University of Florida Digital Collections Home
UFDC Home myUFDC Home  |   Help 
<%BANNER%>

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

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

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


This item has the following downloads:


Full Text

PAGE 1

DETERMINISTICANDRELIABILITYBASEDOPTIMIZATIONOFINTEGRATEDTHERMALPROTECTIONSYSTEMCOMPOSITEPANELUSINGADAPTIVESAMPLINGTECHNIQUESByBHARANIRAVISHANKARADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

PAGE 2

c2012BharaniRavishankar 2

PAGE 3

Tomyparents,RavishankarandParameswariandmysister,Vidhya 3

PAGE 4

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

PAGE 5

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

PAGE 6

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

PAGE 7

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

PAGE 8

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

PAGE 9

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

PAGE 10

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

PAGE 11

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

PAGE 12

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

PAGE 13

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

PAGE 14

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

PAGE 15

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

PAGE 16

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

PAGE 17

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

PAGE 18

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

PAGE 19

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

PAGE 20

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

PAGE 21

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

PAGE 22

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

PAGE 23

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

PAGE 24

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

PAGE 25

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

PAGE 26

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

PAGE 27

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

PAGE 28

Analysis(EGRA)foraccurateestimationoftheoptimizationconstraintswithlimitedcomputationalresources. 2.3.1DeterministicOptimization Deterministicoptimizationinvolvesoptimizingthedesignvariablestominimizetheobjectivefunctionwhilesatisfyingequalityorinequalityconstraintsthatfollowalinearornon-linearformulation.Thegeneralformulationoftheproblemis Minimizew=f(d)suchthatg(d)zh(d)=mdLB
PAGE 29

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

PAGE 30

=)]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

PAGE 31

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

PAGE 32

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

PAGE 33

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

PAGE 34

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

PAGE 35

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

PAGE 36

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

PAGE 37

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

PAGE 38

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

PAGE 39

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

PAGE 40

CrudeMonteCarlo,SeparableMonteCarlomethod,bootstrappingtechniqueandregroupingofrandomvariablestoimproveaccuracyofpfbyapplyingtheseconceptstoacompositepressurevesselproblem. 40

PAGE 41

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

PAGE 42

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

PAGE 43

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

PAGE 44

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

PAGE 45

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

PAGE 46

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

PAGE 47

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

PAGE 48

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

PAGE 49

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

PAGE 50

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

PAGE 51

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

PAGE 52

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

PAGE 53

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

PAGE 54

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

PAGE 55

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

PAGE 56

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

PAGE 57

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

PAGE 58

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

PAGE 59

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

PAGE 60

[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

PAGE 61

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

PAGE 62

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

PAGE 63

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

PAGE 64

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

PAGE 65

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

PAGE 66

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

PAGE 67

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

PAGE 68

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

PAGE 69

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

PAGE 70

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

PAGE 71

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

PAGE 72

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

PAGE 73

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

PAGE 74

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

PAGE 75

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

PAGE 76

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

PAGE 77

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

PAGE 78

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

PAGE 79

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

PAGE 80

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

PAGE 81

method.BootstrappingprovidedreasonableestimatesoftheuncertaintyintheSMCprobabilityoffailureestimates.Bootstrappingalsoallowedestimatingtheindividualcontributionsoftheresponseandcapacitytowardstheuncertaintyintheprobabilityoffailureestimate,thussuggestingadditionalsamplesofinexpensivecapacitywouldproveadvantageous.Furthersubstantialimprovementinaccuracywasachievedbytransferringuncertaintyawayfromexpensivecalculationsbyusingunitloadstressesandgeneratinglargesamplesofloadandstrengths. 81

PAGE 82

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

PAGE 83

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

PAGE 84

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

PAGE 85

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

PAGE 86

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

PAGE 87

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

PAGE 88

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

PAGE 89

FormoreexamplesanddetaileddescriptionofvariousapplicationsoftheEGRAmethodology,thereadersarereferredtoBichonetal.[ 59 86 14 ]. 89

PAGE 90

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

PAGE 91

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

PAGE 92

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

PAGE 93

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

PAGE 94

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

PAGE 95

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

PAGE 96

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

PAGE 97

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

PAGE 98

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

PAGE 99

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

PAGE 100

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

PAGE 101

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

PAGE 102

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

PAGE 103

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

PAGE 104

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

PAGE 105

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

PAGE 106

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

PAGE 107

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

PAGE 108

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

PAGE 109

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

PAGE 110

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

PAGE 111

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

PAGE 112

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

PAGE 113

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

PAGE 114

(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

PAGE 115

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

PAGE 116

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

PAGE 117

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

PAGE 118

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

PAGE 119

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

PAGE 120

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

PAGE 121

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

PAGE 122

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

PAGE 123

REFERENCES [1] M.Biancolini,Evaluationofequivalentstiffnesspropertiesofcorrugatedboard,CompositeStructures,vol.69,no.3,pp.322,July2005. [2] J.F.Davalos,P.Qiao,X.F.Xu,J.Robinson,andK.E.Barth,Modelingandcharacterizationofber-reinforcedplastichoneycombsandwichpanelsforhighwaybridgeapplications,CompositeStructures,vol.52,2001. [3] N.Buannic,P.Cartraud,andT.Quesnel,Homogenizationofcorrugatedcoresandwichpanels,CompositeStructures,vol.59,no.3,pp.299,Feb.2003. [4] J.WallachandL.Gibson,Mechanicalbehaviorofathree-dimensionaltrussmaterial,InternationalJournalofSolidsandStructures,vol.38,no.40-41,pp.7181,Oct.2001. [5] O.A.Martinez,B.V.Sankar,R.T.Haftka,S.K.Bapanapalli,andM.L.Blosser,MicromechanicalAnalysisofCompositeCorrugated-CoreSandwichPanelsforIntegralThermalProtectionSystems,AIAAJournal,vol.45,no.9,pp.2323,Sept.2007. [6] A.Sharma,B.V.Sankar,andR.T.Haftka,Multi-delityanalysisofcorrugated-coresandwichpanelsforintegratedthermalprotectionsystems,in50thAIAAASMEASCEAHSASCStructuresStructuralDynamicsandMaterialsConference,2009,vol.45,pp.1. [7] A.Sharma,Multi-FidelityDesignofanIntegralThermalProtectionSystemforFutureSpaceVehicleduringRe-entry,Ph.D.dissertation,2010. [8] R.C.KuczeraandZ.P.Mourelatos,OnEstimatingtheReliabilityofMultipleFailureRegionProblemsUsingApproximateMetamodels,JournalofMechanicalDesign,vol.131,no.12,pp.121003,2009. [9] H.Arenbeck,S.Missoum,A.Basudhar,andP.Nikravesh,Reliability-BasedOptimalDesignandTolerancingforMultibodySystemsUsingExplicitDesignSpaceDecomposition,JournalofMechanicalDesign,vol.132,no.2,pp.021010,2010. [10] C.TuandR.Barton,ProductionYieldEstimationbytheMetamodelMethodWithaBoundary-FocusedExperimentDesign,inASMEDesignEngineeringTechnicalConference,AmericanSocietyofMechanicalEngineers,1997. [11] P.Ranjan,D.Bingham,C.Va,G.Michailidis,andA.Arbor,SequentialExperimentDesignforContourEstimationfromComplexComputerCodes,Technometrics,pp.527,2008. [12] E.VazquezandJ.Bect,ASequentialBayesianAlgorithmtoEstimateaProbabilityofFailure,inProceedingsof15thIFACSymposiumonSystemIdentication SYSID ,Saint-Malo,France,2009. 123

PAGE 124

[13] V.Picheny,D.Ginsbourger,O.Roustant,R.T.Haftka,andN.-H.Kim,AdaptiveDesignsofExperimentsforAccurateApproximationofaTargetRegion,JournalofMechanicalDesign,vol.132,no.7,pp.071008,2010. [14] B.J.Bichon,EfcientSurrogateModelingforReliabilityAnalysisandDesign,Ph.D.dissertation,2010. [15] B.P.Smarslok,Measuring,Using,andReducingExperimentalandComputationalUncertaintyinReliabilityAnalysisofCompositeLaminates,Ph.D.dissertation,2009. [16] B.P.Smarslok,R.T.Haftka,L.Carraro,andD.Ginsbourger,ImprovingaccuracyoffailureprobabilityestimateswithseparableMonteCarlo,InternationalJournalofReliabilityandSafety,vol.4,no.4,pp.393,2010. [17] B.Ravishankar,B.P.Smarslok,R.T.Haftka,andB.V.Sankar,SeparableSamplingoftheLimitStateforAccurateMonteCarloSimulation,in50thAIAAAS-MEASCEAHSASCStructuresStructuralDynamicsandMaterialsConference,2009,numberMay,pp.1. [18] V.Picheny,N.H.Kim,andR.T.Haftka,Applicationofbootstrapmethodinconservativeestimationofreliabilitywithlimitedsamples,StructuralandMultidisci-plinaryOptimization,vol.41,no.2,pp.205,Aug.2009. [19] B.Ravishankar,B.P.Smarslok,R.T.Haftka,andB.V.Sankar,ErrorEstimationandErrorReductioninSeparableMonteCarloMethod,AIAAJournal,vol.48,no.11,pp.2624,Nov.2010. [20] D.E.Myers,C.J.Martin,andM.L.Blosser,ParametricWeightComparisonofCurrentandProposedThermalProtectionSystem(TPS)Concepts,33rdAIAAThermophysicsConference,June1999. [21] N.Haddard,H.McWilliams,andP.Wagoner,NASAEngineeringDesignChallenges:ThermalProtectionSystems,NASAReport,2007. [22] J.Wilson,NASAFacts,NasaTechnicalMemorandum,1997. [23] M.Blosser,C.Martin,K.Darybeigi,andC.Poteet,ReusableMetallicThermalProtectionSystemsDevelopment,NASATechnicalMemorandum,vol.1,no.11,pp.1829,2004. [24] D.J.Rasky,F.S.Milos,andT.H.Squire,TPSmaterialsandCostsforFutureReusableLaunchVehicles,NasaTechnicalMemorandum,1997. [25] J.Domoulin,SpaceShuttleOrbiterSystems,1988. [26] R.Oakes,SpaceShuttleCeramicTiles,1992. 124

PAGE 125

[27] M.Blosser,DevelopmentofMetallicThermalProtectionSystemsfortheReusableLaunchVehicle,NasaTechnicalMemorandum110296,1996. [28] S.Bapanapalli,DesignofanIntegratedThermalProtectionSystemforFutureSpaceVehicles,Ph.D.dissertation,2007. [29] W.YuandT.Tang,VariationalAsymptoticMethodforUnitCellHomogenizationofPeriodicallyHeterogeneousMaterials,InternationalJournalofSolidsandStructures,vol.44,pp.1,2007. [30] C.-Y.LeeandW.Yu,Homogenizationanddimensionalreductionofcompositeplateswithin-planeheterogeneity,InternationalJournalofSolidsandStructures,vol.48,no.10,pp.1474,May2011. [31] O.A.Martinez,MicromechanicalAnalysisandDesignofanIntegratedThermalProtectionSystemforFutureSpaceVehicles,Ph.D.dissertation,2007. [32] A.Sharma,C.Gogu,O.A.Martinez,B.V.Sankar,andR.T.Haftka,Multi-FidelityDesignofanIntegratedThermalProtectionSystemforSpacecraftReentry,in49thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference,2008,numberApril,pp.1. [33] P.W.ChristensenandK.Anders,AnIntroductiontoStructuralOptimization,2008. [34] Z.Gurdal,R.T.Haftka,andP.Hajela,DesignandOptimizationofCompositeLaminates,JohnWiley&Sons,INC,1999. [35] X.QuandR.T.Haftka,DeterministicandReliability-BasedOptimizationofCompositeLaminatesforCryogenicEnvironments,AIAAJournal,vol.41,no.10,2003. [36] S.VenkataramanandP.Salas,OptimizationofCompositeLaminatesforRobustandPredictableProgressiveFailureResponse,AIAAJournal,vol.45,no.5,pp.1113,May2007. [37] Y.TsompanakisandM.Papadrakakis,Large-scalereliability-basedstructuraloptimization,StructuralandMultidisciplinaryOptimization,vol.26,no.6,pp.429,Apr.2004. [38] O.DitlevsenandH.O.Madsen,StructuralReliabilityMethods,NumberSeptember.2007. [39] R.E.Melchers,StructuralReliabilityAnalysisandPrediction,Wiley,1999. [40] A.HaldarandM.Sankaran,Probability,reliability,andstatisticalmethodsinengineeringdesign,Wiley,2000. [41] H.W.LehetaandA.E.Mansour,Reliability-basedMethodforOptimalStructuralDesignofStiffenedPanelsXp,MarinStructires,pp.323,1997. 125

PAGE 126

[42] M.DiSciuvaandD.Lomario,AcomparisonbetweenMonteCarloandFORMsincalculatingthereliabilityofacompositestructure,CompositeStructures,vol.59,no.1,pp.155,Jan.2003. [43] S.Kumar,D.Villanueva,B.V.Sankar,andR.T.Haftka,ProbabilisticOptimizationofIntegratedThermalProtectionSystem,in12thAIAA/ISSMOMultidisciplinaryAnalysisandOptimizationConference,2008,numberSeptember. [44] D.Villanueva,A.Sharma,R.T.Haftka,andB.V.Sankar,RiskAllocationbyOptimizationofanIntegratedThermalProtectionSystem,in8thWorldCongressonStructuralandMultidisciplinaryOptimization,2009,pp.1. [45] A.Sharma,B.V.Sankar,R.T.Haftka,andN.C.Ebaugh,Multi-FidelityAnalysisofCorrugated-CoreSandwichPanelsforIntegratedThermalProtectionSystems,in50thAIAAASMEASCEAHSASCStructuresStructuralDynamicsandMaterialsConferencethAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference,2009,numberMay,pp.1. [46] D.Villanueva,R.T.Haftka,andB.V.Sankar,IncludingFutureTestsintheDesignofanIntegratedThermalProtectionSystem,in51stAIAAASMEASCEAHSASCStructuresStructuralDynamicsandMaterialsConference,2010,numberApril,pp.1. [47] T.Matsumura,R.T.Haftka,andB.V.Sankar,ReliabilityEstimationIncludingRedesignFollowingFutureTestforanIntegratedThermalProtectionSystem,in9thWorldCongressonStructuralandMultidisciplinaryOptimization,2011,pp.1. [48] A.I.KhuriandJ.A.Cornell,Khuri,A.I.;Cornell,J.A.1996:Responsesurfaces:designsandanalyses.,MarcelDekker,NewYork,2ndeditioedition,1996. [49] R.MyersandD.Montgomery,ResponseSurfaceMethodology:ProcessandProductOptimizationUsingDesignedExperiments,Wiley-Interscience,1995. [50] F.A.C.VianaandR.T.Haftka,Surrogate-basedOptimizationwithParallelSimulationsusingtheProbabilityofImprovement,in13thAIAA/ISSMOMultidis-ciplinaryAnalysisandOptimizationConference,2010,numberSeptember,pp.1. [51] F.A.C.Viana,C.Gogu,andR.T.Haftka,DETC2010-28813,inProceedingsoftheASME2010InternationalDesignEngineeringTechnicalConferences&ComputersandInformationinEngineeringConference,2010,pp.1. [52] F.A.C.Viana,SurrogateToolbox,MATLAB,2009. [53] M.PapilaandR.T.Haftka,ResponseSurfaceApproximations:Noise,ErrorRepair,andModelingErrors,AIAAJournal,vol.38,no.12,pp.2336,Dec.2000. 126

PAGE 127

[54] S.ShanandG.G.Wang,SpaceExplorationandGlobalOptimizationforComputationallyIntensiveDesignProblems:ARoughSetBasedApproach,StructuralandMultidisciplinaryOptimization,vol.28,2004. [55] D.R.Jones,M.Schonlau,andJ.William,EfcientGlobalOptimizationofExpensiveBlack-BoxFunctions,JournalofGlobalOptimization,pp.455,1998. [56] A.Guinta,A,J.M.McFarland,L.P.Swiler,andM.S.Eldred,Thepromiseandperilofuncertaintyquanticationusingresponsesurfaceapproximations,StructureandInfrastructureEngineering:Maintenance,Management,Life-CycleDesignandPerformance,vol.2,no.3-4,2006. [57] L.Wang,D.Beeson,S.Akkaram,andG.Wiggs,GaussianProcessMeta-ModelsforEfcientProbabilisticDesigninComplexEngineeringDesignSpaces,inProc.ofASMEInternationalDesignEngineeringTechnicalConference&ComputersandInformationinEngineeringConference,2005. [58] P.Ranjan,D.Bingham,andG.Michailidis,ErrataSequentialExperimentDesignforContourEstimationFromComplexComputerCodes,Technometrics,vol.50,no.4,pp.4. [59] B.J.Bichon,M.S.Eldred,L.P.Swiler,S.Mahadevan,andJ.M.McFarland,EfcientGlobalReliabilityAnalysisforNonlinearImplicitPerformanceFunctions,AIAAJournal,vol.46,no.10,pp.2459,Oct.2008. [60] L.P.Swiler,T.L.Paez,andR.L.Mayes,EpistemicUncertaintyQuanticationTutorial,inProceedingsoftheIMAC-XXVII,SocietyofExperimentalMechanics,2009. [61] E.Acar,R.T.Haftka,andN.H.Kim,EffectsofStructuralTestsonAircraftSafety,AIAAJournal,vol.48,no.10,pp.2235,Oct.2010. [62] C.C.a.AntonioandL.N.Hoffbauer,Fromlocaltoglobalimportancemeasuresofuncertaintypropagationincompositestructures,CompositeStructures,vol.85,no.3,pp.213,Oct.2008. [63] D.H.OhandL.Librescu,Freevibrationandreliabilityofcompositecantileversfeaturing-uncertainproperties,ReliabilityEngineeringandSystemSafety,vol.56,pp.265,1997. [64] A.K.Noor,J.H.Starnes,andJ.M.Peters,Uncertaintyanalysisofcompositestructures,ComputerMethodsinAppliedMechanicsandEngineering,vol.185,no.2-4,pp.413,May2000. [65] A.K.Noor,J.H.Starnes,andJ.M.Peters,UncertaintyAnalysisofStiffenedCompositePanels,CompositeStructures,vol.51,2001. 127

PAGE 128

[66] E.Acar,R.T.Haftka,andT.F.Johnson,EffectsofUncertaintyReductiononWeightofCompositeLaminatesatCryogenicTemperatures,inASME2005Inter-nationalDesignEngineeringTechnicalConferences&ComputersandInformationinEngineeringConference,2005,pp.1. [67] C.E.PapadopoulosandH.Yeung,UncertaintyestimationandMonteCarlosimulationmethod,FlowMeasurementandInstrumentation,vol.12,pp.291,2001. [68] M.H.KalosandP.A.Whitlock,MonteCarloMethods,Wiley-IntersciencePublications,JohnWileyandSons,NewYork,1986. [69] D.Padmanabhan,H.Agarwal,J.E.Renaud,andS.M.Batill,AstudyusingMonteCarloSimulationforfailureprobabilitycalculationinReliability-BasedOptimization,OptimizationandEngineering,vol.7,no.3,pp.297,Sept.2006. [70] R.Y.Rubinstein,VarianceReductionTechniques:SimulationandtheMonteCarloMethod,JohnWiley&Sons,NewYork,N.Y.,1981. [71] N.H.Kim,P.Ramu,N.V.Queipo,andM.Carlo,TailModelinginReliability-BasedDesignOptimizationforHighlySafeStructuralSystems,47thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference,pp.1,2006. [72] B.P.Smarslok,A.Dylan,andR.T.Haftka,SeparableMonteCarloSimulationAppliedtoLaminatedCompositePlatesReliability,in49thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference,2008,pp.1. [73] T.K.JacobsenandP.Brndsted,MechanicalPropertiesofTwoPlain-WovenChemicalVaporInltratedSiliconCarbide-MatrixComposites,JournalofAmericanCeramicSociety. [74] NASA,CeramicMaterialsDatabase,. [75] NASA,TPSXNASAMaterialsDatabase,. [76] P.Fenici,a.J.FriasRebelo,R.Jones,a.Kohyama,andL.Snead,CurrentstatusofSiC/SiCcompositesR&D,JournalofNuclearMaterials,vol.258-263,pp.215,Oct.1998. [77] ABAQUS,CAEUser'sManual,ABAQUSdocumentation6.9,. [78] ABAQUS,AnalysisUser'sManual,ABAQUSDocumentation6.9,. [79] R.V.MarreyandB.V.Sankar,MicromechanicalModelsforTextileStructuralComposites,Tech.Rep.,1995. 128

PAGE 129

[80] R.V.MarreyandB.V.Sankar,AMicromechanicalModelforTextileCompositePlates,JournalofCompositeMaterials,vol.31,no.12,pp.1187,June1997. [81] R.Gibson,PrinciplesofCompositeMaterials,McGraw-HillInc.,1994. [82] J.Whitney,StructuralAnalysisofLaminatedAnisotropicPlates,Technomic,1987. [83] C.Stamblewski,B.V.Sankar,andD.Zenkert,AnalysisofThree-DimensionalQuadraticFailureCriteriaforThickCompositesusingtheDirectMicromechanicsMethod,JournalofCompositeMaterials,vol.42,no.7,pp.635,Apr.2008. [84] K.PriceandR.Storn,DifferentialEvolutionforContinuousFunctionOptimization,1996. [85] F.Viana,R.Haftka,andL.Watson,SequentialSamplingforContourEstimationwithConcurrentFunctionEvaluations,StructuralandMultidisciplinaryOptimiza-tion,vol.45,no.4,2012. [86] B.J.Bichon,M.Sankaran,andM.Eldred,Reliability-BasedDesignOptimizationusingEfcientGlobalReliabilityAnalysis,in50thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference,2009. 129

PAGE 130

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