Multi-point%20Wing%20Planform%20Optimization%20via%20Control%20Theory

1
Multi-point Wing Planform Optimizationvia
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

2
Typical Drag Break Down of an Aircraft
Mach .85 and CL .52
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
Induced Drag is the largest component
3
Cost Function
Simplified Planform Model
Can be thought of as constraints
4
Choice of Weighting Constants
Minimizing
Maximizing Range
using
5
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

6
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
7
Redesign of Section and Planformusing 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)
8
The Need of Multi-Point Design
Designed Point
9
Cost Function for a Multi-point Design
Gradients
10
Multi-point Design Process
11
Review of Single-Point designusing 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
12
Optimization and Design using Sensitivities
Calculated by the Finite Difference Method
f(x)
13
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
14
Application of Control Theory (Adjoint)
GOAL Drastic Reduction of the Computational
Costs
Drag Minimization
Optimal Control of Flow Equations subject to
Shape(wing) Variations
(for example CD at fixed CL)
(RANS in our case)
15
Application of Control Theory
4230 design variables
One Flow Solution One Adjoint Solution
16
Sobolev Gradient
Continuous descent path
17
Design using the Navier-Stokes Equations
See paper for more detail
18
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 a CW at fixed CL
• (CD and CW are normalized by fixed
  reference area)
• a 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

19
Single-point Redesign using at Cruise condition
20
Isolated Shock Free Theorem
Shock Free solution is an isolated point, away
from the point shocks will develop
Morawetz 1956
21
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.

22
3-Point Design for Sections alone (Planform
fixed)
Condition Mach b
1 2 3 0.84 0.86 0.90 1/3 1/3 1/3
23
Successive 2-Point Design for Sections(Planform
fixed)
Condition Mach b
1 2 0.82 0.92 1/2 1/2
MDD is dramatically improved
24
Lift-to-Drag Ratio of the Final Design
25
Cp at Mach 0.78, 0.79, , 0.92

• Shock free solution no longer exists.
• But overall performance is significantly improved.

26
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.

27
Acknowledgement

• This work has benefited greatly from the support
  of Air Force Office of Science Research under
  grant No. AF F49620-98-2005

Downloadable Publicationshttp//aero-comlab.stan
ford.edu/http//www.stanford.edu/kasidit/