Band structure computation of two-dimensional and three-dimensional phononic crystals using a finite element-least square point interpolation method

Abstract

• FE-LSPIM proposed to calculate band structures of elastic waves in 2D and 3D models. • Shape functions in FE-LSPIM exploit advantages of mesh-free/finite element methods. • FE-LSPIM can achieve excellent accuracy, especially in high-frequency modes. • FE-LSPIM achieves higher accuracy when calculating band gaps for phononic crystals. In the present study, a finite element-least square point interpolation method (FE-LSPIM) is proposed for calculating the band structures of in-plane elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs). This method utilizes new shape functions by combining mesh-free shape functions and finite element shape functions to exploit the specific advantages of the mesh-free method and finite element method (FEM). As a result, FE-LSPIM inherits the completeness properties of the mesh-free method and the compatibility properties of FEM, and thus the solutions obtained tend to be more accurate. Indeed, according to our previous research, the present method obtains excellent accuracy, especially in the high-frequency domain. The proposed FE-LSPIM was combined with Bloch's theory and applied to compute the band gaps (BGs) for 2D PCs and 3D PCs in the present study, where several PCs were investigated to verify the high accuracy when computing the BGs. Numerical analysis showed that the proposed method can predict the BGs more precisely compared with the FEM and modified FEM.