Wave Scattering in Layered Orthotropic Media I: A Stable PML and a High-Accuracy Boundary Integral Equation Method

Abstract

In anisotropic media, the standard perfectly matched layer (PML) technique suffers irrevocable instability in terminating the unbounded problem domains. It remains an open question whether a stable PML-like absorbing boundary condition exists. For wave scattering in a layered orthotropic medium, this question is affirmatively answered for the first time in this paper. In each orthotropic medium, the permittivity tensor uniquely determines a change of coordinates, that transforms the governing anisotropic Helmholtz equation into an isotropic Helmholtz equation in the new coordinate system. This leads us to propose a novel Sommerfeld radiation condition (SRC) to rigorously characterize outgoing waves in the layered orthotropic medium. Naturally, the SRC motivates a regionalized PML (RPML) to truncate the scattering problem, in the sense that a standard PML is set up in the new coordinate system in each orthotropic region. It is revealed that the RPML is unconditionally stable compared with the unstable uniaxial PML. A high-accuracy boundary integral equation (BIE) method is developed to solve the resulting boundary value problem. Numerical experiments are carried out to validate the stability of the RPML and the accuracy of the BIE method, showing exponentially decaying truncation errors as the RPML parameters increase.