Distributed point source modeling of the scattering of elastic waves by a circular cavity in an anisotropic half-space

Abstract

The Distributed Point Source Method (DPSM) is a modeling technique based on superposition of fundamental solutions corresponding to individual pair of source and target points. A collection of source points distributed over the boundaries and interfaces are responsible for transmission, reflection, and refraction of acoustic waves in the solution domain. The strength of the source points may not be known a priory. By imposing the prescribed conditions on the boundaries and interfaces, a system of equations with the source strengths as the unknowns is obtained. After finding the source strengths as the solution to this system of equation, the amount of the solution at any target point in the domain is obtained by superimposing the effect of all source points on that target point. DPSM is an efficient modeling technique for ultrasonic problems since it does not require discretization of the whole solution domain but only the boundaries and interfaces. The fundamental solution, or the Green’s function, between a pair of source and target points serves as the building block for DPSM. For an ideal fluid or a homogeneous isotropic solid the elastodynamic Green’s function is available as closed form algebraic expressions. But for an anisotropic solids, the set of governing equations are considerably more complex and the elastodynamic Green’s function needs to be evaluated numerically. In this study, an anisotropic half-space containing a flaw in the form of a circular hole is considered. The solid half-space is in contact with fluid and a transducer is located in fluid facing the solid half-space. Some efforts have been made to alleviate the computational intensity of the numerical evaluation of anisotropic Green’s function for this problem. Firstly, a technique called “windowing” is used to exploit the repetitive pattern of relative positions of the source and target points in order to considerably reduce the number of Green’s function evaluations. Secondly, the resolution of the integration for evaluation of the anisotropic Green’s function is changed based on the distance between the source and target points, and a calibration technique based on an equivalent isotropic stiffness tensor is suggested. This calibrated multi-resolution integration technique is combined with the windowing technique, and the developed DPSM model is applied to a numerical example containing a transversely isotropic half-space, to show the applicability and effectiveness of DPSM modeling for this class of problems. Important applications like non-destructive evaluation of composite materials may benefit from such modeling capability.