Topological derivatives via one-sided derivative of parametrized minima and minimax

• Published in: Engineering computations 2022-02, Vol.39 (1), p.34-59
• Main Author: Delfour, Michel C
• Format: Article
• Language: English
• Subjects: Derivatives; Equations of state; Euclidean geometry; Mathematical analysis; Methodology; Minimax technique; Parameterization; Partial differential equations; Perturbation; Topology
• ISSN: 0264-4401; 1758-7077
• DOI: 10.1108/EC-06-2021-0318

PurposeThe 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/approachThe 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.FindingsThe 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/implicationsDirect approach that considerably simplifies the analysis and computations.Originality/valueIt 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.