TWO THEOREMS ABOUT PERFORMANCE OF INCOMPATIBLE ELEMENTS

It is well known 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 in this paper. 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.