1

Wing Planform Optimization via an Adjoint Method

Kasidit Leoviriyakit

Department of Aeronautics and AstronauticsStanford University, Stanford CA

Stanford UniversityStanford, CAJune 28, 2005

2

History: Adjoint for Transonic Wing Design

Baseline 747, CD 117 counts Redesigned, CD 103 counts

Redesign for a shock-free wing by modify the wing sections (planform fixed )– Jameson 1995

- Cp

3

Break Down of Drag

Item CD Cumulative CD

Wing Pressure 120 counts 120 counts

(15 shock, 105 induced)

Wing friction 45 165

Fuselage 50 215

Tail 20 235

Nacelles 20 255

Other 15 270

___

Total 270

Boeing 747 at CL ~ .52 (including fuselage lift ~ 15%)

Induced Drag is the largest component

4

Key Concept

Use “shock-free” concept to drive the planform design.

• Conventionally the wing is swept to weaken the shock.

• With the “shock-free” wing capability, it allows more configurations that was previously prohibited by the strong shock.

5

Aerodynamic Design Tradeoffs

DWL

DOD CeAR

CCC

2

intosplit becan t coefficien drag The

L

D is maximized if the two terms are equal.

Induced drag is half of the total drag.

If we want to have large drag reduction, we shouldtarget the induced drag.

Di 2L2

eV 2b2

Design dilemma

Increase bDi decreases

WO increases

Change span by changing planform

6

Can we consider only pure Aerodynamic design?

• Example 1: Vary b to minimize drag

I = CD

As span increases, vortex drag decreases. Infinitely long span

• Example 2: Add a constraint;

b =bmax

There is no need for optimization

• Also true for the sweep variation

maxbb

• Pure aerodynamic design leads to unrealistic results

• Constraints sometimes prevent optimal results

7

Cost Function

I 1CD 2

1

2(p pd )2 dS 3CW

where

CW Structural Weight

qSref

Simplified Planform Model

Wing planform modification can yield largerimprovements BUT affects structural weight.

Can be thoughtof as constraints

8

Choice of Weighting Constants

Breguet range equation

R VL

D

1

sfclog

WO W f

WO

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

can be stated as

R

R

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

9

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 from material of the skin

10

Design Parameters

Using 4224 mesh points on the wing as design variables

Boeing 747

Plus 6 planform variables

Use Adjoint method to calculate both section and planform sensitivities

11

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

III

II

III

I

III

CCwII

xb

xbxf

TT

i

nn

i

iii

i

LD

i

i

ii

f(x)

12

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

13

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

SS

Rw

w

RR

SwR

Sw

R

SS

Iw

w

II

S

SwII

TT

GOAL : Drastic Reduction of the Computational Costs

(for example CD at fixed CL)

(Euler & RANS in our case)

14

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

S

R

S

IG

SGI

w

I

w

R

SS

R

F

Iw

w

R

w

I

SS

Rw

w

RS

S

Iw

w

II

I

R

TT

T

T

T

TT

TT

TTT

One Flow Solution + One Adjoint Solution

15

Outline of the Design Process

Flow solution

Adjoint solution

Gradient calculation

Sobolev gradient

Shape & Grid Modification

Re

pea

ted

unt

il C

on

verg

ence

to

Opt

imu

m S

hap

e

Design Variables• 4224 surface mesh points for the NS design (or 2036 for the Euler design)• 6 planform parameters

-Sweep-Span-Chord at 3span –stations-Thickness ratio

16

Design using the Navier-Stokes Equations

ixT

ijj

jij

jij

jij

vi

i

ii

ii

ii

i

i

vjijvijiji

vii

ii

ku

f

Hu

puu

puupuu

u

f

E

u

uu

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

17

Adjoint Equations

jl

l

k

k

ijl

k

k

i

j

j

i

l

jl

xlj

ljij

xkijxjxilj

xijxxlji

xljp

jiji

SL

S

uuuS

SL

SL

w

fSC

15

1

1

~

~

)(~

2

where

in 0~

1 DLMCT

i

Ti

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Adjoint Boundary Condition

22)cos()cos(

i

1

2

21

22

and cosinedirection theis where

,

,

SndSpI

pnq

qdSqI

ppndSppI

jj

ijijji

kkii

djjd

Cost Function Adjoint Boundary Condition

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Viscous Gradient Comparison: Adjoint Vs Finite Difference

Sweep

Span

cmidcroot ctip t

Sweep

Span

cmid

croot

ctip t

DCx

WCx

• Adjoint gradient in red• Finite-different gradient in blue

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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

g

xg.

Key issue for successful implementation of the Continuous adjoint method.

21

Viscous Results

B747 MD11

BAe MDO Datum

22

B747 Planform Changes Mach .85 Fixed CL .45

baseline

redesigned

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B747 @ Mach .85, Fixed CL .45

CL CD

counts

CW

counts

CM

Boeing 747 .453 137.0

(102.4 pressure, 34.6 viscous)

498

(80,480 lbs)

-.1408

Redesigned 747 .451 116.7

(78.3 pressure, 38.4 viscous)

464

(75,000 lbs)

-.0768

Baseline

Viscous-Redesignedusing Syn107

(RANS Optimization)

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Design Short-CutUse Euler planform optimization as a starting point for the Navier-Stokes Optimization

Euler Optimized

NS Optimized

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Redesigned Planform of Boeing 747

1. Longer span reduces the induced drag2. Less sweep and thicker wing sections reduce the

structural weight3. Section modification keeps the shock drag minimum

• Overall: Drag and Weight Savings

• No constraints posted on planform, but we still get a finite wingwith less than 90 degrees sweep.

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baseline

redesigned

MD11 Planform ChangesMach .83, Fixed CL .50

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MD11 @Mach .83, Fixed CL .5

CL CD

counts

CW

counts

MD 11 .501 179.8

(144.2 pressure, 35.6 viscous)

654

(62,985 lbs)

Redesigned MD11 .500 163.8

(123.9 pressure, 39.9 viscous)

651

(62,696 lbs)

“Same Trend”1. Span increases2. Sweep decreases3. t/c increases4. Shock minimized

Baseline

Redesign

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BAe Planform ChangesMach .85 Fixed CL .45

baseline

redesigned

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BAe MDO Datum @ Mach .85, Fixed CL .45

CL CD

counts

CW

counts

BAe .453 163.9

(120.5 pressure, 43.4 viscous)

574

(87,473 lbs)

Redesigned BAe .452 144.7

(99.3 pressure, 45.4 viscous)

570

(86,863 lbs)

“Same Trend”but not EXTREME

Baseline

Redesign

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Pareto Front: “Expanding the Range of Designs”

WD CCI 31

• The optimal shape depends on the ratio of 3/1

• Use multiple values to capture the Pareto front

(An alternative to solving the optimality condition)

31

Pareto Front of Boeing 747

1

3

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Appendix

33

ConstraintsEnforced in SYN107 and SYN88

For drag minimization

1. Fixed CL

2. Fixed span load• Keep out-board CL low enough to prevent buffet• Fixed root bending moment

3. Maintain specified thickness• Sustain root bending moment with equal structure

weight• Maintain fuel volume

4. Smooth curvature variations via Sobolev gradient

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Point Gradient Calculation for the wing sections

D II

TIIII dDRdNMI

•Use the surface mesh points as the section design variable•Perturb along the mesh line Avoid mesh crossing over

35

Planform Gradient Calculation

D II

TIIII dDRdNMI

E.g.. Gradient with respect to sweep change

36

Planform Gradient Calculation

S SS

D

SSST

IIII RRNMI )(

Surface

Domain

37

References

• Leoviriyakit, K.,"Wing Planform Optimization via an Adjoint Method," Ph.D. Dissertation, Stanford University, March 2005.

• Leoviriyakit, and Jameson, A., "Multi-point Wing Planform Optimization via Control Theory", 43rd Aerospace Sciences Meeting and Exhibit, AIAA Paper 2005-0450, Reno, NV, January 10-13, 2005

• Leoviriyakit, K., Kim, S., and Jameson, A., "Aero-Structural Wing Planform Optimization Using the Navier-Stokes Equations", 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA Paper 2004-4479, Albany, New York, 30 August - 1 September 2004

• Leoviriyakit, K., and Jameson, A., "Case Studies in Aero-Structural Wing Planform and Section Optimization", 22nd Applied Aerodynamics Conference and Exhibit, AIAA Paper 2004-5372, Providence, Rhode Island, 16-19 August 2004

• Leoviriyakit, K. and Jameson, A., "Challenges and Complexity of Aerodynamic Wing Design ", International Conference on Complex Systems (ICCS2004), Boston, MA, May 16-21, 2004.

• Leoviriyakit, K., and Jameson, A., "Aero-Structural Wing Planform Optimization", 42nd AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2004-0029, Reno, Nevada, 5-8 January 2004

• Leoviriyakit, K., Kim, S., and Jameson, A., "Viscous Aerodynamic Shape Optimization of Wings Including Planform Variables", 21st Applied Aerodynamics Conference, AIAA Paper 2003-3498 , Orlando, Florida, 21-22 June 2003

• Kim, S., Leoviriyakit, K., and Jameson, A., "Aerodynamic Shape and Planform Optimization of Wings Using a Viscous Reduced Adjoint Gradient Formula", Second M.I.T. Conference on Computational Fluid and Solid Mechanics at M.I.T., Cambridge, MA, June 17-20, 2003

• Leoviriyakit, K. and Jameson, A., "Aerodynamic Shape Optimization of Wings including Planform Variations", 41st AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2003-0210, Reno, NV, January 6-9, 2003.