Design of arbitrarily shaped acoustic cloaks through partial differential equation-constrained optimization satisfying sonic-metamaterial design requirements

Abstract

We develop an optimization framework for the design of acoustic cloaks, with the aim of overcoming the limitations of usual transformation-based cloaks in terms of microstructure complexity and shape arbitrarity of the obstacle. This is achieved by recasting the acoustic cloaking design as a nonlinear optimal control problem constrained by a linear elliptic partial differential equation. In this setting, isotropic material properties’ distributions realizing the cloak take the form of control functions and a system of the first-order optimality conditions is derived accordingly. Such isotropic media can then be obtained in practice with simple hexagonal lattices of inclusions in water. For this reason, the optimization problem is directly formulated to consider suitable partitions of the control domain. Two types of inclusions are analysed, and long-wavelength homogenization is used to define the feasible set of material properties that is employed as a constraint in the optimization problem. In this manner, we link the stage of material properties optimization with that of microstructure design, aiming at finding the optimal implementable solution. The resulting cloak is numerically tested via coupled structural/acoustic simulations. As a further test benchmark, a concave target and the silhouette of a ship are considered other than the usual axisymmetric cloak.