MULTIDISCPLINARY OPTIMIZATION AND SENSITIVITY ANALYSIS Multidiscplinary优化和灵敏度分析

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MULTIDISCPLINARYOPTIMIZATIONANDSENSITIVITYANALYSISOFFLUTTEROFANAIRCRAFTWINGWITHUNDERWINGSTOREINTRANSONICREGIME

Athesissubmittedinpartialfulfillment

oftherequirementsforthedegreeof

MasterofScienceinEngineering

By

CHAKRADHARBYREDDY

B.Tech.,JawaharlalNehruTechnologicalUniversity,India,1999

2003

WrightStateUniversity

WRIGHTSTATEUNIVERSITY

SCHOOLOFGRADUATESTUDIES

August4,2003

IHEREBYRECOMMENDTHATTHETHESISPREPAREDUNDERMYSUPERVISIONBYChakradharByreddyENTITLEDMultidisciplinaryoptimizationandsensitivityanalysisofflutterofanaircraftwingwithunderwingstoreintransonicregimeBEACCEPTEDINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMasterofScienceinEngineering.

RamanaV.Grandhi,Ph.D.

ThesisDirector

RichardJ.Bethke,Ph.D.

DepartmentChair

CommitteeonFinalExamination

RamanaV.Grandhi,Ph.D.

MitchWolff,Ph.D.

RaviC.Penmetsa,Ph.D.

JosephF.Thomas,Jr.,Ph.D.

Dean,SchoolofGraduateStudies

Abstract

ChakradharR.,Byreddy.M.S.Egr.,DepartmentofMechanicalandMaterialsEngineering,WrightStateUniversity,2003.MultidisciplinaryOptimizationandSensitivityAnalysisofFlutterofanAircraftWingwithUnderwingStoreinTransonicRegime.

Themainobjectiveofthisresearchistoobtainapreliminarydesignofanaircraftwingwithanunderwingstoreforimprovedflutterperformanceinthetransonicregime.Sincethetransonicflowregimeisalreadyhighlynon-linear,thepresenceofanunderwingstorecreatesstore-inducednon-linearitiesinadditiontothosenon-linearitesassociatedwiththewing.Allthesenon-linearitiesarecapturedusingacomputationaltoolcalledComputationalAeroelasticityProgramTransonicSmallDisturbance(CAP-TSD).Inthiswork,amethodologyisdevelopedtoincorporatethenon-linearitiesintoamultidisciplinaryoptimizationalgorithm.Generally,combininganon-linearanalysiswithoptimizationisacomputationallyexpensiveanddifficulttask.Therefore,theparametersthatareinsignificantintheanalysesareidentifiedandexcluded.

Awingwithdifferentstoreconfigurationsismodeledusingfiniteelements.UsingCAP-TSD,severalparametricstudiesontheflutterofvariouswing-underwingstoreconfigurationsareconductedinthetransonicregime.Thesensitivityofflutterisanalyzedforthefollowingstoreparameters:(i)locationofunderwingstorecenterofgravitywithrespecttoaerodynamicrootchord,(ii)locationofunderwingstorealongthespanofthewing,and(iii)underwingclearance(pylonlength).AnalysesarealsoperformedtoidentifytheonsetofLimit-CycleOscillations(LCO)fordifferentconfigurationsoftheunderwingstoreandflightregimes.Inaddition,aninvestigationoftheeffectsofunderwingstoreaerodynamicsontheonsetofflutterisstudied.Thesestudieshelpinexcludingtheparametersthatareconsideredinsignificantfortheoptimizationandinchoosingthecriticalwing-underwingstoreconfiguration.AfinalstudyisconductedtofindthesignificanceofincludingtheparametersassociatedwiththeCAP-TSDintotheoptimization.Thisisdonebyoptimizingthestructuralparameterssuchasthethicknessesofskins,spars,ribsandcross-sectionalareasofthepostsassociatedwiththewingusingAutomatedSTRucturalOptimizationSystem(ASTROS),andthenanalyzingtheflutterforthisoptimizedstructureusingCAP-TSD.Fromthis,thenatureofthefluttersensitivitieshelpsinunderstandingwhetheritisessentialtoconductnon-linearanalysis-basedoptimizationusingCAP-TSD.

Contents

Abstract…………...iii

1Introduction………………….. 1

1.1Introduction…………. 1

1.2LiteratureReview……………………. 4

2AnalysisMethodology……….. 8

2.1GoverningEquations………………… 8

2.2AnalysisProcedure…………………... 10

3ComputationalModels………. 14

3.1StructuralModelingofWingandUnderwingStore… 14

3.2AerodynamicModelingofWingandUnderwingStoreusing

CAP-TSD……………. 17

4EffectofStoreAerodynamicsonFlutter…… 19

4.1DifferentCases………. 19

4.1.1AnalysisofCleanWing……. 20

4.1.2AnalysisofWingwithStore(Massonly)…. 22

4.1.3AnalysisofWingwithStore(StoreAerodynamics)………. 27

5OptimizationofWingwithUnderwingStoreforImprovedFlutter

Performance………………….. 31

5.1OptimizationMethodology………….. 31

5.1.1StructuralAnalysis………….. 33

5.1.2LinearAerodynamicAnalysis……………… 33

5.1.3UnsteadyNon-linearAeroelasticAnalysis… 34

5.1.4MultidisciplinaryDesignOptimization-Formulation…….. 35

5.1.5OptimizationResults……….. 39

6Summary 46

6.1Summary……………… 46

Bibliography……………………… 48

AppendixA……………………….. 52

A.1Example1:ICWwithoutstoreaerodynamics………. 52

A.2Example2:ICWwithstoreaerodynamics………….. 55

AppendixB……………………….. 60

B.1Example1:ICWwithoutstoreaerodynamics………. 60

AppendixC……………………….. 72

C.1Example1:ICWwithstore(AllConstraints)…………72

ListofFigures

1Flowchartforanalysismethodology………….. 12

2ModifiedIntermediateComplexityWingwithunderwingstore……………… 17

3AerodynamicmodelingofwingandunderwingstoreusingCAP-TSD……… 18

4Fluttervelocitiesforacleanwing…………….. 21

5UnsteadyPressureDistribution:IndicatingpresenceofshocksatMach

0.90and0.92respectively…………………….. 21

6Centerofgravityrepresentationforvariousstoreconfigurations…………….. 22

7Sensitivityoffluttervelocitytounderwingstorewithcenterofgravity

(massonly) ……………………. 23

8Sensitivityoffluttervelocitytounderwingstorealongthespanofthe

wing(massonly)forstoreconfiguration2……. 24

9Sensitivityoffluttervelocitytounderwingstorewithunderwing

clearance(massonly)forstoreconfiguration2………………… 25

10Sensitivityoffluttervelocitytounderwingstorewithlocationofspan

withrespecttocenterofgravityofstore(massonly)………... 26

11Sensitivityoffluttervelocitytounderwingstorewithunderwing

clearancewithrespecttocenterofgravityofstore(massonly)……………… 27

12AerodynamicgridofwingwithunderwingstoreusingCAP-TSD…………... 28

13ComparisonofFluttervelocitiesforLinear(ASTROS),Linear

(CAP-TSD)andNon-Linear(CAP-TSD)forstoreconfiguration2………….. 28

14Comparisonoffluttervelocities(knots)forM=0.9andM=0.92

usingNon-Linear(CAP-TSD)forstoreconfiguration2 ………29

15Comparisonofflutterforstoreconfigurations1and2withandwithout

storeaerodynamics……………. 30

16Flowchartdescribingthedesignmethodology……………….. 32

17AerodynamicmodelofwingusingDLM……… 34

18CAP-TSDgridofwing…………35

19Fluttervelocitiesofinitialandoptimizedwing-underwingstore

configurationwithfrequencyconstraints ……... 40

20Fluttervelocitiesofinitialandoptimizedwing-underwingstore

configurationwithflutterconstraint……… 41

21Fluttervelocitiesofinitialandoptimizedwing-underwingstore

configurationwithboththeflutterandfrequencyconstraints… 42

22Percentagechangeinfluttervelocitiesofwing-underwingstore

configurationwithvariousconstraints………… 43

23Thickness(ininches)distributionofinitialandoptimumdesignof

wingstructureatMach0.92…………………… 44

ListofTables

1Differentstoreconfigurations…………………... 16

2Modalfrequenciesfordifferentwing-storeconfigurations……. 25

NOMENCLATURE

inviscidsmalldisturbancevelocitypotential

freestreamMachnumber

γ ratioofspecificheats

ui timevaryinggeneralizeddisplacements

fi verticalcomponentsofthemodeshapes

K structuralstiffness

B structuraldamping

M structuralmass

F externalaerodynamicloads

freestreamdensity

wingreferencechord

freestreamvelocity

liftingpressure

modeshape

q dynamicpressure

generalizedvelocities

W totalstructuralweightofthewingand

nloc numberoflocaldesignvariables

massdensityoftheithstructuralelement

volumeoftheithstructuralelement

tensilestress

compressivestress

shearstress

ω naturalfrequency

requiredlevelofdampingatthejthvelocity

calculateddampingvaluefortheithmodeatjthvelocity.

GFACT valueinordertoscaletheconstraint

non-linearfluttervelocity(CAP-TSD)inKnots

linearfluttervelocity(ASTROS)inKnots

Acknowledgements

TherearemanypeoplewhohavehelpedandmotivatedmeduringmyM.S.Egrdegreeprogram.Myprofessors,familyandfriendshaveallhelpedinonewayoranother.Iamverymuchobligedtoallofthemfortheirconstantsupportandencouragement.

Inparticular,myveryspecialthanksgoestomyadvisor,Dr.RamanaV.Grandhiforhisdiscerningguidanceandalso,fortakingthetimetounderstandmeasanindividualandmotivatingallthetime.IwouldliketothankDr.RaviPenmetsaandDr.MitchWolfffortheirtimeinreviewingthethesisandalso,forbeingpartofmyfinalthesisdefensecommittee.

IwouldalsoliketothankDr.PhilipBeran,Dr.FrankEastep,Dr.NarendraKhotandDr.BrianSandersfromWPAFB,fortheirgenuineinterestinmyresearchworkandforvariousnumerousdiscussionsandmeetingsinthisproject.Furthermore,IwouldliketothankDr.NathanKlingbeilandDr.JosephSlaterfortheirvaluablesuggestions.

Finallyandmostimportantly,Iwouldliketothankmyparents,brother,andmywifefortheirendlesssupportduringmycourseofstudy.

ThisthesisisdedicatedtomyfatherLateMr.RavindraReddy,

motherMrs.PushpaReddy,andbrotherMr.SarathKumar

Chapter1

Introduction

1.1Introduction

Manyfighteraircraftcarryouttheirmissionsinthetransonicregime,andthepresenceofexternalstoresposecomplexanddangerousproblemsinthisregime.Intransonicflowregimes,theeffectofaerodynamicnonlinearitiesbecomessignificantduetothepresenceofshocksonthewingsurface,anddynamicaeroelasticinstabilitiessuchasflutterandLCOareinducedduetothepresenceofexternalstores.AcomputationalmethodbasedontheinviscidTransonicSmallDisturbancetheoryisusedtopredictthenonlinearunsteadyaerodynamicsassociatedwithshockmotionsinthetransonicflowregion[1].Thismethodisusedtosolvethenonlineargoverningequationsinaeroelasticanalysis,andprovidesanefficientbutaccuratealternativetolinearmethodssuchasthedoubletlatticemethod(panelmethod).

Previousliteraturehelpedinunderstandingtheimplicationsofanaircraftwingwithexternalstores(storesconsideredasrigidbodies)onthestaticaeroelasticphenomenaandunsteadypressuredistributionsinthetransonicregime[2].Also,someworkhasbeenperformedontheLCOofanaircraftwing,butnotconsideringtheeffectoftheunderwingstorestructuralparametersonthedynamicaeroelasticphenomenainthetransonicregime.ThepresentworkadvancestheongoingresearchthatisbeingperformedattheAirForceResearchLaboratory(AFRL)byinvestigatingtheeffectsofdynamicaeroelasticphenomenatakingplaceinflightvehiclescarryingstores(missiles,launchers,fueltanks,etc.)[3].Inthepresentwork,differentunderwingstoreconfigurationswerechosensoastounderstandtheinfluenceofthestructuralparametersofstoreonthedynamicaeroelasticinstabilities.PresenceofunderwingstorescausesflutterandstoreinducedLCOinthetransonicregime,whichcanleadtoseveralproblemsassociatedwithtarget-lockingsystem,rollmaneuverabilityetc.Therefore,itplaysanimportantroleinthepreliminarydesignstage.

Theresearchworkhasbeendividedintothreephases.Thefirstphaseinvolvesthevalidationofcomputationofflutterbyconductingananalysisonacleanwing(i.e.,onewithoutstore)usingAutomatedSTRucturalOptimizationSystem(ASTROS)[4]andComputationalAeroelasticityProgramTransonicSmallDisturbance(CAP-TSD)(linearandnonlinearanalysis)inthesubsonicregime.Thesecondphaseoftheworkinvolvesaninvestigationoftheeffectofvariationinthestoreparameterssuchastheunderwingstorecenterofgravity,underwingstorelocationalongthespanofthewingandunderwingclearanceinthetransonicregion.Oneofthecoreissuesinthesecondphaseoftheworkistheinclusionofstoreaerodynamicsinthecalculationoffluttervelocities[5].Therefore,parametricstudiesareconductedbyconsideringtheunderwingstoremassonlyandunderwingstoreaerodynamics.Theaccuracyofcomputedfluttervelocityiscomparedinbothcasestounderstandtheimpactofinclusionofstoreaerodynamics.Thus,identifyingwhethertheeffectofstoreaerodynamicshastobeincludedorneglectedintheoptimizationalgorithms(whichareiterativeinnature).TheseanalysesalsohelpinidentifyingthecriticalparametersthatdirectlyaffecttheflutterandLCOinthetransonicregion.ByobtainingthesensitivitiesoftheseparameterstoflutterandLCO,leastsensitiveparameterscanbeignoredintheanalysis,resultinginreducedcomputationaltimeandcosts.

Withtheresultsobtainedfromthesecondphase,itisviabletoincorporatenonlinearanalysisintothepreliminarydesignprocess,whichisthethirdphaseoftheresearchwork.Basedontheinformationobtainedfromtheaboveanalyses,amultidisciplinaryoptimizationmethodologywasdevelopedtodesignawingstructurewithexternalstorestodelaytheoccurrenceofdynamicaeroelasticphenomenasuchasflutterandLCO.Usingconventionaldesignmethods,itisadifficulttasktofindafeasibledesignthatsatisfiestheconditionsofnonlinearities.Thus,thisstudyhelpsindevelopinganautomatedmethodologytoincorporatethenonlinearitiesassociatedwiththewingandstoreintothemultidisciplinarydesignenvironment.Thisstudyhelpsinthepreliminarydesignofaircraftstructuresforimprovedflutterinthetransonicregimewiththepresenceofstores.Thisresearchworkinvolvesoptimizationwhich,byautomatingtheanalysisforaparticularstoreconfigurationanddifferentflightconditions,helpstodelaytheoccurrenceofflutterofafighteraircraft.Intheoptimizationproblem,awingwithaparticularunderwingstoreconfigurationischosenandmodeled[6].Thisunderwingstoreconfigurationhasacenterofgravityalmostneartheelasticaxisofthewing.Aslightvariationinthestorecenterofgravitywithrespecttotheelasticaxishasasignificanteffectonthedynamicaeroelasticphenomena.Also,thelocationoftheunderwingstorealongthewingspanandtheamountofunderwingclearancehavesignificanteffectsonflutter.

Sincethecombinationofnonlinearanalysiswithoptimizationisdifficultandcomputationallyexpensive,manyparametersthatareinsignificantintheanalysesareexcluded.Onesuchparameteristheadditionofstoreaerodynamicsintheoptimizationproblem[6].Plusaseparatestudyisconductedtofindthesignificanceofincludingtheparametersassociatedwiththenon-linearaerodynamicanalysisofCAP-TSDintotheoptimizationproblem.ThisisconductedbyoptimizingthestructuralparametersassociatedwiththewingusingAutomatedSTRucturalOptimizationSystem(ASTROS),andthenanalyzingtheflutterforthisoptimizedstructureusingthehigherfidelityaerodynamictoolCAP-TSD.Furthermore,thenatureoffluttersensitivitywithMachnumberhelpsinunderstandingwhetheritisessentialtoconductnon-linearaerodynamicanalysis-basedoptimizationusingCAP-TSD.Theoptimizationofthesecriticalparametershelpstoincreasetheairvehiclelife,performance,andflightenvelopeduringthemission.Theinformationobtainedfromthisworkusinghigherfidelitymodelshelpsin“certificationbyanalysis”,therebyreducingcostandtimewhichareprevalentintheflighttests.

1.2LiteratureReview

Manyproblemsassociatedwithfluid-structureinteractionarequitecomplicated,particularlythatofwing-storeinteraction.Differentresearchapproacheshavebeenextensivelystudiedanddevelopedinordertounderstandtheimpactofstructuresandaerodynamicsassociatedwiththewingandstore.HighcomputingpowerledtotheadventofvariousnumericalmethodstosolvetheaeroelasticproblemsforapplicationtorealisticaircraftconfigurationsinthetransonicregimesuchasCAP-TSD[7].Severalresearchersinthelate1980’sandearly1990’semphasizedstructuraloptimizationforimprovedaeroelasticperformance.Theworkduringthisperiodconcentratedonintegratingtheexistingtechniquesofflutteranalysiswithotherdisciplinessuchasstaticstrength,dynamics,staticaeroelasticity,etc.TheunsteadyaerodynamicsconsideredtocalculateflutterwaslinearsuchasDoubletLatticeMethod(DLM)forthesubsonicregime,andstriptheory,Machboxmethodetc.forthesupersonicregime.

Previousliteraturemainlyconcentratedonthesubsonicandsupersonicregimesforwingswithtip/underwingstores.Also,mostoftheresearchworkforwing-bodyconfigurationwascarriedoutineithersubsonicorsupersonicflowregimes[8].AcomprehensiveworkanalyzedbyCattarius[9]onthewing-underwingstoreofaparametricF-16wingconsideredtheeffectofflutterofstores.Inthatwork,theauthordevelopedanunsteadyvortexmethodanduseditinordertounderstandtheeffectsofstoreaerodynamicsinthesubsonicregime.

Inthecaseoftransonicregime,studieswereconductedmainlyonacleanwing.Onesuchstudyincludesthepreliminarydesignofaircraftstructuresforimprovedcontroleffectiveness(steady-staterollperformance)inthetransonicregionforacleanwing[10].Furthermore,Kolonay[11]developedadesignmethodologytoincludetransonicflutterrequirementsforthepreliminarystructuraldesignofacleanwing.Theloadsthatariseinthetransonicregimehaveprofoundinfluenceontheaeroelasticstabilityofafighteraircraft.Meijer[12]developedamethodologyinordertodeterminetheseloads.InclusionoftheseloadsintotheoptimizationprocedurehelpsinbetterunderstandingthemechanismsthatdrivethedynamicaeroelasticinstabilitiessuchasflutterandLCO.

Inthecaseofawingwithexternalstoresinthetransonicregime,theinfluenceofstoreaerodynamicsondifferentwingconfigurationshavebeenstudiedandcomparedwithpresentflightflutterdata[13].ThevariousfactorsthataffectLCOhavebeenextensivelystudied[14]byconsideringthestructuralnonlinearitiesaswellasaerodynamicnonlinearitiesofawingwithtipstore.ThecurrentworkinvolvesacomprehensiveparametricstudyofunderwingstorestructuralparametersonflutterandLCObyusingvariousnonlinearanalysistools.

Toth,etal.[15]conductedvariousstudiesonnon-lineartransonicflutterofF-16aircraftwingwithstoresconfigurationclearance.Thisworkstudiedvariousflutterpredictionmethodsand,hypothesizedthemechanismsthatgoverntheonsetofLimit-CycleOscillations(LCO).ResearchworkthatwasperformedbyBeran[3]involvedtheinvestigationofdynamicaeroelasticinstabilitiesinflightvehiclescarryingstores(missiles,launchers,fueltanks,etc.).TheyusedComputationalFluidDynamics(CFD)methodslinkedwithoptimizationtechniquestodeterminetheaerodynamicsaspectratherthanthestructuralaspects.Therefore,inthisresearchanextensivestudyonthestructuralaspectsofthewingwithunderwingstoreforenhancedflutterperformancewasstudied.

Inthepresentresearchadesignmethodologyisdevelopedbyintegratingvarioustoolsassociatedwithnonlinearanalysisforimprovedairvehicleswithexternalstoresinthetransonicregime.Thisworkcanbeadvancedbyincludingtheeffectsofpylonstiffness[16],i.e.,varioustypesofattachmentsandalsotheflutterofstore.Theflutterofstoreitselfmightcauseextensivefatiguetothepylon.Therefore,studiesinvolvingalltheseeffectshelpinbetterunderstandingthemechanismsandphysicalsignificancethatgoverntheonsetofflutterandLCOduetothepresenceofunderwingstores.Also,theoptimizationofcriticalstorestructuralparametershelpsinincreasingtheairvehiclelife,performanceandflightenvelopeduringtheirmission.Thus,ithelpsinthestudyofthepreliminarydesignofaircraftstructureswithandwithoutstoresforimprovedflutterandLCOperformanceinthetransonicregime.

Chapter2

AnalysisMethodology

2.1GoverningEquations

TheTSDtheoryisbasedontheassumptionthatinthetransonicflowregime,therearesmalldisturbances,orperturbationsaround,athinwing.TheTSDequationinconservationformisgivenas

(2.1.1)

whereistheinviscidsmalldisturbancevelocitypotential.Itisthenonlinearityinthathelpsinmodelingweakshockwavesinthetransonicregime.Intheanalyses,onlytwodifferentformsoftheTSDequationareusedbychoosingeitherthelinearequationcoefficientsortheAMEScoefficients.ThecoefficientsA,B,E,are

(2.1.2) (2.1.3)

(2.1.4)wherethefreestreamMachnumber.ThecoefficientsF,GandHarecalledAMEScoefficients,givenas

(2.1.5)

(2.1.6)

(2.1.7)

whereγistheratioofspecificheats.Thevalueofγusedintheseanalysesis1.4(air)

(2.1.8)

ThenonlinearresultsarecomputedbyusingtheAMEScoefficientsgivenbytheequation

Thelinearresultsarecomputedbysettingthecoefficientsgivenbytheequation(2.1.9)

Whenthelinearequationwasused,thewingandstorewasmodeledasaflatplateinordertoproduceresultssimilartoothermethodssuchasthedoublet-latticemethod.Whenthenonlinearequationwasused,thewingwasmodeledusinganappropriateairfoilsuchasNACA0004,(zerocamber,symmetricandfourpercentthick)sothatthenonlineareffects(suchasmovingshockwaves)canberealisticallycaptured.Howeverthestoreismodeledasaflatplate,forinclusionofstoreaerodynamics.Couplingofthestructuralequationsofmotionwiththeunsteadyaerodynamicsofwingandstoreisimplementedandonlytheverticalcomponentofthemodeshapeisusedforboththewingandstore.

2.2AnalysisProcedure

TheCAP-TSDcodesolvestheunsteadytransonicsmalldisturbanceequationusinganimplicittime–accurateapproximatefactorizationalgorithm[17].Theunsteadyaerodynamicsissimultaneouslyintegratedwiththestructuralequationofmotionsintime.ForthisthevibrationanalysisisperformedusingASTROS[18]andthedisplacementsaresplinedontotheCAP-TSDgridofthewingusingaThinInfinitePlaneSpline(IPS)[19].Thisintegrationisrepresentedbythestructuralresponseintimetosomeinitialperturbations.Thestructureismodeledbyaseriesoforthogonalmodeshapesweightedwithtimevaryingcoefficientscalledthegeneralizeddisplacements.Thegeneralizedcoordinatetransformationrepresentsthephysicaldeformationsofthestructure.Themodaldeflectionsinthestreamwiseandspanwisedirectionsareminuteincomparisontotheverticalmodaldisplacements,andthus,neglected.Therefore,thepositionofthewingatanypointintimeisgivenas

(2.2.1)

whereuiisthetimevaryinggeneralizeddisplacementsandfirepresentstheverticalcomponentsofthemodeshapes.Thestructuralequationsofmotioningeneralizedcoordinatesaregivenas

(2.2.2)

K-StructuralStiffness

B-StructuralDamping

M-StructuralMass

F-Externalaerodynamicloads

where (2.2.3)

-Freestreamdensity

-Wingreferencechord

-Freestreamvelocity

-Liftingpressure

-Modaldisplacementsdescribedinequation10

Equation2.2.2issolvedwithequation2.1.8byusinganimplicittime-marchingaeroelasticsolutionprocedurebasedonapproximatefactorization[20].Inthecurrentwork,theprocedurefortheassessmentofflutterpredictionisdescribedusingtheflowchartinFigure1.Inthismethod,thefluttervelocityiscalculatedbyvaryingfreestreamvelocityanddynamicpressure()whileholdingthedensityconstantatagivenMachnumber(whichiscalledanunmatchedanalysis).TocomputethepointatwhichflutterfirstoccursforagivenMachnumber,severalexecutionsoftheCAP-TSDcodearerequiredatdifferentdynamicpressures.AllCAP-TSDcalculationsincludetheeffectsofshockgeneratedentropyandvorticity.Astaticaeroelasticanalysisisperformedatagivendynamicpressure(thatisassumedtobenearneutralstability)tocreateasteadyflowfieldthatreflectsthewingthickness,camberandmeanangleofattack.Thissteadyflowfieldisessentialforthepropercomputingofthefreedecaytransientsinthedynamicaeroelasticanalysis.

Staticaeroanalysisusing

CAP-TSD

Yes

Staticsolution

converged?

Dynamicaeroelastic

analysis

No

Increasenumberofiterations

Isthesolution

stable?

Usehigherdynamicpressure

Yes

No

Flutteroccurs:

Determineflutterconditions

Sensitivityanalysis,Surrogatemodels&designoptimization

Figure1Flowchartofresearchmethodology

Ifthestaticaeroelasticsolutionisconverged,thenthedynamicaeroelasticanalysisisperformedbyrestartingthecalculationfromtheconvergedstaticaeroelasticsolutionwithsomeinitialdisturbanceontheverticalvelocityofthewing.Ifthesolutionisnotconverged,thenthenumberofiterationsisincreasedtillthestaticsolutionconverges.Afterthedynamicanalysisisrun,thestabilityofthesystem(coefficientoflift)isdetermined.Ifthesystemisstable,theentireprocedureisrepeatedb