MULTIFIDELITY RESPONSE SURFACE MODEL FOR HSCT WING

MULTIFIDELITYRESPONSESURFACEMODELFORHSCTWINGBENDINGMATERIALWEIGHTVladimirBalabanov∗VanderplaatsResearchandDevelopment,Inc.1767South8thStreet,SuiteM-210,ColoradoSprings,CO80906RaphaelT.Haftka†DepartmentofAerospaceEngineering,MechanicsandEngineeringScience,UniversityofFloridaGainesville,Florida32611-6250BernardGrossman‡,WilliamH.Mason§,andLayneT.Watson¶MultidisciplinaryAnalysisandDesign(MAD)CenterforAdvancedVehiclesVirginiaPolytechnicInstituteandStateUniversityBlacksburg,Virginia24061-0203AbstractResponsesurfacetechniquesallowustocombineresultsfromalargenumberofinexpensivelowfidelityanalyseswithasmallnumberofexpensivehighfidelityanalysesforconstructinginexpensiveandaccurateapproximations.Thepaperdemonstratesthisapproachbyconstructingapproximationstowingbendingmaterialweightofahighspeedciviltransport(HSCT).TheapproximationsemployalargenumberofstructuraloptimizationsoffiniteelementmodelsforarangeofHSCTconfigurations.Thousandsofstructuraloptimizationsofcoarsefiniteelementmodelsareusedtoconstructaquadraticresponsesurfacemodel.Thenaboutahundredstructuraloptimizationsofrefinedfiniteelementmodelsareusedtoconstructlinearcorrectionresponsesurfacemodels.TheusefulnessoftheapproximationsisdemonstratedbyperformingaerodynamicoptimizationsoftheHSCTwhileemployingtheresponsesurfacemodelstoestimatewingbendingmaterialweight.TheapproximationsforthefinalHSCTdesignsarecomparedtoresultsofstructuraloptimizationsoftherefinedfiniteelementmodel.1.IntroductionResponsesurfacemethodshavereceivedalotofattentioninthepastfewyearsinthefieldofmultidisciplinaryoptimization(MDO)[1].Thesetechniquesconstructsimplealgebraicapproximations,typicallyquadraticpolynomials,fortheobjectivefunctionand∗SeniorResearchandDevelopmentEngineer,MemberAIAA†Professor,FellowAIAA‡Professor,Dept.Head,Dept.ofAerospace&OceanEng.,AssociateFellowAIAA§Professor,Dept.ofAerospace&OceanEng.,AssociateFellowAIAA¶Professor,Depts.ofComputerSci.andMath.1constraints,basedonthevaluesofthesefunctionsatasetofpointscarefullydistributedthroughoutthedesignspace.Theoptimizationthenproceedsonthebasisoftheseap-proximations.ResponsesurfacemethodsperformseveralimportantfunctionsforMDO.Theysmoothoutnoiseoftenpresentinsomeoftheresponsequantities,theyeasetheintegrationofcodesfromvariousdisciplines,theypermitdisciplinaryexpertstoretaincontrolovertheiranalysiscodesratherthanturnthemovertodesignoptimizationgeneralists,andtheyalloweasyusageofparallelcomputerarchitectures(e.g.,[2],[3]).ApplicationsofMDO,withorwithoutresponsesurface(RS)models,usuallysufferfromthehighcostofthesystemanalysesrequiredforaccurateevaluationoftheobjectivefunctionandconstraints.Ourgrouphaspursuedavariablecomplexitymodelingapproach,involvingthesimultaneoususeofexpensivehighfidelityanalysestogetherwithinexpensivelowfidelityanalysesforalleviatingthisdifficulty.Wehaveusedtheresultsfromthelowerfidelityanalysestoreducethesizeoftheregionindesignspace,wheretheresponsesurfacemodelisconstructedfromtheresultsfromthehigherfidelityanalyses[3],andtoreducethenumberofvariablesintheresponsesurfacemodel[4]orthenumberoftermsused[5].Inthispaperweconsideranotherapproachforcombininglowerfidelityandhigherfidelityanalyses.Thelowerfidelityanalysisisusedtoproducealargenumberofresultstocreateaquadraticresponsesurfacemodel,whiletheexpensivehigherfidelityanalysisisusedtoproduceasmallnumberofresultstocreateaconstantorlinearcorrectionfactor.Thisapproachwassuccessfullyusedinthepastforapproximatingresultsfromstructuralanalyses(e.g.[6,7]).Thepresentpaperdemonstratesitsusefulnessforapproximatingstructuralweightobtainedbystructuraloptimizationofvariousconfigurationsofahighspeedciviltransport(HSCT).2.HSCTDesignProblemInourpaperthedesignproblemisoptimizationofanHSCTconfigurationtominimizetakeoffgrossweightforarangeof5500nauticalmilesandacruiseMachnumberof2.4,whilecarrying250passengers.Thechoiceofgrossweightastheobjectivefunctiondirectlyincorporatesbothaerodynamicandstructuralconsiderations,inthatthestructuraldesigndirectlyaffectsaircraftemptyweightanddrag,whileaerodynamicperformancedictatesthedragandthustherequiredfuelweight.Trimandcontrolrequirementsarealsoexplicitlytreated.Figure1showsatypicalplanformoftheHSCT.Wehavedevelopedasimpledescriptionofthegeometryandtheflighttrajectorythatemploys29designvariables(listedinTable1).Figure1.TypicalplanformoftheHSCT.2Table1.HSCTconfigurationdesignvariablesandbaselinevalues.NumberValueDescription1178.2Wingrootchord(ft)2114.1LEbreakpoint,x(ft)340.7LEbreakpoint,y(ft)4173.4TEbreakpoint,x(ft)512.2TEbreakpoint,y(ft)6146.4LEwingtip,x(ft)79.2Wingtipchord(ft)882.6Wingsemi-span(ft)90.51Chordwisemax.t/clocation102.53LEradiusparameter112.82Airfoilt/catroot(%)121.90Airfoilt/catLEbreak(%)131.70Airfoilt/cattip(%)142.61Fuselagerestraint1,x(ft)150.47Fuselagerestraint1,r(ft)1613.24Fuselagerestraint2,x(ft)172.49Fuselagerestraint2,r(ft)18111.68Fuselagerestraint3,x(ft)195.32Fuselagerestraint3,r(ft)20186.91Fuselagerestraint4,x(ft)215.34Fuselagerestraint4,r(ft)2211.50Nacelle1,y(ft)2328.37Nacelle2,y(ft)24464,743Missionfuel(lbs)2558,403Startingcruisealtitude(ft)2637.97Cruisec