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

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

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

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.

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

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

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

Choice of Weighting Constants Minimizing Maximizing Range using

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

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

Optimization and Design using Sensitivities Calculated by the Finite Difference Method f(x)

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

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)

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

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

Design using the Navier-Stokes Equations

Adjoint Equations

Adjoint Boundary Condition Cost Function Adjoint Boundary Condition

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

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

Viscous Results B747 MD11 BAe MDO Datum

B747 Planform Changes Mach .85 Fixed CL .45 baseline redesigned

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

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

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.

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

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)

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

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)

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)

Pareto Front of Boeing 747

Appendix

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

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

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

Planform Gradient Calculation Surface Domain

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.