Robust design in aerodynamics using third-order sensitivity analysis based on discrete adjoint. Application to quasi-1D flows

摘要

In this paper, the second-order second moment approach, coupled with an adjoint-based steepest descent algorithm, for the solution of the so-called robust design problem in aerodynamics is proposed. Because the objective function for the robust design problem comprises first-order and second-order sensitivity derivatives with respect to the environmental parameters, the application of a gradient-based method, which requires the sensitivities of this function with respect to the design variables, calls for the computation of third-order mixed derivatives. To compute these derivatives with the minimum CPU cost, a combination of the direct differentiation and the discrete adjoint variable method is proposed. This is presented for the first time in the relevant literature and is the most efficient among other possible schemes on condition that the design variables are much more than the environmental ones; this is definitely true in most engineering design problems. The proposed approach was used for the robust design of a duct, assuming a quasi-1D flow model; the coordinates of the Bezier control points parameterizing the duct shape are used as design variables, whereas the outlet Mach number and the Darcy-Weisbach friction coefficient are used as environmental ones. The extension to 2D and 3D flow problems, after developing the corresponding direct differentiation and adjoint variable methods and software, is straightforward.