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

Abstract

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 e ow-sensitivity methods for calculating the gradient of the objective function. The adjoint and e ow-sensitivity equations are solved using a novel preconditioned generalized minimum residual (GMRES)strategy. Together with a complete linearization of the discretized Navier‐Stokes and turbulence model equations, this results in an accurate and efecient evaluation of the gradient. Furthermore, fast e 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,includinginversedesign,lift-constraineddragminimization, liftenhancement, and maximization of lift-to-dragratio. In all examples, the normof the gradientisreduced by several ordersof magnitude, indicating that alocalminimumhasbeen obtained. Bytheuseoftheadjoint method,thegradient isobtained infromone-e fth to one-half of the time required to converge a eow solution.