Wing Planform Optimization via an Adjoint Method

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
Boeing 747 at CL .52 (including fuselage lift
15)
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
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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
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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
Simplified Planform Model
Can be thought of as constraints
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Choice of Weighting Constants
Minimizing
Maximizing Range
using
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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

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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
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Optimization and Design using Sensitivities
Calculated by the Finite Difference Method
f(x)
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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
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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)
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Application of Control Theory
4230 design variables
One Flow Solution One Adjoint Solution
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Outline of the Design Process

• 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

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Design using the Navier-Stokes Equations
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Adjoint Equations
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Adjoint Boundary Condition
Cost Function Adjoint Boundary Condition



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Viscous Gradient Comparison Adjoint Vs Finite
Difference
Sweep
croot
Sweep
Span
cmid
croot
ctip
t
cmid
ctip
t
Span

• Adjoint gradient in red
• Finite-different gradient in blue

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Sobolev Gradient
Continuous descent path
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Viscous Results
B747
MD11
BAe MDO Datum
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B747 Planform Changes Mach .85 Fixed CL .45
baseline
redesigned
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B747 _at_ 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
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Design Short-Cut
Use Euler planform optimization as a starting
point for the Navier-Stokes Optimization
Euler Optimized
NS Optimized
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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.

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MD11 Planform ChangesMach .83, Fixed CL .50
baseline
redesigned
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MD11 _at_Mach .83, Fixed CL .5

• Same Trend
• 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)
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BAe Planform ChangesMach .85 Fixed CL .45
baseline
redesigned
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BAe MDO Datum _at_ 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)
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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)

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Pareto Front of Boeing 747
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Appendix
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ConstraintsEnforced 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

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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

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