ACCELERATING CFD-BASED AEROELASTIC PREDICTIONS USING SYSTEM IDENTIFICATION

Transcription

1 AIAA ACCELERATING CFD-BASED AEROELASTIC PREDICTIONS USING SYSTEM IDENTIFICATION Timohy J. Cowan * and Andrew S. Arena, Jr. Mechanical and Aerospace Engineering Deparmen Oklahoma Sae Universiy Sillwaer, OK 7478 Kajal K. Gupa NASA Dryden Fligh Research Cener Edwards, CA Absrac Sysem idenificaion is evaluaed as an efficien and accurae echnique for modeling unseady aerodynamic forces for use in ime-domain aeroelasic analysis. In he sysem idenificaion mehodology, he consan coefficiens of a linear sysem model are fi o he compued response ime hisories from a 3-D, unseady CFD solver. The resuling model of he unseady CFD soluion is independen of boh dynamic pressure and srucural parameers. Hence, his mehodology has he advanage ha only one CFD flow-field compuaion for each Mach number mus be compleed o deermine he aeroelasic insabiliy boundary. Resuls show ha sysem idenificaion can accuraely model he unseady aerodynamic forces for complex aerospace srucures of pracical ineres. The mehodology resuls in a subsanial savings in compuaional ime when predicing aeroelasic insabiliies. Nomenclaure CFD = Compuaional Fluid Dynamics [C] = generalized damping marix f a = generalized aerodynamic force vecor [K] = generalized siffness marix [M] = generalized mass marix nr = number of roos or mode shapes q = free sream dynamic pressure q = generalized displacemen vecor u = vecor of sysem inpus y = vecor of sysem oupus * Graduae Research Assisan, Suden Member AIAA. Assisan Professor, Senior Member AIAA. Aerospace Engineer, Member AIAA. Copyrigh 998 by Timohy J. Cowan and Andrew S. Arena. Published by he American Insiue of Aeronauics and Asronauics, Inc., wih permission. Inroducion Predicing insabiliies in he aeroelasic behavior of aerospace srucures is imporan in he design of modern aircraf which operae over a wide envelope. Wih recen advances in CPU speeds, curren research has urned oward he applicaion of CFD models o he soluion of hese aeroelasic problems. Using an unseady or Navier-Sokes CFD algorihm coupled wih a srucural dynamics solver, he complee aeroelasic response of he srucure can be prediced. However, he major limiaion in applying such a CFD soluion is he compuaional ime required o run a full aeroelasic simulaion due o he high dimensionaliy of even he simples geomery. Compounding he problem, an aeroelasic insabiliy canno be prediced by jus one such simulaion. Raher, several simulaions are required over he fligh regime in order o predic he crossover from sable o divergen ime hisories. When running hese coupled simulaions, i is he unseady CFD soluion a each ime sep which requires he greaes amoun of CPU ime. The faser srucural dynamics solver is essenially lef waiing on he unseady CFD soluion a each ime sep. Hence, if an accurae and efficien replacemen for he CFD solver could be developed, aeroelasic insabiliy predicions would be much more compuaionally efficien. Such a replacemen migh be found by developing a mahemaical model for he unseady CFD soluion using sysem heory. Concepually, he unseady CFD soluion implemened in an aeroelasic analysis is simply a dynamic sysem which compues an aerodynamic response based on a prescribed moion for he srucure. Furher more, he unseady CFD soluion can be assumed o be a dynamically linear sysem if only small perurbaions abou a nonlinear mean flow are considered. Knowing his, a variey of efficien sysem modeling echniques can be applied which have been developed for linear sysems. American Insiue of Aeronauics and Asronauics

2 One such echnique is sysem idenificaion. As i is defined, sysem idenificaion is a process for obaining a mahemaical model of a dynamic sysem based on a se of measured daa from he sysem. This mehodology is used o fi he parameers of a model srucure o a se of recorded daa from he dynamic sysem. The resul is an algebraic model ha is a mahemaical map beween he inpu and he oupu of he sysem. The success of his echnique is hen dependen on he iniial choice of he model srucure and he amoun and qualiy of daa used o rain he model. The emphasis of he presen work is o deermine he efficacy of using sysem idenificaion echniques o accuraely map he inpu-oupu relaionship for an unseady CFD soluion o an arbirary hreedimensional srucure. This algebraic relaionship hen reduces he oal compuaional ime required for a CFD aeroelasic analysis by replacing he unseady CFD solver in he coupled soluion. Resuls are presened which address he exen o which he sysem idenificaion mehodology is applicable o geomeries and flows of pracical ineres in aerospace applicaions. Mehodology Compuaional analysis for his sudy was performed using he aeroelasic capabiliies of he STARS codes developed a NASA Dryden Fligh Research Cener. STARS 3 is an highly inegraed, finie elemen based code for mulidisciplinary analysis of fligh vehicles including saic and dynamic srucural analysis, compuaional fluid dynamics, hea ransfer, and aeroservoelasic capabiliies. Srucural analysis in STARS is accomplished using he finie elemen mehod o compue he eigenvecors and eigenvalues which describe he elasic modes for a srucure. Any arbirary moion of he srucure can hen be represened by muliplying each eigenvecor by a generalized displacemen and applying modal superposiion. A complee aeroelasic analysis is accomplished by coupling a dynamics solver, using he modal vecors, wih an unseady CFD solver which compues he generalized aerodynamic forces acing on he srucure. The basic aeroelasic equaion of moion solved by STARS is given below in generalized coordinaes. [ M ] q + [ C] q& + [ K] q = ( ) & () The unseady aerodynamics forces on he righ hand side of Eq. () are compued using a imemarched, finie elemen approach o solving he unseady equaions. This CFD soluion is f a performed on a mesh consising of unsrucured erahedra using he ranspiraion mehod o simulae srucural deformaions. Considering he described soluion scheme, he inpu-oupu relaionship for he unseady CFD soluion can be represened by he simple block diagram shown in Figure. q() Inpu(s) Unseady CFD Solver Dynamic Sysem f a () Oupu(s) Figure : Block Diagram Represenaion of STARS Unseady CFD Solver As discussed previously, i is he unseady CFD soluion for he generalized aerodynamic force vecor which requires he greaes proporion of CPU ime. Hence, sysem idenificaion will be applied o his sysem in order o develop an efficien algebraic model for he generalized aerodynamic forces. The basic sysem idenificaion procedure will involve fiing he consan coefficiens of a model srucure o a se of acual ime hisory daa aken from he unseady CFD solver. There are several differen ypes of model srucures ha could be used, bu for he case of a muli-inpu, muli-oupu (MIMO) sysem, he mos common is he auoregressive moving average (ARMA) model. The ARMA model describes he response of a dynamic sysem as a sum of scaled previous oupus and inpus o he sysem. 5 The basic ARMA model srucure for a single-inpu, singleoupu (SISO) sysem is easily vecorized o yield he following model srucure for a MIMO sysem: na nb [ Ai ] y( k i) + [ Bi ] y ( k) = u( k i) () i= i= Noice ha he sysem response a any ime sep, k, is simply a linear combinaion of pas inpus and oupus, making his model very easy o implemen mahemaically. Wih he model srucure of Eq. () in mind, he ask is hen o idenify he marices of consan coefficiens, [A i ] and [B i ], for an assumed model order consising of na pas oupus and nb inpus. This is accomplished by fiing he model coefficiens o a se of ime hisory daa from he unseady CFD solver. In order o obain ime hisory daa suiable for sysem idenificaion, he unseady CFD soluion for a prescribed moion of he srucure is compleed raher han allowing he srucure o move freely in he flow as in a ypical aeroelasic simulaion. The prescribed American Insiue of Aeronauics and Asronauics

3 moion of he srucure is a known inpu which is chosen because i excies a broad specrum of response frequencies which conain he primary flow physics. The only limiaion on choosing he prescribed inpu signal is ha he generalized displacemens and velociies mus be mahemaically consisen. Hisorically, sysem idenificaion has been used exensively in fligh esing o esimae sabiliy derivaives and modal damping parameers from fligh es daa. Thanks o his applicaion, a grea deal of research has been done on he opimal inpu required for successful parameer idenificaion for fligh vehicles. One of he mos widely acceped inpus is he 3 mulisep due o is ease of implemenaion and broad frequency conen. 8 Figure shows he basic forma of his inpu signal. 3 displacemen and velociy inpu for a wo mode sysem which was found o work bes for modeling x x.4. x v x v v v x Figure : 3 Mulisep Inpu Signal Alhough he mulisep inpu signal is easy o implemen experimenally, he sharp corners cause some numerical problems if his inpu is applied o displacemens. As discussed previously, he displacemens and velociies for he prescribed inpu signal mus be mahemaically consisen so ha boh boundary condiions for he unseady CFD soluion are saisfied. Hence, a mulisep inpu applied o displacemens mus be differeniaed o ge he mahemaically consisen velociy inpu. Doing so, one finds ha he velociy inpu would be made up of five infinie spikes. Obviously, such an inpu canno be numerically realized as a CFD boundary condiion. To avoid his numerical problem, he mulisep inpu could be implemened on he velociy boundary condiion and hen inegraed o ge he mahemaically consisen displacemen boundary condiion. This ype of velociy mulisep was esed along wih several oher inpus in he unseady CFD soluion, and he mulisep was found o work bes for idenificaion. Figure 3 shows he prescribed generalized Figure 3: Prescribed Inpu Signal for Unseady CFD Flow Solver Noice in Figure 3 ha he inpu signals for each mode are ou of phase. This is necessary o allow he idenificaion procedure o disinguish beween he effecs of differen inpus in he sysem response. Resuls presened here will show ha he ime hisory daa obained using his inpu signal is sufficien for modeling he aerodynamic response of srucures wih as many as nine modes. Once his inpu signal has been run hrough he unseady CFD soluion, singular value decomposiion (SVD) is used o fi he parameers of he ARMA model srucure o he ime hisory daa. SVD 9 produces a soluion o he sysem of overdeermined equaions ha is he bes approximaion in a leassquares sense. This idenificaion procedure requires an iniial guess for he order of he ARMA model srucure (na and nb). By hen varying he order of he model, one searches for he model which minimizes he error beween he CFD ime hisory and he prediced model ime hisory. Once compleed, he model is used in place of he unseady CFD soluion in he coupled simulaion o predic he full aeroelasic response of he srucure. Figure 4 illusraes concepually how he ARMA model 3 American Insiue of Aeronauics and Asronauics

4 is implemened in a STARS aeroelasic simulaion. The coupled aeroelasic soluion wih he discree-ime model in place can be execued a almos no compuaional cos relaive o he unseady CFD soluion. Furhermore, he model is only dependen on he Mach number for which i was derived. I is no dependen on he free sream dynamic pressure or any of he srucural parameers, such as generalized mass and siffness. This allows one o vary hese parameers and use he fas model soluion o observe heir effec on he aeroelasic response of he srucure. Hence, a subsanial amoun of compuaional ime is saved in he search for aeroelasic insabiliies a each Mach number by running he coupled model soluion raher han he ime-marched CFD soluion. sandard aeroelasic es case ha has been invesigaed experimenally in he Langley Transonic Dynamics unnel. A planform view of he configuraion is shown in Figure 5. This wing geomery is ofen used in he lieraure as a validaion case for compuaional aeroelasic codes in he ransonic flow regime. Recen work has shown ha he STARS aeroelasic analysis module is capable of predicing he experimenal daa for his wing geomery including he ransonic dip in he fluer boundary around Mach.. 9 FEM Solids Analysis Modal Parameers Seady Sae CFD Soluion I.C.'s Aero. Forces Unseady CFD Soluion Global Time Sep Dynamics Solver CFD B.C.'s Sofware Swich Discree Time Model Figure 4: Implemenaion of Sysem Model in STARS Coupled Aeroelasic Simulaion Resuls Unseady CFD soluions for he prescribed srucural moion were run for several geomeries over a wide range of Mach numbers. One such geomery is he AGARD wing configuraion which is a Figure 5: AGARD Tes Wing Geomery and Surface Discreizaion In STARS, he AGARD is modeled srucurally using he wo dominan eigenvecors represening he firs wo naural vibraion modes of he srucure. These mode shapes physically represen wing firs bending and orsion. The CFD mesh for he AGARD consiss of 7,36 nodes and 376,5 erahedral elemens. Before beginning he sysem idenificaion procedure on his geomery, i is imporan o remember ha he modeling procedure assumes ha he sysem is only dynamically linear. In order o make his assumpion, he complee nonlinear mean flow, or seady CFD soluion, mus be compued and used as he iniial condiion for he unseady CFD soluion. This guaranees ime accuracy for he unseady soluion and allows one o assume ha he sysem is dynamically linear for small perurbaions abou he nonlinear mean flow. Following he seady soluion, he mulisep inpu signal show in Figure 3 was analyzed using he unseady solver for he AGARD geomery a Mach.96 and a free sream densiy of slinch/in 3, or a dynamic pressure of.44 psi. Using he resuling generalized aerodynamic force ime hisory, he coefficiens of he ARMA model srucure were compued using he SVD algorihm. The 4 American Insiue of Aeronauics and Asronauics

5 generalized aerodynamic force ime hisory from boh he unseady soluion and he discree-ime model soluion found o bes fi his daa is shown in Figure d.. model hen implemened in an aeroelasic soluion o search for he insabiliy boundary a ha Mach number. Once he poin of insabiliy was found, he soluion was run once o verify ha he model soluion was valid. For each Mach number esed, he unseady and model ime hisories a he fluer boundary were in excellen agreemen. The complee ransonic fluer boundary for he AGARD wing configuraion is given in Figure 8. f d.. model.5..5 x d.. model f Figure 6: Comparison of Unseady and Model Soluions for he AGARD Mulisep Inpu Signal of Figure 3. The new discree-ime model is hen used o search for insabiliies a his Mach number by repeaedly varying he free sream densiy. Once he poin of aeroelasic insabiliy is found, he coupled soluion can hen be run once o verify he accuracy of he coupled model soluion. For Mach.96, a sample comparison beween he aeroelasic ime hisory prediced by he discree-ime model soluion and he unseady soluion is shown in Figure 7 for a free sream densiy near he insabiliy boundary, ρ = slinch/in 3. Noice ha he wo responses are in good agreemen. Furher validaions of he sysem idenificaion procedure where run for he AGARD a five addiional Mach numbers, resuling in a complee ransonic fluer boundary predicion. As wih Mach.96, a new mulisep soluion was compued wih he unseady solver a each differen Mach number. An ARMA model was hen fi o each se of mulisep ime hisory daa, and he resuling discree-ime model was x Figure 7: Comparison of Unseady and Model Soluions for he AGARD Aeroelasic Response a Mach.96 d.. model 5 American Insiue of Aeronauics and Asronauics

6 Fluer Speed Index, Vf Model Soluion STARS CFD Soluion Mach Number, M Figure 8: Comparison of AGARD Fluer Boundary Prediced by STARS CFD and Model Soluions As i will be discussed, he ime savings associaed wih he use of he model is significan. Firs consider he curren mehod for applying CFD o aeroelasic analysis in STARS. For a given Mach number, he full unseady CFD soluion is run a leas four imes a differen densiies in a search for he crossover poin from sable o divergen ime hisories. The resuls from hese ime hisories are hen inerpolaed o deermine he approximae poin a which he sysem is unsable. The oal compuaional ime o run jus one unseady CFD soluion of sufficien lengh o be qualiaively useful is CPU hours on an IBM RS6 3BT for he AGARD. Muliply ha ime by four and i requires days o deermine he approximae sabiliy boundary for he AGARD a one Mach number. The sysem idenificaion echnique requires only one run of he unseady CFD soluion for each Mach number. The lengh of he required mulisep ime hisory is abou one fourh of he lengh required for a full aeroelasic run, so i runs in jus under 3 CPU hours. Following he mulisep soluion, he enire procedure for compuing he bes parameers of he discree-ime model akes less han 3 minues, and hen he discree-ime model can be run repeaedly o predic complee aeroelasic ime hisories in less han 6 CPU seconds. Hence, he oal savings in compuaional ime for each Mach number is over 4 CPU hours, or 94% reducion in oal CPU ime. A comparison of he oal ime required o compue he neural poin of he AGARD a each Mach number is shown graphically in Figure 9. CPU Time (hours) Time Hisories Required o Idenify Sabiliy Boundary 3. d.. model Figure 9: Comparison Beween Required Compuaional Time For Soluion and Discree-Time Model Soluion Anoher ineresing geomery o sudy is ha of he Generic Hypersonic Vehicle (GHV). The GHV is a escase developed by NASA o es he aeroelasic effecs ha migh be observed on a hypersonic vehicle. Figure shows he CFD surface mesh used o model he GHV. The CFD mesh for he GHV consiss of 58,786 nodes and 33,47 erahedral elemens. Figure : GHV Geomery and Surface Discreizaion Srucurally, he GHV is much more complicaed han he AGARD as i is modeled using nine eigenvecors which represen various bending and orsional modes for he wings and he body iself. This geomery was firs analyzed using STARS a Mach. and a free sream densiy of slinch/in 3, which corresponds o a dynamic pressure of 4.5 psi. As wih he AGARD, he seady flow field was firs compued, followed by an unseady CFD soluion o a sequence of saggered mulisep inpus on all nine modes. An aerodynamic model for he GHV is hen compued by fiing he coefficiens of he ARMA 6 American Insiue of Aeronauics and Asronauics

7 model srucure o he mulisep ime hisory daa. Figure presens a comparison beween he unseady soluion and discree-ime model soluion o he mulisep inpu for he GHV. Only six ou of he nine modes are shown here for breviy, however resuls are consisen for all nine. The compued aerodynamic model for he GHV can hen be used in he coupled soluion o compue he aeroelasic response of he GHV. A sample comparison beween he aeroelasic ime hisory prediced by he discree-ime model and he unseady soluion is shown in Figure for modes one hrough six of he GHV. Again, resuls are consisen for all nine modes and can be found in reference. 5 5 d.. model 55 5 d.. model f 48 f d.. model f f d.. model d.. model f 5-4 f d.. model Figure : Comparison of Unseady and Model Soluions for he GHV Mulisep Inpu Signal for Modes hrough 6 a Mach. 7 American Insiue of Aeronauics and Asronauics

8 .5 x x d.. model d.. model d.. model.3 d.. model x x d.. model. d.. model.8.4 x x Figure : Comparison of Unseady and Discree-Time Model Soluion for he GHV Aeroelasic Response a Mach. Conclusions The objecive of his sudy was o develop an accurae and efficien mehod for developing a model of an unseady CFD soluion for use in compuaional aeroelasic analysis of complex aerospace srucures. Research presened here demonsraes ha sysem idenificaion can be used o accuraely map he inpuoupu relaionship of he unseady CFD soluion. The mehodology assumes ha he unseady CFD soluion is dynamically linear for small disurbances abou a nonlinear mean flow. For such a sysem, an aerodynamic model can be developed by fiing he coefficiens of a muli-inpu, muli-oupu ARMA model srucure o a se of CFD ime-hisory daa. The resuling discree-ime aerodynamic model is hen used 8 American Insiue of Aeronauics and Asronauics

9 in place of he unseady CFD soluion in he coupled aeroelasic analysis. This echnique can be exended o differen srucural geomeries over a wide range of Mach numbers including he ransonic regime. The mehodology has he advanage ha only one unseady CFD soluion is required for each Mach number. The aerodynamic model based on ha soluion is independen of boh dynamic pressure and srucural parameers. This allows one o use he same model while searching for aeroelasic insabiliies by varying hese parameers in he coupled soluion. Compuaionally, he discree-ime model is exremely easy o apply and offers a large savings in oal CPU ime when used in an aeroelasic analysis. Hence, his approach may make he use of CFD simulaions rouine in he aeroelasic analysis of aerospace vehicles. Idenificaion, Journal Of Aircraf, Vol. 33, No., 996, pp Press, W.H., Teukolsky, S.A., Veerling, W.T., and Flannery, B.P., Numerical Recipes in Forran 77: The Ar of Scienific Compuing, nd Ediion, Cambridge Universiy Press, Gupa, K.K., Developmen of a Finie Elemen Aeroelasic Analysis Capabiliy, Journal of Aircraf, Vol. 33, No. 5, Sepember-Ocober 996, pp Cowan, T.J., Efficien Aeroelasic CFD Predicions Using Sysem Idenificaion, Masers Thesis, Oklahoma Sae Universiy, May 998. Acknowledgemens Funds for he suppor of his sudy have been allocaed hrough he NASA-Ames Universiy Consorium Office, under Inerchange Number NCC- 55, and Oklahoma Sae Universiy. References. Dowell, E.H., e al, A Modern Course in Aeroelasiciy, 3 rd Revised and Enlarged Ediion, Klewer Academic Publishers, Ljung, L., Sysem Idenificaion: Theory For The User, Prenice Hall, Inc., New Jersey, Gupa, K.K., STARS An Inegraed General-Purpose Finie Elemen Srucural, Aeroelasic, and Aeroservoelasic Analysis Compuer Program, NASA TM-4795, May Hollkamp, J. J., and Baill, S. M., Auomaed Parameer Idenificaion and Order Reducion for Discree Time Series Models, AIAA Journal, Vol. 9, No., 99, pp Pinkelman, J. K., Baill, S. M., and Kehoe, M. W. Toal Leas Squares Crieria in Parameer Idenificaion for Fligh Fluer Tesing, Journal of Aircraf, Vol. 33, No. 4, 996, pp Hollkamp, J. J., and Baill, S. M., A Recursive Algorihm For Discree Time Domain Parameer Idenificaion, AIAA Paper Hamel, P. G., and Jaegaonkar, R. V., Evoluion of Fligh Vehicle Sysem 9 American Insiue of Aeronauics and Asronauics