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基于Chebyshev展开的区间穿孔板超材料分析

刘坚 雷济荣 夏百战

刘坚, 雷济荣, 夏百战. 基于Chebyshev展开的区间穿孔板超材料分析[J]. 力学学报, 2017, 49(1): 137-148. doi: 10.6052/0459-1879-16-254
引用本文: 刘坚, 雷济荣, 夏百战. 基于Chebyshev展开的区间穿孔板超材料分析[J]. 力学学报, 2017, 49(1): 137-148. doi: 10.6052/0459-1879-16-254
Liu Jian, Lei Jirong, Xia Baizhan. THE INTERVAL ANALYSIS OF MULTILAYER-METAMATERIALS WITH PERFORATED APERTURES BASED ON CHEBYSHEV EXPANSION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 137-148. doi: 10.6052/0459-1879-16-254
Citation: Liu Jian, Lei Jirong, Xia Baizhan. THE INTERVAL ANALYSIS OF MULTILAYER-METAMATERIALS WITH PERFORATED APERTURES BASED ON CHEBYSHEV EXPANSION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 137-148. doi: 10.6052/0459-1879-16-254

基于Chebyshev展开的区间穿孔板超材料分析

doi: 10.6052/0459-1879-16-254
基金项目: 

国家自然科学基金 11402083

详细信息
    通讯作者:

    夏百战,副教授,主要研究方向:汽车振动和噪声的不确定数值分析与优化方法.E-mail:xiabz2013@hnu.edu.cn

  • 中图分类号: O241

THE INTERVAL ANALYSIS OF MULTILAYER-METAMATERIALS WITH PERFORATED APERTURES BASED ON CHEBYSHEV EXPANSION

  • 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.

     

  • 图  1  多层穿孔板超材料模型

    Figure  1.  Model of multi-layer metmaterials

    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  5  Michalewicz函数

    Figure  5.  Michalewicz function

    图  6  切比雪夫代理模型平均误差

    Figure  6.  Average error of Chebyshev approximation

    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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出版历程
  • 收稿日期:  2016-09-09
  • 修回日期:  2016-11-20
  • 网络出版日期:  2016-11-24
  • 刊出日期:  2017-01-18

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