| 杆件轴向受迫振动的Galerkin有限元EEP法自适应求解 |
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| 引用本文： | 邢沁妍,杨青浩,陆琛宇,杨杏.杆件轴向受迫振动的Galerkin有限元EEP法自适应求解[J].应用数学和力学,2019,40(9):945-956. |
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| 作者姓名： | 邢沁妍 杨青浩 陆琛宇 杨杏 |
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| 作者单位： | 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 |
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| 基金项目： | 国家自然科学基金（51508305） |
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| 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.
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| 关 键 词： | 受迫振动 时空有限元 自适应 Galerkin法 单元能量投影 |
| 收稿时间： | 2019-02-03 |
An EEP Adaptive Strategy of the Galerkin FEM for Axially Forced Vibration of Bars |
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| Affiliation: | Department of Civil Engineering, Tsinghua University; Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Beijing 100084, P.R.China |
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| 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. |
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