XING Qin-yan, YANG Xing, YUAN Si. An EEP Adaptive Strategy of the Galerkin FEM for Dynamic Equations of Discrete Systems[J]. Applied Mathematics and Mechanics, 2017, 38(2): 133-143. doi: 10.21656/1000-0887.370288
Citation: XING Qin-yan, YANG Xing, YUAN Si. An EEP Adaptive Strategy of the Galerkin FEM for Dynamic Equations of Discrete Systems[J]. Applied Mathematics and Mechanics, 2017, 38(2): 133-143. doi: 10.21656/1000-0887.370288

An EEP Adaptive Strategy of the Galerkin FEM for Dynamic Equations of Discrete Systems

doi: 10.21656/1000-0887.370288
Funds:  The National Natural Science Foundation of China(51508305;51378293;51078199)
  • Received Date: 2016-09-21
  • Rev Recd Date: 2016-11-17
  • Publish Date: 2017-02-15
  • For the solution of structural dynamic equations, generally the accuracy of results and the efficiency of computation both depend on the selection of the time step lengths, which makes the key difficulty for efficient solution of time-dependent problems. With the element energy projection (EEP) super-convergent solution computed at the post-processing stage of the finite element method (FEM) to replace the unknown true solution and then to estimate the error of the conventional FEM solution, the so-called EEP adaptive method can automatically refine the solution mesh and has achieved success in various boundary-value problems with spatial coordinates as the arguments. Based on the Galerkin FEM solution of the weak form, the EEP self-adaptive strategy was introduced and applied to the dynamic equations of discrete systems. As a result, an adaptive mesh was automatically produced in the time domain, and a dynamic displacement solution satisfying the pre-specified error tolerance at any moment was obtained, which leads to a new adaptive computation approach for time-dependent problems.
  • loading
  • [1]
    刘晶波, 杜修力. 结构动力学[M]. 北京: 机械工业出版社, 2005.(LIU Jing-bo, DU Xiu-li. Structural Dynamics [M]. Beijing: China Machine Press, 2005.(in Chinese))
    [2]
    张雄, 王天舒, 刘岩. 计算动力学[M]. 第2版. 北京: 清华大学出版社, 2015.(ZHANG Xiong, WANG Tian-shu, LIU Yan. Computational Dynamics [M]. 2nd ed. Beijing: Tsinghua University Press, 2015.(in Chinese))
    [3]
    钟万勰. 结构动力方程的精细时程积分法[J]. 大连理工大学学报, 1994,34(2): 131-136.(ZHONG Wan-xie. On precise time-integration method for structural dynamics[J]. Journal of Dalian University of Technology,1994,34(2): 131-136.(in Chinese))
    [4]
    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.
    [5]
    Zhong W X. On precise integration method[J]. Journal of Computational and Applied Mathematics,2004,163(1): 59-78.
    [6]
    刘婷婷, 张文首, 林家浩. 基于Householder方法的子域精细积分[J]. 计算力学学报, 2009,26(4): 535-539.(LIU Ting-ting, ZHANG Wen-shou, LIN Jia-hao. Subdomain precise integration method based on Householder method[J]. Chinese Journal of Computational Mechanics,2009,26(4): 535-539.(in Chinese))
    [7]
    谭述君, 高强, 钟万勰. Duhamel项的精细积分方法在非线性微分方程数值求解中的应用[J]. 计算力学学报, 2010,27(5): 752-758.(TAN Shu-jun, GAO Qiang, ZHONG Wan-xie. Applications of Duhamel term’s precise integration method in solving nonlinear differential equations[J]. Chinese Journal of Computational Mechanics,2010,27(5): 752-758.(in Chinese))
    [8]
    高强, 吴锋, 张洪武, 等. 大规模动力系统改进的快速精细积分方法[J]. 计算力学学报, 2011,28(4): 493-498.(GAO Qiang, WU Feng, ZHANG Hong-wu, et al. A fast precise integration method for large-scale dynamic structures[J]. Chinese Journal of Computational Mechanics,2011,28(4): 493-498.(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]
    王枚, 袁驷. Timoshenko梁单元超收敛结点应力的EEP法计算[J]. 应用数学和力学, 2004,25(11): 1124-1134.(WANG Mei, YUAN Si. Computation of super-convergent nodal stresses of Timoshenko beam elements by EEP method[J]. Applied Mathematics and Mechanics,2004,25(11): 1124-1134.(in Chinese))
    [11]
    袁驷, 和雪峰. 基于EEP法的一维有限元自适应求解[J]. 应用数学和力学, 2006,27(11): 1280-1291.(YUAN Si, HE Xue-feng. Self-adaptive strategy for one-dimensional finite element method based on EEP method[J]. Applied Mathematics and Mechanics,2006,27(11): 1280-1291.(in Chinese))
    [12]
    袁驷, 邢沁妍, 王旭, 等. 基于最佳超收敛阶EEP法的一维有限元自适应求解[J]. 应用数学和力学, 2008,29(5): 533-543.(YUAN Si, XING Qin-yan, 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,2008,29(5): 533-543.(in Chinese))
    [13]
    袁驷, 方楠, 王旭, 等. 二维有限元线法自适应分析的若干新进展[J]. 工程力学, 2011,28(3): 1-8.(YUAN Si, FANG Nan, WANG Xu, et al. New progress in self-adaptive analysis of two-dimensional finite element method of lines[J]. Engineering Mechanics,2011,28(3): 1-8.(in Chinese))
    [14]
    袁驷, 徐俊杰, 叶康生, 等. 二维自适应技术新进展: 从有限元线法到有限元法[J]. 工程力学, 2011,28(S2): 1-10.(YUAN Si, XU Jun-jie, YE Kang-sheng, et al. New progress in self-adaptive analysis of 2D problems: from FEMOL to FEM[J]. Engineering Mechanics,2011,28(S2): 1-10.(in Chinese))
    [15]
    徐俊杰. 基于EEP法的二维和三维有限元法自适应分析的研究[D]. 博士学位论文. 北京: 清华大学, 2012.(XU Jun-jie. Research on adaptive FEM analysis for 2D and 3D problems based on EEP super-convergent method[D]. PhD Thesis. Beijing: Tsinghua University, 2012. (in Chinese))
    [16]
    YUAN Si, WANG Yong-liang, YE Kang-sheng. An adaptive FEM for buckling analysis of non-uniform Bernoulli-Euler members via the element energy projection technique[J]. Mathematical Problems in Engineering,2013,2013(7). doi: 10.1155/2013/461832.
    [17]
    YUAN Si, DU Yan, XING Qin-yan, et al. Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method[J]. Applied Mathematics and Mechanics(English Edition),2014,35(10): 1223-1232.
    [18]
    袁驷, 刘泽洲, 邢沁妍. 一维变分不等式问题的自适应有限元分析新探[J]. 工程力学, 2015,32(7): 11-16.(YUAN Si, LIU Ze-zhou, XING Qin-yan. A new approach to self-adaptive FEM for one dimensional variational inequality problems[J]. Engineering Mechanics,2015,32(7): 11-16.(in Chinese))
    [19]
    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))
    [20]
    邢沁妍. 基于EEP法的一维Galerkin有限元自适应分析[D]. 博士学位论文. 北京: 清华大学, 2008.(XING Qin-yan. Adaptive analysis of 1D Galerkin FEM based on EEP super-convergent method[D]. PhD Thesis. Beijing: Tsinghua University, 2008.(in Chinese))
    [21]
    王旭. 基于EEP法的一维有限元与二维有限元线法自适应分析[D]. 博士学位论文. 北京: 清华大学, 2007.(WANG Xu. Adaptive analysis of 1D FEM and 2D FEMOL based on EEP super-convergent method[D]. PhD Thesis. Beijing: Tsinghua University, 2007.(in Chinese))
    [22]
    Chopra A K. Dynamics of Structures: Theory and Applications to Earthquake Engineering [M]. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2001.
    [23]
    Weaver Jr W, Timoshenko S P, Young D H. Vibration Problems in Engineering [M]. Wiley-Interscience, 1990.
  • 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 (1176) PDF downloads(656) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return