不规则频率的消除及其在波浪与结构物作用的高频和多物体干涉问题中的应用
|
摘要
本文使用基于高阶插值函数的边界元方法,在频域内对波浪与单个或多个物体相互作用的非线性问题进行研究,重点研究了“不规则频率”对计算结果的影响及其消除技术在实际计算中的应用。
波浪问题本身具有非线性,线性假设条件下所得的计算结果往往不能满足实际工程的需要,很多问题都必须考虑非线性的影响。波浪与多物体的相互作用时,往往会产生一些与单物体问题不同的物理现象,此时必须同时考虑波浪与多物体的相互作用和多个物体之间的相互影响。
在众多的数值方法中,边界元法很适合用来求解波浪与结构物相互作用的问题。一方面所关注的物理量分布在研究区域的边界;另一方面边界元法所使用的边界积分方程只需要在问题区域边界划分单元,既便于单元划分,又减少了计算量。但是,“不规则频率”问题的存在给边界元法的应用造成了很大的障碍,它使得计算结果在某些频率发生跳跃。本文在前人所提出的消除“不规则频率”的方法基础上,采用扩展积分方程计算区域的方法来消除“不规则频率”。对于扩展区域后的积分方程,使用连续元和部分非连续元对不同区域进行离散,使得计算模型中物理量的连续性与实际问题一致。对于二阶问题中的水面积分强迫项和物面积分强迫项,本文采取有效的方法进行了处理。数值积分计算过程中存在着大量的奇异积分和准奇异积分,根据实际情况本文采取了行之有效的方法消除或者减弱了奇异性。
本文消除“不规则频率”的方法被应用于规则波与单个或多个物体的非线性问题计算当中。文中从理论分析和数值计算两方面研究了“不规则频率”对波浪力、波高和物体运动响应的影响。同时,大量的数值结果表明本文的消除方法有效地消除了“不规则频率”的影响。在分析波浪与多物体相互作用的过程中,发现了一些可能会对海上结构物安全产生重大影响的特殊物理现象。
Boundary element method which is based on higher-order shape functions is applied to the researches of interactions between regular waves and single body or multiple bodies in the frequency domain.The emphasis is put on the effects and removal of irregular frequencies in numerical computation.
The linear assumption can not predict the effects of wave accurately in many situations because of the existence of nonlinearity.Some novel phenomena will arise when waves interact with multiple bodies because of the effect of multiple scattering.
Boundary element method is very suitable to simulate the interactions between waves and structures because the physical quantities of primary interest are only required on the boundaries of stimulation domain.On the other hand,boundary element method only needs integration on the boundaries by Using discrete elements,which has the advantages of generating meshes easily and can reduce the burden of calculation to some extent.However, irregular frequencies become a great obstacle in the application of boundary element method. Irregular frequencies will introduce sharp jump in numerical results.Extended boundary integral equations are adopted to remove the irregular frequencies.Continuous and discontinuous higher order boundary elements are used in different sub-domains according to the continuities on different boundaries.Careful considerations are taken for forcing terms on the free surface and body surface in the integral equations for second order problem. Furthermore,different strategies for numerical integration are applied to deal with singular and nearly singular integrals.
Present method for removing irregular frequencies is applied in the nonlinear analysis of interaction between regular waves and single or multiple bodies.The effect of irregular frequencies on exciting forces,wave elevations and motions of bodies is analyzed by theoretical and numerical method.At the same time,validity of present method for removing irregular frequencies is conformed by extensive numerical results.Some novel physical phenomena which can threaten the safety of structures are found in the analysis of interactions between wave and multiple bodies.
波浪问题本身具有非线性,线性假设条件下所得的计算结果往往不能满足实际工程的需要,很多问题都必须考虑非线性的影响。波浪与多物体的相互作用时,往往会产生一些与单物体问题不同的物理现象,此时必须同时考虑波浪与多物体的相互作用和多个物体之间的相互影响。
在众多的数值方法中,边界元法很适合用来求解波浪与结构物相互作用的问题。一方面所关注的物理量分布在研究区域的边界;另一方面边界元法所使用的边界积分方程只需要在问题区域边界划分单元,既便于单元划分,又减少了计算量。但是,“不规则频率”问题的存在给边界元法的应用造成了很大的障碍,它使得计算结果在某些频率发生跳跃。本文在前人所提出的消除“不规则频率”的方法基础上,采用扩展积分方程计算区域的方法来消除“不规则频率”。对于扩展区域后的积分方程,使用连续元和部分非连续元对不同区域进行离散,使得计算模型中物理量的连续性与实际问题一致。对于二阶问题中的水面积分强迫项和物面积分强迫项,本文采取有效的方法进行了处理。数值积分计算过程中存在着大量的奇异积分和准奇异积分,根据实际情况本文采取了行之有效的方法消除或者减弱了奇异性。
本文消除“不规则频率”的方法被应用于规则波与单个或多个物体的非线性问题计算当中。文中从理论分析和数值计算两方面研究了“不规则频率”对波浪力、波高和物体运动响应的影响。同时,大量的数值结果表明本文的消除方法有效地消除了“不规则频率”的影响。在分析波浪与多物体相互作用的过程中,发现了一些可能会对海上结构物安全产生重大影响的特殊物理现象。
Boundary element method which is based on higher-order shape functions is applied to the researches of interactions between regular waves and single body or multiple bodies in the frequency domain.The emphasis is put on the effects and removal of irregular frequencies in numerical computation.
The linear assumption can not predict the effects of wave accurately in many situations because of the existence of nonlinearity.Some novel phenomena will arise when waves interact with multiple bodies because of the effect of multiple scattering.
Boundary element method is very suitable to simulate the interactions between waves and structures because the physical quantities of primary interest are only required on the boundaries of stimulation domain.On the other hand,boundary element method only needs integration on the boundaries by Using discrete elements,which has the advantages of generating meshes easily and can reduce the burden of calculation to some extent.However, irregular frequencies become a great obstacle in the application of boundary element method. Irregular frequencies will introduce sharp jump in numerical results.Extended boundary integral equations are adopted to remove the irregular frequencies.Continuous and discontinuous higher order boundary elements are used in different sub-domains according to the continuities on different boundaries.Careful considerations are taken for forcing terms on the free surface and body surface in the integral equations for second order problem. Furthermore,different strategies for numerical integration are applied to deal with singular and nearly singular integrals.
Present method for removing irregular frequencies is applied in the nonlinear analysis of interaction between regular waves and single or multiple bodies.The effect of irregular frequencies on exciting forces,wave elevations and motions of bodies is analyzed by theoretical and numerical method.At the same time,validity of present method for removing irregular frequencies is conformed by extensive numerical results.Some novel physical phenomena which can threaten the safety of structures are found in the analysis of interactions between wave and multiple bodies.
引文
[1]Mei C C.Numerical methods in water-wave diffraction and radiation.Annual Review of Fluid Mechanics,1978,10:393-416.
[2]Faltinsen O M.Wave load on the offshore structures.Annual Review of Fluid Mechanics,1990,22:35-56.
[3]Eatock Taylor R,Kernot M P.On Second Order Wave Loading and Response in Irregular Seas.Advances in Coastal and Ocean Engineering,World Scientific,1999:155-212.
[4]Newman J N.Wave effects on multiple bodies.Hydrodynamics in Ship and Ocean Engineering,2001:3-26.
[5]McIver P.Wave interaction with arrays of structures.Applied Ocean Research,2002(24):121-126
[6]John F.On the motion of floating bodies Ⅱ.Comm.Pure Appl.Math.3.1950:45-101.
[7]Hess J L,Smith A M O.Calculation of nonlifting potential flow about arbitrary three-dimensional bodies.J.Ship Res.,1964,8(2):22-44.
[8]Garrison C J.Hydrodynamic loading of large volume offshore structures:three-dimensional source distribution methods.Num.Meth.in Offshore Eng.Chichester:A Wiley-Interscience Publication,1978:87-140.
[9]Korsmeyer F T,Lee C H,Newman J N,Sclavounous P D.The analysis of wave effects on tension-leg platform.Proc.Offshore Mech.& Arctic Eng.,1988,2:1-15.
[10]Lee C.H.Numerical methods for boundary integral equations in wave body interactions,Ph.D.Thesis,Department of Ocean Engineering,MIT,1988.
[11]Maniar H.A three dimensional higher order panel method based on B-splines.,Ph.D.thesis,Dept.of Ocean Engineering,MIT,1995.
[12]Lee C H,Maniar,H D,Newman,J N,Zhu X.Computations of wave loads using a B-spline panel method.Proceedings of 21st Symposium on Naval Hydrodynamics,National Academy Press:Washington,1996:75-92.
[13]Newman J N,Lee C H.Boundary-Element Methods in Offshore Structure Analysis.Proceedings of 20th OMAE Conference,Rio de Janeiro,Brazil 2001 Also published in Journal of Offshore Mechanics and Arctic Engineering,2002,124:81-89.
[14]Liu Y H,Kim M H,Kim C H.The Computation of Second-Order Mean and Double-Frequency Wave Loads on a Compliant TLP by HOBEM.International Journal of Offshore and Polar Engineering,1995,5(2):111-119.
[15]Eatock Taylor R,Chau F P.Wave diffraction theory-some developments in linear and nonlinear theory.Journal of Offshore Mechanics and Arctic Engineering,1992,114:185-194.
[16]Teng B,Eatock Taylor R.New higher order boundary method for wave diffraction/radiation.Applied Ocean Research,1995,17:71-77.
[17]Chau F P.The second order velocity potential for diffraction of waves by fixed offshore structures.Ph.D.thesis,University of London,1989.(Department of Mechanical Engineering,University College London,Rep.OEG/89/1)
[18]Subia S R,Ingber M S,Mitra A K.A comparison of the semidiscontinuous element and multiple node with auxiliary boundary collocation approaches for the boundary element method.Engineering Analysis with Boundary Elements 1995,15:19-27.
[19]Brebbia C A.The boundary element method for engineers.Wley:New York,1978.
[20]Liu Y H,Kim C H,Kim M H.The Computation of mean drift forces and wave run-up by higher-order boundary element method.Proc.Inter.Offshore and Polar Eng.Conf,1990,3:476-481.
[21]Teng B,Gou Y,Ning D Z.A Higher Order BEM for Wave-Current Action on Structures--Direct Computation of Free-Term Coefficient and CPV Integrals.China Ocean Engineering,2006,20(3):395-410.
[22]Newman,J.N.Algorithms for the Free-Surface Green Function.J.Engineering Mathematics,1985,19:57-67.
[23]胡圣荣,陈国华.三维非规则非均匀边界元网格的简便的高精度算法.固体力学学报,1996,17(4):343-347.
[24]威跃龙,嵇醒.关于边界元法中奇异积分的处理.固体力学学报,1994,15(2):161-165.
[25]张效松,叶天麟.二维边界元奇异积分和多域缩聚法分析.计算力学学报,1997,14(2):196-203.
[26]张效松,叶天麟.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.
[27]Li H B,Han G M,Mang H A.A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method,Int.Journal for Numerical Methods in Engineering,1985,21:2071-2098.
[28]Telles J C F.A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals,Int.Journal for Numerical Methods in Engineering,1987,24:959-973.
[29]Zhu X.Irregular frequency removal from the boundary integral equation for the wave-body problem.Master Thesis,Dept.of Ocean Engineering,MIT,1994.
[30]Rokhlin V.Rapid solution of integral equations of classical potential theory.J.Comp.Phys.,1985,60:187-207.
[31]Phillips J R,White J K.A precorrected-FFT method for electrostatic analysis of complicated 3-D structures.IEEE Trans.Comput.-Aided Des.,1997,16(10):059-1072.
[32]Korsmeyer F T,Klemas T J,White J K,Phillips J R.Fast hydrodynamic analysis of large offshore structures.Proc.of Int.Symp.on Offshore and Polar Engineering,ISOPE'99,Brest,France,1999.
[33]GiD User Manual.International Center for Numerical Methods in Engineering.(available from the website http://gid.cimne.upc.es/).
[34]GAMBIT User's Guide.Fluent,Inc.(available from the website http://www.fluent.com/software /gambit/).
[35]MultiSurf User's Manual.AeroHydro,Inc.(available from the website http://www.aerohydro.com /products/marine/multisurf.htm).
[36]Lee C H,Letcher J S,Mack R G,Newman J N,Shook D M,Stanley E.Integration of Geometry Definition and Wave Analysis Software.Proceedings of 21st Internal Conference on Offshore Mech.and Arctic Engineering,ASME:New York,2002.
[37]Lamb H.Hydrodynamics.New York:Dover,1945.
[38]Frank W.Oscillation of cylinders in or below the free surface of deep fluids.Naval Ship Res.Devel.Cent.,Report 2375,1967.
[39]Wu X J,Price W G.An equivalent box approximation to predict irregular frequencies in arbitrarilyshaped three-dimensional marine structures.Applied Ocean Research,1986,8(4):223-231.
[40]Mei C C.The applied dynamics of ocean surface waves.Singapore:World Scientific,1989.
[41]Cline A K,Moler C B,Stewart G W,Wilkinson J H.An estimate for the condition number of a matrix.SIAM Journal on Numerical Analysis.1979,16:368-375.
[42]Ogilvie T F,Shin Y S.Integral-equation solutions for time-dependent free-surface problems.J.Soc.Nav.Arch.Japan,1978,143:86-96.
[43]Ursell F.Irregular frequencies and the motion of floating bodies.J.Fluid Mech.1981,105:143-156.
[44]Ohmatsu S.A new simple method to eliminate the irregular frequencies in the theory water wave radiation problem.Ship Res.Inst.,Japan(Papers,No.70),1983.
[45]段文洋,贺五洲.一种消除水动力求解中不规则频率影响的边界元法.水动力学研究与进展(A 辑),2002,17(2):156-161.
[46]Lee C H,Sclavounos P D.Removing the irregular frequencies from integral equations in wave-body interaction.J.Fluid Mech.,1989,207:393-418.
[47]Kleinman R E.On the mathematical theory of the motion of floating bodies-an update.DTNSRDC Report 82/074,1982.
[48]Lee C H,Newman J N,Zhu X.An extended boundary integral equation method for the removal of irregular frequency effects.Int.Journal of Numerical Methods in Fluids,1996,23:637-660.
[49]Teng B,Li Y C.A unique solvable higher order BEM for wave diffraction and radiation.China Ocean Engineering,1996,10(3):333-342.
[50]Yeung R W.A hybrid integral-equation method for the time-harmonic free-surface flows.In Proceedings:First International Conference on Numerical Ship Hydrodynamics,Gaithersburg,Maryland,1975:581-607.
[51]Bai K J.A variational method in potential flows with a free surface.Ph.D.dissertation,Department of Naval Architecture,University of California,Berkeley,1972.
[52]李玉成,滕斌.波浪对海上建筑物的作用(第2版).北京:海洋出版社,2002.
[53]Kim M H,Yue D K P.The complete second-order diffraction solution for an axisymmetric body.Part 2.Bichromatic incident waves and body motions.Journal of Fluid Mechanics,1990,211:557-593.
[54]Ogilvie T F.Second-order hydrodynamic effects on ocean platforms.Proc.Intl.Workshop on Ship and Platform Motions,ed.Yeung R W.University of California,Berkeley,1983:205-265.
[55]Kim M H,Yue D K P.The complete second-order diffraction solution for an axisymmetric body.Part 1.Monochromatic incident waves.Journal of Fluid Mechanics,1989,200:235-264.
[56]Huang Q,Cruse T A.Some notes on singular integral techniques in boundary element analysis.Int.Journal for Numerical Methods in Engineering,1993,36:2643-2659.
[57]MacCamy R C,Fuchs R A.Wave forces on piles:a diffraction theory.Tech.Mem.,69,US Army Coastal Engineering Research tenter,1954.
[58]滕斌.波浪对三维浮体的二阶作用.水动力学研究与进展:A辑,1995,3:316-327.
[59]Garret C J R.Wave forces on a circular dock.Journal of Fluid Mechanics,1971,46:129-139.
[60]Eatock Taylor R,Hung S M.Second order diffraction forces on a vertical cylinder in regular waves.Applied Ocean Research,1987,9:19-30.
[61]Ghalayini S A,Williams A N.Nonlinear wave forces on vertical cylinder arrays.Journal of Fluids and Structures,1991,5:1-32.
[62]Evans D V,Porter R.Near trapping of waves by circular arrays of vertical cylinders.Applied Ocean Research,1997,19:83-99.
[63]Malenica S,Eatock Taylor R,Huang J B.Second-order water wave diffraction by an array of vertical cylinders.Journal of Fluid Mechanics,1999,390:349-373.
[64]Teigen P,Niedzwecki J M.A computational study of wave effects related to side-by-side LNG offloading.Proc.16th Int.Offshore and Polar Engineering Conf.,San Francisco,USA,2006.
[65]Lee C H,Newman J N.Computation of wave effects using the panel method.In:Chakrabarti S ed.Numerical Models in Fluid-Structure Interaction,Southampton:WIT Press,2005:211-251.
[66]Pauw W H,Huijsmans,R H M,Voogt A.Advances in the hydrodynamics of side-by-side moored vessels.Proc.26th Int.Conf.Offshore Mechanics and Arctic Engineering,OMAE 2007-29374,San Diego,USA,2007.
[67]Kristiansen T,Faltinsen O M.Resonant water motion between a ship and a terminal in shallow water.Proc.26th Int.Conf.Offshore Mechanics and Arctic Engineering,OMAE 2007-29397,San Diego,USA,2007.
[2]Faltinsen O M.Wave load on the offshore structures.Annual Review of Fluid Mechanics,1990,22:35-56.
[3]Eatock Taylor R,Kernot M P.On Second Order Wave Loading and Response in Irregular Seas.Advances in Coastal and Ocean Engineering,World Scientific,1999:155-212.
[4]Newman J N.Wave effects on multiple bodies.Hydrodynamics in Ship and Ocean Engineering,2001:3-26.
[5]McIver P.Wave interaction with arrays of structures.Applied Ocean Research,2002(24):121-126
[6]John F.On the motion of floating bodies Ⅱ.Comm.Pure Appl.Math.3.1950:45-101.
[7]Hess J L,Smith A M O.Calculation of nonlifting potential flow about arbitrary three-dimensional bodies.J.Ship Res.,1964,8(2):22-44.
[8]Garrison C J.Hydrodynamic loading of large volume offshore structures:three-dimensional source distribution methods.Num.Meth.in Offshore Eng.Chichester:A Wiley-Interscience Publication,1978:87-140.
[9]Korsmeyer F T,Lee C H,Newman J N,Sclavounous P D.The analysis of wave effects on tension-leg platform.Proc.Offshore Mech.& Arctic Eng.,1988,2:1-15.
[10]Lee C.H.Numerical methods for boundary integral equations in wave body interactions,Ph.D.Thesis,Department of Ocean Engineering,MIT,1988.
[11]Maniar H.A three dimensional higher order panel method based on B-splines.,Ph.D.thesis,Dept.of Ocean Engineering,MIT,1995.
[12]Lee C H,Maniar,H D,Newman,J N,Zhu X.Computations of wave loads using a B-spline panel method.Proceedings of 21st Symposium on Naval Hydrodynamics,National Academy Press:Washington,1996:75-92.
[13]Newman J N,Lee C H.Boundary-Element Methods in Offshore Structure Analysis.Proceedings of 20th OMAE Conference,Rio de Janeiro,Brazil 2001 Also published in Journal of Offshore Mechanics and Arctic Engineering,2002,124:81-89.
[14]Liu Y H,Kim M H,Kim C H.The Computation of Second-Order Mean and Double-Frequency Wave Loads on a Compliant TLP by HOBEM.International Journal of Offshore and Polar Engineering,1995,5(2):111-119.
[15]Eatock Taylor R,Chau F P.Wave diffraction theory-some developments in linear and nonlinear theory.Journal of Offshore Mechanics and Arctic Engineering,1992,114:185-194.
[16]Teng B,Eatock Taylor R.New higher order boundary method for wave diffraction/radiation.Applied Ocean Research,1995,17:71-77.
[17]Chau F P.The second order velocity potential for diffraction of waves by fixed offshore structures.Ph.D.thesis,University of London,1989.(Department of Mechanical Engineering,University College London,Rep.OEG/89/1)
[18]Subia S R,Ingber M S,Mitra A K.A comparison of the semidiscontinuous element and multiple node with auxiliary boundary collocation approaches for the boundary element method.Engineering Analysis with Boundary Elements 1995,15:19-27.
[19]Brebbia C A.The boundary element method for engineers.Wley:New York,1978.
[20]Liu Y H,Kim C H,Kim M H.The Computation of mean drift forces and wave run-up by higher-order boundary element method.Proc.Inter.Offshore and Polar Eng.Conf,1990,3:476-481.
[21]Teng B,Gou Y,Ning D Z.A Higher Order BEM for Wave-Current Action on Structures--Direct Computation of Free-Term Coefficient and CPV Integrals.China Ocean Engineering,2006,20(3):395-410.
[22]Newman,J.N.Algorithms for the Free-Surface Green Function.J.Engineering Mathematics,1985,19:57-67.
[23]胡圣荣,陈国华.三维非规则非均匀边界元网格的简便的高精度算法.固体力学学报,1996,17(4):343-347.
[24]威跃龙,嵇醒.关于边界元法中奇异积分的处理.固体力学学报,1994,15(2):161-165.
[25]张效松,叶天麟.二维边界元奇异积分和多域缩聚法分析.计算力学学报,1997,14(2):196-203.
[26]张效松,叶天麟.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.
[27]Li H B,Han G M,Mang H A.A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method,Int.Journal for Numerical Methods in Engineering,1985,21:2071-2098.
[28]Telles J C F.A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals,Int.Journal for Numerical Methods in Engineering,1987,24:959-973.
[29]Zhu X.Irregular frequency removal from the boundary integral equation for the wave-body problem.Master Thesis,Dept.of Ocean Engineering,MIT,1994.
[30]Rokhlin V.Rapid solution of integral equations of classical potential theory.J.Comp.Phys.,1985,60:187-207.
[31]Phillips J R,White J K.A precorrected-FFT method for electrostatic analysis of complicated 3-D structures.IEEE Trans.Comput.-Aided Des.,1997,16(10):059-1072.
[32]Korsmeyer F T,Klemas T J,White J K,Phillips J R.Fast hydrodynamic analysis of large offshore structures.Proc.of Int.Symp.on Offshore and Polar Engineering,ISOPE'99,Brest,France,1999.
[33]GiD User Manual.International Center for Numerical Methods in Engineering.(available from the website http://gid.cimne.upc.es/).
[34]GAMBIT User's Guide.Fluent,Inc.(available from the website http://www.fluent.com/software /gambit/).
[35]MultiSurf User's Manual.AeroHydro,Inc.(available from the website http://www.aerohydro.com /products/marine/multisurf.htm).
[36]Lee C H,Letcher J S,Mack R G,Newman J N,Shook D M,Stanley E.Integration of Geometry Definition and Wave Analysis Software.Proceedings of 21st Internal Conference on Offshore Mech.and Arctic Engineering,ASME:New York,2002.
[37]Lamb H.Hydrodynamics.New York:Dover,1945.
[38]Frank W.Oscillation of cylinders in or below the free surface of deep fluids.Naval Ship Res.Devel.Cent.,Report 2375,1967.
[39]Wu X J,Price W G.An equivalent box approximation to predict irregular frequencies in arbitrarilyshaped three-dimensional marine structures.Applied Ocean Research,1986,8(4):223-231.
[40]Mei C C.The applied dynamics of ocean surface waves.Singapore:World Scientific,1989.
[41]Cline A K,Moler C B,Stewart G W,Wilkinson J H.An estimate for the condition number of a matrix.SIAM Journal on Numerical Analysis.1979,16:368-375.
[42]Ogilvie T F,Shin Y S.Integral-equation solutions for time-dependent free-surface problems.J.Soc.Nav.Arch.Japan,1978,143:86-96.
[43]Ursell F.Irregular frequencies and the motion of floating bodies.J.Fluid Mech.1981,105:143-156.
[44]Ohmatsu S.A new simple method to eliminate the irregular frequencies in the theory water wave radiation problem.Ship Res.Inst.,Japan(Papers,No.70),1983.
[45]段文洋,贺五洲.一种消除水动力求解中不规则频率影响的边界元法.水动力学研究与进展(A 辑),2002,17(2):156-161.
[46]Lee C H,Sclavounos P D.Removing the irregular frequencies from integral equations in wave-body interaction.J.Fluid Mech.,1989,207:393-418.
[47]Kleinman R E.On the mathematical theory of the motion of floating bodies-an update.DTNSRDC Report 82/074,1982.
[48]Lee C H,Newman J N,Zhu X.An extended boundary integral equation method for the removal of irregular frequency effects.Int.Journal of Numerical Methods in Fluids,1996,23:637-660.
[49]Teng B,Li Y C.A unique solvable higher order BEM for wave diffraction and radiation.China Ocean Engineering,1996,10(3):333-342.
[50]Yeung R W.A hybrid integral-equation method for the time-harmonic free-surface flows.In Proceedings:First International Conference on Numerical Ship Hydrodynamics,Gaithersburg,Maryland,1975:581-607.
[51]Bai K J.A variational method in potential flows with a free surface.Ph.D.dissertation,Department of Naval Architecture,University of California,Berkeley,1972.
[52]李玉成,滕斌.波浪对海上建筑物的作用(第2版).北京:海洋出版社,2002.
[53]Kim M H,Yue D K P.The complete second-order diffraction solution for an axisymmetric body.Part 2.Bichromatic incident waves and body motions.Journal of Fluid Mechanics,1990,211:557-593.
[54]Ogilvie T F.Second-order hydrodynamic effects on ocean platforms.Proc.Intl.Workshop on Ship and Platform Motions,ed.Yeung R W.University of California,Berkeley,1983:205-265.
[55]Kim M H,Yue D K P.The complete second-order diffraction solution for an axisymmetric body.Part 1.Monochromatic incident waves.Journal of Fluid Mechanics,1989,200:235-264.
[56]Huang Q,Cruse T A.Some notes on singular integral techniques in boundary element analysis.Int.Journal for Numerical Methods in Engineering,1993,36:2643-2659.
[57]MacCamy R C,Fuchs R A.Wave forces on piles:a diffraction theory.Tech.Mem.,69,US Army Coastal Engineering Research tenter,1954.
[58]滕斌.波浪对三维浮体的二阶作用.水动力学研究与进展:A辑,1995,3:316-327.
[59]Garret C J R.Wave forces on a circular dock.Journal of Fluid Mechanics,1971,46:129-139.
[60]Eatock Taylor R,Hung S M.Second order diffraction forces on a vertical cylinder in regular waves.Applied Ocean Research,1987,9:19-30.
[61]Ghalayini S A,Williams A N.Nonlinear wave forces on vertical cylinder arrays.Journal of Fluids and Structures,1991,5:1-32.
[62]Evans D V,Porter R.Near trapping of waves by circular arrays of vertical cylinders.Applied Ocean Research,1997,19:83-99.
[63]Malenica S,Eatock Taylor R,Huang J B.Second-order water wave diffraction by an array of vertical cylinders.Journal of Fluid Mechanics,1999,390:349-373.
[64]Teigen P,Niedzwecki J M.A computational study of wave effects related to side-by-side LNG offloading.Proc.16th Int.Offshore and Polar Engineering Conf.,San Francisco,USA,2006.
[65]Lee C H,Newman J N.Computation of wave effects using the panel method.In:Chakrabarti S ed.Numerical Models in Fluid-Structure Interaction,Southampton:WIT Press,2005:211-251.
[66]Pauw W H,Huijsmans,R H M,Voogt A.Advances in the hydrodynamics of side-by-side moored vessels.Proc.26th Int.Conf.Offshore Mechanics and Arctic Engineering,OMAE 2007-29374,San Diego,USA,2007.
[67]Kristiansen T,Faltinsen O M.Resonant water motion between a ship and a terminal in shallow water.Proc.26th Int.Conf.Offshore Mechanics and Arctic Engineering,OMAE 2007-29397,San Diego,USA,2007.