## Finite-Difference Sensitivity Calculation in Iteratively Solved Problems

20th AIAA Computational Fluid Dynamics Conference 2011, Honolulu, HI, United States Of America, 27 - 30 June 2011

• Publication Type: Conference Paper / Full Text
• Volume:
• Doi Number: 10.2514/6.2011-3071
• City: Honolulu, HI
• Country: United States Of America
• Middle East Technical University Affiliated: Yes

#### Abstract

In this study, the accuracy of the finite-difference sensitivities is examined in iteratively solved problems. The aim of this research is to reduce the error in the finite difference sensitivity calculations. The norm value of the finite-difference sensitivity error in the state variables is minimized with respect to the finite-difference step size. The optimum finitedifference step size is formulated as a function of the norm values of both convergence error and higher order sensitivities. In order to calculate the optimum step size, two methods are introduced. The first method is developed to calculate the convergence error in iteratively solved problem and it is based on the eigenvalue analysis of linear systems, but it can also be used for nonlinear systems. The second method is developed to estimate the higher order sensitivities which are calculated by differentiating the approximately constructed differential equation with respect to the design variables. The results show that with the proposed method, the convergence error can be accurately estimated for both linear and non-linear problems. The accuracy of the method developed for the higher order sensitivity estimation is validated with the finite-difference method. The comparison of the sensitivities calculated with the analytical and the finite-difference methods show that the developed methods can accurately estimate the optimum step size. The effects of the sensitivities calculated with the developed methods on the convergence of inverse design are examined. The results show that estimating the optimum step size with the developed methods improves the convergence of design without significantly increasing the usage of CPU time and memory of computers. © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.