用拟协调元直接构造平面任意四边形单元——进入有限元的禁区

摘要

利用拟协调元方法，在直角坐标系下直接构造了一族平面任意四边形单元，对其收敛性进行了分析，并与平面等参元进行了对比研究．结果证明平面任意四边形单元可采用多项式基函数直接列式，并可以保障单元的收敛性；拟协调元列式可以使平面问题的有限元方法得到统一．与平面等参元相比，单元列式简单，性能稳定，具有显式的刚度阵，计算量小，这说明对于有限元平面问题拟协调元是一个更正确、有效的做法．%The direct formulation of quadrilateral plane element in rectangular Cartesian coordinate system has been a forbidden zone for finite element method. In this paper, the quasi-conforming finite element method is applied on this problem and a bilinear element as well as complete second-order element is constructed. Meanwhile, the convergence anMysis of the elements is carried out with ＂Taylor expansion test＂ and the comparative study with isoparametric element is also considered. The results show the direct formulation of quadrilateral plane elements is feasible within the quasi-conforming framework, and there is not any convergence problem with these elements. Therefore the finite element theory concerning plane element can be unified by quasiconforming framework. Compared with isoparametric elements, the bilinear element present in this paper is easy to formulate, with stable performance and explicit stiffness matrix for the plane problem analysis.