1

Multi-point Wing Planform Optimizationvia Control Theory

Kasidit Leoviriyakitand

Antony Jameson

Department of Aeronautics and AstronauticsStanford University, Stanford CA

43rd Aerospace Science Meeting and ExhibitJanuary 10-13, 2005

Reno Nevada

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Typical Drag Break Down of an Aircraft

Item CD Cumulative CD

Wing Pressure 120 counts 120 counts(15 shock, 105 induced)

Wing friction 45 165Fuselage 50 215

Tail 20 235Nacelles 20 255

Other 15 270___

Total 270

Mach .85 and CL .52

Induced Drag is the largest component

3

Cost Function

I 1CD 212

(p pd )2 dS 3CW

where

CW Structural WeightqSref

Simplified Planform Model

Wing planform modification can yield largerimprovements BUT affects structural weight.

Can be thoughtof as constraints

4

Choice of Weighting Constants

Breguet range equation

R VLD

1sfc

logWO W f

WO

With fixed V , L, sfc, and (WO W f WTO ), the variation of R

can be stated as

RR

CD

CD

1

logWTO

WO

WO

WO

CD

CD

1

logCWTO

CWO

CWO

CWO

Minimizing

I CD 3

1

CW

using

3

1

CD

CWOlog

CWTO

CW0

MaximizingRange

5

Structural Model for the Wing

• Assume rigid wing (No dynamic interaction between Aero and Structure)

• Use fully-stressed wing box to estimate the structural weight

• Weight is calculated based on material of the skin

6

“Trend” for Planform Modification

Increase L/D without any penalty on structural weight by• Stretching span to reduce vortex drag• Decreasing sweep and thickening wing-section to reduce

structural wing weight• Modifying the airfoil section to minimize shock

Boeing 747 -Planform Optimization

Baseline

Suggested

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Redesign of Section and Planformusing a Single-point Optimization

CL CD

counts

CW

counts

Boeing 747 .453 137.0(102.4 pressure, 34.6 viscous)

498(80,480 lbs)

Redesigned 747 .451 116.7(78.3 pressure, 38.4 viscous)

464(75,000 lbs)

Baseline

Redesign

Flight Condition (cruise): Mach .85 CL .45

8

The Need of Multi-Point Design

Undesired characteristics

Designed Point

9

Cost Function for a Multi-point Design

nnIIII 2211

nn gggg 2211

Gradients

10

Multi-point Design Process

11

Review of Single-Point designusing an Adjoint method

Using 4224 mesh points on the wing as design variables

Boeing 747

Plus 6 planform variables-Sweep-Span-Chord at 3span –stations-Thickness ratio

Design Variables

12

Optimization and Design using Sensitivities Calculated by the Finite Difference Method

Newton.-quasi assuch used, bemay search tedsophistica More

is tsimprovemen resulting The

) positive small(with is changes shape theIf

)()( iessensitivit has

)constant at as(such ),( function cost a method, difference finite theusingThen functions shape ofset )(

weight, where

)()( asgeometry thedefine tois approach simplest The

1

IIIIIIII

I

IIICCwII

xb

xbxf

TT

i

nn

i

iii

i

LD

i

i

ii

f(x)

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Disadvantage of the Finite Difference Method

The need for a number of flow calculations proportional to the number of design variables

Using 4224 mesh points on the wing as design variables

Boeing 747

4231 flow calculations ~ 30 minutes each (RANS)

Too Expensive

Plus 6 planform variables

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Application of Control Theory (Adjoint)

Drag Minimization Optimal Control of Flow Equationssubject to Shape(wing) Variations

0 and

0),( as and of dependencd the

expresses which equation governing that theSuppose

change ain results in change a and),(

function cost theDefine

SSRw

wRR

SwRSw

R

SSIw

wII

SSwII

TT

GOAL : Drastic Reduction of the Computational Costs

(for example CD at fixed CL)

(RANS in our case)

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4230 design variables

Application of Control Theory

where

that find weand ,eliminated is first term the

equation adjoint thesatisfy to Choosing

result. thechanging without variation thefrom subtracted and Multiplier Lagrange aby multiplied becan it zero, is variation theSince

SR

SIG

SGI

wI

wR

SSR

FIw

wR

wI

SSRw

wRS

SIw

wII

IR

TT

T

T

T

TT

TT

TTT

One Flow Solution + One Adjoint Solution

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Sobolev Gradient

Continuous descent path

Define the gradient with respect to the Sobolev inner product

I g,f gf g'f ' dxSet

f = g, I g,g This approximates a continuous descent process

dfdt

g

The Sobolev gradient g is obtained from the simple gradient g by the smoothing equation

g x

gx

g.

Key issue for successful implementation of the Continuous adjoint method.

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Design using the Navier-Stokes Equations

ixT

ijj

jij

jij

jij

vi

i

ii

ii

ii

i

i

vjijvijiji

vii

ii

ku

f

Hupuupuupuu

u

f

Euuu

w

fSFfSFwJW

FFt

W

D

3

2

1

33

22

11

3

2

1

0

, ,

, , , where

0

as written becan equations Stokes-Navier the,domain nalcomputatioIn

See paper for more detail

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Test Case

• Use multi-point design to alleviate the undesired characteristics arising form the single-point design result.

• Minimizing at multiple flight conditions;I = CD + CW at fixed CL

(CD and CW are normalized by fixed reference area) is chosen also to maximizing the Breguet range equation

• Optimization: SYN107Finite Volume, RANS, SLIP Schemes, Residual Averaging, Local Time Stepping Scheme,Full Multi-grid

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Single-point Redesign using at Cruise condition

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Isolated Shock Free Theorem

Mach .84

Mach .85

Mach .90

“Shock Free solution is an isolated point, away from the point shocks will develop”

Morawetz 1956

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Design Approach

• If the shock is not too strong, section modification alone can alleviate the undesired characteristics.

• But if the shock is too strong, both section and planform will need to be redesigned.

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3-Point Design for Sections alone (Planform fixed)

Condition Mach

123

0.840.860.90

1/31/31/3

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Successive 2-Point Design for Sections(Planform fixed)

Condition Mach

12

0.820.92

1/21/2

MDD is dramatically improved

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Lift-to-Drag Ratio of the Final Design

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Cp at Mach 0.78, 0.79, …, 0.92

•Shock free solution no longer exists.•But overall performance is significantly improved.

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Conclusion

• Single-point design can produce a shock free solution, but performance at off-design conditions may be degraded.

• Multi-point design can improve overall performance, but improvement is not as large as that could be obtained by a single optimization, which usually results in a shock free flow.

• Shock free solution no longer exists.

• However, the overall performance, as measured by characteristics such as the drag rise Mach number, is clearly superior.

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Acknowledgement

This work has benefited greatly from the support of Air Force Office of Science Research under grant No. AF F49620-98-2005

Downloadable Publications

http://aero-comlab.stanford.edu/http://www.stanford.edu/~kasidit/