Periodic Hamiltonian systems in shape optimization problems with Neumann boundary conditions

TL;DR

This paper extends a Hamiltonian system approach to shape optimization problems with Neumann boundary conditions, enabling simultaneous boundary and topological variations, supported by numerical experiments.

Contribution

It introduces a novel technique for shape optimization with Neumann conditions, incorporating combined cost functionals and extending to nonlinear systems.

Findings

Successful numerical experiments confirm theoretical results.

Method allows boundary and topological variations simultaneously.

Extension to nonlinear state systems is feasible.

Abstract

The recent approach based on Hamiltonian systems and the implicit parametri\-za\-tion theorem, provides a general fixed domain approximation method in shape optimization problems, using optimal control theory. In previous works, we have examined Dirichlet boundary conditions with distributed or boundary observation. Here, we discuss the case of Neumann boundary conditions, with a combined cost functional, including both distributed and boundary observation. Extensions to nonlinear state systems are possible. This new technique allows simultaneous boundary and topological variations and we also report numerical experiments confirming the theoretical results.