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http://aero-comlab.stanford.edu/. Challenges and Complexity of Aerodynamic Wing Design. Kasidit Leoviriyakit and Antony Jameson Stanford University Stanford CA. International Conference on Complex System (ICCS2004) Boston, MA, USA May 16-21, 2004. Airplane is a very complex system. - PowerPoint PPT Presentation
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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
Copyright 2004, K. Leoviriyakit and A. Jameson
Airplane is a very complex system.
Flow is complex.
Everything has to work together:- Aerodynamic- Propulsion- Structure- Control
3
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
9
Copyright 2004, K. Leoviriyakit and A. Jameson
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
10
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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∫
12
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
18
Copyright 2004, K. Leoviriyakit and A. Jameson
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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Copyright 2004, K. Leoviriyakit and A. Jameson
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
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Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
Pareto Front: “Expanding the range of the Designs”
Use multiple αα ==> Multiple Optimal Shapes
Boundary of realizable designs
22
Copyright 2004, K. Leoviriyakit and A. Jameson
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
23
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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
Copyright 2004, K. Leoviriyakit and A. Jameson
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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Copyright 2004, K. Leoviriyakit and A. Jameson
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
29
Copyright 2004, K. Leoviriyakit and A. Jameson
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