Perfectly matched layers for elastic waves in cylindrical and spherical coordinates

Abstract

The perfectly matched layer (PML) for elastic waves in cylindrical and spherical coordinates is developed using an improved scheme of complex coordinates. As is known for electromagnetic waves, Berenger’s original PML scheme does not apply to cylindrical and spherical coordinates. The straightforward extension of the complex coordinates for elastic waves to cylindrical and spherical coordinates requires extra unknowns for time-domain solutions, wasting computer memory and computation time. The main idea of the improved scheme in this work is the use of integrated complex variables. It is shown that for three-dimensional cylindrical and spherical coordinates, this improved PML scheme requires no more unknowns than in Cartesian coordinates. The number of unknowns can be further reduced through the use of symmetry in the partial differential equations. The PML scheme allows an arbitrary inhomogeneity in the medium, and is suitable for numerical solutions of wave equations by finite-difference, finite-element, and pseudospectral methods for elastic waves in inhomogeneous media with cylindrical and spherical structures. Finite-difference time-domain (FDTD) results are shown to demonstrate the efficacy of the PML absorbing boundary condition.