AERO-STRUCTURAL WING PLANFORM OPTIMIZATION

AERO-STRUCTURAL WING PLANFORM

OPTIMIZATION

Kasidit Leoviriyakitand

Antony Jameson

Department of Aeronautics and Astronautics

Stanford University, Stanford, CA

45 th Aero/Astro Industrial Affiliates MeetingStanford, April 27-28, 2004

c© K. Leoviriyakit & A. Jameson 2004 1/15 Aero-Structural Wing Planform Optimization

+ Introduction and Motivation

ã Shock drag and vortex drag are the major fraction of the total drag of thelong-rang transport aircraft.

ã Boeing 747 at CL ∼ .47 (including fuselage lift ∼ 15 %)

Item CD Cumulative CD

Wing pressure 120 counts 120 counts(15 shock,105 vortex)

Wing friction 45 165Fuselage 50 215

Tail 20 235Nacelles 20 255

Other 15 270—

Total 270

ã During the last decade aerodynamic shape optimization methods based oncontrol theory have been used to design shock free wings and the methods areperfected for rigid wings with fixed planforms.

ã Wing planform modifications can also reduce vortex drag.


+ Re-design for a shock free wing using adjoint method

SYMBOL

SOURCE SYN88 DESIGN 17SYN88 DESIGN 0

CL 0.420 0.420

CD 0.00939 0.01077

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSBOEING 747 WING-BODY

REN = 0.00 , MACH = 0.870 , ALPHA = 2.20

Kasidit Leoviriyakit21:03 Thu

1 Jan 04COMPPLOT

JCV 1.13COMPPLOT

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Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 11.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 31.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 50.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 70.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 89.8% Span

ã Modify Sections and Fix Planform.

ã Result: Shock free wing (6% airplane-drag reduction) within 5 minutes(using 1 1.7 gHz processor).


+ Planform Optimization

ã Wing planform modifications can could yield larger improvements in wingperformance.

ã Simplified model

t

b/2C2

C3

C1

ã Design variables: Surface mesh points + six Planform variables.

ã Side effects from planform variation:

- Wing weight.

- Stability & Control issues.

ã The problem becomes a multi-objective optimization to minimize both drag andstructural weight.

ã Minimize the cost functionI = CD + αCW

where CW =Wwing

q∞Sref.


+ Structural Weight Model

ã Choices:

- Depending on geometry ONLY (Last year)

- Depending on BOTH geometry and aerodynamic loading (Current year).

ã Since last year, a more realistic model to calculate the structural wing weighthas been introduced.

ã The wing structure is modeled by a structure box, whose major structuralmaterial is the box skin.

l

A

Ab/2

sc

*zz

Figure 1: Structural model for a swept wing

t

cs

ts

Figure 2: Section A-A

ã The skin thickness (ts) varies along the span and resists the bending momentcaused by the wing lift.

ã Then, the structural wing weight can be calculated based on material of the skin.

ã Adjoint method for this structural model has been developed.


+ Choice of Weighting Constants of I = CD + αCW

ã The choice of α greatly effects the optimum shape.

ã Maximizing the range of an aircraft provides a guide to their values.

ã The simplified range equation can be expressed as

R =V

C

L

Dlog

W1

W2

where W2 is the empty weight of the aircraft. With fixed VC

, W1, and L, thevariation of R can be stated as

δR

R= −

δCD

CD

+1

logW1

W2

δW2

W2

= −

δCD

CD

+1

logCW1

CW2

δCW2

CW2

.

Therefore minimizingI = CD + αCW ,

by choosing

α =CD

CW2log

CW1

CW2

, (1)

corresponds to maximizing the range of the aircraft.


+ Sample of Planform Optimization

ã Test case: Boeing 747 wing-fuselage and modified geometries at the followingflow conditions

M∞ = 0.87

CL = 0.42 (fixed)

Choose α3 according to (1) to maximize range.

ã Test case: MD 11 wing-fuselage and modified geometries at the following flowconditions

M∞ = 0.83

CL = 0.50 (fixed)

Choose α3 according to (1) to maximize range.


+ B747 : Optimized section, Fixed Planform

SYMBOL


CL 0.420 0.420

CD 0.00939 0.01077


REN = 0.00 , MACH = 0.870 , ALPHA = 2.20


1 Jan 04COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

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0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 11.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 31.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 50.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 70.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 89.8% Span

ã a CD is reduced from 107.7 drag counts to 93.9 drag counts (12.8%).

ã a CW remains constant at .0455

ã The redesigned pressure distribution (blue) is shock-free.


+ B747 : Sweepback, Span, Chord, and Section Variations to Maximize Range

Baseline geometry —

Optimized geometry —

Geometry Baseline Optimized Variation(%)Sweep (deg) 42.1 38.8 - 7.8

Span (ft) 191.5 205.7 + 7.4Croot (ft) 48.1 48.6 + 1.0

Cmid (ft) 30.6 30.8 + 0.7Ctip (ft) 10.78 10.75 + 0.3

troot (in) 58.2 62.4 + 7.2tmid (in) 23.7 23.8 + 0.4

ttip (in) 12.98 12.8 - 0.8

SYMBOL


ALPHA 1.930 2.189

CD 0.00872 0.01077


REN = 0.00 , MACH = 0.870 , CL = 0.420


1 Jan 04COMPPLOT

JCV 1.13COMPPLOT

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0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 12.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 33.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 53.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 74.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0C

pX / C

95.8% Span

ã CD is reduced from 107.7 drag counts to 87.2 drag counts (19.0%).

ã Cw is reduced from 0.0455 to 0.0450 (1.1%).


+ MD-11 : Optimized section, Fixed Planform

SYMBOL


ALPHA 2.443 2.523

CD 0.01446 0.01589

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMD-11 WING/BODY + 4.5FT WINGTIP EXTREN = 0.00 , MACH = 0.830 , CL = 0.500

Kasidit Leoviriyakit21:18 Thu25 Mar 04

COMPPLOTJCV 1.13

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0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 13.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 32.9% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 52.2% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 72.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 92.6% Span

ã a CD is reduced from 158.9 drag counts to 144.6 drag counts (9.0%).

ã a CW increases slightly from 0.0345 to 0.0346 (0.3%).

ã The redesigned pressure distribution (blue) is almost shock-free.


+ MD-11 : Sweepback, Span, Chord, and Section Variations to Maximize Range

Baseline geometry —

Optimized geometry —

Geometry Baseline Optimized Variation(%)Sweep (deg) 37.9 36.8 - 2.9

Span (ft) 154.6 163.6 + 5.8Croot (ft) 38.5 39.0 + 1.3

Cmid (ft) 26.3 26.5 + 0.8Ctip (ft) 8.90 8.92 + 0.2

troot (in) 49.2 53.7 + 9.1tmid (in) 22.6 23.8 + 5.3

ttip (in) 9.8 10.1 + 3.1

SYMBOL


ALPHA 2.141 2.523

CD 0.01382 0.01589

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMD-11 WING/BODY + 4.5FT WINGTIP EXTREN = 0.00 , MACH = 0.830 , CL = 0.500

Kasidit Leoviriyakit20:34 Fri

26 Mar 04COMPPLOT

JCV 1.13COMPPLOT

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JCV 1.13MCDONNELL DOUGLAS



0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 13.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 34.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 54.9% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 76.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0C

p

X / C 97.4% Span

ã CD is reduced from 158.9 drag counts to 138.2 drag counts (13.0%).

ã Cw is reduced from 0.0345 to 0.0344 (0.3%).


+ Application of Game Theory

ã To extend the optimal design range, α should not be limited to only one value.

ã Use multiple values of α to capture “Pareto Front” of the problem withoutsolving the Optimality Conditions.

Drag

PInitial geometry

QPareto front

vs.

R

Weight

α 1

α 3

Figure 3: Game strategy with Drag and Weight as players

ã If the optimal shapes are truly optimized, each of them should lie on a curvewhere no reduction in weight can be achieved without an increase in drag andno reduction in drag can be achieved without an increase in weight.

ã This idea is similar to a “game” where one player tries to minimize CD and theother player tries to minimize CW .

ã Eliminate all the dominated points to get Pareto front.


+ B747 : Pareto Front

ã Test case: Boeing 747 wing-fuselage and modified geometries at the followingflow conditions M∞ = 0.87, CL = 0.42 (fixed), multiple α.

80 85 90 95 100 105 1100.038

0.040

0.042

0.044

0.046

0.048

0.050

0.052

CD (counts)

Cw

Pareto front

baseline

optimized section with fixed planform

X = optimized section and planform

maximized range

Figure 4: Pareto front of section and sweep modifications


+ Conclusions

ã By exploring the adjoint method, it is possible to carry out vary rapid wingoptimization for preliminary design. (even on laptop computer)

ã Case studies of wing planform optimization of long range transport aircraftsuggest the possibility of extending the attainable lift-to-drag ratio versus Machnumber significantly beyond historical trends.

ã The results indicate that by stretching the span together with decreasing sweepand thickening the wing sections, the lift-to-drag ratio can be increased withoutany penalty on the structure weight.

ã The intelligent use of optimization techniques may ultimately have an evengreater impact by enabling the exploration of radical departures fromconventional design.

ã The application of game theory provides insight into how to combine drag andwing weight in the cost function, broadening the design range of optimal shapes


+ Acknowledgment

This work has benefited greatly from the support of the Air Force Office ofScience Research under grant No. AF F49620-98-1-2002.


Documents

AERO-STRUCTURAL WING PLANFORM OPTIMIZATION