拟协调有限元与弱形式广义方程

Quasi-conforming element methods and generalized weak formulation equations in elasticity

文章从弱形式基本方程出发,阐述了弱形式弹性力学基本方程是拟协调元的内在本质,指出用弱形式表达的平衡条件既有微分方程也有边界条件,也可看成是变分的出发点,是更根本和原始的条件;从形式上看弱形式对函数的连续性降低了,但对实际的物理问题常常较原始的微分方程更逼近真正解,拟协调元其做法就是广义协调方程的直接解,自然满足平衡对弱连续条件的要求,进而说明拟协调元是有限元发展的必然趋势,它可以解决常规有限元难以适应的领域,是计算力学发展过程中的一个里程碑。

This paper explains that the weak form of the basic equations of elasticity is the inherent nature of the quasi-conforming element on the basis of the basic equation of weak forms. It points out that the weak forms of equilibrium conditions can be differential equations or boundary conditions, which can also be seen as a starting point for variation, and are considered as more fundamental and original conditions. Formally speaking, because of the weak form, the continuity of a function is reduced, while for the actual physical problem, the weak form is often more approximate to the true solution than the original differential equations. The quasi-conforming elernent is a direct solution to the generalized equations, thus satisfying the requirements of the equilibrium for weak continuous conditions. Therefore the quasi-conforming element is the inevitable trend of development of the finite element, and it can be used to solve the problems in the fields where the conventional finite element is difficult to adapt. The quasi-conforming element is a milestone in the development of computational mechanics.