Review and Unification of Methods for Computing Derivatives of Multidisciplinary Systems

This paper presents a comprehensive review of all the options available for computing derivatives of multidisciplinary systems in a unified mathematical framework. The basic building blocks for computing derivatives are first introduced: finite differencing, the complex-step method and symbolic differentiation. A generalized chain rule is derived from which it is possible to derive both algorithmic differentiation and analytic methods. This generalized chain rule is shown to have two forms — a forward form and a reverse form — which correspond to the forward and reverse modes in algorithmic differentiation and the direct and adjoint approaches in analytic methods. Finally, the theory is extended to methods for computing derivatives of multidisciplinary systems, and several new insights are provided.