Stable, high-order computation of impedance–impedance operators for three-dimensional layered medium simulations

Abstract

The faithful modelling of the propagation of linear waves in a layered, periodic structure is of paramount importance in many branches of the applied sciences. In this paper, we present a novel numerical algorithm for the simulation of such problems which is free of the artificial singularities present in related approaches. We advocate for a surface integral formulation which is phrased in terms of impedance-impedance operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet-Neumann operators that appear in classical formulations. We demonstrate a high-order spectral algorithm to simulate these latter operators based upon a high-order perturbation of surfaces methodology which is rapid, robust and highly accurate. We demonstrate the validity and utility of our approach with a sequence of numerical simulations.

On analyticity of scattered fields in layered structures with interfacial graphene

“…For the former, the considerations are the same as those described in Ref. 57 save that we now consider choices which are not necessarily those of Després 66–68 (, ), and the terms in (25) are more involved. However, once the terms and are derived, see (C1) and (C2), these forms are straightforward.…”

“…One approach to eliminating this artificial source of singularity is to utilize “Impedance–Impedance Operators” (IIOs) as advocated by Gillman et al 54 . On interior layers, these IIOs can be constructed to be unitary so that not only are their eigenvalues nonzero, they are restricted to the unit circle in the complex plane giving a very well‐conditioned algorithm 57,58 …”

Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media

“…The transmission problem (2.1) can be formulated via boundary integral equations (BIEs) [1,6] or via non-overlapping Domain Decomposition (DD) [22,19]. Upon discretization, both the BIE and DD amount to solving block tridiagonal linear systems.…”

“…The numerical simulation of interactions between electromagnetic, acoustic, and elastic waves with periodic layered media has numerous applications in the fields of optics, photonics, geophysics [5]. Given the important technological applications of periodic layered media, the simulation of wave propagation in such environments has attracted significant attention [6,15,19,23,22]. Regardless of the type of discretization (finite elements, finite differences, boundary integral operators), iterative solvers are the preferred method of solution especially for high-frequency layered configurations that involve large numbers of layers that may contain inclusions.…”

“…Domain Decomposition Methods are natural candidates for the solution of Helmholtz transmission problems in periodic layered media [23,22,19]. Local subdomain solutions (the subdomains may or may not coincide with the periodic layers) are linked iteratively via Robin type transmission conditions defined on inter domain interfaces.…”