M ULTI-STAGE A ERODYNAMIC D ESIGN OF A IRCRAFT G EOMETRIES BY K RIGING- B ASED M ODELS AND A DJOINT V ARIABLE A PPROACH

沊ࣜ滊ࣜ殶ࣨ^࣭ࣜ決ࣜ懗ࣜ渆ࣨ^࣮ࣜ卆ࣜ涋ࣜ橚^ࣦ࣯

M ULTI-STAGE A ERODYNAMIC D ESIGN OF A IRCRAFT G EOMETRIES BY K RIGING- B ASED M ODELS AND A DJOINT V ARIABLE A PPROACH

J.W. Yim,

¹

B.J. Lee

²

and C. Kim

^*1

An efficient and high-fidelity design approach for wing-body shape optimization is presented. Depending on the size of design space and the number of design of variable, aerodynamic shape optimization process is carried out via different optimization strategies at each design stage. In the first stage, global optimization techniques are applied to planform design with a few geometric design variables. In the second stage, local optimization techniques are used for wing surface design with a lot of design variables to maintain a sufficient design space with a high DOF (Degree of Freedom) geometric change. For global optimization, Kriging method in conjunction with Genetic Algorithm (GA) is used. Asearching algorithm of EI (Expected Improvement) points is introduced to enhance the quality of global optimization for the wing-planform design. For local optimization, a discrete adjoint method is adopted. By the successive combination of global and local optimization techniques, drag minimization is performed for a multi-body aircraft configuration while maintaining the baseline lift and the wing weight at the same time.

Through the design process, performances of the test models are remarkably improved in comparison with the single stage design approach. The performance of the proposed design framework including wing planform design variables can be efficiently evaluated by the drag decomposition method, which can examine the improvement of various drag components, such as induced drag, wave drag, viscous drag and profile drag.

Key Words : Aerodynamic Shape Optimization Global Optimization Local

Optimization GA : Genetic Algorithm Kriging Method Expected

Improvement Adjoint Variables Method Drag Decomposition Method

* Corresponding author, E-mail: chongam@snu.ac.kr

1.

.

.

.

(Local optimization Method), (Global Optimization Method)

.

global

. (Genetic Algorithm.

GA)[1] .

3

.

RSM, Kriging[2] meta-modeling

. (sample point) meta-modeling

GA

.

.

. ,

EI(Expected Improvement)[3] .

, (Gradient Based

Optimization Method, GBOM) (Adjoint Variables method, AV)

. Jameson [4] continuous adjoint approach

/

. Lee [5] discrete adjoint method

.

GBOM ,

local optimum .

multi-stage .

meta-modeling GA Optimizer global optimization .

discrete adjoint approach Local Optimization

. multi-stage

design approach far-field analysis drag decomposition[6]

Induced Drag, Wave Drag .

2.

2.1 Meta Modelling

Meta-modeling

. meta-model

, meta-modeling Response Surface

Methodology(RSM) Kriging model . Kriging

. RSM

.

(1) .

ƗÞƖß á ĸ â³ÞƖß (1)

ĸ RSM

, Ɩ ƌ

. ³ÞƖß Gaussian random function

. Covariance matrix ³ÞƖß (2) .

œƍƔ

ã

³ÞƖ^Ƈßì³ÞƖ^ƈß

ä

á ň^Ï«

ã

«ÞƖ^ƇìƖ^ƈß

ä

(2)

« correlation matrix «ÞƖ^ƇìƖ^ƈß

Ɩ^Ƈ, Ɩ^ƈ (correlation function) . Gaussian correlation function .

Ɩ ýƗÞƖß (3)

.

ýƗÞƖß á ýĸâýƐ^­ÞƖß«^{à Î}ÞƗ à Ƅýĸ ß (3)

ýĸ ĸ , Ɨ

( ) ƌƑ( )

. ýƐ^­ Ɩ ƌ_Ƒ

(correlation vector) . Ƅ 1

ƌ_Ƒ .

.

,

. Ɩ

(4) .

ýƑÞƖß á ýň^Ïƙ

Ɯ

ƚ

Î à Ɛ^­«^{à Î}_{Ɛâ ć} Ƅ^­«^{à Î}Ƅ ÞÎà Ƅ^{à Î}«^{à Î}Ɛß^Ïƛ

Ɲ

ƞ

(4)

2.2 Expected Improvement

,

GA . GA

.

EI(Expected improvement) .

Kriging

. ,

EI (5) (6) .

¢ÞƖß á

å

^{ƄÀ ¼ Áà ý}ƗÞƖßì ¼ ¹ ýƗï Ƅ_{À ¼ Á}

× ì ƍƒƆƃƐƕƇƑƃ á À´ËãƄ_{À ¼ Á à}ýƗÞƖßì×ä

(5)

žã¢ÞƖßä á ÞƄ_{À ¼ Á à}ýƗßij

Þ

^{ćƄÀ ¼ ÁƑà ýƗ}

ß

^{â Ƒŋ}

Þ

^{ćƄÀ ¼ ÁƑà ýƗ}

ß

⁽⁶⁾

Ƅ_À¼Á

, ij ŋ . (7)

EI CFD .

Ƅ_À¼Á Ƅ_À¼Á

. EI

. - EI ,

.

2.3 (GA : Genetic Algorithm)

.

, ,

.

.

(Fitness

value) .

.

.

.

2.4 (Discrete Adjoint Approach)

, GA ,

. ,

.

/ (residual)

,

.

Þªß, Þ±ß Þß

. ª, ±  chain rule

,

(7) .

å

^ćƂ ƂŸ

æ

^á

å

^ćŞª ŞŸ

æ

^­

å

^ćƂ Ƃª

æ

^â

å

^檱 ŞŸ

æ

^­

å

^ćƂ Ƃ±

æ

^â

å

^檝 ŞŸ

æ

â ĩ^­

Þ

^ƙƜƚ

ćŞª Ş«ƛ

Ɲƞ

å

^ćƂ Ƃª

æ

^âƙƜƚ

檱 Ş«ƛ

Ɲƞ

å

^ćƂ Ƃ±

æ

^â

å

^檝 Ş«

æß

⁽⁷⁾

(8)

åƂªîƂæ (8)

.

å

^ćƂ ƂŸ

æ

^á

å

^檱 ŞŸ

æ

^­

å

^ćƂ Ƃ±

æ

^â

å

^檝 ŞŸ

æ

â ĩ^­

Þ

^ƙƜƚ

檱 Ş«ƛ

Ɲƞ

å

^ćƂ Ƃ±

æ

^â

å

^檝 Ş«

æß

â

Þ å

^ćŞª ŞŸ

æ

^{­â ĩƙ}Ɯƚ

ćŞª Ş«ƛ

Ɲ

ƞ

ß å

^ćƂ Ƃª

æ

(8)

Fig. 1 Design process for 2-stage design

Fig. 2 Schematic illustration of 2-stage design , åƂªîƂæ 0

ĩ (9)

(10) .

å

^ćŞª ŞŸ

æ

^{­â ĩ­ƙ}Ɯƚ

ćŞª Ş«ƛ

Ɲ

ƞ_áå×æ^­ ₍₉₎

å

^ćƂ ƂŸ

æ

^á

å

^檱 ŞŸ

æ

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å

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^â

å

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â ĩ^­

Þ

^ƙƜƚ

檱 Ş«ƛ

Ɲƞ

å

^ćƂ Ƃ±

æ

^â

å

^檝 Ş«

æß

⁽¹⁰⁾

2.5 Drag Decomposition Method

,

wing ( )

.

. CFD

. momentum wake

integration method( far field method)

. , (11),

(12) . (11)

, (induced

drag) , (12)

.

á

č

_°š^©tć« ĢƑƂƑ

â

č

_°š^ćÏ Ň_t

ÞƓ^Ïâ Ɣ^ÏßƂƑ â¨ÞĢ^Ïß

(11)

_{ ƂƓ ƁƃƂ}á

č

_°š^ćÏ Ň_t

ÞƓ^Ïâ Ɣ^ÏßƂƑ j

č

_°š^ćÏ

Ň_t ŋńƂƑ

(12)

3. Framework

Fig. 1 wing

2 .

, . Wing planform surface

. planform

. wing

.

.

2 wing

.

Fig. 2 2

. planform

3 . 1

, GBOM 2

. planform

GBOM 3

. 3

. planform

4 GBOM

3 5

Fig. 3 Planform variables of ONERA-M6

Fig. 4 Pressure contour of baseline model

Fig. 5 Pressure contour of Kriging-EI method

Design variables(Dv) Min Base Max Kriging+EI

Dv1 Sweepback (ĩ) 25 30 35 33.6972

Dv2 Half span(ƀîÏ) 1.3240 1.4712 1.6183 1.466160 Dv3 Taper ratio(Ɓ_ÏîƁ_Î) 0.5058 0.5620 0.6182 0.527694

Dv4 Twist angle(ľ) -2 0 2 -0.01857

Table 1 Geometric information of ONERA-M6 and designed wing

Obj. fn. &

Constraint Baseline model

Kriging1 +GA Kriging2 +GA Kriging-EI +GA

Predicted

value Real value Predicted

value Real value Predicted

value Real value

CL 0.261746 0.26174 0.2601053 None None 0.260879 0.2602198

CD 0.011937 0.01059 0.0103969 None None 0.010961 0.0111651

Weight 6.497615 Not included None 6.4926028

Table 2 Comparison of objective functions and constraint value (planform design / ONEAR-M6).

. 5

GBOM

.

3.1 wing planform

Fig. 3 wing planform wing span,

taper ratio, sweepback angle twist angle

. GA

3

GA .

meta modeling . Kriging

EI , CCD(Central

Composite Experimental Design) ,

CFD kriging

. GA

(EI)

. 2

ONERA-M6 wing

. (13) .

œ_ (13)

œ_¥>œ_¥×

°_°×á°_°

œ_¥× = Lift coefficient of baseline model

°_°× = Wing weight of baseline model

Mach number 0.84, 3.06

. 3 Euler

. RoeM[7] ,

MUSCL(Monotonic Upstream Centered Scheme Conservation

Law) 3 .

LU-SGS .

Table. 2

Fig. 6 Pressure contour of 2-stage design model

Fig. 7 2-stage design history of ONERA-M6

Obj. fn. & Constraint Baseline model Planform design only Surface design only (Adjoint)

Multi-stage design

Kriging-EI +GA Kriging-EI + GA + Adjoint

CL 0.261746 0.2602198 0.259997 0.2601049

CD 0.011937 0.0111651 0.007868 0.0075882

Table 3 Comparison of objective functions and constraint value for designed models (ONERA-M6).

Design strategy Prediction method CL CD CD - entropy CD - induced

Baseline model

Surface integration 0.261746 0.011937 N/A N/A

Wake integration 0.267401 0.010837 0.00738432 0.00345286

Planform only (Kriging-EI +GA)

Surface integration 0.260219 0.011165 N/A N/A

Wake integration 0.270864 0.009873 0.00630170 0.00357173

Surface only (AV + BFGS)

Surface integration 0.259997 0.007868 N/A N/A

Wake integration 0.270641 0.007347 0.00360265 0.00374472

2-stage design

Surface integration 0.260104 0.007588 N/A N/A

Wake integration 0.262512 0.007124 0.00362269 0.00350174

Table 4 Comparison of aerodynamic performances evaluated by drag decomposition methods.

. 1 wing

2 3

wing .

2 EI

EI 3

.

Fig. 4 Fig. 5 .

. planform

. 3.2

drag decomposition

.

3.2 2

1 plaform 2

wing

. wing root, mid, tip

10 60 . Wing

Hicks-Henne .

. planform

.

(14) .

œ_

œ_¥>œ_¥

× (14)

Fig. 8 Wing planform variables of DLR-F4

Fig. 9 Surface variables of DLR-F4 wing

Fig. 10 Overset mesh system for DLR-F4

Fig. 11 Pressure contour of baseline model

Fig. 12 Pressure contour of 1st stage design model : Kriging - EI + GA

œ_â°ƒ Zã^×윥×àœ_¥ä

œ_¥

×

°_{ƒ á ć} Şķ Şœ_

îćŞķ Şœ_¥

Table 3 Fig. 7 2

0.6%

0.011937 0.0075882 36.4% .

21.927 34.227 56.3%

.

2 , Drag Decomposition

wing Table 4

.

,

wing

.

( ) . Table 4

.

4. 2

4.1

2

DLR-F4

wing-body . 1 planform

ONERA-M6 . Fig. 8 Fig. 9

. Wing-body

Fig. 10 7 block overset .

122 .

Mach number 0.75, 0 ,

ONERA-M6 .

4.2 1 : DLR-F4 Planform

ONERA-M6 . Table 5 planform

. Table 5

Design variable (Dv) Min Baseline Max Kriging Kriging-EI

Dv1 Sweepback(ȁ) 25° 27.15° 30° 27.187492 27.03516°

Dv2 Kink-Span(Skink) 3733.5 4148.6 4563.2 3886.7324 3908.1826

Dv3 Semi-Span(Ssemi) 11682.0 12980.5 14273.0 12840.069 12990.120

Dv4 Kink-Chord(Ckink) 2731.2 3034.7 3338.1 3138.7873 3160.5550

Dv5 Tip-Chord(Csemi) 1401.0 1556.7 1712.4 1570.5044 1501.5888

Dv6 Twist Angle(ș) -5.0° -4.631° -3.0° -4.445511 -4.603918°

Const. Wing Weight(WW0) 14.3024 14.284285 14.299511

Table 5 Design variables and optimum values of DLR-F4 wing/body

Obj. fn. & Constraint Baseline model Kriging1 +GA Kriging-EI +GA

Predicted value Real value Predicted value Real value

CL 0.710176 0.71020 0.708225 0.71020 0.711111

CD 0.023014 0.02139 0.021064 0.02078 0.020632

Weight 14.3024 14.284285 14.299511

Table 6 Comparison of objective functions/constraint value for design model (planform design / DLR-F4).

Obj. fn. & Constraint Baseline model 1st-stage design 2nd-stage design

Kriging1 +GA Kriging1-EI +GA Kriging1-EI +Adjoint

CL 0.710176 0.708225 0.711111 0.7077625

CD 0.023014 0.021064 0.020632 0.0202102

Table 7 Comparison of objective functions/constraint value for design model (DLR-F4)

Fig. 13 2-stage Design History of DLR-F4

Fig. 14 Pressure Contour of 2nd Stage Design Model

, ONERA-M6

. statistical group weight method

. Table 6 kriging kriging-EI

planform .

EI . Fig. 11 - 12

. wing

.

4.3 2 : DLR-F4 Surface

1 planform 2

ONERA-M6 surface .

.

Table 7 Fig. 13 2 ,

12.3%

Fig. 15 Surface Pressure Distribution (40.9%)

Fig. 16 Surface Pressure Distribution (51.2%)

0.3% .

(¥î) 30.85 35.02 .

DLR-F4 wing-body

wing body

. wing-body

. Fig. 14 ~ Fig. 16 wing

. wing root

mid 2 , tip

.

5.

, 2

. 1

, Kriging GA wing

planform . EI

sample point Kriging model

. 2 1

planform

GBOM wing .

Drag Decomposition .

ONERA-M6 wing DLR-F4 wing-body overset grid, invscid/viscous

(adjoint approach)

. 2

/ 3

. GBOM

.

BK21 2 .

[1] 2004, Chung, H.S. and Alonso, J.J., "Multiobjective optimization using approximation model-based genetic algorithm," 10th AIAA/ISSMO, Multidisciplinary Analysis and Optimization Conference, AIAA 2004-4325 Albany.

[2] 1989, Sacs, J., Welch, W.J., Mitchell, T.J. and Wynn, H.P.,

"Design and analysis of computer experiments," Statistical Science, Vol.4, No.4, pp.409-435.

[3] 2005, Jeong, S., Murayama, M. and Yamamoto, K.,

"Efficient optimization design method using Kriging model,"

Journal of Aircraft, Vol.42, No.2, pp.413-420.

[4] 2005, Leoviriyakit, K. and Jameson, A., "Multi-point wing planform optimization via control theory," 43rd Aerospace Science Meeting and Exhibit, 2005-0450, January 10-13, Reno, Nevada.

[5] 2007, Lee, B.J. and Kim, C., "Viscous aerodynamic shape optimization of wing/body configuration with overset mesh techniques," Infotech@Aerospace 2007 Conference and Exhibit, Invited Session (Design Exploration for Aerospace Systems), AIAA-2007-2872, Rohnert Park, California.

[6] 1999, Giles, M.B. and Cummings, R.M., "Wake integration for three-dimensional flowfield computation: Theoretical development," Journal of Aircraft, Vol.36, No.2, pp.357-365.

[7] 2003, Kim, S., Kim, C., Rho, O. and Hong, S.K., "Cures for the shock instability: Development of shock-stable roe scheme," Journal of Computational Physics, Vol.182, No.2, pp.342-374.