有限元法的误区和拟协调元  

The misconception in finite element method and revision from quasi-conforming

作　　者：唐立民[1] 胡平[2] 夏阳[2] 

机构地区：[1]大连理工大学工程力学系,大连116024 [2]大连理工大学工业装备结构分析国家重点实验室、运载工程与力学学部汽车工程学院,大连116024

出　　处：《中国科学：物理学、力学、天文学》2014年第8期838-850,共13页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金重点项目(批准号:10932003);国家自然科学基金(批准号:11272075)资助项目

摘　　要：现行有限元法的理论和作法存在误区,对有限元函数空间和弱形式导数认识不足,因而阻碍了有限元法的进一步发展.本文给出了拟协调元的理论和作法作为对照.利用函数序列表示有限元函数空间,提出了单元独立性原理,证明协调条件不是单元函数构造时必须考虑的因素.强调多项式基函数在构造单元函数中的作用,讨论了单元函数的构造,指出单元函数应随着单元细化收敛于相应的泰勒级数.证明了使用弱形式平衡方程时,必须同时使用弱形式的协调方程.提出网线弱导数,拓广了有限元中弱导数的内涵.The theory and application of present finite element method have misconceptions. These misconceptions include the insufficient understanding of functional space of finite element and the formulation of weak form derivatives. These misconceptions block the development of finite element method, which lead to the difficulty and failure in solving some problems. The theory and application of quasi-conforming finite element method are displayed as a reference. A sequence of functions is defined to demonstrate the functional space of finite element. The independence principle of elements is given to solve the ＂conforming problem＂. The polynomial basis rather than ＂shape function＂ is emphasized. The formulation of approach functions of finite element is discussed, which should convergence to the corresponding Taylor series. It is proved that the weak form of strain-displacement equation is necessary and sufficient condition of weak form of equilibrium equation. The string net weak derivative is proposed, which expands the connotation of weak form derivative in finite element method.

关 键 词：网线弱导数 有限元函数空间 单元独立性原理 拟协调元 有限元 

分 类 号：O241.82[理学—计算数学]