Topological derivatives via one-sided derivative of parametrized minima and minimax. Issue 1 (28th September 2021) - Delfour, Michel C.
Abstract : Purpose: The object of the paper is to illustrate how to obtain the topological derivative as a semidifferential in a general and practical mathematical setting for d -dimensional perturbations of a bounded open domain in the n -dimensional Euclidean space. Design/methodology/approach: The underlying methodology uses mathematical notions and powerful tools with ready to check assumptions and ready to use formulas via theorems on the one-sided derivative of parametrized minima and minimax. Findings: The theory and the examples indicate that the methodology applies to a wide range of problems: (1) compliance and (2) state constrained objective functions where the coupled state/adjoint state equations appear without a posteriori substitution of the adjoint state. Research limitations/implications: Direct approach that considerably simplifies the analysis and computations. Originality/value: It was known that the shape derivative was a differential. But the topological derivative is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are d -dimensional bounded measures.