THE INTERVAL ANALYSIS OF MULTILAYER-METAMATERIALS WITH PERFORATED APERTURES BASED ON CHEBYSHEV EXPANSION
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摘要: 目前对于声学超材料的传输特性分析和优化大多是基于确定的数值和确定的模型，然而在实际工程和结构设计中存在大量材料自身特性和几何物理参数的不确定性.如果忽略这些不确定变量对声学超材料传输特性分析和优化过程的影响，得到的结果可能不正确.针对这一现状，拟将切比雪夫区间模型引入多层穿孔板超材料，提出多层穿孔板超材料声学透射率的区间切比雪夫展开——蒙特卡洛模拟法（interval Chebyshev expansionMonte Carlo simulation method，ICE-MCSM）.该方法采用截断切比雪夫多项式近似拟合多层穿孔板超材料的声学透射率响应曲线，构造声学透射率响应曲线的切比雪夫代理模型；然后采用蒙特卡洛模拟法（Monte Carlosimulation method，MCSM）随机生成一定数量的不确定区间变量的样本数据点，并将生成的不确定区间变量样本数据点代入切比雪夫代理模型，预测单个不确定区间变量和多个不确定区间变量条件下的多层穿孔板超材料声学透射率区间的上界和下界.数值分析结果表明，ICE-MCSM预测的声学透射率变化区间的上界和下界与直接蒙特卡洛法（direct Monte Carlo simulation method，DMCSM）预测的声学透射率上界和下界的结果非常接近.与DMCSM相比，ICE-MCSM具有更高的计算效率.因此，ICE-MCSM可有效且高效地分析不确定区间变量条件下多层穿孔板超材料声学透射率传输特性，具有良好的工程应用前景.Abstract: Up to now, the response analysis and optimization of acoustic metamaterials are mostly based on deterministic parameters and deterministic models.However, in the real engineering world and structure design fields where there are many uncertainties, such as the uncertain of material properties and geometric parameters.If the effects of uncertain variables on analysis and optimization of acoustic metamaterials are not taken into account, the analysis and optimization results may not true.Aiming at this problem and situation, in this paper where the interval model is introduced into multilayer-metamaterials with perforated apertures, and interval Chebyshev expansion-Monte Carlo simulation method (ICE-MCSM) of multilayer-metamaterials with perforated apertures for transmission characteristic is proposed.In ICE-MCSM, truncated Chebyshev polynomials is applied to surrogate the transmittance of multilayer-metamaterials with perforated apertures;therefore, the surrogate model of transmittance is constructed.The samples of interval variables are produced by Monte Carlo method, then the values of these samples are substituted into the surrogate model of transmittance to predict the interval bounds of multilayer-metamaterials with perforated apertures under single-interval variable and multi-interval variables.The results of numerical analysis show that the upper interval bounds and lower interval bounds calculated by ICE-MCSM match the response yields by direct Monte Carlo simulation method (DMCSM).Compared to DMCSM, ICE-MCSM achieves a higher accuracy in interval bound analysis, so ICE-MCSM can effectively and efficiently analyze the interval bounds of multilayer-metamaterials with perforated apertures under uncertain interval variables.Thus, this proposed method in this paper can have promising prospects in real world engineering applications.
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Key words:
- metamaterials /
- transmittance /
- Chebyshev expansion /
- interval analysis
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图 2 周期数$n$分别等于1,2,3 和4 时声学透射率响应曲线
Figure 2. Response curve of transmittance for periodicity $n =1$,2,3 and 4
图 3 入射波频率分别为11 kHz(a)和15 kHz(b)时的声压分布图(周期数$n =2$)
Figure 3. Sound pressure field for plane wave incidence under the frequency of 11 kHz (a) and 15 kHz (b) ($n=2$)
图 4 DMCSM声学透射率区间边界分析流程
Figure 4. Flow chart of DMCSM for bounds analysis of transmittance
图 7 ICE-MCSM声学透射率区间边界分析流程
Figure 7. Flow chart of ICE-MCSM for bounds analysis of transmittance
图 8 在case 1区间条件下区间变量 $l_{\rm a }$,$l_{\rm b} $,$c_{\rm air}$,$r$的声学透射率区间边界
Figure 8. Interval bounds of transmittance for $l_{\rm a }$,$l_{\rm b} $,$c_{\rm air} $,$r$ in case 1
图 9 在case 2区间条件下区间变量$l_{\rm a }$,$l_{\rm b} $,$c_{\rm air}$,$r$的声学透射率区间边界
Figure 9. Interval bounds of transmittance for $l_{\rm a }$,$l_{\rm b} $,$c_{\rm air}$,$r$ in case 2
图 10 在case 3区间条件下区间变量$l_{\rm a }$,$l_{\rm b} $,$c_{\rm air}$,$r$的声学透射率区间边界
Figure 10. Interval bounds of transmittance for $l_{\rm a }$,$l_{\rm b} $,$c_{\rm air}$,$r$ in case 3
图 11 基于DMCSM声学透射率区间(a)上界(b)下界的收敛图(15 kHz)
Figure 11. The covergence plot of transmittance for interval upper bound (a) and lower bound (b) based on DMCSM (15 kHz)
图 12 Case 1条件下声学透射率区间边界(5阶)
Figure 12. Interval bounds of transmittance in case 1 (5 order)
图 13 Case 2条件下声学透射率区间边界(5阶)
Figure 13. Interval bounds of transmittance in case 2 (5 order)
图 14 Case 3条件下声学透射率区间边界(5阶)
Figure 14. Interval bounds of transmittance in case 3 (5 order)
图 15 Case 3条件下声学透射率区间边界(6阶)
Figure 15. Interval bounds of transmittance in case 3 (6 order)
表 1 多层穿孔板超材料模型结构和材料参数
Table 1. Parameters of multi-layer metmaterials
表 2 不确定区间变量的变化范围
Table 2. Uncertainty range of interval variables
表 3 不同截断阶数条件下ICE-MCSM的相对误差($f=15$ kHz，$f=15.5$ kHz)
Table 3. Relative error of ICE-MCSM for different truncations ($f=15$ kHz，$f=15.5$ kHz)
表 4 单个区间变量下ICE-MCSM和DMCSM声学透射率区间边界计算成本对比
Table 4. Comparison of computational cost between ICE-MCSM and DMCSM for interval bounds of transmittance in the case of single variables
表 5 多个区间变量下ICE-MCSM和DMCSM声学透射率区间边界计算成本对比
Table 5. Comparison of computational cost between ICE-MCSM and DMCSM for interval bounds of transmittance in the case of multi-variables
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[1] Veselago VG. The electrodynamics of substances with simultaneously negative values of ε and μ. Soviet Physics Uspekhi, 1968, 10(4):509 doi: 10.1070/PU1968v010n04ABEH003699 [2] Schurig D, Mock JJ, Justice BJ, et al. Metamaterial electromagnetic cloak at microwave frequencies. Science, 2006, 314(5801):977-980 doi: 10.1126/science.1133628 [3] Smith DR, Padilla WJ, Vier DC, et al. Composite medium with simultaneously negative permeability and permittivity. Physical Review Letters, 2000, 84(18):4184 doi: 10.1103/PhysRevLett.84.4184 [4] Sheng P, Mei J, Liu ZY, et al. Dynamic mass density and acoustic metamaterials. Physica B:Condensed Matter, 2007, 394(2):256-261 doi: 10.1016/j.physb.2006.12.046 [5] Fang N, Xi DJ, Xu JY, et al. Ultrasonic metamaterials with negative modulus. Nature Materials, 2006, 5(6):452-456 doi: 10.1038/nmat1644 [6] Graciá-Salgado R, García-Chocano VM, Torrent D, et al. Negative mass density and ρ-near-zero quasi-two-dimensional metamaterials:design and applications. Physical Review B, 2013, 88(22):224305 doi: 10.1103/PhysRevB.88.224305 [7] Lee SH, Park CM, Seo YM, et al. Composite acoustic medium with simultaneously negative density and modulus. Physical Review Letters, 2010, 104(5):054301 doi: 10.1103/PhysRevLett.104.054301 [8] Christensen J, de Abajo FJG. Negative refraction and backward waves in layered acoustic metamaterials. Physical Review B, 2012, 86(2):024301 doi: 10.1103/PhysRevB.86.024301 [9] Zhu R, Liu XN, Hu GK, et al. Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial. Nature Communications, 2014, 24(5):5510 http://cn.bing.com/academic/profile?id=11c473388907690fc60d25504dde2cd9&encoded=0&v=paper_preview&mkt=zh-cn [10] Qiu CY, Zhang XD, Liu ZY. Far-field imaging of acoustic waves by a two-dimensional sonic crystal. Physical Review B, 2005, 71(5):054302 http://cn.bing.com/academic/profile?id=41e987490cd828c72bac26bb690c9d85&encoded=0&v=paper_preview&mkt=zh-cn [11] 刘宸,孙宏祥,袁寿其等. 基于温度梯度分布的宽频带声聚焦效应. 物理学报, 2016, 65(4):044303 http://www.cnki.com.cn/Article/CJFDTOTAL-WLXB201604024.htmLiu Chen, Sun Hongxiang, Yuan Shouqi, et al. Broadband acoustic focusing effect based on temperature gradient distribution. Acta Physica Sinica, 2016, 65(4):044303(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-WLXB201604024.htm [12] Xia BZ, Dai HQ, Yu DJ. Symmetry-broken metamaterial for blocking, cloaking, and supertunneling of sound in a subwavelength scale. Applied Physics Letters, 2016, 108(25):251902 doi: 10.1063/1.4954711 [13] Kushwaha MS, Halevi P, Dobrzynski L, et al. Acoustic band structure of periodic elastic composites. Physical Review Letters, 1993, 71(13):2022 doi: 10.1103/PhysRevLett.71.2022 [14] Liu Z, Zhang X, Mao Y, et al. Locally resonant sonic materials. Science, 2000, 289(5485):1734-1736 doi: 10.1126/science.289.5485.1734 [15] Oudich M, Djafari-Rouhani B, Pennec Y, et al. Negative effective mass density of acoustic metamaterial plate decorated with low frequency resonant pillars. Journal of Applied Physics, 2014, 116(18):184504 doi: 10.1063/1.4901462 [16] Aközbek N, Mattiucci N, Bloemer MJ, et al. Manipulating the extraordinary acoustic transmission through metamaterial-based acoustic band gap structures. Applied Physics Letters, 2014, 104(16):161906 doi: 10.1063/1.4873391 [17] Wang XL. Theory of resonant sound transmission through small apertures on periodically perforated slabs. Journal of Applied Physics, 2010, 108(6):064903 doi: 10.1063/1.3481434 [18] 温激鸿,韩小云,王刚等. 声子晶体研究概述. 功能材料, 2003, 34(4):364-367 http://www.cnki.com.cn/Article/CJFDTOTAL-GNCL200304004.htmWen Jihong, Han Xiaoyun, Wang Gang, et al. Overview of phononic crystals. Journal of Functional Materials, 2003, 34(4):364-367(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-GNCL200304004.htm [19] Xia BZ, Chen N, Xie LX, et al. Temperature-controlled tunable acoustic metamaterial with active band gap and negative bulk modulus. Applied Acoustics, 2016, 112:1-9 doi: 10.1016/j.apacoust.2016.05.005 [20] 夏百战,覃缘,于德介等. 区间模型下声学超材料的可靠性优化. 机械工程学报, 2016, 13:094-102 http://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201613013.htmXia Baizhan, Qin Yuan, Yu Dejie, et al. Reliability-based optimization of the acoustic metamaterial under the interval model. Journal of Mechanical Engineering, 2016, 13:094-102(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201613013.htm [21] Xia Baizhan. A research on the numerical analysis and optimization method for the nondeterministic acoustic-structural coupled system.[PhD Thesis]. Changsha:Hunan University, 2015(in Chinese) [22] Jin RC, Wei C, Simpson TW. Comparison studies of metamodeling techniques under multiple modelling criteria. Structural and Multidisciplinary Optimization, 2001, 23(1):1-13 doi: 10.1007/s00158-001-0160-4 [23] Xia BZ, Yu DJ. Modified sub-interval perturbation finite element method for 2D acoustic field prediction with large uncertain-but-bounded parameters. Journal of Sound and Vibration, 2012, 331(16):3774-3790 doi: 10.1016/j.jsv.2012.03.024 [24] Edgecombe S, Linse P. Monte Carlo simulation of two interpenetrating polymer networks:structure, swelling, and mechanical properties. Polymer, 2008, 49(7):1981-1992 doi: 10.1016/j.polymer.2008.02.018 [25] Wu JL, Luo Z, Zheng J, et al. Incremental modeling of a new high-order polynomial surrogate model. Applied Mathematical Modelling, 2016, 40(7):4681-4699 http://cn.bing.com/academic/profile?id=56b9d55562be6a696e84cb265e61872a&encoded=0&v=paper_preview&mkt=zh-cn [26] 沈祖和. 区间分析方法及其应用. 应用数学与计算数学学报,1983,2(1):1-29Shen Zuhe. Interval analysis method and its application. Journal of Applied Mathematics and Computational Mathematics, 1983,2(1):1-29(in Chinese) [27] Muscolino G, Sofi A. Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis. Probabilistic Engineering Mechanics, 2012, 28:152-163 doi: 10.1016/j.probengmech.2011.08.011 [28] Xia B, Qin Y, Yu D, et al. Dynamic response analysis of structure under time-variant interval process model. Journal of Sound and Vibration, 2016, 381:121-138 doi: 10.1016/j.jsv.2016.06.030 [29] Wu JL, Luo Z, Zhang YP, et al. An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels. Applied Mathematical Modelling, 2014, 38(5):3706-3723 http://cn.bing.com/academic/profile?id=2a33ca49e8e82bfc03b1ed985dc2ef8c&encoded=0&v=paper_preview&mkt=zh-cn [30] Yin SW, Yu DJ, Huang Y, et al. Hybrid chebyshev interval finite-element and statistical energy analysis method for midfrequency analysis of built-Up systems with interval uncertainties. Journal of Engineering Mechanics, 2016, 142(10):04016071 doi: 10.1061/(ASCE)EM.1943-7889.0001131 [31] 王艳锋. 含共振单元声子晶体的带隙特性及设计.[博士论文]. 北京:北京交通大学, 2015Wang Yanfeng. Bandgap properties and design of phononic crystals with resonant units.[PhD Thesis]. Beijing:Beijing Jiaotong University, 2015(in Chinese) -
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