Copyright 2004, K. Leoviriyakit and A. Jameson Challenges and Complexity of Aerodynamic Wing Design...

Copyright 2004, K. Leoviriyakit and A. Jameson

Challenges and Complexityof

Aerodynamic Wing Design

Kasidit Leoviriyakitand

Antony Jameson

Stanford UniversityStanford CA

http://aero-comlab.stanford.edu/

International Conference on Complex System (ICCS2004)

Boston, MA, USA

May 16-21, 2004

2


Airplane is a very complex system.

Flow is complex.

Everything has to work together:- Aerodynamic- Propulsion- Structure- Control

3


The Navier-Stokes Equationdw + dfi = dfvi

dt dxi dxi

ui 0

u1ui i1

fi = u2ui fvi = i2

u3ui i3

uiH ijuj + k dT/dxi

u1

w = u2

u3

E

p = (-1) {E - 1/2(ui ui)}

Solving the Navier-Stokes is extremely difficult.

4


Levels of CFD

Flow Prediction

Automatic Design

Interactive Calculation

• Integrate the predictive capability into an automatic design method that incorporates computer optimization.

• Attainable when flow calculation can be performed fast enough• But does NOT provide any guidance on how to change the shape if performance is unsatisfactory.

• Predict the flow past an airplane or its important components in different flight regimes such as take-off or cruise and off-design conditions such as flutter.• Substantial progress has been made during the last decade.

HIGHEST

LOWEST

5


Optimization and Design using Sensitivities Calculated by the Finite Difference Method

€

The simplest approach is to define the geometry as

f (x) = α∑ibi(x)

where α i = weight,

bi(x) = set of shape functions

Then using the finite difference method, a cost function

I = I(w,α ) (such as CD at constant CL )

has sensitivities ∂I

∂α i

≈I(α i + δα i) − I(α i)

δα i

If the shape changes is α n +1 = α n − λ∂I

∂α i

(with small positive λ )

The resulting improvements is I + δI = I −∂IT

∂αδα = I − λ

∂IT

∂α

∂I

∂α< I

More sophisticated search may be used, such as quasi - Newton.

f(x)

6


Disadvantage of the Finite Difference Method

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

Using 2016 mesh points on the wing as design variables

Boeing 747

2017 flow calculations ~ 2-5 minutes each (Euler)

Too Expensive

7


Application of Control Theory

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

€

≡

€

Define the cost function

I = I(w,S)

and a change in S results in a change

δI =∂I

∂w

⎡ ⎣ ⎢

⎤ ⎦ ⎥

T

δw +∂I

∂S

⎡ ⎣ ⎢

⎤ ⎦ ⎥

T

δS

Suppose that the governing equation R which expresses

the dependencd of w and S as

R(w,S) = 0

and

δR =∂R

∂w

⎡ ⎣ ⎢

⎤ ⎦ ⎥δw +

∂R

∂S

⎡ ⎣ ⎢

⎤ ⎦ ⎥δS = 0

GOAL : Drastic Reduction of the Computational Costs

e.g. Minimize CD

8


Application of Control Theory

€

Since the variation δR is zero, it can be multiplied by a Lagrange Multiplier ψ

and subtracted from the variation δI without changing the result.

δI =∂IT

∂wδw +

∂IT

∂FδS −ψ T ∂R

∂w

⎡ ⎣ ⎢

⎤ ⎦ ⎥δw +

∂R

∂S

⎡ ⎣ ⎢

⎤ ⎦ ⎥δS

⎛

⎝ ⎜

⎞

⎠ ⎟

=∂IT

∂w−ψ T ∂R

∂w

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

δw +∂IT

∂S−ψ T ∂R

∂S

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

δS

Choosing ψ to satisfy the adjoint equation

∂R

∂w

⎡ ⎣ ⎢

⎤ ⎦ ⎥

T

ψ =∂I

∂w

the first term is eliminated, and we find that

δI =∂I

∂S

T

−ψ T ∂R

∂S

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

δS

One Flow Solution + One Adjoint Solution

(Adjoint)

(Gradient)

2016 design variables

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Advantage of the Adjoint Method:

• Gradient for N design variables with cost equivalent to two flow solutions

• Minimal memory requirement in comparison with automatic differentiation

• Enables shapes to be designed as free surface• No need for user defined shape function• No restriction on the design space

2016 design variables

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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

11


Summary of the ContinuousFlow and Adjoint Equations

€

With computational coordinates ξ i

Euler equations for the flow :

(1) ∂

∂ξ i

Sij f j (w) = 0

where Sij are metrices, f j (w) the fluxes.

Adjoint equation

(2) Ci

∂ψ

∂ξ i

= 0, Ci = Sij

∂f j

∂w

Boundary condition for the Inverse problem

(3) I =1

2( p − pt )

2 ds∫ ψ 2nx +ψ 3ny +ψ 3nz = p − pt

Gradient

(4) δI = −∂ψ T

∂ξ i

δSij f jdD∫ D − δS21ψ 2 + δS22ψ 3 + δS23ψ 4( )pdξ1β w

∫ dξ 3∫

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Sobolev Gradient

Continuous descent path€

Define the gradient with respect to the Sobolev inner product

δI = < g,δf > = gδf + εg'δf '( )dx∫Set

δf = − λ g, δI = − λ < g,g >

This approximates a continuous descent process

df

dt= −g

The Sobolev gradient g is obtained from the simple gradient

g by the smoothing equation

g −∂

∂xε

∂g

∂x= g.

Key issue for successful implementation of the Continuous adjoint method.

13


Computational Costs with N Design Variables(Jameson and Vassberg 2000)

Cost of Search Algorithm

Steepest Descent (N2)

Quasi-Newton (N )

Sobolev Gradient (K )

(Note: K is independent of N)

Total Computational Cost of Design

Finite Difference Gradients

+ Steepest Descent

(N3)

Finite Difference Gradients

+ Quasi-Newton Search or Response surface

(N2)

Adjoint Gradient

+ Quasi-Newton Search

(N )

Adjoint Gradient

+ Sobolev Gradient

(K )

(Note: K is independent of N)

- N~2000- Big Savings- Enables Calculations on a Laptop

14


Redesign of the Boeing 747 Wing at its Cruise Mach NumberConstraints : Fixed CL = 0.42

: Fixed span-load distribution: Fixed thickness 13% wing drag saving

(5 minutes cpu time - 1proc.)

~6% aircraft drag saving

baseline

redesign

Euler Calculation

15


Redesign of the Boeing 747 Wing at its Cruise Mach NumberConstraints : Fixed CL = 0.42

: Fixed span-load distribution: Fixed thickness 10% wing drag saving

(3 hrs cpu time - 16proc.)

~5% aircraft drag saving

baseline

redesign

RANS Calculations

16


Redesign of the Boeing 747 Wing at Mach 0.9 “Sonic Cruiser”

Constraints : Fixed CL = 0.42: Fixed span-load distribution: Fixed thickness Same CD @Cruise

We can fly faster at the same drag.

RANS Calculations

17


Planform and Aero-Structural Optimization

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 ~ .47 (including fuselage lift ~ 15%)

Induced Drag is the largest component

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Wing Planform Optimization

€

I = α 1CD + α 2

1

2(p − pd )2 dS∫ + α 3CW

where

CW =Structural Weight

q∞Sref

Simplified Planform Model

Wing planform modification can yield largerimprovements BUT affects structural weight.

Can be thoughtof as constraints

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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

δR

R= −

δCD

CD

+1

logWTO

WO

δWO

WO

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ = −

δCD

CD

+1

logCWTO

CWO

δCWO

CWO

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

Minimizing

€

I = CD +α 3

α 1

CW

€

≡ using

€

α3

α 1

=CD

CWOlog

CWTO

CW0

MaximizingRange

20


Planform Optimization of Boeing 747

Baseline

Redesign

Constraints : Fixed CL=0.42

CD CW

Baseline 108 455

Optimize Section at Fixed planform 94 455

Optimize both section and planform 87 450

1) Longer span reduces the induced drag2) Less sweep and thicker wing sectionsreduces structure weight3) Section modification keepsshock drag minimum

Over: Drag and Weight Savings

21


Pareto Front: “Expanding the range of the Designs”

Use multiple αα ==> Multiple Optimal Shapes

Boundary of realizable designs

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Conclusion

• Enables aerodynamic design by a small team of experts focusing on the true design issues.

• Significant reduction in time and cost.

• Potential for superior and unconventional designs.

• Aerodynamic wing design is very complex due to the complexity nature of flow around the wing.

• By exploring the adjoint method, aerodynamic wing design can be carried out rapidly and cost efficiently.

Pay-Off

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Acknowledgement

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

• Optimization codes are developed by Intelligent Aerodynamic Inc.

http://aero-comlab.stanford.edu/

24


Aerodynamic Design Process

Preliminary Design Level

Automatic Design

Problems:• Need high level of expertise to improve the design.• Re-generating mesh is time consuming.

Aero-Structural Design

25


Redesign of the Boeing 747: Drag Rise( Three-Point Design )

Improved L/D

Improved MDD

Lower drag at the same MachNumber

Fly faster with the same drag

benefit

benefit

Constraints : Fixed CL = 0.42: Fixed span-load distribution: Fixed thickness

RANS Calculations

26


Planform and Aero-Structural Optimization

• Design tradeoffs suggest an multi-disciplinary design and optimization

€

Range =VL

D

1

sfclog

Wo + W f

Wo

Maximize Minimize

Planform variations can further maximize VL/D but affects WO

27


Aerodynamic Design Tradeoffs

€

The drag coefficient can be split into

CD = CDO +CL

2

πeAR

€

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

πeρV 2b2

Design dilemmaIncrease b

Di decreases

WO increases

Change span by changing planform

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Additional Features Needed

• Structural Weight Estimation• Large scale gradient : span, sweep, etc…• Adjoint gradient formulation for dCw/dx

• Choice of α1, α2, and α3

Use box wing to estimate the structural weight.

Large scale gradient

• Use summation of mapped gradients to be large scale gradient

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Planform Optimization of MD11

Baseline

Redesign

Constraints : Fixed CL=0.45

CD CW

Baseline 159 345

Optimize Section at Fixed planform 145 346

Optimize both section and planform 138 344

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Copyright 2004, K. Leoviriyakit and A. Jameson Challenges and Complexity of Aerodynamic Wing Design...

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