Approximate Hessian for accelerated convergence of aerodynamic shape optimization problems in an adjoint-based framework. (30th May 2018) - Shi-Dong, Doug; Nadarajah, Siva
Highlights: Analytical formulations of the Hessian are derived and compared. Approximations are proposed to reduce computational and implementation cost. Mathematical bounds on the errors incurred are derived. The proposed method is assessed through various aerodynamic test cases. An approximate Hessian results in a significant speedup of the optimization problem. Abstract: 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.