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1
Wing Planform Optimization via an Adjoint Method
Kasidit Leoviriyakit
Department of Aeronautics and AstronauticsStanford University, Stanford CA
Stanford UniversityStanford, CAJune 28, 2005
2
History: Adjoint for Transonic Wing Design
Baseline 747, CD 117 counts Redesigned, CD 103 counts
Redesign for a shock-free wing by modify the wing sections (planform fixed )– Jameson 1995
- Cp
3
Break Down of Drag
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 ~ .52 (including fuselage lift ~ 15%)
Induced Drag is the largest component
4
Key Concept
Use “shock-free” concept to drive the planform design.
• Conventionally the wing is swept to weaken the shock.
• With the “shock-free” wing capability, it allows more configurations that was previously prohibited by the strong shock.
5
Aerodynamic Design Tradeoffs
DWL
DOD CeAR
CCC
2
intosplit becan t coefficien drag The
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
eV 2b2
Design dilemma
Increase bDi decreases
WO increases
Change span by changing planform
6
Can we consider only pure Aerodynamic design?
• Example 1: Vary b to minimize drag
I = CD
As span increases, vortex drag decreases. Infinitely long span
• Example 2: Add a constraint;
b =bmax
There is no need for optimization
• Also true for the sweep variation
maxbb
• Pure aerodynamic design leads to unrealistic results
• Constraints sometimes prevent optimal results
7
Cost Function
I 1CD 2
1
2(p pd )2 dS 3CW
where
CW Structural Weight
qSref
Simplified Planform Model
Wing planform modification can yield largerimprovements BUT affects structural weight.
Can be thoughtof as constraints
8
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
R
R
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
9
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 from material of the skin
10
Design Parameters
Using 4224 mesh points on the wing as design variables
Boeing 747
Plus 6 planform variables
Use Adjoint method to calculate both section and planform sensitivities
11
Optimization and Design using Sensitivities Calculated by the Finite Difference Method
Newton.-quasi assuch used, bemay search tedsophistica More
is tsimprovemen resulting The
) positive small(with is changes shape theIf
)()(
iessensitivit has
)constant at as(such ),(
function cost a method, difference finite theusingThen
functions shape ofset )(
weight, where
)()(
asgeometry thedefine tois approach simplest The
1
III
II
III
I
III
CCwII
xb
xbxf
TT
i
nn
i
iii
i
LD
i
i
ii
f(x)
12
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
13
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
SS
Rw
w
RR
SwR
Sw
R
SS
Iw
w
II
S
SwII
TT
GOAL : Drastic Reduction of the Computational Costs
(for example CD at fixed CL)
(Euler & RANS in our case)
14
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
S
R
S
IG
SGI
w
I
w
R
SS
R
F
Iw
w
R
w
I
SS
Rw
w
RS
S
Iw
w
II
I
R
TT
T
T
T
TT
TT
TTT
One Flow Solution + One Adjoint Solution
15
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
Design Variables• 4224 surface mesh points for the NS design (or 2036 for the Euler design)• 6 planform parameters
-Sweep-Span-Chord at 3span –stations-Thickness ratio
16
Design using the Navier-Stokes Equations
ixT
ijj
jij
jij
jij
vi
i
ii
ii
ii
i
i
vjijvijiji
vii
ii
ku
f
Hu
puu
puupuu
u
f
E
u
uu
w
fSFfSFwJW
FFt
W
D
3
2
1
33
22
11
3
2
1
0
, ,
, , , where
0
as written becan equations Stokes-Navier the,domain nalcomputatioIn
17
Adjoint Equations
jl
l
k
k
ijl
k
k
i
j
j
i
l
jl
xlj
ljij
xkijxjxilj
xijxxlji
xljp
jiji
SL
S
uuuS
SL
SL
w
fSC
15
1
1
~
~
)(~
2
where
in 0~
1 DLMCT
i
Ti
18
Adjoint Boundary Condition
22)cos()cos(
i
1
2
21
22
and cosinedirection theis where
,
,
SndSpI
pnq
qdSqI
ppndSppI
jj
ijijji
kkii
djjd
Cost Function Adjoint Boundary Condition
19
Viscous Gradient Comparison: Adjoint Vs Finite Difference
Sweep
Span
cmidcroot ctip t
Sweep
Span
cmid
croot
ctip t
DCx
WCx
• Adjoint gradient in red• Finite-different gradient in blue
20
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
g
xg.
Key issue for successful implementation of the Continuous adjoint method.
21
Viscous Results
B747 MD11
BAe MDO Datum
22
B747 Planform Changes Mach .85 Fixed CL .45
baseline
redesigned
23
B747 @ Mach .85, Fixed CL .45
CL CD
counts
CW
counts
CM
Boeing 747 .453 137.0
(102.4 pressure, 34.6 viscous)
498
(80,480 lbs)
-.1408
Redesigned 747 .451 116.7
(78.3 pressure, 38.4 viscous)
464
(75,000 lbs)
-.0768
Baseline
Viscous-Redesignedusing Syn107
(RANS Optimization)
24
Design Short-CutUse Euler planform optimization as a starting point for the Navier-Stokes Optimization
Euler Optimized
NS Optimized
25
Redesigned Planform of Boeing 747
1. Longer span reduces the induced drag2. Less sweep and thicker wing sections reduce the
structural weight3. Section modification keeps the shock drag minimum
• Overall: Drag and Weight Savings
• No constraints posted on planform, but we still get a finite wingwith less than 90 degrees sweep.
26
baseline
redesigned
MD11 Planform ChangesMach .83, Fixed CL .50
27
MD11 @Mach .83, Fixed CL .5
CL CD
counts
CW
counts
MD 11 .501 179.8
(144.2 pressure, 35.6 viscous)
654
(62,985 lbs)
Redesigned MD11 .500 163.8
(123.9 pressure, 39.9 viscous)
651
(62,696 lbs)
“Same Trend”1. Span increases2. Sweep decreases3. t/c increases4. Shock minimized
Baseline
Redesign
28
BAe Planform ChangesMach .85 Fixed CL .45
baseline
redesigned
29
BAe MDO Datum @ Mach .85, Fixed CL .45
CL CD
counts
CW
counts
BAe .453 163.9
(120.5 pressure, 43.4 viscous)
574
(87,473 lbs)
Redesigned BAe .452 144.7
(99.3 pressure, 45.4 viscous)
570
(86,863 lbs)
“Same Trend”but not EXTREME
Baseline
Redesign
30
Pareto Front: “Expanding the Range of Designs”
WD CCI 31
• The optimal shape depends on the ratio of 3/1
• Use multiple values to capture the Pareto front
(An alternative to solving the optimality condition)
31
Pareto Front of Boeing 747
1
3
32
Appendix
33
ConstraintsEnforced in SYN107 and SYN88
For drag minimization
1. Fixed CL
2. Fixed span load• Keep out-board CL low enough to prevent buffet• Fixed root bending moment
3. Maintain specified thickness• Sustain root bending moment with equal structure
weight• Maintain fuel volume
4. Smooth curvature variations via Sobolev gradient
34
Point Gradient Calculation for the wing sections
D II
TIIII dDRdNMI
•Use the surface mesh points as the section design variable•Perturb along the mesh line Avoid mesh crossing over
35
Planform Gradient Calculation
D II
TIIII dDRdNMI
E.g.. Gradient with respect to sweep change
36
Planform Gradient Calculation
S SS
D
SSST
IIII RRNMI )(
Surface
Domain
37
References
• Leoviriyakit, K.,"Wing Planform Optimization via an Adjoint Method," Ph.D. Dissertation, Stanford University, March 2005.
• Leoviriyakit, and Jameson, A., "Multi-point Wing Planform Optimization via Control Theory", 43rd Aerospace Sciences Meeting and Exhibit, AIAA Paper 2005-0450, Reno, NV, January 10-13, 2005
• Leoviriyakit, K., Kim, S., and Jameson, A., "Aero-Structural Wing Planform Optimization Using the Navier-Stokes Equations", 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA Paper 2004-4479, Albany, New York, 30 August - 1 September 2004
• Leoviriyakit, K., and Jameson, A., "Case Studies in Aero-Structural Wing Planform and Section Optimization", 22nd Applied Aerodynamics Conference and Exhibit, AIAA Paper 2004-5372, Providence, Rhode Island, 16-19 August 2004
• Leoviriyakit, K. and Jameson, A., "Challenges and Complexity of Aerodynamic Wing Design ", International Conference on Complex Systems (ICCS2004), Boston, MA, May 16-21, 2004.
• Leoviriyakit, K., and Jameson, A., "Aero-Structural Wing Planform Optimization", 42nd AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2004-0029, Reno, Nevada, 5-8 January 2004
• Leoviriyakit, K., Kim, S., and Jameson, A., "Viscous Aerodynamic Shape Optimization of Wings Including Planform Variables", 21st Applied Aerodynamics Conference, AIAA Paper 2003-3498 , Orlando, Florida, 21-22 June 2003
• Kim, S., Leoviriyakit, K., and Jameson, A., "Aerodynamic Shape and Planform Optimization of Wings Using a Viscous Reduced Adjoint Gradient Formula", Second M.I.T. Conference on Computational Fluid and Solid Mechanics at M.I.T., Cambridge, MA, June 17-20, 2003
• Leoviriyakit, K. and Jameson, A., "Aerodynamic Shape Optimization of Wings including Planform Variations", 41st AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2003-0210, Reno, NV, January 6-9, 2003.