Multi-point Wing Planform Optimization via Control Theory
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Transcript of Multi-point Wing Planform Optimization via Control Theory
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Multi-point Wing Planform Optimizationvia Control Theory
Kasidit Leoviriyakitand
Antony Jameson
Department of Aeronautics and AstronauticsStanford University, Stanford CA
43rd Aerospace Science Meeting and ExhibitJanuary 10-13, 2005
Reno Nevada
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Typical Drag Break Down of an Aircraft
Item CD Cumulative CD
Wing Pressure 120 counts 120 counts(15 shock, 105 induced)
Wing friction 45 165Fuselage 50 215
Tail 20 235Nacelles 20 255
Other 15 270___
Total 270
Mach .85 and CL .52
Induced Drag is the largest component
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Cost Function
I 1CD 212
(p pd )2 dS 3CW
where
CW Structural WeightqSref
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 VLD
1sfc
logWO W f
WO
With fixed V , L, sfc, and (WO W f WTO ), the variation of R
can be stated as
RR
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
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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 based on material of the skin
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“Trend” for Planform Modification
Increase L/D without any penalty on structural weight by• Stretching span to reduce vortex drag• Decreasing sweep and thickening wing-section to reduce
structural wing weight• Modifying the airfoil section to minimize shock
Boeing 747 -Planform Optimization
Baseline
Suggested
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Redesign of Section and Planformusing a Single-point Optimization
CL CD
counts
CW
counts
Boeing 747 .453 137.0(102.4 pressure, 34.6 viscous)
498(80,480 lbs)
Redesigned 747 .451 116.7(78.3 pressure, 38.4 viscous)
464(75,000 lbs)
Baseline
Redesign
Flight Condition (cruise): Mach .85 CL .45
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The Need of Multi-Point Design
Undesired characteristics
Designed Point
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Cost Function for a Multi-point Design
nnIIII 2211
nn gggg 2211
Gradients
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Multi-point Design Process
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Review of Single-Point designusing an Adjoint method
Using 4224 mesh points on the wing as design variables
Boeing 747
Plus 6 planform variables-Sweep-Span-Chord at 3span –stations-Thickness ratio
Design Variables
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Optimization and Design using Sensitivities Calculated by the Finite Difference Method
Newton.-quasi assuch used, bemay search tedsophistica More
is tsimprovemen resulting The
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xb
xbxf
TT
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nn
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LD
i
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f(x)
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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
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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
SSRw
wRR
SwRSw
R
SSIw
wII
SSwII
TT
GOAL : Drastic Reduction of the Computational Costs
(for example CD at fixed CL)
(RANS in our case)
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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
SR
SIG
SGI
wI
wR
SSR
FIw
wR
wI
SSRw
wRS
SIw
wII
IR
TT
T
T
T
TT
TT
TTT
One Flow Solution + One Adjoint Solution
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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
gx
g.
Key issue for successful implementation of the Continuous adjoint method.
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Design using the Navier-Stokes Equations
ixT
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jij
jij
jij
vi
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ii
ii
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i
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vjijvijiji
vii
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ku
f
Hupuupuupuu
u
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w
fSFfSFwJW
FFt
W
D
3
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1
33
22
11
3
2
1
0
, ,
, , , where
0
as written becan equations Stokes-Navier the,domain nalcomputatioIn
See paper for more detail
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Test Case
• Use multi-point design to alleviate the undesired characteristics arising form the single-point design result.
• Minimizing at multiple flight conditions;I = CD + CW at fixed CL
(CD and CW are normalized by fixed reference area) is chosen also to maximizing the Breguet range equation
• Optimization: SYN107Finite Volume, RANS, SLIP Schemes, Residual Averaging, Local Time Stepping Scheme,Full Multi-grid
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Single-point Redesign using at Cruise condition
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Isolated Shock Free Theorem
Mach .84
Mach .85
Mach .90
“Shock Free solution is an isolated point, away from the point shocks will develop”
Morawetz 1956
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Design Approach
• If the shock is not too strong, section modification alone can alleviate the undesired characteristics.
• But if the shock is too strong, both section and planform will need to be redesigned.
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3-Point Design for Sections alone (Planform fixed)
Condition Mach
123
0.840.860.90
1/31/31/3
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Successive 2-Point Design for Sections(Planform fixed)
Condition Mach
12
0.820.92
1/21/2
MDD is dramatically improved
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Lift-to-Drag Ratio of the Final Design
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Cp at Mach 0.78, 0.79, …, 0.92
•Shock free solution no longer exists.•But overall performance is significantly improved.
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Conclusion
• Single-point design can produce a shock free solution, but performance at off-design conditions may be degraded.
• Multi-point design can improve overall performance, but improvement is not as large as that could be obtained by a single optimization, which usually results in a shock free flow.
• Shock free solution no longer exists.
• However, the overall performance, as measured by characteristics such as the drag rise Mach number, is clearly superior.
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Acknowledgement
This work has benefited greatly from the support of Air Force Office of Science Research under grant No. AF F49620-98-2005
Downloadable Publications
http://aero-comlab.stanford.edu/http://www.stanford.edu/~kasidit/