Newton–Krylov Algorithm for Aerodynamic Design Using the Navier–Stokes Equations

A Newton–Krylov algorithm is presented for two-dimensional Navier–Stokes aerodynamic shape optimization problems. The algorithm is applied to both the discrete-adjoint and the discrete  ow-sensitivity methods for calculating the gradient of the objective function. The adjoint and  ow-sensitivity equations are solved using a novel preconditioned generalized minimumresidual (GMRES) strategy. Together with a complete linearization of the discretized Navier–Stokes and turbulence model equations, this results in an accurate and efŽ cient evaluation of the gradient. Furthermore, fast  ow solutions are obtained using the same preconditioned GMRES strategy in conjunction with an inexact Newton approach. The performance of the new algorithm is demonstrated for several design examples, including inverse design, lift-constraineddragminimization, lift enhancement, andmaximization of lift-to-drag ratio. In all examples, the norm of the gradient is reduced by several orders of magnitude, indicating that a localminimumhas been obtained.By the use of the adjointmethod, the gradient is obtained in fromone-Ž fth to one-half of the time required to converge a  ow solution.