FIRST BEZIER POINT (SS) R LE LE. φ LE FIRST BEZIER POINT (PS)

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1 Singe- and Muti-Objective Airfoi Design Using Genetic Agorithms and Articia Inteigence A.P. Giotis K.C. Giannakogou y Nationa Technica University of Athens, Greece Abstract Transonic airfoi design probems are soved using a Genetic Agorithm (GA) based optimizer. At the desired operating point, the minimum drag and constant ift targets are achieved through either a scaarized objective function, invoving an arbitrary weighting factor, or the Pareto technique. For the optimization of an airfoi at two operating points, simiar approaches are used. The CPU cost of the optimizer is kept ow through Articia Inteigence. A mutiayer perceptron is trained using aready evauated individuas and provides good, though approximate, tness predictions. With the reguary trained network, the direct ow sover cas are noticeaby reduced. 1 Introduction The search for optimum aerodynamic shapes is a major probem in aeronautics. The airfoi shape optimization requires a geometrica mode fot its contour, which denes the set of contro-parameters. A tness or objective function is then dened, which shoud be driven to its minimum or maximum vaue. The minimization of drag is the usua objective in highspeed civi transport. This shoud be achieved with predened ift and, often, by satisfying constraints reevant to the airfoi cross-section. Computationa Fuid Dynamics (CFD) pays an important roe since it is capabe to enight ow phenomena around compex shapes that are aerodynamicay evauated without resorting to costy experiments. Numerica optimization, usuay through iterative gradient-based methods, is a widey used too. Riva to the aforesaid methods are stochastic optimizers and soft-computing techniques. Their major advantage is the capabiity to ocate goba optima. Among them, GAs, [1], are based on the concept of natura seection and their robustness has been proved in a variety of appications, incuding probems that can be hardy soved using gradient information. However, the shape optimization through standard GAs is time consuming due to the great number of CFD evauations it invoves. In the past, ways to reduce the CPU cost have been proposed by the second author. These were based on (a) the concurrent treatment ofthe individuas on parae computing patforms [2] and/or (b) Articia Inteigence [3]. In the second approach, articia neura networks (ANNs) are rst trained to correate shapes and tness scores and then used to evauate new candidate soutions, without aways resorting to CFD computations. The present paper extends the method previousy used for inverse design turbomachinery probems, to the airfoi shape optimization probem in aeronautics. The imposed requirements (on the ift and the drag) have been used in both the singe- and the muti-objective optimization fashion. Singe- and muti-point designs at transonic ow conditions, have been Postgraduate Student, agiotis@centra.ntua.gr y Assistant Professor, kgianna@centra.ntua.gr 1

2 FIRST BEZIER POINT (SS) R LE LE φ LE FIRST BEZIER POINT (PS) Figure 1: Parameterization near the LE worked out. For the muti-objective probem the Pareto front method has been programmed and its convergence speed was enhanced aso through the ANNs. 2 Design through GAs A GA starts by randomy generating a popuation of individuas, each of which stands for a candidate shape-soution with a binary or rea number coding. Herein, the former was impied by concatenating the binary strings (bits) of the contro-parameters. The popuation size N pop, which does not ater during the genetic evoution, determines its exporation capabiities aong with the computing cost per generation. About individuas are practicay used. These are a evauated using CFD toos and their tness scores are deduced by post-processing the numerica resuts. Using various seection criteria, reproductive trias are aocated to the individuas, with some bias toward the ttests. A mating poo is thus formed and this constitutes the rst step for the evoution to the next generation, by appying the parent recombination operators. The recombination task gives rise to a new generation with possiby improved mean and best tness scores. The evoutions stop if no improvement appears during the ast few generations. 2.1 Proe Parameterization The parameterization of the airfoi contour is carried through Bezier-Bernstein poynomias. The x- (optiona) and y-coordinates of the Bezier points are the free-parameters controed by the GA. Optionay, circuar arcs at the eading (LE) and traiing (TE) edges can be superimposed to the poynomia curves. If the airfoi chord-ength is kept xed, Fig. 1, the use of circuar arcs introduces the anguar position of their centers ( LE and/or TE ) and their radii (R LE and/or R TE ) as extra contro parameters. A geometrica mode that is often used consists of two circuar arcs (2+2 contro parameters) and two Bezier curves with N PS = N SS =5points per airfoi side ( =20 contro parameters, if both Cartesian coordinates are free). Thus, the tota number of contro parameters is N FP =24. Note that, at the junction points between Bezier curves and circuar arcs, curve continuity and smoothness are automaticay preserved. 2.2 Objective Function Optimum or inverse design probems can be soved by the proposed method, provided that an appropriate objective or tness function is dened. In the inverse design probems, the sought for airfoi shoud yied a desired pressure distribution over its contour at specied ow conditions. In this case, the objective is to minimize the deviation between actua and desired pressure distributions. Exampes of inverse designs using dierent acceeration techniques can be found in [2] and [3]. In the optimum airfoi design probem, proes giving maximum or known ift C and minimum drag C d are sought for. 2

3 Herein, we wi be deaing with the re-design probem an initia airfoi is provided which generay denes the search space for the N FP contro-parameters. This proe wi be redesigned aiming at the minimum C d for the same C = C target (at the same ow conditions). Geometrica constraints wi not be impied in the present studies. For a singe-operating point, this can be carried out in two ways: 1. using a singe objective function and the arbitrary factor b, as n o min F = min ) 2 + b C d (C ; C target (1) 2. through a muti-objective optimization, by separatey considering min(c ; C target ) 2 and min(c d ) in order to provide the Pareto front, as discussed in the next section. Muti-point optimizations, at two ow conditions through the Pareto front technique wi be aso demonstrated. The evauation of the candidate shapes is undertaken by a primitive variabe 2-D unstructured grid, ow sover [4]. The inviscid ow equations are soved using a vertex-centered nite-voume technique and a second order upwind scheme. The numerica soution is based on Jacobi-preconditioned GMRES techniques and requires about 40 sec on an Inte Pentium II 400MHz processor (for a typica grid of about 3000 nodes, convergence to 6 orders of magnitude the cost increases to about 60 sec if grid adaptation is used). Prior to the numerica soution, an unstructured grid is generated using an automated procedure. Its computing cost can be safey negected. 2.3 Muti-objective optimization using the Pareto technique In muti-objective optimization, a Pareto front [5] is the subset of candidate soutions obtained by eiminating any other soution for which anabsoutey superior one (with respect to the ensembe of objective functions) can be found. According to the previous denition, the Pareto front consists of the so-caed nondominated (or Pareto-optima) soutions, in the sense that none of them is absoutey superior to any other constituent of the front. Thus, a of them are equay acceptabe soutions to the probem and the choice of one of them requires a deep knowedge of the particuar probem. In order to obtain the Pareto front, a singe-objective GA shoud be enriched by nondominated sorting and sharing. For each candidate soution in the current generation, scores for a of the objectives are rst computed. The soutions are then ranked on the basis of nondomination among them and a number of fronts is thus formed. The rst front is the Pareto-optima front and its members are given a unit dummy cost function (F dum = 1). The reproduction task is based on the F dum vaues, after sharing is appied to each front separatey. Sharing aims to spread the soutions a over the front by penaizing custered ones and this is achieved by modifying the F dum vaues. Starting from the Pareto-front, the sharing factor m i = NX pop j=1 Sh(d(i j)) (2) is computed for its i th member. The function Sh(d) is dened as ( 0 if d(i j) share Sh(d(i j)) = 1 ; d(i j) (3) share if d(i j) < share 3

4 where 0 < share < 1, and i, j are in the same front. Their Eucidian distance d(i j) is computed either in the variabes' or in the objectives' space as foows. d(i j) = vu ux t Nv k=1 v i k ; vj k max i=1 ::: Nf (v i k ) ; min i=1 ::: N f (v i k )! 2 (4) where v stands for the variabes or the objectives respectivey, N v the tota number of variabes (or objectives) and N f the number of individuas beonging to the current front. Since minimization probems are considered, the F dum vaue of each soution is recomputed by mutipying it with m i. On the next front, F dum is rst set equa to the worst score in the previous front pus a sma quantity and then sharing appies, as previousy. In the genetic evoution, parent seection appies on the basis of the so-computed F dum vaues. 3 Acceeration using Articia Neura Networks The overa design cost can be consideraby reduced if some individuas are evauated without resorting to the costy CFD mode. For this purpose ANNs, [6], are used, provided that a training set, i.e. a sucient number of proe shapes - tness scores is avaiabe. Such avaiabiity is certain since the GA continuousy evauates new candidate soutions. An ANN shoud be trained on the input (shape) - output (tness) mapping, through supervised earning. This entais additiona CPU cost but, afterwards, the prediction of the tness of any new shape is straightforward. A mutiayer perceptron (MLP), as the one used herein, is a feedforward ANN with its neurons arranged in ayers [6]. A neurons in a ayer are connected to a neurons in the adjacent ayers using inks associated with synaptic weights. A neuron in a but the rst ayer has many inputs but a singe output. The synaptic weights act as signa mutipiers on the corresponding inks. In our case, as many neurons as the number of free parameters (N FP ) constitute the rst ayer. The output ayer neurons are equa to the number of objectives and practicay, 2-3 intermediate or hidden ayers are used. The number of neurons in these ayers shoud be as high as possibe to increase the ecacy of the network, but as ow as possibe to reduce the training cost. The back error propagation (BEP) agorithm is used to compute the synaptic weights by repetitivey presenting the network with the training set. In genera, the ANN guesses are not dependabe. Nevertheess, they can be used to distinguish the promising soutions in the current generation which wi be then accuratey evauated using the CFD too. Our rue of thumb is to re-examine the top 40%N pop of individuas in each generation through the Euer sover. Considering that the network training costs as much as a direct ow soution, the overa computing cost is expected to reduce about 60%, thanks to the ANN impementation. When the ANN is used in conjunction with the Pareto method, a new individuas in the Pareto-optima front need to be evauated by the ow sover. If these are ess than 40%N pop more individuas that beong to consecutive fronts are evauated. 4 Anaytica test case A simpe muti-objective probem, the simutaneous minimization of two anaytica functions, is rst soved and resuts with and without ANN acceeration are compared with the anaytica soution. It shoud be stressed that, the computationa cost of the anaytica functions is negigibe and the use of ANNs is ony for demonstration purposes. In fact, the economy by 4

5 no means outweighs the cost to train the ANN (buit with one input for x and two outputs for O 1 O 2 ). The two objectives are min(o 1 O 2 ), where: O 1 = 1 p 10 ; x + p x ; 5 O 2 =0:04(x ; 8) 2 +0:3 (5) in the range of x 2 [5 10]. The two objectives aong with the GA soutions, are potted in g. 2. The resuts with or without the use of ANN are identica. The Pareto front (g. 3) is aso the same, though the point distribution is dierent. Instead of using the Pareto technique, a inear combination of the two objectives is possibe, min(o 1 + bo 2 ). Resuts of various singeobjective optimizations are aso incuded in the same gure for various b vaues. Figs. 4and 5 present the increase of the Pareto front members and the tota number of computations performed in each generation. During the evoution, dominated individuas are removed and new nondominated ones are inserted into the Pareto front. With the ANN acceeration, fewer individuas enter the optima front in each generation. As ony a percentage (40% N pop ) is accuratey evauated in each generation determining the maximum number of entries in the optima front, this is reasonabe side-eect. To get the same size of the Pareto front, more generations are needed. Nevertheess, the economy is sti important, as one may see by summing up the accurate evauations used Objective 1 GA soution Objective 2 GA soution Anaytica Objective 1 Anaytica Objective Pareto Front Pareto front using ANN 0 b= Objective b= x Figure 2: The O 1 and O 2 functions aong with the optimum soutions Objective 1 Figure 3: Pareto-optima front Without ANN Front Increase Accurate Evauations Using ANN Front Increase Accurate Evauations Generations Generations Figure 4: Stand-aone GA: Incrementa entries in the optima front and number of evauations per generation. Figure 5: GA couped with ANNs: Incrementa entries in the optima front and number of evauations per generation. 5

6 5 Optimization of a transonic airfoi In this section, the optimization of the RAE-2822 isoated proe, under inviscid ow conditions wi be anayzed. Singe-point Designs: Two dierent probems have been anayzed. In the rst, denoted by A, the ow conditions were M inf =0:73, a =2 o and the C target was that of the standard (starting) proe C target = 0:780. For point A, the foowing computations have been carried out: (a) six computations using the singe-objective function 1 at dierent b vaues (b = ). Some of them used the stand-aone genetic optimizer and some other the ANNbased acceeration. The atter are marked by ANN, in the gures' captions. (b) two mutiobjective optimizations, using the Pareto technique with and without ANNs. The same airfoi was separatey optimized at the B ow conditions (M inf = 0:75, a = 3:19 o ), where C target = 0:997, as a singe-objective optimization with b = 1and as a mutiobjective one with the Pareto technique. For point A, a the obtained resuts are shown in Fig. 6. Ony a part of the Pareto front (the outer eft one) has been drawn. The Paretooptima fronts with and without ANNs are quite cose to each other, especiay in the amost vertica part of this gure, where the C vaue is cose to the target one. Each singe-objective computation added a point in this gure, which was in genera aigned with the Pareto front points. Tabe 1 shows the C and C d vaues of the optimum proes according to the singe-objective optimizations Pareto front Pareto front using ANN b=5 ANN b=5 ANN 0 ANN b=20 ANN Cd (C - Ctar)^2 Figure 6: Operating point A, muti-objective optimization: The Pareto-optima fronts aong with optimum soutions from the singe-objective studies. Case b=5 ANN b=5 ANN 0 ANN RAE2822 C C d Tabe 1: C and C d of the re-designed airfois (Operating point A, singe-objective optimization) Muti-point Design: For the muti-point design, two objective functions for each operating 6

7 point, A and B have been devised, each of them using eq. 1. The case was studied without the ANN acceeration. Fig. 7 iustrates the starting proe and the optimum airfoi shapes obtained using b = It is worth noting that the x- and y- axes are in dierent scaes. Fig. 8 presents the convergence of the singe-objective GA in terms of the ttest individua score. A quite simiar convergence with and without ANNs was obtained, though the costy stand-aone GA optimizer sighty outperforms its economic version (with the ANNs). For the operating point B, simiar pots to those shown for A can be drawn, but these wi be omitted in the interest of space. The two-point design (minimization of eq.1, at both the AandBow conditions) carried out through the Pareto technique is potted in Fig. 9. The Pareto-optima front contains ve entries, which correspond to ve dierent shapes. In the same gure, the computed optimum soutions for points A and B separatey (b = 1 in a cases) are aso incuded. For the optimum proe at point A, two soutions are shown, one with and the other without the ANN acceeration. In a these singe-point optimization resuts the direct ow sover appied to the optimum geometry in order to cacuate the ow ed and the corresponding tness score at the other operating point. As expected, the singe-point optimum soutions are ocated at the top-eft (for point A) and the bottom-right (for point B) parts of the gure. These shoud be conceived as the theoretica edges of the Pareto front RAE2822 b= ANN b=5 b=5 ANN y Min Score x Figure 7: Optimum airfoi shapes for the operating point A (singe-objective optimization) Generations Figure 8: Convergence of the GA (singe-point, singe-objective). No eitism Pareto front Optimum point A, Optimum point B, Optimum point A, ANN D(C) + Cd (point B) D(C) + Cd (point A) Figure 9: Two point design: the Pareto-optima front. 7

8 6 Concusions The goa of this paper was to demonstrate the use of Genetic Agorithms in the airfoi optimization probem and to propose acceeration methods that increase their eciency. The main concusions of this work are isted beow: 1. Mutiayer perceptrons can reduce the CPU cost of design methods based on GAs. With a mutiayer network that has been trained on the shape-tness pairs resuting from the genetic evoution, a great part of the work voume, i.e. of the required CFD computations, is overcome. The CFD too is ony used to re-evauate the ttest individuas on the basis of the ANN predictions. 2. The ANN can be aso used with the Pareto technique, for the purpose of muti-objective optimization. In this method, a singe network with as many outputs as the number of objectives has been used. An aternative way woud be to train as many networks as the number of objectives, a of them with a singe output neuron. 3. The Pareto technique aong with an ANN yieds fewer entries into the Pareto-optima front per generation, compared to the stand-aone GA. Athough more generations are required to reach a front of the same popuation, the tota CPU cost is quite ower. 4. Through the ANNs, economy of the order of 60% can be achieved without damaging the quaity of the obtained resuts. References [1] Godberg D.E. Genetic Agorithms in search, optimization & machine earning. Addison- Wesey, [2] K.C. Giannakogou. A design method for turbine bades using genetic agorithms on parae computers. ECCOMAS 98, John Wiey & Sons, [3] K.C. Giannakogou. Designing turbomachinery bades using evoutionary methods. To appear in ASME Turbo Expo 99, Indianapois, June [4] D.G. Koubogiannis, L.C. Poussouidis, D.V. Rovas, and K.C. Giannakogou. Soution of ow probems using unstructured grids on distributed memory patforms. Comp. Meth. App. Mech. Eng., [5] J. Horn, N. Nafpiotis, and D.E. Godberg. A niched pareto genetic agorithm for mutiobjective optimization. Proceedings of the First IEEE Conference on Evoutionary Computation, 82-87, Voume I, ICEC '94. [6] A. Cichocki and R. Unbehauen. Neura Networks for Optimization and Signa Processing. John Wiey & Sons,