1 Optimal Design f Tansnic Fan Blade Leading Edge Shape Using CFD and Simultaneus Petubatin Stchastic Appximatin Methd X.Q.Xing and M.Damdaan Abstact Simultaneus Petubatin Stchastic Appximatin methd has attacted cnsideable applicatin in many diffeent aeas such as statistical paamete estimatin, feedbac cntl, simulatin-based ptimizatin, signal & image pcessing, and expeimental design. In this pape, its pefmance as a viable ptimizatin tl is demnstated by applying it fist t a simple wing gemety design pblem f which the bjective functin is descibed by an empiical fmula fm aicaft design pactice and then it is used in a tansnic fan blade design pblem in which the bjective functin is nt epesented by any explicit functin but is estimated at each design iteatin by a cmputatinal fluid dynamics algithm f slving the Navie-Stes equatins Keywds glbal ptimizatin; simultaneus petubatin stchastic appximatin methd, simulated annealing, tansnic fan design I. INTRODUCTION he need f slving multivaiate ptimizatin pblems is T pevasive in engineeing, the physical and scial sciences. The chaacteistic featues f such pblems ae the pesence f a lage numbe f design vaiables, cmplex cnstaints, and even discete design paamete values. A numbe f ptimizatin algithms, bth lcal and glbal ptimizatin algithms have been develped f ptimizing such pblems. Besides deteministic methds, stchastic methds such as genetic algithm (GA) and simulated annealing (SA) algithm etc have ecently fund applicatins in pactical engineeing design ptimizatin pblems. These algithms ae all stchastic in natue and easily implemented in bust cmpute cdes as cmpaed with deteministic methds. Hweve, SA and GA methds equie lage numbe f functin evaluatins and elative lng cmputatin time especially in the case f cmplex design pblems. One X.Q. Xing, SMA Reseach Fellw, Cente f Advanced Numeical Engineeing Simulatins, Nanyang Technlgical Univesity, Nanyang Avenue, Singape, 639798 (telephne: 65-7904074, e-mail: xqxing@ ntu.edu.sg). M.Damdaan, Assciate Pfess f Nanyang Technlgical Univesity and SMA Faculty Fellw, Schl f Mechanical and Pductin Engineeing Simulatins, Nanyang Technlgical Univesity, Nanyang Avenue, Singape, 639798 (telephne: 65-7905599, e-mail: mdamdaan@ ntu.edu.sg). appach t educe cmputatinal time wuld be t use paallel GA and paallel SA as utlined in Wang and Damdaan 1. An attactive altenative t SA and GA culd be the Simultaneus Petubatin Stchastic Appximatin (SPSA) methd descibed by Spall 2-4 and which has been applied t difficult multivaiate ptimizatin pblems. The SPSA methd has attacted cnsideable applicatin in many diffeent aeas such as statistical paamete estimatin, feedbac cntl, simulatin-based ptimizatin, signal & image pcessing, and expeimental design. The essential featue f SPSA, which accunts f its pwe and elative ease f implementatin, is the undelying gadient appximatin which equies nly tw measuements f the bjective functin egadless f the dimensin f the ptimizatin pblem. This featue allws f a significant decease in the cst f ptimizatin, especially in pblems with a lage numbe f vaiables t be ptimized. SPSA methds ae biefly utlined and its pefmance as a viable ptimizatin tl is demnstated by applying it fist t a simple wing gemety design pblem f which the bjective functin is descibed by an empiical fmula fm aicaft design pactice and then it is used in a tansnic fan blade design pblem in which the bjective functin is nt epesented by any explicit functin but is estimated at each design iteatin by a cmputatinal fluid dynamics algithm f slving the Navie-Stes equatins. II. SPSA METHOD SPSA is elatively easy t implement and des nt equie gadient infmatin. It is a faily bust methd and has the ability t find a glbal minimum in the pesence f multiple minima. SPSA is an algithm that is based n a simultaneus petubatin gadient appximatin. The simultaneus petubatin appximatin uses nly tw functin measuements independent f the numbe f paametes (say, p) being ptimized. The SPSA algithm ws by iteating fm an initial guess f the ptimal vect X 0. Fist, the cunte index is initialized t a value f 0, an initial guess f the design vaiable vect X and nn-negative empiical cefficients ae set. Next a p-dimensinal andm
2 simultaneus petubatin vect is cnstucted and tw measuements f the bjective functin, namely y( X + c ) and y( X c ) ae btained based n the simultaneus petubatin aund the given vect X. Then geneate the simultaneus petubatin appximatin t m the gadient g ( X ). The paamete c = c0 /( ) whee c 0 is a small psitive numbe taen as 0.01 in this study, is the lp index and m is a cefficient taen as 1/6 in this study. The tem epesents the andm petubatin vect geneated by Mnte-Cal appaches and the cmpnents f this petubatin ae independently geneated fm a zemean pbability distibutin and a simple distibutin that has been used in this study is the Benulli ± 1 distibutin with pbability f ½ f each ± 1 utcme. This is fllwed immediately by the calculatin f the gadient appximatin based n tw measuements f the functin based n the simultaneus petubatin aund the cuent value f the design vaiable vect and the updating f the design vect X t a new value X + 1 using standad SA fm, i.e. X + 1 = X a g( X ). The value f a can be chsen t ensue effective pactical pefmance f the algithm. Finally the algithm is teminated if thee ae insignificant changes in seveal successive iteatins if the maximum allwable numbe f iteatins has been eached. The details f the step-by-step implementatin f the SPSA algithm ae utlined in Spall 2,4. III. WING DESIGN OPTIMIZATION A. Wing Design Pblem The design pblem cncens the design f wing shape such that the aedynamic efficiency f the wing eaches a maximum value duing cuise with the wing weight acting as a cnstaint, i.e. the gal is t detemine the shape f the wing f minimizing D/L (dag-t-lift) maximizing L/D with the wing weight as a cnstaint. Explicit empiical functin f D/L which fm the bjective functin f the ptimizatin pblem and empiical expessins used hee ae defined in Rayme 6. The bjective functin t be ptimized is defined as fllws: Minimize F( x) = D / L (1) Subject t six cnstaints n the design vaiables defined as fllw: 1.0 α 10.0, 10.0 b 50. 0, 3.5 c 10.0, 0.0 λ 35.0, 0.5 A R 15.0, W wing 2473( lb) Whee.0 15 λis the angle f attac, b is the wing span, c is the mean aedynamic chd, λ is the wing sweep, AR is the wing aspect ati and W wing is the wing weight. The cnstaints ae incpated in int a cmpsite bjective functin by way f penalty functins. B. Optimizatin Results and Cmpaisn Table 1 shws the ptimum values f the bjective functin and design vaiables eached by diffeent ptimizatin algithms. It can be seen that the esults f bjective functin and vaiables fm SPSA ae vey simila t thse f Simulated Annealing (SA). Fig. 1 cmpaes the evaluatins f functin t each the ptimal values based n the specified cnvegence citein using SPSA with thse attained by SA methd. It can be seen that SPSA eaches ptimal values abut 9 times faste than SA theeby suggesting that SPSA is a viable and a me efficient stchastic design methd. TABLE I COMPARING OPTIMAL DESIGN RESULTS USING SPSA AND SA OPTIMIZATION METHODS Algithm α ( ) b ( ft ) c ( ft ) λ ( ) Wing Weight ( lb ) bjective Functin D/L SPSA 3.432 44.994 5.986 18.115 2443.93 0.0319 SA 3.199 44.987 5.947 18.002 2443.93 0.0319 SA SPSA Evaluatins f Objective Functin 0 2500 5000 7500 10000 Fig. 1. Evaluatins f Objective Functins IV. TRANSONIC FAN BLADE LEADING EDGE DESIGN PROBLEM Tansnic fan blade design pblems include the design f the blade aifil sectins at diffeent adial psitins and the stacing f these aifils t fm a thee dimensinal blade. Many studies have fcused n the aifil ptimizatin using the Navie-Stes and adjint equatins as in ptimized design f shc-fee aifils and wings epted in Sung and Kwn 8. Catalan 9 pesented an invese design f a thee-dimensinal tub-machiney blade in which bjective functins wee based n inviscid tansnic flw mdel and Tng 10 ppsed a multi-bjective simulated annealing algithm t impve the initial design f tubine blade. Studies f swept ts ae mtivated by the fact that swept wings educed shc wave stength, and the shape f the
3 leading edge f a swept blade can be alteed t cntl the shc stuctue and the migatin f the lw-mmentum flw fluid, which cnsequently decease the lsses induced by the inteactin between shc and bunday laye as well as the inteactin between shc and secnday flw. This mtivates the cuent applicatin pblem dealing with the design f the leading edge cuve t meet design bjectives. The flw fields aund swept wings can be assumed as twdimensinal in which case it is sufficient t descibe the wing leading edge using a sweep angle α s as shwn in Fig. 2(a). Hweve, the leading edges f tansnic fan blades ae theedimensinal as shwn in Fig. 2(b). Ás (a) (b) Fig.2. Diffeence between (a) wing and (b)blade leading edge The design pblem cncens the design f the leading edge cuve such that the adiabatic efficiency f a tansnic fan s t eaches a maximum value duing 100% f the design speed, with the t s stall magin acting as a cnstaint. At the same time, the pessue ati must satisfy a given value. As the stall magin and the pessue ati cannt be expessed by the design vaiables, these wee nt integated int the design pblem as cnstaints but wee nly checed thugh numeical simulatin f the pesent study. A. Blade design pblem The t s leading edge cuve defined hee is same t that f efeence 11 by a cubic thee-dimensinal cuve as fllws: 2 3 R ( t) = a + bt + ct + dt (2) Whee, a = a, a, ], b = b, b, ], a3 b3 c = c, c, ], d = d, d, ] (3) c3 d 3 If the stating pint vect R (0), the cespnding tangent ' vect R (0), the end pint vect R(1) and the cespnding ' tangent vect R (1) ae given, the cuve can be detemined. The equatin f the cubic cuve defined in catesian cdinates is as fllws: x(0) y(0) z(0) 2 3 = x(1) y(1) z(1) x( t), y( t), z( t)] 1, t, t, t ] M x '(0) y '(0) z '(0) x' (1) y' (1) z '(1) (4) Whee, 1 0 0 0 0 0 1 0 M = (5) 3 3 2 1 2 2 1 1 In de t educe design vaiables and t cntl the evlving shape, the ends f the leading edge i.e. x(0), y (0), z(0), x(1), y(1), z(1) ae fixed accding t a baseline design cnfiguatin which is chsen t stat this design ptimizatin study. The gadients at the ends f the leading edge cuve, i.e.. x ' (0), y ' (0), z ' (0), x ' (1), y ' (1), z ' (1) ae alteed duing design iteatin t fm new leading edge cuves s that thee ae the six ptimizatin vaiables. The bjective functin t be maximized is defined as: F = Max ( η ) (6) O the pblem can be cast as a minimizatin pblem as fllws: F = Min( 1./ η ) (7) Whee η is the t's adiabatic efficiency which is estimated using a cmpessible cmputatinal fluid dynamics slve slving the Navie-Stes equatins and this is biefly utlined in the next sectin. B. Numeical Simulatin and Optimizatin The numeical simulatin cde that was used t test the adiabatic efficiency f the t, pessue ati and stall magin is utlined in Wang and Zha 12. F the cuent study, the Navie-Stes equatins ae slved with a finite-vlume, fustep Runge-Kutta, time maching methd, withut cnsideing tip-cleaance effects. The tubulence mdel used hee is a simple algebaic mdel and the gid cnsists f 62 ndes in axial diectin, 25 ndes in the adial diectin and 25 ndes in the blade-t-blade diectin. The whle design pcess is shwn in Fig. 3.
4 Thugh-flw cmputatin Set a baseline leading edge cuve 3D CFD Simulatin Abitay blading Max( η ) NO O satisfied the design demand? YES end Fig. 3. Design Pcess Flw Chat Optimizing the leading edge cuve Based n an initial guess f a cuve defining the shape f the leading edge, it is pssible t design a thee-dimensinal blade pfile using the semi-empiical thugh-flw analysis methd f Wennestm and Putehaugh 13 and the abitay blading methd t detemine the lcal blade cambe in the spanwise diectin and t distibute the thicness aund the cambe. The lcal thicness distibutin at the spanwise statins f the blade is assumed t emain fixed f the cuent study. Afte btaining the appximate blade gemety with lcal spanwise cambe and thicness the flw field f the t can be estimated using CFD. Based n the estimated values f ttal tempeatue and ttal pessue fm CFD analysis, the efficiency η f the t defined as whee in γ 1 γ ut Pin Tut Tin ( P / ) 1 η = / 1 P and P ae the ttal pessues and T and ut in (8) T ut ae the ttal tempeatues at the inflw and utflw bundaies f the t flw field cmputatinal dmain and γ is the ati specific heat f the gas, can be cmputed. This is fllwed by the ptimizatin steps t find a new blade leading edge cuve using the ptimizatin methd and the pcess is cntinued till desied cnvegence is achieved. design iteatins, is fist cnsideed. Figues 4(a)-(c) shws the cnvegence f the design iteatins cespnding t values f the paamete a set t 30, 50 and 100 espectively. The esults shw that the ate f cnvegence is the mst fastest when a =30. When the value f a is inceased, the ate f cnvegence is educed, while if the value f a is deceased t less than 30, thee is a pssibility that the design iteatins may divege duing the initial stages f the ptimizatin. F the case when a=30 the ptimal value is eached in abut 60 iteatins which equies abut 180 functin evaluatins. F the case when a=100 ptimal esult is eached in abut 200 iteatins equiing abut 600 functin evaluatin. The ptimal value f t s efficiency η attained can be baceted between 2 and 5 f all values f a in view f the scillaty manne in which the ptimal value is eached by the SPSA methd. Rt Efficiency (1./bjective functin) Rt's Efficiency (1./Objective Functin) 8 6 4 2 8 6 4 2 SPSA methd 20 40 60 80 100 120 140 160 Iteatins (a) a=30 0 50 100 150 200 N. f Iteatins (b) a=50 SPSA methd C. Results and Discussin F pupse f illustating the applicatin f SPSA methd a swept 3D blade shape with a staight wing leading edge is cnsideed as the baseline cnfiguatin t initiate the ptimizatin pcess. The effect f the paamete a appeaing in the fmulatin f the SPSA methd as discussed in Sectin II n the cnvegence f the bjective functin i.e. the cmputed/ptimized t adiabatic efficiency vs. numbe f
5 Rt's Efficiency (1./Objectin Functin) 8 6 4 2 0 50 100 150 200 250 300 N. f Iteatins SPSA methd (c) a=100 Fig. 4. Effect f SPSA paamete a n cnvegence f bjective functin Fm Fig. 4, it can be seen that the adiabatic efficiency f the t des nt cnvege t a fixed value in view f the scillaty natue f the vaiatin f the bjective functin t the ptimal value. Based n the tends f the bjective functin vaiatin, it is pssible t appximately fit a mean cuve thugh the band f the scillating egin. One easn f this ind f scillaty behavi culd be the manne in which the teatment f the design paametes is dne. It is well nwn that duing the design pcess f a tub-machiney blade, distibutin f lsses and the design paametes within the famew f the empiical methd f abitay blading distibutin shuld be adjusted caefully t get a easnably gd design. In the pesent study the distibutin f the lsses and the design paametes had been limited f all blade shapes t pduce the autmatic seach pcess. The impact f this will be analyzed in a sepaate study as the fcus f this execise is t gauge the feasibility f SPSA methd in shape design. Figues 5 and 6 cmpae the blade shape and the leading edge cuve shape cespnding t baseline and ptimal shape btained by the SPSA methd. (a) (b ) (c) Fig.6. (a) Initial blade shape (b) Optimized blade shape (c) Cmpaisn between (a) and (b) A deteministic methd based n the Byden-Fletche- Gldfab-Shann vaiable metic methd and a stchastic methd based n Simulated Annealing (SA) wee als used t aive at ptimal shape designs f the blade unde the same cnditins cnsideed using the SPSA methd f pupses f cmpaing the utcmes with that f the SPSA methd. Fig. 7 shws the cnvegence f the bjective functin with design iteatins btained by using deteministic methd while Fig. 8 shws the cnvegence f the bjective functin using the simulated annealing ptimizatin methd. Objective Functin 7 6 5 4 3 2 1 0.949 0.947 GB methd 10 20 Iteatin Numbes Fig. 7. Cnvegence f Objective functin Using Deteministic Methd The teminatin citein used t stp the design ptimizatin pcess is 6 f ( x 1) f ( x ) 10. + Fig.5. Cmpaisn f the leading edge shape cespnding t baseline (staight leading edge)and ptimal design (cuved leading edge)
6 Rt's Efficiency (1./Objectin Functin) 8 6 4 2 SA methd 50 100 150 200 250 300 350 400 450 N. f Functin Evaluatins Fig.8. Results f SA methd Fm Fig. 7 it appeas that afte 25 design iteatins the deteministic ptimizatin methd appeas t have been tapped in a lcal minimum as thee des nt appea t be any significant vaiatins in the bjective functin. Fm Fig. 8, it can be seen that cnvegence f the bjective functin using SA methd is vey simila t that f SPSA but the SA methd equies abut 320 t 400 functin evaluatins t achieve a value f η lying between 4 and 6. It is vey clea fm this peliminay investigatin that the SPSA methd is a bust methd f btaining glbal ptimizatin. It is cetainly faste than the SA methd while the gadient-based methd GB methd seems t be tapped in a lcal ptima. REFERENCES 1] X. Wang, M.Damdaan, Cmpaisn f Deteministic and Stchastic Optimizatin Algithms f Geneic Wing Design Pblems, Junal f Aicaft, AIAA, Vl. 37, N.5, 2000. 2] J. C. Spall, Multivaiate Stchastic Appximatin Using a Simultaneus Petubatin Gadient Appximatin, IEEE Tansactin n Autmatic Cntl, Vl.37, N. 3, 1992. 3] J. C. Spall, An Oveview f the Simultaneus Petubatin Methd f Efficient Optimizatin, Jhns Hpins APL Technical Digest, Vl.19, N.4, 1998. 4] J. C. Spall, Implementatin f the Simultaneus Petubatin Algithm f Stchastic Optimizatin, IEEE Tansactins n Aespace and Electnic Systems, Vl. 34, N.3, July 1998. 5] K. Deb, Optimizatin f Engineeing Design, Algithms and Examples, Pentice-Hall f India pivate Limited, New Delhi, 1998. 6] D. P. Rayme, Aicaft Design: A Cnceptual Appach, AIAA Educatin Seies, Seies Edit-in-chief: J.S. Pzemienieci, Published by AIAA, Inc, 370 L Enfant Pmenade, S.W., Washingtn, D.C., 20024, 1989. 7] D. E.Gldbeg, Genetic Algithms in Seach, Optimizatin and Machine Leaning, Addisn Wesley, 1989. 8] C. Sung, J.H. Kwn, Aedynamic Design Optimizatin Using the Navie-Stes and Adjint Equatins, AIAA Pape 2001-0266, 2001. 9] L.A. Catalan, A. Dadne, Pgessive Optimizatin f the Efficient Design f 3D Cascades, AIAA Pape 2001-2578, 2001. 10] T. Tng, Z. P. Feng, Multi-bjective Optimizatin Design f Tansnic Tubine Cascades Using Simulated Annealing Algithm, Submitted t Junal f Xi an Jiatng Univesity. 11] X.Q. Xing, Pbing Int the Cnntatin f Sweep Aedynamics f Tansnic Fans and Cmpesss, ASME Pape 2001-GT-352, 2001. 12] W. Wang and X.L. Zha, Unsteady Numeical Simulatin f the Inteactin between Wae and Dwnsteam Blade Rw, Junal f Engineeing Themphysics, Chinese Academy f Sciences, Vl.17, N. 2, 1996. 13] Wennestm, A. J. and Putehaugh, S. L., A Thee-Dimensinal Mdel f the Pedictin f Shc Lsses in Cmpess Blade Rws, Tans. ASME J. Eng f Gas Tubines and Pwe Vl 106 (1984), pp. 295-299. V. CONCLUSIONS Fm this study fcusing n the design ptimizatin f the blade leading edge cuve, it can be seen that it quite difficult t attain glbal ptimum using deteministic methd. Bth SA and SPSA methd ae able t each ptimal values and that SPSA has the ptential f achieving ptimal esults in much less cmputatinal efft than SA methd. It can be cncluded that SPSA methd is a suitable feasible ptimizatin methd which can be used t handle cmplex design pblems such as the fan blade shape designs. It is als elatively easy t implement. SPSA equies nly tw measuements f the bjective functin egadless f the dimensins f the design space cespnding t the ptimizatin pblem and the cst f ptimizatin deceases. The pesent investigatin fms the basis f futhe study t l at vaius aspects such as eplacing the abitay blading step with a me sphisticated suface gemety epesentatin incpating blade twist, sweep etc theeby inceasing the numbe f design vaiables using paameteic epesentatin f blade suface patches, the impact f the vaius pssible altenative methds f geneating the andm petubatin vect in the SPSA methd and the impact f that n the design f cmplex 3D shape designs.