| Phononic crystals (PCs) are a kind of artificial acoustic functional materials with periodic structures which exhibits acoustic or elastic wave band gaps (BGs). The research of PCs has received a great deal of attention because of their novel and unique features such as band gaps, localization, etc. and perspective applications in the noise and vibration attenuation, acoustic wave control. Large-scale numerical calculations are usually involved in their theoretical analysis and numerical simulations. Therefore, developing fast efficient computational methods has been one of main contents in this research field. In this thesis, boundary integral equation method is developed to compute the band structures and transmission spectra of two-dimensional PCs. Main research is as follows:(1) Boundary element method (BEM) based on the general Green function is formulated to compute the band structures of solid/solid and fluid/fluid PCs; and the characteristics of its efficiency, convergence, stability, adaptivity and computation speed are analyzed. Results show that the efficiency and stability of BEM are influenced by the discretized element number, eigenvalue-selecting error ε, scatterers'shape, wave modes and acoustic impedances. Due to the influence of the discretized element number and eigenvalue-selecting errors, some energy bands may be leaked in certain frequency regions (especially near the high symmetry points of the Brillouin zone), which can be improved by increasing the number of the elements and/or ε value. Compared with other methods such as plane wave expansion method, wavelet method, etc., BEM has a small eigenvalue matrix, fast computation speed and good convergence; and it can consider the interface conditions of the scatterers and the matrix. In addition, BEM can compute the band structures at any frequency intervals and determine the band-gap ranges directly because it solves wave vectors for given frequencies.(2) By considering the fluid-solid interaction and the transverse wave in solid components, BEM is formulated to compute the band structures of mixed fluid-solid PCs. Numerical results show that this method is precise and efficient and not relevant to the scatterers'shapes and acoustic impedances. Especially for solid/fluid systems, BEM can give accurate results because it considers the fluid-solid interaction and the transverse wave, which are important and can not be omitted when the impedance of the solid scatterers is near or less than that of the fluid matrix.(3) BEMs based on the general Green function and periodic Green function are developed to compute the transmission spectra of solid/solid (anti-plane transverse wave), fluid/fluid and solid/fluid systems with finite layers. It is shown that the transmission spectra computed by BEM based on the general Green function are consistent with the corresponding band structures. Only when the acoustic impedance of the scatterers in the solid/fluid system is smaller than that of the matrix, the computed results have some discrepancy. Owing to the slow convergence of the periodic Green function involved, BEM based on the periodic Green function cannot yield the satisfied results and is time-consuming, so it needs to be improved.(4) Boundary integral equation method based on the Dirichlet-to-Neumann map is formulated to compute the band structures of solid/solid (anti-plane transverse wave) and fluid/fluid PCs; and its accuracy, computation speed and adaptivity are discussed. The results show that the method can compute PCs with various acoustic impedance ratios, lattice structures and sctterers'shapes. Moreover, the associate eigenvalue matrix is small; the computation speed is fast; and the cost of the computer memory is low. The method is expected to be extended to compute the vector waves.The boundary integral equation methods developed in this thesis provide an alternative effective numerical tool for the theoretical analysis of wave propagation in PCs.
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