Engineering Mechanics ›› 2018, Vol. 35 ›› Issue (8): 9-13.doi: 10.6052/j.issn.1000-4750.2018.03.0106

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PROOF OF ADAPTIVE TIME-STEP SIZE FORMULA BASED ON MAXIMUM NORM IN TIME INTEGRATION OF LINEAR ELEMENTS

YUAN Quan1, YUAN Si2, LI Yi1, YAN Wei-ming1, XING Qin-yan2   

  1. 1. Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit, Beijing University of Technology, Beijing 100124, China;
    2. Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, China
  • Received:2017-11-12 Revised:2018-06-25 Online:2018-08-29 Published:2018-08-29

Abstract: An algorithm for time integration of motion equations using linear finite elements with adaptive time-step size had been proposed by the authors, in which a formula for the calculation of adaptive time-step size h based on maximum norm, characterized by containing a term of 5/2 order of h, was also presented. This paper gives a mathematical proof for the proposed formula. In addition, the numerical examples of both single and multiple degreed systems are obtained to verify that 5/2 order is optimal.

Key words: FEM, time integration, linear element, adaptive time-step size, maximum norm

CLC Number: 

  • TU311
[1] Clough R W. Dynamics of structures[M]. New York:McGraw-Hill, 1995.
[2] 刘晶波, 杜修力. 结构动力学[M]. 北京:机械工业出版社, 2005. Liu Jingbo, Du Xiuli. Structural dynamics[M]. Beijing:China Machine Press, 2005. (in Chinese)
[3] 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.
[4] Hibbitt H D, Karlsson B I. Analysis of Pipe Whip[R]. Palo Alto:Electric Power Research Institute, 1979.
[5] Bergan P L G, Mollestad E. An automatic time-stepping algorithm for dynamic problems[J]. Computer Methods in Applied Mechanics and Engineering, 1985, 49(3):299-318.
[6] Zienkiewicz C, Xie Y M. A simple error estimator and adaptive time stepping procedure for dynamic analysis[J]. Earthquake Engineering & Structural Dynamics, 1991, 20(7):871-887.
[7] Lages E N, Silveira E S S, Cintra D T, et al. An adaptive time integration strategy based on displacement history curvature[J]. International Journal for Numerical Methods in Engineering, 2013, 93(12):1235-1254.
[8] 袁驷, 袁全, 闫维明, 等. 运动方程自适应步长求解的一个新进展——基于EEP超收敛计算的线性有限元法[J]. 工程力学, 2018, 35(2):13-20. Yuan Si, Yuan Quan, Yan Weiming, et al. New development of solution of equations of motion with adaptive time-step size-linear FEM based on EEP super convergence technique[J]. Engineering Mechanics, 2018, 35(2):13-20. (in Chinese)
[9] 袁驷, 王枚. 一维有限元后处理超收敛解答计算的EEP法[J]. 工程力学, 2004, 21(2):1-9. Yuan Si, Wang Mei. An element-energy-projection method for post-computation of super-convergent solutions in one-dimensional FEM[J]. Engineering Mechanics, 2004, 21(2):1-9. (in Chinese)
[10] 袁驷, 林永静. 二阶非自伴两点边值问题Galerkin有限元后处理超收敛解答计算的EEP法[J]. 计算力学学报, 2007, 24(2):142-147. Yuan Si, Lin Yong-jing. An EEP method for post-computation of super-convergent solutions in one-dimensional Galerkin FEM for second order non-self-adjoint boundary-value problem[J]. Chinese Journal of Computational Mechanics, 2007, 24(2):142-147. (in Chinese)
[11] Yuan Si, Xing Qinyan, Wang Xu, et al. Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order[J]. Applied Mathematics and Mechanics (English version), 2008, 29(5):591-602.
[12] Strang G, Fix G J. An analysis of the finite element method[M]. 2nd ed. Wellesley MA:WellesleyCambridge Press, 2008:39-50.
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