Lumped-mass approach for the calculation of band structure of two-dimensional phononic crystals with high contrast of elastic constant

With each unit cell replaced by a system of finite freedoms of motion, two-dimensional phononic crystals can be simplified to an infinite discrete periodic system. Therefore, the elastic wave band structures of the two-dimensional phononic crystals can be calculated with a straightforward lumped-mass approach, whose computational cost is much lower than the well-known plane wave expansion(PWE) method. The numerical results of the two methods are in reasonable agreements. As the well-known Gibbs oscillations in the PWE can be eliminated with the lumped-mass method, this new approach is insensitive to the sharp variation of elastic constants on the interfaces inside the phononic crystals. Furthermore, the lumped-mass method can also be used to calculate the band structures of two-dimensional phononic crystals with arbitrary unit shapes easily.