Transcript AFOSR 2004 - Stanford University

Wing Planform Optimization
via an Adjoint Method
Kasidit Leoviriyakit
Department of Aeronautics and Astronautics
Stanford University, Stanford CA
Stanford University
Stanford, CA
June 28, 2005
1
History: Adjoint for Transonic Wing Design
Redesign for a shock-free wing by modify the wing sections (planform fixed )
– Jameson 1995
- Cp
Baseline 747, CD 117 counts
Redesigned, CD 103 counts
2
Break Down of Drag
Boeing 747 at CL ~ .52 (including fuselage lift ~ 15%)
Item CD
Wing Pressure 120 counts
Cumulative CD
120 counts
(15 shock, 105 induced)
Wing friction 45
165
Fuselage 50
215
Tail 20
235
Nacelles 20
255
Other 15
270
___
Total 270
Induced Drag is the largest component
3
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.
4
Aerodynamic Design Tradeoffs
T hedrag coefficient can be split into
2
CD  CDO
C
 L  CDW
eAR
L
is maximized if the two terms are equal.
D
Induced drag is half of the total drag.

If we want to have large drag reduction, we should
target the induced drag.
2L2
Di 
eV 2b 2
Change span by
changing planform
Design dilemma
Increase
 b
Di decreases
WO increases
5
Can we consider only pure Aerodynamic design?
• Pure aerodynamic design leads to unrealistic results
• Constraints sometimes prevent optimal results
• 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
 b =bmax
There is no need for optimization
• Also true for the sweep variation
6
Cost Function
Simplified Planform Model
Wing planform modification can yield larger
improvements BUT affects structural weight.
1
I  1CD   2  ( p  pd ) 2 dS   3CW
2
where
Can be thought
Structural Weight
of as constraints
CW 
q Sref
7
Choice of Weighting Constants
Breguet range equation
W Wf
VL 1
log O
D sfc
WO
With fixedV, L, sfc, and (WO  W f  WTO ), t he variat ion of
R
R
can be stated as









C
R
C
1
WO
C
1
WO 
    D 
  D 
CWTO CW 
 CD
R
 CD log WTO WO 
O
log




WO


CWO



Maximizing
Range

Minimizing
I  CD 
3

1

3
C
1 W
CD
CWO log
CWTO
CW0
using
8
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
9
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
10
Optimization and Design using Sensitivities
Calculated by the Finite Difference Method
T hesimplest approach is to define thegeometryas
f ( x)    i bi ( x)
where  i
f(x)
 weight,
bi ( x)  set of shapefunctions
T henusing thefinitedifferencemethod,a cost function
I  I ( w,  )
has sensitivities
If theshapechangesis
(such as C D at constantC L )
I I ( i   i )  I ( i )

 i
 i
I
 n 1   n  
(with small positive )
 i
I T
I T I
T heresultingimprovements is I  I  I 
  I  
I

 
Moresophisticated search may be used, such as quasi - Newton.
11
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
4231 flow calculations
~ 30 minutes each (RANS)
Too Expensive
Boeing 747
Plus 6 planform variables
12
Application of Control Theory (Adjoint)
GOAL : Drastic Reduction of the Computational Costs
Drag Minimization

Optimal Control of Flow Equations
subject to Shape(wing) Variations
Define t hecost funct ion

I  I ( w, S )
(for example CD at fixed CL)
and a changein S result sin a change
 I 
 I 
I    w    S
 w 
 S 
T
T
Suppose t hat t hegoverningequat ion R which expresses
t hedependencdof w and S as
R( w, S )  0
and
(Euler & RANS in our case)
 R 
 R 

w

S  0



 w 
 S 
R  
13
Application of Control Theory
Since the variationR is zero,it can be multipliedby a LagrangeMultiplier
and subtractedfrom the variationI wit hout changing theresult.
  R 
I T
I T
 R  
I 
w 
S  T   w   S 
w
S
 S  
  w 
T
 I T



R

I


T
T  R  

   w  
   S
 w  
 S  
 w
 F
Choosing  to satisfy theadjoint equation
I
 R 


 w 
w
 
thefirst termis eliminated, and we find that
T
I  G T S
where
 I T

T  R 
G 
   
 S  
 S
4230 design
variables
T
One Flow Solution + One Adjoint Solution
14
Outline of the Design Process
Adjoint solution
Gradient calculation
Sobolev gradient
Repeated until Convergence
to Optimum Shape
Flow solution
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
Shape & Grid Modification
15
Design using the Navier-Stokes Equations
In comput at ional domain D, t he Navier- St okesequat ionscan be writ t enas
W



Fi 
Fvi  0
t  i
 i
where
W  J  w, Fi  Sij f j , Fvi  Sij f vj ,
  
 u 
 1 
w   u 2 ,
 u 3 


 E 
u i


 u u  p  
i1 
 i 1

f i   ui u 2  p i 2 ,
 u i u 3  p  i 3 


 ui H

0


  

ij j1


f vi    ij j 2 
  ij j 3 


T
u


k
 j ij
xi 

16
Adjoint Equations

1T ~
C
 M L  0
 i
T
i
in D
where
Ci  S ij
 
 
f j
w
~
L  1   p2 l ( Slj xj )
~

L  i 1  l Slj  xij  xij   ij
 l
 
~
L
5
 
S  u
lj
  ij S lj

1

  l


i x j
uj

xi
k
xk

  u 
k
ij k xk

 l
S  
lj

x j
17
Adjoint Boundary Condition
Cost Function
I
1
2
  p  p  dS,
2
d
I   qi  i dS,
Adjoint Boundary Condition
 j n j  p  pd
 k 1  qk

where qi is thedirectioncosineand i  n j  ij p   ij 
I

cos(  ) 2
 p   dS

 j n j  cos(  )    S 22
2
18
Viscous Gradient Comparison:
Adjoint Vs Finite Difference

 CD
x

Sweep

CW
x
croot
Sweep
Span
croot
cmid
ctip
t
cmid
ctip
t
Span
• Adjoint gradient in red
• Finite-different gradient in blue
19
Sobolev Gradient
Key issue for successful implementation of the Continuous adjoint method.
Define t he gradient wit h respect t o t he Sobolev inner product
I   g,f  
 gf   g'f 'dx
Set
f =   g, I     g,g 
T his approximat es a cont inuous descent process
df
 g
dt
T he Sobolev gradientg is obt ained from t he simple gradient
g by t he smoot hing equat ion
g
 g

 g.
x x
Continuous descent path
20
Viscous Results
B747
MD11
BAe MDO Datum
21
B747 Planform Changes
Mach .85 Fixed CL .45
baseline
redesigned
22
B747 @ Mach .85, Fixed CL .45
Viscous-Redesigned
using Syn107
(RANS Optimization)
Baseline
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
23
Design Short-Cut
Use Euler planform optimization as a starting point
for the Navier-Stokes Optimization
Euler Optimized
NS Optimized
24
Redesigned Planform of Boeing 747
1.
2.
3.
Longer span reduces the induced drag
Less sweep and thicker wing sections reduce the
structural weight
Section modification keeps the shock drag minimum
•
Overall: Drag and Weight Savings
•
No constraints posted on planform, but we still get a
finite wing
with less than 90 degrees sweep.
25
MD11 Planform Changes
Mach .83, Fixed CL .50
baseline
redesigned
26
MD11 @Mach .83, Fixed CL .5
“Same Trend”
1.
2.
3.
4.
Span increases
Sweep decreases
t/c increases
Shock minimized
Redesign
Baseline
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)
27
BAe Planform Changes
Mach .85 Fixed CL .45
baseline
redesigned
28
BAe MDO Datum @ Mach .85, Fixed CL .45
“Same Trend”
but not EXTREME
Redesign
Baseline
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)
29
Pareto Front: “Expanding the Range of Designs”
I  1CD  3CW
• The optimal shape depends on the ratio of 3/1
• Use multiple values 3/1 to capture the Pareto front
(An alternative to solving the optimality condition)
30
Pareto Front of Boeing 747
3
1
31
Appendix
32
Constraints
Enforced 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
33
Point Gradient Calculation for the wing sections
I   M II  N II d    RII dD
T

D
•Use the surface mesh points as the section design variable
•Perturb along the mesh line  Avoid mesh crossing over
34
Planform Gradient Calculation
I   M II  N II d    T RII dD

D
E.g.. Gradient with respect to sweep change
35
Planform Gradient Calculation
I   M II  N II   ( RS S  RS )
T

D
Surface
Domain
S
S  S
36
References
•
•
•
•
•
•
•
•
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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.
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