Min max method, shape, topological derivatives, averaged Lagrangian, homogenization, two scale convergence, Helmholtz equation

TL;DR

This paper rigorously derives shape and topological derivatives for constrained Helmholtz optimization problems using Lagrangian methods, with applications to differential geometry shape functions.

Contribution

It introduces a rigorous derivation of shape and topological derivatives for Helmholtz-based optimization problems using Lagrangian techniques, expanding theoretical understanding.

Findings

Derived explicit formulas for shape derivatives.

Proved topological derivatives under Helmholtz constraints.

Applied derivatives to geometry-related shape functions.

Abstract

In this paper, we perform a rigourous version of shape and topological derivatives for optimizations problems under constraint Helmoltz problems. A shape and topological optimization problem is formulated by introducing cost functional. We derive first by considering the lagradian method the shape derivative of the functional. It is also proven a topological derivative with the same approach. An application to several unconstrained shape functions arising from differential geometry are also given.