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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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