Analysis of the Hessian for aerodynamic optimization: inviscid flow

In this paper, we analyze inviscid aerodynamic shape optimization problems governed by the full potential and the Euler equations in two and three dimensions. The analysis indicates that minimization of pressure-dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and the Euler equations. However, the optimization problems in two and three dimensions are inherently different. While the two-dimensional optimization problems are well posed, the three dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. The analysis also gives an estimate of the Hessian’s condition number which implies that the problems at hand are ill-conditioned. Infinite dimensional approximations for the Hessians are constructed and preconditioners for gradient-based methods are derived from these approximate Hessians. Numerical results are given for the small disturbance potential equation in two and in three space dimensions.