The continuous direct-adjoint approach for second order sensitivities in viscous aerodynamic inverse design problems

Abstract

A novel continuous adjoint approach for the computation of the second order sensitivities of the objective function used in inverse design problems is proposed. In the framework of the Newton method, the proposed approach can be used to efficiently cope with inverse design problems in viscous flows, where the target is a given pressure distribution along the solid walls. It consists of two steps and will, thus, be referred to as the direct-adjoint approach. At the first step, the direct differentiation method is used to compute the first order sensitivities of the flow variables with respect to the design variables and build the gradient of the objective function. At the second step, the adjoint approach is used to compute the second order sensitivities. The final Hessian expression is free of field integrals and its computation requires the solution of N + 1 equivalent flow (system) solutions for N design variables. Since the CPU cost of using the Newton method, with exact gradient and Hessian data at each cycle, becomes prohibitively high, an approach that computes the exact Hessian only once and then updates it in an approximated manner through the BFGS formula, is used instead. The accuracy of the Hessian matrix components, computed using the direct-adjoint approach is demonstrated on the inverse design of a diffuser and a cascade airfoil.