二维数值波浪水槽模式的建立和应用及浪流相互作用研究
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摘要
根据本人在攻读博士学位期间所作的两项海浪研究工作，将学位论文分为二维数值波浪水槽模式的建立和应用，以及浪流相互作用研究两部分。
第一部分在回顾数值波浪水槽发展历程、现状及其应用前景的基础上，重点报告本人独立开发二维非线性数值波浪水槽模式的研究工作，其中包括构建数值波浪水槽的运行原理和具体步骤等全过程。
详细阐述了数值波浪水槽研究的数学物理基础以及建立数值波浪水槽的两大核心技术—二次边界元方法和混合欧拉—拉格朗日方法，给出了数值波浪水槽运行的实时模拟流程图。同时还系统说明了该模式应用的造波技术、消波技术以及特殊数值处理的方法和步骤。其中对阻尼消波技术作了改进，设计了一种纺锤形的阻尼消波器使其具有双向消波功能和低通滤波功能。将其与一种以水位为控制信号的活塞式消波机相结合，获得了更为满意的消波效果。
为检验该波浪水槽模式的性能，对其进行了严格、详细的测试。其中包括驻波的模拟实验，卷波的破碎模拟实验和一组ISOPE的官方测试(Official Benchmark)模拟实验。测试结果表明，该模式具有较完善的造波和消波能力，极好的非线性模拟能力，极佳的计算精度和稳定性，能够较好的满足数值实验的实际需要。
为检验该数值波浪水槽的实用性，对其进行了若干应用试验，以便显示该水槽在波动实验中的应用潜力，并期望能在海洋混合过程及其机理研究中获得应用。其中一项实验是将两层水槽相互耦合，给出模拟界面波的具体方案，实现界面波的模拟尝试。实验证明，在线性框架下该模式是稳定的，但在非线性模拟中却出现了由边界角点引起的弱不稳定现象，必须对模式作进一步修改，为此同时提出了以消波器消除该角点影响的限制性解决方法。另一项实验是借用地形的往复运动产生的波动，模拟往复流中由地形激发的波动现象，找到了往复流地形波与往复频率和地形高度之间的初步关系，发现了由地形引起的高频不稳定和破碎现象并给出了初步的解释。
第二部分根据本人协助导师所作的相关工作，对当前国内外的浪流相互作用研究进展做了系统的回顾，针对海浪对海流的大尺度驱动作用进行了有意义的探索。
在流对浪的作用讨论中，依据波动的射线理论和由变分原理导出的波作用量守恒定律，阐明了稳定波场波数矢量的无旋条件与费马原理的等价性，扼要讨论了背景流场影响海浪的运动学效应和动力学效应，首次指出水位(包括天文潮位与风暴潮位)变化对海浪成长与消衰的影响，为流浪潮联合数值模式的发展提供必要的理论依据。在浪对流作用的讨论中，系统的阐述了海浪在地转作用和粘性条件下所产生的各种作用力，详述了它们的生成机制，并定性的分析了相应的波生流，同时，首次提供了地转条件下浪致雷诺应力张量的完整表达式，为实现海流模式与海浪模式的并行数值计算提供了一种简便的途径。
二维数值波浪水槽模式的建立和应用
在前人有关海浪波生作用力研究工作的基础上，依据f平面近似下线化无粘流
体运动方程的一般形式，并考虑地转角速度水平分量的作用，首次给出真正地转意
义下均匀水底上的小振幅波动解。由此解导出的波生切应力包括波向与横向两个分
量。依地转波生应力和虚拟波生应力计算了海面上波生总应力与风应力的比值，大
致估计了该应力在我国海域环流研究中的重要性。
最后，在上述研究的基础上对海浪的横向彻体切应力进行了定量的具体分析。
使用风浪经验公式和风浪传入近岸的小振幅波动理论，计算并分析了地转条件下波
生横向彻体切应力随风速、风区、水深等因素的变化规律;将其与定常Ekman漂流
中的风生湍粘性力的比较表明，在中、高纬度的通常海况下两者具有相同量级，从
而强调在风生漂流研究中必须考虑由地转引起的波生横向彻体切应力。
According to the two aspects of ocean wave studies that I have made as a P.H.D candidate, the dissertation is divided into two parts: Part I is the development and Applications of a 2-D numerical wave tank(NWT) and Part II the study of wave-current interaction.
In Part I, based on the systematic review of the study history and prospect of NWT, the detailed process of a 2-D NWT model developed independently is reported.
The two key discretization methods, boundary element method (BEM) and mixed Eular-Langurange (MEL)method, are introduced in detail, and an integrated flow chart of the detailed time-domain simulation steps of NWT is presented. Subsequently, the special numerical techniques including the wave generation and absorption that are used in the model is illuminated. Moreover, a spindly shaped damping zone is designed to improve the absorption capability of conditional damping zone, which has the function of two directional wave absorption and low-pass filtering. Combining with a piston wave absorber, the wave absorption efficiency is proved to be powerful.
A series of testing experiments are made to check the performance of the model, including the simulations of a stationary wave and a plunging wave rush on the slope beach. In particular, a group of ISOPE official benchmarks are run so as to compare the developed NWT with others. The results show that the model has perfect functions of wave generation and absorption and is stable and accurate enough for simulating fully nonlinear waves exactly. That is to say, the model can meet the actual demand of common numerical experiments very well.
Several experiments with the developed NWT are carried out to prove its practicability and potential capability, through which it is expected that the developed NWT would find its application in the research on the process of ocean mixing. In the first experiment, a feasible scheme of simulating interfacial wave is presented by coupling two models with each other to simulate some interfacial wave phenomena. The simulating results indicate that the scheme is stable in linear frame, and exists weak instability on the intersecting points of lateral boundary and interface in nonlinear frame. In present stage, a technique of using damping zone to limit the movement of the intersecting points is used to avoid this instability problem, however, to solve it thoroughly, the original model has to be modified in the future.
In the second experiment, the topography stirred waves in a to-and-fro current is simulated, in which the rough relationship of the stirred waves and the topography is made out, a phenomenon of high frequency instability and breaking of surface on curtain condition is detected and analyzed.
In Part II, based on some researches that I have made as an assistant of my tutor, the present advance of international research of wave-current interaction is review systematically. Subsequently, the waves induced forces and their driven effects on currents are studied in detail.
In the discussion of the effects of currents on waves, according to the wave radiation theory and the law of wave action conservation, the equivalence of none vorticity condition of the vector of wave number in constant wave field and Fermat Theory is
testified. The dynamic and kinematics effects of background currents on waves are introduced concisely. The effect of the change of surface elevation on the wave development and elimination is pointed out for the first time, which provides the necessary theoretical basis for the development of united current-wave-tide model; hi the discussion of the effects of waves on currents, a variety of wave-induced forces on the geostrophic and viscous condition and their producing mechanism are introduced systematically, their corresponding wave-induced currents are also analyzed qualitatively. Moreover, a complete expression of wave-induced Reynolds stresses tensor is presented for the first time.
In succession, to amend the preceding studies, a genuine geostrophic small amplitude wave so
第一部分在回顾数值波浪水槽发展历程、现状及其应用前景的基础上，重点报告本人独立开发二维非线性数值波浪水槽模式的研究工作，其中包括构建数值波浪水槽的运行原理和具体步骤等全过程。
详细阐述了数值波浪水槽研究的数学物理基础以及建立数值波浪水槽的两大核心技术—二次边界元方法和混合欧拉—拉格朗日方法，给出了数值波浪水槽运行的实时模拟流程图。同时还系统说明了该模式应用的造波技术、消波技术以及特殊数值处理的方法和步骤。其中对阻尼消波技术作了改进，设计了一种纺锤形的阻尼消波器使其具有双向消波功能和低通滤波功能。将其与一种以水位为控制信号的活塞式消波机相结合，获得了更为满意的消波效果。
为检验该波浪水槽模式的性能，对其进行了严格、详细的测试。其中包括驻波的模拟实验，卷波的破碎模拟实验和一组ISOPE的官方测试(Official Benchmark)模拟实验。测试结果表明，该模式具有较完善的造波和消波能力，极好的非线性模拟能力，极佳的计算精度和稳定性，能够较好的满足数值实验的实际需要。
为检验该数值波浪水槽的实用性，对其进行了若干应用试验，以便显示该水槽在波动实验中的应用潜力，并期望能在海洋混合过程及其机理研究中获得应用。其中一项实验是将两层水槽相互耦合，给出模拟界面波的具体方案，实现界面波的模拟尝试。实验证明，在线性框架下该模式是稳定的，但在非线性模拟中却出现了由边界角点引起的弱不稳定现象，必须对模式作进一步修改，为此同时提出了以消波器消除该角点影响的限制性解决方法。另一项实验是借用地形的往复运动产生的波动，模拟往复流中由地形激发的波动现象，找到了往复流地形波与往复频率和地形高度之间的初步关系，发现了由地形引起的高频不稳定和破碎现象并给出了初步的解释。
第二部分根据本人协助导师所作的相关工作，对当前国内外的浪流相互作用研究进展做了系统的回顾，针对海浪对海流的大尺度驱动作用进行了有意义的探索。
在流对浪的作用讨论中，依据波动的射线理论和由变分原理导出的波作用量守恒定律，阐明了稳定波场波数矢量的无旋条件与费马原理的等价性，扼要讨论了背景流场影响海浪的运动学效应和动力学效应，首次指出水位(包括天文潮位与风暴潮位)变化对海浪成长与消衰的影响，为流浪潮联合数值模式的发展提供必要的理论依据。在浪对流作用的讨论中，系统的阐述了海浪在地转作用和粘性条件下所产生的各种作用力，详述了它们的生成机制，并定性的分析了相应的波生流，同时，首次提供了地转条件下浪致雷诺应力张量的完整表达式，为实现海流模式与海浪模式的并行数值计算提供了一种简便的途径。
二维数值波浪水槽模式的建立和应用
在前人有关海浪波生作用力研究工作的基础上，依据f平面近似下线化无粘流
体运动方程的一般形式，并考虑地转角速度水平分量的作用，首次给出真正地转意
义下均匀水底上的小振幅波动解。由此解导出的波生切应力包括波向与横向两个分
量。依地转波生应力和虚拟波生应力计算了海面上波生总应力与风应力的比值，大
致估计了该应力在我国海域环流研究中的重要性。
最后，在上述研究的基础上对海浪的横向彻体切应力进行了定量的具体分析。
使用风浪经验公式和风浪传入近岸的小振幅波动理论，计算并分析了地转条件下波
生横向彻体切应力随风速、风区、水深等因素的变化规律;将其与定常Ekman漂流
中的风生湍粘性力的比较表明，在中、高纬度的通常海况下两者具有相同量级，从
而强调在风生漂流研究中必须考虑由地转引起的波生横向彻体切应力。
According to the two aspects of ocean wave studies that I have made as a P.H.D candidate, the dissertation is divided into two parts: Part I is the development and Applications of a 2-D numerical wave tank(NWT) and Part II the study of wave-current interaction.
In Part I, based on the systematic review of the study history and prospect of NWT, the detailed process of a 2-D NWT model developed independently is reported.
The two key discretization methods, boundary element method (BEM) and mixed Eular-Langurange (MEL)method, are introduced in detail, and an integrated flow chart of the detailed time-domain simulation steps of NWT is presented. Subsequently, the special numerical techniques including the wave generation and absorption that are used in the model is illuminated. Moreover, a spindly shaped damping zone is designed to improve the absorption capability of conditional damping zone, which has the function of two directional wave absorption and low-pass filtering. Combining with a piston wave absorber, the wave absorption efficiency is proved to be powerful.
A series of testing experiments are made to check the performance of the model, including the simulations of a stationary wave and a plunging wave rush on the slope beach. In particular, a group of ISOPE official benchmarks are run so as to compare the developed NWT with others. The results show that the model has perfect functions of wave generation and absorption and is stable and accurate enough for simulating fully nonlinear waves exactly. That is to say, the model can meet the actual demand of common numerical experiments very well.
Several experiments with the developed NWT are carried out to prove its practicability and potential capability, through which it is expected that the developed NWT would find its application in the research on the process of ocean mixing. In the first experiment, a feasible scheme of simulating interfacial wave is presented by coupling two models with each other to simulate some interfacial wave phenomena. The simulating results indicate that the scheme is stable in linear frame, and exists weak instability on the intersecting points of lateral boundary and interface in nonlinear frame. In present stage, a technique of using damping zone to limit the movement of the intersecting points is used to avoid this instability problem, however, to solve it thoroughly, the original model has to be modified in the future.
In the second experiment, the topography stirred waves in a to-and-fro current is simulated, in which the rough relationship of the stirred waves and the topography is made out, a phenomenon of high frequency instability and breaking of surface on curtain condition is detected and analyzed.
In Part II, based on some researches that I have made as an assistant of my tutor, the present advance of international research of wave-current interaction is review systematically. Subsequently, the waves induced forces and their driven effects on currents are studied in detail.
In the discussion of the effects of currents on waves, according to the wave radiation theory and the law of wave action conservation, the equivalence of none vorticity condition of the vector of wave number in constant wave field and Fermat Theory is
testified. The dynamic and kinematics effects of background currents on waves are introduced concisely. The effect of the change of surface elevation on the wave development and elimination is pointed out for the first time, which provides the necessary theoretical basis for the development of united current-wave-tide model; hi the discussion of the effects of waves on currents, a variety of wave-induced forces on the geostrophic and viscous condition and their producing mechanism are introduced systematically, their corresponding wave-induced currents are also analyzed qualitatively. Moreover, a complete expression of wave-induced Reynolds stresses tensor is presented for the first time.
In succession, to amend the preceding studies, a genuine geostrophic small amplitude wave so
引文
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48． Longuet-Higgins, M. S.(1953) Mass transport in water waves. Phil. Trans. R.Soc.Lond. A245． 535-581
49． Longuet-Higgins, M. S.(1960) Mass transport in the boundary layer at a free oscillating surface. J.F.M.,8, 293-306
50． Longuet-Higgins, M. S. and Cokelet, E. D. (1976) The deformation of steep surface waves on water, I. A numerical method of computation. Proceedings of the Royal Society of London, Series A, 350, 1-26．
51． Maisondieu, C. and Clement, A. (1993) A realizable force feedback-feedforward control loop for a piston wave absorber. Proceedings of the 8th International Workshop on Water Waves and Floating Bodies, St. John's, Newfoundland, Canada, 79-82．
52． Milgram, J. S. (1970) Active water-wave absorbers. Journal of Fluid Mechanics, 43(4) , 845-859．
53． Nabors, K., Kim, S.and White. J. (1991) Fastcap: Amultipole accelerated 3D capacitance extraction program IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. Vol.10 pp1447-1459
54． Noh .F.and Woodward, PR. (1976) Slic (simple line interface calculation). Lecture Notes in Physics, vol. 59, pp 330-340．
55． Ohyama, T. and Nadaoka, K. (1991) Development of a numerical wave tank for analysis of nonlinear and irregular wave field. Fluid Dynamics Research, 8, 231-251．
56． Orlanski, I. (1976) , "A simple boundary condition for unbounded hyperbolic flows", J. of Compt. Phys., Vol.21, pp.251-269
57． Ris, R.C., N. Booij and L.H. Holthuijsen, (1999 ), A third-generation wave model for coastal regions, Part II, Verification, J.Geoph.Research C4,104, 7667-7681．
58． Rienecker.M.M. and Fenton,J.D. (1981) , "A Fourier approximation method for steady water waves", Journal of Fluid Mechanics, Vol.104, pp.119-137
59． Romate, J. E. (1989) The Numerical Simulation of Nonlinear Gravity Waves in Three Dimensions using a Higher Order Panel Method. Ph.D. thesis, University of Twente, The Netherlands.
60． Saad,Y. and Schultz,M.H., (1985) , "A generalized minimal residual algorithm for solving
nonsymmetric linear systems", SIAM J. Sci. and Stat. Comp., Vol.7, pp417-424
61． Schaffer, H. A., Stolborg, T. and Hyllested, P. (1994) Simultaneous generation and active absorption of waves in flumes. Proceedings of the International Symposium: WAVES-Physical and Numerical Modeling, University of British Columbia, Vancouver, Canada, 90-99．
62． She, K., Created, C. A. and Easson, W. J. (1992) Development of a two-dimensional numerical wave tank. Proceedings of the 2nd International Offshore and Polar Engineering Conference, San Francisco, USA, vol. 3,102-109．
63． Skourup, J. (1996) Active absorption in a numerical wave tank. Proceedings of the 6th International Offshore and Polar Engineering Conference, Los Angeles, USA, vol. 3, 31-38．
64． Sugihara,M. and Isshiki,H., (1980) , "Sommerfeld's radiation condition associated with water waves and its application for numerical calculation", J. Kansai Soc.Nav. Arch. Japan, vol.156, pp.67-74
65． Svendsen, I. A. (1971) Mixed boundary value problem for Laplace's equation in domain of arbitrary shape. Basis Research, Progress Report, 23,33-38． Coastal Engineering Laboratory, Technical University of Denmark.
66． Tanizawa,K., (1997b), "Nonlinear theory of wavebody interaction based on acceleration potential and its application to numerical simulation (in Japanese)", Ph.D thesis, Osaka Univ., pp. 1-127
67． Tanizawa, K. and Clement, A.H. (2000) , "Report of the 2nd workshop of ISOPE NWT group, Benchmark test cases of radiation problem", Proc. of 10th ISOPE Conf., Seattle, Vol.3, pp. 175-184
68． Tanizawa, K. and Clement, A.H. (2001) , "Report of the 3nd workshop of ISOPE NWT group, Benchmark test cases of diffraction problem", Proc. of 11th ISOPE Conf., Seattle, Vol.3, pp. 175-184
69． Teng B and Dong G H, 2001, Numerical examination of third-order wave force on axisymmetric bodies, Int. Jour of ISOPE, Vol. 11, No. 4．
70． Teng B and Eatock Taylor R, (1995a), Application of a Higher Order BEM in the Calculation of Wave Run-up on Bodies in a Weak Current, Int. Jour of Offshore and Polar Eng., Vol.5．
71． Teng B and Eatock Taylor R, (1996) , New Higher Order Boundary Element Method for Wave Diffraction/Radiation, Applied Ocean Research (Elsvier Press), Vol. 17(2) ,pp.71-77
72． Teng B and Kato S, 1999, A Method for Second-order Diffraction Potential from an Axisymmitric Body, Ocean Engineering (Pergamon Press), Vol. 26．
73． Tolman, H. L.,( 1997) User manual and system documentation of WAVEWATCH-Ⅲ version 1． 15． NOAA / NWS / NCEP / OMB Technical Note, 151, pp. 97
74． Tolman, H. L.,(1999) User manual and system documentation of WAVEWATCH-Ⅲ version 1． 18． NOAA / NWS / NCEP / OMB Technical Note, 166, pp. 110
75． Trefethen, L. N. and Bau, D. (1997) Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, USA.
76． Ubbink O and Isaa K I. (1999 ) , A method for capturing sharp fluid interfaces on arbitrary meshes. J. Comput. Phys., 153: 26-50
77． Vinje, T. and Brevig, P. (1981) Numerical simulation of breaking waves. Advances in Water Resources, 4,77-82．
78． Wahba,G., (1981) , "Bayesian con.dence intervals for cross validated smoothing spline", Technical Report,Department of statistics, University of Wisconsin,No.645 pp. 1-60
79． Wang, P. (1993) Numerical research on (Ⅰ) ship internal waves, and (Ⅱ) breaking waves. Technical Report 93-93, Ocean Engineering Laboratory, University of California, Santa
Barbara. Part Ⅱ.
80. WAMDIG.(1988): The WAM model-A third generation ocean wave prediction model.Journal of Physical Oceanography, 18, 1775-1810.
81. Wang, P., Yao, Y. and Tulin, M. P. (1994) Wave group evolution, wave deformation, and breaking: Simulations using LONGTANK, a numerical wave tank. International Journal of Offshore and Polar Engineering, 4(3), 200-205.
82. Wang, P., Yao, Y. and Tulin, M. P. (1995) An efficient numerical tank for non-linear water waves based on the multi-subdomain approach with BEM. International Journal for Numerical Methods in Fluids, 20(12), 1315-1336.
83. Xu,H. and Yue.,D.K.P., (1992), "Numerical study of three dimensional overturning waves",Proc. 7th Workshop on Water Waves and Floating Bodies, pp.303-307
84. Yuan Y L et al. (1988) LGFD wave numerical forecast method. Part Ⅰ: Wave Numerical Forecast Model. Journal of Oceanography of Huanghai&Bohai Seas, 6,(2): 1-12
85. Youngs D L. (1982), Time-dependent multi-material flow with large fluid distortion numerical method for fluid dynamics. Academic, New York, 273-285
86. Zhang,S., Yue,D.K.P. and Tanizawa,K., (1996), "Simulation of plunging wave impact on a vertical wall", J. Fluid Mech., Vol.327, pp.221-254
87．柏威，滕斌，邱大洪，(2001)，三维物体上二阶波浪绕射的实时模拟，海洋工程，Vol．19，No．3
88．李瑞遐，(1993)，有限元法与边界元法，上海科技教育出版社，204—270
89．孙大鹏，李玉成，(2002)，非线性波浪变形计算的三维边界元方法，水科学进展，Vol．13，No．4，pp．397-402
90．滕斌，李玉成，董国海，(2000)，轴对称物体上的三阶波浪力，海洋学报，Vol．22，No．2，pp．105-112
91．文圣常，余宙文，(1984)海浪理论与计算原理，科学出版社，13—14
92．忻孝康，刘儒勋，蒋伯诚，(1989)计算流体动力学，国防科技大学出版社，579—635
93．徐萃微，(1985)计算方法引论，高等教育出版社，207—234
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32． Guignard S, Grilli S, Marcer, R Rey V. (1999) . Computation of Shoaling and Breaking Waves in Nearshore Areas by Coupling of BEM and VOF Methods. Proc. ISOPE99 Conf.(Brest,France), pp304-309
33． Harten, A. and Efrony, S. (1978) A partition technique for the solution of potential flow problems by integral equation methods. Journal of Computational Physics, 27,71-87．
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35． Hess, J. L. and Smith, A. M. O. (1964) Calculation of nonlifting potential flow about arbitrary three-dimensional bodies. Journal of Ship Research, 8(2) , 22-44．
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40． Jagannathan, S. (1988) , "Nonlinear free surface flows and an application of the Orlanski
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41． Jagannathan, S. (1992) Energy flux equalization: An open boundary condition for non-linear free surface flows. International Journal for Numerical Methods in Fluids, 14(7) , 793-815．
42． Kim, M. H., Celebi, M. S. and Kim, D. J. (1998) Fully non-linear interactions of waves with a three-dimensional body in uniform currents. Applied Ocean Research, 20(5) , 309-321．
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44． Korsmeyer,F.T., Yue,D.K.P., Nabors.K. and White,J. (1993) , "Multipole accelerated preconditioned iterative methods for three-dimensional potential problems", Proc. of BEM 15,pp.517-527
45． Korsmeyer.F.T., Nabors,K. and White,J. (1996) , "FastLap Version:2", Users guide of FastLap, MIT Research Lab. of Electronics, pp. 1-26
46． Lancaster,P, and Salkauskas,K., (1986) , "Curve and surface fitting", Academic Press, pp.87-111
47． Lin, W.-M., Newman, J. N. and Yue, D. K. P. (1984) Nonlinear forced motion of oating bodies.Preprints: Proceedings of the 15th ONR Symposium on Naval Hydrodynamics, Hamburg, 1-15．
48． Longuet-Higgins, M. S.(1953) Mass transport in water waves. Phil. Trans. R.Soc.Lond. A245． 535-581
49． Longuet-Higgins, M. S.(1960) Mass transport in the boundary layer at a free oscillating surface. J.F.M.,8, 293-306
50． Longuet-Higgins, M. S. and Cokelet, E. D. (1976) The deformation of steep surface waves on water, I. A numerical method of computation. Proceedings of the Royal Society of London, Series A, 350, 1-26．
51． Maisondieu, C. and Clement, A. (1993) A realizable force feedback-feedforward control loop for a piston wave absorber. Proceedings of the 8th International Workshop on Water Waves and Floating Bodies, St. John's, Newfoundland, Canada, 79-82．
52． Milgram, J. S. (1970) Active water-wave absorbers. Journal of Fluid Mechanics, 43(4) , 845-859．
53． Nabors, K., Kim, S.and White. J. (1991) Fastcap: Amultipole accelerated 3D capacitance extraction program IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. Vol.10 pp1447-1459
54． Noh .F.and Woodward, PR. (1976) Slic (simple line interface calculation). Lecture Notes in Physics, vol. 59, pp 330-340．
55． Ohyama, T. and Nadaoka, K. (1991) Development of a numerical wave tank for analysis of nonlinear and irregular wave field. Fluid Dynamics Research, 8, 231-251．
56． Orlanski, I. (1976) , "A simple boundary condition for unbounded hyperbolic flows", J. of Compt. Phys., Vol.21, pp.251-269
57． Ris, R.C., N. Booij and L.H. Holthuijsen, (1999 ), A third-generation wave model for coastal regions, Part II, Verification, J.Geoph.Research C4,104, 7667-7681．
58． Rienecker.M.M. and Fenton,J.D. (1981) , "A Fourier approximation method for steady water waves", Journal of Fluid Mechanics, Vol.104, pp.119-137
59． Romate, J. E. (1989) The Numerical Simulation of Nonlinear Gravity Waves in Three Dimensions using a Higher Order Panel Method. Ph.D. thesis, University of Twente, The Netherlands.
60． Saad,Y. and Schultz,M.H., (1985) , "A generalized minimal residual algorithm for solving
nonsymmetric linear systems", SIAM J. Sci. and Stat. Comp., Vol.7, pp417-424
61． Schaffer, H. A., Stolborg, T. and Hyllested, P. (1994) Simultaneous generation and active absorption of waves in flumes. Proceedings of the International Symposium: WAVES-Physical and Numerical Modeling, University of British Columbia, Vancouver, Canada, 90-99．
62． She, K., Created, C. A. and Easson, W. J. (1992) Development of a two-dimensional numerical wave tank. Proceedings of the 2nd International Offshore and Polar Engineering Conference, San Francisco, USA, vol. 3,102-109．
63． Skourup, J. (1996) Active absorption in a numerical wave tank. Proceedings of the 6th International Offshore and Polar Engineering Conference, Los Angeles, USA, vol. 3, 31-38．
64． Sugihara,M. and Isshiki,H., (1980) , "Sommerfeld's radiation condition associated with water waves and its application for numerical calculation", J. Kansai Soc.Nav. Arch. Japan, vol.156, pp.67-74
65． Svendsen, I. A. (1971) Mixed boundary value problem for Laplace's equation in domain of arbitrary shape. Basis Research, Progress Report, 23,33-38． Coastal Engineering Laboratory, Technical University of Denmark.
66． Tanizawa,K., (1997b), "Nonlinear theory of wavebody interaction based on acceleration potential and its application to numerical simulation (in Japanese)", Ph.D thesis, Osaka Univ., pp. 1-127
67． Tanizawa, K. and Clement, A.H. (2000) , "Report of the 2nd workshop of ISOPE NWT group, Benchmark test cases of radiation problem", Proc. of 10th ISOPE Conf., Seattle, Vol.3, pp. 175-184
68． Tanizawa, K. and Clement, A.H. (2001) , "Report of the 3nd workshop of ISOPE NWT group, Benchmark test cases of diffraction problem", Proc. of 11th ISOPE Conf., Seattle, Vol.3, pp. 175-184
69． Teng B and Dong G H, 2001, Numerical examination of third-order wave force on axisymmetric bodies, Int. Jour of ISOPE, Vol. 11, No. 4．
70． Teng B and Eatock Taylor R, (1995a), Application of a Higher Order BEM in the Calculation of Wave Run-up on Bodies in a Weak Current, Int. Jour of Offshore and Polar Eng., Vol.5．
71． Teng B and Eatock Taylor R, (1996) , New Higher Order Boundary Element Method for Wave Diffraction/Radiation, Applied Ocean Research (Elsvier Press), Vol. 17(2) ,pp.71-77
72． Teng B and Kato S, 1999, A Method for Second-order Diffraction Potential from an Axisymmitric Body, Ocean Engineering (Pergamon Press), Vol. 26．
73． Tolman, H. L.,( 1997) User manual and system documentation of WAVEWATCH-Ⅲ version 1． 15． NOAA / NWS / NCEP / OMB Technical Note, 151, pp. 97
74． Tolman, H. L.,(1999) User manual and system documentation of WAVEWATCH-Ⅲ version 1． 18． NOAA / NWS / NCEP / OMB Technical Note, 166, pp. 110
75． Trefethen, L. N. and Bau, D. (1997) Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, USA.
76． Ubbink O and Isaa K I. (1999 ) , A method for capturing sharp fluid interfaces on arbitrary meshes. J. Comput. Phys., 153: 26-50
77． Vinje, T. and Brevig, P. (1981) Numerical simulation of breaking waves. Advances in Water Resources, 4,77-82．
78． Wahba,G., (1981) , "Bayesian con.dence intervals for cross validated smoothing spline", Technical Report,Department of statistics, University of Wisconsin,No.645 pp. 1-60
79． Wang, P. (1993) Numerical research on (Ⅰ) ship internal waves, and (Ⅱ) breaking waves. Technical Report 93-93, Ocean Engineering Laboratory, University of California, Santa
Barbara. Part Ⅱ.
80. WAMDIG.(1988): The WAM model-A third generation ocean wave prediction model.Journal of Physical Oceanography, 18, 1775-1810.
81. Wang, P., Yao, Y. and Tulin, M. P. (1994) Wave group evolution, wave deformation, and breaking: Simulations using LONGTANK, a numerical wave tank. International Journal of Offshore and Polar Engineering, 4(3), 200-205.
82. Wang, P., Yao, Y. and Tulin, M. P. (1995) An efficient numerical tank for non-linear water waves based on the multi-subdomain approach with BEM. International Journal for Numerical Methods in Fluids, 20(12), 1315-1336.
83. Xu,H. and Yue.,D.K.P., (1992), "Numerical study of three dimensional overturning waves",Proc. 7th Workshop on Water Waves and Floating Bodies, pp.303-307
84. Yuan Y L et al. (1988) LGFD wave numerical forecast method. Part Ⅰ: Wave Numerical Forecast Model. Journal of Oceanography of Huanghai&Bohai Seas, 6,(2): 1-12
85. Youngs D L. (1982), Time-dependent multi-material flow with large fluid distortion numerical method for fluid dynamics. Academic, New York, 273-285
86. Zhang,S., Yue,D.K.P. and Tanizawa,K., (1996), "Simulation of plunging wave impact on a vertical wall", J. Fluid Mech., Vol.327, pp.221-254
87．柏威，滕斌，邱大洪，(2001)，三维物体上二阶波浪绕射的实时模拟，海洋工程，Vol．19，No．3
88．李瑞遐，(1993)，有限元法与边界元法，上海科技教育出版社，204—270
89．孙大鹏，李玉成，(2002)，非线性波浪变形计算的三维边界元方法，水科学进展，Vol．13，No．4，pp．397-402
90．滕斌，李玉成，董国海，(2000)，轴对称物体上的三阶波浪力，海洋学报，Vol．22，No．2，pp．105-112
91．文圣常，余宙文，(1984)海浪理论与计算原理，科学出版社，13—14
92．忻孝康，刘儒勋，蒋伯诚，(1989)计算流体动力学，国防科技大学出版社，579—635
93．徐萃微，(1985)计算方法引论，高等教育出版社，207—234