The main aim of this thesis is to develop an accurate and efficient numerical tool based on the radial basis function collocation method (RBFCM) for the band structure calculations of elastic and acoustic waves in one-dimensional (1D) and two-dimensional (2D) phononic crystals. Some new numerical techniques are proposed to accurately deal with the derivative computations of the field quantities near/on the boundaries/interfaces required by the boundary conditions and the continuity conditions on the interfaces. By using these novel numerical techniques, the stability of the RBFCM can be significantly improved, which leads to an enhanced accuracy and efficiency. Both the global RBFCM (GRBFCM) and the local RBFCM (LRBFCM) are presented and discussed in the thesis. Then, the accuracy and the efficiency of the RBFCM are verified by the numerical results obtained by the finite element method (FEM), and applied to the band structure computations of 1D and 2D solid/solid as well as 2D solid/fluid and fluid/solid phononic crystals with different acoustic impedance mismatches, material combinations, scatterer shapes, and lattice forms. The effects of the key geometrical and material parameters on the band structures especially the bandgaps of 1D and 2D phononic crystals are also investigated and discussed.
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