基于对称性的二维声子晶体带隙特性研究
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摘要
声子晶体(phononic crystal)是一种具有周期性结构并具有声/弹性波带隙的人工新型声学功能材料。由于其奇特的带隙、局域、负折射特征和在振动隔离、噪音控制、声学波导器件及声学透镜等方面的潜在应用,近二十年来,声子晶体的研究引起了各国学者的研究兴趣。
针对声子晶体低频带隙打开的难题,本文基于降低对称性的思想,设计了两种新型晶格结构,利用平面波展开法和有限元法计算二维固/流(气)声子晶体的能带结构、传输谱和压力场分布,打开了低频带隙;基于超声浸水透射技术实验测试了声子晶体的带隙和缺陷态特征。主要工作如下:
(1)正方晶格钢/水声子晶体的聋带研究
对正方晶格钢/水声子晶体,沿着第一Brillouin区的两个高对称方向,实验测量的第一带隙与两种方法的数值结果基本吻合;沿着??方向,在高频段,实验测试的声波衰减频段与平面波法计算的能带结构不符,通过计算压力场证明对应的能带为反对称模态,形成聋带(deaf band),实验证实了在正方晶格钢/水声子晶体中存在聋带,揭示了高频段测试结果与计算结果矛盾的原因。
(2)众所周知,第一Brillouin区高对称点处的能带简并是制约带隙打开的一个重要因素。本文基于降低对称性的思想,在单胞内添加“原子”降低结构的对称性,消除了能带简并,打开了低频带隙。
①混合三角-六角晶格声子晶体的带隙特性研究。在三角晶格的单胞内添加两个“原子”形成混合三角-六角晶格结构,降低了结构对称性。用平面波法计算了混合三角-六角晶格钢/空气声子晶体的能带结构,结果表明,改变不同格点的散射体半径能调控带隙的数目、位置及宽度,且不同格点处的散射体半径变化对带隙的影响不同:三角格点处的散射体半径越大越容易打开高频带隙;六角格点处的散射体半径越大越容易打开低频带隙。
②混合三角晶格声子晶体的带隙特性研究。对混合三角晶格结构,由于在单胞内添加一个“原子”降低了结构的对称性,因此消除了能带在第一Brillouin区高对称点X处的简并,打开了一条新的位于第一、二能带间的低频带隙;改变添加原子的位置可以调控带隙的位置及宽度。实验测试了混合三角晶格钢/水声子晶体的带隙特性,验证了降低对称性是消除能带简并,打开低频带隙的一种有效方法。测试结果与两种方法计算的结果吻合较好。
(3)基于椭圆散射体的声子晶体带隙特性研究
以椭圆散射体取代圆形散射体,对三角晶格钢/空气声子晶体,在相同的晶格常数及填充率下,当散射体由圆柱体改为椭圆柱体时,消除了能带简并,打开了一条新的位于第一、二能带间的低频带隙;并且在一定的填充率范围内打开的第二带隙宽度及位置与圆柱体对应的声子晶体打开的第一带隙的宽度及位置基本一致;散射体截面越接近于圆,越不易打开第一、二能带间的带隙。椭圆截面的旋转角度对第一带隙影响较大。
根据这种带隙特性,把具有不同晶格常数椭圆截面散射体构成的声子晶体并列,在降低结构尺寸的条件下打开了宽的低频带隙。
(4)钢/水声子晶体缺陷态的局域特性研究
基于实验测试和有限元计算研究了多种点、线缺陷的局域特征。在带隙频率范围内,声波在声子晶体中传播时会局域在缺陷处,形成共振腔或波导结构;对直线型线缺陷,声波沿着线缺陷几乎是无损耗地传播,但对点缺陷,局域频率点的透射幅值很小。实验测试的线缺陷的局域频段与计算结果基本吻合;但是对点缺陷,实验测试的局域频率点与计算结果相比有一定的漂移。
基于单点缺陷和直线型线缺陷的局域特征构造了波导-空腔-波导结构,计算表明,该结构极大地提高了点缺陷的局域效果,透射率接近100%。
(5)智能声子晶体的带隙特性探讨
把磁流变弹性体引入声子晶体复合材料,以磁流变弹性体为基体,利用平面波法研究了外加磁场对声子晶体带隙的影响。
Phononic crystal (PC) is a kind of artificial acoustic functional materials with periodic structures which exhibit acoustic or elastic wave band gaps (BGs). For near two decades, the study on PCs has proved to be of great interest due to their novel and unique BGs, localization and negative refraction characteristics, and their potential applications in the vibration isolation, noise control, acoustic filter, waveguide devices acoustic lens, and so on.
In order to obtain low-frequency BG, two kinds of new lattices are designed based on symmetry reduction. Band structure, transmission spectra and pressure field of two-dimensional solid/fluid (air) PCs are investigated with the plane wave expansion (PWE) method and finite element method (FEM), respectively. BG and localization characteristics are measured based on ultrasonic immersion transmission technology. Main results are as follows:
(1) Deaf band for square lattice steel/water PCs
For two-dimensional square lattice steel/water PCs, the measured transmission spectra are in agreement with the numerical results at low-frequency region along the two high symmetry directions in the first Brillouin zone (BZ). While along theгMdirection, the measured transmission spectra are not in accordance with the band structure calculated with the PWE at high-frequency region. The related bands are proved to be anti-symmetry mode, i.e. deaf band through calculating pressure field. Results show that there exists deaf band in square lattice steel/water PCs, and explain the mentioned disagreement.
(2) As well known, bands degeneracy at high symmetry points in the first BZ plays an important role in limiting band gap open. By reducing the symmetry of lattice through introducing an additional“atom(s)”in a unit cell, bands degeneracy is lifted, and an absolute BG is obtained at low-frequency region.
①BG characteristic for hybrid triangular-graphite lattice PCs. By reducing symmetry of structure, hybrid triangular-graphite lattice is considered by placing two additional“atom(s)”averagely at the diagonal of a unit cell of triangular lattice. Band structures for steel/air PCs are computed with the PWE method. Numerical results show that the location and width of stop band can be tuned by varying radii of rods at different positions. Complete BG between high-frequency bands opens easily for large radius of triangular lattice positions. In contrast, there appears wide BG between low-frequency bands easily for large radius of graphite lattice positions.
②BG characteristic for hybrid triangular lattice PCs. For hybrid triangular lattice PCs, due to symmetry reduction breaking bands degeneracy at high symmetry point in the first BZ, a novel low-frequency complete BG occurs between the first and the second bands. The location and width of the BG can be tuned by altering the position of additional“atom”. Measured results show that reducing symmetry is a valid method to lift bands degeneracy and open low-frequency band gap. Experimental results are basically in agreement with numerical results.
(3) BG characteristic for PCs with elliptical scatters
Triangular lattice steel/air PCs are investigated by replacing steel cylinders for elliptical rods. At the same lattice constant and filling fraction, a novel low-frequency stop band occurs between the first and the second bands for elliptical scatters. And the width of the second lowest BG for elliptical scatters is as the same as the lowest BG for steel cylinders at appropriate filling fraction. According to these results, we design a tandem structure to open wide low-frequency band gap under decreasing space size.
(4) Localization characteristic for steel/water PCs defect states
Localization characteristic of several point and line defects are considered by the FEM and experiment. In the stop band frequency, acoustic wave is localized at these defects, and hence forms a resonantor or waveguide. Acoustic wave propagate along the line defect almost losslessly, but for a single point defect, the calculated transmission amplitude is very low at localized frequency. Eexperimental results of line defect are in accordance with numerical results, but for point defect, there exists a little shift for localized frequency between experimental and numerical results. Combining the localization characteristics of single point and line defects, a waveguide–cavity-waveguide structure is designed. Computed transmission spectra show that the transmission ratio of this structure is nearly to 100%, which improves the localization effect of single point defect mostly.
(5) Discusses on intelligent PCs
Magnetorheological elastomer (MRE) is introduced in PCs field. For solid/MRE PCs, tunable BG dependence on external magnet field is considered with the PWE method.
针对声子晶体低频带隙打开的难题,本文基于降低对称性的思想,设计了两种新型晶格结构,利用平面波展开法和有限元法计算二维固/流(气)声子晶体的能带结构、传输谱和压力场分布,打开了低频带隙;基于超声浸水透射技术实验测试了声子晶体的带隙和缺陷态特征。主要工作如下:
(1)正方晶格钢/水声子晶体的聋带研究
对正方晶格钢/水声子晶体,沿着第一Brillouin区的两个高对称方向,实验测量的第一带隙与两种方法的数值结果基本吻合;沿着??方向,在高频段,实验测试的声波衰减频段与平面波法计算的能带结构不符,通过计算压力场证明对应的能带为反对称模态,形成聋带(deaf band),实验证实了在正方晶格钢/水声子晶体中存在聋带,揭示了高频段测试结果与计算结果矛盾的原因。
(2)众所周知,第一Brillouin区高对称点处的能带简并是制约带隙打开的一个重要因素。本文基于降低对称性的思想,在单胞内添加“原子”降低结构的对称性,消除了能带简并,打开了低频带隙。
①混合三角-六角晶格声子晶体的带隙特性研究。在三角晶格的单胞内添加两个“原子”形成混合三角-六角晶格结构,降低了结构对称性。用平面波法计算了混合三角-六角晶格钢/空气声子晶体的能带结构,结果表明,改变不同格点的散射体半径能调控带隙的数目、位置及宽度,且不同格点处的散射体半径变化对带隙的影响不同:三角格点处的散射体半径越大越容易打开高频带隙;六角格点处的散射体半径越大越容易打开低频带隙。
②混合三角晶格声子晶体的带隙特性研究。对混合三角晶格结构,由于在单胞内添加一个“原子”降低了结构的对称性,因此消除了能带在第一Brillouin区高对称点X处的简并,打开了一条新的位于第一、二能带间的低频带隙;改变添加原子的位置可以调控带隙的位置及宽度。实验测试了混合三角晶格钢/水声子晶体的带隙特性,验证了降低对称性是消除能带简并,打开低频带隙的一种有效方法。测试结果与两种方法计算的结果吻合较好。
(3)基于椭圆散射体的声子晶体带隙特性研究
以椭圆散射体取代圆形散射体,对三角晶格钢/空气声子晶体,在相同的晶格常数及填充率下,当散射体由圆柱体改为椭圆柱体时,消除了能带简并,打开了一条新的位于第一、二能带间的低频带隙;并且在一定的填充率范围内打开的第二带隙宽度及位置与圆柱体对应的声子晶体打开的第一带隙的宽度及位置基本一致;散射体截面越接近于圆,越不易打开第一、二能带间的带隙。椭圆截面的旋转角度对第一带隙影响较大。
根据这种带隙特性,把具有不同晶格常数椭圆截面散射体构成的声子晶体并列,在降低结构尺寸的条件下打开了宽的低频带隙。
(4)钢/水声子晶体缺陷态的局域特性研究
基于实验测试和有限元计算研究了多种点、线缺陷的局域特征。在带隙频率范围内,声波在声子晶体中传播时会局域在缺陷处,形成共振腔或波导结构;对直线型线缺陷,声波沿着线缺陷几乎是无损耗地传播,但对点缺陷,局域频率点的透射幅值很小。实验测试的线缺陷的局域频段与计算结果基本吻合;但是对点缺陷,实验测试的局域频率点与计算结果相比有一定的漂移。
基于单点缺陷和直线型线缺陷的局域特征构造了波导-空腔-波导结构,计算表明,该结构极大地提高了点缺陷的局域效果,透射率接近100%。
(5)智能声子晶体的带隙特性探讨
把磁流变弹性体引入声子晶体复合材料,以磁流变弹性体为基体,利用平面波法研究了外加磁场对声子晶体带隙的影响。
Phononic crystal (PC) is a kind of artificial acoustic functional materials with periodic structures which exhibit acoustic or elastic wave band gaps (BGs). For near two decades, the study on PCs has proved to be of great interest due to their novel and unique BGs, localization and negative refraction characteristics, and their potential applications in the vibration isolation, noise control, acoustic filter, waveguide devices acoustic lens, and so on.
In order to obtain low-frequency BG, two kinds of new lattices are designed based on symmetry reduction. Band structure, transmission spectra and pressure field of two-dimensional solid/fluid (air) PCs are investigated with the plane wave expansion (PWE) method and finite element method (FEM), respectively. BG and localization characteristics are measured based on ultrasonic immersion transmission technology. Main results are as follows:
(1) Deaf band for square lattice steel/water PCs
For two-dimensional square lattice steel/water PCs, the measured transmission spectra are in agreement with the numerical results at low-frequency region along the two high symmetry directions in the first Brillouin zone (BZ). While along theгMdirection, the measured transmission spectra are not in accordance with the band structure calculated with the PWE at high-frequency region. The related bands are proved to be anti-symmetry mode, i.e. deaf band through calculating pressure field. Results show that there exists deaf band in square lattice steel/water PCs, and explain the mentioned disagreement.
(2) As well known, bands degeneracy at high symmetry points in the first BZ plays an important role in limiting band gap open. By reducing the symmetry of lattice through introducing an additional“atom(s)”in a unit cell, bands degeneracy is lifted, and an absolute BG is obtained at low-frequency region.
①BG characteristic for hybrid triangular-graphite lattice PCs. By reducing symmetry of structure, hybrid triangular-graphite lattice is considered by placing two additional“atom(s)”averagely at the diagonal of a unit cell of triangular lattice. Band structures for steel/air PCs are computed with the PWE method. Numerical results show that the location and width of stop band can be tuned by varying radii of rods at different positions. Complete BG between high-frequency bands opens easily for large radius of triangular lattice positions. In contrast, there appears wide BG between low-frequency bands easily for large radius of graphite lattice positions.
②BG characteristic for hybrid triangular lattice PCs. For hybrid triangular lattice PCs, due to symmetry reduction breaking bands degeneracy at high symmetry point in the first BZ, a novel low-frequency complete BG occurs between the first and the second bands. The location and width of the BG can be tuned by altering the position of additional“atom”. Measured results show that reducing symmetry is a valid method to lift bands degeneracy and open low-frequency band gap. Experimental results are basically in agreement with numerical results.
(3) BG characteristic for PCs with elliptical scatters
Triangular lattice steel/air PCs are investigated by replacing steel cylinders for elliptical rods. At the same lattice constant and filling fraction, a novel low-frequency stop band occurs between the first and the second bands for elliptical scatters. And the width of the second lowest BG for elliptical scatters is as the same as the lowest BG for steel cylinders at appropriate filling fraction. According to these results, we design a tandem structure to open wide low-frequency band gap under decreasing space size.
(4) Localization characteristic for steel/water PCs defect states
Localization characteristic of several point and line defects are considered by the FEM and experiment. In the stop band frequency, acoustic wave is localized at these defects, and hence forms a resonantor or waveguide. Acoustic wave propagate along the line defect almost losslessly, but for a single point defect, the calculated transmission amplitude is very low at localized frequency. Eexperimental results of line defect are in accordance with numerical results, but for point defect, there exists a little shift for localized frequency between experimental and numerical results. Combining the localization characteristics of single point and line defects, a waveguide–cavity-waveguide structure is designed. Computed transmission spectra show that the transmission ratio of this structure is nearly to 100%, which improves the localization effect of single point defect mostly.
(5) Discusses on intelligent PCs
Magnetorheological elastomer (MRE) is introduced in PCs field. For solid/MRE PCs, tunable BG dependence on external magnet field is considered with the PWE method.
引文
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