2-ijaest-volume-no-3-issue-no-2-aerodynamic-multi-objective-optimization-using-parallel-genetic-algo

Aerodynamic Multi-Objective Optimization

Using Parallel Genetic Algorithm

G MANIKANDAN M ANANDA RAO1

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

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

ISSN: 2230-7818 @ 2011 http://www.ijaest.iserp.org. All rights Reserved. Page 78

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.

Keywords: Parallel Genetic Algorithm;

Cambered Aerofoil; Fitness Function;

Composite Material; Wind Tunnel;

Aerodynamic characteristics.

Nomenclature

A = Set of Scalar Chromosome L = Set of Vector Lift values L/D = Lift by Drag ratio F – Set of Scalar Objective Function f = Scalar Objective Function N

o = 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

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 the ith gene User specified minimum limits on the ith gene ϵ = 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.

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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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. 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] 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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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

(

)… (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 ( )( ) … (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

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

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

(

)

…… (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 ( ) ( )

….. (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

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.

Figure 1: Wind Tunnel Scaled Wing

Model Structure.

Figure 2: Wind Tunnel Scaled Wing

Model S fiber Lamination Procedure

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IV Results and Discussions

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.

Figure 3: Optimized Aerofoil Profiles

Table 1: Upper and Lower Limit of

Design Variables.

Parameter

(Control Point)

Maximum

Ordinate

Minimum

Ordinate

1 0.0 0.0 2 0.0208 0.001000 3 0.037500 0.011400 4 0.051800 0.020800 5 0.0636 0.037500 6 0.072400 0.051800 7 0.078000 0.063600 8 0.0788 0.072400 9 0.078000 0.078000

10 0.078000 0.078800 11 0.076 0.076700 12 0.072600 0.056300 13 0.066100 0.049600 14 0.056300 0.041300 15 0.049600 0.029900 16 0.041300 0.021500 17 0.029900 0.010000 18 0.0 0.0 19 -0.010000 -0.022700 20 -0.016500 -0.030100 21 -0.022700 -0.034600 22 -0.030100 -0.037500 23 -0.034600 -0.041000 24 -0.037500 -0.042300 25 -0.041000 -0.042200 26 -0.041200 -0.041200 27 -0.038000 -0.040000 28 -0.033400 -0.041200 29 -0.027600 -0.038000 30 -0.021400 -0.033400 31 -0.015000 -0.027600 32 -0.008200 -0.021400 33 -0.004800 -0.015000 34 -0.002000 -0.008200 35 0.000000 0.000000

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Table 2: First Generation Lift, Drag and

L/D ratio of 30 Chromosomes.

Airfoil Lift Drag L/D

2412 0.528 0.0084 62.85714 g1c1 0.537 0.0077 69.74026 g1c2 0.171 0.0097 17.62887 g1c3 0.502 0.0077 65.19481 g1c4 0.262 0.0097 27.01031 g1c5 0.547 0.0077 71.03896 g1c6 0.559 0.0077 72.5974 g1c7 0.171 0.0096 17.8125 g1c8 0.276 0.0098 28.16327 g1c9 0.542 0.0077 70.38961 g1c10 0.544 0.0077 70.64935 g1c11 0.171 0.0096 17.8125 g1c12 0.172 0.0096 17.91667 g1c13 0.575 0.0077 74.67532 g1c14 0.504 0.0077 65.45455 g1c15 0.262 0.0097 27.01031 g1c16 0.261 0.0097 26.90722 g1c17 0.524 0.0077 68.05195 g1c18 0.619 0.0077 80.38961 g1c19 0.274 0.0098 27.95918 g1c20 0.509 0.0077 66.1039 g1c21 0.282 0.0098 28.77551 g1c22 0.44 0.0072 61.11111 g1c23 0.263 0.0097 27.1134 g1c24 0.519 0.0077 67.4026 g1c25 0.586 0.0077 76.1039 g1c26 0.272 0.0098 27.7551 g1c27 0.592 0.0077 76.88312 g1c28 0.564 0.0077 73.24675 g1c29 0.171 0.0096 17.8125 g1c30 0.519 0.0077 67.4026

Graph 1: l/d vs. chromosome number of

first generation

Graph 2: Drag vs. Chromosome number

of first generation

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.

Graph 4: Lift vs. Chromosome for 300th

generation

Graph 5: Drag vs. Chromosome for 300

th

generation

Graph 6: L/D vs. Chromosome for 300

th

generation.

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.

Airfoil Lift Drag L/D

g300c1 0.619 0.0077 80.38961 g300c2 0.592 0.0077 76.88312 g300c3 0.586 0.0077 76.1039 g300c4 0.668 0.0072 92.77778 g300c5 0.377 0.008 47.125 g300c6 0.312 0.01 31.2 g300c7 0.432 0.0072 60.00 g300c8 0.506 0.0077 65.71429 g300c9 0.516 0.0077 67.01299 g300c10 0.532 0.0077 69.09091 g300c11 0.536 0.0077 69.61039 g300c12 0.513 0.0077 66.62338 g300c13 0.574 0.0077 74.54545 g300c14 0.523 0.0077 67.92208 g300c15 0.584 0.0077 75.84416 g300c16 0.56 0.0077 72.72727 g300c17 0.526 0.0077 68.31169 g300c18 0.48 0.0078 61.53846 g300c19 0.626 0.0077 81.2987 g300c20 0.585 0.0077 75.97403 g300c21 0.593 0.0077 77.01299 g300c22 0.439 0.0072 60.97222 g300c23 0.456 0.0072 63.33333 g300c24 0.482 0.0072 66.94444 g300c25 0.506 0.0075 67.46667 g300c26 0.507 0.0077 65.84416 g300c27 0.524 0.0077 68.05195 g300c28 0.529 0.0077 68.7013 g300c29 0.533 0.0077 69.22078 g300c30 0.544 0.0077 70.64935

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ISSN: 2230-7818 @ 2011 http://www.ijaest.iserp.org. All rights Reserved. 5

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 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.

Graph 9: Comparative study of

Aerodynamic Characteristics of 2412

Baseline Aerofoil, GA Optimized Aerofoil

with Experimental analysis

V Conclusion

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 – 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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used in each generation will be the further scope of study. Thus the GA optimization procedure implemented is very attractive for parallel computing with at least 45 GB memory.

References

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[5] CZ Janikow and Z Michalewicz, “An

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[8] D Quagliarella and A Vicini, “

Inverse and Direct Airfoil Design Using a Multi Objective Genetic Algorithm”, AIAA Journal, Vol 35, Issue 9, pages 1499-1505, Sept, 1997.

[9] AP Giotis, KC Giannakoglou and

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[12] S.Mulgund, K H Arper, K K Krishna

Kumar and G Zacharias, “Air Compact Tactics Optimization Using

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

[13] M Arakawa, I Hagiwara,

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Back and M Schurtz, “Multi point Airfoil Optimization using Evoluation Strategies”, European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMASS 2000, Barcelona, Spain, Sept 2000

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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 Un-manned 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.

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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