J. of T h e r m a l Science Vol.8, No.3

Genetic Algorithms Development for Multiobjective Design Optimization of Compressor Cascade Jun LI V e n t u r e L a b o r a t o r y , G r a d u a t e School, K y o t o I n s t i t u t e of Technology, M a t s u g a s a k i , S a k y o - k u , K y o t o 606-8585, J a p a n

Koji Morinishi

Nobuyuki Satofuka

D e p a r t m e n t of M e c h a n i c a l a n d S y s t e m E n g i n e e r i n g , K y o t o I n s t i t u t e o f Technology, M a t s u g a s a k i , S a k y o - k u , K y o t o 606-8585, J a p a n

Aerodynamic optimization design of compressor blade shape is a design challenge at present because it is inherently a multiobjective problem. Thus, multiobjective Genetic Algorithms based on the multibranch simulated annealing selection and collection of Pareto solutions strategy have been developed and applied to the optimum design of compressor cascade. The present multiobjective design seeks high pressure rise, high flow turning angle and low total pressure loss at a low inlet Mach number. Pareto solutions obtain the better aerodynamic performance of the cascade than the existing Control Diffusion Airfoil. From the Pareto solutions, the decision maker would be able to find a design that satisfies his design goal best. The results indicate that the feasibility of multiobjective Genetic Algorithms as a multiple objectives optimization tool in the engineering field. Keywords:

multiobjective

pressor cascade,

optimization,

genetic

Pareto

optimal

set, com-

design.

INTRODUCTION Turbomachinery, especially compressor cascade, optimum design presents a grand challenge to numerical optimization. The goal of compressor cascade design is to produce the highest pressure rise with the lowest total pressure loss at the constant flow condition. Parameter of pressure rise is an important performance parameter of compressor. The higher pressure rise at the given flow condition, the better performance of compressor. But, higher pressure rise may lead to large adverse pressure gradients and boundReceived, 1999.

algorithms,

ary layer separation and shock wave. Thus, tradeoff between high pressure rise and low total pressure loss must come to the best comprise. The design of compressor is inherently a multiobjective optimization problem. In addition, aerodynamic performance analysis of compressor cascade is computer intensive and the resulting aerodynamic performance is very sensitive to geometry shape of the cascade, therefore, a robust multiobjective optimization algorithms is indispensable to this research field. Development of aerodynamic design methods for compressor cascade has always been of strong interest. Successful design methods have been reported [1,2] and most of them are based on an inverse design through

Jun LI et al.

Genetic Algorithms Development for Multiobjective Design Optimization of Compressor Cascade

a prescribed pressure distribution. However, this approach leaves the problem of specifying an appropriate pressure distribution to designers at large. An arbitrarily target pressure distribution may correspondingly yield an unrealistic geometry. On the other hand, it is very difficult to decide geometry constraints. So, direct problem design is applied in this paper. Direct problem design is formed by coupling aerodynamic analysis method with numerical optimization method. They optimize a given aerodynamic objective function by iterating directly on the geometry. Genetic Algorithms(GAs) [3] axe known to be robust and have been enjoying increasing popularity in the field of numerical optimization, multiobjective optimization [4,5] in particular, in recent years. Multiobjective problems seek to optimize components of a vector valued objective function. Unlike the single objective optimization, the solution of the multiobjective problem is not a single point, but a family of points known as Pareto optimal set. Each point in this set is optimal in the sense that no improvement can be obtained in one objective component that does not lead to degradation in at least one of the remaining components. GAs can search for many Pareto optimal solutions in parallel by tile Paxeto criteria in the constant population of the solutions [a] . GAs can be very efficient, if they can sample solutions uniformly from the Pareto optimal set. Since GAs are inherently robust, the combination of efficiency and robustness makes them very attractive for solving multiobjective problems. There have many applications in the aerodynamic optimum design using different multiobjective GAs. Such as multidisciplinaxy optimum design of the wing using niching and elitist models of GAs [6] multiobjective optimization of airfoil and wing design through hybrid GAs [7] and direct problem design of compressor cascade using multiobjective GAs based on FonsecaFleming's Paxeto-based ranking and fitness sharing techniques Is] . The Multiple Objectives Genetic Algorithms (MOGAs) based on the multi-branch simulated annealing selection and collection of Paxeto solutions strategy have been developed in this paper. To demonstrate the feasibility of the present algorithms, the two objectives and three objectives of mathematical test functions are used. The obtained Pareto solutions are sam-

159

pled from the analysis Pareto optimal set uniformly. Then, the present MOGAs have been applied to the multiobjective optimum design of the compressor cascade. Pareto solutions obtain the better aerodynamic performance of cascade than the existing Control Diffusion Airfoil (CDA)[9]. From the Paxeto solutions, the decision maker would be able to find a design that satisfies his design goal best. The results indicate that the feasibility of the present MOGAs as a multiple objectives optimization tool in the engineering field.

M U L T I P L E O B J E C T I V E S G E N E T I C ALGORITHMS (MOGAs) Solving a multiple objectives problem generally requires the identification of Pareto optimal set, a conception introduced by V. Pareto, a prominent Italian economist at the end of the past century (1896). A solution is said Paxeto optimal, or non dominated, if, starting from that point in the design space, the value of any of the objective functions can not improved without deteriorating at least one of the others. All potential solutions to the multiple objective problem can thus be classified into dominated and non dominated (Paxeto optimal) solutions. The set of non dominated, solutions of a multiple objectives problem is nanmd as Paxeto front or Pareto optimal set [4]. Therefore, it is the most important for the solution of multiple objectives problem to find the Paxeto optimal set. To define the notion of domination, let us think about the following problem: Min: f ( ~ = {fl(~?), f2(:~), . . . , f~(£)} where ~ is a vector of m decision variables. Suppose xi and 37j are solutions of the f(~?). Zi is said to be dominated by ~?j, there exist Vk C X , . . . , m , f ( ~ i ) ~ f ( £ j ) and 3if(e~) ~ f ( e j ) . We often say that f ( x i ) is partially less than f ( £ j ) . £ i is said to be non dominated if there does not exist any £j in all the potential solutions that dominates £~. All non dominated solutions Z"i of multiple objectives problem can make up Paxeto optimal set. If a solution belongs to the Paxeto optimal set, it is impossible to improve one of the objectives without deteriorating some of the others. GAs are applied to solve multiple objectives problem using Paxeto criteria to drive the evolution of the population. The characterizing feature of multiple objectives GAs is thus the introduction of Pareto criteria

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Journal of Thermal Science, Vol. 8, No.3, 1999

in the m e t h o d used for individuals selection. Then through selecting individuals in reproduction phase according to Pareto criteria, the Pareto optimal set of multiple objectives problem can be developed. In this paper, a new type of Multiple Objectives Genetic Algorithms (MOGAs) have been developed. Various techniques used in the M O G A s are to be introduced briefly in the following: 1. C o d i n g , C r o s s o v e r a n d M u t a t i o n

The binary representation is traditionally used in GAs. The binary alphabet offers the m a x i m u m numbers of the schemata per bit of information of any coding. In addition, the binary coding facilitates theoretical analysis and allows elegant genetic operators. Thus, binary coding m e t h o d is adopted in the MOGAs. In the MOGAs, the one point crossover operator [3] is used. Non-uniform m u t a t i o n [1°] is adopted and work on the word level of the binary code. T h a t is to say, mutation performed directly on the integer range from 0 to 2 n - l , n being the number of bit adopted for each variable. In this case, the binary string is converted into an integer list, m u t a t i o n operator is applied and then the list is converted back into a bit string.

cess, even though individuals are randomly chosen from the population. In each branch of selection, individuals are selected without replacement, so t h a t all individuals will have one chance to compete on each objective. T h a t is to say, all individuals of current generation compete multiple times according to the numbers of objectives. This feature provide the same pressure toward individuals on each objective. This can maintain diversity t h a t is needed to represent a spectrum of design across the Pareto optimal set. 3. P a r e n t s S e l e c t i o n S t r a t e g y

The extended elitism strategy is adopted to parent selection. T h a t is, the assigned percentage of parents come from the actual set of Pareto solutions of local generation. These Pareto solutions can directly turn into parents without undergoing genetic operation. The other percentage of parents come from individuals via genetic operation. After multi-branch simulated annealing selection has been finished, the assigned percentage of parents' pool is full. Individuals in the pool are then m a t e d in crossover and m u t a t i o n operation to form offspring. These children eventually become the parents of next generation. The parent selection strategy is shown in Fig.1.

1.0

2. M u l t i - B r a n c h

Simulated Annealing Selec-

tion

In MOGAs, selection operator plays an i m p o r t a n t role for obtaining the optimization results of the multiobjective problem. The m u l t i - b r a n c h simulated annealing selection [11] has been presented and used for the individuals selection. Multi branch selection approach is organized so that designs compete on one of multiobjective. As the name implies, one branch of the selection measure designs on the first objective, the second branch evaluates individuals on the second objective and so on. To do this, the individuals are selected randomly from the population of current generation on the first fitness using simulated annealing selection. This continues until the numbers of selected individuals come up to the fixed number. T h e n the process is repeated with individuals competing on the second objective. Selection process continues until individuals selected on the last objective. In the multi-branch simulated annealing selection, there are certain guarantees about the selection pro-

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Fig.1 Parents selection strategy for two objectives optimization

4. C o l l e c t i o n s o f t h e P a r e t o S o l u t i o n s Many solutions of multiobjective problem are measured during the run of the MOGAs. The goal of multiobjective design is to find a range of Pareto solutions t h a t approximate the Pareto optimal set of prob-

Jun LI et al.

Genetic Algorithms Development for Multiobjective Design Optimization of Compressor Cascade

161

lem. Therefore, any feasible, Pareto solutions should

A 16 bits string is adopted as a binary coding for

be stored during the run of the MOGAs, not just the Pareto solutions from the last generation. In this way,

the variables representation. Probability of crossover is selected as 1.00 and probability of mutation works

a large number of solutions can be found that represent the Pareto optimal set [12].

at 0.02. The population size equals 100 and evolve 50 generations. The results are shown in Fig.2. Fig.2(a)

The collection of these individuals begins with the

shows the results which are obtained in the last gen-

starting generation. The Pareto solutions which are selected using Pareto criteria in the generation are

eration.

The obtained Pareto solutions are all dis-

stored in special list. In subsequent generation, the

tributed along the Pareto optimal set, but they can not sampled uniformly from the Pareto optimal set.

Pareto solutions are identified. These are then com-

The results are obtained with collection Pareto solu-

pared with the stored list. If individuals currently in the list do not dominate a fresh identified individual, this fresh individual is added to the list. Similarly,

optimal set of the test function is almost uniformly covered by the set of solutions found.

tions strategy and shown in Fig.2(b) and the Pareto

if the fresh identified individual dominate those currently in the list, the dominated individuals are re-

10 - c ~ ' ~ ~ ~ _

moved from the list. In this manner, all Pareto solutions ever encountered during the run of the MOGAs

0.8

are stored in this special list. 0.6 MULTIOBJECTIVE

OPTIMIZATION 0.4

The multiobjective problems were selected to demonstrate the capabilities of the present MOGAs. The mathematical test functions have been tested, then compressor cascade optimization design using MOGAs has been presented in this section.

1. M a t h e m a t i c a l T e s t F u n c t i o n s Multiobjective optimization method with better performance should have the following two characteristics. The first is that the solutions obtained are Pareto solutions. The second is that they are uniformly sampled from the Pareto optimal set [13].

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The use of mathematical test functions for performance evaluation of the algorithms, in the case of multiobjective optimization, is not as clearly established

0.4

as for single objective optimization. As an example,

0.2

let us consider the problem of the maximization of mathematical test function which is taken from the Ref.[14]. Maximize: Subject to:

0.0 0.0

0.2

0.4 fl

f l = x,

f2 = Y

x 2 + y2 < 1

and

0 < x < 1,

0_
(b) Fig.2 Results of the test function with two objectives

In order to increase the difficulty of the problem, let us extend above test function into three objective

162

Journal of Thermal Science, Vol. 8, No.3, 1999

problem, T h a t is, a third objective f3 has been added in the test function. In this way, the dimensions of the search space are increased by one order of magnitude. Maximize: Subject to:

ft =x, f2=Y, fa=z x 2 q_ y2 q_ z 2 < 1 and

0 < x < 1,

0_
T h e same parameters of the M O G A s for above test function are used for this three objective function. But, compare to the previous case, population size was increased to 200 individuals and evolve 80 generations, due to the larger search space of the problem. Fig.3 illustrates the results concerning the set of Pareto optimal obtained. It is apparent t h a t the large range of the exact Pareto optimal set is covered by the obtained solutions. The results of the above test functions indicate the feasibility and excellent search performance for the Pareto solutions of the present MOGAs.

the larger flow turning angle can cause flow separation and corresponding larger total pressure loss. The present multiobjective problem can be described as follows: Function 1: Maximize pressure rise (outlet pressure by inlet pressure) Function 2: Minimize total pressure loss Function 3: Maximize flow turning angle Subject to: De hailer number to be greater t h a n 0.72 (outlet flow velocity by inlet flow velocity) Maxim u m airfoil thickness to be the same as t h a t of CDA [91 To obtain the values of the objective function of compressor cascade, the Euler equations solver has been applied to simulate the cascade flow field. The C - t y p e grid is used, where 151 points are used in the streamwise direction and 25 points are used in the direction normal to the airfoil surface. The given flow condition is set to inlet Mach number of 0.25, inlet flow angle of 40.0 deg., blade stagger angle of 14.4 deg., and the pitch of cascade of 0.5988, similar to Ref.[9]. The present numerical m e t h o d can

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2. Compressor Cascade Optimization Design Goal of the compressor cascade design is to produce the highest pressure rise at the lowest total pressure loss and maximized the flow turning angle in a given flow condition. Flow turning angle is an important par a m e t e r in the design process. In general, the pressure rise increase as the flow turning angle increase. But, there is a limitation of the flow turning angle because

accurately obtain the pressure coefficients of the CDA airfoil and agree with the experiment [9]. The B-spline function has been used for representation of compressor cascade shape. In this work, 13 control points are adopted to change the pressure surface and the other 13 points for the suction surface. The control points for leading edge and trailing edge do not change during the optimization process. We can only change y

Jun LI et al. Genetic Algorithms Development for Multiobjective Design Optimization of Compressor Cascade 163 direction coordinate value of the control points and keep x direction coordinates constant. In this optimization process, the 60 individuals are evolved 60 generations. The optimization experiment work on the Tempest2 Work Station which CPU is the DEC Alpha AXP21164A. The convergence criterion is that the fitness do not improve for 5 generations. Fig.4 shows that the Pareto solutions of the present MOGAs after 45 generations. Each axis is linearly scaled according to the maximum and minimum fitness values. From Fig.4(b), with the pressure rise increase, the total pressure loss also decrease in some extent. However, when the pressure rise reach the fixed high value, the total pressure loss will increase with the pressure rise increase. We can see the pressure rise versus flow turning angle in Fig.4(c), so, it is very difficult to increase both fitness values.

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Fig.5 shows variations of airfoil geometry shape of Pareto solutions. To increase pressure rise, the airfoil tends to have a smaller curvature. However, in order to increase flow turning angle, the airfoil should has a larger curvature. In addition, during numerical analysis of cascade performance, generation of the C-type grid for cascade restrict the limitation of variables and also limit the search space. Above reasons prove the difficulty in meeting both of these objectives. Table 1 summarizes the performance of airfoil designs including the CDA performance. Fig.6 shows the corresponding performance plane. The Pareto solution with the highest pressure rise is found to give better performance than CDA in all three objectives. Fig.7 illustrates the airfoil that has relatively high pressure rise and relatively low total pressure loss. These results confirm that the present MOGAs is able to find improved designs.

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164

Journal of Thermal Science, Vol. 8, No.3, 1999 T a b l e 1 Comparison of cascade airfoil performance CASE

P2/P1

w

B(deg.)

CDA

1.0193

0.0331

38.0

Highest P2/P1

1.0199

0.0288

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Highest B

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Higher P~/P1 & lower w

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Point ( P2/Px )

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F i g . 6 Performance plane of Pareto solutions compared with t h a t of CDA

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X/CL F i g . 7 Comparison of higher P2/P1 & lower w with CDA CONCLUSIONS

B a s e d on t h e m u l t i - b r a n c h s i m u l a t e d a n n e a l i n g sel e c t i o n a n d collection of P a r e t o s o l u t i o n s s t r a t e g y , t h e

As a r e s u l t of t h i s effort, several conclusions have

MOGAs

have b e e n d e v e l o p e d in t h i s p a p e r .

The

b e e n m a d e on t h e M O G A s for m u l t i p l e o b j e c t i v e p r o b -

p r e s e n t M O G A s c a n b e d e m o n s t r a t e d as t h e useful

lem.

t o o l s for t h e m u l t i p l e o b j e c t i v e o p t i m i z a t i o n p r o b -

Jun LI et al.

Genetic Algorithms Development for Multiobjective Design Optimization of Compressor Cascade

lems. The Pareto optimal set approximated for the test functions were fairly good in comparison to the analytical results. In addition, compressor cascade multiobjective design problem also proved their optimization performance. This work has illustrated that the multi-branch simulated annealing selection operator in the MOGAs can provide a practical and successful means for multiobjective design. This method can provide the same selection pressure for each objectives of design problem and can keep diversity of the population. Coupling parent selection strategy, this selection method can efficiently improve optimization performance of the algorithms for multiobjective designs. Collection of the Pareto solutions strategy play an important role for the Pareto solutions of the multiobjective problems. The results shows that obtained Pareto solutions can uniformly covered the analytical Pareto optimal set for the mathematical test functions. The solutions of the compressor cascade design suggests that it is difficult to design an airfoil satisfying high pressure rise and high flow turning angle. From the Pareto solutions computed, the decision maker is able to find a design that meet his design goal best. Acknowledgements The authors would like to thank Professor OBAYASHI, Shigeru who provided many references on the multiobjective GAs applications of his research achievements. The first author are grateful for Professor MATSUMOTO, Tsuguo for supporting him to work in the Venture Laboratory of Kyoto Institute of Technology.

REFERENCES [1] Hobbs, D.E. and Weingold, H.D., "Development of Controlled Diffusion Airfoils for Multistage Compressor Application," Journal of Engineering for Gas Turbines and Power, 106, pp.271-278, (1984). [2] Dunker, R., et al., "Redesign and Performance Analysis of a Transonic Axial Compressor Stator and Equivalent Plane Cascades with Subsonic Controlled Diffusion Blades," Journal of Engineering for Gas Turbines and Power, 106, pp.279-287, (1984).

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[3] Goldberg, D.E., "Genetic Algorithms in Search, Optimization, and Machine Learning," Addison-Wesley, (1989). [4] Quagliarella, D., Vicini A., "Coupling Genetic Algorithms and Gradient Based Optimization Techniques, in Quagliarella, D. et al., editors, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science, John Wiley & Sons Ltd., UK, pp.289-309, (1997). [5] Obayashi, S., "Aerodynamic Optimization with Evolutionary Algorithms," Inverse Design and Optimisation Methods, Von Karman Institute Lecture Series, (1997). [6] Obayashi, S., Takahashi, S. and Takeguchi, Y., Niching and Elitist Models for MOGAs, in A.E. Eiben, M. Schoenauer, and H.- P. Schuefel, editors, Parallel Problem Solving From Nature-PPSNV, Amsterdam, Holland, Springer- Verlag, pp.260-269, (1998). [7] Vicini A., Quagliarella, D., "Airfoil and Wing Design Through Hybrid Optimization Strategies," AIAA paper 98-2729, (1998). [8] Obayashi, S., Tsukahara, T. and Nakamura, T., "Cascade Airfoil Design by Multiobjective Genetic Algorithms," Second International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications, IEE Conference Publication No.446, Glasgow, UK, (1997). [9] Elazar, Y. and Shreeve, R.P., Viscous Flow in a Controlled Diffusion Compressor Cascade with Increasing Incidence," Journal of Turbomachinery, 112, pp.256266, (1990). [10] Michalewicz, Z., Genetic Algorithms -t- Data Structures = Evolution Programs, Springer-Verlag, Berlin, (1992). [11] LI, J. and Satofuka, N., "Optimum Aerodynamic Design Using Whole Annealing Genetic Algorithms," Computational Fluid Dynamics Journal, 8, No.2, Special Issue, (1999). [12] William A. Crossley, Andrea M. Cook, David W. Fanjoy, Vipperla B. Venkayya, "Using the Two-Branch Tournament Genetic Algorithm for Multiobjective Design," AIAA Journal, 37, No.2, pp.261-267, (1999). [13] Obayashi, S., Aerodynamic Inverse Optimisation Problems (chapter 9), in Zalzala, A.M.S. and Fleming, P.J. Eds., Genetic Algorithms in Engineering Systems, IEE, London, pp.203-228, (1997). [14] Obayashi, S., Takahashi, S., Fejtek, Ian, "Transonic Wing Design by Inverse Optimization Using MOGA," Proceedings of the Sixth Annual Conference of the Computational Fluid Dynamics Society of Canada, Quebec, pp.II-41-46, (1998).