弹性力学平面问题的精确元法

An Exact Element Method for Plane Problem

本文在阶梯折算法和精确解析法的基础上,提出构造有限元的新方法——精确元法.该方法不用一般变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法得到弹性力学平面问题的一个非协调任意四边形单元.它具有八个自由度.由于没有采用雅可比变换,该单元可以蜕化为三角形单元,在工程中使用起来较为方便.文中给出收敛性证明.文末给出算例,位移和应力均给出较好的结果,在单元的节点上有较好的数值精度.

In this paper,based on the step reduction method[1] and exact analytic method[2], a new method-exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a quadrilateral noncompatible element with 8 degrees of freedom is derived for the solution of plane problem. Since Jacobi's transformation is not applied, the present element may degenerate into a triangle element. It is convenient to use the element in engineering. In this paper, the convergence is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress and displacements can be obtained and have higher numerical precision in nodes.