A simple and robust quadrilateral non-conforming element with a special 5-point quadrature

Abstract

Abstract A 4-node, 8-DOF non-conforming quadrilateral element with four internal modes, denoted as iQ8, is developed. It takes the four edges' midpoints of the linear element Q4 as the internal virtual nodes, whose shape functions are the same as those of the mid-side nodes of the quadratic element Q8. In order to deal with the rank deficiency of the new element stiffness matrix under 2 × 2 Gaussian quadrature, a new reduced 5-point integration scheme is constructed. By both the eigenvalue analysis of the element stiffness matrix and the example for the patch test, the weight of the central integration point is determined to a value close to 4, correspondingly the 5-point quadrature converges to 1-point quadrature. The non-central integration points are far away from the element domain, which is much different from conventional quadratures. Finally, some typical numerical examples are tested, and the results show that, except for only passing the weak patch test, the new element achieves the best performance of other 4-node quadrilateral elements in the aspects of anti-distortion, being free of trapezoidal locking, and exactness in pure bending.