Novel applications of local optimization semi-Cartesian grid for the complex band structure analysis of phononic crystals

Abstract

In this paper, we propose a finite element method based on the Floquet transform and local optimization semi-Cartesian grid to compute the complex band structure of phononic crystals. Floquet transform is applied to transform the elastic wave equation into a quadratic eigenvalue equation. Value-periodic test function independent of the interface is defined to derive the weak formulation. The local optimization semi-Cartesian grid is proposed to improve the quality of triangular elements near the interface. Our results are in good agreement with ones obtained by other mature methods, and the purely propagative mode coincides with the traditional band diagram. In addition to simple scatterer shapes such as circle or square, the proposed method is able to calculate the complex band structures of phononic crystals with complicated scatterer shapes. Numerical experiments demonstrated the effectiveness of this method in the computation of anisotropic inhomogeneous medium. The complex band structures under different lattice types, scatterer shapes, material properties, rotation angle and modes are obtained, and their effects on the width and number of band gaps are analyzed.