Simulation of the band structure for scalar waves in 2D phononic crystals by the singular boundary method

In this paper, the singular boundary method (SBM) is applied to compute band structures for scalar waves in two-dimensional (2D) phononic crystals (PCs). By employing the SBM to discretize the boundaries, including the periodic boundary and the continuity boundary in the PCs, we can obtain a linear eigenvalue equation which describes the relationship between the Block wave and frequency. By sweeping the frequencies and solving the eigenvalue equation, the band structures can be achieved. Several numerical examples are presented to validate the performance and efficiency of the proposed method. Compared with the PWE and FEM methods, the SBM has an advantage of quick convergence and low computational cost. Moreover, it can easily deal with various shaped scatters, even with sharp corners. Therefore, the present SBM can be considered to be a fast, accurate and stable alternative for band structure calculation of the PCs.