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AircraftFamilyDesignUsingDecomposition-basedMethodsJamesAllison(1),BrianRoth(2),MichaelKokkolaras(1),IlanKroo(2)§andPanosY.
Papalambros(1)(1)UniversityofMichigan,AnnArbor,Michigan,48109,USA(2)StanfordUniversity,Stanford,California,94305,USAThispaperexplorestheuseofdecomposition-basedmethodsforaircraftfamilydesign.
Thetraditionalapproachinmultidisciplinarydesignoptimizationistodecomposeaprob-lemalongdisciplinarylines.
Foraircraftfamilydesignproblems,amorenaturalapproachisdecompositionbyindividualaircraft.
Thisdecompositionfacilitatestheconcurrentde-velopmentofseveralaircraftvariants,providingsubstantialautonomytoindividualaircraftdevelopmentprograms.
Twodecomposition-basedmethodsareappliedtotheaircraftfam-ilyproblem:collaborativeoptimizationandanalyticaltargetcascading.
Thispapermarksthebeginningofacollaborativeeorttounderstandtheessentialdierencesbetweenthesetwomethods,andtheresultingimplications.
Initialproductfamilyresultsillustratehowdecomposition-basedmethodscanbeappliedtotheaircraftfamilyproblem.
I.
IntroductionAproductfamilyisasetofindividualproductsthatsharecommoncomponentsorsubsystemsandaddressarelatedsetofmarketapplications.
1Inanaerospacecontext,aproductfamilyisusuallycomprisedofabaselineaircraftanditsderivativesorvariants.
Aircraftdesignoftentakesplacewithaneyetowardsderivativedevelopment.
Thisisevidencedintheselectionofpowerplantswithgrowthcapabilitiesandinwingdesign,withrespecttothestructuralimplicationsoftipextensionsandwinglets.
Anaerospaceproductfamilyisnotlimitedtoabaselineaircraftandderivatives.
Itcaninvolvetwoormoreaircraftwithdissimilarmissionsthatshareonlyafewkeypartsorsystems.
Strongmotivationexistsforaerospacedesignbasedonproductfamilies.
2,3Astheaerospaceindustryhasmatured,emphasishasshiftedfrom"higher,faster,farther"to"better,faster,cheaper.
"Oneopportunityforcostsavingsisthroughimprovedeciencyinmanufacturing.
Whenmultipleaircraftsharemajorstructuralcomponents,costscanbesavedintoolingandassembly.
Productfamiliesalsoenableaircraftmanufacturerstocatertotheneedsofpotentialcustomersbyoeringawiderselectionofaircraft.
Fromanairline'sperspective,commonalityisalsoanadvantage.
Forexample,avionicscommonalityspeedspilotcross-trainingamongmemberaircraftinaproductfamily.
Additionaladvantagesofcommonalityincludesimplicationofmaintenanceprocedures,exibilityinscheduling,andreducedspare-partsinventory.
Thus,productfamiliesaddvalueforthemanufacturerandthecustomer.
Althoughaproductfamilyapproachcanreducecosts,sharedcomponentsmayleadtoaperformancepenalty.
4–7Commoncomponentsmaynolongerbeoptimalforanyoneaircraftinafamily,sincetheyaredesignedtooptimizesomecollectivemeasureofmerit.
Multidisciplinarydesignoptimization(MDO)providesanaturalcontextinwhichtoconsidertradeosindesignofproductfamilies.
Justasithasbeenusedfortradesbetweenaerodynamicsandstructures,itcanbeemployedtoconsidertradesbetweenperformanceandcost.
DoctoralCandidate,DepartmentofMechanicalEngineering,AIAAStudentMemberDoctoralCandidate,DepartmentofAeronauticsandAstronautics,AIAAStudentMemberAssociateResearchScientist,DepartmentofMechanicalEngineering,AIAASeniorMember§Professor,DepartmentofAeronauticsandAstronautics,AIAAFellowProfessor,DepartmentofMechanicalEngineering,AIAASeniorMemberCopyrightc2006byB.
Roth.
PublishedbytheAmericanInstituteofAeronauticsandAstronautics,Inc.
,withpermission.
1of12AmericanInstituteofAeronauticsandAstronauticsThispaperexplorestheuseofdecomposition-basedmethodsforaircraftfamilydesign.
Recentworkinaerospacefamilydesignconsideredtwoapproachestotheproblem:sequentialandsimultaneousdesign.
8Previousworkindecomposition-basedmethodsextendedtheanalyticaltargetcascading(ATC)formulationtodesignofproductfamilies.
9Here,weillustratetheuseofdecomposition-basedoptimizationtoperformthesimultaneousdesignofanaircraftfamily.
AsillustratedinWillcox&Wakayama,8decompositionisnotnecessaryattheconceptualdesignlevel.
However,duringsubsequentdesignsteps(preliminarydesign),acertainlevelofautonomymaybedesiredbetweenaircraftdevelopmentprograms,particularlyifonlyafewpartsareshared.
Methodssuchascollaborativeoptimization(CO)andATCenabledecisionmakingattheindividualaircraftlevelconsistentwithoverallproductfamilygoals.
Thepaperisorganizedasfollows.
SectionIIprovidesabriefoverviewofthetwodecompositionmethods,comparingsimilaritiesanddierencesintheirformulations.
SectionIIIintroducestheconsideredaircraftfamilydesignproblemandpresentstheformulationsofthetwoapproaches.
SectionIVdiscussestheobtainedresults.
II.
OptimalDesignofDecomposedSystemsBothCO10andATC11aredecomposition-basedmethodsforsolvingcomplexsystemoptimizationprob-lems.
Theyweredevelopedindependentlyinresponsetodierentneedsfordesignandproductdevelopment.
Bothdealwithinteractionsbetweenelementsofapartitionedsystemoptimizationproblem.
Theirbasicmathematicalformulationsexhibitsimilarities,yeteachapproachretainsimportantdistinctions,suchassolutionprocessandcommunicationpatterns.
Thispaperbuildsonrecentwork12toestablishCOandATCasdistinctlydierentmethodswithcomplementarycharacteristicsusingtheaircraftfamilydesignexample.
Whenadesignproblemispartitionedintosmallersubproblems,additionalterminologyisnecessary.
TheterminologypresentedhereiscommontomanyMDOformulations.
Thedesignvectorxcanbepartitionedintolocalvariablesxithatarepertinentonlytosubproblemi,andsharedvariablesxsithatareinputstosubproblemiandatleastoneothersubproblem.
Thevectorxicontainslocalandsharedvariablesrequiredforsubproblemi.
Inaddition,subproblemsareconnectedthroughinteractions,i.
e.
,analysisoutputsofonesubproblemmayberequiredasanalysisinputsforanother.
Thevectorofcouplingvariablesyijisthesetofvaluescomputedbysubproblemjrequiredasinputstosubproblemi.
Thecollectionofallcouplingvariablesyhasnocommoncomponentswithx.
COcoordinatesthesolutionofdisciplinarysubproblemsusingasystemoptimizer.
Ithasbeenappliedsuccessfullytoaerospaceproblems.
13,14Subproblemscanbeexecutedinparallel,andthesubproblemsareconsistentatconvergence.
15ATCwasoriginallyconceivedforproductdevelopment16andhasbeensuccessfullyimplementedinautomotive,17,18architectural,19productdesign,20andmultiple-regimeaircraftdesign.
21ATCconvergencepropertieshavebeenprovenforaspecicclassofcoordinationstrategiesunderstandardconvexityandsmoothnessassumptions.
ATCwasdevelopedasatoolforsettingperformancetargetsforaproductatsystem,subsystem,andcomponentlevelssuchthattop-leveltargetsaremetandtheresultingsystemisconsistent.
LikeCO,ATCalsoprovideseachspecialistorteamwithsubstantialdesignfreedomwhileaccountingforcriticalinteractionsbetweensystemelements.
Whileearlypapers16acknowledgedthesimilaritybetweenCOandATC,onlyrecentlyhaveformalcomparisonsbeenmadebetweenthetwotechniques.
Thesecomparisonswerebasedonsingleproductdesign.
12,22ThispaperexploresCOandATCinthecontextofanaircraftfamilydesignproblem.
A.
AnalyticalTargetCascadingAnalyticaltargetcascadingwasdevelopedbasedonneedsintheautomotiveindustrytotranslatetop-levelproducttargetsintodetaileddesignspecications.
Itisapplicabletosystemsthatpossesshierarchicalrelationships.
AnexampleofahierarchicalsystemisshowninFigure1.
Eachelementinthehierarchycomputesitsownlocalanalysisresponses,andmayrequireasinputsanalysisresponses(couplingvariables)fromlowerlevelelements,inadditiontolocalandsharedvariables.
TheobjectiveoftheATCprocessistodeterminedesignspecicationsforeachelementinthehierarchythataccountforinteractionsothatdesignteamscanproceedwithdetaildesignindependently.
Anopti-mizationproblemisformulatedforeachelement.
Theformulationallowsforalocalobjectiveandobserveslocaldesignconstraints.
ATCallowstheoptimizationalgorithmtochoosecouplingvariablevalues,andusespenaltyfunctions(insteadofequalityconstraints)toensuresystemconsistency.
RecentATCformula-2of12AmericanInstituteofAeronauticsandAstronauticsFigure1.
Hierarchicalsystemdecompositiontions12,23allowmultidirectionalcoupling,andcouplingbetweensame-levelelements.
TheATCformulationforsubproblemPiisminxi,yij,xsCi,yCif(xi,yij)+π(c)subjecttogi(xi,yij)≤0hi(xi,yij)=0wherec=ziziTheobjectiveistominimizethelocalobjective(ifitexists)andafunctionπthatpenalizesnonzerovaluesinthedeviationvectorc,subjecttolocaldesignconstraints.
Thedeviationvectorquantiesthedierencebetweensharedquantitiescomputedlocally,zi,andthecorrespondingsharedquantitiescomputedbyothersubproblems,zi.
Sharedquantitiesforelementiconsistofsharedvariables(xsi),inputandoutputcouplingvariables(yijandyji),andsharedandcouplingvariablesthatlinkelementsthatarechildrenofelementi(xsCiandyCi).
Thecomponentsofziarexedparametersduringtheoptimizationofsubsystemi.
Sinceeachoptimizationproblemisdecoupled,wecansolveallofthesubproblemsataparticularlevelinparallel.
ApopularATCcoordinationstrategyistosolvethetoplevelproblem(withinitialguessesfortop-leveltargets),usetheresultstoupdatethetargetvaluesforthenextleveldown,solvetheproblemsinthesecondlevel,andsoonuntilthebottomlevelisreached.
Thislargeouterloopisrepeateduntilallofthedeviationvectorvaluesstopchanging.
Ecientpenaltyfunctionmethodscanspeedconvergence,andhavebeenshowntoproduceconvergenceinasfewas3outerloopiterations.
21,23B.
CollaborativeOptimizationTheCOmethodisdesignedtopromotedisciplinaryautonomywhileachievinginterdisciplinarycompatibilityinnon-hierarchicalproblems(Figure2).
Problemdecompositiontypicallyismadealonganalysis-convenientboundaries.
Asubspaceoptimizerisintegratedwitheachanalysis-block,andasystemoptimizercoordinatessubspacesolution.
Thisapproachdecouplesthesubspace,whileguidingtheprocesstowardaconsistentsolution.
Eachsubspacehascontroloverlocaldesignvariables,andischargedwithsatisfyingitsowndomain-specicconstraints.
AswithATC,discipline-specicoptimizationalgorithmsmaybeused.
Althoughadirectparallelbetweenthetop-levelprobleminanATCformulationandtheCOsystemoptimizermayseemtoexist,theseelementslldierentroles.
ThetopATCsubproblemissimilartoothersubproblems,exceptthatitstargetsarexed.
Itseekstobringthedesignofitssubsystemintoagreementwiththerestofthesystem;itdoesnotacttocoordinatethesolutionoftheentiresystemdesignproblem.
Aseparatecoordinationalgorithmdetermineswheneachsubproblemshouldbeexecuted,andguidesthesystemtowardconsistency.
TheCOsystemoptimizerisnotassociatedwithananalysisblockfromtheanalysisstructure(Figure2),asisanATCelementandlower-levelCOsubspaces.
ItsroleissimilartothatoftheATCcoordinationalgorithm;itguidestheentiresystemtowardconsistency.
3of12AmericanInstituteofAeronauticsandAstronauticsFigure2.
Non-hierarchicalsystemdecompositionThesystem-leveloptimizerguidesthesystemtowardanoptimalandconsistentsolutionbyminimiz-ingasystemobjectivefunctionf,whileenforcingsystemconsistencyviaauxiliaryconstraints(J=[J1,J2.
.
.
JN]T=0).
Requiringallsubspaceobjectivestobezeroatconvergenceresultsinconsistencybetweenallsharedandcouplingvariables.
Theauxiliaryconstraintsdecouplethesubspaces,facilitatingparallelexecution.
Systemoptimizationisperformedwithrespecttothesystemtargetsz.
Thesystemlevelsendssubspaceithetargetszi,asubsetofthesystemtargetspertinenttosubspacei.
Subspaceireturnsitsbestresponse,zi,tomeetthesesystemtargets.
Thesystemlevelconstraintsarethendenedbythe(squareofthe)dierencebetweentargetvalues,zi,andreturnedvalues,zi.
TheCOformulationisSystemLevelFormulationminzf(z)subjecttoJ(z)=0SubspaceFormulationminxsi,xi,yijJi(xsi,xi,yij)=zizi22subjecttogi(xsi,xi,yij)≤0hi(xsi,xi,yij)=0Atargetinthevectorziexistsforeverysharedvariablexsiusedinsubspaceiandforeveryinputyijandoutputyjicouplingvariable.
Subspacesdeterminethevalueforlocaldesignvariablesxi.
Theentirevariablesetinsubproblemiisxi=[zi,xi]=[xsi,yij,yji,xi].
ThesubspaceobjectiveJimeasuresthediscrepancybetweensubproblemtargetsandthecorrespondingresponses:Ji(xsi,xi,yij)=zizi22.
Localtargetsziarexedparameterssetbythesystemoptimizer,andthesubspaceoptimizerseekstomatchthesetargetsbyvaryingthelocalandshareddesignvariables,andtheinputcouplingvariables,subjecttolocaldesignconstraintsgi(xi)andhi(xi).
Theoutputcouplingvariablesyijarecomputedbasedonthesedecisionvariables,andareincorporatedintoJi.
Ateverysystemleveliteration,theoptimalvalueofthesubspaceobjectivefunctionJiispassedtothesystemoptimizerandusedasasystem-levelauxiliaryconstraint.
Thus,COisimplementedasanestedoptimizationprocess.
C.
DiscussionofCOandATCFormulationsAninitialcomparisonofCOandATCbasedonasingle-productdesignproblem12citeddierencesalongfourimportantdimensions:solutionprocess,targetsandcommunicationpatterns,intendedstructureofcorrespondingdesignproblems,andparadigm.
Thissectionprovidesanupdatedperspective.
ThefundamentaldierencebetweenCOandATCexistsintheoptimizationprocess.
COutilizesnestedoptimization,whileATCsolvesasequenceofoptimizationsubproblemsateachlevel.
InCO,thesystem-leveloptimizationproblemissolvedonlyonce,whilethesubspacesaresolvedmanytimes(onceduringeverysystem-leveliteration).
InATC,acoordinationstrategyinitializesthetop-leveloptimizationproblem(withinitialguessesfortop-leveltargets),usestheresultingsolutiontoupdatethetargetvaluesforthenextleveldown,initializestheproblemsinthesecondlevel,andsoonuntilthebottomlevelisreached.
Thisprocess4of12AmericanInstituteofAeronauticsandAstronauticsisrepeateduntilconvergence.
ThefundamentalprocessdierencebetweenCOandATCleadstoanumberofalgorithmicdistinctions.
Firstly,sinceeachelementinATCissolvedrepeatedly,inexactpenaltyrelaxationsmaybeusedinsteadofequalityconstraintstoensuresystemconsistency.
RelaxationalsohelpstheATCprocessmovemoreecientlytowardthesolution.
Secondly,analysisinATCisconductedatalllevels(includingthesystemlevel),whileanalysisinCOistypicallyconnedtothesubproblems.
ThismakesATCwellsuitedtoobject-baseddecomposition,whereeachelementinahierarchicalmultilevelsysteminvolvesanalysis.
Infact,hierarchicalanalysisstructuresmotivatethesolutionprocessofATC.
Thirdly,eachelementinATCmaypursuealocalobjectiveinadditiontostrivingforcompatibility.
InCO,thesubspaces'soleobjectiveistomatchtargetsprovidedbythesystemlevel.
Inadditiontotheiralgorithmicdistinctions,COandATCalsousedierenttechniquestoimproveeciency.
Forexample,anaugmentedLagrangianATCformulationwasrecentlyproposed,23whileCOoftenemploysresponsesurfacemodelsinthesubproblems.
13,24III.
DesignofanAircraftFamilyThepurposeofaproductfamilyistoreducecostbysharingcommoncomponentsorsystemstoaddressarelatedsetofmarketapplications.
Inspecic,theobjectivefunctionshouldbeabletodierentiatebetweenuniqueaircraftandproductfamilysolutions.
Maximumtakeoweight,oftenusedasanestimateforcost,willnotcapturetheadvantagesofcommonality.
Lifecyclecostisarigorousapproach,butismorecomplexthannecessary.
Theprimarygoalisnottoaccuratelypredicttotalcost,butrathertoquantifythebenetsofaproductfamilyanddenethepreliminarydesignofitsmembers.
Theidealobjectiveshouldincludecostmeasuresthatdistinguishbetweenuniqueaircraftandfamiliesofaircraft,namely,adetailedmodelofacquisitioncostandareasonableestimateoffuelcost.
TheacquisitioncostmodelusedinthispaperisbasedonrecentworkbyMarkish.
25Acquisitioncostissplitintomanufacturinganddevelopmentcosts.
Amanufacturinglearningcurveisappliedsuchthatcostdecreaseswiththenumberofunitsproduced.
Forexample,the100thunitcostslesstomanufacturethanthe1stunit.
Developmentcostisnon-recurringandisaveragedoverthetotalnumberofaircraftproduced.
Foreverypartofanewaircraftdesignthathasalreadybeendevelopedforanotheraircraft(i.
e.
,foranotheraircraftinthefamily),thenon-recurringcostissignicantlylower.
Thus,theeectsofcommonalityarecapturedbytheacquisitioncostmodel.
Manyairlinelaborcosts,suchaspensionplans,arerelativelyunaectedbyanairline'schoiceofaircrafteet.
Otherlaborcosts,however,suchascrewscheduling,training,andmaintenance,aresignicantlyimpactedbychoiceofaircrafteet.
26Thesecostsarediculttomodelandhavenotbeenincludedinthepresentcostmodel.
Thoughnotspecicallyaddressedinthispaper,productfamiliesprovideapotentiallysignicantbenetinthisarea.
Althoughfuelcostpredictionisaworthychallengeinitsownright,thisstudyusesaxedfuelpricepergallonasasimplication.
FuelcostisthencomputedbasedontheBreguetrangeequation.
27Insummary,theobjectivefunctionisacarefullyconstructedcostmeasurethatcapturesthekeydierencesbetweenuniqueaircraftandproductfamilysolutions,includingacquisitionandfuelcosts.
AircraftperformanceisevaluatedusingtheProgramforAircraftSynthesisStudies(PASS),anaircraftconceptualdesigntoolbasedonacollectionofMcDonnell-Douglasmethods,DATCOMcorrelations,andnewanalysesdevelopedspecicallyforconceptualdesignandperformance.
PASShasevolvedovermorethan15years.
28AdetaileddescriptionofthesemethodsmaybefoundonthewebsiteofanaircraftdesigncourseatStanfordUniversity.
29WhileexistingconceptualdesigntoolssuchasPASSarewell-suitedforthedesignofindividualaircraft,amoredetailedstructuralmodelisrequiredforaircraftfamilydesign.
Forexample,wingweightiscomputedusingthefollowingsemi-empiricalequationWwing=4.
22Swg+1.
642106Nultb3√WTOWZFW(1+2λ)(t/c)avgcos2(Λea)Swg(1+λ).
(1)Notethatwingweightisafunctionofwinggeometry(Swg,b,λ,etc.
)aswellasaircraftweight(WTO).
Thus,sharingacommonwinggeometryisnotsucienttoensurewingcommonality.
Anadditionalissueistheneedtocomputetheweightofindividualwingsectionssuchasrootandtipextensions.
Theseissuesassociatedwithwingcommonalitysuggesttheneedforamoredetailedwingweightmodel.
Whileaniteelementmodelwasanoption,thegoalwasalow-delitymodelconsistentwithexistingconceptualdesign5of12AmericanInstituteofAeronauticsandAstronauticstoolsthatcapturedthedesiredeectsandwascomputationallyecient.
Thesolutionwasasimplewing-boxmodelinwhichthewingskincarriedthebendingload.
Ananalysisestimatedtheloaddistributiononthewingandcomputedthematerialnecessarytoresisttheresultingbendingmoment.
Sincehigh-liftsystems,controlsurfaces,andminimumgaugematerialaddtothenalwingweight,anewequationwasdevelopedbasedon"bendingmaterial"andcorrelatedtoexistingaircraft.
Thisequationislistedbelow,whereWstristheweightofmaterialneededtoresistbending,Wministheweightofminimumgaugematerial,andSwingisthewingareaWwing=1.
35(WstrWmin)+4.
9Swing.
(2)Givenawingweightequationappropriateformodelingcommonalitybetweenfamilymembers,thenextstepwastoidentifyanappropriatemeansofparameterizingthewingforuseinadecomposedoptimizationproblem.
Thegoalwastominimizethedimensionalitywhileensuringcommonality.
Itwasnotedthatanapproximatelyquadraticrelationshipexistsbetweenskinthicknessandspanwiselocationinthesimplewingmodel.
Thisenabledathree-termparameterization,wheretheskinthicknesswasdenedatthefollowingspanwiselocations:wingroot(T1),33%span(T2),and67%span(T3)(ofthemainwingsection).
Thewingtipwasintentionallyavoidedinthisparameterizationsinceitisoftensizedbyminimumgaugerequirementsratherthanstressconstraints.
Thisyieldedthefollowingsetofsevenvariablesthatuniquelydenethemainwingsection:Swing,ARwing,λ,Λ,(t/c),T1,T2,andT3.
(Notethatthecurrentinvestigationfocusesoncommonalityofthemainwingsection,witheachaircraftallowedtohaveauniquewingtipextension.
Futureworkwillincludethecapabilityforwingrootandwingtipextensions.
)A.
ProblemStatementTheconsideredproductfamilyincludestwoaircrafttypes,AandB,designedtofulllmissions1and2,respectively.
Mission1requiresarangeof3400nauticalmiles(nmi)andanaircraftcapacityof296passengers.
Mission2requiresarangeof8200nmiandanaircraftcapacityof259passengers.
Forecastssuggestamarketneedfor800typeAaircraft,andaneedfor400typeBaircraft.
Inadditiontomissionrequirements,constraintssuchasbalancedeldlengthandsecondsegmentclimbareincluded.
TofacilitatecomparisonbetweentheCOandATCformulations,thesamebi-leveldecompositionisusedforboth.
Thesystem(productfamily)levelseekstominimizeacostmeasure,subjecttocompatibilityofcommonparts.
Thesubproblem(individualaircraft)levelseekstosatisfycompatibilitywhilemeetingindividualaircraftperformancerequirements.
Localdesignvariablesspecifyallportionsoftheaircraftnotsharedincommonwithotheraircraftinthefamily.
Componentcommonalityinthepresentstudyislimitedtothemainwing.
Eachfamilymemberhasthefreedomtospecifyitsownwingtipextensionarea.
Theaircraftfamilydesignproblemrequiresthespecicationof16designvariablesforeachofthetwoaircrafttypes.
Thedesignvariablesforeachaircraft(x1i.
.
.
x16i,i∈{A,B})aredescribedinTable1.
Theproductfamilydesignproblemimposestheconstraintthatthevariablesx10i.
.
.
x16iareequalforeachaircraft,sincethesepertaintothecommoncomponent—themainwing.
Thevectorofsharedvariablesisxs=[x10A.
.
.
x16A]T=[x10B.
.
.
x16B]T.
ThelocalvariablesforaircraftAandBarexA=[x1A.
.
.
x9A]TandxB=[x1B.
.
.
x9B]T.
Thecompletesetofdesignvariablesfortheproductfamilydesignproblemisx=[xTAxTBxTs]T.
Eachaircraftmustcomplywithasetofveperformanceconstraints,whosenumericvaluesarespecictothemissioneachaircraftisdesignedtoy(seeTable2).
Theobjectiveoftheaircraftfamilydesignproblemistominimizeacompositecostmetricforthefamily,wherethecostmetricforeachmissionisnormalizedbythenumberofaircraftthatyeachmission.
Thecostmetricmodelisbasedonanestimateofdirectandindirectoperatingcosts29withspecicattentiongiventoacquisitioncost.
25ThesystemobjectivefunctionisgiveninEquation(3),wherenAandnBarethenumberofaircraftAandBinthefamily,respectively,andpAandpBarethecostmetricsforeachaircraftf(x)=nAnA+nBpA(xA,xs)+nBnA+nBpB(xB,xs).
(3)6of12AmericanInstituteofAeronauticsandAstronauticsTable1.
DesignvariablesfortheaircraftfamilydesignproblemAircraftAAircraftBVariableNameDescriptionVariableBoundsVariableBoundsx1iWTOtakeoweight300,000-450,000lbs450,000-600,000lbsx2ithrustsealevelstaticthrust50,000-70,000lbs75,000-105,000lbsx3iXwinglocationofwingrootleadingedge0.
20-0.
400.
20-0.
40x4iSh/Srefnondimensionalhorizontaltailarea0.
20-0.
350.
20-0.
35x5iAltIinitialcruisealtitude32,000-45,000ft32,000-45,000ftx6iAltFnalcruisealtitude32,000-45,000ft32,000-45,000ftx7iMachMachnumberatstartofcruise0.
75-0.
920.
75-0.
92x8iflapTOtakeoapdeection0.
0-15.
00.
0-15.
0x9iSwtwingtipextensionarea0-125ft20-125ft2x10iSwmmainwingarea2000-4000ft22000-4000ft2x11iARwmmainwingaspectratio7.
0-12.
07.
0-12.
0x12i(t/c)thicknesstochordratio0.
80-0.
140.
80-0.
14x13iΛwingsweep20.
0-35.
020.
0-35.
0x14iT1skinthicknessatrootofmainwing0.
06-2.
50.
06-2.
5x15iT2skinthicknessat33%spanofmainwing0.
06-2.
00.
06-2.
0x16iT3skinthicknessat67%spanofmainwing0.
06-1.
50.
06-1.
5Table2.
DesignconstraintsfortheaircraftfamilydesignproblemConstraintNameDescriptionAircraftAAircraftBg1Rangeminrange3,400nmi8,200nmig2TOFLmaxtakeoeldlength7,000ft10,000ftg3LFLmaxlandingeldlength5,200ft6,000ftg4γ2min2ndseg.
climbgrad0.
0240.
024g5stabstabilityrequirement≥0≥0g6σ1normalizedstressatwingroot≤0≤0g7σ2normalizedstressat33%span≤0≤0g8σ3normalizedstressat67%span≤0≤0B.
ATCFormulationTheaircraftfamilydesignproblemisdecomposedintoabi-levelATCformulationwiththreeelements.
ThetoplevelproblemP1seekstoattainagreementbetweenthelower-levelsubproblemswithrespecttosharedvariables,whileminimizingtheproblemobjectivef.
Thetwolower-levelproblems,P2andP3,seektomatchtargetssetbyP1,whilemeetinglocaldesignperformanceconstraints.
P2correspondstothedesignofaircraftA,andP3correspondstothedesignofaircraftB.
AlternativeATCdecompositionsexist,butacomparisonoftheseisleftforfuturework.
ForclarityintheATCformulations,asuperscriptinparenthesesindicatesthesubprobleminwhichavalueiscomputed.
ProblemP1isformulatedasminx1=x(1)Tsp(1)Ap(1)BTfp(1)A,p(1)B+π(c1)where:π(c1)=vT1c1+w1c122c1=x(1)Tsx(1)Tsp(1)Ap(1)BTx(2)Tsx(3)Tsp(2)Ap(3)BT7of12AmericanInstituteofAeronauticsandAstronauticsThedeviationvectorc1quantiesthedierencebetweenthetargetssetbyP1andtheachievableresponsesofP2andP3.
TheresponsesarexedparameterswithrespecttoP1.
Notethatfisafunctiononlyoftargetcostmetrics,sincetheseareindependentdecisionvariablesinP1.
Thepenaltyfunctionπ(c1)guidestheATCprocesstowardconsistency.
Thelinearandquadraticpenaltyweights,v1andw1,areupdatedwitheveryexecutionofP1usingtheformulas:23wk+11=βwk1vk+11=vk1+2wk1wk1ck1Typically1<β<3andv01=0.
InthiscaseP1hasnoassociatedanalysis,andtheobjectiveisaquadraticfunction,enablingdirectsolutionwithouttheuseofanoptimizationalgorithm.
P1canbesolvedbyndingx1suchthatx1f1=0,wheref1=f+π.
ProblemP2isformulatedasminx2=x(2)Tsx(2)TATπ(c2)subjecttogA(x2)≤0where:π(c2)=vT2c2+w2c222c2=x(2)Tsp(2)ATx(1)Tsp(1)ATThepenaltyweightvectorsareupdatedusingthesamealgorithmdescribedabove.
TheformulationofProblemP3issimilar(onehassimplytoreplacesubscriptorsuperscript2with3andsubscriptAwithB).
C.
COFormulationTheCOformulationusesthesameproblempartitionasATC.
Eachsubspaceistaskedwithdesigningonememberofthefamily.
Thesubspacesseektomatchtargetssetbythesystemlevel,whilesatisfyinglocaldesignperformanceconstraints.
Thesystemlevelseekstominimizeafamilycostmeasure,whilesatisfyingcompatibilityofthesubspaces.
Thesystemproblemformulationisminz=[xTspApB]Tf(pA,pB)subjecttoJ(z)=0Thesharedvariablesandcostmetricsateachsystem-leveliterationarepassedtotheappropriatesubspaceasxedtargets.
Theformulationforeachsubspaceiisminxs,xiJi(xs,xi)=zizisubjecttogi(xs,xi)≤0where:i∈{A,B}zi=[xspi(xs,xi)]Tzi=[xspi]T(valuessetbysystemoptimizer)IV.
ImplementationandResultsThissectiondescribeshowtheATCandCOformulationswereimplementedtoobtainsolutionstotheaircraftfamilydesignproblem,andpresentsthecorrespondingresults.
AchallengecommontobothimplementationswasthepresenceofgradientdiscontinuitiesintheresponsesofthePASSanalysissoftware,whichprecipitatedslowconvergenceofgradient-basedalgorithmstosuboptimalpoints.
Thismotivatedtheuseofgradient-freealgorithmsforeachimplementation.
OnesourceofgradientdiscontinuitiesinPASSisthecalculationofworst-caseaerodynamicloadsonthewing,whichareafunctionofloadfactor.
Loadfactorisbasedonthelargeroftwoquantities:gustloadandmaneuverload.
Achangeincriticalloadcriteriacantriggerasignicantgradientdiscontinuity.
8of12AmericanInstituteofAeronauticsandAstronauticsA.
ATCImplementationTheATCsubproblemsP2andP3weresolvedusingNOMADm,30animplementationofmeshadaptivedirectsearch.
31,32Thisalgorithmeectivelyhandledthenon-smoothresponsesofthePASSanalysissoftware.
Themeshtoleranceusedindeterminingconvergencewas0.
001,andsubproblemoptimizationstypicallyrequiredbetween400and600functionevaluations.
ATCrequiredbetween8and18NOMADmoptimizationstoobtainasolution,dependingonthevaluechosenforβinthepenaltyupdates.
TheP1subproblemobjectivefunctionisquadratic,andrequiredverylittlecomputationaleorttosolve.
TwoapproacheswereusedtosolveP1:solvingforx1f1=0(wheref1=f+π(c)),andusingagradient-basedalgorithmtominimizef1.
Theformerwasextremelyecient,butthelatterprovedmorerobust.
SystemconsistencywasquantiedusingtherootmeansquareofthecombineddeviationvectorRMS(c)=1|c|cTc,wherec=cT1cT2cT3T,|c|=cardinalityofc.
TheconvergenceofATCisstronglyinuencedbythechoiceofβwhenthepenaltyupdatealgorithmdescribedintheprevioussectionisused.
Alargerβvaluecanhelpforcethesystemintotighterconsistency,butcanresultinastisystemthatrequiresmoreiterationstoconverge.
Theproblemwassolvedusingarangeofdierentβvaluestoillustratethisinuence.
Figure3illustrateshowlargervaluesofβrequiremoreiterationsoftheATCprocess.
Itwasalsoobservedthatlargerβvaluesledtoslightlylargerobjectivefunctionvalues,evenwhensystemconsistencywasapproximatelyequal.
Thisindicatesthatastisolutionprocesscanimpedetheidenticationofbetterdesigns.
Figure3.
InuenceofβonRMS(c)(systemconsistency)B.
COImplementationTheoriginalgoalwastouseSNOPT33asthesubproblem-andsystem-leveloptimizerinCO.
However,SNOPTyieldedsubspacesolutionsthatwereonlylooselyconverged.
GradientaccuracywasnotsucienttoenabletheuseofSNOPTforsystem-leveloptimization,butdidpermittheuseofagradient-freesystem-leveloptimizer.
Arobust(butcomputationallyexpensive)optionwasageneticalgorithm.
Whilenotideal,thisapproachrobustto"noise"fromthesubspacelevelandyieldedproof-of-conceptresultsthateectivelyillustratetheuseofdecomposition-basedmethodsforaircraftfamilydesign.
Futureworkwillfocuson9of12AmericanInstituteofAeronauticsandAstronauticsimplementingamoreecientalternativeforhandlingnon-smoothfunctions,suchasresponsesurfacemodelsofsubspaceresponses.
ResponsesurfaceshavebeensuccessfullyemployedwithCOtoresolveseveralcommonissueswithnon-smoothresponsesandslowsystem-levelconvergence.
13C.
ResultsBothCOandATCprovidereasonabledesigns,asdetailedinTable3.
Table3.
AircraftfamilydesignresultsAircraftAAircraftBNameSharedCOResultsATCResultsCOResultsATCResultspCost$377$390$1016$1072g1Range(nmi)3400340582008157g2TOFL(ft)639468021000010058g3LFL(ft)3197389335764399g4γ20.
0270.
0390.
0330.
031g5stab0.
0000.
0290.
0000.
102g6σ1-0.
560-0.
510-0.
121-0.
006g7σ2-0.
718-0.
500-0.
251-0.
008g8σ3-0.
302-1.
080.
000-0.
296x1WTO(lbs)3.
82·1053.
88·1055.
63·1056.
02·105x2thrust(lbs)5.
50·1046.
61·1040.
88·1051.
03·105x3Xwing0.
230.
250.
250.
28x4Sh/Sref0.
200.
240.
200.
26x5AltI(ft)3.
32·1043.
80·1043.
20·1043.
27·104x6AltF(ft)4.
20·1043.
32·1043.
90·1043.
64·104x7Mach0.
8380.
7890.
8240.
791x8flapTO(deg)7.
19.
21.
514.
9x9Swt(ft2)124.
912.
1104.
820.
7x10Swm(ft2)4.
00·1033.
20·1034.
00·1033.
30·103x11ARwm7.
69.
47.
69.
3x12(t/c)0.
1230.
1140.
1230.
110x13Λ(deg)33.
028.
533.
028.
7x14T1(in)1.
101.
501.
101.
50x15T2(in)1.
001.
101.
001.
13x16T3(in)0.
500.
730.
500.
72Thetwoapproachesyieldsimilar,butnotidentical,results.
Thedierencescanbeattributedtothedierentoptimizationalgorithmsusedforeachmethod,anddonotnecessarilyreectthecapabilitiesoftheCOandATCmethods.
Weemphasizethattheresultsarereportedtodemonstratetheapplicabilityofthetwoapproachesforsolvingthefamilydesignproblem.
Theyarenotpresentedasthesuggesteddesigns.
Additionalworkisrequiredbothinmodelingtheproblem(e.
g.
,accountingforadditionalpartcommonality)andinne-tuningimplementationsandoptimizationalgorithms.
V.
ClosingRemarksThefundamentaldierencebetweencollaborativeoptimization(CO)andanalyticaltargetcascading(ATC)isrelatedtotheoptimizationprocess.
COusesnestedoptimization,whereeachiterationofthesystem-levelproblemrequirescompletesubproblemoptimizations(bi-levelhierarchy).
ATCsolvesasequenceofoptimizationproblems,eachofwhichisassociatedwithanelementofamultilevelhierarchy.
Thisdierence10of12AmericanInstituteofAeronauticsandAstronauticsleadstoseveralalgorithmicdistinctionsandhasanimpactonthetypesofproblemsforwhicheachstrategyisbestsuited.
Aircraftfamilydesignusingdecompositionmethodsoersthesamebenetsaordedbydisciplinarydecomposition.
Whiledecompositionisonlymoderatelybenecialforthesimpleaircraftfamilyprobleminvestigatedinthispaper,itwouldbeadvantageous(ifnotessential)forhigherdelityanalysiswheresubproblem-specicoptimizationtechniquescanbeexploited,whereitisimpossibletointegrateexistingcodes,orwhereorganizationalstructuremayrequiredecomposition.
Thispaperhighlightsneedsandopportunitiesforfutureresearchwork.
WhilecomparativeworkhassoughttoidentifythekeydierencesbetweenCOandATC,additionalstudyisneededtoexploretheimplicationsofthesedierences.
DesignspacediscontinuitiespresentachallengeforMDOtechniques,emphasizingtheneedforbetterapproachestosubspaceoptimization.
Theexampleproblemdetailedhereisjustarststeptowardsaircraftfamilydesign.
Higherlevelsofcommonalityshouldbeconsideredtoenablecomparisonofproductfamilydesignsolutionstoindividualaircraftdesignsolutions.
AcknowledgmentsThisworkwaspartiallysupportedbyUSNSFGraduateResearchFellowshipsandbytheAutomotiveResearchCenter,aUSArmyCenterofExcellenceattheUniversityofMichigan.
TheauthorswouldliketothankMarkAbramsonforhisassistancewithNOMADm.
Anyopinionsexpressedinthispublicationareonlythoseoftheauthors.
References1Meyer,M.
andLehnerd,A.
,ThePowerofProductPlatforms,NewYork:IrwinProfessionalPublishing,1997.
2http://www.
airbus.
com/media/commonality.
asp,(AccessedMay25,2005).
3Bokulich,F.
,"BuildingaFamily,"AerospaceEngineeringOnline,SAEInternational,http://www.
sae.
org/aeromag/features/buildfamily/index.
htm,(AccessedMay25,2005).
4Fellini,R.
,AModel-BasedMethodologyforProductFamilyDesign,Ph.
D.
thesis,UniversityofMichigan,2003.
5Fellini,R.
,Kokkolaras,M.
,Michelena,N.
,Papalambros,P.
,Perez-Duarte,A.
,Saitou,K.
,andFeynes,P.
,"ASensitivity-BasedCommonalityStrategyforFamilyProductsofMildVariation,WithApplicationtoAutomotiveBodyStructures,"StructuralandMultidisciplinaryOptimization,Vol.
27,2004,pp.
89–96.
6Fellini,R.
,Kokkolaras,M.
,Papalambros,P.
Y.
,andPerez-Duarte,A.
,"PlatformSelectionunderPerformanceBoundsinOptimalDesignofProductFamilies,"JournalofMechanicalDesign,TransactionsOftheASME,Vol.
127,No.
4,2005,pp.
524–535.
7Fellini,R.
,Kokkolaras,M.
,andPapalambros,P.
Y.
,"QuantitativePlatformSelectioninOptimalDesignofProductFamilies,withApplicationtoAutomotiveEngineDesign,"JournalofEngineeringDesign,Vol.
17,No.
5,2006,pp.
429–446.
8Willcox,K.
andWakayama,S.
,"SimultaneousOptimizationofaMultiple-AircraftFamily,"JournalofAircraft,Vol.
40,No.
4,July-August2003.
9Kokkolaras,M.
,Fellini,R.
,Kim,H.
,Michelena,N.
,andPapalambros,P.
,"ExtensionoftheTargetCascadingFormulationtotheDesignofProductFamilies,"StructuralandMultidisciplinaryOptimization,Vol.
24,2002,pp.
293–301.
10Braun,R.
,CollaborativeOptimization:AnArchitectureForLarge-ScaleDistributedDesign,Ph.
D.
thesis,StanfordUniversity,April1996.
11Kim,H.
,TargetCascadinginOptimalSystemDesign,Ph.
D.
thesis,UniversityofMichigan,2001.
12Allison,J.
T.
,Kokkolaras,MichaelZawislak,M.
,andPapalambros,P.
Y.
,"OntheUseofAnalyticalTargetCascadingandCollaborativeOptimizationforComplexSystemDesign,"6thWorldConferenceonStructuralandMultidisciplinaryOptimization,May30–June32005.
13Sobieski.
,I.
,MultidisciplinaryDesignUsingCollaborativeOptimization,Ph.
D.
thesis,StanfordUniversity,1998.
14Sobieski,I.
andKroo,I.
,"AircraftDesignUsingCollaborativeOptimization,"34thAerospaceSciencesMeeting,Reno,NV,1996,AIAApaper96-0715.
15Braun,R.
,Gage,P.
,Kroo,I.
,andSobieski,I.
,"ImplementationandPerformanceIssuesinCollaborativeOptimization,"1996,AIAApaper96-4017.
16Kim,H.
M.
,Michelena,N.
F.
,Papalambros,P.
Y.
,andJiang,T.
,"TargetCascadinginOptimalSystemDesign,"JournalofMechanicalDesign,TransactionsOftheASME,Vol.
125,No.
3,2003,pp.
474–480.
17Kim,H.
,Kokkolaras,M.
,Louca,L.
,Delagrammatikas,G.
,Michelena,N.
,Filipi,Z.
,Papalambros,P.
,andAssanis,D.
,"TargetCascadinginVehicleRedesign:AClassVITruckStudy,"InternationalJournalofVehicleDesign,Vol.
29,No.
3,2002,pp.
199–225.
18Kokkolaras,M.
,Louca,L.
S.
,Delagrammatikas,G.
J.
,Michelena,N.
F.
,Filipi,Z.
S.
,Papalambros,P.
Y.
,Stein,J.
L.
,andAssanis,D.
N.
,"Simulation-basedoptimaldesignofheavytrucksbymodel-baseddecomposition:Anextensiveanalyticaltargetcascadingcasestudy,"InternationalJournalofHeavyVehicleSystems,Vol.
11,No.
3-4,2004,pp.
403–433.
19Choudhary,R.
,Malkawi,A.
,andPapalambros,P.
Y.
,"AnalyticTargetCascadinginSimulation-basedBuildingDesign,"AutomationinConstruction,Vol.
14,No.
4,2005,pp.
551–568.
11of12AmericanInstituteofAeronauticsandAstronautics20Michalek,J.
J.
,Feinberg,F.
M.
,andPapalambros,P.
Y.
,"LinkingMarketingandEngineeringProductDesignDecisionsviaAnalyticalTargetCascading,"JournalofProductInnovationManagement,Vol.
22,No.
1,2005,pp.
42–62.
21Allison,J.
,Walsh,D.
,Kokkolaras,M.
,Papalambros,P.
,andCartmell,M.
,"AnalyticalTargetCascadinginAircraftDesign,"44thAIAAAerospaceSciencesMeetingandExhibit,AIAA-2006-1325,Reno,Nevada,January9-122006.
22Allison,J.
,ComplexSystemOptimization:AReviewofAnalyticalTargetCascading,CollaborativeOptimization,andOtherFormulations,Master'sthesis,DepartmentofMechanicalEngineering,UniversityofMichigan,2004.
23Tosserams,S.
,Etman,L.
F.
P.
,Papalambros,P.
Y.
,andRooda,J.
E.
,"AnAugmentedLagrangianRelaxationforAnalyticalTargetCascadingUsingtheAlternatingDirectionMethodofMultipliers,"Vol.
31,2006,pp.
176–189.
24Kroo,I.
andManning,V.
,"CollaborativeOptimization:StatusandDirections,"8thAIAA/USAF/NASA/ISSMOSym-posiumonMultidisciplinaryAnalysisandOptimization,AIAA-2000-4721,LongBeach,California,September6-82000.
25Markish,J.
,ValuationTechniquesforCommercialAircraftProgramDesign,Master'sthesis,MassachusettsInstituteofTechnology,June2002.
26http://www.
jetblue.
com/ar2002/dierence.
html,(AccessedMay25,2005).
27Raymer,Daniel,P.
,AircraftDesign:AConceptualApproach,AIAAEducationalSeries,3rded.
,1999.
28Kroo,I.
,"AnInteractiveSystemforAircraftDesignandOptimization,"AIAAAerospaceDesignConference,Irvine,CA,Feb3-61992,AIAApaper92-1190.
29Kroo,I.
,"AA241—AircraftDesign:SynthesisandAnalysis,"CourseNotes[Online],URL:http://adg.
stanford.
edu/aa241/AircraftDesign.
html,January2005,(AccessedMay27,2005).
30"NOMADm:NonlinearOptimizationforMixedvAriablesandDerivativesforMatlab.
"http://www.
at.
edu/en/ENC/Faculty/MAbramson/nomadm.
html,(AccessedAugust8,2005).
31Audet,C.
andDennis,J.
,"MeshAdaptiveDirectSearchAlgorithmsforConstraintOptimization,"SIAMJournalonOptimization,Vol.
17,No.
1,2006,pp.
188–217.
32Abramson,M.
,Audet,C.
,andDennis,J.
,"NonlinearProgramingwithMeshAdaptiveDirectSearches,"SIAG/OptimizationViews-and-News,Vol.
17,No.
1,2006,pp.
2–11.
33Gill,P.
,Murray,W.
,andSaunders,M.
,"SNOPT:AnSQPAlgorithmforLarge-scaleConstrainedOptimization,"NumericalAnalysisReport97-2,DepartmentofMathematics,UniversityofCalifornia,SanDiego,Vol.
97-2,1997.
12of12AmericanInstituteofAeronauticsandAstronautics