Analysis of general shape optimization problems in nonlinear acoustics

TL;DR

This paper investigates shape optimization in nonlinear acoustics, focusing on high-intensity focused ultrasound, by establishing theoretical foundations for shape derivatives and analyzing specific nonlinear wave equations.

Contribution

It develops a rigorous framework for shape derivatives in nonlinear acoustic wave problems, including the Westervelt and Kuznetsov equations, with proofs of well-posedness and regularity.

Findings

Proved existence of Eulerian shape derivatives for nonlinear acoustic equations.

Established regularity and stability of the forward problem under small data.

Derived explicit expressions for shape derivatives of practical cost functionals.

Abstract

In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theoretical framework, the Westervelt and Kuznetsov equations of nonlinear acoustics are obtained as particular cases. The quadratic gradient nonlinearity, specific to the Kuznetsov equation, requires special attention throughout. To prove the existence of the Eulerian shape derivative, we successively study the local well-posedness and regularity of the forward problem, uniformly with respect to shape variations, and prove that it does not degenerate under the hypothesis of small initial and boundary data. Additionally, we prove H\"older-continuity of the acoustic potential with respect to domain deformations. We then derive and analyze the corresponding adjoint problems for several different cost functionals of practical interest and conclude with the expressions of well-defined shape derivatives.