Boundary Element Method for Analysis of Transverse Wave Resonances in Phononic Crystals

A two-dimensional boundary element method is developed to study the resonances of a periodic array of inclusions in an elastic medium. The system is composed of a series of periodically spaced infinite one-dimensional gratings (or rows) of cavities. Taking into account the periodicity of the scattered field and the configuration, we adopt the periodic Green's function as the fundamental solution, which is used to formulate the boundary integral equations. Then, the resonance properties can be obtained by discretizing these equations. As an example, we compute resonance properties of a system composed of a square lattice with cavities filled in the Aluminum medium. Relations between the resonances and the frequencies are determined, from which the band gaps in the system can be clearly seen. The present results are in good agreement with those obtained by the time domain finite difference method. It is shown that the boundary element method is an effectively numerical method.