Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

Abstract

This paper focuses on discrete and continuous adjoint approaches and direct differentiation methods that can efficiently be used in aerodynamic shape optimization problems. The advantage of the adjoint approach is the computation of the gradient of the objective function at cost which does not depend upon the number of design variables. An extra advantage of the formulation presented below, for the computation of either first or second order sensitivities, is that the resulting sensitivity expressions are free of field integrals even if the objective function is a field integral. This is demonstrated using three possible objective functions for use in internal aerodynamic problems; the first objective is for inverse design problems where a target pressure distribution along the solid walls must be reproduced; the other two quantify viscous losses in duct or cascade flows, cast as either the reduction in total pressure between the inlet and outlet or the field integral of entropy generation. From the mathematical point of view, the three functions are defined over different parts of the domain or its boundaries, and this strongly affects the adjoint formulation. In the second part of this paper, the same discrete and continuous adjoint formulations are combined with direct differentiation methods to compute the Hessian matrix of the objective function. Although the direct differentiation for the computation of the gradient is time consuming, it may support the adjoint method to calculate the exact Hessian matrix components with the minimum CPU cost. Since, however, the CPU cost is proportional to the number of design variables, a well performing optimization scheme, based on the exactly computed Hessian during the starting cycle and a quasi Newton (BFGS) scheme during the next cycles, is proposed.

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

“…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.…”

Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach

“…Their detailed development and validation (separately from the truncated Newton method that this paper is dealing with) can be found in [22,23,25]. For 2D steady flows, the conservative variables U n and the fluxes f nk are given by ⎡…”

Simultaneous identification of moving masses and structural damage

“…The formulation based on (37) and (35) allows the discrete adjoint method to be used (Haftka and Gürdal 1992), which is quicker by one order of magnitude in comparison to the standard direct differentiation method (Papadimitriou and Giannakoglou 2008). For notational simplicity, (35) and (37) and their first derivatives with respect to the variable μ i are stated in the following simple aggregate forms:…”