Applications of adjoint based shape optimization to the ...aero-comlab.stanford.edu/Papers/AIRBUS2013.pdf · Early jet airplanes New generation jet airplanes Noise footprint based

Applications of adjoint based shape

optimization to the design of low drag

airplane wings, including wings to

support natural laminar flow

Antony Jameson and Kui Ou

Aeronautics & Astronautics Department, Stanford University

Airbus UK, Filton, August 22nd, 2013

Challenge of Aeronautical Sciences in the 21st Century

Sustainable aviation is the challenge facing the aerospace community.

Antony Jameson and Kui Ou Aerodynamic Design

Outline of the Presentation

1 Introduction

X Challenge of Sustainable Aviation

2 Fundamentals of Aerodynamic Design

X Minimizing Aerodynamic Drag

3 Aerodynamic Design Methods

X Adjoint Based Aerodynamic Shape OptimizationX Adjoint Method with Transition Prediction

4 Results

5 Conclusion

Antony Jameson and Kui Ou Aerodynamic Design

Introduction

Challenge of Sustainable Aviation

Introduction

Challenge 1 : Modern Efficiency

X Modern airplanes have vastly improved fuel efficiency.

X Relative fuel use per seat-km is reduced by 70%

Environment

Track record of significant progress

Early jet airplanes

New generation jet airplanes

Noise footprint based on 85 dBa

1950s 1990s

Re

lati

ve

fu

el u

se

No

ise

dB

Hig

her

Lo

wer

Mo

reLess

90% Reduction in noise footprint

70% Fuel improvement and CO2 efficiency

Introduction

Challenge 2 : Growing Air Traffic

X Projected 5% annual growth in air traffic

X 3 fold increase in global air travel in the next 30 years

X Each long distance flight of B747 adds 400 tons of CO2 to the atmosphere

X Fastest growing source of greenhouse gases

X Climate impact will overtake cars by the end of the decade

Introduction

Challenge 3 : Resolving Conflicting Objectives

Goal: develop technology that will allow a tripling of capacity with a reduction in

environmental impact

EnvironmentThe commercial aviation challenge–carbon neutral growth

Using less fuel

t Efficient airplanes

t Operational efficiency

Changing the fuel

t Lower lifecycle CO2

t No infrastructure modifications

t “Sustainable biofuel”

2050Carbon neutral timeline2006

Presented to ICAO GIACC/3 February 2009 by Paul Steele on behalf of ACI, CANSO, IATA and ICCAIA

CO

2 E

mis

sio

ns

Forecasted emissions growth without reduction measures

Baseline

Introduction

Innovative Configurations and Optimum Aerodynamics

Challenge for aerodynamicists: maximize aerodynamic efficiency through innovativeconfigurations and minimization of drag.

Introduction

Fundamentals of Aerodynamic Design


Fundamentals

Range Equation for a Jet Aircraft

Wf = W =dW

dt=

dx

dt

dW

dx= V

dW

dx

Wf = −sfc · T = −sfc · D ·W

L

dx = VL

D

1

sfc

dW

W

R = VL

D|{z}

aero

1

sfc|{z}

prop

logW1

W0| {z }

struc


Fundamentals of Supersonic Flight

Drag of Subsonic and Transonic Aircraft

CD = CDP|{z}

parasite

+C2

L

π · AR · e| {z }

vortex

+ ∆CDC| {z }

compressibility

This leads to the need:X to minimize skin friction drag, maximize natural laminar flow

X to minimize transonic wave drag, minimize shock wave strength

X to minimize induced drag for a given lift, increase wing span


Fundamentals

Maximum Lift to Drag Ratio

L

D max=

√π√

e√

AR

2√

CDp

where e is airplane efficiency factor, AR is wing aspect ratio, andCDp

is the parasite drag coefficient.

The optimum aerodynamic design:

X Large wing span

X Extensive natural laminar flow

X Shock free wing

The last two items, i.e. natural laminar wing design andaerodynamic shape optimization are the topics today.


Aerodynamic Shape Optimization


Hierarchy of Computational Aerodynamics

X Prediction of the flow past an airplane in the flight regimes

X Interactive calculations to allow rapid design improvement

X Automatic design optimization


Motivation for Automatic Computer Optimization

X Enable exploration of a large design space → free wing surface

X Some modifications are too subtle to allow trial and errormethods → changes smaller than boundary layer thickness

X Progress in flow solvers and computer hardware have enabledsufficiently rapid turn around time → RANS-base designoptimization under a minute using laptop computer


Aerodynamic Design through Inverse Method

X The airplane aerodynamic performance can be greatlyenhanced by tailoring airfoil shape.

X Inverse airfoil design problem has considerable theoretical andpractical interest.


Lighthill’s Inverse Method

profile P on z plane

⇓profile C on σ plane

1 Let the profile P be conformallymapped to an unit circle C

2 The surface velocity is q = 1h|∇φ|

where φ is the potential in thecircle plane,and h is the mapping modulush =

∣∣ dzdσ

∣∣

3 Choose q = qtarget

4 Solve for the maping modulush = 1

qtarget|∇φ|


Aerodynamic Design through Automatic Optimization

X Inverse design problem can be ill-posed

X Desired airfoil shape does not always exist

X A computationally efficient optimization method can dispensethe inverse method


Gradient for Design Optimization

Optimization gradient

If I is the cost function, and F is the shape function, then

δI = GT δF ,

where

G =∂IT

∂F − ψT

[∂R

∂F

]

.

An improvement can be made with a shape change

δF = −λG

where λ is positive and small enough.

Optimization with steepest descent

δI = −λG2 < 0

Descent towards a lower cost is hence guaranteedAerodynamic Shape Optimization

Automatic Design Optimization: Simple Approach

Define the geometry through a set of design parameters

f (x) =X

αibi (x).

Then a cost function I is selected , for example, to be the drag coefficient or the lift todrag ratio, and I is regarded as a function of the parameters αi .

G =∂I

∂αi

≈I (αi + δαi ) − I (αi )

δαi

.

Make a step in the negative gradient direction by setting

αn+1 = αn + δα,

whereδα = −λG

so that to first order

I + δI = I − GT δα = I − λGTG < I


Disadvantages

% Needs a number of flow calculations proportional to thenumber of design variables to estimate the gradient.% Computational costs prohibitive as the number of designvariables is increased.

X A more efficient method to compute sensitivity gradient isneeded


Aerodynamic Shape Optimization based on Control Theory

Automatic Design Based on Control Theory

X Regard the wing as a device to generate lift by controlling theflow

X Apply the theory of optimal control of PDEs [Lions ]

X Aerodynamic shape optimization via control theory


Abstract Description of the Adjoint Method: Cost Function

Definition of a cost function

I = I (w ,F)

Cost function are functions of flow variables w and wing shape F

Total variation of the cost function

δI =

[∂IT

∂w

]

δw +

[∂IT

∂F

]

δF

A change in F results in a change in the cost function

δI = δI (δw , δF): each variation of δF causes a variation of δw , which requires flows

to be recomputed to obtain the gradient of improvement → extremely expensive


Abstract Adjoint Concept: Constraint

Flow governing equations are the constraints

R (w ,F) = 0

Total variation of the governing equations

δR =

[∂R

∂w

]

δw +

[∂R

∂F

]

δF = 0

Lagrange multiplier

δR = 0 =⇒ ψT δR = 0


Abstract Adjoint Description: Lagrange Multiplier

Augmentation of cost function

δI = δI − ψT δR︸︷︷︸

0

Expansion and rearrangement of cost function

δI =

{∂IT

∂wδw +

∂IT

∂F δF}

− ψT

{[∂R

∂w

]

δw +

[∂R

∂F

]

δF}

=

{∂IT

∂w− ψT

[∂R

∂w

]}

δw +

{∂IT

∂F − ψT

[∂R

∂F

]}

δF

δI = δI (��δw , δF): cost variation is independent of δw , i.e. no additional flow-field

evaluations needed regardless of δF (e.g. infinitely dimensional) → extremely

effective, and attractive for design


Abstract Adjoint Description: Adjoint Equation

Adjoint equation

∂IT

∂w− ψT

[∂R

∂w

]

= 0

Resulting cost function

δI =

{

��:0∂IT

∂w− ψT

[∂R

∂w

]}

δw +

{∂IT

∂F − ψT

[∂R

∂F

]}

δF

δI = δI (δF): cost is associated with the additional solution of the adjoint equation →

independent of design space dimension, which can be extremely large


Abstract Adjoint Description: Optimization gradient

Optimization gradient

δI = GT δF ,where

G =∂IT

∂F − ψT

[∂R

∂F

]

.

An improvement can be made with a shape change

δF = −λG

where λ is positive and small enough.

Optimization with steepest descent

δI = −λG2 < 0

Descent towards a lower cost is hence guaranteedAerodynamic Shape Optimization

Advantages

X The advantage is that the cost function is independent of δw ,

X The cost of solving the adjoint equation is comparable to thatof solving the flow equations.

X When the number of design variables becomes large, thecomputational efficiency of the control theory approach overtraditional approach


Smoothed Design Gradient

Modified Sobolev Gradient

X It is key for successful implementation of adjoint approach

X It enables the generation of smooth shapes

X It leads to a large reduction in design iterations



X The optimization process tends to reduce the smoothness ofthe trajectory at each step

X Sobolev gradient preserves the smoothness class of theredesigned surface



Smoothed Gradient

Original: δI =

∫

G(x)δF(x)dx , δF = −λG

Smoothed: δI =

∫

G(x)δF(x)dx , δF = −λG

Smoothing Equation

G − ∂

∂xǫ∂

∂xG = G

Guaranteed Descent

Original Gradient: δI = −λ∫

G2dx

Sobolev Gradient: δI = −λ∫

(G2 + ǫ

(∂

∂xG)2

)dx



X implicit smoothing may be regarded as a preconditioner

X allows the use of much larger steps for the search procedure

X leads to a large reduction in the number of design iterationsneeded for convergence.



Smoothing Equation using Second-order Central Differencing

Gi − ǫ(Gi+1 − 2Gi + Gi−1

)= Gi , 1 ≤ i ≤ n

G = PGwhere P is the n × n tridiagonal matrix such that

P−1 =

1 + 2ǫ −ǫ 0 . 0ǫ . .

0 . . .

. . . −ǫ0 ǫ 1 + 2ǫ

Then in each design iteration, a step, δF , is taken such that

δF = −λPG


Outline of the Design Process

The design procedure

1 Solve the flow equations for ρ, u1, u2, u3 and p.

2 Solve the adjoint equations for ψ

3 Evaluate G and calculate Sobolev gradient G.

4 Project G into an allowable subspace with constraints.

5 Update the shape based on the direction of steepest descent.

6 Return to 1 until convergence is reached.


Design Cycle

Sobolev Gradient

Gradient Calculation

Flow Solution

Adjoint Solution

Shape & Grid

Repeat the Design Cycleuntil Convergence

Modification


Prediction of Transition and Natural

Laminar Flow

Prediction of Transition and Natural Laminar Flow

Optimization with Prediction of Transition Flows

Boundary-Layer Parameter

X σ∗

X θ

Tollmien-Schlichting Instability

NTS = f (Reθ,Hk ,Tw

Te)

X determine the onset of criticalpoint Reθ ≥ Reθ0

X compute Tollmien-Schlichtingamplification factor NTS

X transition occurred whenNTS ≈ 9



Cross-Flow Instability

Crossflow Reynolds number:

Recf =ρe |wmax |δcf

µe

Critical Reynolds number:

Recritical = 46Tw

Te

Amplification rate

α(Recf ,wmax

Ue

,Hcf ,Tw

Te

)

Crossflow amplification factor

NCF =

Z x

x0

αdx



Viscosity for RANS Baldwin-Lomax Model

τij = (µlaminar + µturbulent )

{∂ui

∂xj

+∂uj

∂xi

− 2

3[∂uk

∂xk

]

}

δij

Transition Prescription

X Create transition linesfrom NTS and NCF

X The wing surface andflow domain are dividedinto the laminar andturbulent regions X

Y

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

turb.

turb.

xtran_upper

xtran_lower

stagnationpoint



Coupling Transition Prediction withAdjoint Solver

1 RANS solution with prescribedtransition locations

2 Converge RANS solutionsufficiently

3 Evoke transition predictionusing converged flows

4 Compute NTS and NCF

5 Iterate until transition locationconverges

6 Continue RANS with updatedviscosity distribution

repeated until

Convergence

Flow Solver

Adjoint Solver

Gradient Calculation

Shape & Grid

Modification

QICTP boundary layer code

Transition Prediction Method

Transition Prediction ModuleCP

Xtran

Design Cycle


Application I: Drag Minimization


Application I: Viscous Optimization of Korn Airfoil

Initial Final

Mesh size=512 x 64, Mach number=0.75, CLtarget = 0.63,Reynolds number=20m


The Effect of Applying Smoothed Gradient

Unsmoothed Smoothed


Application II: Low Sweep Wing Design


Application II: Low Sweep Wing Redesign using RK-SGS

Scheme

Initial Final

Mesh size=256x64x48, Design Steps=15, Design variables=127x33=4191 surfacemesh points


Application III: Natural Laminar Wing

Design

Application III: Natural Laminar Wing Design

Airplane Geometry

X

Y

Z

X

Y

Z

(a) Wing-body Geometry (b) Wing-body Surface Mesh

Mesh size=256x64x48


Airplane Mesh

IAI-NLFP3

GRID 256 X 64 X 48

K = 1

IAI-NLFP3

GRID 256 X 64 X 48

K = 50

Mesh size=256x64x48


Wing Planform and Sectional Profiles

0 0.5 1 1.5 2

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

!5

!4

!3

!2

!1

0

1

(a) Cross Sectional Profiles (b) Wing Planform Shape


Comparison of Turbulent and Laminar Drag

M=.40 M=.50 M=.60 M=.65

CL Transitional Turbulent Transitional Turbulent Transitional Turbulent Transitional Turbulent

.40 112[ 67+45] 152[ 80+72] 113[ 69+44] 150[ 82+68] 120[ 76+44] 151[ 88+63] 128[ 83+45] 156[94+62]

.50 125[ 80+45] 165[ 93+72] 125[ 82+43] 164[ 96+68] 134[ 90+44] 167[103+64] 142[ 98+44] 172[110+62]

.60 141[ 95+46] 182[109+73] 141[ 98+43] 180[112+68] 150[107+43] 184[120+64] 160[117+43] 192[130+62]

.70 159[113+46] 201[128+73] 159[116+43] 200[132+68] 167[125+42] 205[141+64] 184[143+41] 219[158+61]

.80 179[133+46] 223[150+73] 181[137+44] 222[154+68] 190[148+42] 229[165+64] – –

.90 204[157+47] 240[172+68] 205[161+44] 247[179+68] 217[174+43] 255[192+63] – –

1.0 237[186+51] 267[199+68] 238[191+47] 275[208+67] 249[204+45] 286[223+63] – –

1.1 – – 277[226+51] 307[241+66] – – – –


Transitional Flow Prediction: M=.40, CL=.80

XYZ

Transition lines on the wing lower and upper surfaces. Red line: upper surface. Greenline: lower surface.


Design Points: M=0.40, CL=0.4, 0.6, 0.9, 1.0

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(a) M=.40, CL=.40 (b) M=.40, CL=.60

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(c) M=.40, CL=.90 (d) M=.40, CL=1.00

At M = .40, the lower limit of the operating range is set by the appearance of suctionpeak. This suction peak appears at the lower surface when CL < .40, and at theupper surface when CL > .90.



XYZ

Transition lines on the wing lower and upper surfaces. Red line: upper surface. Greenline: lower surface.



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(a) M=.50, CL=.40 (b) M=.50, CL=.60

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(c) M=.50, CL=1.00 (d) M=.50, CL=1.10

At M = .50, the lower limit of the operating range is set by the appearance of suctionpeak. This suction peak appears at the lower surface when CL < .40, and at theupper surface when CL > 1.0.



XYZ

Transition lines on the wing lower and upper surfaces. Red line: upper surface. Green

line: lower surface.Application III: Natural Laminar Wing Design


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(a) M=.60, CL=.40 (b) M=.60, CL=.60

(a) M=.60, CL=.40 (b) M=.60, CL=.60

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.BE-+@0NBMD2--G=43O-+@LB#+7/D

97-P-#;43

962--34>(?----9E2-3433;:=----9L2#34;:(?----------------------------------

)B7-+@0NBMD2--:;4GO-+@LB#+7/D

97-P-#;43

!"!#$%&'(#)*"$+!)!,$------------------------------------------------./012-34533----"671/2-849:;-----------------------------------------------------

<%2--94338----<=2-343:8;;----<.2#949;>>--------------------------------------

=?@ABC2---3----*?@ADE/62--34:F8(G#3(--------------------------------------------

HIAD2-:>JK-5>K-8F---------------------------------------------------------------

<62--34J:J----<D2-343:J>3----<L2#34:F((-----------------------------------

*MMN-+?0NAMC2--984>O-+?LA#+7/C

<7-P-#:43

<62--94:59----<D2-343(5:(----<L2#34>9;:----------------------------------

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<7-P-#:43

<62--34;(>----<D2-3433853----<L2#34(9;:----------------------------------

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<7-P-#:43

(c) M=.60, CL=.90 (d) M=.60, CL=1.00

At M = .60, the lower limit of the operating range is set by the appearance of suctionpeak at the lower surface, and the formation of shock at the upper surface. Thesuction peak appears at the lower surface when CL < .40, and the shock starts toform at the upper surface when CL > .90.


Design Points: M=0.65, CL=0.4, 0.5

!"!#$%&'(#)*"$+!)!,$------------------------------------------------./012-34563----"781/2#34659-----------------------------------------------------

:%2--34;33----:<2-3439=>(----:.2#346?5(--------------------------------------

<@ABCD2---3----*@ABEF/72--349?=GH#3(--------------------------------------------

IJBE2-=6?K-56K-;G---------------------------------------------------------------

:72--34=;5----:E2-3439695----:L2#34959;-----------------------------------

*MMN-+@0NBMD2--9;46O-+@LB#+8/D

:8-P-#=43

:72--346>=----:E2-34396>;----:L2#34(;(?----------------------------------

.BE-+@0NBMD2--6943O-+@LB#+8/D

:8-P-#=43

:72--34=?;----:E2#34339(?----:L2#349>35----------------------------------

)B8-+@0NBMD2--G=46O-+@LB#+8/D

:8-P-#=43

!"!#$%&'(#)*"$+!)!,$------------------------------------------------./012-34563----"781/2-3496:-----------------------------------------------------

;%2--34633----;<2-3439=9>----;.2#345?96--------------------------------------

<@ABCD2---3----*@ABEF/72--349?>9G#3(--------------------------------------------

HIBE2-:6JK-56K-=>---------------------------------------------------------------

;72--34(9>----;E2-3439563----;L2#349?96-----------------------------------

*MMN-+@0NBMD2--9=46O-+@LB#+8/D

;8-P-#:43

;72--34J33----;E2-3439>:J----;L2#34(J??----------------------------------

.BE-+@0NBMD2--6943O-+@LB#+8/D

;8-P-#:43

;72--34(5J----;E2#34339=3----;L2#34:3==----------------------------------

)B8-+@0NBMD2-->:46O-+@LB#+8/D

;8-P-#:43

(a) M=.65, CL=.40 (b) M=.65, CL=.50

At M = .65, the lower limit of the operating range is set by the appearance of suctionpeak at the lower surface, and the formation of shock at the upper surface. Theoperating range has become too narrow to be useful. In particular, the suction peakappears at the lower surface when CL <= 40, and the shock of moderate strength hasalready started to form at the upper surface when CL = .50.


Operating Envelope (Limit) of Natural Laminar Flows

0.4 0.45 0.5 0.55 0.6 0.650.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Mach Number

CL

The figure shows the operating boundaries within which favorablepressure distribution can be maintained.


Closing Remarks

Closing Remarks

Conclusions

Key Features of Adjoint Based Approach

X Efficient computation on affordable computers

X Multi-point optimization

X Variety of cost functions: inverse design, drag minimization,or a combination

Impact of Aerodynamic Shape Optimization

X Achieve highly optimized designs

X Subtle shape changes not achievable through trial and error

X Designer’s input is still important

X Shock free airfoil possible

X Efficient low sweep wing is possible

X Natural laminar flows wing possible over a wide range ofoperating conditions

Closing Remarks

Documents

Applications of adjoint based shape optimization to the ...aero-comlab.stanford.edu/Papers/AIRBUS2013.pdf · Early jet airplanes New generation jet airplanes Noise footprint based