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一种基于Hamel形式的无条件稳定动力学积分算法

顾崴 刘铖 安志朋 史东华

顾崴, 刘铖, 安志朋, 史东华. 一种基于Hamel形式的无条件稳定动力学积分算法. 力学学报, 2022, 54(9): 1-11 doi: 10.6052/0459-1879-22-131
引用本文: 顾崴, 刘铖, 安志朋, 史东华. 一种基于Hamel形式的无条件稳定动力学积分算法. 力学学报, 2022, 54(9): 1-11 doi: 10.6052/0459-1879-22-131
Gu Wei, Liu Cheng, An Zhipeng, Shi Donghua. An unconditionally stable dynamical integration algorithm based on hamel’s formalism. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(9): 1-11 doi: 10.6052/0459-1879-22-131
Citation: Gu Wei, Liu Cheng, An Zhipeng, Shi Donghua. An unconditionally stable dynamical integration algorithm based on hamel’s formalism. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(9): 1-11 doi: 10.6052/0459-1879-22-131

一种基于Hamel形式的无条件稳定动力学积分算法

doi: 10.6052/0459-1879-22-131
基金项目: 国家重点研发计划(2018 YFE0202101), 国家自然科学基金(12102033,12072026,11872107)资助项目
详细信息
    作者简介:

    安志朋, 博士后, 主要研究方向: 计算几何力学与控制. E-mail: 7520200152@bit.edu.cn

AN UNCONDITIONALLY STABLE DYNAMICAL INTEGRATION ALGORITHM BASED ON HAMEL’S FORMALISM

Funds: The project was supported by the (12345678)and (9876543)
  • 摘要: 时间积分算法是求解动力学系统的一个核心问题. 动力学方程的时间积分经常会出现数值不稳定现象, 有限元空间离散也通常会造成伪高频振荡, 因而, 发展解决上述问题的数值积分算法具有重要的理论价值. 本文基于Hamel场变分积分子, 通过新的数值积分算法的构造方法, 提出了一种无条件稳定的Hamel广义$ \alpha $方法, 具体内容包括: 构造特殊的变分形式, 利用变分积分子等工具, 建立无条件稳定的数值积分算法; 在相同框架下, 提出更高精度的数值格式; 结合活动标架法的特性, 将算法的一般形式推广到李群空间, 得到Hamel-广义$ \alpha $方法李群形式; 对算法的收敛性和稳定性等性质进行了讨论, 并通过算例验证了结论. 理论分析的结果表明本文所提出的Hamel广义$ \alpha $方法是无条件稳定的, 具有二阶精度并且能够快速过滤掉虚假的高频振荡. 数值算例的结果显示, 本文所提方法具备了传统方法的精度、耗散和稳定性优势, 既适合一般的线性空间, 也适用于李群空间, 同时还可以发展高阶精度算法. 本文发展了构造变分积分子的新模式.

     

  • 图  1  谱半径

    Figure  1.  Spectral radius

    图  2  数值阻尼比

    Figure  2.  Numerical damping ratio

    图  3  周期延长率

    Figure  3.  Period elongation

    图  4  Hamel广义$ \alpha $方法t = 1时刻的误差

    Figure  4.  Errors of Hamel generalized-$ \alpha $method at time t = 1

    图  5  Hamel广义$ \alpha $方法t = 1时刻的误差

    Figure  5.  Errors of Hamel generalized-$ \alpha $method at time t = 1

    图  6  系统1各种方法数值结果对比图

    Figure  6.  Comparison of numerical results of various methods for system 1

    图  7  系统2各种方法数值结果对比图

    Figure  7.  Comparison of numerical results of various methods for system 2

    图  8  系统1 Hamel广义$ \alpha $方法与广义$ \alpha $方法数值结果误差

    Figure  8.  Errors of numerical results between Hamel generalized-$ \alpha $method and generalized-$ \alpha $method for system 1

    图  9  系统2 Hamel广义$ \alpha $方法与广义$ \alpha $方法数值结果误差

    Figure  9.  Errors of numerical results between Hamel generalized-$ \alpha $method and generalized-$ \alpha $method for system 2

    图  10  柔性双摆系统的初始构型

    Figure  10.  Initial configuration of a flexible double pendulum

    图  11  柔性双摆系统两杆连接处轨迹

    Figure  11.  Trajectory at the joint of a flexible double pendulum

    图  12  柔性双摆系统第二杆末端轨迹

    Figure  12.  Trajectory at the end of a flexible double pendulum

    表  1  Hamel广义$ \alpha $方法与经典算法比较

    Table  1.   Comparison between Hamel generalized-$ \alpha $method and classical algorithm

    $ {B_1} $$ {B_2} $Classical algorithm
    100Newmark
    20$ - \alpha $HHT-$ \alpha $
    3$\dfrac{ { {\alpha _m} } }{ { {\alpha _m} - 1} }$$\dfrac{ { {\alpha _m} - {\alpha _f} } }{ {1 - {\alpha _m} } }$generalized-$ \alpha $
    method
    下载: 导出CSV

    表  2  Hamel广义$ \alpha $方法与经典算法比较

    Table  2.   Comparison between Hamel generalized-$ \alpha $method and classical algorithm

    $ {B_1} $$ {B_2} $classical algorithm
    1$\dfrac{ { {\alpha _m} } }{ { {\alpha _m} - 1} }$$\dfrac{ { {\alpha _m} - {\alpha _f} } }{ {1 - {\alpha _m} } }$Lie group
    generalized-$ \alpha $
    method
    下载: 导出CSV
  • [1] Newmark NM. A method of computation for structural dynamics. Journal of the Engineering Mechanics Division, 1959, 85(3): 67-94 doi: 10.1061/JMCEA3.0000098
    [2] Chung J, Hulbert G. A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. ASME Journal of Applied Mechanics, 1993, 60: 371-375
    [3] Wood WL, Bossak M, Zienkiewicz O C. An alpha modification of Newmark's method. International journal for Numerical Methods in Engineering, 1980, 15(10): 1562-1566 doi: 10.1002/nme.1620151011
    [4] Hilber HM, Hughes TJR, Taylor RL. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics, 1977, 5(3): 283-292 doi: 10.1002/eqe.4290050306
    [5] Bazilevs Y, Calo VM, Cottrell JA, et al. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Computer Methods in Applied Mechanics and Engineering, 2007, 197(1-4): 173-201 doi: 10.1016/j.cma.2007.07.016
    [6] Behnoudfar P, Calo VM, Deng Q, et al. A variationally separable splitting for the generalized-α method for parabolic equations. International Journal for Numerical Methods in Engineering, 2020, 121(5): 828-841 doi: 10.1002/nme.6246
    [7] Gomez H, Hughes TJR, Nogueira X, et al. Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations. Computer Methods in Applied Mechanics and Engineering, 2010, 199(25-28): 1828-1840 doi: 10.1016/j.cma.2010.02.010
    [8] Behnoudfar P, Deng Q, Calo VM. Higher-order generalized-α methods for hyperbolic problems. Computer Methods in Applied Mechanics and Engineering, 2021, 378: 113725 doi: 10.1016/j.cma.2021.113725
    [9] Yen J, Petzold L, Raha S. A time integration algorithm for flexible mechanism dynamics: The DAE α-method. Computer Methods in Applied Mechanics and Engineering, 1998, 158(3-4): 341-355 doi: 10.1016/S0045-7825(97)00261-2
    [10] Arnold M, Brüls O. Convergence of the generalized-α scheme for constrained mechanical systems. Multibody System Dynamics, 2007, 18(2): 185-202 doi: 10.1007/s11044-007-9084-0
    [11] 张雄, 王天书. 计算动力学. 北京: 清华大学出版社, 2007

    Zhang Xiong, Wang Tian-shu. Computational Dynamics. Beijing: Tsinghua University Press, 2007: 236-273(in Chinese))
    [12] 田强. 基于绝对节点坐标方法的柔性多体系统动力学研究与应用. [博士论文]. 武汉: 华中科技大学, 2009

    Tian Qiang. Flexible multibody dynamics research and application based on the absolute nodal coordinate method[phD Thesis]. Wuhan: Huazhong University of Science and Technology, 2009 (in Chinese))
    [13] Behnoudfar P, Deng Q, Calo VM. High-order generalized-α method. Applications in Engineering Science, 2020, 4: 100021 doi: 10.1016/j.apples.2020.100021
    [14] Ji Y, Xing Y, Wiercigroch M. An unconditionally stable time integration method with controllable dissipation for second-order nonlinear dynamics. Nonlinear Dynamics, 2021, 105(4): 3341-3358
    [15] 季奕, 邢誉峰. 一种求解瞬态热传导方程的无条件稳定方法. 力学学报, 2021, 105(4): 3341-3358 (Ji Yi, Xing Yufeng. An unconditionally stable time integration method with controllable dissipation for second-order nonlinear dynamics. Nonlinear Dynamics, 2021, 105(4): 3341-3358 doi: 10.1007/s11071-021-06720-9

    Ji Y, Xing Y, Wiercigroch M. An unconditionally stable time integration method with controllable dissipation for second-order nonlinear dynamics[J]. Nonlinear Dynamics, 2021, 105(4): 3341-3358. doi: 10.1007/s11071-021-06720-9
    [16] Brüls O, Cardona A, Arnold M. Lie group generalized-α time integration of constrained flexible multibody systems. Mechanism and Machine Theory, 2012, 48: 121-137 doi: 10.1016/j.mechmachtheory.2011.07.017
    [17] Arnold M, Brüls O, Cardona A. Error analysis of generalized-α Lie group time integration methods for constrained mechanical systems. Numerische Mathematik, 2015, 129(1): 149-179 doi: 10.1007/s00211-014-0633-1
    [18] 刘铖, 胡海岩. 基于李群局部标架的多柔体系统动力学建模与计算. 力学学报, 2021, 53(1): 213-233 (Liu Cheng, Hu Haiyan. Dynamic modeling and computation for flexible multibody systems based on the local frame of Lie group. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(1): 213-233 (in Chinese)

    Liu Cheng, Hu Haiyan. Dynamic modeling and computation for flexible multibody systems based on the local frame of Lie group. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(1): 213-233(in Chinese))
    [19] Wieloch V, Arnold M. BDF integrators for constrained mechanical systems on Lie groups. Journal of Computational and Applied Mathematics, 2021, 387: 112517 doi: 10.1016/j.cam.2019.112517
    [20] Marsden JE, West M. Discrete mechanics and variational integrators. Acta Numerica, 2001, 10: 357-514 doi: 10.1017/S096249290100006X
    [21] Lew A, Marsden JE, Ortiz M, et al. An overview of variational integrators. In Finite element methods: 1970 s and beyond, CIMNE Barcelona, 2004: 85-146.
    [22] Wendlandt JM, Marsden JE. Mechanical integrators derived from a discrete variational principle. Physica D:Nonlinear Phenomena, 1997, 106(3-4): 223-246 doi: 10.1016/S0167-2789(97)00051-1
    [23] Leyendecker S, Ober-Blöbaum S, Marsden JE, et al. Discrete mechanics and optimal control for constrained systems. Optimal Control Applications and Methods, 2010, 31(6): 505-528 doi: 10.1002/oca.912
    [24] Kane C, Marsden JE, Ortiz M. Symplectic-energy-momentum preserving variational integrators. Journal of mathematical physics, 1999, 40(7): 3353-3371 doi: 10.1063/1.532892
    [25] Cendra H, Marsden JE, Ratiu TS. Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems//Mathematics Unlimited—2001 and Beyond. Berlin, Heidelberg: Springer, 2001: 221-273.
    [26] Hante S, Arnold M. RATTLie: A variational Lie group integration scheme for constrained mechanical systems. Journal of Computational and Applied Mathematics, 2021, 387: 112492 doi: 10.1016/j.cam.2019.112492
    [27] Ober-Blöbaum S, Vermeeren M. Superconvergence of Galerkin variational integrators. IFAC-Papers OnLine, 2021, 54(19): 327-333 doi: 10.1016/j.ifacol.2021.11.098
    [28] Yi Z, Yue B, Deng M. Chebyshev spectral variational integrator and applications. Applied Mathematics and Mechanics, 2020, 41(5): 753-768 doi: 10.1007/s10483-020-2602-8
    [29] Bloch AM, Marsden JE, Zenkov DV. Quasivelocities and symmetries in non-holonomic systems. Dynamical Systems, 2009, 24(2): 187-222 doi: 10.1080/14689360802609344
    [30] Ball KR, Zenkov DV. Hamel’s formalism and variational integrators. Fields Institute Communications, 2015, 73: 477-506
    [31] An Z, Wu H, Shi D. Minimum-time optimal control of robotic manipulators based on Hamel’s integrators. Meccanica, 2019, 54(15): 2521-2537 doi: 10.1007/s11012-019-01093-1
    [32] Shi D, Berchenko-Kogan Y, Zenkov DV, et al. Hamel’s formalism for infinite-dimensional mechanical systems. Journal of Nonlinear Science, 2017, 27(1): 241-283 doi: 10.1007/s00332-016-9332-7
    [33] An Z, Gao S, Shi D, et al. A variational integrator for the Chaplygin– Timoshenko Sleigh. Journal of Nonlinear Science, 2020, 30(4): 1381-1419 doi: 10.1007/s00332-020-09611-2
    [34] Shi D, Zenkov DV, Bloch AM. Hamel’s formalism for classical field theories. Journal of Nonlinear Science, 2020, 30(4): 1307-1353 doi: 10.1007/s00332-020-09609-w
    [35] 王亮, 安志朋, 史东华. 几何精确梁的 Hamel 场变分积分子. 北京大学学报:自然科学版, 2016, 52(4): 692-698 (Wang Liang, An Zhipeng, Shi Donghua. Hamel’s field variational integrator of geometrically exact beam. Acta Scientiarum Naturalium Universitatis Pekinensis, 2016, 52(4): 692-698 (in Chinese)

    Wang Liang, An Zhipeng, Shi Donghua. Hamel’s field variational integrator of geometrically exact beam[J]. Acta Scientiarum Naturalium Universitatis Pekinensis, 2016, 52: 692-698(in Chinese))
    [36] 高山, 史东华, 郭永新. Hamel 框架下几何精确梁的离散动量守恒律. 力学学报, 2021, 53(6): 1712-1719 (Gao Shan, Shi Donghua, Guo Yongxin. Discrete momentum conservation law of geometrically exact beam in Hamel’s framework. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(6): 1712-1719 (in Chinese) doi: 10.6052/0459-1879-21-092

    Gao Shan, Shi Donghua, Guo Yongxin. Discrete momentum conservation law of geometrically exact beam in Hamel’s framework. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(6): 1712-1719(in Chinese)) doi: 10.6052/0459-1879-21-092
    [37] Simo JC, Tarnow N, Wong KK. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 1992, 100(1): 63-116 doi: 10.1016/0045-7825(92)90115-Z
    [38] Kane C, Marsden JE, Ortiz M, et al. Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. International Journal for Numerical Methods in Engineering, 2000, 49(10): 1295-1325 doi: 10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W
    [39] Bardella L, Genna F. Newmark's time integration method from the discretization of extended functionals. Journal of Applied Mechanics, 2005, 72(4): 527-537 doi: 10.1115/1.1934648
    [40] 汤惠颖, 张志娟, 刘铖等. 两类基于局部标架梁单元的闭锁缓解方法. 力学学报, 2021, 53(2): 482-495 (Tang Huiying, Zhang Zhijuan, Liu Cheng, et al. Locking alleviation techniques of two types of beam elements based on the local frame formulation. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 482-495 (in Chinese) doi: 10.6052/0459-1879-20-274

    Tang Huiying, Zhang Zhijuan, Liu Cheng, et al. Locking alleviation techniques of two types of beam elements based on the local frame formulation. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 482-495(in Chinese)) doi: 10.6052/0459-1879-20-274
    [41] Brüls O. Integrated simulation and reduced-order modeling of controlled flexible multibody systems[D]. Liège, Belgique: Université de Liège, 2005
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  • 收稿日期:  2022-03-29
  • 录用日期:  2022-05-27
  • 网络出版日期:  2022-05-28

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