Engineering Applications of Computational Fluid Mechanics Robust ...

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Robust Multi-Objective Wing Design Optimization Via CFD Approximation Model a

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Yu Liang , Xiao-quan Cheng , Zheng-neng Li & Jin-wu Xiang

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School of Aeronautics Science and Engineering, Beihang University, Beijing 100191, China Published online: 19 Nov 2014.

To cite this article: Yu Liang, Xiao-quan Cheng, Zheng-neng Li & Jin-wu Xiang (2011) Robust Multi-Objective Wing Design Optimization Via CFD Approximation Model, Engineering Applications of Computational Fluid Mechanics, 5:2, 286-300, DOI: 10.1080/19942060.2011.11015371 To link to this article: http://dx.doi.org/10.1080/19942060.2011.11015371

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 2, pp. 286–300 (2011)

ROBUST MULTI-OBJECTIVE WING DESIGN OPTIMIZATION VIA CFD APPROXIMATION MODEL Yu Liang, Xiao-quan Cheng*, Zheng-neng Li and Jin-wu Xiang

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School of Aeronautics Science and Engineering, Beihang University, Beijing 100191, China * E-Mail: [email protected] (Corresponding Author) ABSTRACT: A robust multi-objective wing design optimization procedure using CFD is presented in this work. Instead of directly applying CFD to wing design optimization, a Kriging aerodynamic model that approximates CFD result is proposed to save computational cost. By introducing 6σ robust approach as a sub-objective function, multiobjective wing design problems can be solved with aerodynamic robustness. Non-Uniform Rational B-Spline curves (NURBS) are introduced to depict the wing geometry in design optimization process. The most important parameters of the wing geometry can then be screened by design of experiment and treated as design variables due to the local variation property of NURBS. Under this formulation, a multi-objective genetic algorithm based on nondominated sorting is employed to handle multiple flight conditions in wing design. The optimum wing geometry is selected according to trade-off among design objects on the non-dominated front. To validate the proposed approach, two robust design optimization cases are studied for ONERA-M6-Wing. It turns out that the drag coefficient is insensitive to Mach number between Ma0.8–Ma0.9 after robust optimization. The result also verifies the effectiveness of our method in boosting performance and robustness of multiple flight conditions: the lift curve slope (Ma0.3) of optimum wing and the 6σ objective function of transonic drag (Ma0.8–Ma0.9) increase by 11.9% and 25.4%, respectively. Observations in the optimization cases are concluded as follows: 1) For wing aerodynamic design problems, the multi-objective robust optimization can both improve the performance at different flight conditions and provide robustness. 2) The Kriging aerodynamic model derived from CFD result can satisfy the precision requirement of wing design. 3) Even though the optimization result is subject to the weights of the 6σ function in sub-object, the influence is not comparable to the trade-off among design objects. 4) The optimum solution obtained by proposed approach is superior to gradient based optimization method, while the computational cost is acceptable. Keywords:

robustness, NURBS, non-dominated, Kriging model, multi-objective

a given range of Mach numbers can be achieved. Zhong et al. (2008) used a multi-objective estimation of distribution algorithm to minimize mean value and variances of airfoil drag coefficient. Shimoyama et al. (2008) improved 6σ robust approach. Coupled with multi-objective evolutionary algorithms, the novel 6σ approach demonstrates practical and efficient capability to reveal tradeoff information considering both optimality and robustness in design. This method has been applied to airfoil design for Mars Airplane. In recent years, more multi-objective design problems considering practical flight conditions in airplane wing design process are studied. Long et al. (2008) adopted collaborative optimization strategy and weighted method to deal with multiobjective functions of wing aerodynamicstructural design problem. Wang et al. (2008) proposed a multi-objective optimization method for civil transport wing to improve lift-to-drag ratio and inner volume. Although robust design method has been successfully applied in

1. INTRODUCTION Wing aerodynamic design optimization plays an important role in airplane design process. Traditional wing aerodynamic design relies on point-wise design, where designers only focus on performance at a certain operation condition. This method has one congenital disadvantage: it cannot guarantee consistently good performance of aerodynamic force under disturbance of flight condition around the design point. Although multi-point design can obtain better performance for a given set of design points, its effectiveness declines between the regimes of design points (Painchaud-Ouellet et al., 2004). To obtain satisfactory aerodynamic performance around offdesign points, various robust design methods have been proposed in literature. Li et al. (2001) introduced a robust optimization scheme which could adaptively adjust the weights in an optimization formulation to find a drag reduction direction for all design conditions. By using this method, a consistent drag reduction of airfoil over

Received: 15 Sep. 2010; Revised: 29 Dec. 2010; Accepted: 25 Jan. 2011 286

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 2 (2011)

2. WING PARAMETERIZATION BASED ON NURBS

aerodynamic design, there are few cases where multiple flight conditions are simultaneously considered. This problem is challenging because: 1) multi-objective design becomes complicated when taking robustness into account; and 2) many robust approaches are relatively difficult to extend to multi-objective problems. In this paper we propose solutions to two problems in robust multi-objective wing design: 1) how to suppress additional computational cost caused by offdesign conditions; and 2) how to apply robust design method in multiple flight condition, which is common in actual flight. The 6σ robust approach uses the mean value and variance of performance to characterize the design object. Shimoyama et al. (2007, 2008) successfully applied 6σ robust approach in airfoil design. They combined 6σ method with multiobjective algorithm where the mean and variance of performance were optimized through two separate design objects. This method, however, can only satisfy wing (or airfoil) design problem at single flight condition. Complexity degrades its value in problems involving multiple flight condition. In this paper, 6σ robust function is expanded and the whole function combined with mean and variance is treated as a sub-objective function in multi-objective design problems. Applying 6σ robust approach in realistic wing design problems considering multiple flight conditions is also discussed here. To parameterize wing geometry with a set of design variables, an appropriate wing section parametric model is required for robust wing design. Since its precision has direct impact over the design and optimization results, a model which can depict geometry accurately with a few number of design variables is highly desirable. The application of NURBS representation to robust wing design is discussed in this paper. By introducing 6σ robust method and NURBS representation, a genetic algorithm based on nondominated sorting and crowding distance comparison is utilized to solve the wing design problem subject to multiple flight condition and robustness requirement. Kriging approximation model is adopted to reduce the additional computational cost caused by robust design. Two robust wing design cases based on ONERA-M6Wing are investigated to validate the effectiveness of these methods.

In this section, wing parameterization is presented for planform and wing section. Planform parameters are used to describe planform geometry such as sweep angle, aspect ratio, etc. Canonical wing planform parametric depiction is shown in Table 1. Wing geometry is given in Fig. 1. Table 1 Design parameters of a typical wing. Type

Name of parameter

Symbol

Wing planform parameters

Wing area Sweep angle of leading edge Aspect ratio Taper ratio

S ΛLE AR λ

Wing section parameters

Twist angle of section N Thickness ratio of section N Control points and weights in section N

εN (t / c)N

Fig. 1

xN , yN , wN

Part parameters of typical wing geometry. (Cr : Chord length of root; Cr = 2S / [b(1+)]; Ct : Chord length of tip; Ct =  Cr ; b: wing span, b  AR  S ).

There are types of parametric models to represent wing section geometry in literature. Sobieczky (1998) chose basic parameters (e.g., leading edge radius, upper crest location) to characterize airfoil in a 6th-order polynomial function (called PARSEC). Hicks and Henne (1978) proposed to represent airfoil by baseline profile and shape functions. Values of the participation coefficients (design variables) are used to determine the contribution of the shape functions. Both PARSEC and Hick-Henne approaches have a small number of design parameters, but suffer from one significant drawback: they cannot 287

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NURBS sections. Each section is determined by the control points and weights of NURBS respectively. Note that it is desirable to reduce the number of design variables: if all NURBS parameters in each wing section are considered, the total number of design variables will sum up to more than 100, which makes the optimization intractable. For this reason, design of experiment (DOE) is carried out and we only focus on those most important wing section parameters.

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guarantee geometry representation accuracy. NonUniform Rational B-Spline curves (NURBS) technique uses B-Spline and control points to combine and describe complex geometric shape with a relatively small number of variables (Piegl and Tiller, 1995). NURBS has shown far higher accuracy than other methods and it can reduce noise in geometric representation (Lépine J and Trépanier, 2000 ; Lépine et al., 2001; Kumaono et al., 2006), which is important in design optimization. NURBS has been applied to airfoil optimization problems recently (Srinath and Mittal, 2010). Furthermore, NURBS can be embedded in isogeometric analysis to refine mesh and enforce compatibility of the discretization at the interface of rotating components (Hughes et al., 2005, Bazilevs and Hughes, 2008). A pth-degree NURBS curve is defined as: n

C (u )   Ri , p  u Pi

0  u 1

(1)

0  u 1

(2)

i =0

and Ri , p  u  

N i , p (u ) wi n

 N j , p (u )wi j 0

Fig. 2

where {Pi} are the control points with coordinates (xi , yi), {wi} corresponding to control points’ weights, and u is the relative position parameter of the curve which usually resides in interval [0,1]. {N i,p (u)} are the pth-degree B-spline basis functions. In wing section representation, p=3.

Wing section representation by NURBS control parameters.

3. ROBUST WING DESIGN OPTIMIZATION BASED ON 6 APPROACH 3.1 6σ robust approach and wing robust design problem

The basis function N i , p (u ) is founded by knot      vector U   0, ,0, u , , u , 1 , , 1  m  p 1  , which  p 1    p  1 p  1   is calculated by the following equation:

Robust design is defined as a design that is insensitive to external noises or tolerances (Park et al., 2006). Robust design methods can be classified into Taguchi method, robust optimization and robust design with the axiomatic approach. Robust optimization formulates the problem via an objective function and achieves robustness by finding the optimum, which is more suitable for wing design optimization problem in contrast to other methods. Design for six sigma (6σ) method is one such technique. In 6σ method, approaches from structural reliability and robust design with philosophy of 6σ are combined to facilitate comprehensive probabilistic optimization (Koch et al., 2004). Conceptually, 6σ means quality level should satisfy the limitation of defective parts per million parts manufactured. Variance and standard deviation from object mean are used to depict performance variation. The sum of mean and variance is

(3)

In order to achieve higher accuracy, an airfoil’s upper and lower curves are parameterized separately. Both curves use control points and corresponding weights as parameters as shown in Fig. 2. This method can achieve better results especially when the airfoil is complicatedly shaped. The whole parameter set of a typical wing is shown in Table 1, Fig. 1 and Fig. 2. The wing surface is generated by linear lofting among four 288

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treated as the objective function with robustness, as is shown in Eq. (4): minimize:

w  f  w  2f

where CL* is the given lift coefficient in transonic regime. In Eq. (8), there is no lower bound constraint as Eq. (5), as drag coefficient is expected to be as low as possible in wing aerodynamic design.

(4)

where  f is the mean value of the objective function, and  2f is the variance of the objective

3.2 Multi-objective optimization for robust wing

function, w and w are weight factors. Eq. (4) is subject to a quality constraint:

In recent years, more multi-objective problems for wing aerodynamic design have been studied considering actual flight conditions. The problem can be written as:

 f  n f  LSL

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 f  n f  USL

(5)

with USL and LSL as upper and lower bounds of acceptable solution. n is the sigma level defined by users. The sigma level n in Eq. (5) refers to the required robustness quality. Larger value of n means the standard deviation of performance.  f satisfies stricter constraint, i.e. the higher the sigma level n is set, the better the robustness of the optimum should be expected. By the philosophy of 6σ design, the default value of n is 6 in this paper. In the wing aerodynamic design problem, the disturbance of design point (or flight condition) may unpredictably deteriorate the performance of wing. A typical example is the transport jet in which the drag divergent may happen when the flight speed exceeds the design condition in transonic regime. Therefore, it is essential to preserve wing insensitivity to the disturbance of flight condition throughout the design procedure. In this paper, the Mach number Ma of the transonic cruise condition is considered as an uncertain variable in robust design and Ma is assumed to follow a certain distribution in the range [Mamin , Mamax] with a probability density function  [Ma]. Given the drag coefficient under a given cruise lift as the target performance, the mean and variance of CD under the variation of Ma can be expressed as:

C  

Mamax

2 

Mamax

Mamin

D

CD

Mamin

CD ( D, Ma, )v  Ma dMa

(6)

C

(7)

D

( D, Ma, )  CD

 v  Ma dMa 2

w CD  w  C2

subject to:

C  n D

2 CD

x  R m   x g j ( x )  0, j  1, 2, , p

Eq. (4) is expanded to be a sub-objective function  i (x). Each flight condition is expressed by only one objective function. In other words, one objective-function  i (x) stands for the requirement of one flight condition. Hence, it is much easier to optimize and balance among different design objects. In aircraft design, ideally, an airplane should feature short take-off distance, high climbing rate and cruise efficiency. Aiming at these goals, two objects are therefore involved: 1) the lift curve slope at low speed; and 2) 6σ object function of drag coefficient in transonic regime. The design problem is: minimize:

  x   [1 ( x ),2  x ]

where

1 ( x ) 

(10)

1 CL

2 ( x )  w C  w  C2 D

 USLCD

CL  x ,Ma,   C

subject to:

(9)

treated as objective functions separately. This method, however, can only handle single flight condition and it suffers from complexity when multiple flight conditions are involved in the problem: the trade-off between performances at multiple flight conditions and two robust object (  f and  2f ) becomes difficult. In this paper,

(8)

D

  x   [1 ( x ),2  x  ,,n ( x )]

where  (x) is the vector of multi-objective functions and g j (x) is the constraint. In approach proposed by Shimoyama et al. (2007, 2008), 6σ robust design was solved by multiobjective optimization, where  f and  2f were

The robust wing design problem in transonic flight condition then is: minimize:

minimize:

D

subject to: 1) x L  x  xU 2) CD  6 C2D  USLCD

 L

for Mamin  Ma  Mamax

for Mamin  Ma  Mamax 289

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As an appropriate multi-objective optimization algorithm, non-dominated optimization is adopted to solve multi-objective wing design problems. Non-dominated genetic algorithm can accomplish a set of non-dominated solutions, from which the optimum solution can be selected. In contrast to traditional methods, non-dominated method finds optimum solutions that are related to different design tendencies in single optimization process, which avoids redundant computation. Non-dominated solution can be defined as (Coello et al., 2007): if and only if there is no solution U such that i , i (U )  i V 

design process. CFD method based on NavierStokes integration has high precision in evaluating aerodynamic force and ability in accounting viscous effect, which has been applied in many engineering projects (Roberts and Cui, 2010; Yang et al., 2010; Ramakrishna and Govardhan, 2009). In this work, a compressible Navier-Stokes solver with second order upwind scheme is employed in the wing aerodynamic computation. Spalart-Allmaras (S-A) oneequation turbulence model is designed for aerospace application. S-A model has been shown to have satisfactory predictions of boundary layers in pressure gradients and rapidly converge to steady states (Spalart and Allmaras, 1992). Hence, S-A model is selected to account for the viscous effects here. Besides, C-H structural grid is introduced to save computational cost. The grid distribution is 200 (flow direction) × 55 (span direction) × 45 (normal direction). Fluent commercial software is used during the CFD computation. Fast convergence of residual and aerodynamic force is shown in Fig. 3 and Fig. 4 shows the comparison of pressure coefficients between CFD and wind tunnel experiment results of ONERA-M6-Wing (Schmitt and Charpin, 1979) at Ma=0.8395, α=3.06°. Both of the CFD results agree well with experimental data.

(11)

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i , i (U )  i V  , i  1, 2, n where U, V ∈ Rm. The vector V is called nondominated solution or Pareto solution. According to Eq. (11), the non-dominated solution V is in fact an optional optimum for design problem. By genetic algorithm, a set of non-dominated solutions can be found in design space. This aggregate is usually on the frontal edge of feasible solutions. Therefore, the aggregate of non-dominated solutions is also called non-dominated front or Pareto front. By implementing non-dominated optimization, the wing design problem is interpreted as to find Pareto optimums of wing geometry such that objective performances of other geometry solutions are not superior to them. After optimization, design results can be chosen from those wing non-dominated optimums according to design tendency. We adopt the non-dominated algorithm NSGAII (Deb et al., 2002) to reduce computational complexity by fast non-dominated sorting and get better spread of Pareto solutions by crowding distance sorting. Additionally, NSGAII can better converge near the true nondominated (Pareto) front and has lower computational cost compared to other Pareto evolution algorithms. Details concerning the process of NSGAII can be found in the reference (Deb et al., 2002). 4. AERODYNAMIC MODEL AND ROBUST WING DESIGN PROCESS

During wing design process, some complex flows need to be handled, e.g., shock effect in transonic regime. To accurately evaluate the aerodynamic force, high fidelity solver is required during wing

Fig. 3

290

Convergence history of residuals and aerodynamic force in CFD analyses.

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 2 (2011)

Fig. 4

Pressure coefficient comparison of ONERA-M6-Wing.

Since in robust wing design we have to figure out the variance and mean of random variable distributed along a design range, it costs additional computation compared to normal design. Direct application of NS solver in optimization ends in formidable computational cost. Thus an aerodynamic model with high fidelity and low computational cost is expected. Kriging model is used to approximate deterministic computer analysis, which has local approximation property and high accuracy in nonlinear problems (Simpson et al., 1998). This model is integrated in robust wing design to replace the CFD computation of each feasible solution during optimization, which results in significantly suppressed computational cost caused by robust optimization. During initialization of Kriging model, the performances of wing samples determined by design of experiment (Koch et al., 1999) are required. To ensure fidelity, aerodynamic forces of those wing samples are computed by Navier-Stokes solver. These CFD results together with the design variables of wing samples are used to construct wing Kriging model. In Kriging model, the relationship of response and design variables can be defined as (Ryu et al., 2002): y  x   f  x   Z ( x)

E  Z  x    0

(13)

Var  Z  x     2

(14)

Cov  Z  x i  , Z  x j     2 R  R ( x i , x j ) 

(15)

In Eq. (15), R is the correlation matrix, and R(x i, x j ) is the correlation function between two sample data points x i and x j . R(x i, x j ) can usually be expressed by a Gaussian correlation function: R ( x i , x j )  exp[ kdv1 k xki  xkj ] 2

n

(16)

where ndv is the number of design variables,  k are the unknown correlation parameters used to fit the model. The predicted response yˆ is then written as: yˆ  ˆ  r T ( x ) R 1 (y  1ˆ )

(17)

where y is the column vector of sample response. r T ( x ) is the correlation vector between untried x and the sample points x1 , , x ndv . ˆ is a





constant coefficient estimated by:

ˆ   f T R 1f  f T R 1y 1

(12)

(18)

The best Kriging model is found by solving  k in an optimization problem:

where y(x) is the performance response of design variables and f(x) is a known function which is usually set to constant. Z(x) is one realization of a stochastic process, which satisfies the following constraints:

maximize: 

291

[ns ln(ˆ 2 )  ln R ] 2

(19)

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 2 (2011)

where, ˆ 2 is the estimate of variance between global model ˆ and y. Multiple Flight conditions

variables. The sample selection also follows DOE. Evaluated by NS solver, the aerodynamic force of wing samples are utilized to constitute a Kriging model. 6σ robust function is treated as a sub-object in wing multi-object design problem. NSGAII computes objective functions using force obtained from Kriging model and searches for the non-dominated solutions (Pareto front) by nondominated and crowding distance sorting. Finally, the optimum wing is selected from the nondominated solutions according to the design tendency.

Wing parameterization and screening by DOE

Generate of Wing samples according Latin Hypercubes DOE

Aerodynamic forces evaluation by NS solver at each flight condition

5. EXAMPLES OF ROBUST WING DESIGN OPTIMIZATION

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Build Kriging approximation model for each flight condition

Two robust wing design optimization cases are conducted to verify the effectiveness of the proposed methods. Both cases use ONERA-M6Wing (Fig. 6) as original wing during optimization. The wing sections at span positions y/b=0, 1/3, 2/3, 1 are parameterized by NURBS and design variables including platform and four sections’ profiles (see Table 1) are optimized. Because the original section of ONERA-M6Wing is symmetric, only the parameters of half section are considered. By applying design of experiment to the upper surface of each wing section, three weight parameters are selected as design variables (shown in Fig. 7). These weight

Initialize the population of feasible solutions, t=1

For flight condition required robust design, compute 6 function and constrains; for other flight condition, evaluate object functions and constrains

Non-domiated sorting and crowding distance sorting for the population

Selection, crossove and mutation to generate next population

Population t+1

AR = 3.8 λ = 0.56 ΛLE = 30°

If t=Nconvergence

Section 3

No

Section 2

Yes Obtained Pareto front of the design problem

Section 1 Select optimun wing geometry from non-dominated from by designer's tendency

Fig. 5

Section 0

Robust wing design process. Fig. 6

The flowchart for robust wing design process considering multiple flight conditions via Kriging model is shown in Fig. 5. Wing geometry is represented by a set of design parameters using NURBS method. Part of design parameters which have significant effect on wing aerodynamic performance are selected as design variables by the design of experiment (DOE) screening. A set of wing design samples is selected from the design space which is constituted by design

Fig. 7

292

Geometry of ONERA-M6-Wing.

Design variables selected from the NURBS curve of wing section.

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 2 (2011)

parameters play the most significant roles in determining the aerodynamic performance. As the result, for ONERA-M6-Wing, there are a total of 22 design variables selected. These design variables compose 250 wing geometries as samples by design of experiment. The aerodynamic characteristics of these 250 samples are analyzed and formulated by Kriging model. For ONERA-M6-Wing, the drag divergence occurs when the speed is between Ma0.8–Ma0.9. The purpose of robust design is to reduce the drag divergence at given lift coefficient. The random variable Ma is assumed to uniformly distribute in the regime. Therefore Ma=0.8, 0.82, 0.84, 0.86, 0.88 and 0.9 are picked as the samples. A Kriging model is formulated at each sample point of Ma. The mean and variance of aerodynamic performance in Eq. (6) can be simplified by statistical formulation:

C  D

2  CD

ntran 

1 ntran 1 ntran

 C  D, Ma ,  D

ntran

(20)

k

k 1

 C  1  

D

k 1

Both weights for mean and variance are set to 1.0 here as both the mean and variance of CD are regarded as equally important. To solve this single-objective problem, multi-island genetic algorithm is adopted. The population size is 100 (10 islands and 10 populations in each island) and the number of generations is 100. Table 2 and Fig. 8 demonstrate the performance improvement after optimization. The objective function considering mean and variance of drag coefficient decreases by 29.2%. The drag coefficient of optimum has been reduced significantly in the regime Ma0.8–Ma0.9. Since the drag coefficient curve becomes “flat” after optimization, the aerodynamic performance of optimum wing insensitive to Mach number is accomplished. The geometry of single-objective optimization result is shown in Fig. 9, Fig. 10 and Table 3. In contrast to ONERA-M6-Wing, the optimum solution has following geometry properties: 1) the sweep angle and the aspect ratio are higher than ONERA-M6-Wing; 2) the sections near the inner part of the wing becomes thinner; 3) the optimum wing sections have negative twist and the twist angle becomes larger when the position is close to wing tip; and 4) for wing section 3, the shape near the trailing edge is thicker than the original wing.

 ( D, Ma, )  CD  

2

(21)



where C D ( D, Ma, ) is the drag coefficient of Ma samples predicted by Kriging model. n tran is the number of Ma samples, for this robust design problem n tran =6. 5.1 Robust wing design considering only transonic regime

In this single-objective design case, the target is to minimize the drag value and the trend of drag divergence in transonic regime. 6σ function shown in Eq. (16) and Eq. (17) is used to express the objective function in the drag divergence regime. The design problem is then described as: minimize:

C   C2

subject to:

C  6 C2  0.04

D

(22)

D

D

Fig. 8

D

CL  0.26

for 0.8  Ma  0.9

Drag coefficient for single-objective optimum (Ma0.8–Ma0.9 CFD).

Table 2 Optimization result and the prediction error for single-objective optimum. Objective function

ONERA-M6-Wing (CFD)

Single-objective optimum (Kriging model)

Single-objective optimum (CFD)

Kriging prediction error

Performance Improved(CFD)

C   C2

0.02282

0.01615

0.01610

0.3%

29.2%

D

D

(Ma0.8–Ma0.9)

293

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5.2 Robust wing design considering multiple flight condition

Fig. 9

In order to expand the application of the multiobjective method for robust wing design, a wing design problem considering multiple flight conditions is conducted in this subsection. The design objectives are consisted of: 1) lift curve slope at low speed (Ma0.3); 2) 6σ object function (shown in Sec. 5.1) considering drag coefficient in transonic regime. From Eq. (10), the multiobjective problem is:

Wing sections’ geometry for single-objective optimum.

1 ( x ) 

1 CL

(at Ma=0.3)

2 ( x )  1.0C  1.0 C2 D

D

(23)

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subject to: 1) x L  x  xU 2) CD  6 C2D  0.04 for 0.8  Ma  0.9

3) CL  0.26 for 0.8  Ma  0.9 where the values of wμ and wσ are set to 1.0, which is the same as the design case in Sec. 5.1. NSGAII is carried out to solve this multiobjective problem. The population size is 100 and number of generations is set to 50. The optimization history is shown in Fig. 11. Fig. 12 shows the non-dominated solutions (Pareto front) obtained by NSGAII. The optimum solution is selected from the Pareto front. The selected optimum tends toward the robust function of drag coefficient because the robustness of drag coefficient at transonic regime is relatively important than lift curve slope at low speed in this case. Fig. 10 Comparison of planform geometry and twist between optimum wings and ONERA-M6Wing. Table 3 Comparison of design variables (multiobjective optimum). Design Variables

ONERAM6-Wing

ΛLE (°) AR λ ε1 (°) ε2 (°) ε3 (°) (t / c)0 (t / c)1 (t / c)2 (t / c)3

30 3.8 0.562 0 0 0 0.1 0.1 0.1 0.1

Single -objective optimum 34.98 4.1 0.541 -1.484 -2.448 -2.510 0.075 0.075 0.098 0.091

Multi -objective optimum 27.42 4.2 0.566 -4.692 -4.384 -5.020 0.078 0.075 0.075 0.103

Fig. 11 Optimization history of multi-objective robust wing design. 294

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Fig. 12 Pareto solutions for multi-objective optimum.

Fig. 13 Drag coefficient for multi-objective optimum (Ma0.8–Ma0.9, CFD).

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Table 4 Optimization result and the prediction error for multi-objective optimum.

Objective function

ONERA-M6-Wing (CFD)

Multi-objective optimum (Kriging model)

Multi-objective optimum (CFD)

Kriging prediction error

Performance improvement (CFD)

CL /° (Ma0.3)

0.06103

0.06825

0.06832

0.1%

11.9%

0.02282

0.01682

0.01703

1.3%

25.4%

C   C2 D

D

(Ma0.8–Ma0.9)

causes additional transonic drag and larger twist distribution is required to reduce this drag at the given lift coefficient. Fig. 15 shows the pressure distributions at different span positions in transonic regime (Ma0.88, CL =0.26). For both multi-objective and single-objective optimums, although the positions of shock waves move toward upstream a little, the intensities of critical shock waves are reduced which leads to smaller drag coefficient.

The improved performance of multi-objective optimization is shown in Table 4. Both performances at low speed and transonic flight conditions are boosted. The lift curve slope at Ma0.3 has been raised by 11.9%, and the robust function at Ma0.8–Ma0.9 has been improved by 25.4%. The error between CFD and Kriging results is less than 5%, which can satisfy the wing optimization problem even when the number of design variables is up to 22. The drag divergence of optimums and original wing resolved by CFD is shown in Fig. 13. After multi-objective optimization, the drag coefficient of optimum is much more insensitive to ONERA-M6-Wing, which is similar to single-objective solution. The optimum geometry of multi-objective optimum is shown in Fig. 10 and Fig. 14. Different from optimum in single-objective problem, the sweep angle of multi-objective optimum is smaller than that of ONERA-M6Wing. It is due to the fact that the multi-objective optimum must take the requirement of lift curve slope at low speed into account. From Fig. 10 and Table 3, it is observed that the multi-objective optimum has negative twist distribution and the absolute values of twist angle are larger than those of single-objective wing. This can be explained by the smaller sweep angle which

Fig. 14 Wing sections’ geometry for multi-objective optimum.

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The drag coefficient of multi-objective optimum is a little higher than single-objective optimum in transonic regime (Fig. 13). The reason can be found from the geometries comparison in Fig. 10. Unlike ONERA-M6-Wing, the smaller sweep angle of multi-objective optimum improves lift curve slope at low speed, but it also leads to lower critical Mach number and larger critical wave drag. Since the single-objective optimum only needs to reduce the drag coefficient at given lift coefficient in transonic regime, the sweep angle is larger than that of ONERA-M6-Wing, which results in lower critical wave drag. The reason can also be found from the pressure coefficient shown in Fig. 15: due to the smaller sweep angle, the intensity of critical wave for multi-objective optimum is larger than single-objective optimum, which leads to higher critical drag. The different performance between multi-objective and singleobjective optimum reveals that the multiobjective optimum is actually a tradeoff between the two objects.

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(a) y/b =0%

5.3

Computational cost and weight influence

In order to compare the optimization approaches in wing design problem, a typical gradient-based method, sequential quadratic programmingNLPQL (Schittkowski, 1985), is investigated in multi-objective robust wing design. Furthermore, typical wing optimizations without robust design are also evaluated for benchmark comparison. Since NLPQL can only process one objective function in optimization, the objective functions in Eq. (19) are combined into single function by weighted sum method (Ehrgott, 2005) when NLPQL is adopted:

(b) y/b=50%

  w11  w 22

where w 1 and w 2 are weight factors,  1 and  2 are object functions of Eq (19). All computation to solve wing optimization problems in this work is operated on a PC with 8 pieces of Intel Core2 E6300 CPU. The comparison of computational costs among different approaches is shown in Table 5. For the limitation of computation hardware, executing the whole optimization cases without approximation model will become unacceptable, especially those cases adopting genetic algorithm. When the same optimization and robust design method is implemented, the number of iterations for convergence of those approaches without Kriging model is almost identical to those with Kriging model. One method’s computational cost without Kriging model is therefore estimated by the

(c) y/b =99% Fig. 15 Pressure coefficient of wing sections at different span position. Table 5 Computational cost comparison of different optimization strategies. Wing optimization method

Computational cost (hours)

NSGAII+Kring+6σ

627

NSGAII+6σ

12500

NSGAII+Kring

191

NLPQL+Kring+6σ

625

NLPQL+6σ

860

NLPQL+Kriging

190

(24)

296

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multiplication of CFD analysis cost in each feasible solution and the number of iterations from its counterpart with Kriging model. From Table 5, it can be found that: 1) when robust design is included, additional computational cost is unavoidable; 2) Kriging model remarkably reduces the computational cost, especially in genetic algorithm; 3) the computational cost of NSGAII+6σ with Kriging model is less than that of gradient based method without the model; and 4) when using Kriging model, the difference between NSGAII and NLPQL in computational cost is negligible, because the most timeconsuming part is the samples’ analysis during Kriging model’s construction. Although gradient based methods costs less computation, genetic algorithm method based on non-dominated optimization outperforms it in terms of optimum, as shown in Fig. 16 and Fig. 17. Since gradient based optimization is prone to local optimum, the drag coefficient produced by NLPQL cannot obtain robustness even when 6σ method is adopted. From Fig. 17, it can be found that although NLPQL can achieve different optimums by varying weights in weighted sum function, robust optimums discovered by NSGAII are still superior to those by NLPQL. In addition, the Pareto solutions from NSGAII are a set of optimums which stand for different results of trade-off. That is, if the design tendency is changed, designer can re-choose optimums from the Pareto solution without re-optimization. Traditional gradient-based method, however, have to use weighted sum method to deal with multiobjective problem: if designers want to figure out optimums corresponding to different design tendency, re-optimization must be executed and it introduces additional computational costs especially in optimization without approximation model.

Fig. 17 Optimum solution comparison of different optimization approaches.

(a) wμ =0.5, wσ =1.5

(b) wμ =1.5, wσ =0.5 Fig. 18 Pareto solutions of different 6 weight value.

To investigate the influence of weight values of 6σ function, we further conduct two additional multi-objective optimizations from Eq. (10): the design object 1 ( x )  1 / CL (at Ma=0.3) remains unchanged while weights of design object 2 ( x )  w C  w  C2 are altered. The two sets of D

D

weights are: 1) wμ =0.5, wσ =1.5; and 2) wμ =1.5, wσ =0.5. In order to compare the results at the same condition of trade-off, the value of low speed CLα is chosen as 0.06825 when the optimum solutions are selected from Pareto front, which is shown in Fig. 18. The drag coefficients with different weights (predicted by Kriging model) are shown in Fig. 19. With respect to the

Fig. 16 Robustness comparison of different optimization approaches (Kriging model is included). 297

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original problem ( 2 ( x )  1.0C

 1.0 C2D

D

reduced by selecting the most important NURBS parameters at each wing section. Through approximating the relationship between CFD results and variables with Kriging model, directly using CFD in optimization is avoided. Additional computational cost caused by robust design can be reduced to acceptable level. After robust wing optimization with proposed method, the improvements of lift curve slope at low speed and the robustness of drag coefficient in transonic regime have been achieved. The errors between Kriging model and CFD results in cases is less than 5%, which is acceptable in robust wing optimization. The wing optimum solution obtained by proposed method is more superior and robust than that obtained by other wing optimization approaches. By investigating two additional cases in multiobjective robust wing optimization, we found that: weights wμ and wσ in the 6σ function influence the performance and robustness of multi-objective optimum results differently. Larger wμ leads to better performance but worse robustness, while larger wσ means more robustness yet worse performance. The influence of wμ and wσ is not comparable to that by the trade-off on Pareto front.

), the weight

with wμ =0.5, wσ =1.5 better smoothes the drag curve and leads to smaller  C2 ; however the drag D

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value becomes a little higher in the majority of Ma range. In contrast, the weights with wμ =1.5, wσ =0.5 slightly lower the drag value over the original case in the majority of Ma range, yet the drag curve becomes steeper. In summary, at the same condition of trade-off between design objectives, the weight values of wμ and wσ in 6σ function influence the performance and robustness of sub-object respectively. But the effect by weights is not comparable to that of trade-off on Pareto front. Larger value of wμ brings in better performance but worse robustness; while larger value of wσ improves robustness but degrades performance.

NOMENCLATURE

AR b C CD

Fig. 19 Drag coefficient comparison of Pareto optimums with different 6 weight value (CLα =0.06825).



CD

CL CL Cp C(u) f

6. CONCLUSIONS

By adopting six-sigma expression in subobjective function, aerodynamic robust design problems considering multiple flight conditions can be solved. Multi-objective genetic algorithm based on non-dominated sorting can handle robust objective functions without gradient computation, which simplifies the robust multi-objective optimization. The non-dominated solutions (Pareto front) obtained by the genetic algorithm correspond to potential wing optimum solution. So wing design result can be chosen on this nondominated front according to tradeoff among design objects, re-optimization like gradientbased method is not needed. NURBS representation method insures the accuracy of geometry. Combining with design of experiment, wing design variables can be significantly

gj (x) LSL Ma m n ndv nS ntran Ni, p(u) Pi p Ri, p(u) 298

aspect ratio wing span [m] chord length drag coefficient drag coefficient predicted by Kriging model lift coefficient lift curve slope pressure coefficient NURBS curve function column vector filled with ones in Kriging model constraint function lower bound of acceptable solution Mach number dimension of knot vector in NURBS number of control points in NURBS number of design variables in Kriging model number of sample points number of Ma samples basis function in NURBS (order p) control point position in NURBS polynomial order in NURBS NURBS interpolation function

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R(x i, x j) correlation function between point x i, x j in Kriging model correlation matrix in Kriging model R T correlation vector in Kriging model r (x) wing area [m2] S (t/c)i thickness ratio of wing section feasible solution U relative position parameter in NURBS u upper bound of acceptable solution USL non-dominated solution V weight factor of mean value w weight factor of variance value w weight factor of sub-object function w i xi ,yi ,wi coordinates and weight of control point in NURBS sampled data point xi vector of design variables x x coordinate (chord direction) made X/C non-dimensional by chord length on wing section y coordinate (thickness direction) made Y/C non-dimensional by chord length on wing section y/b relative location in span direction predicted response value yˆ column vector of sample’s response y

k  2 ˆ 2

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Abbreviation

CFD DOE NURBS 6σ

computational fluid dynamics design of experiment non-uniform rational B-spline curves six sigma

Subscripts

L D t r tran dv LE

lift drag wing tip wing root transonic regime design variables wing leading edge

Symbols

ˆ i  i     [Ma]

unknown correlation parameters in Kriging model standard deviation variance estimate of variance

predicted constant underlying global portion of Kriging model twist angle of wing section object function sub-object function wing taper ratio wing sweep angle [°] mean value probability density function of Ma 299

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