Bandgap characteristics and optimization of two-dimensional phononic metamaterials with functionally graded rings

This paper systematically investigates the influence of the thickness and power-law exponent of functionally graded rings (FGR) on the bandgap characteristics of two-dimensional (2D) phononic metamaterials (PMs). Based on the plane wave expansion method (PWEM), a spatial variation model for the material parameters of FGR is established, the dispersion relations of elastic waves in 2D PMs are derived, and the evolution of the first bandgap is analyzed. The results indicate that an increase in the thickness of FGR shifts both the upper and lower boundary of the bandgap upward, while the bandgap width initially increases and then decreases, reaching a maximum at a specific thickness. Variations in the power-law exponent significantly affect the width and center frequency of the bandgap, demonstrating a notable nonlinear regulatory effect. On this basis, the single-objective optimization of the bandgap width was achieved by the genetic algorithm combining with PWEM, while the multi-objective optimization of both bandgap width and center frequency was realized using the non-dominated sorting genetic algorithm (NSGA-II) in conjunction with PWEM. The Pareto optimal front indicates that there is a favorable competitive relationship between the bandgap width and the center frequency, and the optimal parameters are not limited to the boundary values.