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European Conference on Computational Fluid Dynamics

ECCOMAS CFD 2006

P. Wesseling, E. Oñate, J. Périaux (Eds)

© TU Delft, The Netherlands, 2006

DESIGN EXPLORATION OF HIGH-LIFT AIRFOIL USING KRIGING

MODEL AND DATA MINING TECHNIQUE

Masahiro Kanazaki*, Kentaro Tanaka

†

, Shinkyu Jeong

‡

, and Kazuomi

Yamamoto

**

* Japan Aerospace Exploration Agency,

7-44-1 Jindaiji-Higashi, Chofu, Tokyo, Japan

e-mail: kanazaki.masahiro@jaxa.jp

Web page: http://www.jaxa.jp/

†

Ryoyu systems Co., Ltd.,

2-19-13, Takanawa, Minato-ku, Tokyo, Japan.

e-mail: kentaro@chofu.jaxa.jp

‡

Institute of Fluid Science, Tohoku University

2-1-1 Katahira Aoba-ku, Sendai, Japan.

e-mail: jeong@edge.ifs.tohoku.ac.jp

**

Japan Aerospace Exploration Agency,

7-44-1 Jindaiji-Higashi, Chofu, Tokyo, Japan

e-mail: yamamoto.kazuomi@jaxa.jp

Key words: High-lift Airfoil, Design Exploration, Data Mining

Abstract. A multi-objective design exploration for a three-element airfoil consisted of a slat,

a main wing, and a flap was carried out. The lift curve improvement is important to design

high-lift system, thus design has to be performed with considered multi-angle. The objective

functions considered here are to maximize the lift coefficient at landing and near stall

conditions simultaneously. Kriging surrogate model which was constructed based on several

sample designs is introduced. The solution space was explored based on the maximization of

Expected Improvement (EI) value corresponding to objective functions on the kriging models.

The improvement of the model and the exploration of the optimum can be advanced at the

same time by maximizing EI value. In this study, a total of 90 sample points are evaluated

using the Reynolds averaged Navier-Stokes simulation (RANS) for the construction of the

kriging model. In order to obtain the information of the design space, two data mining

techniques are applied to design result. One is functional Analysis of Variance (ANOVA)

which can show quantitative information and the other is Self-Organizing Map (SOM) which

can show qualitative information.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

1 INTRODUCTION

A civil aircraft wing is generally designed by considering about a cruise condition. On the

contrary, when an aircraft lands or takes off, its wing should gain enough lift even at low-

speed. In such condition, high-lift system which can increase the wing load at low-speed is

required. Thus, high-lift system is one of the main interests in aircraft design due to its effect

on landing/ take-off performances, and pay-load capacity of an aircraft.

One of a typical high-lift system is a multi-element wing. Flowfield around a multi-

element wing has a complex physics caused by the interaction of each element.

1, 2

The

interactions between the design and its physics have to be examined closely to design high

efficient high-lift system.

In order to obtain the information of the relationship between the design space and the

solution space for realistic design, high quality solutions have to be collected in the multi-

objective design. In Ref. 3, Kriging surrogate model was introduced and perform the efficient

global optimization. In Ref. 5, Analysis of Variance (ANOVA) and Self-Organizing Map

(SOM) were applied to the aerodynamic design exploration. In Ref. 6, these data-mining

techniques are coupled with Kriging model and high efficient design is performed. Moreover,

these techniques are also applied to multi-disciplinary optimization (MDO), successfully.

7

In authors study, Kriging surrogate model and MOGA (multi-objective GA) was applied

to multi-objective design problem for a high-lift airfoil. The three-element airfoil as shown in

Fig. 1 is used as a baseline setting. Generally, a slat increases the stall angle and a flap

produces an upward shift in a lift curve as shown in Fig. 2

1

, thus multi-angle of attack should

be considered. In this study, the multi-objective design of the three-element high lift system

was defined, where objective functions are to maximize C

l

at the angle of attack of 8 degree

which corresponds to landing condition and 20 degree which corresponds to near stall angle

and the design variables are element’ settings. This study obtained many solutions which

achieve higher solution than the baseline settings and Kriging surrogate models which

correspond to each objective functions are constructed.

In this study, data mining techniques are applied to the sample designs which were

collected previous study to obtain circumstantial information about the relation between the

design space and the solution space. To obtain the quantitative information, ANOVA is

applied and to obtain the qualitative information, SOM is applied. Using these results, the

effect of the slat setting and the flap setting are investigated closely using RANS.

Figure 1 :Baseline airfoil and elements’ settings.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

2 FORMULATION

2.1 Flow Solver

Aerodynamic performances of sample designs for Kriging models are evaluated using a

structured multi-block flow solver, UPACS (Unified Platform for Aerospace Computational

Simulation)

9

. UPACS is developed at JAXA as a common-base code for aerodynamic

researchers.

In this study, RANS is applied with Spalart-Allmaras turbulence model. Flux was

evaluated by Roe’s flux difference splitting with MUSCL for third-order spatial accuracy. The

computational grid is decomposed into 35 sub-domains. Number of cells is about 10,000. To

reduce mesh generation time, the deforming mesh method

10

is applied to deform the mesh

around the baseline setting. Mach number is set to 0.2 and Reynolds number is set to

1.24×10

7

.

2.2 Design Variables

As shown in Fig. 3, the overlap, the gap, and the deflection angle between elements are

used as the design variables. Each design variable in limited as follows:

-0.01 c ≤ overlap

slat

≤ 0.01 c

0.01 c ≤ gap

slat

≤ 0.04 c

20.0 ≤ θ

slat

≤ 30.0 (degree)

-0.01 c ≤ overlap

flap

≤ 0.01 c

0.01 c ≤ gap

flap

≤ 0.03 c

30.0 ≤θ

flap

≤40.0 (degree)

Figure2 :High-lift system effect on airfoil lift and ideal design.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

where c is the chord length of airfoil when flap and slat are retracted into the main element.

θ

slat

overlap

slat

- +

gap

slat

θ

flap

overlap

flap

- +

gap

flap

Figure. 3 Design parameters.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

2.3 Objective functions

In this study, the design problem has two objective functions. The objective functions

considered here are to maximize lift co-efficient at angle of attack of 8 degree (C

l

8

) and 20

degree (C

l

20

). Angle of attack of 8 degree is assumed the angle of attack at landing condition

and 20 degree is assumed the stall angle decided from C

l

-α of the baseline setting as

discussed in Ref. 8.

2.4 Procedure of Multi-objective Design Exploration

The procedure of the present design (Fig. 4) is as follows: First, N samples which are

decided by Latin hypercube sampling

11

which is one of the space filling methods are

evaluated using RANS and Kriging surrogate models are constructed. Then, m EI maximum

points are added as sample points, and model accuracy is improved by constructing Kriging

models using N+m samples. This process is iterated until improvement of objective functions

becomes little. Finally, data mining technique can be applied to obtain the information of the

design problem. The detail of each procedure is described in the following sections.

2.4.1 Kriging model

Kriging model

4

expresses the value y(x

i

) at the unknown design point x

i

as:

)()(

ii

xxy

εμ

+=

(i = 1, 2, …., m) (1)

where, m is the number of design variables, μ is a constant global model and ε(x

i

) represents a

local deviation from the global model. The correlation between ε(x

i

) and ε(x

j

) is strongly

related to the distance between the two corresponding point, x

i

and x

j

. In the model, the local

deviation at an unknown point x is expressed using stochastic processes. Some design points

are calculated as sample points and interpolated with Gaussian random function as the

correlation function to estimate the trend of the stochastic process.

2.4.2 Improvement of Kriging model and selection of additional samples

Once the models are constructed, the optimum point can be explored using an arbitrary

optimizer on the model. However, it is possible to miss the global optimum, because the

surrogate model includes uncertainty at the predicted point. This study introduced EI values

3,

4

as the criterion.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

EI for present maximization problem can be calculated as follows:

(2)

where f

max

is the maximum value among sample points and ŷ is the value predicted by Eq.

(1) at an unknown point x. Φ and are the standard distribution and normal density,

respectively. EI consider the predicted function value and its uncertainty, simultaneously.

Thus, the solution that has a large function value and a large uncertainty may be a promising

solution. Therefore, by selecting the point where EI takes the maximum value, as the

additional sample point, robust exploration of the global optimum and improvement of the

model can be achieved simultaneously because this point has a somewhat large probability to

become the global optimum. To apply multi-objective problem, this study considers two EI

values based on two kriging models; EI

Cl8

and EI

Cl20

. Eq. (2) can be written for the present

design problem as follows:

maximize:

(3)

maximize:

Maximizing these objective functions, non-dominated solutions between EI

Cl8

and EI

Cl20

can be obtained. Among these non-dominated solutions, three points are selected as additional

sample points (Fig. 7): i) the point whose EI values of C

l8

is maximum, ii) the mid point in the

non-dominated solutions and iii) the point whose EI values of C

l20

is maximum. Therefore,

the value of m becomes 3 in this study.

2.4.3 Data mining technique

2.4.3.1 Analysis of Variance: ANOVA

An ANOVA

12

which is one of the data mining techniques is carried out to differentiate

the contributions to the variance of the response from the model.

To evaluate the effect of each design variable, the total variance of the model is

decomposed into that of each design variable and their interactions. The decomposition is

accomplished by integrating variables out of the model ŷ. The main effect of design variable

x

i

is as follows:

μμ

−≡

+−

∫∫

niinii

dxdxdxdxxxyx ,..,,,...,),.....,(

ˆ

)(

1111

L

(4)

()

[]

⎟

⎠

⎞

⎜

⎝

⎛

−

+

⎟

⎠

⎞

⎜

⎝

⎛

−

Φ−=

s

fy

s

s

fy

fyIE

maxmax

max

ˆˆ

)

ˆ

(

φ

x

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

Φ−=

s

Cy

s

s

Cy

CyEI

ll

lCl

max_8max_8

max_88

ˆˆ

)

ˆ

(

φ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

Φ−=

s

Cy

s

s

Cy

CyEI

ll

lCl

max_20max_20

max_2020

ˆˆ

)

ˆ

(

φ

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

Two-way interaction effect x

i

and x

j

is written as:

μμμμ

−−−≡

+−+−

∫

∫

)()(,..,,...,,,...,),.....,(

ˆ

)(

111111,, jjiinjjiinjiji

xxdxdxdxdxdxdxxxyx L

(5)

where, total mean μ is as follows:

nn

dxdxxxy ,.....,),.....,(

ˆ

11

∫∫

≡ L

μ

(6)

The variance due to the design variable x

i

is

[]

iii

dxx

2

)(

∫

≡

με

(7)

The proportion of the variance due to design variable x

i

to total variance of model can be

expressed as:

[]

∫∫

≡

−⋅⋅⋅

nn

dxdxxxy

p

...),....,(

ˆ

1

2

1

μ

ε

(8)

The denominator of Eq. (8) means variance of the model. The value obtained by Eq. (8)

indicates the sensitivity of the objective function to the variation of the design variable.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

2.4.3.2 Self-organizing Map: SOM

SOM is an unsupervised learning, nonlinear projection algorithm

13

from high to low

dimensional space. This projection is based on self-organization of a low-dimensional array of

neurons. The weight between the input vector and the array of neurons are adjusted to

represent features of the high dimensional data on low-dimensional map, in the projection

algorithm. The closer two patterns are in the original space, the closer is the response of two

neighboring neurons in the low-dimensional map. Thus, SOM reduces the dimension of input

data while preserving their features. Using SOM, qualitative information can be obtained.

In this study, commercial software Viscovery® SOMine

14

Produced by Eudaptics GmbH

is used. SOMine creates a map in a two dimensional hexagonal grid. Starting from

multivariate data, the neurons on the grid gradually adapt to the intrinsic shape of the data

distribution. Since the order on the grid reflects the neighborhood within the data, features of

the data distribution can be read off from the emerging map on the grid. The trained SOM is

systematically converted into visual information.

It is efficient to group all neurons by the similarity to facilitate SOM for the qualitative

analysis, because number of neurons on the SOM is large is large as a whole. This process f

grouping is called ‘clustering’. Hierarchical agglomerative algorithm is used for the clustering

here. First, ach node itself forms single cluster, and two clusters, which are adjacent in the

map, are merged in each step. The distance between two clusters is calculated by using the

SOM-ward distance. The number of clusters is determined by the hierarchical sequence of

clustering. A relatively small number of clusters are used fir visualization, while a large

number of clusters are used for the generation of weight vectors for respective design

variables.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

3 RESULTS

3.1 Design result

Figure6 shows the solutions obtained based on the present method. From this figure, the

solutions obtained from the initial sampling distributed uniformly in the solution space, on the

other hand, the solutions obtained from 15th-20th additional samplings achieve the better

performance than that of the initial samplings. The non-dominated front gradually advances to

the optimum direction as the improving process is preceded. These results show that the

present method selects the additional samples properly.

Figure4 Procedure of multi-objective

global exploration.

Selected samples

Figure5 Selection of additional samples

based on EI maximization.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

3.2 Data Mining Result

3.2.1 Result of ANOVA

Total variances of models were decomposed into the variance due to each design variable.

The proportion to the total variable of design variables and their interactions are shown in Fig.

16. According to Fig. 16(a), the flap setting gives over 70% effect on the Cl8. Moreover,

according to this figure, the two-way interaction between overlap

flap

and gap

flap

has a large

effect on C

l8

. This result suggests that overlap

flap

and gap

flap

should be designed with

considering their interaction carefully. Besides, θ

flap

has a relative small effect because the

maximum point of C

l8

existent over the upper bound of θ

flap

(See Fig. 8(b)). Generally the

design space should be adapted in such case, however, the design space was determined based

on practical use in this case. Therefore, elements’ settings should design in this design space.

According to Fig. 16(b), the slat and the flap setting both give effect on the C

l20

. This result

suggests that the proper setting of elements for C

l20

is more difficult than that for C

l8

.

According to this figure, the gap of flap is also important design variable for each objective.

Generally, a slat is set to increase stall angle, however, this result suggest that the flap setting

has also important to the aerodynamic performance near stall condition. Not only slat but also

flap should be designed carefully for near stall condition.

3.2.2 Result of SOM

To obtain quantitative information among the design space and the solution space from

design results, SOM is employed. Once Kriging models are constructed, function’s value at

unknown points can be predicted. Using these Kriging models, the non-dominated solutions

can be also obtained. Using sample points collected by the prediction of the non-dominated

solutions, clustering is performed by SOM.

Figure9 (a) and (b) show SOM colored by each objective functions. In Fig (a), good C

l8

performances are clustered in right hand side on the map and bad C

l8

performance are

clustered in left hand side. On the other hand, in Fig (a), good C

l20

performances are clustered

in left hand side on the map and bad C

l8

performance are clustered in left hand side. This

result suggests that two objective functions considered in this study have a strict trade-off.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

Figure10 shows SOM colored by design variables. The SOM by

θ

flap

is thoroughly

colored by red. It suggests that many solutions on the trade-off have to have highest value of

θ

flap

in the design space. The SOM by

θ

slat

is thoroughly colored by green. It suggest that

θ

slat

have to be mid-value (about 23 degree in this study) on the design space to obtain better

solutions. The SOM by overlap

slat

is thoroughly colored by blue. It suggests that overlap

slat

have to be minimum value on the design space. Other maps are spotted patterns. It suggests

that their design variables have interaction among other design variables.

3.2.3 Slat effect

Generally, they say that the slat has an influence on high angle of attack and the flap has an

influence on low angle of attack. However, according to ANOVA result shown in 3.2.1, the

interaction between the slat and the flap setting has an effect on C

l20

. To invest the slat effect

and its interaction with the flap, the slat only setting are designed by the procedure expressed

in 2.4. Figure11 shows the comparison of Slat-Flap design and Slat only designs (7

th

samplings). According to this result, slat can only improve the lift at high angle of attack. It is

agree with the general theory about high-lift airfoil. However, many solutions which obtained

by Slat-Flap design achieve better C

l20

than solutions which obtained by Slat only design.

This result suggests that the flap can also improve lift at high angle of attack and they have

interaction.

3.2.4 Flap effect

According to SOM result, flap deflection angle of many solutions achieving higher C

l

is

near upper bound (40 degree) in the design space. To invest the highest C

l

obtained by flap

deflection angle, the deflection angles, 40, 45, and 50 degree which out of design range are

also calculated by RANS. Figure12 shows C

l8

-flap deflection angle. According to this result,

C

l8

shows maximum value at flap deflection angle of 40-45 degree. This result suggests that

the flap deflection angle should be less than 45 degree and the high-lift airfoil should stall if

flap deflection angle becomes over 45 degree.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

overlap_flap-gap_flap

overlap_flap

gap_flap

θflap

θslat

θslat-overlap_flap

θflap-gap_flap

θslat-gap_flap

θflap-overlap_flap

others

gap_flap

overlap_flap-gap_flap

gap_slat

overlap_flap

gap_slat-overlap_flap

θslat-gap_flap

overlap_slat

θslat-gap_slat

gap_slat-gap_flap

overlap_slat-gap_slat

θflap-gap_flap

others

(a) (b)

Figure8 Total proportion to the total variance of models: (a) C

l8

, (b) C

l20.

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Cl8

Cl20

After 5th sampling

After 10th sampling

Afrer 15th sampling

After 20th sampling

Initial samplings

Baseline

Figure6 Sample points obtained based on EI maximization.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

Figure10 SOM colored by design variables

.

(a) (b)

Figure9 SOM: (a) colored by C

l8

, (b) colored by C

l20.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

4 CONCLUSIONS

Multi-objective design exploration for the elements’ settings of the high-lift airfoil

consisted of a slat, a main wing, and a flap was performed. There were two objective

functions: maximizing lift coefficient at a landing condition (C

l8

), maximizing lift coefficient

near stall condition (C

l20

). Flowfields were simulated by solving the Navier-Stocks equations

with Spalart-Allmaras turbulent model using the multi-block structured grid method. The

computational grids were deformed automatically for each design.

2.5

2.7

2.9

3.1

3.3

3.5

3.7

35.0 40.0 45.0 50.0

Flap deflection angle

Cl8

Figure12 Effect of flap deflection angle

4.0

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.0

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

Cl8

Cl20

Slat-Flap design

Baseline

Slat design

Figure11 The comparison of Slat-Flap design and Slat design.

M. Kanazaki, K. Tanaka, S. Jeong, and K. Yamamoto.

In this study, the objective functions, C

l8

and C

l20

, were transformed to the corresponding

EI values on the kriging model and global optimization was performed based on maximizing

their values. Using kriging surrogate model, the computational cost can be reduced and EI

value permit to carry out high efficient design on the Kriging model. The resulting designs

were also used as the additional samples to update the Kriging models.

Through the present method, the solutions based on the EI maximization advanced to the

optimum direction in the solution space. As the result, element settings that give higher

performance than that of baseline were successfully obtained. This result suggests that the

present method can be applied to the multi-objective problem while reducing computational

time drastically.

In order to obtain the information about design space, ANOVA which produces

quantitative information and SOM which produces qualitative information by projecting the

multi-dimensional data into two dimensional data are applied to the sampling result. This

result shows the useful information for the design. From their data mining results, slat and

flap effect are studied closely. According to these results, not only the slat but also the flap

has to be designed carefully to obtain higher C

l20

. To obtain higher C

l8

, the flap deflection

angle has to be decided with considering stall at the flap.

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