杆件轴向受迫振动的Galerkin有限元EEP法自适应求解

邢沁妍; 杨青浩; 陆琛宇; 杨杏

doi:10.21656/1000-0887.400051

杆件轴向受迫振动的Galerkin有限元EEP法自适应求解

doi: 10.21656/1000-0887.400051

清华大学土木工程系; 土木工程安全与耐久教育部重点实验室, 北京 100084

基金项目: 国家自然科学基金（51508305）

详细信息

作者简介:
邢沁妍（1981—），女，讲师，博士 (通讯作者. E-mail: xingqy@tsinghua.edu.cn);杨青浩(1994—)，男，回族，硕士生(E-mail: yangqh16@mails.tsinghua.edu.cn);陆琛宇(1992—)，男，硕士生(E-mail: lcy577@163.com);杨杏(1988—)，男，硕士生(E-mail: xihuanyuye@126.com).

中图分类号: O242.21
计量
- 文章访问数: 1178
- HTML全文浏览量: 128
- PDF下载量: 508
- 被引次数: 0
出版历程
- 收稿日期: 2019-02-03
- 修回日期: 2019-07-12
- 刊出日期: 2019-09-01

An EEP Adaptive Strategy of the Galerkin FEM for Axially Forced Vibration of Bars

Department of Civil Engineering, Tsinghua University; Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Beijing 100084, P.R.China

Funds: The National Natural Science Foundation of China（51508305）

摘要

摘要: 基于单元能量投影(element energy projection,EEP)法自适应分析在杆件静力问题以及离散系统运动方程组中所取得的成果，以直杆轴向受迫振动为例，研究并建立了一种在时间域和一维空间域同时实现自适应分析的方法.该方法在时间和空间两个维度都采用连续的Galerkin有限元法(finite element method，FEM)进行求解，根据半离散的思想，由空间有限元离散将模型问题的偏微分控制方程转化为离散系统运动方程组，对该方程组进行时域有限元自适应求解；然后再基于空间域超收敛计算的EEP解对空间域进行自适应，直至最终的时空网格下动位移解答的精度逐点均满足给定误差限要求.文中对其基本思想、关键技术和实施策略进行了阐述，并给出了包括地震波输入下的典型算例以展示该法有效可靠.
- 受迫振动 /
- 时空有限元 /
- 自适应 /
- Galerkin法 /
- 单元能量投影
Abstract: Based on the successful applications of the element energy projection (EEP) adaptive method for the static problems of bars and the dynamic equations for discrete systems, a strategy was proposed to adaptively solve the axially forced vibration problems of bars in both the time dimension and the space dimension. In this strategy, the continuous space-time Galerkin finite element method (FEM) was used. Based on the idea of semi-discretization, through discretization in space first, the governing partial differential equations of the model problem were transformed into a system of 2nd-order ordinary differential equations with initial boundary conditions, which were called dynamic equations for discrete systems hereinafter. These dynamic equations for discrete systems were then solved with the proposed EEP adaptive FEM in the time domain. After that, the EEP super-convergent formula for dynamic displacements in the space direction was derived, with which errors of the conventional Galerkin FEM solutions were estimated and the corresponding adaptive analysis method was established. Finally, the presented EEP adaptive strategy gave dynamic displacements with high accuracy point-wisely satisfying the pre-specified error tolerance, together with the automatically produced space-time mesh. The basic idea, the key technologies and the implementation strategy were elaborated. Representative numerical examples including seismic wave input demonstrate effectiveness and reliability of the method.
- forced vibration /
- space-time finite element /
- adaptivity /
- Galerkin method /
- element energy projection

HTML全文

参考文献(25)

[1]	〖JP2〗BABUSKA I, RHEINBOLDT W C. Error estimates for adaptive finite element computations[J]. SIAM Journal on Numerical Analysis,1989,15(4): 746-754.
[2]	ZIENKIEWICZ O C, ZHU J Z. A simple error estimator and adaptive procedure for practical engineering analysis[J]. International Journal for Numerical Methods in Engineering,1987,24(2): 337-357.
[3]	ZIENKIEWICZ O C, TAYLOR R L, ZHU J Z. The Finite Element Method: Its Basis and Fundamentals [M]. 7th ed. Singapore: Elsevier, 2013.
[4]	BANGERTH W, RANNACHER R. Adaptive Finite Element Methods for Differential Equations [M]. Springer, 2013.
[5]	LI X D, WIBERG N E. Implementation and adaptivity of a space-time finite element method for structural dynamics[J]. Computer Methods in Applied Mechanics & Engineering,1998, 156(1/4): 211-229.
[6]	THOMPSON L L, HE D T. Adaptive space-time finite element methods for the wave equation on unbounded domains[J]. Computer Methods in Applied Mechanics & Engineering,2005,194(18): 1947-2000.
[7]	张雄, 王天舒, 刘岩. 计算动力学[M]. 2版. 北京: 清华大学出版社, 2015.(ZHANG Xiong, WANG Tianshu, LIU Yan. Computational Dynamics [M]. 2nd ed. Beijing: Tsinghua University Press, 2015.(in Chinese))
[8]	BLUM H, RADEMACHER A, SCHRDER A. Space adaptive finite element methods for dynamic Signorini problems[J].Computational Mechanics,2008, 44(4): 481-491.
[9]	MAYR M, WALL W A, GEE M W. Adaptive time stepping for fluid-structure interaction solvers[J]. Finite Elements in Analysis and Design,2018,141: 55-69.
[10]	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))
[11]	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(English Edition),2004,25(11): 1124-1134.(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 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 Edition),2008,29(5): 533-543.(in Chinese))
[13]	〖JP2〗YUAN S, DU Y, XING Q Y, 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.
[14]	YUAN Si, DONG Yiyi, XING Qinyan, et al. Adaptive finite element method of lines with local mesh refinement in maximum norm based on element energy projection method[J]. International Journal of Computational Methods,2019,18(3): 195008.
[15]	YUAN S, WU Y, XING Q Y. Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique[J]. Applied Mathematics and Mechanics (English Edition),2018,39(7): 1031-1044.
[16]	LIU P F, XING Q Y, DONG Y Y, et al. Application of finite layer method in pavement structural analysis[J].Applied Sciences,2017,7(6): 611.
[17]	邢沁妍, 杨杏, 袁驷. 离散系统运动方程的Galerkin有限元EEP法自适应求解[J]. 应用数学和力学, 2017,38(2): 133-143.(XING Qinyan, 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.(in Chinese))
[18]	邢向华, 张雄, 陆明万. 基于Galerkin法弱形式的时间积分法[J]. 工程力学, 2006,23(7): 8-12.(XING Xianghua, ZHANG Xiong, LU Mingwan. A time integration method based on the weak form Galerkin method [J]. Engineering Mechanics,2006,23(7): 8-12.(in Chinese))
[19]	BORRI M, GHIRINGHELLI G L, LANZ M, et al. Dynamic response of mechanical systems by a weak Hamilton formulation[J]. Computers & Structures,1985,20(1/3): 495-508.
[20]	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. A 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))
[21]	袁全, 袁驷, 李易, 等. 线性元时程积分按最大模自适应步长公式的证明[J]. 工程力学, 2018,35(8): 9-13.(YUAN Quan, YUAN Si, LI Yi, et al. Proof of adaptive time-step size formula based on maximum norm in time integration of linear elements[J]. Engineering Mechanics,2018,35(8): 9-13.(in Chinese))
[22]	杨杏. 基于EEP法的杆件受迫振动有限元自适应分析[D]. 硕士学位论文. 北京: 清华大学, 2016.(YANG Xing. Adaptive analysis of FEM for forced vibrations of bars based on EEP super-convergent method[D]. Master Thesis. Beijing: Tsinghua University, 2016.(in Chinese))
[23]	陆琛宇. 基于EEP法的平面直杆系受迫振动自适应分析的研究[D]. 硕士论文. 北京: 清华大学, 2018.(LU Chenyu. Adaptive analysis of forced vibrations for skeletal systems based on EEP super-convergent method[D]. Master Thesis. Beijing: Tsinghua University, 2018.(in Chinese))
[24]	XING Q Y, LU C Y, YANG X, et al. Adaptive finite element analysis for forced vibration of Euler beams in transverse direction with EEP method[C]// Proceeding of the 15th East Asia-Pacific Conference on Structural Engineering and Constrcution . Xi’an, China, 2017.
[25]	袁驷, 邢沁妍. 一维Ritz有限元超收敛计算的EEP法简约格式的误差估计[J]. 工程力学, 2014,31(12): 1-3.(YUAN Si, XING Qinyan. A direct derivation and proof of super-convergence of EEP displacement of simplified form in one-dimensional Ritz FEM[J]. Engineering Mechanics,2014,31(12): 1-3.(in Chinese))