局域共振声子晶体的优化设计与模拟
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摘要
三组元局域共振型声子晶体的晶格尺寸可以比布拉格散射产生的禁带波长小1～2个数量级,从而获得了利用小尺寸有效控制大波长的能力,可广泛应用于新型声波波导,滤波器以及声开关等器件的设计与制造。本文主要计算了二维及三维局域共振型声子晶体的共振频率、禁带位置以及相应薄层材料透射谱受共振单元改变的影响。共振单元的改变主要包括三种组分几何形状和填充率的改变,以及单元对称性降低后元胞结构的变化。本文主要通过对材料共振单元形状及晶格排列方式的优化设计,探索其对共振及透射特性的影响,获得了调节薄层局域共振型声子晶体材料的共振频率及透射率的有效途径。
第一章简要介绍声学以及空气中声速的推导,引出流体中声波的方程;接着将流体中声场推广为固体中的弹性波场,详细介绍了声子晶体的起源、进展及研究现状。回顾传统声学中噪音、消声器以及两个共振现象之后,引出局域共振型声子晶体,讨论禁带形成机制,略述其在隔音方面的应用前景。第二章给出坐标转换公式,以此为基础由常见的直角坐标系中的弹性介质方程推导出柱坐标及球坐标中相应方程的表达式。在简要讨论了几种流体与固体的边界条件之后,给出一个球形振子算例,说明局域共振型声子晶体中弹性场的计算流程。
第三章将最小二乘配点法及拉格朗日乘子法结合的思想引入到具有不规则边界振子的共振单元计算中,使其与位移势的单极子多阶展开结合形成半解析方法。该方法兼有解析解和数值解的优点:较短的计算时间及较广的应用范围,因此比较适用于实际问题。提出用动力学有效质量密度来确定其禁带位置及宽度,缩短了能带计算时间。用半解析方法对三维轴对称共振单元的共振频率及能带进行优化。计算结果表明,共振频率及归一化的带宽都可以得到优化。
第四章研究了二维三组元局域共振声子晶体的共振模式及透射谱。其元胞由共轭放置的两个具有不规则振子的共振单元构成。共振模式通过第三章中提出的半解析法得到,随后用有限元模拟验证。椭圆柱振子沿短轴和长轴两振动模式会造成弹性波的全反射。在这两种振动模式之间,存在一种沿层面方向反相振动的模式,会使有效质量密度为零,从而造成弹性波的全透射。这种新振动模式的共振频率,会随着椭圆振子的指向角连续变化。
第五章将单极多阶方法加以扩展成多极多阶方法,用以计算形状更一般的振子的声子晶体共振频率。虽然这会使势函数由一组非正交基展开,但是半解析法的收敛性会有明显增强;随形状改变产生的矩阵奇异性也被削弱,有利于求解。文中还简略叙述了有限元方法的数学基础,并尝试对二维形状作网格剖分。
The lattice constant of ternary locally resonant phononic crystal(PC) canbe 1～2 orders of magnitude lower than wavelength in its spectral gaps whichinduced by Bragg scattering. This characteristic can be used to control longclassical elastic wave by composite material of small dimensions. Hence it possesspotential for manufacturing of new wave guide, filters and switches. This thesiscompute the e?ect on resonance frequency, position of bandgap and transmissionspectrum of 2D and 3D locally resonant phononic crystal by changing the shapeof resonant unit. The change of resonant unit mainly includes the shape, fillingratio of three components and the cell structure for lower symmetry unit. Bystudying the units’shape and structure dependence of resonance frequency andtransmission characteristic, we obtain the way to tune the resonance frequencyand transmission spectrum of PC slab.
In the first chapter, acoustics and brief deduction of sound velocity weregiven, then were generalized to the wave equation in ?uid. The concept ofelastic wave field was proposed to usher in the description of PC. The origin,progress and the current research of PC were introduced in detail. The tradi-tional noise, noise reducer and two kind of classical resonance phenomena werereviewed brie?y. Then the formalism of band gap in locally resonant PC and theprospect in noise reduce field was discussed. The orthogonal curvilinear coordi-nates transformation and the elastic wave equation in rectangular coordinates wasintroduced in Chapter two, from which the equations in cylindrical and sphericalcoordinates can be obtained. After a brief description of several kinds of bound-ary conditions in solid-?uid coupled problems, a spherical resonator example wasdiscussed to illustrate the elastic wave field computation.
In chapter 3, both the least square collocation method and the Lagrange mul- tiplier method were ushered in to the computation of resonance unit with irreg-ular boundaries. Combined with the single-pole multiple expansion of displace-ment potential, the semi-analytical method was formulated. The semi-analyticalway possessed merits of both analytical method and numerical method: shortercomputation time and more general application scope, and hence is suit for prac-tical problems computation. The dynamic e?ective mass density was proposedto figure out the position and width of band gap, which can save time. Both theresonance frequency and normalized band gap width of resonance unit can beoptimized by adjusting the shape of the unit.
In chapter 4, the resonance modes and the transmission spectrum of 2Dternary PC were studied. The unit cell consists of two conjugate placed asym-metric elliptic cylinders coated with silicon rubber and embedded in a rigid ma-trix. The modes are obtained by the semi-analytic method in the least squarecollocation scheme and confirmed by the finite element method simulations. Tworesonance modes, corresponding to the vibration of the cylinder along the longand short axes, give rise to resonance re?ections of elastic waves. One mode inbetween the two modes, related to the opposite vibration of the two cylindersin the unit cell in the direction along the layer, results in the total transmissionof elastic waves due to zero e?ective mass density at the frequency. The reso-nance frequency of this mode, which has not yet been identified before, changescontinuously with the orientation angle of the elliptic resonator.
The multiple expansion with single pole was extended to multiple multipolemethod, so as to obtain the resonance frequency of PC unit with more generalresonator shape. Although the wave fields are expanded into a set of nonorthogo-nal basis functions, the convergence was enhanced obviously, and the sti? matrixattribute to the irregular shape can be solved more accurately. The second partof this chapter was dedicated to the mathematical basis of the finite elementmethod, and we tried to generate the two dimensional mesh for irregular shapedresonator.
第一章简要介绍声学以及空气中声速的推导,引出流体中声波的方程;接着将流体中声场推广为固体中的弹性波场,详细介绍了声子晶体的起源、进展及研究现状。回顾传统声学中噪音、消声器以及两个共振现象之后,引出局域共振型声子晶体,讨论禁带形成机制,略述其在隔音方面的应用前景。第二章给出坐标转换公式,以此为基础由常见的直角坐标系中的弹性介质方程推导出柱坐标及球坐标中相应方程的表达式。在简要讨论了几种流体与固体的边界条件之后,给出一个球形振子算例,说明局域共振型声子晶体中弹性场的计算流程。
第三章将最小二乘配点法及拉格朗日乘子法结合的思想引入到具有不规则边界振子的共振单元计算中,使其与位移势的单极子多阶展开结合形成半解析方法。该方法兼有解析解和数值解的优点:较短的计算时间及较广的应用范围,因此比较适用于实际问题。提出用动力学有效质量密度来确定其禁带位置及宽度,缩短了能带计算时间。用半解析方法对三维轴对称共振单元的共振频率及能带进行优化。计算结果表明,共振频率及归一化的带宽都可以得到优化。
第四章研究了二维三组元局域共振声子晶体的共振模式及透射谱。其元胞由共轭放置的两个具有不规则振子的共振单元构成。共振模式通过第三章中提出的半解析法得到,随后用有限元模拟验证。椭圆柱振子沿短轴和长轴两振动模式会造成弹性波的全反射。在这两种振动模式之间,存在一种沿层面方向反相振动的模式,会使有效质量密度为零,从而造成弹性波的全透射。这种新振动模式的共振频率,会随着椭圆振子的指向角连续变化。
第五章将单极多阶方法加以扩展成多极多阶方法,用以计算形状更一般的振子的声子晶体共振频率。虽然这会使势函数由一组非正交基展开,但是半解析法的收敛性会有明显增强;随形状改变产生的矩阵奇异性也被削弱,有利于求解。文中还简略叙述了有限元方法的数学基础,并尝试对二维形状作网格剖分。
The lattice constant of ternary locally resonant phononic crystal(PC) canbe 1～2 orders of magnitude lower than wavelength in its spectral gaps whichinduced by Bragg scattering. This characteristic can be used to control longclassical elastic wave by composite material of small dimensions. Hence it possesspotential for manufacturing of new wave guide, filters and switches. This thesiscompute the e?ect on resonance frequency, position of bandgap and transmissionspectrum of 2D and 3D locally resonant phononic crystal by changing the shapeof resonant unit. The change of resonant unit mainly includes the shape, fillingratio of three components and the cell structure for lower symmetry unit. Bystudying the units’shape and structure dependence of resonance frequency andtransmission characteristic, we obtain the way to tune the resonance frequencyand transmission spectrum of PC slab.
In the first chapter, acoustics and brief deduction of sound velocity weregiven, then were generalized to the wave equation in ?uid. The concept ofelastic wave field was proposed to usher in the description of PC. The origin,progress and the current research of PC were introduced in detail. The tradi-tional noise, noise reducer and two kind of classical resonance phenomena werereviewed brie?y. Then the formalism of band gap in locally resonant PC and theprospect in noise reduce field was discussed. The orthogonal curvilinear coordi-nates transformation and the elastic wave equation in rectangular coordinates wasintroduced in Chapter two, from which the equations in cylindrical and sphericalcoordinates can be obtained. After a brief description of several kinds of bound-ary conditions in solid-?uid coupled problems, a spherical resonator example wasdiscussed to illustrate the elastic wave field computation.
In chapter 3, both the least square collocation method and the Lagrange mul- tiplier method were ushered in to the computation of resonance unit with irreg-ular boundaries. Combined with the single-pole multiple expansion of displace-ment potential, the semi-analytical method was formulated. The semi-analyticalway possessed merits of both analytical method and numerical method: shortercomputation time and more general application scope, and hence is suit for prac-tical problems computation. The dynamic e?ective mass density was proposedto figure out the position and width of band gap, which can save time. Both theresonance frequency and normalized band gap width of resonance unit can beoptimized by adjusting the shape of the unit.
In chapter 4, the resonance modes and the transmission spectrum of 2Dternary PC were studied. The unit cell consists of two conjugate placed asym-metric elliptic cylinders coated with silicon rubber and embedded in a rigid ma-trix. The modes are obtained by the semi-analytic method in the least squarecollocation scheme and confirmed by the finite element method simulations. Tworesonance modes, corresponding to the vibration of the cylinder along the longand short axes, give rise to resonance re?ections of elastic waves. One mode inbetween the two modes, related to the opposite vibration of the two cylindersin the unit cell in the direction along the layer, results in the total transmissionof elastic waves due to zero e?ective mass density at the frequency. The reso-nance frequency of this mode, which has not yet been identified before, changescontinuously with the orientation angle of the elliptic resonator.
The multiple expansion with single pole was extended to multiple multipolemethod, so as to obtain the resonance frequency of PC unit with more generalresonator shape. Although the wave fields are expanded into a set of nonorthogo-nal basis functions, the convergence was enhanced obviously, and the sti? matrixattribute to the irregular shape can be solved more accurately. The second partof this chapter was dedicated to the mathematical basis of the finite elementmethod, and we tried to generate the two dimensional mesh for irregular shapedresonator.
引文
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[1] Goffaux C. and S′anchez-Dehesa J.,“Two-dimensional phononic crystalsstudied using a variational method: Application to lattices of locally res-onant materials”, Physical Review B, 2003, 67(14), 144301.
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