1. 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

2. 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

3. Break Down of Drag Item CD Cumulative CD
Boeing 747 at CL ~ .52 (including fuselage lift ~ 15%)
Item CD
Cumulative CD
Wing Pressure
120 counts
(15 shock, 105 induced)
Wing friction 45
165
Fuselage 50
215
Tail 20
235
Nacelles
255
Other 15
270
___
Total
Induced Drag is the largest component

4. Use “shock-free” concept to drive the planform design.
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
If we want to have large drag reduction, we should
target the induced drag.
Change span by
changing planform
Design dilemma
Di decreases
Increase b
WO increases

6. 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
There is no need for optimization
Also true for the sweep variation

7. Cost Function Wing planform modification can yield larger
Simplified Planform Model
Wing planform modification can yield larger
improvements BUT affects structural weight.
Can be thought
of as constraints

8. Choice of Weighting Constants
Minimizing
Maximizing
Range
using

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
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
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
(for example CD at fixed CL)
(Euler & RANS in our case)

14. Application of Control Theory
4230 design
variables
One Flow Solution + One Adjoint Solution

15. Outline of the Design Process
Flow solution
Adjoint solution
Gradient calculation
Sobolev gradient
Shape & Grid Modification
Repeated until Convergence
to Optimum Shape
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

17. Adjoint Equations

18. Adjoint Boundary Condition
Cost Function
Adjoint Boundary Condition

19. Viscous Gradient Comparison: Adjoint Vs Finite Difference
Sweep
croot
Sweep
Span
croot
cmid
ctip t
cmid
ctip t
Span
Adjoint gradient in red
Finite-different gradient in blue

20. Sobolev Gradient
Key issue for successful implementation of the Continuous adjoint method.
Continuous descent path

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 Viscous-Redesigned Baseline CL CD counts
using Syn107
(RANS Optimization)
Baseline CL CD
counts CW 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

24. Design Short-Cut Use Euler planform optimization as a starting point
for the Navier-Stokes Optimization
Euler Optimized
NS Optimized

25. Redesigned Planform of Boeing 747
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.

26. MD11 Planform Changes Mach .83, Fixed CL .50
baseline
redesigned

27. MD11 @Mach .83, Fixed CL .5 “Same Trend” Redesign Baseline
Span increases
Sweep decreases
t/c increases
Shock minimized
Redesign
Baseline CL CD
counts CW
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)

28. BAe Planform Changes Mach .85 Fixed CL .45
baseline
redesigned

29. BAe MDO Datum @ Mach .85, Fixed CL .45
“Same Trend”
but not EXTREME
Redesign
Baseline CL CD
counts CW
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)

30. Pareto Front: “Expanding the Range of Designs”
The optimal shape depends on the ratio of a3/a1
Use multiple values a3/a1 to capture the Pareto front
(An alternative to solving the optimality condition)

31. Pareto Front of Boeing 747

32. Appendix

33. Constraints Enforced in SYN107 and SYN88
For drag minimization
Fixed CL
Fixed span load
Keep out-board CL low enough to prevent buffet
Fixed root bending moment
Maintain specified thickness
Sustain root bending moment with equal structure weight
Maintain fuel volume
Smooth curvature variations via Sobolev gradient

34. Point Gradient Calculation for the wing sections
Use the surface mesh points as the section design variable
Perturb along the mesh line  Avoid mesh crossing over

35. Planform Gradient Calculation
E.g.. Gradient with respect to sweep change

36. Planform Gradient Calculation
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 , 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 , 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 , Providence, Rhode Island, 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 , 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 , Orlando, Florida, 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 , Reno, NV, January 6-9, 2003.