材料参数对三维固体声子晶体带隙的影响
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摘要
声子晶体(Phononic Crystal)是一种具有空间周期性结构并呈现弹性波带隙的复合材料。由于其丰富的物理特性,以及在隔音、降噪、减震等领域的应用前景,有关声子晶体的研究目前已成为力学、凝聚态物理、机械工程等众多领域的重要方向之一。本文发展了计算三维声子晶体能带结构的时域有限差分(Finite difference time domain. FDTD)方法,并验证了该方法的精确性与效率。基于所发展的FDTD方法,进行了如下研究：
1.通过弹性波波动方程的FDTD差分格式得到了直接影响三维固/固声子晶体带隙的材料参数,包括散射体和基体的横波波速比ct1/ct2、波阻抗比zt1/zt2和泊松比v1、v2。
2.研究了材料参数对三维固/固体系带隙的影响,考虑了简立方、体心立方和面心立方三种晶格的体系。结果表明：散射体和基体的波速比和波阻抗比是决定声子晶体带隙的主要材料参数,泊松比相对于这两个参数对带隙的影响较小；带隙仅存在于波阻抗比较大且波速差异相对较小,或波速比较小且波阻抗差异相对较小的两类材料区域内；声子晶体的带隙中心频率随波速比的减小而降低,波速比是调控声子晶体类型(布拉格散射或局域共振)和带隙中心频率的重要材料参数；相对于简立方,体心立方和面心立方晶格的声子晶体更容易产生带隙,且在填充率和材料相同的情况下得到的带隙更宽。
3.研究了三维孔隙固体声子晶体的能带特性。由理论分析得到,泊松比是直接决定体系能带结构的材料参数。计算表明,在简立方、体心立方、面心立方三种点阵的体系中,能带结构随泊松比的变化不明显,且不存在完全带隙。
Phononic crystals (PNCs) are a kind of composite materials which have periodic structures and exhibit elastic wave band gaps where the propagation of acoustic/elastic waves is fully forbidden. Due to their unique properties and potential applications in sound insulation, noise control, shock attenuation, etc. they have become one of the most important areas in mechanics. condensed matter physics and mechanical engineering, etc. In this thesis, the finite difference time domain (FDTD) method for calculating the band structures of three-dimensional (3D) PNCs is developed, and the accuracy and efficiency of the method are verified. Based on the self-developed FDTD method, we carry out the studies in following aspects:
1. The material parameters which determine the band gaps in the 3D solid/solid PNCs are obtained directly from the finite difference form of the elastic wave equations. These material parameters include the transverse wave velocity ratio (ct1/ct2). the acoustic impedance ratio(zt1/zt2) and the Poisson's ratios v1 and v2.
2. The effects of the material parameters on the band gaps in 3D solid/solid PNCs with the simple-cubic (SC). body-centered (BCC) and face-centered (FCC) lattices are studied. The numerical results show that:the transverse wave velocity ratio and the acoustic impedance ratio play more important roles in determining the band gaps than the Poisson's ratios do:the band gaps appear in the system with large impedance ratio and small mismatch in the wave velocities, or with small wave velocity ratio and small mismatch in the acoustic impedances; the mid-gap frequency decreases as the wave velocity ratio decreases; the wave velocity ratio is the most important parameter which determines the mechanisms (Bragg scattering or local resonance) of the band gaps and the mid-gap frequencies; band gaps are easily opened in the BCC and FCC systems than in the SC system:with the same filling fraction and the material parameters, the band gaps in the BCC-and FCC-latticed systems are generally wider than that corresponding to the SC-latticed system.
3. The band structures of the 3D porous PNCs are studied. It is shown that the Poisson's ratio of the solid host is the unique material parameter which has direct effect on the band structures of a 3D porous PNC. The numerical results show that. in the svstems with the SC, BCC or FCC lattices. the band structure is not sensitive to the Poisson's ratio. and no full band gap can be generated.
1.通过弹性波波动方程的FDTD差分格式得到了直接影响三维固/固声子晶体带隙的材料参数,包括散射体和基体的横波波速比ct1/ct2、波阻抗比zt1/zt2和泊松比v1、v2。
2.研究了材料参数对三维固/固体系带隙的影响,考虑了简立方、体心立方和面心立方三种晶格的体系。结果表明：散射体和基体的波速比和波阻抗比是决定声子晶体带隙的主要材料参数,泊松比相对于这两个参数对带隙的影响较小；带隙仅存在于波阻抗比较大且波速差异相对较小,或波速比较小且波阻抗差异相对较小的两类材料区域内；声子晶体的带隙中心频率随波速比的减小而降低,波速比是调控声子晶体类型(布拉格散射或局域共振)和带隙中心频率的重要材料参数；相对于简立方,体心立方和面心立方晶格的声子晶体更容易产生带隙,且在填充率和材料相同的情况下得到的带隙更宽。
3.研究了三维孔隙固体声子晶体的能带特性。由理论分析得到,泊松比是直接决定体系能带结构的材料参数。计算表明,在简立方、体心立方、面心立方三种点阵的体系中,能带结构随泊松比的变化不明显,且不存在完全带隙。
Phononic crystals (PNCs) are a kind of composite materials which have periodic structures and exhibit elastic wave band gaps where the propagation of acoustic/elastic waves is fully forbidden. Due to their unique properties and potential applications in sound insulation, noise control, shock attenuation, etc. they have become one of the most important areas in mechanics. condensed matter physics and mechanical engineering, etc. In this thesis, the finite difference time domain (FDTD) method for calculating the band structures of three-dimensional (3D) PNCs is developed, and the accuracy and efficiency of the method are verified. Based on the self-developed FDTD method, we carry out the studies in following aspects:
1. The material parameters which determine the band gaps in the 3D solid/solid PNCs are obtained directly from the finite difference form of the elastic wave equations. These material parameters include the transverse wave velocity ratio (ct1/ct2). the acoustic impedance ratio(zt1/zt2) and the Poisson's ratios v1 and v2.
2. The effects of the material parameters on the band gaps in 3D solid/solid PNCs with the simple-cubic (SC). body-centered (BCC) and face-centered (FCC) lattices are studied. The numerical results show that:the transverse wave velocity ratio and the acoustic impedance ratio play more important roles in determining the band gaps than the Poisson's ratios do:the band gaps appear in the system with large impedance ratio and small mismatch in the wave velocities, or with small wave velocity ratio and small mismatch in the acoustic impedances; the mid-gap frequency decreases as the wave velocity ratio decreases; the wave velocity ratio is the most important parameter which determines the mechanisms (Bragg scattering or local resonance) of the band gaps and the mid-gap frequencies; band gaps are easily opened in the BCC and FCC systems than in the SC system:with the same filling fraction and the material parameters, the band gaps in the BCC-and FCC-latticed systems are generally wider than that corresponding to the SC-latticed system.
3. The band structures of the 3D porous PNCs are studied. It is shown that the Poisson's ratio of the solid host is the unique material parameter which has direct effect on the band structures of a 3D porous PNC. The numerical results show that. in the svstems with the SC, BCC or FCC lattices. the band structure is not sensitive to the Poisson's ratio. and no full band gap can be generated.
引文
[1]Kushwaha M S. Halevi P. Martinez G. Dobrzynski L. Djafarirouhani B. Acoustic band structure of periodic elastic composites [J]. Physical Review Letters.1993.71:2022-2025.
[2]Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics [J]. Physical Review Letters.1987,58:2059-2062.
[3]John S. Strong localization of photons in certain disordered dielectric superlattices [J]. Physical Review Letters,1987,58:2486-2489.
[4]温熙森.光子/声子晶体理论与技术[M].北京：科学出版社.2006.
[5]温熙森,温激鸿,郁殿龙.声子晶体[M].北京：国防工业出版社，2009.
[6]Sigalas M M. Economou E N. Elastic and acoustic wave band structure [J]. Journal of Sound and Vibration.1992,158:377-382.
[7]Martines-Sala R. Sancho J, Sanchez J V. Gomez V. Llinares J. Meseguer F. Sound-attenuation by sculpture [J]. Nature.1995,378:241-241.
[8]温激鸿,韩小云,王刚,赵宏刚,刘耀宗.声子晶体研究概述[J].功能材料.2003，34：364-367.
[9]Grosso G, Parravicini G P. Solid State Physics[M]. Academic Press,2000.
[10]Liu Z Y. Zhang X X. Mao Y W. Zhu Y Y, Yang Z Y. Chan C T. Sheng P. Locally resonant sonic materials [J]. Science.2000,289:1734-1736.
[11]Miklowitz J. The Theory of Elastic Waves and Waveguides [M]. New York: North-Holland Publishing Company, 1978.
[12]冯端,金国钧.凝聚态物理学(上卷)[M].北京：高等教育出版社,2003.
[13]Farhat M. Guenneau S, Enoch S. Movchan A. All-angle-negative-refraction and ultra-refraction for liquid surface waves in 2D phononic crystals [J]. Journal of Computational and Applied Mathematics.2010.234:2011-2019.
[14]Vasseur J O. Deymier P A. Beaugeois M. Pennec Y. Djafari-Rouhani B, Prevost D. Experimental observation of resonant filtering in a two-dimensional phononic crystal waveguide [J]. Zeitschrift Fur Kristallographie.2005.220:829-835.
[15]Castanier M P. Pierre C. Predicting localization via Lyapunov exponent statistics [J]. Journal of Sound and Vibration.1997.203:151-157.
[16]Li F M. Xu M Q. Wang Y S. Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals [J]. Solid State Communications.2007.141: 296-301.
[17]Chen J J. Bonello B. Hou Z L. Plate-mode waves in phononic crystal thin slabs: mode conversion [J]. Physical Review E.2008,78:036609.
[18]Brunet T. Vasseur J. Bonello B. Djafari-Rouhani B. Hladky-Hennion A C. Lamb waves in phononic crystal slabs with square or rectangular symmetries [J]. Journal of Applied Physics, 2008.104:043506.
[19]Hao G J. Fu X J. Hou Z L. Band structure of phononic crystal constructed by fibonaccisuper-cell on square lattice [J]. Acta Physica Sinica.2009.58:8484-8488.
[20]Robillard J F. Matar O B. Vasseur J O. Devmier P A, Stippinger M. Hladky-Hennion A C. Pennec Y, Djafari-Rouhani B. Tunable magnetoelastic phononic crystals [J]. Applied Physics Letters,2009,95:124104.
[21]Zhong L H. Wu F G. Propagation of water wave over a periodically perforated bottom and the band structure [J]. Acta Physica Sinica,2009.58:6363-6368.
[22]Lin S. Huang T J. Acoustic mirage in two-dimensional gradient-index phononic crystals [J]. Journal of Applied Physics, 2009, 106:053529.
[23]Zhou X Z, Wang Y S. Zhang C Z. Effects of material parameters on elastic band gaps of two-dimensional solid phononic crystals [J]. Journal of Applied Physics.2009, 106:014903.
[24]Mu Z F. Wu F G. Acoustic band gaps tuned by translation group symmetry in two-dimensional periodic composites [J]. Modern Physics Letters B.2009.23:1687-1694.
[25]Romero-Garcia V. Sanchea-Perez J V, Garcia-Raffi L M. Propagating and evanescent properties of double-point defects in sonic crystals [J]. New Journal of Physics.2010,12:083024.
[26]Manzanares-Martinez B. Ramos-Mendieta F. Baltazar A. Ultrasonic elastic modes in solid bars: an application of the plane wave expansion method [J]. Journal of the Acoustical Society of America, 2010.127:3503-3510.
[27]Zhan Z Q. Wei P J. Influences of anisotropy on band gaps of 2d phononic crystal [J]. Acta Mechanica Solida Sinica,2010, 23:181-188.
[28]Mohammadi S. Eftekhar A A. Khelif A. Adibi A. Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs [J]. Optics Express.2010,18:9164-9172.
[29]Zhu X F, Liu S C. Xu T. Wang T H. Cheng J C. Investigation of a silicon-based one-dimensional phononic crystal plate via the super-cell plane wave expansion method [J]. Chinese Physics B.2010,19:044301.
[30]Romero-Garcia V, Sanchez-Perez J V, Castineira-Ibanez S. Garcia-Raffi L M. Evidences of evanescent bloch waves in phononic crystals [J]. Applied Physics Letters,2010,96:124102.
[31]Liu Z Y, Chan C T, Sheng P, Goertzen A L, Page J H. Elastic wave scattering by periodic structures of spherical objects:theory and experiment [J]. Physical Review B,2000,62: 2446-2457.
[32]Kafesaki M. Economou E N. Multiple-scattering theory for three-dimensional periodic acoustic composites [J]. Physical Review B.1999.60:11993-12001.
[33]Zhao H G. Wen J H. Yu D L. Wen X S. Low-frequency acoustic absorption of localized resonances:experiment and theory [J]. Journal of Applied Physics.2010.107:023519.
[34]Tanaka Y. Tomoyasu Y. Tamura S. Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch [J]. Physical Review B,2000.62: 7387-7392.
[35]Sigalas M M. Soukoulis C M. Elastic-wave propagation through disordered and/or absorptive layered systems [J]. Physical Review B.1995,51:2780-2789.
[36]Hou Z L. Fu X J, Liu Y Y. Calculational method to study the transmission properties of phononic crystals [J]. Physical Review B.2004,70:14304.
[37]Chen A L. Wang Y S. Yu G L. Guo Y F. Wang Z D. Elastic wave propagation in two-dimensional phononic crystals with one-dimensional quasi-periodicity and random disorder [J]. Acta Mechanica Solida Sinica.2008.21:517-528.
[38]Sigalas M M. Garcia N. Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method [J]. Journal of Applied Physics, 2000,87:3122-3125.
[39]王刚,温激鸿,韩小云,赵宏刚.二维声子晶体带隙计算的时域有限差分法[J].物理学报，2003.52：1943-1947.
[40]Cao Y. Hou Z Liu Y. Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals [J]. Solid State Communications.2004,132:539-543.
[41]刘丰铭,二维声子晶体中的FDTD理论计算方法及其应用实例[D].武汉：武汉大学.2005.
[42]Hsieh P F. Wu T T, Sun J H. Three-dimensional phononic band gap calculations using the FDTD method and a PC cluster system [J]. IEEE Transactions on Ultrasonics. Ferroelectrics, and Frequency Control.2006.53:148-158.
[43]熊伟.考虑材料阻尼的声子晶体仿真算法研究[D].长沙：国防科技大学，2006.
[44]Tanaka Y. Yano T. Tamura S. Surface guided waves in two-dimensional phononic crystals [J]. Wave Motion.2007.44:501-512.
[45]Sun J H. Wu T T. Propagation of acoustic waves in phononic-crystal plates and waveguides using a finite-difference time-domain method [J]. Physical Review B.2007.76:104304.
[46]苏晓星,汪越胜.用FDTD与高分辨率谱估计相结合计算二维声子晶体的能带[J].人工晶体学报.2009，38：1375-1379.
[47]Su X X. Li J B. Wang Y S. A postprocessing method based on high-resolution spectral estimation for FDTD calculation of phononic band structures [J]. Physica B,2010,405: 2444-2449.
[48]Langlet P. Hladkyhennion A C. Decarpigny J N. Analysis of the propagation of plane acoustic-waves in passive periodic materials using the finite-element method [J]. Journal of the Acoustical Society of America.1995.98:2792-2800.
[49]Khelif A. Aoubiza B, Mohammadi S. Adibi A. Laude V. Complete band gaps in two-dimensional phononic crystal slabs [J]. Physical Review E.2006,74:0466104.
[50]Li J B. Wang Y S, Zhang C Z. Finite element method for analysis of band structures of three dimensional phononic crystals [C].2008 IEEE International Ultrasonics Symposium Proceedings.2008.1468-1471.
[51]Eichenfield M, Chan J, Safavi-Naeini A H. Vahala K J. Painter O. Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals [J]. Optics Express.2009.17:20078.
[52]Safavi-Naeini A H. Painter O. Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab [J]. Optics Express.2010,18:14926.
[53]李建宝,汪越胜,张传增.二维声子晶体微腔能带结构的有限元分析与设计[J].人工晶体学报.2010，39：649-655.
[54]Checoury X, Lourtioz J M. Wavelet method for computing band diagrams of 2d photonic crystals [J]. Optics Communications.2006,259:360-365.
[55]Yan Z Z. Wang Y S. Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals [J]. Physical Review B.2006,74:224303.
[56]闫志忠.基于小波理论的二维声子晶体带隙结构分析[D].北京：北京交通大学.2007.
[57]Wang G. Wen J H. Liu Y Z. Wen X S. Lumped-mass method for the study of band structure in two-dimensional phononic crystals [J]. Physical Review B.2004.69:184302.
[58]Li F L. Wang Y S. Band gap analysis of two-dimensional phononic crystals based on boundary element method [C].2008 IEEE International Ultrasonics Symposium Proceedings.2008, 245-248.
[59]Kushwaha M S. Halevi P. Band-gap engineering in periodic elastic composites [J]. Applied Physics Letters.1994.64:1085-1087.
[60]Kushwaha M S. Halevi P. Martine G. Theory of acoustic band structure of periodic elastic composites [J]. Physical Review B.1994.49:2313-2322.
[61]Kafesaki M. Sigalas M M. Economou E N. Elastic wave band gaps in 3-d periodic polymer matrix composites [J]. Solid State Communications,1995.96:285-289.
[62]Kushwaha M S, DjafariRouhani B. Complete acoustic stop bands for cubic arrays of spherical liquid balloons [J]. Journal of Applied Physics,1996.80:3191-3195.
[63]Kushwaha M S. Halevi P. Stop bands for cubic arrays of spherical balloons [J]. Journal of The Acoustical Society of America.1997,101:619-622.
[64]Vasseur J O. Djafari-Rouhani B. Dobrzynski L. et al. Complete acoustic band gaps in periodic fibre reinforced composite materials:the carbon/epoxy composite and some metallic systems [J]. Journal of Physics:Condensed Matter.1994:8759-8770.
[65]Goffaux C. Vigneron J P. Theoretical study of a tunable phononic band gap system [J]. Physical Review B,2001.64:075118.
[66]Wu F G. Liu Z Y. Liu Y Y. Acoustic band gaps created by rotating square rods in a two-dimensional lattice [J]. Physical Review E,2002.66:046628.
[67]Kuang W M, Hou Z L, Liu Y Y. The effect of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals [J]. Physics Letters A,2004,332:481-490.
[68]Kuang W M, Hou Z L. Liu Y Y. et al. The band gaps of cubic phononic crystals with different shapes of scatterers [J]. Journal of Physics D.2006.39:2067-2071.
[69]Sigalas M, Economou E N. Band structure of elastic waves in two dimensional systems [J]. Solid State Communications,1993,86:141-143.
[70]Kee C S, Kim J E. Park H Y, Chang K J, Lim H. Essential role of impedance in the formation of acoustic band gaps [J]. Journal of Applied Physics,2000,87:1593-1596.
[71]Zhou X Z. Wang Y S, Zhang C Z. Three-dimensional sonic band gaps tunned by material parameters [J]. Applied Mechanics and Materials.2010,29-32:1797-1802.
[72]Yee K. Numerical solution initial boundary value problems involving Maxwell's Equations in isotropic media [J]. IEEE Transactions on Antennas and Propagation.1966,14:302-307.
[73]葛德彪,闫玉波.电磁波时域有限差分方法(第二版)[M].西安：西安电子科技大学出版社，2005.
[74]杨桂通,张善元.弹性动力学[M].北京：中国铁道出版社，1988.
[75]李建宝.声子晶体带隙调控的数值与实验研究[D].北京：北京交通大学,2011.
[76]余文华,苏涛,Mittra R,刘永峻.并行时域有限差分[M].北京：中国传媒大学出版社.2005.
[77]Gerber R. Smith K B. Bik A. Tian X著.王涛,单久龙,孙广中译.软件优化技术[M].北京：电子工业出版社.2007.
[78]Akhter S. Roberts J著.李宝峰,谢弘毅,李韬译.多核程序设计技术[M].北京：电子工业出版社.2007.
[79]都志辉.高性能计算并行编程技术[M].北京：清华大学出版社.2001.
[80]王鹏,吕爽,聂治,谢千河.并行计算应用及实战[M].北京：机械工业出版社，2008.
[81]卢天健,何德平,陈常青,赵长颖,方岱宁,王晓林.超轻多孔金属材料的多功能特性及应用[J].力学进展.2006.36：517-535.
[82]卢天健,刘涛,邓子辰.多孔金属材料多功能化设计的若干进展[J].力学与实践,2008，30：1-9.
[83]杨亚政,杨嘉陵,曾涛,方岱宁.轻质多孔材料研究进展[J].力学季刊.2007，28：503-516.
[84]Ruzzene M. Scarpa F. Soranna F. Wave beaming effects in two-dimensional cellular structures [J]. Smart Material and Structure.2003.12:363-372.
[85]甄妮,闫志忠,汪越胜.蜂窝材料的弹性波传播特性[J].力学学报.2008.40：769-775.
[86]Liu Y. Su J Y. Gao L T. The influence of the micro-topology on the phononic band gaps in 2D porous phononic crystals [J]. Physics Letters A,2008,372:6784-6789.
[87]Liu Y, Su J Y, Xu Y L. Zhang X C. The influence of pore shapes on the band structures in phononic crystals with periodic distributed void pores [J]. Ultrasonics.2009,49:276-280.
[2]Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics [J]. Physical Review Letters.1987,58:2059-2062.
[3]John S. Strong localization of photons in certain disordered dielectric superlattices [J]. Physical Review Letters,1987,58:2486-2489.
[4]温熙森.光子/声子晶体理论与技术[M].北京：科学出版社.2006.
[5]温熙森,温激鸿,郁殿龙.声子晶体[M].北京：国防工业出版社，2009.
[6]Sigalas M M. Economou E N. Elastic and acoustic wave band structure [J]. Journal of Sound and Vibration.1992,158:377-382.
[7]Martines-Sala R. Sancho J, Sanchez J V. Gomez V. Llinares J. Meseguer F. Sound-attenuation by sculpture [J]. Nature.1995,378:241-241.
[8]温激鸿,韩小云,王刚,赵宏刚,刘耀宗.声子晶体研究概述[J].功能材料.2003，34：364-367.
[9]Grosso G, Parravicini G P. Solid State Physics[M]. Academic Press,2000.
[10]Liu Z Y. Zhang X X. Mao Y W. Zhu Y Y, Yang Z Y. Chan C T. Sheng P. Locally resonant sonic materials [J]. Science.2000,289:1734-1736.
[11]Miklowitz J. The Theory of Elastic Waves and Waveguides [M]. New York: North-Holland Publishing Company, 1978.
[12]冯端,金国钧.凝聚态物理学(上卷)[M].北京：高等教育出版社,2003.
[13]Farhat M. Guenneau S, Enoch S. Movchan A. All-angle-negative-refraction and ultra-refraction for liquid surface waves in 2D phononic crystals [J]. Journal of Computational and Applied Mathematics.2010.234:2011-2019.
[14]Vasseur J O. Deymier P A. Beaugeois M. Pennec Y. Djafari-Rouhani B, Prevost D. Experimental observation of resonant filtering in a two-dimensional phononic crystal waveguide [J]. Zeitschrift Fur Kristallographie.2005.220:829-835.
[15]Castanier M P. Pierre C. Predicting localization via Lyapunov exponent statistics [J]. Journal of Sound and Vibration.1997.203:151-157.
[16]Li F M. Xu M Q. Wang Y S. Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals [J]. Solid State Communications.2007.141: 296-301.
[17]Chen J J. Bonello B. Hou Z L. Plate-mode waves in phononic crystal thin slabs: mode conversion [J]. Physical Review E.2008,78:036609.
[18]Brunet T. Vasseur J. Bonello B. Djafari-Rouhani B. Hladky-Hennion A C. Lamb waves in phononic crystal slabs with square or rectangular symmetries [J]. Journal of Applied Physics, 2008.104:043506.
[19]Hao G J. Fu X J. Hou Z L. Band structure of phononic crystal constructed by fibonaccisuper-cell on square lattice [J]. Acta Physica Sinica.2009.58:8484-8488.
[20]Robillard J F. Matar O B. Vasseur J O. Devmier P A, Stippinger M. Hladky-Hennion A C. Pennec Y, Djafari-Rouhani B. Tunable magnetoelastic phononic crystals [J]. Applied Physics Letters,2009,95:124104.
[21]Zhong L H. Wu F G. Propagation of water wave over a periodically perforated bottom and the band structure [J]. Acta Physica Sinica,2009.58:6363-6368.
[22]Lin S. Huang T J. Acoustic mirage in two-dimensional gradient-index phononic crystals [J]. Journal of Applied Physics, 2009, 106:053529.
[23]Zhou X Z, Wang Y S. Zhang C Z. Effects of material parameters on elastic band gaps of two-dimensional solid phononic crystals [J]. Journal of Applied Physics.2009, 106:014903.
[24]Mu Z F. Wu F G. Acoustic band gaps tuned by translation group symmetry in two-dimensional periodic composites [J]. Modern Physics Letters B.2009.23:1687-1694.
[25]Romero-Garcia V. Sanchea-Perez J V, Garcia-Raffi L M. Propagating and evanescent properties of double-point defects in sonic crystals [J]. New Journal of Physics.2010,12:083024.
[26]Manzanares-Martinez B. Ramos-Mendieta F. Baltazar A. Ultrasonic elastic modes in solid bars: an application of the plane wave expansion method [J]. Journal of the Acoustical Society of America, 2010.127:3503-3510.
[27]Zhan Z Q. Wei P J. Influences of anisotropy on band gaps of 2d phononic crystal [J]. Acta Mechanica Solida Sinica,2010, 23:181-188.
[28]Mohammadi S. Eftekhar A A. Khelif A. Adibi A. Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs [J]. Optics Express.2010,18:9164-9172.
[29]Zhu X F, Liu S C. Xu T. Wang T H. Cheng J C. Investigation of a silicon-based one-dimensional phononic crystal plate via the super-cell plane wave expansion method [J]. Chinese Physics B.2010,19:044301.
[30]Romero-Garcia V, Sanchez-Perez J V, Castineira-Ibanez S. Garcia-Raffi L M. Evidences of evanescent bloch waves in phononic crystals [J]. Applied Physics Letters,2010,96:124102.
[31]Liu Z Y, Chan C T, Sheng P, Goertzen A L, Page J H. Elastic wave scattering by periodic structures of spherical objects:theory and experiment [J]. Physical Review B,2000,62: 2446-2457.
[32]Kafesaki M. Economou E N. Multiple-scattering theory for three-dimensional periodic acoustic composites [J]. Physical Review B.1999.60:11993-12001.
[33]Zhao H G. Wen J H. Yu D L. Wen X S. Low-frequency acoustic absorption of localized resonances:experiment and theory [J]. Journal of Applied Physics.2010.107:023519.
[34]Tanaka Y. Tomoyasu Y. Tamura S. Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch [J]. Physical Review B,2000.62: 7387-7392.
[35]Sigalas M M. Soukoulis C M. Elastic-wave propagation through disordered and/or absorptive layered systems [J]. Physical Review B.1995,51:2780-2789.
[36]Hou Z L. Fu X J, Liu Y Y. Calculational method to study the transmission properties of phononic crystals [J]. Physical Review B.2004,70:14304.
[37]Chen A L. Wang Y S. Yu G L. Guo Y F. Wang Z D. Elastic wave propagation in two-dimensional phononic crystals with one-dimensional quasi-periodicity and random disorder [J]. Acta Mechanica Solida Sinica.2008.21:517-528.
[38]Sigalas M M. Garcia N. Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method [J]. Journal of Applied Physics, 2000,87:3122-3125.
[39]王刚,温激鸿,韩小云,赵宏刚.二维声子晶体带隙计算的时域有限差分法[J].物理学报，2003.52：1943-1947.
[40]Cao Y. Hou Z Liu Y. Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals [J]. Solid State Communications.2004,132:539-543.
[41]刘丰铭,二维声子晶体中的FDTD理论计算方法及其应用实例[D].武汉：武汉大学.2005.
[42]Hsieh P F. Wu T T, Sun J H. Three-dimensional phononic band gap calculations using the FDTD method and a PC cluster system [J]. IEEE Transactions on Ultrasonics. Ferroelectrics, and Frequency Control.2006.53:148-158.
[43]熊伟.考虑材料阻尼的声子晶体仿真算法研究[D].长沙：国防科技大学，2006.
[44]Tanaka Y. Yano T. Tamura S. Surface guided waves in two-dimensional phononic crystals [J]. Wave Motion.2007.44:501-512.
[45]Sun J H. Wu T T. Propagation of acoustic waves in phononic-crystal plates and waveguides using a finite-difference time-domain method [J]. Physical Review B.2007.76:104304.
[46]苏晓星,汪越胜.用FDTD与高分辨率谱估计相结合计算二维声子晶体的能带[J].人工晶体学报.2009，38：1375-1379.
[47]Su X X. Li J B. Wang Y S. A postprocessing method based on high-resolution spectral estimation for FDTD calculation of phononic band structures [J]. Physica B,2010,405: 2444-2449.
[48]Langlet P. Hladkyhennion A C. Decarpigny J N. Analysis of the propagation of plane acoustic-waves in passive periodic materials using the finite-element method [J]. Journal of the Acoustical Society of America.1995.98:2792-2800.
[49]Khelif A. Aoubiza B, Mohammadi S. Adibi A. Laude V. Complete band gaps in two-dimensional phononic crystal slabs [J]. Physical Review E.2006,74:0466104.
[50]Li J B. Wang Y S, Zhang C Z. Finite element method for analysis of band structures of three dimensional phononic crystals [C].2008 IEEE International Ultrasonics Symposium Proceedings.2008.1468-1471.
[51]Eichenfield M, Chan J, Safavi-Naeini A H. Vahala K J. Painter O. Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals [J]. Optics Express.2009.17:20078.
[52]Safavi-Naeini A H. Painter O. Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab [J]. Optics Express.2010,18:14926.
[53]李建宝,汪越胜,张传增.二维声子晶体微腔能带结构的有限元分析与设计[J].人工晶体学报.2010，39：649-655.
[54]Checoury X, Lourtioz J M. Wavelet method for computing band diagrams of 2d photonic crystals [J]. Optics Communications.2006,259:360-365.
[55]Yan Z Z. Wang Y S. Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals [J]. Physical Review B.2006,74:224303.
[56]闫志忠.基于小波理论的二维声子晶体带隙结构分析[D].北京：北京交通大学.2007.
[57]Wang G. Wen J H. Liu Y Z. Wen X S. Lumped-mass method for the study of band structure in two-dimensional phononic crystals [J]. Physical Review B.2004.69:184302.
[58]Li F L. Wang Y S. Band gap analysis of two-dimensional phononic crystals based on boundary element method [C].2008 IEEE International Ultrasonics Symposium Proceedings.2008, 245-248.
[59]Kushwaha M S. Halevi P. Band-gap engineering in periodic elastic composites [J]. Applied Physics Letters.1994.64:1085-1087.
[60]Kushwaha M S. Halevi P. Martine G. Theory of acoustic band structure of periodic elastic composites [J]. Physical Review B.1994.49:2313-2322.
[61]Kafesaki M. Sigalas M M. Economou E N. Elastic wave band gaps in 3-d periodic polymer matrix composites [J]. Solid State Communications,1995.96:285-289.
[62]Kushwaha M S, DjafariRouhani B. Complete acoustic stop bands for cubic arrays of spherical liquid balloons [J]. Journal of Applied Physics,1996.80:3191-3195.
[63]Kushwaha M S. Halevi P. Stop bands for cubic arrays of spherical balloons [J]. Journal of The Acoustical Society of America.1997,101:619-622.
[64]Vasseur J O. Djafari-Rouhani B. Dobrzynski L. et al. Complete acoustic band gaps in periodic fibre reinforced composite materials:the carbon/epoxy composite and some metallic systems [J]. Journal of Physics:Condensed Matter.1994:8759-8770.
[65]Goffaux C. Vigneron J P. Theoretical study of a tunable phononic band gap system [J]. Physical Review B,2001.64:075118.
[66]Wu F G. Liu Z Y. Liu Y Y. Acoustic band gaps created by rotating square rods in a two-dimensional lattice [J]. Physical Review E,2002.66:046628.
[67]Kuang W M, Hou Z L, Liu Y Y. The effect of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals [J]. Physics Letters A,2004,332:481-490.
[68]Kuang W M, Hou Z L. Liu Y Y. et al. The band gaps of cubic phononic crystals with different shapes of scatterers [J]. Journal of Physics D.2006.39:2067-2071.
[69]Sigalas M, Economou E N. Band structure of elastic waves in two dimensional systems [J]. Solid State Communications,1993,86:141-143.
[70]Kee C S, Kim J E. Park H Y, Chang K J, Lim H. Essential role of impedance in the formation of acoustic band gaps [J]. Journal of Applied Physics,2000,87:1593-1596.
[71]Zhou X Z. Wang Y S, Zhang C Z. Three-dimensional sonic band gaps tunned by material parameters [J]. Applied Mechanics and Materials.2010,29-32:1797-1802.
[72]Yee K. Numerical solution initial boundary value problems involving Maxwell's Equations in isotropic media [J]. IEEE Transactions on Antennas and Propagation.1966,14:302-307.
[73]葛德彪,闫玉波.电磁波时域有限差分方法(第二版)[M].西安：西安电子科技大学出版社，2005.
[74]杨桂通,张善元.弹性动力学[M].北京：中国铁道出版社，1988.
[75]李建宝.声子晶体带隙调控的数值与实验研究[D].北京：北京交通大学,2011.
[76]余文华,苏涛,Mittra R,刘永峻.并行时域有限差分[M].北京：中国传媒大学出版社.2005.
[77]Gerber R. Smith K B. Bik A. Tian X著.王涛,单久龙,孙广中译.软件优化技术[M].北京：电子工业出版社.2007.
[78]Akhter S. Roberts J著.李宝峰,谢弘毅,李韬译.多核程序设计技术[M].北京：电子工业出版社.2007.
[79]都志辉.高性能计算并行编程技术[M].北京：清华大学出版社.2001.
[80]王鹏,吕爽,聂治,谢千河.并行计算应用及实战[M].北京：机械工业出版社，2008.
[81]卢天健,何德平,陈常青,赵长颖,方岱宁,王晓林.超轻多孔金属材料的多功能特性及应用[J].力学进展.2006.36：517-535.
[82]卢天健,刘涛,邓子辰.多孔金属材料多功能化设计的若干进展[J].力学与实践,2008，30：1-9.
[83]杨亚政,杨嘉陵,曾涛,方岱宁.轻质多孔材料研究进展[J].力学季刊.2007，28：503-516.
[84]Ruzzene M. Scarpa F. Soranna F. Wave beaming effects in two-dimensional cellular structures [J]. Smart Material and Structure.2003.12:363-372.
[85]甄妮,闫志忠,汪越胜.蜂窝材料的弹性波传播特性[J].力学学报.2008.40：769-775.
[86]Liu Y. Su J Y. Gao L T. The influence of the micro-topology on the phononic band gaps in 2D porous phononic crystals [J]. Physics Letters A,2008,372:6784-6789.
[87]Liu Y, Su J Y, Xu Y L. Zhang X C. The influence of pore shapes on the band structures in phononic crystals with periodic distributed void pores [J]. Ultrasonics.2009,49:276-280.