  In this work, two fast multipole boundary element formulations for the linear
time-harmonic acoustic analysis of finite periodic structures are presented.
Finite periodic structures consist of a bounded number of unit cell
replications in one or more directions of periodicity. Such structures can be
designed to efficiently control and manipulate sound waves and are referred to
as acoustic metamaterials or sonic crystals. Our methods subdivide the geometry
into boxes which correspond to the unit cell. A boundary element discretization
is applied and interactions between well separated boxes are approximated by a
fast multipole expansion. Due to the periodicity of the underlying geometry,
certain operators of the expansion become block Toeplitz matrices. This allows
to express matrix-vector products as circular convolutions which significantly
reduces the computational effort and the overall memory requirements. The
efficiency of the presented techniques is shown based on an acoustic scattering
problem. In addition, a study on the design of sound barriers is presented
where the performance of a wall-like sound barrier is compared to the
performance of two sonic crystal sound barriers.
