A sparse polynomial surrogate model for phononic crystals with uncertain parameters

Abstract The traditional analysis and design of the phononic crystals (PhCs) are based on the deterministic physical models. However, uncertainties, unavoidably existing in the PhCs, seriously affect their physical properties. To quantify the influence of uncertainties on the band structures of the PhCs, three uncertain models, including an interval model, a random model and a hybrid uncertain model, are considered. In order to overcome the considerably computational cost of the evaluation of the band structures of the PhCs, a new high-order sparse Gegenbauer polynomial surrogate model (HOSGPSM) is proposed. In the HOSGPSM, a new spare sequential sampling scheme based on the maximin principle is employed in the sampling process, greatly reducing the potential samples. And then, the least square method is introduced to calculate the unknown coefficients of the HOSGPSM. According to the orthogonal property of the Gegenbauer polynomial, the expectation and variance of the HOSGPSM with respect to random variables can be directly yielded. The bounds of the HOSGPSM with respect to the interval model and the hybrid uncertain model can be obtained by the Monte Carlo method. The numerical results of three typical PhCs show that the HOSGPSM can accurately and efficiently evaluate the band structures of the PhCs under three uncertain models, providing a unified platform for the numerical analysis and the optimal design of various uncertain PhCs.

[1]  B. Djafari-Rouhani,et al.  Acoustic band structure of periodic elastic composites. , 1993, Physical review letters.

[2]  Gil-Yong Lee,et al.  Wave mixing in coupled phononic crystals via a variable stiffness mechanism , 2016, 1603.05316.

[3]  Dejie Yu,et al.  Modified sub-interval perturbation finite element method for 2D acoustic field prediction with large uncertain-but-bounded parameters , 2012 .

[4]  Jian-chun Cheng,et al.  Existence of broad acoustic bandgaps in three-component composite , 2003 .

[5]  Brian P. Mann,et al.  Uncertainty Propagation in the Band Gap Structure of a 1D Array of Magnetically Coupled Oscillators , 2013 .

[6]  F. Tin-Loi,et al.  Probabilistic interval analysis for structures with uncertainty , 2010 .

[7]  M. D. McKay,et al.  A comparison of three methods for selecting values of input variables in the analysis of output from a computer code , 2000 .

[8]  Dejie Yu,et al.  Hybrid uncertain analysis of acoustic field with interval random parameters , 2013 .

[9]  David Moens,et al.  A survey of non-probabilistic uncertainty treatment in finite element analysis , 2005 .

[10]  Marcin Kamiński,et al.  Uncertainty in effective elastic properties of particle filled polymers by the Monte-Carlo simulation , 2015 .

[11]  Baizhan Xia,et al.  Dynamic response analysis of structure under time-variant interval process model , 2016 .

[12]  Dejie Yu,et al.  Acoustic Metamaterial With Fractal Coiling Up Space for Sound Blocking in a Deep Subwavelength Scale , 2018 .

[13]  M. E. Johnson,et al.  Minimax and maximin distance designs , 1990 .

[14]  Dejie Yu,et al.  Transformed perturbation stochastic finite element method for static response analysis of stochastic structures , 2014 .

[15]  Dejie Yu,et al.  Temperature-controlled tunable acoustic metamaterial with active band gap and negative bulk modulus , 2016 .

[16]  P. Sheng,et al.  Locally resonant sonic materials , 2000, Science.

[17]  R. Mullen,et al.  Interval Monte Carlo methods for structural reliability , 2010 .

[18]  V. Kouznetsova,et al.  Visco-elastic effects on wave dispersion in three-phase acoustic metamaterials , 2016 .

[19]  Z. Qiu,et al.  Non-probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty , 2009 .

[20]  Sheng Zhang,et al.  Sparse regression Chebyshev polynomial interval method for nonlinear dynamic systems under uncertainty , 2017 .

[21]  T. J. Mitchell,et al.  Exploratory designs for computational experiments , 1995 .

[22]  V. Kouznetsova,et al.  Towards optimal design of locally resonant acoustic metamaterials , 2014 .

[23]  I. Elishakoff,et al.  Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters , 1993 .

[24]  B. Djafari-Rouhani,et al.  Evidence of fano-like interference phenomena in locally resonant materials. , 2002, Physical review letters.

[25]  Dirk Gorissen,et al.  A Novel Hybrid Sequential Design Strategy for Global Surrogate Modeling of Computer Experiments , 2011, SIAM J. Sci. Comput..

[26]  Nong Zhang,et al.  Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties , 2014 .

[27]  Liping Chen,et al.  Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions , 2013 .

[28]  Z. Qiu,et al.  Hybrid uncertain analysis for steady-state heat conduction with random and interval parameters , 2015 .

[29]  Xiaolei Leng,et al.  Evolutionary earthquake response of uncertain structure with bounded random parameter , 2004 .

[30]  Zhen Luo,et al.  Incremental modeling of a new high-order polynomial surrogate model , 2016 .

[31]  Hui Yin,et al.  A unified method for the response analysis of interval/random variable models of acoustic fields with uncertain‐but‐bounded parameters , 2017 .

[32]  Massimo Ruzzene,et al.  Topology design and optimization of nonlinear periodic materials , 2013 .

[33]  Eunho Kim,et al.  Wave propagation in single column woodpile phononic crystals: Formation of tunable band gaps , 2014 .

[34]  G. Stefanou.  The stochastic finite element method: Past, present and future , 2009 .

[35]  Zhen Luo,et al.  An arbitrary polynomial chaos expansion approach for response analysis of acoustic systems with epistemic uncertainty , 2018 .

[36]  Kazem Abhary,et al.  Optimum design of phononic crystal perforated plate structures for widest bandgap of fundamental guided wave modes and maximized in-plane stiffness , 2016 .

[37]  Yuesheng Wang,et al.  Topology optimization of anisotropic broadband double-negative elastic metamaterials , 2017 .

[38]  Liping Chen,et al.  A Chebyshev interval method for nonlinear dynamic systems under uncertainty , 2013 .

[39]  Yunlong Li,et al.  Hybrid uncertainty propagation in structural-acoustic systems based on the polynomial chaos expansion and dimension-wise analysis , 2017 .

[40]  Qibai Huang,et al.  A new hybrid uncertain analysis method for structural-acoustic systems with random and interval parameters , 2016 .

[41]  Feng-Ming Li,et al.  The propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals , 2008 .

[42]  Liping Li,et al.  Fractal labyrinthine acoustic metamaterial in planar lattices , 2018 .

[43]  Dejie Yu,et al.  Symmetry-broken metamaterial for blocking, cloaking, and supertunneling of sound in a subwavelength scale , 2016 .

[44]  Nong Zhang,et al.  An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels , 2014 .

[45]  Mehdi Dehghan,et al.  Solving a laminar boundary layer equation with the rational Gegenbauer functions , 2013 .