Transcript AFOSR 2004 - Stanford University

Multi-point Wing Planform Optimization
via Control Theory
Kasidit Leoviriyakit
and
Antony Jameson
Department of Aeronautics and Astronautics
Stanford University, Stanford CA
43rd Aerospace Science Meeting and Exhibit
January 10-13, 2005
Reno Nevada
1
Typical Drag Break Down of an Aircraft
Mach .85 and CL .52
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
2
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
3
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
4
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
5
“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
Suggested
Baseline
Boeing 747 -Planform Optimization
6
Redesign of Section and Planform
using a Single-point Optimization
Redesign
Baseline
Flight Condition (cruise): Mach .85 CL .45
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)
7
The Need of Multi-Point Design
Designed Point
Undesired characteristics
8
Cost Function for a Multi-point Design
I  1I1  2 I 2   n I n
Gradients
g  1g1  2 g2   n gn
9
Multi-point Design Process
10
Review of Single-Point design
using an Adjoint method
Design Variables
Using 4224 mesh points
on the wing as design variables
Boeing 747
Plus 6 planform variables
-Sweep
-Span
-Chord at 3span –stations
-Thickness ratio
11
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.
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
4231 flow calculations
~ 30 minutes each (RANS)
Too Expensive
Boeing 747
Plus 6 planform variables
13
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
(RANS in our case)
 R 
 R 

w

S  0



 w 
 S 
R  
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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
15
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
16
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

See paper for more detail
0


  

ij j1


f vi    ij j 2 
  ij j 3 


T
u


k
 j ij
xi 

17
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: SYN107
Finite Volume, RANS, SLIP Schemes,
Residual Averaging, Local Time Stepping Scheme,
Full Multi-grid
18
Single-point Redesign using
at Cruise condition
19
Isolated Shock Free Theorem
Mach .90
Mach .84
“Shock Free solution is an isolated point, away
Mach from
.85 the
point shocks will develop”
Morawetz 1956
20
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.
21
3-Point Design for Sections alone
(Planform fixed)
Condition Mach
1
2
3
0.84
0.86
0.90

1/3
1/3
1/3
22
Successive 2-Point Design for Sections
(Planform fixed)
MDD is dramatically improved
Condition
Mach

1
2
0.82
0.92
1/2
1/2
23
Lift-to-Drag Ratio of the Final Design
24
Cp at Mach 0.78, 0.79, …, 0.92
•Shock free solution no longer exists.
•But overall performance is significantly improved.
25
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.
26
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/
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