Inverse Design of Airfoil Using Vortex Element Method

International Journal of Fluid Machinery and Systems Vol. 11, No. 2, April-June 2018

DOI: http://dx.doi.org/10.5293/IJFMS.2018.11.2.163 ISSN (Online): 1882-9554

Original Paper

Inverse Design of Airfoil Using Vortex Element Method Avinash G.S.1 and S. Anil Lal2 1

Department of Mechanical Engineering, Vidya Academy of Science and Technology, Thiruvananthapuram, [email protected] 2 Department of Mechanical Engineering, Government Engineering College Barton Hill Thiruvananthapuram, 695035, India, [email protected] Abstract A methodology for the parameterization and inverse design of airfoils, for obtaining a given target surface pressure distribution is presented. The airfoil parameterization is carried out using ordered pairs representing the x-y coordinates of ten control points of Bezier curve as parameters. The forward model consists of analysis of flow over airfoils carried out using vortex element method, which involve discretisation of the airfoil curve alone, in contrast to complicated grid generation over the region of flow. The airfoil parameters are selected by global search using a Genetic Algorithm code. Examples to illustrate the parameterization and design of airfoils are presented. A good matching between the target and designed airfoil shows that present methodology can be used as a tool for the design of airfoils. Keywords: Bezier parameterization, Inverse design, Airfoil, Vortex element method, Genetic algorithm.

1. Introduction Inverse aerodynamic design of airfoils has been a challenging task and subject of investigation for a long time, date back to the beginning of airfoil research. It involves determination of airfoil shapes to meet specified aerodynamic requirements. Early developments in inverse design techniques were based on the theory of potential flow with the shape determination done using either conformal mapping [1-4] or method of distribution of singularities [5]. Eppler [1] developed an airfoil design theory based on conformal mapping. In this method airfoil is divided into segments and each segment of the airfoil is designed for different value of design angle of attacks and corresponding to constant velocity on the surface. Selig and Maughmer [2] modified Eppler's original theory, to prescribe a non constant velocity along a segment at the design angle of attack associated with that segment. The design of multi element airfoil and complex aerodynamic systems using conformal mapping is not possible since the complex variable are defined only in two dimensions. Therefore the applicability of conformal mapping is limited to the design of two dimensional airfoils and the singularity method has a restriction to design of only thin airfoils. Levenberg-Marquardt, conjugate gradient and response surface [6] method are techniques belonging to another class of optimization, which use the gradient of an objective function with respect to a selected number of independent variable parameters relevant to the geometry. Important disadvantages of gradient-based methods are the development cost, ineffectiveness under difficulties such as noisy objective function spaces, inaccurate gradients, categorical variables, and topology optimization. Another often mentioned disadvantage of gradient-based methods is the difficulty connected with convergence to any local extrema rather than global extrema [7]. However gradient based methods are found to be effective for fine tuning of the results obtained from global optimization tool. On the other hand, advancement in global optimization techniques such as Artificial Neural Network (ANN) and Genetic Algorithm (GA) have resulted in improved and accurate strategies for inverse design of airfoils. ANN [8] is a data processing system consisting of a large number of simple, highly interconnected processing elements (artificial neurons) in an architecture inspired by structure of cerebral cortex of the brain. Genetic algorithm [9] is an emergent optimization algorithm that mimic the natural evolution of a biological population to adapt to an environment by selection, recombination and mutation, from one generation to another. Genetic algorithms are versatile optimization tools suitable for solving multidisciplinary optimization problems in aerodynamics, where the design parameters may exhibit non-smooth variations. However, the fitness evaluation phase of the algorithms casts a large overhead on the computational requirement and is particularly acute in aerodynamic problems where time-consuming CFD methods are needed for evaluating performances. It is a known fact that conventional Genetic Algorithm perform poorly with small population sizes, due to insufficient information processing, and that they converge prematurely to non-optimal results. Methods and strategies to improve the performance of basic genetic algorithms are important to enable the method to be useful for complicated three-dimensional or multidisciplinary problems. Population size in conventional GA usually varies from tens to hundreds and such a high population can Received July 19 2017; revised October 10 2017; accepted for publication October 29 2017: Review conducted by Young-Seok Choi. (Paper number O17054K) Corresponding author: S. Anil Lal, Professor, [email protected]

163

lead to formidable computational cost. In this context, it is essential to note that the efficiency of micro GA [10, 11] results from use of small population size and the GA operators are applied within the small population, which leads to a more rapid convergence. One of the essential elements which characterize the inverse design method is a procedure to describe the geometry through control variables, which is otherwise known as parameterization. A geometry defined using minimum number of parameters that can yield an accurate definition of a surface model is an ideal shape parameterization. Such shape parameterization can reduce significantly the cycle times in inverse design. Geometry parameterization is done using curves such as Bezier curves, B-Spline curves and non-uniform rational b-splines. Bezier parameterization [12] is a method of parameterizing arbitrary shaped curves and surfaces including airfoils. The geometry is expressed using coordinates of an arbitrary number of points called control points which forms the parameters. PARSEC [13] is another method of parameterization that uses parameters directly relevant to airfoils, such as leading edge radius, upper and lower crest location including curvatures, trailing edge ordinate, thickness, direction, and wedge angles for both top as well as bottom curves. B spline functions are well known type of polynomial functions which are also being used to parameterize airfoil shapes. Srinath and Mittal [14] have used a fourth order non-uniform rational b-spline (NURBS) to parameterize airfoil shapes. Shahrokhi and Jahangirian [15] have reported the details of an improved parameterization technique to represent accurately the trailing edge shape of airfoils based on modification of Sobieczky method [13] and compared its performance over conventional PARSEC method. Ref [15] uses an initial profile to start the optimization and small changes in the shape is introduced in every iteration till the final required shape is obtained. Navier-Stokes solver applied on regenerated computational mesh is used as the forward model for a GA optimization tool. The algorithm used by Obayashi and Takanashi [16] also uses an initial airfoil, grid generation and a number of trials of a high fidelity flow solver for Navier-Stokes solution to design an airfoil that develop a given pressure distribution. Recent trend followed in Tomas et al. [17], Bian et al. [18] and Afzal et al. [19] is to apply a combination of design of experiments for selecting CFD sample solutions, surrogate models and optimization tools to design engineering systems. Afzal et al. have used surrogate models to replace high fidelity CFD models. They have carried out optimization of two thermal fluid design problems and two analytical functions. The optimization tool used is Particle Swarm Optimization (PSO). Derksen and Rogalsky [20] introduced a combination of Bezier and PARSEC method for parameterizing the airfoil. They reported that PARSEC method is superior due to its ability to minimise the non-linear dependency of objective function with design parameters. Phelivanoglu [9] has given a comparison between PARSEC and Bezier parameterization and reported that PARSEC method has a better performance in subsonic flow conditions, whereas Bezier representation is more suitable in transonic regimes. This method uses a vibrational genetic algorithm as the tool for optimization. One popular alternative of inverse design is based on relating a change in pressure distribution to a change in stream line curvature. A disadvantage of this approach is that it requires an initial airfoil with a pressure distribution that roughly approximate the target pressure distribution. Surface singularity has been also considered as an efficient tool for inverse design. In this approach the airfoil shape is represented by a distribution of vortices and/or sources and sinks. The strength of these singularities determines the shape of airfoil. The above review of literature shows that design of airfoils uses a starting profile, high fidelity CFD models, and simplified substitute models derived from a designed number of CFD trials. Commonly used CFD solution techniques require a mesh generator and intensive computations. The motivation behind the present work is to design airfoils using a low fidelity fluid dynamic model without requiring (i) the specification of any initial shape, (ii) mesh generation and (iii) substitute models. The method used for inverse design in this work consists of three steps. The first step is called parameterization, where a set of parameters involved in the mathematical equation of the surface or curve of the airfoil are determined. Different shapes for airfoil are obtained by controlling these parameters. In the second step the aerodynamic performance of a selected profile is found out by applying a suitable numerical technique. In this study Vortex Element Method (VEM) is used as the numerical technique for solving potential flow over bodies. Third step is to carryout global search using Genetic Algorithm to find a new set of geometric parameters with an objective to maximize the fitness function. This paper also reports a review and comparison of different methods for parameterization and numerical techniques for the aerodynamic evaluation of airfoils. The present methodology has been applied to illustrate the parameterization and design of airfoils. Vortex element method used as forward model restricts the accuracy of designed airfoils to flows which can be closely approximated as incompressible, inviscid and irrotational. For other flow situations such as viscous flows, the designed airfoils enable as the starting profile in conjunction with methods presented in Obayashi and Takanashi [16], Afzal et al. [19].

2. Airfoil parameterization The following is a brief review of three methods commonly applied for the parameterization of airfoils. 2.1 Bezier Curve Bezier curves [20] are polynomial curves defined by using a set of control points. The ordered pair representing the x(t) and y(t) coordinates of points on the curve, where t is a real value in [0,1] is given by the following equations.

x(t ) = y(t ) = where

n

Cni t i (1 − t )n−1 xi  i =0

(1)

n

C ni t i (1 − t )n−1 yi  i =0

(2)

Cni = i! (nn−! i )! , n ! =1.2.3... n and (xi , yi ) represents the x and y coordinates of n control points. The two control

points at the leading edge and trailing edge are fixed as (0,0) and (1,0). Bezier parameterization has many properties that are

164

attractive for airfoil design. The tangent to the curve is along the vector joining the endpoint and the closest control point. The curve always lies within the convex figure defined by the extreme points of the polygon. The curve is nth order continuous throughout and never oscillates away from its defining control points. Figure 1 is a schematic sketch showing the parameterization of airfoil using Bezier curve. 2.2 PARSEC PARSEC [4] is also an effective method for airfoil parameterization. It uses eleven basic parameters as illustrated in Fig. 2. These are the leading edge radius (rle), upper and lower crest location (Xup, Zup, Xlo, Zlo) and curvature (ZXXup, ZXXlo), trailing edge coordinate (ZTE) and direction (αTE), trailing edge wedge angle (βTE) and thickness ΔZTE. The eleven parameters is closed under six coefficients as given below.

Fig. 1 Representation of an Airfoil using Bezier parameters. The top and bottom surfaces are parameterized using 9 and 8 control points, respectively. 6

Zk =

Fig. 2 Illustration of airfoil parameters used in PARSEC Method

n−1 2

ank X k  n =1

(3)

The values of subscript k equal to 1 and 2 denote the upper and lower surfaces. The coefficient ank is determined from the set of values of the eleven geometric parameters. These eleven parameters help to control the maximum curvature of the upper and lower surfaces and location of maximum curvature which is very useful in reducing the shock wave strength or delaying its occurrence. However, at the trailing edge of the airfoil, PARSEC fits a smooth curve between the maximum thickness point and the trailing edge which in turn disables the necessary changes in the curvature close to the trailing edge. Even though PARSEC offers benefits of controlling the important parameters on the upper and lower surfaces, it does not provide enough control over the trailing edge shape. 2.3 Hicks-Henne Bump Functions

Fig. 3 Hicks-Henne Bump functions Hicks-Henne shape functions recently became popular for modeling small or moderate perturbations of "baseline" airfoil shapes for solving various optimization problems such as inverse design. In this method the shape of the curve is assumed to be the sum of a basis shape and sum of suitably defined and weighted sine functions. This is given by,

165

M

y = ybasis +



 (

j

f j (x )

j =1

f j (x ) = sin  x log 0.5

log t1

(4)

)

t2

, 0  x 1

(5)

where t1 locates the maximum point of bump and t2 controls the width of the bump. The design variables are the weights αj multiplying each Hicks-Henne bump functions. This flexibility allows one to place the bump at strategic points where a redesign is preferred while leaving other parts of the airfoil intact. Figure 3 shows a set of typical Hicks-Henne Bump functions with parameter t2 =10.

3. Aerodynamic Evaluation using Vortex Element Method A number of methods can be found in literature for the aerodynamic evaluation of airfoils. The type of method depends on the level of accuracy required. For low Reynolds number flow equations of potential flow can be solved in conjunction with laminar boundary layer solver to determine the aerodynamic quantities. For higher Reynolds number flows solution of non-linear governing equations with the help of suitable turbulence model becomes necessary. However potential flow solutions are very simple and found to predict the pressure distribution with a reasonable accuracy level. A theoretical background of Vortex Element method is given below. Assuming fully attached flow, the boundary layer around an airfoil is approximated as an infinitesimally thin vorticity sheet. The inviscid velocity is then found in terms of the vorticity strength on the boundary. Consider a small vorticity element  (s ) ds , where  (s ) is defined as the vorticity strength per unit length at point s. Since the thickness of the element (normal to the surface) is infinitesimal, the circulation around it is just (vs − vi ) ds , where vs and vi are the fluid velocity just outside and inside the sheet. This can be equated to the total amount of vorticity enclosed by the contour. Therefore it is possible to write

(vs − vi )ds =  (s )ds

(6)

Since the no-slip condition on the body surface require that vi = 0, thus

v s =  (s )

(7)

Fig. 4 Discrete vorticity model for a 2-Dimensional body. The symbol Sn denotes the coordinate of the point n, which is the distance along the curve from a starting point with Sn = 0 The body surface is represented discretely by a finite number of short, straight panels as shown in Fig. 4. The Matenesen’s boundary integral equation is

1 1 − m + 2 4



im X (( n X rmn ) X im ) ds 3 rmn

s

+ im X (W X im ) = 0

(8)

By referring to Fig. 4(b), Matenesen’s boundary integral equation for two dimensional flows can be approximated numerically as

1 −  (S m ) + 2

 k (S

m , Sn

) (S n ) dS n + W (cos   cos  m + sin   sin  m ) = 0

where the last term is the component of W∞ resolved parallel to body surface at m and the coupling coefficient k (S m , S n )

166

(9)

 ( y − y ) cos  − (x − x )sin   n m m n m (10)  m    (xm − xn )2 + ( y m − y n )2 The resulting linear system of equation is solved for  (s ) on each panel, which is then exactly equal to the inviscid velocity. Clearly, VEM requires the discretization of only the curve of the airfoil. This is very easy compared to other conventional numerical techniques, where a complicated grid generation over the region of fluid flow is required. Such grid generation processes for large number of GA iterated off-spring airfoils becomes exceedingly tedious. k (S m , S n ) =

1 2

4. Optimization using Genetic Algorithm (GA) Genetic Algorithm is an emergent optimization algorithm mimicking the natural evolution, where a biological population evolves over generation to adapt to an environment by selection, recombination and mutation. GA is an attractive method for aerodynamic design, since it is able to find the design variables corresponding to the global optimum condition. The algorithm encode a potential solution to a specific problem on a simple chromosome like data structure and apply recombination operators to these structures so as to preserve critical information. Genetic algorithms are often viewed as function optimizer. In the present study simple Genetic algorithm is applied to the optimization of an airfoil. Thus, fitness, chromosomes and genes are corresponding to the objective function, design candidates and design variables, respectively. The evolution of the randomly selected individuals is carried through the following genetic operations. 1) Parent Selection - Reproductive trials are allocated to each individual, according to their fitness score. These trials define the number of copies of each individual in the mating pool. The fitter individuals are likely to receive more than one copy. Some of the less fit individuals are likely to stay out of the mating pool. First, all individuals are mapped onto a roulette wheel with slots. The size of the slot corresponding to the individual is proportional to the difference between its score and the score of the worst individual in this generation. Using the roulette wheel, the mating pool for the next generation is formed. 2) Crossover - Pairs of individuals are selected from this pool at random and their chromosomes are cut at a point selected at random; the parts after and before the cuts are mutually exchanged. Here, a 1-point crossover per free-parameter is used, which corresponds to a point crossover for the full chromosome of length. So, the substring of any free-parameter contains a mixture of genes from the parental substrings. The crossover possibility for each pair in the mating pool is kept very high (90%). 3) Mutation - Mutation is applied to each individual after crossover. It randomly alters bits with a small probability (usually less than 1%)

5. Results and Discussion 5.1 Parameterization of a given airfoil In this section the method and results of parameterization of a given airfoil is described. The data of airfoil to be parameterized inputted as the discrete ya coordinates values of the upper and lower curve corresponding to a set of x-coordinates. The methodology consists of generating Bezier curve using a set of control points, provided by the GA Code. The fitness function is selected such as to maximize

1

1+

 (y

a

)

− yg 2

(11)

where yg is the y coordinate of the airfoil generated from the Bezier curve obtained using parameters of GA, corresponding to the same value of x, where ya is defined. The maximum value of fitness function is equal to 1. As iteration proceeds, the value of fitness function increases, with a consequent reduction in the difference between the y coordinates of the original and parameterized airfoil. Two different Bezier curves are generated for the top and bottom surfaces of the airfoil.

Fig. 5 Comparison of parameterized and standard RAE 2822

167

The airfoil RAE 2822 is taken as a test case in this study. The Bezier curve parameters are the co-ordinates of control points. The upper and lower curves have been parameterized using seven control points each, out of which two control point are the end point of the curve itself, making a total of ten parameters for GA code to optimize. The number of control points equal to seven was decided after a sensitivity analysis performed by varying the number of control points. It is found that parameterization using seven control points result in the highest value for the fitness function. The control points and the resulted curve computed corresponding to the input data of RAE 2822 are shown in Fig. 5. This figure also shows the data points used as input for parameterizing airfoil. The comparison shows that the fitted Bezier equation with ten unknown parameters exactly matches with the actual airfoil. A similar curve computed for another airfoil named Fauvel with ten control points is shown in Fig. 6. The parameterized control points almost exactly representing the standard profiles of RAE 2822 and Fauvel are tabulated in Table 1.

Fig. 6 Parameterized Fauvel Airfoil Table 1 Coordinates of the Bezier control points for the standard airfoils RAE2822 and Fauvel Airfoil

RAE2822

FAUVEL

Top Surface x/c y/c 0.0000000 0.0000000 0.0500000 0.0763736 0.4375004 0.0453124 0.5000005 0.0938113 0.6249987 0.0472640 0.8138607 0.0423762 1.0000000 0.0000000 0.0000000 0.0000000 0.0499994 0.1847793 0.5856109 0.0780597 0.5000110 0.0249296 0.9141673 -0.0000048 0.7500322 0.0250000 1.0000000 0.0000000

Bottom Surface x/c y/c 0.0000000 0.0000000 0.0500000 -0.0750033 0.4702153 -0.0624938 0.3800052 -0.0234377 0.1874992 -0.1410160 0.8046854 0.0234375 1.0000000 0.0000000 0.0000000 0.0000000 0.0023804 -0.1000262 0.8658446 0.0000017 0.0861894 -0.0484641 0.4999251 -0.1000171 0.9960813 0.0015562 1.0000000 0.0000000

5.2 Inverse design of a given airfoil Inverse design of airfoil uses VEM as the forward model to predict the pressure distribution and Genetic Algorithm for optimization. The desired pressure distribution is given as an input for the inverse design. The pressure distribution over different airfoils, taken from the search space of GA is compared with the desired pressure distribution for selecting the final optimum shape. The target pressure for the inverse has been taken as the same computed using VEM for flow over RAE 2822 airfoil with an angle of attack of 2.79 degree. The fitness function to be maximized is

1

1+

 (Pt − Pg )2

(12)

where Pg is the pressure corresponding to the same x locations of Pt for the airfoil generated in GA code. The final shape of airfoil for the given target pressure distribution is shown in Fig. 7 corresponding to RAE 2822 airfoil. The convergence history of the restarted GA in 500 generations consisting of 100 populations in each generation making a total of 50000 iterations is shown in two graphs in Fig. 8. Clearly, the value of fitness function increases from zero and settles at a

168

maximum value of 0.99534 after 40000 iterations. The oscillations in the fitness function is due to restarting of the GA when the average value of fitness function becomes equal to the maximum value. Increase of value of fitness function beyond 0.99534 could not be obtained even after restarting the GA iterations few number of times. Having obtained a designed profile corresponding to this global maximum fitness function, fine turning can be followed using local search tools to obtain the airfoil that develop exactly the same pressure distribution.

Fig. 7 Comparison of pressure distributions over the designed airfoil and the reference airfoil, RAE 2822

Fig. 8 Convergence history of GA iterations for the design of airfoil

Fig. 9 Comparison between the design and reference profile corresponding to RAE 2822 airfoil

169

Figure 9 shows a comparison between the airfoil generated using GA and the original RAE 2822 airfoil. Both the airfoil sections match well except over a small region after the mid chord on the upper surface and towards the trailing edge on the bottom surface. Corresponding to the small deviation in the shape of the airfoil, the pressure distribution over the upper & lower surfaces of the airfoil close to the trailing edge region is different. A good agreement in the pressure distribution of the generated profile can be found up to 60% chord as shown in Fig.7.

6. Conclusion A methodology for parameterization and inverse design of airfoils has been developed. The important conclusions derived from the study are 1) Parameterization using Bezier curve provides an accurate mathematical representation of airfoil shapes. It is found that an optimum number of control points is required for obtaining the best representation of the shape. 2) Vortex Element Method is a highly suitable numerical technique for the aerodynamic evaluations of a number of off-spring designs of Genetic Algorithm. 3) The methodology of design consisting of Bezier parameterization, VEM and GA is found to generate the expected airfoil shape reasonably well and therefore, it can be used as a tool for the aerodynamic design of airfoils. 4) The maximum value of fitness function obtainable using GA has a limit. However it is very close to one. 5) The profile developed using the present methodology can be used as an initial profile for design considering more complex physics involved in the flow.

References [1] Eppler, R, 1974, “Direct calculation of airfoil from pressure distribution,” NASA TT F-15, Vol. 93, pp. 247-248. [2] Selig, M. S., Maughmer, M. D., 1992, “Generalized multipoint inverse airfoil design,” AIAA Journal Vol. 30, No. 11, pp. 2618-2625. [3] Selig, M. S., 1994, “Multipoint inverse design of an infinite cascade of airfoils,” AIAA Journal Vol. 32, No. 4, pp. 774-782. [4] Gopalarathnam, A., Selig, M. S., 2002, “Hybrid inverse airfoil design method for complex three dimensional lifting surfaces,” Journal of Aircraft, Vol. 39, No. 3, pp. 409-417. [5] Lewis, R., 1991, “Vortex element method for fluid dynamic analysis of engineering systems,” Cambridge University Press, Cambridge. [6] Gupta, S., Manohar, C. S., 2004, “An improved response surface method for the determination of failure probability and importance measures,” Structural Safety, Vol. 26, No. 2, pp. 123-139. [7] Zingg, D. W., Nemec, M., 2008, “A comparative evaluation of genetic and gradient-based algorithms applied to aerodynamic optimization,” REMN-Shape design in aerodynamics, pp. 103-126. [8] Giannakoglou, K. C., 2002, “Design of optimal aerodynamic shapes using stochastic optimization methods and computational intelligence,” Progress in Aerospace sciences, Vol. 38, No. 1, pp. 43-76. [9] Pehlivanoglu, Y. V., 2009, “Representation method effects on vibrational genetic algorithm in 2D airfoil design,” Journal of Aeronautic and Space Technologies, Vol. 4, pp. 7-13. [10] Krishnakumar, K., 1989, “Micro-genetic algorithms for stationary and non-stationary function optimization,” SPIE proceedings: Intelligent Control and Adaptive Systems, pp. 289-296. [11] Szlls, A., Hájek, J., and Šmíd, M., 2009, “Aerodynamic optimization via multi-objective micro-genetic algorithm with range adaptation, knowledge-based reinitialization, crowding and e-dominance,” Advances in Engineering Software, Vol. 40, No. 6, pp. 419-430. [12] Venkataraman, P., 1995, “A new procedure for airfoil definition,” 13th Applied Aerodynamic Conference, AIAA 95-1875-CP. [13] Sobieczky, H., 1998, “Parametric airfoils and wings,” Notes on Numerical Fluid Mechanics, Vol. 68, pp. 71-88. [14] Srinath, D. N., Mittal, S., 2010, “Optimal aerodynamic design of airfoils in unsteady viscous flows,” Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 1976-1991. [15] Shahrokhi, A., Jahangirian, A., 2007, “Airfoil shape parameterization for optimum Navier-Stokes design with genetic algorithm,” Aerospace Science and Technology, Vol. 11, No. 6, pp. 443-450. [16] Obayashi, S. and Takanashi, S., 1995, Genetic Optimization of Target Pressure Distributions for Inverse Design Methods, AIAA-95-1649-CP. [17] Tomas, K., Zavadil, L., and Doubrava, V., 2016, “CFD and surrogates-based inducer optimization,” International Journal of Fluid Machinery and Systems, Vol. 9 No.3, pp. 213-221. [18] Bian, T., Han, Q., and Bhole, M., 2017, “A Design Method for Cascades Consisting of Circular Arc Blades with Constant Thickness,” International Journal of Fluid Machinery and Systems, Vol. 10, No.1, pp. 63-75. [19] Afzal, A., Kim, K., Y., and Seo, J., 2017, “Effects of Latin hypercube sampling on surrogate modeling and optimization,” International Journal of Fluid Machinery and Systems, Vol. 10, No.3, pp. 240-253. [20] Derksen, R. W, Rogalsky, T., 2010, “Bezier-parsec; an optimized airfoil parameterization for design,” Advances in Engineering Software, Vol. 41, No. 7-8, pp. 923-930.

170