Three-dimensional acoustic scattering by layered media: a novel surface formulation with operator expansions implementation

Abstract

The scattering of acoustic waves by irregular structures plays an important role in a wide range of problems of scientific and technological interest. This contribution focuses on the rapid and highly accurate numerical approximation of solutions of Helmholtz equations coupled across irregular periodic interfaces meant to model acoustic waves incident upon a multi-layered medium. We describe not only a novel surface formulation for the problem in terms of boundary integral operators (Dirichlet-Neumann operators), but also a Boundary Perturbation methodology (the Method of Operator Expansions) for its numerical simulation. The method requires only the discretization of the layer interfaces (so that the number of unknowns is an order of magnitude smaller than volumetric approaches), while it avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of Boundary Integral/Element Methods. The approach is a generalization to multiple layers of Malcolm & Nicholls' Operator Expansions algorithm for dielectric structures with two layers. As with this precursor, this approach is efficient and spectrally accurate.

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

“…We began by demonstrating the validity of our algorithm by conducting experiments using the method of manufactured solutions (MMS) [58,59]. We then showed comparisons between TFE simulations of a three-layer configuration with the DNO formulation [45] and our new IIO version (3.8). More specifically, we considered a configuration far from singularities of the innerlayer DNO, H, and a structure exactly (up to machine precision) at a singularity.…”

“…Our numerical approach to solving the layered medium problems presented in this section is to use either the DNO formulation of the problem [45] or its IIO counterpart (3.8), with the relevant operators (DNOs and IIOs, respectively) simulated using the TFE methodology. For brevity, we discuss how this is accomplished for the interior layer IIO, R, described in detail in §5.…”

“…For the latter of these, we demand the 'upward propagating Rayleigh expansion radiation condition' (URC) and its 'downward propagating' analogue (DRC) as specified in [48] (which we make precise in §2). In [45], we detailed a restatement of the classical governing equations in terms of DNOs, which we revise in this contribution (see §3).…”

“…While extremely useful for many configurations of interest, our previous formulation of the scattering problem above [39,45] in terms of surface operators suffers from the fact that interior layer DNOs, H, do not exist at the 'Dirichlet eigenvalues', i.e. choices of k (m) for which (2.1) does not have a unique solution.…”

“…In §2, we recall the governing equations for scattering of linear waves by a periodic layered medium in three dimensions, with a particular discussion of transparent boundary conditions. In §3, we describe an interfacial reformulation of these equations in terms of surface quantities and IIOs that generalizes our previous formulation [39,45]. In §4, we begin a detailed discussion of these IIOs by explicitly computing their action on domains with infinitesimal (flat) interfaces.…”