周期性结构声学特性及其算法研究
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摘要
具有周期性排列结构的复合材料被称为声子晶体,当声波或者弹性波在该结构中传播时,会形成一定的频率禁带,使得某些频率处于禁带范围内的声波或者弹性波不能在该结构中传播。对于这种周期性结构的研究具有广泛的应用价值,例如可以设计全新的减振降噪结构等。
本文主要研究了以下三方面的内容:
(1)采用平面波展开法对下述多种周期结构的声波(弹性波)禁带进行研究,包括1)正方形截面散射体的二维液相和固相周期结构中,考虑截面旋转时,结构中压力波(声波)和横向剪切波的传播特性和禁带的产生规律。2)在二维液相周期结构中引入了椭圆截面散射体这种新的结构形式,研究了椭圆截面长短半径的变化以及绕中心轴线旋转时,声波禁带的生成规律。3)在三维液相周期结构中引入了椭球散射体这种新的结构形式,研究了椭球半径变化时,声波禁带的生成规律。
研究结果表明,组成周期性结构的各种材料的密度对声波频率禁带的影响最大,此外,在填充率保持一定的情况下,二维椭圆截面的半径和旋转角度、三维椭球散射体的半径都对声波禁带的产生有较大的影响。
(2)提出了基于间接Trefftz法的波数法进行声场强度的预测。波数法将声场的动力学响应近似分解为两部分,一部分为一组精确满足齐次Helmholtz方程的通解(波函数),另一部分为外部激励产生的满足自由空间非齐次Helmholtz方程的特解。各个通解的系数可以通过采用加权余量公式,强迫该近似解在平均意义上满足边界条件来得到。通过对二维和三维非耦合稳态声场,二维耦合稳态声场和多层介质稳态声场的声压进行预测,证明了该方法的有效性,结果表明,波数法在计算的精度和收敛性方面要优于有限元和边界元方法。
(3)应用波数法对有限周期性结构的声波禁带特性进行研究。通过对一维层状周期结构和二维正方形截面散射体周期结构的声压频率响应函数曲线来确定其不完全声波禁带。将波数法的结果与平面波展开法和有限元法的结果进行比较,验证该方法的有效性。
The composite material with periodic structure is called Phononic crystal. It can give rise to complete acoustic band gaps within which sound and vibration are forbidden. The motivation for these studies is their numerous engineering applications such as kinds of new equipments which can attenuate the noise and vibration.
The main content of this dissertation are listed as follow:
Firstly, the plane wave expansion method is used to calculate the acoustic(elastic) wave band gap of some specific periodic structures, for example (1)In the two dimensional liquid or solid periodic structure with square cross section, the characteristics of the band gap structures of the longitudinal wave or transverse wave are studied respectively as the cross section rotates. (2) The elliptic cross section is introduced in the two dimensional liquid periodic structure, and the influences of the elliptical radii and the rotation angel on acoustic wave band gaps are studied. (3)The ellipsoid cross section is introduced in the three dimensional liquid periodic structure, and the influences of the ellipsoid radii on acoustic wave band gap are studied.
The results show that, the material density has more influence on the acoustic wave band gap structures than the other parameters, and the radii and rotation angle of the ellipse or ellipsoid have influence also if the filling fraction is unchanged.
Secondly, a wave number method is proposed to deal with the acoustic problem. The method is based on an indirect Trefftz approach, in which the dynamic pressure response variable is approximated by a set of wave functions exactly satisfying the Helmholtz equation. The set of wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions, which arise from the external excitations. The weighting coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions and it is performed by applying the weighted residual formulation. The two and three dimensional uncoupled, the two dimensional coupled and the multilayer inhomogeneous acoustic problems are calculated. Comparing with FEM and BEM, the wave number method has better accuracy and convergence.
Thirdly, the wave number method is used to calculate the acoustic wave band gap of periodic structure with finite size. From the acoustic pressure distribution and the pressure frequency response function of the one dimensional finite multilayer structure and the two dimensional finite periodic structure with square scatterers, the directional acoustic wave band gap structures are obtained. Comparing the results of the wave number method with the FEM and the plane wave expansion method, the prior method shows more efficiency than the other two.
本文主要研究了以下三方面的内容:
(1)采用平面波展开法对下述多种周期结构的声波(弹性波)禁带进行研究,包括1)正方形截面散射体的二维液相和固相周期结构中,考虑截面旋转时,结构中压力波(声波)和横向剪切波的传播特性和禁带的产生规律。2)在二维液相周期结构中引入了椭圆截面散射体这种新的结构形式,研究了椭圆截面长短半径的变化以及绕中心轴线旋转时,声波禁带的生成规律。3)在三维液相周期结构中引入了椭球散射体这种新的结构形式,研究了椭球半径变化时,声波禁带的生成规律。
研究结果表明,组成周期性结构的各种材料的密度对声波频率禁带的影响最大,此外,在填充率保持一定的情况下,二维椭圆截面的半径和旋转角度、三维椭球散射体的半径都对声波禁带的产生有较大的影响。
(2)提出了基于间接Trefftz法的波数法进行声场强度的预测。波数法将声场的动力学响应近似分解为两部分,一部分为一组精确满足齐次Helmholtz方程的通解(波函数),另一部分为外部激励产生的满足自由空间非齐次Helmholtz方程的特解。各个通解的系数可以通过采用加权余量公式,强迫该近似解在平均意义上满足边界条件来得到。通过对二维和三维非耦合稳态声场,二维耦合稳态声场和多层介质稳态声场的声压进行预测,证明了该方法的有效性,结果表明,波数法在计算的精度和收敛性方面要优于有限元和边界元方法。
(3)应用波数法对有限周期性结构的声波禁带特性进行研究。通过对一维层状周期结构和二维正方形截面散射体周期结构的声压频率响应函数曲线来确定其不完全声波禁带。将波数法的结果与平面波展开法和有限元法的结果进行比较,验证该方法的有效性。
The composite material with periodic structure is called Phononic crystal. It can give rise to complete acoustic band gaps within which sound and vibration are forbidden. The motivation for these studies is their numerous engineering applications such as kinds of new equipments which can attenuate the noise and vibration.
The main content of this dissertation are listed as follow:
Firstly, the plane wave expansion method is used to calculate the acoustic(elastic) wave band gap of some specific periodic structures, for example (1)In the two dimensional liquid or solid periodic structure with square cross section, the characteristics of the band gap structures of the longitudinal wave or transverse wave are studied respectively as the cross section rotates. (2) The elliptic cross section is introduced in the two dimensional liquid periodic structure, and the influences of the elliptical radii and the rotation angel on acoustic wave band gaps are studied. (3)The ellipsoid cross section is introduced in the three dimensional liquid periodic structure, and the influences of the ellipsoid radii on acoustic wave band gap are studied.
The results show that, the material density has more influence on the acoustic wave band gap structures than the other parameters, and the radii and rotation angle of the ellipse or ellipsoid have influence also if the filling fraction is unchanged.
Secondly, a wave number method is proposed to deal with the acoustic problem. The method is based on an indirect Trefftz approach, in which the dynamic pressure response variable is approximated by a set of wave functions exactly satisfying the Helmholtz equation. The set of wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions, which arise from the external excitations. The weighting coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions and it is performed by applying the weighted residual formulation. The two and three dimensional uncoupled, the two dimensional coupled and the multilayer inhomogeneous acoustic problems are calculated. Comparing with FEM and BEM, the wave number method has better accuracy and convergence.
Thirdly, the wave number method is used to calculate the acoustic wave band gap of periodic structure with finite size. From the acoustic pressure distribution and the pressure frequency response function of the one dimensional finite multilayer structure and the two dimensional finite periodic structure with square scatterers, the directional acoustic wave band gap structures are obtained. Comparing the results of the wave number method with the FEM and the plane wave expansion method, the prior method shows more efficiency than the other two.
引文
[1]黄昆,韩汝琦.固体物理学.北京:高等教育出版社,1988.
[2]陈洗.固体物理基础.武汉:华中工学院出版社.1986.
[3]韦丹.固体物理.北京:清华大学出版社.2003.
[4]温熙森.光子\声子晶体理论与技术.北京:科学出版社,2006.
[5]庄飞,徐婕.固体物理能带论类比法教学研究.杭州师范学院学报.2001,18(5): 56-58
[6] Yablonovitch E. Inhibited Spontaneous Emission in Solid-State Physics and Electronics. Physical Review Letters, 1987, 58(20): 2059~2062.
[7] John S. Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 1987, 58(23): 2486~2489.
[8] Kushwaha M S, Djafari-Rouhani B. Band-gap engineering in two-dimensional periodic photonic crystals. Journal of Applied Physics, 2000, 88(5): 2877~2884.
[9]温维佳,沈平.局域共振的光子、声子功能材料.物理, 2004,33(2): 106~110.
[10]赵辉,张其锦.光子晶体研究进展.高分子通报,2001(5):60～65.
[11]快素兰,张俞之,胡行方.光子晶体的能带结构、潜在应用和制备方法.无机材料学报,2001,16(2):193～199.
[12]庄飞,何赛灵,何江平等.大带隙的二维各向异性椭圆介质柱光子晶体.物理学报,2002,51(2):355~360.
[13] Sigalas M M, Economou E N. Elastic and acoustic wave band structure. Journal of Sound and Vibration, 1992, 158 (2): 377~380.
[14] Kushwaha M S, Halevi P, Dobrzynsi L, et al. Acoustic band structure of periodic elastic composites. Physical Review Letters, 1993, 71(13): 2022~2025.
[15] Vasseur J O, Djafari-Rouhani B, Dobrzynski L, et al. Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite andsome metallic systems. Journal of Physics: Condensed Matter, 1994, 6: 8759~8770.
[16] Vasseur J O, Djafari-Rouhani B, Dobrzynski L, et al. Acoustic band gaps in fibre composite materials of boron nitride structure. Journal of Physics: Condensed Matter, 1997, 9 (35): 7327~7341.
[17] Munjal M L. Response of a multi-layered infinite plate to an oblique plane wave by means of transfer matrices. Journal of Sound and Vibration, 1993, 162(2): 333~343.
[18] Martinez-Sala R, Sancho J, Sanchez J V, et al. Sound attenuation by sculpture. Nature, 1995, 378: 241.
[19] Economou E N, Sigalas M M. Classical wave propagation in periodic structures: Cermet versus network topology. Physical Review B, 1993, 48(18):13434~ 13438.
[20] Sigalas M M, Economou E N. Elastic waves in plates with periodically placed inclusions. Journal of Applied Physics, 1994, 75(6): 2845~2850.
[21] Economou E N, Sigalas M M. Stop bands for elastic waves in periodic composite materials. Journal of the Acoustical Society of America, 1994, 95(4): 1734~1740.
[22] Kafesaki M, Sigalas M M, Economou E N. Elastic wave band gaps in 3-D periodic polymer matrix composites. Solid State Communications, 1995, 96(5): 285~289.
[23] Sigalas M M, Soukoulis C M. Elastic-wave propagation through disordered and/or absorptive layered systems. Physical Review B, 1995, 51(5): 2780~2789.
[24] Sigalas M M. Elastic wave band gaps and defect states in two-dimensional composites. Journal of the Acoustic Society of America, 1997, 101(3): 1256~ 1261.
[25] Sigalas M M. Defect states of acoustic waves in a two-dimensional lattice of solid cylinders. Journal of Applied Physics, 1998, 84(6): 3026~3030.
[26] Sigalas M M and Garcia N. Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method. Journal of Applied Physics, 2000, 87(6): 3122~3125.
[27] Sigalas M M, Garcia N. Importance of coupling between longitudinal and transverse components for the creation of acoustic band gaps: The aluminum in mercury case. Applied Physics Letters, 2000, 76(16): 2307~2309.
[28] Kafesaki M, Sigalas M M, Garcfa N. Frequency Modulation in the Transmittivity of Wave Guides in Elastic-Wave Band-Gap Materials. Physical Review Letters, 2000, 85(19): 4044~4047.
[29] García-Pablos D, Sigalas M, Montero de Espinosa F R, et al. Theory and Experiments on Elastic Band Gaps. Physical Review Letters, 2000, 84(19): 4349~4352.
[30] Vasseur J O, Deymier P A, Frantziskonis G, et al. Experimental evidence for the existence of absolute acoustic band gaps in two-dimensional periodic composite media. Journal of Physics: Condensed Matter, 1998, 10(27): 6051~6064.
[31] Vasseur J O, Deymier P A, Chenni B, et al. Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals. Physical Review Letters, 2001, 86(14): 3012~3015.
[32] H. Chandra, P. A. Deymier, O. Vasseur. Elastic wave propagation along waveguides in three-dimensional phononic crystals. Physical Review B, 2004, 70(5): 054302(6)
[33] Kushwaha M S, Halevi P, Martinez G, et al. Theory of acoustic band structure of periodic elastic composites. Physical Review B, 1994, 49(4): 2313~2322.
[34] Kushwaha M S, Halevi P. Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders. Applied Physics Letters, 1996, 69(1): 31~33.
[35] Kushwaha M S, Djafari-Rouhani B. Complete acoustic stop bands for cubic arrays of spherical liquid balloons. Journal of Applied Physics, 1996, 80(6): 3191~ 3195.
[36] Kushwaha M S, Halevi P. Stop bands for cubic arrays of spherical balloons. Journal of the Acoustical Society of America, 1997, 101 (1): 619-622.
[37] Kushwaha M S, Halevi P. Ultrawideband filter for noise control. Japanese Journal of Applied Physics, 1997, 36: 1043~1044.
[38] Kushwaha M S. Stop-bands for periodic metallic rods: Sculptures that can filter the noise. Physics Letter, 1997, 70(24): 3218~3220.
[39] Kushwaha M S, Djafari-Rouhani B. Giant sonic stop bands in two-dimensional periodic system of fluids. Journal of Applied Physics, 1998, 84(9): 4677~4683.
[40] Kushwaha M S, Halevi P. Band-gap engineering in periodic elastic composites. Applied Physics Letters, 1994, 64(9): 1085~1087.
[41]陈源,黄其柏,师汉民.基于振动与噪声控制的声子晶体研究进展.噪声与振动控制.2005,5:1-3.
[42] Kafesaki M, Economou E N. Interpretation of the band-structure results for elasticand acoustic waves by analogy with the LCAO approach. Physical Review B, 1995, 52(18): 13317~13331.
[43] Kafesaki M, Economou E N. Multiple-scattering theory for three-dimensional periodic acoustic composites. Physical Review B, 1999, 60(17): 11993~12001.
[44] Kafesaki M, Penciu R S and Economou E N. Air Bubbles in Water: Strongly Mutiple Scattering Medium for Acoustic Waves. Physical Review Letters, 2000, 84(26): 6050~6053.
[45] Klironomos A D, Economou E N. Elastic wave band gaps and single scattering. Solid State Communications, 1998, 105(5): 327~332.
[46] Yang Suxia, Page J H, Liu Zhengyou, et al. Focusing of Sound in a 3D Phononic Crystal. Physical Review Letters, 2004, 93(2): 024301(4).
[47] Gorishnyy T, Ullal C K, Maldovan M, et al. Hypersonic Phononic Crystals. Phys. Rev. Lett., 2005, 94: 115501.
[48] Lu Yanqing, Zhu Yongyuan, Chen Yanfeng, et al. Optical properties of an ionic-type phononic crystal. Science, 1999, 284:1822~1824.
[49]陆延青,朱永元,陈延峰等.离子型声子晶体的光学性质.物理,2000,29(4): 193~195.
[50] Liu Zhengyou, Zhang Xixiang, Mao Yiwei, et al. Locally resonant sonic materials. Science, 2000, 289: 1734~1736.
[51] Liu Zhengyou, Chan C T, Sheng Ping, et al. Elastic wave scattering by periodic structures of spherical objects: Theory and experiment. Physical Review B, 2000, 62(4): 2446~2457.
[52] Wu Fugen, Hou Zhilin, Liu Zhengyou, et al. Point defect states in two-dimensional phononic crystals. Physics Letter A, 2001,292 (3): 198~202.
[53] Wu Fugen, Liu Zhengyou, Liu Youyan. Stop Gaps and Single Defect States of Acoustic Waves in Two-Dimensional Lattice of Liquid Cylinders. Chinese Physics Letters, 2001, 18(6): 785~787.
[54] Liu Zhengyou, Chan C T, Sheng Ping. Three-component elastic wave band-gap material. Physical Review B, 2002, 65(16): 165116.
[55] Wu Fugen, Liu Zhengyou, Liu Youyan. Acoustic band gaps in 2D liquid phononic crystals of rectangular structure. Journal of Physics D: Applied Physics, 2002, 35(2): 162~165.
[56] Yang Suxia, Page J H, Liu Zhengyou, et al. Ultrasound Tunneling through 3D Phononic Crystals. Physics Review Letter, 2002, 88(10): 104301.
[57]吴福根,刘正猷,刘有延.二维周期性复合介质中弹性波的能带结构.声学学报,2001,26(4):319~323.
[58]齐共金,杨盛良,赵恂.新型功能材料——声子晶体.物理,2002,31(9): 568~571.
[59]齐共金,杨盛良,白书欣等.二维正方点阵液态声子晶体的带隙计算.国防科技大学学报,2003,25(1):103~106.
[60]齐共金,杨盛良,白书欣等.基于平面波算法的二维声子晶体带结构的研究.物理学报,2003,52(3):668~671.
[61]温激鸿,韩小云,王刚等.声子晶体研究概述.功能材料,2003,24(4):364~367.
[62]王刚,温激鸿,刘耀宗等.任意散射体形状二维周期结构的弹性波带结构计算方法研究.功能材料,2004(增刊),35:2257-2260.
[63]赵宏刚,韩小云,温激鸿等.新型声学功能材料——声子晶体.材料科学与工程学报,2003,21(1):153~156.
[64]王刚,温激鸿,韩小云等.二维声子晶体带隙计算中的时域有限差分方法.物理学报,2003,52(8):1943~1947.
[65]赵宏刚,韩小云,温激鸿等.空气中周期管阵列的声波禁带研究.材料科学与工程学报,2004,22(1):68~70.
[66]温激鸿,刘耀宗,郁殿龙等.基于散射单元的声子晶体振动带隙研究.人工晶体学报,2004,33(3):358~362.
[67]温激鸿,王刚,刘耀宗等.基于集中质量法的一维声子晶体弹性波带隙计算.物理学报, 2004, 53( 10) :3384-3388.
[68] Wang Gang, Wen Jihong, Liu Yaozong, et al. Lumped-mass method for the study of band structure in two-dimensional phononic crystals. Phys. Rev. B, 2004, 69: 184302.
[69] Cervera F, Sanchis L, Sanchez-Perez J V, et al. Refractive Acoustic Devices for Airborne Sound. Physical Review Letters, 2002, 88(2): 023902.
[70] Martin Hirsekorn. Small-size sonic crystals with strong attenuation bands in theaudible frequency range. Applied Physics Letters, 2004, 84(17): 3364-3366.
[71] Torres M, Montero de Espinosa F R, Garcia-Pablos D, et al. Sonic Band Gaps in Finite Elastic Media: Surface States and Localization Phenomena in Linear and Point Defects.. Physical Review Letters, 1999, 82 (15): 3054~3057.
[72] Psarobas I E, Stefanou N, Modinos A. Phononic crystals with planar defects. Physical Review B, 2000, 62(9): 5536~5540.
[73] Esquivel-Sirvent R, Noguez C. Theory of the acoustic signature of topological and morphological defects in SiC/porous SiC laminated ceramics. Journal of Applied Physics, 1997, 82(7): 3618~3620.
[74] Psarobas I E, Stefanou N, Modinos A. Scattering of elastic waves by periodic arrays of spherical bodies. Physical Review B, 2000, 62(1): 278~291.
[75] Manzanares-Martinez B, Ramos-Mendieta F. Transverse elastic waves in superlattices The Brewster acoustic angle. Physical Review B, 2000, 61(19): 12877~12881.
[76] Suzuki T, Yu P K L. Suppression and enhancement of elastodynamic radiation from a point source load in elastic wave band structures. Journal of Applied Physics, 1996, 80(10): 5665~5673.
[77] Acoustic Bandgap Materials and Devices. DAILPA. 99-034; 992-034.
[78] Diez A, Kakarantzas G, Birks T A, et al. Acoustic stop-bands in periodically microtapered optical fibers. Applied Physics Letters, 2000, 76(23): 3481~3483.
[79]吴福根,刘有延.二维周期性复合介质中声波带隙结构及其缺陷态.物理学报, 2002,51(7):1434~1438.
[80]张荣英,姜根山,王璋奇,等.声子晶体的研究进展及应用前景.声学技术.2006, 25(2): 35-42.
[81]钱斯文,杨盛良,赵恂.新型光学/声学带隙材料研究进展.材料科学与工程学报, 2004,22(3):436~441.
[82]张舒,程建春.二维三元周期复合结构的声带隙特性.自然科学进展,2003, 13(10): 1095~1098.
[83] Yukihiro Tanaka, Shin-ichiro Tamura. Acoustic stop bands of surface and bulk modes in two-dimensional phononic lattices consisting of aluminum and a polymer.Physical Review B, 1999, 60(19): 13294~13297.
[84] Yukihiro Tanaka, Yoshinobu Tomoyasu, Shin-ichiro Tamura. Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch. Physical Review B, 2000, 62(11): 7387~7392.
[85] Ye Zhen and Hoskinson E. Band gaps and localization in acoustic propagation in water with air cylinders. Applied Physics Letters, 2000, 77(26): 4428~4430.
[86] Sanchez-Perez J V, Caballero D, Martinez-Sala R, et al. Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders. Physical Review Letters, 1998, 80(24): 5325~5328.
[87]莫尔斯.振动与声.(第二版).南京大学<振动与声>翻译组译.北京:科学出版社, 1974.
[88]莫尔斯,英格特.理论声学.吕如榆,杨训仁等译.北京:科学出版社,1984.
[89]许肖梅.声学基础.北京:科学出版社, 2003.92-101.
[90] Atalla N, Bernhard R J. Review of numerical solutions for low-frequency structural-acoustic problems. Applied Acoustics, 1994, 43: 271~294.
[91] Deraemaeker, Babuska I, Ph. Bouillard. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. International Journal for Numerical Methods in Engineering, 1999, 46: 471~499.
[92] Bouillard P, Ihlenburg F. Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications. Computer methods in applied mechanics and engineering, 1999, 176:147-163.
[93] Astley R J. Infinite elements for wave problems: a review of current formulations and an assessment of accuracy. International Journal for Numerical Methods in Engineering , 2000, 49: 951~976
[94] Desmet W. Mid-frequency vibro-acoustic modelling challenges and potential solutions. Proceedings of ISMA2002 (2), 835~862.
[95]王勖成,邵敏.有限单元法基本原理和数值方法(第二版).北京:清华大学出版社,1997.
[96]布瑞比亚,泰勒斯,诺贝尔.边界单元法的理论和工程应用.龙述尧等译.北京:国防工业出版社,1988
[97] Trefftz E. Ein Gegenstück zum Ritzschen Verfahren. Proceedings of the 2nd International Congress. Applied Mechanics, Zurich, 1926: 131-137
[98] Kita E, Kamiya N. Trefftz method: an overview. Advances in Engineering Software, 1995, 24: 3-12.
[99] Zielinski A P. On trial functions applied in the generalized Trefftz method. Advances in Engineering Software, 1995, 24, 147~155.
[100] Pluymers B, Desmet W, Vandepitte, D, et al. A Trefftz-based prediction technique for multi-domain steady-state acoustic problems. Proceedings of the Tenth International Congress on Sound and Vibration, Stockholm, Sweden, 2003, 2833 - 2840.
[101] Desmet W, Sas P, Vandepitte D. a new wave based prediction technique for coupled vibro-acoustic analysis theoretical description and application to a double wall structure. ISMA21: Noise and vibration Engineering.1996: 105-134.
[102] Desmet W. A wave based prediction technique for coupled vibro-acoustic analysis. K U Leuven, division PMA, Leuven, 1998, 65-192.
[103] Desmet W, Sas P, Vandepitte D. On the numerical properties of a wave based prediction technique for coupled vibro-acoustic analysis. International Conference on noise and vibation engineering, ISMA23, 1998, 587-594.
[104] Van Hal B, Desmet W, Vandepitte D, et al. Application of the efficient wave based prediction technique for the steady-state dynamic analysis of flat plates. Proceedings ISMA25 - Noise and Vibration Engineering, Leuven, Belgium, 2000, 607 - 614.
[105] Hepberger A, Priebsch H H, Desmet W, et al. Application of the wave based method for the steady-state acoustic response prediction of a Car cavity in the mid-frequency range. Proceedings of ISMA2002, Leuven, Belgium, 2002(2): 877-884.
[106] Van Hal B, Hepberger A, Priebsch H H, et al. High performance implementation and conceptual development of the wave based method for steady-state dynamic analysis of acoustic problems. Proceedings of ISMA2002 (2), 817~826.
[107] Desmet W. mid-frequency vibro-acoustic modelling challenges and potential solutions. Proceedingg of ISMA2002(2): 835-862.
[108] Desmet W, Van Hal B, Sas P. A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems. Advances inEngineering Software, 2002 (33): 527-540.
[109] Pluymers B, Desmet W, Vandepitte D. Application of the wave based prediction technique for the analysis of the coupled vibro-acoustic behaviour of a 3D cavity. Proceedingg of ISMA2002(2): 891-900.
[110] Pluymers B, Desmet W, Vandepitte D, et al. Application of the wave based prediction technique for the steady-state dynamic analysis of multi-domain acoustic problems. Proceedings of the 6th National Congress on Applied and Theoretical Mechanics, Gent, Belgium, 2003.
[111] Pluymers B, Desmet W, Vandepitte D, et al. A Trefftz-based prediction technique for multi-domain steady-state acoustic problems. Proceedings of the Tenth International Congress on Sound and Vibration, Stockholm, Sweden, 2003.
[112] Pluymers B, Desmet W, Vandepitte D, et al. On the use of a wave based prediction technique for steady-state structural-acoustic radiation analysis. International Workshop on MeshFree Methods 2003:1-7.
[113] Pluymers B, Hepberger A, Desmet W. Experimental validation of the wave based prediction technique for the analysis of the coupled vibro-acoustic behaviour of a 3D cavity. Proceedings of the Second MIT Conference on Solid and Fluid Mechanics (MIT2), Boston, Massachusetts, USA, 2003, 1483 - 1487
[114] Van Hal B. Automation and performance optimization of the wave based method for interior structural-acoustic problems. Ph.D. dissertation, K.U. Leuven. 2004, 61-90.
[115] Pluymers B, Desmet W, Vandepitte D, et al. Feasibility study of the wave based method for high frequency steady-state acoustic analysis. Proceedings of ISMA2004, 1555~1574.
[116] Pluymers B, Desmet W, Vandepitte D, et al. Application of an efficient wave-based prediction technique for the analysis of vibro-acoustic radiation problems. Journal of Computational and Applied Mathematics archive. 2004(168): 353-364
[117] Hepberger A, Pluymers B, Jalics K, et al. Validation of a wave based technique for the analysis of a multi-domain 3D acoustic cavity with interior damping and loudspeaker excitation. Inter Noise 2004, Prague, Czech Republic, 2004.4.
[118] Van Hal B, Vanmaele C, Desmet W, et al. Hybrid finite element– wave based method for steady-state acoustic analysis. Proceeding of ISMA2004: 1629-1642.
[119] Vanmaele C, Desmet W, Vandepitte D. On the use of the wave based method for the steady-state dynamic analysis of three-dimensional plate assemblies. Proceeding of ISMA2004: 1643-1657.
[120] Neittaanm?ki P, Rossi T, Majava K, et al. A wave based prediction technique for the steady-state dynamic analysis of flat assemblies. in: Neittaanm?ki P. European Congress of Computational Methods in Applied Sciences and Engineering. Jyv?skyl?: 2004.1-11.
[121] Van Hal B, Desmet W, Vandepitte D. Hybrid finite element—wave-based method for steady-state interior structural-acoustic problems. Computers and Structures 2005(83): 167-180.
[122] Hepberger A, Diwoky F, Jalics K, et al. Application of Wave Based Technique for a cavity considering forced excitation at boundaries and effects of absorption material. Proceedings of the 2004 International Conference on Modal Analysis Noise and Vibration Engineering, Leuven, Belgium, 2004: 1501–1512
[123] Varah J M. Pitfalls in the numerical solution of linear ill-posed problems. SIAM J. Sci. Stat. Comput. 1983(4), 164-176 .
[2]陈洗.固体物理基础.武汉:华中工学院出版社.1986.
[3]韦丹.固体物理.北京:清华大学出版社.2003.
[4]温熙森.光子\声子晶体理论与技术.北京:科学出版社,2006.
[5]庄飞,徐婕.固体物理能带论类比法教学研究.杭州师范学院学报.2001,18(5): 56-58
[6] Yablonovitch E. Inhibited Spontaneous Emission in Solid-State Physics and Electronics. Physical Review Letters, 1987, 58(20): 2059~2062.
[7] John S. Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 1987, 58(23): 2486~2489.
[8] Kushwaha M S, Djafari-Rouhani B. Band-gap engineering in two-dimensional periodic photonic crystals. Journal of Applied Physics, 2000, 88(5): 2877~2884.
[9]温维佳,沈平.局域共振的光子、声子功能材料.物理, 2004,33(2): 106~110.
[10]赵辉,张其锦.光子晶体研究进展.高分子通报,2001(5):60～65.
[11]快素兰,张俞之,胡行方.光子晶体的能带结构、潜在应用和制备方法.无机材料学报,2001,16(2):193～199.
[12]庄飞,何赛灵,何江平等.大带隙的二维各向异性椭圆介质柱光子晶体.物理学报,2002,51(2):355~360.
[13] Sigalas M M, Economou E N. Elastic and acoustic wave band structure. Journal of Sound and Vibration, 1992, 158 (2): 377~380.
[14] Kushwaha M S, Halevi P, Dobrzynsi L, et al. Acoustic band structure of periodic elastic composites. Physical Review Letters, 1993, 71(13): 2022~2025.
[15] Vasseur J O, Djafari-Rouhani B, Dobrzynski L, et al. Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite andsome metallic systems. Journal of Physics: Condensed Matter, 1994, 6: 8759~8770.
[16] Vasseur J O, Djafari-Rouhani B, Dobrzynski L, et al. Acoustic band gaps in fibre composite materials of boron nitride structure. Journal of Physics: Condensed Matter, 1997, 9 (35): 7327~7341.
[17] Munjal M L. Response of a multi-layered infinite plate to an oblique plane wave by means of transfer matrices. Journal of Sound and Vibration, 1993, 162(2): 333~343.
[18] Martinez-Sala R, Sancho J, Sanchez J V, et al. Sound attenuation by sculpture. Nature, 1995, 378: 241.
[19] Economou E N, Sigalas M M. Classical wave propagation in periodic structures: Cermet versus network topology. Physical Review B, 1993, 48(18):13434~ 13438.
[20] Sigalas M M, Economou E N. Elastic waves in plates with periodically placed inclusions. Journal of Applied Physics, 1994, 75(6): 2845~2850.
[21] Economou E N, Sigalas M M. Stop bands for elastic waves in periodic composite materials. Journal of the Acoustical Society of America, 1994, 95(4): 1734~1740.
[22] Kafesaki M, Sigalas M M, Economou E N. Elastic wave band gaps in 3-D periodic polymer matrix composites. Solid State Communications, 1995, 96(5): 285~289.
[23] Sigalas M M, Soukoulis C M. Elastic-wave propagation through disordered and/or absorptive layered systems. Physical Review B, 1995, 51(5): 2780~2789.
[24] Sigalas M M. Elastic wave band gaps and defect states in two-dimensional composites. Journal of the Acoustic Society of America, 1997, 101(3): 1256~ 1261.
[25] Sigalas M M. Defect states of acoustic waves in a two-dimensional lattice of solid cylinders. Journal of Applied Physics, 1998, 84(6): 3026~3030.
[26] Sigalas M M and Garcia N. Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method. Journal of Applied Physics, 2000, 87(6): 3122~3125.
[27] Sigalas M M, Garcia N. Importance of coupling between longitudinal and transverse components for the creation of acoustic band gaps: The aluminum in mercury case. Applied Physics Letters, 2000, 76(16): 2307~2309.
[28] Kafesaki M, Sigalas M M, Garcfa N. Frequency Modulation in the Transmittivity of Wave Guides in Elastic-Wave Band-Gap Materials. Physical Review Letters, 2000, 85(19): 4044~4047.
[29] García-Pablos D, Sigalas M, Montero de Espinosa F R, et al. Theory and Experiments on Elastic Band Gaps. Physical Review Letters, 2000, 84(19): 4349~4352.
[30] Vasseur J O, Deymier P A, Frantziskonis G, et al. Experimental evidence for the existence of absolute acoustic band gaps in two-dimensional periodic composite media. Journal of Physics: Condensed Matter, 1998, 10(27): 6051~6064.
[31] Vasseur J O, Deymier P A, Chenni B, et al. Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals. Physical Review Letters, 2001, 86(14): 3012~3015.
[32] H. Chandra, P. A. Deymier, O. Vasseur. Elastic wave propagation along waveguides in three-dimensional phononic crystals. Physical Review B, 2004, 70(5): 054302(6)
[33] Kushwaha M S, Halevi P, Martinez G, et al. Theory of acoustic band structure of periodic elastic composites. Physical Review B, 1994, 49(4): 2313~2322.
[34] Kushwaha M S, Halevi P. Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders. Applied Physics Letters, 1996, 69(1): 31~33.
[35] Kushwaha M S, Djafari-Rouhani B. Complete acoustic stop bands for cubic arrays of spherical liquid balloons. Journal of Applied Physics, 1996, 80(6): 3191~ 3195.
[36] Kushwaha M S, Halevi P. Stop bands for cubic arrays of spherical balloons. Journal of the Acoustical Society of America, 1997, 101 (1): 619-622.
[37] Kushwaha M S, Halevi P. Ultrawideband filter for noise control. Japanese Journal of Applied Physics, 1997, 36: 1043~1044.
[38] Kushwaha M S. Stop-bands for periodic metallic rods: Sculptures that can filter the noise. Physics Letter, 1997, 70(24): 3218~3220.
[39] Kushwaha M S, Djafari-Rouhani B. Giant sonic stop bands in two-dimensional periodic system of fluids. Journal of Applied Physics, 1998, 84(9): 4677~4683.
[40] Kushwaha M S, Halevi P. Band-gap engineering in periodic elastic composites. Applied Physics Letters, 1994, 64(9): 1085~1087.
[41]陈源,黄其柏,师汉民.基于振动与噪声控制的声子晶体研究进展.噪声与振动控制.2005,5:1-3.
[42] Kafesaki M, Economou E N. Interpretation of the band-structure results for elasticand acoustic waves by analogy with the LCAO approach. Physical Review B, 1995, 52(18): 13317~13331.
[43] Kafesaki M, Economou E N. Multiple-scattering theory for three-dimensional periodic acoustic composites. Physical Review B, 1999, 60(17): 11993~12001.
[44] Kafesaki M, Penciu R S and Economou E N. Air Bubbles in Water: Strongly Mutiple Scattering Medium for Acoustic Waves. Physical Review Letters, 2000, 84(26): 6050~6053.
[45] Klironomos A D, Economou E N. Elastic wave band gaps and single scattering. Solid State Communications, 1998, 105(5): 327~332.
[46] Yang Suxia, Page J H, Liu Zhengyou, et al. Focusing of Sound in a 3D Phononic Crystal. Physical Review Letters, 2004, 93(2): 024301(4).
[47] Gorishnyy T, Ullal C K, Maldovan M, et al. Hypersonic Phononic Crystals. Phys. Rev. Lett., 2005, 94: 115501.
[48] Lu Yanqing, Zhu Yongyuan, Chen Yanfeng, et al. Optical properties of an ionic-type phononic crystal. Science, 1999, 284:1822~1824.
[49]陆延青,朱永元,陈延峰等.离子型声子晶体的光学性质.物理,2000,29(4): 193~195.
[50] Liu Zhengyou, Zhang Xixiang, Mao Yiwei, et al. Locally resonant sonic materials. Science, 2000, 289: 1734~1736.
[51] Liu Zhengyou, Chan C T, Sheng Ping, et al. Elastic wave scattering by periodic structures of spherical objects: Theory and experiment. Physical Review B, 2000, 62(4): 2446~2457.
[52] Wu Fugen, Hou Zhilin, Liu Zhengyou, et al. Point defect states in two-dimensional phononic crystals. Physics Letter A, 2001,292 (3): 198~202.
[53] Wu Fugen, Liu Zhengyou, Liu Youyan. Stop Gaps and Single Defect States of Acoustic Waves in Two-Dimensional Lattice of Liquid Cylinders. Chinese Physics Letters, 2001, 18(6): 785~787.
[54] Liu Zhengyou, Chan C T, Sheng Ping. Three-component elastic wave band-gap material. Physical Review B, 2002, 65(16): 165116.
[55] Wu Fugen, Liu Zhengyou, Liu Youyan. Acoustic band gaps in 2D liquid phononic crystals of rectangular structure. Journal of Physics D: Applied Physics, 2002, 35(2): 162~165.
[56] Yang Suxia, Page J H, Liu Zhengyou, et al. Ultrasound Tunneling through 3D Phononic Crystals. Physics Review Letter, 2002, 88(10): 104301.
[57]吴福根,刘正猷,刘有延.二维周期性复合介质中弹性波的能带结构.声学学报,2001,26(4):319~323.
[58]齐共金,杨盛良,赵恂.新型功能材料——声子晶体.物理,2002,31(9): 568~571.
[59]齐共金,杨盛良,白书欣等.二维正方点阵液态声子晶体的带隙计算.国防科技大学学报,2003,25(1):103~106.
[60]齐共金,杨盛良,白书欣等.基于平面波算法的二维声子晶体带结构的研究.物理学报,2003,52(3):668~671.
[61]温激鸿,韩小云,王刚等.声子晶体研究概述.功能材料,2003,24(4):364~367.
[62]王刚,温激鸿,刘耀宗等.任意散射体形状二维周期结构的弹性波带结构计算方法研究.功能材料,2004(增刊),35:2257-2260.
[63]赵宏刚,韩小云,温激鸿等.新型声学功能材料——声子晶体.材料科学与工程学报,2003,21(1):153~156.
[64]王刚,温激鸿,韩小云等.二维声子晶体带隙计算中的时域有限差分方法.物理学报,2003,52(8):1943~1947.
[65]赵宏刚,韩小云,温激鸿等.空气中周期管阵列的声波禁带研究.材料科学与工程学报,2004,22(1):68~70.
[66]温激鸿,刘耀宗,郁殿龙等.基于散射单元的声子晶体振动带隙研究.人工晶体学报,2004,33(3):358~362.
[67]温激鸿,王刚,刘耀宗等.基于集中质量法的一维声子晶体弹性波带隙计算.物理学报, 2004, 53( 10) :3384-3388.
[68] Wang Gang, Wen Jihong, Liu Yaozong, et al. Lumped-mass method for the study of band structure in two-dimensional phononic crystals. Phys. Rev. B, 2004, 69: 184302.
[69] Cervera F, Sanchis L, Sanchez-Perez J V, et al. Refractive Acoustic Devices for Airborne Sound. Physical Review Letters, 2002, 88(2): 023902.
[70] Martin Hirsekorn. Small-size sonic crystals with strong attenuation bands in theaudible frequency range. Applied Physics Letters, 2004, 84(17): 3364-3366.
[71] Torres M, Montero de Espinosa F R, Garcia-Pablos D, et al. Sonic Band Gaps in Finite Elastic Media: Surface States and Localization Phenomena in Linear and Point Defects.. Physical Review Letters, 1999, 82 (15): 3054~3057.
[72] Psarobas I E, Stefanou N, Modinos A. Phononic crystals with planar defects. Physical Review B, 2000, 62(9): 5536~5540.
[73] Esquivel-Sirvent R, Noguez C. Theory of the acoustic signature of topological and morphological defects in SiC/porous SiC laminated ceramics. Journal of Applied Physics, 1997, 82(7): 3618~3620.
[74] Psarobas I E, Stefanou N, Modinos A. Scattering of elastic waves by periodic arrays of spherical bodies. Physical Review B, 2000, 62(1): 278~291.
[75] Manzanares-Martinez B, Ramos-Mendieta F. Transverse elastic waves in superlattices The Brewster acoustic angle. Physical Review B, 2000, 61(19): 12877~12881.
[76] Suzuki T, Yu P K L. Suppression and enhancement of elastodynamic radiation from a point source load in elastic wave band structures. Journal of Applied Physics, 1996, 80(10): 5665~5673.
[77] Acoustic Bandgap Materials and Devices. DAILPA. 99-034; 992-034.
[78] Diez A, Kakarantzas G, Birks T A, et al. Acoustic stop-bands in periodically microtapered optical fibers. Applied Physics Letters, 2000, 76(23): 3481~3483.
[79]吴福根,刘有延.二维周期性复合介质中声波带隙结构及其缺陷态.物理学报, 2002,51(7):1434~1438.
[80]张荣英,姜根山,王璋奇,等.声子晶体的研究进展及应用前景.声学技术.2006, 25(2): 35-42.
[81]钱斯文,杨盛良,赵恂.新型光学/声学带隙材料研究进展.材料科学与工程学报, 2004,22(3):436~441.
[82]张舒,程建春.二维三元周期复合结构的声带隙特性.自然科学进展,2003, 13(10): 1095~1098.
[83] Yukihiro Tanaka, Shin-ichiro Tamura. Acoustic stop bands of surface and bulk modes in two-dimensional phononic lattices consisting of aluminum and a polymer.Physical Review B, 1999, 60(19): 13294~13297.
[84] Yukihiro Tanaka, Yoshinobu Tomoyasu, Shin-ichiro Tamura. Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch. Physical Review B, 2000, 62(11): 7387~7392.
[85] Ye Zhen and Hoskinson E. Band gaps and localization in acoustic propagation in water with air cylinders. Applied Physics Letters, 2000, 77(26): 4428~4430.
[86] Sanchez-Perez J V, Caballero D, Martinez-Sala R, et al. Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders. Physical Review Letters, 1998, 80(24): 5325~5328.
[87]莫尔斯.振动与声.(第二版).南京大学<振动与声>翻译组译.北京:科学出版社, 1974.
[88]莫尔斯,英格特.理论声学.吕如榆,杨训仁等译.北京:科学出版社,1984.
[89]许肖梅.声学基础.北京:科学出版社, 2003.92-101.
[90] Atalla N, Bernhard R J. Review of numerical solutions for low-frequency structural-acoustic problems. Applied Acoustics, 1994, 43: 271~294.
[91] Deraemaeker, Babuska I, Ph. Bouillard. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. International Journal for Numerical Methods in Engineering, 1999, 46: 471~499.
[92] Bouillard P, Ihlenburg F. Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications. Computer methods in applied mechanics and engineering, 1999, 176:147-163.
[93] Astley R J. Infinite elements for wave problems: a review of current formulations and an assessment of accuracy. International Journal for Numerical Methods in Engineering , 2000, 49: 951~976
[94] Desmet W. Mid-frequency vibro-acoustic modelling challenges and potential solutions. Proceedings of ISMA2002 (2), 835~862.
[95]王勖成,邵敏.有限单元法基本原理和数值方法(第二版).北京:清华大学出版社,1997.
[96]布瑞比亚,泰勒斯,诺贝尔.边界单元法的理论和工程应用.龙述尧等译.北京:国防工业出版社,1988
[97] Trefftz E. Ein Gegenstück zum Ritzschen Verfahren. Proceedings of the 2nd International Congress. Applied Mechanics, Zurich, 1926: 131-137
[98] Kita E, Kamiya N. Trefftz method: an overview. Advances in Engineering Software, 1995, 24: 3-12.
[99] Zielinski A P. On trial functions applied in the generalized Trefftz method. Advances in Engineering Software, 1995, 24, 147~155.
[100] Pluymers B, Desmet W, Vandepitte, D, et al. A Trefftz-based prediction technique for multi-domain steady-state acoustic problems. Proceedings of the Tenth International Congress on Sound and Vibration, Stockholm, Sweden, 2003, 2833 - 2840.
[101] Desmet W, Sas P, Vandepitte D. a new wave based prediction technique for coupled vibro-acoustic analysis theoretical description and application to a double wall structure. ISMA21: Noise and vibration Engineering.1996: 105-134.
[102] Desmet W. A wave based prediction technique for coupled vibro-acoustic analysis. K U Leuven, division PMA, Leuven, 1998, 65-192.
[103] Desmet W, Sas P, Vandepitte D. On the numerical properties of a wave based prediction technique for coupled vibro-acoustic analysis. International Conference on noise and vibation engineering, ISMA23, 1998, 587-594.
[104] Van Hal B, Desmet W, Vandepitte D, et al. Application of the efficient wave based prediction technique for the steady-state dynamic analysis of flat plates. Proceedings ISMA25 - Noise and Vibration Engineering, Leuven, Belgium, 2000, 607 - 614.
[105] Hepberger A, Priebsch H H, Desmet W, et al. Application of the wave based method for the steady-state acoustic response prediction of a Car cavity in the mid-frequency range. Proceedings of ISMA2002, Leuven, Belgium, 2002(2): 877-884.
[106] Van Hal B, Hepberger A, Priebsch H H, et al. High performance implementation and conceptual development of the wave based method for steady-state dynamic analysis of acoustic problems. Proceedings of ISMA2002 (2), 817~826.
[107] Desmet W. mid-frequency vibro-acoustic modelling challenges and potential solutions. Proceedingg of ISMA2002(2): 835-862.
[108] Desmet W, Van Hal B, Sas P. A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems. Advances inEngineering Software, 2002 (33): 527-540.
[109] Pluymers B, Desmet W, Vandepitte D. Application of the wave based prediction technique for the analysis of the coupled vibro-acoustic behaviour of a 3D cavity. Proceedingg of ISMA2002(2): 891-900.
[110] Pluymers B, Desmet W, Vandepitte D, et al. Application of the wave based prediction technique for the steady-state dynamic analysis of multi-domain acoustic problems. Proceedings of the 6th National Congress on Applied and Theoretical Mechanics, Gent, Belgium, 2003.
[111] Pluymers B, Desmet W, Vandepitte D, et al. A Trefftz-based prediction technique for multi-domain steady-state acoustic problems. Proceedings of the Tenth International Congress on Sound and Vibration, Stockholm, Sweden, 2003.
[112] Pluymers B, Desmet W, Vandepitte D, et al. On the use of a wave based prediction technique for steady-state structural-acoustic radiation analysis. International Workshop on MeshFree Methods 2003:1-7.
[113] Pluymers B, Hepberger A, Desmet W. Experimental validation of the wave based prediction technique for the analysis of the coupled vibro-acoustic behaviour of a 3D cavity. Proceedings of the Second MIT Conference on Solid and Fluid Mechanics (MIT2), Boston, Massachusetts, USA, 2003, 1483 - 1487
[114] Van Hal B. Automation and performance optimization of the wave based method for interior structural-acoustic problems. Ph.D. dissertation, K.U. Leuven. 2004, 61-90.
[115] Pluymers B, Desmet W, Vandepitte D, et al. Feasibility study of the wave based method for high frequency steady-state acoustic analysis. Proceedings of ISMA2004, 1555~1574.
[116] Pluymers B, Desmet W, Vandepitte D, et al. Application of an efficient wave-based prediction technique for the analysis of vibro-acoustic radiation problems. Journal of Computational and Applied Mathematics archive. 2004(168): 353-364
[117] Hepberger A, Pluymers B, Jalics K, et al. Validation of a wave based technique for the analysis of a multi-domain 3D acoustic cavity with interior damping and loudspeaker excitation. Inter Noise 2004, Prague, Czech Republic, 2004.4.
[118] Van Hal B, Vanmaele C, Desmet W, et al. Hybrid finite element– wave based method for steady-state acoustic analysis. Proceeding of ISMA2004: 1629-1642.
[119] Vanmaele C, Desmet W, Vandepitte D. On the use of the wave based method for the steady-state dynamic analysis of three-dimensional plate assemblies. Proceeding of ISMA2004: 1643-1657.
[120] Neittaanm?ki P, Rossi T, Majava K, et al. A wave based prediction technique for the steady-state dynamic analysis of flat assemblies. in: Neittaanm?ki P. European Congress of Computational Methods in Applied Sciences and Engineering. Jyv?skyl?: 2004.1-11.
[121] Van Hal B, Desmet W, Vandepitte D. Hybrid finite element—wave-based method for steady-state interior structural-acoustic problems. Computers and Structures 2005(83): 167-180.
[122] Hepberger A, Diwoky F, Jalics K, et al. Application of Wave Based Technique for a cavity considering forced excitation at boundaries and effects of absorption material. Proceedings of the 2004 International Conference on Modal Analysis Noise and Vibration Engineering, Leuven, Belgium, 2004: 1501–1512
[123] Varah J M. Pitfalls in the numerical solution of linear ill-posed problems. SIAM J. Sci. Stat. Comput. 1983(4), 164-176 .