Single and multi-point aerodynamic optimizations of a supersonic transport aircraft using strategies involving adjoint equations and genetic algorithm

Transcription

1 Single and multi-point aerodynamic optimizations of a supersonic transport aircraft using strategies involving adjoint equations and genetic algorithm Prepared by : G. Carrier (ONERA, Applied Aerodynamics/Civil Transport) For : ERCOFTAC 2006 April 5-7, 2006, Las Palmas, Gran Canaria, Spain Office National d Études et de Recherches Aérospatiales

2 Contents Introduction Problem formulation, parameterization, tools Overview of the optimization framework Page 2 Multi-point optimizations Single point optimizations Conclusion

3 Introduction Significant progress achieved in CFD More robust and faster CFD codes are available Used routinely in aerospace industry for design Mature numerical optimization techniques available: Local (gradient based) Global (stochastic) CFD based optimizations may help the aerodynamic design of aerospace products At preliminary design stage At detailed design stage Motivation: Apply advanced CFD based optimization techniques to the aerodynamic design of a High-Speed Commercial Transport (HSCT) aircraft: Evolutionary algorithm Gradient based algorithm using the adjoint equations Page 3

4 Problem formulation Objective : Optimize the wing geometry of a HSCT for optimum performance at supersonic (M=2.0) and transonic (M=0.9) Mach Mach0.95 Mach range HSCT flight profile Multi-point optimization problem approach (weighted objectives) Minimize: k 1. CD M=2.0 (α) + k 2. CD M=0.9 (α) s.c. CL M=2.0 (α) 0.11 CL M=0.9 (α) 0.21 Geometrical constraints Page 4 with α=(α 1, α N ): wing shape design variables + A.o.A. at each flight points

5 Configuration to optimize Generic HSCT configuration Wing planform and thickness distribution are kept frozen Reference design with a «flat wing» : no twist, no camber Page 5

6 Parameterization 1 Simple parameterization: 6 geometry parameters 2 profil section Camber law thickness law 0.1 Twist and camber control sections on the wing z y z Page 6 Bezier curves (camber) Cubic splines (for spanwise splining) + parameters for A.o.A at the different flight points In-house analytical shape deformation module (+ sensitivities calculation) Mode 1: mid twist Mode 2: tip twist Mode 3: mid camber Mode 4: tip camber -2 Root section airfoil x Mode 5: tip max thickness position Mode 6: tip max thickness position

7 Physical & numerical modeling CFD code elsa (ONERA) RANS/Euler equations, structured multiblocs grid Performance cruise condition inviscid flow: Euler equations for evaluation of aerodynamic coefficients Discrete Euler adjoint equations for gradient Discretizations: Roe-Harten scheme for inviscid fluxes Backward-Euler + LU-SSOR + Multigrid CFD grid : cells Page 7 Residuals Residuals convergence of state variables Mach adjoint residual 10 Mach adjoint residual (baseline geometry) 12 CPU min iterations iterations Mach CL Mach CL Mach residual Mach residual CL 0.05 convergence of adjoint variable (baseline geometry) 10 CPU min.

8 Optimization loop: architecture and components 2D surface deformation + Surface sensitivities (analytical) Mesh deformation (TFI ) 3D mesh + Mesh sensitivities Design variables x r : Optimization software DAKOTA (SANDIA Page 8 lab., GNU) Computer 1 r r f ( x), g ( x) Objective, constraints + Gradients ( r f, r gi...) elsa Flow Mach 2.0 elsa/opt Adjoint (if requested) i Computer 2 Analysis/Adjoint failures cases handling and Objectives (+gradient) assembling elsa Flow Mach 0.9 elsa/opt Adjoint (if requested)

9 Multi-point optimization history (Mach 2.0 & Mach 0.9) Gradient optimization : CONMIN (Method of Feasible Direction, Vanderplaats) Gradients assembled from adjoint solutions 90 Multi-points Optimization (M2.0 and M0.9) Multi-points Optimization M2.0 and M Objective Page 9 Objective CL M2.0 CL M eval_id Optimization functions history CL Shape1, Shape Shape1 Shape2 Shape3 Shape4 Shape5 Shape6 Angle_of_Attack_1 Shape4, Shape Shape3, Shape eval_id Design parameters history Angle_of_Attack_1

10 Multi-point optimization results (Mach 2.0 & Mach 0.9) Initial Optimized Page 10 Mach CD Initial CD Final dc 70.1 dc -13% dc 85.7 dc -8%

11 Single point Mach 2.0 optimization GA Gradient strategy Single point optimization problem: Minimise: CD M=2.0 (α) Page 11 s.c. CL M=2.0 (α) 0.11 Geometry (design space: 7 variables) Optimization strategy=2 steps: Genetic Algorithm Gradient 1. Genetic algorithm (GADO*) exploration of design space 2. Gradient based approach (adjoint) efficient and accurate convergence (*GADO: Genetic Algorithm for Design Optimization original code: K. Rasheed, Rutgers University) CD (x10 4 ) CL 0.11 Population size: 40 Final gradient optimisation Drag coefficient (x10 4 ) evaluations evaluations Mach 2.0 optimization

12 Single point Mach 2.0 optimization GA Gradient strategy Single point optimization problem: Minimise: CD M=2.0 (α) Page 12 s.c. CL M=2.0 (α) 0.11 Geometry (design space: 7 variables) Optimization strategy=2 steps: Genetic Algorithm Gradient 1. Genetic algorithm (GADO*) exploration of design space 2. Gradient based approach (adjoint) efficient and accurate convergence (*GADO: Genetic Algorithm for Design Optimization original code: K. Rasheed, Rutgers University) CD (x10 4 ) CL Population size: 40 Final gradient optimisation Drag coefficient (x10 4 ) Reference: CD Mach 2.0 = 81 d.c. evaluations evaluations Mach 2.0 optimization Optimized: CD Mach 2.0 = 69 d.c.

13 Single point Mach 0.9 optimization GA Gradient strategy Population size: CL CL CL CD CD eval_id eval_id Page Genetic Algorithm Gradient eval_id Mach 0.9 optimization: GA gradient Reference: CD Mach 0.9 = 93 d.c. Optimized: CD Mach 0.9 = 74 d.c.

14 Hybrid Genetic/Gradient optimization Overview of the algorithm Rational: try to combine advantages of both global and local optimization techniques Page 14 Use GA for robust and wide design space exploration (robustness) Perform gradient descent each time a new local optimum is found (efficiency) Ingredients: Genetic algorithm : GADO Gradient SQP method Ind 1 Ind 2 Ind 3 Ind i Ind j Ind k Ind N-2 Ind N-1 Ind N Selection Selection Cross-over No New New best? Mutation Yes Gradient descent (SQP)

15 Hybrid Genetic/Gradient optimization Validation & application M2.0 GADO M2.0 GADO+gradient M2.0 gradient (SQP) Aircraft Range Page parameters Population size: 50 CFSQP (60 runs)-average (MoM) CFSQP (60 runs)-median (MoM) CFSQP (60 runs)-best_of_all_runs (MoM) CFSQP (60 runs)-worst_of_all_runs (MoM) GADO (30 runs) - average (MoM) GADO+grad (30 runs)-average (MoM) iter Validation on an analytical problem: Preliminary design (MDO) of a SSBJ based on analytical models (Sobieski) Statistical convergence analysis CD (x10 4 ) CDmin = 68.7 d.c. 7 parameters Population size: 40 CDmin = 68.7 d.c evaluations CDmin = 69.2 d.c. Application to the «real world» HSCT problem (single point M 2.0)

16 Conclusions Gradient is a very valuable information for optimization. It can be evaluated efficiently by CFD-adjoint approach (at fraction of the cost of FD). Simple multi-point optimization approach are of interest for realistic aircraft design application. Page 16 Optimization strategies based on combination/hybridization of evolutionary and gradient algorithm offer interesting performance and flexibility for bridging the gap between conceptual and detailed design. Ongoing work focus on: Use of surrogate models (SVM) in optimization Application of viscous adjoint techniques MDO