Phononic Band Structures and Transmission Coefficients: Methods and Approaches

The purpose of this chapter is first to recall some fundamental notions from the theory of crystalline solids (such as direct lattice, unit cell, reciprocal lattice, vectors of the reciprocal lattice, Brillouin zone, etc.) applied to phononic crystals and second to present the most common theoretical tools that have been developed by several authors to study elastic wave propagation in phononic crystals and acoustic metamaterials. These theoretical tools are the plane wave expansion method, the finite-difference time domain method, the multiple scattering theory, and the finite element method. Furthermore, a model reduction method based on Bloch modal analysis is presented. This method applies on top of any of the numerical methods mentioned above. Its purpose is to significantly reduce the size of the final matrix model and hence enable the computation of the band structure at a very fast rate without any noticeable loss in accuracy. The intention in this chapter is to give to the reader the basic elements necessary for the development of his/her own calculation codes. The chapter does not contain all the details of the numerical methods, and the reader is advised to refer to the large bibliography already devoted to this topic.