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Turbomachinery Airfoil Design Optimization

Using Differential Evolution

Nateri K. Mada,'an

NASA Advanced -;upercomputing Division,NASA Ames Res_ _rch Center, Moffett Field, CA 94035-1000, USA

Abstract. An a( r(,dynamic design optimization procedure that is based on a evolu-tionary algorithm known at Differential Evolution is described. Differential Evolutionis a simple, fast, md robust evolutionary strategy that has been proven effective indetermining the gkd)al optimum for several difficult optimization problems, includinghighly nonlinear s .:stems with discontinuities and multiple local optima. The method iscombined with a %vier-Stokes solver that evaluates the various intermediate designs

and provides inp_ ts to the optimization procedure. An efficient constraint handlingmechanism is als(b incorporated. Results are presented for the inverse design of a tur-bine airfoil from ._ modern jet engine. The capability of the method to search largedesign spaces and obtain the optimal airfoils in an automatic fashion is demonstrated.Substantial redu( ions in the overall computing time requirements are achieved byusing the algoritb n in conjunction with neural networks.

1 Introduc t.ion

Remarkable prot ress has been made in recent years in the ability to design tur-

bomachinery airfoil shapes that are optimal with regard to certain desired char-

acteristics. This m)gress has been achieved by combining improved methods for

predicting the c(,mplicated flow fields in turbomachinery with efficient numerical

optimization tec!miques and by harnessing the powerful capabilities of modern

computers. Both steady and unsteady Navier-Stokes and Euler solvers have beencombined with v._rious optimization techniques (gradient-based methods [1], [2],

response surface:., _tc.) to optimize the design of turbomachinery airfoils.

More recentl>, there has been considerable interest in the development of tur-

bomachinery airtoit design optimization techniques that are based on nontradi-

tional approache _such as evolutionary algorithms and neural networks. Variousapproaches base, I on neural networks (see, for example, [3], [4], [5]), neural net-

works in conjun, tion with response surfaces [6], [7], genetic algorithms [8], [9],

and genetic alg(,cithms in conjunction with neural networks [10],[11],[12] have

been reported in the literature. These techniques offer several advantages over

traditional optin iz._tion methods.

This paper d ',als with the development of a turbomachinery airfoil design

optimization pro :edure that is based on a relatively new evolutionary algorithm

known as Differ(' lt.ial Evolution [13] developed for single-objective optimizationin continuous se_r(:h spaces. It is conceptually simple and possesses good conver-

gence properties _hat have been demonstrated in a variety of applications. Dif-

ferential Evoluti,,n (DE) is best characterized as an evolutionary strategy (ES)

https://ntrs.nasa.gov/search.jsp?R=20020073236 2020-05-02T16:28:24+00:00Z

2 NateriK Madavan

ratherthan as a _enetic algorithm (GA), although the distinction between GAs

and ESs has blm red in recent years. Perhaps the main ideological difference lies

in the relative i_t_portance given to the two main evolutionary operators, recom-

bination (crosso'er) and mutation, with GA-based approaches relying heavily

on the former at,d ES-based approaches on the latter. DE has proven to be an

effective approa_ h in determining the global optimum for several difficult opti-

mization problems in a variety of applications. Its application in aeronautics,

however, has be,,n rather limited. It has been used in the predictive control of

aircraft dynamic s 114]. DE has been used in conjunction with a potential flowsolver in the imerse design of turbomachinery airfoils [15]; the same authors

have also presem;ed a hybridized version [16] that combined DE with a localsearch method t _minimize the number of objective function evaluations using

the potential flow solver.In this paper the DE algorithm is combined with a Navier-Stokes solver

that provides inl,uls to the optimization procedure. An efficient constraint han-

dling mechanisr_ is also incorporated in the algorithm. An airfoil geometryparametrization hat uses a minimal number of variables is also used to minimize

the number of (,!_.jective function evaluations. The procedure is also combined

with neural neta, otks that are incrementally trained on the Navier-Stokes sim-ulation data an( (an then be used in the objective function evaluation. This

results in subst>ntial reductions in the overall computing time. Additionally,

the procedure h._.s been implemented on a distributed parallel computer in a

straightforward manner that relies on the simultaneous computation of multi-

pie, independent aerodynamic simulations on separate processors. The procedureis primarily scril_t-based and allows for a variable number of processors to be

used depending ,,n the size of the population used in the DE algorithm. Details

of the method aud its implementation and results for the inverse design of a

turbine airfoil to demonstrate its capabilities are described.

2 Design Optimization Method

The main ingredim_ts of the design optimization method are discussed in this sec-tion. The DE alg, ,rithm, the airfoil geometry parametrization procedure, and the

CFD flow solver _lsed for evahmting the objective function are discussed briefly.

Some details reg_ rding the implementation of the method on parallel distributed

computers are al _o provided. A hybrid version that helps reduce computational

cost by combiniHg he DE algorithm with a neural network approach is alsodescribed.

2.1 Differential Evolution

Differential Evoh lion is an ES-based approach developed for single-objective op-

timization in coJ tinuous search spaces. It is conceptually simple and possesses

good convergenc, properties that have been demonstrated in a variety of appli-

cations [18]. Details of the algorithm can be found elsewhere [13], [17]; only itsmain features arc summarized here.

TurbomachineryAirfoilDesign Optimization 3

The approach uses a population PQ that contains N K-dimensional real-

valued paramete "vectors in generation Q, where K is the number of parametersor decision varia!,l(s:

PQ = P/Q =P_k, i= 1,..,N ; k = 1,...,K (1)

The populati, _n is usually initialized in a random fashion and the population

size N is maintained constant throughout the optimization process. Differentialevolution is thu_ similar to a (# + _) ES [19] with tt and _ equal to N [20].

The method hov, _'_verdiffers from standard ES approaches in several respects as

described below.

As with all E_;-based approaches, mutation is the key ingredient of differential

evolution. The b_ sic idea is to generate new parameter vectors for the subsequent

generation by us ng weighted differences between two (or more) parameter vec-tors selected raJJdomly from the current population to provide appropriately

scaled perturbati ms that modify another parameter vector (or, comparison vec-

tor) selected fron the same population. This idea has been implemented invarious forms bu the form discussed and used here is the classical implementa-

tion where new lcial parameter vectors Pi?+1 for the next generation Q + 1 aregenerated according to the following mutation scheme:

where

j}Q+I _Q Q pQi,k = t_,,k +F'(pb,,k- _,,k) , i=I,...,N; k=l,...,K (2)

a, b, ce {1,...,N} ; aiCbi¢ciCi

The integers t.,, hi, and ci represent three random individuals of the popu-

lation that are n utually different from each other and from the running index

i. The mutation parameter F E [0, 2] is a real, constant, user-supplied param-

eter that control,; the amplification of the differential variation. Other variantsthat either use tl:e difference between more than two parameter vectors or keep

track of the best mrameter vector at each generation and use it in the mutation

scheme have als( been developed [9] and used with varying success in specific

applications. The;s, differential evolution differs from other ESs most notably inthat the mutatio _ operator is derived from the current population and not by

probability densi,y functions that are defined separately.DE is similar to other reeombinative ES approaches in that it also uses dis-

crete recombination. While various recombination strategies exist [19] the strat-

egy adopted in , igerentlal evolution is to modify the trial parameter vectorsg_)C_+ I

i as follows:

/3Q+a =/5 Q+I if ri,k <Cr or k=dk (4)i,k i,k --

and

ii_Q+l ----- PQi.k otherwise (5)i,k

4 Nateri K. Madavan

In the above,, 7" E [0,1] is a uniform random number, CT C [0,1] is the

crossover parant, t_er, and dk is a randomly chosen index in {1, ..., K} that ensures

that/5 Q+I gets a least one parameter from the mutated/5 Q+t and not all fromi,k i,k

P_k" Note also tttat the mutation and recombination operations described abovecan lead to new v(.ctors that may fall outside the boundaries of the variables.

Various repair r,lles can be used to ensure that these inadmissible vectors do

not enter the poimlation. A simple strategy, which is the one adopted here, isto delete these imLdmissible vectors and form new ones until the population is

filled.The selectiot_ s,:heme used in DE is deterministic but differs from methods

usually employec i in standard ES approaches. Selection is based on local com-

petition only, wil h the child trial vector (or, child)/SiO+l competing against one

population member P/Q (the comparison vector) and the survivor entering the

new population ,_¢!+1 . In other words, if/5/Q+1 yields a better objective func-

tion value than ,'_,_' then pQ+l is set to /SiQ+1. Otherwise, the old value PiQ is

retained. This gr.,edy selection criterion results in fast convergence; the adaptivenature of the mutation operator, in general, helps safeguard against premature

convergence and al!ows the process to extricate itself from local optima.An efficient c )nstraint handling mechanism has been incorporated into the

algorithm. Detaits are given elsewhere; briefly, it is a parameter-less penaltyfunction approa, h where infeasible solutions are penalized and help guide the

algorithm away from these regions. Physical constraints, e.g., maximum airfoil

thickness, etc., ae imposed, as well as aerodynamic constraints (wavy surfaces,

etc.). Airfoil geo,_mtries that meet the constraints but for which the CFD solver

is unable to con 'erge or "blows up" are also deemed infeasible and penalized

accordingly.

2.2 Airfoil Ch_ometry Parametrization

Geometry parametrization and prudent selection of design variables are amongthe most critical :_.sl)eets of any shape optimization procedure. The ability to rep-

resent various airfoil geometries with a common set of geometrical parameters

is essential. Vari,tions of the airfoil geometry can be obtained then by smoothly

varying these pa_ ameters. Geometrical constraints imposed for various reasons,

such as structural, aerodynamic (e.g., to eliminate flow separation), etc., shouldbe included in t}tis parametric representation as much as possible. Addition-

ally, the smallesl n71mber of parameters should be used to represent the familyof airfoils. Here, the airfoil geometry parametrization method described in [6]

that uses a total )t" 13 parameters to define the turbine airfoil geometry is used.

Figure 1 illustrai es the method for a generic airfoil. The geometric parametersused are the lea(ling edge and trailing edge airfoil metal angles (2 parameters),

eccentricity of ul [mr leading edge ellipse (1 parameter), angles defining the ex-

tent of the leadh_g edge ellipses (2 parameters), semi-minor axes values at the

leading edge (2 p.trameters), angles defining the extent of the trailing edge circle

(2 parameters), _ Moil y-coordinate values at about 50% chord on the upper and

TurbomachineryAirfoilDesignOptimization 5

lowersurfaces(2parameters),andairfoily-coordinatevaluesatabout25%and75% chord on th., _pper surface (2 parameters).

2.3 Flow Soher for Objective Function Evaluation

A two-dimensional Navier-Stokes solver is used to perform the flow simulations

(direct function ,v_luations) that serve as inputs to the optimization process.Multiple grids at,, used to discretize the flow domain; an inner "0" grid that con-

tains the airfoil ;rod an outer "H" grid that conforms to the external boundaries

as shown in Fig. :3 The flow parameters that are specified are the turbine pres-

sure ratio, inlet t_mlperature and flow angle, flow coefficient, and unit Reynoldsnumber based oi_ inlet conditions.

2.4 Parallel l:nplementation Details

In order to redm e overall design time, the procedure has been implemented on

distributed paratle[ computers. The results in this article were obtained on the

SGI Origin 3000 _md the Cray SV1 at NASA Ames Researach Center. The imple-mentation of the method is quite straightforward and relies on the simultaneous

computation of z_m]tiple, independent aerodynamic simulations on separate pro-cessors. The proc,'.dure is primarily script-based and allows for a variable number

of processors to be used depending on processor availability and the size of the

population used in the DE algorithm. The number of processors can also be

adjusted as the design proceeds. The current setup is based on a "master-slave"

arrangement, wi_ h the master handling the tasks of setting up the simulations,neural network lraining for the hybrid method, and farming out of the aero-

dynamic compul ttions to the other "slave" processors. Since the aerodynamic

computations ar, it_.dependent of each other, no communication between the pro-

cessors is requir_ 3 until the computations are completed. The slave processorsthen communicaT e 1,heir results to the master which then performs the necessary

calculations to d,'Wrmine the members of the next population.

2.5 Hybrid Difl'erential Evolution-Neural Network (DE-NN)

Approach

While the DE al,:orithm is quite efficient in terms of exploring the entire design

space in its searct for the optimum solution, there is a tendency for the algorithmto slow down aft_ r it approaches the vicinity of the optimal solution and several

function evaluati, m_ are required in order to obtain the exact optimal solution.

This can be reme, tied by a hybrid approach that combines the DE algorithm withneural networks l hat are incrementally trained on the Navier-Stokes simulation

data. The traine l neural network can then be used to evaluate the objective

function evaluati,_n with little or no computational expense instead of using the

Navier-Stokes s_l _,_r. While hybridization is always useful, it must be done with

caution. Here we r_ort to it only in the latter stages after the entire population

6 Nateri14.k_adavan

hasevolvedto _hegeneralvicinityof the optimalsolution.Thusthe neuralnetworkisused+,sa"local"responsesurfacewithvalidityonlyinasmallregionofthedesignspa,e.Thismakesit easiertotraintheneuralnetworkandimprovesits generalization+abilities.

Theresultso_tainedin thisarticlewereobtainedusinga feed-forwardneu-ral networkwith twohiddenlayersasshownin Fig.3. Thefirst nodein theinputlayerisabia,,',node(inputof1.0)andtheremaininginputnodesareusedto specifythewriousdesignparameters.Forthe inversedesignoptimizationproblem,theneur_dnetworkis trainedonthesum-of-squareserrorbetweentheactualpressureamdthetargetpressureat variouspointsontheairfoil.Forthesakeofbrevity,d+_tailssuchasthenumberofneuronsineachlayer,etc.arenotincludedhere.

3 Results

The design met] od was used in the inverse design of a turbine airfoil with a

specified pressm,, distribution. The target pressure distribution was obtained

at the midspan ,,f a turbine vane from a modern jet engine and was supplied

by Pratt and WI,itney (Private Communication, F. Huber, 1997). Several flow

and geometry pa_ ameters were also supplied and used in the design process. The

design objective tunction was formulated as the equally-weighted sum-of-squareserror between th, target and actual pressure obtained during the optimization

process at 45 lo( Ltions on the airfoil.The initial dedgn space was chosen to be quite large to allow for a wide range

of airfoil shapes 1 ) be explored. In order to hold the CFD function evaluations to

a reasonable nun,her, 6 (instead of 13) design variables were used in the initial

stages of the desi 4n. The population of 50 members were then evolved using the

DE algorithm. _f!t('+DE mutation and crossover parameters were both chosen tobe 0.8 based on ,rior experience with other problems; no attempts were made

to optimize thes, _parameters. Figure 4 shows the pressure distribution for an

intermediate airfl,il that represents the best airfoil obtained after 10 generations

using the DE optimization method. The algorithm is able to approach the tar-

get distribution wi1_hin roughly 500 function evaluations (note that the actualnumber of functi+)n evaluations is much less, because many of the initial airfoil

geometries that _,ere infeasible were not evaluated by the CFD solver. After 10generations, the m_mber of design variables is increased to 13 and the popu-

lation evolved fu ther. As is typical with other genetic algorithms, once in the

vicinity of the ol_timal solution, convergence of the DE algorithm slows down

considerably. Th,, tinal design shown in Fig. 5 was obtained after another 10

generations.In order to r, duce the computational cost of the CFD function evaluations

the population atl.er the first 10 generations was used to train a neural network.

Figure 6 shows lh_ band around the target pressure distribution in which allthe data used to _rain the neural network lie. Note, however, that the network

was trained dire( 1_ on the sum-square-error and not on the individual pressure

Turbomachinery Airfoil Design Optimization 7

data. The data ,and in Fig. 6 gives an idea of the local nature of the neural

network respon', '. surface in the vicinity of the optimal solution. The trained

neural network ,vas used in lieu of the CFD solver to perform the function

evaluation with negligible computational cost. The final design obtained was

close to that shoa'n in Fig. 5. Finally, Fig. 7 shows the optimal airfoil geometry

obtained by the DE algorithm.

Attempts to ,luantify the computational costs and obtain more detailed eval-

uation of the vat i(,us parameters are currently underway.

4 Summary

An aerodynamic design optimization procedure that is based on a evolutionary

algorithm know_ al; Differential Evolution is described. The method is combined

with a Navier-S1 )kes solver that evaluates the various intermediate designs and

provides inputs 1,(_ the optimization procedure. Results are presented for the

inverse design ot a turbine airfoil from a modern jet engine. The capability of

the method to s _arch large design spaces and obtain the optimal airfoils with

a reasonable number of CFD function evaluations in an automatic fashion is

demonstrated. _1he airfoil geometry parametrization and constraint handling

procedure helps infit the number of CFD function evaluations required by dis-

allowing airfoil !;eometries that are infeasible from a physical or aerodynamic

standpoint. Sub>tantial reductions in the overall computing time requirements

are achieved by 1he hybrid DE-NN algorithm that uses a neural network trained

on the CFD dat L to evaluate the objective function in the latter stages of the

design evolution

References

1. S. Y.Lee, K. 5 . Kim: 'Design Optimization of Axial Flow Compressor Blades with

a Three-Dimensional Navier-Stokes Solver'. Proceedings of the ASME Turbo Expo

2000, Munich, Germany, May 8-11, 2000

2. J. M. Janus, J C. Newman: 'Aerodynamic and Thermal Design Optimization for

2-_rbine Airfoi s'. AIAA Paper No. 2000-08_0, Jan. 2001

3. J. M. Sanz: 'D,.vdopment of a Neural Network Design System for Advanced Turbo-

Engines'. _th t r.S. National Congress on Computational Mechanics, San Francisco,

CA, Aug. 199"

4. M. M. Raft 'A Rapid Aerodynamic Design Procedure Based on Artificial Neural

Networks'. Alia Paper No. 2001-0315, Jan. 2001

5. S. Pierret, R. \. V. Braembussche: Journal of T_rbomachinery, 121, 326 (1999)

6. M. M. Rai, N. K Madavan: AIAA Journal, 38, 173 (2000)

7. N. Papila, W. Shyy, L. Griffin, D. J. Dorney: 'Shape optimization of supersonic

turbines using response surface and neural network methods'. AIAA Paper No.

200I-i065, Ja_. 2001

8. S. Obayashi, S Takanashi: AIAA Journal, 34, 881 (1996)

9. B.H. Dennis, J N. Egorov, Z. X. Han, G. S. Dulikravich, C. Poloni: 'Multi-Objective

Optimization ( f _Purbomachinery Cascades for Minimum Loss, Maximum Loading,

and Maximum (;ap-to-Chord Ratio'. AIAA Paper No. 2000-_,876, Sept. 2000

8 Nateri K Madavan

10. D. Quagliarel a, A. D. Cioppa: 'Genetic Algorithms Applied to the Aerodynamic

Design of Tra,,s(,nic Airfoils'. AIAA Paper 94-1896-CP, 1994

11. M. Uelschen. M. Lawerenz: 'Design of Axial Compressor Airfoils with Artificial

Neural Netwo_ ks and Genetic Algorithms'. AIAA Paper No. 2000-25_6, June 2000

12. C. Poloni, A. 2iurgevich, L. Onesti, V. Pediroda: Comp. Meth. Appl. Mech. Engr.,

186, 403 (200))

13. 1_. Storn_ K. :)rice: Dr. Dobb's Journal, 22, 18 (1997)

14. K. Nho, R. I_; Agarwal: 'Fuzzy Logic Model-Based Predictive Control of Aircraft

Dynamics Usi Lg ANFIS'. AIAA Paper 2001-0316, Jan. 2001

15. T. Rogalsky. [_. W. Derksen, S. Kocabiyik: Can. Aero. Space Inst. Jour., 46, 183

(2000)

16. T. Rogalsky, R W. Derksen: 'Hybridization of Differential Evolution for Aero-

dynamic Desi _,n. Proceedings 8th Ann. Conf. Comp. Fluid Dyn. Soe. Can., pp.

729-736, Jun, ti-13, 2000

17. K. V. Price: ' Differential Evolution: A Fast and Simple Numerical Optimizer'. In:

Biennial Con./erence of the North American Fuzzy Information Processing Society,

(NAFIPS), ,l¢_rm 1996, ed. by M. Smith, M. Lee, J. Keller, J. Yen (IEEE Press,

New York 1996) pp. 524-527

18. J. Lampinen: A Bibliography of Differential Evolution Algorithm. Technical Re-

port. Lappeei, vanta University of Technology, Department of Information Technol-

ogy, Laboratoy of Information Processing, Lappeenranta, Finland (2001)

19. T. Back, U. t ammel, H.-P. Schwefel: IEEE Transactions on Evolutionary Compu-

tation, 1, 3 (1[}97)

20. M. A. Shokr, Ilahi, R. Storm 'Design of Efficient Erasure Codes with Differential

Evolution'. In ['roceedings of ISIT 2000, International Symposium on Information

Theory, Sorr_ ,_to, Italy, June 25-30, 2000

9 Nateri K. M; d_wan

.. x, Yu

x, (Q 2 I 1 I 7 I

O. 0 0.75 1.00 1.25-0.25 t 5)) 0.25 x./Sc_

Figure 1. Schen atic of a generic airfoil showing location of control points on the

airfoil surface a id the defining angles used in the parameterization of the airfoil

geometry.

_--_ i i_ !_:_:_i_,.---___Airfoil

ii-_-__- I !iiiiii=. _ _=- -- i_lll I I 1 l

()uter H-Grid __

Inner O-Grid9-Grid "_ :,,,_::,,,,,,,_ _ I--

llh"llIl', I 1 1

Fig_re 2. Representative turbine airfoil geometry and

con putational grid used in the CFD simulations.

10 Nateri K. M_davan

Layer 1 Layer 2

Relevant yer 3

De,,ign ---'_Pap ameters ;'_

] J/Output/ / Layerof

Node _ NeuronsBia ,

InputNodes

Hidden Layers

Fig_re 3. Schematic of the three-layer feed-forward

neu_ al network used in this study.

II 1 I I I ' ' I I I .... 1 _ ' '

0 96

[)9

085

• ..2 ll8,k.

::,-075

(1.7

oss • Target-- Intermediate Design

(,,_ _- .... 0i5. .... 0 |.5 ......... 0.75

x/cFigt re 4. AiHbil surthce pressure distributions obtainedfron_ CFD simulations for the intermediate design

(using 6 design variables).

11 Nateri K. M: davan

0.9

0.8

0.7

• Target-- Opttmal Design

0.6 i t i i I i i , , I , , , , I , i i J0.25 0.5 0.75

x/c

Fig Lre 5. Airfoil surface pressure distributions obtained

fro_ 1 CFD simulations for the final optimal design using

13, lesign variables.

LL95

0,9 •

C.85

.-: 0.8

E, C.75

D,7

o85 • Target• --Neural Net Training Data Band

_'6i' " '0'.,' ' _0'.2" 0'_;' _'_i'__;'._' "01_" ;'._" '0'.8";'.9"x/c

Fig,re 6. Band of airfoil surface pressure distributions

obt;_ined from CFD simulations used in training the

neu -al network for the hybrid approach.

12 NateriK,M;d_van

-- Optimal Design

I b I r l I I I , , , , I , , _ , I ,

x/c

Fig_re 7. Airfoil geometry for the optimal design.