 )fornfuturealternatives.Notethatthedesignvariablesareunderlinedtoshowtheirpositionintheprocess. futureredesigntochoosethesafetymarginsandredesigncriteriontominimizemassandprobabilityofredesign.Todothis,weformulateanoptimizationproblemthatminimizesthemeanmassmandprobabilityofredesignpresubjecttoconstraintsonthefuturemeanprobabilityoffailurepf,andthe95thpercentileoftheprobabilityoffailureP95(pf).Thedesignvariablesarethesafetymarginsandredesigncriterion.Theformulationisshownin 64

PAGE 65

Eq.( 4 ). minSini;SL;SU;Srem;presubjectto(pf)BeforeRedesign0:1%(P95(pf))BeforeRedesign0:5%(pf)AfterRedesign0:01%(P95(pf))AfterRedesign0:05%35Sini;Sre65Sini)]TJ /F12 11.955 Tf 10.26 0 Td[(35SLSiniSiniSUSini+351:24mmtw1:77mm4:94mmtB7:06mm49:9mmdS71:3mm(4)TheconstraintsonSiniandSrerestrictthetwovaluestobewithinthewindowof35to65K,andtheyarenotconstrainedtohaveequalvalues.Thelowerlimitisintendedtoreectaregulatorymandate,but,justincase,boundsonthebeforeredesignprobabilityoffailurearepresenttopreventdesignsthatarelargelyunsafebeforeredesign.TheconstraintsonSLandSUrestricttheacceptablevaluesofthesafetymarginaftercorrectiontowithin35KofSini.Notethatinthischapterthedesignandredesignpolicyisoptimizedonthebasisofasinglepanel.Ifanoptimizationlikethatiscarriedoutinpractice,weassumethatcompromisevalueswillbeusedbasedonsimilaroptimizationforseveralcases.Forthisproblem,thecomputationalandexperimentalerrorsweredistributedasdescribedinTable 4-3 .Giventhesedistributions,thecorrectionfactorrangedfrom0.85to1.15.ThedistributionsofthevariableswithuncertaintyduetovariabilityareprovidedinAppendix D .Toreducethecomputationalcostofsimulatingafuturetest,surrogatesofthemassandreliabilityindexweredeveloped.Thereliabilityindexisrelatedtotheprobabilityoffailure 65

PAGE 66

Table4-3.Boundsofcomputationalandexperimentalerrors ErrorDistributionBounds ecUniform0:12exUniform0:03 bypf=()]TJ /F4 11.955 Tf 7.61 0 Td[(),whereisthestandardnormalcumulativedensityfunction.Forexample,foraprobabilityoffailureof0.1%,thereliabilityindexis3.72.ThedevelopmentofthesesurrogatesisdescribedinAppendix C .TheprobleminEq.( 4 )wassolvedbyformingacloudof10,000pointsusingLatinHypercubesamplingofthedesignvariablesSini,SL,SU,andSre.Foreachsetofdesignvariables,10,000alternativefuturesweresampledtoobtainthedistributionsofthemassandprobabilityoffailure,andtheprobabilityofredesign.Thesetofpointsthatsatisedtheconstraintsontheprobabilityoffailurewasfound,and,fromthissetoffeasiblepoints,weformedtheParetofrontforminimumprobabilityofredesignandmeanmassafterredesign. 4.7.2ResultsAsapointofcomparison,werstfoundtheoptimumdesignforminimummassthatsatisedthebeforeredesignconstraintsontheprobabilityoffailure.Sinceredesignwasnotperformed,theonlyvalueofinterestisSini.TheminimumvalueofSini=48:9Kwhichsatisedtheprobabilityconstraintsofameanof0.1%and95thpercentileof0.5%ledtoamassof24.7kg=m2.Inaddition,wefoundtheminimumSinidesignthatsatisedtheafterredesignprobabilityoffailureconstraintswithoutactuallyperformingredesign(i.e.,theminimumSinithatsatisedpf0:01%andP95(pf)0:05%withoutanyredesign).Inthiscase,theminimumSiniwas62.5Kforamassof25.3kg=m3forpf=0:01%andP95(pf)=0:05%.Plotsoftheprobabilitydensityofthesafetymarginaftercorrection(i.e.,Scorr=Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(Tcalc;corr)fortheSini=48:9KandSini=62:5KcasesareshowninFig. 4-5 .Figure 4-5 showsthedistributionofthecorrectedsafetymarginswiththetwovaluesofSini.ThegurealsoshowsthevalueofSneededtoachievethedesiredprobabilitiesoffailure(Sini=29:7Kforpf=0:1%inFig. 4-5(a) andSini=41:3Kforpf=0:01%inFig. 4-5(b) )intheabsenceofepistemicuncertainty.Weobservedthat79%ofScorrvalueswere 66

PAGE 67

(a)Sini=48:9K (b)Sini=62:5K Figure4-5.Probabilitydensityfunctionofthesafetymarginaftercorrectionfor (a) Sini=48:9KwhichalsodisplaystheSinirequiredintheabsenceofepistemicuncertaintyforamassof23:9kg=m2and (b) Sini=62:5KwhichalsodisplaystheSinirequiredintheabsenceofepistemicuncertaintyforamassof24:3kg=m2 greaterthan29.7KforSini=48:9Kand84%greaterthan62.5KforforSini=41:3K.Thiswasbecausethemeanprobabilityoffailurewasinuenceddisproportionallybyafewlargevaluesasthemedianprobabilityoffailurebeforeredesignwas7.3e-4%forSini=48:9Kand3.2e-6%forSini=62:5K.Thegurecaptionalsonotesthatthemassrequiredtoachievethedesiredprobabilityoffailureintheabsenceofepistemicuncertaintieswas23.9kg=m2for0.1%and24.3kg=m2for0.01%.Withtheepistemicuncertainty,werequired25:3kg=m2tocompensateforthecomputationalerror,andthis1kg=m2or4%penaltywaswhatcanbereducedbymoreaccuratecomputationortests.Allowingredesign,theParetofrontforminimumprobabilityofredesignandmeanmassafterredesignisdisplayedinFig. 4-6 thatsatisestheconstraintsoftheprobleminEq.( 4 ).Weobservedreductionsinmeanmasswithincreasingprobabilitiesofredesign.Themeanmassvaluesafterredesignatthesepointswerelessthantheminimummassof25.3kg=m3obtainedwhenredesignwasnotallowed.At40%probabilityofredesign,themeanmasswasevenlessthan24.7kg=m3;themassoftheoptimumdesignthat 67

PAGE 68

satisedtherelaxedbeforeredesignconstraintsonprobabilityoffailure(pf0:1%andP95(pf)0:5%). Figure4-6.Paretofrontforminimumprobabilityofredesignandmeanmassafterredesign.Feasiblepointsinthedesignspaceareshown,alongwiththebefore-redesignmassofthepointsontheParetofront ThevaluesofthesafetymarginsforthedesignsontheParetofrontaredisplayedinFig. 4-7 .WeobservedthattheinitialsafetymarginSiniwasnearlyconstantatapproximately63K.ThelowerboundoftheacceptablesafetymarginwithcorrectionSLremainedbetween28to32K,forwhichthedifferencefromSiniisneartheupperboundof35K(i.e.,theconstraintonthelowerboundofSLisactiveornearlyactive).Thisresultedinthesmallprobabilityofredesignofunconservativedesigns.InFig. 4-8 ,whichshowsthepercentageofthetotalprobabilityofredesignthatisconservativeandunconservative,weobservedthatthiswasindeedthecase,andthatlessthan5%ofthetotalprobabilityofredesignwasattributedtounconservativeredesignforallpointsontheParetofront.FortheupperboundonacceptablesafetymarginwithcorrectionSU,weobservedthatthevalueswerelarge(nearly100K)butgraduallyreducedtovaluesnearSiniat65K.Thisledtothegradualincreaseinprobabilityofconservativeredesignastheprobabilityofunconservativeredesignremainedatlowvalues.Thus,theprobabilityofconservative 68

PAGE 69

(a)Initialandredesignsafetymargins (b)Boundsofcorrectedsafetymargin Figure4-7.ForParetofrontforminimumprobabilityofredesignandmeanmassafterredesign, (a) initialandredesignsafetymarginsversustotalprobabilityofredesignand (b) boundsoftheacceptablecorrectedsafetymarginsversustotalprobabilityofredesign Figure4-8.PercentageofconservativeandunconservativeredesignsofthepointsontheParetofront redesigncomprisedthemajorityofthetotalprobabilityofredesignforthedesignsontheParetofront.Atthesametime,weobservedthatthesafetymarginSreoftheredesignwassettovaluesbelowSiniandatvalueslessthantheminimumvaluewithouttestsandredesignof63.5K.Thatis,afterthetest,theredesignhasasmallersafetymarginthanpossiblefortheoriginaldesign.ThisvalueisevenlessthatthesafetymarginrequiredtosatisfytherelaxedbeforeredesignconstraintsofSini=48:9K.Thecombinedeffectof 69

PAGE 70

redesigningconservativedesignsforareducedsafetymarginwasareductioninthemeanmasswhilesatisfyingmorestringentconstraintsontheprobabilityoffailure.Theresultsshowthattheoptimalchoicesafetymarginsandredesigncriterioncanbechosenbasedontheprobabilityoffailurethataccountsforfutureredesign.Weobservethatcompaniescanbenetbyhavingdesignersconsiderconservativesafetymarginsfortheinitialdesign,whichcorrespondtothesafetymarginrequiredtosatisfytheprobabilisticconstraints.Theredesigncriterionshouldthenmostlyresultintheredesignofoverlyconservativedesignstotrimmassbyallowingasmallersafetymarginforredesign(becauseofadditionalknowledgeduetothetestinthecorrectionfactor),withafewunsafedesignsredesignedforsafety. 4.7.3UnconservativeInitialDesignApproachWhiletheParetooptimaldesignsshowedthattheinitialdesignshouldbeconservativewithredesignperformedtotrimmass,weexaminedthetrade-offinprobabilityofredesignandmasswhenstartingwithaninitiallyunconservativedesign(i.e.,aninitialdesignthatdoesnotsatisfytheconstraintsofpf).Inthisapproach,thedesignerusesasmallersafetymargintoachieveaminimalweightdesign,relyingonthetestandredesigntocorrectanydangerousdesigns.Inthisproblem,theinitialsafetymarginwasxedat48.9K(correspondingtoamassof24.7kg=m2)andtheremainingsafetymargins(Sre,SL,andSU)werethedesignvariables.ThesameconstraintsasinEq.( 4 )wereused.Figure 4-9 displaystheParetofrontfoundwiththeunconservativeapproach,andcomparestheresulttothepreviouslyfoundresultsthatusedaconservative-rstapproachfoundinSec. 4.7.2 .Itwasobservedthattomeettheprobabilityoffailurerequirements,theprobabilityofredesignwasatleast27%fortheunconservativeapproachwithSini=48:9K.Thatis,thedesignermustacceptatleasta27%probabilityofredesign,whichwouldleadtoameanmassofapproximately25.2kg=m2.Thisvalueofthemassisonly0.4%smallerthantheinitialmassrequiredtosatisfytheprobabilityconstraintswithoutredesignwiththeinitiallyconservativedesign. 70

PAGE 71

Figure4-9.ParetofrontshowingtheParetofrontfoundfortheunconservativeapproach(withSini=48:9K)incomparisontotheParetofrontwiththeconservativeapproach(withSini=62:5K) Figure 4-10 displaysthevaluesofthedesignvariablesoftheParetooptimalsolutions,andFig. 4-11 displaysthebreakdownofthetotalprobabilityofredesignduetoconservativeandunconservativedesigns.Itwasobservedthatredesignwasprimarilyperformedtoincreasesafetyatthesmallestprobabilitiesofredesign(27%),increasingtheredesignofconservativedesignswithincreasingprobabilityofredesign.Thehistogramofthemassfor10,000alternativefuturesafterredesignisdisplayedinFig. 4-12(a) .Theapproximately2%increaseinthemeanmassafterredesignisattributedtothelargeprobabilitiesoffailureassociatedwithredesignofunconservativedesigns.AbreakdownofthealternativefuturesthatresultedinthemeanmassisshowninTable 4-4 .Itwasobservedthattheredesignofunconservativedesignsresultedinaincreaseof8%inthemeanmass.Incontrast,thesamemeanmassof25.2kg=m2afterredesigncanbeachievedwiththeconservative-rstapproachwithaprobabilityofredesignaround8%,andforaprobabilityofredesignof27%,themeanmassisnearly24.7kg=m2.Inthiscase,thereductioninmean 71

PAGE 72

(a)Initialandredesignsafetymargins (b)Boundsofcorrectedsafetymargin Figure4-10.ForParetofrontforminimumprobabilityofredesignandmeanmassafterredesignwiththeunconservative-rstapproach, (a) initialandredesignsafetymarginsversustotalprobabilityofredesignand (b) boundsoftheacceptablecorrectedsafetymarginsversustotalprobabilityofredesign Figure4-11.PercentageofconservativeandunconservativeredesignsofthepointsontheParetofrontwiththeunconservative-rstapproach massisduetolargereductionsinmassinthecasesthatrequiredredesignofconservativedesigns.Thehistogramofthemassfor10,000alternativefuturesisshowninFig. 4-12(b) andthemassandprobabilityofredesignisdetailedinTable 4-4 .Itwasobservedthattheredesignofoverlyconservativedesignsresultedina10%reductioninthemeanmass. 72

PAGE 73

(a)InitiallyUnconservative (b)InitiallyConservative Figure4-12.Histogramsofmassafterredesignfor10,000alternativefuturesfor (a) initiallyunconservativedesignwith27%probabilityofredesignand (b) initiallyconservativedesignwith8%probabilityofredesign. Table4-4.Breakdownofalternativefuturesfortheunconservativeinitialdesignwith27%probabilityofredesignandconservativeinitialdesignwith8%probabilityofredesign Outcomepre(%)MeanMass(kg=m2) InitiallyUnconservativeNoredesign7324.7Unconservative25.526.8Conservative1.523.5Total25.21InitiallyConservativeNoredesign93.525.3Unconservative2.125.8Conservative4.422.7Total25.21 1Calculatedaspnoredesignm0+(prem)conservative+(prem)unconservative Comparingthisvaluealongwiththe8%increaseinmassseenintheinitiallyunconservativecase,weobservedthatthechangeinmassduetoredesignismuchlargerthanthe2%differenceinmassofthetwoinitialdesigns.However,withtheinitiallyconservativedesignmostredesignsacttoreducethemass,whereasthemassismostlyincreasedintheinitiallyunconservativeredesigncases.Therefore,thedesignerhasachoice: 73

PAGE 74

1. useasmallerinitialsafetymarginforaninitiallysmallmassandaccepta27%proba-bilityofredesignthatwillincreasethemass,or 2. usealargerinitialsafetymarginforaninitiallylargermassthatcanachievelessthanorequaltothesamemasswithprobabilitiesofredesigngreaterthan8%.Ifthetestshowsthecomponentdoesnothavetoberedesigned,therewouldbeanearly2%masspenaltyinusingtheconservativesafetymargin. 4.7.4DiscussionUsingtheminimalsafetymarginfortheinitialdesigncanbethoughtofasusingsafetymarginsgivenbyregulatoryagencies,whichprovideminimumvaluesofsafetymarginsandsafetyfactors.Forexample,theFederalAviationAdministrationhasrecommendedminimumdesignandtestfactorsforstructuresonreusablelaunchvehicles[ 72 ].Inthiswork,thevaluesofSini(andSre)of35Kmaybetheminimumvalueimposedbyanagency,andthevalueof48.9Kmaybethecurrentminimumvalueimposedbyacompanybasedonhistoryorexperience.Theresultspresentedinthechaptershowthatacompanymayhaveanincentivetoimposetheirownsafetymargins,andsetthedesignandredesignrulestobalancedevelopmentcosts.TheresultsinSec. 4.7.2 showedthatprobabilisticconstraintscanbesatisedbyrstusingaconservativesafetymarginandacceptingariskofincreaseddevelopmentcostthroughincreasedredesigntotrimexcessmass.Thisdirectlycontraststheapproachofusingminimalsafetymarginvaluesandredesigningbasedonthetestresulttoincreasesafety.Consideringthepossiblefutureredesignanditscostallowsthecompanytomakebetterdecisionsatthedesignstage. 4.8SummaryandDiscussiononPossibleFutureResearchDirectionsInthischapter,weusedthemodelingoffutureredesigntoprovideawayofbalancingdevelopmentcosts(testandredesigncosts)andperformance(mass)bydesigningthedesignandredesignrules.Weobservedthatthepresenceofepistemicuncertaintyledtoamasspenalty,whichcouldbereducedbyatestandredesign.Sincedeterministicdesignemployingsafetymarginsandsafetyfactorsiscommonpracticeinindustry,weshowedthatsafetymarginsandredesigncriteriacanbechosenusingtheprobabilityoffailurewith 74

PAGE 75

futureredesign.Astudyonanintegratedthermalprotectionsystemshowedthataminimummassdesignthatsatisedprobabilisticconstraintscanbeachievedbyhavinganinitiallyconservativedesignandaredesigncriterionsuchthatredesignismainlyperformedonoverlyconservativedesignstotrimexcessmass.Incontrast,weexaminedthetrade-offinstartingwithaninitiallysmallsafetymargin,whichmaybeaminimumvaluerecommendedbyaregulatoryagency,andusingthetestandredesigntocorrectdangerousdesigns.Therefore,inthisexample,acompanywouldhaveanincentivetouseconservativesafetymarginsattheinitialdesignstage,whileincreasingperformancebyimplementingaredesigncriterionaimedatdiscoveringoverlyconservativedesigns.Thisalsoprovidesabalancebetweenprobabilisticdesignandthemoretraditionaldeterministicapproach.Possiblefutureresearchincludesconsideringtheuncertaintyreductionmethodsthatoftentakeplaceafteracomponentisdesignedbutbeforeacomponentistested.Forexample,lowerdelitymethodsmaybeusedtondastartingpointfortheinitialdesign.Beforeadesignistested,itmaybebettercharacterizedthroughhigherdelitymodelingoroptimizationinasmallerdesignspaceaboutthisdesign.Boththehigherdelitymodelingandre-optimizationcanreducetheuncertaintyinthedesignbeforeatestisevenperformed.Therefore,astudythatmodelstheseactionsandconsidersthesubsequentuncertaintyreductionwouldbeusefulinndingtheoptimalbalanceindesignanddevelopmentcostsandperformance. 75

PAGE 76

CHAPTER5DYNAMICDESIGNSPACEPARTITIONINGFORLOCATINGMULTIPLEOPTIMA:ANAGENT-INSPIREDAPPROACHInthischapter,weexploretheuseofdesignspacepartitioningtotackleoptimizationproblemsinwhicheachpointisexpensivetoevaluateandtherearemultiplelocaloptima.Theoverarchinggoalofthemethodpresentedistolocatealllocaloptimaratherthanjusttheglobalone.Locatingmultipledesignsprovidesinsuranceagainstdiscoveringthatlateinthedesignprocessadesignispoorduetomodelingerrorsoroverlookedobjectivesorconstraints.Theproposedstrategytolocatemultiplecandidatedesignsdynamicallypartitionsthesearchspaceamongseveralagentsthatapproximatetheirsub-regionlandscapeusingsurrogates.Agentscoordinatebyexchangingpointstoformanapproximationoftheobjectivefunctionorconstraintsinthesub-regionandbymodifyingtheboundariesoftheirsub-regions.Throughaself-organizedprocessofcreationanddeletion,agentsadaptthepartitionastoexploitpotentiallocaloptimaandexploreunknownregions.Thisideaisdemonstratedonasix-dimensionalanalyticalfunction,andapracticalengineeringexample,thedesignofanintegratedthermalprotectionsystem.Aspartofahistoryonthisresearch,theideaofworkinginpartitioneddesignsspaceswasbornoutofanideatodecomposetheproblemamongseveralagents,whichacttosolvetheirownsub-problemsandcoordinatetosolvetheglobalproblem.Combinedwiththeideaofsolvingthesesub-problemscheaply,wethenturnedtotheuseofsurrogatestoapproximateexpensiveobjectivefunctionsorconstraints.Thus,wedevelopedamethod-ology,presentedinthischapter,inwhichanagentusesitsownlocalsurrogatetosolvethesub-probleminitssub-region.Thegoalwastolimittheamountofinformationsharedbetweentheagents,whereeachagentwouldndthebestsolutioninitsregion.Lateron,aspresentedinthenextchapter,weexploredtheeffectivenessofusinglocalsurrogatesratherthanglobalsurrogatesandalsotheuseofpartitionsratherthanstartingoptimizationrunsfromdifferentpointsofthedesignspace. 76

PAGE 77

5.1MotivationandBackgroundonLocatingMultipleOptimaManycontemporaryapplicationscanbemodeledasdistributedoptimizationproblems(ambientintelligence,machine-to-machineinfrastructures,collectiverobotics,complexproductdesign,etc.).Optimizationprocessesiterativelychoosenewpointsinthesearchspaceandevaluatetheirperformancesuntilasolutionisfound.However,apracticalandcommondifcultyinoptimizationproblemsisthattheevaluationsofnewpointsrequireex-pensivecomputations.Forinstance,ifonewantstocomputeapropertyofacomplexobject(e.g.largedeectionsofanaircraftwingundersomeloading),ahighdelitycomputation(e.g.niteelementanalysis)mayberequired.Therefore,manyresearchersintheeldofoptimizationhavefocusedonthedevelopmentofoptimizationmethodsadaptedtoexpensivecomputations.Themainideasunderlyingsuchmethodsareoftentheuseofsurrogates,problemdecomposition,andparallelcomputation.Theuseofsurrogatestoreplaceex-pensivecomputationsandexperimentsinoptimizationhasbeenwelldocumented[ 40 43 ].Moreover,inoptimization,acommonwaytodecomposeproblemsistopartitionthesearchspace[ 73 74 ].Furthermore,totakeadvantageofparallelcomputing,manyhaveproposedstrategiesforusingmultiplesurrogatesinoptimization[ 9 12 ].Thegoalofmostofthesealgorithmsistolocatetheglobaloptimum,whiletryingtominimizethenumberofcallstotheexpensivefunctions.Likethesealgorithms,wetrytoreducethenumberthenumberofcallstotheexpensivefunctions,butourmaingoalistolocatemultipleoptimaasmultiplecandidatedesigns.Forreal-worldproblems,theabilitytolocatemanyoptimainalimitedcomputationalbudgetisdesirableastheglobaloptimummaybetooexpensivetond,andbecauseitprovidestheuserwithadiversesetofacceptablesolutionsasinsuranceagainstlateformulationchanges(e.g.,newconstraints)inthedesignprocess.Besidestheaforementionedtechniquesfordistributingthesolvingprocess,multi-agentoptimizationisanactiveresearcheldproposingsolutionsfordistinctagentstocooperativelyndsolutionstodistributedproblems[ 75 ].Theymainlyrelyonthedistribution(anddecomposition)oftheformulationoftheproblem.Generally,theoptimizationframework 77

PAGE 78

consistsofdistributingvariablesandconstraintsamongseveralagentsthatcooperatetosetvaluestovariablesthatoptimizeagivencostfunction,likeintheDCOPmodel[ 14 ].Anotherapproachistodecomposeortransformproblemsintodualformsthatcanbesolvedbyseparateagents[ 15 ](forproblemswithspecicproperties,suchaslinearity).Here,wedescribeamulti-agentmethodinwhichthesearchspaceisdynamicallypartitioned(andnottheproblemformulation)intosub-regionsinwhicheachagentevolvesandperformsasurrogate-basedcontinuousoptimization.Thenoveltyofthisapproachcomesfromthejointuseof(i)surrogate-basedoptimizationtechniquesforexpensivecomputationand(ii)self-organizationtechniquesforpartitioningthesearchspaceandndingallthelocaloptima.Coordinationbetweenagents,throughexchangeofpointsandself-organizedevolutionofthesub-regionboundariesallowstheagentstostabilizearoundlocaloptima.Likesomenature-inspirednichingmethods[ 76 77 ],suchasparticleswarm[ 78 80 ]andgeneticalgorithms[ 81 82 ],orclusteringglobaloptimizationalgorithms[ 83 ],ourgoalistolocatemultipleoptima,butunlikethesealgorithms,ourapproachaimstosparinglycallthetrueobjectivefunctionandconstraints.Ourmulti-agentapproachfurther(i)usesthecreationofagentsforexploringthesearchspaceand,(ii)mergesordeletesagentstoincreaseefciency.Additionally,theuseofsurrogatescombinedwithpartitioningofthedesignspaceallowsustotakeadvantageoflocalsearchmethodsbyaimingtooptimizeusingthesurrogatepredictionsratherthanthetruefunctions.Here,weproposeamethodologythataimstorstexploitthesurrogateprediction(i.e.,byseekingrsttominimizethefunction)andonlyexplorewhennopointsthatleadtofurtherimprovementscanbefound.Thus,inthismethodwedonotaimtoexplicitlybalanceexploitationversusexplorationiterationsaswithmostglobaloptimizationalgorithms,butinsteadprimarilyexploit,andexplorethroughdesignspacepartitioningandtheoccasionaladditionofspacellingpoints.Inthefollowingsection,wediscussmoredeeplythemotivationformultiplecandidatedesigns.Next,weprovidesomebackgroundonsurrogate-basedoptimization.InSec. 5.4 78

PAGE 79

wedescribetheautonomousagentsthatperformthecooperativeoptimizationprocess.InSec. 5.5 ,wepresentthemethodsofspacepartitioningandpointallocationthatareintendedtodistributelocaloptimaamongthepartitionswhilemaximizingtheaccuracyofthesurrogateinaself-organizedwaythroughagentcreationanddeletion.InSec. 5.6 asix-dimensionalproblemistackledusingourmulti-agentoptimizer,andinSec. 5.7 apracticalengineeringexample,thedesignofanintegratedthermalprotectionsystem,ispresented. 5.2MotivationforMultipleCandidateDesignsInoptimizationcourses,weoftentellstudentsthatdeninganoptimizationproblemproperlyisthemostimportantstepforobtainingagooddesign.However,evenexperiencedhandsoftenoverlookimportantobjectivefunctionsandconstraints.Therearealsoepistemicuncertainties,suchasmodelingerrors,intheobjectivefunctionsandconstraintsdenitionsthatwilltypicallyperturbtheirrelativevaluesthroughoutthedesignspace.Inonecase,Na-gendraetal.[ 81 ]usedageneticalgorithmtondseveralstructuraldesignswithcomparableweightandidenticalloadcarryingcapacity.However,whenthreeofthesedesignswerebuiltandtested,theirloadcarryingcapacitywasfoundtodifferby10%.Forsuchreasons,alternativelocaloptimamaybebetterpracticalsolutionstoagivenoptimizationproblemthanasingleidealizedglobaloptimum.Asadvancesincomputerpowerhavemadeitpossibletomovefromsettlingonlocaloptimatondingtheglobaloptimum,whendesigningengineer-ingsystems.Thisusuallyrequiressearchinmultipleregionsofdesignspace,expendingmostofthecomputationneededtodenemultiplealternatedesigns.Thus,focusingsolelyonlocatingthebestdesignmaybewasteful.Inengineeringdesign,thesimulatedbehaviorofobjectivefunctionsandconstraintsusuallyhasmodelingerror,orepistemicuncertainty,duetotheinabilitytoperfectlymodelphenomena.Modelingerrorscandegradelocaloptimaorevencausethemtodisappear.Letusnowturntoapracticalengineeringdesignexampletodemonstratethepresence,diversity,andfragilityofcandidatedesigns.Largeportionsoftheexteriorsurfaceofmanyspacevehiclesaredevotedtoprovidingprotectionfromthesevereaerodynamicheating 79

PAGE 80

experiencedduringascentandatmosphericreentry.Aproposedintegratedthermalpro-tectionsystem(ITPS)providesstructuralloadbearingfunctioninadditiontoitsinsulationfunctionandindoingsoprovidesachancetoreducelaunchweight.Figure 5-1 displaysthecorrugated-coresandwichpanelconceptofanITPS,whichconsistsoftopandbottomfacesheets,webs,andinsulationfoambetweenthewebsandfacesheets. Figure5-1.Integratedthermalprotectionsystemprovidesbothinsulationandloadcarryingcapacity,andconsequentlycanleadtoalternativeoptimaofsimilarmassbutdifferentwayofaddressingthethermalandstructuralrequirements. Thethermalandstructuralrequirementsoftenconictduetothenatureofthemecha-nismsthatprotectagainstthefailureinthedifferentmodes.Forexample,thinwebspreventtheowofheattothebottomfacebutaremoresusceptibletobucklingandstrengthfailure.Areductioninfoamthickness(paneldepth)improvesresistanceagainstbucklingofthewebsbutincreasesheatow.Athickbottomfacesheetactsasaheatsinkandreducesthetemperatureatthebottomfacebutincreasesstressintheweb.Theseconictsmayberesolvedbycandidatedesignsindifferentways.Figure 5-2 displaystheinfeasibleandfeasibleregionswithconstraintsonthetemperatureofthebottomfacesheetandstressinthewebforthethree-dimensionalexample,inwhichtheweb,bottomface,andfoamthicknesseswerethedesignvariables.Threeislandsoffeasibilityareobserved,inwhichthedesignisdriventoprotectagainstdifferentfailuremodes.ThetwominimummassdesignsareinRegions1and2.InRegion1,thefailureisprimarilyfromstress,becausethewebsarethin.Thisiscompensatedby 80

PAGE 81

Figure5-2.Feasibleregions1-3andinfeasibleregions(left)andmassobjectivefunctionrepresentedbycolor(right).ThetwocompetitiveoptimainRegions1and2relyondifferentconcepts.InRegion1thethermalfunctionissatisedbyathickbottomfacesheetactingasheatsink,whileinRegion2itissatisedbythickinsulation. reducingthelengthofthewebbyreducingthefoamthickness,andtocompensatefortheincreaseheatow,thebottomfacesheetisincreasedtoprovidealargerheatsink.InRegion2,bothfailuremodesarepresent,sothattheinsulationisthickertopreventthermalfailureandthebottomfaceisthinnertoreducestressintheweb.Withbothfeasibleregionsbeingrathernarrow,modelingerrorscaneasilywipeoutoneoftheseregions,sothathavingbothdesignsprovidesvaluableinsurance.Furthermore,evenifmodelingerrorsdonotwipeoutRegion2butonlynarrowit,thiscanresultinsubstantialincreaseinmass,whileRegion1ismorerobust.Thisexampledemonstratesthebenetofmultiplecandidatedesignsduetothefragilitydesignsfromerrors.Inthefollowingsections,thedynamicdesignspacepartitioningalgorithmisdescribed,andasixdimensionalanalyticalexampleispresentedbeforereturningtotheITPSexampleinSec. 5.7 81

PAGE 82

5.3Surrogate-BasedOptimizationAsurrogateisamathematicalfunctionthat(i)approximatesoutputsofastudiedmodel(e.g.themassorthestrengthortherangeofanaircraftasafunctionofitsdimensions),(ii)isoflowcomputationcostand(iii)aimsatpredictingnewoutputs[ 84 ].Thesetofinitialcandidatesolutions,orpoints,usedtotthesurrogateiscalledthedesignofexperiments(DOE).Knownexamplesofsurrogatesarepolynomialresponsesurface,splines,neuralnetworksorkriging.Letusconsiderthegeneralformulationofaconstrainedoptimizationproblem, minimizex2S
PAGE 83

performedagain.Therefore,wedenotetheDOEatatimetasXtandtheassociatedsetofobjectivefunctionvaluesandconstraintvaluesasFtandGt,respectively.Thesurrogate-basedoptimizationprocedureissummarizedinAlgorithm 2 (whichalsoreferstoaglobaloptimizationprocedureinAlgorithm 3 ). Algorithm2Overallsurrogate-basedoptimization 1: t=1(initialstate) 2: whilettmaxdo 3: Buildsurrogatesfandgfrom(Xt;Ft;Gt) 4: Optimizationtondx(seeAlgorithm 3 ) 5: Calculatef(x)andg(x) 6: Updatedatabase(Xt;Ft;Gt)[(x;f(x);g(x)) 7: t=t+1 8: endwhile Algorithm3Constrainedoptimizationprocedure 1: Input:f;g,Xt,L 2: Output:x 3: x argminx2Lf(x)subjecttog(x)0 4: ifxisnearXtoroutofthesearchdomainLthen 5: x argmaxx2Sdistance(Xt) 6: endif 5.4AgentOptimizationBehaviorAsstatedintheintroduction,ourapproachconsistsinsplittingthespaceinsub-regionsandassigningagentstoeachofthesesub-regionsaspresentedinFig. 5-3 .Therefore,Algorithm 6 canbethoughtofastheprocedurefollowedbyasingleagenttondonepoint,thatwillberepeatinguntiltermination.However,inthemulti-agentapproachwedescribehere,eachagentisrestrictedtoonlyasub-regionofthedesignspace,i.e.,SisreplacedbyapartofS.Therationalebehindthisideaisthateachagenthasaneasieroptimizationsubproblemtosolvebecauseitsearchesasmallerspace,whichwedenoteasPifortheithagent,andconsidersasimplerfunction.Eachagentmustconsideronlythepointsinitssub-region,whichareavailableinitsinternaldatabase(Xt;Ft;Gt)i.Thesub-regionofanagentisdenedbythepositionofitscenterc.Apointinthespacebelongstothesub-regionwith 83

PAGE 84

thenearestcenter,wherethedistanceistheEuclideandistance.Thiscreatessub-regionsthatareVoronoicells[ 85 ].Thechoiceofwheretoplacethecenterisdiscussedinthenextsection.Figure 5-3 illustratesthepartitionofatwo-dimensionalspaceintofoursub-regionsforfouragents,whichrequiresfourcenters.Inthisexample,weplacethecentersrandomly.spacepartitionswillbediscussedinSection 5.5 Figure5-3.Multi-agentSystemoverview:agentsperformsurrogate-basedoptimizationindifferentsub-regionsofthepartitionedsearchspacebasedonpersonalsurrogates(dashedl.)andexchangepointswiththeirdirectneighbors(dottedl.) TheprocedureofasingleagentisgiveninAlgorithm 4 .Assumingthatsub-regionsaredened,eachagenttsseveralsurrogatesitknows(asmanydifferentwaystoapproximate)andchoosestheonethatmaximizestheaccuracyinitssub-region(line5-9).Toavoidill-conditioning,ifmorepointsareneededthanareavailabletoanagent,theagentasksneighboringagentsforpoints.Theneighboringagentsthencommunicatetheinformationassociatedwiththesepoints(lines6).Wedenethebestsurrogateastheonewiththeminimumcross-validationerror,thepartialpredictionerrorsumofsquaresPRESSRMS.Thisisfoundbyleavingoutapoint,rettingthesurrogate,andmeasuringtheerroratthatpoint.Theoperationisrepeatedforppointsintheagent'ssub-region(disregardinganypointsreceivedfromotheragents)toformavectorofthecross-validationerrorseXV.Thevalueof 84

PAGE 85

PRESSRMSisthencalculatedby PRESSRMS=s 1 peTXVeXV(5) Algorithm4Agentioptimizationinitssub-region. 1: t=1(initialstate) 2: whilettmaxdo 3: UpdatePi=fx2Ss.t.jjx)]TJ /F5 11.955 Tf 10.26 0 Td[(cijj2jjx)]TJ /F5 11.955 Tf 10.26 0 Td[(cjjj2;j,ig 4: Updateinternaldatabasefromthenewspacepartition 5: Buildsurrogatesfandgfrom(Xt;Ft;Gt)i 6: ifNotsufcientnumberofpointsininternaldatabasetobuildasurrogatethen 7: Getpointsfromotheragentsclosesttoci 8: Buildsurrogates 9: endif 10: ChoosebestsurrogatebasedonpartialPRESSRMSerror 11: Optimizationtondx[withAlgorithm 3 (f;g;Xt;Pi)] 12: Calculatef(x)andg(x) 13: (Xt+1;Ft+1;Gt+1)i (Xt;Ft;Gt)i[(x;f(x);g(x)) 14: Updatecenterci(seeSection 5.5.1 ) 15: Checkformerge,splitorcreate(seeSection 5.5.2 ) 16: t=t+1 17: endwhile Oncetheagentshavechosensurrogates(line10),theoptimizationisperformedtosolvetheprobleminEq.( 5 )insidethesub-region(line11).Iftheoptimizergivesaninfeasiblepoint(i.e.,thepointdoesnotsatisfytheconstraintinEq.( 5 )orisoutofthesub-region)orrepeatsanexistingpoint,theagentthenexplorestondanalternatepointinthesub-region.Toexplore,theagentaddsapointtothedatabasethatmaximizestheminimumdistancefromthepointsalreadyinitsinternaldatabase(seeAlgorithm 3 ).Thetruevaluesfandgoftheiteratearethencalculated(line12),and(x;f(x);g(x))isaddedtotheinternaldatabase(lines12). 5.5DynamicDesignSpacePartitioningTheprevioussectionexpoundsthecooperativeoptimizationprocessperformedbyagentsinapre-partitionedspace.Thegoalofthismethodistohaveeachagentlocateasingleoptimum,suchthatthepartitioningstronglydependsonthetopologyofthespace. 85

PAGE 86

Therefore,asapartofthecooperativeoptimizationprocess,weproposeaself-organizingmechanismtodynamicallypartitionthespacewhichadaptstothesearchspace.Byself-organizing,wemeanthatagents(andthereforesub-regions)willbecreatedanddeleteddependingonthecooperativeoptimizationprocess.Agentswillsplitwhenpointsareclusteredinsideasingleregion(creation),andwillbemergedwhenlocaloptimaconverge(deletion). 5.5.1MovingtheSub-regions'CentersThemethodofspacepartitioningweproposefocusesonmovingthesub-regions'centerstodifferentlocaloptima.Asaresult,eachagentcanchooseasurrogatethatisaccuratearoundthelocaloptimum,andtheagentcanalsoexplorethesub-regionaroundthelocaloptimum.Atthebeginningoftheprocess,onlyoneagentexistsandisassignedtothewholesearchspace.Thenitbeginsoptimizationbychoosingasurrogate,ttingitandoptimizingonthissurrogate.Asaresulttheagentcomputesanewpointxt)]TJ /F15 7.97 Tf 5.06 0 Td[(1.Then,thecenterofthesub-regionismovedtothebestpointinthesub-regionintermsoffeasibilityandobjectivefunctionvalue(line14).Thisisdonebycomparingthecenteratthelastiterationct)]TJ /F15 7.97 Tf 5.07 0 Td[(1tothelastpointaddedbytheagentxt)]TJ /F15 7.97 Tf 5.07 0 Td[(1.Thecenterismovedtothelastpointaddedbytheagentifitisbetterthanthecurrentcenter.Otherwise,thecenterremainsatthepreviouscenter.Forconvenience,incomparingtwopointsxmandxn,weusethenotationxmxntorepresentxmisbetterthanxn.Fortwocenters,insteadofpointsxwewouldconsiderthecentersc.TheconditionstodeterminethebetteroftwopointsaregiveninAlgorithm 5 5.5.2Merge,SplitandCreateSub-regionsOnceanagenthasaddedanewpointinitsdatabase(line13)andmoveditscentertothebestpoint(line14),itwillcheckwhethertosplit,ortomergewithotherones(line15).Mergingagents(andtheirsub-regions)preventsagentsfromcrowdingthesamearea,allowingoneagenttocapturethebehaviorinaregion.Splittinganagentisawaytoexplorethespaceasitrenesthepartitioningofthespaceinadditiontothesearchthateachagent 86

PAGE 87

Algorithm5Algorithmtodeterminingif,fortwopointsxmandxn,xmisbetterthanxn(xmxn)andviceversa.Notethatforthealgorithmbelow,thetime(superscriptt)isomittedasthealgorithmisvalidforthecomparisonofanytwopointsatanytime. 1: Givenf(xm);f(xn);max(g(xm));max(g(xn)) 2: ifmax(g(xm))0&max(g(xn))0then 3: //bothpointsarefeasible 4: iff(xm)f(xn)then 5: xmxn 6: else 7: xnxm 8: endif 9: elseifmax(g(xm))0&max(g(xn))>0then 10: //onlyxmisfeasible 11: xmxn 12: elseifmax(g(xm))>0&max(g(xn))0then 13: //onlyxnisfeasible 14: xnxm 15: elseifmax(g(xm))>0&max(g(xn))>0then 16: //bothareinfeasible 17: ifmax(g(xm))max(g(xn))then 18: //maximumconstraintviolationofxmislessthanxn 19: xmxn 20: else 21: xnxm 22: endif 23: endif canperforminitssub-region.Splitandmergeoccursattheendofeachiteration(line15):agentsarerstmerged(ifnecessary),thepointsbelongingtothemergedagent(s)aredistributedtotheremainingagentsbasedondistancefromthecenteroftheremainingagents'sub-regions,andtheneachremainingagentsexaminestodeterminewhethertosplitornot. 5.5.2.1MergeconvergingagentsAgentsaremerged(deleted)ifthecentersoftheagents'sub-regionsaretoocloseasmeasuredbytheEuclideandistancebetweenthecenters.WemeasuretheminimumEu-clideandistancebetweentwocentersasapercentageofthemaximumpossibleEuclideandistancebetweenpointsinthedesignspace.Whenexaminingtheagents,theagentwiththe 87

PAGE 88

centerwiththelowestperformanceisdeleted.Forexample,foragents1and2,ifc1c2,agent2isdeleted.Beforedeletion,thedeletedagentdistributesitsinternaldatabasepointstoclosestneighbors. 5.5.2.2Splitclusteredsub-regionsItisdesirabletocreateanagentifitisfoundthatpointsareclusteredintwoseparateareasofasingleagent'ssub-region,asillustratedinFig. 5-4 (a) .Suchasituationcanoccuriftherearetwooptimainasubregion. . . . . . . . . . . . . Potentialclusters Presentbestsolution&Agenti'scenter (a)Potentialclusterswithinasub-region . . . . . . . . . . . Centersfromk-means Presentbestsolution&Agenti'scenter . c1 . c2 (b)Clustersfromk-means . . . . . . . . . . . . . . . . . Newagentj'scenter Presentbestsolution&Agenti'scenter . c2 (c)Finalclustersaftermovingcenterstopresentbestsolutionandnearestdatapoint Figure5-4.Illustrationofprocessusedtocreateanagentjgivenpointsinasingleagenti'ssub-region. Agentsarecreatedbyusingk-meansclustering[ 86 ]fortwoclusters(k=2)giventhepointsinthesub-region,wheretheinitialguessesofthecentersarethepresentbestsolution(thecurrentcenter)andthemeanofthedataset.Sincek-meansclusteringgivescentersthatarenotcurrentdatapointsasillustratedinFig. 5-4 (b) ,wemovethecenterstoavailabledatapointstoavoidmorecallstoevaluatetheexpensivefunctions.Thisisdonebyrst 88

PAGE 89

measuringthedistanceofthecentersfromk-meanstothepresentbestsolution,andmovingtheclosestcentertothepresentbestsolution,aswewanttopreservethissolution.Fortheothercenter,wemeasurethedistanceofthecurrentdatapointstotheothercenter,andmaketheclosestdatapointtheothercenter.ThenalclusteringisillustratedinFig. 5-4 (c) .Theresultisanewagentwithacenteratanalreadyexistingdatapoint,wherethecreatingagentretainsitscenteratitspresentbestsolution.Thisnalclusteringisvalidatedusingthemeansilhouettevalueofthepointsinthesub-region.Thesilhouette,introducedbyRousseeuw[ 87 ],isusedtovalidatethenumberofclusters,byprovidingameasureofthewithin-clustertightnessandseparationfromotherclustersforeachdatapointiforasetofpoints.Thesilhouettevalueforeachpointisgivenas s(i)=b(i))]TJ /F5 11.955 Tf 10.26 0 Td[(a(i) maxfa(i);b(i)g(5)whereaiistheaveragedistancebetweenpointiandallotherpointsintheclustertowhichpointibelongs,andbiistheminimumoftheaveragedistancesbetweenpointiandthepointsintheotherclusters.Thevaluesofsirangefrom-1to1.Forsinearzero,thepointcouldbeassignedtoanothercluster.Ifsiisnear-1,thepointismisclassied,and,ifallvaluesarecloseto1,thedatasetiswell-clustered.Theaveragesilhouetteofthedatapointsisoftenusedtocharacterizeaclustering.Inthiswork,weaccepttheclusteringifallsiaregreaterthan0andtheaveragevalueofthesilhouetteisgreaterthansomevalue. 5.5.2.3CreatenewagentsTheagentsmayreachapointwherethereisnoimprovementmadebytheoverallsysteminseveraliterations(i.e.,thecentersofallagentshaveremainedatthesamepoints).Forexample,thiscanoccurwheneachagenthaslocatedthebestpointinitssub-region,theareaaroundeachbestpointispopulatedbypoints,eachagentisdriventoexploreforseveraliterations,andnootherpotentiallocaloptimaarelocated.Thiscanalsooccuratearlyiterationsinwhichthesurrogatesarenotwell-trainedinthesub-region.Inordertoimproveexploration,anewagentiscreatedinthedesignspacewhenthereis 89

PAGE 90

noimprovementforniterations(i.e.,thecentersofthesub-regionshavenotmovedforniterations).Wecallthisparameterthestagnationthreshold.Tocreateanewagent,anewcenteriscreatedatanalreadyexistingdatapointthatmaximizestheminimumdistancefromthealreadyexistingcenters,thusforminganewagent.Thedesignspaceisthenrepartitioned. 5.6Six-DimensionalAnalyticalExampleInthissection,weexaminethesix-dimensionalHartmanfunction(Hartman6)thatisoftenusedtotestglobaloptimizationalgorithms. minimizexfhart(x)=)]TJ /F7 7.97 Tf 15.51 15.23 Td[(qXi=1aiexp0BBBBBB@)]TJ /F7 7.97 Tf 14.62 14.63 Td[(mXj=1bij(xj)]TJ /F5 11.955 Tf 10.26 0 Td[(dij)21CCCCCCAsubjectto0xj1;j=1;2;:::;m=6(5)InthisinstanceofHartman6,q=4anda=1:01:23:03:2whereB=2666666666666666666666666666410:03:017:03:51:78:00:0510:017:00:18:014:03:03:51:710:017:08:017:08:00:0510:01:014:037777777777777777777777777775D=266666666666666666666666666640:13120:16960:55690:01240:82830:58860:23290:41350:83070:37360:10040:99910:23480:14510:35220:28830:30470:30470:40470:88280:87320:57430:10910:038137777777777777777777777777775Aswewishtolocatemultipleoptima,wemodiedHartman6tocontain4distinctlocaloptimabydrillingtwoadditionalGaussianholesattwolocationstoformtwolocaloptima,inadditiontotheglobaloptimumandonelocaloptimumprovidedintheliterature[ 39 ].ThemodiedHartman6functionis f(x)=fhart(x))]TJ /F12 11.955 Tf 10.26 0 Td[(0:521(x))]TJ /F12 11.955 Tf 10.26 0 Td[(0:182(x)(5) 90

PAGE 91

wherethemeanandstandarddeviationassociatedwith1and1=0:660:070:270:950:480:13and=0:3(alldirections),respectively.Themeanandstandarddeviationassociatedwith2and2=0:870:520:910:040:950:55and2=0:25(alldirections),respectively.TheoptimaaredisplayedinTable 5-1 .Toobtainanapproximatemeasureofthesizeofthebasinsofattractionthatcontaintheoptima,wemeasuredthepercentageoflocaloptimiza-tionrunsthatconvergedtoeachoptimum.Todothis,twenty-thousandpointsweresampledusingLatinHypercubesamplingandalocaloptimizationwasperformedstartingateachoneofthesepointsusingaSQPalgorithm. Table5-1.ModiedHartman6optimaandthepercentageofrunsthatfoundeachoptimumwithmultiplestartsandaSQPoptimizer Optimumfxpercentageofruns Global-3.33(0:200:150:480:280:310:66)50Local1-3.21(0:400:880:790:570:160:04)21Local2-3.00(0:870:520:910:040:950:55)9Local3-2.90(0:640:070:270:950:480:13)20 Thepercentageofstartsthatconvergedtoanoptimumisalsoameasureofthevolumeofitsbasinofattractionincomparisontootherbasins.SinceLocal2hasthesmallestpercentageofruns,itwasexpectedthatitwouldbethemostdifcultoptimumtolocatebytheagents.Note,however,thatinsix-dimensionalspacearatioof9% 50%involumewouldbeproducedbyaratioof0.75incharacteristicdimension. 5.6.1ExperimentalSetupSincetherearenononlinearconstraints,onlytheobjectivefunctionisapproximatedbysurrogates..Thethreepossiblesurrogates,whicharekrigingsurrogateswithdifferenttrendfunctions,aredescribedinTable 5-2 .Fromthisset,eachagentchosethebestsurrogatebasedonPRESSRMS.ThesetofsurrogatesandtheminimumnumberofpointsusedtoteachsurrogateareprovidedinTable 5-2 .Iftheminimumnumberofpointsarenotavailable,pointsareborrowedfromneighboringsub-regionsintheorderofincreasingdistancetotheagentcenter,and,iftherequirementisstillnotmet,thenallavailablepointsareused. 91

PAGE 92

Table5-2.Surrogatesconsideredinthisstudy IDDescriptionminimum#ofptsfort 1Kriging(quadratictrend)1.5*#coefcientsforquadraticresponsesurface2Kriging(lineartrend)3Kriging(constanttrend) TheparametersinTable 5-3 wereusedforallresults.Theseparametersincludemaximumnumberofagents(e.g.,themaximumnumberofcomputingnodesavailable),parametersthatdictatehowclosepointsandcenterscanbe,andparametersthatdeneifanewagentshouldbecreated.Sincewearesimulatingexpensivefunctionevaluations,wealsoxedacomputationalbudgetto400functionevaluations.Beyondthisnumber,thesystemstops:thisisouronlyterminationcriterion.Finally,westartthemulti-agentsystemwithasingleagentabletosplitandmergewithtime. Table5-3.Multi-AgentParametersformodiedHartman6 ParameterValue Max#offunctionevaluations400Max#ofagents8Initial/Min#ofagents1Mindistancebetweenagentcenters10%ofmaxpossibledistanceinspaceMinimumdistancebetweenpoints1e-3(absoluteforeachdimension)Minaveragesilhouette0.25Min#ofpointsineachagentaftercreation4Stagnationthreshold3 Thesuccessandefciencyofthemulti-agentapproachiscomparedtoasingleagentsystemwhichperformsaclassicalsurrogate-basedoptimizationprocedureasdescribedinAlgorithm 6 .However,thissingleagentisunabletoperformdynamicpartitioningandoptimizesoverthewholespace.Thissingleagenthasalsoacomputationbudgetof400callstotheexpensivefunction.Thesingleagentcongurationisastandardtowhichwecompareourmulti-agentoptimizer.Ineachcase,(multi-orsingleagent),astoevaluatethecapabilityofthealgorithmstoexplorethesearchspace,wealsoranseveralexperimentsfordifferentinitialDOEsizes 92

PAGE 93

(351,56,and100)thatstillaccountforthenumberoffunctionevaluations.Therefore,foralargerinitialDOE,thesystemexecutesfewersteps.Foreachofthecasesthatwerestudied,theresultsshownarethemedianof50repetitions(i.e,50differentinitialDOEs).Thelocaloptimizationproblemsweresolvedwithasequentialquadraticprogramming(SQP)algorithm[ 88 ].DOEsareobtainedusingLatinHypercubesamplingandthemaximincriterionforveiterations. 5.6.2SuccessestoLocateOptimaFor50repetitions,thepercentageofrepetitionsthatsuccessfullylocatedasolutiona1%distancefromtheoptimumwithasingleagentandamulti-agentsystemisshowninFig. 5-5 .ThisdistanceistheEuclideandistancenormalizedbythemaximumpossibledistancebetweenpointsinthedesignspace(here,p 6).Itwasobservedthatthesingleagenthadfewersuccesseswithanequivalentnumberoffunctionevaluationscomparedtothemulti-agentcase.Inboththesingleandmulti-agentcases,Local2wastheoptimumthatwasthemostdifculttolocatewithlessthan10successeswithasingleagentand32successeswithamulti-agentsystem.BasedonthesmallpercentageofrunsthatlocatedLocal2withmultiplestartsandthetruefunctionvalueswiththeSQPoptimizer(c.f.Table 5-1 ),thiswasnotunexpected. 5.6.3AgentEfciencyandDynamicsThemedianobjectivefunctionvalueofthesolutionclosesttoeachoptimumisshowninFig. 5-6 .FortheglobaloptimumandLocal1,itwasobservedthattheefciencyisnearlyequalinthesingleandmulti-agentcases.ItwasalsoobservedthatthesmallerDOEsrequiredfewerfunctionevaluationstondtheseoptima.ForLocal2andLocal3,themulti-agentsystemhasaclearadvantageinndingsolutionswiththeobjectivefunctionnearthe 1Thisdoesnotsatisfytheminimumrequirednumberofpointsforthet,soallpointsareused(asinglesurrogatespanstheentiredesignspace)untilasufcientnumberofpointsareobtained. 93

PAGE 94

Figure5-5.ForthemodiedHartman6example,thepercentageofrepetitionsthatlocatedasolutionwithin1%distancefromeachoptimum. trueoptimumvalue.WhileitisclearfortheglobaloptimumandLocal1thatsmallerDOEsaremoreefcient,thereisnoclearrelationshipbetweenDOEsizeandefciency(considerLocal2).RecallthatLocal2wasexpectedtobethemostdifcultoptimumtondjudgingbythesmallpercentageofrunsofmultiplestartswiththeSQPoptimizerthatweresuccessful.TheseresultsconrmthatexplorationisrequiredtolocateLocal2,andthemulti-agentsystem,inwhichexplorationisaninherentfeature,ismorecapableofndingthisoptimum.Explorationbythemulti-agentsystemwasmeasuredbythepercentageofcallstothetrueobjectivefunctioninwhichexploitationorexplorationoccurredasshowninFig. 5-7 .Wedeneanexploitationcallaswhentheagentaddsapointthatminimizestheobjectivefunction.Notethatweconstrainedtheminimumdistanceanewpointshouldbefromanalreadyexistingdatapointsothatmultiplelocaloptimacouldbelocated.Explorationiswhenarandompointisaddedbyanagent.Explorationgenerallyoccurswhenexploitationhasfailed,meaningallstartsinthesub-regionresultedinpointsthatwerenotfarenoughfromexistingdatapointsorwereoutsideofthesub-region. 94

PAGE 95

Figure5-6.Medianobjectivefunctionvalueofsolutionclosesttoeachoptimumwithnumberoffunctionevaluations.ThesingleagentcaseisdenotedbySandthemulti-agentcasebyM,withtheinitialDOEsizerepresentedbythenumber. Figure5-7.ForthemodiedHartman6example,thepercentageofpointsthatwereaddedinexploitationorexploration. Weobservedthatthemulti-agentcasemostlyperformedexploitationwithafewexplorations,whereasthesingleagentperformedexplorationonly1%ofthetime.ThiscouldbeduetothesingleagentseekingtotunearoundtheglobaloptimumandLocal1,whichitlocateswiththefewestfunctionevaluations.Thenumberoftimesinwhichthean 95

PAGE 96

agentputspointtotunearoundtheoptimumcanbereducedbyincreasingtheminimumdistancebetweenpoints.Inaddition,itwasobservedthatthesingleagentwasslowlyaddingexploitationpointsinthevicinityofLocal3asshowninFig. 5-6 .Figure 5-8 showsthemediannumberofagents.Whileupto8agentscouldbecreated,itwasobservedthatthemediannumberofagentsstabilizedaround4,thenumberoflocaloptima.Thisisbecauseonceallthelocaloptimaarelocated,newagentsarecreatedbutaresoondeletedastheyconvergetothebasinsofattractionofthealreadyfoundoptima. Figure5-8.Foramulti-agentsystem,themediannumberofagents TheaccuracyofthesurrogateswasmeasuredbythepartialPRESSRMSandtheerrorat1000testpointsbytheerms.PRESSRMSisaleave-one-outcross-validationerror.InthesingleagentcasethecalculationofPRESSisstraightforward,butforthemulti-agentcaseitistakenbycalculatingPRESSRMSineachsub-regionandtakingthemeanofthevalues.ThevaluesofPRESSRMSandeRMSaredisplayedinFig. 5-9 .ThePRESSRMSindicatedthattheerrorwasdecreasingforboththesingleandmulti-agentcases,withthesingleagentcaseslightlymoreaccurate.However,theeRMSprovidedamoreglobalindicationoftheaccuracyofthesurrogateandshowedthattheapproximationmadebythemultipleagentsimprovedwithfunctionevaluationsmorethantheapproximationofthesingleagentdid.ThisisduetothesingleagentputtingmanypointsneartheglobaloptimumandLocal1,makingthesurrogateaccurateintheselocationsbutlessaccurateglobally.TheslowlocationofLocal3 96

PAGE 97

bythesingleagentcanalsobepartiallyexplainedbythepooraccuracyofthesingleagent'ssurrogate. (a)PRESS (b)erms Figure5-9.ForthemodiedHartman6example, (a) thePRESSRMSand (b) errorat1000testpoints. 5.7EngineeringExample:IntegratedThermalProtectionSystemInthissection,weillustratethemulti-agentmethodandtheimportanceoflocatingmultiplecandidatedesignsonanintegratedthermalprotectionsystem(ITPS).Figure 2-1 showstheITPSpanelthatisstudied,whichisacorrugatedcoresandwichpanelconcept.Thedesignconsistsofatopfacesheetandwebsmadeoftitaniumalloy(Ti-6Al-4V),andabottomfacesheetmadeofberyllium.SaflRfoamisusedasinsulationbetweenthewebs.TherelevantgeometricvariablesoftheITPSdesignarealsoshownontheunitcellinFigure 5-10 .Thesevariablesarethetopfacethickness(tT),bottomfacethickness(tB),thicknessoftheinsulationfoam(dS),webthickness(tw),andcorrugationangle().ThemassperunitareaiscalculatedusingEq.( 2 ) f=TtT+BtB+wtwdS psin( 2 )whereT,B,andwarethedensitiesofthematerialsthatmakeupthetopfacesheet,bottomfacesheet,andweb,respectively. 97

PAGE 98

Figure5-10.CorrugatedcoresandwichpanelITPSconcept TheoptimizationproblemtominimizethemasssubjecttoconstraintsonthemaximumbottomfacesheettemperatureTBandmaximumstressinthewebwisshowninEq.( 5 ). minimizex=ftw;tB;dS;tT;gf(x)subjecttoTB(x))]TJ /F5 11.955 Tf 10.26 0 Td[(TallowB0w(x))]TJ /F4 11.955 Tf 10.26 0 Td[(alloww0xL;ixixU;ifori=1:::5wherexL=1:316:0060:61:1375:3andxU=1:969:0060:61:2784:8(5)Thebottomfacesheettemperatureandthemaximumstress,whicharebothfunctionsofthedesignvariablesallvedesignvariables,areconstrainedtobytheirmaximumallowablevalues.AsdescribedinSec. 5.2 ,the3-Dproblem,wherethebottomface,web,andfoamthicknesseswerethedesignvariables,hadthreedistinctfeasibleregionscontainingthreelocaloptima.Forthe5-Dproblem,wefoundthetrueoptimabysolvingthetrueoptimizationproblemwith1000randominitialpointswiththeSQPoptimizer.Table 5-4 liststheoptimathatwerefoundandgivesthepercentageoftherunsthatlocatedeachoptimum.Asthepercentageofrunsthatconvergetoeachoptimumisameasureofthedifcultytolocatetheoptimum,weobservedthatoptimum3wouldbethemostdifculttolocatebytheagents. 98

PAGE 99

Table5-4.5-DITPSexampleoptimaandthepercentageofrunsthatfoundeachoptimumwithmultiplestartsandaSQPoptimizer Optimumfxpercentageofruns 129.27(1:316:0076:81:1381:5)50229.29(1:316:0075:61:1376:5)30329.30(1:316:0077:51:1384:8)4431.30(1:318:2960:61:2784:8)8534.65(1:319:0074:01:1375:3)6638.06(1:849:0065:91:2584:8)1Other(pointsthatwerenottruelocaloptima)1 5.7.1ExperimentalSetupInthisexample,wefollowthesameexperimentalsetupasforthemodiedHartman6example,withtheparametersprovidedinTable 5-3 .However,thecomputationalbudgetisxedat120evaluationsoftheexpensivefunctionsandthemaximumnumberofagentsisraisedto10.Astheobjectivefunction,themass,calculatedbythesimpleexpressioninEq.( 2 )theagentsonlyapproximatethetwolimitstatesg1andg2withsurrogates.TheinitialDOEsizewasvariedat25,42,and84points. 5.7.2SuccessestoLocateOptimaFor50repetitions,thepercentageofrepetitionsthatweresuccessfulatlocatingafeasiblesolutionwithin1%distancefromeachoptimumisprovidedinFig. 5-11 .Thedifferencesinthenumberofsuccessesbetweenthesingleandmulti-agentcasesforalloptimaweresmallpartitcularlyforoptima1and2,whilethesingleagentwasclearlymoreefcientandsuccessfulinlocatingoptimum3.Itwasobservedthatthemulti-agentsystemwaslesssuccessfulatlocatingoptimum3,particularlywiththeinitialDOEsizeof84.Incomparingthesuccessoflocatingalloptimainasinglerepetition,itwasclearthatthesuccessinlocatingalloptimawasdictatedbythesuccessinlocatingoptimum3. 5.7.3AgentsEfciencyandDynamicsThemedianobjectivefunctionvalueoftheclosestsolutiontoeachoptimumisshowninFig. 5-12 .ItshouldbenotedthatallsolutionswerefeasibleandthatinallcasesthesmallestinitialDOEsizeof25wasthemostefcient.Foroptima1,2,and4,itwasobservedthatthe 99

PAGE 100

Figure5-11.Forthe5-DITPSexample,thepercentageofrepetitionsthatlocatedasolutionwithin1%distancefromeachoptimum. differencesbetweenthesingleandmulti-agentcaseswithvaryinginitialDOEsweresmall.Within30functionevaluations,boththesingleandmulti-agentcaseswereabletolocateeachoptimum.Foroptima5and6,itisclearthatthesingleagentismoreefcientthatthemulti-agent,locatingtheoptimumwith5-10fewerfunctionevaluations.Foroptimum3,whichbothagentshaddifcultylocating,weobservedthatthesingleagentisclearlymuchmoreefcientthanthemultipleagents.NotethatforallinitialDOEsizes,theclosestsolutiontooptimum3isatoptimum1(f=29:27),whichisnotunexpectedasthetwooptimaareonlyadistanceof0.16(inthenormalizeddesignspaceandnormalizedbyp 5)apartmakingthesetheclosestpairofalltheoptima.Figure 5-13 displaysthemediannumberofagents.Thoughupto10agentscouldbecreated,themediannumberofagentsstabilizedaround3.Figure 5-14 comparesthenum-berofexploitationsandexplorationsforthesingleandmulti-agentcasesfordifferentinitialDOEsizes.Weobservedthat,althoughthenumberofagentsstabilizesat3,exploitationisstillperformedmorebythemultipleagentsconsideringtheconstraintontheminimum 100

PAGE 101

(a)Opt1 (b)Opt2 (c)Opt3 (d)Opt4 (e)Opt5 (f)Opt6 Figure5-12.Forthe5-DITPSexample,themedianfofthesolutionnearesttoeachoptimum.ThesingleagentcaseisdenotedbySandthemulti-agentcasebyM,withtheinitialDOEsizerepresentedbythenumber. distancebetweenpoints.Inallmulti-agentcases,therewasmoreexploitationthanexplo-ration.Forthesingleagent,therewasmoreexploration,exceptfortheinitialDOEof84points.Thiswasduetotheabilityofthesingleagenttolocatethemultipleoptimaquicklywithexploitationiterationsduetheconstraintontheminimumdistancebetweenpoints.Afterthisoccurred,thesingleagentperformedmoreexploration,whichaidedinthelocationofthe 101

PAGE 102

mostdifcultoptimumtolocate,optimum3.FortheinitialDOEof84,thesingleagenthadfewerevaluationsinwhichtolocatealloptima,soexploitationwasdominant. (a)Numberofactiveagents (b)Numberofiterationsvsfunctionevaluations Figure5-13.Forthe5-DITPSexample,thenumberofactiveagentsandthenumberofiterations. Figure5-14.Forthe5-DITPSexample,thepercentageofpointsthatwereaddedinexploitationorexploration. Thesuperiorperformancebythesingleagentwasalsoattributedtotheaccuracyofitssurrogateapproximations.Fig. 5-15 comparesthePRESSRMSofeachsurrogateforthesingleandmulti-agentcases.ThePRESSRMSdecreasedwithincreasingnumberoffunctionevaluationsafteritinitiallyincreased.Thiswasduetotheplacementofpointsaroundtheoptima,whichmadethesurrogatelessaccuratefurtherfromtheoptima.Thisledtolargeerrorswhenapointthatwasfarawayfromotherpointswasleftoutincalculatingthe 102

PAGE 103

cross-validationerror.Asmoreexplorationpointswereadded,thePRESSRMSwasreduced.Forthemulti-agentcase,weobservedthatthePRESSRMSincreasedthroughthefunctionevaluations.Thiswasduetothelargenumberofpointsputaroundtheoptimainexploitationiterations,whichoutnumberedtheexplorationiterations. Figure5-15.MedianPRESSRMSofthesurrogatesofthelimitstatesforthe5-DITPSexample Figure 5-16 displaystheerrorat1000testpointseRMS.WeobservedmuchofthesametrendsaswithPRESSRMS,withthesurrogatesinthemulti-agentcasedecreasinginaccuracywhilethesurrogatesofthesingleagentcasesincreasedaccuracy. 5.8DiscussionThesingleagentapproachshowedaclearadvantageoverthemulti-agentmethodintheITPSexampleinSec. 5.7 .Thesingleagentapproachissimple:aglobalsurrogateisusedandaconstraintontheminimumdistancebetweenpointsistheonlywaytoinstigateexplorationofthedesignspace.Itisadvantageousinitssimplicityandeaseofimplemen-tation,butitssuccessdependsonhowwellasinglesurrogatecanapproximatetheglobalbehavior.TheaccuracymeasuresofthelimitstatesfortheITPSexampleinFig. 5-16 show 103

PAGE 104

Figure5-16.Medianerrorat1000testpointseRMSofthesurrogatesofthelimitstatesforthe5-DITPSexample thattheglobalsurrogateofthesingleagentisindeedmoreaccuratethanseverallocalsur-rogates.Uponfurtherinvestigationofthetemperatureandstresstrendsinthedesignspace,itwasfoundthatsimplequadraticresponsesurfacesoverthedesignspaceweresufcientapproximations.Onthecontrary,themodiedHartman6functionpresentedinSec. 5.6 ,forwhichtheunmodiedversionisoftenusedasabenchmarkfunctionforsurrogate-basedglobaloptimizationalgorithms,isthoughttobemorecomplexcomparedtotheITPSexample.Theerrorattestpoints(c.f.Fig. 5-9(b) )showsthattheaccuracyofthelocalsurrogatesisslightlybetterthanthatofthesingleagent'sglobalsurrogate.Inthisexample,weobservedthatthemulti-agentmethodcanbesuccessfulandefcient.Withthatsaid,thecomparisonsmadehereintermsofefciencyarebasedonthenumberoffunctionevaluations.Ideally,themulti-agentpartitioningmethodwouldbeparallelizedsuchthatasingleiterationwouldinvolvesimulatenousoptimizationineachsub-region.Thatis,asingleiterationcouldaccountforfourfunctionevaluationsifthere 104

PAGE 105

arefouragentsworkinginfoursub-regions.Therefore,intermsofefciencytheiterations,whichgiveanideaofwallclocktime,shouldalsobecompared.FortheITPSexample,thesingleagentwasshowntobemoreefcientatlocatingsomeoptimabyvetotenfunctionevaluations,butintermsofiterationsthemultipleagentsaremoreefcientifthenumberoffunctionevaluationsistranslatedintoiterationsbyFig. 5-13(b) .Otherwise,whatdoesthissayaboutthismulti-agentalgorithm?Basedonthesetwoexamples,thesuccessrateandefciencyofthemulti-agentmethodmaybedependentonhavinghigheraccuracylocalsurrogatescomparedtoaglobalsurrogate.Otherwise,simpleralgorithmsmaybemoreefcient.Furtherinvestigationontheneedforlocalsurrogatesisrequired,andastudythatusesaglobalsurrogatewiththeagent-baseddynamicdesignspacepartitioningisplanned.Itwillallowustostudyseparatelytwoingredientsthatmakeupthemethodinvestigatedhere:localversusglobalsurrogateandspacepartitioningtoincreasechancesofvisitingmanybasinsofattractionleadingtodifferentlocaloptima.Additionally,theefciencygainsfromparallelizationshouldbeinvestigated. 5.9SummaryandDiscussiononPossibleFutureResearchDirectionsThischapterintroducedamulti-agentmethodologyforoptimizationthatdynamicallypartitionsthedesignspaceastondmultipleoptima.Multipledesignsprovideinsuranceagainstdiscoveringthatlateinthedesignprocessadesignispoorduetomodelingerrorsoroverlookedobjectivesorconstraints.Themethodusedsurrogatestoapproximateexpensivefunctionsandagentsoptimizedusingthesurrogatesinthesub-regions.Thecentersoftheagentssubregionsmovedtostabilizearoundoptima,andagentswerecreatedanddeletedatrun-timeasameansofexplorationandefciency,respectively.Themethodwasappliedtotwoexamples,ananalyticaltestfunctionandapracticalengineeringexample.Itwasobservedthatforproblemsinwhichthebehaviorissimpletoapproximatewithaglobalsurrogate,thesimplersingleagentismoreefcientandsuccessfulthanthemultipleagents.Forthemorecomplicatedtestfunction,inwhichlocalsurrogateswereslightlymoreaccurate,themultipleagentsoutperformedthesingleagent.These 105

PAGE 106

resultsleadustobelievethatthesuccessofthecurrentagentalgorithmisdependentonlocalsurrogatesbeingmoreaccuratethanaglobalsurrogate.Thismethodaimstooexploitthesurrogatepredictionsbypurelyminimizingthefunctionratherthanexploringthedesignspaceasevidencedbythecomparisonofthepercentageofiterationsthatputpointsneartheoptima.Thereisafocusonusingsurrogatepredictionstoaidmultiplecheaplocalsearchesthroughoutthedesignspace.Thisisanimportantdifferentiationbetweenthisalgorithmandmanyglobaloptimizationalgorithmsthataimforandtoutthebalanceofexploitationandexploration.Continuingresearchcanfocusonusingaglobalsurrogatewiththedynamicpartitioningstillinplace.Thereasonforthisistwo-fold:(i)intheauthors'experience,therearefewsituationsinwhichlocalsurrogatesaresignicantlymoreaccuratethanaglobalsurrogate,(ii)thecomplicationofmanagingpointsbetweensub-regionstocreatesurrogatesisremoved.EfcientGlobalOptimization[ 89 ](EGO)isapopularglobaloptimizationalgorithmthatusesaglobalsurrogateandaddspointsbasedonthepresentbestsolution(thepresentbestdatapoint).ItisplannedtomodifytheEGOalgorithmforusewiththedynamicdesignspacepartitioning,inwhicheachsub-regionhasitsownpresentbestsolution.Theproposedagentoptimizationmethodalsohasagreatpotentialforparallelcom-puting.Asthenumberofcomputingnodesnincreases,thecalculationoftheexpensiveobjectiveandconstraintsfunctionsscaleswith1=nintermsofwall-clocktime.Butthespeedatwhichproblemscanbesolvedthenbecomeslimitedbythetimetakenbytheoptimizer,i.e.,theprocessofgeneratinganewcandidatesolution.Inthealgorithmwehavedeveloped,theoptimizationtaskitselfcanbedividedamongthennodesthroughagents.Weplantoexplorehowagentscanprovideausefulparadigmforoptimizinginparallel,distributed,asynchronouscomputingenvironments. 106

PAGE 107

CHAPTER6FURTHERINVESTIGATIONONTHEUSEOFSURROGATE-BASEDOPTIMIZATIONTOLOCATEMULTIPLECANDIDATEDESIGNSThepreviouschapterpresentedamethodtodynamicallypartitionthedesignspacetolocatemultiplecandidatedesigns.Theoverallconclusionwasthatthesuccessofthepartitioningmethoddependedonthedegreeofdifcultyinapproximatingtheproblemwithsurrogates.Theworkinpreviouschapterledustoquestioniflocalsurrogateswereactuallybetterthanusingasingleglobalsurrogate.Afterall,wedidndthattheerrorinthesingleagent'sglobalsurrogatewasgenerallylessthanthatofthemultipleagents.Asaresult,weperformedastudyinwhichthedesignspacewaspartitionedandglobalandlocalsurrogateswerettothedesignspaceandsub-regions.Wemeasuredtheerrorattestpointsandconcludedthatusinglocalsurrogatesweregenerallynotlessaccuratethanasingleglobalsurrogate,particularlywhenthenumberofdesignpointswassmall.ThisstudycanbefoundinAppendix E .Inthischapter,wecomparethismethodtoamethodthatperformslocaloptimizationwithaglobalsurrogatethataddsmultiplepointsatatimeandtheEfcientGlobalOptimiza-tionalgorithm.Weobservedthesurprisingresultthatpartitioningofthedesignspacemaynotholdasimportantofaroleasspreadingoutmanylocalsearchesinthedesignspace.Itwasobservedthatexistingglobaloptimizationalgorithmshavepotentialtobeadaptedtolo-catemultiplecandidatedesigns,butthekeytoefciencyliesinparallelizationofoptimizationprocesses. 6.1MotivationforInvestigatingSurrogate-BasedTechniquesLocatingmultipleoptimaisoftendonewithnature-inspiredalgorithmsusingnichingmethods(Beasleyetal.,1993[ 76 ];HocaogluandAnderson,1997[ 77 ]).Forexample,Na-gendraetal.[ 81 ]usedageneticalgorithmtondseveralstructuraldesignswithcomparableweightandidenticalloadcarryingcapacity.However,whenthreeofthesedesignswerebuiltandtested,theirloadcarryingcapacitywasfoundtodifferby10%.ParsopoulosandVrahasusedparticleswarmoptimization[ 90 ].Restartedlocaloptimizationmethodswith 107

PAGE 108

clustering(TornandZilinkas[ 83 ],TornandViitanen[ 91 ])havealsobeenproposedforndingmanylocaloptima.Thesemethodsrequirealargenumberoffunctionevaluations,whichisprohibitivelycostlyifthefunctionsareexpensivetoevaluate.Toreducethecostofoptimization,surrogatemodelsareoftenused(e.g.Jonesetal.,1998[ 89 ],Alexandrovetal.,1998[ 92 ])toapproximatetheoutputofthesimulations.Asurrogate(ormetamodel)isanalgebraicexpressionttoanumberofsimulations.Traditionally,thelocationswheresimulationsarecarriedoutwereselectedindependentlyoftheoptimization,sothatasurrogatettingphaseprecededtheoptimizationphase.Morerecently,globaloptimizationalgorithmsthatcombinesurrogatettingandoptimizationhavegainedpopularity,mostnotablytheEfcientGlobalOptimization(EGO)algorithm(Schonlau,1997[ 93 ],Jonesetal.,1998[ 89 ]).Theseuseadaptivesequentialsamplingwithpointsaddedatlocationswithhighpotentialofimprovingthedesign.Theresearchinthepreviouschapterdescribedamethodologythatsoughttoextendadaptivesamplingsurrogatetechniquestolocatemultipleoptima.Theproposedapproachwasbasedontheconjunctionoftwoprinciplestoidentifymanycandidateoptima:i)dynamicpartitioningofthesearchspaceandii)localsurrogateapproximations.Inthischapter,weexaminetheeffectivenessofthisapproachinlocatingmultipleoptimaalongwithtwomethods:multiplestartingpointsforlocaloptimizationandEGO,whichisperhapsthecurrentlyfavoredsurrogate-basedglobaloptimizationalgorithm.Additionally,wecomparetheuseofglobalsurrogateapproximationsinourpreviouslydevelopedapproachinplaceoflocalsurrogates.Thenextsectionofthischapterbrieysummarizesthegeneralideabehindsurrogate-basedoptimizationtomotivatethetwoapproachesusedascomparisonsinthisstudy,multiplestartingpointsofmultiplelocaloptimizationsandtheEGOalgorithm,anddescribeswhytheyareinterestingforthisstudy.Section 6.3 comparesthesemethodsontwotwo-dimensionalnumericalexamples,minimizationoftheBranin-HooandSasenafunctions. 108

PAGE 109

6.2Surrogate-BasedOptimizationAsdescribedinSec. 5.3 ,asurrogateisamathematicalfunctionthat(i)approximatesoutputsofastudiedmodel(e.g.themassorthestrengthortherangeofanaircraftasafunctionofitsdimensions),(ii)isoflowcomputationcostand(iii)aimsatpredictingnewoutputs[ 84 ].Thesetofinitialcandidatesolutions,orpoints,usedtotthesurrogateiscalledthedesignofexperiments(DOE).Well-knownexamplesofsurrogatesarepolynomialresponsesurface,splines,neuralnetworks,andkriging.Usingthesurrogate,anewpointisaddedbasedonsomesamplingcriterion,andthesurrogateisupdatedwiththenewpoint.Thisprocesscontinuesasdictatedbysomestoppingcriterion.ThegeneralprocedureforaproblemwithinequalityconstraintsdescribedbythelimitstatesgproceedsasinAlgorithm 6 .Thechoiceofhowtousethesurrogatepredictiontondtheoptimumxisconsideredin Algorithm6Overallsurrogate-basedoptimization 1: t=1(initialstate) 2: whilettmaxdo 3: Buildsurrogatesfandgfrom(Xt;Ft;Gt) 4: Optimizationtondx(seeAlgorithm 3 ) 5: Calculatef(x)andg(x) 6: Updatedatabase(Xt+1;Ft+1;Gt+1)[(x;f(x);g(x)) 7: t=t+1 8: endwhile thenexttwosub-sections.Thereisthechoiceofsimplysolvingoriginaloptimizationproblem(i.e,minimizingf)bysolvingEq.( 5 ),buttherearemanypopularin-llsamplingcriteriathathavebeendeveloped.Thefollowingtwosub-sectionsexpandontwooptionstondtheoptimumx. minimizex2S
PAGE 110

Thechoiceofthestartingpointorinitialguesscanaffecttheabilityofalocalalgorithmtondtheglobaloptimum.Notethatstochasticmethodsarealsoaffectedbytheinitialcongurations(e.g.,theinitialpopulationingeneticalgorithms).Anoptimizerthatusesgradientsmaynotndtheglobaloptimumifitsstartingpointisinthebasinofattractionofapoorerlocaloptimum.Toovercomethis,multiplestartingpointsareoftenused,inordertotakemultipletrajectoriestondasolution.Thus,theuseofmultiplestartingpointsisgoodpracticewhenusingsuchalgorithms.Thereisthepossibilityofobtainingasmanysolutionsasstartingpoints,thoughnotallsolutionsmaybeuniqueassomestartsmayndthesamesolution.Thechoiceofstartingpointsisimportant,andtypicallycomesfromsamplingmethods(e.g.,randomsampling,gridsampling,LatinHypercubesampling,etc.).Thereisalsotheoptionofhaltingtheoptimizationforpoortrajectories.Usingmultiplestartingpointsisarelativelysimpleapproachthatcanbecomepro-hibitivelycostlyiftheobjectivefunctionorconstraintsisexpensive.Inproblemswherethecostofttingandevaluatingasurrogateismuchlessthanthecostofevaluatingthetrueob-jectivefunctionorconstraints,usingmultiplestartingpointsinconjunctionwithsurrogatesisaviableapproachtondmultipleoptima.Thus,wecaninvestigatetheuseofmultiplestart-ingpointstondmultipleoptimabysolvingEq.( 5 ).Inaniterativeoptimizationscheme,thesinglebestsolutionofthestartsorthemultiplepointsresultingfrommultiplestartingpointscanaddedperiteration.Somerandomsamplingisneededtopreventprematureconvergence.Thisresearchinvestigatesbothsurrogateandmulti-startapproaches. 6.2.2EfcientGlobalOptimizationTheEfcientGlobalOptimization(EGO)algorithm[ 89 ]isasequentialsamplingglobaloptimizationmethod.Itstartsbyttingasurrogatethatcomeswithapredictionuncertainty.Afterttingthesurrogate,thealgorithmiterativelyaddspointstothedatasetinanefforttoimproveuponthepresentbestsample.Ineachcycle,thenextpointtobesampledistheonethatmaximizestheexpectedimprovement,E[I(x)].ThoughanothervariantoftheEGOalgorithmusesmaximizestheprobabilityofimprovementonatargetedsolution(EGO-AT 110

PAGE 111

[ 51 ]),thisworkfocusesonE[I(x)].E[I(x)]isameasureofhowmuchimprovementuponthepresentbestsampleweexpecttoachieveifweaddapoint.Ratherthanonlysearchingfortheoptimumpredictedbythesurrogate,EGOwillalsofavorpointswheresurrogatepredictionshavehighuncertainty.Therefore,EGOisabletobalanceexploitationofareaswithsmallobjectivefunctionvaluesandexplorationofareaswithhighuncertainty.ForfurtherdetailsontheEGOalgorithm,thereadermayseekoutoneofthemanypapersonEGO,notably[ 89 93 ].SinceEGOtriestoimproveuponthepresentbestsolution,itmaynotbepossibleforittolocatemultipleoptimaifthevaluesoftheoptimaareverydifferentorwhensomeoptimaareverypoor.Forexample,ifalocaloptimumofafunctionisalreadyfound,theexpectedimprovementmaynotbelargeenoughtoputapointinanareathatcontainsanotheroptimumwithapoorerobjectivefunctionvalue.Forthisreason,forlocatingmultipleoptimawerestrictthecomparisonsoftheEGOmethodtooptimathatarecloseinobjectivefunctionvalue.Additionally,EGOiscapableofaddingmorethanonepointperiteration[ 50 94 ],butthisresearchonlyconsiderstheuseofEGOinaddingasinglepointperiteration. 6.3NumericalExamplesThissectioncomparesthesuccessandefciencyofthemethodspresentedinthischapteronlocatingmultiplecandidatedesignsfortwonumericalexamples.Fivemethodsarecompared:Twomulti-agentmethodsthatusedynamicpartitioningbutdifferentsurrogatesetups,twosingleagentmethodsthataddoneorthreepointsperiteration,andEGO.ThemethodsconsideredaredescribedinTable 6-1 .TherstcaselistedinTable 6-1 usesthedynamicpartitioningdescribedinCh. 5 ,butsharesaglobalsurrogateamongtheagentsratherthanttingalocalsurrogatetoeachagent.Usingaglobalsurrogateremovesthecomplicationofexchangingpointsbetweensub-regionsinordertoavoidill-conditioningwhenttingalocalsurrogates.Additionally,itmaybemoreaccuratethanseverallocalsurrogatesasshowninthepreviouschapter.ThemethodthatuseslocalsurrogateswithpartitioningisstillstudiedandlistedasthesecondmethodinTable 6-1 111

PAGE 112

Table6-1.Descriptionofmethods MethodDescription AgentsandPartitioning:Global(Sec. 5.4 )Usesmulti-agentswithdynamicpartitioning.Agentsusethesameglobalsurrogate.Asmanypointsasagentsareaddedperiteration.AgentsandPartitioning:Local(Sec. 5.4 )Usesmulti-agentswithdynamicpartitioning.Agentuseslocalsurrogateforitssub-region.Asmanypointsasagentsareaddedperiteration.Single(Sec. 6.2.1 )Singleglobalsurrogateagentaddingonepointperiteration.Multiplestartsareusedandthebestpointintermsoffea-sibilityandobjectivefunctionvalueischosen.Explorationoccurswhenastartgivesasolutiontoonearanalreadyexistingpoint.Explorationaddsapointthatmaximizestheminimumdistancefromexistingpoints.Single:MultiplePointsPerIteration(MPPI)(Sec. 6.2.1 )Singleglobalsurrogateadding3pointsperiterationby3startpointsforlocaloptimization.Explorationoccurswhenastartgivesasolutiontooneartoanalreadyexistingpoint.Explorationaddsapointthatmaximizestheminimumdis-tancefromexistingpoints.EGO(Sec. 6.2.2 )GlobalsurrogateusingEGOalgorithmtoaddonepointperiteration Inallcases,thenumberoffunctionevaluationswasxedat100,includingthoserequiredfortheinitialdesignofexperiments,forbothexamples.ThenumberofpointsaddedfortheSingle:MPPIcasewassettothreebecauseitwasobservedthatthiswasthemeannumberofpointsaddedinthemulti-agentcasesforthesameexamples.Thus,wexedthenumberofpointsperiterationtothreetoprovideafairercomparisonbetweenthetwomethods.AsdifferentmethodslistedinTable 6-1 addadifferentnumberofpointsperiteration(i.e.,theSingle:MPPImethodaddsthreepointsperiteration,multi-agentmethodaddsasmanypointsasagents,andSingleandEGOcasesonlyaddonepointperiteration),weexaminetwovalueswhencomparingefciency:numberoffunctionevaluationsandnumberofiterations.TheadvantageofaddingmultiplepointsperiterationasintheSingle:MPPIandmulti-agentmethodscomesfromparallelizationbetweentheprocessesthataddthemultiplepoints.Thus,comparingthesemethodstoonesthataddonlyasinglepointperiterationshouldbedonebasedonthenumberofiterationsratherthanfunctionevaluations.Inthe 112

PAGE 113

resultspresentedforthetwoexamplesinthissection,wepresentbothvalues.NotethatfortheSingleAgentandEGOcasestheiterationsandfunctionevaluationsareequalasbothonlyaddasinglepointpercycle. 6.3.1ExperimentalSetupFortheexampleconsideredinthisstudy,therearenononlinearconstraints,soonlytheobjectivefunctionisapproximatedbysurrogates..Thethreepossiblesurrogatesarekrigingsurrogateswithquadratic,linear,orconstanttrendfunction.Fromthisset,eachagentchosethebestsurrogatebasedonPRESSRMS.Thesetofsurrogatesandtheminimumnumberofpointsusedtoteachsurrogateis1.5xthenumberofcoefcientsofaquadraticresponsesurface.Iftheminimumnumberofpointsarenotavailablewhenttinglocalsurrogatestothesub-regions,pointsareborrowedfromneighboringsub-regionsintheorderofincreasingdistancetotheagentcenter,and,iftherequirementisstillnotmet,thenallavailablepointsareused.Fortheagentcaseswithpartitioning,theparametersareprovidedinTable 6-2 .For Table6-2.Multi-agentparametersforexampleproblems ParameterValue Max#offunctionevaluations100Max#ofagents6Initial/Min#ofagents1Mindistancebetweenagentcenters10%ofmaxpossibledistanceinspaceMindistancebetweenpoints0.2%ofmaxpossibledistanceinspaceMinaveragesilhouette0.25Min#ofpointsineachagentaftercreation4Stagnationthreshold3Numberofstartingpointsforlocaloptimization10 distances,weconsiderthedistanceinthenormalizedspaceasafractionofthemaximumpossibledistancebetweentwopointsinthedesignspace(e.g.,fortwo-dimensionalprob-lems,wenormalizedbyp 2).Forexample,weusethisdistancewhenconsideringtheminimumdistancebetweencentersandtheminimumdistancebetweendatapointsasgiveninTable 6-2 113

PAGE 114

TheinitialsizeoftheDOEforbothexampleproblemswas12,withthepointssampledbyLatinHypercubeSampling.IneachcasegiveninTable 6-1 ,theresultsshownarethemedianof50repetitions(i.e,50differentinitialDOEs).Thelocaloptimizationproblemsweresolvedwithasequentialquadraticprogramming(SQP)algorithm[ 88 ].DOEsareobtainedusingLatinHypercubesamplingandthemaximincriterionforveiterations. 6.3.2Branin-HooTestFunctionTherstexampleistheminimizationoftheBranin-Hootestfunction,acommonbenchmarktestfunctionusedinsurrogate-basedglobaloptimization.ItisgiveninEq.( 6 ). f(x)= x2)]TJ /F12 11.955 Tf 12.35 8.1 Td[(5:1 42x21+5 x1)]TJ /F12 11.955 Tf 10.26 0 Td[(6!+10 1)]TJ /F12 11.955 Tf 14.59 8.1 Td[(1 8!cos(x1)+80(6)Thedomainofthefunctionis)]TJ /F12 11.955 Tf 7.6 0 Td[(5x110and0x115.TherearethreeglobaloptimaoftheBranin-Hoofunction,forwhichf=0:40.AcontourplotoftheBranin-HoofunctionisshowninFig. 6-1 Figure6-1.ContourplotofBranin-Hoofunctionshowingthreeoptima.Foralloptima,f=0:40. 114

PAGE 115

For50repetitions,thepercentageofrepetitionsthatsuccessfullylocatedasolutiona1%distancefromtheoptimumisshowninFig. 6-2 ,wherethenumberoffunctionevalua-tionsshowndoesnotincludetheevaluationsrequiredfortheinitalDOEof12points.ThisdistanceistheEuclideandistancenormalizedbythemaximumpossibledistancebetweenpointsinthedesignspace(here,p 2). Figure6-2.FortheBranin-Hooexample,thepercentageof50repetitionsthatfoundasolutionwithin1%distancefromeachoptimum.Forthecasesthatmultiplepointsperiteration(agentsandSingle:MPPI)thevalueintermsofiterationsisgivenbythesolidlinesanddashedlinesforfunctionevaluations. 115

PAGE 116

First,ifwemakethecomparisonintermsofiterations,weobservedthattheSin-gle:MPPImethodisquicklysuccessfulin80%oftherepetitions.However,toreachpercent-agesgreaterthan90%,themulti-agentmethodwiththeglobalsurrogatelocatedalloptimawiththefewestiterations.Intermsofthenumberofiterations,thegeneraltrendwasthatthemulti-agentmethodwithaglobalsurrogatereached100%successwiththefewestiterationsfollowedbyeitherthemulti-agentmethodwithlocalsurrogatesorSingle:MPPI,thenthesingleagentaddingonepointperiteration,andnallyEGO.However,itshouldbenotedthatEGOactuallyfoundsolutionsneartotheoptimawithasmallnumberoffunctionevaluations,butrequiredmorefunctionevaluationstoputapointwithin1%ofalloptimaasitwasdriventosearchotherregionswithhigherexpectedimprovementduetolargeruncertaintyinthesurrogate.Whenconsideringthenumberoffunctionevaluations,whichdoesnotaccountfortheparallelizationintheadditionofmultiplepointsperiteration,thesingleagentisthemostefcientwhilethemulti-agentmethodwithlocalsurrogatesistheleastefcientwiththeothermethodsfallinginbetween.Figure 6-3 displaysthemedianobjectivefunctionvaluewithiterationsandfunctionevaluationsforthesolutionnearesttoeachoptimum.WeobservedthattheSingle:MPPImethodhadamedianfvalueclosesttotheglobaloptimumoff=0:39withthefewestiterations,andeventhefewestfunctionevaluationsascomparedtotheothercases.Infact,theSingle:MPPImethodconsideringfunctionevaluationswasevenmoreefcientthanagentsintermsofiterations.Otherwise,weobservedmuchofthesametrendsbetweentheone-point-periterationmethodsandtheagentmethods. 116

PAGE 117

Figure6-3.FortheBranin-Hooexample,themedianobjectivefunctionofthesolutionnearesttoeachoptimum.Forthecasesthatmultiplepointsperiteration(agentsandSingle:MPPI)thevalueintermsofiterationsisgivenbythesolidlinesanddashedlinesforfunctionevaluations. Figure 6-4 displaystheplacementofpointsinthedesignspaceafter100totalfunctionevaluationsforeachmethodforasinglerepetition.Itisobservedthatthesingleagentmethodputmanypointsaroundtheoptimaandonlyfewpointsintherestofthedesignspaceinexploration.Incontrast,theSingle:MPPImethodbothclusteredpointsaroundtheoptimaandlledthedesignspace.TheEGOmethoddidnotputnearlyasmanypointsaroundtheoptimaandputmanypointsinexplorationinthedesignspace,whichisconsistentwithitsgoalofsearchinginareaswithpromisingimprovement. 117

PAGE 118

Figure6-4.FortheBranin-Hooexample,theplotofthepointsfoundbydifferentmethodsforonerepetition Theplacementofpointsforthemulti-agentcaseswasquitedifferentwhenusingglobalandlocalsurrogates.Whilebothputmanypointsaroundeachoptimum,localsurrogatesresultedinmanypointsawayfromtheoptima.Thiswaspartiallyduetotheerrorinthesurrogate,whichpredictedgoodobjectivefunctionvaluesawayfromtheoptima.Tomeasuretheerror,wecalculatedtheerrorat1000testpointsbyerms.TheermsnormalizedbytheestimatedrangeoftheBranin-HoofunctionisprovidedinFig. 6-5 118

PAGE 119

Figure6-5.FortheBranin-Hooexample,theerrorat1000testpointswithermsasthepercentageoftheestimatedrangeofthefunction Itwasobservedthatlocalsurrogatesweretheleastaccurate,whilethesingleglobalsurrogateintheSingle:MPPImethodhadlessthan1%errorafter20functionevaluations.Afterobservingthatagentmethodusingtheglobalsurrogatewasonlyslightlylessefcientthanusingasingleagentadding3pointsperiteration,weexaminedthecasewithaconstant3agents(3sub-regions)andcomparedittotheSingle:MPPImethodaddingthreepointsperiteration.Thissimulatesacasewhenthreecomputingnodesareavailable,andallresourcesareusedbyassigningoneagenttoonenode.Recall,thatwepreviouslysettheinitialnumberofagentsatone,andletthenumberofagentsevolveovertimewhilesettingthemaximumnumberofagentsatsix.Figure 6-6 displaysthepercentageof50repetitionsthatlocatedasolutionwith1%distancefromeachoptimumwithiterationsandfunctionevaluations. 119

PAGE 120

Figure6-6.FortheBranin-Hooexample,acomparisonbetweenthepercentageof50repetitionsthatfoundasolutionwithin1%oftheeachoptimumfortheaconstant3agentswithaglobalsurrogateandtheSingle:MPPIapproach.Notethatbothmethodsaddthreepointsperiteration. Itwasobservedthattherateatwhich100%successwasachievedwasquitesimilarbetweenthetwocases,withtheagentmethodslightlymoreefcient.Therefore,efciencymaybeincreasedbymaximizingtheuseofcomputationalresourcesbysettingaconstantnumberofagentsratherthanlettingthenumberofagentsevolveovertime.However,itshouldbenotedthatSingle:MPPIonlyallowsthreestartingpointsinthedesignspace,whereasthethreeagentseachhavetenstartingpointsinthedesignspace,foratotalof30startingpoints.Thus,itisnotentirelyunexpectedthatthethreeagentsachievesuccessataslightlyhigherrateastheychoosefrom10pointstondthebestpointperiterationascomparedtoone.Thisshowsthatspreadingoutalargenumberoflocalsearchesinthedesignspace,whichisdonebyaddingmultiplepointsperiterationwiththesingleglobalsurrogate,maysimplybeaseffectiveaspartitioningofthedesignspace. 120

PAGE 121

6.3.3SasenaTestFunctionThesecondexampleproblem,istheminimizationoftheSasenafunction,whichwasusedbySasenaunderthenamemysteryfunctionduetoitsunknownorigin[ 95 ]. f(x)=2+0:01(x2)]TJ /F5 11.955 Tf 10.86 0 Td[(x21)2+(1)]TJ /F5 11.955 Tf 10.86 0 Td[(x1)2+2(2)]TJ /F5 11.955 Tf 10.85 0 Td[(x2)2+7sin(0:5x1)sin(0:7x1x2)(6)Thedomainofthefunctionis0x1;x25.AcontourplotoftheSasenafunctionisshowninFig. 6-7 .Therearefouroptima,asshowninthegure,butthevaluesofoptima3(f=12:7)and4(f=33:2)are40%and90%fromtheglobaloptimum(f=)]TJ /F12 11.955 Tf 7.61 0 Td[(1:46)intermsoftherangeofthefunction(38.6).Optimum2(f=2:87),whichhas11%differencefromtheglobaloptimumistheonlycompetitiveoptimum.Therefore,itisexpectedthatthemethodscomparedhereareonlyeffectiveatlocatingoptimum1and2. Figure6-7.ContourplotofSasenafunctionshowingfouroptima 121

PAGE 122

Inadditiontoitspoorobjectivefunctionvalue,optimum4hasasmallbasinofattraction.Incaseswherethemodelthatisbeingoptimizedhaserror,suchsmallregionsaremaybewipedoutbyerrorssoadesigninthisregioncouldbevulnerable.For50repetitions,thepercentageofrepetitionsthatsuccessfullylocatedasolutiona1%distancefromtheoptimumisshowninFig. 6-8 .Weobservedverylittlesuccessatlocatingalloptimabyallmethods,whichwasnotunexpectedasoptima3and4arepoorincomparisontothetoptwooptima. Figure6-8.FortheSasenaexample,thepercentageof50repetitionsthatfoundasolutionwithin1%distancefromeachoptimum 122

PAGE 123

Whenthecomparisonismadeintermsofnumberofiterations,themethodsthatcallforparallelizationoftheoptimizationprocesses(i.e,themulti-agentmethodsandSingle:MPPI)outperformtheone-point-per-iterationsingleagentandEGOalgorithm.TheSingle:MPPImethodagainachievesahighpercentageofsuccesseswithonlyafewiterationsforoptima1and2,buttherateofsuccessforbothmulti-agentcasesiscomparablefortheglobaloptimum.Anotherthingtonoteisthecomparableperformanceofthemulti-agentmethodwithlocalsurrogatescomparedtotheglobalsurrogate,whichwasnotobservedintheBranin-Hooexample.ItwasalsoobservedthatEGOwasefcientinlocatingtheglobaloptimum,butonlyhadunder10%successinlocatingoptimum2.Basedonthisexample,thecurrentimplementa-tionofEGOhaslesspotentialtobesuccessfulinlocatingmultipleoptimawhentheoptimaarenotalmostequal.Whencomparingtheefciencyintermsofnumberoffunctionevaluations,theone-point-per-iterationsingleagentisthemostefcient.Infact,thesingleagentisabletolocateoptimum3in90%oftherepetitions.Thisisbecausethesingleagentlocatesoptima1and2inearlyiterations,andisabletoputpointsintheotherpartsofthespacetolocateoptimum3.Figure 6-9 displaysthemedianobjectivefunctionvaluewithiterationsandfunctionevaluationsforthesolutionnearesttoeachoptimum.WeobservedthattheSingle:MPPImethodhadamedianfvalueclosesttooptima1,2,and3withthefewestiterations,andinmostcases,thefewestfunctionevaluations.Additionally,itwasobservedthatforoptimum3,themedianffortheSingle:MPPImethodisquiteclosetothetruevalue,whichwasnotclearwhenexaminingthesuccesspercentage.Foroptimum4,itwasobservedthatthenearestsolutionforallcasesexceptSingle:MPPIhadamedianfof35.9,whichcorrespondstoaspace-llingpointatthecornerofthedesignspaceat(5;5). 123

PAGE 124

Figure6-9.FortheSasenaexample,themedianobjectivefunctionofthesolutionnearesttoeachoptimum.Forthecasesthatmultiplepointsperiteration(agentsandSingle:MPPI)thevalueintermsofiterationsisgivenbythesolidlinesanddashedlinesforfunctionevaluations. Theerrorofthesurrogateapproximationsat1000testpointsisshowninFig. 6-10 .AsintheBranin-Hooexample,itisobservedthatthesinglesurrogatewithmultiplepointsperiterationswasthemostaccurateandthelocalsurrogatesfromthemulti-agentsystemweretheleastaccurate. Figure6-10.FortheSasenaexample,theerrorat1000testpointswithermsasthepercentageoftheestimatedrangeofthefunction 124

PAGE 125

Finally,wecomparedthesuccessinlocatingasolutionwithin1%distancefromeachoptimumusingaconstant3agentswithaglobalsurrogateandtheSingle:MPPImethod.Inbothcases,threepointsareaddedperiteration.TheresultsareshowninFig. 6-11 Figure6-11.FortheSasenaexample,acomparisonbetweenthepercentageof50repetitionsthatfoundasolutionwithin1%oftheeachoptimumfortheaconstant3agentswithaglobalsurrogateandtheSingle:MPPIapproach.Notethatbothmethodsaddthreepointsperiteration. Fortheoptimum1,itwasobservedthatthemulti-agentmethodwasslightlymoreefcientinachieving100%success,butforoptimum2,theSingle:MPPImethodwasonlyslightlymoreefcient.Foroptimum3,theagentmethodhadaslightlyhighersuccesspercentage,whilethesuccesspercentagewasbelow10%forbothmethodsforoptimum4.AsintheBranin-Hooexample,thisshowedthatefciencyoftheagentmethodmaybeincreasedbymaximizingtheuseofcomputationalresourcesbysettingaconstantnumberofagents.Moreinterestingly,weobservedagainthatspreadingoutlocalsearcheswasnearlyassuccessfulandefcientasusingagentsandpartitioning. 125

PAGE 126

6.4DiscussionandSummaryThischapterprovidedacomparisonofthesuccessandefciencyinlocatingmultipleoptimabydifferentsurrogate-basedoptimizationmethods.Ourpreviouslydevelopedmulti-agentmethodthatdynamicallypartitionsthedesignspacewascomparedagainsttheEGOalgorithmandasimplemethodthatusesmultiplestartsinthedesignspacetoeitheraddoneorseveralpointsinaniteration.ItwasobservedthatEGOhasthepotentialtolocatemultipleoptimawhenoptimafunctionsvaluesaresimilar,whiletheothermethodspresentedherehavethisabilityforoptimathatarewithin11%oftherangeofthefunction.Inpractice,thisisanidealscenarioasonewouldnotwanttowasteresourcesonsearchingforpooroptima.Themostefcientmethodsofthosestudiedhereaimedtotakeadvantageofparallelcomputingforoptimization.Theuseofmultiplestartingpointsforlocaloptimizationandaddingmultiplepointspercycleprovedtobeasimpleyetefcientmethodthatwarrantsfur-therresearch.Themulti-agentapproach,whichinvolvesoptimizationinseveraldynamicallychangingsub-regionsinparallel,wasalsoshowntobeefcientinlocatingcompetitiveop-tima.Thisshowsthatthebenetofpartitioningthedesignspaceisthatithelpsspreadlocalsearchesthroughoutthedesignspace,andalsoincreasesthepotentialforparallelization.Weobservedthattheerrorinlocalsurrogateapproximationsbythemultipleagentswaslargercomparedtoaglobalsurrogate.Additionally,wedidnotobservethatlocalsurrogatesoutperformedtheglobalsurrogateineithertestproblem,whichsupportsthestudyinAppendix E thatcomparedtheaccuracyofglobaltolocalsurrogatesforseveraltestfunctions.Forthesereasons,itmaybepossibletoonlyuseaglobalsurrogate,whichremovesthecomplicationofexchangingpointsbetweenagentstotlocalsurrogates.Inthefuture,amorein-depthlookattheadvantagesofusingmultiplepointsperiterationcanbestudied.Thereisthepossibilitytoexploreasynchronousagentsthatpartitionthestartingpointsforlocaloptimizationinthespaceandupdatetheglobalsurrogateasnewpointsareadded. 126

PAGE 127

CHAPTER7CONCLUSIONSTheinitialstagesofdesignincludetheformulationoftheoptimizationproblem,includingobjectivefunctionsandconstraints,andoftenincludebuildingacomputationalmodelwithwhichtoperformtheinitialdesignoptimization.However,thereisuncertaintyintheprocess,whichstemsfromtheinabilitytoperfectlyformulatetheoptimizationproblem,inherentuncertaintiesinthedesign,andtheuncertaintiesinthecomputationalmodel.Testsandredesignareoftenperformedoncandidatedesigns,whichallowsfortheidenticationofdangerousdesignsthatcanberedesignedandalsoprovidesmeasuresbywhichtocalibratecomputationalmodels.Thisresearchconsiderstwoareasofthedesignofengineeringsystems:1)thetrade-offoftheeffectofatestandpost-testredesignonreliabilityandcostand2)thesearchformultiplecandidatedesignsasinsuranceagainstunforeseenfaultsinsomedesigns.Themaincontributionsofthisresearchareasfollows: 1. Amethodologytoquantifytheeffectofasinglefuturetestandredesignonperfor-manceandcost 2. Aninvestigationonhowtotradeoffperformanceanddevelopmentcostsbyinclud-ingtheeffectofasinglefuturetestandredesign,andadditionallyhowthisallowscompaniesprobabilisticallysetdesignandre-designrules 3. AdynamicpartitioningmethodofthedesignspacethatcombinessurrogatesandlocalsearchtolocatemultiplecandidatedesignsFirst,amethodologytoquantifytheeffectofasinglefuturetestandredesignonperformanceandcostwaspresentedforxeddesignandredesignrules.Thismethodwasbasedonsamplingcomputationalandexperimentalerrorstosimulatealternativefuturetestoutcomes,forwhichthedecisiontodesignorredesignwasmade.Twomethodsofcalibrationandredesignwerepresented.Inonemethod,asimplecorrectionfactorbasedontheratioofthesimulatedexperimentalmeasurementtothepredictedvaluefromthecomputationalmodelwasusedincalibration,andredesignwasperformeddeterministicallytorestoretheinitiallevelofsafetyofthedesignasdictatedbyarequiredsafetymargin.The 127

PAGE 128

secondmethodusedBayesianupdatingtoupdatetheinitialcomputationalerrordistributionandredesignedtomeetatargetedreliabilitylevelgiventheupdatederrordistribution.Itwasobservedthesemethodsprovidedestimatesofthedistributionoftheprobabilityoffailureandperformanceofadesignaftertestandredesign.Asanextensionofthepreviousresearch,itwasshownthattheprobabilisticquanti-cationoftheeffectofthefuturetestandredesigncouldbeusedtotradeoffperformanceanddevelopmentcostsbysettingdesignandredesignrules.Thisresearchconsidereddeterministicdesignandredesignrules,whicharerepresentativeofcurrentdesignpracticeusedpresentlyinindustry.Inthisstudy,itwasshownthattheoptimaltrade-offcalledforinitiallyconservativedesignswithlargesafetymargins,whichweremadelessconservativebutwithincreasedperformancewithincreasingredesign(development)costs.Thisresultwascomparedtotheoppositeapproachinwhichaminimumrequiredsafetymarginisgiven,whichreectsthepracticeofregulatoryagenciesprovidingminimalrequiredsafetymarginsandfactors.Thethirdareaofresearchfocusedonlocatingmultiplecandidatedesignsbyacombi-nationofdynamicdesignspacepartitioningandsurrogate-basedoptimizationbymutliplelocalsearchesinsub-regions.Thisresearchfocusedonhowtopartitionthedesignspacesuchthatthecenterofeachregionwaslocatedonalocaloptimum,whilecreatingregionstoexplorethedesignspaceandmergingregionsthatconvergedtothesamearea.Thecoordinationbetweenregionsforsurrogate-tting,optimization,andexchangeofdesignpointswasinspiredbymulti-agentapproachesseenindistributedoptimizationalgorithmsthattakeadvantageofthedecompositionoftheoptimizationformulation.Thisresearchmainlyexploredtheuseoflocalandglobalsurrogates,wherealocalsurrogatewasusedineachregionofthedesignspaceinordertoprovideamoreaccurateapproximationofthelocalbehavior.Thismethodwascomparedtoarelativelysimpleapproachinwhichasinglesurrogateusingmultiplestartingpointsforlocaloptimizationovertheentiredesignspace.Itwasobservedthatthesuccessofthepartitioningmethodwasprimarilyduetotheuse 128

PAGE 129

ofmultiplestartingpointsforlocalsearchesinthespace.Ineffect,thisledthemulti-agentmethodtoexploitregionsofpredictedlowfunctionvalueandonlyexploresparselypopulatedregionsofthedesignspacewhenthiswasnotpossible.Additionally,itwasfoundthatthesuccessandefciencyofthepartitioningwithlocalversusglobalsurrogatesmethodmaybedependentonthedegreeofdifcultyinapproximatingthebehaviorwithsurrogates.Forseveraltestproblems,itwasfoundthatusinglocalsurrogateswasnotasadvantageousasasingleglobalsurrogate.Itwasalsoobservedthattheremaybelargegainsinefciencyiftheoptimizationintheregionsisparallelized. 7.1PerspectivesBasedontheresearchpresentedinthisdissertation,futureresearchincludesthreetasks:1)efcientidenticationofindividuallocaloptima,2)establishingtherangeofaccept-abledesignsbasedonthevulnerabilityofthebestoptima3)distributedimplementationofthemethods. 7.1.1EfcientIdenticationofIndividualLocalOptimaWhiletheresearchinCh. 5 haveshownthatlocatingmultipleoptimawithsurrogatesisapromisingdirectionofresearch,thereareremainingformidablechallengesthatcanbeaddressed. 7.1.1.1IsolatingbasinsofattractionThegoalistoisolateandcharacterizethebasinsofattractionofeachlocaloptimum.Inpastimplementations,onlytheperformanceofdesignpointswasconsideredtocentertheagents.Yet,onesub-regionmayspantwoormorebasinsofattraction,suchthatalocaloptimummaybemissedbecausetheoptimizationrepeatedlyndstheoptimumwiththelowerobjectivefunctionvalue.Thereisaneedtodevelopanefcientmethodtodetectthatasub-regioncontainsmorethanoneacceptableoptimum. Proposedapproach.Sincethecenterofthesub-regionisatthebestpointinthesub-regionintermsoftheobjectivefunctionvalue(andfeasibilityinconstrainedproblems),thepoorerotheroptimaarenotexplicitlyidentied.Theproposedimprovementistotag 129

PAGE 130

thepotentiallocaloptimathatarefoundusingmulti-startlocalmethodsonthesurrogatesaccordingtowhetherornottheysatisfyoptimalityconditions.Thereshouldalsobeadistinctionbetweenoptimainsideasub-regionandoptimaattheboundaryofasub-region,astheboundariesbetweensub-regionsareartifactsofthemethodbutnotrealdesignconstraints.Forexample,iftherearethreeoptimaintwosub-regions,twocentersmaybelocatedatthetwocenters,butonebasinofattractionmayspanbothsub-regions.Recognizingthepositionofthepointsattheboundaryalongwithtaggingthepotentialoptimamayaidinthecreationofsub-regionsthatwillleadtotheisolationofbasinsofattraction. 7.1.1.2Suspendingorallocatingfewresourcestounpromisingsub-regionsSomesub-regionsmayappeartohavelocaloptimathataretoopoortobeworthwhile.Itisimportanttonotwrite-offtheseregionscompletelyastheymaycontainagoodoptimuminaverynarrowbasinofattraction.However,searchinthesub-regionmaybesuspendedorfewerexpensivefunctionevaluationsmaybeallocatedtothisregionuntilmorepromisingsub-regionsareexplored.Developingsuchacriterionisanimportantsteptomakethistypeofmethodmoreefcient. Proposedapproach.Acommonstoppingcriterionfoundinglobaloptimizationisbasedontheconvergenceoftheobjectivefunctionvalue[ 96 97 ].Forthisresearch,itisproposedthatthecriteriontosuspendsearchinasub-regionorallocatefewerresourcestotheregionisbasedonobjectivefunctionvalue,feasibility,sizeofthedomain,anditsvulnerabilitytomodelingerrors.Suchacriterionwoulddifferentiatebetweenrelativelysmallregionswithpoorobjectivefunctionvaluesinwhichsearchmaybesuspended,andlargeregionswithsmallfunctionvalueswhichmaystillbenetfromfurtherexplorationtoidentifymoreoptima.IntheDIRECTmethod,Jonesetal.[ 98 ]allocatedresourcesintheregionsofthedesignspacethatrepresentedthebestcompromises,accordingtoParetodominance,betweentheobjectivefunctionandthesizeoftheunexploredneighborhood.Itisproposedtoextendsuchamulti-criterionrationaleinguidingthesearchconsideringi)theperformanceofthedesigns 130

PAGE 131

andii)thedensityofthestartingpointsintheagentsub-regions,and(iii)vulnerabilityindicesasthecriteriaforallocatingcomputingresources.Withsuchanapproach,narrowbasinsofattractionwillbefoundearlyiftheyareinaregionofgoodobjectivefunctions,andviceversa,whichmatchesthepracticalinteresttheymayhave(anoptimuminanarrowvalleysurroundedbypoorperformancedesignsislikelytobetoounstableforpracticalpurposes). 7.1.2VulnerabilityAnalysisandRangeofAcceptableObjectiveFunctionsChapters 3 and 4 presentedamethodologythatsoughttomodeltheeffectoferrorsontheeventualmassofadesignbysimulatingfutureswheretheerrorsforceredesignbasedonboundsonmodelingerrors.Thiswasdoneassumingthattheerrorsaresmalltomoderateandfutureteststhatwillrevealthemtorequireonlyre-calibrationoftheanalysis.Theresearchdemonstratedthatadesigncansignicantlychangewhenconsideringerrorsincombinationwithtestsandredesigntocheckforandcompensatefortheseerrors.Byconsideringtheeffectoferrorsandsimulatingpossiblefutures,thevulnerabilityofadesigntobeaffectedbyerrorscanbemeasuredbytheprobabilityofredesign(theprobabilitythatthedesigndoesnotmeetrequirements),whileperformancemeasures,suchasmass,giveanindicationoftheextentofthechangestothedesignthathavetobemadetocompensatefortheerrors.Additionalresearchinthisareahasextendedthismethodologytomultiplefailuremodes[ 68 ]andalsosoughttomodeltheeffectofunexpectedlargeerrors[ 99 ].Thisworkcanbeextendedtoallowthecomparisonofmultipledesignsandtoestimatetheprobabilityofislandsoffeasibilitybeingwipedout. Proposedapproachthattakesadvantageofmodelingoftheeffectoffuturetestsandredesign.ThepresentapproachisbasedonMonteCarlosimulationsofmultiplepossiblefuturesassociatedwithdifferenterrormagnitudes.Theassociatedcomputationalcostishigh,andthereforenotfeasibletobeincorporatedinsideaglobaloptimizationalgorithm.Weproposetodevelopapproximateestimatesofthemeanobjectivefunctionincrementafterfutureredesignaswellasitsstandarddeviation.Thiswillbebasedona 131

PAGE 132

simplere-calibrationapproachthatwehaveexploredinCh. 3 .Itisalsoproposedtodevelopestimatesoftheprobabilitythatanislandoffeasibilityassociatedwithagivenlocaloptimumwillcompletelydisappear.Thiscanhappenwhentheislandisnarrowduetosmallerrorsthatarewithintheexpectedrangeanticipatedbythedesigner.Forexample,fortheexamplepresentedinCh. 5 ,anerrorof12KinthetemperaturemodeloftheITPSwillwipeoutRegion2ofFigure3duetothermalfailure.However,feasibleregionsmayalsodisappearduetounexpectedappearanceofoverlookedfailuremodesorobjectivefunctions.Itisproposedtoallowdesignerstoassignsuchprobabilitiesbasedontheclosenessofthedesigntotheirpastexperience.Designsthatappeartobemererenementsofpreviousexperiencemaybeassignedlowprobabilitywhiledesignsthatlookverydifferentmaybeassignedhigherprobabilities. 132

PAGE 133

APPENDIXACOMPARISONOFBAYESIANFORMULATIONSInarigorousformulationofthelikelihoodfunction,wewouldcalculatetheconditionalprobabilityofobtainingthemeasuredtemperaturewhenthetruetemperatureofthetestarticleisT,asshowninEq.( A ) ltest(T)=8>>>><>>>>:1 0:14TifT)]TJ /F7 7.97 Tf 5.07 0 Td[(Tmeas T0:07;0otherwise:(A)IntheillustrativeexampleinSection 3.3.2.1 ,wesimpliedthisformulationsothatwecalculatedtheconditionalprobabilityofobtainingTgivenTmeas,asshowninEq.( 3 ).InFig. A-1 ,wecomparethetwolikelihoodfunctionsandtheresultingupdateddistributionoffupdtest;Ptrueforthecaseintheexample. FigureA-1.IllustrativeexampleofBayesianupdatingusingthelikelihoodaboutTmeas(top),andthelikelihoodaboutT(bottom) Theguresshowonlyasmalldifferenceintheboundsoftheupdatedtemperaturedistributionandthevaluesofthepdf.AcomparisonisshowninTable A-1 133

PAGE 134

TableA-1.Comparisonoffupdtest;Ptruewithdifferentformulationsofthelikelihoodfunction Comparisonltest(T)aboutTmeasltest(T)aboutT Boundswhereupdateddistributionisnonzero[0.9765,1.1][0.9813,1.1]Maxfupdtest;Ptrueandlocation8.1on[0.9765,1.1]8.9atT=0.9813 134

PAGE 135

APPENDIXBEXTRAPOLATIONERRORInthisresearch,itwasassumedthevariationinthemagnitudeoftheextrapolationerroreextrapwaslinearwiththedistanceofthedesignfromthetestdesign.Thechoiceofthisextrapolationerrorisverymuchuptotheanalyst,asitisameasureinthevariationoftheerrorsfromtheupdatedBayesianestimateawayfromthetestdesign.Here,weexaminetheeffectofanassumptionthattheextrapolationerrorisquadratic,asexpressedinEq.( B ). eextrap=(eextrap)max kd)]TJ /F5 11.955 Tf 10.26 0 Td[(dtestk dlim!2(B)FortheexampleprobleminSection 3.5 ,weestimatedeextraptobe2%whendischangedby10%fromdtest.Withthequadraticextrapolationerror,thisisexpressedasinEq.( B ).Becauseofthisrequirement,themagnitudeofthequadraticextrapolationerrorissmallerfordesignsatadistancelessthat10%awayfromthetestdesignbutlargeratgreaterdistancescomparedtothelinearvariation.WepresentthiscomparisoninFig.( B-1 ).Examiningthesame10000possibleoutcomesofthefuturetestwithprobabilisticredesign(pf;target=0:01%),theresultsinTable B-1 wereobtained. eextrap=0:02 kd)]TJ /F5 11.955 Tf 10.26 0 Td[(dtestk 0:1kdtestk!2(B)Theresultsshowthatthereisimprovedagreementbetweenthetrueandanalystestimatedprobabilitiesoffailure,aswellasaslightlydecreasedmassandvariationinthemass,withthequadraticvariationinextrapolationerror.Sincetheextrapolationerrorissmalleratadistancelessthan10%awayfromthetestdesign,theagreementbetweenthetrueandanalyst-estimatedprobabilitiesoffailureisbetterwiththequadraticextrapolationerror.However,theagreementstillsuffersduetothelargemagnitudeoftheextrapolationerroratdistancesgreaterthan10%. 135

PAGE 136

FigureB-1.Comparisonoftheeextrapwithlinearandquadraticvariationwiththedistanceofthedesignfromthetestdesign(testdesignisd=71:3mm) 136

PAGE 137

TableB-1.CalibrationbytheBayesianupdatingapproachwithprobabilityoffailurebasedredesign(pf;target=0:01%),quadraticextrapolationerror,andnoboundsonredesigndS VariationineextrapwithdsParameterOriginalMeanStandardDeviationMinimumMaximum LineardS(mm)71.365.38.947.577.7mass(kg=m2)35.133.72.129.536.5pf;true(%)0.120.0030.01600.100pf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr(%)0.120.0070.00400.015 QuadraticdS(mm)71.366.47.354.477.1mass(kg=m2)35.133.91.731.136.4pf;true(%)0.120.0040.01900.100pf;analyst)]TJ /F7 7.97 Tf 5.07 0 Td[(corr(%)0.120.0070.00400.015 137

PAGE 138

APPENDIXCSIMULATINGATESTRESULTANDCORRECTIONFACTORAsdescribedinSec. 4.3 ,atestisperformedtoverifyadesign,andthetestisperformedonatestarticledenotedbydtestandrtesttondtheexperimentallymeasuredtemperatureTmeas.Forthisdesign,wecancalculateTcalc(dtest;rtest).Wecanrelateboththemea-suredandcalculatedtemperaturestothetruetemperatureofthetestarticlebythetrueexperimentalandcomputationalerrorsas Ttest;true=T0+Tmeas(dtest;rtest)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ex;true)=T0+Tcalc(dtest;rtest)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ec;true)(C)Rearrangingthisequation,wearriveatthecorrectionfactor=1)]TJ /F7 7.97 Tf 5.07 0 Td[(ec;true 1)]TJ /F7 7.97 Tf 5.07 0 Td[(ex;true.Inthissection,itisshownthatthemassbeforeandafterredesigncanbefoundusingasurrogatethatisafunctionofsafetymarginanddifferencebetweentheallowabletemperatureTallowandinitialtemperatureT0.Asurrogateofoftheprobabilityoffailurethatisafunctionofthesametwovariablesandthecomputationalerroreccanbemadeaswell.AsshowninEq.( 4 ),theinitialdesignsatises T0+Tcalc(d;r)+S1=Tallow(C)RearrangedsothatTcalc(d;r)isonthelefthandside,thisbecomes Tcalc(d;r)=(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0))]TJ /F5 11.955 Tf 10.26 0 Td[(S1(C)ByEq.( 4 )theredesignshouldsatisfy T0+Tcalc(d;r)+S4=Tallow(C)whichrearrangedsothatTcalc(d;r)isonthelefthandsideis Tcalc(d;r)=(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0)=)]TJ /F5 11.955 Tf 10.26 0 Td[(S4=(C) 138

PAGE 139

ByEqs.( C )and( C ),thetwoareequivalentif(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0)=[(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0)=]afterredesignandS1=S4=.Therefore,Tcalc,alongwithitscorrespondingmassandprobabilityoffailure,isafunctionof(Tallow)]TJ /F5 11.955 Tf 9.46 0 Td[(T0)andS,wherethevalueswithandwithoutredesignarerelatedthrough.Thisallowsthemasstobecalculatedsimplyusingsurrogateswiththeinputs(Tallow)]TJ /F5 11.955 Tf 10.46 0 Td[(T0)andS.Asurrogatetoobtaintheprobabilityoffailurecanalsobeobtainedbyincludingthecomputationalerrorecasaninput.NotethatTcalc(d;r)doesnotneedtobecalculatedbecause,foragiven(Tallow)]TJ /F5 11.955 Tf 10.75 0 Td[(T0)andS1,wecanndTcalc(d;r)by (Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0))]TJ /F5 11.955 Tf 10.26 0 Td[(S1=Tcalc(d;r)(C)Whenthecorrectionisapplied,thenweevaluateifredesignisnecessaryby Redesignif:(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0))]TJ /F4 11.955 Tf 10.26 0 Td[([(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0))]TJ /F5 11.955 Tf 10.26 0 Td[(S1]S2or(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0))]TJ /F4 11.955 Tf 10.26 0 Td[([(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0))]TJ /F5 11.955 Tf 10.26 0 Td[(S1]S3(C)whichsimpliesto Redesignif:(Tallow)]TJ /F5 11.955 Tf 10.27 0 Td[(T0)(1)]TJ /F4 11.955 Tf 10.26 0 Td[()+S1S2or(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0)(1)]TJ /F4 11.955 Tf 10.26 0 Td[()+S1S3(C)Krigingsurrogates(quadratictrendfunctionwithaGaussiancorrelationmodel)wereusedforthesurrogatesofthemassandreliabilityindex.TheaccuracyofthesurrogateswasmeasuredbythePRESSRMS,aleave-one-outcrossvalidationerrormeasure,andtheeRMSat50testpoints.AsummaryofthesurrogatesisprovdiedinTable C-1 TableC-1.Summaryofsurrogates SurrogateInputs#ofPointsforFittingPRESSRMS1(%)TesteRMS2(%) (Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0),S,ec40117m(Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(T0),S200.50.1 1PRESSRMS=q 1 peTXVeXV,wherepisthenumberofpointsusedforttingandeXVisthevectorofthedifferencebetweenthetruevalueandthesurrogateprediction 2eRMS=q 1 qeTtestetest,whereqisthenumberoftestpointsandetestisathevectorofthedifferencebetweenthetruevalueandthesurrogateprediction 139

PAGE 140

APPENDIXDEFFECTOFADDITIONALUNCERTAINTIESInCh. 4 weformulatedtheprobabilityoffailureiscalculatedwiththelimitstategas gtrue=Tallow)]TJ /F5 11.955 Tf 10.26 0 Td[(Ttrue(d;r;v0) ( 4 )where Ttrue(d;r;v0)=T0(1)]TJ /F5 11.955 Tf 10.26 0 Td[(v0)+(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ec;true)Tcalc(d;r)( 4 )Giventheuncertaintiesinv0,Tcalc,ec,andTallow,wecancalculatethevarianceofthelimitstateas 2g;current=T202v0+2Tcalc+T2calc2ec+2Tcalce2c+2ec2Tcalc+2Tallow(D)WeusethesubscriptcurrenttodenotethisasthelimitstatethatisusedinCh. 4 .InCh. 3 andin[ 100 ],thelimitstatewasformulatedas gprevious=Tdetallow)]TJ /F5 11.955 Tf 10.26 0 Td[(Tcalc(d;r)(1)]TJ /F5 11.955 Tf 10.26 0 Td[(ec)(D)forwhichthevarianceis gprevious=2Tcalc+T2calc2ec+2Tcalce2c+2ec2Tcalc(D)InCh. 4 ,weincludedtheadditionaluncertaintiesintheinitialtemperature,calculatedchangeandtemperature,andallowabletemperaturetoformamorerealisticproblem.LetusconsidertwocaseswhereredesignthecombinationofthetestandredesignreducesthestandarddeviationofecforthedesignlistedinTable D-1 .ThevaluesoftheuncertainvariablesaregiveninTable 3-1 ,forwhichthevariablesinvolvedinthecalculationofTcalcandTcalcresultinastandarddeviationof12.4Kinthesevalues.UsingEqs.( D )and( D ),wecalculatethestandarddeviationofthelimitstategasshownintTable D-2 .Itwasobservedthattheadditionaluncertainties,particularlytheuncertaintyinTallow,reducedtheeffectofthetestandredesign'sreductionofeconthereductionofthestandarddeviationofthelimitstate.Thereductionsweremorethantwo 140

PAGE 141

TableD-1.Valuesoftheuncertainvariablesinthelimitstates. DistributionBeforeRedesignAfterRedesignCase1AfterRedesignCase2 T0300(deterministic)TcalcN(550,12:42)TcalcN(250,12:42)TallowLN(660,162)ecN(0,0:0692)1N(0,0:06212)2N(0,0:0352)3 1Thisisthestandarddeviationofthenormaldistributionthatisequivalenttotheuniformdistributionofecbetween0:12(i.e.,0:12 p 3). 2Incase1,redesigncausesa10%reductioninstandarddeviationofec. 3Incase2,redesigncausesa50%reductioninstandarddeviationofec. TableD-2.Standarddeviationofthelimitstatesbeforeandafterredesign.Notethatthenominalvalueofecis0. DistributionBeforeRedesignAfterRedesignCase1AfterRedesignCase2(%change)(%change) gprevious39.936.4(-9%)22.7(-43%)gcurrent26.825.7(-4%)22.2(-17%) timeslargerusingthepreviousformulation,whichaccountsforthedifferencesinmeanand95thpercentileoftheprobabilityoffailureweobservedinCh. 4 andtheworkintheCh. 3 141

PAGE 142

APPENDIXEGLOBALVSLOCALSURROGATESInthisappendix,theaccuracyofasingleglobalsurrogateoverthedesignregioniscomparedtoseverallocalsurrogatestinsub-regions.Inthisstudyweexaminevetwo-dimensionalfunctionsandonesix-dimensionalfunction.Fourofthetwo-dimensionalfunctionsweretakenfromastudybyXiongetal.[ 101 ],inwhichsomefunctionswereexaminedbecausetheyhadvisiblenon-stationarybehavior(varyinglevelsofsmoothnessorbumpinessinthespace)suchthatthestationaryassumptionofastationarycovariancestructurethatunderlieskrigingdoesnothold.Thesixfunctionsstudiedarelistedbelow,wherethefunctionsthatcomefromthestudy[ 101 ]arelabeled. 1. Branin-Hoo: f(x)= x2)]TJ /F12 11.955 Tf 12.35 8.09 Td[(5:1 42x21+5 x1)]TJ /F12 11.955 Tf 10.26 0 Td[(6!+10 1)]TJ /F12 11.955 Tf 14.59 8.09 Td[(1 8!cos(x1)+80x1[)]TJ /F12 11.955 Tf 7.6 0 Td[(5;10],x2[0;15]( 6 ) 2. Sasena(mysteryfunctionin[ 101 ]): f(x)=2+0:01(x2)]TJ /F5 11.955 Tf 10.86 0 Td[(x21)2+(1)]TJ /F5 11.955 Tf 10.86 0 Td[(x1)2+2(2)]TJ /F5 11.955 Tf 10.85 0 Td[(x2)2+7sin(0:5x1)sin(0:7x1x2)x1,x2[0;1]( 6 ) 3. Function3[ 101 ]: f(x)=sin(1 x1x2),x1,x2[0:3;1](E) 4. Function4[ 101 ]: f(x)=x1exp()]TJ /F5 11.955 Tf 8.2 0 Td[(x21)]TJ /F5 11.955 Tf 10.85 0 Td[(x22),x1,x2[)]TJ /F12 11.955 Tf 7.61 0 Td[(2:5;2:5](E) 5. Function5[ 101 ]: f(x)=cos(6(x1)]TJ /F12 11.955 Tf 10.26 0 Td[(0:5))+3:1jx1)]TJ /F12 11.955 Tf 10.26 0 Td[(0:7j+2(x1)]TJ /F12 11.955 Tf 10.26 0 Td[(0:5)+7sin(1 jx1)]TJ /F12 11.955 Tf 10.26 0 Td[(0:5j+0:31)+0:5x2x1,x2[0;1](E) 6. Hartman6: f(x)=)]TJ /F7 7.97 Tf 15.51 15.23 Td[(qXi=1aiexp0BBBBBB@)]TJ /F7 7.97 Tf 14.63 14.62 Td[(mXj=1bij(xj)]TJ /F5 11.955 Tf 10.26 0 Td[(dij)21CCCCCCAxi[0;1](E)InthisinstanceofHartman6,q=4anda=h1:01:23:03:2iwhere 142

PAGE 143

B=26666666666666410:03:017:03:51:78:00:0510:017:00:18:014:03:03:51:710:017:08:017:08:00:0510:01:014:0377777777777775D=2666666666666640:13120:16960:55690:01240:82830:58860:23290:41350:83070:37360:10040:99910:23480:14510:35220:28830:30470:30470:40470:88280:87320:57430:10910:0381377777777777775Thetestprocedurecanbedescribedasfollows: 1. GenerateaDOEusingLHS 2. Fitaglobalsurrogate 3. Fitlocalsurrogates:PartitionthedesignspaceintonregionsbychoosingcrandomcentersfromtheDOEandpartitionthespacebasedonthedistanceofapointtothenearestcenter.Fitasurrogateineachregion.Repeatfor10randomsetsofcenters. 4. Calculateerrorat500testpointsandthencalculateeRMSforglobalandlocalsurro-gatesThisprocessisrepeatedfor50DOEsforthesizeof12,22,31,41,and50pointsforthetwo-dimensionalfunctionsand56,80,103,127,and150forHartman6.Duetotherandomnessinchoosingcenters,itispossiblethatsomesub-regionsmaybesmallandholdonlyasmallnumberofpoints.Inordertoavoidill-conditioninginthesecases,thenearestpointsfromneighboringsub-regionsusedtobuildthesurrogate.Forboththequadraticresponsesurfaceandkriging,theminimumnumberofpointsusedtotasurrogatewas12forthetwo-dimensionalproblemsand56forthesix-dimensionalproblem.Figure E-1 showsthetesterrorat500testpoints,measuredbytherootmeansquareoftheerror,andnormalizedbythemeanoftheavailabledata.Overall,itwasobservedthatthekrigingsurrogatewasmoreaccuratethanthequadraticresponsesurface.ItwasobservedthataglobalkrigingsurrogatewasclearlymoreaccuratefortheSasenafunctionandfunction4.Fortheothertestproblems,therewasonlyasmallobservabledistanceinaccuracybetweenglobalandlocalkriging,withglobalkrigingseeminglyslightlymoreaccurate.AtlargeDOEs,thereislessofadifferenceinaccuracybetweenglobalandlocalsurrogates. 143

PAGE 144

FigureE-1.For4equalsizedregions,acomparisonofthetesterrorforglobalandlocalkrigingsurrogates. Fromthisstudy,wecanconcludethatseverallocalsurrogatesgenerallydonotcarryanadvantageoverlocalsurrogates.However,inthisstudy,thesub-regionsdidnotresultfromapartitioningschemethatconsideredfunctionvalue;theyweretheresultsofrandomlyplacedcentersinthedesignspace.Itisnotclearifdividingthedesignspaceintopartitionsthatcapturedlocalbehaviorwouldimprovetheaccuracyofthesurrogate.Additionally,knowinghowtosmartlysplitthedesignspacewouldrequireanaccurateestimationoftheglobalbehavior,whichwouldcomefromanaccurateglobalsurrogate.Intermsofoptimization,accurateglobalsurrogatescouldleadtoaswitchfromoptimizationwithasurrogatetolocaloptimizationwiththetruefunctionininterestingregionsratherthanformingalocalsurrogateattheinterestingregionandcontinuingoptimizationwithasurrogate. 144

PAGE 145

REFERENCES [1] Oberkampf,W.L.,Deland,S.M.,Rutherford,B.M.,Diegert,K.V.,andAlvin,K.F.,ErrorandUncertaintyinModelingandSimulation,ReliabilityEngineeringandSystemSafety,Vol.75,2002,pp.333. [2] Acar,E.,Haftka,R.T.,andKim,N.H.,EffectsofStructuralTestsonAircraftSafety,AIAAJournal,Vol.48,No.10,2010,pp.2235. [3] Melchers,R.E.,StructuralReliabilityAnalysisandPrediction,NewYork:Wiley,1999. [4] Fujimoto,Y.,C.,K.S.,Hamada,K.,andHuang,F.,InspectionPlanningUsingGeneticAlgorithmforFatigueDeterioratingStructures,InternationalOffshoreandPolarEngineeringConference,Vol.4,InternationalSocietyofOffshoreandPolarEngineers,Golden,CO,1998,pp.99. [5] Toyoda-Makino,M.,Cost-basedOptimalHistoryDependentStrategyforRandomFatigueCracksGrowth,ProbabilisticEngineeringMechanics,Vol.14,No.4,Oct.1999,pp.339. [6] Garbatov,Y.andSoares,C.G.,CostandReliabilityBasedStrategiesforFatigueMaintenancePlanningofFloatingStructures,ReliabilityEngineeringandSystemsSafety,Vol.73,No.3,2001,pp.293. [7] Kale,A.,Haftka,R.T.,andSankar,B.V.,EfcientReliability-BasedDesignandInspectionofPanelsAgainstFatigue,JournalofAircraft,Vol.45,No.1,2008,pp.8696. [8] Sankararaman,S.,McLemore,K.,Liang,C.,Bradford,S.C.,andPeterson,L.,Testresourceallocationforuncertaintyquanticationofmulti-levelandcoupledsystems,52ndAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference,Denver,CO,2011. [9] Voutchkov,I.andKeane,A.J.,Multiobjectiveoptimizationusingsurrogates,7thInternationalConferenceonAdaptiveComputinginDesignandManufacture,Bristol,UK,2006,pp.167. [10] Samad,A.,Kim,K.,Goel,T.,Haftka,R.T.,andShyy,W.,Multiplesurrogatemodelingforaxialcompressorbladeshapeoptimization,JournalofPropulsionandPower,Vol.25,No.2,2008,pp.302. [11] Viana,F.A.C.andHaftka,R.T.,Usingmultiplesurrogatesformetamodeling,7thASMO-UK/ISSMOInternationalConferenceonEngineeringDesignOptimization,Bath,UK,2008. [12] Glaz,B.,Goel,T.,Liu,L.,Friedmann,P.,andHaftka,R.T.,Multiple-surrogateapproachtohelicopterrotorbladevibrationreduction,AIAAJournal,Vol.47,No.1,2009,pp.271. 145

PAGE 146

[13] Viana,F.A.C.andHaftka,R.T.,Surrogate-basedoptimizationwithparallelsimula-tionsusingprobabilityofimprovement,13thAIAA/ISSMOMultidisciplinaryAnalysisandOptimizationConference,FortWorth,TX,2010. [14] Modi,P.J.,Shen,W.,Tambe,M.,andYokoo,M.,ADOPT:AsynchronousDistributedConstraintOptimizationwithQualityGuarantees,ArticialIntelligence,Vol.161,No.2,2005,pp.149. [15] Holmgren,J.,Persson,J.A.,andDavidsson,P.,AgentBasedDecompositionofOptimizationProblems,FirstInternationalWorkshoponOptimizationinMulti-AgentSystems,2008. [16] Welcomme,J.-B.,Gleizes,M.-P.,andGlize,P.,Resolutiondeproblemesmultidisci-plinairesmulti-objectifsparsystemesmulti-agentsadaptifs:Applicationalaconceptionpreliminaireavion,JourneesFrancophonessurlesSystemesMulti-Agents(JFSMA2007),Carcassonne,2007,pp.1. [17] Picard,G.andGlize,P.,Modelandanalysisoflocaldecisionbasedoncooperativeself-organizationforproblemsolving,MultiagentandGridSystems,Vol.2,No.3,2006,pp.253. [18] Tu,J.,Choi,K.K.,andPark,Y.H.,ANewStudyonReliabilityBasedDesignOpti-mization,JournalofMechanicalDesign,Vol.121,No.4,1999,pp.557. [19] Y.-T.,W.,ComputationalMethodsforEfcientStructuralReliabilityandReliabilitySensitivityAnalysis,AIAAJournal,Vol.32,No.8,1994,pp.1717. [20] Grandhi,R.V.andWang,L.P.,Reliability-BasedStructuralOptimizationUsingImprovedTwo-PointAdaptiveNonlinearApproximations,FiniteElementAnalysisandDesign,Vol.29,No.1,1998,pp.35. [21] Qu,X.andHaftka,R.T.,Reliability-basedDesignOptimizationUsingProbabilisticSufciencyFactor,StructuralandMutlidisciplinaryOptimization,Vol.27,No.5,2004,pp.314. [22] Liang,J.,Mourelatos,J.P.,andNikolaidis,E.,ASingle-LoopApproachforSystemReliability-BasedDesignOptimization,JournalofMechanicalDesign,Vol.129,No.12,2007,pp.1215. [23] Ba-abbad,M.A.,Nikolaidis,E.,andKapania,R.K.,NewApproachforSystemReliability-BasedDesignOptimization,AIAAJournal,Vol.44,No.5,2006,pp.10871096. [24] Wu,Y.-T.,Shin,Y.,Sues,R.,andCesare,M.,Safety-FactorBasedApproachforProbabilistic-BasedDesignOptimization,42ndAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference,Seattle,WA,2001. 146

PAGE 147

[25] Castillo,E.,Minguez,R.,Teran,A.R.,Fernandez-Canteli,A.,Minguez,andR.Teran,A.R.,Designandsensitivityanalysisusingtheprobability-safety-factormethod.Anapplicationtoretainingwalls,StructuralSafety,Vol.26,No.2,2004,pp.159. [26] Du,X.andChen,W.,SequentialOptimizationandReliabilityAssessmentMethodforEfcientProbabilisticDesign,JournalofMechanicalDesign,Vol.126,No.2,2004,pp.225. [27] Rosenblatt,M.,RemarksonaMultivariateTransformation,TheAnnalsofMathemati-calStatistics,Vol.23,No.3,1952,pp.470. [28] Smarslok,B.P.,Haftka,R.T.,Carraro,L.,andGinsbourger,D.,ImprovingaccuracyoffailureprobabilityestimateswithseparableMonteCarlo,InternationalJournalofReliabilityandSafety,Vol.4,No.4,2010,pp.393. [29] Kale,A.andHaftka,R.T.,TradeoffofWeightandInspectionCostinReliability-BasedStructuralOptimization,JournalofAircraft,Vol.45,No.1,2008,pp.77. [30] Acar,E.,Haftka,R.T.,Kim,N.H.,andBuchi,D.,IncludingtheEffectsofFutureTestsinAircraftStructuralDesign,8thWorldCongressforStructuralandMultidisciplinaryOptimization,Lisbon,Portugal,June2009. [31] Meyers,R.H.andMontgomery,D.C.,Responsesurfacemethodology:processandproductoptimizationusingdesignexperiments,JohnWileyandSons,1995. [32] Meyers,R.H.,Classicalandmodernregressionwithapplications,DuxburyPress,2000. [33] Kleijnen,J.P.C.,Krigingmetamodelingandsimulation:areview,EuropeanJournalofOperationalResearch,Vol.192,No.3,2009,pp.707. [34] Simpson,T.W.,Mauery,T.M.,Korte,J.J.,andMistree,F.,Krigingmodelsforglobalapproximationinsimulation-basedmultidisciplinarydesignoptimization,AIAAJournal,Vol.39,No.12,2001,pp.2233. [35] Stein,M.L.,InterpolationofSpatialData:sometheoryforkriging,SpringerVerlag,1999. [36] Smola,A.J.andScholkopf,B.,Atutorialonsupportvectorregression,Statisticsandcomputing,Vol.14,No.3,2004,pp.199. [37] Park,J.andSandberg,I.W.,Universalapproximationusingradial-basis-functionnetworks,NeuralComputation,Vol.3,No.2,1991,pp.246. [38] Smith,M.,NeuralNetworksforStatisticalModeling,VonNostrandReinhold,1993. [39] Viana,F.A.C.,MultipleSurrogatesforPredictionandOptimization,Ph.D.thesis,UniversityofFlorida,2011. 147

PAGE 148

[40] Jin,R.,Du,X.,andChen,W.,Theuseofmetamodelingtechniquesforoptimizationunderuncertainty,StructuralandMultidisciplinaryOptimization,Vol.25,No.2,2003,pp.99. [41] Queipo,N.V.,Haftka,R.T.,Shyy,W.,andGoel,T.,Surrogate-basedanalysisandoptimization,ProgressinAerospaceSciences,Vol.41,No.1,2005,pp.1. [42] Sacks,J.,Welch,W.J.,J.,M.T.,andWynn,H.P.,Designandanalysisofcomputerexperiments,StatisticalScience,Vol.4,No.4,1989,pp.409. [43] Simpson,T.W.,Peplinski,J.D.,Koch,P.N.,andAllen,J.K.,Metamodelsforcom-puterbasedengineeringdesign:surveyandrecommendations,EngineeringwithComputers,Vol.17,No.2,2001,pp.129. [44] Zerpa,L.E.,Queipo,N.V.,Pintos,S.,andSalager,J.-L.,Anoptimizationmethod-ologyofalkaline-surfactant-polymeroodingprocessesusingeldscalenumericalsimulationandmultiplesurrogates,JournalofPetroleumScienceandEngineering,Vol.47,No.3,2005,pp.197. [45] Goel,T.,Haftka,R.T.,Shyy,W.,andQueipo,N.V.,EnsembleofSurrogates,StructuralandMultidisciplinaryOptimization,Vol.33,No.3,2007,pp.199. [46] Acar,E.andRais-Rohani,M.,Ensembleofmetamodelswithoptimizedweightfactors,StructuralandMultidisciplinaryOptimization,Vol.37,No.3,2010,pp.279294. [47] Acar,E.,Variousapproachesforconstructinganensembleofmetamodelsusinglocalmeasures,StructuralandMultidisciplinaryOptimization,Vol.42,No.6,2010,pp.879. [48] Ginsbourger,D.,LeRiche,R.,andCarraro,L.,ComputationalIntelligenceinExpen-siveOptimizationProblems,chap.Krigingis,SpringerseriesinEvolutionaryLearningandOptimization,2010,pp.131. [49] Queipo,N.V.,Verde,A.,Pintos,S.,andHaftka,R.T.,AssessingthevalueofanothercycleinGaussianprocesssurrogate-basedoptimization,StructuralandMultidisci-plinaryOptimization,Vol.39,No.5,2009,pp.1. [50] Viana,F.A.C.,Haftka,R.T.,andWatson,L.T.,Whynotruntheefcientglobaloptimizationalgorithmwithmultiplesurrogates?51thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference,Orlando,FL,USA,2010. [51] Chaudhuri,A.,Haftka,R.,andViana,F.,EfcientGlobalOptimizationwithAdaptiveTargetforProbabilityofTargetedImprovement,8thAIAAMultidisciplinaryDesignOp-timizationSpecialistConference,AmericanInstituteofAeronauticsandAstronautics,Honolulu,HI,April2012,pp.1. 148

PAGE 149

[52] Kumar,S.,Villanueva,D.,Haftka,R.T.,andSankar,B.V.,ProbabilisticOptimizationofanIntegratedThermalProtectionSystem,12thAIAA/ISSMOMultidisciplinaryAnalysisandOptimizationConference,Victoria,BC,2008. [53] Villanueva,D.,Sharma,A.,Haftka,R.T.,andSankar,B.V.,RiskAllocationbyOptimizationofanIntegratedThermalProtectionSystem,8thWorldCongressforStructuralandMultidisciplinaryOptimization,Lisbon,Portugal,2009. [54] Villanueva,D.,Haftka,R.T.,andSankar,B.V.,IncludingFutureTestsintheDesignandOptimizationofanIntegratedThermalProtectionSystem,12thAIAANon-DeterministicApproachesConference,Orlando,FL,2010. [55] Villanueva,D.,Haftka,R.T.,andSankar,B.V.,ProbabilisticOptimizationofanIntegratedThermalProtectionSystemIncludingtheEffectofFutureTests,2ndInternationalConferenceonEngineeringOptimization,Lisbon,Portugal,2010. [56] Villanueva,D.,LeRiche,R.,Picard,P.,andHaftka,R.T.,AMulti-AgentSystemApproachToReliabilityBasedDesignOptimizationIncludingFutureTests,12eCongrfegsdelaSocifegtfegFrancaisedeRechercheOpfegrationnelleetd'AidefaglaDecision(ROADEF'11),Saint-Etienne,France,2011. [57] Villanueva,D.,LeRiche,R.,Picard,G.,Haftka,R.T.,andSankar,B.V.,Decomposi-tionofSystemLevelReliability-BasedDesignOptimizationtoReducetheNumberofSimulations,ASME2011InternationalDesignEngineeringTechnicalConferencesandComputersandInformationinEngineeringConference,Washington,DC,USA,2011. [58] Freeman,D.C.,Talay,T.A.,andAustin,R.E.,ReusableLaunchVehicleTechnologyProgram,ActaAstronautica,Vol.41,No.11,1997,pp.777. [59] Blosser,M.L.,DevelopmentofAdvanceMetallic,Thermal-Protection-SystemPrototypeHardware,JournalofSpacecraftandRockets,Vol.41,No.2,2004,pp.183. [60] Poteet,C.C.,Abu-Khajeel,H.,andHsu,S.-Y.,Preliminarythermal-mechanicalsizingofametallicthermalprotectionsystem,JournalofSpacecraftandRockets,Vol.41,No.2,2004,pp.173. [61] Bapanapalli,S.K.,DesignofanIntegratedThermalProtectionSystemforFutureSpaceVehicles,Ph.D.thesis,UniversityofFlorida,2007. [62] Gogu,C.,Haftka,R.T.,Bapanapalli,S.K.,andSankar,B.V.,DimensionalityReduc-tionApproachforResponseSurfaceApproximations:ApplicationtoThermalDesign,AIAAJournal,Vol.47,No.7,2009,pp.1700. [63] Sharma,A.,Multi-delitydesignofanintegralthermalprotectionsystemforfuturespacevehicleduringre-entry,Ph.D.thesis,UniversityofFlorida,2010. [64] Viana,F.A.C.,SURROGATESToolboxUser'sGuide,Version2.1,2010. 149

PAGE 150

[65] Golden,P.,Millwater,H.,Dubinsky,C.,andSingh,G.,ExperimentalResourceAllocationforStatisticalSimulationofFrettingFatigueProblem(Preprint),Tech.rep.,DTICDocument,2012. [66] Venter,G.andScotti,S.J.,AccountingforProofTestDatainaReliability-BasedDesignOptimizationFramework,AIAAJournal,Vol.50,No.10,2012,pp.2159. [67] DurgaRao,K.,Kushwaha,H.,Verma,A.,andSrividya,A.,Quanticationofepistemicandaleatoryuncertaintiesinlevel-1probabilisticsafetyassessmentstudies,ReliabilityEngineering&SystemSafety,Vol.92,No.7,July2007,pp.947. [68] Matsumura,T.,Haftka,R.T.,andSankar,B.V.,ReliabilityEstimationIncludingRedesignFollowingFutureTestforanIntegratedThermalProtectionSystem,9thWorldCongressonStructuralandMultidisciplinaryOptimization,Shizuoka,Japan,2011. [69] Birge,J.R.andLouveaux,F.,IntroductiontoStochasticProgramming,SpringerSeriesinOperationsResearchandFinancialEngineering,SpringerNewYork,NewYork,NY,2011. [70] Liu,M.L.andSahinidis,N.V.,OptimizationinProcessPlanningunderUncertainty,Industrial&EngineeringChemistryResearch,Vol.5885,No.95,1996,pp.41544165. [71] Gupta,A.andMaranas,C.D.,ATwo-StageModelingandSolutionFrameworkforMultisiteMidtermPlanningunderDemandUncertainty,Industrial&EngineeringChemistryResearch,Vol.39,No.10,Oct.2000,pp.3799. [72] GuidetoVerifyingSafety-CriticalStructuresforReusableLaunchandReentryVehi-cles,Version1,Tech.Rep.November,FederalAviationAdministration,Washington,DC,USA,2005. [73] Zhao,D.andXue,D.,Amulti-surrogateapproximationmethodformetamodeling,EngineeringwithComputers,Vol.27,No.2,2005,pp.139. [74] Wang,G.G.andSimpson,T.W.,FuzzyClusteringBasedHierarchicalMetamodelingforDesignSpaceReductionandOptimization,EngineeringOptimization,Vol.36,No.3,2004,pp.313. [75] Shoham,Y.andLeyton-Brown,K.,MultiagentSystems:Algorithmic,Game-Theoric,andLogicalFoundations,CambridgeUniversityPress,2009. [76] Beasley,D.,Bull,D.R.,andMartin,R.R.,Asequentialnichetechniqueformultimodalfunctionoptimization,Evolutionarycomputation,Vol.1,No.2,1993,pp.101. [77] Hocaoglu,C.andSanderson,A.C.,Multimodalfunctionoptimizationusingminimalrepresentationsizeclusteringanditsapplicationtoplanningmultipaths,EvolutionaryComputation,Vol.5,No.1,1997,pp.81. 150

PAGE 151

[78] Brits,R.,Engelbrecht,A.P.,andvandenBergh,F.,Locatingmultipleoptimausingparticleswarmoptimization,AppliedMathematicsandComputation,Vol.189,No.2,2007,pp.1859. [79] Parsopoulos,K.E.andVrahatis,M.N.,ArticialNeuralNetworksandGeneticAlgorithms,chap.Modicati,Springer,2001,pp.324. [80] Li,X.,AdaptivelyChoosingNeighbourhoodBestsUsingSpeciesinaParticleSwarmOptimizerforMultimodalFunctionOptimization,GeneticandEvolutionaryComputation(GECCO2004),Vol.3102ofLectureNotesinComputerScience,2004,pp.105. [81] Nagendra,S.,Jestin,D.,Gurdal,Z.,Haftka,R.T.,andWatson,L.T.,Improvedgeneticalgorithmforthedesignofstiffenedcompositepanels,Computers&Structures,Vol.58,No.3,1996,pp.543. [82] Li,J.P.,Balazas,M.E.,Parks,G.,andClarkson,P.J.,ASpeciesConservingGeneticAlgorithmforMultimodalFunctionOptimization,EvolutionaryComputation,Vol.10,No.3,2002,pp.207. [83] Torn,A.andZilinskas,A.,GlobalOptimization,LectureNotesinComputerScience350,SpringerVerlag,1989. [84] Kleijen,J.,Designandanalysisofsimulationexperiments,Springer,2008. [85] Aurenhammer,F.,Voronoidiagrams:asurveyofafundamentalgeometricdatastructure,ACMComputingSurveys(CSUR),Vol.23,No.3,1991,pp.345. [86] Hartigan,J.A.andWong,M.A.,AlgorithmAS136:AK-MeansClusteringAlgorithm,JournaloftheRoyalStatisticalSociety.SeriesC(AppliedStatistics),Vol.28,No.1,1979,pp.100. [87] Rousseeuw,P.J.,Silhouettes:aGraphicalAidtotheInterpretationandValidationofClusterAnalysis,ComputationalandAppliedMathematics,Vol.20,1987,pp.53. [88] MATLAB,version7.9.0.529(R2009b),chap.fmincon,TheMathWorksInc.,Natick,Massachusetts,2009. [89] Jones,D.R.,Schonlau,M.,andWelch,W.J.,EfcientGlobalOptimizationofExpensiveBlack-BoxFunctions,JournalofGlobalOptimization,Vol.13,No.4,1998,pp.455. [90] Parsopoulos,K.E.andVrahatis,M.N.,ModicationoftheParticleSwarmOptimizerforlocatingalltheglobalminima,ArticialNeuralNetworksandGeneticAlgorithms,2001,pp.324. [91] Torn,A.andViitanen,S.,Topographicalglobaloptimizationusingpre-sampledpoints,JournalofGlobalOptimization,Vol.5,No.3,Oct.1994,pp.267. 151

PAGE 152

[92] Alexandrov,N.M.,DennisJr,J.E.,Lewis,R.M.,andTorczon,V.,Atrust-regionframeworkformanagingtheuseofapproximationmodelsinoptimization,StructuralOptimization,Vol.15,No.1,1998,pp.16. [93] Schonlau,M.,Computerexperimentsandglobaloptimization,Ph.D.thesis,UniversityofWaterloo. [94] Janusevskis,J.,LeRiche,R.,Ginsbourger,D.,andGirdziusas,R.,Expectedim-provementsfortheasynchronousparallelglobaloptimizationofexpensivefunctions:Potentialsandchallenges,LearningandIntelligentOptimization,Springer,2012,pp.413. [95] Sasena,M.J.,FlexibilityandEfciencyEnhancementsforConstrainedGlobalDesignOptimizationwithKrigingApproximationsby,Ph.D.thesis,UniversityofMichigan,2002. [96] Zielinski,K.andLaur,R.,Stoppingcriteriaforaconstrainedsingle-objectiveparticleswarmoptimizationalgorithm,INFORMATICA-LJUBLJANA-,Vol.31,No.1,2007,pp.51. [97] Schwefel,H.-P.P.,Evolutionandoptimumseeking:thesixthgeneration,JohnWiley&Sons,Inc.,1993. [98] Jones,D.R.,Perttunen,C.D.,andStuckman,B.E.,LipschitzianoptimizationwithouttheLipschitzconstant,JournalofOptimizationTheoryandApplications,Vol.14,No.3,2004,pp.199. [99] Matsumura,T.,Haftka,R.,andKim,N.,TheContributionofBuildingBlockTesttoDiscoverUnexpectedFailureModes,52thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference,AmericanInstituteofAeronauticsandAstronautics,2011. [100] Villanueva,D.,Haftka,R.T.,andSankar,B.V.,IncludingtheEffectofaFutureTestandRedesigninReliabilityCalculations,AIAAJournal,Vol.49,No.12,2011,pp.2760. [101] Xiong,Y.,Chen,W.,Apley,D.,andDing,X.,Anon-stationarycovariance-basedKrigingmethodformetamodellinginengineeringdesign,InternationalJournalforNumericalMethodsinEngineering,Vol.71,No.6,2007,pp.733. 152

PAGE 153

BIOGRAPHICALSKETCH DianeVillanuevawasborninJacksonville,Floridain1986.ShegraduatedwithaBachelorofScienceinAerospaceEngineeringfromtheUniversityofFloridain2008.Thesameyear,shejoinedtheStructuralandMultidisciplinaryOptimizationGroupattheUniversityofFlorida.Duringthesummerof2009,shewasavisitingresearcheratNASALangleyResearchCenterinHampton,Virginia.In2010,sheenteredintoajointPh.D.programwiththeEcoledesMinesdeSaint-EtienneinFrance.ShewasawardedtheAlumniFellowshipfromtheUniversityofFloridaandtheAmeliaEarhartFellowshipfromZontaInternational.Herresearchinterestsincludeuncertaintyanalysis,non-deterministicmethods,anddistributedoptimizationtechniques. 153