Discrete Wave‐Number Boundary‐Element Method for 3‐D Scattering Problems

Abstract

The discrete wave-number boundary element method is extended to apply to three-dimensional elastodynamic problems. The method—which was first introduced and applied to two-dimensional problems—combines the direct boundary-element method with the discrete wave-number Green’s function. The advantage of this method is its accuracy and flexibility for boundary configurations. As a demonstration of the applicability of the method, the responses of a hemispherical canyon for incident SH, SV, and P waves in a three-dimensional half-space are presented in time domain as well as in frequency domain. Time histories of seismic motion along the surface in and around the canyon are studied for incident SH, SV, and P waves with the shape of a Ricker wavelet. Many of the physical phenomena—such as diffracted waves (called creeping waves) propagating inside the canyon and Rayleigh waves generated at the edges of the canyon and propagating outward, away from it—which have been observed in the response of the two-dimensional model (i.e., semicircular canyon in an elastic half-space), are also observed in the response of the three-dimensional model (i.e., hemispherical canyon in an elastic half-space). The mathematical formulation presented is general. Nevertheless, the axisymmetric geometry of the hemispherical canyon is exploited to develop a very efficient computational algorithm to solve the problem. Since a nonaxisymmetric scatterer may always be assumed to be embedded in a region that is defined by an axisymmetric imaginary boundary (such as a hemisphere), application of this algorithm can be extended to nonaxisymmetric scatterers.