二维自由振动问题的自适应有限元分析初探  

A PRELIMINARY STUDY ON ADAPTIVE FINITE ELEMENT ANALYSIS OF TWO-DIMENSIONAL FREE VIBRATION PROBLEMS

作　　者：袁驷[1] 孙浩涵 YUAN Si;SUN Hao-han(Department of Civil Engineering,Tsinghua University,Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry,Beijing 100084,China)

机构地区：[1]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

出　　处：《工程力学》2020年第1期17-25,共9页Engineering Mechanics

基　　金：国家自然科学基金项目(51878383,51378293)

摘　　要：自由振动反映结构动力特性,是抗震分析和结构设计的重要基础。近年来,基于单元能量投影(EEP)法的自适应有限元分析已在一系列线弹性及非线性问题中取得成功,而有限元线法(FEMOL)自适应分析在二维自由振动问题中的应用也被证实是有效的。在此基础上,该文进一步提出二维自由振动问题的自适应有限元分析方法。通过将特征值问题线性化,合理引入二维线性问题的 EEP 超收敛计算和自适应求解技术,该法可得到满足精度要求的自振频率和按最大模度量满足用户给定误差限的振型。该文以弹性薄膜为例,介绍了这一进展,并给出数值算例以表明该方法的有效性和可靠性。Free vibration reflects the dynamic characteristics of structures, serving as a basis for seismic analysis and structural design. In recent years, the adaptive finite element (FE) analysis of both linear elastic and nonlinear problems based on element energy projection (EEP) super-convergent technique has achieved remarkable success and a strategy for adaptive finite element method of lines (FEMOL) has also been developed for 2D free vibration problems. On this basis, a new adaptive strategy for FE analysis of 2D free vibration problems is initially proposed in this paper. By linearizing eigenvalue problems into a series of linear problems, the existing EEP-based adaptive strategy for 2D linear problems has been well incorporated. The algorithm yields natural frequencies with required accuracy and mode functions with the accuracy satisfying the user-preset tolerance in maximum norm. Taking elastic membrane as a model problem, the paper gives an up-to-date report of the new progress and presents a number of typical numerical examples to show the validity and reliability of the proposed method.

关 键 词：自由振动 超收敛 最大模 自适应有限元法 单元能量投影 

分 类 号：O32[理学—一般力学与力学基础] O241.82[理学—力学]