高阶形式广义节点有限元法及其应用

所发展的广义节点有限元法是将传统有限元法中的节点广义化,在不增加节点个数的前提下,仅通过提高广义节点插值函数的阶次,达到提高有限元解精度的目的。传统有限元法是这种方法当广义节点阶数退化为０时的特例。主要讨论了这一新 高阶形式。重点分析了广义节点阶次的提高对计算精度以及计算量的影响,并与低阶方法以及传统有限元法进行了比较。对受弯悬臂梁和半无限平面受集中力作用两个算例的数值分析表明：１）广义节点阶次的

High-order generalized-node finite element method

The generalized node finite element method (GFEM) is a novel numerical method developed recently. In this method, the node is generalized and can have more than two or three generalized degrees of freedom which are not necessarily required to have their definite physical meaning. At each generalized node, a generalized type of nodal interpolation function (GNIF) which can take a polynomial or series chosen by a user is defined. When a zero order GNIF is used, the method would be reduced to conventional finite element method (FEM). In the paper, the fundamental and formulation with second order polynomial GNIF have been presented and its generalized nodal stiffness matrix has been given. Then the method is applied to analyze two benchmark problems, i.e., bending of a cantilever beam under shear loading at the free end and a half plane foundation under a vertical point force. It has been shown that both displacements and stresses computed by this new method with only a few elements agree fairly well with theoretical solutions. Compared with the conventional FEM and lower order GFEM, the smoothing characteristics of stress solutions among elements could be substantially improved when a high order GNIF is used.