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

Abstract

SUMMARYIn 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.

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

“…The idea of gradient smoothing using Sobolev gradients [23] has extended to more complex approximate Hessians via shape calculus and Fourier analysis [2,48]. With the advent of automatic differentiation (AD), exact second-order sensitivities [51,38,14] have been employed in ASO problems. However, the high cost of computing the exact Hessian has lead to truncated-Newton methods [35,19] that use conjugate-gradient or Newton-Krylov methods to approximate the search direction with matrix-vector products of the Hessian.…”

“…It has since been expanded to be used in different areas such as optimization, extrapolation, and uncertainty analysis. The work of Papadimitriou and Giannakoglou [38,37,36,15,39] is more closely tied to ASO and explores the use of an exactly-initialized BFGS algorithm to optimize two-dimensional aerodynamic shapes. Although the Hessian is only evaluated once in the exactly-initialized BFGS algorithm, the initial cost is still too large to be effective.…”

“…Furthermore, the proposed method is implemented in a three-dimensional framework which introduces additional implementation complexity such as computational scalability and memory usage. The same initialized-BFGS framework as Papadimitriou and Giannakoglou [38] is used, but with an approximate initial Hessian. The approximation is recovered by partially solving first-order flow sensitivities, which are the most computationally intensive terms.…”

“…Chapter 4 describes how the first-order and second-order sensitivities are derived and evaluated. Exact analytical formulations of the direct-direct, adjoint-direct, adjoint-adjoint, and direct-adjoint Hessian are presented based on previous works [51,38].…”

Partitioned solution of an unsteady adjoint for strongly coupled fluid-structure interactions and application to parameter identification of a one-dimensional problem

“…It can be calculated using finite differences, the direct method and the adjoint method [10,11]. These techniques will be described in detail later in this work.…”