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

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.

Citation

Delfour, M.C. (2022), "Topological derivatives via one-sided derivative of parametrized minima and minimax", Engineering Computations, Vol. 39 No. 1, pp. 34-59. https://doi.org/10.1108/EC-06-2021-0318