基于B样条的三维船体水动力数值计算研究
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摘要
随着我国加入WTO以及经济体制改革的深入进行，航运事业必然会得到蓬勃发展，而面对航道中日益增多的船舶，人类迫切需要了解船体上所受到的水动力情况，尤其是船体通过桥墩、两船相遇时的干扰水动力变化趋势，这些对避免船舶碰撞等海难事故的发生具有重要意义。
计算流体动力学的发展和商业化，为数值模拟船舶流场提供了条件。与试验方法相比，数值计算方法具有自身的优点与发展潜力。作者在对与船体水动力尤其是干扰水动力的研究相关的文献的阅读与分析的基础上，结合作为定义工业产品几何形状的唯一数学描述的非均匀有理B样条方法，选择B样条(统一成非均匀有理B样条形式)来进行有关三维船体水动力的数值计算研究工作。它不仅为船体水动力的研究带来了新的处理方法，而且在船舶计算机辅助设计与船舶流体动力学之间起到了无缝的接口作用。
作为数值计算的基础性工作，第2章采用人机交互的方式对船体型线与船体曲面进行几何造型，并全部统一成非均匀有理B样条形式，所生成的网格顶点直接为后续水动力的数值计算利用，这样避免了传统水动力计算程序中的许多数据输入和处理的过程。为了考查本文方法的可行性，第3章以圆球体绕流、圆柱绕流以及两个等圆球体在流场中的附加质量作为例子，通过在物体表面布置强度以非均匀有理B样条表达的源汇和着重对奇异积分的处理后完成了相应的计算，从与理论解的比较可以看出本方法的优越：在数目很少的配置点下数值解可取得十分良好的精度。第4章对Wigley船型的线性兴波阻力、不同弗氏数下的三维波形和等高线图作了数值计算，也取得了令人满意的结果。
由于有升力的存在，第5章直接从速度势格林公式出发，并采用了一种近似的Kutta条件，首次采用B样条边界元方法对作斜航船
武汉理}_人学博十学位论文
体的操纵水动力开展了数值计豹.训一究工作，所得到的结果在小角度
斜航时精度要好于作大角度斜航时的精度，这可能因为所采纳的
Kutta条件所导致的。第6和7资卡卜对船体之间的干扰水动力问题，
分别从两个层次上予以研究，即进行拟定常化处理和完全在时域内
作三维非定常计算，从与己公开发表的文献试验资料比较来看，所
得的结果可以在定性和数量级上很好地反映干扰水动力的变化趋
势，从而为了解船舶碰撞的运动机理提供了有力的理论根据，同时
也为如何实
避碰提供了一定的决策支持。
第8章选择日本MMG的操纵运动数学模型对船舶作回转运动、
Z型操纵试验的情况进行了数值模拟，并着重考虑了航道水深对船舶
操纵性能的影响问题，其目的上要是为了课题组今后进一步开发诸
如内河船舶操纵模拟器、船舶避碰计算机动态演示的需要。最后第9
章对全文进行了总结并提出了今后进一步的研究方向
The cause of shipping must have a flourish development with the access to World Trade Organization and further reformation of economy system of our country. It is imperious need to understand the hydrodynamic forces acted on ship hull while the amount of ships in the sea-routes is increasing. These forces, especially the changing tendency of hydrodynamics when the two ships are meeting or a ship is passing a pier are of great importance to avoid the shipwrecks such as ship collision.
Computational Fluid Dynamics (( FD) provides a condition for numerically simulating ship fluid flow field with the development and commercialization. CFD has its superiority and potential for developing with contrast to trial method. Incorporating the non uniform rational B spline (NURBS) which is the only definition of the form of industry products, the author selects the method of B spline to do the numerical computation research work of the hydrodynamics of three dimensional ship hull based on the reading and analysis of the documents about hydrodynamics, especially the interaction hydrodynamics. The method not only brings a new method for the calculation of the hydrodynamics of ship hull, but also provides an interface effect having no sew between the ship Computer Aided Design (CAD) and ship Computational fluid Dynamics (CFD).
As a fundamental work of the numerical calculation, Chapter 2 geometrically sculpts the body lines and surface of ship hull using an interactive method and adopts the format of NURBS. The vertexes are directly used by the latter numerical calculation, which avoids the tedious work of data inputting and processing in the traditional CFD programs. In order to verify the feasibility of the present method, Chapter 3 takes the fluid flow of sphere, cylinder and the additive mass of two spheres as three examples, finishes the calculation by disposing the source or sink whose strength is expressed as the format of NURBS and emphatically the processing of the singular integral. It can be drawn from the comparison of the results of present method and
the theoretical solution, i.e. a good precision can be gotten under a few collocation points with the method. Chapter 4 computes the wave-making resistance, the 3D wave figure and its contour figure of Wigley ship and makes satisfying results.
Chapter 5 directly sets out from the Green formulae of velocity potential due to the existence of lifting force, adopts an approximate Kutta condition and firstly calculates the maneuvering hydrodynamics of a Wigley ship with an attack angle using the present method of B spline. The precision of the numerical results at a small attack angle is better than that at a big attack angle. It may owe to the approximate Kutta condition adopted here. Chapter 6 and 7 are divided into two parts for the interaction hydrodynamics, i.e. the quasi-steady and the unsteady calculation fully in time domain. The results qualitatively and quantitatively agree with the trial results published in the documents. These not only provide theoretical basis for the motion mechanism of ship collisions, but also give the decision support on avoiding the ship collisions.
Chapter 8 simulates the ship maneuvering motion of turning and zigzag test and takes into account the effect of water depth on ship maneuverability by use of the Maneuvering Model Group (MMG) of Japan. The main aim is to satisfy the need such as developing the maneuvering simulator of freshwater, the dynamic demonstration of avoiding ship collisions. Finally, Chapter 9 summarizes the whole paper and puts forward the further research interests.
计算流体动力学的发展和商业化，为数值模拟船舶流场提供了条件。与试验方法相比，数值计算方法具有自身的优点与发展潜力。作者在对与船体水动力尤其是干扰水动力的研究相关的文献的阅读与分析的基础上，结合作为定义工业产品几何形状的唯一数学描述的非均匀有理B样条方法，选择B样条(统一成非均匀有理B样条形式)来进行有关三维船体水动力的数值计算研究工作。它不仅为船体水动力的研究带来了新的处理方法，而且在船舶计算机辅助设计与船舶流体动力学之间起到了无缝的接口作用。
作为数值计算的基础性工作，第2章采用人机交互的方式对船体型线与船体曲面进行几何造型，并全部统一成非均匀有理B样条形式，所生成的网格顶点直接为后续水动力的数值计算利用，这样避免了传统水动力计算程序中的许多数据输入和处理的过程。为了考查本文方法的可行性，第3章以圆球体绕流、圆柱绕流以及两个等圆球体在流场中的附加质量作为例子，通过在物体表面布置强度以非均匀有理B样条表达的源汇和着重对奇异积分的处理后完成了相应的计算，从与理论解的比较可以看出本方法的优越：在数目很少的配置点下数值解可取得十分良好的精度。第4章对Wigley船型的线性兴波阻力、不同弗氏数下的三维波形和等高线图作了数值计算，也取得了令人满意的结果。
由于有升力的存在，第5章直接从速度势格林公式出发，并采用了一种近似的Kutta条件，首次采用B样条边界元方法对作斜航船
武汉理}_人学博十学位论文
体的操纵水动力开展了数值计豹.训一究工作，所得到的结果在小角度
斜航时精度要好于作大角度斜航时的精度，这可能因为所采纳的
Kutta条件所导致的。第6和7资卡卜对船体之间的干扰水动力问题，
分别从两个层次上予以研究，即进行拟定常化处理和完全在时域内
作三维非定常计算，从与己公开发表的文献试验资料比较来看，所
得的结果可以在定性和数量级上很好地反映干扰水动力的变化趋
势，从而为了解船舶碰撞的运动机理提供了有力的理论根据，同时
也为如何实
避碰提供了一定的决策支持。
第8章选择日本MMG的操纵运动数学模型对船舶作回转运动、
Z型操纵试验的情况进行了数值模拟，并着重考虑了航道水深对船舶
操纵性能的影响问题，其目的上要是为了课题组今后进一步开发诸
如内河船舶操纵模拟器、船舶避碰计算机动态演示的需要。最后第9
章对全文进行了总结并提出了今后进一步的研究方向
The cause of shipping must have a flourish development with the access to World Trade Organization and further reformation of economy system of our country. It is imperious need to understand the hydrodynamic forces acted on ship hull while the amount of ships in the sea-routes is increasing. These forces, especially the changing tendency of hydrodynamics when the two ships are meeting or a ship is passing a pier are of great importance to avoid the shipwrecks such as ship collision.
Computational Fluid Dynamics (( FD) provides a condition for numerically simulating ship fluid flow field with the development and commercialization. CFD has its superiority and potential for developing with contrast to trial method. Incorporating the non uniform rational B spline (NURBS) which is the only definition of the form of industry products, the author selects the method of B spline to do the numerical computation research work of the hydrodynamics of three dimensional ship hull based on the reading and analysis of the documents about hydrodynamics, especially the interaction hydrodynamics. The method not only brings a new method for the calculation of the hydrodynamics of ship hull, but also provides an interface effect having no sew between the ship Computer Aided Design (CAD) and ship Computational fluid Dynamics (CFD).
As a fundamental work of the numerical calculation, Chapter 2 geometrically sculpts the body lines and surface of ship hull using an interactive method and adopts the format of NURBS. The vertexes are directly used by the latter numerical calculation, which avoids the tedious work of data inputting and processing in the traditional CFD programs. In order to verify the feasibility of the present method, Chapter 3 takes the fluid flow of sphere, cylinder and the additive mass of two spheres as three examples, finishes the calculation by disposing the source or sink whose strength is expressed as the format of NURBS and emphatically the processing of the singular integral. It can be drawn from the comparison of the results of present method and
the theoretical solution, i.e. a good precision can be gotten under a few collocation points with the method. Chapter 4 computes the wave-making resistance, the 3D wave figure and its contour figure of Wigley ship and makes satisfying results.
Chapter 5 directly sets out from the Green formulae of velocity potential due to the existence of lifting force, adopts an approximate Kutta condition and firstly calculates the maneuvering hydrodynamics of a Wigley ship with an attack angle using the present method of B spline. The precision of the numerical results at a small attack angle is better than that at a big attack angle. It may owe to the approximate Kutta condition adopted here. Chapter 6 and 7 are divided into two parts for the interaction hydrodynamics, i.e. the quasi-steady and the unsteady calculation fully in time domain. The results qualitatively and quantitatively agree with the trial results published in the documents. These not only provide theoretical basis for the motion mechanism of ship collisions, but also give the decision support on avoiding the ship collisions.
Chapter 8 simulates the ship maneuvering motion of turning and zigzag test and takes into account the effect of water depth on ship maneuverability by use of the Maneuvering Model Group (MMG) of Japan. The main aim is to satisfy the need such as developing the maneuvering simulator of freshwater, the dynamic demonstration of avoiding ship collisions. Finally, Chapter 9 summarizes the whole paper and puts forward the further research interests.
引文
[1] L.Larsson. CFD as a Tool in Ship Design. CFD Workshop Tokyo.1994,1
[2] 赵成璧．船舶曲面设计现代方法与软件系统的研究：[博士学位论文]．武汉：武汉交通科技大学，1999
[3] 沈泓萃．船舶与海洋工程水动力学集成软件的开发与应用．第7届全国内河船舶及航运学术会议论文集：372～387
[4] E.O.Tuck. A Systematic Asymptotic Expansion Procedure for Slender Ship. Journal of ship Research, 1964,8(1)
[5] E.O.Tuck,J.N.Newman. Hydrodynamic Interactions between Ships.10th Symposium on Hydrodynamics, 1977
[6] E.O.Tuck. Some Methods for Flows past Blunt Slender Bodies. Journal of Fluid Mechanics, 1964,18
[7] J.N.Newman. Lateral Motion of a Slender Body between two Parallel Walls. Journal of Fluid Mechanics, 1973,39(Part 4)
[8] R.W. Yeung. On the Interactions of Slender Ship in Shallow Water. Journal of Fluid Mechanics, 1978,85(Part 1)
[9] G.W. King. Unsteady Hydrodynamic Interactions Between Ships.Journal of Ship Research, 1977,21(3)
[10] R.W. Yeung,W.Y. Hwang.Nearfield Hydrodynamic Interactions of Ship in Shallow Water. Journal of Hydronautics, 1977,11
[11] R.W. Yeung,W.T. Tan. Hrdrodynmic Interactions of Ships with Fixed Obstacles. Journal of Ship Research, 1980,24(1)
[12] R.W. Yeung. Applications of Slender-body Theory to Ships Moving in Restricted Shallow Water.Proc. Symposium on Aspects of Navigability of Constrained Waterways, 1978
[13] A.M.J.Davis. Hydrodynamic Effects of Fixed Obstacles on Ships in Shallow Water. Journal of Ship Research, 1986,30(2)
[14] 何青．船体动流体力及倾斜角影响．西部造船会会报，第79号，1988
[15] M.A.Abkowitz,G.M.Ashe,R.M.Fortson. Interaction Effects of Ships Operating in Proximity in Deep and Shallow Water. Proc. 11th Symposium
on Naval Hydrodynamics, March, 1976
[16] 郑才土．浅水中船舶相互作用的一种计算方法．船舶工程，1994．6
[17] 贵岛胜郎，安川宏纪．狭水路中航行船操纵性能．日本造船学会论文集，第156号
[18] 张乐文．限制水域船体间水动干扰力计算．中国造船，1990
[19] 贵岛胜郎，古川芳孝，何青．侧壁近旁航行船间相互影响．西部造船会会报，第81号
[20] Q.Y. Gui,J.M.Chuang,C.C.Hsiung. Computation of Ship Interaction Forces and Moments in Restricted Waterways Using the Numerical Conformal Mapping Method. International Shipbuilding Progress, 1990,37(412)
[21] C.C.Hsiung,Q.Y. Gui.Computing Interaction Forces and Moments on a Ship in Restricted Waterways. International Shipbuilding Progress, 1988,35(403)
[22] Q.Y.Gui,J.M.Chuang,C.C.Hsiung. A Discussion on Irregularities Which Occur in Computation of Ship Manoeuvring in a Restricted Waterway Using the Numerical Conformal Mapping Method. International Shipbuilding Progress, 1992,39(417)
[23] 许汉珍，王刚，陆小刚，等．复杂几何边界限制水域中船舶水动力的计算．华中理工大学学报，No．24，1996(7)：6-10
[24] 王刚，张戎军等．一种基于数值保角变换的计算限制水域中船舶水动力的方法．水动力学研究与进展，1996
[25] P.J.Taylor. The Blockage Coefficient for Flow About an Arbitrary Body Immersed in a Channel. Journal of Ship Research, 1973,17(6)
[26] R.F. Beck. Forces and Moments on a Ship Moving in a Shallow Channel. Journal of Ship Research,1977,21(2)
[27] A.M.J.Davis,J.F. Geer. The Application of Uniform-slender-body theory to the Motion of two Ships in Shallow Water. Journal of Fluid Mechanics, 1982,114
[28] W. Bollay. A Nonlinear Wing Theory and its Application to Rectangular Wings of Small Aspect Ratio. ZAMM,1939
[29] 井上正佑，贵岛胜郎．时船体操纵性微系数推定．西部造船会会报，1979，第55号
[30] 井上正佑，平野雅祥．等吃水船体操纵性微系数．西部造船会
会报，1979，第57号
[31] 曾效松．船体和舵相互干扰的水动力理论预报及试验研究：[硕士学位论文]．武汉：武汉水运工程学院，1990
[32] 张耀东．船体操纵运动水动力计算中的小展弦比机翼理论之改进及其应用．中国造船，1989．3
[33] 汤室彰规．An Approximate Calculation of Maneuvring Interaction Forces Between Two Ships．关西造船协会志，第212号平成元年9月
[34] 汤室彰规．An Approximate Calculation of Maneuvring Interaction Forces Between Two Ships(Continued)．关西造船协会志，第216号平成3年9月
[35] John L. Hess and A.M.O.Smith. Calculation of Non-Lifting Potential Flow about Arbitrary Three-Dimensional Bodies. Douglas Aircraft Division Report No.E.S.40622,1962
[36] John L. Hess. Calculation of Potential Flow about Arbitrary Three-Dimensional Bodies. Douglas Report NO019-71-C-0524,1972
[37] Hironori Yasukawa. Bank Effect on Ship Maneuverability in a Channel with Varying Width，西部造船会会报，1990，第81号
[38] 邹早建，黄兰萍．一种实用的浅水非线性兴波阻力计算方法．北京：人民交通出版社，1999．484～490
[39] 邹早建．穿透自由面平板横荡和首摇水动力计算．武汉交通科技大学学报，1996(6)：641～648
[40] 张谢东．限制航道中的船舶水动力研究：[硕士学位论文]．武汉：武汉水运工程学院，1989
[41] 张谢东．浅水中两船之间的相互作用力理论研究．第3届全国海事技术研讨会文集
[42] F.T. Korsmeyer, C.-H.Lee,J.N.Newman. Computation of Ship Interaction Forces in Restricted Waters. Journal of Ship Research, 1993,37(4)
[43] M.G. Nagati,J.D.Iversent,J.M.Vogel. Vortex Sheet Modeling with Curved Higher-Order Panels.Journal of Aircraft,Vol.24(11)
[44] Chiu Y. Chan,et al. Second-Order Boundary Element Method Calculations of Hydrodynamic Interactions between Particles in Close Proximity. International Journal for Numerical Methods in Fluids, 1992,14
[45] 李谊乐，刘应中，缪国平．Rankine 源高阶边界元法求解势流问题．水动力学研究与进展，1999，14(1)
[46] M.Landrini, G.Grytyr, O.M.Faltinsen.A B-Spline Based BEM for Unsteady Free-Surface Flows. Journal of Ship Research,1999,43(1)
[47] H.D.Maniar. A Three Dimensional Higher Order Panel Method Based on B-spline:[Ph.D thesis].MIT, 1995
[48] Markov, N.E.,Suzuki,K. Fundamental Studies on Rankine Source Panel Method Fully Based on B-splines. Journal of the Society of Naval Architects of Japan,Vol. 187:13～23
[49] C.-H.Lee,H.Maniar, J.Newman,X.Zhu. Computations of Waves Loads Using a B-Spline Panel Method. 21st Symposium on Naval Hydrodynamics,Washington,USA, 1997:75～92
[50] 藤斌，李玉成．波浪问题中唯一可解的高阶边界元方法．海洋工程，1996(1)：30～38
[51] 柏威，藤斌，邱大洪．基于 B-样条的边界元方法及其在波浪力计算中的应用．海洋工程，2000(4)：27～35
[52] 杜双兴，殷学文，李琪华．B-Spline 高阶元方法在三维水弹性力学中的应用．船舶力学，2000(1)：17～23
[53] 熊新民，吴秀恒．自由面和岸壁对限制航道中船舶操纵性水动力的影响，中国造船，1994(1)：34～44
[54] 余志兴，吴秀恒，张乐文．限制水域中三维船舶操纵运动水动力计算．武汉交通科技大学学报，1996(6)：677～682
[55] Tetsuji Hoshino. Hydrodynamic Analysis of Propellers in Steady Flow Using a Surface Panel Method. Journal of The Society of Naval Architects of Japan,Vol.165
[56] Tetsuji Hoshino. Hydrodynamic Analysis of Propellers in Unsteady Flow Using a Surface Panel Method. Journal of The Society of Naval Architects of Japan,Vol.174
[57] 谭廷寿，熊鹰，王德峋．面元法预报螺旋桨表面非定常压力分布．中国造船，2000，41(2)
[58] 郑伟涛．划船桨叶水动力性能的数值模拟与试验研究：[博士学位论文]．武汉：武汉交通科技大学，1999
[59] L.Bardis,M.Vafiadou. Ship-hull Geometry representation with B-spline Surface Patches. Computer Aided Design, 1992,24(4)
[60] L.Piegl, W. Tiller. Algorithm for Approximate NURBS Skinning. Computer
Aided Design, 1996,28(9)
[61] W. Tiller. Knot-removal Algorithms for NURBS Curves and Surfaces. Computer Aided Design,1992,24(8)
[62] 胡晓明，刘祖源．船舶相遇过程中船间效应的试验研究．第8届全国船舶操纵性学术讨论会论文集，2000
[63] 张兢．船舶在浅水中固定障碍物附近航行时的受力分析．武汉造船，1999(4)
[64] Haw L.Wong,Sander M.Calisal.Numerical Algorithms for Slender Bodies with Vortex Shedding and Density Stratification. Journal of Ship Research, 1996,40(1)
[65] XuQi.Numerical Simulation of 3-D Nonlinear Water Wave by Boundary Element Method. Journal Society of Naval Architects of Japan, 1989, 165
[66] Telles,J.C.F.A Self-Adaptive Co-ordinate Transformation for Efficient Numerical Evaluation of General Boundary Element Integrals. International Journal for Numerical Methods in Engineering, 1987,24
[67] Hironori Yasukawa. A Rankine Panel Method to Calculate Unsteady Ship Hydrodynamic Forces. Journal Society of Naval Architects of Japan, 1990,168
[68] Tuck,E.O. Shallow Water Flow Past Slender Bodies. Journal of Fluid Mechanics, 1966,26:81-95
[69] 水野俊明：二船并行时相互作用．日本造船学会论文集，第110号，昭 36．12
[70] 贵岛胜郎，何青：侧壁近航行船操纵运动．日本造船学会论文集，1987，第162号
[71] 余志兴．限制水域中三维船舶操纵运动水动力计算：[硕士学位论文]．武汉：武汉交通科技大学，1996
[72] 熊新民．限制水域中考虑自由表面影响的三维船舶操纵运动水动力研究：[博士学位论文]．武汉：武汉交通科技大学，1992
[73] 高高．船波问题中的辐射条件、自由面条件的研究及面元法误差分析：[博士学位论文]．武汉：武汉交通科技大学，1997
[74] 贾欣乐，杨盐生，船舶运动数学模型．大连：大连海事大学出版社，1999
[75] 吴秀恒，张乐文，王仁康．船舶操纵性与耐波性．北京：人民交通出版社，1989
[76] 范余明．船舶在波浪中的操纵运动预报：[博士学位论文]．上海：中国船舶及海洋工程设计研究院，2000
[77] 王献孚．计算流体力学．武汉水运工程学院，1981
[78] 王献孚等．计算船舶流体力学．上海：上海交通大学出版社，1992
[79] 王献孚等．机翼理论．北京：人民交通出版社，1987
[80] 王献孚．船用翼理论．北京：国防工业出版社，1998
[81] J.N. 纽曼著．周树国译．船舶流体动力学．北京：人民交通出版社，1986
[82] 徐华舫．亚、超音速定常位流的面元法．北京：国防工业出版社，1981
[83] 李美菁．船舶在风浪流及限制航道中多工况操纵运动模拟计算与操纵性能综合评判：[博士学位论文]．武汉：武汉水运工程学院，1988
[84] 戴遗山．舰船在波浪中运动的频域与时域势流理论．北京：国防工业出版社，1998
[85] 刘岳元，冯铁城，刘应中．水动力学基础．上海：上海交通大学出版社，1990
[86] 刘应中，缪国平．船舶在波浪上的运动理论．上海：上海交通大学出版社，1987
[87] 施法中．计算机辅助几何设计与非均匀有理B样条．北京：北京航空航天大学出版社，1994
[88] 孙家广．计算机图形学．北京：清华大学出版社，1998
[89] 朱心雄．自由曲线曲面造型技术，北京：科学出版社，2000
[90] 苏步青，刘鼎元．计算几何，上海：上海科学技术出版社，1980
[91] 孙家昶．样条函数与计算几何．北京：科学出版社，1982
[92] 方逵，周经伦等．曲线曲面设计技术与显示原理．长沙：国防科技大学出版社，1997
[93] [法]吉贝·德芒热，让皮尔·晡热．曲线与曲面的数学．北京：商务印书馆，2000
[94] 彭芳瑜，周济，周艳红等．基于最小二乘法的曲面生成算法研究．工程图学学报，1999(3)：41～46
[95] 张乐年，周来水，周儒荣．基于复杂曲线的 NURBS 蒙皮曲面．工程图学学报，1996(2)：58～63
[96] 张丽艳，周来水，周儒荣．裁剪 NURBS 曲面面积计算的扫描分片积分方法．工程图学学报，1997(2-3)：52～58
[97] 康宝生．有理曲面的截面设计方法．工程图学学报，1992(2)：6～12
[98] 王勤．B-Spline 曲线顶点的反算．工程图学学报，1993(2)：32～39
[99] Nakatake et al. On the Flow Field and the Hydrodynamic Forces of an
Obliquing Ship. Transcations of The West-Japan Society of Naval Architects (in Japanese), No.80,1990
[100] Klaus-Peter Beier. Highlight-line algorithm for real-time surface-quality assessment. Computer Aided Design, 1994(4):268～277
[101] Weiyin Ma, J.P. Kruth. Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces. Computer Aided Design, 1995(9):663～675
[102] Boehm W. Inserting new knots into B-spline curve.Computer Aided Design, 1980(4):199～201
[103] Boehm W., Prautzsh H. The inserting algorithm. Computer Aided Design, 1985(2):58～59
[104] H.Park, K.Kim and S-C Lee. A method for approximate NURBS curve compatibility based on multiple curve refitting. Computer Aided Design, 2000(4):237～252
[105] Guiggiani,M. A general algorithm for multi-dimensional cauchy principal value integrals in the boundary element method. Journal Applied Mechanics. Vol.57, 1990:906～915
[106] Guiggiani,M. A general algorithm for the numerical solution of hypersingular boundary integral equations. Journal Applied Mechanics. Vol.59,1992:604～614
[107] 董国祥．船舶多工况操纵运动仿真及其在初步设计中的应用．武汉造船 (增刊)，2000：13～18
[108] 乐美龙，陆惠生，汪希龄，等．船舶操纵模拟器操纵仿真数学模型．中国航海，1999(1)：75～83
[109] 於健．浅狭航道中船舶操纵性能的研究．中国航海，1996(1)：13～18
[110] C.G. Biancardi. Practical Calculation Method of Ship Manoeuvring Characteristics at the Design Stage. International Shipbuilding Progress,Vol.37,No.411(1990):221～245
[111] 邬吉明，余德浩．区间上强奇异积分的一种近似计算方法．数值计算与计算机应用，1998(2)：118～126
[112] 张效松，叶天麒．三维位势问题边界元奇异积分分析．数值计算与计算机应用，1998(4)：271～275
[113] 郭森林，党耀国．二重奇异积分近似计算的一种优化算法．工科数
学，1998(2)：84～87
[114] 蒋和理．无穷域二重积分优化中心数值算法．安徽工学院学报，1989(3)：10～17
[115] 蒋和理．二重积分优化复化梯形与辛卜生算法，合肥工业大学学报，1987(6)：34～43
[116] 蒋和理．n 重积分优化中心数值算法．合肥工业大学学报，1990(1)：15～19
[117] 古文贤．船舶操纵．大连：大连海运学院出版社，1993．139～143
[118] 黎在良，王元汉等．断裂力学中的边界数值方法．北京：地震出版社，1996
[119] 嵇醒，臧跃龙，程玉民．边界元法进展及通用程序．上海：同济大学出版社，1997
[120] 秦荣．样条边界元法，南宁：广西科学技术出版社，1988
[2] 赵成璧．船舶曲面设计现代方法与软件系统的研究：[博士学位论文]．武汉：武汉交通科技大学，1999
[3] 沈泓萃．船舶与海洋工程水动力学集成软件的开发与应用．第7届全国内河船舶及航运学术会议论文集：372～387
[4] E.O.Tuck. A Systematic Asymptotic Expansion Procedure for Slender Ship. Journal of ship Research, 1964,8(1)
[5] E.O.Tuck,J.N.Newman. Hydrodynamic Interactions between Ships.10th Symposium on Hydrodynamics, 1977
[6] E.O.Tuck. Some Methods for Flows past Blunt Slender Bodies. Journal of Fluid Mechanics, 1964,18
[7] J.N.Newman. Lateral Motion of a Slender Body between two Parallel Walls. Journal of Fluid Mechanics, 1973,39(Part 4)
[8] R.W. Yeung. On the Interactions of Slender Ship in Shallow Water. Journal of Fluid Mechanics, 1978,85(Part 1)
[9] G.W. King. Unsteady Hydrodynamic Interactions Between Ships.Journal of Ship Research, 1977,21(3)
[10] R.W. Yeung,W.Y. Hwang.Nearfield Hydrodynamic Interactions of Ship in Shallow Water. Journal of Hydronautics, 1977,11
[11] R.W. Yeung,W.T. Tan. Hrdrodynmic Interactions of Ships with Fixed Obstacles. Journal of Ship Research, 1980,24(1)
[12] R.W. Yeung. Applications of Slender-body Theory to Ships Moving in Restricted Shallow Water.Proc. Symposium on Aspects of Navigability of Constrained Waterways, 1978
[13] A.M.J.Davis. Hydrodynamic Effects of Fixed Obstacles on Ships in Shallow Water. Journal of Ship Research, 1986,30(2)
[14] 何青．船体动流体力及倾斜角影响．西部造船会会报，第79号，1988
[15] M.A.Abkowitz,G.M.Ashe,R.M.Fortson. Interaction Effects of Ships Operating in Proximity in Deep and Shallow Water. Proc. 11th Symposium
on Naval Hydrodynamics, March, 1976
[16] 郑才土．浅水中船舶相互作用的一种计算方法．船舶工程，1994．6
[17] 贵岛胜郎，安川宏纪．狭水路中航行船操纵性能．日本造船学会论文集，第156号
[18] 张乐文．限制水域船体间水动干扰力计算．中国造船，1990
[19] 贵岛胜郎，古川芳孝，何青．侧壁近旁航行船间相互影响．西部造船会会报，第81号
[20] Q.Y. Gui,J.M.Chuang,C.C.Hsiung. Computation of Ship Interaction Forces and Moments in Restricted Waterways Using the Numerical Conformal Mapping Method. International Shipbuilding Progress, 1990,37(412)
[21] C.C.Hsiung,Q.Y. Gui.Computing Interaction Forces and Moments on a Ship in Restricted Waterways. International Shipbuilding Progress, 1988,35(403)
[22] Q.Y.Gui,J.M.Chuang,C.C.Hsiung. A Discussion on Irregularities Which Occur in Computation of Ship Manoeuvring in a Restricted Waterway Using the Numerical Conformal Mapping Method. International Shipbuilding Progress, 1992,39(417)
[23] 许汉珍，王刚，陆小刚，等．复杂几何边界限制水域中船舶水动力的计算．华中理工大学学报，No．24，1996(7)：6-10
[24] 王刚，张戎军等．一种基于数值保角变换的计算限制水域中船舶水动力的方法．水动力学研究与进展，1996
[25] P.J.Taylor. The Blockage Coefficient for Flow About an Arbitrary Body Immersed in a Channel. Journal of Ship Research, 1973,17(6)
[26] R.F. Beck. Forces and Moments on a Ship Moving in a Shallow Channel. Journal of Ship Research,1977,21(2)
[27] A.M.J.Davis,J.F. Geer. The Application of Uniform-slender-body theory to the Motion of two Ships in Shallow Water. Journal of Fluid Mechanics, 1982,114
[28] W. Bollay. A Nonlinear Wing Theory and its Application to Rectangular Wings of Small Aspect Ratio. ZAMM,1939
[29] 井上正佑，贵岛胜郎．时船体操纵性微系数推定．西部造船会会报，1979，第55号
[30] 井上正佑，平野雅祥．等吃水船体操纵性微系数．西部造船会
会报，1979，第57号
[31] 曾效松．船体和舵相互干扰的水动力理论预报及试验研究：[硕士学位论文]．武汉：武汉水运工程学院，1990
[32] 张耀东．船体操纵运动水动力计算中的小展弦比机翼理论之改进及其应用．中国造船，1989．3
[33] 汤室彰规．An Approximate Calculation of Maneuvring Interaction Forces Between Two Ships．关西造船协会志，第212号平成元年9月
[34] 汤室彰规．An Approximate Calculation of Maneuvring Interaction Forces Between Two Ships(Continued)．关西造船协会志，第216号平成3年9月
[35] John L. Hess and A.M.O.Smith. Calculation of Non-Lifting Potential Flow about Arbitrary Three-Dimensional Bodies. Douglas Aircraft Division Report No.E.S.40622,1962
[36] John L. Hess. Calculation of Potential Flow about Arbitrary Three-Dimensional Bodies. Douglas Report NO019-71-C-0524,1972
[37] Hironori Yasukawa. Bank Effect on Ship Maneuverability in a Channel with Varying Width，西部造船会会报，1990，第81号
[38] 邹早建，黄兰萍．一种实用的浅水非线性兴波阻力计算方法．北京：人民交通出版社，1999．484～490
[39] 邹早建．穿透自由面平板横荡和首摇水动力计算．武汉交通科技大学学报，1996(6)：641～648
[40] 张谢东．限制航道中的船舶水动力研究：[硕士学位论文]．武汉：武汉水运工程学院，1989
[41] 张谢东．浅水中两船之间的相互作用力理论研究．第3届全国海事技术研讨会文集
[42] F.T. Korsmeyer, C.-H.Lee,J.N.Newman. Computation of Ship Interaction Forces in Restricted Waters. Journal of Ship Research, 1993,37(4)
[43] M.G. Nagati,J.D.Iversent,J.M.Vogel. Vortex Sheet Modeling with Curved Higher-Order Panels.Journal of Aircraft,Vol.24(11)
[44] Chiu Y. Chan,et al. Second-Order Boundary Element Method Calculations of Hydrodynamic Interactions between Particles in Close Proximity. International Journal for Numerical Methods in Fluids, 1992,14
[45] 李谊乐，刘应中，缪国平．Rankine 源高阶边界元法求解势流问题．水动力学研究与进展，1999，14(1)
[46] M.Landrini, G.Grytyr, O.M.Faltinsen.A B-Spline Based BEM for Unsteady Free-Surface Flows. Journal of Ship Research,1999,43(1)
[47] H.D.Maniar. A Three Dimensional Higher Order Panel Method Based on B-spline:[Ph.D thesis].MIT, 1995
[48] Markov, N.E.,Suzuki,K. Fundamental Studies on Rankine Source Panel Method Fully Based on B-splines. Journal of the Society of Naval Architects of Japan,Vol. 187:13～23
[49] C.-H.Lee,H.Maniar, J.Newman,X.Zhu. Computations of Waves Loads Using a B-Spline Panel Method. 21st Symposium on Naval Hydrodynamics,Washington,USA, 1997:75～92
[50] 藤斌，李玉成．波浪问题中唯一可解的高阶边界元方法．海洋工程，1996(1)：30～38
[51] 柏威，藤斌，邱大洪．基于 B-样条的边界元方法及其在波浪力计算中的应用．海洋工程，2000(4)：27～35
[52] 杜双兴，殷学文，李琪华．B-Spline 高阶元方法在三维水弹性力学中的应用．船舶力学，2000(1)：17～23
[53] 熊新民，吴秀恒．自由面和岸壁对限制航道中船舶操纵性水动力的影响，中国造船，1994(1)：34～44
[54] 余志兴，吴秀恒，张乐文．限制水域中三维船舶操纵运动水动力计算．武汉交通科技大学学报，1996(6)：677～682
[55] Tetsuji Hoshino. Hydrodynamic Analysis of Propellers in Steady Flow Using a Surface Panel Method. Journal of The Society of Naval Architects of Japan,Vol.165
[56] Tetsuji Hoshino. Hydrodynamic Analysis of Propellers in Unsteady Flow Using a Surface Panel Method. Journal of The Society of Naval Architects of Japan,Vol.174
[57] 谭廷寿，熊鹰，王德峋．面元法预报螺旋桨表面非定常压力分布．中国造船，2000，41(2)
[58] 郑伟涛．划船桨叶水动力性能的数值模拟与试验研究：[博士学位论文]．武汉：武汉交通科技大学，1999
[59] L.Bardis,M.Vafiadou. Ship-hull Geometry representation with B-spline Surface Patches. Computer Aided Design, 1992,24(4)
[60] L.Piegl, W. Tiller. Algorithm for Approximate NURBS Skinning. Computer
Aided Design, 1996,28(9)
[61] W. Tiller. Knot-removal Algorithms for NURBS Curves and Surfaces. Computer Aided Design,1992,24(8)
[62] 胡晓明，刘祖源．船舶相遇过程中船间效应的试验研究．第8届全国船舶操纵性学术讨论会论文集，2000
[63] 张兢．船舶在浅水中固定障碍物附近航行时的受力分析．武汉造船，1999(4)
[64] Haw L.Wong,Sander M.Calisal.Numerical Algorithms for Slender Bodies with Vortex Shedding and Density Stratification. Journal of Ship Research, 1996,40(1)
[65] XuQi.Numerical Simulation of 3-D Nonlinear Water Wave by Boundary Element Method. Journal Society of Naval Architects of Japan, 1989, 165
[66] Telles,J.C.F.A Self-Adaptive Co-ordinate Transformation for Efficient Numerical Evaluation of General Boundary Element Integrals. International Journal for Numerical Methods in Engineering, 1987,24
[67] Hironori Yasukawa. A Rankine Panel Method to Calculate Unsteady Ship Hydrodynamic Forces. Journal Society of Naval Architects of Japan, 1990,168
[68] Tuck,E.O. Shallow Water Flow Past Slender Bodies. Journal of Fluid Mechanics, 1966,26:81-95
[69] 水野俊明：二船并行时相互作用．日本造船学会论文集，第110号，昭 36．12
[70] 贵岛胜郎，何青：侧壁近航行船操纵运动．日本造船学会论文集，1987，第162号
[71] 余志兴．限制水域中三维船舶操纵运动水动力计算：[硕士学位论文]．武汉：武汉交通科技大学，1996
[72] 熊新民．限制水域中考虑自由表面影响的三维船舶操纵运动水动力研究：[博士学位论文]．武汉：武汉交通科技大学，1992
[73] 高高．船波问题中的辐射条件、自由面条件的研究及面元法误差分析：[博士学位论文]．武汉：武汉交通科技大学，1997
[74] 贾欣乐，杨盐生，船舶运动数学模型．大连：大连海事大学出版社，1999
[75] 吴秀恒，张乐文，王仁康．船舶操纵性与耐波性．北京：人民交通出版社，1989
[76] 范余明．船舶在波浪中的操纵运动预报：[博士学位论文]．上海：中国船舶及海洋工程设计研究院，2000
[77] 王献孚．计算流体力学．武汉水运工程学院，1981
[78] 王献孚等．计算船舶流体力学．上海：上海交通大学出版社，1992
[79] 王献孚等．机翼理论．北京：人民交通出版社，1987
[80] 王献孚．船用翼理论．北京：国防工业出版社，1998
[81] J.N. 纽曼著．周树国译．船舶流体动力学．北京：人民交通出版社，1986
[82] 徐华舫．亚、超音速定常位流的面元法．北京：国防工业出版社，1981
[83] 李美菁．船舶在风浪流及限制航道中多工况操纵运动模拟计算与操纵性能综合评判：[博士学位论文]．武汉：武汉水运工程学院，1988
[84] 戴遗山．舰船在波浪中运动的频域与时域势流理论．北京：国防工业出版社，1998
[85] 刘岳元，冯铁城，刘应中．水动力学基础．上海：上海交通大学出版社，1990
[86] 刘应中，缪国平．船舶在波浪上的运动理论．上海：上海交通大学出版社，1987
[87] 施法中．计算机辅助几何设计与非均匀有理B样条．北京：北京航空航天大学出版社，1994
[88] 孙家广．计算机图形学．北京：清华大学出版社，1998
[89] 朱心雄．自由曲线曲面造型技术，北京：科学出版社，2000
[90] 苏步青，刘鼎元．计算几何，上海：上海科学技术出版社，1980
[91] 孙家昶．样条函数与计算几何．北京：科学出版社，1982
[92] 方逵，周经伦等．曲线曲面设计技术与显示原理．长沙：国防科技大学出版社，1997
[93] [法]吉贝·德芒热，让皮尔·晡热．曲线与曲面的数学．北京：商务印书馆，2000
[94] 彭芳瑜，周济，周艳红等．基于最小二乘法的曲面生成算法研究．工程图学学报，1999(3)：41～46
[95] 张乐年，周来水，周儒荣．基于复杂曲线的 NURBS 蒙皮曲面．工程图学学报，1996(2)：58～63
[96] 张丽艳，周来水，周儒荣．裁剪 NURBS 曲面面积计算的扫描分片积分方法．工程图学学报，1997(2-3)：52～58
[97] 康宝生．有理曲面的截面设计方法．工程图学学报，1992(2)：6～12
[98] 王勤．B-Spline 曲线顶点的反算．工程图学学报，1993(2)：32～39
[99] Nakatake et al. On the Flow Field and the Hydrodynamic Forces of an
Obliquing Ship. Transcations of The West-Japan Society of Naval Architects (in Japanese), No.80,1990
[100] Klaus-Peter Beier. Highlight-line algorithm for real-time surface-quality assessment. Computer Aided Design, 1994(4):268～277
[101] Weiyin Ma, J.P. Kruth. Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces. Computer Aided Design, 1995(9):663～675
[102] Boehm W. Inserting new knots into B-spline curve.Computer Aided Design, 1980(4):199～201
[103] Boehm W., Prautzsh H. The inserting algorithm. Computer Aided Design, 1985(2):58～59
[104] H.Park, K.Kim and S-C Lee. A method for approximate NURBS curve compatibility based on multiple curve refitting. Computer Aided Design, 2000(4):237～252
[105] Guiggiani,M. A general algorithm for multi-dimensional cauchy principal value integrals in the boundary element method. Journal Applied Mechanics. Vol.57, 1990:906～915
[106] Guiggiani,M. A general algorithm for the numerical solution of hypersingular boundary integral equations. Journal Applied Mechanics. Vol.59,1992:604～614
[107] 董国祥．船舶多工况操纵运动仿真及其在初步设计中的应用．武汉造船 (增刊)，2000：13～18
[108] 乐美龙，陆惠生，汪希龄，等．船舶操纵模拟器操纵仿真数学模型．中国航海，1999(1)：75～83
[109] 於健．浅狭航道中船舶操纵性能的研究．中国航海，1996(1)：13～18
[110] C.G. Biancardi. Practical Calculation Method of Ship Manoeuvring Characteristics at the Design Stage. International Shipbuilding Progress,Vol.37,No.411(1990):221～245
[111] 邬吉明，余德浩．区间上强奇异积分的一种近似计算方法．数值计算与计算机应用，1998(2)：118～126
[112] 张效松，叶天麒．三维位势问题边界元奇异积分分析．数值计算与计算机应用，1998(4)：271～275
[113] 郭森林，党耀国．二重奇异积分近似计算的一种优化算法．工科数
学，1998(2)：84～87
[114] 蒋和理．无穷域二重积分优化中心数值算法．安徽工学院学报，1989(3)：10～17
[115] 蒋和理．二重积分优化复化梯形与辛卜生算法，合肥工业大学学报，1987(6)：34～43
[116] 蒋和理．n 重积分优化中心数值算法．合肥工业大学学报，1990(1)：15～19
[117] 古文贤．船舶操纵．大连：大连海运学院出版社，1993．139～143
[118] 黎在良，王元汉等．断裂力学中的边界数值方法．北京：地震出版社，1996
[119] 嵇醒，臧跃龙，程玉民．边界元法进展及通用程序．上海：同济大学出版社，1997
[120] 秦荣．样条边界元法，南宁：广西科学技术出版社，1988