平面广义四节点等参元GQ4及其性能探讨

广义节点有限元是将传统有限元方法中的节点广义化,在不增加节点个数的前提下,仅通过提高广义节点的插值函数的阶次,从而达到提高有限元解精度的目的.与现有的p型和hp型有限元不同,在这种新的有限元中,节点自由度全部定义在节点处,在理论与程序实现上与传统有限元方法具有很好的相容性,传统有限元方法是这种新方法的广义节点退化为0阶时的特殊情形.文中主要讨论了这一新方法的四节点等参元（记为GQ4）的形式.对GQ4进行的各种数值试验表明,所发展的广义四节点等参单元具有精度高且无剪切自锁与体积自锁等的特点.

GENERALIZED 4-NODE QUADRILATERAL ISOPARAMETRIC FINITE ELEMENTS

In general, stresses between elements is discontinuous with jump in the conventional finite element methods such as the FEM with three node triangle elements. The main reason is that the element displacement function is smooth only with zero-order. If high-order element displacement functions could be constructed, stress discontinuity between elements would be im-proved.Different from that in p-type and hp-type finite elements, a new procedure for constructing displacement functions of element is proposed by the authors recently. In this method, the dis-placements of element nodes are represented by a product of the nodal interpolation functionsand generalized degrees of freedom. Therefore the resulting shape function is formed by com-bining shape functions of the conventional FEM with the so-called nodal interpolation functions, which can be chosen by users as polynomial functions, trigonometric functions or even Taylor series according to characteristics of local displacement field. The order of shape function can be enhanced very easily through adoption of high-order nodal interpolation functions while the number of nodes is kept unchanged. The order of global functions would then be increased and high precise approximation of field function is gained finally. In the new FEM, a node can have multi-degrees-of-freedoms more than two for a plane problem or three for a space problem. Such a node is defined as the generalized node. Therefore, the new FEM is termed as generalized-node finite element method (GNFEM).In the new FEM, because of influence of the conventional shape function, consistency of the new shape function can be naturally fulfilled. By adoption of high-order nodal interpolation functions, the GNFEM can not only increase the precision of displacements and stresses of the conventional FEM, but also smooth away the stress jump between two adjacent elements, leading to that efforts in stress smoothing in the FEM can be saved.If nodal interpolation functions are chosen as constant functions, the GNFEM is reduced to the conventional FEM.The GNFEMs with triangle plane elements have been discussed by the authors. Presented in the paper are the fundamental and formulation of generalized 4-node quadrilateral isoparametric finite element with first-order and second-order polynomial nodal interpolation functions. Numerical analyses for the benchmark problems indicates that this type of elements is superior over the conventional isoparametric finite elements in its high accuracy, no shear locking and no volume locking, high efficiency and good stability.