Uncertain Analysis For Phononic Crystals Based On Orthogonal Polynomials

Researches carried by scholars on phononic crystals and metamaterials are based on deterministic model without considering the uncertainties.However,uncertainties are unavoidably in the presence of phononic crystals and metamaterials.For example,the variation of the surrounding temperature leads to the variation of material properties,the variation of dimension,external loads and boundary conditions.Generally,Phononic crystals and metamaterials are sensitive to the uncertainties,subtle variations will lead to huge changes of the system responses.As a result,when designing or applying theses attractive and new properties,uncertainties should be taken into consideration.In order to discuss the effects of uncertainties on phononic crystals and metamaterials,two cases,the interval uncertainties and stochastic uncertainties,are deeply discussed in this paper.Based on the interval model and stochastic model,the numerical models are proposed for phononic crystals and metamaterials,and then corresponded numerical methods are presented to discuss the uncertainties existing in the phononic crystals and metamaterials.The main contents are listed as follows:(1)An interval Chebyshev expansion-Monte Carlo Simulation method(ICE-MCSM)is proposed for interval models.In ICE-MCSM,the transmittance of multi-layer perforated metamaterial is approximated by truncated Chebyshev polynomial expansion,thereby constructing the surrogate model of transmittance,which replaces the calculations of directly employing the finite element method.Then,substituting the random points produced by Monte Carlo Simulation method into the surrogate model.Thus,the variational ranges of transmittance under single and multi-interval parameters are calculated.The results show that the interval bounds calculated by ICE-MCSM match the interval bounds calculated by DMCSM.(2)An improved Chebyshev polynomial expansion-Monte Carlo Simulation method(ICPE-MCSM)is proposed for interval models.This method is strongly suitable for high dimensional interval problems.In ICPE-MCSM,band structures of phononic crystals are approximated by an improved Chebyshev polynomial expansion(ICPE).Compared with the conventional Chebyshev polynomial expansion(CCPE),the ICPE not only retains the accuracy of the CCPE,but also greatly improves the computational efficiency by employing a new sampling strategy in the construction of Chebyshev surrogate model.Based on the Chebyshev surrogate model,the samples yielded by the Monte Carlo Simulation Method(MCSM)are used to calculate the variational ranges of the band structures of phononic crystals.Three numerical examples are employed to verify the effectiveness and efficiency of ICPE-MCSM.The results show that the interval bounds of band structures calculated by ICPE-MCSM perfectly match the results calculated by DMCSM.The efficiency of ICPE-MCSM is significantly higher than that of the DMCS.(3)An improved Gegenbauer polynomial expansion(IGPE)is proposed for stochastic models when the probabilities of stochastic parameters are precisely defined.In IGPE,band structures of phononic crystals are approximated by an improved Chebyshev Gegenbauer expansion,thereby constructing the surrogate model of transmittance,which substitutes the calculations of directly employing by finite element method.Then,the surrogate model is used for calculating the expectation and variance of band structures of phononic crystals.The results show that the expectation and variance calculated by IGPE match the results calculated by DMCSM.