弹性力学有限元分析中的平衡与协调理论  

RESEARCH ON EQUILIBRIUM AND CONFORMING THEORY OF THE FINITE ELEMENT METHOD IN ELASTICITY

作　　者：傅向荣[1] 陈璞[2] 孙树立[2] 袁明武[2] FU Xiang-rong;CHEN Pu;SUN Shu-li;YUAN Ming-wu(Department of Civil Engineering,China Agricultural University,Beijing 100083,China;Department of Mechanics and Engineering Science,Peking University,Beijing 100871,China)

机构地区：[1]中国农业大学土木工程系,北京100083 [2]北京大学力学与工程科学系,北京100871

出　　处：《工程力学》2023年第2期8-16,共9页Engineering Mechanics

基　　金：国家自然科学基金项目(11672362,11472014)。

摘　　要：有限元分析中的单元可以遵循不同的方法构造,该文提出以单元模型的平衡性与协调性进行分类,并对弹性力学平面问题中的几种经典单元进行了分析比较,总结了协调元、非协调元和超协调元的协调性方法,以及基于解析试函数法的平衡型方法。单元的协调性理论思路包含单纯形格式协调元和非单纯形格式协调元,以及相应的非协调和超协调元格式,关注的重点是单元边界的协调。单元的平衡性理论思路包含解析试函数和权函数的高阶完备性,关注的重点研究是单元内部及边界平衡性。研究表明:针对弹性力学中平衡性和协调性要求,两类理论给出的不同单元格式各具特点,而既能保证单元内部平衡性,又能考虑单元界面协调性的单元类型给出了更精确、合理的计算结果。Element formulations in finite element analysis may follow different approaches.A classification of elements in equilibrium and compatibility is presented.Analysis and comparison of some classic plane elements show that,non-conforming,conforming and super conforming elements can be constructed based on the compatibility theory,and analytical trial functions can be employed to help the construction of equilibrium elements.The researches of compatibility focus on the boundary conforming between elements,which conform bases of simplex conforming,non-simplex conforming,non-conforming and super conforming elements.The researches of equilibrium focus on the equilibrium principle in element or between elements,which emphasize high-precise completeness of analytical trial functions and weight functions.The benchmark tests show that although all kinds of elements have their own advantages and shortcomings,elements that satisfy both the equilibrium inside elements and the compatibility between elements deliver generally better performance.

关 键 词：特征微分方程解法 平衡型单元 协调型单元 超协调元 解析试函数法 

分 类 号：O343[理学—固体力学]