| Citation: | XING Yufeng, JI Yi, ZHANG Huiminet al. Advances and challenges in time integration methods[J]. Journal of Beijing University of Aeronautics and Astronautics, 2022, 48(9): 1692-1701. doi: 10.13700/j.bh.1001-5965.2022.0288(in Chinese)
|
Time integration methods are a powerful tool for solving transient responses, which have been widely used to solve dynamic problems in aerospace, civil engineering, machinery manufacturing, and other fields. This paper reviews the advances in time integration methods in the past decades. Firstly, some classical methods, such as the series expansion method, the Runge-Kutta method and the Newmark method, are introduced. To solve the drawbacks involved in the classical methods, several time integration methods with more desirable numerical properties, including accuracy, efficiency, dissipation and stability, have been developed. Moreover, the advanced methods, including parameters methods, higher-order unconditionally stable methods, energy-conserving methods, linear multistep methods, composite methods and BN-stable methods, are introduced in this paper. Finally, the numerical properties and application scope of the existing time integration methods are compared, and some issues worthy of attention are given.
| [1] |
HUGHES T J R. The finite element method: Linear static and dynamic finite element analysis[M]. Upper Saddle River: Prentice-Hall Press, 1987.
|
| [2] |
BUTCHER J C. Numerical methods for ordinary differential equations[M]. New York: John Wiley & Sons, Ltd, 2016.
|
| [3] |
DAHLQUIST G G. A special stability problem for linear multistep methods[J]. BIT Numerical Mathematics, 1963, 3(1): 27-43. doi: 10.1007/BF01963532
|
| [4] |
KUHL D, CRISFIELD M A. Energy-conserving and decaying algorithms in nonlinear structural dynamics[J]. International Journal for Numerical Methods in Engineering, 1999, 45(5): 569-599. doi: 10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A
|
| [5] |
PARK K C. An improved stiffly stable method for direct integration of nonlinear structural dynamics[J]. Journal of Applied Mechanics-Transactions of the ASME, 1975, 42(2): 464-470. doi: 10.1115/1.3423600
|
| [6] |
邵慧萍, 蔡承文. 结构动力学方程数值积分的三参数算法[J]. 应用力学学报, 1988, 5(4): 77-81. https://www.cnki.com.cn/Article/CJFDTOTAL-YYLX198804009.htm
SHAO H P, CAI C W. A three parameters algorithm for numerical integration of structural dynamic equations[J]. Chinese Journal of Applied Mechanics, 1988, 5(4): 77-81(in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-YYLX198804009.htm
|
| [7] |
CHUNG J, HULBERT G. A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method[J]. Journal of Applied Mechanics-Transactions of the ASME, 1993, 60(2): 371-375. doi: 10.1115/1.2900803
|
| [8] |
CHUNG J, LEE J M. A new family of explicit time integration methods for linear and nonlinear structural dynamics[J]. International Journal for Numerical Methods in Engineering, 1994, 37(23): 3961-3976. doi: 10.1002/nme.1620372303
|
| [9] |
HILBER H M, HUGHES T J R, TAYLOR R L. Algorithms in structural dynamics[J]. Earthquake Engineering and Structural Dynamics, 1977, 5: 283-292. doi: 10.1002/eqe.4290050306
|
| [10] |
WOOD W L, BOSSAK M, ZIENKIEWICZ O C. An alpha modification of Newmark's method[J]. International Journal for Numerical Methods in Engineering, 1980, 15(10): 1562-1566. doi: 10.1002/nme.1620151011
|
| [11] |
ZHANG H M, XING Y F. A three-parameter single-step time integration method for structural dynamic analysis[J]. Acta Mechanica Sinica, 2019, 35(1): 112-128. doi: 10.1007/s10409-018-0775-y
|
| [12] |
XING Y F, ZHANG H M. An efficient non-dissipative higher-order single-step integration method for long-term dynamics simulation[J]. International Journal of Structural Stability and Dynamics, 2018, 18(9): 1850113. doi: 10.1142/S0219455418501134
|
| [13] |
FUNG T C. Solving initial value problems by differential quadrature method: Part 1-First-order equations[J]. International Journal for Numerical Methods in Engineering, 2001, 50(6): 1411-1427. doi: 10.1002/1097-0207(20010228)50:6<1411::AID-NME78>3.0.CO;2-O
|
| [14] |
FUNG T C. Solving initial value problems by differential quadrature method: Part 2-Second-and higher-order equations[J]. International Journal for Numerical Methods in Engineering, 2001, 50(6): 1429-1454. doi: 10.1002/1097-0207(20010228)50:6<1429::AID-NME79>3.0.CO;2-A
|
| [15] |
KIM W, LEE J H. A comparative study of two families of higher-order accurate time integration algorithms[J]. International Journal of Computational Methods, 2020, 17(8): 1950048. doi: 10.1142/S0219876219500488
|
| [16] |
FUNG T C. Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms: Part 1: First-order equations[J]. International Journal for Numerical Methods in Engineering, 1999, 45(8): 941-971. doi: 10.1002/(SICI)1097-0207(19990720)45:8<941::AID-NME612>3.0.CO;2-S
|
| [17] |
JI Y, XING Y F. An improved higher-order time integration algorithm for structural dynamics[J]. CMES-Computer Modelling in Engineering & Sciences, 2021, 126(2): 549-575.
|
| [18] |
FUNG T C. Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms: Part 2: Second-order equations[J]. International Journal for Numerical Methods in Engineering, 1999, 45(8): 971-1006. doi: 10.1002/(SICI)1097-0207(19990720)45:8<971::AID-NME613>3.0.CO;2-M
|
| [19] |
BELYTSCHKO T, SCHOEBERLE D F. On the unconditional stability of an implicit algorithm for nonlinear structural dynamics[J]. Journal of Applied Mechanics-Transactions of the ASME, 1975, 42(4): 865-869. doi: 10.1115/1.3423721
|
| [20] |
LAVRENÇIÇ M, BRANK B. Comparison of numerically dissipative schemes for structural dynamics: Generalized-alpha versus energy-decaying methods[J]. Thin-Walled Structures, 2020, 157: 107075. doi: 10.1016/j.tws.2020.107075
|
| [21] |
ZHANG H M, XING Y F, JI Y. An energy-conserving and decaying time integration method for general nonlinear dynamics[J]. International Journal for Numerical Methods in Engineering, 2020, 121(5): 925-944. doi: 10.1002/nme.6251
|
| [22] |
HUGHES T J R, CAUGHEYV T K, LIU W K. Finite-element methods for nonlinear elastodynamics which conserve energy[J]. Journal of Applied Mechanics-Transactions of the ASME, 1978, 45(2): 366-370. doi: 10.1115/1.3424303
|
| [23] |
WU B, PAN T L, YANG H W, et al. Energy-consistent integration method and its application to hybrid testing[J]. Earthquake Engineering and Structural Dynamics, 2020, 49(5): 415-433. doi: 10.1002/eqe.3246
|
| [24] |
ZHANG R, ZHONG H Z. A quadrature element formulation of an energy-momentum conserving algorithm for dynamic analysis of geometrically exact beams[J]. Computers & Structures, 2016, 165: 96-106.
|
| [25] |
KRENK S. Global format for energy-momentum based time integration in nonlinear dynamics[J]. International Journal for Numerical Methods in Engineering, 2014, 100(6): 458-476. doi: 10.1002/nme.4745
|
| [26] |
GONZALEZ O. Exact energy and momentum conserving algorithms for general models in nonlinear elasticity[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 190(13-14): 1763-1783. doi: 10.1016/S0045-7825(00)00189-4
|
| [27] |
MAMOURI S, HAMMADI F, IBRAHIMBEGOVIC A. Decaying/conserving implicit scheme and non-linear instability analysis of 2D frame structures[J]. International of Journal Non-Linear Mechanics, 2014, 67: 144-152. doi: 10.1016/j.ijnonlinmec.2014.08.011
|
| [28] |
ORDEN J C. Energy and symmetry-preserving formulation of nonlinear constraints and potential forces in multibody dynamics[J]. Nonlinear Dynamics, 2019, 95: 823-837. doi: 10.1007/s11071-018-4598-y
|
| [29] |
KUHL D, EKKELARD R. Constraint energy momentum algorithm and its application to non-linear dynamics of shells[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 136(3-4): 293-315. doi: 10.1016/0045-7825(95)00963-9
|
| [30] |
SIMO J C, WONG K K. Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum[J]. International Journal for Numerical Methods in Engineering, 1991, 31(1): 19-52. doi: 10.1002/nme.1620310103
|
| [31] |
JI Y, XING Y F. A two-step time integration method with desirable stability for nonlinear structural dynamics[J]. European Journal of Mechanics-A/Solids, 2022, 94: 104582. doi: 10.1016/j.euromechsol.2022.104582
|
| [32] |
ZHANG J. A-stable linear two-step time integration methods with consistent starting and their equivalent single-step methods in structural dynamics analysis[J]. International Journal for Numerical Methods in Engineering, 2021, 122(9): 2312-2359. doi: 10.1002/nme.6623
|
| [33] |
ZHANG H M, ZHANG R S, MASARATI P. Improved second-order unconditionally stable schemes of linear multi-step and equivalent single-step integration methods[J] Computational Mechanics, 2021, 67: 289-313. doi: 10.1007/s00466-020-01933-y
|
| [34] |
DONG S. BDF-like methods for nonlinear dynamic analysis[J]. Journal of Computational Physics, 2010, 229(8): 3019-3045. doi: 10.1016/j.jcp.2009.12.028
|
| [35] |
ZHANG J. A-stable two-step time integration methods with controllable numerical dissipation for structural dynamics[J]. International Journal for Numerical Methods in Engineering, 2020, 121(1): 54-92. doi: 10.1002/nme.6188
|
| [36] |
YANG C, WANG X, LI Q, et al. An improved explicit integration algorithm with controllable numerical dissipation for structural dynamics[J]. Archive of Applied Mechanics, 2020, 90: 2413-2431. doi: 10.1007/s00419-020-01729-9
|
| [37] |
REZAIEE-PAJAND M, SARAFRAZI S R. A mixed and multi-step higher-order implicit time integration family[J]. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2010, 224(10): 2097-2108. doi: 10.1243/09544062JMES2093
|
| [38] |
ZHOU X, TAMMA K K. Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics[J]. International Journal for Numerical Methods in Engineering, 2004, 59(5): 597-668. doi: 10.1002/nme.873
|
| [39] |
LI J Z, YU K P. Development of composite sub-step explicit dissipative algorithms with truly self-starting property[J]. Nonlinear Dynamics, 2021, 103: 1-26. doi: 10.1007/s11071-020-06053-z
|
| [40] |
JI Y, XING Y F. Optimization of a class of n-sub-step time integration methods for structural dynamics[J] International Journal of Applied Mechanics, 2021, 13(6): 2150064. doi: 10.1142/S1758825121500642
|
| [41] |
LIU T H, HUANG F L, WEN W B, et al. Further insights of a composite implicit time integration scheme and its performance on linear seismic response analysis[J]. Engineering Structures, 2021, 241: 112490. doi: 10.1016/j.engstruct.2021.112490
|
| [42] |
XING Y F, JI Y, ZHANG H M. On the construction of a type of composite time integration methods[J] Computers & Structures, 2019, 221: 157-178.
|
| [43] |
JI Y, XING Y F. An optimized three-sub-step composite time integration method with controllable numerical dissipation[J] Computers & Structures, 2020, 231: 106210.
|
| [44] |
NOH G, BATHE K J. The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method[J] Computers & Structures, 2019, 212: 299-310.
|
| [45] |
KIM W. An improved implicit method with dissipation control capability: The simple generalized composite time integration algorithm[J]. Applied Mathematical Modelling, 2020, 81: 910-930. doi: 10.1016/j.apm.2020.01.043
|
| [46] |
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.
|
| [47] |
LI J Z, ZHAO R, YU K P, et al. Directly self-starting higher-order implicit integration algorithms with flexible dissipation control for structural dynamics[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 389: 114274. doi: 10.1016/j.cma.2021.114274
|
| [48] |
WEN W B, WEI K, LEI H S, et al. A novel sub-step composite implicit time integration scheme for structural dynamics[J]. Computers & Structures, 2017, 182: 176-186. doi: 10.11897/SP.J.1016.2017.00176
|
| [49] |
NOH G, HAM S, BATHE K J. Performance of an implicit time integration scheme in the analysis of wave propagations[J]. Computers & Structures, 2013, 123: 93-105.
|
| [50] |
CHANDRA Y, ZHOU Y, STANCIULESCU I, et al. A robust composite time integration scheme for snap-through problems[J]. Computational Mechanics, 2015, 55: 1041-1056. doi: 10.1007/s00466-015-1152-3
|
| [51] |
KOLAY C, RICLES J M. Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation[J]. Earthquake Engineering and Structural Dynamics, 2014, 43(9): 1361-1380. doi: 10.1002/eqe.2401
|
| [52] |
KOLAY C, RICLES J M. Improved explicit integration algorithms for structural dynamic analysis with unconditional stability and controllable numerical dissipation[J]. Journal of Earthquake Engineering, 2019, 23(5): 771-792. doi: 10.1080/13632469.2017.1326423
|
| [53] |
DU X Q, YANG D X, ZHOU J L, et al. New explicit integration algorithms with controllable numerical dissipation for structural dynamics[J]. International Journal of Structural Stability and Dynamics, 2018, 18(3): 1850044. doi: 10.1142/S021945541850044X
|
| [54] |
LI J Z, YU K P. Noniterative integration algorithms with controllable numerical dissipations for structural dynamics[J]. International Journal of Computer Mathematics, 2018, 15(3): 1850111.
|
| [55] |
BURRAGE K, BUTCHER J C. Stability criteria for implicit Runge-Kutta methods[J]. SIAM Journal on Numerical Analysis, 1979, 16(1): 46-57. doi: 10.1137/0716004
|
| [56] |
JI Y, XING Y F, WIERCIGROCH M. An unconditionally stable time integration method with controllable dissipation for second-order nonlinear dynamics[J]. Nonlinear Dynamics, 2021, 105: 3341-3358. doi: 10.1007/s11071-021-06720-9
|
| [57] |
HUANG C, FU M H. A composite collocation method with low-period elongation for structural dynamics problems[J]. Computers & Structures, 2018, 195: 74-84.
|
| [58] |
KIM W, REDDY J N. A new family of higher-order time integration algorithms for the analysis of structural dynamics[J]. International Journal for Numerical Methods in Engineering, 2017, 84: 071008.
|
| [59] |
HOUBOLT J C. A recurrence matrix solution for the dynamic response of elastic aircraft[J]. Journal of the Aeronautical Sciences, 1950, 17(9): 540-550. doi: 10.2514/8.1722
|
| [60] |
BANK R E, COUGHRAN W M, FICHTNER W, et al. Transient simulations of silicon devices and circuits[J]. IEEE Transactions on Electro Devices, 1985, 32(10): 1992-2007. doi: 10.1109/T-ED.1985.22232
|
| [61] |
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.
|
| [62] |
ZAKIAN P, BATHE K J. Transient wave propagations with the propagations with the Noh-Bathe scheme and the spectral element method[J]. Computers & Structures, 2021, 254: 106531.
|
| [63] |
TAMMA K K, HAR J, ZHOU X M, et al. An overview and recent advances in vector and scalar formalisms: Space/time discretizations in computational dynamics-A unified approach[J]. Archives of Computational Methods in Engineering, 2011, 18: 119-283. doi: 10.1007/s11831-011-9060-y
|
| [64] |
邢誉峰, 张慧敏, 季奕. 动力学常微分方程的时间积分方法[M]. 北京: 科学出版社, 2022.
XING Y F, ZHANG H M, JI Y. Time integration methods in dynamic ordinary differential equations[M]. Beijing: Science Press, 2022(in Chinese).
|
| [65] |
ZHONG W X, WILLIAMS F W. A 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. doi: 10.1243/PIME_PROC_1994_208_148_02
|
| [66] |
XING Y F, ZHANG H M, WANG Z K. Highly precise time integration method for linear structural dynamic analysis[J]. International Journal for Numerical Methods in Engineering, 2018, 116(8): 505-529. doi: 10.1002/nme.5934
|
| [67] |
JI Y, XING Y F. Highly precise and efficient solution strategy for linear heat conduction and structural dynamics[J]. International Journal for Numerical Methods in Engineering, 2022, 123(2): 366-395. doi: 10.1002/nme.6859
|
| [68] |
邢誉峰, 郭静. 与结构动特性协同的自适应Newmark方法[J]. 力学学报, 2012, 44(5): 904-911. https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201205013.htm
XING Y F, GUO J. A self-adaptive Newmark method with parameters dependent upon structural dynamic characteristics[J] Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(5): 904-911(in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201205013.htm
|
| [69] |
邢誉峰, 杨蓉. 单步辛算法的相位误差分析及修正[J]. 力学学报, 2007, 39(5): 668-671. doi: 10.3321/j.issn:0459-1879.2007.05.013
XING Y F, YANG R. Phase errors and their correction in symplectic implicit single-step algorithm[J]. Chinese Journal of Theoretical and Applied Mechanics 2007, 39(5): 668-671(in Chinese). doi: 10.3321/j.issn:0459-1879.2007.05.013
|
| [70] |
FENG K, QIN M Z. Hamiltonian algorithm for Hamiltonian dynamical systems[J]. Progress in Natural Science, 1991, 1(2): 105-116.
|
Figures(1)
Copyright © Journal of Beijing University of Aeronautics and Astronautics
Address: Editorial Department of Journal of Beijing University of Aeronautics and Astronautics, 37 Xueyuan Road, Haidian District, Beijing Post Code: 100191 Email: jbuaa@buaa.edu.cn
Supported by:
Beijing Renhe Information Technology Co., Ltd.
DownLoad: