非协调元性能分析的两个定理

TWO THEOREMS ABOUT PERFORMANCE OF INCOMPATIBLE ELEMENTS

在构造非协调元的过程中,必须遵守一定的构造规律。本文从基本力学观点出发,提出并证明了两个定理。定理一、如果某种类型的有限单元共有n个独立参与整体刚度运算的自由度,则该单元最多只能精确模拟n种弹性力学基本解。该定理说明了单元的精度从根本上受自身自由度限制的,并指出了现有的四边形四结点单元发展空间不大,而四边形八结点Q8单元以及三维八结点H8单元仍然具有较大的发展余地。定理二则认为四边形四结点内参型非协调元如果能够通过小片试验,则不可能在任意畸变状态下精确表示纯弯场。该定理表明了畸变问题的尝试是有限制的。以上的结论虽然是针对非协调元的构造来提出的,但从论证过程看,应对其它类型的有限单元也适用。定理一和定理二对于今后新型有限元的发展可以起到一定的指导作用。

It is well know that some rules have to be followed when an incompatible element is formulated. From the basic mechanical concepts, two new theorems are presented and proved. Theorem 1: If there exist n independent DOFs, in the computation of stiffness matrix of any kind of finite elements, this element should only simulate n kinds of elasticity basic solutions at the very best. This theorem illustrates that the accuracy of element is limited by its DOF essentially, thus a deduction predicts the quadrilateral 4-node element cannot be improved too much and the quadrilateral 8-node Q8 element and the 3-dimensional 8-node H8 element may be well improved. Theorem 2: If a quadrilateral 4-node incompatible λ-type element passes the patch test, then it should not simulate the pure bending state in arbitrary meshes. This theorem shows that the attempt to overcome distortion is limited. These theorems also hold for other types of elements in addition to incompatible elements. In a word, they would help to formulate new elements.