气动声学问题研究的Hamilton方法及其应用
|
摘要
得益于计算机技术发展,科学计算得到了飞速的发展。目前,人们研究噪声机理的手段有实验、理论分析以及数值计算。气动声学研究一般是从宏观的角度展开的,本论文在Hamilton体系下,从宏观和微观相关联的角度研究声波在气体介质中的传播问题,并在求解波动方程过程中,
引入了与现有差分算法不同的辛算法。辛算法相对传统算法有其独特的优越性,因为保守体系可用Hamilton体系的方法描述,其特点是保辛。保辛给出保体系结构最重要的特性。而对于某些非保守系统则也可通过转化为保守系统进行分析。与现有差分算法不同的是,辛算法具有保辛性,保辛性也是保体系结构最重要的特性。而对于非保守系统,则可通过转化,将它专为保守系统进行研究。数值算例分析表明,与同阶的有限差分格式相比较,本文给出的辛算法在效率上和精度上有较大的优势。
在Hamilton体系下,利用辛算法分析并求解了气动声学经典波动方程。首先建立一个在离散化网格上的准粒子体系,引入准粒子间相互作用势,用Hamilton力学描述这个在时间上连续而空间上离散的体系,准粒子按照Hamilton正则方程运动。推导演绎了Hamilton描述、辛算法及波动方程之间的联系及准粒子体系的互作用系数。通过数值算例分析,验证了本文算法的正确性和稳定性。文中应用辛算法数值模拟了声在空气中的传播。从数值模拟的结果来看,用Hamilton系统方法来描述声波的传播是有效的。同时对应的保结构辛算法也可以直接应用于数值模拟,它比耗散型格式的计算结果更为符合物理实质,也更为精确。
由于微观体系粒子的能级与粒子配分函数之间存在特定的关系,而由配分函数可以求出体系的内能、熵、自由能等等热力学量,进而可获得气动声场的声压等宏观参数。在Hamilton体系下,分别以量子力学的不同表述形式构建模型研究声的传播问题。即分别以波动力学核心的Schrodinger方程为粒子运动控制方程,结合群论、配分函数以及路径积分方法关联配分函数,构建新模型来研究声的传播。数值计算结果表明,这两种模型都能用来分析气动声场。本文的方法研究及其应用,为气动声学的研究提供了新的途径。
Due to the development of computer ability, The computational seience got a fast developing. There are the three means to understand the mechanism of aeroacoustics that its the method of theoretical study, experiment and numerical simulation.Conventional study of aeroacoustics is conducted under the system of Newtonian mechanics. The doctoral thesis study the propagation of sound waves in a gas medium by contacting the macro and micro perspectives in the Hamilton framework, and solve the wave equation in the process of introducing symplectic. Difference from the traditional algorithms, the symplectic algorithm has mang advantages. the most of these advantages is the conservation of symplecticness, which is the most important feature of conservative systems can be described with the Hamilton system, and its characteristic is the conservation of symplecticness. Symplectic algorithms can handle some non-conservative systems through converting inio the conservative system. Numerical examples show that: comparing with the same order finite difference scheme, symplectic algorithms used by this doctoral thesis have a greater advantage in efficiency and accuracy,
the Hamilton theory framework is adopted in this dissertation to analyze the problems of acoustic wave propagation in air, to study the propagation of stimulation acoustic waves, and to explore new approaches to research on aeroacoustics. A quasi-particle system on a discrete lattice is built up first, and the interaction potential among quasi-particles is then determined. This system, which is continuous in time while discrete in space, is expressed in terms of Hamilton mechanics, and the motion of quasi-particles is governed by Hamilton canonical equations. Based on the basic principle of Hamilton mechanics, the relationships between the symplectic algorithms and the the wave equation and interaction coefficient s of quasi-particle system are derived. The numerical simulation results suggest that the Hamilton method is effective in describing the propagation of acoustic waves. The corresponding conserving symplectic algorithm can also be directly applied in numerical simulations, the results of which are not only more consistent with physical essence but are also more accurate than those of a dissipative scheme.
Since there is a specific relationship between the energy levels of the micro-system particle and particle distribution function,the thermodynamic quantities which contains the internal energy of the system, entropy,the free energy, etc,can be calculated by the partition function, and then macroscopic quantities of the sound field can be got by the partition function. In Hamilton system, sound propagation is studied by different model with different expressions of the quantum mechanics. Combining with group theory and the partition function,a new model is built by considering the Schrodinger equation as particle motion control equation to study sound propagation. symplectic algorithm is introduced to solve the Schrodinger equation with the complex potential. Another new model is built through the method of Feynman path integral and the partition function. The numerical simulation results show that the two model are effective in studying the propagation of acoustic waves. Hamilton Method and its Application for Aeroacoustics in this doctoral thesis paves a new way for the study of wave propagation and aeroacoustics.
引入了与现有差分算法不同的辛算法。辛算法相对传统算法有其独特的优越性,因为保守体系可用Hamilton体系的方法描述,其特点是保辛。保辛给出保体系结构最重要的特性。而对于某些非保守系统则也可通过转化为保守系统进行分析。与现有差分算法不同的是,辛算法具有保辛性,保辛性也是保体系结构最重要的特性。而对于非保守系统,则可通过转化,将它专为保守系统进行研究。数值算例分析表明,与同阶的有限差分格式相比较,本文给出的辛算法在效率上和精度上有较大的优势。
在Hamilton体系下,利用辛算法分析并求解了气动声学经典波动方程。首先建立一个在离散化网格上的准粒子体系,引入准粒子间相互作用势,用Hamilton力学描述这个在时间上连续而空间上离散的体系,准粒子按照Hamilton正则方程运动。推导演绎了Hamilton描述、辛算法及波动方程之间的联系及准粒子体系的互作用系数。通过数值算例分析,验证了本文算法的正确性和稳定性。文中应用辛算法数值模拟了声在空气中的传播。从数值模拟的结果来看,用Hamilton系统方法来描述声波的传播是有效的。同时对应的保结构辛算法也可以直接应用于数值模拟,它比耗散型格式的计算结果更为符合物理实质,也更为精确。
由于微观体系粒子的能级与粒子配分函数之间存在特定的关系,而由配分函数可以求出体系的内能、熵、自由能等等热力学量,进而可获得气动声场的声压等宏观参数。在Hamilton体系下,分别以量子力学的不同表述形式构建模型研究声的传播问题。即分别以波动力学核心的Schrodinger方程为粒子运动控制方程,结合群论、配分函数以及路径积分方法关联配分函数,构建新模型来研究声的传播。数值计算结果表明,这两种模型都能用来分析气动声场。本文的方法研究及其应用,为气动声学的研究提供了新的途径。
Due to the development of computer ability, The computational seience got a fast developing. There are the three means to understand the mechanism of aeroacoustics that its the method of theoretical study, experiment and numerical simulation.Conventional study of aeroacoustics is conducted under the system of Newtonian mechanics. The doctoral thesis study the propagation of sound waves in a gas medium by contacting the macro and micro perspectives in the Hamilton framework, and solve the wave equation in the process of introducing symplectic. Difference from the traditional algorithms, the symplectic algorithm has mang advantages. the most of these advantages is the conservation of symplecticness, which is the most important feature of conservative systems can be described with the Hamilton system, and its characteristic is the conservation of symplecticness. Symplectic algorithms can handle some non-conservative systems through converting inio the conservative system. Numerical examples show that: comparing with the same order finite difference scheme, symplectic algorithms used by this doctoral thesis have a greater advantage in efficiency and accuracy,
the Hamilton theory framework is adopted in this dissertation to analyze the problems of acoustic wave propagation in air, to study the propagation of stimulation acoustic waves, and to explore new approaches to research on aeroacoustics. A quasi-particle system on a discrete lattice is built up first, and the interaction potential among quasi-particles is then determined. This system, which is continuous in time while discrete in space, is expressed in terms of Hamilton mechanics, and the motion of quasi-particles is governed by Hamilton canonical equations. Based on the basic principle of Hamilton mechanics, the relationships between the symplectic algorithms and the the wave equation and interaction coefficient s of quasi-particle system are derived. The numerical simulation results suggest that the Hamilton method is effective in describing the propagation of acoustic waves. The corresponding conserving symplectic algorithm can also be directly applied in numerical simulations, the results of which are not only more consistent with physical essence but are also more accurate than those of a dissipative scheme.
Since there is a specific relationship between the energy levels of the micro-system particle and particle distribution function,the thermodynamic quantities which contains the internal energy of the system, entropy,the free energy, etc,can be calculated by the partition function, and then macroscopic quantities of the sound field can be got by the partition function. In Hamilton system, sound propagation is studied by different model with different expressions of the quantum mechanics. Combining with group theory and the partition function,a new model is built by considering the Schrodinger equation as particle motion control equation to study sound propagation. symplectic algorithm is introduced to solve the Schrodinger equation with the complex potential. Another new model is built through the method of Feynman path integral and the partition function. The numerical simulation results show that the two model are effective in studying the propagation of acoustic waves. Hamilton Method and its Application for Aeroacoustics in this doctoral thesis paves a new way for the study of wave propagation and aeroacoustics.
引文
[1]J. W. S. Baron Rayleigh, The Theory of Sound (Macmillan, London,1877-1878); 2nd ed. (1894); repr. (Dover, New York,1945).
[2]居鸿宾,钟芳源,沈孟育.涡、声干扰研究的某些进展.力学进展,1997,27(3)：358～371.
[3]Xiaoxian Chen, Xun Huang, Xin Zhang, Sound Radiation from a Bypass Duct with Bifurcations, AIAA J.,2009,47(2):429～436.
[4]Delfs, J. and H. Heller, Aeroacoustics Research in Europe-1996 Highlights. Aerospace Science and Technology,1997,204(4):609～622.
[5]Rienstra, S.W., Aeroacoustics Research in Europe:the Ceas-asc Report on 1997 Highlights. Journal of Sound and Vibration,1998.214(1):139～164.
[6]Juve, D., Aeroacoustic research in europe: The CEAS-ASC report on 1998 highlights. Journal of Sound and Vibration,1999.227(2):321～342.
[7]Boden, H. and S. Sarin, Aeroacoustic research in Europe:the CEAS-ASC report on 1999 highlights. Journal of Sound and Vibration,2000.237(3):477～482.
[8]Ianniello, S., Aeroacoustic research in Europe: The CEAS-ASC report on 2000 highlights. Journal of Sound and Vibration,2002.249(2):351～370.
[9]Fisher, M.J. and R.H. Self, Aeroacoustics research in Europe:The CEAS-ASC report on 2001 highlights. Journal of Sound and Vibration,2002.258(1):1～30.
[10]Campos, E., Aeroacoustics research in Europe:the CEAS-ASC report on 2002 highlights. Journal of Sound and Vibration,2003.268(4):809～824.
[11]Schroder, W., Aeroacoustics research in Europe:The CEAS-ASC report on 2003 highlights. Journal of Sound and Vibration,2004.278(1-2):1～19.
[12]Fitzpatrick, J.A., Aeroacoustics research in Europe: The CEAS-ASC report on 2004 highlights. Journal of Sound and Vibration,2005.288(1-2):1～32.
[13]Voutsinas, S.G., Aeroacoustics research in Europe:The CEAS-ASC Report on 2005 highlights. Journal of Sound and Vibration,2007.299(3):419～459.
[14]Brouwer, H.H. and S.W. Rienstra, Aeroacoustics research in Europe:The CEAS-ASC report on 2007 highlights. Journal of Sound and Vibration,2008.318(4-5): 625～654.
[15]Ventres, C.S., A Computer Program to Calculate Cascade 2D Kernel. NASA Technical Memorandum,1980.
[16]Montgomery, M.D. and J.M. Verdon, A three-dimensional linearized unsteady Euler analysis for turbomachinery blade rows. NASA CR-4770,1997.
[17]Lighthill, M.J., Jet noise. AIAA Journal,2003.41(7):282-292.
[18]Engblom, W.A., A. Khavaran, and J. Bridges, Numerical prediction of chevron nozzle noise reduction using WIND-MGBK methodology. AIAA paper,2004.2004～2979.
[19]Bogey, C. and C. Bailly, Effects of inflow conditions and forcing on subsonic jet flows and noise. AIAA journal,2005.43(5):1000～1007.
[20]Crichton, D., et al., Design and Operation for ultra low noise takeoff. AIAA,2007. 456:45.
[21]Hernandez-Cruz, G., et al., Noise-induced fluctuations of period lengths of stable periodic orbits. Physical Review E,2003.67(036210):1-5.
[22]Leng, X.L. Numerical Analysis of Bifurcation and Chaos Response in a Cracked Rotor System Under White Noise Disturbance. Springer,2007,77～86.
[23]M.James Lighthill. On sound generated aerodynamically I. General theory. Proceedings of the Royal Society A,1952,211:564～587.
[24]A. Powell, Theory of Vortex Sound, Journal of the Acoustic Society of America, 1964,36(1):177～195.
[25]Howe, M.S. The Theory of Vortex Sound. Cambridge: Cambridge University Press, 2003.
[26]高执棣,郭国霖.统计热力学导论.北京：北京大学出版社,2004.
[27]K.Feng,M.Z.Qin, Symplectic Geometric Algorithms for Hamiltonian System. Science and Technology Press, zhe jiang,2003.
[28]N. Curle. The Inuence of Solid Boundaries upon Aerodynamic Sound. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,1955, 231:505-514.
[29]Ffowcs Williams J. E., Hawkings D.L.. Sound generation by turbulence and surfaces in arbitrary motion. Philosophical Transactions of the Royal Society of London, 1969,No.1151 264:321-342.
[30]M. E. Goldstein and B. Rosenbaum. Effect of Anisotropic Turbulence on Aerodynamic Noise. Journal of the Acoustical Society of America,1973,54:630～ 645.
[31]Goldstein, M. E., Aeroacoustics, McGraw-Hill Book Co.,1976.
[32]Farassat,F.,Theory of noise generation from moving bodies with an application to helicopter rotors. NASA TR R-451,1975.
[33]Lilley, G. M., "On the noise from jets," Tech. Rep. CP-131,AGARD,1974
[34]F. Farassat, Theory of noise generation from moving bodies with an application to helicopter rotors, NASA TR R-451,1975.
[35]F. Farassat, R. J. Pegg, D. A. Hilton, Thickness noise of helicopter rotors at high tip speeds,1975,AIAA Paper:75～453.
[36]F. Farassat, Linear acoustic formulas for calculation of rotating blade noise,AIAA Journal,1981,19(9):1122～1130.
[37]Lowson M.V., The sound field for singulantities in motion. Proeeedings of the Royal Soeiety of London.Series A,1965,286:559～572.
[38]Lowson M.V, Theoretical study of helicoPter rotor noise.Journal of Sound and Vibration,1969,9:197～222.
[39]Lowson M.V, Theoretical analysis of compressor noise. The Journal of the Acoustical Society of Ameriea,1970,47:371～385.
[40]Hanson D.B., Unified analysis of fan stator noise, The Journal of the Acoustical Society of Ameriea,1973,54:1571～1591.
[41]Hanson D.B., Spectrum of rotor noise caused by inlet guide vane wakes.The Journal of the Acoustical Society of Anlerica,1974,55:1247～1251.
[42]Khavaran, A., "Application of Geometrical Acoustics to Propagation of High Frequency Jet Noise," Computational Aeroacoustics, (J. C. Harding & M. Y. Hussaini, eds.), Springer-Verlag,1993:456～480.
[43]Kim, C. M., Khavaran, A., Krejsa, E. A., "The Significance of Shock Structures on Supersonic Jet Mixing Noise of Axisymmetric Nozzles," AIAA Journal,1994,Vol.32, No.9:1920～1923.
[44]Khavaran, A., Krejsa, E. A., Kim, C. M., "Computation of Supersonic Jet Mixing Noise for an Axisymmetric Convergent-Divergent Nozzle," AIAA Journal of Aircraft, 1994, Vol.31, No.3:603～609.
[45]Das, I. S., Khavaran, A., Krejsa, E. A., "A Computational Study of a Contoured Plug-Nozzle Jet Noise," J. Sound and Vibration,1997, Vol.206(2):169～194.
[46]Khavaran, A., "Role of Anisotropy in Turbulent Mixing Noise," AIAA Journal,1999, Vol.37(7),832～841.
[47]Wundrow,D.W.,Khavaran, A., "On the applicability of High-Frequency Approxima- tion to Lilley's Equation," J. Sound and Vibration,2004, Vol.272:793-830.
[48]Koch, L. D., Bridges, J. E., Brown, C., Khavaran, A., "Experimental and Analytical Determination of the Geometric Far-field for Round Jets," Noise Control Engineering Journal,2005,53(1):20～28.
[49]Khavaran, A., Bridges, J. E., "Modeling of Fine-Scale Turbulence Noise," J. Sound and Vibration,2005,Vol.279:1131～1154.
[50]Khavaran, A., Kenzakowski, D. C., Mielke-Fagan, A. F., "Hot Jets and Sources of Jet Noise," International J. Aeroacoustics, June 2010, Vol.9(4,5):491～532.
[51]Bechara, W., Bailly, C., Lafon, P.& Candel, S., Stochastic approach to noise modeling for free turbulent flows,1994, AIAA Journal,32(3):455～463.
[52]Bailly C. and Juve D., "A Stochastic Approach to Compute Subsonic Noise Using Linearized Euler's Equations," AIAA-paper,1999,496～506.
[53]Bailly C., Bogey C., and "Computation of Flow Noise Using Source Terms in Linearized Euler's Equations,"AIAA-paper,2000,2000～2047.
[54]Bogey, C., Bailly, C.& Juve, D., Computation of flow noise using source terms in linearized Euler's equations,2002, AIAA Journal,40(2):235～243.
[55]Bogey, C., Gloerfelt, X.& Bailly, C., An illustration of the inclusion of sound-flow interactions in Lighthill's equation,2003, AIAA Journal,41(8):1604～1606.
[56]Bailly, C.& Bogey, C., Contributions of computational aeroacoustics to jet noise research and prediction, International Journal of Computational Fluid Dynamics,2004, 18(6):481～491
[57]Bailly, C., Bogey, C. and Gloerfelt, X., Some useful hybrid approaches for predicting aerodynamic noise, C. R. Me., Acad. Sci. Paris,2005,333(9):666～675.
[58]Bogey, C.& Bailly, An analysis of the correlations between the turbulent flow and the sound pressure field of turbulent jets, J. Fluid Mech.2007,583,71～97.
[59]T. Colonius, S.K. Lele, and P. Moin, Direct computation of the sound generated by a two-dimensional shear layer,1993, AIAA Paper,93-4328.
[60]T. Colonius, S.K. Lele, and P. Moin, The Scattering of Sound Waves by a Vortex: Numerical Simulations and Analytical Solutions, Journal of Fluid Mechanics,1994, 260:271～298.
[61]T. Colonius, S.K. Lele, and P. Moin, The sound generated by a two-dimensional shear layer:A comparison of direct computations and acoustic analogies,CEAS/AIAA Paper,1995,95-036.
[62]T. Colonius, S.K. Lele, and P. Moin, The sound generated by a two-dimensional shear layer: The far field directivity from computations and acoustic analogies, Computational Aeroacoustics, volume 219, ASME,1995,219:47～52.
[63]T. Colonius, Aeroacoustics, Aerospace America,1995,33(12):8～8.
[64]T. Colonius, S.K. Lele, and P. Moin, Sound generation in a mixing layer, Journal of Fluid Mechanics,1997,330:375～409.
[65]T. Colonius,Direct Numerical Simulation of Sound Generation in Turbulent Shear Flows,J. Acoust. Soc.Am.,1999,105,Issue 2,1007～1007.
[66]T. Colonius, A.J. Basu, and C.W. Rowley,Computation of sound generation and flow/acoustic instabilities in the flow past an open cavity, Third ASME/JSME Joint Fluids Engineering Conference, number FEDSM99-7228. ASME,1999.
[67]T. Colonius, Computation of the sources of sound in turbulent flow,J. Acoust. Soc. Am.,2001,109, Issue 5,2488～2488.
[68]H. Ran and T. Colonius, Numerical Simulation of Sound Radiated from a Turbulent Vortex Ring,2004, AIAA Paper.
[69]T. Colonius, Computational Aeroacoustics:overview and numerical methods, Computational Aeroacoustics, Lecture Series 2006-05. von Karman Institute for Fluid Dynamics,2001.
[70]T. Colonius and S.K. Lele.,Computational Aeroacoustics:Progress in Nonlinear Problems of Sound Generation, Progress in Aerospace Sciences,2004, 40(6):345～416.
[71]吴介之.波涡相互作用研究的某些进展(Ⅰ).力学进展,1991,21(4)：430-443.
[72]吴介之.波涡相互作用研究的某些进展(Ⅱ).力学进展,1992,22(1)：35～45.
[73]Lighthill M. J., On sound generated aerodynamically II. turbulenceas a source of sound. Proc R Soc,1954, A222:1～32.
[74]Atassi H. M., Grzedzinski J., Unsteady disturbances of streaming motions around bodies. J Fluid Mech,1989,209:385 ～403.
[75]Fang J., Atassi H. M., Numerical solutions for unsteady subsonic vortical flows around loaded cascades. ASM E J Turbomachinery,1993,115:810～816.
[76]Atassi H. M, Fang J, Patrick S. Direct calculation of sound radiated from bodies in nonuniform flows. ASM E J Fluids Engineering,1993,115:573～579.
[77]Caruthers J. E., Dalton W. W., Unsteady aerodynamic response of a cascade to nonuniform in flow. ASM E J Turbomachi nery,1993,115:76～84.
[78]Blade W. K., Mechanics of Flow-induced Sound and Vibration, Volume I. General Concepts and Elementary Sources. Florida: Academic Press,1986.421.
[79]Blade W. K., Mechanics of Flow-induced Sound and Vibration, Volume II. Complex Flow-Structure Interactions. Florida:A2 cademic Press,1986,953.
[80]Zaman K. B. M. Q.,EfFect of acoustic excitation on stalled flows over an airfoil. A IAA J,1992,30 (6):1492～1499.
[81]Balsa T.F., On the spatial instability of piecewise linear free shear layers. J. Fluid Mech.,1987,174:553～558.
[82]Proudman I. The generation of noise by isotropic turbulence. Proc Roy Soc L ondon, Ser A,1952,214:119.
[83]Lighthill M. J., An estimate of the covariance of Txx without using statistical assumptions. In:Hardin J C, Hussaini M Y (eds),Computational Aeroacoustics, New York:Springer-Verlag,1993.
[84]Lilley G. M., The radiated noise from isotropic turbulence revisited. Theoret Comput Fl uid Dynamics,1994,6:281.
[85]Sarkar S., Hussaini M. Y., Computation of the sound generated by isotropic turbulence. ICASE Rept 93-74.1993.
[86]Zhou Y, Proskovsky A.,Oncley S., On the Lighthill relationship and sound generation from isotropic turbulence. Theoret Comput Fluid Dynamics,1995,7:355～361.
[87]Moyal J. E., The Spectra of turbulence in a compressible fluid, eddy turbulence and random noise. Math Proc Cambridge Philos Soc,1952,48 (1):329
[88]Tokunaga H., Tatsumi T., Interaction of plane nonlinear waves in a compressible fluid and two-dimensional shock-turbulence. J Phys Soc Japan,1975,38:1167.
[89]Doak P. E., Analysis of internally generated sound in continuous materials:The momentum potential field description of fluctuating fluid motions as a basis for unified theory of internally generated sound. J Sound & Vib,1973,26 (1):91～120.
[90]Li, X.M., R.C.K. Leung, and R.M.C. So, One-step aeroacoustics simulation using lattice Boltzmann method. AIAA journal,2006.44(1):78～89.
[91]Crighton D G. Basic principles of aerodynamic noise generation. Progress in Aerospace Sciences,1975,16:31-96.
[92]Hardin J C, Ristorcelli J, Tam C K W, Eds. ICASE/LaRC workshop on benchmark problems in computational aeroacoustics (CAA)[R]. NASA-CP-3300,1995.
[93]Leung,R.C.K.,So,R.M.C.,Kam,E.W.S. et al. Attempt to calculate acosutic directivety using LBM. AIAA 2006-2574.
[94]黄雪芬,薛永飞,涂运冲,吴克启.基于路径积分的声传播微观特性的研究.工程热物理学报.2012,33(4)：603～606
[95]涂运冲谢军龙薛永飞王嘉冰吴克启.基于群论及谐振子模型的声传播方法的研究,工程热物理学报,2013,34(3)：450～453.
[96]Marilyn,J.S.,Robert,W.S. Extension of CFD Techniques to computational aeroacoustics (CAA): a comparative evaluation.31st AIAA Aerospace Sciences Meeting and Exhibit.1993,1.
[97]Lele, S.K. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics,1992,103:16～42.
[98]Tam, C.K.W., Webb, J.C. Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics. Journal of Computational Physics.1993,107:262～281.
[99]Hirsh, R. Higher order accurate difference solutions of fluid mechanics problewms by a compact differencing technique. J. Comput. Phys.,1975,19:90～109.
[100]Adams, N., Shariff, K. A high resolution hybrid compact-ENO scheme for shock turbulence interaction problems, J. Comput. Phys.,1996,127:27-51.
[101]Mahesh, K. A Family of High Order Finite Difference Schemes with Good Spectral Resolution, J. Comput. Phys.,1998,145:332～358.
[102]Visbal, M. R., Rizzetta, D. P. Large-Eddy Simulation on General Geometries Using Compact Differencing and Filtering Schemes. AIAA Aerospace Sciences Meeting & Exhibit,40th, Reno, NV; UNITED STATES,14-17 Jan.2002.
[103]Kim, J. W., Lee, D. J. Optimized Compact Finite Difference Schemes with Maximum Resolution. AIAA Journal,1996,34(5):887～893.
[104]傅德薰,马延文.高精度差分格式及多尺度流场特性的数值模拟.空气动力学学报,1998,16(1)：24-35
[105]高慧,马延文,傅德薰.六阶精度的群速度直接控制紧致格式及其应用.航空动力学报,2003,18(1)：24～31.
[106]A. Uzun, G.A. Blaisdell, A.S. Lyrintzis, Application of compact schemes to large eddy simulation of turbulent jets, Journal of Scientific Computing,2004,21 (3): 283～319.
[107]S. C. Chang and W. M. To, A New Numerical Framework for Solving Conservation Laws-The Method of Space-Time Conservation Element and Solution Element, NASA TM 104495 (NASA, August 1991).
[108]刘凯欣,王景焘,张德良.时-空守恒元解元(CE/SE)方法简述.第三届全国计算 爆炸力学会议,2005,21～28.
[109]Chang, S. C. The method of space-time conservation element and solution element-a new approach for solving the Navier Stokes and Euler equations. Computational Physics,1995,119:295～324.
[110]Wang, X.Y., "Computational Fluid Dynamics based on the Method of Space-Time Conservation Element and Solution Element," Ph.D. Dissertation, Univ. of Colorado at Boulder,1995.
[111]Zhang, Z.C., "A Modified Space-Time Conservation Element and Solution Element Schemes," Report at the Beijing Workshop on CFD (in Chinese), 1996.10(8):50～54.
[112]Zhang, Z.C., Shen, M.Y., "New Approach to Obtain Space-Time conservation Schemes," Chinese J. of Aeronautics,1997,10(2):87～90
[113]Chang, S.C., Himansu, A., Loh, C.Y., Wang,X.Y, Yu S.T., Jorgenson, P., "Robust and Simple Nonreflecting Boundary Conditions for the Space-Time Conservation Element and Solution Element Method," AIAA Paper 97-2077, the 13th AIAA CFD Conference, June 1997, Snow Mass, CO.
[114]Zhang, Z.C., Shen, M.Y., "Improved Scheme of Space-Time Conservation Element and Solution Element," J. of Tsinghua University (Sci & Tech),1997, (in Chinese), 37(8):65-68.
[115]Zhang, Z.C., "Modified Space-Time Method for Solving Conservation Laws," Chinese J. of Computational Mechanics (in Chinese),1997,,14 (special issue):741-744.
[116]Zhang, Z.C., "A New General Space-Time Conservation Scheme for 2D Euler Equations," Chinese J. of Computational Mechanics (in Chinese),14(3),1997, 377～381.
[117]Zhang, Z.C., Shen, M.Y. and Li, H.D., "Modified Space-Time Conservation Schemes for 2D Euler Equations," 7th Inter. Symposium on CFD, Beijing, China,1997, 253～258,.
[118]Zhang, Z.C., Li, H.D. and Shen, M.Y, "Sapce-Time Conservation Scheme for 3D Euler Equations," 7th Inter. Symposium on CFD, Beijing, China,1997,259-262.
[119]Chang, S.C., Yu, S.T., Himansu, A., Wang,X.Y, Chow, C.Y. and Loh, C.Y, "The Method of Space-Time Conservation Element andSolution Element - A New Paradigm for Numerical Solution of Conservation Laws,"Computational Fluid Dynamics Review,1998, Vol.1,206-240.
[120]Zhang, Z.C. and Shen, M.Y., "Modified Space-Time Conservation Scheme for Solving Conservation Laws with Stiff Source Terms," Proceedings of ICFM-III, Beijing, China,1998,821～826.
[121]Zhang, Z.C. and Yu, S.T., "Shock Capturing without Riemann Solver-A Modified Space-Time CE/SE Method for Conservation Laws," AIAA Paper 99-0904, Jan. 1999.
[122]S. C. Chang, New Developments in the Method of Space-Time Conservation Element and Solution Element--Applications to the Euler and Navier-Stokes Equations, NASA TM 106226 (NASA, August 1993).
[123]S. C. Chang, X. Y. Wang, and C. Y. Chow, New Developments in the Method of Space-Time Conservation Element and Solution Element-Applications to Two-Dimensional Time-Marching Problems, NASA TM 106758 (NASA, December 1994).
[124]E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Second Edition. Springer, Berlin Heidelberg,1999.
[125]E. F. Toro, R. C. Millington, and L. A. M. Nejad. Towards very high order godunov schemes. In E. F. Toro, editor, Godunov Methods:Theory and Applications. Kluwer Academic Publishers,2001.
[126]E. F. Toro. Anomalies of conservative methods:Analysis, numerical evidence and possible cures. Computational Fluid Dynamics Journal,2002,11(2):128～143.
[127]E. F. Toro. Riemann solvers with evolved initial conditions. Technical Report NI05003-NPA,Isaac Newton Institute for Mathematical Sciences,2005.
[128]Z.J. Wang, Y. Liu, A high-order spectral (Finite) volume method for conservation laws on unstructured grids. AIAA Paper,2002,1～13.
[129]Z.J. Wang, Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation::Basic Formulation,Journal of Computational Physics, 2002,178(1):210～251.
[130]Z.J. Wang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids:II. Extension to two-dimensional scalar equation, Journal of Computational Physics,2002,179 (2):665～697.
[131]Z.J. Wang, Y. Liu, L. Zhang, High-order spectral volume method for 2D Euler equations Proceedings of the 16th AIAA Computational Fluid Dynamics Conference.2003.
[132]Y. Sun, Z.J. Wang, High-order spectral volume method for the Navier-Stokes equations on unstructured grids, AIAA Paper 2133,2004.
[133]Z.J. Wang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids Ⅲ: One dimensional systems and partition optimization, Journal of Scientific Computing,2004,20 (1):137～157.
[134]Z.J. Wang, L.Zhang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids Ⅳ:extension to two-dimensional systems, Journal of Computational Physics,2004,194 (2):716～741.
[135]Y. Sun, Z.J. Wang, Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids, International journal for numerical methods in fluids,2004,45 (8):819～838.
[136]Y. Sun, Z.J. Wang, Formulations and analysis of the spectral volume method for the diffusion equation, Communications in numerical methods in engineering,2004, 20 (12):927-937.
[137]Y. Sun, Z.J. Wang,Numerical Experiments of the Spectral Volume Method for Viscous Flows, Modern Physics Letters B,2005,19 (28n29):1439～1442.
[138]Z.J. Wang, Y. Liu, Extension of the spectral volume method to high-order boundary representation, Journal of Computational Physics,2006,211 (1):154～178.
[139]Z.J. Wang, Y. Liu, Y. Sun, M. Georg, A. Jameson, Spectral volume and spectral difference methods for unstructured grids, Lecture SeriesS-Von Karman Institute for Fluid Dynamics 1,10,2006.
[140]Y. Liu, M. Vinokur, Z.J. Wang, Spectral (finite) volume method for conservation laws on unstructured grids V:extension to three-dimensional systems, Journal of Computational Physics,2006,212 (2):454～472.
[141]Y. Sun, Z.J. Wang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids Ⅵ:extension to viscous flow, Journal of Computational Physics, 2006,215 (1):41～58.
[142]K. Van den Abeele, C. Lacor, Z.J.Wang, Short Note: On the connection between the spectral volume and the spectral difference method, Journal of Computational Physics,2007,227 (2):877～885.
[143]T. Haga, K. Sawada, Z.J. Wang, An implicit LU-SGS scheme for the spectral volume method on unstructured tetrahedral grids, Communications in Computational Physics,2009,6 (5):978-996.
[144]R.E. Harris, Z.J. Wang, High-order adaptive quadrature-free spectral volume method on unstructured grids, Computers & Fluids,2009,38 (10),2006～2025.
[145]R. Kannan, Z.J. Wang, The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method, Computers & Fluids,2010,39 (10), 2007～2021.
[146]Wang, Z. J., High-order spectral volume method for benchmark aeroacoustic problems.41st Aerospace Sciences Meeting and Exhibit 6-9 January, Reno Nevada AIAA-2003-0880,2003.
[147]Huang, X., Zhang, X., A Fourier Pseudospectral Method for Some Computational A-eroacoustics Problems. International Journal of Aeroacoustics,2006, 5,(3):279～294.
[148]Proudman,I.The generation of noise by isotropic turbulence. Proceedings of the Royal Society of London A, Mathematical and Physical Sciences,1952,214 (1116):119～132.
[149]Lilley, G. M., The source of aerodynamic noise, Aeroacoustics,2003,2 (3 & 4), 241～254.
[150]Lilley, G. M., The radiated noise from isotropic turbulence with applications to the theory of jet noise, J. Sound Vib.,1996,190(3),463～476.
[151]Lilley, G. M., The generation and radiation of supersonic jet noise. Vol. IV-Theory of turbulence generated jet noise, noise radiation from upstream sources, and combustion noise. Part Ⅱ:Generation of sound in a mixing region, Air Force Aero Propulsion Laboratory, AFAPL-TR-72-53,1972.
[152]Mankbadi,R.R.,Hixon,R.,Shih,S.H. et al. Use of linearized euler equations for supersonic jet noise prediction. AIAA Journal,1998,36 (2):140-147.
[153]Djambazovm,G., Lai, C. H.,Pericleous,K. Development of numerical techniques for near field aerodynamic computations. International Journal for Numerical Method in Fluids,1999,29:719～731.
[154]李晓东,高军辉.二维平行剪切层声波产生和辐射的数值模拟.航空学报,2003,24(1)：5-9.
[155]Morgans, R. P., "The Kirchhoff Formula Extended to a Moving Surface," Philosophical Magazine,1930, Vol.9 (s.7, No.55),141～161.
[156]Ffowcs Williams, J.E.& Gordon, C. G., Noise of highly turbulent jets at low exhaust speeds, AIAA Journal,1965,3(4),791～793.
[157]Ffowcs Williams, J.E., Jet noise at very low and very high speed, Afosrutias Symposium on Aerodynamic Noise, Toronto,1968,131～145.
[158]Ffowes Williams, J. E., Hawkings, D. L.,Sound generation by turbulence and Surface in Arbitrary Motion," Philosophical. Transaction of the Royal SocietyLond A,1969, 264:321～342.
[159]Ffowes Williams, J. E., Hawkings, D. L., "Theory Relating to the Noise of Rotating Machinery,"J. Sound and Vib.,1969, Vol.10, No.1,10～21.
[160]Ffowes Williams, J. E., Hall, L. H., "Aerodynamic Sound Generation by Turbulent Flow in the Vicinity of a Scattered Half Plane,"J. Fluid Mech.,1970, Vol.40, No.4, pp.657～670.
[161]Ffowcs Williams, J. E.& Howe, M. S., The generation of sound by density inhomogeneities in low Mach number nozzle flows, J. Fluid Mech.,1975,70(3), 605～622.
[162]Ffowcs Williams, J. E., Sound sources in aerodynamics Fact and fiction, AIAA Journal,1982,20(3),307～315.
[163]Farassat, F. and Myers, M. K., "Extension of Kirchhoff s Formula to Radiation from Moving Surfaces," Journal of Sound and Vibration,451～460, Vol.123, No.3, June 1988.
[164]A. S. Lyrintzis Review:The use of Kirchhoffs method in computational aeroacoustics Journal of fluids engineering,1994,116 (4),665-676.
[165]Y. Xue, A.S. Lyrintzis, Rotating Kirchhoff method for three-dimensional transonic blade-vortex interaction hover noise, AIAA journal,1994,32 (7),1350-1359.
[166]A.S. Lyrintzis, R. Mankbadi, Prediction of the far-field jet noise using Kirchhoffs formulation AIAA journal,1996,34 (2),413～416.
[167]R.C. STRAWN, R. Biswas, A.S. Lyrintzis, Helicopter noise predictions using Kirchhoff methods, Journal of Computational Acoustics,1996,4 (03),321～339.
[168]A.R. Pilon, A.S. Lyrintzis, Integral methods for computational aeroacoustics, AIAA paper, AIAA paper,20,1997.
[169]A.R. Pilon, A.S. Lyrintzis, Development of an improved Kirchhoff method for jet aeroacoustics, AIAA journal,199836 (5),783～790.
[170]A.S. Lyrintzis, Surface integral methods in computational aeroacoustics-From the (CFD) near-field to the (Acoustic) far-field International journal of aeroacoustics,2003,2 (2),95～128.
[171]A. Uzun, AS Lyrintzis, GA Blaisdell, Coupling of integral acoustics methods with LES for jet noise prediction, International Journal of Aeroacoustics,2004,3(4), 297～346.
[172]Holevo, A.S., Bounds for the quantity of information transmitted by a quantum communication channel, Problemy Peredachi Informatsii,1973,9(3):3～11.
[173]Ingarden, R.S., Quantum information theory, Rep. Math. Phys.,1976,10:43～72.
[174]Manin, Y., Computable and Uncomputable, Moscow: Sovetskoye Radio,1980.
[175]Poplavskii, R.P, Thermodynamical models of information processing, (in Russian). Uspekhi Fizicheskikh Nauk,1975,115(3):465～501.
[176]Wiesner, S., Conjugate coding, Sigact news,1983,18:78～88.
[177]David Deutsch, Quantum theory, the Church-Turingprinciple and the universal quantum computer, Proc. R. Soc. Lond.
[178]Peter Shor, Algorithms for Quantum Computation:Discrete Logarithms and Factoring, IEEE Symposium on Foundations of Computer Science,1994:124-134.
[179]冯康,秦孟兆.Hamilton系统的辛几何算法.杭州：浙江科学技术出版社,2003.
[180]Feng K. On Difference Schemes and Symplectic Geometry. Proceedings of the 5th international symposium on differential geometry and differential equations, Beijing, 1984.
[181]Feng K, Wu H M, Qin M Z. Symplectic Difference Schemes for Hamiltonian Systems in General Symplectic Structures. Journal of Computation and Mathematics, 1991,9(1):86～96.
[182]Qin M Z. Canonical Difference Schemes for the Hamiltonian Equation. Mathematical Methods in the Applied Sciences,1989,11(3):543～557.
[183]Feng K, Qin M Z. Hamiltonian Algorithms for Hamiltonian Systems and a Comparative Numerical Study. Computer Physics Communications,1991, 65(1-3):173～187.
[184]Feng K, Wang D L. A Note on Conservation Laws of Symplectic Difference Schemes for Hamiltonian Systems. Journal of Computation and Mathematics,1991, 9(3):229-237.
[185]Feng K. The Step-transition Operator for Multi-step Methods of ODEs. Journal of Computation and Mathematics,1998,16(3):193～202.
[186]Arnold V I. Mathematical Methods of Classical Mechanics. New York:Springer-Verlag,1989.
[187]Hairer E, Lubich C, Wanner G Geometric Numerical Integration: Structure-Preserving Algorithm for Ordinary Differential Equations.2ed. New York:Springer, 2006.
[188]姚伟岸,钟万勰.辛弹性力学.北京：高等教育出版社,2002.
[189]U. Frish, B. Hassalacher, Y. Pomeau, Lattice Gas automata for the Navier-Stokes Equation,Phys.Rev.Lett.,1986,56(14):1505～1508.
[190]M.Henon, Implementation of the FCHC Lattice Gas Model on the Connection Machine, J.Stat.Phys.,1992,68:353-377.
[191]G.R.McNamara and Zanetti G., Use of Boltzmann Equation to Simulate Lattice Gas Automata, Phys.Rev.Lett.,1988,61(20):2332～2335.
[192]Z. L. Guo, C. G Zheng, A coupled LBGK model for the Boussinesq equations, Int. J. Num. Meth. Fluids.,2002,39:325～342.
[193]M. A. Moussaoui, M. Jami, A. Mezrhab, H. Naji, MRT-Lattice Boltzmann simulationof forced convection in a plane channel with an inclined square cylinder, Int. J. Therm. Sci.,2010,49(1):131～142.
[194]Y.Qian, D.d'Humieres, P.Lallemand, LBGK models for N-S equations, Europhys.Lett.,1992,17:479～484
[195]Q. Zou, S. Hou, S. Chen, G. D. Doolen, An improved incompressible lattice Boltzmann model for time-independent, J.Stat.Phys.,1995,81:35-48.
[196]Y. Chen, H. Ohashi, LBGK methods for simulating incompressible fluid flows,Int.J.Mod.Phys.C.,1997,8:793～803.
[197]X.He, L.S.Luo, LBM for the incompressible Navier-Stokes equation, J.Stat.Phys., 1997,88(3-4):927-944.
[198]Guo Z.L., B.C.Shi, N.C.Wang, Lattice BGK model for incompressible Navier-Stokes equation, J.Comput.Phys.,2000,165:288～306.
[199]Q. S. Zou, X. Y. He, On Pressure and veloeity boundary conditions for the lattiee Boltzmann BGK model, Phys.Fluids,1997, vol.9, No.6:1591～1598.
[200]E. Laurendeau, J. P. Bonnet, P. Jordan, and J. Delville. "Impact of fluidic chevrons on the turbulence structure of a subsonic jet," AIAA Paper No.2006-3510
[201]米建春,冯宝平,Deo Ravinesh C, etc.出口雷诺数对平面射流自保持性的影响.物理学报,2009,58(11)：7756～7764
[202]黄昆,韩汝琦.固体物理学.北京：高等教育出版社,1988(10)：78-152.
[203]吴克启.宏观气动声场量子分子动力学讲义,2010
[204]John F.. Finite amplitude waves in a homogeneous isotropic elastic solid. Commun. Pure Appl. Math 1977; 30:421～446.
[205]Feng K, Qin M.Z. Sympleetic geometric algorithms for Hamiltonian systems. Science and Technology Press, zhe jiang,2003.
[206]Peter M.The lattice Boltzmann phononic lattice solid. Journal of Statistical Physics 1992; 68:591～609.
[207]Struwe M. Variational methods:Applications to nonlinear partial differential equations and Hamiltonian systems. Berlin, New-York:Springer Verlag; 1990.
[208]Sun G. A class of explicitly symplectic schemes for waveequations. Comput.Math.(in Chinese) 1997; 1:1-10.
[209]Zhai, P.W., Kattawar, G.W., Yang P., Li C.H. Application of the symplectic finite-difference time-domain method to light scattering by small particles, Applied Optics 2005; 44:1650～1656.
[210]Sha, W., Huang Z.X., Wu X.L., Chen M.S. Application of the symplectic finite difference time-domain scheme to electromagnetic simulation, Journal of Computational Physics,2007; 225:33～50.
[211]Feng, K. Collected works of Feng Kang. Beijing:Science Press; 1985.
[212]Yefet A., Petropoulos P.G. A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. J. Comput. Phys.2001 April; 168: 286～315.
[213]Joe L., Doru M., Gilbert S. Modeling of the seat dip effect using the finite-difference time-domain method. J.Acoust. Soc. Am.1996; 100(4):2204～2212.
[214]Botteldooren Finite-difference time-domain simulation of low frequency room acoustic problems. J.Acoust. Soc. Am.1995; 98(6):3302～3308.
[215]罗民秋,刘洪,李幼铭,韩汝琦.地震波传播的哈密顿表述及辛几何算法.地球物理学报,2001,44(1)：120～128
[216]Zeng J Y. Quantum Mechanics, volume II (3rd edition). Beijing:Science Press.2000.
[217]Duan L M,Guo G G, Energy Level Excitations--the Elementary Excitations in the Crystal-optical-field Strong Coupled System. Acta Physica Sinica,1997,46, (6):1114～1117
[218]Li X D, Gao J H, Numerical Simulation of Sound Generation and Radiation from a 2D Parallel Shear Layer, Acta Aeronautica et Astronaut Ica Sinica,
[219]Xie X D, Jiang P, Lu F. Group Theory and its Application to Physics. Science Press, 1986.
[220]Zhou G D, Duan L Y.Bases of Structural Chemistry, (2rd edition). Peking University Press,1995:26～39
[221]Yan J. The Energy Transfer Between Two Diatomc Molecules in Collinear Collisions. Jinan:Shangdong University,2007
[222]Shen J, Sha W, Huang Z X, Chen MS, Wu X L,High-oder Symplectic FDTD Scheme for Solving Time-dependent SchrSdinger Equation, Acta Physica Sinica, 2012,61(19):190202-1-190202-7
[223]Wang S J, Zhang H. The Fidelity and Algebraic Dynamics Algorithm of Physical Computing:I Algebraic Dynamics Solvings and Algebraic Dynamics Algorithm of the Dynamic System. Science in China Series G:Physcis,Mechanics & Astronomy, 2005,35(6):573----608
[224]Liu J J,Zhai L X,Using the Finite Difference Mmethod to Solve the Energy Eigenvalue Equation. Journal of Beijing Univesity of Technology,2008, (34): 325-328.
[225]Cheng X P,The Direct Perturbation Method and Approximate Solutions of Nonlinear Schrodinger Equation Hierarchy, zhejiang:zhejiang Normal University,2006.
[226]Kosloff R, Rice S A, Gaspars P, et al. Wavepacket Dancing:Achieving Chemical Selectivity by Shaping Light Pulses.Chemical Physics,1989,139(1):201～220.
[227]Feynman R P, Hibbs A R(Write).ZHANG Banggu (Translation). Quantum mechanics and path integrals.Beijing:Science Press.1986.
[2]居鸿宾,钟芳源,沈孟育.涡、声干扰研究的某些进展.力学进展,1997,27(3)：358～371.
[3]Xiaoxian Chen, Xun Huang, Xin Zhang, Sound Radiation from a Bypass Duct with Bifurcations, AIAA J.,2009,47(2):429～436.
[4]Delfs, J. and H. Heller, Aeroacoustics Research in Europe-1996 Highlights. Aerospace Science and Technology,1997,204(4):609～622.
[5]Rienstra, S.W., Aeroacoustics Research in Europe:the Ceas-asc Report on 1997 Highlights. Journal of Sound and Vibration,1998.214(1):139～164.
[6]Juve, D., Aeroacoustic research in europe: The CEAS-ASC report on 1998 highlights. Journal of Sound and Vibration,1999.227(2):321～342.
[7]Boden, H. and S. Sarin, Aeroacoustic research in Europe:the CEAS-ASC report on 1999 highlights. Journal of Sound and Vibration,2000.237(3):477～482.
[8]Ianniello, S., Aeroacoustic research in Europe: The CEAS-ASC report on 2000 highlights. Journal of Sound and Vibration,2002.249(2):351～370.
[9]Fisher, M.J. and R.H. Self, Aeroacoustics research in Europe:The CEAS-ASC report on 2001 highlights. Journal of Sound and Vibration,2002.258(1):1～30.
[10]Campos, E., Aeroacoustics research in Europe:the CEAS-ASC report on 2002 highlights. Journal of Sound and Vibration,2003.268(4):809～824.
[11]Schroder, W., Aeroacoustics research in Europe:The CEAS-ASC report on 2003 highlights. Journal of Sound and Vibration,2004.278(1-2):1～19.
[12]Fitzpatrick, J.A., Aeroacoustics research in Europe: The CEAS-ASC report on 2004 highlights. Journal of Sound and Vibration,2005.288(1-2):1～32.
[13]Voutsinas, S.G., Aeroacoustics research in Europe:The CEAS-ASC Report on 2005 highlights. Journal of Sound and Vibration,2007.299(3):419～459.
[14]Brouwer, H.H. and S.W. Rienstra, Aeroacoustics research in Europe:The CEAS-ASC report on 2007 highlights. Journal of Sound and Vibration,2008.318(4-5): 625～654.
[15]Ventres, C.S., A Computer Program to Calculate Cascade 2D Kernel. NASA Technical Memorandum,1980.
[16]Montgomery, M.D. and J.M. Verdon, A three-dimensional linearized unsteady Euler analysis for turbomachinery blade rows. NASA CR-4770,1997.
[17]Lighthill, M.J., Jet noise. AIAA Journal,2003.41(7):282-292.
[18]Engblom, W.A., A. Khavaran, and J. Bridges, Numerical prediction of chevron nozzle noise reduction using WIND-MGBK methodology. AIAA paper,2004.2004～2979.
[19]Bogey, C. and C. Bailly, Effects of inflow conditions and forcing on subsonic jet flows and noise. AIAA journal,2005.43(5):1000～1007.
[20]Crichton, D., et al., Design and Operation for ultra low noise takeoff. AIAA,2007. 456:45.
[21]Hernandez-Cruz, G., et al., Noise-induced fluctuations of period lengths of stable periodic orbits. Physical Review E,2003.67(036210):1-5.
[22]Leng, X.L. Numerical Analysis of Bifurcation and Chaos Response in a Cracked Rotor System Under White Noise Disturbance. Springer,2007,77～86.
[23]M.James Lighthill. On sound generated aerodynamically I. General theory. Proceedings of the Royal Society A,1952,211:564～587.
[24]A. Powell, Theory of Vortex Sound, Journal of the Acoustic Society of America, 1964,36(1):177～195.
[25]Howe, M.S. The Theory of Vortex Sound. Cambridge: Cambridge University Press, 2003.
[26]高执棣,郭国霖.统计热力学导论.北京：北京大学出版社,2004.
[27]K.Feng,M.Z.Qin, Symplectic Geometric Algorithms for Hamiltonian System. Science and Technology Press, zhe jiang,2003.
[28]N. Curle. The Inuence of Solid Boundaries upon Aerodynamic Sound. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,1955, 231:505-514.
[29]Ffowcs Williams J. E., Hawkings D.L.. Sound generation by turbulence and surfaces in arbitrary motion. Philosophical Transactions of the Royal Society of London, 1969,No.1151 264:321-342.
[30]M. E. Goldstein and B. Rosenbaum. Effect of Anisotropic Turbulence on Aerodynamic Noise. Journal of the Acoustical Society of America,1973,54:630～ 645.
[31]Goldstein, M. E., Aeroacoustics, McGraw-Hill Book Co.,1976.
[32]Farassat,F.,Theory of noise generation from moving bodies with an application to helicopter rotors. NASA TR R-451,1975.
[33]Lilley, G. M., "On the noise from jets," Tech. Rep. CP-131,AGARD,1974
[34]F. Farassat, Theory of noise generation from moving bodies with an application to helicopter rotors, NASA TR R-451,1975.
[35]F. Farassat, R. J. Pegg, D. A. Hilton, Thickness noise of helicopter rotors at high tip speeds,1975,AIAA Paper:75～453.
[36]F. Farassat, Linear acoustic formulas for calculation of rotating blade noise,AIAA Journal,1981,19(9):1122～1130.
[37]Lowson M.V., The sound field for singulantities in motion. Proeeedings of the Royal Soeiety of London.Series A,1965,286:559～572.
[38]Lowson M.V, Theoretical study of helicoPter rotor noise.Journal of Sound and Vibration,1969,9:197～222.
[39]Lowson M.V, Theoretical analysis of compressor noise. The Journal of the Acoustical Society of Ameriea,1970,47:371～385.
[40]Hanson D.B., Unified analysis of fan stator noise, The Journal of the Acoustical Society of Ameriea,1973,54:1571～1591.
[41]Hanson D.B., Spectrum of rotor noise caused by inlet guide vane wakes.The Journal of the Acoustical Society of Anlerica,1974,55:1247～1251.
[42]Khavaran, A., "Application of Geometrical Acoustics to Propagation of High Frequency Jet Noise," Computational Aeroacoustics, (J. C. Harding & M. Y. Hussaini, eds.), Springer-Verlag,1993:456～480.
[43]Kim, C. M., Khavaran, A., Krejsa, E. A., "The Significance of Shock Structures on Supersonic Jet Mixing Noise of Axisymmetric Nozzles," AIAA Journal,1994,Vol.32, No.9:1920～1923.
[44]Khavaran, A., Krejsa, E. A., Kim, C. M., "Computation of Supersonic Jet Mixing Noise for an Axisymmetric Convergent-Divergent Nozzle," AIAA Journal of Aircraft, 1994, Vol.31, No.3:603～609.
[45]Das, I. S., Khavaran, A., Krejsa, E. A., "A Computational Study of a Contoured Plug-Nozzle Jet Noise," J. Sound and Vibration,1997, Vol.206(2):169～194.
[46]Khavaran, A., "Role of Anisotropy in Turbulent Mixing Noise," AIAA Journal,1999, Vol.37(7),832～841.
[47]Wundrow,D.W.,Khavaran, A., "On the applicability of High-Frequency Approxima- tion to Lilley's Equation," J. Sound and Vibration,2004, Vol.272:793-830.
[48]Koch, L. D., Bridges, J. E., Brown, C., Khavaran, A., "Experimental and Analytical Determination of the Geometric Far-field for Round Jets," Noise Control Engineering Journal,2005,53(1):20～28.
[49]Khavaran, A., Bridges, J. E., "Modeling of Fine-Scale Turbulence Noise," J. Sound and Vibration,2005,Vol.279:1131～1154.
[50]Khavaran, A., Kenzakowski, D. C., Mielke-Fagan, A. F., "Hot Jets and Sources of Jet Noise," International J. Aeroacoustics, June 2010, Vol.9(4,5):491～532.
[51]Bechara, W., Bailly, C., Lafon, P.& Candel, S., Stochastic approach to noise modeling for free turbulent flows,1994, AIAA Journal,32(3):455～463.
[52]Bailly C. and Juve D., "A Stochastic Approach to Compute Subsonic Noise Using Linearized Euler's Equations," AIAA-paper,1999,496～506.
[53]Bailly C., Bogey C., and "Computation of Flow Noise Using Source Terms in Linearized Euler's Equations,"AIAA-paper,2000,2000～2047.
[54]Bogey, C., Bailly, C.& Juve, D., Computation of flow noise using source terms in linearized Euler's equations,2002, AIAA Journal,40(2):235～243.
[55]Bogey, C., Gloerfelt, X.& Bailly, C., An illustration of the inclusion of sound-flow interactions in Lighthill's equation,2003, AIAA Journal,41(8):1604～1606.
[56]Bailly, C.& Bogey, C., Contributions of computational aeroacoustics to jet noise research and prediction, International Journal of Computational Fluid Dynamics,2004, 18(6):481～491
[57]Bailly, C., Bogey, C. and Gloerfelt, X., Some useful hybrid approaches for predicting aerodynamic noise, C. R. Me., Acad. Sci. Paris,2005,333(9):666～675.
[58]Bogey, C.& Bailly, An analysis of the correlations between the turbulent flow and the sound pressure field of turbulent jets, J. Fluid Mech.2007,583,71～97.
[59]T. Colonius, S.K. Lele, and P. Moin, Direct computation of the sound generated by a two-dimensional shear layer,1993, AIAA Paper,93-4328.
[60]T. Colonius, S.K. Lele, and P. Moin, The Scattering of Sound Waves by a Vortex: Numerical Simulations and Analytical Solutions, Journal of Fluid Mechanics,1994, 260:271～298.
[61]T. Colonius, S.K. Lele, and P. Moin, The sound generated by a two-dimensional shear layer:A comparison of direct computations and acoustic analogies,CEAS/AIAA Paper,1995,95-036.
[62]T. Colonius, S.K. Lele, and P. Moin, The sound generated by a two-dimensional shear layer: The far field directivity from computations and acoustic analogies, Computational Aeroacoustics, volume 219, ASME,1995,219:47～52.
[63]T. Colonius, Aeroacoustics, Aerospace America,1995,33(12):8～8.
[64]T. Colonius, S.K. Lele, and P. Moin, Sound generation in a mixing layer, Journal of Fluid Mechanics,1997,330:375～409.
[65]T. Colonius,Direct Numerical Simulation of Sound Generation in Turbulent Shear Flows,J. Acoust. Soc.Am.,1999,105,Issue 2,1007～1007.
[66]T. Colonius, A.J. Basu, and C.W. Rowley,Computation of sound generation and flow/acoustic instabilities in the flow past an open cavity, Third ASME/JSME Joint Fluids Engineering Conference, number FEDSM99-7228. ASME,1999.
[67]T. Colonius, Computation of the sources of sound in turbulent flow,J. Acoust. Soc. Am.,2001,109, Issue 5,2488～2488.
[68]H. Ran and T. Colonius, Numerical Simulation of Sound Radiated from a Turbulent Vortex Ring,2004, AIAA Paper.
[69]T. Colonius, Computational Aeroacoustics:overview and numerical methods, Computational Aeroacoustics, Lecture Series 2006-05. von Karman Institute for Fluid Dynamics,2001.
[70]T. Colonius and S.K. Lele.,Computational Aeroacoustics:Progress in Nonlinear Problems of Sound Generation, Progress in Aerospace Sciences,2004, 40(6):345～416.
[71]吴介之.波涡相互作用研究的某些进展(Ⅰ).力学进展,1991,21(4)：430-443.
[72]吴介之.波涡相互作用研究的某些进展(Ⅱ).力学进展,1992,22(1)：35～45.
[73]Lighthill M. J., On sound generated aerodynamically II. turbulenceas a source of sound. Proc R Soc,1954, A222:1～32.
[74]Atassi H. M., Grzedzinski J., Unsteady disturbances of streaming motions around bodies. J Fluid Mech,1989,209:385 ～403.
[75]Fang J., Atassi H. M., Numerical solutions for unsteady subsonic vortical flows around loaded cascades. ASM E J Turbomachinery,1993,115:810～816.
[76]Atassi H. M, Fang J, Patrick S. Direct calculation of sound radiated from bodies in nonuniform flows. ASM E J Fluids Engineering,1993,115:573～579.
[77]Caruthers J. E., Dalton W. W., Unsteady aerodynamic response of a cascade to nonuniform in flow. ASM E J Turbomachi nery,1993,115:76～84.
[78]Blade W. K., Mechanics of Flow-induced Sound and Vibration, Volume I. General Concepts and Elementary Sources. Florida: Academic Press,1986.421.
[79]Blade W. K., Mechanics of Flow-induced Sound and Vibration, Volume II. Complex Flow-Structure Interactions. Florida:A2 cademic Press,1986,953.
[80]Zaman K. B. M. Q.,EfFect of acoustic excitation on stalled flows over an airfoil. A IAA J,1992,30 (6):1492～1499.
[81]Balsa T.F., On the spatial instability of piecewise linear free shear layers. J. Fluid Mech.,1987,174:553～558.
[82]Proudman I. The generation of noise by isotropic turbulence. Proc Roy Soc L ondon, Ser A,1952,214:119.
[83]Lighthill M. J., An estimate of the covariance of Txx without using statistical assumptions. In:Hardin J C, Hussaini M Y (eds),Computational Aeroacoustics, New York:Springer-Verlag,1993.
[84]Lilley G. M., The radiated noise from isotropic turbulence revisited. Theoret Comput Fl uid Dynamics,1994,6:281.
[85]Sarkar S., Hussaini M. Y., Computation of the sound generated by isotropic turbulence. ICASE Rept 93-74.1993.
[86]Zhou Y, Proskovsky A.,Oncley S., On the Lighthill relationship and sound generation from isotropic turbulence. Theoret Comput Fluid Dynamics,1995,7:355～361.
[87]Moyal J. E., The Spectra of turbulence in a compressible fluid, eddy turbulence and random noise. Math Proc Cambridge Philos Soc,1952,48 (1):329
[88]Tokunaga H., Tatsumi T., Interaction of plane nonlinear waves in a compressible fluid and two-dimensional shock-turbulence. J Phys Soc Japan,1975,38:1167.
[89]Doak P. E., Analysis of internally generated sound in continuous materials:The momentum potential field description of fluctuating fluid motions as a basis for unified theory of internally generated sound. J Sound & Vib,1973,26 (1):91～120.
[90]Li, X.M., R.C.K. Leung, and R.M.C. So, One-step aeroacoustics simulation using lattice Boltzmann method. AIAA journal,2006.44(1):78～89.
[91]Crighton D G. Basic principles of aerodynamic noise generation. Progress in Aerospace Sciences,1975,16:31-96.
[92]Hardin J C, Ristorcelli J, Tam C K W, Eds. ICASE/LaRC workshop on benchmark problems in computational aeroacoustics (CAA)[R]. NASA-CP-3300,1995.
[93]Leung,R.C.K.,So,R.M.C.,Kam,E.W.S. et al. Attempt to calculate acosutic directivety using LBM. AIAA 2006-2574.
[94]黄雪芬,薛永飞,涂运冲,吴克启.基于路径积分的声传播微观特性的研究.工程热物理学报.2012,33(4)：603～606
[95]涂运冲谢军龙薛永飞王嘉冰吴克启.基于群论及谐振子模型的声传播方法的研究,工程热物理学报,2013,34(3)：450～453.
[96]Marilyn,J.S.,Robert,W.S. Extension of CFD Techniques to computational aeroacoustics (CAA): a comparative evaluation.31st AIAA Aerospace Sciences Meeting and Exhibit.1993,1.
[97]Lele, S.K. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics,1992,103:16～42.
[98]Tam, C.K.W., Webb, J.C. Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics. Journal of Computational Physics.1993,107:262～281.
[99]Hirsh, R. Higher order accurate difference solutions of fluid mechanics problewms by a compact differencing technique. J. Comput. Phys.,1975,19:90～109.
[100]Adams, N., Shariff, K. A high resolution hybrid compact-ENO scheme for shock turbulence interaction problems, J. Comput. Phys.,1996,127:27-51.
[101]Mahesh, K. A Family of High Order Finite Difference Schemes with Good Spectral Resolution, J. Comput. Phys.,1998,145:332～358.
[102]Visbal, M. R., Rizzetta, D. P. Large-Eddy Simulation on General Geometries Using Compact Differencing and Filtering Schemes. AIAA Aerospace Sciences Meeting & Exhibit,40th, Reno, NV; UNITED STATES,14-17 Jan.2002.
[103]Kim, J. W., Lee, D. J. Optimized Compact Finite Difference Schemes with Maximum Resolution. AIAA Journal,1996,34(5):887～893.
[104]傅德薰,马延文.高精度差分格式及多尺度流场特性的数值模拟.空气动力学学报,1998,16(1)：24-35
[105]高慧,马延文,傅德薰.六阶精度的群速度直接控制紧致格式及其应用.航空动力学报,2003,18(1)：24～31.
[106]A. Uzun, G.A. Blaisdell, A.S. Lyrintzis, Application of compact schemes to large eddy simulation of turbulent jets, Journal of Scientific Computing,2004,21 (3): 283～319.
[107]S. C. Chang and W. M. To, A New Numerical Framework for Solving Conservation Laws-The Method of Space-Time Conservation Element and Solution Element, NASA TM 104495 (NASA, August 1991).
[108]刘凯欣,王景焘,张德良.时-空守恒元解元(CE/SE)方法简述.第三届全国计算 爆炸力学会议,2005,21～28.
[109]Chang, S. C. The method of space-time conservation element and solution element-a new approach for solving the Navier Stokes and Euler equations. Computational Physics,1995,119:295～324.
[110]Wang, X.Y., "Computational Fluid Dynamics based on the Method of Space-Time Conservation Element and Solution Element," Ph.D. Dissertation, Univ. of Colorado at Boulder,1995.
[111]Zhang, Z.C., "A Modified Space-Time Conservation Element and Solution Element Schemes," Report at the Beijing Workshop on CFD (in Chinese), 1996.10(8):50～54.
[112]Zhang, Z.C., Shen, M.Y., "New Approach to Obtain Space-Time conservation Schemes," Chinese J. of Aeronautics,1997,10(2):87～90
[113]Chang, S.C., Himansu, A., Loh, C.Y., Wang,X.Y, Yu S.T., Jorgenson, P., "Robust and Simple Nonreflecting Boundary Conditions for the Space-Time Conservation Element and Solution Element Method," AIAA Paper 97-2077, the 13th AIAA CFD Conference, June 1997, Snow Mass, CO.
[114]Zhang, Z.C., Shen, M.Y., "Improved Scheme of Space-Time Conservation Element and Solution Element," J. of Tsinghua University (Sci & Tech),1997, (in Chinese), 37(8):65-68.
[115]Zhang, Z.C., "Modified Space-Time Method for Solving Conservation Laws," Chinese J. of Computational Mechanics (in Chinese),1997,,14 (special issue):741-744.
[116]Zhang, Z.C., "A New General Space-Time Conservation Scheme for 2D Euler Equations," Chinese J. of Computational Mechanics (in Chinese),14(3),1997, 377～381.
[117]Zhang, Z.C., Shen, M.Y. and Li, H.D., "Modified Space-Time Conservation Schemes for 2D Euler Equations," 7th Inter. Symposium on CFD, Beijing, China,1997, 253～258,.
[118]Zhang, Z.C., Li, H.D. and Shen, M.Y, "Sapce-Time Conservation Scheme for 3D Euler Equations," 7th Inter. Symposium on CFD, Beijing, China,1997,259-262.
[119]Chang, S.C., Yu, S.T., Himansu, A., Wang,X.Y, Chow, C.Y. and Loh, C.Y, "The Method of Space-Time Conservation Element andSolution Element - A New Paradigm for Numerical Solution of Conservation Laws,"Computational Fluid Dynamics Review,1998, Vol.1,206-240.
[120]Zhang, Z.C. and Shen, M.Y., "Modified Space-Time Conservation Scheme for Solving Conservation Laws with Stiff Source Terms," Proceedings of ICFM-III, Beijing, China,1998,821～826.
[121]Zhang, Z.C. and Yu, S.T., "Shock Capturing without Riemann Solver-A Modified Space-Time CE/SE Method for Conservation Laws," AIAA Paper 99-0904, Jan. 1999.
[122]S. C. Chang, New Developments in the Method of Space-Time Conservation Element and Solution Element--Applications to the Euler and Navier-Stokes Equations, NASA TM 106226 (NASA, August 1993).
[123]S. C. Chang, X. Y. Wang, and C. Y. Chow, New Developments in the Method of Space-Time Conservation Element and Solution Element-Applications to Two-Dimensional Time-Marching Problems, NASA TM 106758 (NASA, December 1994).
[124]E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Second Edition. Springer, Berlin Heidelberg,1999.
[125]E. F. Toro, R. C. Millington, and L. A. M. Nejad. Towards very high order godunov schemes. In E. F. Toro, editor, Godunov Methods:Theory and Applications. Kluwer Academic Publishers,2001.
[126]E. F. Toro. Anomalies of conservative methods:Analysis, numerical evidence and possible cures. Computational Fluid Dynamics Journal,2002,11(2):128～143.
[127]E. F. Toro. Riemann solvers with evolved initial conditions. Technical Report NI05003-NPA,Isaac Newton Institute for Mathematical Sciences,2005.
[128]Z.J. Wang, Y. Liu, A high-order spectral (Finite) volume method for conservation laws on unstructured grids. AIAA Paper,2002,1～13.
[129]Z.J. Wang, Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation::Basic Formulation,Journal of Computational Physics, 2002,178(1):210～251.
[130]Z.J. Wang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids:II. Extension to two-dimensional scalar equation, Journal of Computational Physics,2002,179 (2):665～697.
[131]Z.J. Wang, Y. Liu, L. Zhang, High-order spectral volume method for 2D Euler equations Proceedings of the 16th AIAA Computational Fluid Dynamics Conference.2003.
[132]Y. Sun, Z.J. Wang, High-order spectral volume method for the Navier-Stokes equations on unstructured grids, AIAA Paper 2133,2004.
[133]Z.J. Wang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids Ⅲ: One dimensional systems and partition optimization, Journal of Scientific Computing,2004,20 (1):137～157.
[134]Z.J. Wang, L.Zhang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids Ⅳ:extension to two-dimensional systems, Journal of Computational Physics,2004,194 (2):716～741.
[135]Y. Sun, Z.J. Wang, Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids, International journal for numerical methods in fluids,2004,45 (8):819～838.
[136]Y. Sun, Z.J. Wang, Formulations and analysis of the spectral volume method for the diffusion equation, Communications in numerical methods in engineering,2004, 20 (12):927-937.
[137]Y. Sun, Z.J. Wang,Numerical Experiments of the Spectral Volume Method for Viscous Flows, Modern Physics Letters B,2005,19 (28n29):1439～1442.
[138]Z.J. Wang, Y. Liu, Extension of the spectral volume method to high-order boundary representation, Journal of Computational Physics,2006,211 (1):154～178.
[139]Z.J. Wang, Y. Liu, Y. Sun, M. Georg, A. Jameson, Spectral volume and spectral difference methods for unstructured grids, Lecture SeriesS-Von Karman Institute for Fluid Dynamics 1,10,2006.
[140]Y. Liu, M. Vinokur, Z.J. Wang, Spectral (finite) volume method for conservation laws on unstructured grids V:extension to three-dimensional systems, Journal of Computational Physics,2006,212 (2):454～472.
[141]Y. Sun, Z.J. Wang, Y. Liu, Spectral (finite) volume method for conservation laws on unstructured grids Ⅵ:extension to viscous flow, Journal of Computational Physics, 2006,215 (1):41～58.
[142]K. Van den Abeele, C. Lacor, Z.J.Wang, Short Note: On the connection between the spectral volume and the spectral difference method, Journal of Computational Physics,2007,227 (2):877～885.
[143]T. Haga, K. Sawada, Z.J. Wang, An implicit LU-SGS scheme for the spectral volume method on unstructured tetrahedral grids, Communications in Computational Physics,2009,6 (5):978-996.
[144]R.E. Harris, Z.J. Wang, High-order adaptive quadrature-free spectral volume method on unstructured grids, Computers & Fluids,2009,38 (10),2006～2025.
[145]R. Kannan, Z.J. Wang, The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method, Computers & Fluids,2010,39 (10), 2007～2021.
[146]Wang, Z. J., High-order spectral volume method for benchmark aeroacoustic problems.41st Aerospace Sciences Meeting and Exhibit 6-9 January, Reno Nevada AIAA-2003-0880,2003.
[147]Huang, X., Zhang, X., A Fourier Pseudospectral Method for Some Computational A-eroacoustics Problems. International Journal of Aeroacoustics,2006, 5,(3):279～294.
[148]Proudman,I.The generation of noise by isotropic turbulence. Proceedings of the Royal Society of London A, Mathematical and Physical Sciences,1952,214 (1116):119～132.
[149]Lilley, G. M., The source of aerodynamic noise, Aeroacoustics,2003,2 (3 & 4), 241～254.
[150]Lilley, G. M., The radiated noise from isotropic turbulence with applications to the theory of jet noise, J. Sound Vib.,1996,190(3),463～476.
[151]Lilley, G. M., The generation and radiation of supersonic jet noise. Vol. IV-Theory of turbulence generated jet noise, noise radiation from upstream sources, and combustion noise. Part Ⅱ:Generation of sound in a mixing region, Air Force Aero Propulsion Laboratory, AFAPL-TR-72-53,1972.
[152]Mankbadi,R.R.,Hixon,R.,Shih,S.H. et al. Use of linearized euler equations for supersonic jet noise prediction. AIAA Journal,1998,36 (2):140-147.
[153]Djambazovm,G., Lai, C. H.,Pericleous,K. Development of numerical techniques for near field aerodynamic computations. International Journal for Numerical Method in Fluids,1999,29:719～731.
[154]李晓东,高军辉.二维平行剪切层声波产生和辐射的数值模拟.航空学报,2003,24(1)：5-9.
[155]Morgans, R. P., "The Kirchhoff Formula Extended to a Moving Surface," Philosophical Magazine,1930, Vol.9 (s.7, No.55),141～161.
[156]Ffowcs Williams, J.E.& Gordon, C. G., Noise of highly turbulent jets at low exhaust speeds, AIAA Journal,1965,3(4),791～793.
[157]Ffowcs Williams, J.E., Jet noise at very low and very high speed, Afosrutias Symposium on Aerodynamic Noise, Toronto,1968,131～145.
[158]Ffowes Williams, J. E., Hawkings, D. L.,Sound generation by turbulence and Surface in Arbitrary Motion," Philosophical. Transaction of the Royal SocietyLond A,1969, 264:321～342.
[159]Ffowes Williams, J. E., Hawkings, D. L., "Theory Relating to the Noise of Rotating Machinery,"J. Sound and Vib.,1969, Vol.10, No.1,10～21.
[160]Ffowes Williams, J. E., Hall, L. H., "Aerodynamic Sound Generation by Turbulent Flow in the Vicinity of a Scattered Half Plane,"J. Fluid Mech.,1970, Vol.40, No.4, pp.657～670.
[161]Ffowcs Williams, J. E.& Howe, M. S., The generation of sound by density inhomogeneities in low Mach number nozzle flows, J. Fluid Mech.,1975,70(3), 605～622.
[162]Ffowcs Williams, J. E., Sound sources in aerodynamics Fact and fiction, AIAA Journal,1982,20(3),307～315.
[163]Farassat, F. and Myers, M. K., "Extension of Kirchhoff s Formula to Radiation from Moving Surfaces," Journal of Sound and Vibration,451～460, Vol.123, No.3, June 1988.
[164]A. S. Lyrintzis Review:The use of Kirchhoffs method in computational aeroacoustics Journal of fluids engineering,1994,116 (4),665-676.
[165]Y. Xue, A.S. Lyrintzis, Rotating Kirchhoff method for three-dimensional transonic blade-vortex interaction hover noise, AIAA journal,1994,32 (7),1350-1359.
[166]A.S. Lyrintzis, R. Mankbadi, Prediction of the far-field jet noise using Kirchhoffs formulation AIAA journal,1996,34 (2),413～416.
[167]R.C. STRAWN, R. Biswas, A.S. Lyrintzis, Helicopter noise predictions using Kirchhoff methods, Journal of Computational Acoustics,1996,4 (03),321～339.
[168]A.R. Pilon, A.S. Lyrintzis, Integral methods for computational aeroacoustics, AIAA paper, AIAA paper,20,1997.
[169]A.R. Pilon, A.S. Lyrintzis, Development of an improved Kirchhoff method for jet aeroacoustics, AIAA journal,199836 (5),783～790.
[170]A.S. Lyrintzis, Surface integral methods in computational aeroacoustics-From the (CFD) near-field to the (Acoustic) far-field International journal of aeroacoustics,2003,2 (2),95～128.
[171]A. Uzun, AS Lyrintzis, GA Blaisdell, Coupling of integral acoustics methods with LES for jet noise prediction, International Journal of Aeroacoustics,2004,3(4), 297～346.
[172]Holevo, A.S., Bounds for the quantity of information transmitted by a quantum communication channel, Problemy Peredachi Informatsii,1973,9(3):3～11.
[173]Ingarden, R.S., Quantum information theory, Rep. Math. Phys.,1976,10:43～72.
[174]Manin, Y., Computable and Uncomputable, Moscow: Sovetskoye Radio,1980.
[175]Poplavskii, R.P, Thermodynamical models of information processing, (in Russian). Uspekhi Fizicheskikh Nauk,1975,115(3):465～501.
[176]Wiesner, S., Conjugate coding, Sigact news,1983,18:78～88.
[177]David Deutsch, Quantum theory, the Church-Turingprinciple and the universal quantum computer, Proc. R. Soc. Lond.
[178]Peter Shor, Algorithms for Quantum Computation:Discrete Logarithms and Factoring, IEEE Symposium on Foundations of Computer Science,1994:124-134.
[179]冯康,秦孟兆.Hamilton系统的辛几何算法.杭州：浙江科学技术出版社,2003.
[180]Feng K. On Difference Schemes and Symplectic Geometry. Proceedings of the 5th international symposium on differential geometry and differential equations, Beijing, 1984.
[181]Feng K, Wu H M, Qin M Z. Symplectic Difference Schemes for Hamiltonian Systems in General Symplectic Structures. Journal of Computation and Mathematics, 1991,9(1):86～96.
[182]Qin M Z. Canonical Difference Schemes for the Hamiltonian Equation. Mathematical Methods in the Applied Sciences,1989,11(3):543～557.
[183]Feng K, Qin M Z. Hamiltonian Algorithms for Hamiltonian Systems and a Comparative Numerical Study. Computer Physics Communications,1991, 65(1-3):173～187.
[184]Feng K, Wang D L. A Note on Conservation Laws of Symplectic Difference Schemes for Hamiltonian Systems. Journal of Computation and Mathematics,1991, 9(3):229-237.
[185]Feng K. The Step-transition Operator for Multi-step Methods of ODEs. Journal of Computation and Mathematics,1998,16(3):193～202.
[186]Arnold V I. Mathematical Methods of Classical Mechanics. New York:Springer-Verlag,1989.
[187]Hairer E, Lubich C, Wanner G Geometric Numerical Integration: Structure-Preserving Algorithm for Ordinary Differential Equations.2ed. New York:Springer, 2006.
[188]姚伟岸,钟万勰.辛弹性力学.北京：高等教育出版社,2002.
[189]U. Frish, B. Hassalacher, Y. Pomeau, Lattice Gas automata for the Navier-Stokes Equation,Phys.Rev.Lett.,1986,56(14):1505～1508.
[190]M.Henon, Implementation of the FCHC Lattice Gas Model on the Connection Machine, J.Stat.Phys.,1992,68:353-377.
[191]G.R.McNamara and Zanetti G., Use of Boltzmann Equation to Simulate Lattice Gas Automata, Phys.Rev.Lett.,1988,61(20):2332～2335.
[192]Z. L. Guo, C. G Zheng, A coupled LBGK model for the Boussinesq equations, Int. J. Num. Meth. Fluids.,2002,39:325～342.
[193]M. A. Moussaoui, M. Jami, A. Mezrhab, H. Naji, MRT-Lattice Boltzmann simulationof forced convection in a plane channel with an inclined square cylinder, Int. J. Therm. Sci.,2010,49(1):131～142.
[194]Y.Qian, D.d'Humieres, P.Lallemand, LBGK models for N-S equations, Europhys.Lett.,1992,17:479～484
[195]Q. Zou, S. Hou, S. Chen, G. D. Doolen, An improved incompressible lattice Boltzmann model for time-independent, J.Stat.Phys.,1995,81:35-48.
[196]Y. Chen, H. Ohashi, LBGK methods for simulating incompressible fluid flows,Int.J.Mod.Phys.C.,1997,8:793～803.
[197]X.He, L.S.Luo, LBM for the incompressible Navier-Stokes equation, J.Stat.Phys., 1997,88(3-4):927-944.
[198]Guo Z.L., B.C.Shi, N.C.Wang, Lattice BGK model for incompressible Navier-Stokes equation, J.Comput.Phys.,2000,165:288～306.
[199]Q. S. Zou, X. Y. He, On Pressure and veloeity boundary conditions for the lattiee Boltzmann BGK model, Phys.Fluids,1997, vol.9, No.6:1591～1598.
[200]E. Laurendeau, J. P. Bonnet, P. Jordan, and J. Delville. "Impact of fluidic chevrons on the turbulence structure of a subsonic jet," AIAA Paper No.2006-3510
[201]米建春,冯宝平,Deo Ravinesh C, etc.出口雷诺数对平面射流自保持性的影响.物理学报,2009,58(11)：7756～7764
[202]黄昆,韩汝琦.固体物理学.北京：高等教育出版社,1988(10)：78-152.
[203]吴克启.宏观气动声场量子分子动力学讲义,2010
[204]John F.. Finite amplitude waves in a homogeneous isotropic elastic solid. Commun. Pure Appl. Math 1977; 30:421～446.
[205]Feng K, Qin M.Z. Sympleetic geometric algorithms for Hamiltonian systems. Science and Technology Press, zhe jiang,2003.
[206]Peter M.The lattice Boltzmann phononic lattice solid. Journal of Statistical Physics 1992; 68:591～609.
[207]Struwe M. Variational methods:Applications to nonlinear partial differential equations and Hamiltonian systems. Berlin, New-York:Springer Verlag; 1990.
[208]Sun G. A class of explicitly symplectic schemes for waveequations. Comput.Math.(in Chinese) 1997; 1:1-10.
[209]Zhai, P.W., Kattawar, G.W., Yang P., Li C.H. Application of the symplectic finite-difference time-domain method to light scattering by small particles, Applied Optics 2005; 44:1650～1656.
[210]Sha, W., Huang Z.X., Wu X.L., Chen M.S. Application of the symplectic finite difference time-domain scheme to electromagnetic simulation, Journal of Computational Physics,2007; 225:33～50.
[211]Feng, K. Collected works of Feng Kang. Beijing:Science Press; 1985.
[212]Yefet A., Petropoulos P.G. A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. J. Comput. Phys.2001 April; 168: 286～315.
[213]Joe L., Doru M., Gilbert S. Modeling of the seat dip effect using the finite-difference time-domain method. J.Acoust. Soc. Am.1996; 100(4):2204～2212.
[214]Botteldooren Finite-difference time-domain simulation of low frequency room acoustic problems. J.Acoust. Soc. Am.1995; 98(6):3302～3308.
[215]罗民秋,刘洪,李幼铭,韩汝琦.地震波传播的哈密顿表述及辛几何算法.地球物理学报,2001,44(1)：120～128
[216]Zeng J Y. Quantum Mechanics, volume II (3rd edition). Beijing:Science Press.2000.
[217]Duan L M,Guo G G, Energy Level Excitations--the Elementary Excitations in the Crystal-optical-field Strong Coupled System. Acta Physica Sinica,1997,46, (6):1114～1117
[218]Li X D, Gao J H, Numerical Simulation of Sound Generation and Radiation from a 2D Parallel Shear Layer, Acta Aeronautica et Astronaut Ica Sinica,
[219]Xie X D, Jiang P, Lu F. Group Theory and its Application to Physics. Science Press, 1986.
[220]Zhou G D, Duan L Y.Bases of Structural Chemistry, (2rd edition). Peking University Press,1995:26～39
[221]Yan J. The Energy Transfer Between Two Diatomc Molecules in Collinear Collisions. Jinan:Shangdong University,2007
[222]Shen J, Sha W, Huang Z X, Chen MS, Wu X L,High-oder Symplectic FDTD Scheme for Solving Time-dependent SchrSdinger Equation, Acta Physica Sinica, 2012,61(19):190202-1-190202-7
[223]Wang S J, Zhang H. The Fidelity and Algebraic Dynamics Algorithm of Physical Computing:I Algebraic Dynamics Solvings and Algebraic Dynamics Algorithm of the Dynamic System. Science in China Series G:Physcis,Mechanics & Astronomy, 2005,35(6):573----608
[224]Liu J J,Zhai L X,Using the Finite Difference Mmethod to Solve the Energy Eigenvalue Equation. Journal of Beijing Univesity of Technology,2008, (34): 325-328.
[225]Cheng X P,The Direct Perturbation Method and Approximate Solutions of Nonlinear Schrodinger Equation Hierarchy, zhejiang:zhejiang Normal University,2006.
[226]Kosloff R, Rice S A, Gaspars P, et al. Wavepacket Dancing:Achieving Chemical Selectivity by Shaping Light Pulses.Chemical Physics,1989,139(1):201～220.
[227]Feynman R P, Hibbs A R(Write).ZHANG Banggu (Translation). Quantum mechanics and path integrals.Beijing:Science Press.1986.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |