Computationally efficient, numerically exact design space derivatives via the complex Taylor's series expansion method

Abstract

Within numerical design optimization, discrete sensitivity analysis is often used to estimate the derivative of an objective function with respect to the design variables. Discrete sensitivity analysis estimates these derivatives by taking advantage of additional derivative information available in an implicit computational fluid dynamics (CFD) solver of the discretized governing partial differential equations. The key benefits of steady-state discrete sensitivity analysis are its computational efficiency and numerical accuracy. More recently, the complex Taylor's series expansion (CTSE) method has been used to generate these design space derivatives to machine accuracy, by analyzing a complex perturbation of the objective function. For fortran codes, this method is quite easy to implement, for both implicit and explicit codes; unfortunately, the CTSE method can be quite time consuming, because it requires a complex solution of the governing partial differential equations. In this paper, the authors demonstrate that the direct formulation of discrete sensitivity analysis and the CTSE method solve the same iterative sensitivity equation, which sheds light on the most efficient use of the CTSE method. Finally, these methods are demonstrated via application to numerical simulations of one-dimensional and two-dimensional open-channel flows.