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Reuther J., Jameson A., Farmer J., Martinelli L., Saunders D. Aerodinamix Shape Optimization of Complex Aircraft Configurations 1996

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NASA-CR-203275
ResearchInstituteforAdvancedComputerScience
NASAAmesResearchCenter
AerodynamicShapeOptimizationof
ComplexAircraftConfigurationsviaan
AdjointFormulation
JamesReuther,AntonyJameson,JamesFarmer,LuigiMartinelliandDavidSaunders
RIACSTechnicalReport96.02January1996
PresentedattheAIAA34thAerospaceSciencesMeetingandExhibit,January1996,
AIAApaper96-0094
AerodynamicShapeOptimizationof
ComplexAircraftConfigurationsviaan
AdjointFormulation
JamesReuther,AntonyJameson,JamesFarmer,LuigiMartinelliandDavidSaunders
TheResearchInstituteofAdvancedComputerScienceisoperatedbyUniversitiesSpaceResearch
Association,TheAmericanCityBuilding,Suite212,Columbia,MD21044,(410)730-2656
WorkreportedhereinwassponsoredbyNASAundercontractNAS2-13721betweenNASAandtheUniversities
SpaceResearchAssociation0dSRA).
AerodynamicShapeOptimizationof
ComplexAircraftConfigurationsviaanAdjointFormulation
J.Reuther*
ResearchInstituteforAdvancedComputerScience
NASAAmesResearchCenter,MS227-6
MoffettField,California94035,U.S.A.
A.Jamesont
DepartmentofMechanicalandAerospaceEngineering
PrincetonUniversity
Princeton,NewJersey08544,U.S.A.
J.Farmer
AdvancedCombustionandEngineeringResearchCenter
BrighamYoungUniversity
75JCTBP.O.Box24214ProvoUtah84602,U.S.A.
L.Martinelli_
DepartmentofMechanicalandAerospaceEngineering
PrincetonUniversity
Princeton,NewJersey08544,U.S.A.
D.Saunders
SterlingSoftware
NASAAmesResearchCenter,MS227-6
MoffettField,California94035,U.S.A.
ABSTRACT
ThisworkdescribestheImplementationofoptimizationtechniques
basedoncontroltheoryforcomplexaircraftconfigurations.Here
controltheoryisemployedtoderivetheadjointdifferentialequa-
tions,thesolutionofwhichallowsforadrasticreductionincom-
putationalcostsoverpreviousdesignmethods[13,12,43,38].In
ourearlierstudies[19,20,22,23,39,25,40,41,42]itwasshown
thatthismethodcouldbeusedtodeviseeffectiveoptimizationpro-
ceduresforairfoils,wingsandwing-bodiessubjecttoeitheranalytic
orarbitrarymeshes.Designformulationsforbothpotentialflows
andflowsgovernedbytheEulerequationshavebeendemonstrated,
showingthatsuchmethodscanbedevisedforvariousgoverning
equations[39,25].Inourmostrecentworks[40,42]themethod
wasextendedtotreatwing-bodyconfigurationswithalargenumber
ofmeshpoints,verifyingthatsignificantcomputationalsavingscan
begamedforpracticaldesignproblems.Inthispaperthemethodis
extendedfortheEulerequationstotreatcompleteaircraftconfigura-
tionsviaanewmultiblockimplementation.Newelementsinclude
amultiblock-multigridflowsolver,amultiblock-multigridadjoint
solver,andamultiblockmeshperturbationscheme.Twodesignex-
amplesarepresentedinwhichthenewmethodisusedforthewing
redesignofatransonicbusinessjet.
"StudentMemberAIAA
*JamesS.McDonnellDistinguishedUniversityProfessorofAerospaceEngineering,
A/AAFellow
*AIAAMember
INTRODUCTION
ToallowthefuUreaLizationofthepotentialofComputationalFluid
Dynamics(CFD)toproducesuperiordesigns,thereisaneednot
onlyforaccurateaerodynamicpredictionmethodsforgivencon-
figurations,butalsofordesignmethodscapableofcreatingnew
optimumconfigurations.Yet,whileflowanalysishasmaturedto
theextentthatNavier-Stokescalculationsareroutinelycannedout
oververycomplexconfigurations,directCFDbaseddesignisonly
justbeginningtobeusedinthetreatmentofmoderatelycomplex
three-dimensionalconfigurations.
ExistingCFDanalysismethodscanbeusedtotreatthedesign
problembycouplingthemwithnumericalopttmizationmethods.
Theessenceofthesemethods,whichmayincurheavycomputa-
tionalexpenses,isverysimple:anumericaloptimizationprocedure
isusedtoextremizeachosenaerodynamicfigureofmeritwhich
isevaluatedbythegivenCFDcode.Theconfigurationissystem-
aticaUymodifiedthroughuserspecifieddesignvariables.Mostof
theseoptimizationproceduresrequirethegradientofthecostfunc-
tionwithrespecttochangesinthedesignvariables.ThesImplestof
themethodstoobtainthesenecessarygradientsisthefinitedifference
method.Inthistechnique,thegradientcomponentsareestimated
byindependentlyperturbingeachdesignvariablewithafinitestep,
calculatingthecorrespondingvalueoftheobjectivefunctionusing
CFDanalysis,andformingtheratioofthedifferences.Thegradi-
entisusedbythenumericalopttmizationalgorithmtocalculatea
searchdirectionusingsteepestdescent,conjugategradient,orquasi-
Newtontechniques.Afterfindingtheminimumormaximumofthe
objectivefunctionalongthesearchdirection,theentireprocessis
repeateduntilthegradientapproacheszeroandfurtherimprovement
isnotpossible.
Thefinitedifferencebasedoptimizationstrategyiscomputation-
allyexpensivebecausetheflowmustberecalculatedforperturba-
tionsmeverydesignvari_ibletodeterminethegradient.Never-
theless,itisattractivewhencomparedwithothertraditionaldesign
strategiessuchasinversemethods,sinceitpermitsanychoiceof
theaerodynamicfigureofmerit.Theuseofnumericaloptimization
fortransonicaerodynamicshapedesignwaspioneeredbyHicks,
MurmanandVanderplaats[13].Theyappliedthemethodtotwo-
dimensionalprofiledesignsubjecttothepotentialflowequation.
ThemethodwasquicklyextendedtowingdesignbyHicksand
Henne[12].Later,intheworkofReuther,Cliff,HicksandVan
Dam,themethodwassuccessfullyusedforthedesignofsupersonic
wing-bodytransportconfigurations[38].Inallofthesecases,fi-
nitedifferencemethodswereusedtoobtaintherequiredgradient
mformation.
Recentlythroughworkbybothourselvesandothergroups,al-
ternative,lessexpensivemethodsforobtainingdesignsensitivities
havebeendevelopedwhichgreatlyreducethecomputationalcosts
ofoptimization.Themostpromisingoftheseemergingapproaches
istheadjointformulationwherebythesensitivitywithrespecttoan
arbitrarynumberofdesignvariablesisobtainedwiththeequivalent
ofonlyoneadditionalflowcalculation.
FORMULATIONOFTHEADJOINTEQUATIONS
TheaerodynamicpropertieswhichdefinethecostfunctionIare
functionsoftheflowfieldvariables(w)andthephysicallocation
oftheboundary,whichmayberepresentedbythefunction_,say.
Then
1=1(w,.7r)
andachangeinf"resultsinachange
151=°qlT_sw"k
OlT
Ow"ff_6_r(11
inthecostfunction.ThegoverningequationRanditsfirstvariation
expressthedependenceofwand.T"withintheflowfielddomainD:
R(w,_-)=o,6R=7ww_w+__-=0.(2)
Next,introducingaLagrangemultiplier_,wehave
61-oIT6w-kOIT_T
Choosing¢tosatisfytheadjointequation
0=0---_(3)
thefirsttermiseliminated,andwefindthatthedesiredgradientis
givenby
_ToITy)T[OR]
=0.7:"_.(4)
Since(4)isindependentof6w,thegradientofIwithrespectto
anarbitrarynumberofdesignvariablescanbedeterminedwithout
theneedforadditionalflowfieldevaluations.Themaincostisin
solvingtheadjointequation(3).Ingeneral,theadjointproblem
isaboutascomplexasaflowsolution.Ifthenumberofdesign
variablesislarge,itbecomescompellingtotakeadvantageofthe
costdifferentialbetweenoneadjointsolutionandthelargenumber
offlowfieldevaluationsrequiredtodeterminethegradientbyfinite
differences.Onceequation(4)isobtained,Gcanbefedintoany
numericaloptimizationalgorithmtoobtainanimproveddesign.
ISSUESOFIMPORTANCEFORDESIGNPROBLEMS
Thedevelopmentofaerodynamicdesignproceduresthatemployan
adjointequationformulationiscurrentlybeinginvestigatedbymany
researchers.Thesemethodspromisetoallowcomputationalfluid
dynamicsmethodstobecometrueaerodynamicdesignmethods.
References[1,2,3,5,4,7,8,6,32,29,16,35,30,28,34,45,31,
36,37,47,15,33,14]representapartiallistofrecentworksinthis
developingfield.However,asisthecaseinanynewresearchfield,
manyquestionsremain.Probablythemostsalientissuesofconcern
arethefollowing:
1.Discretevs.continuoussensitivities
2.Choiceofoplamizationprocedures
3.Treatmentofgeometricandaerodynamicconstraints
4.Thelevelofcouplingbetweendesignandanalysis
5.Theparameterizationofthedesignspace
Thesetopicsstillrequirefurtherinvestigation.Withregardtothe
firstitem,itishistoricallyinterestingthatJamesonm1988[19]first
developedtheequationsnecessaryforacontinuoussensitivityap-
proachtotreatthedesignofairfoilsandwingssubjecttotransonic
flows.Thistechniquewaslaterimplementedbothbyourgroupand
independentlybyLewisandAgarwal[33].Bycontinuoussensitiv-
itiesitisimpliedthatthestepsrepresentedbyequations(1)-(4)are
appliedtothegoverningdifferentialequations.Theadjointdifferen-
tialequationswiththeappropriateboundaryconditionsmaythenbe
discretizedandsolvedinamannersimilartothatusedfortheflow
solutionalgorithm.Onemayalternativelyderiveasetofdiscretead-
jointequationsdirectlyfromthediscreteapproximationtotheflow
equationsbyfollowingtheprocedureoutlinedinequations(1)-(4).
Theresultingdiscreteadjointequationsareoneofthepossibledis-
cretizationsofthecontinuousadjointequations.Thisalternativeis
mentionedin[20],butwasnotadoptedinthatworkbecauseofthe
complexityoftheresultingdiscreteadjointsystem.Theapproach
hasbeenfavoredbyTayloretal.[29,16,35,30,28]andBaysalet
al.[1,2,3,5,4,7,8,6,321.
Itseemsthatbothalternativeshavesomeadvantages.Thecon-
tinuousapproachgivestheresearchersomehopeforanintuitive
understandingoftheadjointsystemanditsrelatedboundarycondi-
tions.Thediscreteapproach,intheory,maintainsperfectalgebraic
consistencyatthediscretelevel.Ifproperlyimplemented,itwill
givegradientswhichcloselymatchthoseobtainedthroughfinitedif-
ferences.Thecontinuousformulationproducesslightlyinaccurate
gradientsduetodifferencesinthediscretization.However,these
inaccuraciesareofcomparablemagnitudetotheinaccuraciesinthe
flowsolutionitself,andmustvanishasthemeshwidthisreduced.
Thediscreteadjointequationsformalinearsystem,whetherde-
riveddirectlybythediscreteapproachorbydiscretizatonofthe
continuousadjointequation.Thesizeandcomplexityofthesystem,
however,makestheuseofdirectsolutionmethodsunrealisticfor
allbutthesmallestproblems.Theapplicationofthecontinuous
sensitivityanalysisfostersastraightforwardrecyclingoftheflow
solutionalgorithmforthesolutionoftheadjointequations,sincethe
stepsappliedtotheoriginalgoverningdifferentialequationscanbe
duplicatedfortheadjointdifferentialequations.Whenthediscrete
approachisused,theadjointequationshaveacomplexitywhich
makesithardtofinddecompositionstofacilitatetheirsolutionun-
lessthestructurefromthecontinuousadjointisusedasaguide.The
discretemethodissubject,moreover,tothedifficultythatthediscrete
flowequationsoftencontainnonlinearfluxlimitingfunctionswhich
arenotdifferentiable.Italsolimitstheflexibilitytouseadaptive
discretizationtechniqueswithorderandmeshrefinement,suchas
theh-pmethod,becausetheadjointdiscretizationisfixedbythe
flowdiscretization.Itiscrucialforthesuccessofagradientmethod
thatthecostfunctiondependscontinuouslyonthedesignvariables.
However,eventhoughthetrueflowsolutiondependscontinuously
onthedesignvariables,theuseofadaptivediscretizationornonlin-
earfluxlimitersmaycausethediscretesolutiontoceasetodepend
continuouslyonthedesignvariablesbecauseofsuddenchangesin
thediscretization.
Itturnsoutthatthedeterminationofitems2-4intheabovelist
stronglyhingeonthechoicefor5.InJameson'sfirstworksinthe
area[19,20,22],everysurfacemeshpointwasusedasadesign
variable.Inthree-dimensionalwingdesigncasesthisledtoasmany
as4,224designvariables[22].Theuseoftheadjointmethodelim-
inatedtheunacceptablecoststhatsuchalargenumberofdesign
variableswouldincurfortraditionalfinitedifferencemethods.Ifthe
approachwereextendedtotreatcompleteaircraftconfigurations,at
leasttensofthousandsofdesignvariableswouldbenecessary.Such
alargenumberofdesignvariablesalsoprecludestheuseofdescent
algorithmssuchasNewtonorquasi-Newtonapproachessimplybe-
causeofthehighcostofmatrixoperationsforsuchmethods.The
useofasimpledescentprocedure,suchassteepestdescent,hasthe
advantage,moreover,thatsignificanterrorscanbetoleratedinthe
gradientevaluationduringtheearlysteps.Suchmethodstherefore
favortightercouplingoftheflowsolver,theadjointsolver,and
theoveralldesignproblemtoaccelerateconvergence.Ta'asanet
al.[31,45]havetakenadvantageofthisbyformulatingthedesign
problemasa"oneshot"procedurewhereallthreesystemsaread-
vancedsimultaneously.Theuseofsuchadesignspacecanlead
topoorlyconditioneddesignproblems.Thiscanbeillustratedby
thecasewhereonlyonepointonthesurfaceofthegeometryis
moved,resultinginahighlynonlineardesignresponse.Inhisorig-
inalwork,Jameson[20,21]addressedthispoorconditioningofthe
designproblembysmoothingthecontrol(surfaceshape)andthus
attenuatingthehighfrequencycontentinthedevelopingsolution
shape.
Hicksandothers[13,12,43,38]haveinthepastparameterized
thedesignspaceusingsetsofsmoothfunctionsthatperturbthe
initialgeometry.Byusingsuchaparameterizationitispossible
toworkwithconsiderablyfewerdesignvariablesthanthechoice
ofeverymeshpoint.AndsinceHicks"originalworksexclusively
usedfinitedifferencegradients,theinherentrestrictiontoasmall
numberofdesignvariablesallowedfortheuseofmoreefficient
searchslrategies.Onesimplechoiceofdesignvariablesforairfoils
suggestedbyHicksandHenne[12]hasthefollowing"sinebump"
form:
Herehlocatesthemaximumofthebumpintherange0<z<1
atz=h,sincethemaximumoccurswhenx'_=1wherea=
5'
log½/logtl,oralogtl=log½.Parametert2controlsthewidthof
thebump.
Whendistributedovertheentireconfiguration,suchanalyticper-
turbationfunctionsadmitalargereachabledesignspace.Theycan
bechosensuchthatsymmetry,thickness,orvolumecanbeexplicitly
constrained,thusavoidingtheuseofelaborateconstrainedoptimiza-
tionalgorithmstoimposegeometricconstraints.Further,particular
choicesofthesevariableswillconcentratethedesigneffortinregions
whererefinementisneeded,whileleavingtherestofthegeometry
virtuallyundisturbed.Thedisadvantageofthesefunctionsisthat
theyarenotorthogonal,andthereisnosimplewaytoformabasis
fromthesefunctionswhichiscompleteforthespaceofcontinuous
functionswhichvanishatar=0andz=1.Thus,theydonot
guaranteethatasolution,forexample,oftheinverseproblemfora
realizabletargetpressuredistributionwillbeattained.Nevertheless,
theyhaveprovedtobequiteeffectiveinrealizingdesignimprove-
mentswithalimitednumberofdesignvariables.Thedesignprocess
thatusesthesebasisspaces,canbeacceleratedtowardconvergence
eitherbytightercouplingoftheindividualdesignelements,aswas
thecasewhenusingthemeshpointsthemselves,orthroughtheuse
ofhigherorderoptimizationalgorithms.Finally,theyhavethead-
vantagethatthereisnoneedtosmooththeresultingsolutionsasthe
designproceeds,sincebyconstructionhigherfrequenciesarenot
admitted,andthusthedesignspacesarenaturallywellposed.
AnotherapproachistheuseofB-splinecontrolpointsasdesign
variables[5,47,15].TheseinturndefineB-splinecurvesand
surfaceswhichrepresentthegeometricconfigurations.Likethe
Hicks-Hennefunctions,B-splinesallowforagreatlyreducednumber
ofdesignvariables,andthuspermittheuseof,say,aquasi-Newton
designprocedure.Ifforexampletheupperandlowersurfaces
ofanairfoilaretreatedseparately,themethodadmitscamberor
thicknessconstraintsexplicitlywithinthedesignspace.Further,
localcontrolisalsopossiblebychoosingonlyalimitednumberof
controlpointsasactivedesignvariables.Thismethodinpractice
seemstohaveanadvantageovertheHicks-Hennefunctionsinthat
amorecompletebasisspaceofadmitteddesignsispermittedfora
givennumberofdesignvariables.Finally,sincethesecurvesand
surfacesarethenaturalentitiesusedinCADenvironments,they
provideastraightforwardwayofintegratingCADandaerodynamic
design.Despitetheseadvantages,itwasshowninourrecentstudy
[41]that,incontrasttoHicks-Hennefunctions,B-splinecon_ol
pointshaveatendencytoproducewavysolutionsinproblemsthat
requirealargenumberofcontrolpoints.Thisdifficultycanbe
attributedtothefactthattheyadmithighfrequencycomponentsin
thedesignvariations,ratherlikethosewhichareencounteredwhen
themeshpointsareusedasdesignvariables.Apossiblesolutionto
thisdifficultywouldbetousethesametypeofimplicitsmoothing
procedurethatprovedeffectivewhentheshapewasdefinedbythe
meshpoints.
MULTIBLOCKFLOWSOLUTION
Theextensionofthemethodspresentedinourearlierworksinthree-
dimensions,suchthattheymaytreatcompleteaircraftconfigura-
tions,requiresthereplacementofthesingleblockflowsolverusedin
References[22,40,42].InordertouseCFDinanautomateddesign
environment,theflowsolvermustmeetfundamentalrequirements
ofaccuracy,efficiency,androbustconvergence.Highaccuracyis
requiredsincethepredictedimprovementsinthedesignrealizedby
themethodcanonlybeasgoodastheaccuracyoftheflowanalysis.
Efficiencyoftheflowsolverisalsocriticalsincetheoptimization
ofthedesignwillgenerallyrequirethecomputationofmanyflow
solutionsorothersolutionsofcomparablecomplexity.Thelastas-
peckrobustconvergence,isalsoofsignificantimportance.Inhighly
refinedaerodynamicdesignapplications,themainbenefitofaero-
dynamicoptimizationisinobtainingthelastfewpercentagepoints
inimprovedefficiency.Insuchcasesthesolutionsmustbehighly
convergedsothatthenoiseinthefigureofmerit,sayofdragata
fixedlift,iswellbelowthelevelofrealizableimprovement.Thusin
contrasttoflowanalysiswhere3ordersofmagnitudeconvergence
inRMSresidualisusuallyconsideredadequate,aflowsolverused
indesignapplicationsmustbetypicallyabletoconverge7ordersin
R.MSresidual.
Inourthree-dimensionalsingleblockapplicationstheFLO87
codewrittenbythesecondauthoreasilymetalloftheabovecriteria.
FLO87achievesfastconvergencewiththeaidofmultigriddingand
residualsmoothing.Itisnormallyeasytoobtainsolutionsthatcon-
vergetomachineaccuracy.Thechallengeinthepresentworkwas
tomeettheseslrictconvergencerequirementswithintheframework
ofamultiblockflowsolver.
Thegeneralstrategyindevelopingthemultiblockflowsolveris
toconstructandupdateahaloaroundeachblocksuchthattheflow
solutioninsideeachblockistransparenttotheblockboundaries.
Thistaskrequiresestablishingthesizeandlocationofhalocells
adjacenttoblockboundaries,andloadingthehalocellvalueswith
appropriateflowfielddataattheappropriatetime.Toaccomplish
thistask,atwo-levelhaloisconstructedaroundeachblock.At
interiorboundarieswheretwoormoreblocksmeet,thevaluesofthe
statevectorsinthehalocellsareidenticaltothevaluesintheinternal
cellsfromadjacentblocks.Halocellsontheexternalboundaryof
theentirecomputationaldomainareconstructedandupdatedby
extrapolationandreflection.Oncethehaloconfigurationissetup
foreachblock,standardmethodsforspatialdiscretizationandtime
integration(includingartificialdissipation,residualaveraging,and
multigridding)areemployedtocomputetheflowsolutionwithin
eachindividualblock.
Thestrategyforacompleteflowsolutionproceedsasfollows:
First,theblocksthatcomprisetheflowfieldmesharereadfroman
externalfile.Then,thedoublehaloconfigurationisestablished,for
eachindividualblock,byinsertingintohalocelllocationsvaluesfor
gridmetrics,etc.,takenfromtheinteriorceilsofadjacentblocks.
Forthecoarsegiidsrequiredinthemultigridprocedure,theprocess
isrepeatedwithcoarsegridhalocellsdefinedbytheinternalceils
ofadjacentcoarsegridblocks.Forblockfacesthatlieonsolid,
symmetryorfarfieldboundaries,standardsingle-blocktechniques
areusedtodefinethehalocells.Asanexample,considerthesimple
4-blockgriddepictedinFigure1.ThehaloceilsforblockIwillbe
obtainedfromtheinternalceilsofblocks1-1,1]],andIV,andfrom
solidorfarfieldboundarytechniquesforthefacesnotadjacentto
otherblocks.Coarsegridsarecomputedintheusualfashion,by
aggregatinggroupsofeightcellsandthenrepeatingtheabovehalo
cellprocess.Oncethehaloconfigurationiscompleteforthefineand
allcoarsegrids,theflowsolutioncommences.
Thesystemofequationssolvedhereaswellasthesolutionstrategy
followsthatpresentedinmanyearlierworks[26,18,17].Thethree-
dimensionalEulerequationsmaybewrittenas
OwOf,
O---_-+_-z=0inD,(6)
whereitisconvenienttodenotetheCartesiancoordinatesandve-
locitycomponentsbyxl,z2,z3andul,u2,u3,andwandf,are
definedas
///
pul]puiul+p6,1
W=put',f,=pUiU2"[-pSi2(7)
pu3]puiu3-_-pt_i3
pEpuiH
with6ubeingtheKroneckerdeltafunction.Also,
p=(7--1)pE-_
and
oH=pE+p(9)
where7istheratioofthespecificheats.Consideratransformation
tocoordinates_1,(2,_3where
K,,:J=det(K),n,,'l=
lO=_j•
Introducescaledcontravariantvelocitycomponentsas
UI_Qi3B5
where
Q=jK-t.
TheEulerequationscannowbewrittenas
OWOF,
o-7-+Tff=oinm,
with
(10)
W=J_
ppU,
pulpU,ul+Qilp
pU2,ft=Quf.7=pUiu2@Qi2p"
pu3pU,u3+Qi3p
pE,pU,H
(ll)
Forthemultiblockcase,theabovenotationappliestoeachblock
inturn.Theflowisthusdeterminedasthesteadystatesolutionto
equation(10)inallblockssubjecttotheflowtangencyconditions
onallsolidboundaryfacesofallblocks:
U,=0onailBs(12)
where_is1,2,or3dependingonthedirectionthatisnormalto
faceBswhereasolidsurfaceisassumedtoexist.Atthefarfield
boundaryfaces,BF,freestreamconditionsaxespecifiedforincom-
ingwaves,whileoutgoingwavesaredeterminedbythesolution.
Thetimeintegrationschemefollowsthatusedinthesingleblock
slxategy[26].Thesolutionproceedsbyperformingthecellflux
balance,updatingtheflowvariables,andsmoothingtheresiduals,at
eachstageofthetamesteppingschemeandeachlevelofthemultigrid
cycle.Themaredifferenceintheintegrationstrategyistheneedto
loopoverallblocksduringeachstageoftheintegrationprocess.The
useofthedouble-haloconfigurationpermitsstandardsingle-block
subroutinestobeused,withoutmodification,forthecomputationof
theflowfieldwithineachindividualblock.Thisincludesthesingle-
blocksubroutinesforconvectiveanddissipativefluxdiscretization,
multistagetimestepping,andmultigridconvergenceacceleration.
Theonlydifferencebetweentheintegrationstrategiesisinthe
implementationoftheresidualaveraging.Inthesingle-blocksolu-
tionstrategy,atridiagonalsystemofequationsissetupandsolved
usingflowinformationfromtheentiregrid.Thus,eachresidualis
replacedbyanaverageofitselfandtheresidualsoftheentiregrid.
Inthemultiblockstrategy,thesupportfortheresidualsmoothingis
reducedtothesizeofeachblock,inordertoeliminatetheneedto
solvetridiagonalsystemsspanningtheblocks,whichwouldincura
penaltyincommunicationcosts.Thischangehasnoeffectonthe
finalconvergedsolution,andinthepresentapplicationhasnotled
toanysignificantreductionintherateofconvergence.
THEADJOINTFORMULATIONFORTHEEULEREQUA-
TIONS
Theapplicationofcontroltheorytoaerodynamicdesignproblems
isillustratedbytreatingthecaseofthree-dimensionaldesign,using
theEulerequationsdiscussedaboveasthemathematicalmodelfor
compressibleflow.Inourpreviousworks,theillustrativeproblem
mostoftenusedspecifiedthecostfunctionasameasureofthedif-
ferencebetweenthecurrentandsomedesiredpressuredislribution.
Forvariety,thedevelopmentherewillusedragatafixedliftasthe
costfunction.
I=CD
=CAcosa+CNsina
Srefs
whereS_andSudefineprojectedsurfaceareas,Srefisthereference
area,andd_landd_2arethetwocoordinateindicesthatareinthe
planeofthefaceinquestion.Notethattheintegralinthefinal
expressionaboveisintegratedoverallsolidboundaryfaces.The
designproblemisnowtreatedasacontrolproblemwherethecontrol
functionisthegeometryshape,whichistobechosentominimize
I,subjecttotheconstraintsdefinedbytheflowequations(6-11).A
variationintheshapewillcauseavariation8pinthepressureand
consequentlyavariationinthecostfunction
8I=_CACOS_+6Cysina
+{-CASino_+CNCOSO_}$ot
OCNsino_'_8a
+{_aAcosa+0_
)
where6CAand6Cyarevariationsduetochangesinthedesign
parameterswith_fixed.Totreattheinterestingproblemofpractical
design,dragmustbeminimizedatafixedliftcoefficient.Thusan
additionalconsl_aintisgivenby
whichgives
6CL=0,
6Cycosa-$CAsino
+{-CNsinc_-CAcosa}8c_
{OCt,OCA}
+---0_--ccosc_---_--ff-csin_8_=0
Combiningthesetwoexpressionstoeliminate&rgives
61=_CAcosc_+6Cysin
+_{_Cycos,_-_C_sina'}
(13)
where_isgivenby
_=(CL+_COS_+%_--e_'sin_)
oca.c0(CD--_coso_+o_sm
Sincepdependsonwthroughtheequationofstate(8-9),thevaria-
tion8pcanbedeterminedfromthevariation8w.Ifafixedcomputa-
tionaldomainisused,thevariationsintheshaperesultinvariations
inthemappingderivatives.DefinetheJacobianmatrices
Of,
A,=_w'C,=Q,sAj.(14)
Thentheequationfor6winthesteadystatebecomes
0
0_--_(SF,)=o,
where
8F,=c,8 +(Q,)f,.
Now,multiplyingbyavectorco-statevariable_b,assumingtheresult
isdifferentiable,andintegratingbypartsovertheentiredomain,
/D(o_TfF'_d"=/B(fi_TfFi)d'''\O'i/I(15)
wherefiiarecomponentsofaunitvectornormaltotheboundary.
Thevariationinthecostfunctioncanalsobeexpressedintermsof
8pafter(13)and(15)axesummedtogive,
61=
_-¢Mo_Srefs
+_'_(Svcosc_-SxsinoL)}d_ld_2
Srefs
(16)
OnthesolidsurfacesBs,fil=fi2=0.Itfollowsfromequation
(12)that
/SF,=
o{o
Q,__p/5(Q,,)
Q,:tv+v6(Q,:)
Q,_3/Sp,5(Q.3)
00
onanyBs.
(17)
Supposenowthat_/,isthesteadystatesolutionoftheadjoint
equation
a__ra¢
atv,o_i=0inD.(18)
Atinternalblockboundaries,thefaceintegralscancelfromthead-
jacentblocks.Atthefarfieldthechoiceoftheadjointboundary
conditionsdependsonwhethertheflowissubsonicorsupersonic.
Forsubsonicflow,solongastheouterdomainisveryfarfromthe
configurationofinterest,wemayset
_,l-s=OonallBr.
Ifhowevertheflowissubsonicandtheboundaryisfairlyclose,
thenfarfieldfacesmaybesetbytb_-s=0forincomingwaves,
whileoutgoingwavesaredeterminedbythesolution.Itisnoted
thatthewavesintheadjointproblempropagateintheopposite
directiontothoseintheflowproblemduetothela'ansposeinequation
(18).Forsupersonicflows,thechoiceofboundaryconditionsatthe
outerdomaincanbedevelopedfromphysicalintuitionaswellas
mathematics.Foragivengeometry,sayawing,achangeinthe
surfaceatanyparticularpoint,"P,willincurchangesintheflow
fieldandhencetheperformanceintheregiondefinedbytheMach
coneoriginatingatP.Similarly,itispossibletodeterminethe
regionoverwhichsurfacechangesaffecttheflowconditionata
givenpoint.Thisregionwouldalsoformaconethatwouldpoint
roughlymtheoppositedirectionoftheMachcone,dependingon
localconditions.Itisthesolutionofthisreverseproblemthatthe
adjointrepresents.Thecontributionto,say,dragatagivenpointis
influencedbychangestothesurfaceatallpointswithinthereverse
cone.Thecorrectsupersonicfarfieldboundaryconditionsforthe
adjointequationthatareconsistentwiththisreversedcharacterare:
_/,1-5=0attheexit;
_b1-5extrapolatedfromtheinteriorattheenlxance.
Thenffthecoordinatetransformationissuchthat/5(JK-_)isneg-
ligibleinthefarfield,thelastintegralin(16)reducesto
-//CT6Fnd_ld&.(19)
JJB
Thusbylettingq_satisfytheboundaryconditions.
(02Q.1+_3Qn2+y54Qn3)=QonanBs,(20)
where
1
Q-,2{(S_cos,+Susin,)
-rMooSref
+n(S_cos_-Sxsino)},
wefindafterintegratingbypartsagainthat
1//.61=_Cv{(SSxcosa+6Svsinc_)
S
+f
(21)
MULTIBLOCKMESHVARIATIONSANDDESIGNVARI-
ABLES
Inordertoconstruct/51inequation(21),thevariationinthemetric
termsmustbeobtainedineachblock.Onewaytoaccomplishthisis
tousefinitedifferencestocalculatethenecessaryinformation.This
approachavoidstheuseofmultipleflowsolutionstodeterminethe
gradient,butitunfortunatelystillrequiresthemeshgeneratortobe
usedrepeatedly.Thenumberofmeshsolutionsrequiredispropor-
tionaltothenumberofdesignvariables.Theinherentdifficultyin
theapproachistwo-fold.First,forcomplicatedthree-dimensional
configurations,ellipticorhyperbolicpartialdifferentialequations
mustoftenbesolvediterativelyinordertoobtainacceptablysmooth
meshes.Theseiterativemeshgenerationproceduresareoftencom-
putationallyexpensive.Intheworstcasetheyapproachthecostof
theflowsolutionprocess.Thustheuseoffinitedifferencemeth-
odsforobtainingmetricvariationsincombinationwithaniterative
meshgeneratorleadstocomputationalcostswhichs_onglyhingeon
thenumberofdesignvariables,despitetheuseofanadjointsolver
toeliminatetheflowvariablevariations.Second,multiblockmesh
generationisbynomeansaa'ivialtask.Infactnomethodcurrently
existsthatallowsthistobeaccomplishedasacompletelyautomatic
processforcomplexthree-dimensionalconfigurations.
Inourearlierworks[40,39,25,19,20,22],twomethodshave
beenexploredwhichavoidthesedifficulties.Inthefirstmethod,
acompletelyanalyticmappingprocedurewasusedforthemesh
generation.Thistechniqueisnotonlyfullyautomatacandresultsin
smoothconsistentmeshes,butitalsoallowsforcompleteelimination
offinitedifferenceinformationforthemeshmela-icterms.Since
themappingfunctionfullydeterminestheentiremeshbasedonthe
surfaceshape,thisanalyticrelationshipmaybedirectlydifferentiated
inordertoobtaintherequiredinformationwithoutconsideringa
finitestep.Ananalyticmappingmethodrequiresthegeometry
topologytobebuiltdirectlyintotheformulation,andonlyworksfor
simpleconfigurations.Nevertheless,withintheselimitationsithas
proventobehighlyeffective[19,20,221.
Thesecondmethodthatwehaveexploredistheuseofananalytic
meshperturbationtechnique.Inthisapproach,ahighqualitymesh
appropriatefortheflowsolverisfirstgeneratedbyanyavailablepro-
cedurepriortothestartofthedesign.Inexamplestobeshownlater,
thesemesheswerecreatedusingtheGridgensoftwaredevelopedby
thecompanyPointwise[44].Thisinitialmeshbecomesthebasisfor
allsubsequentmesheswhicharedevelopedbyanalyticalperturba-
tions.Inthemethodthatwaspreviouslydevelopedforwing-body
configurationsithadbeenassumedthatonlyonesurface,saythe
wing,wasperturbedduringadesigncase.Thispermittedtheuseof
averysimplealgebraicmeshperturbationalgorithm.Newmeshes
arecreatedbymovingallthemeshpointsonanindexlineprojecting
fromthesurfacebyanamountwhichisattenuatedasthearclength
fromthesurfaceincreases.Iftheouterboundaryofthegriddomain
isheldconstantthemodificationtothegridhastheform
....ldSo_a"_-z°_d)(22)
Xi_T,i-_-(X,,.'1
wherez,representsthevolumegridpoints,x.,,representsthesurface
gridpointsandSrepresentsthearclengthalongtheradialmeshline
measuredfromtheouterdomain,normalizedsothatS=1atthe
innersurface.Unfortunatelythissimplelogicbreaksdowninthe
casewheremultiplefacessharingcommonedgesareallowedto
move.Thusinordertouseanalyticmeshperturbationsforthe
treatmentofthemoregeneralproblemwheremultiplefacesofa
givenblockmaybesimultaneouslydeformed,equation(22)hadto
bemodifiedinawaythatresemblestransfiniteinterpolation(TFI)
[46].UnlikeTFI,wherethereisnopriorknowledgeoftheinterior
mesh,theperturbationalgorithmdevelopedhere(WARP3D)does
makeuseoftherelativeinteriorpointdistributionsintheinitialmesh.
WARP3Dmaybethoughtofasatwostageprocedurethatoperates
withineachblock.Thefirststageshiftstheinternalmeshpointsto
produceareferenceblockthatisdeterminedentirelybythenew
locationsofthe8cornerpointsdefiningtheblock.Corresponding
tothemotionofeachcornerpoint,eachinteriorpointisshiftedby
adisplacementproportionaltooneminusthenormalizeddistance
alongtheindexlinesawayfromthatcornerpoint.Thesecond
stagecheckstheperturbationofeachpointinallsixfacesrelative
tothepositionofthecorrespondingpointinthereferenceblock.If
theperturbationofthedomaininvolvesasimpletranslationofall
boundarypoints,theserelativechangesofthefacepointswillbe
zeroandalltheperturbationwillbeaccomplishedbythefirststage.
If,however,facepointsareperturbedrelativetothereferenceblock,
thenthesechangesarepropagatedtotheinteriorpointsthrough
relativearclength-basedperturbationsalongprojectingindexlines
asdescribedintheoriginalsinglefacealgorithm(22).Forthis
secondpart,eachinteriorpointisdependentupontherelativemotion
ofonepointoneachofthesixfacesthatisdefinedbytheindex
markersofthepointinquestion.TheideaofWARP3Distousean
initialmeshwithgoodqualityattributesasastartingpoint,andthen
systematicallyperturbthismeshinamannersuchthattheoriginal
gridqualityismaintained,withouttheneedforexpensiveelliptic
smoothing.
Sinceourcurrentflowsolveranddesignalgorithmassumeapoint-
to-pointmatchbetweenblocks,eachblockmaybeindependently
perturbedbyWARP3D,providedthatperturbedsurfacesaretreated
continuouslyacrossblockboundaries.Theentiremethodofcreating
anewmeshisgivenbythefollowingalgorithm.
1.Allfacesofallblocksthatarecoincidentwithperturbedsur-
facesareexplicitlyperturbed.
2.Thefacesinallblocksthatshareanedgewithanexplicitly
perturbedfaceareimplicitlyperturbedbyaquasi-3Dformof
WARP3D.
3.WARP3Disusedoneachblockthathasoneormoreexplicitly
orimplicitlyperturbedfacestodeterminetheadjustedinterior
points.
Notethatmuchofthemesh,especiallyawayfromthesurfaces,will
notrequiremeshperturbationsandthusmayremainfixedthrough
theentiredesignprocess.Sincethismeshperturbationalgorithm
isanalyticitispossibletoworkouttheanalyticalvariationsinthe
metrictermsrequiredforequation(21).Thisapproachwasfollowed
inreference[40].Howeversincethemeshperturbationalgorithm
thatwasusedinthecurrentpaperwassignificantlymorecomplex,
anditwasdiscoveredthatthecomputationalcostofrepeatedlyus-
ingtheblockpemn-bationalgorithmwasminimal,finitedifferences
wereusedtocalculate8Qi:insteadofderivingtheexactanalytical
relationships.Evenincaseswithhundredsofdesignvariables,the
computationalcostofrepeatedlyre-evaluating_Q,3forallnecessary
blocksisstillinsignificantcomparedwiththecostofasingleflow
solution.Theconclusionisthattheanalyticalmeshperturbation
algorithm,WARP3D,unlikeanellipticalmeshgenerationmethod,
isefficienttotheextentthatthecostofremeshingcanbeneglected.
Itremainstochooseasetofdesignvariableswhichsmoothly
modifiestheoriginalshape,saybi.Thegradientcanthenbedefined
withrespecttothesedesignvariablesas
bl
G(b,)=bb"_'(23)
where_Iiscalculatedby(21)andeachtermb,isindependently
perturbedbyafinitestep.Therefore,toconstructG,abasisspaceof
independentperturbationfunctionsb,,i=-1,2,...,n(n=number
ofdesignvariables)mustbechosentoallowfortheneededfreedom
ofthedesignspace.Inthiswork,designvariableswerechosenasa
setofHicks-Hennefunctionssimplyfortheireaseofimplementaion
andtheirprovenreliability.Togeneralizetheapplicationofthe
Hicks-Hennefunctions,whichhavetraditionallybeenusedforthe
modificationofairfoilsectionswherethexinequation(5)refersto
thechordwiseposition,theyareapplieddirectlytotheparametric
(fi,_)surfacesthatrepresentthemeshfaces.Thustheymaybe
appliedasfunctionsineitherthefi,the_,orbothdirections.The
designcodeisfurtherstructuredsothatthesevariablesmaybe
appliedtoanysubsetoftheparametricsurface.Alternativesare
providedsuchthatthesevariablesmaybelinearlyloftedinthe
seconddirectionasopposedtosayHicks-Hennefunctioninboth
directions.Alloftheseoptionsmaybeprescribedattheinputlevel,
leadingtoahighlyversatiledesigncodeinwhichoneormorefacesin
themultiblockdomainmaybeperturbedbythedesignvariables.To
enforcegeometricconstraints,eachdesignvariablemaybeactivated
onmorethanoneface.Forexample,ifthethicknessofawingis
tobepreservedandtheupperandlowerhalvesofthewingare
inseparateblocks,thenthedesignvariablesneedtobeappliedat
theproperlocationswiththeproperweightsandontheappropriate
facesinbothblockssuchthatthicknessdoesnotchangewhileboth
surfacesareallowedtobemodified.
IMPLEMENTATIONOFTHEDESIGNALGORITHM
Withallthenecessarycomponentsdefinedforthemultiblockadjoint
baseddesign,itisnowpossibletooutlinethecompleteprocedure:
1.Solvetheflowfieldgoverningequations(6-11).
2.Solvetheadjointequations(18)subjecttotheboundarycondi-
tion(20).
3.Foreachofthendesignvariablesrepeatthefollowing:
•Perturbthedesignvariablebyafinitestepaccordingto
(5,etc.).
•Explicitlyperturballfacesaffectedbythedesignvariable.
•Implicitlyperturballfacesthatshareanedgewithan
explicitlyperturbeddesignvariable.
•Obtainthenewinternalmeshpointlocationsvia
WARP3Dforthoseblockswithperturbedfaces.
•Calculateallthedeltametricterms,_Q,,j,withinthose
blocksthatwereperturbedbyfinitediffferencing.
•Integrateequation(21)toobtain_51forthoseblocksthat
containnonzero_SQI,j.
•Determinethegradientcomponentbyequation(23).
4.Calculatethesearchdirectionandperformalinesearch.
5.Returnto(1)ifminimumhasnotbeenreached.
Thebasicmethodherebuildsonthatusedinreference[40]withthe
properextensionstotreatmultiblockdomains.Inordertoimple-
mentthemethod,equation(18)andboundarycondition(20)must
bediscretizedonthemultiblockdomain.Inthecurrentimplemen-
tation,acellcentered,centraldifferencestencilthatmimicstheflux
balancingusedfortheflowsolutionisused.Sincethischoiceof
discretizationdiffersfromtheoneobtainedifthediscreteflowequa-
tionJacobianmatrixwereactuallytransposedtoformtheadjoint
system,thegradientsobtainedbythepresentmethodwillnotbe
exactlyequaltothegradientscalculatedbyfinitedifferencingthe
discreteflowsolutions.However,asthemeshisrefinedthesediffer-
encesshouldvanish.Continuing,theadjointsystemsodiscretized
issolvedonthemultiblockdomaininanidenticalfashiontothat
usedfortheflowsolution.Therefore,theadjointsolver,likethe
flowsolver,usesanexplicitmultistageRunge-Kutta-likealgorithm
acceleratedbyresidualsmoothingandmultigridding.Intra-block
communicationisagainhandledthroughadoublehalowhichallows
forthefulltransferofinformationacrossboundariesexceptforthe
stencilofsupportfortheimplicitresidualsmoothing.
Step(3)intheaboveprocedureistheportionofthemethodthat
isstilltreatedbyfinitedifferences.Fortunately,allofthesesteps
incuronlyatrivialcomputationalcostcomparedwithevenasingle
flowanalysistimestep.Itisthereforepossible,withoutsignificant
penalty,toleavethisinfinitedifferenceformevenforcaseswhere
manyhundredsofdesignvariablesareused.
Thepresentimplementationusesthequasi-Newtonalgorithm,
QNMDIF,developedbyGill,MurrayandPitfield[10]anden-
hancedbyKennelly[27],tocalculatethesearchdirection.Itisan
unconstrainedoptimizationalgorithmthatusesBroyden-Fletcher-
Goldfarb-Shanno(BFGS)updatestotheCholeskyfactoredHessian
maU'ix.Acompletetreatmentofthequasi-Newtonandotheropti-
mizationstrategiesisgivenbyGill,MurrayandWright[111.
NUMERICALTESTSANDRESULTS
Allofthedesigntestcasestobepresentedinthispaperuseabusiness
jetconfigurationthatisthesubjectofanotherpaperatthisconference
(seeReference[9]).Theresultspresentedin[9]wereobtained
throughtheuseofthesingleblockdesignmethoddevelopedbyour
groupandpresentedlastyear[40].Therefore,thischoiceallowedus
tovalidatethequalitativeresultsofboththemultiblockflowsolver
andthemultiblockdesignmethod.
Todemonstratetheutilityofthenewflowsolver(FLO87-MB)an
initialflowsolutionona72blockmeshwithatotalof750Kceilsis
showninFigure(2).ThesolutionwascarriedoutataMachnumber
of0.80andaCLof0.3.Itcanbeseenfromthefigurethatthisis
awing-body-nacellegeometry.Theactualsolutionwascantedout
onthelefthalfoftheconfigurationwithasymmetryplaneboundary
conditionenforcedtoobtainacompletesolution.Theempennage
andnacellepylonswerenotmodeledheresimplytoensurethat
resultscouldbeobtainedbytheconferencedate.Thesurfaceofthe
geometryshowninFigure(2)isanisometricviewcoloredbythe
localMachnumber.Thenacelleismodeledasflowthrough,witha
singleH-blocktraversingitsentireinterior.AscanbeseeninFigure
(3),whereacutofthesolutionistakenthroughsomeoftheblocks
(darklinesindicateblockboundaries)thegenerallayofthe72block
meshisC-O.Thiswaschosenforconveniencesinceanytopology
isallowedwithinthemultiblockframework.Figure(3)alsoshows
thatthecontourlines,coloredwithconstantMachnumber,a-averse
theblockboundarieswithoutanyevidenceofsolutionmismatches.
Thisvalidatestheeffectivenessofthebasicmultiblockflowsolution
strategy.Fourmultigridlevelswereusedforthissolutiononall
blockssincethelimiting(smallest)blockcontainedonly8cells
inonecoordinatedirection.Thesolutionpresentedheretook100
multigridcyclestoconverge4.5ordersintheRMSresidualfromthe
startingresidual.Thisrequiredabout18minutesofCPUtimeona
singleprocessorofaCrayC90.
Thefirststepinestablishingthevalidityoftheadjointbaseddesign
methodistoperformacheckofthegradientsitproducesascom-
paredwiththoseobtainedbyfinitedifferences.Figure(4)showsa
comparisonofadjointgradientsvs.finitedifferencegradientsfor
testcase1tobediscussedindetailbelow.Thefinitedifference
gradientswerecalculatedusingforwarddifferencingforonly24of
the250designvariablesusedfortestcase1sincethesecalculations
werequitecomputationaUyexpensive.Althoughtheadjointmethod
calculatedallthegradientsforall250designvariables,onlythose
correspondingtothosecalculatedforfinitedifferencesareshownin
Figure(4).Forthefinitedifferenceresultstheflowsolutionwas
alwaysconverged4.5ordersfromtheinitialstartingresidual.In
ordertoreducethecomputationalcostoftheseevaluationstheflow
solutionsforthegradientcomponentswererestartedfromneighbor-
ingsolutions.Theadjointgradientswereobtainedusing4.5orders
ofconvergenceintheflowsolutionand2.0ordersintheadjointso-
lution.ThedesignvariablesconsistedofHicks-Hennefunctionsin
thechordwisedirectionandlinearloftinginthespanwisedirection.
TheresultsinFigure(4)show24designvariablesthatspanfromthe
leadingtothetrailingedgealongtheuppersurfaceataconstantspan
stationof0.44.Theexcellentagreementbetweenthetwocurves
showsthatthecontinuoussensitivityapproach,ifproperlyapplied,
cangiveveryaccurateresults.Theremainingdiscrepanciesbetween
themethodscouldbeproducedbythelimitedconvergenceineither
theflowortheadjomtsolutionsforeitherofthetwomethods,or
simplyfromthediscretizationmismatchforthecontinuousadjoint
approach.Thekeypointinthiscomparisonisthatthecalculationof
theadjointgradientforall250designvariablestook37minutesof
CrayC90singleprocessorCPUtime.Thecomparablefinitediffer-
encecalculationwouldhavetakenanestimatedminimumof1,000
minutes.Thusthereisafactorof27betweenthecomputational
costsofthetwoapproaches.
Thefirsttestcaseusedthemeansquaredeviationfromatarget
pressureastheobjectivefunction.Wingstationsintheoriginal
geometryshowninFigures(2)and(3)werereplacedbythickness-
scaledNACA0012airfoilsinallareasexceptforthetipandthe
root.ThesolutionfromFigures(2)and(3)wasthenusedasa
targetpressureontheentirewingsurface.Thedesigntherefore
attemptedtorecovertheoriginalwingstartingfarfromthesolution.
Thedesignwasruninconstant_asopposedtoconstantCzmode
withaMachnumberof0.80.250designvariablesweredistributed
overtheentirewingsurfacetoallowforpossibledesignchanges.
ThesevariablesassumedtheformofHicks-Hennefunctionsinthe
chordwisedirectionandlinearloftinginthespanwisedirection.
50variablesspreadoverboththeupperandlowersurfaceswere
specifiedateachof5definingstationsacrossthewing.The72
blockmeshhasthepropertythatthreeblocksabuttothewingupper
surfaceandthreeblocksabuttothewinglowersurface.Thussome
ofthedesignvariablesIraversedmorethanonefaceinseparate
blocks.Figure(5)showsthepressuredistributionsandthetargetsat
variousstationsalongthewingforthestartingpointinthedesign.
Figure(6)showsthesolutionandthetargetatthesamestations
after12designcycles.Notethatthedesignalmostfullyrecovered
theoriginalpressuredistributions.Thisrepresentsadropinthe
calculatedcostfunctionfrom6.02to0.10.Thedesignalgorithm
wasstoppedafter12cyclestoconservecomputerbudget,though
furtherreductionsinthecostfunctionwerewithinreach.Itmust
alsoberememberedthattheHicks-Hennefunctionsdonotadmita
completedesignspace,sothatevenifconvergencetoaminimumis
achieved,slightmismatcheswillremain.
Thesecondtestcaseattemptstoimprovetheoriginalconfiguration
byreducingitsdragatahigherMachnumberthantheoriginaldesign
Machnumber.InthiscasetheMachnumberwassetat0.82andCL
at0.30.ThedesignwasruninfixedliftmodewithCoasthecost
functiontobeminimized.Sincetheflowsolverwasinviscid,this
dragconsistedofformdragandinduceddrag.Inthiscase105design
variablesoftheHicks-Henneforminthechordwisedirectionand
linearloftinginthespanwisedirectionwereused.Tokeepthewing
fromviolatingrealisticconstraints,thedesignvariableswerechosen
sothatthethicknessdisWibutionovertheentirewingwaspreserved.
Thisrequiredsomeofthedesignvariablestooperateonuptofour
facesindifferentblockssimultaneously.Againthedesignvariables
wereallowedtomodifymostofthewingexceptfornearthetipand
neartheroot.Theinitialandfinaldesignsafter6cyclesareshownin
Figure(7).Notethattheshockonthewinguppersurfacehasbeen
largelyeliminatedovertheentirespan.Thisisevenachievednear
therootwherenogeometricchangeswereallowed.Toexplainthis
phenomenonitmustberememberedthatthedesignwasruninfixed
liftmode.Itturnsoutthattheincidenceoftheentiregeometryis
reducedbytheredesign,thuscausingtheuppersurfacenottohave
toworksohard.Asaconsequenceitcanbeseenthatthestrengthof
theshockonthewinglowersurfaceisslightlyincreasedovermuch
ofthespan.Apparentlythistrade-offproducesadragreduction.
Mostoftheuppersurfacepressuredistributionhasbeenmodified
intonearfiatroof-topdesignswithweaktonoshocks.Thefinal
configurationdragwasreducedby19%,mostofwhichwasdueto
thereductionofthedragonthewing.
CONCLUSIONSANDRECOMMENDATIONS
Intheperiodsincethisapproachtooptimalshapedesignwasfirst
proposedbythesecondauthor[19],themethodhasbeenverifiedby
numericalimplementationforbothpotentialflowandflowsmodeled
bytheEulerequations[20,39,25,23].Ithasbeendemonstrated
thatitcanbesuccessfullyusedwithafinitevolumeformulation
toperformcalculationswitharbitrarynumericallygeneratedgrids
[39,25].Further,resultshavebeenpresentedforthree-dimensional
calculationsusingboththeanalyticmappingandgeneralfinitevol-
umeimplementations[40].Inthelastyearthetechniquehasbeen
adoptedbysomeindustryparticipantstoperformtheaerodynamic
designoffutureconfigurations[9].Nowwiththeextensiontomulti-
blockmeshesdevelopedhere,thetechnologyhasadvancedtothe
degreethataerodynamicshapedesignofcompleteaircraftconfigu-
rationsispossible.Toachievethiscapabilitytheflowsolver,adjoint
solver,andmeshperturbationalgorithmwereallextendedtotreat
multiblockmeshes.Theuseoftheadjointbaseddesignmethod
reducedthecomputationaltimeoverthatrequiredforthefinitedif-
ferencebasedmethodbyatleastafactorof27.Anaccompanying
paperdiscussestheimplementationofthemethodforparallelcom-
puterarchitectures[24].Infutureeffortsthetechniqueswillbe
extendedtoaddressbothunstructuredmeshesandflowsgoverned
bytheNavier-Stokesequations.
ACKNOWLEDGMENTS
Thisresearchhasbenefitedgreatlyfromthegeneroussupportofthe
AFOSRundergrantnumberAFOSR-91-0391,ARPAundergrant
numberN00014-92-J-1976,USRAthroughRIACS,theHighSpeed
ResearchbranchofNASAAmesResearchCenter,andIBM.Con-
siderablethanksalsogoestoMarkRimlingerofCarnegieMellon
Universityforhisassistanceintheinitialmultiblockgridgeneration.
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11
BlockII
/
Center
."::
:
:.:.:
I"-i
I!!
--I"=l""1
III
Center
BlockIVCenter
II
"1""1""1
III
"1''1""1
III
BlockI
BlockICenter
SolidBoundary
IncludingDoubleHalo
Figure1:4Blockinterfaceusingadoublehaloofcellsaroundeachblock.
Eachblock'sdoublehaloofcellscontainsinformationfrominternalcellsin
adjacentblocks.
12
20.0
---C]---FiniteDifferenceGradient
AdjointGradient
10.0
0.0
"_-10.0
-20.0
-30.0
-40.0
12345678910111213141516
Variable
1718192021222324
Figure4:ComparisonofGradientsfor24DiscreteDesignVariables
Adjointvs.FiniteDifferences
DesignVariablesSpantheUpperSurfaceataSpanStaionof0.44
BeginningattheLeadingEdgeandEndingatTheTrailingEdge
14
SYN87-MBSolutiononatypicalbusinessjet
72Blocks-750kmeshpoints-Mach-0.80-CL=0.30
Figure2:Initialflowsolutionforwing-body-nacellegeometry.
SurfacecoloredbyMachnumber.
Figure3:Initialflowsolutionshowingblockboundariesasdarklines
andcontourscoloredbyMachnumber.
°°°°°°
5a:spanstationz=0.125
<_..
5b:spanstationz=0.312
°°°°.
<_
5c:spanstationz=0.559
5d:spanstationz=0.764
Figure5:SYN87-MBInverseTargetPressureDesign.
72BlockMesh,750Kmashcells,M=0.8
250Hicks-Hennevariables.
•,TargetPressures
--,InitialPressures.
15
-_--I°'°'"""
6a:spanstationz=0.125
f
6b:spanstationz=0.312
6c:spanstationz=0.559
f
6d:spanstationz=0.764
Figure6:SYN87-MBInverseTargetPressureDesign.
72BlockMesh,750Kmeshcells,M=0.8
250Hicks-Hennevariables.
•,TargetPressures
--,DesignPressuresAfter12DesignCycles.
16
-er-
7a:spanstationz=0.125
7b:spanstationz=0.312
7c:spanstationz=0.559
7d:spanstationz=0.764
Figure7:SYN87-MB,FixedLiftDragMinimization.
72BlockMesh,750Kmeshcells,M=0.82,CL=0.3
105Hicks-Hennevariables.
---,InitialPressures
m,PressuresAfter6DesignCycles.
17
Автор
Redmegaman
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