Generation of controlled acoustic waves by optimal design of surface loads with constrained forms

Abstract

Model calculations are presented for the optimal design of surface force patterns to generate acoustic waves that come to a focus within the bulk of a homogeneous elastic solid. The optimal design consists of achieving a high level of energy at the target at a prescribed time by applying a relatively minimal surface force while aiming for a minimal system disturbance away from the focal target. Such optimal designs were derived in an earlier paper, in which no restriction was imposed on the functional form of the applied stress. In this paper we examine the importance of the fine detail in the earlier derived forcing functions in achieving efficient acoustic focusing. We repeat the optimal design calculations with the surface stress constrained to be in the form of rings of variable radius, with cross sectional profiles made by the superposition of two Gaussians. The optimality conditions are secured via the conjugate gradient algorithm (CGA) and the mechanics of the elastic medium are treated by the finite element method along with using the half space Green's function matrix. We use a criterion for focusing efficiency of the ratio of acoustic energy in the target volume to the total work done on the surface, at a prescribed time. The calculations show the high levels of focusing efficiency derived in earlier work with unconstrained force patterns also can be achieved with constrained and simplified force patterns. This observation is encouraging in terms of the robustness of the optimal solution as well as the possibility of laboratory realizations of the designed force patterns for generating focused acoustic waves.