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Multidisciplinary Design Optimization with a New Effective Method

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Multidisciplinary Design Optimization with a New Effective Methoda,A,with,New

    Multidisciplinary Design Optimization with

    a New Effective Method

    CHINESEJOURNALOFMECHANICALENGINEERTNG

    Vot.23,No.4,2010?505?

    DOI:1O.3901/CJME.2010.04.505.availableonlineatwww.cjmenet.com;www.cjmenet.corn.cn

    MultidisciplinaryDesignOptimizationwithaNewEffectiveMethod

    CHENXiaokai,LIBangguo

    ,

    andLINYi

    NationalEngineeringLaboratoryofElectricVehicle.BeijingInstituteofTechnology,Beijing100081.China

    ReceivedOctober28,2009;revisedMay24,2010;acceptedJune3,2010;publishedelectronicallyAugust20,2010

    Abstract:Collaborativeoptimization(CO)isoneofthemostwidelyusedmethodsinmultidisciplinarydesignoptimization(MDO),

    whichisaneffectivemethodologytosolvemodemcomplexengineeringproblems.COconsistsoftwoleveloptimizationproblems

    whicharesystemoptimizationproblemandsubspaceoptimizationproblem.ThearchitectureofCOcanreservetheautonomyof

    individualdisciplinesinmaximum,whileprovidingamechanismforcoordinatingdesignproblem.However,COhaslowcomputation

    efficiencyandiseasytodiverge.Forthepurposeofsolvingtheseproblems,theformerimprovedmethodswerestudied.Therelaxation

    factorswereusedtochangethesystemconsistencyconstraintstoinequalityconstraints,ortheresponsesurfaceestimationwasusedto

    surrogatethesystemconsistencyconstraints.However,thesemethodsdidn'tavoidthecomp

utationaldifficultiesverywell,furthermore,

    somenewproblemsarose.Theconceptofoptimumconstraintsensitivitywasproposed,andthequadraticconstraintsinsystemlevel

    werereforrned.Hence,anewcollaborativeoptimizationwasdeveloped.whichiscalledsystemleveldynamicconstraintcollaborative

    optimization(DCCO).Thenovelmethodisabletoincreasetheexchangeofinformationbetweensystemlevelanddisciplinaryleve1.

    Theoptimizationresultsofeachdisciplinaryoptimizationcanbefeedbacktosystemlevelwiththeoptimumconstraintsensitivity.On

    thebasisoftheinformation,thenewsystemlevellineardynamicconstraintscanbeconstructed;itisbettertoreflecttheeffectof

    disciplinaryleveloptimizations.Thesystemleveloptimizercanclearlycapturetheboundarywheredisciplinaryobjectivefunctions

    becomezero,andconsiderablyenhancetheconvergence.TwostandardMDOexampleswereconductedtoverifythefeasibilityand

    effectivenessofDCCO.TheresultsshowthatDCCOcansavethesolvingtime,andismuchbetterintermsofconvergenceand

    robustness,hence,thenewmethodismoreefficient.

    Keywords:multidisciplinarydesignoptimization(MDO),collaborativeoptimization(co),dynamicconstraint

    1Introduction

    Collaborativeoptimization(C0)isanewdesign

    architecturetotacklethelarge-scale,distributed-analysis

    applicationoftenfoundinindustryJJ_COwasoriginally

    proposedin1994.Itisoneofseveraldecompositionbased

    methodsthatdivideadesignproblemalongdisciplinary(or

    otherconvenient)boundaries.Itconsistsoftwolevel

    optimizationproblemswhicharesystemoptimization

    problemandsubspaceoptimizationproblem.System

    optimizeroptimizesthemultidisciplinaryvariable(system

    leveltarget1ztosatisfytheinterdisciplinaryconstraints whileminimizingthesystemobjective.Subspaceoptimizer minimizestheinterdisciplinarycompatibilityconstraints, whilesatisfyingthesubspaceconstraints.Relativetoother decompositionbasedmethods,COprovidesthe

    disciplinarysubspacewithanunusuallyhighlevelof autonomy[.

    ThebasicCOformulationiscomposedofsystemlevel andsubspacelevel,thesystemlevelisgivenbyEq.(1) Correspondingauthor.Email:peking008@126.corn

    ThisprojectissupposedbyNationalHitechResearchandDevelop

    mentProgramofChina(863Program,GrantNo.2006AA04Z119) =

    0,1,2,,,

    whereF(z)isglobalobjective,zisvariable(i.e.,system leveltargetsforsharedvariables),xs'issubspacetarget responsethatprovideseachsubspace'sbestattempttomeet thesystemleveltargets(z),anditisaparameterinsystem level,,2isthenumberofsubspaces.

    ThelowersubspacelevelisillustratedinEq.(2): =

    IIxs

    ci(x,x1)?0,

    (2)

    wherexisanindependentsharedvariable.x1isalocal variable.whichisrelativeonlytothelocalsubspace.On thebasisofanalyzingY=y(x,x1),Yiscouplingvariable, xs(xs=[x,Y])issharedvariable,zisaparameter, c(x1)isalocalconstraint.

    Thesubspaceobjectivetriestomatchtargetsforthe

    sharedvariablesthathavebeensentbythesystemlevelt. Thedependentvariablesinsubspaceleve1includeshared variabless)andlocalvariables1).Thesharedvariables ,

    ?

    I_

    __

    F

    Lm

    ,.....................

    .

    m

    m'

    ?506?CHENXiaokai,etal:MultidisciplinaryDesignOptimizationwithaNewEffectiveMe

    thod

    includebothindependentvariables(x)andcouplingproblemisgivenbyEq.(4)

    variables).

    COhasbeensuccessfullyappliedtoavarietyof mathematicalproblemsandengineeringdesignproblems, andusedfortheconceptualdesignoflaunchvehicles[,

    highspeedciviltransports[,andtmmannedaerial vehicles[.

    However,themethodalsosuffersfromsome

    challenges,whichhasbeendocumentedby

    ALEXANDROV,DEMIGUEL.etal[681.Theyhighlighted thefeaturesofC0thathasanadverseeffectonrobustness andcomputationalemciency.

    Threedifficultiesofthebi.1eveloptimizationproblem statedinEqs.(1)and(2)areconsidered.

    r1,ThesystemlevelJacobianissingularatthe

    solutiont.Thiscanbeseenbynotingthattheconstraint gradientsaregivenby=2.0(z一《).Evenwitha

    robustoptimizer,thishasanadverseimpactontherateof convergence.

    (2)TheLagrangemultipliersinthesubspaceproblem areeitherzeroesorconvergetozeroesaszconverges to.[9.Thisgreatlyaffectssubspaceconvergence. (3)Thesubspaceresponse()is,ingeneral,nonsmooth functionsofthetargetszAsaresult.thesystemlevel consaintsarenonsmooth.hinderinglocalandglobal convergenceproofsforthesystemlevelproblem. InCO,thesystemcompatibilityconstraintsareequality constraintsofquadraticforms.whichoften1eadtosome problemsofconvergence.Becauseofthequadraticequality constraints.COalsostronglydependsontheinitial conditionforconvergence.Inecientconvergenceisoften

    causedwhengradient.basedmethodisused.

    ThebasicconcepttoenhanceCOistomodifythe systemconstraints,whichcausetheconvergence difhcultiestJ.Thecurrentresearchisfocusedonusingthe natureofthesubspaceproblem,thereforetheoptimum consaintssensitivityispresentedtofindtheclosetpoint fromthetargetpoint,whilesatisfyingalldisciplinary constraints.

    2DescriptionoftheMethod

    AZARMandLI["gavetheformulationofatwolevel

    designoptimizationwithanseparableobjectiveand separableconstraints.TheformationisgivenbyEq.(3): m

in,)=f0(z)+?(z,),:?

    s.tel(z)?0,,=1,2,,

    .,(z,xi)?0,i=1,2,,;J:1,2,,,

    wherefisanintegratedobjective

    objectivefunctioninsubspacei.

    TheKarush?Kuhn-Tucker(KKT)

    (3)

    function,isan

    +

    L

    Oz)

    :++盟堕

    8z9z8z=:8x8z

    (4)

    Accordingtothetwoleveldesignoptimizationproblem, COCallbewrittenasanotherform.Systemlevelproblemis

    givenbyEq.(5),andsubspaceproblemsaregivenbyEq. f6):

    f,

    f(z,X):fo(+,

    (5)

    ls.t.el?0,,=1,2,,.

    23

    (6)

    TheKKToptimalityconditionforsubspacelevel optimizationproblemcanbewrittenasfollows: -o,

    lu?0,i=1,2,

    InCO,zisfixedandisvariedinsubspaceproblem, weshouldhave

塑堕一0xi:00z8x8Z(8)

    Likewise,theKKTconditionsforthesystemlevel optimizationproblemcanbewrittenasfollows: Ocl

    ~

    O,Ul_l,2,,,

    0Z

    (9)

    ForCO,thevariablesindisciplinaryoptimization problemconsistofsharedvariablesandlocalvariables,the

    KKTconditionsforsharedvariablesandlocalvariables canbewrittenas

    +0,

    8t

    +aa

    (10)

    conditionforthiswheresissharedvariableindisciplinaryproblemi,

    

    挑瓦

    ,

    ?

    ,

    ?

    CHINESEJOURNALOFMECHANICALENGINEERING?507? islocalvariableindisciplinaryproblemi. COsynergizesthedisciplinaryproblemvia variables,accordingtoEqs.(4)(9),aformulation

    obtainedasfcIllows:

    OCi

    8

    subspaceoptimizationproblemsandtreatedasasetof sharedfixedparameters.Thesubspaceoptimizationproblemsare canbethensolvedwhilesatisfyingthesubspaceconstraint. TheparameterxsandxIareoptimizedinthis

    optimization.

    onthebasisofxsandx,theoptimumconstraints f121sensitivityisobtainedas

    Oncethesharedvariableshavebeenidentified,Eq.(12) canbeusedt..btain.Likewise,Eq.(12)canbeused o'z

    byanoptimizationmethodwhichdoesnotyieldthevalue ofu.

    InCO,tomodifythesystemlevelconstraints,wedefine thederivativeofloca1constraintswhilethevariables are,Xlastheoptimumsensitivityofdisciplinary constraintsaccordingtotheideadescribedabove.Thatis ,

    Theoptimumsensitivityofdisciplina~_

    constraintscan

    reflectthechanginginformationofdisciplinaryconstraints. whichenablethesystemleveloptimizertoknowthe boundarywherethesubspaceobjectivesarezeroes. Throughtheoptimumconstraintssensitivity,thelinear dynamicconstraintsofsystemlevelcanbeconstructedby Taylorexpansionaroundthesubspaceoptimumasfollows: c(zl,z2,,z)=ci(x1,x2,,xn;Xll,l2,,l)+,,十?+},

    Oci

    OX1

    (z1?z)+c

    OX2

(z~'I(Oci()

    Wherefisthenumberofdisciplinaryoptimization problems.misthedimensionofloca1variablesx1.nis thedimensionofindependentsharedvariablesx. Thesenewconstraintsarelinearconstraintsof variablezinsystemleve1.whichcanavoidthe

    computationaldifficultiescausedbytheorigina1quadratic equationconstraints.cistheCOBStraintvaluewhen x=xandxl=x.whichisoptimalvalueofeach

    disciplinaryoptimization.Throughtheselineardynamic constraints,theoptimizedinforlrlationofsubspace optimizationcanbesenttothesystem1eve1.which reinforcestheexchangebetweensystemlevelandsubspace leve1.ThereformedC0jsreferredtoassystemlevel1inear dynamicconstraintscollaborativeoptimization(DCCO1. 3FlowofDCC0

    Thesolutionprocessbeginswithaninitialsetofsystem leveldesignvariablez0.Thisvariableissenttothe Thenonthebasisof

    ,

    ?

    1

    ,

    xs,xandTaylorexpansion,thelineardynamicsystem levelconstraintsareobtained.

    Thesystemleveloptimizerdetermineswhetherthe designvariablez0satisfiesthenewconstraints.Untilnow onewholeoptimizationisfinished.Theprocessisrepeated untilzreachestheoptimum.

    4AnalyticTestCaseandApplication

    ThissectionillustratestheapplicationofDCCO.The resultsofatypicalfunetionaloptimizationproblemanda gearreduceroptimizationproblemarecomparedwiththose obtainedviatheorigina1versionofCO.Al1problemswere solvedbysequentialquadraticprogramming(SQP)method basedonoptimizer:NPS0L.

    4.1Typicalfunctionoptimizationproblem

    BRAUN[solvedthistypicalfunctionoptimization problemviaoriginalversionofCO.Thisproblemisa constraintnonlinearproblem,anditsmathematicalmodel +,

    +Px24<0.

    

    PXlx2<0,

    where6isaparameter,and8=01Thisproblemis decomposedinthefollowingmanner.Thesystemlevel problemandsubspacelevelproblemaredescribed respectively.

    TheproblemissolvedbyoriginalversionofCO,and systemlevelproblemisasfollows:

    lminF(z)=z+z22,{Zl,z2

    S.t.J1<0.0001,J2<0.0001

    ,fJ.........J..

    

    ,

    ?

    ,

    ?

    

    =

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