Stochastic approach to optimal aerodynamic shape design

Abstract

The purpose of this study was to show that stochastic methods can be applied effectively to optimal aerodynamic shape design problems, and that optimal solutions within relatively complex design spaces can be located in a reasonable amount of computation time. The design methodology presented is based on a modified simulated annealing algorithm and is global in nature, i.e., relative large complex design spaces can be automatically investigated. Current approaches that typically employ gradient-based optimization schemes tend to get caught in the numerous real and false local minima common in the design spaces of practical aerodynamics shape design problems. Within the context of the optimal shape design of a minimum drag axisymmetric forebody problem, a comparison was made between the proposed stochastic methodology and a gradient-based optimization approach. The results obtained clearly demonstrated the ability of this methodology to locate optimal designs in relatively complex design spaces where gradient-based optimization approaches experience difficulties. Further, the computation time required by the stochastic approach compared favorably to that of the gradient-based method. Nomenclature A = cross-sectional area of forebody, m2 Cd = coefficient of drag Fd - force of drag, N m = length of Markov chain n = number of cross sections P = Metropolis criterion R = random number uniformly distributed over the range [0, 1] Rp = baseline parabolic shape r, = radius at the /th cross section, ft T = control parameter Ux - freestream air velocity, m/s z = distance from the nose of the forebody measured along the z axis, ft £max = length of forebody, ft a. = nose angle, deg /3 = control parameter reduction coefficient A£ = change in objective function value p — freestream air density, kg/m3