A phase-field approach to shape and topology optimization of acoustic waves in dissipative media

TL;DR

This paper develops a phase-field method for optimizing the shape and topology of acoustic lenses in dissipative media, ensuring well-posedness and deriving optimality conditions for the design problem.

Contribution

It introduces a novel phase-field formulation for acoustic lens optimization in dissipative media, linking diffuse interface models to sharp interface shape optimization.

Findings

The formulation is well-posed and admits a minimizer.

First-order optimality conditions are rigorously derived.

The relation between diffuse and sharp interface models is established via $ ext{Gamma}$-convergence.

Abstract

We investigate the problem of finding the optimal shape and topology of a system of acoustic lenses in a dissipative medium. The sound propagation is governed by a general semilinear strongly damped wave equation. We introduce a phase-field formulation of this problem through diffuse interfaces between the lenses and the surrounding fluid. The resulting formulation is shown to be well-posed and we prove that the corresponding optimization problem has a minimizer. By analyzing properties of the reduced objective functional and well-posedness of the adjoint problem, we rigorously derive first-order optimality conditions for this problem. Additionally, we consider the $\Gamma$-limit of the reduced objective functional and in this way establish a relation between the diffuse interface problem and a perimeter-regularized sharp interface shape optimization problem.