Optimal shape design of aerodynamic configurations is a challenging problem due to the
nonlinear effects of complex flow features such as shock waves, boundary layers, and separation.
A Newton–Krylov algorithm is presented for aerodynamic design using gradient-based
numerical optimization. The flow is governed by the two-dimensional compressible Navier–
Stokes equations in conjunction with a one-equation turbulence model, which are discretized
on multi-block structured grids. The discrete-adjoint method is applied to compute the objective
function gradient. The adjoint equation is solved using the preconditioned generalized
minimal residual (GMRES) method. A novel preconditioner is introduced, and together with
a complete differentiation of the discretized Navier–Stokes and turbulence model equations,
this results in an accurate and efficient evaluation of the gradient. The gradient is obtained in
just one-fifth to one-half of the time required to converge a flow solution. Furthermore, fast
flow solutions are obtained using the same preconditioned GMRES method in conjunction with
an inexact-Newton approach. Optimization constraints are enforced through a penalty formulation,
and the resulting unconstrained problem is solved via a quasi-Newton method. The
performance of the new algorithm is demonstrated for several design examples that include lift
enhancement, where the optimal position of a flap is determined within a high-lift configuration,
lift-constrained drag minimization at multiple transonic operating points, and the computation
of a Pareto front based on competing objectives. In all examples, the gradient is reduced by
several orders of magnitude, indicating that a local minimum has been obtained. Overall, the
results show that the new algorithm is among the fastest presently available for aerodynamic
shape optimization and provides an effective approach for practical aerodynamic design.
