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

Symplectic Runge-Kutta Method for Structural Dynamics

doi: 10.3879/j.issn.1000-0887.2014.01.002
Funds:  The National Natural Science Foundation of China(11172046;11172028;11372021)
  • Received Date: 2013-07-15
  • Rev Recd Date: 2013-10-21
  • Publish Date: 2014-01-15
  • An explicit and efficient implementation of the symplectic implicit Gauss-Legendre Runge-Kutta (RK) method of stage s and order 2s,was presented for solution of the dynamical ordinary differential equation with physical damping and external loads. The analytical explicit spectral radii and single-step phase errors of the implicit Gauss-Legendre RK method were given and compared with those of the explicit classical RK method of stage 4 and order 4. Numerical comparisons through the dynamical solution of a linear multi-degree-of-freedom (MDOF) system and a nonlinear Rayleigh system were made to validate the present study and showed the advantages of the symplectic RK method over the classical RK method with numerical dissipation, especially in aspects of the kinematic properties and long time numerical simulation.
  • loading
  • [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.
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (1230) PDF downloads(1248) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return