The HalfSpace Matching (HSM) method was developed a few years ago to solve scattering problems in unbounded complex media. The complexity of the media can result from the type of equations (Maxwell’s equations or elasticity equations), the physical properties of the media (periodic, stratified, or anisotropic coefficients) and/or even their geometry (infinite 2D or 3D media or 3D plates). All classical methods for solving scattering problems in homogeneous media, which are based on integral equation techniques, transparent boundary conditions involving Dirichlet-to-Neumann operators or perfectly matched layer techniques, either cannot be applied to complex media or only with an enormous computational effort.

In contrast, the HSM method is based on a simple and quite general idea: the solution of half-space problems can be expressed via an adapted transformation in the transverse direction thanks to its trace at the edge of the half-space (for example the Fourier transform in the homogeneous case or the Floquet-Bloch transform in the periodic case). The method then consists in coupling representations of the solution in several half-spaces with the solution itself in a bounded domain. The system of equations is obtained by imposing that these representations coincide wherever they coexist and it couples the traces of the solution at the edges of the half-spaces with the restriction of the solution to a square.

In this talk, we will first use a toy problem, namely the Helmholtz equation with constant coefficients (up to possibly compactly supported perturbations), to explain the derivation of the HSM formulation, its mathematical analysis and its discretization. I will then explain how it extends to more complex situations, in particular acoustic or elastodynamic problems in anisotropic or periodic media.

In the second part of the talk, we will show how the HSM method can be extended to solve the 2D Helmholtz equation in media corresponding to the junctions of open waveguides. For this purpose, we will use a generalization of the Fourier transform for the half-space representations. The analysis is complete for the dissipative case. While numerical simulations converge even without dissipation, we explain the difficulties arising from the analysis.

This talk is the result of several collaborations: Anne-Sophie Bonnet Ben-Dhia (POEMS), Simon Chandler-Wilde ( Reading University), Christophe Hazard (POEMS), Patrick Joly (POEMS), Karl-Mikael Perfekt (NTNU), Yohanes Tjandrawidjaja and Antoine Tonnoir (INSA Rouen).