2-IJAEST-Volume-No-3-Issue-No-2-Aerodynamic-Multi-Objective-Optimization-Using-Parallel-Genetic-Algo by ISERP ISERP

G MANIKANDAN et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 3, Issue No. 2, 078 - 088

Aerodynamic Multi-Objective Optimization Using Parallel Genetic Algorithm M ANANDA RAO1

G MANIKANDAN

Professor and Principal SS Institute of Technology Dundigal, Hyderabad Andhra Pradesh, India profanandarao @yahoo.com

Professor SS Institute of Technology Dundigal, Hyderabad Andhra Pradesh, India manii731@yahoo.co.in

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R2= User specified parameter which controls the probability test of global random number mutation operator R (0, 1) = Random number generator which returns a random value between 0 and 1. ith gene from the jth chromosome from the nth GA generation. jth chromosome from nth GA generation User specified maximum limits on th the i gene User specified minimum limits on th the i gene Ďľ = User specified parameter which controls the size of perturbation mutation parameters Subscripts i = Gene Index j = Chromosome Index k = Objective function index m = No of scalar objective function Superscripts n = Population Index t = Temporary chromosome and gene values obtained after initial selection and before modification operator.

Abstract - Shape optimization of airfoil for the aerodynamic analysis of a low speed and low Reynolds number unmanned aerial vehicle wing is performed using parallel Genetic Algorithm. NACA 2412 chambered airfoil is chosen as zero generation airfoil. Real number coding is implemented for inputting seed value. Four modification operators are applied in this design space search method. The design space genes are control points of airfoil. Multiple fitness functions are utilized. Genetic Algorithm optimized airfoil profiles are used for the fabrication of composite material wing and are tested in the subsonic wind tunnel. The aerodynamic characteristics gleaned from experimental analysis are compared with base line airfoil and genetic algorithm optimized airfoil.

Nomenclature

Keywords: Parallel Genetic Algorithm; Cambered Aerofoil; Fitness Function; Composite Material; Wind Tunnel; Aerodynamic characteristics.

A = Set of Scalar Chromosome L = Set of Vector Lift values L/D = Lift by Drag ratio F â&#x20AC;&#x201C; Set of Scalar Objective Function f = Scalar Objective Function No = nth GA generation M= User specified vector with four elements that controls modification operators mpt = Pass through operator mc = Random average cross over operator mpm= Perturbation mutation operator mm = Random mutation operator R1= User specified parameter which controls the probability test of perturbation mutation operator

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I Introduction The objective of airfoil design optimization is to enhance the lift and L/D ratio and minimize the drag. There is a tradeoff between drag and lift because one of the drag components called Induced drag increases in proportion to the square of lift. Therefore the design airfoil profile is a challenging problem. Very precise shape optimization using very sensitive control points is needed. Aerodynamic evaluation using high fidelity model using Navier Stroke equation leads to very expensive function evolution. Gradient based numerical method for optimizing the airfoil shape was in practice for many years. The efficiency of gradient based optimization generally requires a smooth design space

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Profound knowledge and quite essential idea of optimization by genetic algorithm is delivered by DE Goldberg [1]. Parallel computing for genetic algorithm optimization is used for the fitness evaluation of computational fluid dynamics analysis because of large computational effort required for aerodynamic optimization. In aerospace most of the airfoil optimization have however employed sequential genetic algorithm than the parallel. [2, 3, 4]. In this paper parallel computing method and real number coding is implemented. The chromosomes are coded as finite length string of real numbers corresponding to the design variables. The real coded genetic algorithm outperformed binary coded genetic algorithm in many design problems [5, 6]. Hybrid genetic algorithms have been one of the advanced techniques adapted for improving GA performance. It requires care in balancing various elements of search space. It adversely affects population and force the evaluation in wrong direction if the high rated solutions are injected in the population at the earlier evolution stage. Moreover it requires special care in encoding. Therefore it is worthwhile to extend the optimization by genetic algorithm. Coupling genetic algorithms on gradient based optimization techniques gives flexibility in design airfoil

design optimization [7]. Direct and inverse airfoil design is carried out using multi objective genetic algorithm [8]. Multi objective optimization based on Pareto front technique and neural network by reduced cost was developed [9]. Shape optimization by the use of Voxel (N dimensional pixel) based presentation using series of binary number was proposed by Peter Baron, Robert Fischer and R Smith [10]. To solve problems with large number of real design parameters, Stochastic Genetic Algorithm are used effectively and efficiently [11]. For Air Combat Tactics optimization, Stochastic GA has been successfully applied [12]. The dynamic coding for binary coded GAs to treat continuous design space is a novel technique adopted by Adaptive Range GA (ARGA) [13]. The aerodynamic airfoil shape optimization can be performed better than real coded GA by ARGA [14]. Airfoil shape optimization was carried out by multi objective optimization technique [15]. Missile aerodynamic shape optimization and Wing shape optimization was also carried out by multi objective optimization [16], [17]

and single extreme or initial guess very close to global extreme for quick and proper convergence. The number of function evaluation required for the convergence of Genetic Algorithm (GA) optimization process exceeds the finite difference based gradient optimization. Genetic algorithm has capability of finding a global optimum from multiple design variables effectively because it does not use any derivative information. Therefore in this paper a promising GA approach is used for airfoil shape optimization.

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II Parallel Genetic Algorithm for Airfoil Shape Optimization For a single objective optimization problem involving lesser number of design variables for the airfoil shape optimization generally follows sequential genetic algorithm. For the optimization problem involving more than one objective is a very difficult situation because each objective must be simultaneously optimized and each objective plays a vital role in deriving optimal solution. In multi objective airfoil optimization the concept of dominance is utilized. Three vectors lift, drag, and L/D are used as scalar objective functions. The vector lift (L1)=L(l1, …, li ,…, lN) is said to dominate another vector lift (L2)=L(m1,…,

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)… (1)

(

The j subscript indicates chromosome number and n superscript indicates genetic algorithm generation number. Real number encoding is used to represent all genes. The initial generation using the real number genes is represented by (

)

... (2)

The population size considered is 30. Each gene with each chromosome is assigned with an initial real number value by random number generation between fixed upper and lower limits. The ith gene in an arbitrary chromosome is computed using (

number sequence is reset. The fitness values for each chromosome are calculated by fitness function evaluation denoted by …. (4)

)

(

There are three fitness functions used namely lift, drag and lift by drag ratio are defined by )

(

…. (5)

(

)

…. (6)

(

)

…. (7)

The function represents quantitative evaluation of lift, drag and lift-drag ratio. The chromosome with highest fitness i.e. high lift and L/D ratio and low drag is ranked 1 with the second highest fitness ranked 2 and so on. The highest fitness function chromosome is passed through the next generation. In this paper four modification operators-pass through, random average cross over, perturbation mutation and mutation are used. The number of chromosomes modified with each operator is controlled by M vector. The vector consists of 4 parameters . The value of each M vector element ranges from 0 to 1 and the sum of all four elements is equal to 1. The M vectors are in the ratio 1:3:3:3. The pass through operator is performed first and then the other operator until the airfoil shape optimization is converged. The highest individual fitness valued chromosome is passed to the next generation. Thereby guaranteeing that none of the maximum fitness valued chromosomes will get dropped during GA iteration. The random average cross over operator is applied on randomly selected two chromosomes from the population. The gene

mi,…, mN ) if and only if li ≥ mi for all i and there exist at least one value of i such that li>mi. The airfoil multi objective optimization problem is defined by F=F (f1(A),… fk(A),… fm(A)). The decision variable vector A consists of 35 independent co-ordinates. The multi objective optimization of airfoil shape profoundly involve in finding the set of A= ̅ that produce non dominated values of F= ̅ ; ̅ is known as Pareto Front. The idea behind Pareto Front is for many events; roughly 80% of the effects come from 20% of the causes. In GA optimization design space is discreetly described by decision variables i.e. control points Ai. These parameters are called genes in GA parlance. The decision variable vector, A is known as chromosome and is denoted by

)(

)

… (3)

The random number generator used in this paper provides an integer input seed value. If the integer is positive the current random number sequence is selected or else random number sequence is reset. If same seed value is selected then also the random

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)

( …… (5)

The perturbation mutation operator is applied by first selecting a random chromosome from the population. Probability test is performed on each gene in the selected chromosome Aj using random number generator. If the random number is greater than the user defined random number R1 then the gene is not modified or else it is modified by )[ (

) …. (8)

( ]

The value of „ϵ‟, a user specified tolerance which controls the perturbation operator lies between 0 and 1.0. The random number mutation is applied by selecting a random chromosome from the population and a probability test is performed on each gene in the selected chromosome Ai. If the random number is less than the user specified random number R2 then the gene is modified by (

ribs (one in the root and tip chord and two at the mid chord). The skin is made of two layers. First layer is 2mm balsa sheet and the second layer is 1 mm fiber glass reinforced with epoxy resins. Pressure tapings are provided in the mid chord for the investigation of the pressure distribution over the wing model. Load cells are used to find the aerodynamic characteristics such as lift, drag, etc. The composite wing model is tested at an angle of attack of 2 deg and Mach number of 0.06. The total weight of the wing is 500 gram. Various stages of wing fabrication are shown in figure 1 to 2.

by gene basis combination of the two selected chromosomes is achieved by:

) (

Figure 1: Wind Tunnel Scaled Wing Model Structure.

)

….. (9) In this paper ϵ is assumed as 0.9, R1 as .9and R2 as .6. III Wind Tunnel Model preparation, Testing and Analysis

The optimized aerofoil profile by GA is used for the fabrication of scaled wing model of Unmanned Aerial Vehicle (UAV) having a chord of 15 cm and span of 21 cm using Balsa wood reinforced by S fiber glass with epoxy resins. The wing is a single spar multi rib type having I section spar and 4

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Figure 2: Wind Tunnel Scaled Wing Model S fiber Lamination Procedure

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Parameter (Control Point) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Maximum Ordinate 0.0 0.0208 0.037500 0.051800 0.0636 0.072400 0.078000 0.0788 0.078000 0.078000 0.076 0.072600 0.066100 0.056300 0.049600 0.041300 0.029900 0.0 -0.010000 -0.016500 -0.022700 -0.030100 -0.034600 -0.037500 -0.041000 -0.041200 -0.038000 -0.033400 -0.027600 -0.021400 -0.015000 -0.008200 -0.004800 -0.002000 0.000000

Genetic Algorithms are stochastically based search algorithms which produce results with statistical variations from generation to generation. The initial seed depends on random number generator. A total of 9999 random number valued chromosomes are generated out of which 30 chromosomes are selected as initial seed based on their fitness function ranking. The maximum and minimum range of values for the design space variables (control points) are fixed based on the co-ordinates of the base line aerofoil NACA 2412. The optimized aerofoil profile obtained in each generation is tested for aerodynamic characteristics by panel method using Design Foil software. The leading edge, maximum thickness location and trailing edge genes are fixed and rest 31 genes are altered by parallel GA as shown in table 1. The aerodynamic characteristics of 30 chromosomes for the first generation are shown in the graph 1 to 3. It was found that the highest lift was produced by g1c18 chromosome and lowest by g1c29, highest L/D ratio was achieved by g1c18 and lowest by g1c2 and lowest drag is obtained by g1c22 and highest drag is achieved by g1c26. The comparative study of lift, drag and L/D ratio of first generation is presented in table 2. The generation wise optimized aerofoil profile generated is shown in figure 3.

Table 1: Upper and Lower Limit of Design Variables. Minimum Ordinate 0.0 0.001000 0.011400 0.020800 0.037500 0.051800 0.063600 0.072400 0.078000 0.078800 0.076700 0.056300 0.049600 0.041300 0.029900 0.021500 0.010000 0.0 -0.022700 -0.030100 -0.034600 -0.037500 -0.041000 -0.042300 -0.042200 -0.041200 -0.040000 -0.041200 -0.038000 -0.033400 -0.027600 -0.021400 -0.015000 -0.008200 0.000000

IV Results and Discussions

Figure 3: Optimized Aerofoil Profiles

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Drag 0.0084 0.0077 0.0097 0.0077 0.0097 0.0077 0.0077 0.0096 0.0098 0.0077 0.0077 0.0096 0.0096 0.0077 0.0077 0.0097 0.0097 0.0077 0.0077 0.0098 0.0077 0.0098 0.0072 0.0097 0.0077 0.0077 0.0098 0.0077 0.0077 0.0096

g1c30

0.519

0.0077

L/D 62.85714 69.74026 17.62887 65.19481 27.01031 71.03896 72.5974 17.8125 28.16327 70.38961 70.64935 17.8125 17.91667 74.67532 65.45455 27.01031 26.90722 68.05195 80.38961 27.95918 66.1039 28.77551 61.11111 27.1134 67.4026 76.1039 27.7551 76.88312 73.24675 17.8125

Graph 1: l/d vs. chromosome number of first generation

Lift 0.528 0.537 0.171 0.502 0.262 0.547 0.559 0.171 0.276 0.542 0.544 0.171 0.172 0.575 0.504 0.262 0.261 0.524 0.619 0.274 0.509 0.282 0.44 0.263 0.519 0.586 0.272 0.592 0.564 0.171

Graph 2: Drag vs. Chromosome number of first generation

Airfoil 2412 g1c1 g1c2 g1c3 g1c4 g1c5 g1c6 g1c7 g1c8 g1c9 g1c10 g1c11 g1c12 g1c13 g1c14 g1c15 g1c16 g1c17 g1c18 g1c19 g1c20 g1c21 g1c22 g1c23 g1c24 g1c25 g1c26 g1c27 g1c28 g1c29

Table 2: First Generation Lift, Drag and L/D ratio of 30 Chromosomes.

67.4026 Graph 3: Lift vs. Chromosome number for first generation

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The aerodynamic characteristics of 30 chromosomes for 300th generations are shown in the graph 4 to 6.

It was found that the highest lift was produced by g300c4 chromosome and lowest by g300c6, highest L/D ratio was achieved by g300c4 and lowest by g300c6 and lowest drag is obtained by g300c3 and highest drag is achieved by g300c6. The comparative study of lift, drag and L/D ratio is presented in table 3. Table 3: 300th Generation Lift, Drag and L/D ratio of 30 Chromosomes. Drag 0.0077 0.0077 0.0077 0.0072 0.008 0.01 0.0072 0.0077 0.0077 0.0077 0.0077 0.0077 0.0077 0.0077 0.0077 0.0077 0.0077 0.0078 0.0077 0.0077 0.0077 0.0072 0.0072 0.0072 0.0075 0.0077 0.0077 0.0077 0.0077 0.0077

Graph 5: Drag vs. Chromosome for 300th generation

Graph 6: L/D vs. Chromosome for 300th generation.

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L/D 80.38961 76.88312 76.1039 92.77778 47.125 31.2 60.00 65.71429 67.01299 69.09091 69.61039 66.62338 74.54545 67.92208 75.84416 72.72727 68.31169 61.53846 81.2987 75.97403 77.01299 60.97222 63.33333 66.94444 67.46667 65.84416 68.05195 68.7013 69.22078 70.64935

Lift 0.619 0.592 0.586 0.668 0.377 0.312 0.432 0.506 0.516 0.532 0.536 0.513 0.574 0.523 0.584 0.56 0.526 0.48 0.626 0.585 0.593 0.439 0.456 0.482 0.506 0.507 0.524 0.529 0.533 0.544

Graph 4: Lift vs. Chromosome for 300th generation

Airfoil g300c1 g300c2 g300c3 g300c4 g300c5 g300c6 g300c7 g300c8 g300c9 g300c10 g300c11 g300c12 g300c13 g300c14 g300c15 g300c16 g300c17 g300c18 g300c19 g300c20 g300c21 g300c22 g300c23 g300c24 g300c25 g300c26 g300c27 g300c28 g300c29 g300c30

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Lift increases from 0.171 for the first generation to 0.668 for the 300th generation for a fixed speed of 0.06 Mach, Reynolds Number 90000 and angle of attack 2 deg. The L/D ratio also increases from 17.62 to 92.7 from first to 300th generation as shown in graph 7 and 8. No remarkable reduction in drag is achieved from first to 300th generation.

Graph 9: Comparative study of Aerodynamic Characteristics of 2412 Baseline Aerofoil, GA Optimized Aerofoil with Experimental analysis V Conclusion

Graph 7: L/D vs. Generation

Graph 8: Lift vs. Generation

Three parameters , R1 and R2 variation effect on the GA convergence is analyzed. The effect of different M vector on GA convergence for number of function evolutions is also analyzed. The comparative study of aerodynamic characteristics of baseline, GA optimized aerofoil with experimental analysis is shown in graph 9.

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A parallel GA optimization procedure is developed for the multi objective optimization of airfoil shape. It uses real number coding for the representation of design space of 35 decision variables as genes and 30 populations to go from generation to generation. 4 modification operators â&#x20AC;&#x201C; Pass through, Random average cross over, Perturbation mutation and Mutation are utilized to advance from one generation to another. The best solution for each objective is a parato front. For each case attempted global parato front optimum is achieved by the convergence of GA optimization algorithm. Over 300 generations are considered to study the convergence efficiency. In some cases convergence was achieved quickly and in other cases it was much slower. One value of caused early convergence and the other value caused late convergence. R1 has small effect on convergence and R2 has negligible effect on convergence for all the generation considered. The M vector has moderate effect on the convergence. The effect of number of chromosomes used in each generation and the effect of number of genes

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[7]

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G. Manikandan was born on 12th January 1969 from the famous big temple city Thanjavur, Tamil Nadu. He obtained his Engineering Graduation (Mech) in the year 1994 from Institution of Engineers (India), Calcutta and M.Tech (CAD/CAM) in the year 2002 from JNTU, Hyderabad. He put up 16 years of colorful service in Indian Air Force. In his credit, he overhauled 365 Rolls Royce Viper Turbojet Engine fitted on Kiran Aircraft and Carried out Structural Repairs and maintenance of Cheetah and Chetak helicopters and Kiran aircraft. He was team leader for several Structural re-fabrications of Ardhra and Rohini Gliders. He developed many Unmanned Aerial Vehicles (UAV). Presently, his contributions are in the area of aerofoil shape optimization and flutter analysis. He was awarded best in trade and all-rounder for Kiran Aircraft in the year 2000.

Stochastic Genetic Algorithms” Proceedings of IEEE International Conference on Systems , Man and Cyber tics, pages 3136-3141, 1998

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M. Ananda Rao obtained B.E (Mech) in 1968, M.Tech (Machine Design) in 1970 and M.Tech (Industrial Engg) in 1984. He was awarded PhD from IIT, Madras in the area of “Machine Dynamics”. He worked over 33 years in Andhra University at various capacities. He worked in the Link Interchange Program with UK Universities for about 03 years by British Council and Government of India. He was awarded three times “The Best Researcher Award” in the year 1992, 1999 and 2001. He worked as a technical adviser for Altair Company for the development of software in the domain of solvers. He is one of the renowned researchers in the area of Vibration and Condition Monitoring in the World. He was the nucleus in the starting of Condition Monitoring Society of India.

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