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Deterministic and Reliability Based Optimization of Integrated Thermal Protection System Composite Panel using Adaptive ...

Permanent Link: http://ufdc.ufl.edu/UFE0043944/00001

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Title: Deterministic and Reliability Based Optimization of Integrated Thermal Protection System Composite Panel using Adaptive Sampling Techniques
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Ravishankar, Bharani P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: Conventional space vehicles have thermal protection systems (TPS) that provide protection to an underlying structure that carries the flight loads. In an attempt to save weight, there is interest in an integrated TPS (ITPS) that combines the structural function and the TPS function. This has weight saving potential, but complicates the design of the ITPS that now has both thermal and structural failure modes. The main objectives of this dissertation was to optimally design the ITPS subjected to thermal and mechanical loads through deterministic and reliability based optimization. The optimization of the ITPS structure requires computationally expensive finite element analyses of 3D ITPS (solid) model. To reduce the computational expenses involved in the structural analysis, finite element based homogenization method was employed, homogenizing the 3D ITPS model to a 2D orthotropic plate. However it was found that homogenization was applicable only for panels that are much larger than the characteristic dimensions of the repeating unit cell in the ITPS panel. Hence a single unit cell was used for the optimization process to reduce the computational cost. Deterministic and probabilistic optimization of the ITPS panel required evaluation of failure constraints at various design points. This further demands computationally expensive finite element analyses which was replaced by efficient, low fidelity surrogate models. In an optimization process, it is important to represent the constraints accurately to find the optimum design. Instead of building global surrogate models using large number of designs, the computational resources were directed towards target regions near constraint boundaries for accurate representation of constraints using adaptive sampling strategies. Efficient Global Reliability Analyses (EGRA) facilitates sequentially sampling of design points around the region of interest in the design space. EGRA was applied to the response surface construction of the failure constraints in the deterministic and reliability based optimization of the ITPS panel. It was shown that using adaptive sampling, the number of designs required to find the optimum were reduced drastically, while improving the accuracy. System reliability of ITPS was estimated using Monte Carlo Simulation (MCS) based method. Separable Monte Carlo method was employed that allowed separable sampling of the random variables to predict the probability of failure accurately. The reliability analysis considered uncertainties in the geometry, material properties, loading conditions of the panel and error in finite element modeling. These uncertainties further increased the computational cost of MCS techniques which was also reduced by employing surrogate models. In order to estimate the error in the probability of failure estimate, bootstrapping method was applied. This research work thus demonstrates optimization of the ITPS composite panel with multiple failure modes and large number of uncertainties using adaptive sampling techniques.
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 Bharani P Ravishankar.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Sankar, Bhavani V.
Local: Co-adviser: Haftka, Raphael T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043944:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043944/00001

Material Information

Title: Deterministic and Reliability Based Optimization of Integrated Thermal Protection System Composite Panel using Adaptive Sampling Techniques
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Ravishankar, Bharani P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: Conventional space vehicles have thermal protection systems (TPS) that provide protection to an underlying structure that carries the flight loads. In an attempt to save weight, there is interest in an integrated TPS (ITPS) that combines the structural function and the TPS function. This has weight saving potential, but complicates the design of the ITPS that now has both thermal and structural failure modes. The main objectives of this dissertation was to optimally design the ITPS subjected to thermal and mechanical loads through deterministic and reliability based optimization. The optimization of the ITPS structure requires computationally expensive finite element analyses of 3D ITPS (solid) model. To reduce the computational expenses involved in the structural analysis, finite element based homogenization method was employed, homogenizing the 3D ITPS model to a 2D orthotropic plate. However it was found that homogenization was applicable only for panels that are much larger than the characteristic dimensions of the repeating unit cell in the ITPS panel. Hence a single unit cell was used for the optimization process to reduce the computational cost. Deterministic and probabilistic optimization of the ITPS panel required evaluation of failure constraints at various design points. This further demands computationally expensive finite element analyses which was replaced by efficient, low fidelity surrogate models. In an optimization process, it is important to represent the constraints accurately to find the optimum design. Instead of building global surrogate models using large number of designs, the computational resources were directed towards target regions near constraint boundaries for accurate representation of constraints using adaptive sampling strategies. Efficient Global Reliability Analyses (EGRA) facilitates sequentially sampling of design points around the region of interest in the design space. EGRA was applied to the response surface construction of the failure constraints in the deterministic and reliability based optimization of the ITPS panel. It was shown that using adaptive sampling, the number of designs required to find the optimum were reduced drastically, while improving the accuracy. System reliability of ITPS was estimated using Monte Carlo Simulation (MCS) based method. Separable Monte Carlo method was employed that allowed separable sampling of the random variables to predict the probability of failure accurately. The reliability analysis considered uncertainties in the geometry, material properties, loading conditions of the panel and error in finite element modeling. These uncertainties further increased the computational cost of MCS techniques which was also reduced by employing surrogate models. In order to estimate the error in the probability of failure estimate, bootstrapping method was applied. This research work thus demonstrates optimization of the ITPS composite panel with multiple failure modes and large number of uncertainties using adaptive sampling techniques.
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 Bharani P Ravishankar.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Sankar, Bhavani V.
Local: Co-adviser: Haftka, Raphael T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043944:00001


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DETERMINISTICANDRELIABILITYBASEDOPTIMIZATIONOFINTEGRATEDTHERMALPROTECTIONSYSTEMCOMPOSITEPANELUSINGADAPTIVESAMPLINGTECHNIQUESByBHARANIRAVISHANKARADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012BharaniRavishankar 2

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Tomyparents,RavishankarandParameswariandmysister,Vidhya 3

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ACKNOWLEDGMENTSFirstandforemostIwouldliketothankmyadvisorsDr.BhavaniSankarandDr.RaphaelHaftkaattheDepartmentofMechanicalandAerospaceEngineering.Allofyourinsights,guidance,andpatiencehasbeengreatlyappreciated.Iwouldalsoliketothankthemembersofmyadvisorycommittee,Dr.AshokKumarattheDepartmentofMechanicalandAerospaceEngineeringandDr.GaryConsolazioattheDepartmentofCivilandCoastalEngineering.IwouldlikethankmyfriendsattheCenterforAdvancedCompositeslab,Anurag,Prasanna,MinSong,Marlana,TimandSayan.IwouldalsoliketothankthemembersoftheStructuralandMultidiscplinaryGroup,Ben,Felipe,Diane,AnirbanandTaikiforalltheirhelp,valuableinputsandsuggestions.IwouldalsoliketothankmyfriendsAnirudh,SriramandNarenforallthehelpandsupportthroughoutmyPhDprogram,makingthisexperiencememorable.Iwouldliketothankmyfamilyfortheirsupportandpatience. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 11 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1OutlineofDissertation ............................. 19 1.1.1MotivationandObjectives ....................... 19 1.1.2ThesisOrganization .......................... 20 2BACKGROUNDSTUDYANDLITERATUREREVIEW .............. 21 2.1OrbiterThermalProtectionSystem-30yearsLegacy ............ 21 2.1.1TileTPS ................................. 23 2.1.2Non-TileTPS ............................. 23 2.2FiniteElementbasedHomogenization .................... 24 2.3DeterministicandReliabilitybasedoptimization ............... 27 2.3.1DeterministicOptimization ....................... 28 2.3.2StructuralReliability .......................... 29 2.4UseofSurrogateModels ........................... 31 2.4.1ApplicationofSurrogatestoImproveConstraintBoundaries .... 32 2.5AdaptiveSampling ............................... 33 2.5.1EfcientGlobalReliabilityAnalysis .................. 34 2.6UncertaintyModeling ............................. 35 2.7EstimationofProbabilityofFailure ...................... 36 2.7.1MomentbasedMethods ........................ 37 2.7.2SamplingMethods ........................... 38 3DESIGNANDANALYSISOFINTEGRATEDTHERMALPROTECTIONSYSTEMPANEL ........................................ 41 3.1DesignofITPSCompositePanel ....................... 41 3.1.1KeyDimensionsoftheITPSstructure ................ 41 3.1.2ITPSMaterials ............................. 42 3.1.3BoundaryConditions .......................... 43 3.1.4Loads .................................. 44 3.2FiniteElementAnalyses ............................ 44 3.2.1TransientHeatTransferAnalysis ................... 45 3.2.2StressAnalysis ............................. 47 5

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3.2.3BucklingAnalysis ............................ 48 3.3Summary .................................... 50 4FINITEELEMENTBASEDHOMOGENIZATION ................. 51 4.1A,B,Dmatrices ................................. 51 4.2TransverseShearStiffness .......................... 53 4.3AccuracyoftheHomogenizationMethod .................. 57 4.3.1DeectionComparison ......................... 57 4.3.2StressComparison ........................... 57 4.4ResultsandDiscussion ............................ 58 4.4.1A,B,Dmatrices-Equivalentmaterialvs.Equivalentstructure ... 58 4.4.2TransverseShearStiffnessofITPS .................. 60 4.4.3AccuracyoftheHomogenizationMethod ............... 63 4.5Summary .................................... 66 5MONTECARLOSIMULATIONS-RELIABILITYESTIMATION ......... 67 5.1CrudeMonteCarloMethod(CMC) ...................... 67 5.2SeparableMonteCarlomethod(SMC) .................... 68 5.3ErrorintheProbabilityofFailureEstimate .................. 69 5.4SMCwithregroupingandseparablesamplingofthelimitstaterandomvariables .................................... 71 5.5ApplicationtoFailureAnalysisofCompositeLaminate ........... 72 5.6ResultsandDiscussion ............................ 75 5.6.1CrudeandSeparableMonteCarloMethod .............. 75 5.6.2Regroupingandseparablesamplingofthelimitstatevariablesforimprovingaccuracy ........................... 78 5.6.3Summary ................................ 80 6EFFICIENTGLOBALRELIABILITYANALYSIS(EGRA) ............. 82 6.1EGRAalgorithm ................................ 82 6.2IllustrationofEGRA .............................. 83 6.3Summary .................................... 88 7DETERMINISTICANDRELIABILITYBASEDOPTIMIZATION ......... 90 7.1DeterministicOptimization ........................... 90 7.2UncertaintyModeling ............................. 93 7.3EstimationofSystemReliability ........................ 94 7.4ReliabilityBasedOptimization ......................... 96 7.5ResultsandDiscussion ............................ 98 7.5.1DeterministicOptimization ....................... 98 7.5.2SensitivityAnalysis ........................... 104 7.5.3ReliabilityoftheDeterministicOptimum ............... 107 7.5.4Reliabilitybasedoptimization ..................... 110 6

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7.5.5Errorintheprobabilityoffailureestimate ............... 113 7.5.6Summary ................................ 115 8CONCLUSIONS ................................... 116 8.1Conclusion ................................... 116 8.2FutureWork ................................... 117 ATHERMALPROPERTIESOFITPSCOMPONENTS ............... 118 BSTRENGTHPROPERTIESOFITPSMATERIALS ................ 120 CDESIGNOFEXPERIMENTSADDEDBYEGRASAMPLINGTECHNIQUE .. 121 REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 130 7

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LISTOFTABLES Table page 3-1KeyDimensionsoftheITPSpanel ......................... 41 3-2ComponentsofITPS,materialsandberorientation ............... 43 3-3MaterialpropertiesofthecomponentsofITPS .................. 43 3-4Heatuxloadstepsinthetransientheattransferanalysis ............ 46 4-1Periodicboundaryconditionsforthesixdeformations .............. 52 4-2VariationofA44andA55ofITPSpanelwithn ................... 61 4-3Tipdeectionratioalongwithcontributionofbendingandsheardeformationtowardsdeection .................................. 64 5-1Materialpropertiesanduncertaintyoftherandomvariables ........... 74 5-2EmpiricalandbootstrappingestimatesofprobabilityoffailureusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitions ....... 76 5-3StandarddeviationandcoefcientofvariationofempiricalandbootstrappingpfestimatesusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitionsfororiginallimitstate ...................... 77 5-4Relativecontributionsofresponse(stresses)andcapacity(strengths)towardstheuncertaintyinpfthroughbootstrappingandalsocomparedwithempiricalresults ......................................... 77 5-5StandarddeviationandcoefcientofvariationofCMCandSMCforincreasingsamplesizeofMandN=500 ............................ 79 5-6StandarddeviationandcoefcientofvariationofCMC,SMCandSMCregroupedforincreasingsamplesizeofMandN=500 .................... 79 7-1Failure,safetyfactorsappliedonconstraints .................... 91 7-2Lowerandupperboundsofdesignvariablesfordeterministicoptimization ... 92 7-3CoefcientofvariationofinputrandomvariablesincludedintheITPSdesign 93 7-4Nominalallowablevaluesofcapacityandcoefcientofvariation ........ 94 7-5OptimumdesignvariablesandminimizedstructuralmassthroughglobalandEGRADOEfordeterministicoptimization ..................... 101 7-6ComparisonbetweentheaccuracyofglobalsurrogateandEGRAsurrogateatthedeterministicoptimumusingcorrespondingsurrogates .......... 103 8

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7-7Comparisonofresponsesatallthedeterministicoptimausingthe60DOEsurrogate ....................................... 103 7-8Comparisonofresponsesatallthedeterministicoptimausingthe20DOEsurrogate ....................................... 104 7-9ComparisonofresponsesatallthedeterministicoptimausingtheEGRAupdatedsurrogate ....................................... 104 7-10Differenceinuncertaintyinresponsesbeforeandaftersensitivityanalysis ... 106 7-11Uncertaintyinresponseduetoinputuncertainty ................. 108 7-12Individualprobabilitiesoffailureofthedeterministicoptimum .......... 109 7-13Lowerandupperboundsofdesignvariablesforprobabilisticoptimization ... 110 7-14OptimumdesignvariablesandminimizedstructuralmassthroughEGRAupdatedDOEforreliabilitybasedoptimization ........................ 112 7-15Individualprobabilitiesoffailureoftheprobabilisticoptimum ........... 112 7-16Individualprobabilitiesoffailureofthedeterministicoptimumusingprobabilisticoptimumsurrogate .................................. 112 7-17Errorintheprobabilitiesoffailureofthedeterministicoptimumusingbootstrappingandrepetitions .................................... 114 7-18Errorintheprobabilitiesoffailureoftheprobabilisticoptimumusingbootstrappingandrepetitions .................................... 114 A-1DensityoftheITPSComponents .......................... 118 A-2ThermalPropertiesofthebottomfacesheet-GraphiteEpoxy ......... 118 A-3ThermalPropertiesofthetopfacesheetandWrap-SiC/SiC .......... 118 A-4ThermalConductivityoftheinsulationfoam-AETB ............... 119 A-5Specicheatoftheinsulationfoam-AETB .................... 119 B-1DensityoftheITPSComponents .......................... 120 C-1DesignpointsacquiredusingEGRAtoimprovetheaccuracyofTemperatureboundary ....................................... 121 C-2DesignpointsacquiredusingEGRAtoimprovetheaccuracyofstressboundary 121 C-3DesignpointsacquiredusingEGRAtoimprovetheaccuracyofbucklingloadboundary ....................................... 121 9

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C-4DesignpointsacquiredusingEGRAtoimprovetheaccuracyofreliabilityindexboundary ....................................... 122 10

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LISTOFFIGURES Figure page 2-1ThermalProtectionSystem(TPS)usedinspaceshuttles ............ 22 2-2Sinusoidalcoresandwichedbetweenlaminates .................. 26 2-3IntegratedThermalProtectionSystemwithcorrugatedcore ........... 26 3-1SchematicviewoftheITPSpanel ......................... 42 3-2DimensionsoftheITPSpanel ............................ 42 3-3BoundaryconditionsappliedontheITPSpanel .................. 44 3-4HeatuxproleincidentontheITPSpanel .................... 45 3-5Boundaryconditionsforthetransientheattransferanalysis ........... 46 3-6TemperaturevariationonTFSandBFSwithrespecttoreentrytime ...... 47 3-7BuckledITPSduetothermalloads ........................ 49 4-1SchematicviewoftheITPSunitcell(RVE) .................... 52 4-2Periodicboundaryconditionappliedontheunitcellforx=1 .......... 54 4-3Schematicviewoftheunitcelldeformationsduetostrainsandcurvatures ... 54 4-41-DITPSmodelwithunitcellsinx-direction ................... 55 4-51DmodeloftheITPSwithunitcellsinx-directionsubjectedtopressureloadandxedBCs ..................................... 58 4-6TransverseshearstiffnessA55oforthotropicpanel ................ 61 4-7VariationofA44andA55ofITPSpanelwithn ................... 62 4-8ConvergedA44andA55ofITPSpanelwithn ................... 62 4-9Ratiobetweentipdeectionofthehomogenizedmodelandthe3-DITPSmodelalongthexandy-direction ............................. 63 4-10Stress(11,22)ratiobetweenthehomogenizedmodelandthe3DITPSmodel 65 4-11Stress(11,22)atn=10comparisonbetweenthe3-DITPSmodelandthehomogenizedmodel ................................. 65 4-12Verticaldeectionalongx-axisoftheoriginalITPSandhomogenizedmodelduetopressureload ................................. 66 5-1IllustrationofcrudeandseparableMonteCarloMethodcomparisons ...... 69 11

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5-2Schematicrepresentationofbootstrappingwhenonlyresponseissampled. .. 70 5-3Illustrationofseparablesamplingwithunitloads ................. 72 5-4Compositepressurevesselwithinternalpressureof100kPaandstressesactinginasmallelementofthevessel. ....................... 73 5-5Distributionofstressandstrengthinthe2-direction(2)showingtheprobablefailureregion ..................................... 75 5-6StandardDeviationofCMC,SMCandregroupedlimitstateSMCwhereN=500(xed)andMisvaryingfor10,000repetitions ................... 80 6-1TruecontoursoftheBraninhoofunction ...................... 83 6-2TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction 84 6-3ExpectedFeasibilityFunctionevaluatedusingtheinitialkrigingapproximation 85 6-4Krigingmodelupdatedwiththenewdesignandexpectedfeasibilityfunction .. 86 6-5TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction 86 6-6TargetcontouraccuratelyapproximatedbytheEGRAmethodology ...... 87 7-1VariationofresponseswrtthicknessofthewraptW ............... 99 7-2VariationofcriticalresponseswrtthicknessofthebottomfacesheettB .... 100 7-3VariationofcriticalresponseswrtthicknessofthetopfacesheettT ....... 101 7-4Variationofcriticalresponseswrtheightofthefoamh .............. 102 7-5SensitivityanalysisofmaximumBFStemperature ................ 106 7-6SensitivityanalysisofWrapstress11 ....................... 107 7-7Sensitivityanalysisofbucklingloadfactor ..................... 108 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDETERMINISTICANDRELIABILITYBASEDOPTIMIZATIONOFINTEGRATEDTHERMALPROTECTIONSYSTEMCOMPOSITEPANELUSINGADAPTIVESAMPLINGTECHNIQUESByBharaniRavishankarMay2012Chair:BhavaniV.SankarCochair:RaphaelT.HaftkaMajor:MechanicalEngineering Conventionalspacevehicleshavethermalprotectionsystems(TPS)thatprovideprotectiontoanunderlyingstructurethatcarriestheightloads.Inanattempttosaveweight,thereisinterestinanintegratedTPS(ITPS)thatcombinesthestructuralfunctionandtheTPSfunction.Thishasweightsavingpotential,butcomplicatesthedesignoftheITPSthatnowhasboththermalandstructuralfailuremodes.ThemainobjectivesofthisdissertationwastooptimallydesigntheITPSsubjectedtothermalandmechanicalloadsthroughdeterministicandreliabilitybasedoptimization. TheoptimizationoftheITPSstructurerequirescomputationallyexpensiveniteelementanalysesof3DITPS(solid)model.Toreducethecomputationalexpensesinvolvedinthestructuralanalysis,niteelementbasedhomogenizationmethodwasemployed,homogenizingthe3DITPSmodeltoa2Dorthotropicplate.HoweveritwasfoundthathomogenizationwasapplicableonlyforpanelsthataremuchlargerthanthecharacteristicdimensionsoftherepeatingunitcellintheITPSpanel.Henceasingleunitcellwasusedfortheoptimizationprocesstoreducethecomputationalcost. DeterministicandprobabilisticoptimizationoftheITPSpanelrequiredevaluationoffailureconstraintsatvariousdesignpoints.Thisfurtherdemandscomputationallyexpensiveniteelementanalyseswhichwasreplacedbyefcient,lowdelitysurrogate 13

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models.Inanoptimizationprocess,itisimportanttorepresenttheconstraintsaccuratelytondtheoptimumdesign.Insteadofbuildingglobalsurrogatemodelsusinglargenumberofdesigns,thecomputationalresourcesweredirectedtowardstargetregionsnearconstraintboundariesforaccuraterepresentationofconstraintsusingadaptivesamplingstrategies. EfcientGlobalReliabilityAnalyses(EGRA)facilitatessequentiallysamplingofdesignpointsaroundtheregionofinterestinthedesignspace.EGRAwasappliedtotheresponsesurfaceconstructionofthefailureconstraintsinthedeterministicandreliabilitybasedoptimizationoftheITPSpanel.Itwasshownthatusingadaptivesampling,thenumberofdesignsrequiredtondtheoptimumwerereduceddrastically,whileimprovingtheaccuracy. SystemreliabilityofITPSwasestimatedusingMonteCarloSimulation(MCS)basedmethod.SeparableMonteCarlomethodwasemployedthatallowedseparablesamplingoftherandomvariablestopredicttheprobabilityoffailureaccurately.Thereliabilityanalysisconsidereduncertaintiesinthegeometry,materialproperties,loadingconditionsofthepanelanderrorinniteelementmodeling.TheseuncertaintiesfurtherincreasedthecomputationalcostofMCStechniqueswhichwasalsoreducedbyemployingsurrogatemodels.Inordertoestimatetheerrorintheprobabilityoffailureestimate,bootstrappingmethodwasapplied. ThisresearchworkthusdemonstratesoptimizationoftheITPScompositepanelwithmultiplefailuremodesandlargenumberofuncertaintiesusingadaptivesamplingtechniques. 14

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CHAPTER1INTRODUCTION Whenaspacevehiclereenterstheatmosphereathypersonicspeeds,thevehicle'sexteriorissubjectedtosevereaerodynamicheatingandpressure.Toprotectthevehiclefromsuchextremetemperatures,aThermalProtectionSystem(TPS),isaddedontothemainloadbearingstructure.But,theTPSisincompatiblewiththemainstructureduetothemismatchintheirthermalproperties.Thisincompatibility,alongwiththeactionofnumerousotherloadslikeaerodynamicpressure,impactloadsandotherin-planeinertialloadsmayleadtothestructure'sfailurewithcatastrophicconsequences.Moreover,theTPSencompassesasignicantportionofthevehicle'sexteriorconstitutingmajorpartofthelaunchweight.Thus,itisimperativethatapartfrommakingtheTPSsuitableforprotectionpurposes,itshouldbelightweightinordertoincreasepayloadsthatcanbecarriedaboard. Toaccommodatetheserequirements,amultifunctionalIntegratedThermalProtectionSystem(ITPS)couldbedesigned,inwhichtheloadbearingstructureandtheTPSareintegratedintoacompositesandwichpanel.ThegreatestchallengeinsuchadesignwouldbeincombiningthevariousconictingrequirementsoftheTPSandtheloadbearingmemberintoasinglestructure.Thestructuralrequirementsfavorusingmetals,butthiscangreatlyincreasetheconductionofheatthroughtheITPS.Ontheotherhandmostinsulationmaterialssuchasceramicsandfoamshavelowstrength.Materialselectionisimportantaswellasdimensioningthestructureproperly,transformingthisdesignintoamultidisciplinaryoptimizationproblem.Evenassumingthatthematerialpropertiesandtheirexactbehaviorareknownaccurately,thisproblemiscomputationallyintensiveinvolvinglargenumberofniteelementanalyses.IfcompositesareusedforITPS,itfurtherincreasesthenumberofuncertaintiesintheproblem.Whentheseuncertaintiesareincludedinthedesign,thecomputationalexpensebecomesprohibitivelylarge.Thiscouldbesolvedusingtraditionaldeterministic 15

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methods,whicharecomputationallyfeasiblebutlessaccurate.Thisproblemcouldbedenitivelyaddressedthroughprobabilisticoptimization,whichisaccuratebutcomputationallyintractable. TheoverallobjectiveofthisdissertationistoemployefcientandcosteffectivemethodstooptimizetheITPSpanel.Varioustechniqueshavebeenappliedalongthedifferentphasesofthisdissertationtoincreasethecomputationalefciencyandreducethecostandtimeinvolved.ToreducethecomputationalcostofstructuralanalysisofITPS,niteelementbasedhomogenizationisinvestigated.Homogenizationisaprocessofapproximatingthebehaviorofheterogeneousstructuresashomogeneousbydeterminingtheirequivalentstiffnessproperties.Pastresearcheffortshaveshownthathomogenizationofthecompositestructureasanequivalenttwo-dimensionalorthotropicplatewouldreducethecomputationalcostassociatedwithstructuralanalysistremendously[ 1 2 3 4 5 ].Sharmaetal.[ 6 7 ]appliedthehomogenizationtoanITPSpanelwithcorrugatedcore.Owingtonegligiblefoamdensity,thepanelwasmodeledusingshellelements.TheITPSpanelconsideredinthisdissertationhasdenserinsulationfoamrequiring3-Dbrickelements.Thisresearchthusaimstoextendthehomogenizationtechniqueto3-Dsandwichpanels. Inhomogenization,theequivalentstiffnesspropertiesofacompositesandwichpanelarethein-planestiffness[A],couplingstiffness[B],bendingstiffness[D]whicharecombinedandrepresentedasthe[ABD]stiffnessmatrixandtransverseshearstiffness(A44andA55).Theequivalentstiffnesspropertiesweredeterminedbytakingadvantageoftherepeatingstructureofthepanel.Therepetitiveblockofthepanelisreferredasunitcellorrepresentativevolumeelement(RVE).The[ABD]stiffnessmatrixwasdeterminedbysubjectingtheunitcelltounitstrainsandcurvatures.Thetransverseshearstiffnesswasdeterminedbyanalyzingaone-dimensionalplateoftheITPSmodelunderendloads.However,itwasfoundthathomogenizationofthecompositesandwichpanelwasaccurateonlywhenthepaneldimensionsaremuch 16

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largerthanthecharacteristicdimensionsoftheunitcell.HenceoneunitcellofITPSpanelisconsideredandanalyzedduringtheoptimizationprocess. Thesecondobjectiveofthisresearchistoperformdesignoptimizationthroughdeterministicandreliabilitybasedapproaches.OneofthemainpurposesofoptimizationistominimizetheweightoftheITPSwhilesatisfyingseveralconictingdesignconstraints.Designoptimizationthroughatraditional,safetyfactorapproachwithoutconsideringtheuncertaintiesinvolvedwouldprovidelessaccurateandconservativedesigns.Thecompositepanelunderconsiderationcontributestothemajorpartoftheweightontheexteriorofthespacevehicle.ThusitisimperativetoreducethestructuralweightofITPSwithoutresultinginconservativeandunsafedesigns.Hence,itisdesiredtooptimizethedesignoftheITPSusingreliabilitybasedapproachbyincludingalltheuncertaintiesinthedesign.Howeveritisdesiredtoperformaninitialdeterministicoptimizationasitwouldhelpinnarrowingtotheappropriatedesignspacetoperformreliabilitybasedoptimization. Likemostengineeringstructures,theITPScouldundergomultiplefailuremodes:temperature,stressesandbucklingloads.Fordeterministicoptimization,thedesignvariablesareoptimizedtominimizethestructuralmass.Theoptimizationprocessinvolvesevaluationoftheaforementionedfailureinducersatlargenumberofdesignpointswhichincreasesthecomputationalcost.Constructionofsurrogatemodels(interchangeablyreferredasresponsesurfaceapproximation)oftheconstraintsatlimiteddesignpointsserveasanefcientalternativetohighdelityniteelementanalyses.Reliabilitybasedoptimizationhasanadditionalreliabilityconstraintthatisafunctionofdesignvariablesandrandomvariables.ThedesignvariablesaredimensionsoftheITPSpanelandtherandomvariablesconsideredarematerialproperties,geometry,thermalandmechanicalloads.Materialpropertiesarecomplextomodelastheydependontemperature.Geometryisanothersourceofuncertaintyaffectingboththestructuralbehaviorandthermalloading.Henceitisimportantto 17

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analyzethefailureinducerssuchstresses,temperatureandbucklingduetotheabovementioneduncertainties.Withlargenumberofuncertaintiesinvolved,evenconstructionofqualitysurrogatemodelsrequiresniteelementanalyzesatextremelylargenumberofrandominputs. Failureconstraintsthatseparatethefeasibledesignsfromtheinfeasibleoneshavetorepresentedaccuratelyinordertodetermineanaccurateoptimum.Commonlytheconstraintsareevaluatedusingglobalsurrogatemodelsusingspacellingdesignofexperiments.Thiswouldresultinwastingalargenumberofhighdelityniteelementanalyses.Earlierresearcheffortshaveintroducedsamplingmethodstoimproveaccuracyofthetargetregionswithlessnumberofdesigns[ 8 9 10 11 12 13 14 ].ThesamplingmethoddevelopedbyBichonetal.[ 14 ],EfcientGlobalReliabilityAnalyses(EGRA)sequentiallysamplesdesignpointsinthevicinityofthetargetregionusingagaussianprocessresponsesurfaceconstructedinitiallywithfewerdesignpoints.Thiscoulddrasticallyreducethenumberofcomputationallyexpensiveniteelementsimulationswhileresultinginaccurateoptimumdesign.TheITPSoptimizationprobleminvolvinghighdimensionaldesignspaceseekspolynomialresponsesurface.ThisresearchworkextendsthecurrentEGRAmethodologytoadapttothecurrentITPSproblembyincorporatingpolynomialsurrogatesinthealgorithm. Thegoalofreliabilityanalysisistodeterminetheprobabilitythatasystemwillfailinservice,giventhatitsbehaviorisdependentonrandominputs.Tocalculatethesystemreliability(probabilityoffailure),MonteCarlosimulation(MCS),acommonlyusedmethodformultiplefailuremodesisemployed.Toefcientlyhandlenumerousuncertaintiesandmultiplefailuremodes,SeparableMonteCarlo(SMC)developedbySmarsloketal.[ 15 ]wasadopted.Whentheuncertainrandomvariablesarestatisticallyindependent,theconceptofSMCistogrouptheuncertaintiesinvolvedintheproblemandsamplethemappropriatelytohaveanaccuratefailurepredictionforagivencomputationalbudget.PreviousworkwithSMConlyexploredsimplelimitstates 18

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expressedasadifferencebetweenarandomresponseandarandomcapacity[ 16 ].Thisresearchworklooksatamoregenerallimitstatefunctionthatcombinessetsofrandomresponseandcapacitycomponents[ 17 ].Furthertheerrorintheprobabilityoffailureestimateforthesimplelimitstatewasderivedintermsofthenumberofsamplesoftheresponseandcapacity.Thisisnotapplicableforthegeneralizedlimitstate.Thisresearchaimsatestimatingtheerrorintheprobabilityoffailureforseparablecaseusingthebootstrappingmethod[ 18 ],aresamplingtechnique,whichinvolvestakingthesamplesofresponse(expensive)andresamplingthemwithreplacement[ 19 ]. 1.1OutlineofDissertation 1.1.1MotivationandObjectives Tosummarizetheabovediscussion,themainobjectivesofthisresearcharegivenbelow. OptimizationofITPSiscomputationallyexpensive,becauseof Structuralanalysisofhighdelity3Dniteelementmodel Probabilisticanalysisincludingalluncertainties-aleatory(materialproperties,geometryandloadingconditions)andepistemicuncertainties(errorinmodelingandsimulation). MultiplefailuremodesofITPSdemandsMonteCarlosimulationsforreliabilitycalculations. TosuccessfullyoptimizetheITPSstructurethroughdeterministicandreliabilitybasedoptimization,thisdissertationproposes TodevelopareducedbutefcientmodelofITPStolowerthecomputationcostinvolvedintheniteelementanalyses. Toinvestigatelowdelitysurrogatemodels(ResponseSurfaceApproximationoftheconstraints)toreplacehighdelityniteelementanalyses. ToadaptEfcientGlobalReliabilityAnalysisthatfacilitatessequentiallysamplingaroundtheconstraintsrequiringfewernumberofsamplestoimprovetargetregionandpredictaccurateoptimum. 19

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ToutilizeseparableMonteCarlomethodthatallowsseparablesamplingoftheexpensiveandinexpensivevariableswhichwillreducecomputationalcostandalsoprovideimprovedaccuracy. Thisdissertationessentiallydemonstratesthemethodologyfordeterministicandreliabilitybasedoptimizationofcomplexstructureswithlargenumberofuncertaintiesusingsurrogatemodels,EGRA,separableMonteCarlotechniqueandbootstrappingmethod. 1.1.2ThesisOrganization Chapter 2 reviewsthethermalprotectionsystemusedinspacevehiclessuchasChallenger,DiscoverandColumbia.Thischapterpresentsareviewontheniteelementbasedhomogenizationtechnique.Further,presentsareviewonstructuraloptimizationandthedifferentaspectsofdeterministicandreliabilitybasedoptimization.Chapter 3 presentsdetaileddescriptiononthegeometry,components,loadsandboundaryconditionsofthemulti-functionalIntegratedThermalProtectionSystemfollowedbyadiscussiononniteelementanalysesofITPSinvolvedinthedesignandtheoptimizationprocess.Chapter 4 presenttheniteelementbasedhomogenizationmethodandresultsobtainedfromthemethod.Chapter 5 presentsadetaileddiscussiononMonteCarlosimulationtechniquestoestimateprobabilityoffailurewhichincludescrudeMonteCarloandseparableMonteCarlomethod.bootstrappingmethodtoestimateerrorintheprobabilityoffailureapplyingittoapressurevesselproblem.Chapter 6 illustratestheworkingofEfcientGlobalReliabilityAnalysis(EGRA)usingatwovariableproblem.EGRAappliedtotheoptimizationoftheITPSpanelisdiscussedinChapter 7 .ThischapterpresentsthedeterministicandreliabilitybasedoptimizationoftheITPSpanelandotherimportantcomponents(uncertaintymodelingandpropagation,sensitivityanalyses)associatedwiththeoptimizationprocess.ThedissertationisconcludedwithsummaryofresultsandpossiblefutureworkinChapter 8 20

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CHAPTER2BACKGROUNDSTUDYANDLITERATUREREVIEW Thermalprotectionsystemshavebeenanareaofcontinuousresearchanddevelopmentinspacevehicledesign.Withatmosphericheatingduringreentrybeingthemajorcauseofconcernforaspacevehicle'ssafeoperation,severaldesignandanalysistechniqueshavebeenputforthbyresearches.ThischapterreviewsthedifferenttypesofThermalProtectionsystemusedsofarinspacevehicles.FurtheritalsointroducesandreviewsthevarioustechniquesthatareaimedatreducingthecomputationalcostassociatedwiththedeterministicandreliabilitybasedoptimizationofthestructurallyintegratedThermalProtectionSystem. 2.1OrbiterThermalProtectionSystem-30yearsLegacy Whenaspaceshuttleorbiterre-enterstheEarth'satmosphere,itistravelinginexcessof17,000mph(7600m/s).Whileslowingdowntolandingspeed,frictionwiththeatmosphereproducesexternalsurfacetemperaturesashighas1900K.Specialthermalshieldsontheexteriorsurfaceprotectthevehicleanditsoccupantsduringlaunchandreentry[ 20 21 22 ].Earliermannedspacecraft,suchasMercury,GeminiandApollo,wereprotectedduringre-entrybyaheatshieldconstructedofphenolicepoxyresinsinanickel-alloyhoneycombmatrix.Theheatshieldwascapableofwithstandingveryhighheatingrates.Duringthereentry,theheatshieldwouldablate,orcontrollablyburnwiththecharlayerprotectingthelayersbelow.Despitetheadvantages,ablativeheatshieldshadsomemajordrawbacks.Theywerebondeddirectlytothevehicle,theywereheavy,andtheywerenotreusable.Withadesignlifeof100missions,spaceshuttleorbiterrequiredalightweightreusablethermalprotectionsystem(TPS)notonlytoprotecttheorbiterfromthesearingheatofreentry,butalsotoprotecttheairframeandmajorsystems[ 23 24 ]. Thevehicle'scongurationandentrytrajectorydenesthetemperaturedistributiononthevehicle.Overthepast30years,theSpaceShuttle'sThermalProtectionsystem 21

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hasbeenbuiltwithmaterialswithahightemperaturecapabilityandunderlyingthermalinsulationtoinhibittheconductionofheattotheinteriorofthevehicle.Duringtheentiremission,differentlocationsontheorbitergetheatedtodifferenttemperatures.Theleadingedgesofwingsandthenosecaparethehighesttemperatureregions.DuetothewidevariationofthesetemperaturestheTPSselectedforspaceshuttlewascomposedofmanydifferentmaterials.Eachmaterial'stemperaturecapability,durabilityandweightdeterminedtheextentofitsapplicationonthevehicle.TherearebasicallytwocategoriesofTPSbeingused,thetileTPSandnon-tileTPS.AdetaileddescriptionofeachcategorywouldprovideabetterunderstandingontheTPSselectionfordifferentregionsonthevehicle[ 25 26 ]. Figure2-1.ThermalProtectionSystem(TPS)usedinspaceshuttles[ 26 ] 22

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2.1.1TileTPS ThedifferenttypesoftileTPSare High-temperaturereusablesurfaceinsulation(HRSI)tilesweredevelopedtoprovideprotectionagainsttemperaturesupto1533K.Theywereusedinareasontheupperforwardfuselage,verticalstabilizerleadingedge,andupperbodyapsurface.Thetileiscomposedofhighpuritysilicabers.Ablackcoating,reactioncuredglass(RCG)wasappliedtoallbutonesideofthetiletoprotecttheporoussilicaandtoincreasetheheatsinkproperties.HRSIwasprimarilydesignedtowithstandtransitionfromareasofextremelylowtemperature,about-270Ctothehightemperaturesofre-entrytypicallyaround1600Cthusmaximizingheatrejectionduringthehotphaseofreentry. Fibrousrefractorycompositeinsulationtiles(FRCI)weresimilarformoftheHRSItiles.TheFRCItileshadhigherstrengthderivedbyaddingalumina-borosilicateber,calledNextel,tothepuresilicatileslurry.Thoughdevelopedforthesamepurpose,FRCIandHRSIhaddifferentphysicalpropertiesbecauseof20%Nextelinit.FRCItileswerelighterthanthebasicHRSItiles.Furthermore,theFRCItilesalsohadtensilestrengththatwasatleastthreetimesgreaterthanthatoftheHRSItilesandcouldbeusedatatemperaturealmost100CofhigherthanthatofHRSItiles.Inanutshell,theyprovidedabetterstrength,durability,crackingresistance,andweightreduction. Tougheneduni-piecebrousinsulation(TUFI)isanimprovedlowdensityrigidceramiccomposite,withveryhighimpactresistance(20-100timesmorethanRCGcoating).TUFIwasusedinregionswheretemperaturesreachashighas1260degreesCelsius.TUFItileswerebuiltashightemperatureblackversionsforuseintheorbiter'sundersideprovidingsufcientheatinsulationfortheorbiter'sunderside.Andlowertemperaturewhiteversionsforuseontheupperbodyconductingmoreheatwhichlimitstheirusetotheorbiter'supperbodyapandmainenginearea. Low-temperaturereusablesurfaceinsulation(LRSI)werewhiteincolorpossessinghighthermalreectivity.Thesetileswereusedtoprotectareaswherereentrytemperaturesarebelow649C(1200F).Theyaregenerallyinstalledontheuppersurfaceofthevehicle,maximizingsolargainwhentheorbiterisontheilluminatedpartoftheorbit.Theywerealsousedtoprotecttheupperwingneartheleadingedgeandalsoareasoftheforward,mid,andaftfuselage,verticaltail. 2.1.2Non-TileTPS Thedifferenttypesofnon-tileTPSare ReinforcedCarbon-Carbon(RCC)laminatedcompositematerialwereprimarilyusedforcoveringthewingleadingedgesandnosecapwherethetemperatures 23

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reachamaximumof1510Cduringreentry.Thiscompositecoveringhadveryhighfatigueresistancewhichisessentialduringascentandreentry.GenerallyallTPScomponents(tilesandblankets)weremountedontostructuralmaterialsthatsupportthem,mainlythealuminumframeandskinoftheorbiter.RCCistheonlyTPSmaterialthatalsoservedasasupportforpartsoftheorbitersaerodynamicshape,wingleadingedgesandthenosecap. FlexibleInsulationBlankets(FIB)werewhitelow-densitybroussilicamaterial.TheseblanketsweredevelopedasreplacementforLRSItiles.TheyrequireverylessmaintenancethanLRSItilesyettheypossessedthesamethermalproperties. FeltReusableSurfaceInsulation(FRSI)aregenerallywhite,exiblefabricofferingprotectionupto371C(700F).FRSIcoveredtheOrbiter'swinguppersurface,theupperpayloadbaydoors,andaftfuselage. Gapllerswereplacedatdoorsandmovingsurfacestominimizetheheatcreatedopengapsintheheatprotectionsystem.Thesematerialswereusedaroundtheleadingedgeoftheforwardfuselagenosecaps,windshields,sidehatch,wing,verticalstabilizer,therudder,bodyap,andheatshieldoftheshuttle'smainengines.ThellermaterialsaremadeofeitherwhiteAB312bersorblackAB312clothcovers(whichcontainaluminabers). Forfurtherindepthdiscussiononthermalprotectionconceptsusedinspacevehicles,thereaderisreferredtoBlosser[ 27 ]andBapanapalli[ 28 ]. 2.2FiniteElementbasedHomogenization Micromechanicalanalyseshavebeentraditionallyusedtoestimatetheeffectivestiffnesspropertiesofcompositematerials.Someofthesemethodsarealsosuitabletoobtainhomogenizedpropertiesforplatelikestructureswithperiodicity.Thereareseveralapproachestohomogenization:mechanicsofmaterialsapproach,elasticityapproach,energymethodsandniteelementanalysis.Allmethodsassumethatthereisarepresentativevolumeelement(RVE)orunitcellthatrepeatsitselftoformthestructure.Theunitcellissubjectedtosixlinearlyindependentdeformationstodeterminetheequivalentstiffnesspropertiesofthepanel.Thedeformationsareappliedasperiodicdisplacementboundaryconditions(PBC)whichincludesthreemid-planestrainsandthreecurvatures. 24

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Biancolini[ 1 ]usedanenergyequivalenceapproachtohomogenizeacorrugatedcorepaneltoanequivalentanisotropiclamina.Thehomogenizationprocedurewasappliedusingstaticcondensationinwhichtheinternalnodesofthemicrogeometryareremovedandexternalnodesattheboundaryofthemodelrepresenttheedgesoftheequivalentlamina.Theeffectivestiffnesspropertiesobtainedusingthismethodwasvalidatedwithaseriesofnumericaltests.Yuetal.[ 29 30 ]variationalasymptoticmethodforunitcellhomogenization(VAMUCH)topredicteffectivepropertiesofperiodicallyheterogeneousmaterials.Davalosetal.[ 2 ]evaluatedtheequivalentpropertiesofberreinforcedhoneycombsandwichpanels(Figure 2-2 ).Thepanelconsistedsinusoidalcoreintheplaneandextendedverticallybetweenthelaminates.Thehomogenizationtechniquewasacombinationofenergymethodandmechanicsofmaterialapproachtopredicttheequivalentpropertiesofthepanel.Forasimilarsinusoidalcorepanel,Buannicetal.[ 3 ]usedanasymptoticexpansionmethodforestimatingtheequivalentstiffnessproperties.WallachandGibson[ 4 ]usedtheunitcellapproachtocalculatethestiffnessproperties,compressivestrengthandshearstrengthofsandwichstructureshavingpyramidaltrusscores;employingtwomethods,atruss-analysisprogrambasedonmatrixmethodsandacommercialniteelementanalysis(FEA)programusingABAQUS.Theniteelementapproachconrmedresultsfromthetrussanalysesandallowedtoextendtheanalysistoincludenonlineareffectssuchasmaterialyieldandlargedeformation. PasteffortsonthehomogenizationoftheITPSpanel(Figure 2-3 )includeananalyticalapproachandniteelementbasedapproach.Martinezetal.[ 5 31 ]followedstrainenergyapproachandsheardeformableplatetheorytodevelopananalyticalmodelforthehomogenizationofacorrugatedsandwichpanel(Figure 2-3 )oftheITPS.Thoughanalyticalmodelsprovidereasonablygoodestimateofstiffnessproperties,theyinvolveseveralassumptionscompromisingtheaccuracyofthestructure.Moreovermostoftheanalyticalapproachesrequireaniteelementbasedvalidation.Sharmaetal. 25

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Figure2-2.Sinusoidalcoresandwichedbetweenlaminates[ 1 ] [ 32 6 ]employedaniteelementbasedhomogenizationtechniqueforhomogenizingcorrugatedcoresandwichpanelasanorthotropicplateandshowedthattheresultsagreedwellwiththethreedimensionalmodel.Henceaniteelementmethodbasedhomogenizationprocedureisadoptedheretoobtaintheequivalentplateproperties.Inniteelementbasedhomogenizationperiodicboundaryconditionsareimposedontherepresentativeunitcell(RVE)thatcorrespondstoagivenstateofmid-planestrainsandcurvaturesoftheequivalentplatetoobtaintheeffectivestiffnessproperties.TheniteelementbasedhomogenizationoftheITPSpanel(Figure 3-1 )consideredinthisstudywillbediscussedinChapter 4 Figure2-3.IntegratedThermalProtectionSystemwithcorrugatedcore[ 28 31 ] 26

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2.3DeterministicandReliabilitybasedoptimization Anefcientstructuraldesignplaysanimportantroleinvariousengineeringdisciplines(foreg.inautomobiledesign,aerospaceengineeringandinmarineapplications).Akeyareaofcontinuousresearchisinndinganoptimum,efcientstructure-makingastructurelightweightyethavingadequateloadcarryingproperties,isapopularexample.Recentdevelopmentinoptimizationtheoryandcomputationaltoolshavefacilitatedwaystondoptimalstructures[ 33 ].Further,designandoptimizationofcompositestructureshavegainedspecialattention[ 34 35 36 ]. Themethodologiesdiscussedinthisresearcharedeterministicoptimizationandreliabilitybasedoptimization.Deterministicoptimizationinvolvesminimizingtheweightwhileapplyingsafetyfactorsonfailureconstraints.Reliabilitybasedoptimizationincludesalltherandomnessinthedesignprocessandhasanadditionalconstraintreferredasreliabilityorprobabilityoffailureconstraint.Reliabilitybasedoptimizationhasgainedincreasingpopularitymainlyduetotheuseofcompositematerialsandthetypicaldesigndriversbeingminimumstructuralmassandhighreliability.Howeveritprovesbenecialtoperformdeterministicoptimizationinitiallyasitfacilitatesinchoosingappropriatedesignspaceforreliabilitybasedoptimization.Theoptimizationprocessinvolvesevaluationofthefailureconstraintsusingniteelementanalysesthatarecomputationallyexpensive.Thisresearchproposestoperformdeterministicandreliabilitybasedoptimizationoftheintegratedthermalprotectionsystemcompositepanelandaimsatreducingthecomputationalcostatdifferentphasesofoptimization,usingtechniquessuchasresponsesurfacemethodology,EfcientGlobalReliabilityAnalysis(EGRA),separableMonteCarlo(SMC)methodandbootstrappingmethod. Thefollowingsectionspresentssomebackgroundandreviewondeterministicoptimization,useofsurrogatemodelsforoptimization,typesofuncertaintiesincludedtoestimatereliability,methodstoevaluateprobabilityoffailure,methodstoestimateaccuracyofprobabilityoffailure.FurtheritalsodiscussesEfcientGlobalReliability 27

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Analysis(EGRA)foraccurateestimationoftheoptimizationconstraintswithlimitedcomputationalresources. 2.3.1DeterministicOptimization Deterministicoptimizationinvolvesoptimizingthedesignvariablestominimizetheobjectivefunctionwhilesatisfyingequalityorinequalityconstraintsthatfollowalinearornon-linearformulation.Thegeneralformulationoftheproblemis Minimizew=f(d)suchthatg(d)zh(d)=mdLB
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77Konthetemperatureconstraintand1.5onboththestressandbucklingconstraints.Sincethedeterministicoptimumwilllieoneitheroralloftheconstraints,itseparatesthefeasibledesignspacefromtheinfeasibleone.ThishelpsinnarrowingthedesignspacetothefeasibleonesforperformingprobabilisticoptimizationandthusavoidingexpensiveFEanalysisontheinfeasiblespace. 2.3.2StructuralReliability Reliabilitybasedoptimizationhasgrownratherquicklyduringthelastfewdecadesandstructuralreliabilitymethodshavedevelopedrapidlyandhavebeenwidelyappliedinthepracticaldesignofstructures[ 37 ].Theaimofreliabilityanalysisisthequanticationandtreatmentofuncertaintiesandthentheevaluationofameasureofsafetyorreliabilitytobeusedindesign[ 38 ].Thetermsreliabilityandprobabilityoffailurearecomplementary,inthatthemorereliablethedesign,thelowertheprobabilityoffailure.Instructuralapplications,reliabilitybasedoptimizationinvolvesminimizingweightofthesystemwhilesatisfyingareliability/probabilityoffailureconstraint.Thegeneralformulationoftheproblemis Minimizew=f(d)suchthatpf(g(d;X)Cr(2) wheretheobjectivefunctionfisafunctionofonlythedeterministicdesignvariablesd,buttheresponsefunctionintheprobabilityoffailureconstraintgisafunctionofdandX,avectorofrandomvariablesdenedbyknownprobabilitydistributions[ 39 40 ].pf;Cristhetargetprobabilityoffailure.Sinceprobabilitymeasuresarehighlynon-linear,thisconstraintisoftenreplacedbyreliabilityconstraintwherethereliabilityindexiscalculatedas 29

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=)]TJ /F9 11.955 Tf 9.3 0 Td[()]TJ /F4 7.97 Tf 6.59 0 Td[(1(pf)(2) whereisthecumulationdistributionfunctionofstandardnormaldistribution.Cristhetargetreliabilityindex. ReliabilityanalysescanbeperformedusingMoment-basedmethodssuchastherst-order-reliability-method(FORM)orsecond-order-reliability-method(SORM)orsamplingmethodssuchasMonteCarlosimulations(MCS).Researchershaveusedthesemethodstooptimizecompositestructuresaswell.LehetaandMansour[ 41 ]carriedoutlimitstateanalysis,rstorderreliabilityanalysisandreliability-basedstructuraloptimizationofshipstiffenedpanels.QuandHaftka[ 35 ]demonstratedreliabilitybasedoptimizationofcompositesatcryogenictemperaturesusingresponsesurfaceapproximationsandMonteCarloSimulations.ScuivaandLomario[ 42 ]performedacomparisonbetweenMonteCarlosimulationsandFORMsincalculatingthereliabilityofacompositestructure.AsignicantamountofresearchhasdealtwithoptimizationandreliabilityanalysesofthecorrugatedcoreITPSpanel(Figure 2-3 ).Kumaretal.[ 43 ]andVillanuevaetal.[ 44 ]studiedthedifferenceinriskallocationbetweenstructuralandthermalfailuresbydeterministicandprobabilisticoptimization.Sharmaetal.[ 45 ]performedamulti-delityanalysisoftheITPSpanelusingCorrectionResponseSurface.FurtherVillanuevaetal.[ 46 ]studiedtheeffectsofsinglefuturetestinthedeterministicandprobabilisticdesignoftheITPSpanel.TheyconsideredonlythethermalfailuremodeoftheITPSpanel.Matsmuraetal.[ 47 ]consideredmultiplefailuremodesoftheITPSpaneltostudytheeffectoffuturetestsinthereliabilityestimationoftheITPSpanel.ThisresearchconsidersadifferentconceptofITPSandaimsatdeterministicallyandprobabilisticallyoptimizingtheITPSpanelusingcosteffectiveadaptivesamplingtechniques. Momentbasedmethodscanbecost-efcientastheyinvolveapproximationofresponsefunctiontondthemostprobablepoint(MPP),however,theyareinaccurate. 30

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Samplingmethodsareaccurateifadequatesamplesareusedbutcomputationallyintensive.Inengineeringapplicationsreliabilityestimationmayinvolveindividualfailuremodesuchasstressorsystemlevelfailurewithmultiplefailuremodeswhichcouldbe,say,stressanddeection.Moment-basedmethodscouldbeusedforsystemwithsinglefailuremodes.Howeverwhentheprobleminvolveshigherdimensionswithmultiplefailuremodesthatareexpensivetoevaluate,theresponsefunctionscannotbeapproximatedasanalyticalderivatives[ 42 ].SuchsituationsareaddressedbynumericalapproximationofresponsesusingsurrogatemodelsandsamplingmethodssuchasMonteCarlosimulationstoestimatereliability. Asmentionedearlier,theITPSpanelconsideredhasmultiplefailuremodesandinvolvescalculationofsystemlevelreliability.Systemlevelfailureisdenedeitherasparallelfailureorseriesfailure.Whensystemisassumedtohavefailedifallthemodesfailitisreferredasaparallelsystemfailure,wheneitheroneofthemodesfail,itisreferredseriessystemfailure.IntheITPSproblem,ifeitherofthemodes,temperature,stressorbucklingmodesfail,thepanelisconsideredtohavefailed. 2.4UseofSurrogateModels Inordertoreducethecomputationcostassociatedwithniteelementsimulations,qualitysurrogatemodelsofthevariousconstraintsareconstructed.TheyarealternatelyreferredasResponsesurfaceapproximations(RSA)ormetamodels.Surrogatemodelsusuallyemployloworderpolynomialstothestructuralresponsewithalimitednumberofinputdesignpoints[ 48 49 ].Theinitialdesignpointsareobtainedusingdesignofexperimentssuchasfullfactorialdesign,Centralcompositedesign,Latinhypercube,A-optimalorD-optimaldesigns. Inthecaseofexpensiveresponseevaluations,surrogatemodelshavebeenusedforevaluatingconstraintsinthedeterministicandprobabilisticoptimizationprocess.Vianaetal.[ 50 51 ]havedemonstratedtheuseofsurrogatemodelsandsurrogatebaseddesignoptimization.FurthertheyalsodevelopedaSurrogateToolbox 31

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inMATLABthatmakestheoptimizationprocesseasier[ 52 ].ThisresearchmainlyusestheSurrogateToolboxforttingresponsesurfacemodelsandfmincon()aMATLABoptimizerfortheoptimizationprocess.Moreover,theprobabilitycalculatedfromMonteCarlosimulationoftenintroducesrandomerrorsduetolimitedsamplesizeresultinginunsafeoptimum[ 35 53 ].Theuseofresponsesurfaceapproximationreducessucherrors. Thoughsurrogatemodelsarebenecialinthecaseofcomputationallyexpensiveproblemssuchasthosedemandingcomplexniteelementanalyses(deterministicoptimization),withlargenumberofuncertaintiesandmultipleconstraints,theprobabilityoffailurecalculationsevenwithresponsesurfacemodelsaredifculttohandle.Traditionallytheresponsesurfacemodelsconstructedwereaglobalapproximationoftheresponsewithdesignofexperimentsthatareindependentoftheresponsefunction.Sincefailureisdenedusingaconstraints/limitstatesthatseparatethefeasibleandinfeasibledesigns,itisimportanttoconstructaccuratecontoursoftheconstraintswhileitisacceptabletohavelargeerrorsatotherregionsinthedesignspace.Accurateapproximationofconstraintswouldleadtoaccuratepredictionofoptimum. 2.4.1ApplicationofSurrogatestoImproveConstraintBoundaries Indeterministicandreliabilitybasedoptimization,itisimportantthattheconstraintboundariesthatseparatethefeasibledesignsfromtheinfeasibleonesareestimatedaccuratelywithminimalerror.Globalsurrogatemodelscanbeusedbutwouldrequirelargenumberofdesignpointsandwouldnotensureaccurateapproximationofconstraintboundaries.Tobuildsurrogatemodelsthatapproximatesconstraintboundaryaccurately,researchershaveintroducedvariousmethods.KuczeraandMourelatos[ 8 ]usedacombinationofglobalandlocalsurrogatemodelstorstdetectthecriticalregionsandthenobtainalocallyaccurateapproximation.Arenbecketal.[ 9 ]usedsupportvectormachineandadaptivesamplingtoapproximatefailureregions.TuandBarton[ 10 ]usedamodiedD-optimalstrategyforboundary-focused 32

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polynomialregression.VazquezandBect[ 12 ]proposedaniterativestrategyforaccuratecomputationofprobabilityoffailurebasedonkriging.ShanandWang[ 54 ]developedaroughsetbasedapproachtoidentifysub-regionsofthedesignspacethatareexpectedtohaveperformancevaluesequaltoatargetvalue.Forconstrainedoptimizationandreliabilityestimation,Pichenyetal.[ 13 ]developedtargetedIntegratedMeanSquarecriterion(IMSE)toconstructdesignofexperimentssuchthatthemetamodelaccuratelyapproximatethevicinityofaboundaryindesignspacewhichiseitherdenedbyatargetvalueorgaussiandistribution. 2.5AdaptiveSampling Whenoptimizationproblemsinvolvenon-linearandmultimodalobjectivefunctions,theyrequirelargenumberoffunctionsevaluationstondtheglobaloptimum.Andwhentheevaluationsarelimitedbyexpensivecomputations,ndinganaccurateglobaloptimumbecomesdifcult.Surrogatemodelscouldbeemployedbuttheiraccuracyisalsocompromisedwhencomplexfunctionsareconstructedwithfewerdesignpoints.Jonesetal.[ 55 ]developedEfcientGlobaloptimization(EGO),anunconstrainedoptimizerthatfocusesonaddingpointstothedesignspacetoaccuratelymodelcomplexfunctionsandhencendtheglobaloptimumaccurately. Inthismethod,aninitialGaussianprocessmodel[ 56 57 ]isbuiltasaglobalsurrogatefortheresponsefunction.EGOthenadaptivelyselectsadditionalsamplestobeaddedtothedesignspacetoformanewGaussianprocessmodelinsubsequentiterations.EGOusesaspecicformulationknownastheExpectedImprovementthatidentiesthenewdesignpointsbasedonhowmuchtheyareexpectedtoimprovethecurrentbestsolutiontotheoptimizationproblem.Whenthisexpectedimprovementisacceptablysmall,thegloballyoptimalsolutionhasbeenfound. EfcientGlobalOptimizationwasfurtherextendedtoimprovetargetregionsinthedesignspace.Ranjanetal.[ 11 58 ]developedasequentialdesignmethodologybasedon(EGO)thatexploresthedesignalongthecontourofinterest.Bichonetal.[ 59 ] 33

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alsodevelopedasimilarmethodbasedonEGOreferredasEfcientGlobalReliabilityAnalyses(EGRA).TheyillustratedtheapplicationofEGRAbyimprovinglimitstatesforaccuratereliabilityestimation.ThisresearchproposestoextendtheapplicationofEGRAtoimproveconstraintboundariesindeterministicoptimizationandreliabilitybasedoptimizationoftheITPSpanel.Theconceptofsamplingdesignsiteratively,reducesthecomputationalcostofexpensiveFEsimulationsandatthesametimeincreasestheaccuracyofthefailureboundary. 2.5.1EfcientGlobalReliabilityAnalysis Themethodhastwomainfeatures,aninitialgaussianprocessmodelandtheexpectedfeasibilityfunctiontoidentifytheadditionalsamplesiterativelyatthevicinityofthelimitstate.Theinitialsurrogatemodel^Gisconstructedusinglimitednumberofdesignsusingaknownsamplingmethod,inthiscase,Latinhypercubesampling.Theexpectedfeasibilityfunctionisoptimizedtondthenextdesignthatwouldimprovethetargetboundaryz.Thefunctionisgivenas EF(^G(X))=(G)]TJ /F9 11.955 Tf 12.67 0 Td[(z)2z)]TJ /F3 11.955 Tf 11.95 0 Td[(G G)]TJ /F9 11.955 Tf 11.95 0 Td[(z)]TJ /F2 11.955 Tf 9.75 -4.34 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(G G+z+)]TJ /F3 11.955 Tf 11.95 0 Td[(G G)]TJ /F3 11.955 Tf 9.3 0 Td[(G2z)]TJ /F3 11.955 Tf 11.96 0 Td[(G G)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F2 11.955 Tf 9.74 -4.34 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(G G+z+)]TJ /F3 11.955 Tf 11.95 0 Td[(G G+z+)]TJ /F3 11.955 Tf 11.96 0 Td[(G G)]TJ /F9 11.955 Tf 11.95 0 Td[(z+)]TJ /F3 11.955 Tf 11.95 0 Td[(G G(2) TheresponsepredictedbythegaussianmodelfollowsthedistributionN(G,G),=GistheerrorbandaroundthelimitstatewhichisafunctionofthestandarddeviationGand,afactorappliedonthestandarddeviation.z+=z+andz)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(=z-.andarethestandardnormalpdf(probabilitydensityfunction)andcdf(cumulativedistributionfunction)respectively.WhenGisclosetothetargetcontour,thersttermdominatestheexpression.Thesepointsareclosetotheband.IfGisfarawayfromtarget,thesecondtermtendstodominate.Thistermfacilitatessamplinginregionsoftheinputspacewheretheestimatedresponseisoutsidetheband,buttheuncertainty 34

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ofpredictionishigh.Thethirdtermisrelatedtothevariabilityofthepredictedresponseintheneighborhoodof[ 11 58 ].Itsupportssamplinginregionsneartheestimatedcontourbutwherethepredictionvarianceisquitehigh. CurrentlyeithergaussianprocessorkrigingarethedefaultsurrogatemodelsforimplementingEGRA.SincetheITPSinvolvesalargenumberofrandomvariables,theuseofpolynomialresponsesurfaceisfavoredovergaussianprocessorkriging.HencetoadapttotheoptimizationoftheITPSpanelthisresearchproposestoimplementpolynomialresponsesurfaceinEGRAmethodology. 2.6UncertaintyModeling Modelingandanalysisofinputuncertaintiesaidinunderstandingthepropagationofuncertaintyintheoutputs.Uncertaintiesinanengineeringsystemcanbeclassiedasepistemicuncertaintyandaleatoryuncertaintyalsocalledvariability.Epistemicuncertaintygenerallyrepresentsalackofknowledgeofaquantityorprocessofasystemorenvironment.Thisuncertaintyisalsoreferredasreducibleuncertaintyasitcanbereduced(orincreased)fromincreasedunderstandingoftheuncertainvariableorfrommorerelevantexperimentaldata.Aleatoryuncertaintyisgenerallycharacterizedbyinherentrandomnessinthephysicalsystemorenvironmentwhichcannotbereducedbyfurtherdata[ 60 ]. Instructuralapplications,aleatoryvariabilitycanbeintroducedbymanufacturingimperfectionssuchasvariabilityinmaterialpropertiesandgeometricdimensions,variabilityinloadingandepistemicuncertaintiesbyerrorsinmodelingandsimulation.Uncertainvariablesareincludedinthedesignprocessbytheirrespectiveprobabilisticdistributions.Inadditiontothevariabilityintheinputparameters,errorinmodelingandsimulation(epistemicuncertainty)isalsointroducedintheoutputparameterstoestimatetheprobabilityoffailure.Theprobabilitydistributionoftherandomvariablescouldfollowuniform,normal,lognormalorWeibulldistribution[ 61 15 ].However,thisresearchmainlyassumesuniformandnormaldistributionfortheinputrandomvariables. 35

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Estimatessuchasvariance,standarddeviationorcoefcientofvariationoftheresponsesampleswouldprovidetheuncertaintyintheresponseduetoinputuncertainties. Theimportanceofuncertaintycharacterizationanduncertaintypropagationincompositeshasbeenillustratedbyvariousresearchers[ 62 63 64 65 66 ].AntonioandHoffbauer[ 62 ]studiedtheeffectsofdeviationsinmechanicalproperties,plyangles,plythicknessandappliedloadsforalaminatedshellcomposite.Theydemonstratedthatuncertaintyanalysisisveryusefulindesigninglaminatedcompositestructuresminimizingtheunavoidableeffectsofinputparameteruncertaintiesonstructuralreliability.OhandLibrescu[ 63 ]addressedtheproblemofalleviatingtheeffectsofuncertaintiesforfreevibrationofcompositecantileversunderuncertaintiessuchaslayerthickness,elasticconstantsandplyangle.Nooretal.[ 64 ]studiedthevariabilityofnon-linearresponseofstiffenedcompositepanelsduetovariationsingeometricandmaterialparametersusinghierarchicalsensitivityanalysisandfuzzysetanalysisapproach. FortheITPSoptimizationproblem,therandomvariablesaredimensions,mechanicalloads,thermalloads,thermo-mechanicalpropertiessuchasYoung'smodulus,shearmodulus,poisson'sratio,coefcientofthermalexpansion,thermalconductivityandspecicheat.Inadditiontothis,uncertaintyisconsideredintheallowablelimitssuchasmaterialstrength,temperature.Theerrorinniteelementmodelingandsimulations(transientheattransfer,staticstressandbucklinganalyses)isalsoincluded. 2.7EstimationofProbabilityofFailure TheprobabilityoffailureistheprobabilitythattherandomvariablesX=fx1,x2,...xigareinthefailureregionthatisdenedbylimitstatefunctionG(X)<0.IfthejointpdfofXisfx(X),theprobabilityoffailureisevaluatedwiththeintegral pf=PfG(X)>0g=ZG(X)<0fx(X)dx(2) 36

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Thereliabilityiscomputedby R=1)]TJ /F3 11.955 Tf 11.95 0 Td[(pf=PfG(X)>0g=1)]TJ /F10 11.955 Tf 11.96 16.28 Td[(ZG(X)<0fx(X)dx(2) Thedirectevaluationoftheprobabilityintegrationstateaboveisextremelydifcultwhen alargenumberofrandomvariablesareinvolved.Insuchcasestheprobabilityintegrationbecomesmultidimensionalwhichistypicalofengineeringapplications. theintegrandfx(X)isthejointpdfofXandisgenerallyanonlinearmultidimensionalfunction. thelimitstateboundaryG(X)=0isalsomultidimensional,nonlinearfunction.Instructuralapplications,G(X)isoftenablack-boxmodel(orniteelementsimulation),andtheevaluationofG(X)iscomputationallyexpensive. Hencemomentbasedmethodsandsamplingmethodscouldbeappliedtoevaluatetheprobabilityintegration. 2.7.1MomentbasedMethods MomentbasedmethodssuchasFirst-OrderReliabilityMethod(FORM)andSecond-OrderReliabilityMethod(SORM)easethecomputationaldifcultiesbysimplifyingtheintegrandfx(X)andapproximatingthelimitstatefunctionG(X)toestimateprobabilityoffailure(Equation 2 ).ThemethodsimpliesthejointdistributionfunctionbytransformingtheoriginalrandomvariablesfromX-spacetostandardnormalU-space.Thelimitstatefunctionisapproximatedusingrst-orderTaylorseriesexpansion.Inthestandardnormalspace,thepointonthelimitstatefunctionwhereG(U)=0attheminimumdistancefromtheoriginisthemostprobablepoint(MPP)offailure.ThereliabilityismeasuredasthedistancefromtheorigintotheMPP.Thismeasureisreferredasreliabilityindex().TheMPPisdeterminedas Minimize=p UTUsuchthatG(U)=0(2) 37

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whereUisthevectorofvariablesinstandardnormalspace.FORMisfairlyaccuratewhenthelimitstatefunctioncanbeapproximatedasalinearfunction.Secondordermethodscanbeusedwhenthelimitstatefunctionhasahigherorderofcurvature.Thismethodapproximatesthelimitstateasaquadratic,andprovidesamoreaccurateapproximationinsuchcases. 2.7.2SamplingMethods MonteCarlosimulation(MCS)isoneofthepowerfulandeasytoimplementmethodstopropagatetheuncertaintyininputrandomvariablestotheuncertaintyinfailure[ 67 ].Reliability-baseddesignofastructuralsystemisoftenaddressedusingMonteCarlosimulations[ 68 39 69 ],especiallywhenthesystemunderconsiderationfailsduetomultiplefailuremodes.ProbabilityoffailurepfisdeterminedthroughalimitstatefunctionGthatseparatesthefeasibledesignsfromtheinfeasibleones.Thelimitstateisgenerallyafunctionofrandomvariables,responseRandcapacityC.Whentheresponseexceedsthecapacity,itisconsideredfailure,example,maximumstressfailuretheorywheretheresponsewillbecalculatedstressandcapacitywouldbestrengthofthematerial.ThecapacityandresponseareassumedtobefunctionsofstatisticallyindependentrandomvariablesX1andX2,respectively.Equation 2 showstheseparablecasewherefailureoccurswhenasinglecomponentofresponseexceedsasinglecomponentofcapacity. G(X1;X2)=R(X1))]TJ /F3 11.955 Tf 11.95 0 Td[(C(X2)(2) FailureoccurswhenG0andthesystemissafewhenG<0.Inthemoregeneralcase,thecapacityandtheresponseinthelimitstatecannotbeexplicitlyseparated,andthelimitstatefunctionmayberepresentedas G(X1;X2)=G(R(X1);C(X2))(2) 38

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WhereRandCmaybescalarorvectorquantities.ThelimitstatecouldbeahigherorderpolynomialfunctionintermsofresponseandcapacitysuchasTsai-HillorTsai-Wucriteriongenerallyemployedtopredictfailureincompositestructures.Statisticalestimatessuchasvariance,standarddeviationandcoefcientofvariationofprobabilityoffailurewouldprovideaccuracyintheestimate. Thetraditional,crudeMonteCarlotechnique(CMC)issimple,butitlacksaccuracywhensamplingislimitedduetocomputationallyexpensivestructuralanalysis,suchasfromniteelementanalysis(FEA).TherearevarioustechniquestoimprovetheaccuracyorefciencyofCMC,includingtailmodeling,conditionalexpectation,importancesamplingandtheuseofsurrogates[ 70 71 ].However,anothertechniquereferredasseparableMonteCarlowasdevelopedbySmarsloketal.[ 72 16 ],whichisapplicableincombinationwiththeabovementionedmethodstofurtherimproveaccuracyorefciency.Whentheresponseandcapacityrandomvariablesarestatisticallyindependent,accuracycanbeimprovedbytheseparableMonteCarlomethod(SMC).Thisfacilitatesimprovedaccuracyofthecalculationoftheprobabilityoffailureforthesamecomputationalbudget. TheerrorinprobabilityoffailureestimateforcrudeMonteCarlocanbeobtainedfrombinomiallaw.ForSMCinvolvingsimplelimitstate(Equation 2 ),thevarianceestimatorwasderivedusingconditionalcalculusandvalidatedusingsimulationestimates[ 15 16 ].ThesimulationestimatesofvariancewereshowntobeofcomparableaccuracytothoseobtainedforCMC.Howeverthisisnotapplicableinthecaseofcomplexnon-separablelimitstate(Equation 2 ).ForSMCwithnon-separablelimitstates,bootstrappingtechniqueisproposed[ 18 ].Theerrorinthestandarddeviationestimateofthebootstrappedprobabilityoffailureprovidesameasureoftheaccuracyofbootstrapping.Furtheritwasalsodemonstratedthatthevariabilityestimateofcapacityandresponsecanhelpinchoosingthesamplesizeneededforgivenaccuracy[ 17 19 ].Chapter 5 onestimationofprobabilityoffailureprovidesadetaileddescriptionofthe 39

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CrudeMonteCarlo,SeparableMonteCarlomethod,bootstrappingtechniqueandregroupingofrandomvariablestoimproveaccuracyofpfbyapplyingtheseconceptstoacompositepressurevesselproblem. 40

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CHAPTER3DESIGNANDANALYSISOFINTEGRATEDTHERMALPROTECTIONSYSTEMPANEL 3.1DesignofITPSCompositePanel Theprimaryobjectiveofthisresearch,asstatedearlier,istodeterminetheoptimaldesignofanITPScompositepanel.ThischapterdiscussesthegeometryandmaterialselectioninvolvedinatypicalITPSconstruction.Further,theniteelementanalysesoftheITPSmodelisalsodiscussed. TheintegratedThermalProtectionSytempanelconsideredhereconsistsofstackedrigidAluminaEnhancedThermalBarrierfoam(AETB)insulationbarsthatarespirallywrappedwithaSiliconCarbide/SiliconCarbide(SiC/SiC)laminate.Thebarsarestackedorthogonallyintwolayersina0/90conguration.ThebarsarethensupportedbyatopfacesheetandabottomfacesheetmadeofSiC/SiCandGraphite/Epoxylaminate,respectivelyformingasandwichstructure(Figure 3-1 ).Thevariouscomponents,stackorientationandmaterialpropertiesaregiveninTables 3-2 and 3-3 [ 73 74 75 ].Theconceptnotonlyhasmaterialasymmetryaboutthemid-planebutalsogeometricasymmetryastherigidinsulationbarsarestackedorthogonally.Further,thedesigniscomplexbecauseofthelaminatewrappedaroundtheAETBinsulationbar.Itisimportanttonotethatinsuchadesign,transverseshearoftheinsulationandthewebs(wraps)wouldhaveapronouncedeffectonthestructure. 3.1.1KeyDimensionsoftheITPSstructure ThekeydimensionsoftheITPSarethewidthwf,andheighthoftheinsulationfoam,thicknessofthetopfacesheettT,bottomfacesheettBandwraptWandthenumberofbarsn(Figure 3-2 ).ThenominalvaluesofthedimensionsaregiveninTable 3-1 Table3-1.KeyDimensionsoftheITPSpanel wmmhmmtTmmtBmmtWmmn 20.020.02.02.00.512 41

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Figure3-1.SchematicviewoftheITPSpanel Figure3-2.DimensionsoftheITPSpanel 3.1.2ITPSMaterials ThecomponentsofITPSaremadeofceramicmatrixcomposites,polymermatrixcompositeandhighdensityinsulationfoam.ThetopfacesheetismadeofSiC/SiCplainwoventextilecompositelaminatewhichisknownforitshighmechanicalstrengthatelevatedtemperatures,highthermalstabilityandlowdensity[ 76 ].Thebottomfacesheetisgraphite/epoxylaminate(polymermatrixcomposite).Thenotablepropertiesofpolymermatrixcompositesarehightensilestrengthandstiffness,highfracturetoughnessandgoodcorrosionresistance.Moreovertheyarepopularduetotheir 42

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lowcostandsimplefabricationmethods.Theinsulation,madeofaluminaenhancedthermalbarrier(AETB)foamdemonstratehigherstrength,addeddurability,andhaveamaximumoperationaltemperatureof1700K[ 26 ].Thematerialpropertiesofthecomponents,theberorientationandthethicknessofeachcomponentofITPSarepresentedinTable 3-2 andTable 3-3 below. Table3-2.ComponentsofITPS,materialsandberorientation ComponentMaterialFiberorientationThickness TopfacesheetSiC/SiClaminate[(0/90)4]S16layers-0.125mmeachInsulationfoamAETB8-35mmBottomfacesheetGr/Eplaminate[(0/90)4]S16layers-0.125mmeachWrapSiC/SiClaminate[0/90]S4layers-0.125mmeach Table3-3.MaterialpropertiesofthecomponentsofITPS[ 75 74 73 ] PropertiesSiC/SiCGraphite/EpoxyAETBfoam E1GPa1701380.153E2=E3GPa12490.153v12=v130.260.30.25v230.200.3420.25G12=G13GPa566.90.062G23GPa603.580.0621x10)]TJ /F4 7.97 Tf 6.59 0 Td[(6/K40.00321.772x10)]TJ /F4 7.97 Tf 6.59 0 Td[(6/K40.1131.77 3.1.3BoundaryConditions InthepastresearchonITPSpanels,thecommonboundaryconditionsappliedontheITPSpanelweresimplysupportedboundaryconditionsandclampedboundaryconditions.Also,sincethepanelissymmetricinthexandydirections,onlyonequarterofthepanelisconsideredandsymmetryboundaryconditionswereapplied.Inthisresearchwork,simplysupportedboundaryconditionsandxedboundaryconditionsareconsideredforthehomogenizationofthepanel(Chapter4).HoweveritwasfoundthathomogenizationcouldnotbesatisfactorilyappliedtotheITPSpanel(detaildiscussioninChapter4).HenceanewsetofboundaryconditionswhichsimulatesastructurallyintegratedTPSpanelwasanalyzed.Itsimulatesthepanelmountedontheframeofthevehicleandallowedtoexpandequallyinthex)]TJ /F3 11.955 Tf 11.95 0 Td[(yplane. 43

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Figure3-3.BoundaryconditionsappliedontheITPSpanel 3.1.4Loads Fromtakeofftolanding,thespacevehiclesaresubjectedtovariousloadingconditionssuchasdrag,aerodynamicheating(thermalloads),in-planeinertialloads(mechanicalloads),pressureandforeignobjectimpactloads.Theseloadscouldactindividuallyatanytimeduringthemission,ortheycouldacttogetherleadingtomultiplecomponentfailure.Thisresearchworkprimarilyexploresandanalyzestheeffectofthermalloadscausedbytemperaturegradientandmismatchinthermo-mechanicalpropertiesofdifferentmaterialcombinations,andmechanicalloads(pressure)throughtransientheattransferanalyses,thermalstressanalyses,pressureanalysesandthermalandmechanicalbucklinganalyses. 3.2FiniteElementAnalyses Thissectionpresentsadescriptionoftheniteelementmodelsandanalysesforheattransfer,stressanalysisandbucklinganalysisoftheITPSpanels.Themodelconsideredinthisresearchisconstructedusingacommercialniteelementsoftware,ABAQUS(Figure 3-3 ).ABAQUS.Twenty-nodethree-dimensionalbrickelementswereusedtomodeltheITPS.The3Dsolidelementmodelhas3displacementdegreesoffreedomateachnode. 44

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3.2.1TransientHeatTransferAnalysis Itwasobservedthatthereentryheatingratesaremoreseverethantheheatingratesduringascent,thatis,theheatingratesincreasemoresteeplyandthetotalintegratedheatloadismuchlargerduringreentry.Thus,itcanbeinferredthatthereentryheatingrateswouldbemostinuentialintheITPSdesignand,therefore,theywereusedintheheattransferFEanalysisforthedesignprocess. AtypicalheatingrateusedfordesignisshowninFigure 3-4 .Theinitialtemperatureofthestructureisassumedas295K(72F)[ 28 ].Alargeportionofthisheatisradiatedouttotheambientbythetopsurface.TheremainingheatisconductedintotheITPS.Somepartofthisheatisconductedtothebottomfacesheetbytheinsulationmaterialandsomebythewebs. Figure3-4.HeatuxproleincidentontheITPSpanel LoadsandboundaryconditionsfortheheattransferproblemareschematicallyillustratedinFigure 3-5 .Thefourouterfacesandbottomsurfaceofthebottomfacesheetareassumedtobeperfectlyinsulated.Thisisaworstcasescenariowherethebottomfacesheettemperaturewouldrisetoamaximumasitcannotdissipatetheheat.Itisalsoassumedthatthereisnolateralheatowoutoftheunitcell.Theheatuxincidentonaunitcelliscompletelyabsorbedbythatunitcellonly,thoughinanactual 45

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ITPSpanelheatwouldowintothefacesheetsandwrapswhichcouldactasathermalmassandtherewouldbealateralowofheatinthepanelfromoneunitcelltoanother.TheloadstepsaretabulatedinTable 3-4 Figure3-5.Boundaryconditionsforthetransientheattransferanalysis Table3-4.Heatuxloadstepsinthetransientheattransferanalysis LoadTimeHeatFluxTimeStepAmbientStepPeriodInputSizeTemperature Step10-4500.034,069W=m230sec213KRamplinearlyStep2450-157534,06939,748W=m225sec243KRamplinearlyStep31575-217539,7480.0W=m230sec273KRamplinearlyStep42175-5175-50sec295K Therstthreeloadstepsconsideredintheheattransferanalysis,theheatuxisrampedlinearlyfrom0to34,069to39,748W=m2.Thestep4representsthetimeperiodaftertouchdownandtheFEanalysiscontinuesforanother50secondsinordertocapturethetemperatureriseofthebottomfacesheet.Duringthisperiod,alongwithradiativeheattransfer,convectiveheattransferboundaryconditionsareimposedon 46

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thetopsurfacetosimulatetheheattransfertothesurroundingswhilethevehicleisstandingontherunway.Themaximumbottomfacesheettemperatureisrecordedfromthisanalysis.Also,alongthereentrytimethetemperaturedistributionwithmaximumdifferencebetweenthetopfacesheetandbottomfacesheetwouldinducemaximumstressesinthepanel(Figure 3-6 ).Hencethisdistributionalongtheheightofthepanelisrecordedwhichactsasthethermalloadforthestressandbucklinganalyses. Figure3-6.TemperaturevariationonTFSandBFSwithrespecttoreentrytime 3.2.2StressAnalysis StressesdevelopedintheITPSduetothermalandpressureloadsareanalyzedindividuallybyperformingastaticstressanalysis. Inpreviousstudies[ 7 28 31 ]theITPSpanelwasassumedtobesimplysupportedalongitsedges.Itwasfoundthatthedeformationsduetothermalstresseswerenotrealisticastherewasnocompatibilityofdisplacementsbetweenadjacentpanels.InorderforanITPStobeeffectivetheyhavetobestructurallyintegrated(S-ITPS). 47

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Inthisconcepttheattachmentswillbedesignedsuchthatthethermalforceswillbetransferredfromonepaneltonextadjoiningpanel.ThusonecanimposeperiodicBCsbetweenpanelsaslongasthesurfaceheatingisuniform.TheproblemcanbefurthersimpliedbyanalyzingoneunitcellwithperiodicBCs.Thisisispossiblebecauseweassumeeachpaneliscomposedofrepeatingunitcells.TheboundaryconditionsofastructurallyintegratedTPSpanelisshowninFigure 3-3 .Thenodesononeofthefacesonthex)]TJ /F3 11.955 Tf 12.59 0 Td[(zplaneareallowedhaveequaldisplacementsinthey-directionandtheadjacentfaceonthey)]TJ /F3 11.955 Tf 12.83 0 Td[(zplaneisallowedtohaveequaldisplacementsinthex-direction.Thebottomedgesofthesetwoadjacentfaceswereconstrainedfrommovingintheverticaldirectionsimulatingthatthepaneledgesweremountedontheframeofthevehicle.TheothertwoadjacentedgesareassignedsymmetryBCs. Thetemperaturedistributionsareappliedalongtheheightofthepanel(z-coordinate)oneachnodeofthemodel.Thisimpliesthatthetopandbottomfacesheettemperaturesareuniformthroughoutthelengthandwidthofthepanel.Althoughthetemperaturevariesslightlyinthexandy-directions,thisvariationisverysmallandcanbeneglected.Thestressesdevelopedinthevariouscomponents(facesheets,foamandwrap)areextractedandinvestigatedforthefailureofthepanel.Thestressesinthewraparehighercomparedtotheothercomponentscontributingtowardsfailureofthepanel.Forpressureanalysis,aunitcellcannotbeanalyzedwithperiodicBCs.Hence,aquarterpanelwithsixunitcellsinthexandydirectionwasanalyzed.Apressureofoneatmosphere(1atm.)isappliedonthetopfacesheet.Itwasobservedthatstressesunderpressureloadswereatleasttwoorderslesscomparedtothethermallyinducedstresses.Sincepressureloadsdonotcausefailureofthepanel,theywillnotbeincludedintheoptimizationprocess. 3.2.3BucklingAnalysis ThecomplexdesignoftheITPSwithfacesheetsandwebsleadstolocalandglobalbucklingofthelaminatesunderthermalstressesandmechanicalstresses.Though 48

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thewebsinthepanelaresupportedbytheinsulationfoam,thebucklingofthewebscouldbeduetopoorbondingbetweenthefoamandthewebandalsothepossibilityofdamagedfoam(duetothermalloads)notbeingabletosupportthewebsfrombuckling.ThebucklingoftheITPScanbecarriedoutontheunitcellorasalarge3-Dshellmodelwithoutanyfoam(assumingdamagedfoam).Theseanalyseswouldprovideanestimateofthebucklingload.Whilelocalbuckling,byitself,maynotalwaysleadtocatastrophicfailure,itcouldcontributeindirectlyalongwithhightemperaturesandstresses. BucklingoftheITPSinABAQUSismodeledasaeigenvaluebucklingpredictionproblemwheretheeigenvaluecanbeconsideredastheloadfactoratwhichthestructurewouldfailduetobuckling.Theanalysisisverysimilartothestressanalysisunderthermalloadsexceptfortheloadstepwhichislinearperturbationprocedure.Theloadconsideredwouldbetemperaturedistributionifthereisthermalbucklingandpressureloading,ifthereisbucklingduetomechanicalloads. Figure3-7.BuckledITPSduetothermalloads InitiallyonlybucklingoftheITPSunitcellwithoutthefoamwasanalyzed(Figure 3-7 ).Thebucklingeigenvaluepredictedshouldbegreaterthan1.0.Eigenvalueslessthan1.0suggestthatthestructurefailsatloadslesserthantheappliedload.For 49

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instance,iftheeigenvalueisequalto0.7,itimpliesthatat70%oftheappliedloadthestructurewouldbuckleaccordingtothecorrespondingbucklingmode.Ifthesmallesteigenvalueisaboveunity,thenthestructurewillnotbuckleundertheappliedloads.BucklinganalysesoftheITPSunitcellwithoutfoamshowstopfacesheetbuckleswhichleadstothebucklingofthetopweb. 3.3Summary Thedifferentniteelementanalysesincorporatedintheoptimizationprocessispresentedinthischapter.Furtherformoredetailsonthevariousaspectsoftheniteelementmodelingandaforementionedstructuralanalysis,thereaderisreferredtoABAQUSdocumentation[ 77 78 ]. PastresearchincludedthestructuralanalysisoftheITPSpanelwithcorrugatedcore[ 28 ]usingshellelementsneglectingthefoam.Thisresearchconsidersacomplexdesignwithlaminateswrappedaroundadenserfoamthatcannotbeneglectedthusrequiring3-DbrickelementstomodeltheITPSpanel.Astheanalysisofthe3-Dniteelementmodeliscomputationallyintensive,thisresearchproposestohomogenizethe3-DITPSpanelasanequivalent2-Dorthotropicplate,discussedindetailinChapter 4 50

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CHAPTER4FINITEELEMENTBASEDHOMOGENIZATION Structuralresponses(temperature,stressandbucklingload)arecomputedusinghighdelityniteelementanalysesofthe3-DITPSpanelthatarecomputationallyexpensive.Thecomplexdesignofthepanelconsistingoffacesheets,wrapsandinsulationfoamsandthefactthatthesecomponentsaremadeofcompositematerialsmakethecomputationalexpensesprohibitivelylarge.Moreover,thepresenceoffoaminthepanelrequires3-Dniteelementstomodelthepanel.A3-Dniteelementmodel(12x12unitcells)oftheITPSconsistsof114,000nodesandastaticthermo-mechanicalanalysisofthemodeltakesapproximately40minutes.ToreducethecomputationalcostinvolvedintheanalysisofthecompositeITPSpanel,thisresearchproposeshomogenizationofa3-Dplatelikestructureintoa2-Dequivalentorthotropicplate. Thehomogenizationmethodisbasedonmicromechanicsapproach[ 79 ]wherearepresentativevolumeelement(RVE)ofthestructureissubjectedtosixlinearlyindependentdeformationstopredicttheequivalentplatestiffnessproperties.Aunitcellisusedtoestimatethein-planestiffnessproperties([ABD]matrix)anda1-DplatemodelisemployedtoestimatethetransverseshearstiffnessoftheITPSpanel.TheniteelementmodeloftheITPSismodeledusing20-nodebrickelements(C3D20R)inthecommercialniteelementsoftwareABAQUS.Theaccuracyofthemethodisinvestigatedbycomparingtheresponses(displacementsandstresses)ofthe3-DmodeloftheITPSpaneltotheequivalenthomogenizedplatemodel. 4.1A,B,Dmatrices Theunitcellissubjectedtosixlinearlyindependentdeformationstodeterminetheequivalentstiffnesspropertiesofthepanel.Thedeformationsareappliedasperiodicdisplacementboundaryconditions(Table 4-1 )whichincludesthreemid-planestrains("x0,"y0andxy0)andthreecurvatures(x,yandxy)[ 80 ]. 51

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Figure4-1.SchematicviewoftheITPSunitcell(RVE) Table4-1.Periodicboundaryconditionsforthesixdeformations [u(a,y)-[v(a,y)-[w(a,y)-[u(x,b)-[v(x,b)-[w(x,b)-u(0,y)]v(0,y)]w(0,y)]u(x,0)]v(x,0)]w(x,0)] "x=1a00000"y=10000b0xy=10a/20b/200x=1az0-a2/2000y=10000bz-b2/2xy=10az/2-ay/2bz/20-bx/2 Thesedeformationscreatein-planeforces(Nx,NyandNxy)andmoments(Mx,MyandMxy)intheunitcell.Theconstitutiverelationbetweenthein-planeforce,momentsandstrainsandcurvaturegivesthestiffnesspropertiesofthestructure[ 81 ].Theconstitutiverelationisgivenas 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:NxNyNxyMxMyMxy9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;=2666666666666664A11A120B11B120A12A220B12B22000A6600B66B11B120D11D120B12B220D12D22000B6600D6637777777777777758>>>>>>>>>>>>>><>>>>>>>>>>>>>>:"x0"y0xy0xyxy9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;(4) 52

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Forinstancewhentheunitcellissubjectedtoin-planestrain"x0inthex-direction,thefacey)]TJ /F3 11.955 Tf 10.43 0 Td[(zatx=aoftheunitcellisdisplacedbyadistanceainthex-directionrelativetothefacex=0.Oneofthecornernodesisxedtopreventrigidbodymotion.Fromthenodalforcesactingonthenodesonthefacesoftheunitcell,thein-planeforcesandmomentscanbecalculatedasfollows: Nx=1 bnnPi=1F(i)x(a;y;z)Ny=1 annPi=1F(i)y(x;b;z)Nxy=1 bnnPi=1F(i)y(a;y;z)Mx=1 bnnPi=1zF(i)x(a;y;z)My=1 annPi=1zF(i)y(x;b;z)Mxy=1 bnnPi=1zF(i)y(a;y;z)(4) Thein-planeforceswouldprovidetherstcolumnofthein-planestiffnessmatrix[A]andthemomentswouldgivetherstcolumnofthecouplingstiffness[B](Equation 4 ).Theaboveprocedureisrepeatedwiththeothermid-planestrainsandcurvaturestopopulatetheentire[ABD]matrix.Whenthisprocedureisimplementedcorrectly,thecalculated[ABD]matrixshouldbeasymmetricmatrix. Figure 4-3 showsthedeformationoftheunitcellforeachcaseofunitstrainapplied.Figure 4-3 ashowshowtheunitcellwoulddeformduetotheperiodicboundaryconditionsappliedinFigure 4-2 .Theseguresareintendedtoshowhowtheunitcellsdeformduetotheperiodicboundaryconditions. 4.2TransverseShearStiffness InthecurrentdesignoftheITPSpanel,theinsulationbarsactasthecoreofthesandwichpanelandthewrapsactastheshearwebsprovidingconsiderableshearstiffness.TheunitcellmodelwithperiodicBCscannotbeusedforcalculatingthetransverseshearstiffness.Whenatransverseshearforce,say,Qxispresent,itgives 53

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Figure4-2.Periodicboundaryconditionappliedontheunitcellforx=1 Figure4-3.Schematicviewoftheunitcelldeformationsduetostrainsandcurvatures risetothebendingmomentMxwhichvarieslinearlyalongthex-direction.Thustheconditionsforperiodicityareviolatedandonecannotcomeupwithboundaryconditions 54

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thatreectthisvariationexactly.Hence,analternatemethodwithan1-DITPSplatemodelisconsidered.Thismethodcombines3Dniteelementanalysisandbeam/platetheorytocalculatethetransverseshearstiffness.Whiletheformerisaccurate,itcarrieswithittheeffectsoftheboundaryconditionsonbothendsofthemodel.Whenthemodelislongtheendeffectswillbecomeinsignicant.Thebeamtheoryisgoodforlongpanels.Thesetwofactorstogetherrequirelongbeam/1DplateforthepresentITPSmodel. Forinstance,todeterminethetransverseshearstiffnessA55the1-Dplatemodelwithunitcellsinthex-directionisconsidered(Figure 4-4 ). Figure4-4.1-DITPSmodelwithunitcellsinx-direction Theboundaryconditionsonthe1-DITPSmodelareatx=0,u(0;y;z)=0,Bottomedgew(0;y;0)=0;atx=L,u(L;;y;z)=0,w(L;y;z))]TJ /F3 11.955 Tf 10.18 0 Td[(w(0;y;z)=C(C=Constant).Figure 4-4 showsthedisplacementboundaryconditionsandplanestrainboundaryconditionsappliedonthex)]TJ /F3 11.955 Tf 12.08 0 Td[(zplane[ 82 ].Theaimoftheanalysisistodeectthe1-Dplatemodelvertically(transversedirection)usingatipforceF.Thetotaldeectionduetobendingandsheardeformationcanbecalculatedanalyticallyas 55

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w(x)=Fx A55)]TJ /F3 11.955 Tf 20.82 8.09 Td[(F 2D011x3 3)]TJ /F3 11.955 Tf 13.16 8.09 Td[(Lx2 2(4) D011=D11)]TJ /F3 11.955 Tf 13.15 8.09 Td[(B211 A11(4) Thersttermissheardeformationandthesecondtermisbendingdeformation.Fisthenetverticalforceduetothedisplacementboundarycondition,Listhelengthofthe1-DpanelandD011isthereducedbendingstiffness[ 82 ].ThebendingdeectioncanbecalculatedanalyticallyusingtipforceFandthe[ABD]matrixthatisalreadydetermined.ThetotaldeectioncanbecalculatedusingniteelementanalysisandthustheshearstiffnesscanbecalculatedusingEquation 4 Transverseshearstiffnessisacrosssectionalpropertyandforahomogenousstructurewhichcanbeconsideredtobemadeofinnitenumberofunitcells,thetransverseshearstiffnessisindependentofthelength.However,theITPSpanelconsideredherehasnitenumberofunitcells.Theshearstiffnessisoverestimatedduetotheboundaryeffects.Becauseofthelimitedcomputationalresources,itisnotaffordabletondtheshearstiffnessusing1-Dniteelementplatemodelsconsistingoflargenumberofunitcells,hencethestiffnessvalueiscomputedforsmallerlengths,10,20and40unitcellsandthenaconvergencecriterionisappliedtondtheconvergedvalueofthestiffnessvalue.Theconvergenceratecanbeexpressedas k3)]TJ /F3 11.955 Tf 11.96 0 Td[(k2 k2)]TJ /F3 11.955 Tf 11.96 0 Td[(k1=N1 N2(4) wherek1,k2,k3arethevaluesofstiffnessforsayn=10,20and40respectively.N1=N2istheratioofthelengthofthepanel(ratioofnumberofunitcells,hereN1=N2=10/20=20/40=0.5).istheconvergencefactorwhichshouldbegreaterthanunityforconvergenceofthestiffness.Calculatingtheconvergencefactorfromn=10,20and40,thenusingEquation 4 ,thestiffnesscouldbepredictedforlongerpanelstoobtaina 56

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convergedvalueofthestiffnessproperty.SimilarprocedureiscarriedoutforestimationofA44.Theconvergedtransverseshearstiffnessalongwiththe[ABD]matrixisusedtoestimatetheaccuracyofthehomogenizedmodel. 4.3AccuracyoftheHomogenizationMethod 4.3.1DeectionComparison TheITPSpanelissubjectedtovariouscombinationsofloadswheninstalledontheexteriorofthespacevehicle.Pressure,temperature,in-planeinertialloadsandimpactloadsarethevariousloadsthatactonthepanelduringight.Inthischapter,pressureloadsareappliedtocomparetheresponse(displacementandstresses)ofthe3-DITPSpaneltothatofthehomogenizedplatetoevaluatetheaccuracyofhomogenizationprocedure.Sinceitiscomputationallyexpensivetoconstructaniteelementmodelofthe3-DITPSpanelofincreasedlengths(Figure 3-1 ),an1-Dmodelconsistingofseriesofunitcellsineither(xory)directionisconsidered.The3-DITPSmodelisxedatoneendandfreeattheotherendwithplanestrainboundaryconditionsparalleltothex-zplane.Itissubjectedtoapressureloadp=1000Pa(Figure( 4-5 )).Foranequivalent1Dorthotropicplatewithsimilarboundaryconditionsandload,aclosedformsolutionofthedeectionisavailableasshowninEquation 4 w(x)=)]TJ /F3 11.955 Tf 16.43 8.09 Td[(p A55Lx)]TJ /F3 11.955 Tf 13.15 8.09 Td[(x2 2)]TJ /F3 11.955 Tf 22.48 8.09 Td[(p 2D011L2x2 2+x4 12)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Lx3 3(4) Thehomogenizedandthe3-DITPSmodel(Figure( 4-5 ))deectionsarecomparedtoevaluatetheaccuracyoftheequivalentstiffnesspropertiesestimated. 4.3.2StressComparison ThestressesarecomparedbetweentheequivalenthomogenizedplateandITPSpanelbyemployingthereversehomogenizationprocedure.TheoriginalITPSmodelissubjectedtopressureloadandstressesatvariouscrosssectionsaredetermined.Forthehomogenizedmodelsubjectedtopressureload,atanydistancex,mid-planestrainsandcurvaturescanbecalculatedas 57

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Figure4-5.1DmodeloftheITPSwithunitcellsinx-directionsubjectedtopressureloadandxedBCs Mx=p(L)]TJ /F3 11.955 Tf 11.96 0 Td[(x)2 2(4) x=Mx1 D011(4) "x0=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B11 A11x(4) Themid-planestrainsandcurvaturesofthehomogenizedplateareappliedtotheITPSunitcellusingtheperiodicboundaryconditions(Table 4-1 )andthestressesaredetermined.Thisisreferredasreversehomogenizingtheequivalenthomogenizedplatetothe3-DITPSmodel.ThestressesintheunitcellarecomparedtothatoftheITPSatvariouspointsalongitslength. 4.4ResultsandDiscussion 4.4.1A,B,Dmatrices-Equivalentmaterialvs.Equivalentstructure Someresearchershaveconsideredhomogenizingthesandwichpanelasaplatemadeofahomogeneousmaterial.Thiswillbeapplicablewhenthepanelissubjectedtoonlyin-planeforces.However,whentherearetransverseforcesduetosurfacepressure 58

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orvibrations,thebendingstiffnesshastobeconsideredandaswillbeshownbelowthe[D]matrixhastobeevaluateddirectly.Further,forthecaseofaplatemadeofhomogeneousmaterial,[B]0.Howeverbending-stretchingcouplingmaybepresentduetoasymmetryandthismayplayacrucialroleinbucklingofthepanel. Homogenizingtoequivalentmaterialbasicallysmearsthepropertiesthroughoutthepanelbytreatingitasahomogenousorthotropicmaterial.Thein-planestiffnessmatrix[A]canbecalculatedfromelasticconstantsandusingtheconstitutiverelationinEquation 4 .Fromtheclassicallaminationplatetheory,the[D]matrix(bendingstiffness)isequalto(h2/12)[A]andforisotropichomogenousmaterial,[B][0].The[A]and[D]matrixcalculatedassumingthepanelisanequivalentmaterialisgiveninEquation 4 andEquation 4 [A]eqv:matl:=2666649141061431060143106913106000241106377775(4) [D]eqv:matl:=2666641611032510302510316010300043103377775(4) However,theITPSconstructionhasdifferentmaterialsforthetopfacesheet(TFS)andbottomfacesheet(BFS)andsothe[B]matrixcannotbezero.ItisinterestingtocomparetheITPSpropertiesasequivalentstructure(Equation 4 )versussmearingthepropertiestoformanequivalentmaterial(Equation 4 )and(Equation 4 ).Ifhomogenizedasanequivalentstructure,anon-zero[B]matrixisobtainedandthebendingstiffnessseemstobetwicethatofequivalentmaterial.ThusitisimportanttonotethathomogenizingtheITPSpanelusingequivalentstructureapproachcapturesthebehaviorofsandwichstructuresaccurately(Equation 4 )whencomparedtoequivalentmaterialapproach. 59

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[ABD]eqv:str:=266666666666666491410614310602:11061:4106014310691310601:41065106000241106002:21062:11061:410603101035010301:410651060501033101030002:210600861033777777777777775(4) Moreover,classicallaminationtheoryassumesinniteshearstiffness(A44=A55=innity)whichwouldmaketheITPSastiff,rigidstructure,butitwillbeshowninthenextsectionthatthepanelisasheardeformablesandwichstructure.Thenextsub-sectiondiscussesthesignicanceoftransverseshearstiffnessA44andA55ofthepanel. 4.4.2TransverseShearStiffnessofITPS ThetransverseshearstiffnessA44andA55arecrosssectionalpropertiesandhencearesupposedtobeaconstantforagivenmicrostructure.Theanalysisofthe1-Dplatemodeltodeterminetransverseshearstiffnesswasvalidatedconsideringamodelmadeentirelyoforthotropicmaterialwhosetransverseshearstiffnesscouldbecalculatedanalyticallyas A55=5 6Gh(4) WhereGistheshearmodulusinthex)]TJ /F3 11.955 Tf 12.33 0 Td[(zplaneandhistheheightofthemodel.Theorthotropicmaterialconsideredwasgraphite/epoxycomposite,G=6:9GPa(Table 3-3 ).ThelengthofthemodelwasL=420mm(n=20),widthw=21mmandheighth=46mm.ThetransverseshearstiffnessoftheorthotropicITPSpanelwasestimatedandcomparedwiththeanalyticalstiffnessvalue(Equation 4 ).Ideallyforsuchapanelmadeofinnitenumberofunitcells,thestiffnessvalueisindependentofthelength,howeverduetotheboundaryconditionsappliedonthepanelends,it 60

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approximatelytakes0.2mtoconvergeandagreewiththeanalyticalsolution.AstheITPSpanelconsistsofnitenumberofunitcells,thiseffectwouldbemorepronouncedandwouldrequirelargenumberofunitcellsforthestiffnessvaluetostabilizetoaconvergedvalue. Figure4-6.TransverseshearstiffnessA55oforthotropicpanel NowtheabovemethodisappliedtotheITPSmodelwithvaryingmaterialconguration.Owingtothecomputationalexpense,modelconsistingofunitcellsineitherxorydirectionisconsidered.Unitcellsarrangedinthex(A11,B11andD11stiffnessesconsidered)andy-direction(A22,B22andD22stiffnessesconsidered)providesA55andA44respectively. Table4-2.VariationofA44andA55ofITPSpanelwithn nA55N/mA44N/m 1016.910620.61062010.110611.8106408.11068.8106 61

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Figure4-7.VariationofA44andA55ofITPSpanelwithn Figure 4-7 showsthevariationofthetransverseshearstiffnessasafunctionofnumberofunitcells.Itcanbeseenthatthetransverseshearstiffnessishighwhenfewerunitcellsareconsidered,itdecreasesasthenumberofunitcellsincreases.Sincetheniteelementanalysesoflargeplatemodelswaslimitedbytheavailablecomputationalresources,theaforementionedconvergencecriterionisusedtoestimatetheshearstiffness.UsingthestiffnessvaluesfromTable 4-2 andEquation 4 theconvergencefactorisestimated.Asmentionedearliershouldbegreaterthanunityforconvergence.Thevaluesofwerefoundtobe2.2and1.9forA55andA44respectively,whichsuggestsconvergenceofthestiffnessvalue. Figure4-8.ConvergedA44andA55ofITPSpanelwithn 62

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Thetransverseshearstiffnesscanbepredictedforlargerpanelsusingtheabovevalueof.Consideringk1andk2correspondingtoN=10and20respectively,thevalueofk3(N=40)canbecalculatedusingEquation 4 (hereNi=Ni+1isalwaysequalto0.5).Thususingsuccessivecalculationsstiffnessforverylargenumberofunitcellscanbeestimated.Foraverylargenumberofunitcells(500unitcells),thetransverseshearstiffnessconvergedto7.4106(A55)and7.1106(A44)alsoshowninFigure 4-7 .Itisimportanttonotethatitrequireslargenumberofunitcellsfortheshearstiffnesstoconvergetoaconstantvalue.ThissuggeststhatdeterminedshearstiffnesscannotbeusedtohomogenizesmallerpanelssuchastheITPSpanelconsistingof12x12unitcells.However,theconvergedtransverseshearstiffnessandthecalculated[ABD]matrixcouldbeusedtopredicttheminimumlength(numberofunitcells)requiredforhomogenizationmethodtoyieldacceptableresults. 4.4.3AccuracyoftheHomogenizationMethod ThetipdeectionfromobtainedniteelementanalysisoftheITPSmodeliscomparedwiththeanalyticaldeectionofthehomogenizedplate(Equation 4 )forvariouslengthsofthecantileverbeammodel.Whenthetipdeectionratiobetweenthe2-Dmodeland3-Dmodelisclosetounity,thecorrespondinglengthwouldbetheminimumsizerequiredforhomogenizationtobeapplicable. Figure4-9.Ratiobetweentipdeectionofthehomogenizedmodelandthe3-DITPSmodelalongthexandy-direction 63

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Table4-3.Tipdeectionratioalongwithcontributionofbendingandsheardeformationtowardsdeection L/aTipdeectionratioalongShear%Bending%x-direction(w2D=w3D) 121.367228201.124852401.051981601.01991701.00793 UsingEquation( 4 )thecontributionofsheardeformationtowardstotaldeectionwasdetermined.FortheITPSpaneltypicallyconsistingof12unitcells,thecontributionofshearis72%thusconrmingthepanelishighlysheardeformable.Thetablealsoprovidesthecontributionofsheardeformationforlongerpanels.Figure 4-9 showsthetipdeectionratioforvariouslengthsoftheITPSmodel.Thex-axisoftheplotisL=aratio(a-widthofunitcell)denotingthenumberofunitcells.ItisevidentfromFigure 4-9 thattheminimumlengthofthepanelmustbe1.5m(70unitcells)tohomogenizetheITPSstructuretoanequivalentorthotropicplate.Evenforsuchalargepanel,thereis7%sheardeformationandneglectingtransverseshearstiffnesswouldyieldin7%discrepancyinthehomogenizationofthepanelfordeections. Nowthepanelwithlargerlength(1.5m)isanalyzedunderpressureloadforstresscomparison.Thestressesinthehomogenizedmodelaredeterminedbyapplyingmid-planestrainsandcurvatures(Equation 4 and 4 )onaunitcell.Thestressratiobetweenhomogenizedandoriginal3-Dmodeliscomparedatcrosssectionsatvariousdistancesalongthelengthofthepanel(Figure 4-10 ).Itcanbenotedthatthestressratioclosetofreeendisgreaterthanunity.ThiscanbeattributedtotheSt.Venant'seffectsuggestingcertaindistanceintothemodelisrequiredforthefreeedgeeffectstodissipate.Thestressratiodecreasesandconvergestounity(approximately)atL=a=10. ThestressvaluesatL=a=10alongthewidthofthepanelisshowninFigure( 4-11 ).Furthertoshowthathomogenizationcannotbeappliedforsmallpanels,the3-DITPSandhomogenizedplatemodelsconsistingof12x12unitcellsare 64

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Figure4-10.Stress(11,22)ratiobetweenthehomogenizedmodelandthe3DITPSmodel Figure4-11.Stress(11,22)atn=10comparisonbetweenthe3-DITPSmodelandthehomogenizedmodel analysedunderpressureloadandxedboundaryconditions.Aquarteroftheplateisconsideredwithsymmetryboundaryconditions.Figure 4-12 showsalargediscrepancyinthedeectionsbetweenthe2-Dhomogenizedplatemodeland3-DITPSpanel(w2D=w3D=3.8). Itcanbeconcludedthatacompositesandwichpanelhastobesufcientlylargetohomogenizeittoanorthotropicplate.ForthecurrentITPSpanelwithafairlycomplexdesignandmadeofcomposites,aminimumof70unitcellsarerequired. 65

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Figure4-12.Verticaldeectionalongx-axisoftheoriginalITPSandhomogenizedmodelduetopressureload 4.5Summary Arepresentativevolumeelement/unitcelloftheITPSpanelwasanalyzedtodeterminetheequivalentstiffnesspropertiesofthe2Dplate.Theunitcellwassubjectedtosixlinearlyindependentdeformationsandtheequivalentstiffnesspropertiesofthecompositestructurewereestimated.Analysisofthe1-DITPSplatemodelthatcombinesniteelementsandsheardeformableplatetheorywasproposedtoestimatethetransverseshearstiffnessoftheequivalenthomogenizedplatemodel.Toevaluatetheaccuracyofthehomogenization,thehomogenized2-Dplatemodelwasanalyzedunderpressureloadandcomparedwiththe3-DITPSpanel.Thevaluableinsightsofthisresearchworkare:a)TransverseshearstiffnessofITPSsandwichpanelsareimportantandcannotbeneglected.b)Homogenizationcannotbeappliedtothesmallpanelswithfewerunitcells.HenceasingleunitcelloftheITPSpanelwillbeusedforoptimizationprocess. Thehomogenizationmethodisapplicabletopanelsoflengthmuchlargerthantheunitcelldimensions.ItwasestimatedthattohomogenizethecurrentdesignofITPSpaneltoanequivalentorthotropicplate,thepanelmusthaveaminimumof70unitcells. 66

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CHAPTER5MONTECARLOSIMULATIONS-RELIABILITYESTIMATION ThischapterprovidesadetaileddescriptionofthecrudeMonteCarloandseparableMonteCarlomethod,there-allocationofuncertaintytopredictaccurateestimateofprobabilityoffailureandbootstrappingtechniquetoestimatetheaccuracyofthepfestimate. ThemeritsofusingseparableMonteCarloovercrudeMonteCarlomethodisillustratedusinganon-separablelimitstatefunction(theresponseandcapacitycomponentsareintegratedinthelimitstate).TheprobleminvolvesestimatingtheprobabilityoffailureofcompositepressurevesselusingTsai-Wufailurecriterion(non-separablelimitstate).TheaccuracyofthemethodsiscomparedfordifferentgroupingsoftherandomvariablesfortheseparableMonteCarlomethodfromtheestimateofcoefcientofvariation/standarddeviation.ItisdemonstratedthatusingSMCwiththeregroupedlimitstatereducestheerrorassociatedwiththeprobabilityoffailureestimate.Thisisaccomplishedbyreallocatingtherandomvariablesasinexpensivevariableswithnolimitonsamplesizeandexpensivevariableswithlimitedsamples.Furthertheexpensivevariablesarebootstrappedandtheaccuracyofthebootstrappedprobabilityoffailureestimateiscomparedwiththatoftheempiricalestimate. 5.1CrudeMonteCarloMethod(CMC) Acommonsampling-basedmethodforcalculatingtheprobabilityoffailurepf,istraditional,crudeMonteCarlo.Theprobabilityoffailureisestimatedbycomparingpairsofrandomlygeneratedresponseandcapacitysamples,asshowninEquation 5 ^pcmc=1 NNXi=1I[G(Ri;Ci)0](5) WhereIistheindicatorfunction,whichequals1iftheconditionistrueand0iftheconditionisfalse.Thus,thisessentiallysumsthenumberoffailuresforNcomparisons. 67

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Therootmeansquare(RMS)errorintheestimatemaybemeasuredbythestandarddeviationforcrudeMCgivenas stdev(^pcmc)=r pf(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pf) N(5) CV(^pcmc)=stdev(^pcmc) N=s (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pf) pfNr 1 pfN(5) Forexample,foraprobabilityoffailureofoneinamillion,100millionsimulationsareneededfor10%error.Thecoefcientofvariationcalculatedfromstandarddeviationandmeanprovidesameasureoftherelativerootmeansquare(RMS)errorintheprobabilityoffailureestimateaboutitsmeanvalueEquation 5 .Notethatbecausewehaveonlyanestimateoftheprobabilityoffailure,wegetonlyanestimateoftheerror.ThecostofresponsecalculationisoftenthelimitingfactorinthenumberofsamplesN,becauseresponsecalculationsoftenrequireexpensiveniteelementsimulations. 5.2SeparableMonteCarlomethod(SMC) Whenthecapacityandresponsearestatisticallyindependentinthelimitstate,thentheycanbesampledseparatelyusingamethodcalledseparableMonteCarlo(SMC).SeparableMChasalreadybeeninvestigatedforthesimplelimitstateasshowninEquation 2 .ThisstudylooksatSMCforthegenerallimitstateinEquation 2 ,theseparableMCmethodisshowninEquation 5 ^pcmc=1 NNXi=1I[G(Ri;Ci)0](5) WhereNisthenumberofresponsesamplesandMisthenumberofcapacitysamples.Sincetheresponseandcapacityaresampledseparately,allofthepossiblecombinationscanbeconsideredtoestimatepf.Figure 5-1 illustratestheresultingdifferencebetweenCMCandSMC. 68

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ACrudeMonteCarlo BSeparableMonteCarlo Figure5-1.IllustrationofcrudeandseparableMonteCarloMethodcomparisonsA.CrudeMonteCarlomethodB.SeparableMonteCarlomethodwhereeverysampledresponseiscomparedwitheverysampledcapacity InFigure 5-1 a,thedirectone-to-onecomparisonsofcrudeMCareshownforNsamples.Whereas,Figure 5-1 bshowsthatseparableMClooksatallofthepossiblecombinationsofrandomsamples,whichmakesitinherentlymoreaccuratethanCMC.Inaddition,differentsamplesizescanbeusedtoenhancetheaccuracy,dependingontherelativecomputationalexpenseoftheresponseandcapacitywhichwasdemonstratedusinganon-separablelimitstatetopredicttheprobabilityoffailureofacompositepressurevessel(Section 5.5 ). 5.3ErrorintheProbabilityofFailureEstimate ForCMC,theRMSerrorintheprobabilityestimateisprovidedbyEquation 5 .ForseparableMCwiththesimplelimitstateasinEquation 2 ,Smarsloketal.[ 15 ]derivedanalyticalestimatesofthestandarddeviationviaexpectationcalculus.ForthemoregeneralcaseofEquation 2 ,weproposebootstrappingthecomponentsofthelimitstate[ 18 ].Forestimatingtheerrorintheprobabilityoffailureestimate,weusebootstrapping,are-samplingtechnique,whichinvolvestakingthesamplesofresponse(expensive)andre-samplingthemwithreplacement(sothatthesamplesmaycontainduplicates).Whenweperformthere-samplingbtimes,weobtainbprobabilityoffailure 69

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Figure5-2.Schematicrepresentationofbootstrappingwhenonlyresponseissampled. estimates^pboot.Withbestimatesof^pboot,wecanobtainthestandarddeviationdenotingtheerrorintheestimate.Aswillbeshowninthelatterpart,thebootstrappingerrorestimatesappearstobecomparabletotheCMCestimateofEquation 5 Furthermore,wecanobtain^pbootestimatesbybootstrappingcapacityatmeanvaluesofresponseandviceversa.Theknowledgeoftheindividualcontributionsoftheresponseandcapacitytowardstheuncertaintyaidsinchoosingtheappropriatesamplesizeforresponse(N)andcapacity(M)whichwouldprovideanaccurateestimateofthevariationinthepfestimate.Whenwere-sampleboththeresponseandcapacity,weobtainthetotaluncertaintystdev(^psmc;boot).Butfortheindividualcontributionsoftheresponseandcapacity,theresponsehastobootstrappedatmeancapacityandviceversatoobtainstdevR(^psmc;boot)andstdevC(^psmc;boot). Inthenumericalresults,thebootstrappingvaluesarecomparedwiththeempiricalvalues,stdevR(^psmc;boot)andstdevC(^psmc;boot)todemonstratetheaccuracyofthebootstrappingmethod.Inordertomeasuretheerrorinthebootstrappedestimate,theuncertaintyinthestandarddeviationofthebootstrappedprobabilityoffailureestimate(stdev(stdev(^psmc;boot)))isalsocalculated.Thebootstrappingestimateswouldallowus 70

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tojudgewhetherthenumberofexpensivesimulationsissufcientforadesiredlevelofaccuracy. 5.4SMCwithregroupingandseparablesamplingofthelimitstaterandomvariables Whenthebootstrappingestimatesshowthattheaccuracyoftheprobabilityoffailureisnotgoodenough,andthattheculpritistoofewsamplesoftheexpensivesimulation,wemayhaveapainfulchoicebetweenveryhighcomputationalcostandaccuracy.Oftenthereisanalternativetoincreasingthenumberofexpensivesamplesbyreformulatingthelimitstate,whichisdescribedinthissection.Thenumberofsamplesrequiredforaccuratemodelingoftheresponseandcapacitydependsontherelativecontributionsoftherandomcomponentsinthelimitstatefunction.Wherethelargertheuncertaintycontribution,moresamplesarerequiredforaccuraterepresentationofthedistribution.Assumingacomputationallimitonthenumberofsamplesoftheexpensiveresponse(N),itisdesirabletoreducetheuncertaintyintheresponse(i.e.obtainnarrowerdistributionoftheresponse)toachieveimprovedaccuracy. Inmoststructuralproblems,failureofthesystemdependsonthestrengthofthematerialS(e.g.capacity),andstressesthestructuresustains(e.g.response).SothelimitstateinEquation 2 maybecomeG(;S).Inlinearproblems,stresses,arealinearfunctionoftheloadp,asinEquation 5 =Up(5) WhereUarestressesperunitload.Therandomnessintheloadp,isoftenindependentoftherandomvariablesthataffectU(geometryandmaterialproperties),butpaddsuncertaintytothecomputationallyexpensivestresscalculation.SoitwouldbeadvantageoustoseparatetheloadsfromthestressesanddeterminestressesperunitloadU.Thisalsoenablesalargersamplesizeoftheload.Evenwithalimited 71

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numberofsamplesofstressesperunitload,theprobabilityoffailurecanbemoreaccuratelyestimatedasboththestrengthandtheloadcanbecheaplysampled.ThentheexpensiveunitloadresponseUissampledandcombinedwiththeloadinEquation 5 .Finally,thelimitstateisreformulatedas G(;S)=G(U;p;S)(5) TheprobabilityoffailurecanbedeterminedfromalargesampleofloadsandstrengthscomparedtoalimitedsampleofstresseswhichisillustratedinFigure 5-3 Awithstresses Bwithunitloadstresses Figure5-3.IllustrationofseparablesamplingwithunitloadsA.withstressesB.withunit-loadstresses TheseparableMonteCarloformulathatcorrespondstoFigure 5-1 bisshowninEquation 5 ^pUsmc=1 M1 NMXj=1NXi=1I[GU(Ui;Pj;Sj)0](5) AsimilarformofEquation 5 couldbewrittenforFigure 5-3 ,butwithdifferentindices. 5.5ApplicationtoFailureAnalysisofCompositeLaminate ThecrudeMonteCarloandseparablesamplingsimulationmethodsandtheirerrorestimatesarecomparedandillustratedbyapplyingthemtoanon-separablelimitstateproblem(Equation 2 ).Forcomplexstructures,thestressiscalculated 72

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Figure5-4.Compositepressurevesselwithinternalpressureof100kPaandstressesactinginasmallelementofthevessel. throughniteelementanalysisanditmaybecomputationallyexpensive.InordertoallowustoperformthousandsofMonteCarlosimulationsneededtovalidatethemethod,weselectedanexamplethatrequiresthecalculationofstressesatasinglepointusingClassicalLaminationTheory(CLT).TheprobleminvolvespredictionoffailureofcompositepressurevesselaccordingtotheTsai-Wufailurecriterion. Acompositelaminatepressurevessel(Figure 5-4 )ismadeofagraphite/epoxy[+25/-25]ssymmetriclaminatewitheachlayerbeing125mthickandissubjectedtoaninternalpressureof100kPa.ThematerialpropertiesofthecompositeareshowninTable 5-1 .Inthispaper,alloftheinputrandomvariablesareassumedtobenormallydistributed.However,theperformanceofSMCdependsonthedistributionoftheresponseandcapacity,whichisnotnecessarilynormal.Previousresearchhasusedotherdistributions,suchaslognormalanduniform,withseparableMonteCarlomethod[ 15 61 ]. Thestressesgeneratedareafunctionoftheinternalpressurepandmaterialpropertiesofthelaminatewhichareindependentofeachother. 8>>>><>>>>:12129>>>>=>>>>;=[T][Q][A])]TJ /F4 7.97 Tf 6.59 0 Td[(1=8>>>><>>>>:pd=2pd=409>>>>=>>>>;=[T][Q][A])]TJ /F4 7.97 Tf 6.59 0 Td[(18>>>><>>>>:d=2d=409>>>>=>>>>;p=8>>>><>>>>:U1U2U129>>>>=>>>>;p(5) 73

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Thestresses1,2(normal)and12(shear)actingineachplyofthelaminatearefunctionofin-planestiffnessmatrixofthelaminate[A],reducedstiffnessmatrixofeachlamina[Q],transformationmatrixofeachlamina[T],thepressureloadpandthediameterd(1m)ofthevesselasinEquation 5 .Failureofcompositelaminateispredictedfromthestrengthofthecompositeandstressesgeneratedusingasuitablefailurecriterion.ThemostwidelyusedcriterionforcompositesistheTsai-Wucriterion.ThecriterionisafunctionofthestrengthsS(showninTable 5-1 alongwiththeiruncertainties),normalstresses,intheberandtransversedirection(e.g.1and2direction,respectively)andshearstress,. Table5-1.Materialpropertiesanduncertaintyoftherandomvariables(normallydistributed)[ 83 ] PropertiesMeanCV%StrengthMeanCV% E1GPa159.15%S1T231210%E2GPa8.35%S1C180910%G12GPa3.35%S2T39.210%v12GPa0.2535%S2C97.210%PkPa10015%S1233.210% Accordingtothecriterion,alayerofthelaminateisassumedtohavefailedwhenthelimitstateinEquation 5 isgreaterthanorequaltozero. G(;S)=F1121+F2222+F66212+F11+F22+F1212)]TJ /F9 11.955 Tf 11.96 0 Td[(1(5) F11=1 S1TS1CF1=1 S1T)]TJ /F9 11.955 Tf 19.49 8.09 Td[(1 S1CF22=1 S2TS2CF2=1 S2T)]TJ /F9 11.955 Tf 19.49 8.09 Td[(1 S2CF12=1 S212F12=p F11F22 2(5) UncertaintiesintheTsai-Wucoefcients(F11;F22;F66;F1;F2;F12)areduetorandomnessintheunidirectionaltensile,compressive,andshearstrengthsSofthecomposite.Uncertaintyinthestressesisduetorandomnessinmaterialproperties(orU)andpressureloadp.Thein-planenormalandshearstresses(1,2,12T),are 74

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Figure5-5.Distributionofstressandstrengthinthe2-direction(2)showingtheprobablefailureregion uniqueineachlayerofthelaminate.Aspreviouslymentioned,thelaminateismadeof[+25/-25]spliesandtheanalysisshowsthattheinner(-25)pliesfailandthenouterpliesfail.Amongthenormalandshearstressesactingontheply,thestressinthetransversedirection2causesthefailureofthelaminate(theoverlapofthestressesandstrengths)whichcanbeseeninFigure 5-5 ThestressesareafunctionofmaterialpropertiesandinternalpressurepasinEquation 5 .Forunitpressureload(p=1),stressesareequaltoU.Therefore,theoriginallimitstatefunction(G(;S))canbereorganizedasindicatedbyEquations 5 5 and 5 5.6ResultsandDiscussion 5.6.1CrudeandSeparableMonteCarloMethod TheprobabilityoffailureofacompositepressurevesselwascalculatedusingcrudeMonteCarloandseparableMonteCarlomethod.Itwasassumedthatourcomputationalbudgetonlypermitted500stresscalculations.Therefore,forcrudeMonteCarlo,anequalnumberofrandomresponseandcapacity(S)variables(N=500)weresampledforcomparison.InthecaseofseparableMonteCarlo,theresponsesamples(N)werecomparedagainstallthecapacitysamples(M=500)resultingin250,000evaluationsofthelimitstate.Theactualprobabilityoffailureispf=0.0121(Asthisisasimpleproblem,theactualprobabilityoffailurewasestimatedbyCrudeMonteCarlomethodby 75

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generating5millionsamplesofstressandstrengthvalues).TherelativeerrorincrudeMonteCarlowasmeasuredbycalculatingthestandarddeviationfromEquation 5 andhencethecoefcientofvariation.Thisvalueprovidesameasureofhowaccurateistheprobabilityoffailureestimate.Forasimplelimitstate(asinEquation 2 ),theaccuracyofseparableMonteCarlocanbeestimatedasderivedbySmarsloketal.[ 16 ].Sincethisproblemisdenedbyagenerallimitstate(e.g.Tsai-Wu),bootstrappingwasperformedtoassesstheaccuracyoftheprobabilityoffailureestimateofSMCwithab=1,000bootstraprepetitions.Sincethestressesarecomputationallyexpensiveanditischeapertosamplethestrengths,randomsamplesofstresseswerebootstrapped,butthestrengthsweresampledanewratherthanbootstrapped.Thatis,ineachofthe1,000bootstraprepetitions,thestresseswerere-sampledfromthesame500samples,whilethestrengthshadafreshsampleeverytime. Inthisstudy(wheresimpleCLTcalculationswereused)theaccuracyoftheseparableMonteCarlomethodwasalsoassessedbyanempiricalcoefcientofvariationobtainedbyperformingn=10,000repetitions.TheprobabilitiesoffailureestimatesarelistedinTableandtheestimatesoftheerrorintheprobabilityoffailurearetabulatedinTable 5-3 .Itshowsthatthecoefcientofvariationisreducedfrom40%to21.0%bySMC.OnaveragethebootstrappingerrorestimateoftheSMCprobability(asmeasuredbythestandarddeviation)isabout20%low(0.002comparedto0.0025),withastandarddeviationwhichisvetimeslowerthanthataverage.Thusinthelargemajorityofcasestheerrorestimateiswithin50%oftheempiricalerror. Table5-2.EmpiricalandbootstrappingestimatesofprobabilityoffailureusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitions CMCSMCoriginallimitstateSMCRegroupedEmpiricalBootstrappingEmpiricalBootstrappingmean(^pcmc)mean(^psmc)mean(^psmc;boot)mean(^pusmc)mean)]TJ /F9 11.955 Tf 6.46 -9.69 Td[(^pusmc;boot 0.01210.01210.01210.01200.0122 Nextwedemonstrateobtainingtheindividualcontributionoftheresponseandcapacitytotheuncertaintyintheprobabilityoffailureestimateobtainedby 76

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Table5-3.StandarddeviationandcoefcientofvariationofempiricalandbootstrappingpfestimatesusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitionsfororiginallimitstate.Standarddeviationofthebootstrappingerrorisalsoshown CMCSMCoriginallimitstateSMCRegroupedEmpiricalBootstrappingstdev(^pcmc)CV(^pcmc)stdev(^psmc)CV(^psmc)mean(stdev(stdev(stdev( ^pusmc;boot)^pusmc;boot))0.004840%0.002521%0.00220.0004 bootstrappingtheresponseatmeanvaluesofthecapacity,andviceversa.Theindividualcontributionswouldhelpwhentheoverallerrorestimateislargeandneedstobereducedbyincreasedsamplesize.Thevaluesofmean(stdevR)]TJ /F9 11.955 Tf 6.46 -9.69 Td[(^pusmc;boot)andmean(stdevC)]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pusmc;boot)inTable 5-3 providetheuncertaintyinpfestimateduetothestressandstrengthrespectively.Thesevaluesarealsocomparedwithrelativecontributionsofthestressandstrengthobtainedempiricallytoillustratetheaccuracyofthebootstrappingmethod. Table5-4.Relativecontributionsofresponse(stresses)andcapacity(strengths)towardstheuncertaintyinpfthroughbootstrappingandalsocomparedwithempiricalresults EmpiricalBootstrapping stdevR(^psmc)0.0017mean(stdevR(^psmc;boot))0.0019CVR(^psmc)15.4%stdev(stdevR(^psmc;boot))0.0004stdevC(^psmc)0.0012mean(stdevC(^psmc;boot))0.0014CVC(^psmc)9.8%stdev(stdevC(^psmc;boot))0.0002 FromTable 5-4 wecanseethatthecontributionoftheresponsetotheuncertaintyinthepfestimateishigherthanthecontributionofthecapacity.Itispossibletoreducetheresponseuncertaintybyusingalargersample.However,sinceresponsecalculationisusuallyexpensive,welookinsteadtoreducetheuncertaintyintheresponsecontributionbyothermeans.Theresponsecontainstheloadwithitslargeuncertainty(CV(p)=15%).CalculatingstressperunitloadU,isolatestheexpensiveCLTcalculation(orFEA)fromthelargeuncertaintyintheload.Thenextsection 77

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exploreshowtherandomcomponentsinthelimitstatecanbereformulatedbyusingunitstressestoreducetheerrorinthepfestimate. 5.6.2Regroupingandseparablesamplingofthelimitstatevariablesforimprov-ingaccuracy Intheoriginallimitstate(G(,S)),thestresscalculationcontainsthelargeuncertaintyfromtheload.Therefore,rearrangingtheresponsetoobtainstressperunitloadU,andloadppermitscalculatingresponsethatdoesnotincludetheuncertaintyintheload.ThisarrangementwillenableseparablesamplingoftheindependentvariablesofthelimitstateGu(U,p,S),similartothatshowninEquation 5 .Thisregroupingshiftsthelargeuncertaintyintheloadawayfromtheexpensivestresscalculation)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(U.ForN=M=500(N=numberofstressperunitloadsamples,M=numberofsamplesofstrengthandload),theuncertainty(mean(stdev()]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pusmc;boot))inthebootstrappedestimateforthereformulatedlimitstateduetotheresponse(unitloadstresses)reducestofrom0.0019to8.6x10)]TJ /F4 7.97 Tf 6.59 0 Td[(5.Ontheotherhand,thecapacity/loaduncertaintyincreasestofrom0.0014to0.0044.Itwouldappearthatwemadethesituationworse,butnowwecanreducetheerrorinpfestimatebyincreasingM,whichisnormallycheap.ThevalueofMwasvariedfrom500-10,000samplesandtheuncertaintyintheestimateforreformulatedlimitstateisshowninTable 5-5 .Itisclearlyseenthattheregroupingallowsustokeepasmallnumberofresponsecalculationsandreducetheuncertaintybyhavingaverylargenumberofinexpensivecapacity(load)calculations.Thestandarddeviationoftheprobabilityoffailurefortheregroupedlimitstatestdev)]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pUsmcwasalsoestimatedempiricallyandshowninTable 5-6 .ThestandarddeviationsobtainedareplottedinFigure 5-6 Figure 5-6 clearlyillustratestheeffectofregroupingoftheinexpensiverandomvariablesofthelimitstate.InthecrudeMonteCarlomethod,theprobabilityoffailureiscalculatedusinganequalnumberofresponseandcapacitysamples.Inthiscase,500 78

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capacitysamplesand500responsesampleswereused,whichcorrespondstoasinglevalueontheplotinFigure 5-6 Table5-5.StandarddeviationandcoefcientofvariationofCMCandSMCforincreasingsamplesizeofMandN=500.Bootstrappedandempiricalestimatesareshown. CMCSMC(Empirical)SMC(Bootstrapping)Mstdev(^pcmc)CV(^pcmc)stdev(^psmc)CV(^pcmc)mean(stdev(stdev(stdev(^psmc;boot)^psmc;boot) 5000.004840%0.002521%0.0020.000450000.002117.6%-*-100000.002016.8%-Table5-6.StandarddeviationandcoefcientofvariationofCMC,SMCandSMCregroupedforincreasingsamplesizeofMandN=500.Bootstrappedandempiricalestimatesareshown. CMCSMC(Empirical)SMC(Bootstrapping)Mstdev(^pcmc)CV(^pcmc)stdev)]TJ /F9 11.955 Tf 6.46 -9.69 Td[(^pUsmcCV)]TJ /F9 11.955 Tf 6.47 -9.69 Td[(^pUsmcmean(stdev(stdev(stdev(^pUsmc;boot)^pUsmc;boot) 5000.004840%0.004537.2%0.00460.000150000.001512.6%--100000.00097.9%-Incontrast,theseparableMonteCarlomethodcanusedifferentsamplesizes(M)fortherandomvariables.ObservethatthestandarddeviationfromtheoriginallimitstateofSMClevelsofftoanearlyconstantvalueof0.002forMsamplesgreaterthan5000.Ontheotherhand,thestandarddeviationfortheregroupedlimitstatecontinuallydecreaseswiththenumberofMsamplesallthewaytostdev)]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pUsmc0.0009(or7.9%CV).Inotherwords,inCMCtheerrorestimateofthefailureprobabilityis40%,buttheerrorassociatedinSMCisonly16.8%withtheoriginallimitstateor7.9%withtheregroupedlimitstate.Thatis,fornearlythesamecomputationalcost,separableMCwithregroupingcanestimatethefailureprobabilitymoreaccuratelythancrudeMonteCarlo. Figure 5-6 showsthatthemagnitudeoftheuncertaintyreduceswithincreaseinnumberofMsamples.Thusbyincreasingthesamplesizeoftheinexpensive 79

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Figure5-6.StandardDeviationofCMC,SMCandregroupedlimitstateSMCwhereN=500(xed)andMisvaryingfor10,000repetitions components(strengthandtheload),wecouldreducetheuncertaintyinthepfestimate.ForverylargeM,itwouldreachanasymptoticvalueduetothenitevalueofN.Figure 5-6 showsthatbytransferringsomeoftheuncertaintyfromtheresponsetothecapacity,wecantakeadvantageofincreasedMtofurtherreducetheerrorintheprobabilityestimate. 5.6.3Summary TheseparableMonteCarlo(SMC)methodcanprovidesubstantialimprovementsinaccuracyoverthecrudeMonteCarlomethodwhenresponseandcapacityaregovernedbyindependentrandomvariables.ObtainingestimatesoftheaccuracyofSMCiscriticaltotakingfulladvantageofthemethod.HereweproposedusingbootstrappinginordertoobtainestimatesoftheerrorintheSMCestimates,aswellasthecontributionsofthecapacityandtheresponsetothaterror.TheapproachwasdemonstratedthroughanexampleproblemoffailureanalysisofacompositepressurevesselusingTsai-Wufailurecriterion.Becauseofthelowcomputationalcostoftheexample,itwaspossibletoconductmultiplesimulationsandassesstheaccuracyofthebootstrappedestimateempirically.SeparableMonteCarloledtosubstantialaccuracyimprovementindeterminingtheprobabilityoffailurecomparedtocrudeMonteCarlo 80

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method.BootstrappingprovidedreasonableestimatesoftheuncertaintyintheSMCprobabilityoffailureestimates.Bootstrappingalsoallowedestimatingtheindividualcontributionsoftheresponseandcapacitytowardstheuncertaintyintheprobabilityoffailureestimate,thussuggestingadditionalsamplesofinexpensivecapacitywouldproveadvantageous.Furthersubstantialimprovementinaccuracywasachievedbytransferringuncertaintyawayfromexpensivecalculationsbyusingunitloadstressesandgeneratinglargesamplesofloadandstrengths. 81

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CHAPTER6EFFICIENTGLOBALRELIABILITYANALYSIS(EGRA) ThenameofthemethodincludesReliabilityAnalysis,howeverEGRAisasequentialsamplingmethodthataimstoimprovesurrogatesinvolvedinconstrainedoptimizationandreliabilityanalysis.Thismethodcanbeusedimprovethetargetboundariesinsurrogatemodelsfacilitatingestimationaccurateoptimum(orreliability).TheEGRAalgorithmcodedinthesurrogatetoolboxinMATLABwasusedtoillustratetheadvantagesofusingthismethod. TheITPSoptimizationprobleminvolvesconstructionofsurrogatesinfourormoredimensions,wherevisualizingtheworkingofEGRAisnotpossible.ThischapterillustratesthemethodologyofEGRAusingBranin-Hoofunctionintwodimensionalspacefortheeaseofvisualization.Asmentionedinsection 2.5.1 EGRAhastwoimportantfeatures-aninitialresponsesurfaceconstructedwithfewerdesignpointsandtheExpectedFeasibilityfunctioncriterion.Withagiventargetlevel,thefeasibilityfunctionusestheexistingresponsesurfacetoidentifyanewdesignpointthatwouldimprovethecontourofthetargetboundary. 6.1EGRAalgorithm Bichonetal.[ 59 14 ]hasprovidedadetaildescriptionoftheEGRAalgorithm. 1. Generateasmallnumberofrandomsamplesfromthetrueresponsefunction.(a)Thisinitialselectionisarbitrary.However,oneoftheoptionsareconsideringnumberofsamplesrequiredtodeneaquadraticpolynomialisusedhereasaconvenientruleofthumb((n+1)(n+2)/2sampleswherenisthenumberofrandomvariables).(b)Latinhypercubesampling(LHS)isusedtogeneratethesamplesinthedesignspace. 2. BuildaninitialGaussianprocessmodel(interchangeablyreferredaskrigingmodel)fromthegeneratedsamples. 3. Findthepointwithmaximumexpectedfeasibility.(a)Theexpectedfeasibilityfunctionisbuiltwith=2G.(b)Tolocatethemaximumexpectedfeasibilityanappropriateoptimizerisused.ThesurrogatetoolboxusesDifferentialEvolutionaryOptimizer[ 84 ]. 4. Evaluatethetrueresponsefunctionatthispoint 82

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5. Addthisnewsampletotheprevioussetoftrainingdataandbuildanewgaussianprocessmodel.Gotostep3. 6. Thisupdatedsurrogatemodelisthenusedanapproximationtoevaluateconstraintsinconstrainedoptimization 6.2IllustrationofEGRA TheBranin-Hoofunctionisastandardtestfunctionusedinoptimizationproblems.Thefunctionhastwovariableswithbounds-5x110and0x215andadequatelynon-linear.Thefunctionisdenedas g(x)=x2)]TJ /F9 11.955 Tf 13.15 8.08 Td[(5:1x21 42+5x1 )]TJ /F9 11.955 Tf 11.95 0 Td[(62+101)]TJ /F9 11.955 Tf 16.68 8.08 Td[(1 8cos(x1)+1050(6) ContoursofthetruefunctionisshowninFigure 6-1 A.Thecontourg(x)=50ischosenasthetargetboundary(Figure 6-1 B).ThegoalistorepresentthisboundaryasaccuratelyaspossibleusingtheEGRAmethodology.ThetruecontoursoftheBranin-HoofunctionareshowninFigure 6-1 A.Alsothetargetboundaryg(x)=50isshowninFigure 6-1 B. ATrueFunction Bg(x)=50ofTruefunction Figure6-1.TruecontoursoftheBraninhoofunctionA.allcontoursB.targetcontourg(x)=50 83

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Aninitialsurrogatemodelisbuilt.Thesurrogatemodelavailablebydefaultiseitherkrigingorgaussianprocessmodel.Theinitialsurrogatemodelwastted10LHSdesignofexperimentsasshowninFigure 6-2 A[ 85 ].Usingtheresponsesurfaceapproximationandthepredictionuncertainty,theexpectedfeasibilityimprovementfunction(Equation 2 )ismaximizedusingtheDifferentialEvolutionaryoptimizerinthesurrogatetoolbox.Figure 6-3 showsexpectedfeasibilityfunctioncomputedandplottedinthedesignspaceforthepurposeofvisualizingtheprocess.Itcanbeseenthatthefunctionhasamaximumatupperleftcornermarkedbyacircle. Figure6-2.TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction.TotalDOE=10points Theoptimizerfoundthemaximumtobeattheupperleftcorner(0,15)andthisdesignisevaluatedusingthetrueresponsefunctionandincludedintheDOEtoconstructthenewresponsesurface.ThenewdesignandthetargetcontourofthenewthekrigingmodelisshowninFigure 6-4 Theexpectedfeasibilityfunctionofthenewsurrogatemodelshowsthattheoptimizerwouldsuggestapointatlowerrightcorner(markedbyacircle)wherethe 84

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Figure6-3.ExpectedFeasibilityFunctionevaluatedusingtheinitialkrigingapproximation functionismaximum.ThenewdesignpointacquiredattheendofthisEGRAcycleis(10,0).TheexistingsurrogatemodelisupdatedwiththisdesignandthenewmodelisshowninFigure 6-5 Thisprocessisrepeatedforanother18cyclestoimprovethetargetboundaryg(x)=50.Thedifferentialevolutionaryoptimizer[ 84 ]maximizesthefeasibilityfunctiontondtheadditionaldesigns,onepercycle.ItshouldbenotedthatLHSsampling,oftentimesdonotplacepointsattheboundariesofthedesignspace.Thiscauseshighuncertaintyinthepredictionmodel.Theexpectedfeasibilityfunctioninitiallyaddspointsatregionswherepredictionuncertaintydominates.Thisisreferredtoasexplorationofthedesignspace.Whenthepredictionuncertaintyinthemodelisreducedduetotheadditionalpoints,theexpectedfeasibilityfunctionimprovesthecontourbyplacingpointsnearthetargetboundary(Figure 6-6 ).Thisisreferredasexploitation.Thusbalancingtheexplorationandexploitationthissamplingmethodsuccessfullyapproximates 85

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AKrigingmodelupdatedbyEGRADOE BExpectedFeasibilityFunction Figure6-4.Krigingmodelupdatedwiththenewdesignandexpectedfeasibilityfunction.DesignupdatedbyEGRADOEisshowninRed.TotalDOE=11points Figure6-5.TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction.TotalDOE=12points 86

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Figure6-6.TargetcontouraccuratelyapproximatedbytheEGRAmethodology.InitialDOE=10,EGRAupdatedDOE=20 Figure6-7.Krigingsurrogatemodelconstructedusingglobaldesignofexperiments.TotalDOE=30 87

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thetargetboundarieswithfewernumberofdesignpointsthanrequiredbyaglobalsurrogatewiththesameaccuracy. Thenalkrigingmodelwith10initialdesignsand20designsupdatedbyEGRAinshowninFigure 6-6 .Itcanbeseenthereismuchlessdiscrepancybetweenthetrueresponseandtheupdatedsurrogateresponsethusdemonstratingtheefciencyofthemethodology. Thisupdatedsurrogateisfurthercomparedwithasurrogateconstructedfrom30designofexperimentsdistributedglobally.ThedesignsarealsocreatedusingLHSformulation.TheglobalsurrogateshowninFigure 6-7 isgrosslyinaccurateincomparisonwithtrueresponsefunction.WiththesamenumberofpointsEGRAefcientlyprovidedanaccurateapproximationoftheresponse. ThismethodwasincorporatedtoestimatethedeterministicoptimumandprobabilisticoptimumoftheITPSpanel.TheITPSoptimizationproblemseekspolynomialsurrogatesasitinvolveshighdimensions.ThisresearchworkincludedpolynomialresponseapproximationintheexistingEGRAalgorithm. 6.3Summary BoundariesofcomplexfunctionscanbeapproximatedefcientlyusingEfcientGlobalReliabilityAnalysis.ItwasshownthatatargetboundaryoftheBranin-Hoofunctionwasaccuratelyapproximatedusing10initialLHSdesignsand20additionalpointsacquiredusingEGRA.Theimprovementcriterionbalancesbetweenexploitingregionsofthedesignspace(wheregoodsolutionshavebeendiscovered)andexploringregionsthathavenotbeenwellsampled(andthushavegreateruncertainty)tosuccessfullyapproximatetargetregions.EGRAreducesthecomputationalcostwhendealingwithevaluationofresponsefunctionthroughexpensiveniteelementanalyses.Itefcientlyfacilitatesaccurateoptimumbyaccuratelyapproximatingthetargetboundariesofconstraintswhileminimizingthecomputationalcostinvolved. 88

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FormoreexamplesanddetaileddescriptionofvariousapplicationsoftheEGRAmethodology,thereadersarereferredtoBichonetal.[ 59 86 14 ]. 89

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CHAPTER7DETERMINISTICANDRELIABILITYBASEDOPTIMIZATION ThischapterpresentsthedeterministicandprobabilisticoptimizationprocessoftheITPSpanel.Thecomputationalexpensesassociatedwiththehighdelityniteelementanalyseswasreplacedbyefcientsurrogatemodels.FurthertheaccuracyofthemodelsattargetregionswereimprovedusingEfcientGlobalReliabilityAnalysis. Asurrogateofthefailureresponsewasconstructedwithfewdesignpoints(eg.numberofdesignsequaltonumberofcoefcientswhenthesurrogatewasapolynomialapproximation).Withatargetlevelgiven,EfcientGlobalReliabilityAnalysis(EGRA)usedtheexistingresponseapproximationandtheexpectedfeasibilityfunctiontoidentifythenewdesignpointthatimprovedtheconstraintboundary(temperature,stressandbucklingloadindeterministicoptimizationandreliabilityindexconstraintintheprobabilisticoptimization).Further,eachdesigngivenbyEGRAwereinputtotheniteelementanalysesandtheresponseswereevaluated.Thiswasaddedtotheexistingdesignofexperimentstoconstructanupdatedsurrogate.Thenewdesignswereaddedsequentiallyandthesurrogatemodelwasupdatedlikewise.Theupdatedsurrogatemodelimprovedtheboundaryofthetargetconstraint.Thetotalnumberofnewdesignsrequiredtoapproximateatargetboundarygenerallydependsonthecomplexityofthefunctionandtheinitialresponsesurface.Thismethodologyimprovedtargetboundariesoftheconstraintsbothindeterministicandprobabilisticoptimization. FurtherthischapterpresentssystemreliabilityofITPSfoundthroughseparableMonteCarlomethodandtheerrorintheindividualprobabilitiesoffailureestimatedusingthebootstrappingmethod. 7.1DeterministicOptimization AsarststepinoptimizingthedesignoftheITPS,adeterministicoptimizationofthepanelisdesired.Thisresultedinanappropriatenarroweddowndesignspaceandprovedtobeausefulpointofreferencefortheprobabilisticoptimization.Furtheritwould 90

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beinterestingtocomparetheminimizedstructuralmassbetweenthedeterministicandtheprobabilisticoptimum.Thedesignvariables-thegeometryparameters,tW,tB,tTandh(Figure 3-2 )wereoptimizedforminimummassoftheITPSunitcell.Theformulationoftheoptimizationproblemis Minimizem=f(d)=f(tW;tB;tT;h)suchthatTmax;BFSallowLBdUB(7) wheremisthestructuralmass,Tmax;BFSisthemaximumbottomfacesheettemperature,=1,2,12arethethermalstressesintheITPScomponentsandisthethermalbucklingload.Thefailurethresholds,theallowablevaluesandthesafetyfactorsforeachconstraintisgiveninTable 7-1 .Thesafetyfactorsandmarginscanbeconsideredastheriskallocatedforeachmodeoffailure. Table7-1.Failure,safetyfactorsappliedonconstraints ConstraintsFailureSafetyFactorAllowabletargetconstraints Tallow550K77K(safetymargin)473KWrapstrengthS1T323MPa1.5215.3MPaBucklingloadallow1.01.51.5 Asmentionedinsection 3.2.2 ,sincepressureloadsdonotcausefailure,theITPSisanalyzedonlyfortemperatureloads.Itwasfoundthatofallthecomponents,thewrapstresseswerecriticaland1;Wofthewrapcausedfailureofthepanel.Henceonlythewrapstressesareincludedasconstraintsintheoptimizationofthepanel.TheboundsofthegeometricdesignvariablesfordeterministicoptimizationaregiveninTable 7-2 91

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Table7-2.Lowerandupperboundsofdesignvariablesfordeterministicoptimization DesignVariableLowerBound(LB)UpperBound(UB) Wrapthickness,tWmm0.41.0Bottomfacesheetthickness,tBmm2.06.0Topfacesheetthickness,tTmm1.03.0Heightofthefoam,hmm25.040.0 Withtheresponseapproximationsconstructedforeachfailureconstraint,theseapproximationswereusedalongwithMATLABoptimizerfmincon()toevaluatetheconstraintsatvariousiterationsfordeterminingtheoptimaldesignoftheITPS. Traditionallylargenumberofdesignswouldbeusedforconstructingglobalresponsesurfaceoftheconstraints.Thenumberofdesignswouldgenerallydependonthenumberofdesignvariables,thedesiredpolynomialapproximationandthenumberofsimulationsaffordable.Forexample,consideringasecondorderpolynomialapproximation,fora4designvariableproblem,thereare15polynomialcoefcientsandaminimumof30designswouldberequired.Generallytoensureaccuracyofthesurrogate,morethan30simulationscouldbeconsidered. Theoptimumislikelytofallonsomeoftheconstraintboundaries.Insteadofusingalargenumberofdesignofexperiments(DOE)toconstructglobalresponseapproximations,theavailablecomputationalresourcesweretargetedtowardsregionsinthedesignspacetoapproximatetheconstraintboundariesaccurately.Thiswasaccomplishedbyupdatingthesurrogatesusingadaptivesamplingtechnique,EfcientGlobalReliabilityAnalysis. EGRAfacilitatedestimationofanaccurateoptimumwhilereducingthecomputationalcostassociatedwiththeoptimizationprocess.SincethepolynomialsurrogatesweremoresuitableforthehighdimensionalITPSproblem,polynomialresponsesurfaceswereincorporatedintheEGRAalgorithminthesurrogatetoolbox.TheoptimumdeterminedusingtheEGRAupdatedsurrogatesiscomparedwiththeoptimumdeterminedusingglobalsurrogates.ItwasfoundthatusingEGRAupdatedsurrogates,theaccurateoptimumcouldbeobtainedwithone-thirdnumberofdesignpointsrequired 92

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tobuildglobalsurrogates.Furthermoreitwasfoundthatevenwithlargenumberofdesigns,theoptimumevaluatedusingglobalsurrogateswasinaccurate. Further,itwasdesiredtopredictthesystemreliabilityofthedeterministicoptimum.Thiswouldserveasaconstraintfortheprobabilisticoptimization.Topredictthesystemreliability,uncertaintymodelinganduncertaintyanalysiswascarriedout.Theuncertaintyintheresponsesduetotheuncertaintyininputrandomvariableswouldaidinselectingappropriatesamplesizefortherandomvariablesleadingtoaccuratereliabilityestimation. 7.2UncertaintyModeling ThealeatoryvariabilityconsideredintheITPSdesignaregeometry,materialproperties(Young'smodulus,Poisson'sratio,shearmodulus,thermalexpansioncoefcient,thermalconductivityandspecicheat)andloads.Theepistemicuncertaintiesincludedaretheerrorinmodelingandsimulationoftheniteelementanalyses. EachcomponentoftheITPSismadeofcompositematerial(Table 3-2 )increasingthenumberuncertaintiesdrastically.ForthecompositeITPSpanel,atotalof33uncertainmaterialpropertiesvariablesareconsidered.ThevariabilityintheinputparametersandthecapacityvariablesareshowninTables 7-3 and 7-4 .Inadditiontothevariabilityintheinputparameters,errorinmodelingandsimulationisalsointroducedintheresponsestoestimatetheprobabilityoffailure.Theerrorinheattransfereth,staticstressestrandthermalbucklinganalysisebkwasassumedtobe3.0%,uniformlydistributed. Table7-3.CoefcientofvariationofinputrandomvariablesincludedintheITPSdesign RandomNo.ofUncertaintyProbabilityVariablesvariables%distribution WrapthicknesstW16.25NormalBFSthicknesstB16.25NormalTFSthicknesstT16.25NormalHeightoffoamh13NormalMaterialProperties335NormalLoad,heatuxq1;q2215Normal 93

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Table7-4.Nominalallowablevaluesofcapacityandcoefcientofvariation CapacityNominalvalueCoefcientofTypeofvaluevariation%distribution WrapStrengthS1T323MPa3%NormalAllowableTmax;BFS55015KUniformAllowable1.0-ThetotalnumberofinputuncertainvariablesintheITPSdesignis39.Thevariablesareheatuxloads:q1,q2,geometry:tW;tB;tT;h,thermalpropertiesofthecomponents:thermalconductivitykandspecicheatCpofWrap,materialpropertiesofBFS:E1;E2;v12;nu23;G12;G23;1;2,foam:E;v;,TFS:E1;E3;v12;nu13;G12;G13;andWrap:E1;E3;v12;nu13;G12;G13;.Althoughthereare39uncertaintiesinthedesignofITPS,theresponseswouldbenotsensitivetoalltheuncertainvariables.Henceasimplesensitivityanalyseswasperformedusinglinearpolynomialsurrogates.ApolynomialsurrogateformaximumBFStemperature,stressandbucklingwasconstructedasfunctionofthe39variablestondthemostsensitivevariables.FurtherthemostsensitivevariableswereusedtoestimatesystemreliabilityoftheITPSpanelusinglinearsurrogatesofthecorrespondingresponses. 7.3EstimationofSystemReliability Thesystemreliabilityisthecombinedprobabilityoffailureduetotheaforementionedfailureconstraints(temperature,stressandbuckling).Itisaknownfactthatlargesamplesofresponsesandcapacityarerequiredtoestimatetheprobabilityoffailureaccurately.ThiswasaccomplishedbyconstructingresponsesurfaceswithlimitednumberofFEsimulations.Thuslargenumberofresponsesampleswereevaluatedwithminimumcomputationalexpense.Failureisdenedbythelimitstatefunctionthatseparatesthefeasibledesignsfromtheinfeasibleones.Limitstatefunctionforstresses,thermalandbucklingfailureareshowninEquation 7 .Thelimitstatesinthisproblemaredenedasdifferencebetweentherandomresponseandrandomcapacity(Equation 2 ). 94

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Gth=Tmax;BFS)]TJ /F3 11.955 Tf 11.96 0 Td[(TallowGstr=11;W)]TJ /F3 11.955 Tf 11.96 0 Td[(S1T;WGbk=allow)]TJ /F3 11.955 Tf 11.95 0 Td[((7) Whenevertheresponseexceedsthecapacitythepanelisassumedtohavefailed.Systemreliabilityofthepaneliscomputedwhenthepanelfailsineitherofthemodes.Sincethecapacitiesareindependentoftheresponses,andfollowagivenstatisticaldistribution,thecapacitiesandresponsescouldhavedifferentsamplesizeandallpossiblecombinationscouldbeconsidered.ConsideringNsamplesofresponseandMsamplesofcapacity,theprobabilityoffailurepfcouldbeestimatedusingseparableMonteCarlomethodas ^pth=1 NMMXj=1NXi=1I[Gth(Cj;Ri)>0]^pstr=1 NMMXj=1NXi=1I[Gstr(Cj;Ri)>0]^pbk=1 NNXi=1I[Gstr(C;Ri)>0](7) Thecapacitiesforthermalandstressfailuresfollowadistribution(Table 7-4 ),resultinginNMcomparisonstoestimateprobabilityoffailure.Thecapacityforbucklingloadfactorisdeterministic(allow=1.0)andhenceresultedinNcomparisons.Thissuggestedthatthebucklingprobabilityoffailure^pbkshouldbecalculatedwithlargesamplesizesofbucklingloadresponses. Furthertheniteelementmodelingerrorwasintegratedtotheresponseforeachcaseas Gth=Tmax;BFS(1+eth))]TJ /F3 11.955 Tf 11.95 0 Td[(TallowGstr=(1+estr))]TJ /F3 11.955 Tf 11.95 0 Td[(SGbk=allow)]TJ /F3 11.955 Tf 11.95 0 Td[((1+ebk)(7) 95

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Sincetheerrorsarealsoindependentoftheresponse,theycouldbeassignedadifferentsamplesize(Q).Foreacherror,theNsamplesoftheresponseandMsamplesofthecapacityaresampledtoestimateasinglevalueofprobabilityoffailure.QerrorvalueswouldprovideQprobabilityoffailureestimates.ThemeanoftheQestimatesisconsideredasthemeanprobabilityoffailureforeachfailuremode.Thesumoftheprobabilitiesoffailureofthethreemodesisconsideredasthesystemreliabilityofthepanel. Furthertheerrorintheindividualprobabilitesoffailurewasestimatedusingthebootstrappingtechnique,describedindetailinSection 5.3 .Furthertheerrorsestimatedwouldbecomparedwiththeerrorsestimatedfromnumerousrepetitionstovalidatethebootstrappingmethod. 7.4ReliabilityBasedOptimization Thereliabilitybasedoptimizationinvolvesoneadditionalconstraint,theprobabilityoffailureconstraint.Theprobabilityoffailureofthedeterministicoptimumactsastheconstraintfortheprobabilisticoptimization.TheoptimizationproblemoftheITPSpanelisformulatedas Minimizem=f(d)=f(tW;tB;tT;h)suchthat(g(d;X)Cr(7) whereiscalculatedusingEquation 2 .Itdependsonthefailureconstraintsg(d;X)evaluatedasafunctionofdesignandrandomvariables.Cristhedesiredlevelofreliabilityfortheoptimum.ThisvaluewasassignedthedeterministicreliabilityindextoanalyzehowthefailureconstraintsdifferfromthedeterministicoptimizationandwhetherthishasaidedinfurtherreducingorincreasingthemassoftheITPSpanel.Furtherthisanalysisontheriskallocationmightsuggestwhatsafetyfactorscouldbesetforthedeterministicoptimizationthatwouldfurtherminimizethemassleadingtononconservative,safeoptimumdesign. 96

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Thereliabilitybasedoptimizationrequiredcalculationofreliabilityindexinthedesignvariablespace.Hencearesponsesurfaceofthereliabilityindexasafunctiondesignvariableswasconstructed.Inthedesignvariablespace,ateachdesignsetd,probabilityoffailureandhencethereliabilityindexwascomputed.Thisinvolvedconstructionoffailureresponseapproximationsateachdesignpoint.Usingtheresponsesurfaces,therandomvariablesweresampledaccordingtotheirrespectiveprobabilitydistributiontoestimatetheprobabilityoffailure.Henceateachdesignpoint,adesignofexperimentswascreatedintherandomvariablespace,failureresponsesurfaceswereconstructedandsamplesoftherandomvariablesweregeneratedtoestimatetheprobabilityoffailure.Thisprocesswascarriedoutatdesignpointsinthedesignvariablespacetoobtainareliabilityindexresponsesurfacemakingthisprocesscomputationallyintractable. Sincetheprobabilisticoptimizationinvolvessatisfyingthereliabilityconstraint,EfcientGlobalReliabilityAnalysiswasusedtoimprovetheaccuracyofthesurrogateatthedesiredreliabilitylevelwithfewernumberofdesignpoints.Initiallyasecondorderpolynomialsurfaceofthereliabilityindexwasconstructed.HoweverafteraddingadditionaldesignpointsusingEGRA,thetargetboundarywasinaccurate.Itwasfoundthatthereliabilityindexresponsewashighlynon-linearandsecondorderpolynomialwasapoorapproximationoftheresponse.Theorderofthepolynomialwaslimitedbytheavailablecomputationalresources.ThusthereliabilityindexwasttedusingkrigingsurrogatemodelwhichapproximatedtheresponseandthenEGRAwasusedtoimprovetheboundaryofdesiredreliabilitylevel.ThispreventedwastingoflargeofFEsimulationsatunimportantregionsandfocusthecomputationalresourcesinapproximatingthedesiredreliabilitylevelaccuratelytodeterminetheaccurateprobabilisticoptimum. 97

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7.5ResultsandDiscussion 7.5.1DeterministicOptimization ThedesignvariablesforthedeterministicoptimizationwerethedimensionsoftheITPS.Thepanelwasoptimizedforminimummasssatisfyingthermal,structuralandbucklingconstraints.ThemaximumBFStemperatureandthermalbucklingloadfactorwerethecriticalresponsesconsideredasfailureconstraints.Bucklinganalysesoftheunitcellshowedthattopfacesheetwasthecomponentthatbuckledleadingtobucklingofthewebs.ThermalstressanalysisoftheITPSunitcellshowedthatthewrapstresseswerecriticalandwrapstress1;Wwasalsooneoftheactiveconstraintsintheoptimizationprocess. Astudyonthevariationoftheresponseswithrespecttothedesignvariableswasperformedtondtheorderofthepolynomialthatwouldcloselyapproximatetheresponsewiththeresultsfromtheniteelementanalyses.Finiteelementsimulationswereperformedwhereoneofthedesignvariableswasvariedbetweentheirrespectiveranges(Table 7-2 )andotherswereheldconstant(tW=0.4tB=2.5,tT=2.0andh=32.5mm).Thevariationofthebucklingloadfactorincreasestremendouslywiththeincreaseinthethicknessoftopfacesheetwhilethevariationisnotpronouncedwiththeincreaseinwrapthickness.Thisconrmsthatbucklingofthetopfacesheet(Figure 7-3 )leadstobucklingofthetopweb(Figure 3-7 ). WiththevariationoftheresponsesshowninFigures 7-1 7-2 7-3 and 7-4 ,secondorderpolynomialresponsesurfacesoftheconstraintswereconstructed. Thedesignpointsfortheresponsesurfacegenerationwerecreatedusingthelatinhypercubesamplingwithinthebounds.Asstatedearlier,foraquadraticresponsewith4designvariables,thenumberofpolynomialcoefcientsare15.Asaruleofthumb,designpointsmorethantwicethenumberofcoefcientsareconsidered.Inthisproblematotalof60designpointswereinputtotheniteelementanalyses,togeneratetheresponsesurfaces.Thisrequired180niteelementanalysesfor3responses(transient 98

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Figure7-1.VariationofresponseswrtthicknessofthewraptW.(tB,tTandhwereheldconstant) heattransfer,thermalstressandthermalbucklinganalyses).ThetimetakentoperformallthethreeFEsimulationsandextractallthedesiredresponseatasingledesignpointis8.5minutessummingto8.5hoursfor60designpoints.Thisisassumedastheavailablecomputationalbudgettondthedeterministicoptimum. Insteadofusing60designstobuildaglobalresponsesurface,thecomputationalresourceswereusedtoapproximatetheconstraintboundariesaccuratelytoestimateanaccurateoptimum.Itwasfoundthatallthefollowingconstraints,temperature,stress1andbucklingloadwereactiveintheoptimization.Hencetoimprovetheaccuracyofeachactiveconstraint,EfcientGlobalReliabilityAnalyseswasapplied.EGRAbalancesbothexplorationandexploitationofthedesignspacetoadddesignpointsthatimprovedtheaccuracyofthetargetboundary.Aninitialresponsesurfacewasconstructedusing16LHSdesignpointsandsixadditionalpointswereacquiredforeachconstraintusingEGRA.Sincetemperaturegradientactsastheloadsforthermalstressandbuckling 99

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Figure7-2.VariationofcriticalresponseswrtthicknessofthebottomfacesheettB.(tW,tTandhwereheldconstant) analyses,foreachstressorbucklinganalysis,atransientheattransferanalysishadtobeperformed.ThedesignsupdatedbyEGRAandthecorrespondingresponsesareprovidedinAppendix C .Thetrueresponsesattheadditionaldesignsareveryclosetothevalueofthetargetconstraint. Adesignpointonthetargetboundarywasevaluatedwiththeupdatedsurrogate.Whenthedifferencebetweenthesurrogatepredictionandthetrueresponsewas10)]TJ /F4 7.97 Tf 6.59 0 Td[(3,EGRAsamplingwasstopped.Whenthisprodedurewasfollowed,Itwasfoundthat4pointsupdatedbyEGRA,togetherwiththeinitial16pointsapproximatedthetargetboundaryaccuratelyyieldingaccurateoptimum.Thusthesimulationsrequiredwereonly68initialFEsimulationscomparedto180FEsimulations(globalsurrogate)andthecomputationaltimewas3hoursand15minutes.Thecomputationaltimewasreducedto40%whencomparedtothatofglobalsurrogates. 100

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Figure7-3.VariationofcriticalresponseswrtthicknessofthetopfacesheettT.(tW,tBandhwereheldconstant) Furthera20DOEglobalresponsewasusedforoptimizationtoprovideareasonablecomparisonwiththesamenumberofDOE.Theoptimumdesignvariablesd,theminimizedmassestimatedusing60DOE,20DOEand16+4EGRADOEsurrogatesaretabulatedinTable 7-5 Table7-5.OptimumdesignvariablesandminimizedstructuralmassthroughglobalandEGRADOEfordeterministicoptimization DOEtWmmtBmmtTmmhmmMasskg/m2 60DOE0.42.841.4132.321.420DOE0.42.871.3832.721.816+4EGRA0.42.851.3731.721.3 Theoptimizationthroughthesurrogatesyieldedoptimawithalltheconstraintsbeingactive.Further,theconstraintsevaluatedusingthesurrogatesarecomparedwithniteelementanalysestoassesstheaccuracyofthesurrogateattheirrespectiveoptimuminTable 7-6 .The60DOEoptimumwasapparentlynottheoptimumasthe 101

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Figure7-4.Variationofcriticalresponseswrtheightofthefoamh.(tW,tTandtBwereheldconstant) constraintsdeterminedbythesurrogatedifferfromtheniteelementanalyses.Theniteelementanalysesareconsideredastruevalueshere.Thus,itcanbeseenthatthoughtheglobalDOEhas60designpoints,itdoesnotapproximatetheconstraintboundaryaccuratelyleadingtoerroneousresults.Forexample,thoughtheTmax;BFSglobalsurrogatepredicted473Katthe(60DOE)optimum,thetrueresponseis469.6K.Evenwithlargenumberofdesigns,thediscrepancyisabout4K.HoweverthesurrogateconstructedusingthesequentialsamplingofEGRAhasimprovedthetargetboundary(472.9K)topredictaccurateoptimum.TheconstraintvaluespredictedbytheEGRAimprovedsurrogateagreeswellwiththetruevaluesoftheniteelementanalyseswithlessthanapercenterror.Theoptimumdeterminedfromthe20DOEsurrogateprovedthatasurrogatewith20globaldesignsisgoingtobegrosslyinaccuratewhereasthe16+4=20DOEconstructedusingEGRAwashighlyaccurate.Thisshowsthat 102

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whenthecomputationalresourcesareefcientlydirectedtowardstargetregions,thecomputationalcostcanbereducedto40%oftheavailablecomputationalbudget. Table7-6.ComparisonbetweentheaccuracyofglobalsurrogateandEGRAsurrogateatthedeterministicoptimumusingcorrespondingsurrogates.Thepredictedstandarddeviationisgivenin() SurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE473(0.63)1.5(0.05)215.3(0.65)469.61.54211.320DOE473(3.4)1.5(0.78)215.3(1.6)467.51.77217.316+4EGRA473(9.1)1.5(2.5)215.3(2.4)472.91.51214.6 Threeoptimaweredeterminedusing60DOE,20DOEandEGRAupdatedsurrogates.Theresponsesatthethreeoptimawereevaluatedusingallthesurrogatestoanalyzehowthesurrogatesperformedattheotheroptima.Table 7-7 showstheresponsesofalltheoptimaevaluatedusingthe60DOEglobalsurrogate.Amaximumdifferenceof8%wasobservedinthebucklingloadofthe20DOEoptimum.Itwasfoundthattheerrorbetweentheresponseswerelessthan8%,howeverthecomputationaltimerequiredfortheanalyseswas60%morethanthatofEGRA. Table7-7.Comparisonofresponsesatallthedeterministicoptimausingthe60DOEsurrogate OptimaSurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE4731.5215.3469.61.54211.320DOE470.91.62214.8467.51.77217.316+4EGRA475.71.45212.7472.91.51214.5 Inasimilarfashion,thethreeoptimawereevaluatedusingthe20DOEsurrogateandtabulatedinTable 7-8 .Thebucklingloadsofthe60DOEandEGRAoptimahadanerrorof13%whencomparedtothetrueresponse.ThevalueofTmax;BFSoftheEGRAoptimumdifferedby5K.Althoughbypercentitisalittleover1%error,thisisthelargestdifferencebetweenpredictionandtrueresponseusinganysurrogate.Tosumup,Itwasfoundthat20DOEsurrogatewasinaccurateattheotheroptimaaswell. 103

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Table7-8.Comparisonofresponsesatallthedeterministicoptimausingthe20DOEsurrogate OptimaSurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE474.21.35214.7469.61.54211.320DOE4731.5215.3467.51.77217.316+4EGRA478.11.31210.3472.91.51214.5 TheresponsesofthethreeoptimaevaluatedusingtheEGRAupdatedsurrogatealongwiththetrueresponsesisshowninTable 7-9 .ItwasfoundthatEGRAupdatedsurrogatepredictedresponsesat60DOEwithlessthan4%error.Howeveratthe20DOEoptima,therewas16%errorinthebucklingloadresponses.ThiscouldattributedtothesignicantdifferenceintheheightofthefoambetweentheEGRAand20DOEoptimawhichisawayfromthetargetboundary.EGRAupdatedsurrogateseemstohaveminimalerroratalltheoptimawhencomparedtoothersurrogatesexceptforthebucklingresponseofthe20DOEoptimum.Alsothewrapstresseswerepredictedwithmaximumof3%errorbyallthesurrogates.Thus,whenanalysedonthebasisoncomputationaltimerequiredandaccuracyofthesurrogate,EGRAupdatedsurrogatesseemstoperformbetterthan60and20DOEglobalsurrogates. Table7-9.ComparisonofresponsesatallthedeterministicoptimausingtheEGRAupdatedsurrogate OptimaSurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE470.61.56213.4469.61.54211.320DOE465.91.46215.6467.51.77217.316+4EGRA4731.5215.3472.91.51214.5 Toestimatethereliabilityoftheoptimumdesign,uncertaintymodelingandanalysisofthefailureresponsesisrequiredwhichisdiscussedinthenextsection. 7.5.2SensitivityAnalysis Thetotalnumberofuncertaintiesconsideredinthedesignwere39inputrandomvariables(Table 7-3 ).Tostudytheuncertaintypropagationandestimatereliability, 104

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responsesurfaceapproximationoftheoutputparametersasafunctionoftheinput39uncertaintieswasconstructed.Aminimumof40designsarerequiredtoconstructalinearresponsesurface(numberofcoefcients=40).Sinceallthevariableswouldnotbesensitivetotheresponses,sensitivityanalysiswasdesiredtostudythesensitivityoftherandomvariablestowardsthefailureresponses.Thiswouldaidinndingthemostsensitivevariablesandthusreducingthenumberofniteelementsimulationsrequiredtoconstructresponsesurfaces.Designpointstwicethenumberofcoefcientswerecreatedtoanalyzethesensitivevariables.Thecoefcientsofthelinearresponsewereusedasweightstondthemostsensitivevariablestotheresponses,maximumBFStemperature,WrapStressandbucklingloadfactor. AstheBFStemperaturedoesnotdependonthethermo-mechanicalpropertiesofthecomponents,therandomvariablesconsideredareheatuxloads,geometryandthermalpropertiesofthevariouscomponentsoftheITPS.Alinearpolynomialwastwith30points(15coefcients)wasconstructedasfunctionofthe14randomvariables.Figure 7-5 showsthelinearcoefcientsoftherandomvariablesasabarplot.Thehigherthemagnitudeofthecoefcient,thehigherthesensitivity.TheBFStemperaturewashighlysensitivetoq1andq2,tW,handkandCpofthewrap.Inanefforttoreducethenumberofvariablesfurtherq2wascalculatedasConstant+q2.Thetotalof5inputrandomvariablesweresensitivetoBFStemperature(q1,tW,h,kandCp). Inthecaseofwrapstressandbucklingload,inadditiontotheloads,geometryandthermo-mechanicalproperties,thethermalconductivityandspecicheatofthewrapwasalsoconsideredastemperaturegradientsactasloadforthethermalstressandbucklinganalysis.Withatotalof33uncertainvariables,66LHSdesignsweregeneratedandlinearresponseforwrapstress11andbucklingloadwereconstructed. Figure 7-6 and 7-7 showstheproductofuncertaintyandcoefcientofeachrandomvariableasbarplot.Fromthemagnitudeofthecoefcients,themostsensitivevariableswerechosenforeachresponse.Thesensitivevariablesofwrapstresswereq1,tW,tB, 105

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Figure7-5.SensitivityanalysisofmaximumBFStemperature,OrderofthevariableskandCpofWrap,BFS,TFS,Foam tT,E1,ofTFS,E1,ofthewrap.Furtherthesensitivevariablesofbucklingloadwereq1,tW,tT,h,ofTFS,ofwrap. Inordertomakesurethatsomeofthesensitivevariablesarenotneglectedintheprocess,coefcientofvariationoftheresponseasfunctionsoftheinitialnumberofuncertaintiesandreducednumberofuncertaintiesarecalculatedfromtheconstructedpolynomialequation.Iffistheresponse,theuncertaintyinresponseduetouncertaintyininputvariablesx1;x2;::;xiiscalculatedas f=s X@f @xi22xi(7) Table7-10.Differenceinuncertaintyinresponsesbeforeandaftersensitivityanalysis ResponseDifferenceinuncertainty Tmax;BFS4%Wrapstress117%Bucklingload1% 106

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Figure7-6.SensitivityanalysisofWrapstress11.Orderofthevariables,heatuxloads:q1,q2,geometry:tW;tB;tT;h,materialpropertiesofBFS:E1;E2;v12;v23;G12;G23;1;2,foam:E;v;,TFS:E1;E3;v12;v13;G12;G13;andWrap:E1;E3;v12;v13;G12;G13;;k;Cp ThedifferenceinuncertaintyintheresponseduetoinitialnumberofuncertainvariablesandreducednumberisprovidedinTable 7-10 .Themaximumdifferenceis7%variationinstressresponse.ThisisanacceptabledifferenceasasignicantnumberofexpensiveFEsimulationsaresavedasresultofthesensitivityanalysis.Thusthetotalnumberofsensitivevariablesreducedfrom39to11cumulatively. 7.5.3ReliabilityoftheDeterministicOptimum Thedesignspaceformedbytheuncertainvariablesaresmallandthevariationoftheresponsesisfairlylinearinthisrandomvariablespace.Hencelinearsurrogatesoftheresponseswereconstructedasfunctionoftherandomvariables.TheuncertaintyinthesevariablesisshowninTable 7-3 .Foratotalof11randomvariables,13LHSdesignsareconsidered.Heretheresponsesare:bottomfacesheettemperature,stressesandbucklingload.Thecorrespondingcapacitiesare:allowableBFStemperature, 107

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Figure7-7.Sensitivityanalysisofbucklingloadfactor.Orderofthevariables,heatuxloads:q1,q2,geometry:tW;tB;tT;h,materialpropertiesofBFS:E1;E2;v12;v23;G12;G23;1;2,foam:E;v;,TFS:E1;E3;v12;v13;G12;G13;andWrap:E1;E3;v12;v13;G12;G13;;k;Cp strengthvaluesandallowablebucklingload.Thesesurrogateswerealsousedtodeterminetheuncertaintypropagationfromtheinputrandomvariablestotheoutputresponses.Theuncertaintyintheresponseprovidedinsightontheappropriatesamplesizestoestimateaccurateprobabilityoffailure. Table7-11.Uncertaintyinresponseduetoinputuncertainty ResponseCoefcientofVariation% Tmax;BFS3.5Wrapstress119.1Bucklingload15.1 Theuncertaintyintheresponseisestimatedfromthestandarddeviationorcoefcientofvariation(CV)of1000samples.Thisprocesswassimulatedfornumerousrepetitions(1000)tondnominalcoefcientofvariationsforeachresponse.ThemeanoftheCVestimatesisgiveninTable 7-11 .ItcanbeseenthattheuncertaintyinthebucklingloadfactorishighcomparedtotheBFStemperatureandwrapstress.This 108

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kindofsituationwasaddressedbyrearrangingtherandomvariablesinthelimitstateandassigningappropriatesamplesize.However,capacityisdeterministicandtheerrorebkwasonlyvariablethatcouldbemovedtothecapacitytoreducetheuncertainty.Howeverthisdidnothelpisreducingtheuncertainty,hencethebucklingprobabilityoffailurewascalculatedusinglargenumberofsamples(Nbk=5000). Forthedeterministicoptimum(Table 7-5 ),theprobabilityoffailurewasestimatedforthethreefailuremodesusingthecorrespondinglimitstates(Equation 7 )Randomsamplesofthematerialproperties,loadandgeometryaregeneratedusingMonteCarlosimulations.Withthegeneratedsamplesofinputrandomvariables,theoutputresponsessuchascomponentstresses,maximumBFStemperatureandbucklingloadswereevaluatedfromthecorrespondinglinearresponseapproximations.TheprobabilityoffailureforeachfailuremodeofwascalculatedusingseparableMonteCarlomethod 7 .Itwasassumedthatourcomputationalbudgetonlypermitted500samplesfortheresponse,capacityanderror(N=M=Q=500)withanexceptiononthesamplessizeforbucklingload(Nbk=5000).InseparableMonteCarlomethod,theresponsesamples(N)werecomparedagainstallthecapacitysamples(M=500)resultingin250,000evaluationsofthelimitstate.Henceforeacherrorsample,250,000evaluationsresultedinasingleprobabilityoffailure.Themeanofthe500estimatesofprobabilityoffailureisconsideredtheprobabilityoffailureduetoerrorandvariability. Table7-12.Individualprobabilitiesoffailureofthedeterministicoptimum.Thetotalprobabilityoffailure=2.7%. FailuremodeTmax;BFSWrapstress11Bucklingload probabilityoffailure^p0.00180.00260.022Response472.9K214.5MPa1.51SafetyFactor77K1.51.5 TheindividualprobabilitiesoffailurearegiveninTable 7-12 .Itisassumedthatthethreefailuremodesareapproximatelyindependentandthetotalprobabilityoffailure(systemreliability)iscalculatedasthesumofalltheprobabilitiesoffailure.For 109

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thedeterministicoptimumthesystemprobabilityoffailurewas2.7%amountingtoareliabilityindexof1.93.Themassofthedeterministicoptimumwas21.3kg=m2. Oftentimesdeterministicoptimizationyieldconservativedesignsastheyarebasedonsafetyfactors.ThusreliabilitybasedoptimizationoftheITPSpanelwasperformed. 7.5.4Reliabilitybasedoptimization Deterministicoptimizationinvolvesapplyingsafetyfactorsonfailureresponses.Thesesafetyfactorscanbeconsideredasriskallocatedtoeachconstraint.Optimizationusingsafetyfactorsoftenleadstoconservativedesigns.Reliabilitybasedoptimizationdoesnotinvolvesafetyfactors,howeverincludesareliabilityconstraintthatisafunctionoftheresponses.Sincetherenohardconstraintsontheresponses,theoptimizationprocessmightyieldnon-conservativedesigns. Reliabilitybasedoptimizationinvolvesminimizingthestructuralmassofthepanelbyconsideringalltheuncertaintiesinthedesign.Inthisproblemthepanelwasoptimizedtohaveareliabilityequaltoorgreaterthanthereliabilityofthedeterministicoptimum.Itwasalsodesiredtoanalyzetheriskallocationtothefailureinducingresponsesincomparisontothedeterministicoptimum. Table7-13.Lowerandupperboundsofdesignvariablesforprobabilisticoptimization DesignVariableLowerBound(LB)UpperBound(UB) Wrapthickness,tWmm0.41.0Bottomfacesheetthickness,tBmm2.04.0Topfacesheetthickness,tTmm1.03.0Heightofthefoam,hmm29.036.0 TheoptimizationfortheprobabilisticoptimumwasalsoperformedusingtheMATLABoptimizerfmincon()withconstraintonthedeterministicreliabilityindex1.93.Thedeterministicoptimizationandtheknowledgeoftheresponsesatthedeterministicaidedinshrinkingthedesignvariablespace(Table 7-13 ).Thispreventedsamplingatregionswithlargeprobabilitiesoffailure. Similartotheresponsesurfaceapproximationofthedeterministicconstraints,30LHSdesignofexperimentswerecreatedtobuildaasecondorderpolynomialforthe 110

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reliabilityindex.Ateachdesignpoint,therandomnessintheuncertainvariableswereintroducedanddesignpointswerecreatedintherandomvariablespace.Therandomvariablespacewassmallmakingthevariationoftheresponseslinear.Foratotalof11uncertainvariablesconsidered,linearresponsesurfacesconstructedwith13DOEwereaccurate.ThesepointswereinputintheniteelementanalysestocomputethemaximumBFStemperature,wrapstressandbucklingload.Theseresponseswereusedtoconstructalinearapproximationoftheresponses.Further,largenumberofsamplesofuncertaintiesweregeneratedandtheresponseswereevaluatedusingtheaforementionedlinearapproximation.Usingthegeneratedsamplesofresponse,capacityanderrorsamples,theprobabilityoffailurewasestimatedusingtheseparableMonteCarlomethod.Thisprocesswasrepeatedforeverypointcreatedinthedesignvariablespace.Theniteelementanalysesat13designsapproximatelytook1hourand50minutes.FurtherthetimetoestimatetheprobabilityoffailurethroughseparableMonteCarlomethodwasapproximately25minutes.Thetotalcomputationaltimerequiredtoestimateprobabilityoffailureatasingleinthedesignvariablespacewas2hourand15minutes. FurtherEGRAwasemployedtoimprovethetargetboundaryequalto1.93(2.7%pf).Foreveryadditionalpointacquired,theentireprocessmentionedabovewasrepeatedtondtheprobabilityoffailurewhichwasthenincludedtothedesignofexperimentstoupdatethesurrogate.Itwasfoundthatevenafteradding25newdesignpoints,theapproximationofthetargetboundarywasinaccuratesuggestingthatasecondorderpolynomialcouldbeapoorapproximationtobeginwith.Toincreasetheorderofthepolynomial,largernumberofinitialdesignswererequired.Forexample,toconstructa3rdorderpolynomialasfunctionof4designvariables,thenumberofcoefcientsequals35andaminimumof70designpointsarerequired.Consideringthetimerequiredtoestimatetheprobabilityoffailureofonedesign,alargerDOEwascomputationallyintractable. 111

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Thuskrigingmodelwasconsideredforthereliabilityindexresponsesurfaceasitdoesnotrequireminimumnumberofdesignstoconstructaresponsesurface.ToaffordenoughadditionaldesignsthroughEGRA,a20DOEwascreatedtobuildtheinitialresponsesurface.Furtheranadditional27designswereacquiredtoimprovethedesiredlevelofreliabilityusingEGRA.Sowithatotalof47designpoints,thecomputationaltimerequiredwasapproximately105hoursand45minutes.GlobalsurrogatesofthereliabilityindexcouldbecreatedtocomparewiththeEGRAupdatedsurrogateduetolimitedcomputationalresources.Theprobabilisticoptimizationwasperformedwiththeupdatedsurrogate.TheoptimizeddesignvariablesandtheminimizedmassoftheprobabilisticoptimumisgiveninTable 7-14 Table7-14.OptimumdesignvariablesandminimizedstructuralmassthroughEGRAupdatedDOEforreliabilitybasedoptimization DOEtWmmtBmmtTmmhmmMasskg/m2 20+27EGRA0.42.01.331.119.4 Table7-15.Individualprobabilitiesoffailureoftheprobabilisticoptimum.Thetotalprobabilityoffailurepf=2.6%. FailuremodeTmax;BFSWrapstress11Bucklingload probabilityoffailure^p0.00340.0040.019Response492K245.3MPa1.7SafetyFactor55K1.241.7 Table7-16.Individualprobabilitiesoffailureofthedeterministicoptimumusingprobabilisticoptimumsurrogate.Thetotalprobabilityoffailurepf=2.8%. FailuremodeTmax;BFSWrapstress11Bucklingload probabilityoffailure^p0.00270.0030.023Response483.2K224.6MPa1.6 Atotalof13designswerecreatedthroughLatinHypercubesamplingaroundtheoptimizeddesignandwasinputtoFiniteElementanalyses.Linearsurrogatesoftheresponseswereconstructedandtheprobabilityoffailurewasestimated.Thereliabilityindexwasestimatedas1.95whichamountedto2.6%probabilityoffailure.TheprobabilisticoptimumobtainedusingtheEGRAupdatedresponsesurfacewas 112

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accuratewithanerrorof2%inreliabilityindex.Thisamounttoa3%differenceintheprobabilityoffailure.Furtherusingtheprobabilisticoptimumresponsesurrogates(temperature,stressandbuckling),theprobabilityoffailureofthedeterministicoptimumwasestimateforthepurposeofvalidcomparison.Thetotalprobabilityoffailurewas2.8%.ThisincreaseinthepfestimatewasduetothediscrepancyintheresponseevaluationasthesomeofthedeterministictBandhwereoutoftheprobabilisticrandomvariablespace. Withthereliabilityindexsurrogate,thereliabilityindexofthedeterministicoptimumwasfoundtobe1.97withastandarddeviationof0.07.Thisamountstoaprobabilityoffailureof2.4%whilethedeterministicprobabilityoffailureestimatedthroughseparableMonteCarlowas2.7%(=1.93). Themassminimizedthroughdeterministicoptimizationwas21.3kg/m2.ThesafetyfactorsforthevariousconstraintsisshowninTable 7-12 .Fromthevalueoftheresponsesattheprobabilisticoptimum,thesafetyfactorswerecalculated,showninTable 7-16 .Theprobabilisticoptimizationprocesshasallocatedriskdifferentlythatreducesthemassfrom21.3to19.4kg/m2.Theoptimumdesignseemstohavehigherriskallocatedtotemperatureandstresswhencomparedtodeterministicsafetyfactor.However,theincreaseinthesafetyfactorofthebucklingloadfactorhasyieldedreducedmasswiththesameprobabilityoffailure.Thisdemonstratesthatreliabilitybasedoptimizationyieldsnon-conservativebutasafedesign. Further,itwasdesiredtondtheaccuracyoftheprobabilitiesoffailure.Thiswasaccomplishedbyemployingbootstrappingmethod.Thismethodwasappliedearlierforcomplexlimitstateandwasvalidatedbyestimatingtheerrorthroughlargenumberofrepetitionsinprobabilityoffailureestimationprocess. 7.5.5Errorintheprobabilityoffailureestimate Forasimplelimitstate(asinEquation 2 ),theaccuracyofseparableMonteCarlocanbeestimatedasderivedbySmarsloketal.[ 16 ].Inthecaseagenerallimitstate 113

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(Equation 2 ),bootstrappingwasproposedtoassesstheaccuracyoftheprobabilityoffailure. Bootstrappingmethodismoresuitablewhencomputationoftheresponseinthelimitstateisexpensive.Moreover,thismethodcanbeemployedincaseoflowprobabilitiesoffailureoftheorder10)]TJ /F4 7.97 Tf 6.59 0 Td[(6-10)]TJ /F4 7.97 Tf 6.59 0 Td[(8.SincetheITPSprobleminvolvessimplelimitstates,surrogatesareusedtoevaluateresponsesfortheprobabilityoffailurecalculation,andthefacttheprobabilityoffailureisashighas3%,bootstrappingisnotrequiredtoestimatetheaccuracyoftheprobabilityoffailure.However,themethodisemployedinthisproblemtoshowthatthemethodworksforsuchsituationstoo. Theaccuracyoftheprobabilityoffailureestimatewascalculatedwithab=1,000bootstraprepetitions.Randomsamplesofresponseswerebootstrapped,butthecapacityanderrorsweresampledforeverybootstraprepetition.Thatis,ineachofthe1,000bootstraprepetitions,theresponseswerere-sampledfromthesame500samples,whilethecapacityanderrorshadafreshsampleeverytime. Table7-17.Errorintheprobabilitiesoffailureofthedeterministicoptimumusingbootstrappingandrepetitions.Bootstrapsamples=1000,Repetitions=1000 RepetitionsBootstrappingProbabilityoffailuremean(pf)std(pf)mean(^pboot)std(^pboot) ^pth0.0018510)]TJ /F4 7.97 Tf 6.59 0 Td[(40.0019610)]TJ /F4 7.97 Tf 6.59 0 Td[(4^pstr0.0026110)]TJ /F4 7.97 Tf 6.59 0 Td[(30.0026910)]TJ /F4 7.97 Tf 6.59 0 Td[(4^pbk0.022710)]TJ /F4 7.97 Tf 6.59 0 Td[(30.023810)]TJ /F4 7.97 Tf 6.59 0 Td[(3 Table7-18.Errorintheprobabilitiesoffailureoftheprobabilisticoptimumusingbootstrappingandrepetitions.Bootstrapsamples=1000,Repetitions=1000 RepetitionsBootstrappingProbabilityoffailuremean(pf)std(pf)mean(^pboot)std(^pboot) ^pth0.0034810)]TJ /F4 7.97 Tf 6.59 0 Td[(40.003710)]TJ /F4 7.97 Tf 6.59 0 Td[(4^pstr0.004110)]TJ /F4 7.97 Tf 6.59 0 Td[(30.003110)]TJ /F4 7.97 Tf 6.59 0 Td[(3^pbk0.019710)]TJ /F4 7.97 Tf 6.59 0 Td[(30.021910)]TJ /F4 7.97 Tf 6.59 0 Td[(3 TheaccuracyoftheseparableMonteCarlomethodwasalsoassessedbyanempiricalstandarddeviationobtainedbyperformingn=1000repetitions.Themeanandthestandarddeviationoftheprobabilitiesoffailureestimatedthroughrepetitionsand 114

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bootstrappingaretabulatedinTable 7-17 forthedeterministicoptimumandTable 7-18 fortheprobabilisticoptimum.Theerrorintheprobabilityoffailureisapproximately3timeslessthanthemeanestimates.Themeanprobabilitiesoffailureoftheprobabilisticoptimumdifferby7%forbucklingloadwhileotherpfestimatesagreewell.Thebootstrappingmethodiscomparablewiththeempiricalestimatesdeterminedthroughrepetitionswithmaximumdiscrepancyinthebucklingprobabilitiesoffailure.Themajordifferenceisintheerrorestimatesfor^pbkwhichwas14%and21%fordeterministicandprobabilisticoptimumrespectively.Thiscanbeattributedtohighuncertaintyinthebucklingloads(15%). 7.5.6Summary ThreecriticalresponsescausingfailureoftheITPSpanelwasincludedasconstraintsintheoptimizationprocesstodeterminethedeterministicandprobabilisticoptimumoftheITPSpanel.Surrogatemodelsefcientlyreplacedcomputationallyexpensiveniteelementanalyses.Inadditiontothis,theevaluationofdeterministicconstraintsusingEGRAupdatedsurrogatesreducedthecomputationalrequiredto40%oftheavailablecomputationalbudget.Itwasalsofoundthatglobalsurrogateswithlargenumberofdesignsdonotalwaysprovidetheaccurateoptimum.EGRAwasalsosuccessfullyusedtoupdatereliabilityindexsurrogatesforreliabilitybasedoptimization. ThesystemreliabilityofthepanelwasestimatedusingMonteCarlosimulations.ThestatisticalindependenceoftherandomvariablesallowedSeparableMonteCarlomethodtobeemployedtoestimatetheprobabilityoffailure.Thesafetyfactorbaseddeterministicoptimumyieldedanoptimaldesignwithstructuralmassof21.3kg/m2with2.7%probabilityoffailure.Furthermassoftheprobabilisticoptimumwas19.4kg/m2withdeterministicprobabilityoffailure.Reliabilitybasedoptimizationdidnotinvolvehardconstraints(safetyfactors)thusdemonstratingthatriskallocationplaysanimportantroleinndinganon-conservativebutsafedesign.Bootstrappingprovidedreasonableestimatesofuncertaintyintheprobabilityoffailureestimates. 115

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CHAPTER8CONCLUSIONS 8.1Conclusion ThehighcomputationalcostofthedeterministicandprobabilisticoptimizationoftheITPSpanelwasefcientlyaddressedusingvarioustechniquesatdifferentphasesoftheoptimizationprocess.Toreducethecomputationalexpensesinvolvedinthestructuralanalysis,niteelementbasedhomogenizationmethodwasemployed,homogenizingthe3DITPSmodeltoa2Dorthotropicplate.HoweveritwasfoundthathomogenizationwasapplicableonlyforpanelsthataremuchlargerthanthecharacteristicdimensionsoftherepeatingunitcellintheITPSpanel.Henceasingleunitcellwasusedfortheoptimizationprocesstoreducethecomputationalcost.Thefailureconstraintsinvolvedintheoptimizationprocessdemandedcomputationallyexpensiveniteelementanalyseswhichwasreplacedbyefcient,lowdelitysurrogatemodels.Thelimitedcomputationalresourcesweredirectedtowardstargetregionsforaccuraterepresentationofconstraintsusingadaptivesamplingstrategies. Itwasfoundthatcomputationalcostofthedeterministicoptimizationwasreducedto40%oftheavailablecomputationalbudgetbyemployingEfcientGlobalReliabilityAnalysis.Whenitwasfoundthatthecomputationalcostwasdrasticallylowered,theadaptivesamplingmethodwasalsoemployedtoimprovetheapproximationofthereliabilityindexinthereliabilitybasedoptimizationprocess.Thesafetyfactorbaseddeterministicoptimumyieldedanoptimaldesignwithstructuralmassof21.3kg/m2with2.7%probabilityoffailure.Reliabilitybasedoptimizationwithoutsafetyfactorbasedconstraintsallocatedhigherrisktothefailureresponsesthatfurtherminimizedmass.Themassoftheprobabilisticoptimumwas19.4kg/m2withdeterministicprobabilityoffailure.Thisshowedthattheriskallocationinprobabilisticoptimizationplayedanimportantroleindeterminingnon-conservativebutsafedesigns.Theaccuracyofthe 116

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probabilityoffailurewasestimatedusingthebootstrappingmethod.Thebootstrappingestimateswerecomparabletotheempiricalestimatesdeterminedthroughrepetitions. 8.2FutureWork Possiblefutureworkinthestructuralanalysis Thestressfailureinthepanelwaspredictedthroughmaximumstresstheoryasthemainfocuswasonthemethodologyinvolvedintheoptimizationprocess.ThiscanbereplacedwithamoresuitablecompositefailurecriterionsuchasTsai-HillorTsai-Wucriteriontopredictfailure. Asbucklingoccursinthepanelandnotinaunitcell,bucklingcouldbeanalysedforanentirepanel. 117

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APPENDIXATHERMALPROPERTIESOFITPSCOMPONENTS TableA-1.DensityoftheITPSComponents ITPScomponentmaterialDensitykg/m3 BFSGraphite/epoxy1576.2TFS&WrapSiC/SiC2500FoamAETB86.1 ThethermalconductivityandspecicheatcapacityoftheITPScomponentsaretemperaturedependentproperties. TableA-2.ThermalPropertiesofthebottomfacesheet-GraphiteEpoxy TemperatureThermalConductivitykW/m/kSpecicheatCpW/m2 2880.026512.93060.027554.85580.0441172.36480.0501381.67380.0541591.010080.0702177.111880.0762533.013680.0842899.415480.0913223.817280.0973558.819080.1033967.0 TableA-3.ThermalPropertiesofthetopfacesheetandWrap-SiC/SiC TemperatureThermalConductivitykW/m/kSpecicheatCpW/m2 2939.50150012737.603000 118

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TableA-4.ThermalConductivityoftheinsulationfoam-AETB TemperatureThermalConductivitykW/m/k 0.0045255.50.0055394.40.0067533.30.0083672.20.0102811.00.0121949.90.01441088.80.01681227.70.01941366.50.02201505.40.02441644.30.02491783.2 TableA-5.Specicheatoftheinsulationfoam-AETB TemperatureSpecicheatCpW/m2 1884.0255.52637.6394.43165.2533.33454.0672.23617.3811.03717.8949.93768.11088.83805.71227.73805.71783.2 119

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APPENDIXBSTRENGTHPROPERTIESOFITPSMATERIALS TableB-1.DensityoftheITPSComponents Strength(MPa)SiC/SiCGraphite/EpoxyAETB Tensilestrengthinthe1-directionS1T3232,3120.689Tensilestrengthinthe2-directionS2T32339.20.689Compressivestrengthinthe1-directionS1C63218090.689Compressivestrengthinthe2-directionS2C63297.20.689ShearstrengthS1217633.20.379 120

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APPENDIXCDESIGNOFEXPERIMENTSADDEDBYEGRASAMPLINGTECHNIQUE TableC-1.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofTemperatureboundary.Target=473K tWmmtBmmtTmmhmmTmax;BFS 0.46.03.027.42474.31.02.03.039.20475.60.726.01.031.49474.81.02.01.039.10473.50.46.01.027.38470.70.42.03.033.33472.0 TableC-2.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofTemperatureboundary.Target=215.3MPa tWmmtBmmtTmmhmm1;W 0.42.01.7125.0225.41061.02.01.7140.0213.41061.05.153.040.0214.21060.44.503.025.0214.71060.46.03.031.4218.31060.46.02.3840.0217.8106 TableC-3.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofbucklingloadboundary.Target=1.5 tWmmtBmmtTmmhmm 1.06.01.040.01.650.42.01.025.01.700.626.01.025.01.850.42.01.6540.01.440.46.01.8840.01.291.02.01.040.01.61 121

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TableC-4.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofreliabilityindexboundary.Target=1.93 tWmmtBmmtTmmhmmReliabilityindex 0.523.811.6134.502.110.423.292.9535.401.770.442.142.6435.101.480.983.542.9130.701.960.702.652.6831.101.780.632.992.2829.801.820.842.491.9333.902.020.404.001.4629.002.130.402.001.2429.001.940.864.001.0036.001.850.402.001.5335.781.700.402.763.0029.001.801.002.001.0033.151.320.594.003.0036.003.131.002.003.0036.002.280.584.003.0029.001.750.702.003.0036.002.150.404.002.0736.001.931.004.001.9831.861.650.404.001.2932.031.720.402.002.4529.001.451.002.993.0034.092.010.403.161.2229.001.441.004.003.0032.591.900.622.001.0036.001.691.004.001.0033.992.380.403.051.6836.001.83 122

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BIOGRAPHICALSKETCHBharaniRavishankarwasbornin1985inthestateofTamilNadu,India.ShegraduatedwithaBachelorofEngineeringinAeronauticalEngineeringfromMadrasInstituteofTechnologyin2006.ShestartedhergraduatestudiesatUniversityofFloridaasamaster'sdegreestudentinthedepartmentofMechanicalandAerospaceEngineeringinJanuary2007.ShejoinedtheCenterforAdvancedCompositesLabinMay2007wheresheworkedasResearchAssistantwithDr.BhavaniSankar.InAugust2008,shecontinuedtopursueherPhDdegreeinMechanicalEngineering.ShewasawardedtheAmeliaEarhartFellowshipfromZontaInternational. 130