Band structure analysis of phononic crystals based on the Chebyshev interval method.

Due to environmental changes and manufacturing errors, uncertainties inherently exist in phononic crystals, especially in the material properties and geometric parameters. To handle the uncertainties with limited information, two different methods are proposed for band structure analysis of phononic crystals with uncertainties. One is the Monte Carlo method (MCM). The main shortcoming of it is the high computational cost. The other is the simplex Chebyshev polynomial expansion (SPCE) method. In addition, the computational efficiency of the SPCE method is much higher. In the SPCE method, the simplex Chebyshev is employed to estimate the band structures of phononic crystals. Meanwhile, the simplified incremental sampling strategy is introduced for the simplex Chebyshev to retain calculation accuracy and improve computational efficiency simultaneously. In the Chebyshev surrogate model, the samples yielded with the MCM are used to calculate the interval ranges of the band structures in phononic crystals. Three numerical examples, including a two-dimensional (2D) Helmholtz resonator phononic crystal, a 2D solid-solid phononic crystal, and a three-dimensional phononic crystal, are introduced to verify the effectiveness and efficiency of the proposed method.