非线性动力学系统的精细逐块积分求解方法
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摘要
针对非线性动力学系统提出了一种精细逐块求解的积分方法。通过引入逐块积分格式和精细积分算法,典型的非线性动力微分方程最终可被转换为容易求解的逐块代数方程组。由于这种隐式积分格式具有高精度和稳定性,相比于四阶Runge-Kutta方法和Newmark方法,此方法可以对非线性动力系统应用较大的步长。此外,此方法对具有奇异或接近奇异的系统矩阵的动力学系统仍然有效。数值算例验证了此方法的有效性。
This paper proposes a precise block-by-block integration method for nonlinear dynamic systems.By applying the block-by-block method and the precision integration scheme,the classic nonlinear differential governing equations are converted into algebraic equations within each block,which is favorable to solve.Due to the high accuracy and stability of this implicit integration scheme,a large step size can be utilized for nonlinear dynamic systems comparing with those of the fourth order Runge-Kutta method and the Newmark method.In addition,the proposed method is also effective for dynamic systems with a singular or close to singular system matrix.Numerical examples are given to verify the proposed method.
This paper proposes a precise block-by-block integration method for nonlinear dynamic systems.By applying the block-by-block method and the precision integration scheme,the classic nonlinear differential governing equations are converted into algebraic equations within each block,which is favorable to solve.Due to the high accuracy and stability of this implicit integration scheme,a large step size can be utilized for nonlinear dynamic systems comparing with those of the fourth order Runge-Kutta method and the Newmark method.In addition,the proposed method is also effective for dynamic systems with a singular or close to singular system matrix.Numerical examples are given to verify the proposed method.
引文
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[16]WANG M F,AU F T K.Precise integration method without inverse matrix calculation for structural dynamic equations[J].Earthquake Engineering and Engineering Vibration,2007,6(1):57-64.
[17]GU Y X,CHEN B S,ZHANG H W,et al.Precise time-integration method with dimensional expanding for structural dynamic equations[J].AIAA Journal,2001,39(12):2394-2399.
[18]LI K,DARBY A P.A high precision direct integration scheme for nonlinear dynamic systems[J].Journal of Computational and Nonlinear Dynamics,2009,4(4):1-10.
[19]BERRUT J P,TREFETHEN L N.Barycentric Lagrange interpolation[J].SIAM Review,2004,46(3):501-517.
[20]GARCIA C B,ZANGWILL W I.Pathways to solutions,fixed points,and equilibria[M].New Jersey:PrenticeHall,1981:67.
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[22]WANG M F,ZHOU X Y.Modified precise time step integration method of structural dynamic analysis[J].Earthquake Engineering and Engineering Vibration,2005,4(2):287-293.
[23]HAIRER E,LUBICH C,WANNER G.Geometric Numerical Integration:Structure-preserving algorithms for ordinary differential equations(The 2nd edtion)[M].SIAM Review,2003,45(4):817-821.
[24]URABE M.Galerkins procedure for nonlinear periodic systems[J].Archive for Rational Mechanics and Analysis,1965,20(2):120-152.
[2]KANG J S,LEE J K,BAE D S.A chassis-based kinematic formulation for flexible multi-body vehicle dynamics[J].Mechanics Based Design of Structures and Machines,2016,45(1):12-24.
[3]BORNEMANN P B,GALVANETTO U,CRISFIELD MA.Some remarks on the numerical time integration of nonlinear dynamical systems[J].Journal of Sound and Vibration,2002,252(5):935-944.
[4]CHUNG J,HULBERT G M.A time integration algorithm for structural dynamics with improved numerical dissipation:the generalized-αmethod[J].Journal of Applied Mechanics,Transactions ASME,1993,60(2):371-375.
[5]HUGHES T J R,CAUGHEY T K,LIU W K.Finite-element methods for nonlinear elastodynamics which conserve energy[J].Journal of Applied Mechanics,Transactions ASME,1978,45(2):366-370.
[6]SIMO J C,TARNOW N,WONG K K.Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics[J].Computer Methods in Applied Mechanics and Engineering,1992,100(1):63-116.
[7]ANDERS D,UHLAR S,KRGER M,et al.Investigating a flexible wind turbine using consistent time-stepping schemes[J].Engineering Computations(Swansea,Wales),2012,29(7):661-688.
[8]BATHE K J,BAIG M M I.On a composite implicit time integration procedure for nonlinear dynamics[J].Computers&Structures,2005,83(31-32):2513-2524.
[9]BATHE K J.Conserving energy and momentum in nonlinear dynamics:A simple implicit time integration scheme[J].Computers&Structures,2007,85(7-8):437-445.
[10]BATHE K J,NOH G.Insight into an implicit time integration scheme for structural dynamics[J].Computers&Structures,2012,98-99:1-6.
[11]LINZ P.Analytical and numerical methods for volterra equations[M].Philadelphia:SIAM,1985:265-267.
[12]ZHONG W X,WILLIAMS F W.Precise time step integration method[J].Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,1994,208(6):427-430.
[13]ZHONG W X.On precise integration method[J].Journal of Computational and Applied Mathematics,2004,163(1):59-78.
[14]LIN J H,SHEN W P,WILLIAMS F W.A high precision direct integration scheme for structures subjected to transient dynamic loading[J].Computers&Structures,1995,56(1):113-120.
[15]WANG M F,AU F T K.Assessment and improvement of precise time step integration method[J].Computers&Structures,2006,84(12):779-786.
[16]WANG M F,AU F T K.Precise integration method without inverse matrix calculation for structural dynamic equations[J].Earthquake Engineering and Engineering Vibration,2007,6(1):57-64.
[17]GU Y X,CHEN B S,ZHANG H W,et al.Precise time-integration method with dimensional expanding for structural dynamic equations[J].AIAA Journal,2001,39(12):2394-2399.
[18]LI K,DARBY A P.A high precision direct integration scheme for nonlinear dynamic systems[J].Journal of Computational and Nonlinear Dynamics,2009,4(4):1-10.
[19]BERRUT J P,TREFETHEN L N.Barycentric Lagrange interpolation[J].SIAM Review,2004,46(3):501-517.
[20]GARCIA C B,ZANGWILL W I.Pathways to solutions,fixed points,and equilibria[M].New Jersey:PrenticeHall,1981:67.
[21]BATHE K J,WILSON E L.Numerical methods in finite element analysis[M].USSR Computational Mathematics and Mathematical Physics,1980,20(2):290.
[22]WANG M F,ZHOU X Y.Modified precise time step integration method of structural dynamic analysis[J].Earthquake Engineering and Engineering Vibration,2005,4(2):287-293.
[23]HAIRER E,LUBICH C,WANNER G.Geometric Numerical Integration:Structure-preserving algorithms for ordinary differential equations(The 2nd edtion)[M].SIAM Review,2003,45(4):817-821.
[24]URABE M.Galerkins procedure for nonlinear periodic systems[J].Archive for Rational Mechanics and Analysis,1965,20(2):120-152.