Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation

This work describes the implementation of optimization techniques based on control theory for complex aircraft configurations. Here control theory is employed to derive the adjoint differential equations, the solution of which allows for a drastic reduction in computational costs over previous design methods [13, 12, 43, 38]. In our earlier studies [19, 20, 22, 23, 39, 25, 40, 41, 42] it was shown that this method could be used to devise effective optimization procedures for airfoils, wings and wing-bodies subject to either analytic or arbitrary meshes. Design formulations for both potential flows and flows governed by the Euler equations have been demonstrated, showing that such methods can be devised for various governing equations [39, 25]. In our most recent works [40, 42] the method was extended to treat wing-body configurations with a large number of mesh points, verifying that significant computational savings can be gained for practical design problems. In this paper the method is extended for the Euler equations to treat complete aircraft configurations via a new multiblock implementation. New elements include a multiblock-multigrid flow solver, a multiblock-multigrid adjoint solver, and a multiblock mesh perturbation scheme. Two design examples are presented in which the new method is used for the wing redesign of a transonic business jet. * Student Member AIAA flames S. McDonnellDistinguishedUniversityProfcssorof Aerospace Engineering, AIAA Fellow *AIAA Member INTRODUCTION To allow the full realization of the potential of Computational Fluid Dynamics (CFD) to produce superior designs, there is a need not only for accurate aerodynamic prediction methods for given configurations, but also for design methods capable of creating new optimum configurations. Yet, while flow analysis has matured to the extent that Navier-Stokes calculations are routinely carried out over very complex configurations, direct CFD based design is only just beginning to be used in the treatment of moderately complex three-dimensional configurations. Existing CFD analysis methods can be used to treat the design problem by coupling them with numerical optimization methods. The essence of these methods, which may incur heavy computational expenses, is very simple: a numerical optimization procedure is used to extremize a chosen aerodynamic figure of merit which is evaluated by the given CFD code. The configuration is systematically modified through user specified design variables. Most of these optimization procedures require the gradient of the cost function with respect to changes in the design variables. The simplest of the methods to obtain these necessary gradients is the finite difference method. In this technique, the gradient components are estimated by independently perturbing each design variable with a finite step, calculating the corresponding value of the objective function using CFD analysis, and forming the ratio of the differences. The gradient is used by the numerical optimization algorithm to calculate a search direction using steepest descent, conjugate gradient, or quasiNewton techniques. After finding the minimum or maximum of the objective function along the search direction, the entire process is repeated until the gradient approaches zero and further improvement is not possible. The finite difference based optimization strategy is computationally expensive because the flow must be recalculated for perturbations in every design variable to determine the gradient Nevertheless, it is attractive when compared with other traditional design strategies such as inverse methods, since it permits any choice of the aerodynamic figure of merit The use of numerical optimization for transonic aerodynamic shape design was pioneered by Hicks, Murman and Vanderplaats [13]. They applied the method to twodimensional profile design subject to the potential flow equation. The method was quickly extended to wing design by Hicks and Henne [12]. Later, in the work of Reuther, Cliff, Hicks and Van Dam, the method was successfully used for the design of supersonic wing-body transport configurations [38]. In all of these cases, finite difference methods were used to obtain the required gradient information. Recently through work by both ourselves and other groups, alternative, less expensive methods for obtaining design sensitivities have been developed which greatly reduce the computational costs of optimization. The most promising of these emerging approaches is the adjoint formulation whereby the sensitivity with respect to an arbitrary number of design variables is obtained with the equivalent of only one additional flow calculation. FORMULATION OF THE ADJOINT EQUATIONS The aerodynamic properties which define the cost function / are functions of the flow field variables (w) and the physical location of the boundary, which may be represented by the function J-, say. Then (D and a change in T results in a change