Transparent boundary conditions for locally perturbed infinite hexagonal periodic media

TL;DR

This paper introduces a method to compute the Dirichlet-to-Neumann operator for infinite hexagonal periodic media with local perturbations, using a factorization involving non-local operators and Floquet-Bloch transform techniques.

Contribution

It presents a novel factorization approach for the DtN operator in complex media, combining half-space and symmetry-based operators with new integral equation formulations.

Findings

Explicit characterization of the half-space DtN operator via Floquet-Bloch transform

Derivation of an affine operator equation for the DtD operator

Framework applicable to lossy, locally perturbed hexagonal media

Abstract

In this paper, we propose a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy and locally perturbed hexagonal periodic media. We obtain a factorization of this operator involving two non local operators. The first one is a DtN type operator and corresponds to a half-space problem. The second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem. The half-space DtN operator is characterized via Floquet-Bloch transform, a family of elementary strip problems and a family of stationary Riccati equations. The DtD operator is the solution of an affine operator valued equation which can be reformulated as a non standard integral equation.