applications of adjoint based shape optimization to the...
TRANSCRIPT
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
ASDOM, August 17th, 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 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 Aerodynamic Design
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 of Aerodynamic Design
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.
Fundamentals of Aerodynamic Design
Aerodynamic Shape Optimization
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
Aerodynamic Shape 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 Shape Optimization
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.
Aerodynamic Shape Optimization
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 Shape Optimization
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
Aerodynamic Shape Optimization
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
Aerodynamic Shape Optimization
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
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
Aerodynamic Shape Optimization
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
Aerodynamic Shape Optimization
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
Aerodynamic Shape Optimization
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
Aerodynamic Shape Optimization
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
Aerodynamic Shape Optimization
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
Aerodynamic Shape Optimization
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
Aerodynamic Shape Optimization
Smoothed Design Gradient
X The optimization process tends to reduce the smoothness ofthe trajectory at each step
X Sobolev gradient preserves the smoothness class of theredesigned surface
Aerodynamic Shape Optimization
Smoothed Design Gradient
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
Aerodynamic Shape Optimization
Smoothed Design Gradient
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.
Aerodynamic Shape Optimization
Smoothed Design Gradient
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
Aerodynamic Shape Optimization
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.
Aerodynamic Shape Optimization
Design Cycle
Sobolev Gradient
Gradient Calculation
Flow Solution
Adjoint Solution
Shape & Grid
Repeat the Design Cycleuntil Convergence
Modification
Aerodynamic Shape Optimization
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
Prediction of Transition and Natural Laminar Flow
Optimization with Prediction of Transition Flows
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
Prediction of Transition and Natural Laminar Flow
Optimization with Prediction of Transition Flows
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
Prediction of Transition and Natural Laminar Flow
Optimization with Prediction of Transition Flows
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
Prediction of Transition and Natural Laminar Flow
Application I: Drag Minimization
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
Application I: Drag Minimization
The Effect of Applying Smoothed Gradient
Unsmoothed Smoothed
Application I: Drag Minimization
Application II: Low Sweep Wing Design
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 II: Low Sweep Wing Design
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
Application III: Natural Laminar Wing Design
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
Application III: Natural Laminar Wing Design
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
Application III: Natural Laminar Wing Design
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] – – – –
Application III: Natural Laminar Wing Design
Transitional Flow Prediction: M=.40, CL=.80
XYZ
Transition lines on the wing lower and upper surfaces. Red line: upper surface. Greenline: lower surface.
Application III: Natural Laminar Wing Design
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.
Application III: Natural Laminar Wing Design
Transitional Flow Prediction: M=.50, CL=.80
XYZ
Transition lines on the wing lower and upper surfaces. Red line: upper surface. Greenline: lower surface.
Application III: Natural Laminar Wing Design
Design Points: M=0.50, CL=0.4, 0.6, 0.9, 1.0
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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.
Application III: Natural Laminar Wing Design
Transitional Flow Prediction: M=.60, CL=.80
XYZ
Transition lines on the wing lower and upper surfaces. Red line: upper surface. Green
line: lower surface.Application III: Natural Laminar Wing Design
Design Points: M=0.60, CL=0.4, 0.6, 0.9, 1.0
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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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(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.
Application III: Natural Laminar Wing Design
Design Points: M=0.65, CL=0.4, 0.5
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(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.
Application III: Natural Laminar Wing Design
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.
Application III: Natural Laminar Wing Design
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