A finite element method towards acoustic phononic crystals by weak formulation

A weak formulation method is presented to analyze the propagation of acoustic waves in periodic crystal-like systems called phononic crystals. First, a compact circle and double split ring resonators with high impedance contrast to air background are employed to compute band structures with not only real parts of wavevectors but also imaginary parts of them. Calculating the imaginary part, which directly predicts a propagation length in band gaps, is not readily available in previous studies. It analytically enables us to more understand the behavior of evanescent mode in band gaps. Moreover, we present that the weak formulation method is advantageous in calculating equi-frequency contour.