Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems

In this paper, four approaches to compute the Hessian matrix of an objective function used often in aerodynamic inverse design problems are presented. The computationally less expensive among them is selected and applied to the reconstruction of cascade airfoils that reproduce a prescribed pressure distribution over their walls, under inviscid and viscous flow considerations. The selected approach is based on the direct sensitivity analysis method for the computation of first derivatives, followed by the discrete adjoint method for the computation of the Hessian matrix. The applications presented in this paper show that the Newton method, based on exact Hessian matrices, outperforms other gradient‐based algorithms such as steepest descent or BFGS algorithm. Copyright © 2007 John Wiley & Sons, Ltd.