A Numerical Mode Matching Method for Wave Scattering in a Layered Medium with a Stratified Inhomogeneity

Abstract

Numerical mode matching (NMM) methods are widely used for analyzing wave propagation and scattering in structures that are piecewise uniform along one spatial direction. For open structures that are unbounded in transverse directions (perpendicular to the uniform direction), the NMM methods use the perfectly matched layer (PML) technique to truncate the transverse variables. When incident waves are specified in homogeneous media surrounding the main structure, the total field is not always outgoing, and the NMM methods rely on reference solutions for each uniform segment. Existing NMM methods have difficulty handing gracing incident waves and special incident waves related to the onset of total internal reflection, and are not very efficient at computing reference solutions for non-plane incident waves. In this paper, a new NMM method is developed to overcome these limitations. A Robin-type boundary condition is proposed to ensure that non-propagating and non-decaying wave field components are not reflected by truncated PMLs. Exponential convergence of the PML solutions based on the hybrid Dirichlet-Robin boundary condition is established theoretically. A fast method is developed for computing reference solutions for cylindrical incident waves. The new NMM is implemented for two-dimensional structures and polarized electromagnetic waves. Numerical experiments are carried out to validate the new NMM method and to demonstrate its performance.

Time-Harmonic Acoustic Scattering from a Nonlocally Perturbed Trapezoidal Surface

“…We remark that there are some major differences between the current work and [32]. In [32], an NMM method has been developed for the scattering problem in a two-layer medium with a stratified inhomogeneity, where an outgoing wavefield is much easier to extract in terms of directly subtracting the background reference wavefield from the total wavefield so that the aformentioned truncation techniques can be easily applied.…”

“…We remark that there are some major differences between the current work and [32]. In [32], an NMM method has been developed for the scattering problem in a two-layer medium with a stratified inhomogeneity, where an outgoing wavefield is much easier to extract in terms of directly subtracting the background reference wavefield from the total wavefield so that the aformentioned truncation techniques can be easily applied. Moreover, a hybrid Robin-Dirichlet boundary condition was proposed therein on the PML boundary to make the mode expansion procedure applicable, but this in fact is unnecessary in some standard methods like FEM methods since one can simply put zero Dirichlet boundary condition on the whole PML boundary to terminate the outgoing wavefield.…”

“…If β is close to 0, this plane wave can propagate nearly parallel to the PML entrance x 2 = L, which numerically causes inefficiency of the PML absorption. To resolve this issue, our previous work [32] suggests to eliminate the plane wave by the following Robin-type relation,…”

“…Since u sc differs from outgoing waves by only reflected plane waves propagating parallel to d , we impose a homogeneous Robin-type boundary condition to eliminate u re and u re h on the boundary of the PML by following [32]. As the medium structure can be decomposed into two x 1 -uniform segment, the numerical mode matching (NMM) method of [32] will be adapted to our scattering problem. More precisely, we expand u sc in terms of eigenmodes in each uniform segment by separating variables in the governing equations, and then match the two mode expansions on the common interface separating the two regions.…”

“…The NMM method [14,32], a.k.a mode expansion method or modal methods [6,29,37], and its numerous numerical invariants [16,20,21,25,27,30,33,35,38,19,4,5] are applicable when the structure can be divided into a number of segments, where the medium becomes uniform along one spatial variable. The classical mode matching method solves the eigenmodes analytically while the NMM methods solve the eigenmodes by numerical methods, and they are easier to implement and applicable to more general structures.…”