  Multiple scattering methods are widely used to reduce the computational
complexity of acoustic or electromagnetic scattering problems when waves
propagate through media containing many identical inclusions. Historically,
this numerical technique has been limited to situations in which the inclusions
(particles) can be covered by nonoverlapping disks in two dimensions or spheres
in three dimensions. This allows for the use of separation of variables in
cylindrical or spherical coordinates to represent the solution to the governing
partial differential equation. Here, we provide a more flexible approach,
applicable to a much larger class of geometries. We use a Green's
representation formula and the associated layer potentials to construct
incoming and outgoing solutions on rectangular enclosures. The performance and
flexibility of the resulting scattering operator formulation in two-dimensions
is demonstrated via several numerical examples for multi-particle scattering in
free space as well as in layered media. The mathematical formalism extends
directly to the three dimensional case as well, and can easily be coupled with
several commercial numerical PDE software packages.
