Hearing Protection for High-Noise Environments. Atachment 1: Development of Elastoacoustic Integral-Equation Solver: Surface and Volumetric Integral Equations

摘要

Following an introductory section, Section 2 provides a complete account of the integral-equation formulations as they are being implemented in our solver for elasticity problems. Section 3 presents a discussion of several equivalent forms of the Lame differential equation in elasticity, including its first-order representation as a set of coupled equations for the displacement and the stress tensor. In addition, we derive forms of the differential equations with separated terms describing solution in the 'background medium' (such as air) and interaction terms describing effects of the deviations of the medium properties from those of the background material. In Section 4, we discuss the Green functions associated with the Lame equation for the displacement field. In particular, we obtain a form of the Green function that explicitly exhibits a nonsingular behavior of its dyadic-derivative part, which facilitates discretization of the resulting integral equations. In the following Sections, 5 and 6, we are concerned with integral-equation formulations in elasticity in three cases: (1) purely surface (boundary- element) equations, (2) purely volumetric (Lippmann-Schwinger, or L-S) equations, and (3) a coupled system of volume and surface equations. The last of these is a novel approach we developed, mostly to be able to efficiently model geometry components -- in our case, the middle and inner ear -- characterized by small sizes and intricate surfaces, and embedded in a larger volume of an inhomogeneous material.