Convergence problem of plane-wave expansion method for phononic crystals

A new formulation of eigenproblem for phononic crystals is developed. The convergence of the new formulation in the band-structure calculations is examined in detail and compared with that of the conventional plane wave expansion (CPWE) method. Numerical results show that the slow convergence of the CPWE method is not due to the slow convergence of the Fourier series for the elastic coefficients (or displacement fields) in the interfaces of different materials, but to the inappropriate formulation of the eigenproblem used in the calculations. Numerical calculations also show that the new formulation can provide much more accurate numerical results than the CPWE method for the systems of either very high or very low filling fractions, or of large elastic mismatch