重调和方程的 Argyris 元法

The Argyris ElementMethod of BiharmonicEquation

摘要: 本文考虑求解四阶椭圆边值问题的协调有限元方法: Argyris 元。重调和方程有着非常重要的物理背景和有广泛的应用。例如，在薄板弯曲问题中，用来描述其弯曲程度，是一个经典的问题。Argyris 元是四阶问题的协调有限元，也即是 Argyris 元基函数构成的有限元空间是解空间的子空间。而 Argyris 元的基函数是 5 次多项式，推导而言比较麻烦。本文利用每个单元上的单元信息求解出来每个单元上的基函数，再利用单元上的基函数构造多项式逼近原函数，这样便不必要推出其基函数的一般表达式，减少了麻烦的计算。

Abstract: In this paper, we consider the coordination finite element method for the fourth order elliptic boundary value problem: Argyris element. Reharmonic equations have a very important physical background and wide applications. For example, in the problem of sheet bending, used to describe the degree of bending, is a classic problem. The Argyris element is the coordinated finite element of the fourth order problem, that is, the finite element space formed by the Argyris element basis function is a subspace of the solution space. But the basis function of Argyris element is a polynomial of 5 times, which is more troublesome to derive. In this paper, the unit information on each cell is used to solve the basis function on each cell, and the basis function on the unit is used to construct polynomial approximating the original function, which is not necessary to introduce the general expression of its basis function, which reduces the troublesome calculation.