Abstract
In this paper, we introduce a novel localized collocation solver for two-dimensional (2D) phononic crystal analysis. In the proposed collocation solver, the displacement at each node is expressed as a linear combination of T-complete functions in each stencil support and the sparse linear system is obtained by satisfying the considered governing equation at interior nodes and boundary conditions at boundary nodes. As compared with finite element method (FEM) results and the analytical solutions, the efficiency and accuracy of the proposed localized collocation solver are verified under a benchmark example. Then, the proposed method is applied to 2D phononic crystals with various lattice forms and scatterer shapes, where the related band structures, transmission spectra, and displacement amplitude distributions are calculated as compared with the FEM.
Highlights
In recent decades, more and more attention has been paid to a new kind of artificial periodic composite structures, which are well known as phononic crystals [1,2,3,4,5,6]
This paper is organized as follows: The mathematical formulation of the anti-plane transverse elastic wave in the 2D phononic crystal is described in Section 2; the corresponding discretization formulation based on the localized collocation Trefftz method (LCTM) is clearly introduced in Section 3; in Section 4, as compared with the COMSOL simulation, the efficiency and accuracy of the proposed LCTM are verified under a benchmark example, and the wave propagation behavior is investigated by calculation of band structures, transmission spectra, and displacement amplitude distributions in phononic crystals with various lattice forms and scatterer shapes; and in Section 5, our conclusions are summarized
A novel localized collocation scheme based on T-complete functions is applied, for the first time, to calculate the band structures, transmission spectra, and displacement amplitude distribution for anti-plane transverse elastic waves in 2D solid phononic crystals
Summary
More and more attention has been paid to a new kind of artificial periodic composite structures, which are well known as phononic crystals [1,2,3,4,5,6]. This paper is organized as follows: The mathematical formulation of the anti-plane transverse elastic wave in the 2D phononic crystal is described in Section 2; the corresponding discretization formulation based on the localized collocation Trefftz method (LCTM) is clearly introduced in Section 3; in Section 4, as compared with the COMSOL simulation, the efficiency and accuracy of the proposed LCTM are verified under a benchmark example, and the wave propagation behavior is investigated by calculation of band structures, transmission spectra, and displacement amplitude distributions in phononic crystals with various lattice forms and scatterer shapes; and, our conclusions are summarized. A perfectly phononic crystal withsemi-infinite semi-infinite periodic ofof (a)(a) square lattices and and (b) triangular latticeslattices with with the related boundary conditions
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