Optimal dynamic normal surface loading of a homogeneous elastic solid for interior acoustic-energy focusing to a specific target volume is treated in this paper. The goal is achieved by additionally imposing the criteria of applying a relatively minimal total surface force and system disturbance except at the target. By utilizing the calculus of variations, the optimality conditions are obtained, and the approach is implemented through the finite-element method along with half-space dyadic Green’s functions. The optimization procedure is done via the conjugate-gradient method. Numerical results show that the optimal normal surface loads consist of two radially shrinking concentric rings of special structure. For a positively constrained surface force, the earlier of the two surface load streams produces a self-focusing shear volume wave that moves to the target volume to create a high acoustic-energy density. An additional following longitudinal load adds further energy by compression of the surface directly over the target volume. In a second example without any positive constraint on the surface force, however, the earlier of the two surface load streams now produces a concentric surface wave as well as a self-focusing shear volume wave. An additional longitudinal load then converts the surface wave into bulk volume waves for achieving the objective with a minimum of total surface force and of total system disturbance. The overall approach of optimal design of dynamic surface loading for bulk volume objectives is flexible and capable of treating a variety of complex problems.

• Received 29 October 1990

DOI:https://doi.org/10.1103/PhysRevB.44.4892