Chinese Journal of Aeronautics · Keywords: optimization; genetic algorithm; double-stage...

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Chinese Journal of Aeronautics

journal homepage: www.elsevier.com/locate/cja

Chinese Journal of Aeronautics 24 (2011) 568-576

Double-stage Metamodel and Its Application in Aerodynamic Design Optimization

ZHANG Dehua, GAO Zhenghonga,*, HUANG Likenga, WANG Mingliangb aNational Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical

University, Xi’an 710072, China

bNational Key Laboratory of Science and Technology on Test Physics & Numerical Mathematical, Beijing 100076, China

Received 28 September 2010; revised 20 November 2010; accepted 22 March 2011

Abstract

Constructing metamodel with global high-fidelity in design space is significant in engineering design. In this paper, a dou-ble-stage metamodel (DSM) which integrates advantages of both interpolation metamodel and regression metamodel is con-structed. It takes regression model as the first stage to fit overall distribution of the original model, and then interpolation model of regression model approximation error is used as the second stage to improve accuracy. Under the same conditions and with the same samples, DSM expresses higher fidelity and represents physical characteristics of original model better. Besides, in order to validate DSM characteristics, three examples including Ackley function, airfoil aerodynamic analysis and wing aerody-namic analysis are investigated. In the end, airfoil and wing aerodynamic design optimizations using genetic algorithm are pre-sented to verify the engineering applicability of DSM.

Keywords: optimization; genetic algorithm; double-stage metamodel; Kriging; BP neural network; Latin hypercube; Parsec method

1. Introduction1

In engineering design, the increasing demand of high-fidelity computational simulations yields complex simulation codes with a growing requirement of com-putational cost. Despite the considerably improved computer power over the past few decades, high- fidelity computational simulations are still prohibitive for design optimization. To address this problem, there have been many researches of metamodel [1-2] .

Metamodel, called ‘model of the model’, is widely used in engineering design to replace the existing analysis code while providing a better understanding of the relationship between design variables and per-formance parameters, easier integration of domain *Corresponding author. Tel.: +86-29-88495971. E-mail address:[email protected] 1000-9361 © 2011 Elsevier Ltd.doi: 10.1016/S1000-9361(11)60066-6

dependent computer codes, and fast analysis tools for optimization and exploration of the design space by using approximations in lieu of the computationally expensive analysis codes [2].

Metamodeling fidelity is mainly affected by complexity of physical model, metamodeling strategy, size and dimensions of design space, amount and distribution of sample points. To construct higher fidelity model with fewer samples, many researches of meta-model application in design and optimization have been conducted in past decades [3-10].

Research contributions of metamodel can be roughly divided into three categories: multiple metamodel, multi-fidelity metamodel and metamodel correction. Multiple metamodel, also called ensemble metamodel sometimes, integrates two or more metamodeling strategies by combing optimized weight factors [6] or aggregating average value of each metamodel [7]. Multi-fidelity metamodel also called variable-fidelity or variable-complexity model captures wide attention Open access under CC BY-NC-ND license.

http://creativecommons.org/licenses/by-nc-nd/4.0/

No.5 ZHANG Dehu et al. / Chinese Journal of Aeronautics 24(2011) 568-576 · 569 ·

in recent research. It improves low-fidelity physics- based model by metamodeling its error to high-fidelity model [8-9]. Metamodel correction can be further di-vided based on the order of correction function and the second-order correction is demonstrated to be more flexible [10]. However, most researches mentioned above are focused on applying metamodel with trust- region model management (TRMM) while fewer re-searches are about how to improve global fidelity in the whole design space. In engineering design, such as design space exploration and global optimization, computational efficient global model with high fidelity is required.

According to this situation, a double-stage metamodel (DSM) is constructed by interpolation and regression models in this paper. In the whole design space, DSM expresses higher prediction accuracy with the same samples and reflects physical characteristics of original model better.

2. Optimization Strategies

There are three different types of metamodel-based design optimization (MBDO) strategies [3-4] as illus- trated in Fig.1: sequential approach, adaptive MBDO and direct sampling approach. Sequential approach is the traditional approach which fits a global metamodel and then uses the metamodel to calculate fitness value within optimization. Adaptive MBDO involves re- sampling and re-modeling in the loop, additional sam-ples are generated iteratively to update the metamodel to improve accuracy. The third approach directly gen-erates new sample points towards the optimum with the guidance of metamodel. The optimization is rea- lized by adaptive sampling alone and there is no formal optimization process. This method needs to be further tested. At the end of the paper, applying adap-tive MSDO approach, airfoil and wing aerodynamic design optimizations utilizing DSM and genetic algo-rithm [11-12] are presented to demonstrate the practica-bility of DSM.

Fig. 1 MBDO strategies.

3. Metamodel

Metamodeling involves three steps: (1) Sample points; the points sampled should reflect

general information of design space. (2) Calculate data values using physics-based model. (3) Construct metamodel and test its fidelity. Metamodeling strategies include neural networks [2,13],

Kriging [2,14-15], radial basis functions (RBF) [16], Gaus-sian process (GP) [17], polynomial response surface approximations (PRS) [2], support vector regression (SVR) [18] and so on.

In terms of the error at sample points, there are two types of metamodel: interpolation model and regression model. Fig. 2 shows their schematic diagrams.

Fig. 2 Schematic diagrams of two types of metamodels.

3.1. Interpolation model

During interpolation model construction, no error is demanded at sample points. Taking one-dimensinal polynomial interpolation model [19-22] as an example, the formula of the error at a given point x is as follows:

( 1)

c c1

( )( ) ( ) ( ) ( )( 1)!

n n

ii

fe x f x f x x xn

ξ+

=

= − = −+ ∏ (1)

(for some ξ∈(a,b)). Where n is the total number of sample points, xi∈(a,b) is the ith sample point (i= 1, 2, …, n; (a, b) is the domain). When domain and samples are determinate, ec(x) depends on the nonlinearity of the original model. Though there is no error of interpolation model at sample points, its approxima-tion error will appear large when high-order interpola-tion is needed on condition that original model has strong nonlinearity, and it may change local characters of original model (see Fig. 2(a)).

3.2. Regression model

The formula of one-dimensional linear regression model [21-23] is as follows:

· 570 · ZHANG Dehu et al. / Chinese Journal of Aeronautics 24(2011) 568-576 No.5

h0

( ) ( )m

j jj

f x c xφ=

= ∑ (2)

where 0{ ( )}mj jxφ = is linearly independent basal func-

tions. Select parameters 0{ }mj jc = to minimize

12 2

20

|| ||n

i ii

rω=

= =∑r

122

0 0( )

n m

i j j i ii j

c x yω φ= =

⎡ ⎤−⎢ ⎥

⎢ ⎥⎣ ⎦∑ ∑ (3)

where n is the number of sample points, ωi and ri = fh (xi) − f (xi) (i=0, 1, 2, …, n) are weight and ap-proximation error at the ith sample point respectively. The minimization problem is solved to get the values of 0{ }m

j jc = and then the model construction is fin-ished. The whole approximation error at all sample points is considered during regression model construc-tion, so regression model can reflect the main distribu-tion of the original model more fully. But as regression model does not require no error at sample points, it cannot represent local characters well (see Fig. 2(b)).

3.3. DSM

In engineering design, not only a high prediction accuracy but also accurate reflection of the basic char-acteristics of the original physical model is needed to meet requirements of design optimization.

DSM is constructed by interpolation model and re-gression model, and it integrates advantages of both types of metamodel. It expresses lower prediction error and reflects the characteristics of original model more accurately. Fig. 3 shows its construction framework. After sampling points in design space, regression model is used as the first stage to fit original model according to its characteristics of fitting main distribu-tion well. Its approximation error

h h( ) ( ) ( )e f f= −x x x (4)

has the local characteristics that the regression model cannot reflect. Compared with the original model, dis-tribution of eh(x) is simpler and the value is lower. Because interpolation model can fit samples accu-rately, it is applied as the second stage to surrogate regression model approximation error to improve ac-curacy, and then DSM is composed by combining the two stages. If DSM prediction accuracy satisfies de-mand, DSM construction comes to end. Otherwise, new sample points are needed. Since local characters are appended to regression model through interpola-tion model of its approximation error, DSM integrates advantages of both types of metamodel. It can not only reflect the overall distribution of original model well but also ensure local accuracy.

Fig. 3 DSM construction framework.

From DSM construction process, it can be known that DSM approximation error is as follows:

DSM DSM( ) ( ) ( )e f f= =−x x x

h hc( ) ( ( ) ( ))f f e− + =x x x

h hc h hc( ( ) ( )) ( ) ( ) ( )f f e e e− − = −x x x x x (5)

ehc(x) is interpolation model of eh(x), so there is no DSM approximation error at sample points:

eDSM(xi)=eh(xi)−ehc(xi)=0 ( i=1, 2, …, n) (6)

The equations above show that DSM approximation error is the same as second stage approximation error. It greatly depends on whether the regression model can fit the original model well. If regression model fits the original model better, distribution of eh(x) is simpler, ehc(x) more accurately, and DSM prediction accuracy is higher.

4. Validation of DSM Characteristics

In this section, three examples including Ackley function, airfoil aerodynamic analysis and wing aero-dynamic analysis are investigated to validate DSM characteristics.

Interpolation and regression models used here are Kriging and back propagation (BP) neural network respectively. Kriging is a representative of interpolation model. It is well-established, easy to use and


best suited for applications with random error, but it is just appropriate for applications with fewer factors [2]. Second-order Kringing model is employed here, as second-order correction is demonstrated to be flexible in metamodel correction [10]. BP neural network is a typical regression model. It is good for highly nonlinear or very large problems, best suited for determinis-tic applications, best for repeated application, but the computational expense is high [2].

Validation criteria used here are maximum absolute error (MAX), mean absolute error (MEAN) and % root mean square error (%RMSE).

MAX max | ( ) | ( 1, 2, , )if i k= − = Lx

(7)

1| ( ) |

MEAN

k

ii

f

k=

−=∑ x

(8)

2

1

1

1100 ( ( ) )%RMSE

1 | ( ) |

k

ik

i

fk

fk

=

=

−=

∑

∑

x

x (9)

where k is the number of validation samples, f(x) observed value and predicted value.

4.1. Ackley function

Mathematical expression of Ackley function is as follows:

2

1

1 1( ) 20 e 20exp5

M

ii

f xM =

= + − − −∑x

1

1exp cos(2 )M

ii

xM =

− π∑ (10)

where x=[x1 x2 … xM], and M the number of vari-

ables. Fig. 4(a) is the graph for M=2 with domain [−5 −5, 5 5]. This function is strongly nonlinear and has many local minimums.

1 100 points are sampled in domain [−5 −5, 5 5] by Latin hypercube sampling (LHS) [24] method. 600 points are taken as training data randomly and the other 500 ones left are regarded as validation data. Kriging, BP neural network and DSM are used to sur-rogate Ackley function respectively. Graphs of meta-models are shown in Fig. 4. Table 1 lists their ap-proximation error. Kriging can not fit overall distribu-tion well (see Fig. 4(b)), BP neural network can not represent local characters (see Fig. 4(c)) and DSM approximation error is the lowest and can reflect basi-cal distribution of Ackley function (see Fig. 4(d)). Therefore, DSM is better than Kriging and BP neural network.

Fig. 4 Graphs of Ackley function and metamodels


Table 1 Approximation error comparison of metamdels

Error Kriging BP neural network DSM

MAX 3.368 263 1.833 042 2.777 611 MEAN 0.312 660 0.511 924 0.311 007

%RMSE 5.252 162 6.705 477 5.183 605

4.2. Airfoil aerodynamic analysis

Simplified Parsec method [25] (trailing edge coordi-nate and thickness are zero) is chosen here to parame- terize airfoils, as shown in Fig. 5. The nine variables and their ranges are listed in Table 2.

Fig. 5 Simplified Parsec method.

Table 2 Design space of simplified Parsec method

Parameter Minimum Maximum

Leading edge radius rLE/m 0.006 0.010

Upper crest chordwise location xup/m 0.41 0.46

Upper crest thickness value zup/m 0.061 0.066

Upper crest curvature zxxup/m−1 −0.47 −0.42

Lower crest chordwise location xlw/m 0.28 0.38

Lower crest thickness values zlw/m 0.055 0.060

Lower crest curvature zxxlw/m−1 −0.62 −0.57

Trailing edge direction αTE/(°) 6.9 8.9

Trailing edge wedge angle βTE/(°) 6.4 8.4

Aerodynamic performance parameters are analyzed at free stream Mach number Ma∞=0.73, angle of attack α=2.79°, Reynolds number Re=6 500 000. Reynolds averaged Navier-Stokes (RANS) [26] equations are solved to get the aerodynamic performance parame-ters, including lift coefficient CL, drag coefficient CD and pitching moment coefficient Cm.

280 LHS points are sampled in design space, 250 of them are taken as training data and the others left are used as validation data. Fig. 6 compares accuracy of

Fig. 6 Comparison of prediction accuracy of three meta-models (airfoil).

Kriging, BP neural network and DSM. For any pa-rameter, DSM represents the highest fidelity according to every validation criterion. This shows the same characteristics with Ackley function and indicates that DSM prediction accuracy is higher than both types of metamodel.

4.3. Wing aerodynamic analysis

Root airfoil and tip airfoil of the wing are para- meterized by simplified Parsec method. The twist an-gle of tip airfoil and the leading edge sweep angle are considered as design variables too. So there are 20 design variables. The aerodynamic condition is as fol-lows:

Ma=0.839 5, α=3.06°, Re=1.171×107

The original physical model is numerical solution of Navier-Stokes (N-S) equations. Aerodynamic performance parameters are lift coefficient, drag coefficient and lift drag ratio K. 180 points are sampled in design space by LHS. 160 of them are selected as training data and the other 20 ones are used as validation data.

Prediction accuracy of Kriging, BP neural network and DSM is depicted in Fig. 7. All the characters shown in it is similar with Fig. 6. DSM approximation error is lower than Kriging and BP neural network. It indicates that DSM is adaptable to different-dimensional engineering problems.


Fig. 7 Comparison of prediction accuracy of three meta-models (wing).

5. Design Optimization and Result Analysis

DSMs constructed in Section 4.2 and Section 4.3 are applied in this section to optimize the airfoil and wing respectively. Standard genetic algorithm of MATLAB optimization toolbox is used here. Cross-over fraction is 0.8 and migration fraction is 0.1 within search. Fig. 8 shows the flowchart of design optimiza-tion. Two prior individuals of each generation are checked by numerical solution of N-S equations. They are appended to training data and then DSM is up-dated.

5.1. Airfoil aerodynamic design optimization

To reduce drag of the airfoil, DSM and genetic al-gorithm are used to optimize wing here. Design space is the same as that in Table 2. The optimization model

Fig. 8 Flowchart of design optimization.

is as follows: min CD s.t. CL ≥ CL0, |Cm| ≤ |Cm0|, d ≥ d0

where CL0, Cm0, d0 are lift coefficient, pitching moment coefficient and maximum airfoil thickness of initial airfoil respectively. Population size is 90 and genera-tions is 50.

Optimized airfoil and its pressure coefficient Cp dis-tribution are shown in Fig. 9 and Fig. 10 respectively. The strength of shock wave has been reduced. Aero-

Fig. 9 Comparison of airfoil shapes.


Fig. 10 Comparison of Cp distribution.

dynamic performance parameters of the airfoils are listed in Table 3. It could be seen that design result reduces the drag coefficient by 26.32 counts while satisfying all constrains.

Table 3 Comparison of performance parameters

Airfoil CL CD Cm d

Initial 0.786 770 0.017 750 −0.089 80 0.119 620

Optimized 0.787 044 0.015 118 −0.083 89 0.119 763

Δ 0.000 274 −0.002 632 0.005 91 0.000 143

5.2. Wing aerodynamic design optimization

DSM constructed in Section 4.3 is used here to maximize lift-drag ratio of the wing. The initial con-figuration has strong shock wave on the upper surface (Fig. 11(a)). The optimization model is as follows:

max K s.t. CL≥CL0, dmax root> dmax root0, dmax tip> dmax tip0

where CL0 is the lift coefficient of the initial shape, dmax root0 and dmax tip0 are relative maximum thicknesses of initial root and tip airfoils respectively. Population size is 200 and generations is 100.

Fig. 11 compares upper surface Cp contour and

Fig. 11 Comparison of Cp contour of upper surface.

Fig. 12 shows distributions of Cp at 20% and 95% sec-tion. Shapes of root airfoil and tip airfoil are compared in Fig. 13. Table 4 and Table 5 show comparisons of

Fig. 12 Distribution of Cp at 20% and 95% section.


Fig. 13 Comparison of airfoil shapes of each section.

Table 4 Comparison of aerodynamic performance pa-rameters

Airfoil CL CD K Initial 0.268 4 0.017 21 15.595 6

Optimized 0.281 3 0.015 31 18.373 6 Δ 0.012 9 −0.001 9 2.778 0

Table 5 Comparison of geometrical angles

Parameter Initial Optimized

Leading edge sweep angle/(°) 30 33

Tip airfoil twist angle/(°) 0 −2.43

aerodynamic performance parameters and geometrical angles between initial and optimized configurations. The optimized wing has a larger leading edge sweep angle and a lower tip twist angle. Both root and tip airfoils have larger camber while keeping relative maximum airfoil thickness not decreasing. Upper sur-face shock wave strength of optimized wing is greatly reduced and the lift-drag ratio is improved by 17.8 %.

6. Conclusions

(1) DSM integrates advantages of interpolation

model and regression model. It reflects original model physical characters more accurately, and it has higher prediction accuracy based on the same samples.

(2) Mathematical function and different-dimensional engineering models are investigated to validate DSM characteristics. Under three validation criteria, ap-proximation error of DSM is lower than each type of metamodel. This demonstrates that DSM has general adaptability.

(3) Airfoil and wing aerodynamic design optimiza-tions are presented in the end. Satisfactory results are obtained in both design optimizations, which indicates that DSM has strong engineering practicality.

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Biographies:

ZHANG Dehu Born in 1986, he is a Ph.D. candidate. His main research interest is aircraft design. E-mail: [email protected]

GAO Zhenghong Born in 1960, she is a professor in Northwestern Polytechnical University. Her main research interest lies in flight vehicle design and flight control. E-mail:[email protected]

Chinese Journal of Aeronautics · Keywords: optimization; genetic algorithm; double-stage...

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