Bandgap Analysis Of Uncertain Phononic Crystals Based On Orthogonal Polynomials

Locally resonant phononic crystal(PhC)can control large wavelengths with a small lattice sizes to produce bandgaps in the low frequency range.There are some significant prospects for low-frequency damping and noise reduction.In practical engineering applications,uncertainties widely exist in PhCs systems due to manufacturing errors,measurement errors,changeable environmental factors and unpredictable external excitations.The uncertainties seriously affect the physical properties of the PhCs.However,there are few studies on the uncertainties in PhCs.Considering the great influence of uncertainties,there is a great prospect for studying the uncertainties of PhCs.The main research contents of this article are as follows:(1)In view of the fact that uncertainties exists widely in PhCs,the interval model was introduced into the PhCs.Then,the Chebyshev polynomial is employed to construct the surrogate model of the band structure of the PhCs.Based on the surrogate model,the influence of the uncertain parameters on the bandgaps of the PhCs is investigated.Finally,an optimization model of the PhCs based on the Chebyshev surrogate model is constructed.In this optimization model,the maximization of the bandgaps of the PhCs is set as an objective function and the variational range of the bandgaps is considered as a constraint condition.The genetic algorithm is used to solve this optimization model.Numerical results show that the Chebyshev surrogate model can accurately evaluate the bandgaps of the PhCs based on the interval model.(2)Random variables and interval variables exist in uncertain PhCs respectively,and they may exist simultaneously in some cases.Based on the Gegenbauer polynomial,three uncertain models are constructed for the response analysis of the bands of the PhCs.The three uncertain models are an interval model,a random model and a hybrid uncertain model,respectively.Owing to the properties of Gegenbauer polynomial,the response of these three uncertain models of the PhCs can be approximated by a unified Gegenbauer polynomial.Based on the orthogonal property of Gegenbauer polynomial,expectations and variances of the band response related to the random variables can be readily obtained.The bounds of Gegenbauer polynomial with respect to the interval variables can be determined by the Monte-Carlo method.The numerical results show that the three uncertain models can achieve a great accuracy and good efficiency for uncertain PhCs.(3)High-order Gegenbauer polynomials can obtain high approximation accuracy.However,high-order polynomials may produce expensive computational cost for multi-variable nonlinear problems.Based on the simplex format of polynomial and a new sparse sequential sampling scheme,we propose a new uniform high-order sparse Gegenbauer polynomial surrogate model(HOSGPSM).In the sampling process,the maximin principle is employed to select the most representative samples in the candidate set,which greatly reduces the sample data.The numerical results of three typical PhCs show that HOSGPSM can accurately and efficiently evaluate the band structure of PhCs under three uncertain models,providing a unified platform for the numerical prediction and the optimal design of the PhCs with different types of uncertain parameters.