Fourier Series for Accurate Stable ReducedOrder Models for Linear CFD Applications - PDF document

FourierSeriesforAccurate,Stable,Reduced-OrderModelsforLinearCFDApplications.K.WillcoxandA.MegretskiMassachusettsInstituteofTechnologyCambridge,MA02139Anewmethod,Fouriermodelreduction(FMR),forobtainingstable,accurate,low-ordermodelsofverylargelinearsystemsispresented.Thetechniquedrawsontraditionalcontrolanddynamicalsystemconceptsandutilizesthemin AssistantProfessorofAeronauticsandAstronau-tics,Room37-447,MIT,Cambridge,MA02139,kwill-cox@mit.edu,MemberAIAAAssociateProfessorofElectricalEngineeringandCom-puterScience,Room35-418,MIT,Cambridge,MA02139,Copyright2003byK.WillcoxandA.Megretski.PublishedbytheAmericanInstituteofAeronauticsandAstronautics,Inc.withpermission.casewhencouplingbetweendisciplinarymodels AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 reductionistoprojecttheCFDgoverningequa-tionsontothespacespannedbythe“rstvectors.Whilethereconstructionoftheorigi-nalsnapshotsmightbeŽoptimalŽusingthePODbasis,onecanmakenostatementabouttheac-curacyoftheresultingreduced-orderdynamicalsystem.Infact,onecannotevenguaranteethestabilitypropertiesofthissystem.Despitethis,thePODisfoundtoworkwellinmanycaseshasbeensuccessfullyappliedforabroadrangeofCFDapplications(see,forexample,reviewsinDowellandHallandBeranandSilvaAnotherclassofreductiontechniquesthatarebasedonmatchingmomentsofthesystemtrans-ferfunctionhasbeendevelopedforanalysisoflargelinearcircuits.Forinstance,theArnoldialgorithmcanbeusedtogeneratevectorswhichformanorthonormalbasisfortheKrylovsub-space,andhasbeenappliedtoanalysisofRLCturbomachineryaeroelasticbehaviorandactivecontroldesignforasupersonicdif-TheArnoldimethodiscomputation-allymoree
cientthanthePOD,butalsoof-fersnoguaranteesastotheaccuracyorstabilityofthereduced-ordermodel.Moreover,neitherthePODnorArnolditakesaccountofsystemoutputswhenperformingthereduction,hencethereduced-ordermodelsproducedmaybeinef-“cient.ThePODhasbeensuggestedasameanstoobtainanapproximatebalancedtruncationforlargesystemsusingbothinputsandoutputs;12,13however,thereductionapproachiscomputation-allyexpensiveandagainoersnostabilityguar-antees.InWillcoxandMegretski,anewtechnique,Fouriermodelreduction(FMR),forperformingmodelreductionofverylargesystemswaspre-sented.Thismethoddrawsonclassicaldynamicalsystemandcontroltheoryconcepts,andappliesthemusinganiterativeprocedurethatisverye
cientforlargesystems.Theresultingreduced-ordermodelsareguaranteedtopreservethesta-bilitypropertiesoftheoriginalsystemandhaveanassociatederrorboundthatdependsonthesmoothnessoftheoriginaltransferfunction.Inthispaper,theFMRalgorithmisreviewedandthenappliedtotwodierentCFDapplications,bothofwhichrequire”uidmodelsofloworder.The“rstexampleinvestigatestheunsteadymo-tionofatwo-dimensionalsubsonicairfoil,whichformsthe”uidcomponentofanaeroelasticanal-ysis.Thesecondexamplederivesamodelofthe”owdynamicsofasupersonicinletthatwillbeusedtoderiveactivecontrolstrategiesandisusedtodemonstratethemultiple-input,multiple-output(MIMO)capabilitiesoftheFMRalgo-rithm.AcomparisonisalsomadewithreducedmodelsderivedusingArnoldiandPOD.Theoutlineofthepaperisasfollows.Inthefollowingsection,thedynamicalsystemarisingfromimplementationofCFDisbrie”ydescribed.TheFMRtechniqueisthenpresented,alongwiththeextensiontotheMIMOcase,andcomparedwithArnoldiandPODapproaches.ThetwoCFDexamplesarethenconsidered,and“nally,conclu-sionsaredrawn.ComputationalFluidDynamicModelConsideragenerallinearizedCFDmodel,whichcanbewrittenas Bu,containstheunknownpertur-bation”owquantitiesateachpointinthecompu-tationalgrid.Forexample,fortwo-dimensional,compressible,inviscid”ow,whichisgovernedbytheEulerequations,theunknownsateachgridpointaretheperturbationsin”owdensity,Cartesianmomentumcomponentsand”owen-ergy.Thevectors)and)in(1)containthesysteminputsandoutputsrespectively.Thede“-nitionofinputsandoutputswilldependupontheproblemathand.Inaeroelasticanalysisofawing,inputsconsistofwingmotionwhileoutputsofin-terestaretheforcesandmomentsgenerated.Forcontrolpurposes,theoutputmightmonitora”owconditionataparticularlocationwhichvariesinresponsetoadisturbanceintheincoming”ow.Thelinearizationmatricesin(1)areevaluatedatsteady-stateconditions.Thematrixisincludedforgenerality,andmaycontainsomezerorows,whicharisefromimple-mentationof”owboundaryconditions.Onsolidwalls,aconditionisimposedonthe”owvelocity,whileatfar“eldboundariescertain”owparam-etersarespeci“ed,dependingonthenatureoftheboundary(in”ow/out”ow)andthelocal”owconditions(subsonic/supersonic).Althoughthese2of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 prescribedquantitiescouldbecondensedoutof(1)toobtainasmallerstate-spacesystem,suchamanipulationisoftencomplicatedandcande-stroythesparsityofthesystem.Themoregeneralformofthesystemisthereforeconsidered.Thesystem(1)ise
cientfortimecomputa-tionssinceatimediscretization,suchasbackwardEuler,canbeappliedandtheresultinglargesystemmatrixinvertedjustonce.However,theorderofthesystemisstillprohibitivelyhighformanyapplications,suchasaeroelasticityandac-tive”owcontrol.Inthenextsection,wepresentane
cientmethodwithqualityguaranteestore-ducethesizeofthesystemwhileretaininganaccuraterepresentationofimportant”owdynam-FourierModelReductionWeconsiderthetaskof“ndingalow-order,sta-ble,continuoustime,lineartimeinvariant(LTI),state-spacemodel )(2)whichapproximateswellthegivenstablestate-spacemodel(1).Weconsider“rstthecaseofsingleinput,singleoutput(SISO),thendiscusslatertheMIMOcase.Typically,in(1)aresparse,squarematricesofverylargedimen-,andthedesiredorderislessthan50.Thequalityofasanapproximationofde“nedastheH-In“nitynormofthedierencebetweentheirtransferfunctions:=supwhichinturnequalsthesquarerootofthemax-imalenergyofthedierence,whichcanbegener-atedwhentestingbothwithanarbitraryunitenergyinputasshowninFigure1.Withthismeasureofmodelreductionerror,ifagoodreducedmodelisfound,canberepresented,fordesignoranalysispurposes,asaseriescon-nection(i.e.asum)ofandasmallŽuncertainŽ G G




Fig.1Comparingerrorsystem=,andthestandardre-sultsfromrobustnessanalysiscanbeappliedtopredicttheeectofreplacinginevenlargerscalesystems.FMRisalowcomplexityalgorithmthatallowsonetoperformmodelreductionofverylargesys-tems.Whiletheresultisnotanoptimalreducedmodelof,itsatis“esanattractiveguaranteedH-In“nitybound.FourierSeriesofDiscreteTimeSystemsConsiderthefulldiscretetime(DT)LTIsystemmodelde“nedbythedierenceequations+1)=a,b,c,daregivenmatricesofcoe
cients,isthesystemstate,and)arescalarinputandoutput.Itwillbeassumedthatisstable,i.e.1,where)denotesthespectralradiusof,de“nedasthemaximalabsolutevalueofitseigenvalues.ThetransferfunctionhastheFourierdecompositiond,g)(8)TheFourierexpansionconvergesexponentially).Notethatthe“rstFouriercoef-“cientsareeasytocalculateusingthecheapŽiterativeprocesswhichisexpectedtobestableŽsince3of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 LetdenotetheorderapproximationofbasedontheFourierseriesexpansion:Wenotethattheapproximationisguaranteedtobestableandthattheerrorisrelatedtothesmoothnessof ,whereisthederivativeofwithrespecttoFourierSeriesofContinuousTimeSystemsConsiderthefullcontinuoustimeLTIsys-temmodelde“nedbythesystem(1),where)arescalarinputandoutput.Ifisin-vertiblethen)isthesystemstate.Itwillbeassumedthatisstable,i.e.thatallrootsofthecharacteristicequationdet(havenegativerealpart,andthatremainsboundedas0bea“xedpositiverealnumber.ThetransferfunctionhastheFourierdecomposition d,GTheFourierexpansionisderivedfromtheidentity whichallowsonetoapplytheobservationsandtheoremfromtheprevioussubsectiontothiscase.Notethatbycomparing(7)and(13),itcanbeseenthatReducedModelConstructionToconstructan-orderreducedmodel,one“rstcalculatestheFouriercoe
cients,,...,g.TheDTreducedmodelisthengivenent+1]=t]+bu[t]y[t]=t]+du[t],(19)wherea=00001000010000100g1g2...gAneectiveapproachistousethee
cientiterativeproceduretocalculateseveralhundredcoe
cients,resultinginanintermediatereducedmodeloftheform(19).Asecondreductionstepusingbalancedtruncationcannowbeperformedeasily,sincetheexpressionsforthegrammiansareknownexplicitly.FortheDTreducedmodel(19),thecontrollabilitymatrixistheidentitymatrixandtheobservabilitymatrixistheHankelmatrixthathasasits“rstrow.Thebalancingvec-torscanthereforebeobtainedbycomputingthesingularvectorsofthe-orderHankelmatrix...g...gTheHankelsingularvalues,,...,m,oftheintermediatereducedsystemaregivenbythesingularvaluesof.The“nalreducedsystem,iscomputedasisamatrixwhosecolumnsarethe“rstsingularvectorsof,andischosenaccord-ingtothedistributionofHankelsingularvalues.Speci“cally,itcanbeshownthatthefollowingerrorboundholds:4of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 Itshouldbenotedthattheapplicationofopti-malHankelmodelreductiontotheintermediatereducedmodelwouldyieldaboundontheer-rorlowerthanthatin(23);however,balancedtruncationcanbeappliedverye
cientlysincethegrammiansareknownexplicitly.Thisalsoavoidstheexplicitconstructionoftheintermediatesys-tem,ascanbeseeninthefollowingalgorithm.FourierModelReductionAlgorithmTheFouriermodelreductionalgorithmissum-marizedinthefollowingsteps.1.Chooseavalueof.Thevalueofre”ectthefrequencyrangeofinterest.Thenominalvalueisunity;however,ifthere-sponseathighfrequenciesisofinterest,ahighervalueofshouldbechosen.Onecanvisualizethetransformationfromcon-tinuoustodiscretefrequencyasamappingoftheimaginaryaxisintheplanetotheunitcircleintheplane.Thevalueofthendescribesthecompressionoffrequenciesaroundtheunitcircle.2.Calculate+1Fouriercoe
cientsusing(14)-(18).Usingtheiterativeprocedure,anynumberofcoe
cientscanbecalculatedwithasingle-ordermatrixinversionorfactor-3.Using(21),calculatethe-orderHankelmatrix.Calculateitssingularvaluesandsin-gularvectors.4.Using(22),constructa-orderDTsystem,.ThevalueofischosenaccordingtothedistributionofHankelsingularvaluesoftheintermediatesystem.5.Convertthe-order,DTreducedmodeltoacontinuous-timemodelusingtherelation-Multiple-Input,Multiple-OutputSystemsAlthoughtheabovealgorithmwasderivedforSISOsystems,itcanbeeasilyextendedtotheMIMOcase.Infact,thestepsaboveareessen-tiallyunchanged.Considerthegeneralcaseofinputsandoutputs.ForStep2,onecanstilluse(14)-(18)tocalculatetheFouriercoe
cients,al-thoughnoweachisamatrix.Theiterativeprocedureisstillverye
cient,sinceconsiderationofmultipleinputsandoutputsrequiresonlyad-ditionalmatrixvectorproducts.InStep3,oneagainuses(21)toconstructtheHankelmatrix.Thesizeofthismatrixwillnow.ThebalancedtruncationinStep4proceedsasbefore,notingonlythateachscalarentryofin(20)isreplacedwithablockentry(zeromatrixoridentitymatrix,asappro-priate)ofsize.Finally,thetransformationsinStep5arevalidfortheMIMOcase.Intheresultssection,twoexampleswillbepre-sentedthatdemonstratetheeectivenessofthisapproach.We“rstcomparetheadvantagesofFMRwithothercommonlyusedtechniques.ComparisonwithAlternativeReductionMethodsThePODisthemostwidelyusedreductionmethodinthe”uiddynamicscommunity.basicideaistocollectasetofsnapshots,whicharesolutionsofthelinearizedCFDmodelatselectedtimeinstantsorfrequencies.Thesesnapshotsarethencombinedtoformane
cientbasis,whichisoptimalinthesensethatitminimizestheer-rorbetweentheexactandprojectedCFDdata.Whilethemethodresultsinreduced-ordermodelswhichaccuratelyreproducethosedynamicsin-cludedinthesamplingprocess,themodelsarenotvalidoutsidethesampledrange.Ifsnapshotsareobtainedinthetimedomain,itcanbedi
-culttochooseanappropriateforcingfunctionandtypicallymanysnapshotsarerequiredtoobtainaccurateresults.Usingthefrequencydomaintocomputesnapshotsisoftenmoreconvenient,how-evereachfrequencysampledrequiresanmatrixinversion.Moreover,thereduced-ordermodelsobtainedviathePODoernoguaranteeofstability.TheArnoldimethodhasalsobeenusedformodelreductionoflargeCFDsystems.goalofmoment-matchingtechniques,suchastheArnoldimethod,istodetermineareduced-ordermodelbymatchingmoments(orTaylorseriesco-e
cients)ofthehigh-ordersystemtransferfunc-tion.TheArnoldimethodhasanadvantageover5of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 thePODinthatasequenceofbasisvectorscanbegeneratedinthefrequencydomainwithjustasin--ordermatrixinversion,andtheapproachofgeneratinganintermediatemodelwhichcanbesubsequentlyfurtherreducedviaHankelmodelreductionalsoworkseectivelyforthistechnique.However,theArnoldimethoddoesnotconsidersystemoutputswhenformingthebasisvectorsandtheresultingreduced-ordermodelsarenotguaranteedtobestable.SincetheArnoldibasisvectorsarederivedfromanexpansionaboutzerofrequency,forsomeap-plicationstheresultingreduced-ordermodelcanbelargeiffrequenciesfarawayfromzeroareofinterest.ThemultiplefrequencypointArnoldimethodattemptstoaddressthisissuebyconsid-eringtransferfunctionexpansionsaboutmultiplefrequencypoints.Inthisway,accuratemodelscanbederivedoveraspeci“edfrequencyrange.Thereductioncostinthiscaseisproportionaltothenumberoffrequencypointsconsidered.BygeneratingmultipleArnoldivectorsateachfre-quencypoint,thesamplescanbeplacedfurtherapartwithoutlossinaccuracy.Themulti-pointArnoldimethodthereforeprovidesawaytotradebetweenthelowreductioncostofArnoldiandthefrequencyspanofPOD.Table1comparestheattributesofeachofthesereductiontechniqueswiththenewFMRap-proachdescribedinthispaper.Reductioncostreferstothenumberof-ordersysteminver-sionsrequired.(ThisisthedominatingfactorinreductionoflargeCFDsystems.)Frequencyrangereferstotherangeofvalidityoftheresult-ingreduced-ordermodels.ThisrangeisselectedaprioriforthePODandmulti-pointArnoldiapproaches.ThevalidityoftheArnoldi-basedreduced-ordermodelisrestrictedtofrequenciesclosetotheexpansionpoint(usuallytakentobezero),whilethenewapproachallowsawiderrangeoffrequenciestobeconsideredthroughthechoiceof.Finally,wenotethatnoneoftheal-ternativemethodsuseinformationpertainingtosystemoutputswhenderivingthereducedmodel;moreover,noneareguaranteedtoproduceasta-blereduced-ordermodel.Inpractice,thePODandArnoldiapproachescanoftengenerateun-stablemodelseventhoughtheoriginalsystemisThenewmethodologyoutperformsothertech-niquesinallcategorieslistedinTable1.TheaccuracyoftheFMRreducedmodelcanalsobequanti“edifthesmoothnessoftheoriginalsys-temtransferfunctionisknown.Wenowpresenttwotestcasesthatdemonstratethenewmethod-ology.ResultswillalsobeshowntocomparethenewmethodagainstArnoldiandPOD.TwoCFDapplicationswillbeconsidered.Eachhasverydierent”owdynamicsandusesadif-ferentCFDformulation;however,FMRwillbeshowntoworkveryeectivelyforboth.SubsonicAirfoilThe“rstexampleisatwo-dimensionalNACA0012airfoiloperatinginunsteadyplungingmo-tionwithasteady-stateMachnumberof0.755.The”owisassumedtobeinviscid,sothegovern-ingequationsarethelinearizedEulerequations,whichhavefourunknownspergridpoint.A“nitevolumeCFDformulationisusedwithaCFDmeshcontaining3482gridpoints,whichcorre-spondstoatotalof=13928unknownsinthelinearstate-spacesystem.Theinputtothissys-temisarigidplungingmotion(verticalmotionoftheairfoil),whiletheoutputofinterestistheliftforcegenerated.Thisinputandoutputaretypicalforanaeroelasticanalysis,whichtypicallywouldalsoincludepitching(angular)motionasaninputandairfoilpitchingmomentasanout-Fouriercoe
cientsweregeneratedusing(14)-(18)forseveraldierentvaluesof.The“rst201Fouriercoe
cientsareplottedinFigure2for=1.Fordemonstrationpurposes,the=200intermediatemodelsfor10werecon-structedandtheresultingtransferfunctionsarecomparedwiththeCFDinFigure3.Onewouldnotexpecttheplungedynamicstorequiresuchalargenumberofstates;however,sincethecostofcomputingadditionalFouriercoe
cientsissmallandtheprocessisguaranteedtobestable,onecanchoosetobelarge.As(11)shows,asthenumberofcoe
cientsincreases,thereductioner-rordecreases.Indeed,ascanbeenseeninFigure3,thetransferfunctionsmatchverycloselyoveralargefrequencyrange.SincetheoriginalCFDmodelisstable,thesereduced-ordermodelsarealsoallguaranteedtobestable.Moreover,thechoiceofdoesnothavealargeeectinthis6of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 Table1ComparisonofreductiontechniquesforCFDsystems. Method Reduction Frequency Stability Preserved TimePOD O(1) FrequencyPOD O(No.ofsnapshots) O(1) Restricted No Multi-PointArnoldi O(No.offreq.points) Selected FMR Moregeneral Yes InputsandOutputs 5 10 15 20 25 30 3 5 0 0.5 1 1.5 2 2.5 3 3.5 4 Hankel singular value Fig.2First201Fouriercoe
cientsofthetransferfunctionfromplungingmotiontoliftforceforsubsonicairfoil.Figure3isshowntodemonstratethemethodol-ogy;inpracticetheseintermediatemodelswouldnotbecomputed.Rather,theFouriercoe
cientsareusedtoconstructtheHankelmatrixforfur-therreductionviabalancedtruncation.The“rstthirtyHankelsingularvaluesoftheintermediate=1systemareshowninFigure4.Thesedataindicatethatafurtherreductionto“vestatescanbeachievedwithvirtuallynolossinaccu-racy.Thetransferfunctionofa“ve-statemodelisshowninFigure5,andcanbeseentomatchtheCFDdataextremelywell.Inordertocomparetheperformanceofthenewmethod,reduced-ordermodelsforthisproblemwerecomputedusingthePODandArnoldimeth-ods.PODsnapshotswereobtainedbycausingtheairfoiltoplungeinsinusoidalmotionatselectedfrequencies.Frequencieswereselectedat0.1in-crementsfrom=0to0,requiring21complexinversionsandsolves.Fromthesesnap-shotsasetofPODbasisvectorswasobtained. 1 2 3 4 5 6 7 8 9 10 12 10 8 6 4 frequencyreal(G(jw)) w0=1 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 frequencyimag(G(jw)) Fig.3Transferfunctionfromplungingmo-tiontoliftforceforsubsonicairfoil.Re-sultsfromCFDmodel(=13)arecom-paredtoreduced-ordermodelsderivedusing201Fouriercoe
cientswith 5 10 15 20 25 30 3 5 0 0.5 1 1.5 2 2.5 3 3.5 4 Hankel singular value Fig.4Hankelsingularvaluesforsubsonicairfoilcase.Valuesarecalculatedusing201Fouriercoe
cients,evaluatedwith7of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 0 1 2 3 4 5 6 7 8 9 10 12 10 8 6 4 frequencyreal(G(jw)) FMR: k=5 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 frequencyimag(G(jw)) Fig.5Transferfunctionfromplungingmotiontoliftforceforsubsonicairfoil.ResultsfromCFDmodel(=13)arecomparedtoaFMRmodelderivedusing=200Arnoldivectorsweregeneratedabout=0.Us-ingthesameapproachasforthenewmethod,200Arnoldivectorsweregenerated(withthecostofasingle-orderinversion),andbalancedtrun-cationwasappliedtotheresulting200system.Figure6showsthetransferfunctionsof“fth-orderreduced-ordermodelsconstructedus-ingPOD,ArnoldiandthenewFMRapproach.Itcanbeenseenthateventhoughitismuchmoreexpensivetocomputethantheothermethods,thePODhastheworstperformance.Inpar-ticular,the“gureshowsthat,asexpected,thePOD-basedreduced-ordermodelhasalargeer-roroutsidethefrequencyrangeincludedinthesnapshots.Forthisexample,theArnoldi-basedreduced-ordermodelshowssimilaraccuracytothenewapproach.SupersonicDiuserForthesecondexample,weconsiderunsteady”owthroughasupersonicdiuserasshowninFig-ure7.TheCFDmodelisdevelopedbylinearizingtheEulerequationsaboutasteady-statesolution,theMachcontoursforwhichareplottedinFig-ure7.Thenominalin”owMachnumberforthissteadycaseis2.2.TheCFDformulationforthisproblemresultsinasystemoftheform(1)andisdescribedfullyinLassaux.TheCFDmodelhas3078gridpointsand11,730unknowns.Areduced-ordermodelisrequiredforthedif-fuserdynamicsinordertoderiveactivecontrolstrategies.Thesestrategieswillbeusedtocoun- 1 2 3 4 5 6 7 8 9 10 12 10 8 6 4 frequencyreal(G(jw)) FMR 1 2 3 4 5 6 7 8 9 10 1 0 1 2 3 4 frequencyimag(G(jw)) Fig.6Transferfunctionfromplungingmo-tiontoliftforceforsubsonicairfoil.ResultsfromCFDmodel(=13)arecomparedreduced-ordermodelsderivedusingFMR,ArnoldiandPOD.BoththeArnoldiandFMRmodelshavebeencreatedusingbalancedtruncationfroman=200 Fig.7Machcontoursforsteady”owthroughsupersonicdiuser.Steady-statein”owMachnumberis2.2.teracttheeectofvariationsintheincoming”ow.Innominaloperation,thereisastrongshockdownstreamofthediuserthroat,ascanbeseeninFigure7.Incomingdisturbancescancausetheshocktomoveforwardtowardsthethroat.Whentheshocksitsatthethroat,theinletisunsta-ble,sinceanydisturbancethatmovestheshockslightlyupstreamwillcauseittomoveforwardrapidly,leadingtounstartoftheinlet.Thisisextremelyundesirable,sinceunstartresultsinalargelossofthrust.Inordertopreventunstartfromoccurring,”owbleedingupstreamofthedif-fuserthroatwillbeusedtoactivelycontrolthepositionoftheshockasshowninFigure8.Therearethereforetwoinputsofinteresttothesystem:theincoming”owperturbationandthebleedactuation.Forthisexample,the”owperturbationofinterestisavariationindensitythatisspatiallyconstantacrossthediuserin-8of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 Pressure sensingUpstream bleedShockAverageInlet disturbance (density) Engine compresso r Fig.8Supersonicdiuseractive”owcontrolproblemsetup.Upstreambleedisusedtocontrolthepositionoftheshockinthepresenceofincoming”owdisturbances.letplane.TheoutputofinterestistheaverageMachnumberatthethroat,whichcanbemoni-toredtoensurethattheshockremainssu
cientlyfardownstream.Thereduced-ordermodelshouldcapturethedynamicsaccuratelyforfrequencies,wherethereferencefrequencyisde-“nedasthefreestreamspeedofsounddividedbythediuserheight.Thisfrequencyrangeissu
cienttocapturetypicaldisturbancesduetoatmosphericvariations.InWillcoxandMegretski,theFMRtech-niquewasappliedtoeachofthesetransferfunc-tionsindependently.Itwasfoundthattenstatesweresu
cienttocapturethebleeddynamicsac-curately,butmorestateswererequiredforthedisturbancedynamicssincetheycontainadelay.Inthelattercase,atleasttwentystateswerere-quiredtoobtainagood“tforandvalues10yieldedmoreaccuratemodelsthan=1.TheFMRmodelsforeachsetofdynamicsrepresentaconsiderablereductioninorderfromtheoriginalCFDsystem,howeveritisdesirabletoobtainasinglereduced-ordermodelthatcap-turestheeectsofbothinputs.TheFMRalgorithmwasappliedtothistwo-input,single-outputproblemwith=5.201setsofFouriercoe
cientswerecalculated,whereeachsetcontainedtwocoe
cients(onecorrespondingtoeachinput/outputpair).The400Hankelmatrixwasthenconstructedandusedtoperformafurtherreductionviabal-ancedtruncation.The“rst“ftyHankelsingu-larvaluesoftheintermediatesystemareplottedinFigure9.Basedonthisdistribution,twentystateswereretainedforthe“naltwo-input,single-outputreduced-ordermodel.Thetransferfunc-tionsofthistwenty-statemodelarecomparedwithCFDresultsinFigures10and11.Asthe“guresshow,withonlytwentystatestheMIMOreduced-ordermodelcapturesthedynamicsveryaccuratelyoverthefrequencyrangeofinterest. 10 20 30 40 50 6 0 0 0.5 1 1.5 2 2.5 Fig.9HankelsingularvaluesofintermediateFMRsupersonicdiusersystem.201coe
-cientswereusedforeachinputandIfabettermatchathigherfrequenciesisre-quiredforthedensitydisturbancedynamics,moreFouriercoe
cientscouldbecomputedtoformtheintermediatemodel,orahighervalueofbechosen.Finally,acomparisonismadeforthebleeddy-namicsusingFMR,ArnoldiandPODmodels,eachofsize=10.Aten-stateSISOFMRmodelwasderivedusing=201coe
cients=5.FortheArnoldimethod,thetwostepreductionapproachcouldnotbeusedinthiscase,sinceincludingahighnumberofArnoldivectorsinthebasisresultedinanunstablein-termediatemodel.TheArnoldireducedmodelwasthereforecomputedbyprojectingtheCFDmodelontothespacespannedbythe“rsttenArnoldivectors.ThePODmodelwasobtainedbycomputing41snapshotsat21equally-spacedfrequenciesfromf/f=0tof/f=2.Thisre-quiredtheinversionofonerealand20complex-ordermatrices.Theresultingtransferfunc-tionsareplottedinFigure12.ItcanbeseenfromFigure12thattheFMR9of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 1 0 1 2 f/f0real(G(jw)) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 1.5 1 0.5 0 0.5 1 f/f0imag(G(jw)) FMR Fig.10TransferfunctionfrombleedactuationtoaveragethroatMachnumberforsupersonicdiuser.ResultsfromCFDmodel(=11arecomparedtoMIMOFMRreduced-ordermodelwithtwentystates. 0.5 1 1.5 2 2.5 3 3.5 4 1 0.5 0 0.5 1 1.5 2 2.5 f/f0real(G(jw)) 0.5 1 1.5 2 2.5 3 3.5 4 2 1.5 1 0.5 0 0.5 1 f/f0imag(G(jw)) FMR Fig.11Transferfunctionfromincomingden-sityperturbationtoaveragethroatMachnum-berforsupersonicdiuser.ResultsfromCFDmodel(=11)arecomparedtoMIMOFMRreduced-ordermodelwithtwentystates.modelmatchestheCFDresultswellovertheen-tirefrequencyrangeplotted,withasmalldiscrep-ancyathigherfrequencies.TheArnoldimodelmatcheswellforlowfrequencies,butshowscon-siderableerrorforf/f3.ThePODmodelhassomeundesirableoscillationsatlowfrequen-cies,andstrictlyisonlyvalidoverthefrequencyrangesampledinthesnapshotensemble(f/f2).TheperformanceofthePODandArnoldimodelscanbeimprovedbyincreasingthesizeofthereduced-ordermodels,however,inprac- 0.5 1 1.5 2 2.5 3 3. 5 0 0.5 1 1.5 2 2.5 f/f0|G(jw)| POD 0.5 1 1.5 2 2.5 3 3. 5 4 2 0 2 4 f/f0phase(G(jw)) Fig.12TransferfunctionfrombleedactuationtoaveragethroatMachnumberforsupersonicdiuser.ResultsfromCFDmodel(=11arecomparedtoFMR,PODandArnoldimod-elswith=10tice,itisfoundthatiftoomanybasisvectorsareincluded,thesetechniquesresultinunstablemodels.ForthePOD,onecouldalsoconsideraddingmoresnapshotstotheensemble.AmajordisadvantageofthesemethodsisthatthechoiceofthenumberofPODorArnoldibasisvectorstobeincludedandthesnapshotlocationsmustbedeterminedinanadhocmannerthatbalancesstabilityandaccuracy.Moreover,asinthedif-fuser”owcaseconsideredhere,stabilityofthereducedmodelisoftenatroublesomeissue.Fouriermodelreduction(FMR)isanewmethodformodelreductionofverylargelinearsystems.Themethodyieldsaccurate,guaran-teedstablereduced-ordermodels,whichcanbederivedusingane
cientiterativeprocedure.AneectiveuseofthemethodistoderivetheHan-kelmatrixofanintermediatereducedsystem,withmorestatesthandesiredbutwhichcapturestherelevantdynamicsveryaccurately.Balancedtruncationcanthenbeappliede
cientlytoob-taina“nalreduced-ordermodelwhosesizeischosenaccordingtothedistributionofHankelsin-gularvalues.Resultshavebeenpresentedfortwolargedy-namicalsystems,arisingfromimplementationofcomputational”uiddynamicmethods.Whilethe”owdynamicsofthetwosystemsareverydier-ent-oneisasubsonicexternal”ow,theother10of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235 asupersonicinternal”ow-thenewmethodisshowntoworkextremelyeectivelyinbothcases.Thecosttoevaluatethereduced-ordermodelsismuchlowerthanforothercommonlyusedmeth-odssuchastheproperorthogonaldecomposition.Moreover,thenewmethodyieldsreduced-ordermodelsthatarenotrestrictedtoaparticularfrequencyrange,accountforsysteminputsandoutputs,andhaveguaranteedstability.Adamjan,V.,Arov,D.,andKrein,M.,“AnalyticPropertiesofSchmidtPairsforaHankelOperatorandtheGeneralizedSchur-TakagiProblem,”Math.USSRSbornikVol.15,1971,pp.31–73.Bettayeb,M.,Silverman,L.,andSafonov,M.,“Op-timalApproximationofContinuous-TimeSystems,”Pro-ceedingsofthe19thIEEEConferenceonDecisionandControl,Vol.1,December1980.Kung,S.-Y.andLin,D.,“OptimalHankel-NormModelReductions:MultivariableSystems,”IEEETrans-actionsonAutomaticControl,Vol.AC-26,No.1,August1981,pp.832–52.Moore,B.,“PrincipalComponentAnalysisinLinearSystems:Controllability,Observability,andModelReduc-IEEETransactionsonAutomaticControl,Vol.AC-26,No.1,August1981,pp.17–31.Holmes,P.,Lumley,J.,andBerkooz,G.,Turbulence,CoherentStructures,DynamicalSystemsandSymmetryCambridgeUniversityPress,Cambridge,UK,1996.Sirovich,L.,“TurbulenceandtheDynamicsofCoher-entStructures.Part1:CoherentStructures,”ofAppliedMathematics,Vol.45,No.3,October1987,pp.561–571.Dowell,E.andHall,K.,“Modelingofuid-structureinteraction,”AnnualReviewofFluidMechanics,Vol.33,2001,pp.445–90.Beran,P.andSilva,W.,“Reduced-OrderModeling:NewApproachesforComputationalPhysics.”AIAAPaper2001-0853,presentedat39thAerospaceSciencesMeeting&Exhibit,Reno,NV,January2001.Silveira,L.,Kamon,M.,Elfadel,I.,andWhite,J.,“ACoordinate-TransformedArnoldiAlgorithmforGener-atingGuaranteedStableReduced-OrderModelsofRLCComputerMethodsinAppliedMechanicsandEngineering,Vol.169,No.3-4,February1999,pp.377–Willcox,K.,Peraire,J.,andWhite,J.,“AnArnoldiapproachforgenerationofreduced-ordermodelsforturbo-machinery,”ComputersandFluids,Vol.31,No.3,2002,pp.369–89.Lassaux,G.andWillcox,K.,“Modelreductionforac-tivecontroldesignusingmultiple-pointArnoldimethods.”AIAAPaper2003-0616,2003.Lall,S.,Marsden,J.,andGlavaski,S.,“EmpiricalModelReductionofControlledNonlinearSystems,”Pro-ceedingsoftheIFACWorldCongress,1999.Willcox,K.andPeraire,J.,“BalancedModelReduc-tionviatheProperOrthogonalDecomposition,”,Vol.40,No.11,November2002,pp.2323–30.Willcox,K.andMegretski,A.,“FourierModelReduc-tionforLarge-ScaleApplicationsinComputationalFluidDynamics,”TobepresentedatSIAMConferenceonAp-pliedLinearAlgebra,Williamsburg,VA,July2003.Willcox,K.,Reduced-OrderAerodynamicModelsforAeroelasticControlofTurbomachines,Ph.D.thesis,Dept.ofAeronauticsandAstronautics,MIT,February2000.Lassaux,G.,High-FidelityReduced-OrderAerody-namicModels:ApplicationtoActiveControlofEngine,Master’sthesis,Dept.ofAeronauticsandAstro-nautics,MIT,June2002.11of11 AmericanInstituteofAeronauticsandAstronauticsPaper2003-4235