Approximate Hessian for accelerated convergence of aerodynamic shape optimization problems in an adjoint-based framework

The current work explores the use of an approximate Hessian to accelerate the convergence of an adjoint-based aerodynamic shape optimization framework. Exact analytical formulations of the direct-direct, adjoint-direct, adjoint-adjoint, and direct-adjoint Hessian approaches are presented and the equivalence between the adjoint-adjoint and direct-adjoint formulations is demonstrated. An approximation of the Hessian is obtained from the analytical formulation by partially solving first-order sensitivities to reduce computational time, while neglecting second-order sensitivities to ease implementation. Error bounds on the resulting approximation are presented for the first-order sensitivities through perturbation analysis. The proposed method is first assessed using an inverse pressure problem for a quasi-one-dimensional Euler flow. Additionally, three-dimensional inviscid transonic test cases are used to demonstrate the effectiveness of the method.