In design, analysis is used repetitively to answer the “What If” questions, whether asked by a person or by a search algorithm in a formal optimization. Answers to such questions may be quantified in the form of derivatives of a set of dependent behavior or state variables y with respect to the design variables x. Hence, sensitivity analysis emerges as a tool necessary for design in general and for optimization in particular. This chapter reviews the fundamentals of the sensitivity analysis based on an analytical approach as opposed to the finite difference (FD) techniques, the latter being explained in any text on numerical methods.
7.1 Analytical Method
Direct differentiation of closed-form functions is the preferred way to obtain the derivatives of y(x) with respect to x. However, in most large-scale engineering problems, the direct differentiation approach is not practical because the analyses used yield numerical rather than analytical results or the functional expressions are too complicated. Nevertheless, the analytical method can be used providing it is applied to differentiate the governing equations of the problem employing the Implicit Function Theorem (Rektorys, 1969 ).
Consider a set of governing equations of a problem expressed by means of a vector of functions f (x, y) of the vector arguments x and y
Hidden by the vector notation above ...