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| Citation |
- Permanent Link:
- https://ufdc.ufl.edu/UFE0043944/00001
Material Information
- Title:
- Deterministic and Reliability Based Optimization of Integrated Thermal Protection System Composite Panel using Adaptive Sampling Techniques
- Creator:
- Ravishankar, Bharani P
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2012
- Language:
- english
- Physical Description:
- 1 online resource (130 p.)
Thesis/Dissertation Information
- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Mechanical Engineering
Mechanical and Aerospace Engineering
- Committee Chair:
- Sankar, Bhavani V
- Committee Co-Chair:
- Haftka, Raphael T
- Committee Members:
- Kumar, Ashok V
Consolazio, Gary R
- Graduation Date:
- 5/5/2012
Subjects
- Subjects / Keywords:
- Boundary conditions ( jstor )
Buckling ( jstor )
Design optimization ( jstor )
Estimate reliability ( jstor )
Estimation methods ( jstor )
Finite element analysis ( jstor )
Modeling ( jstor )
Monte Carlo methods ( jstor )
Random variables ( jstor )
Stiffness ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
deterministic -- homogenization -- optimization -- reliability-based
- Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm )
Mechanical Engineering thesis, Ph.D.
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. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2012.
- Local:
- Adviser: Sankar, Bhavani V.
- Local:
- Co-adviser: Haftka, Raphael T.
- Statement of Responsibility:
- by Bharani P Ravishankar.
Record Information
- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Ravishankar, Bharani P. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 864879914 ( OCLC )
- Classification:
- LD1780 2012 ( lcc )
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DETERMINISTICANDRELIABILITYBASEDOPTIMIZATIONOFINTEGRATEDTHERMALPROTECTIONSYSTEMCOMPOSITEPANELUSINGADAPTIVESAMPLINGTECHNIQUESByBHARANIRAVISHANKARADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012
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c2012BharaniRavishankar 2
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Tomyparents,RavishankarandParameswariandmysister,Vidhya 3
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ACKNOWLEDGMENTSFirstandforemostIwouldliketothankmyadvisorsDr.BhavaniSankarandDr.RaphaelHaftkaattheDepartmentofMechanicalandAerospaceEngineering.Allofyourinsights,guidance,andpatiencehasbeengreatlyappreciated.Iwouldalsoliketothankthemembersofmyadvisorycommittee,Dr.AshokKumarattheDepartmentofMechanicalandAerospaceEngineeringandDr.GaryConsolazioattheDepartmentofCivilandCoastalEngineering.IwouldlikethankmyfriendsattheCenterforAdvancedCompositeslab,Anurag,Prasanna,MinSong,Marlana,TimandSayan.IwouldalsoliketothankthemembersoftheStructuralandMultidiscplinaryGroup,Ben,Felipe,Diane,AnirbanandTaikiforalltheirhelp,valuableinputsandsuggestions.IwouldalsoliketothankmyfriendsAnirudh,SriramandNarenforallthehelpandsupportthroughoutmyPhDprogram,makingthisexperiencememorable.Iwouldliketothankmyfamilyfortheirsupportandpatience. 4
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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 11 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1OutlineofDissertation ............................. 19 1.1.1MotivationandObjectives ....................... 19 1.1.2ThesisOrganization .......................... 20 2BACKGROUNDSTUDYANDLITERATUREREVIEW .............. 21 2.1OrbiterThermalProtectionSystem-30yearsLegacy ............ 21 2.1.1TileTPS ................................. 23 2.1.2Non-TileTPS ............................. 23 2.2FiniteElementbasedHomogenization .................... 24 2.3DeterministicandReliabilitybasedoptimization ............... 27 2.3.1DeterministicOptimization ....................... 28 2.3.2StructuralReliability .......................... 29 2.4UseofSurrogateModels ........................... 31 2.4.1ApplicationofSurrogatestoImproveConstraintBoundaries .... 32 2.5AdaptiveSampling ............................... 33 2.5.1EfcientGlobalReliabilityAnalysis .................. 34 2.6UncertaintyModeling ............................. 35 2.7EstimationofProbabilityofFailure ...................... 36 2.7.1MomentbasedMethods ........................ 37 2.7.2SamplingMethods ........................... 38 3DESIGNANDANALYSISOFINTEGRATEDTHERMALPROTECTIONSYSTEMPANEL ........................................ 41 3.1DesignofITPSCompositePanel ....................... 41 3.1.1KeyDimensionsoftheITPSstructure ................ 41 3.1.2ITPSMaterials ............................. 42 3.1.3BoundaryConditions .......................... 43 3.1.4Loads .................................. 44 3.2FiniteElementAnalyses ............................ 44 3.2.1TransientHeatTransferAnalysis ................... 45 3.2.2StressAnalysis ............................. 47 5
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3.2.3BucklingAnalysis ............................ 48 3.3Summary .................................... 50 4FINITEELEMENTBASEDHOMOGENIZATION ................. 51 4.1A,B,Dmatrices ................................. 51 4.2TransverseShearStiffness .......................... 53 4.3AccuracyoftheHomogenizationMethod .................. 57 4.3.1DeectionComparison ......................... 57 4.3.2StressComparison ........................... 57 4.4ResultsandDiscussion ............................ 58 4.4.1A,B,Dmatrices-Equivalentmaterialvs.Equivalentstructure ... 58 4.4.2TransverseShearStiffnessofITPS .................. 60 4.4.3AccuracyoftheHomogenizationMethod ............... 63 4.5Summary .................................... 66 5MONTECARLOSIMULATIONS-RELIABILITYESTIMATION ......... 67 5.1CrudeMonteCarloMethod(CMC) ...................... 67 5.2SeparableMonteCarlomethod(SMC) .................... 68 5.3ErrorintheProbabilityofFailureEstimate .................. 69 5.4SMCwithregroupingandseparablesamplingofthelimitstaterandomvariables .................................... 71 5.5ApplicationtoFailureAnalysisofCompositeLaminate ........... 72 5.6ResultsandDiscussion ............................ 75 5.6.1CrudeandSeparableMonteCarloMethod .............. 75 5.6.2Regroupingandseparablesamplingofthelimitstatevariablesforimprovingaccuracy ........................... 78 5.6.3Summary ................................ 80 6EFFICIENTGLOBALRELIABILITYANALYSIS(EGRA) ............. 82 6.1EGRAalgorithm ................................ 82 6.2IllustrationofEGRA .............................. 83 6.3Summary .................................... 88 7DETERMINISTICANDRELIABILITYBASEDOPTIMIZATION ......... 90 7.1DeterministicOptimization ........................... 90 7.2UncertaintyModeling ............................. 93 7.3EstimationofSystemReliability ........................ 94 7.4ReliabilityBasedOptimization ......................... 96 7.5ResultsandDiscussion ............................ 98 7.5.1DeterministicOptimization ....................... 98 7.5.2SensitivityAnalysis ........................... 104 7.5.3ReliabilityoftheDeterministicOptimum ............... 107 7.5.4Reliabilitybasedoptimization ..................... 110 6
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7.5.5Errorintheprobabilityoffailureestimate ............... 113 7.5.6Summary ................................ 115 8CONCLUSIONS ................................... 116 8.1Conclusion ................................... 116 8.2FutureWork ................................... 117 ATHERMALPROPERTIESOFITPSCOMPONENTS ............... 118 BSTRENGTHPROPERTIESOFITPSMATERIALS ................ 120 CDESIGNOFEXPERIMENTSADDEDBYEGRASAMPLINGTECHNIQUE .. 121 REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 130 7
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LISTOFTABLES Table page 3-1KeyDimensionsoftheITPSpanel ......................... 41 3-2ComponentsofITPS,materialsandberorientation ............... 43 3-3MaterialpropertiesofthecomponentsofITPS .................. 43 3-4Heatuxloadstepsinthetransientheattransferanalysis ............ 46 4-1Periodicboundaryconditionsforthesixdeformations .............. 52 4-2VariationofA44andA55ofITPSpanelwithn ................... 61 4-3Tipdeectionratioalongwithcontributionofbendingandsheardeformationtowardsdeection .................................. 64 5-1Materialpropertiesanduncertaintyoftherandomvariables ........... 74 5-2EmpiricalandbootstrappingestimatesofprobabilityoffailureusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitions ....... 76 5-3StandarddeviationandcoefcientofvariationofempiricalandbootstrappingpfestimatesusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitionsfororiginallimitstate ...................... 77 5-4Relativecontributionsofresponse(stresses)andcapacity(strengths)towardstheuncertaintyinpfthroughbootstrappingandalsocomparedwithempiricalresults ......................................... 77 5-5StandarddeviationandcoefcientofvariationofCMCandSMCforincreasingsamplesizeofMandN=500 ............................ 79 5-6StandarddeviationandcoefcientofvariationofCMC,SMCandSMCregroupedforincreasingsamplesizeofMandN=500 .................... 79 7-1Failure,safetyfactorsappliedonconstraints .................... 91 7-2Lowerandupperboundsofdesignvariablesfordeterministicoptimization ... 92 7-3CoefcientofvariationofinputrandomvariablesincludedintheITPSdesign 93 7-4Nominalallowablevaluesofcapacityandcoefcientofvariation ........ 94 7-5OptimumdesignvariablesandminimizedstructuralmassthroughglobalandEGRADOEfordeterministicoptimization ..................... 101 7-6ComparisonbetweentheaccuracyofglobalsurrogateandEGRAsurrogateatthedeterministicoptimumusingcorrespondingsurrogates .......... 103 8
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7-7Comparisonofresponsesatallthedeterministicoptimausingthe60DOEsurrogate ....................................... 103 7-8Comparisonofresponsesatallthedeterministicoptimausingthe20DOEsurrogate ....................................... 104 7-9ComparisonofresponsesatallthedeterministicoptimausingtheEGRAupdatedsurrogate ....................................... 104 7-10Differenceinuncertaintyinresponsesbeforeandaftersensitivityanalysis ... 106 7-11Uncertaintyinresponseduetoinputuncertainty ................. 108 7-12Individualprobabilitiesoffailureofthedeterministicoptimum .......... 109 7-13Lowerandupperboundsofdesignvariablesforprobabilisticoptimization ... 110 7-14OptimumdesignvariablesandminimizedstructuralmassthroughEGRAupdatedDOEforreliabilitybasedoptimization ........................ 112 7-15Individualprobabilitiesoffailureoftheprobabilisticoptimum ........... 112 7-16Individualprobabilitiesoffailureofthedeterministicoptimumusingprobabilisticoptimumsurrogate .................................. 112 7-17Errorintheprobabilitiesoffailureofthedeterministicoptimumusingbootstrappingandrepetitions .................................... 114 7-18Errorintheprobabilitiesoffailureoftheprobabilisticoptimumusingbootstrappingandrepetitions .................................... 114 A-1DensityoftheITPSComponents .......................... 118 A-2ThermalPropertiesofthebottomfacesheet-GraphiteEpoxy ......... 118 A-3ThermalPropertiesofthetopfacesheetandWrap-SiC/SiC .......... 118 A-4ThermalConductivityoftheinsulationfoam-AETB ............... 119 A-5Specicheatoftheinsulationfoam-AETB .................... 119 B-1DensityoftheITPSComponents .......................... 120 C-1DesignpointsacquiredusingEGRAtoimprovetheaccuracyofTemperatureboundary ....................................... 121 C-2DesignpointsacquiredusingEGRAtoimprovetheaccuracyofstressboundary 121 C-3DesignpointsacquiredusingEGRAtoimprovetheaccuracyofbucklingloadboundary ....................................... 121 9
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C-4DesignpointsacquiredusingEGRAtoimprovetheaccuracyofreliabilityindexboundary ....................................... 122 10
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LISTOFFIGURES Figure page 2-1ThermalProtectionSystem(TPS)usedinspaceshuttles ............ 22 2-2Sinusoidalcoresandwichedbetweenlaminates .................. 26 2-3IntegratedThermalProtectionSystemwithcorrugatedcore ........... 26 3-1SchematicviewoftheITPSpanel ......................... 42 3-2DimensionsoftheITPSpanel ............................ 42 3-3BoundaryconditionsappliedontheITPSpanel .................. 44 3-4HeatuxproleincidentontheITPSpanel .................... 45 3-5Boundaryconditionsforthetransientheattransferanalysis ........... 46 3-6TemperaturevariationonTFSandBFSwithrespecttoreentrytime ...... 47 3-7BuckledITPSduetothermalloads ........................ 49 4-1SchematicviewoftheITPSunitcell(RVE) .................... 52 4-2Periodicboundaryconditionappliedontheunitcellforx=1 .......... 54 4-3Schematicviewoftheunitcelldeformationsduetostrainsandcurvatures ... 54 4-41-DITPSmodelwithunitcellsinx-direction ................... 55 4-51DmodeloftheITPSwithunitcellsinx-directionsubjectedtopressureloadandxedBCs ..................................... 58 4-6TransverseshearstiffnessA55oforthotropicpanel ................ 61 4-7VariationofA44andA55ofITPSpanelwithn ................... 62 4-8ConvergedA44andA55ofITPSpanelwithn ................... 62 4-9Ratiobetweentipdeectionofthehomogenizedmodelandthe3-DITPSmodelalongthexandy-direction ............................. 63 4-10Stress(11,22)ratiobetweenthehomogenizedmodelandthe3DITPSmodel 65 4-11Stress(11,22)atn=10comparisonbetweenthe3-DITPSmodelandthehomogenizedmodel ................................. 65 4-12Verticaldeectionalongx-axisoftheoriginalITPSandhomogenizedmodelduetopressureload ................................. 66 5-1IllustrationofcrudeandseparableMonteCarloMethodcomparisons ...... 69 11
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5-2Schematicrepresentationofbootstrappingwhenonlyresponseissampled. .. 70 5-3Illustrationofseparablesamplingwithunitloads ................. 72 5-4Compositepressurevesselwithinternalpressureof100kPaandstressesactinginasmallelementofthevessel. ....................... 73 5-5Distributionofstressandstrengthinthe2-direction(2)showingtheprobablefailureregion ..................................... 75 5-6StandardDeviationofCMC,SMCandregroupedlimitstateSMCwhereN=500(xed)andMisvaryingfor10,000repetitions ................... 80 6-1TruecontoursoftheBraninhoofunction ...................... 83 6-2TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction 84 6-3ExpectedFeasibilityFunctionevaluatedusingtheinitialkrigingapproximation 85 6-4Krigingmodelupdatedwiththenewdesignandexpectedfeasibilityfunction .. 86 6-5TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction 86 6-6TargetcontouraccuratelyapproximatedbytheEGRAmethodology ...... 87 7-1VariationofresponseswrtthicknessofthewraptW ............... 99 7-2VariationofcriticalresponseswrtthicknessofthebottomfacesheettB .... 100 7-3VariationofcriticalresponseswrtthicknessofthetopfacesheettT ....... 101 7-4Variationofcriticalresponseswrtheightofthefoamh .............. 102 7-5SensitivityanalysisofmaximumBFStemperature ................ 106 7-6SensitivityanalysisofWrapstress11 ....................... 107 7-7Sensitivityanalysisofbucklingloadfactor ..................... 108 12
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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDETERMINISTICANDRELIABILITYBASEDOPTIMIZATIONOFINTEGRATEDTHERMALPROTECTIONSYSTEMCOMPOSITEPANELUSINGADAPTIVESAMPLINGTECHNIQUESByBharaniRavishankarMay2012Chair:BhavaniV.SankarCochair:RaphaelT.HaftkaMajor:MechanicalEngineering Conventionalspacevehicleshavethermalprotectionsystems(TPS)thatprovideprotectiontoanunderlyingstructurethatcarriestheightloads.Inanattempttosaveweight,thereisinterestinanintegratedTPS(ITPS)thatcombinesthestructuralfunctionandtheTPSfunction.Thishasweightsavingpotential,butcomplicatesthedesignoftheITPSthatnowhasboththermalandstructuralfailuremodes.ThemainobjectivesofthisdissertationwastooptimallydesigntheITPSsubjectedtothermalandmechanicalloadsthroughdeterministicandreliabilitybasedoptimization. TheoptimizationoftheITPSstructurerequirescomputationallyexpensiveniteelementanalysesof3DITPS(solid)model.Toreducethecomputationalexpensesinvolvedinthestructuralanalysis,niteelementbasedhomogenizationmethodwasemployed,homogenizingthe3DITPSmodeltoa2Dorthotropicplate.HoweveritwasfoundthathomogenizationwasapplicableonlyforpanelsthataremuchlargerthanthecharacteristicdimensionsoftherepeatingunitcellintheITPSpanel.Henceasingleunitcellwasusedfortheoptimizationprocesstoreducethecomputationalcost. DeterministicandprobabilisticoptimizationoftheITPSpanelrequiredevaluationoffailureconstraintsatvariousdesignpoints.Thisfurtherdemandscomputationallyexpensiveniteelementanalyseswhichwasreplacedbyefcient,lowdelitysurrogate 13
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models.Inanoptimizationprocess,itisimportanttorepresenttheconstraintsaccuratelytondtheoptimumdesign.Insteadofbuildingglobalsurrogatemodelsusinglargenumberofdesigns,thecomputationalresourcesweredirectedtowardstargetregionsnearconstraintboundariesforaccuraterepresentationofconstraintsusingadaptivesamplingstrategies. EfcientGlobalReliabilityAnalyses(EGRA)facilitatessequentiallysamplingofdesignpointsaroundtheregionofinterestinthedesignspace.EGRAwasappliedtotheresponsesurfaceconstructionofthefailureconstraintsinthedeterministicandreliabilitybasedoptimizationoftheITPSpanel.Itwasshownthatusingadaptivesampling,thenumberofdesignsrequiredtondtheoptimumwerereduceddrastically,whileimprovingtheaccuracy. SystemreliabilityofITPSwasestimatedusingMonteCarloSimulation(MCS)basedmethod.SeparableMonteCarlomethodwasemployedthatallowedseparablesamplingoftherandomvariablestopredicttheprobabilityoffailureaccurately.Thereliabilityanalysisconsidereduncertaintiesinthegeometry,materialproperties,loadingconditionsofthepanelanderrorinniteelementmodeling.TheseuncertaintiesfurtherincreasedthecomputationalcostofMCStechniqueswhichwasalsoreducedbyemployingsurrogatemodels.Inordertoestimatetheerrorintheprobabilityoffailureestimate,bootstrappingmethodwasapplied. ThisresearchworkthusdemonstratesoptimizationoftheITPScompositepanelwithmultiplefailuremodesandlargenumberofuncertaintiesusingadaptivesamplingtechniques. 14
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CHAPTER1INTRODUCTION Whenaspacevehiclereenterstheatmosphereathypersonicspeeds,thevehicle'sexteriorissubjectedtosevereaerodynamicheatingandpressure.Toprotectthevehiclefromsuchextremetemperatures,aThermalProtectionSystem(TPS),isaddedontothemainloadbearingstructure.But,theTPSisincompatiblewiththemainstructureduetothemismatchintheirthermalproperties.Thisincompatibility,alongwiththeactionofnumerousotherloadslikeaerodynamicpressure,impactloadsandotherin-planeinertialloadsmayleadtothestructure'sfailurewithcatastrophicconsequences.Moreover,theTPSencompassesasignicantportionofthevehicle'sexteriorconstitutingmajorpartofthelaunchweight.Thus,itisimperativethatapartfrommakingtheTPSsuitableforprotectionpurposes,itshouldbelightweightinordertoincreasepayloadsthatcanbecarriedaboard. Toaccommodatetheserequirements,amultifunctionalIntegratedThermalProtectionSystem(ITPS)couldbedesigned,inwhichtheloadbearingstructureandtheTPSareintegratedintoacompositesandwichpanel.ThegreatestchallengeinsuchadesignwouldbeincombiningthevariousconictingrequirementsoftheTPSandtheloadbearingmemberintoasinglestructure.Thestructuralrequirementsfavorusingmetals,butthiscangreatlyincreasetheconductionofheatthroughtheITPS.Ontheotherhandmostinsulationmaterialssuchasceramicsandfoamshavelowstrength.Materialselectionisimportantaswellasdimensioningthestructureproperly,transformingthisdesignintoamultidisciplinaryoptimizationproblem.Evenassumingthatthematerialpropertiesandtheirexactbehaviorareknownaccurately,thisproblemiscomputationallyintensiveinvolvinglargenumberofniteelementanalyses.IfcompositesareusedforITPS,itfurtherincreasesthenumberofuncertaintiesintheproblem.Whentheseuncertaintiesareincludedinthedesign,thecomputationalexpensebecomesprohibitivelylarge.Thiscouldbesolvedusingtraditionaldeterministic 15
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methods,whicharecomputationallyfeasiblebutlessaccurate.Thisproblemcouldbedenitivelyaddressedthroughprobabilisticoptimization,whichisaccuratebutcomputationallyintractable. TheoverallobjectiveofthisdissertationistoemployefcientandcosteffectivemethodstooptimizetheITPSpanel.Varioustechniqueshavebeenappliedalongthedifferentphasesofthisdissertationtoincreasethecomputationalefciencyandreducethecostandtimeinvolved.ToreducethecomputationalcostofstructuralanalysisofITPS,niteelementbasedhomogenizationisinvestigated.Homogenizationisaprocessofapproximatingthebehaviorofheterogeneousstructuresashomogeneousbydeterminingtheirequivalentstiffnessproperties.Pastresearcheffortshaveshownthathomogenizationofthecompositestructureasanequivalenttwo-dimensionalorthotropicplatewouldreducethecomputationalcostassociatedwithstructuralanalysistremendously[ 1 2 3 4 5 ].Sharmaetal.[ 6 7 ]appliedthehomogenizationtoanITPSpanelwithcorrugatedcore.Owingtonegligiblefoamdensity,thepanelwasmodeledusingshellelements.TheITPSpanelconsideredinthisdissertationhasdenserinsulationfoamrequiring3-Dbrickelements.Thisresearchthusaimstoextendthehomogenizationtechniqueto3-Dsandwichpanels. Inhomogenization,theequivalentstiffnesspropertiesofacompositesandwichpanelarethein-planestiffness[A],couplingstiffness[B],bendingstiffness[D]whicharecombinedandrepresentedasthe[ABD]stiffnessmatrixandtransverseshearstiffness(A44andA55).Theequivalentstiffnesspropertiesweredeterminedbytakingadvantageoftherepeatingstructureofthepanel.Therepetitiveblockofthepanelisreferredasunitcellorrepresentativevolumeelement(RVE).The[ABD]stiffnessmatrixwasdeterminedbysubjectingtheunitcelltounitstrainsandcurvatures.Thetransverseshearstiffnesswasdeterminedbyanalyzingaone-dimensionalplateoftheITPSmodelunderendloads.However,itwasfoundthathomogenizationofthecompositesandwichpanelwasaccurateonlywhenthepaneldimensionsaremuch 16
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largerthanthecharacteristicdimensionsoftheunitcell.HenceoneunitcellofITPSpanelisconsideredandanalyzedduringtheoptimizationprocess. Thesecondobjectiveofthisresearchistoperformdesignoptimizationthroughdeterministicandreliabilitybasedapproaches.OneofthemainpurposesofoptimizationistominimizetheweightoftheITPSwhilesatisfyingseveralconictingdesignconstraints.Designoptimizationthroughatraditional,safetyfactorapproachwithoutconsideringtheuncertaintiesinvolvedwouldprovidelessaccurateandconservativedesigns.Thecompositepanelunderconsiderationcontributestothemajorpartoftheweightontheexteriorofthespacevehicle.ThusitisimperativetoreducethestructuralweightofITPSwithoutresultinginconservativeandunsafedesigns.Hence,itisdesiredtooptimizethedesignoftheITPSusingreliabilitybasedapproachbyincludingalltheuncertaintiesinthedesign.Howeveritisdesiredtoperformaninitialdeterministicoptimizationasitwouldhelpinnarrowingtotheappropriatedesignspacetoperformreliabilitybasedoptimization. Likemostengineeringstructures,theITPScouldundergomultiplefailuremodes:temperature,stressesandbucklingloads.Fordeterministicoptimization,thedesignvariablesareoptimizedtominimizethestructuralmass.Theoptimizationprocessinvolvesevaluationoftheaforementionedfailureinducersatlargenumberofdesignpointswhichincreasesthecomputationalcost.Constructionofsurrogatemodels(interchangeablyreferredasresponsesurfaceapproximation)oftheconstraintsatlimiteddesignpointsserveasanefcientalternativetohighdelityniteelementanalyses.Reliabilitybasedoptimizationhasanadditionalreliabilityconstraintthatisafunctionofdesignvariablesandrandomvariables.ThedesignvariablesaredimensionsoftheITPSpanelandtherandomvariablesconsideredarematerialproperties,geometry,thermalandmechanicalloads.Materialpropertiesarecomplextomodelastheydependontemperature.Geometryisanothersourceofuncertaintyaffectingboththestructuralbehaviorandthermalloading.Henceitisimportantto 17
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analyzethefailureinducerssuchstresses,temperatureandbucklingduetotheabovementioneduncertainties.Withlargenumberofuncertaintiesinvolved,evenconstructionofqualitysurrogatemodelsrequiresniteelementanalyzesatextremelylargenumberofrandominputs. Failureconstraintsthatseparatethefeasibledesignsfromtheinfeasibleoneshavetorepresentedaccuratelyinordertodetermineanaccurateoptimum.Commonlytheconstraintsareevaluatedusingglobalsurrogatemodelsusingspacellingdesignofexperiments.Thiswouldresultinwastingalargenumberofhighdelityniteelementanalyses.Earlierresearcheffortshaveintroducedsamplingmethodstoimproveaccuracyofthetargetregionswithlessnumberofdesigns[ 8 9 10 11 12 13 14 ].ThesamplingmethoddevelopedbyBichonetal.[ 14 ],EfcientGlobalReliabilityAnalyses(EGRA)sequentiallysamplesdesignpointsinthevicinityofthetargetregionusingagaussianprocessresponsesurfaceconstructedinitiallywithfewerdesignpoints.Thiscoulddrasticallyreducethenumberofcomputationallyexpensiveniteelementsimulationswhileresultinginaccurateoptimumdesign.TheITPSoptimizationprobleminvolvinghighdimensionaldesignspaceseekspolynomialresponsesurface.ThisresearchworkextendsthecurrentEGRAmethodologytoadapttothecurrentITPSproblembyincorporatingpolynomialsurrogatesinthealgorithm. Thegoalofreliabilityanalysisistodeterminetheprobabilitythatasystemwillfailinservice,giventhatitsbehaviorisdependentonrandominputs.Tocalculatethesystemreliability(probabilityoffailure),MonteCarlosimulation(MCS),acommonlyusedmethodformultiplefailuremodesisemployed.Toefcientlyhandlenumerousuncertaintiesandmultiplefailuremodes,SeparableMonteCarlo(SMC)developedbySmarsloketal.[ 15 ]wasadopted.Whentheuncertainrandomvariablesarestatisticallyindependent,theconceptofSMCistogrouptheuncertaintiesinvolvedintheproblemandsamplethemappropriatelytohaveanaccuratefailurepredictionforagivencomputationalbudget.PreviousworkwithSMConlyexploredsimplelimitstates 18
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expressedasadifferencebetweenarandomresponseandarandomcapacity[ 16 ].Thisresearchworklooksatamoregenerallimitstatefunctionthatcombinessetsofrandomresponseandcapacitycomponents[ 17 ].Furthertheerrorintheprobabilityoffailureestimateforthesimplelimitstatewasderivedintermsofthenumberofsamplesoftheresponseandcapacity.Thisisnotapplicableforthegeneralizedlimitstate.Thisresearchaimsatestimatingtheerrorintheprobabilityoffailureforseparablecaseusingthebootstrappingmethod[ 18 ],aresamplingtechnique,whichinvolvestakingthesamplesofresponse(expensive)andresamplingthemwithreplacement[ 19 ]. 1.1OutlineofDissertation 1.1.1MotivationandObjectives Tosummarizetheabovediscussion,themainobjectivesofthisresearcharegivenbelow. OptimizationofITPSiscomputationallyexpensive,becauseof Structuralanalysisofhighdelity3Dniteelementmodel Probabilisticanalysisincludingalluncertainties-aleatory(materialproperties,geometryandloadingconditions)andepistemicuncertainties(errorinmodelingandsimulation). MultiplefailuremodesofITPSdemandsMonteCarlosimulationsforreliabilitycalculations. TosuccessfullyoptimizetheITPSstructurethroughdeterministicandreliabilitybasedoptimization,thisdissertationproposes TodevelopareducedbutefcientmodelofITPStolowerthecomputationcostinvolvedintheniteelementanalyses. Toinvestigatelowdelitysurrogatemodels(ResponseSurfaceApproximationoftheconstraints)toreplacehighdelityniteelementanalyses. ToadaptEfcientGlobalReliabilityAnalysisthatfacilitatessequentiallysamplingaroundtheconstraintsrequiringfewernumberofsamplestoimprovetargetregionandpredictaccurateoptimum. 19
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ToutilizeseparableMonteCarlomethodthatallowsseparablesamplingoftheexpensiveandinexpensivevariableswhichwillreducecomputationalcostandalsoprovideimprovedaccuracy. Thisdissertationessentiallydemonstratesthemethodologyfordeterministicandreliabilitybasedoptimizationofcomplexstructureswithlargenumberofuncertaintiesusingsurrogatemodels,EGRA,separableMonteCarlotechniqueandbootstrappingmethod. 1.1.2ThesisOrganization Chapter 2 reviewsthethermalprotectionsystemusedinspacevehiclessuchasChallenger,DiscoverandColumbia.Thischapterpresentsareviewontheniteelementbasedhomogenizationtechnique.Further,presentsareviewonstructuraloptimizationandthedifferentaspectsofdeterministicandreliabilitybasedoptimization.Chapter 3 presentsdetaileddescriptiononthegeometry,components,loadsandboundaryconditionsofthemulti-functionalIntegratedThermalProtectionSystemfollowedbyadiscussiononniteelementanalysesofITPSinvolvedinthedesignandtheoptimizationprocess.Chapter 4 presenttheniteelementbasedhomogenizationmethodandresultsobtainedfromthemethod.Chapter 5 presentsadetaileddiscussiononMonteCarlosimulationtechniquestoestimateprobabilityoffailurewhichincludescrudeMonteCarloandseparableMonteCarlomethod.bootstrappingmethodtoestimateerrorintheprobabilityoffailureapplyingittoapressurevesselproblem.Chapter 6 illustratestheworkingofEfcientGlobalReliabilityAnalysis(EGRA)usingatwovariableproblem.EGRAappliedtotheoptimizationoftheITPSpanelisdiscussedinChapter 7 .ThischapterpresentsthedeterministicandreliabilitybasedoptimizationoftheITPSpanelandotherimportantcomponents(uncertaintymodelingandpropagation,sensitivityanalyses)associatedwiththeoptimizationprocess.ThedissertationisconcludedwithsummaryofresultsandpossiblefutureworkinChapter 8 20
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CHAPTER2BACKGROUNDSTUDYANDLITERATUREREVIEW Thermalprotectionsystemshavebeenanareaofcontinuousresearchanddevelopmentinspacevehicledesign.Withatmosphericheatingduringreentrybeingthemajorcauseofconcernforaspacevehicle'ssafeoperation,severaldesignandanalysistechniqueshavebeenputforthbyresearches.ThischapterreviewsthedifferenttypesofThermalProtectionsystemusedsofarinspacevehicles.FurtheritalsointroducesandreviewsthevarioustechniquesthatareaimedatreducingthecomputationalcostassociatedwiththedeterministicandreliabilitybasedoptimizationofthestructurallyintegratedThermalProtectionSystem. 2.1OrbiterThermalProtectionSystem-30yearsLegacy Whenaspaceshuttleorbiterre-enterstheEarth'satmosphere,itistravelinginexcessof17,000mph(7600m/s).Whileslowingdowntolandingspeed,frictionwiththeatmosphereproducesexternalsurfacetemperaturesashighas1900K.Specialthermalshieldsontheexteriorsurfaceprotectthevehicleanditsoccupantsduringlaunchandreentry[ 20 21 22 ].Earliermannedspacecraft,suchasMercury,GeminiandApollo,wereprotectedduringre-entrybyaheatshieldconstructedofphenolicepoxyresinsinanickel-alloyhoneycombmatrix.Theheatshieldwascapableofwithstandingveryhighheatingrates.Duringthereentry,theheatshieldwouldablate,orcontrollablyburnwiththecharlayerprotectingthelayersbelow.Despitetheadvantages,ablativeheatshieldshadsomemajordrawbacks.Theywerebondeddirectlytothevehicle,theywereheavy,andtheywerenotreusable.Withadesignlifeof100missions,spaceshuttleorbiterrequiredalightweightreusablethermalprotectionsystem(TPS)notonlytoprotecttheorbiterfromthesearingheatofreentry,butalsotoprotecttheairframeandmajorsystems[ 23 24 ]. Thevehicle'scongurationandentrytrajectorydenesthetemperaturedistributiononthevehicle.Overthepast30years,theSpaceShuttle'sThermalProtectionsystem 21
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hasbeenbuiltwithmaterialswithahightemperaturecapabilityandunderlyingthermalinsulationtoinhibittheconductionofheattotheinteriorofthevehicle.Duringtheentiremission,differentlocationsontheorbitergetheatedtodifferenttemperatures.Theleadingedgesofwingsandthenosecaparethehighesttemperatureregions.DuetothewidevariationofthesetemperaturestheTPSselectedforspaceshuttlewascomposedofmanydifferentmaterials.Eachmaterial'stemperaturecapability,durabilityandweightdeterminedtheextentofitsapplicationonthevehicle.TherearebasicallytwocategoriesofTPSbeingused,thetileTPSandnon-tileTPS.AdetaileddescriptionofeachcategorywouldprovideabetterunderstandingontheTPSselectionfordifferentregionsonthevehicle[ 25 26 ]. Figure2-1.ThermalProtectionSystem(TPS)usedinspaceshuttles[ 26 ] 22
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2.1.1TileTPS ThedifferenttypesoftileTPSare High-temperaturereusablesurfaceinsulation(HRSI)tilesweredevelopedtoprovideprotectionagainsttemperaturesupto1533K.Theywereusedinareasontheupperforwardfuselage,verticalstabilizerleadingedge,andupperbodyapsurface.Thetileiscomposedofhighpuritysilicabers.Ablackcoating,reactioncuredglass(RCG)wasappliedtoallbutonesideofthetiletoprotecttheporoussilicaandtoincreasetheheatsinkproperties.HRSIwasprimarilydesignedtowithstandtransitionfromareasofextremelylowtemperature,about-270Ctothehightemperaturesofre-entrytypicallyaround1600Cthusmaximizingheatrejectionduringthehotphaseofreentry. Fibrousrefractorycompositeinsulationtiles(FRCI)weresimilarformoftheHRSItiles.TheFRCItileshadhigherstrengthderivedbyaddingalumina-borosilicateber,calledNextel,tothepuresilicatileslurry.Thoughdevelopedforthesamepurpose,FRCIandHRSIhaddifferentphysicalpropertiesbecauseof20%Nextelinit.FRCItileswerelighterthanthebasicHRSItiles.Furthermore,theFRCItilesalsohadtensilestrengththatwasatleastthreetimesgreaterthanthatoftheHRSItilesandcouldbeusedatatemperaturealmost100CofhigherthanthatofHRSItiles.Inanutshell,theyprovidedabetterstrength,durability,crackingresistance,andweightreduction. Tougheneduni-piecebrousinsulation(TUFI)isanimprovedlowdensityrigidceramiccomposite,withveryhighimpactresistance(20-100timesmorethanRCGcoating).TUFIwasusedinregionswheretemperaturesreachashighas1260degreesCelsius.TUFItileswerebuiltashightemperatureblackversionsforuseintheorbiter'sundersideprovidingsufcientheatinsulationfortheorbiter'sunderside.Andlowertemperaturewhiteversionsforuseontheupperbodyconductingmoreheatwhichlimitstheirusetotheorbiter'supperbodyapandmainenginearea. Low-temperaturereusablesurfaceinsulation(LRSI)werewhiteincolorpossessinghighthermalreectivity.Thesetileswereusedtoprotectareaswherereentrytemperaturesarebelow649C(1200F).Theyaregenerallyinstalledontheuppersurfaceofthevehicle,maximizingsolargainwhentheorbiterisontheilluminatedpartoftheorbit.Theywerealsousedtoprotecttheupperwingneartheleadingedgeandalsoareasoftheforward,mid,andaftfuselage,verticaltail. 2.1.2Non-TileTPS Thedifferenttypesofnon-tileTPSare ReinforcedCarbon-Carbon(RCC)laminatedcompositematerialwereprimarilyusedforcoveringthewingleadingedgesandnosecapwherethetemperatures 23
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reachamaximumof1510Cduringreentry.Thiscompositecoveringhadveryhighfatigueresistancewhichisessentialduringascentandreentry.GenerallyallTPScomponents(tilesandblankets)weremountedontostructuralmaterialsthatsupportthem,mainlythealuminumframeandskinoftheorbiter.RCCistheonlyTPSmaterialthatalsoservedasasupportforpartsoftheorbitersaerodynamicshape,wingleadingedgesandthenosecap. FlexibleInsulationBlankets(FIB)werewhitelow-densitybroussilicamaterial.TheseblanketsweredevelopedasreplacementforLRSItiles.TheyrequireverylessmaintenancethanLRSItilesyettheypossessedthesamethermalproperties. FeltReusableSurfaceInsulation(FRSI)aregenerallywhite,exiblefabricofferingprotectionupto371C(700F).FRSIcoveredtheOrbiter'swinguppersurface,theupperpayloadbaydoors,andaftfuselage. Gapllerswereplacedatdoorsandmovingsurfacestominimizetheheatcreatedopengapsintheheatprotectionsystem.Thesematerialswereusedaroundtheleadingedgeoftheforwardfuselagenosecaps,windshields,sidehatch,wing,verticalstabilizer,therudder,bodyap,andheatshieldoftheshuttle'smainengines.ThellermaterialsaremadeofeitherwhiteAB312bersorblackAB312clothcovers(whichcontainaluminabers). Forfurtherindepthdiscussiononthermalprotectionconceptsusedinspacevehicles,thereaderisreferredtoBlosser[ 27 ]andBapanapalli[ 28 ]. 2.2FiniteElementbasedHomogenization Micromechanicalanalyseshavebeentraditionallyusedtoestimatetheeffectivestiffnesspropertiesofcompositematerials.Someofthesemethodsarealsosuitabletoobtainhomogenizedpropertiesforplatelikestructureswithperiodicity.Thereareseveralapproachestohomogenization:mechanicsofmaterialsapproach,elasticityapproach,energymethodsandniteelementanalysis.Allmethodsassumethatthereisarepresentativevolumeelement(RVE)orunitcellthatrepeatsitselftoformthestructure.Theunitcellissubjectedtosixlinearlyindependentdeformationstodeterminetheequivalentstiffnesspropertiesofthepanel.Thedeformationsareappliedasperiodicdisplacementboundaryconditions(PBC)whichincludesthreemid-planestrainsandthreecurvatures. 24
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Biancolini[ 1 ]usedanenergyequivalenceapproachtohomogenizeacorrugatedcorepaneltoanequivalentanisotropiclamina.Thehomogenizationprocedurewasappliedusingstaticcondensationinwhichtheinternalnodesofthemicrogeometryareremovedandexternalnodesattheboundaryofthemodelrepresenttheedgesoftheequivalentlamina.Theeffectivestiffnesspropertiesobtainedusingthismethodwasvalidatedwithaseriesofnumericaltests.Yuetal.[ 29 30 ]variationalasymptoticmethodforunitcellhomogenization(VAMUCH)topredicteffectivepropertiesofperiodicallyheterogeneousmaterials.Davalosetal.[ 2 ]evaluatedtheequivalentpropertiesofberreinforcedhoneycombsandwichpanels(Figure 2-2 ).Thepanelconsistedsinusoidalcoreintheplaneandextendedverticallybetweenthelaminates.Thehomogenizationtechniquewasacombinationofenergymethodandmechanicsofmaterialapproachtopredicttheequivalentpropertiesofthepanel.Forasimilarsinusoidalcorepanel,Buannicetal.[ 3 ]usedanasymptoticexpansionmethodforestimatingtheequivalentstiffnessproperties.WallachandGibson[ 4 ]usedtheunitcellapproachtocalculatethestiffnessproperties,compressivestrengthandshearstrengthofsandwichstructureshavingpyramidaltrusscores;employingtwomethods,atruss-analysisprogrambasedonmatrixmethodsandacommercialniteelementanalysis(FEA)programusingABAQUS.Theniteelementapproachconrmedresultsfromthetrussanalysesandallowedtoextendtheanalysistoincludenonlineareffectssuchasmaterialyieldandlargedeformation. PasteffortsonthehomogenizationoftheITPSpanel(Figure 2-3 )includeananalyticalapproachandniteelementbasedapproach.Martinezetal.[ 5 31 ]followedstrainenergyapproachandsheardeformableplatetheorytodevelopananalyticalmodelforthehomogenizationofacorrugatedsandwichpanel(Figure 2-3 )oftheITPS.Thoughanalyticalmodelsprovidereasonablygoodestimateofstiffnessproperties,theyinvolveseveralassumptionscompromisingtheaccuracyofthestructure.Moreovermostoftheanalyticalapproachesrequireaniteelementbasedvalidation.Sharmaetal. 25
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Figure2-2.Sinusoidalcoresandwichedbetweenlaminates[ 1 ] [ 32 6 ]employedaniteelementbasedhomogenizationtechniqueforhomogenizingcorrugatedcoresandwichpanelasanorthotropicplateandshowedthattheresultsagreedwellwiththethreedimensionalmodel.Henceaniteelementmethodbasedhomogenizationprocedureisadoptedheretoobtaintheequivalentplateproperties.Inniteelementbasedhomogenizationperiodicboundaryconditionsareimposedontherepresentativeunitcell(RVE)thatcorrespondstoagivenstateofmid-planestrainsandcurvaturesoftheequivalentplatetoobtaintheeffectivestiffnessproperties.TheniteelementbasedhomogenizationoftheITPSpanel(Figure 3-1 )consideredinthisstudywillbediscussedinChapter 4 Figure2-3.IntegratedThermalProtectionSystemwithcorrugatedcore[ 28 31 ] 26
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2.3DeterministicandReliabilitybasedoptimization Anefcientstructuraldesignplaysanimportantroleinvariousengineeringdisciplines(foreg.inautomobiledesign,aerospaceengineeringandinmarineapplications).Akeyareaofcontinuousresearchisinndinganoptimum,efcientstructure-makingastructurelightweightyethavingadequateloadcarryingproperties,isapopularexample.Recentdevelopmentinoptimizationtheoryandcomputationaltoolshavefacilitatedwaystondoptimalstructures[ 33 ].Further,designandoptimizationofcompositestructureshavegainedspecialattention[ 34 35 36 ]. Themethodologiesdiscussedinthisresearcharedeterministicoptimizationandreliabilitybasedoptimization.Deterministicoptimizationinvolvesminimizingtheweightwhileapplyingsafetyfactorsonfailureconstraints.Reliabilitybasedoptimizationincludesalltherandomnessinthedesignprocessandhasanadditionalconstraintreferredasreliabilityorprobabilityoffailureconstraint.Reliabilitybasedoptimizationhasgainedincreasingpopularitymainlyduetotheuseofcompositematerialsandthetypicaldesigndriversbeingminimumstructuralmassandhighreliability.Howeveritprovesbenecialtoperformdeterministicoptimizationinitiallyasitfacilitatesinchoosingappropriatedesignspaceforreliabilitybasedoptimization.Theoptimizationprocessinvolvesevaluationofthefailureconstraintsusingniteelementanalysesthatarecomputationallyexpensive.Thisresearchproposestoperformdeterministicandreliabilitybasedoptimizationoftheintegratedthermalprotectionsystemcompositepanelandaimsatreducingthecomputationalcostatdifferentphasesofoptimization,usingtechniquessuchasresponsesurfacemethodology,EfcientGlobalReliabilityAnalysis(EGRA),separableMonteCarlo(SMC)methodandbootstrappingmethod. Thefollowingsectionspresentssomebackgroundandreviewondeterministicoptimization,useofsurrogatemodelsforoptimization,typesofuncertaintiesincludedtoestimatereliability,methodstoevaluateprobabilityoffailure,methodstoestimateaccuracyofprobabilityoffailure.FurtheritalsodiscussesEfcientGlobalReliability 27
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Analysis(EGRA)foraccurateestimationoftheoptimizationconstraintswithlimitedcomputationalresources. 2.3.1DeterministicOptimization Deterministicoptimizationinvolvesoptimizingthedesignvariablestominimizetheobjectivefunctionwhilesatisfyingequalityorinequalityconstraintsthatfollowalinearornon-linearformulation.Thegeneralformulationoftheproblemis Minimizew=f(d)suchthatg(d)zh(d)=mdLB
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77Konthetemperatureconstraintand1.5onboththestressandbucklingconstraints.Sincethedeterministicoptimumwilllieoneitheroralloftheconstraints,itseparatesthefeasibledesignspacefromtheinfeasibleone.ThishelpsinnarrowingthedesignspacetothefeasibleonesforperformingprobabilisticoptimizationandthusavoidingexpensiveFEanalysisontheinfeasiblespace. 2.3.2StructuralReliability Reliabilitybasedoptimizationhasgrownratherquicklyduringthelastfewdecadesandstructuralreliabilitymethodshavedevelopedrapidlyandhavebeenwidelyappliedinthepracticaldesignofstructures[ 37 ].Theaimofreliabilityanalysisisthequanticationandtreatmentofuncertaintiesandthentheevaluationofameasureofsafetyorreliabilitytobeusedindesign[ 38 ].Thetermsreliabilityandprobabilityoffailurearecomplementary,inthatthemorereliablethedesign,thelowertheprobabilityoffailure.Instructuralapplications,reliabilitybasedoptimizationinvolvesminimizingweightofthesystemwhilesatisfyingareliability/probabilityoffailureconstraint.Thegeneralformulationoftheproblemis Minimizew=f(d)suchthatpf(g(d;X) Cr(2) wheretheobjectivefunctionfisafunctionofonlythedeterministicdesignvariablesd,buttheresponsefunctionintheprobabilityoffailureconstraintgisafunctionofdandX,avectorofrandomvariablesdenedbyknownprobabilitydistributions[ 39 40 ].pf;Cristhetargetprobabilityoffailure.Sinceprobabilitymeasuresarehighlynon-linear,thisconstraintisoftenreplacedbyreliabilityconstraintwherethereliabilityindexiscalculatedas 29
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=)]TJ /F9 11.955 Tf 9.3 0 Td[()]TJ /F4 7.97 Tf 6.59 0 Td[(1(pf)(2) whereisthecumulationdistributionfunctionofstandardnormaldistribution.Cristhetargetreliabilityindex. ReliabilityanalysescanbeperformedusingMoment-basedmethodssuchastherst-order-reliability-method(FORM)orsecond-order-reliability-method(SORM)orsamplingmethodssuchasMonteCarlosimulations(MCS).Researchershaveusedthesemethodstooptimizecompositestructuresaswell.LehetaandMansour[ 41 ]carriedoutlimitstateanalysis,rstorderreliabilityanalysisandreliability-basedstructuraloptimizationofshipstiffenedpanels.QuandHaftka[ 35 ]demonstratedreliabilitybasedoptimizationofcompositesatcryogenictemperaturesusingresponsesurfaceapproximationsandMonteCarloSimulations.ScuivaandLomario[ 42 ]performedacomparisonbetweenMonteCarlosimulationsandFORMsincalculatingthereliabilityofacompositestructure.AsignicantamountofresearchhasdealtwithoptimizationandreliabilityanalysesofthecorrugatedcoreITPSpanel(Figure 2-3 ).Kumaretal.[ 43 ]andVillanuevaetal.[ 44 ]studiedthedifferenceinriskallocationbetweenstructuralandthermalfailuresbydeterministicandprobabilisticoptimization.Sharmaetal.[ 45 ]performedamulti-delityanalysisoftheITPSpanelusingCorrectionResponseSurface.FurtherVillanuevaetal.[ 46 ]studiedtheeffectsofsinglefuturetestinthedeterministicandprobabilisticdesignoftheITPSpanel.TheyconsideredonlythethermalfailuremodeoftheITPSpanel.Matsmuraetal.[ 47 ]consideredmultiplefailuremodesoftheITPSpaneltostudytheeffectoffuturetestsinthereliabilityestimationoftheITPSpanel.ThisresearchconsidersadifferentconceptofITPSandaimsatdeterministicallyandprobabilisticallyoptimizingtheITPSpanelusingcosteffectiveadaptivesamplingtechniques. Momentbasedmethodscanbecost-efcientastheyinvolveapproximationofresponsefunctiontondthemostprobablepoint(MPP),however,theyareinaccurate. 30
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Samplingmethodsareaccurateifadequatesamplesareusedbutcomputationallyintensive.Inengineeringapplicationsreliabilityestimationmayinvolveindividualfailuremodesuchasstressorsystemlevelfailurewithmultiplefailuremodeswhichcouldbe,say,stressanddeection.Moment-basedmethodscouldbeusedforsystemwithsinglefailuremodes.Howeverwhentheprobleminvolveshigherdimensionswithmultiplefailuremodesthatareexpensivetoevaluate,theresponsefunctionscannotbeapproximatedasanalyticalderivatives[ 42 ].SuchsituationsareaddressedbynumericalapproximationofresponsesusingsurrogatemodelsandsamplingmethodssuchasMonteCarlosimulationstoestimatereliability. Asmentionedearlier,theITPSpanelconsideredhasmultiplefailuremodesandinvolvescalculationofsystemlevelreliability.Systemlevelfailureisdenedeitherasparallelfailureorseriesfailure.Whensystemisassumedtohavefailedifallthemodesfailitisreferredasaparallelsystemfailure,wheneitheroneofthemodesfail,itisreferredseriessystemfailure.IntheITPSproblem,ifeitherofthemodes,temperature,stressorbucklingmodesfail,thepanelisconsideredtohavefailed. 2.4UseofSurrogateModels Inordertoreducethecomputationcostassociatedwithniteelementsimulations,qualitysurrogatemodelsofthevariousconstraintsareconstructed.TheyarealternatelyreferredasResponsesurfaceapproximations(RSA)ormetamodels.Surrogatemodelsusuallyemployloworderpolynomialstothestructuralresponsewithalimitednumberofinputdesignpoints[ 48 49 ].Theinitialdesignpointsareobtainedusingdesignofexperimentssuchasfullfactorialdesign,Centralcompositedesign,Latinhypercube,A-optimalorD-optimaldesigns. Inthecaseofexpensiveresponseevaluations,surrogatemodelshavebeenusedforevaluatingconstraintsinthedeterministicandprobabilisticoptimizationprocess.Vianaetal.[ 50 51 ]havedemonstratedtheuseofsurrogatemodelsandsurrogatebaseddesignoptimization.FurthertheyalsodevelopedaSurrogateToolbox 31
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inMATLABthatmakestheoptimizationprocesseasier[ 52 ].ThisresearchmainlyusestheSurrogateToolboxforttingresponsesurfacemodelsandfmincon()aMATLABoptimizerfortheoptimizationprocess.Moreover,theprobabilitycalculatedfromMonteCarlosimulationoftenintroducesrandomerrorsduetolimitedsamplesizeresultinginunsafeoptimum[ 35 53 ].Theuseofresponsesurfaceapproximationreducessucherrors. Thoughsurrogatemodelsarebenecialinthecaseofcomputationallyexpensiveproblemssuchasthosedemandingcomplexniteelementanalyses(deterministicoptimization),withlargenumberofuncertaintiesandmultipleconstraints,theprobabilityoffailurecalculationsevenwithresponsesurfacemodelsaredifculttohandle.Traditionallytheresponsesurfacemodelsconstructedwereaglobalapproximationoftheresponsewithdesignofexperimentsthatareindependentoftheresponsefunction.Sincefailureisdenedusingaconstraints/limitstatesthatseparatethefeasibleandinfeasibledesigns,itisimportanttoconstructaccuratecontoursoftheconstraintswhileitisacceptabletohavelargeerrorsatotherregionsinthedesignspace.Accurateapproximationofconstraintswouldleadtoaccuratepredictionofoptimum. 2.4.1ApplicationofSurrogatestoImproveConstraintBoundaries Indeterministicandreliabilitybasedoptimization,itisimportantthattheconstraintboundariesthatseparatethefeasibledesignsfromtheinfeasibleonesareestimatedaccuratelywithminimalerror.Globalsurrogatemodelscanbeusedbutwouldrequirelargenumberofdesignpointsandwouldnotensureaccurateapproximationofconstraintboundaries.Tobuildsurrogatemodelsthatapproximatesconstraintboundaryaccurately,researchershaveintroducedvariousmethods.KuczeraandMourelatos[ 8 ]usedacombinationofglobalandlocalsurrogatemodelstorstdetectthecriticalregionsandthenobtainalocallyaccurateapproximation.Arenbecketal.[ 9 ]usedsupportvectormachineandadaptivesamplingtoapproximatefailureregions.TuandBarton[ 10 ]usedamodiedD-optimalstrategyforboundary-focused 32
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polynomialregression.VazquezandBect[ 12 ]proposedaniterativestrategyforaccuratecomputationofprobabilityoffailurebasedonkriging.ShanandWang[ 54 ]developedaroughsetbasedapproachtoidentifysub-regionsofthedesignspacethatareexpectedtohaveperformancevaluesequaltoatargetvalue.Forconstrainedoptimizationandreliabilityestimation,Pichenyetal.[ 13 ]developedtargetedIntegratedMeanSquarecriterion(IMSE)toconstructdesignofexperimentssuchthatthemetamodelaccuratelyapproximatethevicinityofaboundaryindesignspacewhichiseitherdenedbyatargetvalueorgaussiandistribution. 2.5AdaptiveSampling Whenoptimizationproblemsinvolvenon-linearandmultimodalobjectivefunctions,theyrequirelargenumberoffunctionsevaluationstondtheglobaloptimum.Andwhentheevaluationsarelimitedbyexpensivecomputations,ndinganaccurateglobaloptimumbecomesdifcult.Surrogatemodelscouldbeemployedbuttheiraccuracyisalsocompromisedwhencomplexfunctionsareconstructedwithfewerdesignpoints.Jonesetal.[ 55 ]developedEfcientGlobaloptimization(EGO),anunconstrainedoptimizerthatfocusesonaddingpointstothedesignspacetoaccuratelymodelcomplexfunctionsandhencendtheglobaloptimumaccurately. Inthismethod,aninitialGaussianprocessmodel[ 56 57 ]isbuiltasaglobalsurrogatefortheresponsefunction.EGOthenadaptivelyselectsadditionalsamplestobeaddedtothedesignspacetoformanewGaussianprocessmodelinsubsequentiterations.EGOusesaspecicformulationknownastheExpectedImprovementthatidentiesthenewdesignpointsbasedonhowmuchtheyareexpectedtoimprovethecurrentbestsolutiontotheoptimizationproblem.Whenthisexpectedimprovementisacceptablysmall,thegloballyoptimalsolutionhasbeenfound. EfcientGlobalOptimizationwasfurtherextendedtoimprovetargetregionsinthedesignspace.Ranjanetal.[ 11 58 ]developedasequentialdesignmethodologybasedon(EGO)thatexploresthedesignalongthecontourofinterest.Bichonetal.[ 59 ] 33
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alsodevelopedasimilarmethodbasedonEGOreferredasEfcientGlobalReliabilityAnalyses(EGRA).TheyillustratedtheapplicationofEGRAbyimprovinglimitstatesforaccuratereliabilityestimation.ThisresearchproposestoextendtheapplicationofEGRAtoimproveconstraintboundariesindeterministicoptimizationandreliabilitybasedoptimizationoftheITPSpanel.Theconceptofsamplingdesignsiteratively,reducesthecomputationalcostofexpensiveFEsimulationsandatthesametimeincreasestheaccuracyofthefailureboundary. 2.5.1EfcientGlobalReliabilityAnalysis Themethodhastwomainfeatures,aninitialgaussianprocessmodelandtheexpectedfeasibilityfunctiontoidentifytheadditionalsamplesiterativelyatthevicinityofthelimitstate.Theinitialsurrogatemodel^Gisconstructedusinglimitednumberofdesignsusingaknownsamplingmethod,inthiscase,Latinhypercubesampling.Theexpectedfeasibilityfunctionisoptimizedtondthenextdesignthatwouldimprovethetargetboundaryz.Thefunctionisgivenas EF(^G(X))=(G)]TJ /F9 11.955 Tf 12.67 0 Td[(z)2z)]TJ /F3 11.955 Tf 11.95 0 Td[(G G)]TJ /F9 11.955 Tf 11.95 0 Td[(z)]TJ /F2 11.955 Tf 9.75 -4.34 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(G G+z+)]TJ /F3 11.955 Tf 11.95 0 Td[(G G)]TJ /F3 11.955 Tf 9.3 0 Td[(G2z)]TJ /F3 11.955 Tf 11.96 0 Td[(G G)]TJ /F3 11.955 Tf 11.95 0 Td[(z)]TJ /F2 11.955 Tf 9.74 -4.34 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(G G+z+)]TJ /F3 11.955 Tf 11.95 0 Td[(G G+z+)]TJ /F3 11.955 Tf 11.96 0 Td[(G G)]TJ /F9 11.955 Tf 11.95 0 Td[(z+)]TJ /F3 11.955 Tf 11.95 0 Td[(G G(2) TheresponsepredictedbythegaussianmodelfollowsthedistributionN(G,G),=GistheerrorbandaroundthelimitstatewhichisafunctionofthestandarddeviationGand,afactorappliedonthestandarddeviation.z+=z+andz)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(=z-.andarethestandardnormalpdf(probabilitydensityfunction)andcdf(cumulativedistributionfunction)respectively.WhenGisclosetothetargetcontour,thersttermdominatestheexpression.Thesepointsareclosetotheband.IfGisfarawayfromtarget,thesecondtermtendstodominate.Thistermfacilitatessamplinginregionsoftheinputspacewheretheestimatedresponseisoutsidetheband,buttheuncertainty 34
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ofpredictionishigh.Thethirdtermisrelatedtothevariabilityofthepredictedresponseintheneighborhoodof[ 11 58 ].Itsupportssamplinginregionsneartheestimatedcontourbutwherethepredictionvarianceisquitehigh. CurrentlyeithergaussianprocessorkrigingarethedefaultsurrogatemodelsforimplementingEGRA.SincetheITPSinvolvesalargenumberofrandomvariables,theuseofpolynomialresponsesurfaceisfavoredovergaussianprocessorkriging.HencetoadapttotheoptimizationoftheITPSpanelthisresearchproposestoimplementpolynomialresponsesurfaceinEGRAmethodology. 2.6UncertaintyModeling Modelingandanalysisofinputuncertaintiesaidinunderstandingthepropagationofuncertaintyintheoutputs.Uncertaintiesinanengineeringsystemcanbeclassiedasepistemicuncertaintyandaleatoryuncertaintyalsocalledvariability.Epistemicuncertaintygenerallyrepresentsalackofknowledgeofaquantityorprocessofasystemorenvironment.Thisuncertaintyisalsoreferredasreducibleuncertaintyasitcanbereduced(orincreased)fromincreasedunderstandingoftheuncertainvariableorfrommorerelevantexperimentaldata.Aleatoryuncertaintyisgenerallycharacterizedbyinherentrandomnessinthephysicalsystemorenvironmentwhichcannotbereducedbyfurtherdata[ 60 ]. Instructuralapplications,aleatoryvariabilitycanbeintroducedbymanufacturingimperfectionssuchasvariabilityinmaterialpropertiesandgeometricdimensions,variabilityinloadingandepistemicuncertaintiesbyerrorsinmodelingandsimulation.Uncertainvariablesareincludedinthedesignprocessbytheirrespectiveprobabilisticdistributions.Inadditiontothevariabilityintheinputparameters,errorinmodelingandsimulation(epistemicuncertainty)isalsointroducedintheoutputparameterstoestimatetheprobabilityoffailure.Theprobabilitydistributionoftherandomvariablescouldfollowuniform,normal,lognormalorWeibulldistribution[ 61 15 ].However,thisresearchmainlyassumesuniformandnormaldistributionfortheinputrandomvariables. 35
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Estimatessuchasvariance,standarddeviationorcoefcientofvariationoftheresponsesampleswouldprovidetheuncertaintyintheresponseduetoinputuncertainties. Theimportanceofuncertaintycharacterizationanduncertaintypropagationincompositeshasbeenillustratedbyvariousresearchers[ 62 63 64 65 66 ].AntonioandHoffbauer[ 62 ]studiedtheeffectsofdeviationsinmechanicalproperties,plyangles,plythicknessandappliedloadsforalaminatedshellcomposite.Theydemonstratedthatuncertaintyanalysisisveryusefulindesigninglaminatedcompositestructuresminimizingtheunavoidableeffectsofinputparameteruncertaintiesonstructuralreliability.OhandLibrescu[ 63 ]addressedtheproblemofalleviatingtheeffectsofuncertaintiesforfreevibrationofcompositecantileversunderuncertaintiessuchaslayerthickness,elasticconstantsandplyangle.Nooretal.[ 64 ]studiedthevariabilityofnon-linearresponseofstiffenedcompositepanelsduetovariationsingeometricandmaterialparametersusinghierarchicalsensitivityanalysisandfuzzysetanalysisapproach. FortheITPSoptimizationproblem,therandomvariablesaredimensions,mechanicalloads,thermalloads,thermo-mechanicalpropertiessuchasYoung'smodulus,shearmodulus,poisson'sratio,coefcientofthermalexpansion,thermalconductivityandspecicheat.Inadditiontothis,uncertaintyisconsideredintheallowablelimitssuchasmaterialstrength,temperature.Theerrorinniteelementmodelingandsimulations(transientheattransfer,staticstressandbucklinganalyses)isalsoincluded. 2.7EstimationofProbabilityofFailure TheprobabilityoffailureistheprobabilitythattherandomvariablesX=fx1,x2,...xigareinthefailureregionthatisdenedbylimitstatefunctionG(X)<0.IfthejointpdfofXisfx(X),theprobabilityoffailureisevaluatedwiththeintegral pf=PfG(X)>0g=ZG(X)<0fx(X)dx(2) 36
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Thereliabilityiscomputedby R=1)]TJ /F3 11.955 Tf 11.95 0 Td[(pf=PfG(X)>0g=1)]TJ /F10 11.955 Tf 11.96 16.28 Td[(ZG(X)<0fx(X)dx(2) Thedirectevaluationoftheprobabilityintegrationstateaboveisextremelydifcultwhen alargenumberofrandomvariablesareinvolved.Insuchcasestheprobabilityintegrationbecomesmultidimensionalwhichistypicalofengineeringapplications. theintegrandfx(X)isthejointpdfofXandisgenerallyanonlinearmultidimensionalfunction. thelimitstateboundaryG(X)=0isalsomultidimensional,nonlinearfunction.Instructuralapplications,G(X)isoftenablack-boxmodel(orniteelementsimulation),andtheevaluationofG(X)iscomputationallyexpensive. Hencemomentbasedmethodsandsamplingmethodscouldbeappliedtoevaluatetheprobabilityintegration. 2.7.1MomentbasedMethods MomentbasedmethodssuchasFirst-OrderReliabilityMethod(FORM)andSecond-OrderReliabilityMethod(SORM)easethecomputationaldifcultiesbysimplifyingtheintegrandfx(X)andapproximatingthelimitstatefunctionG(X)toestimateprobabilityoffailure(Equation 2 ).ThemethodsimpliesthejointdistributionfunctionbytransformingtheoriginalrandomvariablesfromX-spacetostandardnormalU-space.Thelimitstatefunctionisapproximatedusingrst-orderTaylorseriesexpansion.Inthestandardnormalspace,thepointonthelimitstatefunctionwhereG(U)=0attheminimumdistancefromtheoriginisthemostprobablepoint(MPP)offailure.ThereliabilityismeasuredasthedistancefromtheorigintotheMPP.Thismeasureisreferredasreliabilityindex().TheMPPisdeterminedas Minimize=p UTUsuchthatG(U)=0(2) 37
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whereUisthevectorofvariablesinstandardnormalspace.FORMisfairlyaccuratewhenthelimitstatefunctioncanbeapproximatedasalinearfunction.Secondordermethodscanbeusedwhenthelimitstatefunctionhasahigherorderofcurvature.Thismethodapproximatesthelimitstateasaquadratic,andprovidesamoreaccurateapproximationinsuchcases. 2.7.2SamplingMethods MonteCarlosimulation(MCS)isoneofthepowerfulandeasytoimplementmethodstopropagatetheuncertaintyininputrandomvariablestotheuncertaintyinfailure[ 67 ].Reliability-baseddesignofastructuralsystemisoftenaddressedusingMonteCarlosimulations[ 68 39 69 ],especiallywhenthesystemunderconsiderationfailsduetomultiplefailuremodes.ProbabilityoffailurepfisdeterminedthroughalimitstatefunctionGthatseparatesthefeasibledesignsfromtheinfeasibleones.Thelimitstateisgenerallyafunctionofrandomvariables,responseRandcapacityC.Whentheresponseexceedsthecapacity,itisconsideredfailure,example,maximumstressfailuretheorywheretheresponsewillbecalculatedstressandcapacitywouldbestrengthofthematerial.ThecapacityandresponseareassumedtobefunctionsofstatisticallyindependentrandomvariablesX1andX2,respectively.Equation 2 showstheseparablecasewherefailureoccurswhenasinglecomponentofresponseexceedsasinglecomponentofcapacity. G(X1;X2)=R(X1))]TJ /F3 11.955 Tf 11.95 0 Td[(C(X2)(2) FailureoccurswhenG0andthesystemissafewhenG<0.Inthemoregeneralcase,thecapacityandtheresponseinthelimitstatecannotbeexplicitlyseparated,andthelimitstatefunctionmayberepresentedas G(X1;X2)=G(R(X1);C(X2))(2) 38
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WhereRandCmaybescalarorvectorquantities.ThelimitstatecouldbeahigherorderpolynomialfunctionintermsofresponseandcapacitysuchasTsai-HillorTsai-Wucriteriongenerallyemployedtopredictfailureincompositestructures.Statisticalestimatessuchasvariance,standarddeviationandcoefcientofvariationofprobabilityoffailurewouldprovideaccuracyintheestimate. Thetraditional,crudeMonteCarlotechnique(CMC)issimple,butitlacksaccuracywhensamplingislimitedduetocomputationallyexpensivestructuralanalysis,suchasfromniteelementanalysis(FEA).TherearevarioustechniquestoimprovetheaccuracyorefciencyofCMC,includingtailmodeling,conditionalexpectation,importancesamplingandtheuseofsurrogates[ 70 71 ].However,anothertechniquereferredasseparableMonteCarlowasdevelopedbySmarsloketal.[ 72 16 ],whichisapplicableincombinationwiththeabovementionedmethodstofurtherimproveaccuracyorefciency.Whentheresponseandcapacityrandomvariablesarestatisticallyindependent,accuracycanbeimprovedbytheseparableMonteCarlomethod(SMC).Thisfacilitatesimprovedaccuracyofthecalculationoftheprobabilityoffailureforthesamecomputationalbudget. TheerrorinprobabilityoffailureestimateforcrudeMonteCarlocanbeobtainedfrombinomiallaw.ForSMCinvolvingsimplelimitstate(Equation 2 ),thevarianceestimatorwasderivedusingconditionalcalculusandvalidatedusingsimulationestimates[ 15 16 ].ThesimulationestimatesofvariancewereshowntobeofcomparableaccuracytothoseobtainedforCMC.Howeverthisisnotapplicableinthecaseofcomplexnon-separablelimitstate(Equation 2 ).ForSMCwithnon-separablelimitstates,bootstrappingtechniqueisproposed[ 18 ].Theerrorinthestandarddeviationestimateofthebootstrappedprobabilityoffailureprovidesameasureoftheaccuracyofbootstrapping.Furtheritwasalsodemonstratedthatthevariabilityestimateofcapacityandresponsecanhelpinchoosingthesamplesizeneededforgivenaccuracy[ 17 19 ].Chapter 5 onestimationofprobabilityoffailureprovidesadetaileddescriptionofthe 39
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CrudeMonteCarlo,SeparableMonteCarlomethod,bootstrappingtechniqueandregroupingofrandomvariablestoimproveaccuracyofpfbyapplyingtheseconceptstoacompositepressurevesselproblem. 40
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CHAPTER3DESIGNANDANALYSISOFINTEGRATEDTHERMALPROTECTIONSYSTEMPANEL 3.1DesignofITPSCompositePanel Theprimaryobjectiveofthisresearch,asstatedearlier,istodeterminetheoptimaldesignofanITPScompositepanel.ThischapterdiscussesthegeometryandmaterialselectioninvolvedinatypicalITPSconstruction.Further,theniteelementanalysesoftheITPSmodelisalsodiscussed. TheintegratedThermalProtectionSytempanelconsideredhereconsistsofstackedrigidAluminaEnhancedThermalBarrierfoam(AETB)insulationbarsthatarespirallywrappedwithaSiliconCarbide/SiliconCarbide(SiC/SiC)laminate.Thebarsarestackedorthogonallyintwolayersina0/90conguration.ThebarsarethensupportedbyatopfacesheetandabottomfacesheetmadeofSiC/SiCandGraphite/Epoxylaminate,respectivelyformingasandwichstructure(Figure 3-1 ).Thevariouscomponents,stackorientationandmaterialpropertiesaregiveninTables 3-2 and 3-3 [ 73 74 75 ].Theconceptnotonlyhasmaterialasymmetryaboutthemid-planebutalsogeometricasymmetryastherigidinsulationbarsarestackedorthogonally.Further,thedesigniscomplexbecauseofthelaminatewrappedaroundtheAETBinsulationbar.Itisimportanttonotethatinsuchadesign,transverseshearoftheinsulationandthewebs(wraps)wouldhaveapronouncedeffectonthestructure. 3.1.1KeyDimensionsoftheITPSstructure ThekeydimensionsoftheITPSarethewidthwf,andheighthoftheinsulationfoam,thicknessofthetopfacesheettT,bottomfacesheettBandwraptWandthenumberofbarsn(Figure 3-2 ).ThenominalvaluesofthedimensionsaregiveninTable 3-1 Table3-1.KeyDimensionsoftheITPSpanel wmmhmmtTmmtBmmtWmmn 20.020.02.02.00.512 41
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Figure3-1.SchematicviewoftheITPSpanel Figure3-2.DimensionsoftheITPSpanel 3.1.2ITPSMaterials ThecomponentsofITPSaremadeofceramicmatrixcomposites,polymermatrixcompositeandhighdensityinsulationfoam.ThetopfacesheetismadeofSiC/SiCplainwoventextilecompositelaminatewhichisknownforitshighmechanicalstrengthatelevatedtemperatures,highthermalstabilityandlowdensity[ 76 ].Thebottomfacesheetisgraphite/epoxylaminate(polymermatrixcomposite).Thenotablepropertiesofpolymermatrixcompositesarehightensilestrengthandstiffness,highfracturetoughnessandgoodcorrosionresistance.Moreovertheyarepopularduetotheir 42
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lowcostandsimplefabricationmethods.Theinsulation,madeofaluminaenhancedthermalbarrier(AETB)foamdemonstratehigherstrength,addeddurability,andhaveamaximumoperationaltemperatureof1700K[ 26 ].Thematerialpropertiesofthecomponents,theberorientationandthethicknessofeachcomponentofITPSarepresentedinTable 3-2 andTable 3-3 below. Table3-2.ComponentsofITPS,materialsandberorientation ComponentMaterialFiberorientationThickness TopfacesheetSiC/SiClaminate[(0/90)4]S16layers-0.125mmeachInsulationfoamAETB8-35mmBottomfacesheetGr/Eplaminate[(0/90)4]S16layers-0.125mmeachWrapSiC/SiClaminate[0/90]S4layers-0.125mmeach Table3-3.MaterialpropertiesofthecomponentsofITPS[ 75 74 73 ] PropertiesSiC/SiCGraphite/EpoxyAETBfoam E1GPa1701380.153E2=E3GPa12490.153v12=v130.260.30.25v230.200.3420.25G12=G13GPa566.90.062G23GPa603.580.0621x10)]TJ /F4 7.97 Tf 6.59 0 Td[(6/K40.00321.772x10)]TJ /F4 7.97 Tf 6.59 0 Td[(6/K40.1131.77 3.1.3BoundaryConditions InthepastresearchonITPSpanels,thecommonboundaryconditionsappliedontheITPSpanelweresimplysupportedboundaryconditionsandclampedboundaryconditions.Also,sincethepanelissymmetricinthexandydirections,onlyonequarterofthepanelisconsideredandsymmetryboundaryconditionswereapplied.Inthisresearchwork,simplysupportedboundaryconditionsandxedboundaryconditionsareconsideredforthehomogenizationofthepanel(Chapter4).HoweveritwasfoundthathomogenizationcouldnotbesatisfactorilyappliedtotheITPSpanel(detaildiscussioninChapter4).HenceanewsetofboundaryconditionswhichsimulatesastructurallyintegratedTPSpanelwasanalyzed.Itsimulatesthepanelmountedontheframeofthevehicleandallowedtoexpandequallyinthex)]TJ /F3 11.955 Tf 11.95 0 Td[(yplane. 43
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Figure3-3.BoundaryconditionsappliedontheITPSpanel 3.1.4Loads Fromtakeofftolanding,thespacevehiclesaresubjectedtovariousloadingconditionssuchasdrag,aerodynamicheating(thermalloads),in-planeinertialloads(mechanicalloads),pressureandforeignobjectimpactloads.Theseloadscouldactindividuallyatanytimeduringthemission,ortheycouldacttogetherleadingtomultiplecomponentfailure.Thisresearchworkprimarilyexploresandanalyzestheeffectofthermalloadscausedbytemperaturegradientandmismatchinthermo-mechanicalpropertiesofdifferentmaterialcombinations,andmechanicalloads(pressure)throughtransientheattransferanalyses,thermalstressanalyses,pressureanalysesandthermalandmechanicalbucklinganalyses. 3.2FiniteElementAnalyses Thissectionpresentsadescriptionoftheniteelementmodelsandanalysesforheattransfer,stressanalysisandbucklinganalysisoftheITPSpanels.Themodelconsideredinthisresearchisconstructedusingacommercialniteelementsoftware,ABAQUS(Figure 3-3 ).ABAQUS.Twenty-nodethree-dimensionalbrickelementswereusedtomodeltheITPS.The3Dsolidelementmodelhas3displacementdegreesoffreedomateachnode. 44
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3.2.1TransientHeatTransferAnalysis Itwasobservedthatthereentryheatingratesaremoreseverethantheheatingratesduringascent,thatis,theheatingratesincreasemoresteeplyandthetotalintegratedheatloadismuchlargerduringreentry.Thus,itcanbeinferredthatthereentryheatingrateswouldbemostinuentialintheITPSdesignand,therefore,theywereusedintheheattransferFEanalysisforthedesignprocess. AtypicalheatingrateusedfordesignisshowninFigure 3-4 .Theinitialtemperatureofthestructureisassumedas295K(72F)[ 28 ].Alargeportionofthisheatisradiatedouttotheambientbythetopsurface.TheremainingheatisconductedintotheITPS.Somepartofthisheatisconductedtothebottomfacesheetbytheinsulationmaterialandsomebythewebs. Figure3-4.HeatuxproleincidentontheITPSpanel LoadsandboundaryconditionsfortheheattransferproblemareschematicallyillustratedinFigure 3-5 .Thefourouterfacesandbottomsurfaceofthebottomfacesheetareassumedtobeperfectlyinsulated.Thisisaworstcasescenariowherethebottomfacesheettemperaturewouldrisetoamaximumasitcannotdissipatetheheat.Itisalsoassumedthatthereisnolateralheatowoutoftheunitcell.Theheatuxincidentonaunitcelliscompletelyabsorbedbythatunitcellonly,thoughinanactual 45
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ITPSpanelheatwouldowintothefacesheetsandwrapswhichcouldactasathermalmassandtherewouldbealateralowofheatinthepanelfromoneunitcelltoanother.TheloadstepsaretabulatedinTable 3-4 Figure3-5.Boundaryconditionsforthetransientheattransferanalysis Table3-4.Heatuxloadstepsinthetransientheattransferanalysis LoadTimeHeatFluxTimeStepAmbientStepPeriodInputSizeTemperature Step10-4500.034,069W=m230sec213KRamplinearlyStep2450-157534,06939,748W=m225sec243KRamplinearlyStep31575-217539,7480.0W=m230sec273KRamplinearlyStep42175-5175-50sec295K Therstthreeloadstepsconsideredintheheattransferanalysis,theheatuxisrampedlinearlyfrom0to34,069to39,748W=m2.Thestep4representsthetimeperiodaftertouchdownandtheFEanalysiscontinuesforanother50secondsinordertocapturethetemperatureriseofthebottomfacesheet.Duringthisperiod,alongwithradiativeheattransfer,convectiveheattransferboundaryconditionsareimposedon 46
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thetopsurfacetosimulatetheheattransfertothesurroundingswhilethevehicleisstandingontherunway.Themaximumbottomfacesheettemperatureisrecordedfromthisanalysis.Also,alongthereentrytimethetemperaturedistributionwithmaximumdifferencebetweenthetopfacesheetandbottomfacesheetwouldinducemaximumstressesinthepanel(Figure 3-6 ).Hencethisdistributionalongtheheightofthepanelisrecordedwhichactsasthethermalloadforthestressandbucklinganalyses. Figure3-6.TemperaturevariationonTFSandBFSwithrespecttoreentrytime 3.2.2StressAnalysis StressesdevelopedintheITPSduetothermalandpressureloadsareanalyzedindividuallybyperformingastaticstressanalysis. Inpreviousstudies[ 7 28 31 ]theITPSpanelwasassumedtobesimplysupportedalongitsedges.Itwasfoundthatthedeformationsduetothermalstresseswerenotrealisticastherewasnocompatibilityofdisplacementsbetweenadjacentpanels.InorderforanITPStobeeffectivetheyhavetobestructurallyintegrated(S-ITPS). 47
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Inthisconcepttheattachmentswillbedesignedsuchthatthethermalforceswillbetransferredfromonepaneltonextadjoiningpanel.ThusonecanimposeperiodicBCsbetweenpanelsaslongasthesurfaceheatingisuniform.TheproblemcanbefurthersimpliedbyanalyzingoneunitcellwithperiodicBCs.Thisisispossiblebecauseweassumeeachpaneliscomposedofrepeatingunitcells.TheboundaryconditionsofastructurallyintegratedTPSpanelisshowninFigure 3-3 .Thenodesononeofthefacesonthex)]TJ /F3 11.955 Tf 12.59 0 Td[(zplaneareallowedhaveequaldisplacementsinthey-directionandtheadjacentfaceonthey)]TJ /F3 11.955 Tf 12.83 0 Td[(zplaneisallowedtohaveequaldisplacementsinthex-direction.Thebottomedgesofthesetwoadjacentfaceswereconstrainedfrommovingintheverticaldirectionsimulatingthatthepaneledgesweremountedontheframeofthevehicle.TheothertwoadjacentedgesareassignedsymmetryBCs. Thetemperaturedistributionsareappliedalongtheheightofthepanel(z-coordinate)oneachnodeofthemodel.Thisimpliesthatthetopandbottomfacesheettemperaturesareuniformthroughoutthelengthandwidthofthepanel.Althoughthetemperaturevariesslightlyinthexandy-directions,thisvariationisverysmallandcanbeneglected.Thestressesdevelopedinthevariouscomponents(facesheets,foamandwrap)areextractedandinvestigatedforthefailureofthepanel.Thestressesinthewraparehighercomparedtotheothercomponentscontributingtowardsfailureofthepanel.Forpressureanalysis,aunitcellcannotbeanalyzedwithperiodicBCs.Hence,aquarterpanelwithsixunitcellsinthexandydirectionwasanalyzed.Apressureofoneatmosphere(1atm.)isappliedonthetopfacesheet.Itwasobservedthatstressesunderpressureloadswereatleasttwoorderslesscomparedtothethermallyinducedstresses.Sincepressureloadsdonotcausefailureofthepanel,theywillnotbeincludedintheoptimizationprocess. 3.2.3BucklingAnalysis ThecomplexdesignoftheITPSwithfacesheetsandwebsleadstolocalandglobalbucklingofthelaminatesunderthermalstressesandmechanicalstresses.Though 48
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thewebsinthepanelaresupportedbytheinsulationfoam,thebucklingofthewebscouldbeduetopoorbondingbetweenthefoamandthewebandalsothepossibilityofdamagedfoam(duetothermalloads)notbeingabletosupportthewebsfrombuckling.ThebucklingoftheITPScanbecarriedoutontheunitcellorasalarge3-Dshellmodelwithoutanyfoam(assumingdamagedfoam).Theseanalyseswouldprovideanestimateofthebucklingload.Whilelocalbuckling,byitself,maynotalwaysleadtocatastrophicfailure,itcouldcontributeindirectlyalongwithhightemperaturesandstresses. BucklingoftheITPSinABAQUSismodeledasaeigenvaluebucklingpredictionproblemwheretheeigenvaluecanbeconsideredastheloadfactoratwhichthestructurewouldfailduetobuckling.Theanalysisisverysimilartothestressanalysisunderthermalloadsexceptfortheloadstepwhichislinearperturbationprocedure.Theloadconsideredwouldbetemperaturedistributionifthereisthermalbucklingandpressureloading,ifthereisbucklingduetomechanicalloads. Figure3-7.BuckledITPSduetothermalloads InitiallyonlybucklingoftheITPSunitcellwithoutthefoamwasanalyzed(Figure 3-7 ).Thebucklingeigenvaluepredictedshouldbegreaterthan1.0.Eigenvalueslessthan1.0suggestthatthestructurefailsatloadslesserthantheappliedload.For 49
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instance,iftheeigenvalueisequalto0.7,itimpliesthatat70%oftheappliedloadthestructurewouldbuckleaccordingtothecorrespondingbucklingmode.Ifthesmallesteigenvalueisaboveunity,thenthestructurewillnotbuckleundertheappliedloads.BucklinganalysesoftheITPSunitcellwithoutfoamshowstopfacesheetbuckleswhichleadstothebucklingofthetopweb. 3.3Summary Thedifferentniteelementanalysesincorporatedintheoptimizationprocessispresentedinthischapter.Furtherformoredetailsonthevariousaspectsoftheniteelementmodelingandaforementionedstructuralanalysis,thereaderisreferredtoABAQUSdocumentation[ 77 78 ]. PastresearchincludedthestructuralanalysisoftheITPSpanelwithcorrugatedcore[ 28 ]usingshellelementsneglectingthefoam.Thisresearchconsidersacomplexdesignwithlaminateswrappedaroundadenserfoamthatcannotbeneglectedthusrequiring3-DbrickelementstomodeltheITPSpanel.Astheanalysisofthe3-Dniteelementmodeliscomputationallyintensive,thisresearchproposestohomogenizethe3-DITPSpanelasanequivalent2-Dorthotropicplate,discussedindetailinChapter 4 50
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CHAPTER4FINITEELEMENTBASEDHOMOGENIZATION Structuralresponses(temperature,stressandbucklingload)arecomputedusinghighdelityniteelementanalysesofthe3-DITPSpanelthatarecomputationallyexpensive.Thecomplexdesignofthepanelconsistingoffacesheets,wrapsandinsulationfoamsandthefactthatthesecomponentsaremadeofcompositematerialsmakethecomputationalexpensesprohibitivelylarge.Moreover,thepresenceoffoaminthepanelrequires3-Dniteelementstomodelthepanel.A3-Dniteelementmodel(12x12unitcells)oftheITPSconsistsof114,000nodesandastaticthermo-mechanicalanalysisofthemodeltakesapproximately40minutes.ToreducethecomputationalcostinvolvedintheanalysisofthecompositeITPSpanel,thisresearchproposeshomogenizationofa3-Dplatelikestructureintoa2-Dequivalentorthotropicplate. Thehomogenizationmethodisbasedonmicromechanicsapproach[ 79 ]wherearepresentativevolumeelement(RVE)ofthestructureissubjectedtosixlinearlyindependentdeformationstopredicttheequivalentplatestiffnessproperties.Aunitcellisusedtoestimatethein-planestiffnessproperties([ABD]matrix)anda1-DplatemodelisemployedtoestimatethetransverseshearstiffnessoftheITPSpanel.TheniteelementmodeloftheITPSismodeledusing20-nodebrickelements(C3D20R)inthecommercialniteelementsoftwareABAQUS.Theaccuracyofthemethodisinvestigatedbycomparingtheresponses(displacementsandstresses)ofthe3-DmodeloftheITPSpaneltotheequivalenthomogenizedplatemodel. 4.1A,B,Dmatrices Theunitcellissubjectedtosixlinearlyindependentdeformationstodeterminetheequivalentstiffnesspropertiesofthepanel.Thedeformationsareappliedasperiodicdisplacementboundaryconditions(Table 4-1 )whichincludesthreemid-planestrains("x0,"y0andxy0)andthreecurvatures(x,yandxy)[ 80 ]. 51
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Figure4-1.SchematicviewoftheITPSunitcell(RVE) Table4-1.Periodicboundaryconditionsforthesixdeformations [u(a,y)-[v(a,y)-[w(a,y)-[u(x,b)-[v(x,b)-[w(x,b)-u(0,y)]v(0,y)]w(0,y)]u(x,0)]v(x,0)]w(x,0)] "x=1a00000"y=10000b0xy=10a/20b/200x=1az0-a2/2000y=10000bz-b2/2xy=10az/2-ay/2bz/20-bx/2 Thesedeformationscreatein-planeforces(Nx,NyandNxy)andmoments(Mx,MyandMxy)intheunitcell.Theconstitutiverelationbetweenthein-planeforce,momentsandstrainsandcurvaturegivesthestiffnesspropertiesofthestructure[ 81 ].Theconstitutiverelationisgivenas 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:NxNyNxyMxMyMxy9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;=2666666666666664A11A120B11B120A12A220B12B22000A6600B66B11B120D11D120B12B220D12D22000B6600D6637777777777777758>>>>>>>>>>>>>><>>>>>>>>>>>>>>:"x0"y0xy0xyxy9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;(4) 52
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Forinstancewhentheunitcellissubjectedtoin-planestrain"x0inthex-direction,thefacey)]TJ /F3 11.955 Tf 10.43 0 Td[(zatx=aoftheunitcellisdisplacedbyadistanceainthex-directionrelativetothefacex=0.Oneofthecornernodesisxedtopreventrigidbodymotion.Fromthenodalforcesactingonthenodesonthefacesoftheunitcell,thein-planeforcesandmomentscanbecalculatedasfollows: Nx=1 bnnPi=1F(i)x(a;y;z)Ny=1 annPi=1F(i)y(x;b;z)Nxy=1 bnnPi=1F(i)y(a;y;z)Mx=1 bnnPi=1zF(i)x(a;y;z)My=1 annPi=1zF(i)y(x;b;z)Mxy=1 bnnPi=1zF(i)y(a;y;z)(4) Thein-planeforceswouldprovidetherstcolumnofthein-planestiffnessmatrix[A]andthemomentswouldgivetherstcolumnofthecouplingstiffness[B](Equation 4 ).Theaboveprocedureisrepeatedwiththeothermid-planestrainsandcurvaturestopopulatetheentire[ABD]matrix.Whenthisprocedureisimplementedcorrectly,thecalculated[ABD]matrixshouldbeasymmetricmatrix. Figure 4-3 showsthedeformationoftheunitcellforeachcaseofunitstrainapplied.Figure 4-3 ashowshowtheunitcellwoulddeformduetotheperiodicboundaryconditionsappliedinFigure 4-2 .Theseguresareintendedtoshowhowtheunitcellsdeformduetotheperiodicboundaryconditions. 4.2TransverseShearStiffness InthecurrentdesignoftheITPSpanel,theinsulationbarsactasthecoreofthesandwichpanelandthewrapsactastheshearwebsprovidingconsiderableshearstiffness.TheunitcellmodelwithperiodicBCscannotbeusedforcalculatingthetransverseshearstiffness.Whenatransverseshearforce,say,Qxispresent,itgives 53
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Figure4-2.Periodicboundaryconditionappliedontheunitcellforx=1 Figure4-3.Schematicviewoftheunitcelldeformationsduetostrainsandcurvatures risetothebendingmomentMxwhichvarieslinearlyalongthex-direction.Thustheconditionsforperiodicityareviolatedandonecannotcomeupwithboundaryconditions 54
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thatreectthisvariationexactly.Hence,analternatemethodwithan1-DITPSplatemodelisconsidered.Thismethodcombines3Dniteelementanalysisandbeam/platetheorytocalculatethetransverseshearstiffness.Whiletheformerisaccurate,itcarrieswithittheeffectsoftheboundaryconditionsonbothendsofthemodel.Whenthemodelislongtheendeffectswillbecomeinsignicant.Thebeamtheoryisgoodforlongpanels.Thesetwofactorstogetherrequirelongbeam/1DplateforthepresentITPSmodel. Forinstance,todeterminethetransverseshearstiffnessA55the1-Dplatemodelwithunitcellsinthex-directionisconsidered(Figure 4-4 ). Figure4-4.1-DITPSmodelwithunitcellsinx-direction Theboundaryconditionsonthe1-DITPSmodelareatx=0,u(0;y;z)=0,Bottomedgew(0;y;0)=0;atx=L,u(L;;y;z)=0,w(L;y;z))]TJ /F3 11.955 Tf 10.18 0 Td[(w(0;y;z)=C(C=Constant).Figure 4-4 showsthedisplacementboundaryconditionsandplanestrainboundaryconditionsappliedonthex)]TJ /F3 11.955 Tf 12.08 0 Td[(zplane[ 82 ].Theaimoftheanalysisistodeectthe1-Dplatemodelvertically(transversedirection)usingatipforceF.Thetotaldeectionduetobendingandsheardeformationcanbecalculatedanalyticallyas 55
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w(x)=Fx A55)]TJ /F3 11.955 Tf 20.82 8.09 Td[(F 2D011x3 3)]TJ /F3 11.955 Tf 13.16 8.09 Td[(Lx2 2(4) D011=D11)]TJ /F3 11.955 Tf 13.15 8.09 Td[(B211 A11(4) Thersttermissheardeformationandthesecondtermisbendingdeformation.Fisthenetverticalforceduetothedisplacementboundarycondition,Listhelengthofthe1-DpanelandD011isthereducedbendingstiffness[ 82 ].ThebendingdeectioncanbecalculatedanalyticallyusingtipforceFandthe[ABD]matrixthatisalreadydetermined.ThetotaldeectioncanbecalculatedusingniteelementanalysisandthustheshearstiffnesscanbecalculatedusingEquation 4 Transverseshearstiffnessisacrosssectionalpropertyandforahomogenousstructurewhichcanbeconsideredtobemadeofinnitenumberofunitcells,thetransverseshearstiffnessisindependentofthelength.However,theITPSpanelconsideredherehasnitenumberofunitcells.Theshearstiffnessisoverestimatedduetotheboundaryeffects.Becauseofthelimitedcomputationalresources,itisnotaffordabletondtheshearstiffnessusing1-Dniteelementplatemodelsconsistingoflargenumberofunitcells,hencethestiffnessvalueiscomputedforsmallerlengths,10,20and40unitcellsandthenaconvergencecriterionisappliedtondtheconvergedvalueofthestiffnessvalue.Theconvergenceratecanbeexpressedas k3)]TJ /F3 11.955 Tf 11.96 0 Td[(k2 k2)]TJ /F3 11.955 Tf 11.96 0 Td[(k1=N1 N2(4) wherek1,k2,k3arethevaluesofstiffnessforsayn=10,20and40respectively.N1=N2istheratioofthelengthofthepanel(ratioofnumberofunitcells,hereN1=N2=10/20=20/40=0.5).istheconvergencefactorwhichshouldbegreaterthanunityforconvergenceofthestiffness.Calculatingtheconvergencefactorfromn=10,20and40,thenusingEquation 4 ,thestiffnesscouldbepredictedforlongerpanelstoobtaina 56
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convergedvalueofthestiffnessproperty.SimilarprocedureiscarriedoutforestimationofA44.Theconvergedtransverseshearstiffnessalongwiththe[ABD]matrixisusedtoestimatetheaccuracyofthehomogenizedmodel. 4.3AccuracyoftheHomogenizationMethod 4.3.1DeectionComparison TheITPSpanelissubjectedtovariouscombinationsofloadswheninstalledontheexteriorofthespacevehicle.Pressure,temperature,in-planeinertialloadsandimpactloadsarethevariousloadsthatactonthepanelduringight.Inthischapter,pressureloadsareappliedtocomparetheresponse(displacementandstresses)ofthe3-DITPSpaneltothatofthehomogenizedplatetoevaluatetheaccuracyofhomogenizationprocedure.Sinceitiscomputationallyexpensivetoconstructaniteelementmodelofthe3-DITPSpanelofincreasedlengths(Figure 3-1 ),an1-Dmodelconsistingofseriesofunitcellsineither(xory)directionisconsidered.The3-DITPSmodelisxedatoneendandfreeattheotherendwithplanestrainboundaryconditionsparalleltothex-zplane.Itissubjectedtoapressureloadp=1000Pa(Figure( 4-5 )).Foranequivalent1Dorthotropicplatewithsimilarboundaryconditionsandload,aclosedformsolutionofthedeectionisavailableasshowninEquation 4 w(x)=)]TJ /F3 11.955 Tf 16.43 8.09 Td[(p A55Lx)]TJ /F3 11.955 Tf 13.15 8.09 Td[(x2 2)]TJ /F3 11.955 Tf 22.48 8.09 Td[(p 2D011L2x2 2+x4 12)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Lx3 3(4) Thehomogenizedandthe3-DITPSmodel(Figure( 4-5 ))deectionsarecomparedtoevaluatetheaccuracyoftheequivalentstiffnesspropertiesestimated. 4.3.2StressComparison ThestressesarecomparedbetweentheequivalenthomogenizedplateandITPSpanelbyemployingthereversehomogenizationprocedure.TheoriginalITPSmodelissubjectedtopressureloadandstressesatvariouscrosssectionsaredetermined.Forthehomogenizedmodelsubjectedtopressureload,atanydistancex,mid-planestrainsandcurvaturescanbecalculatedas 57
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Figure4-5.1DmodeloftheITPSwithunitcellsinx-directionsubjectedtopressureloadandxedBCs Mx=p(L)]TJ /F3 11.955 Tf 11.96 0 Td[(x)2 2(4) x=Mx1 D011(4) "x0=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(B11 A11x(4) Themid-planestrainsandcurvaturesofthehomogenizedplateareappliedtotheITPSunitcellusingtheperiodicboundaryconditions(Table 4-1 )andthestressesaredetermined.Thisisreferredasreversehomogenizingtheequivalenthomogenizedplatetothe3-DITPSmodel.ThestressesintheunitcellarecomparedtothatoftheITPSatvariouspointsalongitslength. 4.4ResultsandDiscussion 4.4.1A,B,Dmatrices-Equivalentmaterialvs.Equivalentstructure Someresearchershaveconsideredhomogenizingthesandwichpanelasaplatemadeofahomogeneousmaterial.Thiswillbeapplicablewhenthepanelissubjectedtoonlyin-planeforces.However,whentherearetransverseforcesduetosurfacepressure 58
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orvibrations,thebendingstiffnesshastobeconsideredandaswillbeshownbelowthe[D]matrixhastobeevaluateddirectly.Further,forthecaseofaplatemadeofhomogeneousmaterial,[B]0.Howeverbending-stretchingcouplingmaybepresentduetoasymmetryandthismayplayacrucialroleinbucklingofthepanel. Homogenizingtoequivalentmaterialbasicallysmearsthepropertiesthroughoutthepanelbytreatingitasahomogenousorthotropicmaterial.Thein-planestiffnessmatrix[A]canbecalculatedfromelasticconstantsandusingtheconstitutiverelationinEquation 4 .Fromtheclassicallaminationplatetheory,the[D]matrix(bendingstiffness)isequalto(h2/12)[A]andforisotropichomogenousmaterial,[B][0].The[A]and[D]matrixcalculatedassumingthepanelisanequivalentmaterialisgiveninEquation 4 andEquation 4 [A]eqv:matl:=2666649141061431060143106913106000241106377775(4) [D]eqv:matl:=2666641611032510302510316010300043103377775(4) However,theITPSconstructionhasdifferentmaterialsforthetopfacesheet(TFS)andbottomfacesheet(BFS)andsothe[B]matrixcannotbezero.ItisinterestingtocomparetheITPSpropertiesasequivalentstructure(Equation 4 )versussmearingthepropertiestoformanequivalentmaterial(Equation 4 )and(Equation 4 ).Ifhomogenizedasanequivalentstructure,anon-zero[B]matrixisobtainedandthebendingstiffnessseemstobetwicethatofequivalentmaterial.ThusitisimportanttonotethathomogenizingtheITPSpanelusingequivalentstructureapproachcapturesthebehaviorofsandwichstructuresaccurately(Equation 4 )whencomparedtoequivalentmaterialapproach. 59
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[ABD]eqv:str:=266666666666666491410614310602:11061:4106014310691310601:41065106000241106002:21062:11061:410603101035010301:410651060501033101030002:210600861033777777777777775(4) Moreover,classicallaminationtheoryassumesinniteshearstiffness(A44=A55=innity)whichwouldmaketheITPSastiff,rigidstructure,butitwillbeshowninthenextsectionthatthepanelisasheardeformablesandwichstructure.Thenextsub-sectiondiscussesthesignicanceoftransverseshearstiffnessA44andA55ofthepanel. 4.4.2TransverseShearStiffnessofITPS ThetransverseshearstiffnessA44andA55arecrosssectionalpropertiesandhencearesupposedtobeaconstantforagivenmicrostructure.Theanalysisofthe1-Dplatemodeltodeterminetransverseshearstiffnesswasvalidatedconsideringamodelmadeentirelyoforthotropicmaterialwhosetransverseshearstiffnesscouldbecalculatedanalyticallyas A55=5 6Gh(4) WhereGistheshearmodulusinthex)]TJ /F3 11.955 Tf 12.33 0 Td[(zplaneandhistheheightofthemodel.Theorthotropicmaterialconsideredwasgraphite/epoxycomposite,G=6:9GPa(Table 3-3 ).ThelengthofthemodelwasL=420mm(n=20),widthw=21mmandheighth=46mm.ThetransverseshearstiffnessoftheorthotropicITPSpanelwasestimatedandcomparedwiththeanalyticalstiffnessvalue(Equation 4 ).Ideallyforsuchapanelmadeofinnitenumberofunitcells,thestiffnessvalueisindependentofthelength,howeverduetotheboundaryconditionsappliedonthepanelends,it 60
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approximatelytakes0.2mtoconvergeandagreewiththeanalyticalsolution.AstheITPSpanelconsistsofnitenumberofunitcells,thiseffectwouldbemorepronouncedandwouldrequirelargenumberofunitcellsforthestiffnessvaluetostabilizetoaconvergedvalue. Figure4-6.TransverseshearstiffnessA55oforthotropicpanel NowtheabovemethodisappliedtotheITPSmodelwithvaryingmaterialconguration.Owingtothecomputationalexpense,modelconsistingofunitcellsineitherxorydirectionisconsidered.Unitcellsarrangedinthex(A11,B11andD11stiffnessesconsidered)andy-direction(A22,B22andD22stiffnessesconsidered)providesA55andA44respectively. Table4-2.VariationofA44andA55ofITPSpanelwithn nA55N/mA44N/m 1016.910620.61062010.110611.8106408.11068.8106 61
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Figure4-7.VariationofA44andA55ofITPSpanelwithn Figure 4-7 showsthevariationofthetransverseshearstiffnessasafunctionofnumberofunitcells.Itcanbeseenthatthetransverseshearstiffnessishighwhenfewerunitcellsareconsidered,itdecreasesasthenumberofunitcellsincreases.Sincetheniteelementanalysesoflargeplatemodelswaslimitedbytheavailablecomputationalresources,theaforementionedconvergencecriterionisusedtoestimatetheshearstiffness.UsingthestiffnessvaluesfromTable 4-2 andEquation 4 theconvergencefactorisestimated.Asmentionedearliershouldbegreaterthanunityforconvergence.Thevaluesofwerefoundtobe2.2and1.9forA55andA44respectively,whichsuggestsconvergenceofthestiffnessvalue. Figure4-8.ConvergedA44andA55ofITPSpanelwithn 62
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Thetransverseshearstiffnesscanbepredictedforlargerpanelsusingtheabovevalueof.Consideringk1andk2correspondingtoN=10and20respectively,thevalueofk3(N=40)canbecalculatedusingEquation 4 (hereNi=Ni+1isalwaysequalto0.5).Thususingsuccessivecalculationsstiffnessforverylargenumberofunitcellscanbeestimated.Foraverylargenumberofunitcells(500unitcells),thetransverseshearstiffnessconvergedto7.4106(A55)and7.1106(A44)alsoshowninFigure 4-7 .Itisimportanttonotethatitrequireslargenumberofunitcellsfortheshearstiffnesstoconvergetoaconstantvalue.ThissuggeststhatdeterminedshearstiffnesscannotbeusedtohomogenizesmallerpanelssuchastheITPSpanelconsistingof12x12unitcells.However,theconvergedtransverseshearstiffnessandthecalculated[ABD]matrixcouldbeusedtopredicttheminimumlength(numberofunitcells)requiredforhomogenizationmethodtoyieldacceptableresults. 4.4.3AccuracyoftheHomogenizationMethod ThetipdeectionfromobtainedniteelementanalysisoftheITPSmodeliscomparedwiththeanalyticaldeectionofthehomogenizedplate(Equation 4 )forvariouslengthsofthecantileverbeammodel.Whenthetipdeectionratiobetweenthe2-Dmodeland3-Dmodelisclosetounity,thecorrespondinglengthwouldbetheminimumsizerequiredforhomogenizationtobeapplicable. Figure4-9.Ratiobetweentipdeectionofthehomogenizedmodelandthe3-DITPSmodelalongthexandy-direction 63
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Table4-3.Tipdeectionratioalongwithcontributionofbendingandsheardeformationtowardsdeection L/aTipdeectionratioalongShear%Bending%x-direction(w2D=w3D) 121.367228201.124852401.051981601.01991701.00793 UsingEquation( 4 )thecontributionofsheardeformationtowardstotaldeectionwasdetermined.FortheITPSpaneltypicallyconsistingof12unitcells,thecontributionofshearis72%thusconrmingthepanelishighlysheardeformable.Thetablealsoprovidesthecontributionofsheardeformationforlongerpanels.Figure 4-9 showsthetipdeectionratioforvariouslengthsoftheITPSmodel.Thex-axisoftheplotisL=aratio(a-widthofunitcell)denotingthenumberofunitcells.ItisevidentfromFigure 4-9 thattheminimumlengthofthepanelmustbe1.5m(70unitcells)tohomogenizetheITPSstructuretoanequivalentorthotropicplate.Evenforsuchalargepanel,thereis7%sheardeformationandneglectingtransverseshearstiffnesswouldyieldin7%discrepancyinthehomogenizationofthepanelfordeections. Nowthepanelwithlargerlength(1.5m)isanalyzedunderpressureloadforstresscomparison.Thestressesinthehomogenizedmodelaredeterminedbyapplyingmid-planestrainsandcurvatures(Equation 4 and 4 )onaunitcell.Thestressratiobetweenhomogenizedandoriginal3-Dmodeliscomparedatcrosssectionsatvariousdistancesalongthelengthofthepanel(Figure 4-10 ).Itcanbenotedthatthestressratioclosetofreeendisgreaterthanunity.ThiscanbeattributedtotheSt.Venant'seffectsuggestingcertaindistanceintothemodelisrequiredforthefreeedgeeffectstodissipate.Thestressratiodecreasesandconvergestounity(approximately)atL=a=10. ThestressvaluesatL=a=10alongthewidthofthepanelisshowninFigure( 4-11 ).Furthertoshowthathomogenizationcannotbeappliedforsmallpanels,the3-DITPSandhomogenizedplatemodelsconsistingof12x12unitcellsare 64
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Figure4-10.Stress(11,22)ratiobetweenthehomogenizedmodelandthe3DITPSmodel Figure4-11.Stress(11,22)atn=10comparisonbetweenthe3-DITPSmodelandthehomogenizedmodel analysedunderpressureloadandxedboundaryconditions.Aquarteroftheplateisconsideredwithsymmetryboundaryconditions.Figure 4-12 showsalargediscrepancyinthedeectionsbetweenthe2-Dhomogenizedplatemodeland3-DITPSpanel(w2D=w3D=3.8). Itcanbeconcludedthatacompositesandwichpanelhastobesufcientlylargetohomogenizeittoanorthotropicplate.ForthecurrentITPSpanelwithafairlycomplexdesignandmadeofcomposites,aminimumof70unitcellsarerequired. 65
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Figure4-12.Verticaldeectionalongx-axisoftheoriginalITPSandhomogenizedmodelduetopressureload 4.5Summary Arepresentativevolumeelement/unitcelloftheITPSpanelwasanalyzedtodeterminetheequivalentstiffnesspropertiesofthe2Dplate.Theunitcellwassubjectedtosixlinearlyindependentdeformationsandtheequivalentstiffnesspropertiesofthecompositestructurewereestimated.Analysisofthe1-DITPSplatemodelthatcombinesniteelementsandsheardeformableplatetheorywasproposedtoestimatethetransverseshearstiffnessoftheequivalenthomogenizedplatemodel.Toevaluatetheaccuracyofthehomogenization,thehomogenized2-Dplatemodelwasanalyzedunderpressureloadandcomparedwiththe3-DITPSpanel.Thevaluableinsightsofthisresearchworkare:a)TransverseshearstiffnessofITPSsandwichpanelsareimportantandcannotbeneglected.b)Homogenizationcannotbeappliedtothesmallpanelswithfewerunitcells.HenceasingleunitcelloftheITPSpanelwillbeusedforoptimizationprocess. Thehomogenizationmethodisapplicabletopanelsoflengthmuchlargerthantheunitcelldimensions.ItwasestimatedthattohomogenizethecurrentdesignofITPSpaneltoanequivalentorthotropicplate,thepanelmusthaveaminimumof70unitcells. 66
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CHAPTER5MONTECARLOSIMULATIONS-RELIABILITYESTIMATION ThischapterprovidesadetaileddescriptionofthecrudeMonteCarloandseparableMonteCarlomethod,there-allocationofuncertaintytopredictaccurateestimateofprobabilityoffailureandbootstrappingtechniquetoestimatetheaccuracyofthepfestimate. ThemeritsofusingseparableMonteCarloovercrudeMonteCarlomethodisillustratedusinganon-separablelimitstatefunction(theresponseandcapacitycomponentsareintegratedinthelimitstate).TheprobleminvolvesestimatingtheprobabilityoffailureofcompositepressurevesselusingTsai-Wufailurecriterion(non-separablelimitstate).TheaccuracyofthemethodsiscomparedfordifferentgroupingsoftherandomvariablesfortheseparableMonteCarlomethodfromtheestimateofcoefcientofvariation/standarddeviation.ItisdemonstratedthatusingSMCwiththeregroupedlimitstatereducestheerrorassociatedwiththeprobabilityoffailureestimate.Thisisaccomplishedbyreallocatingtherandomvariablesasinexpensivevariableswithnolimitonsamplesizeandexpensivevariableswithlimitedsamples.Furthertheexpensivevariablesarebootstrappedandtheaccuracyofthebootstrappedprobabilityoffailureestimateiscomparedwiththatoftheempiricalestimate. 5.1CrudeMonteCarloMethod(CMC) Acommonsampling-basedmethodforcalculatingtheprobabilityoffailurepf,istraditional,crudeMonteCarlo.Theprobabilityoffailureisestimatedbycomparingpairsofrandomlygeneratedresponseandcapacitysamples,asshowninEquation 5 ^pcmc=1 NNXi=1I[G(Ri;Ci)0](5) WhereIistheindicatorfunction,whichequals1iftheconditionistrueand0iftheconditionisfalse.Thus,thisessentiallysumsthenumberoffailuresforNcomparisons. 67
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Therootmeansquare(RMS)errorintheestimatemaybemeasuredbythestandarddeviationforcrudeMCgivenas stdev(^pcmc)=r pf(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pf) N(5) CV(^pcmc)=stdev(^pcmc) N=s (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pf) pfNr 1 pfN(5) Forexample,foraprobabilityoffailureofoneinamillion,100millionsimulationsareneededfor10%error.Thecoefcientofvariationcalculatedfromstandarddeviationandmeanprovidesameasureoftherelativerootmeansquare(RMS)errorintheprobabilityoffailureestimateaboutitsmeanvalueEquation 5 .Notethatbecausewehaveonlyanestimateoftheprobabilityoffailure,wegetonlyanestimateoftheerror.ThecostofresponsecalculationisoftenthelimitingfactorinthenumberofsamplesN,becauseresponsecalculationsoftenrequireexpensiveniteelementsimulations. 5.2SeparableMonteCarlomethod(SMC) Whenthecapacityandresponsearestatisticallyindependentinthelimitstate,thentheycanbesampledseparatelyusingamethodcalledseparableMonteCarlo(SMC).SeparableMChasalreadybeeninvestigatedforthesimplelimitstateasshowninEquation 2 .ThisstudylooksatSMCforthegenerallimitstateinEquation 2 ,theseparableMCmethodisshowninEquation 5 ^pcmc=1 NNXi=1I[G(Ri;Ci)0](5) WhereNisthenumberofresponsesamplesandMisthenumberofcapacitysamples.Sincetheresponseandcapacityaresampledseparately,allofthepossiblecombinationscanbeconsideredtoestimatepf.Figure 5-1 illustratestheresultingdifferencebetweenCMCandSMC. 68
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ACrudeMonteCarlo BSeparableMonteCarlo Figure5-1.IllustrationofcrudeandseparableMonteCarloMethodcomparisonsA.CrudeMonteCarlomethodB.SeparableMonteCarlomethodwhereeverysampledresponseiscomparedwitheverysampledcapacity InFigure 5-1 a,thedirectone-to-onecomparisonsofcrudeMCareshownforNsamples.Whereas,Figure 5-1 bshowsthatseparableMClooksatallofthepossiblecombinationsofrandomsamples,whichmakesitinherentlymoreaccuratethanCMC.Inaddition,differentsamplesizescanbeusedtoenhancetheaccuracy,dependingontherelativecomputationalexpenseoftheresponseandcapacitywhichwasdemonstratedusinganon-separablelimitstatetopredicttheprobabilityoffailureofacompositepressurevessel(Section 5.5 ). 5.3ErrorintheProbabilityofFailureEstimate ForCMC,theRMSerrorintheprobabilityestimateisprovidedbyEquation 5 .ForseparableMCwiththesimplelimitstateasinEquation 2 ,Smarsloketal.[ 15 ]derivedanalyticalestimatesofthestandarddeviationviaexpectationcalculus.ForthemoregeneralcaseofEquation 2 ,weproposebootstrappingthecomponentsofthelimitstate[ 18 ].Forestimatingtheerrorintheprobabilityoffailureestimate,weusebootstrapping,are-samplingtechnique,whichinvolvestakingthesamplesofresponse(expensive)andre-samplingthemwithreplacement(sothatthesamplesmaycontainduplicates).Whenweperformthere-samplingbtimes,weobtainbprobabilityoffailure 69
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Figure5-2.Schematicrepresentationofbootstrappingwhenonlyresponseissampled. estimates^pboot.Withbestimatesof^pboot,wecanobtainthestandarddeviationdenotingtheerrorintheestimate.Aswillbeshowninthelatterpart,thebootstrappingerrorestimatesappearstobecomparabletotheCMCestimateofEquation 5 Furthermore,wecanobtain^pbootestimatesbybootstrappingcapacityatmeanvaluesofresponseandviceversa.Theknowledgeoftheindividualcontributionsoftheresponseandcapacitytowardstheuncertaintyaidsinchoosingtheappropriatesamplesizeforresponse(N)andcapacity(M)whichwouldprovideanaccurateestimateofthevariationinthepfestimate.Whenwere-sampleboththeresponseandcapacity,weobtainthetotaluncertaintystdev(^psmc;boot).Butfortheindividualcontributionsoftheresponseandcapacity,theresponsehastobootstrappedatmeancapacityandviceversatoobtainstdevR(^psmc;boot)andstdevC(^psmc;boot). Inthenumericalresults,thebootstrappingvaluesarecomparedwiththeempiricalvalues,stdevR(^psmc;boot)andstdevC(^psmc;boot)todemonstratetheaccuracyofthebootstrappingmethod.Inordertomeasuretheerrorinthebootstrappedestimate,theuncertaintyinthestandarddeviationofthebootstrappedprobabilityoffailureestimate(stdev(stdev(^psmc;boot)))isalsocalculated.Thebootstrappingestimateswouldallowus 70
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tojudgewhetherthenumberofexpensivesimulationsissufcientforadesiredlevelofaccuracy. 5.4SMCwithregroupingandseparablesamplingofthelimitstaterandomvariables Whenthebootstrappingestimatesshowthattheaccuracyoftheprobabilityoffailureisnotgoodenough,andthattheculpritistoofewsamplesoftheexpensivesimulation,wemayhaveapainfulchoicebetweenveryhighcomputationalcostandaccuracy.Oftenthereisanalternativetoincreasingthenumberofexpensivesamplesbyreformulatingthelimitstate,whichisdescribedinthissection.Thenumberofsamplesrequiredforaccuratemodelingoftheresponseandcapacitydependsontherelativecontributionsoftherandomcomponentsinthelimitstatefunction.Wherethelargertheuncertaintycontribution,moresamplesarerequiredforaccuraterepresentationofthedistribution.Assumingacomputationallimitonthenumberofsamplesoftheexpensiveresponse(N),itisdesirabletoreducetheuncertaintyintheresponse(i.e.obtainnarrowerdistributionoftheresponse)toachieveimprovedaccuracy. Inmoststructuralproblems,failureofthesystemdependsonthestrengthofthematerialS(e.g.capacity),andstressesthestructuresustains(e.g.response).SothelimitstateinEquation 2 maybecomeG(;S).Inlinearproblems,stresses,arealinearfunctionoftheloadp,asinEquation 5 =Up(5) WhereUarestressesperunitload.Therandomnessintheloadp,isoftenindependentoftherandomvariablesthataffectU(geometryandmaterialproperties),butpaddsuncertaintytothecomputationallyexpensivestresscalculation.SoitwouldbeadvantageoustoseparatetheloadsfromthestressesanddeterminestressesperunitloadU.Thisalsoenablesalargersamplesizeoftheload.Evenwithalimited 71
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numberofsamplesofstressesperunitload,theprobabilityoffailurecanbemoreaccuratelyestimatedasboththestrengthandtheloadcanbecheaplysampled.ThentheexpensiveunitloadresponseUissampledandcombinedwiththeloadinEquation 5 .Finally,thelimitstateisreformulatedas G(;S)=G(U;p;S)(5) TheprobabilityoffailurecanbedeterminedfromalargesampleofloadsandstrengthscomparedtoalimitedsampleofstresseswhichisillustratedinFigure 5-3 Awithstresses Bwithunitloadstresses Figure5-3.IllustrationofseparablesamplingwithunitloadsA.withstressesB.withunit-loadstresses TheseparableMonteCarloformulathatcorrespondstoFigure 5-1 bisshowninEquation 5 ^pUsmc=1 M1 NMXj=1NXi=1I[GU(Ui;Pj;Sj)0](5) AsimilarformofEquation 5 couldbewrittenforFigure 5-3 ,butwithdifferentindices. 5.5ApplicationtoFailureAnalysisofCompositeLaminate ThecrudeMonteCarloandseparablesamplingsimulationmethodsandtheirerrorestimatesarecomparedandillustratedbyapplyingthemtoanon-separablelimitstateproblem(Equation 2 ).Forcomplexstructures,thestressiscalculated 72
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Figure5-4.Compositepressurevesselwithinternalpressureof100kPaandstressesactinginasmallelementofthevessel. throughniteelementanalysisanditmaybecomputationallyexpensive.InordertoallowustoperformthousandsofMonteCarlosimulationsneededtovalidatethemethod,weselectedanexamplethatrequiresthecalculationofstressesatasinglepointusingClassicalLaminationTheory(CLT).TheprobleminvolvespredictionoffailureofcompositepressurevesselaccordingtotheTsai-Wufailurecriterion. Acompositelaminatepressurevessel(Figure 5-4 )ismadeofagraphite/epoxy[+25/-25]ssymmetriclaminatewitheachlayerbeing125mthickandissubjectedtoaninternalpressureof100kPa.ThematerialpropertiesofthecompositeareshowninTable 5-1 .Inthispaper,alloftheinputrandomvariablesareassumedtobenormallydistributed.However,theperformanceofSMCdependsonthedistributionoftheresponseandcapacity,whichisnotnecessarilynormal.Previousresearchhasusedotherdistributions,suchaslognormalanduniform,withseparableMonteCarlomethod[ 15 61 ]. Thestressesgeneratedareafunctionoftheinternalpressurepandmaterialpropertiesofthelaminatewhichareindependentofeachother. 8>>>><>>>>:12129>>>>=>>>>;=[T][Q][A])]TJ /F4 7.97 Tf 6.59 0 Td[(1=8>>>><>>>>:pd=2pd=409>>>>=>>>>;=[T][Q][A])]TJ /F4 7.97 Tf 6.59 0 Td[(18>>>><>>>>:d=2d=409>>>>=>>>>;p=8>>>><>>>>:U1U2U129>>>>=>>>>;p(5) 73
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Thestresses1,2(normal)and12(shear)actingineachplyofthelaminatearefunctionofin-planestiffnessmatrixofthelaminate[A],reducedstiffnessmatrixofeachlamina[Q],transformationmatrixofeachlamina[T],thepressureloadpandthediameterd(1m)ofthevesselasinEquation 5 .Failureofcompositelaminateispredictedfromthestrengthofthecompositeandstressesgeneratedusingasuitablefailurecriterion.ThemostwidelyusedcriterionforcompositesistheTsai-Wucriterion.ThecriterionisafunctionofthestrengthsS(showninTable 5-1 alongwiththeiruncertainties),normalstresses,intheberandtransversedirection(e.g.1and2direction,respectively)andshearstress,. Table5-1.Materialpropertiesanduncertaintyoftherandomvariables(normallydistributed)[ 83 ] PropertiesMeanCV%StrengthMeanCV% E1GPa159.15%S1T231210%E2GPa8.35%S1C180910%G12GPa3.35%S2T39.210%v12GPa0.2535%S2C97.210%PkPa10015%S1233.210% Accordingtothecriterion,alayerofthelaminateisassumedtohavefailedwhenthelimitstateinEquation 5 isgreaterthanorequaltozero. G(;S)=F1121+F2222+F66212+F11+F22+F1212)]TJ /F9 11.955 Tf 11.96 0 Td[(1(5) F11=1 S1TS1CF1=1 S1T)]TJ /F9 11.955 Tf 19.49 8.09 Td[(1 S1CF22=1 S2TS2CF2=1 S2T)]TJ /F9 11.955 Tf 19.49 8.09 Td[(1 S2CF12=1 S212F12=p F11F22 2(5) UncertaintiesintheTsai-Wucoefcients(F11;F22;F66;F1;F2;F12)areduetorandomnessintheunidirectionaltensile,compressive,andshearstrengthsSofthecomposite.Uncertaintyinthestressesisduetorandomnessinmaterialproperties(orU)andpressureloadp.Thein-planenormalandshearstresses(1,2,12T),are 74
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Figure5-5.Distributionofstressandstrengthinthe2-direction(2)showingtheprobablefailureregion uniqueineachlayerofthelaminate.Aspreviouslymentioned,thelaminateismadeof[+25/-25]spliesandtheanalysisshowsthattheinner(-25)pliesfailandthenouterpliesfail.Amongthenormalandshearstressesactingontheply,thestressinthetransversedirection2causesthefailureofthelaminate(theoverlapofthestressesandstrengths)whichcanbeseeninFigure 5-5 ThestressesareafunctionofmaterialpropertiesandinternalpressurepasinEquation 5 .Forunitpressureload(p=1),stressesareequaltoU.Therefore,theoriginallimitstatefunction(G(;S))canbereorganizedasindicatedbyEquations 5 5 and 5 5.6ResultsandDiscussion 5.6.1CrudeandSeparableMonteCarloMethod TheprobabilityoffailureofacompositepressurevesselwascalculatedusingcrudeMonteCarloandseparableMonteCarlomethod.Itwasassumedthatourcomputationalbudgetonlypermitted500stresscalculations.Therefore,forcrudeMonteCarlo,anequalnumberofrandomresponseandcapacity(S)variables(N=500)weresampledforcomparison.InthecaseofseparableMonteCarlo,theresponsesamples(N)werecomparedagainstallthecapacitysamples(M=500)resultingin250,000evaluationsofthelimitstate.Theactualprobabilityoffailureispf=0.0121(Asthisisasimpleproblem,theactualprobabilityoffailurewasestimatedbyCrudeMonteCarlomethodby 75
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generating5millionsamplesofstressandstrengthvalues).TherelativeerrorincrudeMonteCarlowasmeasuredbycalculatingthestandarddeviationfromEquation 5 andhencethecoefcientofvariation.Thisvalueprovidesameasureofhowaccurateistheprobabilityoffailureestimate.Forasimplelimitstate(asinEquation 2 ),theaccuracyofseparableMonteCarlocanbeestimatedasderivedbySmarsloketal.[ 16 ].Sincethisproblemisdenedbyagenerallimitstate(e.g.Tsai-Wu),bootstrappingwasperformedtoassesstheaccuracyoftheprobabilityoffailureestimateofSMCwithab=1,000bootstraprepetitions.Sincethestressesarecomputationallyexpensiveanditischeapertosamplethestrengths,randomsamplesofstresseswerebootstrapped,butthestrengthsweresampledanewratherthanbootstrapped.Thatis,ineachofthe1,000bootstraprepetitions,thestresseswerere-sampledfromthesame500samples,whilethestrengthshadafreshsampleeverytime. Inthisstudy(wheresimpleCLTcalculationswereused)theaccuracyoftheseparableMonteCarlomethodwasalsoassessedbyanempiricalcoefcientofvariationobtainedbyperformingn=10,000repetitions.TheprobabilitiesoffailureestimatesarelistedinTableandtheestimatesoftheerrorintheprobabilityoffailurearetabulatedinTable 5-3 .Itshowsthatthecoefcientofvariationisreducedfrom40%to21.0%bySMC.OnaveragethebootstrappingerrorestimateoftheSMCprobability(asmeasuredbythestandarddeviation)isabout20%low(0.002comparedto0.0025),withastandarddeviationwhichisvetimeslowerthanthataverage.Thusinthelargemajorityofcasestheerrorestimateiswithin50%oftheempiricalerror. Table5-2.EmpiricalandbootstrappingestimatesofprobabilityoffailureusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitions CMCSMCoriginallimitstateSMCRegroupedEmpiricalBootstrappingEmpiricalBootstrappingmean(^pcmc)mean(^psmc)mean(^psmc;boot)mean(^pusmc)mean)]TJ /F9 11.955 Tf 6.46 -9.69 Td[(^pusmc;boot 0.01210.01210.01210.01200.0122 Nextwedemonstrateobtainingtheindividualcontributionoftheresponseandcapacitytotheuncertaintyintheprobabilityoffailureestimateobtainedby 76
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Table5-3.StandarddeviationandcoefcientofvariationofempiricalandbootstrappingpfestimatesusingseparableandcrudeMonteCarlowithN=M=500andn=10,000repetitionsfororiginallimitstate.Standarddeviationofthebootstrappingerrorisalsoshown CMCSMCoriginallimitstateSMCRegroupedEmpiricalBootstrappingstdev(^pcmc)CV(^pcmc)stdev(^psmc)CV(^psmc)mean(stdev(stdev(stdev( ^pusmc;boot)^pusmc;boot))0.004840%0.002521%0.00220.0004 bootstrappingtheresponseatmeanvaluesofthecapacity,andviceversa.Theindividualcontributionswouldhelpwhentheoverallerrorestimateislargeandneedstobereducedbyincreasedsamplesize.Thevaluesofmean(stdevR)]TJ /F9 11.955 Tf 6.46 -9.69 Td[(^pusmc;boot)andmean(stdevC)]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pusmc;boot)inTable 5-3 providetheuncertaintyinpfestimateduetothestressandstrengthrespectively.Thesevaluesarealsocomparedwithrelativecontributionsofthestressandstrengthobtainedempiricallytoillustratetheaccuracyofthebootstrappingmethod. Table5-4.Relativecontributionsofresponse(stresses)andcapacity(strengths)towardstheuncertaintyinpfthroughbootstrappingandalsocomparedwithempiricalresults EmpiricalBootstrapping stdevR(^psmc)0.0017mean(stdevR(^psmc;boot))0.0019CVR(^psmc)15.4%stdev(stdevR(^psmc;boot))0.0004stdevC(^psmc)0.0012mean(stdevC(^psmc;boot))0.0014CVC(^psmc)9.8%stdev(stdevC(^psmc;boot))0.0002 FromTable 5-4 wecanseethatthecontributionoftheresponsetotheuncertaintyinthepfestimateishigherthanthecontributionofthecapacity.Itispossibletoreducetheresponseuncertaintybyusingalargersample.However,sinceresponsecalculationisusuallyexpensive,welookinsteadtoreducetheuncertaintyintheresponsecontributionbyothermeans.Theresponsecontainstheloadwithitslargeuncertainty(CV(p)=15%).CalculatingstressperunitloadU,isolatestheexpensiveCLTcalculation(orFEA)fromthelargeuncertaintyintheload.Thenextsection 77
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exploreshowtherandomcomponentsinthelimitstatecanbereformulatedbyusingunitstressestoreducetheerrorinthepfestimate. 5.6.2Regroupingandseparablesamplingofthelimitstatevariablesforimprov-ingaccuracy Intheoriginallimitstate(G(,S)),thestresscalculationcontainsthelargeuncertaintyfromtheload.Therefore,rearrangingtheresponsetoobtainstressperunitloadU,andloadppermitscalculatingresponsethatdoesnotincludetheuncertaintyintheload.ThisarrangementwillenableseparablesamplingoftheindependentvariablesofthelimitstateGu(U,p,S),similartothatshowninEquation 5 .Thisregroupingshiftsthelargeuncertaintyintheloadawayfromtheexpensivestresscalculation)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(U.ForN=M=500(N=numberofstressperunitloadsamples,M=numberofsamplesofstrengthandload),theuncertainty(mean(stdev()]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pusmc;boot))inthebootstrappedestimateforthereformulatedlimitstateduetotheresponse(unitloadstresses)reducestofrom0.0019to8.6x10)]TJ /F4 7.97 Tf 6.59 0 Td[(5.Ontheotherhand,thecapacity/loaduncertaintyincreasestofrom0.0014to0.0044.Itwouldappearthatwemadethesituationworse,butnowwecanreducetheerrorinpfestimatebyincreasingM,whichisnormallycheap.ThevalueofMwasvariedfrom500-10,000samplesandtheuncertaintyintheestimateforreformulatedlimitstateisshowninTable 5-5 .Itisclearlyseenthattheregroupingallowsustokeepasmallnumberofresponsecalculationsandreducetheuncertaintybyhavingaverylargenumberofinexpensivecapacity(load)calculations.Thestandarddeviationoftheprobabilityoffailurefortheregroupedlimitstatestdev)]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pUsmcwasalsoestimatedempiricallyandshowninTable 5-6 .ThestandarddeviationsobtainedareplottedinFigure 5-6 Figure 5-6 clearlyillustratestheeffectofregroupingoftheinexpensiverandomvariablesofthelimitstate.InthecrudeMonteCarlomethod,theprobabilityoffailureiscalculatedusinganequalnumberofresponseandcapacitysamples.Inthiscase,500 78
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capacitysamplesand500responsesampleswereused,whichcorrespondstoasinglevalueontheplotinFigure 5-6 Table5-5.StandarddeviationandcoefcientofvariationofCMCandSMCforincreasingsamplesizeofMandN=500.Bootstrappedandempiricalestimatesareshown. CMCSMC(Empirical)SMC(Bootstrapping)Mstdev(^pcmc)CV(^pcmc)stdev(^psmc)CV(^pcmc)mean(stdev(stdev(stdev(^psmc;boot)^psmc;boot) 5000.004840%0.002521%0.0020.000450000.002117.6%-*-100000.002016.8%-Table5-6.StandarddeviationandcoefcientofvariationofCMC,SMCandSMCregroupedforincreasingsamplesizeofMandN=500.Bootstrappedandempiricalestimatesareshown. CMCSMC(Empirical)SMC(Bootstrapping)Mstdev(^pcmc)CV(^pcmc)stdev)]TJ /F9 11.955 Tf 6.46 -9.69 Td[(^pUsmcCV)]TJ /F9 11.955 Tf 6.47 -9.69 Td[(^pUsmcmean(stdev(stdev(stdev(^pUsmc;boot)^pUsmc;boot) 5000.004840%0.004537.2%0.00460.000150000.001512.6%--100000.00097.9%-Incontrast,theseparableMonteCarlomethodcanusedifferentsamplesizes(M)fortherandomvariables.ObservethatthestandarddeviationfromtheoriginallimitstateofSMClevelsofftoanearlyconstantvalueof0.002forMsamplesgreaterthan5000.Ontheotherhand,thestandarddeviationfortheregroupedlimitstatecontinuallydecreaseswiththenumberofMsamplesallthewaytostdev)]TJ /F9 11.955 Tf 6.47 -9.68 Td[(^pUsmc0.0009(or7.9%CV).Inotherwords,inCMCtheerrorestimateofthefailureprobabilityis40%,buttheerrorassociatedinSMCisonly16.8%withtheoriginallimitstateor7.9%withtheregroupedlimitstate.Thatis,fornearlythesamecomputationalcost,separableMCwithregroupingcanestimatethefailureprobabilitymoreaccuratelythancrudeMonteCarlo. Figure 5-6 showsthatthemagnitudeoftheuncertaintyreduceswithincreaseinnumberofMsamples.Thusbyincreasingthesamplesizeoftheinexpensive 79
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Figure5-6.StandardDeviationofCMC,SMCandregroupedlimitstateSMCwhereN=500(xed)andMisvaryingfor10,000repetitions components(strengthandtheload),wecouldreducetheuncertaintyinthepfestimate.ForverylargeM,itwouldreachanasymptoticvalueduetothenitevalueofN.Figure 5-6 showsthatbytransferringsomeoftheuncertaintyfromtheresponsetothecapacity,wecantakeadvantageofincreasedMtofurtherreducetheerrorintheprobabilityestimate. 5.6.3Summary TheseparableMonteCarlo(SMC)methodcanprovidesubstantialimprovementsinaccuracyoverthecrudeMonteCarlomethodwhenresponseandcapacityaregovernedbyindependentrandomvariables.ObtainingestimatesoftheaccuracyofSMCiscriticaltotakingfulladvantageofthemethod.HereweproposedusingbootstrappinginordertoobtainestimatesoftheerrorintheSMCestimates,aswellasthecontributionsofthecapacityandtheresponsetothaterror.TheapproachwasdemonstratedthroughanexampleproblemoffailureanalysisofacompositepressurevesselusingTsai-Wufailurecriterion.Becauseofthelowcomputationalcostoftheexample,itwaspossibletoconductmultiplesimulationsandassesstheaccuracyofthebootstrappedestimateempirically.SeparableMonteCarloledtosubstantialaccuracyimprovementindeterminingtheprobabilityoffailurecomparedtocrudeMonteCarlo 80
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method.BootstrappingprovidedreasonableestimatesoftheuncertaintyintheSMCprobabilityoffailureestimates.Bootstrappingalsoallowedestimatingtheindividualcontributionsoftheresponseandcapacitytowardstheuncertaintyintheprobabilityoffailureestimate,thussuggestingadditionalsamplesofinexpensivecapacitywouldproveadvantageous.Furthersubstantialimprovementinaccuracywasachievedbytransferringuncertaintyawayfromexpensivecalculationsbyusingunitloadstressesandgeneratinglargesamplesofloadandstrengths. 81
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CHAPTER6EFFICIENTGLOBALRELIABILITYANALYSIS(EGRA) ThenameofthemethodincludesReliabilityAnalysis,howeverEGRAisasequentialsamplingmethodthataimstoimprovesurrogatesinvolvedinconstrainedoptimizationandreliabilityanalysis.Thismethodcanbeusedimprovethetargetboundariesinsurrogatemodelsfacilitatingestimationaccurateoptimum(orreliability).TheEGRAalgorithmcodedinthesurrogatetoolboxinMATLABwasusedtoillustratetheadvantagesofusingthismethod. TheITPSoptimizationprobleminvolvesconstructionofsurrogatesinfourormoredimensions,wherevisualizingtheworkingofEGRAisnotpossible.ThischapterillustratesthemethodologyofEGRAusingBranin-Hoofunctionintwodimensionalspacefortheeaseofvisualization.Asmentionedinsection 2.5.1 EGRAhastwoimportantfeatures-aninitialresponsesurfaceconstructedwithfewerdesignpointsandtheExpectedFeasibilityfunctioncriterion.Withagiventargetlevel,thefeasibilityfunctionusestheexistingresponsesurfacetoidentifyanewdesignpointthatwouldimprovethecontourofthetargetboundary. 6.1EGRAalgorithm Bichonetal.[ 59 14 ]hasprovidedadetaildescriptionoftheEGRAalgorithm. 1. Generateasmallnumberofrandomsamplesfromthetrueresponsefunction.(a)Thisinitialselectionisarbitrary.However,oneoftheoptionsareconsideringnumberofsamplesrequiredtodeneaquadraticpolynomialisusedhereasaconvenientruleofthumb((n+1)(n+2)/2sampleswherenisthenumberofrandomvariables).(b)Latinhypercubesampling(LHS)isusedtogeneratethesamplesinthedesignspace. 2. BuildaninitialGaussianprocessmodel(interchangeablyreferredaskrigingmodel)fromthegeneratedsamples. 3. Findthepointwithmaximumexpectedfeasibility.(a)Theexpectedfeasibilityfunctionisbuiltwith=2G.(b)Tolocatethemaximumexpectedfeasibilityanappropriateoptimizerisused.ThesurrogatetoolboxusesDifferentialEvolutionaryOptimizer[ 84 ]. 4. Evaluatethetrueresponsefunctionatthispoint 82
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5. Addthisnewsampletotheprevioussetoftrainingdataandbuildanewgaussianprocessmodel.Gotostep3. 6. Thisupdatedsurrogatemodelisthenusedanapproximationtoevaluateconstraintsinconstrainedoptimization 6.2IllustrationofEGRA TheBranin-Hoofunctionisastandardtestfunctionusedinoptimizationproblems.Thefunctionhastwovariableswithbounds-5x110and0x215andadequatelynon-linear.Thefunctionisdenedas g(x)=x2)]TJ /F9 11.955 Tf 13.15 8.08 Td[(5:1x21 42+5x1 )]TJ /F9 11.955 Tf 11.95 0 Td[(62+101)]TJ /F9 11.955 Tf 16.68 8.08 Td[(1 8cos(x1)+1050(6) ContoursofthetruefunctionisshowninFigure 6-1 A.Thecontourg(x)=50ischosenasthetargetboundary(Figure 6-1 B).ThegoalistorepresentthisboundaryasaccuratelyaspossibleusingtheEGRAmethodology.ThetruecontoursoftheBranin-HoofunctionareshowninFigure 6-1 A.Alsothetargetboundaryg(x)=50isshowninFigure 6-1 B. ATrueFunction Bg(x)=50ofTruefunction Figure6-1.TruecontoursoftheBraninhoofunctionA.allcontoursB.targetcontourg(x)=50 83
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Aninitialsurrogatemodelisbuilt.Thesurrogatemodelavailablebydefaultiseitherkrigingorgaussianprocessmodel.Theinitialsurrogatemodelwastted10LHSdesignofexperimentsasshowninFigure 6-2 A[ 85 ].Usingtheresponsesurfaceapproximationandthepredictionuncertainty,theexpectedfeasibilityimprovementfunction(Equation 2 )ismaximizedusingtheDifferentialEvolutionaryoptimizerinthesurrogatetoolbox.Figure 6-3 showsexpectedfeasibilityfunctioncomputedandplottedinthedesignspaceforthepurposeofvisualizingtheprocess.Itcanbeseenthatthefunctionhasamaximumatupperleftcornermarkedbyacircle. Figure6-2.TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction.TotalDOE=10points Theoptimizerfoundthemaximumtobeattheupperleftcorner(0,15)andthisdesignisevaluatedusingthetrueresponsefunctionandincludedintheDOEtoconstructthenewresponsesurface.ThenewdesignandthetargetcontourofthenewthekrigingmodelisshowninFigure 6-4 Theexpectedfeasibilityfunctionofthenewsurrogatemodelshowsthattheoptimizerwouldsuggestapointatlowerrightcorner(markedbyacircle)wherethe 84
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Figure6-3.ExpectedFeasibilityFunctionevaluatedusingtheinitialkrigingapproximation functionismaximum.ThenewdesignpointacquiredattheendofthisEGRAcycleis(10,0).TheexistingsurrogatemodelisupdatedwiththisdesignandthenewmodelisshowninFigure 6-5 Thisprocessisrepeatedforanother18cyclestoimprovethetargetboundaryg(x)=50.Thedifferentialevolutionaryoptimizer[ 84 ]maximizesthefeasibilityfunctiontondtheadditionaldesigns,onepercycle.ItshouldbenotedthatLHSsampling,oftentimesdonotplacepointsattheboundariesofthedesignspace.Thiscauseshighuncertaintyinthepredictionmodel.Theexpectedfeasibilityfunctioninitiallyaddspointsatregionswherepredictionuncertaintydominates.Thisisreferredtoasexplorationofthedesignspace.Whenthepredictionuncertaintyinthemodelisreducedduetotheadditionalpoints,theexpectedfeasibilityfunctionimprovesthecontourbyplacingpointsnearthetargetboundary(Figure 6-6 ).Thisisreferredasexploitation.Thusbalancingtheexplorationandexploitationthissamplingmethodsuccessfullyapproximates 85
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AKrigingmodelupdatedbyEGRADOE BExpectedFeasibilityFunction Figure6-4.Krigingmodelupdatedwiththenewdesignandexpectedfeasibilityfunction.DesignupdatedbyEGRADOEisshowninRed.TotalDOE=11points Figure6-5.TargetcontourapproximatedbytheinitialkrigingmodeloftheBraninFunction.TotalDOE=12points 86
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Figure6-6.TargetcontouraccuratelyapproximatedbytheEGRAmethodology.InitialDOE=10,EGRAupdatedDOE=20 Figure6-7.Krigingsurrogatemodelconstructedusingglobaldesignofexperiments.TotalDOE=30 87
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thetargetboundarieswithfewernumberofdesignpointsthanrequiredbyaglobalsurrogatewiththesameaccuracy. Thenalkrigingmodelwith10initialdesignsand20designsupdatedbyEGRAinshowninFigure 6-6 .Itcanbeseenthereismuchlessdiscrepancybetweenthetrueresponseandtheupdatedsurrogateresponsethusdemonstratingtheefciencyofthemethodology. Thisupdatedsurrogateisfurthercomparedwithasurrogateconstructedfrom30designofexperimentsdistributedglobally.ThedesignsarealsocreatedusingLHSformulation.TheglobalsurrogateshowninFigure 6-7 isgrosslyinaccurateincomparisonwithtrueresponsefunction.WiththesamenumberofpointsEGRAefcientlyprovidedanaccurateapproximationoftheresponse. ThismethodwasincorporatedtoestimatethedeterministicoptimumandprobabilisticoptimumoftheITPSpanel.TheITPSoptimizationproblemseekspolynomialsurrogatesasitinvolveshighdimensions.ThisresearchworkincludedpolynomialresponseapproximationintheexistingEGRAalgorithm. 6.3Summary BoundariesofcomplexfunctionscanbeapproximatedefcientlyusingEfcientGlobalReliabilityAnalysis.ItwasshownthatatargetboundaryoftheBranin-Hoofunctionwasaccuratelyapproximatedusing10initialLHSdesignsand20additionalpointsacquiredusingEGRA.Theimprovementcriterionbalancesbetweenexploitingregionsofthedesignspace(wheregoodsolutionshavebeendiscovered)andexploringregionsthathavenotbeenwellsampled(andthushavegreateruncertainty)tosuccessfullyapproximatetargetregions.EGRAreducesthecomputationalcostwhendealingwithevaluationofresponsefunctionthroughexpensiveniteelementanalyses.Itefcientlyfacilitatesaccurateoptimumbyaccuratelyapproximatingthetargetboundariesofconstraintswhileminimizingthecomputationalcostinvolved. 88
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FormoreexamplesanddetaileddescriptionofvariousapplicationsoftheEGRAmethodology,thereadersarereferredtoBichonetal.[ 59 86 14 ]. 89
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CHAPTER7DETERMINISTICANDRELIABILITYBASEDOPTIMIZATION ThischapterpresentsthedeterministicandprobabilisticoptimizationprocessoftheITPSpanel.Thecomputationalexpensesassociatedwiththehighdelityniteelementanalyseswasreplacedbyefcientsurrogatemodels.FurthertheaccuracyofthemodelsattargetregionswereimprovedusingEfcientGlobalReliabilityAnalysis. Asurrogateofthefailureresponsewasconstructedwithfewdesignpoints(eg.numberofdesignsequaltonumberofcoefcientswhenthesurrogatewasapolynomialapproximation).Withatargetlevelgiven,EfcientGlobalReliabilityAnalysis(EGRA)usedtheexistingresponseapproximationandtheexpectedfeasibilityfunctiontoidentifythenewdesignpointthatimprovedtheconstraintboundary(temperature,stressandbucklingloadindeterministicoptimizationandreliabilityindexconstraintintheprobabilisticoptimization).Further,eachdesigngivenbyEGRAwereinputtotheniteelementanalysesandtheresponseswereevaluated.Thiswasaddedtotheexistingdesignofexperimentstoconstructanupdatedsurrogate.Thenewdesignswereaddedsequentiallyandthesurrogatemodelwasupdatedlikewise.Theupdatedsurrogatemodelimprovedtheboundaryofthetargetconstraint.Thetotalnumberofnewdesignsrequiredtoapproximateatargetboundarygenerallydependsonthecomplexityofthefunctionandtheinitialresponsesurface.Thismethodologyimprovedtargetboundariesoftheconstraintsbothindeterministicandprobabilisticoptimization. FurtherthischapterpresentssystemreliabilityofITPSfoundthroughseparableMonteCarlomethodandtheerrorintheindividualprobabilitiesoffailureestimatedusingthebootstrappingmethod. 7.1DeterministicOptimization AsarststepinoptimizingthedesignoftheITPS,adeterministicoptimizationofthepanelisdesired.Thisresultedinanappropriatenarroweddowndesignspaceandprovedtobeausefulpointofreferencefortheprobabilisticoptimization.Furtheritwould 90
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beinterestingtocomparetheminimizedstructuralmassbetweenthedeterministicandtheprobabilisticoptimum.Thedesignvariables-thegeometryparameters,tW,tB,tTandh(Figure 3-2 )wereoptimizedforminimummassoftheITPSunitcell.Theformulationoftheoptimizationproblemis Minimizem=f(d)=f(tW;tB;tT;h)suchthatTmax;BFSallowLBdUB(7) wheremisthestructuralmass,Tmax;BFSisthemaximumbottomfacesheettemperature,=1,2,12arethethermalstressesintheITPScomponentsandisthethermalbucklingload.Thefailurethresholds,theallowablevaluesandthesafetyfactorsforeachconstraintisgiveninTable 7-1 .Thesafetyfactorsandmarginscanbeconsideredastheriskallocatedforeachmodeoffailure. Table7-1.Failure,safetyfactorsappliedonconstraints ConstraintsFailureSafetyFactorAllowabletargetconstraints Tallow550K77K(safetymargin)473KWrapstrengthS1T323MPa1.5215.3MPaBucklingloadallow1.01.51.5 Asmentionedinsection 3.2.2 ,sincepressureloadsdonotcausefailure,theITPSisanalyzedonlyfortemperatureloads.Itwasfoundthatofallthecomponents,thewrapstresseswerecriticaland1;Wofthewrapcausedfailureofthepanel.Henceonlythewrapstressesareincludedasconstraintsintheoptimizationofthepanel.TheboundsofthegeometricdesignvariablesfordeterministicoptimizationaregiveninTable 7-2 91
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Table7-2.Lowerandupperboundsofdesignvariablesfordeterministicoptimization DesignVariableLowerBound(LB)UpperBound(UB) Wrapthickness,tWmm0.41.0Bottomfacesheetthickness,tBmm2.06.0Topfacesheetthickness,tTmm1.03.0Heightofthefoam,hmm25.040.0 Withtheresponseapproximationsconstructedforeachfailureconstraint,theseapproximationswereusedalongwithMATLABoptimizerfmincon()toevaluatetheconstraintsatvariousiterationsfordeterminingtheoptimaldesignoftheITPS. Traditionallylargenumberofdesignswouldbeusedforconstructingglobalresponsesurfaceoftheconstraints.Thenumberofdesignswouldgenerallydependonthenumberofdesignvariables,thedesiredpolynomialapproximationandthenumberofsimulationsaffordable.Forexample,consideringasecondorderpolynomialapproximation,fora4designvariableproblem,thereare15polynomialcoefcientsandaminimumof30designswouldberequired.Generallytoensureaccuracyofthesurrogate,morethan30simulationscouldbeconsidered. Theoptimumislikelytofallonsomeoftheconstraintboundaries.Insteadofusingalargenumberofdesignofexperiments(DOE)toconstructglobalresponseapproximations,theavailablecomputationalresourcesweretargetedtowardsregionsinthedesignspacetoapproximatetheconstraintboundariesaccurately.Thiswasaccomplishedbyupdatingthesurrogatesusingadaptivesamplingtechnique,EfcientGlobalReliabilityAnalysis. EGRAfacilitatedestimationofanaccurateoptimumwhilereducingthecomputationalcostassociatedwiththeoptimizationprocess.SincethepolynomialsurrogatesweremoresuitableforthehighdimensionalITPSproblem,polynomialresponsesurfaceswereincorporatedintheEGRAalgorithminthesurrogatetoolbox.TheoptimumdeterminedusingtheEGRAupdatedsurrogatesiscomparedwiththeoptimumdeterminedusingglobalsurrogates.ItwasfoundthatusingEGRAupdatedsurrogates,theaccurateoptimumcouldbeobtainedwithone-thirdnumberofdesignpointsrequired 92
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tobuildglobalsurrogates.Furthermoreitwasfoundthatevenwithlargenumberofdesigns,theoptimumevaluatedusingglobalsurrogateswasinaccurate. Further,itwasdesiredtopredictthesystemreliabilityofthedeterministicoptimum.Thiswouldserveasaconstraintfortheprobabilisticoptimization.Topredictthesystemreliability,uncertaintymodelinganduncertaintyanalysiswascarriedout.Theuncertaintyintheresponsesduetotheuncertaintyininputrandomvariableswouldaidinselectingappropriatesamplesizefortherandomvariablesleadingtoaccuratereliabilityestimation. 7.2UncertaintyModeling ThealeatoryvariabilityconsideredintheITPSdesignaregeometry,materialproperties(Young'smodulus,Poisson'sratio,shearmodulus,thermalexpansioncoefcient,thermalconductivityandspecicheat)andloads.Theepistemicuncertaintiesincludedaretheerrorinmodelingandsimulationoftheniteelementanalyses. EachcomponentoftheITPSismadeofcompositematerial(Table 3-2 )increasingthenumberuncertaintiesdrastically.ForthecompositeITPSpanel,atotalof33uncertainmaterialpropertiesvariablesareconsidered.ThevariabilityintheinputparametersandthecapacityvariablesareshowninTables 7-3 and 7-4 .Inadditiontothevariabilityintheinputparameters,errorinmodelingandsimulationisalsointroducedintheresponsestoestimatetheprobabilityoffailure.Theerrorinheattransfereth,staticstressestrandthermalbucklinganalysisebkwasassumedtobe3.0%,uniformlydistributed. Table7-3.CoefcientofvariationofinputrandomvariablesincludedintheITPSdesign RandomNo.ofUncertaintyProbabilityVariablesvariables%distribution WrapthicknesstW16.25NormalBFSthicknesstB16.25NormalTFSthicknesstT16.25NormalHeightoffoamh13NormalMaterialProperties335NormalLoad,heatuxq1;q2215Normal 93
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Table7-4.Nominalallowablevaluesofcapacityandcoefcientofvariation CapacityNominalvalueCoefcientofTypeofvaluevariation%distribution WrapStrengthS1T323MPa3%NormalAllowableTmax;BFS55015KUniformAllowable1.0-ThetotalnumberofinputuncertainvariablesintheITPSdesignis39.Thevariablesareheatuxloads:q1,q2,geometry:tW;tB;tT;h,thermalpropertiesofthecomponents:thermalconductivitykandspecicheatCpofWrap,materialpropertiesofBFS:E1;E2;v12;nu23;G12;G23;1;2,foam:E;v;,TFS:E1;E3;v12;nu13;G12;G13;andWrap:E1;E3;v12;nu13;G12;G13;.Althoughthereare39uncertaintiesinthedesignofITPS,theresponseswouldbenotsensitivetoalltheuncertainvariables.Henceasimplesensitivityanalyseswasperformedusinglinearpolynomialsurrogates.ApolynomialsurrogateformaximumBFStemperature,stressandbucklingwasconstructedasfunctionofthe39variablestondthemostsensitivevariables.FurtherthemostsensitivevariableswereusedtoestimatesystemreliabilityoftheITPSpanelusinglinearsurrogatesofthecorrespondingresponses. 7.3EstimationofSystemReliability Thesystemreliabilityisthecombinedprobabilityoffailureduetotheaforementionedfailureconstraints(temperature,stressandbuckling).Itisaknownfactthatlargesamplesofresponsesandcapacityarerequiredtoestimatetheprobabilityoffailureaccurately.ThiswasaccomplishedbyconstructingresponsesurfaceswithlimitednumberofFEsimulations.Thuslargenumberofresponsesampleswereevaluatedwithminimumcomputationalexpense.Failureisdenedbythelimitstatefunctionthatseparatesthefeasibledesignsfromtheinfeasibleones.Limitstatefunctionforstresses,thermalandbucklingfailureareshowninEquation 7 .Thelimitstatesinthisproblemaredenedasdifferencebetweentherandomresponseandrandomcapacity(Equation 2 ). 94
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Gth=Tmax;BFS)]TJ /F3 11.955 Tf 11.96 0 Td[(TallowGstr=11;W)]TJ /F3 11.955 Tf 11.96 0 Td[(S1T;WGbk=allow)]TJ /F3 11.955 Tf 11.95 0 Td[((7) Whenevertheresponseexceedsthecapacitythepanelisassumedtohavefailed.Systemreliabilityofthepaneliscomputedwhenthepanelfailsineitherofthemodes.Sincethecapacitiesareindependentoftheresponses,andfollowagivenstatisticaldistribution,thecapacitiesandresponsescouldhavedifferentsamplesizeandallpossiblecombinationscouldbeconsidered.ConsideringNsamplesofresponseandMsamplesofcapacity,theprobabilityoffailurepfcouldbeestimatedusingseparableMonteCarlomethodas ^pth=1 NMMXj=1NXi=1I[Gth(Cj;Ri)>0]^pstr=1 NMMXj=1NXi=1I[Gstr(Cj;Ri)>0]^pbk=1 NNXi=1I[Gstr(C;Ri)>0](7) Thecapacitiesforthermalandstressfailuresfollowadistribution(Table 7-4 ),resultinginNMcomparisonstoestimateprobabilityoffailure.Thecapacityforbucklingloadfactorisdeterministic(allow=1.0)andhenceresultedinNcomparisons.Thissuggestedthatthebucklingprobabilityoffailure^pbkshouldbecalculatedwithlargesamplesizesofbucklingloadresponses. Furthertheniteelementmodelingerrorwasintegratedtotheresponseforeachcaseas Gth=Tmax;BFS(1+eth))]TJ /F3 11.955 Tf 11.95 0 Td[(TallowGstr=(1+estr))]TJ /F3 11.955 Tf 11.95 0 Td[(SGbk=allow)]TJ /F3 11.955 Tf 11.95 0 Td[((1+ebk)(7) 95
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Sincetheerrorsarealsoindependentoftheresponse,theycouldbeassignedadifferentsamplesize(Q).Foreacherror,theNsamplesoftheresponseandMsamplesofthecapacityaresampledtoestimateasinglevalueofprobabilityoffailure.QerrorvalueswouldprovideQprobabilityoffailureestimates.ThemeanoftheQestimatesisconsideredasthemeanprobabilityoffailureforeachfailuremode.Thesumoftheprobabilitiesoffailureofthethreemodesisconsideredasthesystemreliabilityofthepanel. Furthertheerrorintheindividualprobabilitesoffailurewasestimatedusingthebootstrappingtechnique,describedindetailinSection 5.3 .Furthertheerrorsestimatedwouldbecomparedwiththeerrorsestimatedfromnumerousrepetitionstovalidatethebootstrappingmethod. 7.4ReliabilityBasedOptimization Thereliabilitybasedoptimizationinvolvesoneadditionalconstraint,theprobabilityoffailureconstraint.Theprobabilityoffailureofthedeterministicoptimumactsastheconstraintfortheprobabilisticoptimization.TheoptimizationproblemoftheITPSpanelisformulatedas Minimizem=f(d)=f(tW;tB;tT;h)suchthat(g(d;X)Cr(7) whereiscalculatedusingEquation 2 .Itdependsonthefailureconstraintsg(d;X)evaluatedasafunctionofdesignandrandomvariables.Cristhedesiredlevelofreliabilityfortheoptimum.ThisvaluewasassignedthedeterministicreliabilityindextoanalyzehowthefailureconstraintsdifferfromthedeterministicoptimizationandwhetherthishasaidedinfurtherreducingorincreasingthemassoftheITPSpanel.Furtherthisanalysisontheriskallocationmightsuggestwhatsafetyfactorscouldbesetforthedeterministicoptimizationthatwouldfurtherminimizethemassleadingtononconservative,safeoptimumdesign. 96
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Thereliabilitybasedoptimizationrequiredcalculationofreliabilityindexinthedesignvariablespace.Hencearesponsesurfaceofthereliabilityindexasafunctiondesignvariableswasconstructed.Inthedesignvariablespace,ateachdesignsetd,probabilityoffailureandhencethereliabilityindexwascomputed.Thisinvolvedconstructionoffailureresponseapproximationsateachdesignpoint.Usingtheresponsesurfaces,therandomvariablesweresampledaccordingtotheirrespectiveprobabilitydistributiontoestimatetheprobabilityoffailure.Henceateachdesignpoint,adesignofexperimentswascreatedintherandomvariablespace,failureresponsesurfaceswereconstructedandsamplesoftherandomvariablesweregeneratedtoestimatetheprobabilityoffailure.Thisprocesswascarriedoutatdesignpointsinthedesignvariablespacetoobtainareliabilityindexresponsesurfacemakingthisprocesscomputationallyintractable. Sincetheprobabilisticoptimizationinvolvessatisfyingthereliabilityconstraint,EfcientGlobalReliabilityAnalysiswasusedtoimprovetheaccuracyofthesurrogateatthedesiredreliabilitylevelwithfewernumberofdesignpoints.Initiallyasecondorderpolynomialsurfaceofthereliabilityindexwasconstructed.HoweverafteraddingadditionaldesignpointsusingEGRA,thetargetboundarywasinaccurate.Itwasfoundthatthereliabilityindexresponsewashighlynon-linearandsecondorderpolynomialwasapoorapproximationoftheresponse.Theorderofthepolynomialwaslimitedbytheavailablecomputationalresources.ThusthereliabilityindexwasttedusingkrigingsurrogatemodelwhichapproximatedtheresponseandthenEGRAwasusedtoimprovetheboundaryofdesiredreliabilitylevel.ThispreventedwastingoflargeofFEsimulationsatunimportantregionsandfocusthecomputationalresourcesinapproximatingthedesiredreliabilitylevelaccuratelytodeterminetheaccurateprobabilisticoptimum. 97
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7.5ResultsandDiscussion 7.5.1DeterministicOptimization ThedesignvariablesforthedeterministicoptimizationwerethedimensionsoftheITPS.Thepanelwasoptimizedforminimummasssatisfyingthermal,structuralandbucklingconstraints.ThemaximumBFStemperatureandthermalbucklingloadfactorwerethecriticalresponsesconsideredasfailureconstraints.Bucklinganalysesoftheunitcellshowedthattopfacesheetwasthecomponentthatbuckledleadingtobucklingofthewebs.ThermalstressanalysisoftheITPSunitcellshowedthatthewrapstresseswerecriticalandwrapstress1;Wwasalsooneoftheactiveconstraintsintheoptimizationprocess. Astudyonthevariationoftheresponseswithrespecttothedesignvariableswasperformedtondtheorderofthepolynomialthatwouldcloselyapproximatetheresponsewiththeresultsfromtheniteelementanalyses.Finiteelementsimulationswereperformedwhereoneofthedesignvariableswasvariedbetweentheirrespectiveranges(Table 7-2 )andotherswereheldconstant(tW=0.4tB=2.5,tT=2.0andh=32.5mm).Thevariationofthebucklingloadfactorincreasestremendouslywiththeincreaseinthethicknessoftopfacesheetwhilethevariationisnotpronouncedwiththeincreaseinwrapthickness.Thisconrmsthatbucklingofthetopfacesheet(Figure 7-3 )leadstobucklingofthetopweb(Figure 3-7 ). WiththevariationoftheresponsesshowninFigures 7-1 7-2 7-3 and 7-4 ,secondorderpolynomialresponsesurfacesoftheconstraintswereconstructed. Thedesignpointsfortheresponsesurfacegenerationwerecreatedusingthelatinhypercubesamplingwithinthebounds.Asstatedearlier,foraquadraticresponsewith4designvariables,thenumberofpolynomialcoefcientsare15.Asaruleofthumb,designpointsmorethantwicethenumberofcoefcientsareconsidered.Inthisproblematotalof60designpointswereinputtotheniteelementanalyses,togeneratetheresponsesurfaces.Thisrequired180niteelementanalysesfor3responses(transient 98
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Figure7-1.VariationofresponseswrtthicknessofthewraptW.(tB,tTandhwereheldconstant) heattransfer,thermalstressandthermalbucklinganalyses).ThetimetakentoperformallthethreeFEsimulationsandextractallthedesiredresponseatasingledesignpointis8.5minutessummingto8.5hoursfor60designpoints.Thisisassumedastheavailablecomputationalbudgettondthedeterministicoptimum. Insteadofusing60designstobuildaglobalresponsesurface,thecomputationalresourceswereusedtoapproximatetheconstraintboundariesaccuratelytoestimateanaccurateoptimum.Itwasfoundthatallthefollowingconstraints,temperature,stress1andbucklingloadwereactiveintheoptimization.Hencetoimprovetheaccuracyofeachactiveconstraint,EfcientGlobalReliabilityAnalyseswasapplied.EGRAbalancesbothexplorationandexploitationofthedesignspacetoadddesignpointsthatimprovedtheaccuracyofthetargetboundary.Aninitialresponsesurfacewasconstructedusing16LHSdesignpointsandsixadditionalpointswereacquiredforeachconstraintusingEGRA.Sincetemperaturegradientactsastheloadsforthermalstressandbuckling 99
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Figure7-2.VariationofcriticalresponseswrtthicknessofthebottomfacesheettB.(tW,tTandhwereheldconstant) analyses,foreachstressorbucklinganalysis,atransientheattransferanalysishadtobeperformed.ThedesignsupdatedbyEGRAandthecorrespondingresponsesareprovidedinAppendix C .Thetrueresponsesattheadditionaldesignsareveryclosetothevalueofthetargetconstraint. Adesignpointonthetargetboundarywasevaluatedwiththeupdatedsurrogate.Whenthedifferencebetweenthesurrogatepredictionandthetrueresponsewas10)]TJ /F4 7.97 Tf 6.59 0 Td[(3,EGRAsamplingwasstopped.Whenthisprodedurewasfollowed,Itwasfoundthat4pointsupdatedbyEGRA,togetherwiththeinitial16pointsapproximatedthetargetboundaryaccuratelyyieldingaccurateoptimum.Thusthesimulationsrequiredwereonly68initialFEsimulationscomparedto180FEsimulations(globalsurrogate)andthecomputationaltimewas3hoursand15minutes.Thecomputationaltimewasreducedto40%whencomparedtothatofglobalsurrogates. 100
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Figure7-3.VariationofcriticalresponseswrtthicknessofthetopfacesheettT.(tW,tBandhwereheldconstant) Furthera20DOEglobalresponsewasusedforoptimizationtoprovideareasonablecomparisonwiththesamenumberofDOE.Theoptimumdesignvariablesd,theminimizedmassestimatedusing60DOE,20DOEand16+4EGRADOEsurrogatesaretabulatedinTable 7-5 Table7-5.OptimumdesignvariablesandminimizedstructuralmassthroughglobalandEGRADOEfordeterministicoptimization DOEtWmmtBmmtTmmhmmMasskg/m2 60DOE0.42.841.4132.321.420DOE0.42.871.3832.721.816+4EGRA0.42.851.3731.721.3 Theoptimizationthroughthesurrogatesyieldedoptimawithalltheconstraintsbeingactive.Further,theconstraintsevaluatedusingthesurrogatesarecomparedwithniteelementanalysestoassesstheaccuracyofthesurrogateattheirrespectiveoptimuminTable 7-6 .The60DOEoptimumwasapparentlynottheoptimumasthe 101
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Figure7-4.Variationofcriticalresponseswrtheightofthefoamh.(tW,tTandtBwereheldconstant) constraintsdeterminedbythesurrogatedifferfromtheniteelementanalyses.Theniteelementanalysesareconsideredastruevalueshere.Thus,itcanbeseenthatthoughtheglobalDOEhas60designpoints,itdoesnotapproximatetheconstraintboundaryaccuratelyleadingtoerroneousresults.Forexample,thoughtheTmax;BFSglobalsurrogatepredicted473Katthe(60DOE)optimum,thetrueresponseis469.6K.Evenwithlargenumberofdesigns,thediscrepancyisabout4K.HoweverthesurrogateconstructedusingthesequentialsamplingofEGRAhasimprovedthetargetboundary(472.9K)topredictaccurateoptimum.TheconstraintvaluespredictedbytheEGRAimprovedsurrogateagreeswellwiththetruevaluesoftheniteelementanalyseswithlessthanapercenterror.Theoptimumdeterminedfromthe20DOEsurrogateprovedthatasurrogatewith20globaldesignsisgoingtobegrosslyinaccuratewhereasthe16+4=20DOEconstructedusingEGRAwashighlyaccurate.Thisshowsthat 102
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whenthecomputationalresourcesareefcientlydirectedtowardstargetregions,thecomputationalcostcanbereducedto40%oftheavailablecomputationalbudget. Table7-6.ComparisonbetweentheaccuracyofglobalsurrogateandEGRAsurrogateatthedeterministicoptimumusingcorrespondingsurrogates.Thepredictedstandarddeviationisgivenin() SurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE473(0.63)1.5(0.05)215.3(0.65)469.61.54211.320DOE473(3.4)1.5(0.78)215.3(1.6)467.51.77217.316+4EGRA473(9.1)1.5(2.5)215.3(2.4)472.91.51214.6 Threeoptimaweredeterminedusing60DOE,20DOEandEGRAupdatedsurrogates.Theresponsesatthethreeoptimawereevaluatedusingallthesurrogatestoanalyzehowthesurrogatesperformedattheotheroptima.Table 7-7 showstheresponsesofalltheoptimaevaluatedusingthe60DOEglobalsurrogate.Amaximumdifferenceof8%wasobservedinthebucklingloadofthe20DOEoptimum.Itwasfoundthattheerrorbetweentheresponseswerelessthan8%,howeverthecomputationaltimerequiredfortheanalyseswas60%morethanthatofEGRA. Table7-7.Comparisonofresponsesatallthedeterministicoptimausingthe60DOEsurrogate OptimaSurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE4731.5215.3469.61.54211.320DOE470.91.62214.8467.51.77217.316+4EGRA475.71.45212.7472.91.51214.5 Inasimilarfashion,thethreeoptimawereevaluatedusingthe20DOEsurrogateandtabulatedinTable 7-8 .Thebucklingloadsofthe60DOEandEGRAoptimahadanerrorof13%whencomparedtothetrueresponse.ThevalueofTmax;BFSoftheEGRAoptimumdifferedby5K.Althoughbypercentitisalittleover1%error,thisisthelargestdifferencebetweenpredictionandtrueresponseusinganysurrogate.Tosumup,Itwasfoundthat20DOEsurrogatewasinaccurateattheotheroptimaaswell. 103
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Table7-8.Comparisonofresponsesatallthedeterministicoptimausingthe20DOEsurrogate OptimaSurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE474.21.35214.7469.61.54211.320DOE4731.5215.3467.51.77217.316+4EGRA478.11.31210.3472.91.51214.5 TheresponsesofthethreeoptimaevaluatedusingtheEGRAupdatedsurrogatealongwiththetrueresponsesisshowninTable 7-9 .ItwasfoundthatEGRAupdatedsurrogatepredictedresponsesat60DOEwithlessthan4%error.Howeveratthe20DOEoptima,therewas16%errorinthebucklingloadresponses.ThiscouldattributedtothesignicantdifferenceintheheightofthefoambetweentheEGRAand20DOEoptimawhichisawayfromthetargetboundary.EGRAupdatedsurrogateseemstohaveminimalerroratalltheoptimawhencomparedtoothersurrogatesexceptforthebucklingresponseofthe20DOEoptimum.Alsothewrapstresseswerepredictedwithmaximumof3%errorbyallthesurrogates.Thus,whenanalysedonthebasisoncomputationaltimerequiredandaccuracyofthesurrogate,EGRAupdatedsurrogatesseemstoperformbetterthan60and20DOEglobalsurrogates. Table7-9.ComparisonofresponsesatallthedeterministicoptimausingtheEGRAupdatedsurrogate OptimaSurrogateFiniteElementAnalysesTmax;BFSBucklingWrapStressTmax;BFSBucklingWrapStressKload1MPaKload1MPa 60DOE470.61.56213.4469.61.54211.320DOE465.91.46215.6467.51.77217.316+4EGRA4731.5215.3472.91.51214.5 Toestimatethereliabilityoftheoptimumdesign,uncertaintymodelingandanalysisofthefailureresponsesisrequiredwhichisdiscussedinthenextsection. 7.5.2SensitivityAnalysis Thetotalnumberofuncertaintiesconsideredinthedesignwere39inputrandomvariables(Table 7-3 ).Tostudytheuncertaintypropagationandestimatereliability, 104
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responsesurfaceapproximationoftheoutputparametersasafunctionoftheinput39uncertaintieswasconstructed.Aminimumof40designsarerequiredtoconstructalinearresponsesurface(numberofcoefcients=40).Sinceallthevariableswouldnotbesensitivetotheresponses,sensitivityanalysiswasdesiredtostudythesensitivityoftherandomvariablestowardsthefailureresponses.Thiswouldaidinndingthemostsensitivevariablesandthusreducingthenumberofniteelementsimulationsrequiredtoconstructresponsesurfaces.Designpointstwicethenumberofcoefcientswerecreatedtoanalyzethesensitivevariables.Thecoefcientsofthelinearresponsewereusedasweightstondthemostsensitivevariablestotheresponses,maximumBFStemperature,WrapStressandbucklingloadfactor. AstheBFStemperaturedoesnotdependonthethermo-mechanicalpropertiesofthecomponents,therandomvariablesconsideredareheatuxloads,geometryandthermalpropertiesofthevariouscomponentsoftheITPS.Alinearpolynomialwastwith30points(15coefcients)wasconstructedasfunctionofthe14randomvariables.Figure 7-5 showsthelinearcoefcientsoftherandomvariablesasabarplot.Thehigherthemagnitudeofthecoefcient,thehigherthesensitivity.TheBFStemperaturewashighlysensitivetoq1andq2,tW,handkandCpofthewrap.Inanefforttoreducethenumberofvariablesfurtherq2wascalculatedasConstant+q2.Thetotalof5inputrandomvariablesweresensitivetoBFStemperature(q1,tW,h,kandCp). Inthecaseofwrapstressandbucklingload,inadditiontotheloads,geometryandthermo-mechanicalproperties,thethermalconductivityandspecicheatofthewrapwasalsoconsideredastemperaturegradientsactasloadforthethermalstressandbucklinganalysis.Withatotalof33uncertainvariables,66LHSdesignsweregeneratedandlinearresponseforwrapstress11andbucklingloadwereconstructed. Figure 7-6 and 7-7 showstheproductofuncertaintyandcoefcientofeachrandomvariableasbarplot.Fromthemagnitudeofthecoefcients,themostsensitivevariableswerechosenforeachresponse.Thesensitivevariablesofwrapstresswereq1,tW,tB, 105
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Figure7-5.SensitivityanalysisofmaximumBFStemperature,OrderofthevariableskandCpofWrap,BFS,TFS,Foam tT,E1,ofTFS,E1,ofthewrap.Furtherthesensitivevariablesofbucklingloadwereq1,tW,tT,h,ofTFS,ofwrap. Inordertomakesurethatsomeofthesensitivevariablesarenotneglectedintheprocess,coefcientofvariationoftheresponseasfunctionsoftheinitialnumberofuncertaintiesandreducednumberofuncertaintiesarecalculatedfromtheconstructedpolynomialequation.Iffistheresponse,theuncertaintyinresponseduetouncertaintyininputvariablesx1;x2;::;xiiscalculatedas f=s X@f @xi22xi(7) Table7-10.Differenceinuncertaintyinresponsesbeforeandaftersensitivityanalysis ResponseDifferenceinuncertainty Tmax;BFS4%Wrapstress117%Bucklingload1% 106
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Figure7-6.SensitivityanalysisofWrapstress11.Orderofthevariables,heatuxloads:q1,q2,geometry:tW;tB;tT;h,materialpropertiesofBFS:E1;E2;v12;v23;G12;G23;1;2,foam:E;v;,TFS:E1;E3;v12;v13;G12;G13;andWrap:E1;E3;v12;v13;G12;G13;;k;Cp ThedifferenceinuncertaintyintheresponseduetoinitialnumberofuncertainvariablesandreducednumberisprovidedinTable 7-10 .Themaximumdifferenceis7%variationinstressresponse.ThisisanacceptabledifferenceasasignicantnumberofexpensiveFEsimulationsaresavedasresultofthesensitivityanalysis.Thusthetotalnumberofsensitivevariablesreducedfrom39to11cumulatively. 7.5.3ReliabilityoftheDeterministicOptimum Thedesignspaceformedbytheuncertainvariablesaresmallandthevariationoftheresponsesisfairlylinearinthisrandomvariablespace.Hencelinearsurrogatesoftheresponseswereconstructedasfunctionoftherandomvariables.TheuncertaintyinthesevariablesisshowninTable 7-3 .Foratotalof11randomvariables,13LHSdesignsareconsidered.Heretheresponsesare:bottomfacesheettemperature,stressesandbucklingload.Thecorrespondingcapacitiesare:allowableBFStemperature, 107
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Figure7-7.Sensitivityanalysisofbucklingloadfactor.Orderofthevariables,heatuxloads:q1,q2,geometry:tW;tB;tT;h,materialpropertiesofBFS:E1;E2;v12;v23;G12;G23;1;2,foam:E;v;,TFS:E1;E3;v12;v13;G12;G13;andWrap:E1;E3;v12;v13;G12;G13;;k;Cp strengthvaluesandallowablebucklingload.Thesesurrogateswerealsousedtodeterminetheuncertaintypropagationfromtheinputrandomvariablestotheoutputresponses.Theuncertaintyintheresponseprovidedinsightontheappropriatesamplesizestoestimateaccurateprobabilityoffailure. Table7-11.Uncertaintyinresponseduetoinputuncertainty ResponseCoefcientofVariation% Tmax;BFS3.5Wrapstress119.1Bucklingload15.1 Theuncertaintyintheresponseisestimatedfromthestandarddeviationorcoefcientofvariation(CV)of1000samples.Thisprocesswassimulatedfornumerousrepetitions(1000)tondnominalcoefcientofvariationsforeachresponse.ThemeanoftheCVestimatesisgiveninTable 7-11 .ItcanbeseenthattheuncertaintyinthebucklingloadfactorishighcomparedtotheBFStemperatureandwrapstress.This 108
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kindofsituationwasaddressedbyrearrangingtherandomvariablesinthelimitstateandassigningappropriatesamplesize.However,capacityisdeterministicandtheerrorebkwasonlyvariablethatcouldbemovedtothecapacitytoreducetheuncertainty.Howeverthisdidnothelpisreducingtheuncertainty,hencethebucklingprobabilityoffailurewascalculatedusinglargenumberofsamples(Nbk=5000). Forthedeterministicoptimum(Table 7-5 ),theprobabilityoffailurewasestimatedforthethreefailuremodesusingthecorrespondinglimitstates(Equation 7 )Randomsamplesofthematerialproperties,loadandgeometryaregeneratedusingMonteCarlosimulations.Withthegeneratedsamplesofinputrandomvariables,theoutputresponsessuchascomponentstresses,maximumBFStemperatureandbucklingloadswereevaluatedfromthecorrespondinglinearresponseapproximations.TheprobabilityoffailureforeachfailuremodeofwascalculatedusingseparableMonteCarlomethod 7 .Itwasassumedthatourcomputationalbudgetonlypermitted500samplesfortheresponse,capacityanderror(N=M=Q=500)withanexceptiononthesamplessizeforbucklingload(Nbk=5000).InseparableMonteCarlomethod,theresponsesamples(N)werecomparedagainstallthecapacitysamples(M=500)resultingin250,000evaluationsofthelimitstate.Henceforeacherrorsample,250,000evaluationsresultedinasingleprobabilityoffailure.Themeanofthe500estimatesofprobabilityoffailureisconsideredtheprobabilityoffailureduetoerrorandvariability. Table7-12.Individualprobabilitiesoffailureofthedeterministicoptimum.Thetotalprobabilityoffailure=2.7%. FailuremodeTmax;BFSWrapstress11Bucklingload probabilityoffailure^p0.00180.00260.022Response472.9K214.5MPa1.51SafetyFactor77K1.51.5 TheindividualprobabilitiesoffailurearegiveninTable 7-12 .Itisassumedthatthethreefailuremodesareapproximatelyindependentandthetotalprobabilityoffailure(systemreliability)iscalculatedasthesumofalltheprobabilitiesoffailure.For 109
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thedeterministicoptimumthesystemprobabilityoffailurewas2.7%amountingtoareliabilityindexof1.93.Themassofthedeterministicoptimumwas21.3kg=m2. Oftentimesdeterministicoptimizationyieldconservativedesignsastheyarebasedonsafetyfactors.ThusreliabilitybasedoptimizationoftheITPSpanelwasperformed. 7.5.4Reliabilitybasedoptimization Deterministicoptimizationinvolvesapplyingsafetyfactorsonfailureresponses.Thesesafetyfactorscanbeconsideredasriskallocatedtoeachconstraint.Optimizationusingsafetyfactorsoftenleadstoconservativedesigns.Reliabilitybasedoptimizationdoesnotinvolvesafetyfactors,howeverincludesareliabilityconstraintthatisafunctionoftheresponses.Sincetherenohardconstraintsontheresponses,theoptimizationprocessmightyieldnon-conservativedesigns. Reliabilitybasedoptimizationinvolvesminimizingthestructuralmassofthepanelbyconsideringalltheuncertaintiesinthedesign.Inthisproblemthepanelwasoptimizedtohaveareliabilityequaltoorgreaterthanthereliabilityofthedeterministicoptimum.Itwasalsodesiredtoanalyzetheriskallocationtothefailureinducingresponsesincomparisontothedeterministicoptimum. Table7-13.Lowerandupperboundsofdesignvariablesforprobabilisticoptimization DesignVariableLowerBound(LB)UpperBound(UB) Wrapthickness,tWmm0.41.0Bottomfacesheetthickness,tBmm2.04.0Topfacesheetthickness,tTmm1.03.0Heightofthefoam,hmm29.036.0 TheoptimizationfortheprobabilisticoptimumwasalsoperformedusingtheMATLABoptimizerfmincon()withconstraintonthedeterministicreliabilityindex1.93.Thedeterministicoptimizationandtheknowledgeoftheresponsesatthedeterministicaidedinshrinkingthedesignvariablespace(Table 7-13 ).Thispreventedsamplingatregionswithlargeprobabilitiesoffailure. Similartotheresponsesurfaceapproximationofthedeterministicconstraints,30LHSdesignofexperimentswerecreatedtobuildaasecondorderpolynomialforthe 110
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reliabilityindex.Ateachdesignpoint,therandomnessintheuncertainvariableswereintroducedanddesignpointswerecreatedintherandomvariablespace.Therandomvariablespacewassmallmakingthevariationoftheresponseslinear.Foratotalof11uncertainvariablesconsidered,linearresponsesurfacesconstructedwith13DOEwereaccurate.ThesepointswereinputintheniteelementanalysestocomputethemaximumBFStemperature,wrapstressandbucklingload.Theseresponseswereusedtoconstructalinearapproximationoftheresponses.Further,largenumberofsamplesofuncertaintiesweregeneratedandtheresponseswereevaluatedusingtheaforementionedlinearapproximation.Usingthegeneratedsamplesofresponse,capacityanderrorsamples,theprobabilityoffailurewasestimatedusingtheseparableMonteCarlomethod.Thisprocesswasrepeatedforeverypointcreatedinthedesignvariablespace.Theniteelementanalysesat13designsapproximatelytook1hourand50minutes.FurtherthetimetoestimatetheprobabilityoffailurethroughseparableMonteCarlomethodwasapproximately25minutes.Thetotalcomputationaltimerequiredtoestimateprobabilityoffailureatasingleinthedesignvariablespacewas2hourand15minutes. FurtherEGRAwasemployedtoimprovethetargetboundaryequalto1.93(2.7%pf).Foreveryadditionalpointacquired,theentireprocessmentionedabovewasrepeatedtondtheprobabilityoffailurewhichwasthenincludedtothedesignofexperimentstoupdatethesurrogate.Itwasfoundthatevenafteradding25newdesignpoints,theapproximationofthetargetboundarywasinaccuratesuggestingthatasecondorderpolynomialcouldbeapoorapproximationtobeginwith.Toincreasetheorderofthepolynomial,largernumberofinitialdesignswererequired.Forexample,toconstructa3rdorderpolynomialasfunctionof4designvariables,thenumberofcoefcientsequals35andaminimumof70designpointsarerequired.Consideringthetimerequiredtoestimatetheprobabilityoffailureofonedesign,alargerDOEwascomputationallyintractable. 111
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Thuskrigingmodelwasconsideredforthereliabilityindexresponsesurfaceasitdoesnotrequireminimumnumberofdesignstoconstructaresponsesurface.ToaffordenoughadditionaldesignsthroughEGRA,a20DOEwascreatedtobuildtheinitialresponsesurface.Furtheranadditional27designswereacquiredtoimprovethedesiredlevelofreliabilityusingEGRA.Sowithatotalof47designpoints,thecomputationaltimerequiredwasapproximately105hoursand45minutes.GlobalsurrogatesofthereliabilityindexcouldbecreatedtocomparewiththeEGRAupdatedsurrogateduetolimitedcomputationalresources.Theprobabilisticoptimizationwasperformedwiththeupdatedsurrogate.TheoptimizeddesignvariablesandtheminimizedmassoftheprobabilisticoptimumisgiveninTable 7-14 Table7-14.OptimumdesignvariablesandminimizedstructuralmassthroughEGRAupdatedDOEforreliabilitybasedoptimization DOEtWmmtBmmtTmmhmmMasskg/m2 20+27EGRA0.42.01.331.119.4 Table7-15.Individualprobabilitiesoffailureoftheprobabilisticoptimum.Thetotalprobabilityoffailurepf=2.6%. FailuremodeTmax;BFSWrapstress11Bucklingload probabilityoffailure^p0.00340.0040.019Response492K245.3MPa1.7SafetyFactor55K1.241.7 Table7-16.Individualprobabilitiesoffailureofthedeterministicoptimumusingprobabilisticoptimumsurrogate.Thetotalprobabilityoffailurepf=2.8%. FailuremodeTmax;BFSWrapstress11Bucklingload probabilityoffailure^p0.00270.0030.023Response483.2K224.6MPa1.6 Atotalof13designswerecreatedthroughLatinHypercubesamplingaroundtheoptimizeddesignandwasinputtoFiniteElementanalyses.Linearsurrogatesoftheresponseswereconstructedandtheprobabilityoffailurewasestimated.Thereliabilityindexwasestimatedas1.95whichamountedto2.6%probabilityoffailure.TheprobabilisticoptimumobtainedusingtheEGRAupdatedresponsesurfacewas 112
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accuratewithanerrorof2%inreliabilityindex.Thisamounttoa3%differenceintheprobabilityoffailure.Furtherusingtheprobabilisticoptimumresponsesurrogates(temperature,stressandbuckling),theprobabilityoffailureofthedeterministicoptimumwasestimateforthepurposeofvalidcomparison.Thetotalprobabilityoffailurewas2.8%.ThisincreaseinthepfestimatewasduetothediscrepancyintheresponseevaluationasthesomeofthedeterministictBandhwereoutoftheprobabilisticrandomvariablespace. Withthereliabilityindexsurrogate,thereliabilityindexofthedeterministicoptimumwasfoundtobe1.97withastandarddeviationof0.07.Thisamountstoaprobabilityoffailureof2.4%whilethedeterministicprobabilityoffailureestimatedthroughseparableMonteCarlowas2.7%(=1.93). Themassminimizedthroughdeterministicoptimizationwas21.3kg/m2.ThesafetyfactorsforthevariousconstraintsisshowninTable 7-12 .Fromthevalueoftheresponsesattheprobabilisticoptimum,thesafetyfactorswerecalculated,showninTable 7-16 .Theprobabilisticoptimizationprocesshasallocatedriskdifferentlythatreducesthemassfrom21.3to19.4kg/m2.Theoptimumdesignseemstohavehigherriskallocatedtotemperatureandstresswhencomparedtodeterministicsafetyfactor.However,theincreaseinthesafetyfactorofthebucklingloadfactorhasyieldedreducedmasswiththesameprobabilityoffailure.Thisdemonstratesthatreliabilitybasedoptimizationyieldsnon-conservativebutasafedesign. Further,itwasdesiredtondtheaccuracyoftheprobabilitiesoffailure.Thiswasaccomplishedbyemployingbootstrappingmethod.Thismethodwasappliedearlierforcomplexlimitstateandwasvalidatedbyestimatingtheerrorthroughlargenumberofrepetitionsinprobabilityoffailureestimationprocess. 7.5.5Errorintheprobabilityoffailureestimate Forasimplelimitstate(asinEquation 2 ),theaccuracyofseparableMonteCarlocanbeestimatedasderivedbySmarsloketal.[ 16 ].Inthecaseagenerallimitstate 113
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(Equation 2 ),bootstrappingwasproposedtoassesstheaccuracyoftheprobabilityoffailure. Bootstrappingmethodismoresuitablewhencomputationoftheresponseinthelimitstateisexpensive.Moreover,thismethodcanbeemployedincaseoflowprobabilitiesoffailureoftheorder10)]TJ /F4 7.97 Tf 6.59 0 Td[(6-10)]TJ /F4 7.97 Tf 6.59 0 Td[(8.SincetheITPSprobleminvolvessimplelimitstates,surrogatesareusedtoevaluateresponsesfortheprobabilityoffailurecalculation,andthefacttheprobabilityoffailureisashighas3%,bootstrappingisnotrequiredtoestimatetheaccuracyoftheprobabilityoffailure.However,themethodisemployedinthisproblemtoshowthatthemethodworksforsuchsituationstoo. Theaccuracyoftheprobabilityoffailureestimatewascalculatedwithab=1,000bootstraprepetitions.Randomsamplesofresponseswerebootstrapped,butthecapacityanderrorsweresampledforeverybootstraprepetition.Thatis,ineachofthe1,000bootstraprepetitions,theresponseswerere-sampledfromthesame500samples,whilethecapacityanderrorshadafreshsampleeverytime. Table7-17.Errorintheprobabilitiesoffailureofthedeterministicoptimumusingbootstrappingandrepetitions.Bootstrapsamples=1000,Repetitions=1000 RepetitionsBootstrappingProbabilityoffailuremean(pf)std(pf)mean(^pboot)std(^pboot) ^pth0.0018510)]TJ /F4 7.97 Tf 6.59 0 Td[(40.0019610)]TJ /F4 7.97 Tf 6.59 0 Td[(4^pstr0.0026110)]TJ /F4 7.97 Tf 6.59 0 Td[(30.0026910)]TJ /F4 7.97 Tf 6.59 0 Td[(4^pbk0.022710)]TJ /F4 7.97 Tf 6.59 0 Td[(30.023810)]TJ /F4 7.97 Tf 6.59 0 Td[(3 Table7-18.Errorintheprobabilitiesoffailureoftheprobabilisticoptimumusingbootstrappingandrepetitions.Bootstrapsamples=1000,Repetitions=1000 RepetitionsBootstrappingProbabilityoffailuremean(pf)std(pf)mean(^pboot)std(^pboot) ^pth0.0034810)]TJ /F4 7.97 Tf 6.59 0 Td[(40.003710)]TJ /F4 7.97 Tf 6.59 0 Td[(4^pstr0.004110)]TJ /F4 7.97 Tf 6.59 0 Td[(30.003110)]TJ /F4 7.97 Tf 6.59 0 Td[(3^pbk0.019710)]TJ /F4 7.97 Tf 6.59 0 Td[(30.021910)]TJ /F4 7.97 Tf 6.59 0 Td[(3 TheaccuracyoftheseparableMonteCarlomethodwasalsoassessedbyanempiricalstandarddeviationobtainedbyperformingn=1000repetitions.Themeanandthestandarddeviationoftheprobabilitiesoffailureestimatedthroughrepetitionsand 114
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bootstrappingaretabulatedinTable 7-17 forthedeterministicoptimumandTable 7-18 fortheprobabilisticoptimum.Theerrorintheprobabilityoffailureisapproximately3timeslessthanthemeanestimates.Themeanprobabilitiesoffailureoftheprobabilisticoptimumdifferby7%forbucklingloadwhileotherpfestimatesagreewell.Thebootstrappingmethodiscomparablewiththeempiricalestimatesdeterminedthroughrepetitionswithmaximumdiscrepancyinthebucklingprobabilitiesoffailure.Themajordifferenceisintheerrorestimatesfor^pbkwhichwas14%and21%fordeterministicandprobabilisticoptimumrespectively.Thiscanbeattributedtohighuncertaintyinthebucklingloads(15%). 7.5.6Summary ThreecriticalresponsescausingfailureoftheITPSpanelwasincludedasconstraintsintheoptimizationprocesstodeterminethedeterministicandprobabilisticoptimumoftheITPSpanel.Surrogatemodelsefcientlyreplacedcomputationallyexpensiveniteelementanalyses.Inadditiontothis,theevaluationofdeterministicconstraintsusingEGRAupdatedsurrogatesreducedthecomputationalrequiredto40%oftheavailablecomputationalbudget.Itwasalsofoundthatglobalsurrogateswithlargenumberofdesignsdonotalwaysprovidetheaccurateoptimum.EGRAwasalsosuccessfullyusedtoupdatereliabilityindexsurrogatesforreliabilitybasedoptimization. ThesystemreliabilityofthepanelwasestimatedusingMonteCarlosimulations.ThestatisticalindependenceoftherandomvariablesallowedSeparableMonteCarlomethodtobeemployedtoestimatetheprobabilityoffailure.Thesafetyfactorbaseddeterministicoptimumyieldedanoptimaldesignwithstructuralmassof21.3kg/m2with2.7%probabilityoffailure.Furthermassoftheprobabilisticoptimumwas19.4kg/m2withdeterministicprobabilityoffailure.Reliabilitybasedoptimizationdidnotinvolvehardconstraints(safetyfactors)thusdemonstratingthatriskallocationplaysanimportantroleinndinganon-conservativebutsafedesign.Bootstrappingprovidedreasonableestimatesofuncertaintyintheprobabilityoffailureestimates. 115
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CHAPTER8CONCLUSIONS 8.1Conclusion ThehighcomputationalcostofthedeterministicandprobabilisticoptimizationoftheITPSpanelwasefcientlyaddressedusingvarioustechniquesatdifferentphasesoftheoptimizationprocess.Toreducethecomputationalexpensesinvolvedinthestructuralanalysis,niteelementbasedhomogenizationmethodwasemployed,homogenizingthe3DITPSmodeltoa2Dorthotropicplate.HoweveritwasfoundthathomogenizationwasapplicableonlyforpanelsthataremuchlargerthanthecharacteristicdimensionsoftherepeatingunitcellintheITPSpanel.Henceasingleunitcellwasusedfortheoptimizationprocesstoreducethecomputationalcost.Thefailureconstraintsinvolvedintheoptimizationprocessdemandedcomputationallyexpensiveniteelementanalyseswhichwasreplacedbyefcient,lowdelitysurrogatemodels.Thelimitedcomputationalresourcesweredirectedtowardstargetregionsforaccuraterepresentationofconstraintsusingadaptivesamplingstrategies. Itwasfoundthatcomputationalcostofthedeterministicoptimizationwasreducedto40%oftheavailablecomputationalbudgetbyemployingEfcientGlobalReliabilityAnalysis.Whenitwasfoundthatthecomputationalcostwasdrasticallylowered,theadaptivesamplingmethodwasalsoemployedtoimprovetheapproximationofthereliabilityindexinthereliabilitybasedoptimizationprocess.Thesafetyfactorbaseddeterministicoptimumyieldedanoptimaldesignwithstructuralmassof21.3kg/m2with2.7%probabilityoffailure.Reliabilitybasedoptimizationwithoutsafetyfactorbasedconstraintsallocatedhigherrisktothefailureresponsesthatfurtherminimizedmass.Themassoftheprobabilisticoptimumwas19.4kg/m2withdeterministicprobabilityoffailure.Thisshowedthattheriskallocationinprobabilisticoptimizationplayedanimportantroleindeterminingnon-conservativebutsafedesigns.Theaccuracyofthe 116
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probabilityoffailurewasestimatedusingthebootstrappingmethod.Thebootstrappingestimateswerecomparabletotheempiricalestimatesdeterminedthroughrepetitions. 8.2FutureWork Possiblefutureworkinthestructuralanalysis Thestressfailureinthepanelwaspredictedthroughmaximumstresstheoryasthemainfocuswasonthemethodologyinvolvedintheoptimizationprocess.ThiscanbereplacedwithamoresuitablecompositefailurecriterionsuchasTsai-HillorTsai-Wucriteriontopredictfailure. Asbucklingoccursinthepanelandnotinaunitcell,bucklingcouldbeanalysedforanentirepanel. 117
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APPENDIXATHERMALPROPERTIESOFITPSCOMPONENTS TableA-1.DensityoftheITPSComponents ITPScomponentmaterialDensitykg/m3 BFSGraphite/epoxy1576.2TFS&WrapSiC/SiC2500FoamAETB86.1 ThethermalconductivityandspecicheatcapacityoftheITPScomponentsaretemperaturedependentproperties. TableA-2.ThermalPropertiesofthebottomfacesheet-GraphiteEpoxy TemperatureThermalConductivitykW/m/kSpecicheatCpW/m2 2880.026512.93060.027554.85580.0441172.36480.0501381.67380.0541591.010080.0702177.111880.0762533.013680.0842899.415480.0913223.817280.0973558.819080.1033967.0 TableA-3.ThermalPropertiesofthetopfacesheetandWrap-SiC/SiC TemperatureThermalConductivitykW/m/kSpecicheatCpW/m2 2939.50150012737.603000 118
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TableA-4.ThermalConductivityoftheinsulationfoam-AETB TemperatureThermalConductivitykW/m/k 0.0045255.50.0055394.40.0067533.30.0083672.20.0102811.00.0121949.90.01441088.80.01681227.70.01941366.50.02201505.40.02441644.30.02491783.2 TableA-5.Specicheatoftheinsulationfoam-AETB TemperatureSpecicheatCpW/m2 1884.0255.52637.6394.43165.2533.33454.0672.23617.3811.03717.8949.93768.11088.83805.71227.73805.71783.2 119
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APPENDIXBSTRENGTHPROPERTIESOFITPSMATERIALS TableB-1.DensityoftheITPSComponents Strength(MPa)SiC/SiCGraphite/EpoxyAETB Tensilestrengthinthe1-directionS1T3232,3120.689Tensilestrengthinthe2-directionS2T32339.20.689Compressivestrengthinthe1-directionS1C63218090.689Compressivestrengthinthe2-directionS2C63297.20.689ShearstrengthS1217633.20.379 120
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APPENDIXCDESIGNOFEXPERIMENTSADDEDBYEGRASAMPLINGTECHNIQUE TableC-1.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofTemperatureboundary.Target=473K tWmmtBmmtTmmhmmTmax;BFS 0.46.03.027.42474.31.02.03.039.20475.60.726.01.031.49474.81.02.01.039.10473.50.46.01.027.38470.70.42.03.033.33472.0 TableC-2.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofTemperatureboundary.Target=215.3MPa tWmmtBmmtTmmhmm1;W 0.42.01.7125.0225.41061.02.01.7140.0213.41061.05.153.040.0214.21060.44.503.025.0214.71060.46.03.031.4218.31060.46.02.3840.0217.8106 TableC-3.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofbucklingloadboundary.Target=1.5 tWmmtBmmtTmmhmm 1.06.01.040.01.650.42.01.025.01.700.626.01.025.01.850.42.01.6540.01.440.46.01.8840.01.291.02.01.040.01.61 121
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TableC-4.DesignpointsacquiredusingEGRAtoimprovetheaccuracyofreliabilityindexboundary.Target=1.93 tWmmtBmmtTmmhmmReliabilityindex 0.523.811.6134.502.110.423.292.9535.401.770.442.142.6435.101.480.983.542.9130.701.960.702.652.6831.101.780.632.992.2829.801.820.842.491.9333.902.020.404.001.4629.002.130.402.001.2429.001.940.864.001.0036.001.850.402.001.5335.781.700.402.763.0029.001.801.002.001.0033.151.320.594.003.0036.003.131.002.003.0036.002.280.584.003.0029.001.750.702.003.0036.002.150.404.002.0736.001.931.004.001.9831.861.650.404.001.2932.031.720.402.002.4529.001.451.002.993.0034.092.010.403.161.2229.001.441.004.003.0032.591.900.622.001.0036.001.691.004.001.0033.992.380.403.051.6836.001.83 122
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BIOGRAPHICALSKETCHBharaniRavishankarwasbornin1985inthestateofTamilNadu,India.ShegraduatedwithaBachelorofEngineeringinAeronauticalEngineeringfromMadrasInstituteofTechnologyin2006.ShestartedhergraduatestudiesatUniversityofFloridaasamaster'sdegreestudentinthedepartmentofMechanicalandAerospaceEngineeringinJanuary2007.ShejoinedtheCenterforAdvancedCompositesLabinMay2007wheresheworkedasResearchAssistantwithDr.BhavaniSankar.InAugust2008,shecontinuedtopursueherPhDdegreeinMechanicalEngineering.ShewasawardedtheAmeliaEarhartFellowshipfromZontaInternational. 130
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