Some inverse problems in structural acoustics

Fundamental problems of direct and inverse solutions of the multidimensional Helmholtz equation in structural acoustics are discussed. Inversion of an obstacle from the far-field pattern is compared to that of a medium and/or an inhomogeneity. An overview of various methods of solving these problems is presented. The linear techniques for a medium and/or inhomogeneity problem and some intrinsically nonlinear methods of inversion for an obstacle are described. Four nonlinear methods are compared, namely, the ansatz of potential over an internal curve, the dual space method using the Herglotz wave functions, methods based on the Rayleigh hypothesis, and the most recent method of shape differentiation combined with the Padé approximation. The efficacy and advantages of the last technique are illustrated by numerically reconstructing both penetrable and impenetrable obstacles of various shapes and orientations. The robustness of the method in regard to the initial conditions and regularization are demonstrated and its extension to elasticity and an acoustical shell structure are presented. Finally, some open questions and future directions are discussed.