## 留言板

 引用本文: 郭静, 邢誉峰. 结构动力学方程的辛RK方法[J]. 应用数学和力学, 2014, 35(1): 12-21.
GUO Jing, XING Yu-feng. Symplectic Runge-Kutta Method for Structural Dynamics[J]. Applied Mathematics and Mechanics, 2014, 35(1): 12-21. doi: 10.3879/j.issn.1000-0887.2014.01.002
 Citation: GUO Jing, XING Yu-feng. Symplectic Runge-Kutta Method for Structural Dynamics[J]. Applied Mathematics and Mechanics, 2014, 35(1): 12-21.

## 结构动力学方程的辛RK方法

##### doi: 10.3879/j.issn.1000-0887.2014.01.002

###### 作者简介:郭静（1983—），女，石家庄人，工程师，博士(通讯作者. Tel: +86-10-68384847; E-mail: guojing2662632@126.com
• 中图分类号: O342;TU311.3

## Symplectic Runge-Kutta Method for Structural Dynamics

Funds: The National Natural Science Foundation of China(11172046;11172028;11372021)
• 摘要: 针对有阻尼和外载荷的线性动力学常微分方程，给出了s级2s阶隐式Gauss-Legendre辛RK(Gauss-Legendre symplectic Runge-Kutta, GLSRK)方法的一种显式高效的执行格式，首次给出了Gauss-Legendre辛RK方法和经典RK方法(classical RK, CRK)的谱半径和单步相位误差的显式表达式，并将两者进行了比较．线性多自由度系统和非线性Rayleigh系统数值算例表明，对结构动力学系统而言，辛RK方法远比经典RK方法优越，在运动学特性和长时间数值模拟方面尤为明显.
•  [1] de Vogelaere R. Methods of integration which preserve the contact transformation property of the Hamiltonian equations[R]. Notre Dame: Department of Mathematics, University of Notre Dame, Report No 4, N7-ONR-43906, 1956. [2] Ruth R. A canonical integration technique[J]. IEEE Transactions on Nuclear Science,1983,30(4): 2669-2671. [3] FENG Kang. On difference schemes and symplectic geometry[C]// Proceedings of the 5th International Symposium on Differential Geometry and Differential Equations.Beijing, 1984: 42-58. [4] Sanz-Serna J M, Calvo M P. Numerical Hamiltonian Problems [M]. London: Chapman and Hall Press, 1994. [5] Lasagni F M. Canonical Runge-Kutta methods[J]. Journal of Applied Mathematics and Physis(ZAMP),1988,39(6): 952-953. [6] Sanz-Serna J M. Runge-Kutta schemes for Hamiltonian systems[J]. BIT Numerical Mathematics,1988,28(4): 877-883. [7] Suris Y B. On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems[C]//Filippov S S ed. Numerical Solution of Ordinary Differential Equations.Moscow: Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, 1988: 148-160.(in Russian) [8] Burrage K, Butcher J C. Stability criteria for implicit Runge-Kutta methods[J]. SIAM Journal on Numerical Analysis,1979,16(1): 46-57. [9] Crouzeix M. Sur la B-stabilité des méthods de Runge-Kutta[J].Numerische Mathematik,1979,32(1): 75-82. [10] Saito S, Sugiura H, Mitsui T. Family of symplectic implicit Runge-Kutta formulae[J]. BIT Numerical Mathematics,1992,32(3): 539-543. [11] Sanz-Serna J M, Abia L. Order conditions for canonical Runge-Kutta schemes[J]. SIMA Journal on Numerical Analysis,1991,28(4): 1081-1096. [12] Abia L, Sanz-Serna J M. Partitioned Runge-Kutta methods for separable Hamiltonian problems[J]. Mathematics of Computation,1993,60(202): 617-634. [13] Sun G. A simple way constructing symplectic Runge-Kutta methods[J]. Journal of Computational Mathematics,2000,18(1): 61-68. [14] Grimm V, Scherer R. A generalized W-transformation for constructing symplectic partitioned Runge-Kutta methods[J]. BIT Numerical Mathematics,2003,43(1): 57-66. [15] Monovasilis T, Kalogiratou Z, Simos T E. Symplectic partitioned Runge-Kutta methods with minimal phase-lag[J]. Computer Physics Communications,2010,181(7): 1251-1254. [16] Suris Y B. On the canonicity of mappings that can be generated by methods of Runge-Kutta type for integrating system x〖DD(-1〗¨〖DD)〗=U/x[J].Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki,1989,29(2): 202-211, 317.(in Russian) [17] Okunbor D, Skeel R D. An explicit Runge-Kutta-Nystrom method in canonical if and only if its adjoint is explicit[J]. SIAM Journal on Numerical Analysis,1992,29(2): 521-527. [18] Ramaswami G. Perturbed collocation and symplectic RKN methods[J]. Advances in Computational Mathematics,1995,3(1/2): 23-40. [19] Tsitouras C H. A tenth order symplectic Runge-Kutta-Nystrom method[J]. Celestial Mechanics and Dynamical Astronomy,1999,74(4): 223-230. [20] Van de Vyrer H. A symplectic Runge-Kutta-Nystrom method with minimal phase lag[J]. Physics Letters A,2007,367(1/2): 16-24. [21] Iserles A. Efficient Runge-Kutta methods for Hamiltonian equations[J]. Bulletin Greek Mathematical Society,1991,32: 3-20. [22] Sun G. Construction of high order symplectic Runge-Kutta methods[J]. Journal of Computational Mathematics,1993,11(3): 250-260. [23] Hairer E, Wanner G. Scientific notes: symplectic Runge-Kutta methods with real eigenvalues[J]. BIT Numerical Mathematics,1994,34(2): 310-312. [24] Simos T E, Vigo-Aguiar J. Exponentially fitted symplectic integrator[J]. Physical Review E,2003,67(1): 016701. [25] Monovasilis T, Kalogiratou Z, Simos T E. Exponentially fitted symplectic methods for the numerical integration of the Schrodinger equation[J]. Journal of Mathematical Chemistry,2005,37(3): 263-270. [26] Tocino A, Vigo-Aguiar J. Symplectic conditions for exponential fitting Runge-Kutta-Nystrom methods[J]. Mathematical and Computer Modelling,2005,42(7/8): 873-876. [27] Van de Vyver H. A fourth order symplectic exponentially fitted integrator[J]. Computer Physics Communications,2006,174(4): 255-262. [28] Gladman B, Duncan M, Candy J. Symplectic integrators for long-term integrations in celestial mechanics[J]. Celestial Mechanics and Dynamical Astronomy,1991,52(3): 221-240. [29] Kinoshita H, Yoshida H, Nakai H. Symplectic integrators and their application to dynamical astronomy[J]. Celestial Mechanics and Dynamical Astronomy,1991,50(1): 59-71. [30] Gray S, Manolopoulos D E. Symplectic integrators tailored to the time-dependent Schrodinger equation[J]. Journal of Chemical Physics,1996,104(18): 7099-7112. [31] Cary J R, Doxas J. An explicit symplectic integration scheme for plasma simulations[J]. Journal of Computational Physics,1993,107(1): 98-104. [32] Dragt A J. Computation of maps for particle and light optics by scaling, splitting and squaring[J]. Physical Review Letters,1995,75(10): 1946-1948. [33] Channell P J, Scovel C. Symplectic integration of Hamiltonian systems[J]. Nonlinearity,1990,3(2): 231-259. [34] 邢誉峰, 杨蓉. 动力学平衡方程的Euler中点辛差分求解格式[J]. 力学学报, 2007,39(1): 100-105.(XING Yu-feng, YANG Rong. Application of Euler midpoint symplectic integration method for the solution of dynamic equilibrium equations[J]. Chinese Journal of Theoretical and Applied Mechanics,2007,39(1): 100-105.(in Chinese)) [35] Bathe K J, Wilson E L. Numerical Methods in Finite Element Analysis [M]. New Jersey, Englewood Cliffs: Prentice-Hall, 1976. [36] Hughes T J R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis[M]. New Jersey, Englewood Cliffs: Prentice-Hall, 1987.
##### 计量
• 文章访问数:  1326
• HTML全文浏览量:  150
• PDF下载量:  1249
• 被引次数: 0
##### 出版历程
• 收稿日期:  2013-07-15
• 修回日期:  2013-10-21
• 刊出日期:  2014-01-15

/

• 分享
• 用微信扫码二维码

分享至好友和朋友圈