Efficient Aerodynamic Shape Optimization

Efficient Aerodynamic Shape Optimization Antony Jameson Department of Aeronautics and Astronautics Stanford University, Stanford, CA 13 th Conference on Finite Elements for Flow Problems University of Wales at Swansea, United Kingdom April 4-6, 2005 c  A. Jameson 2004 Stanford University, Stanford, CA 1/90 Efficient Aerodynamic Shape Optimization ☞ Aerodynamic Design Tradeoffs A good first estimate of performance is provided by the Breguet range equation: Range = V L D 1 SF C log W 0 + W f W 0 . (1) Here V is the speed, L/D is the lift to drag ratio, SF C is the specific fuel consumption of the engines, W 0 is the loading weight(empty weight + payload+ fuel resourced), and W f is the weight of fuel burnt. Equation (1) displays the multidisciplinary nature of design. A light structure is needed to reduce W 0 . SF C is the province of the engine manufacturers. The aerodynamic designer should try to maximize V L D . This means the cruising speed V should be increased until the onset of drag rise at a Mach Number M = V C ∼ .85. But the designer must also consider the impact of shape modifications in structure weight. c  A. Jameson 2004 Stanford University, Stanford, CA 2/90 Efficient Aerodynamic Shape Optimization ☞ Aerodynamic Design Tradeoffs The drag coefficient can be split into an approximate fixed component C D 0 , and the induced drag due to life. C D = C D 0 + C 2 L πεAR (2) where AR is the aspect ratio, and ε is an efficiency factor close to unity. C D 0 includes contributions such as friction and form drag. It can be seen from this equation that L/D is maximized by flying at a lift coefficient such that the two terms are equal, so that the induced drag is half the total drag. Moreover, the actual drag due to lift D v = 2L 2 περV 2 b 2 varies inversely with the square of the span b. Thus there is a direct conflict between reducing the drag by increasing the span and reducing the structure weight by decreasing it. c  A. Jameson 2004 Stanford University, Stanford, CA 3/90 Efficient Aerodynamic Shape Optimization ☞ Weight Tradeoffs a σ t d The bending moment M is carried largely by the upper and lower skin of the wing structure box. Thus M = σ t d a For a given stress σ, the required skin thickness varies inversely as the wing depth d. Thus weight can be reduced by increasing the thickness to chord ratio. But this will increase shock drag in the transonic region. c  A. Jameson 2004 Stanford University, Stanford, CA 4/90 Efficient Aerodynamic Shape Optimization ☞ Overall Design Process 15−30 engineers 1.5 years $6−12 million $60−120 million 6000 engineers Weight, performance Preliminary sizing Defines Mission $3−12 billion 5 years 2.5 years Final Design 100−300 engineers Design Preliminary Design Conceptual Figure 1: The Overall Design Process c  A. Jameson 2004 Stanford University, Stanford, CA 5/90 Efficient Aerodynamic Shape Optimization ☞ Cash flow −12 b 400 aircraft 80 b sales Year Economic Projection (Jumbo Jet) Preliminary Design 9 15 (if atleast 100 orders) Launch Conceptual Design −300 m Decisions here decide final cost and performance Leads to performance guarantees Detailed Design and certification −12 −2 −4 −6 −8 −10 4 Cash Flow $ billion 1.5 c  A. Jameson 2004 Stanford University, Stanford, CA 6/90 Efficient Aerodynamic Shape Optimization ☞ Aerodynamic Design Process { Propulsion Noise Stability Control Loads Structures Fabrication Conceptual Design CAD Definition Mesh Generation CFD Analysis Visualization Performance Evaluation Multi−Disciplinary Evaluation Wind Tunnel Testing Central Database Detailed Final Design Release to Manufacturing Model Fabrication Outer Loop Major Design Cycle Inner Loop Figure 2: The Aerodynamic Design Process c  A. Jameson 2004 Stanford University, Stanford, CA 7/90 Efficient Aerodynamic Shape Optimization ☞ Vision Effective Simulation Simulation−based Design c  A. Jameson 2004 Stanford University, Stanford, CA 8/90 Efficient Aerodynamic Shape Optimization ☞ Automatic Design Based on Control Theory ➣ Regard the wing as a device to generate lift (with minimum drag) by controlling the flow ➣ Apply theory of optimal control of systems governed by PDEs (Lions) with boundary control (the wing shape) ➣ Merge control theory and CFD c  A. Jameson 2004 Stanford University, Stanford, CA 9/90 Efficient Aerodynamic Shape Optimization ☞ Automatic Shape Design via Control Theory ➣ Apply the theory of control of partial differential equations (of the flow) by boundary control (the shape) ➣ Find the Frechet derivative (infinite dimensional gradient) of a cost function (performance measure) with respect to the shape by solving the adjoint equation in addition to the flow equation ➣ Modify the shape in the sense defined by the smoothed gradient ➣ Repeat until the performance value approaches an optimum c  A. Jameson 2004 Stanford University, Stanford, CA 10/90 Efficient Aerodynamic Shape Optimization [...]... respect to the shape by solving the adjoint equation in addition to the flow equation ¢ Modify the shape in the sense defined by the smoothed gradient ¢ Repeat until the performance value approaches an optimum c A Jameson 2004 Stanford University, Stanford, CA 10/90 Efficient Aerodynamic Shape Optimization Aerodynamic Shape Optimization: Gradient Calculation For the class of aerodynamic optimization problems... Stability Control Loads Structures Fabrication Model Fabrication Figure 2: The Aerodynamic Design Process c A Jameson 2004 Stanford University, Stanford, CA 7/90 Efficient Aerodynamic Shape Optimization Vision Effective Simulation Simulation−based Design c A Jameson 2004 Stanford University, Stanford, CA 8/90 Efficient Aerodynamic Shape Optimization Automatic Design Based on Control Theory ¢ Regard the wing... Stanford University, Stanford, CA 11/90 Efficient Aerodynamic Shape Optimization Aerodynamic Shape Optimization: Gradient Calculation Computing the gradient of a cost function for a complex system can be a numerically intensive task, especially if the number of design parameters is large and the cost function is an expensive evaluation The simplest approach to optimization is to define the geometry through... systems governed by PDEs (Lions) with boundary control (the wing shape) ¢ Merge control theory and CFD c A Jameson 2004 Stanford University, Stanford, CA 9/90 Efficient Aerodynamic Shape Optimization Automatic Shape Design via Control Theory ¢ Apply the theory of control of partial differential equations (of the flow) by boundary control (the shape) ¢ Find the Frechet derivative (infinite dimensional gradient)... Year 400 aircraft Detailed Design −4 15 and certification 80 b sales −6 −8 −10 Launch (if atleast 100 orders) −12 −12 b c A Jameson 2004 Stanford University, Stanford, CA 6/90 Efficient Aerodynamic Shape Optimization Aerodynamic Design Process Conceptual Design CAD Definition Central Database Inner Loop Outer Loop Major Design Cycle Mesh Generation Detailed Final Design CFD Analysis Visualization Performance... Jameson 2004 Stanford University, Stanford, CA 1 2 jpq 16/90 ∂xp ∂xq irs ∂ξr ∂ξs (7) Efficient Aerodynamic Shape Optimization Design using the Euler Equations Then 1 ∂ Sij = jpq ∂ξi 2 = 0 irs     2 2 ∂ xp ∂xq ∂xp ∂ xq   +  ∂ξr ∂ξi ∂ξs ∂ξr ∂ξs∂ξi  (8) Also in the subsequent analysis of the effect of a shape variation it is useful to note that ∂xp ∂xq , S1j = jpq ∂ξ2 ∂ξ3 ∂xp ∂xq , S2j = jpq ∂ξ3... δF δI = ∂w ∂F The flow field equation and its first variation are R(w, F) = 0 ∂R  ∂R      δw +   δF δR = 0 = ∂w ∂F  c A Jameson 2004 Stanford University, Stanford, CA  13/90   Efficient Aerodynamic Shape Optimization Symbolic Development of the Adjoint Method (cont.) Introducing a Lagrange Multiplier, ψ, and using the flow field equation as a constraint       ∂R   ∂I T ∂I T ∂R   T ... arbitrarily large number of design variables at a single design point to δI =     One Flow Solution + One Adjoint Solution c A Jameson 2004 Stanford University, Stanford, CA 14/90 Efficient Aerodynamic Shape Optimization Design using the Euler Equations The three-dimensional Euler equations may be written as ∂w ∂fi + = 0 in D, ∂t ∂xi where         ρ         ρui          ... function Also,  1 2 p = (γ − 1) ρ E − ui  ,   2    and ρH = ρE + p where γ is the ratio of the specific heats c A Jameson 2004 Stanford University, Stanford, CA 15/90  (5) (6) Efficient Aerodynamic Shape Optimization Design using the Euler Equations In order to simplify the derivation of the adjoint equations, we map the solution to a fixed computational domain with coordinates ξ1, ξ2, ξ3 where... in each design ∂αi parameter in turn and recalculating the flow to obtain the change in I Then I(αi + δαi) − I(αi) ∂I ≈ ∂αi δαi c A Jameson 2004 Stanford University, Stanford, CA 12/90 Efficient Aerodynamic Shape Optimization Symbolic Development of the Adjoint Method Let I be the cost (or objective) function I = I(w, F) where w = flow field variables F = grid variables The first variation of the cost function . University, Stanford, CA 10/90 Efficient Aerodynamic Shape Optimization ☞ Aerodynamic Shape Optimization: Gradient Calculation For the class of aerodynamic optimization problems under consideration, the design. minimum. c  A. Jameson 2004 Stanford University, Stanford, CA 11/90 Efficient Aerodynamic Shape Optimization ☞ Aerodynamic Shape Optimization: Gradient Calculation Computing the gradient of a cost function. also consider the impact of shape modifications in structure weight. c  A. Jameson 2004 Stanford University, Stanford, CA 2/90 Efficient Aerodynamic Shape Optimization ☞ Aerodynamic Design Tradeoffs The