A new numerical integration over arbitrary quadrilateral element based on smoothed strain energy and Richardson extrapolation

Abstract

This article presents a new numerical quadrature based on smoothed strain energy and Richardson extrapolation with conventional one-point quadrature to considerably increase the accuracy of computing element stiffness matrices in finite element method. It is a fast algorithm to obtain the stiffness matrix of an arbitrary quadrilateral element. The computation of the stiffness matrix is sampled in two stages over the same domain. For the first approximation with one sub-cell, unstable one-point sampling is applied for the entire element. Then, One-point sampling is applied independently to each of the four sub-squares of the mapped 2-square of the element quadrilateral and assembled. The matrix is recast in terms of its corner node variables using transition-element type adaptation yielding a stable second approximation. Weighted addition of the approximations by Richardson extrapolation produces a stable stiffness matrix, in a third of the time for, and as good as by, Gauss quadrature and smoothed finite element method. The numerical examples show that the present scheme is free from zero-energy mode, shears locking and produces better accuracy with optimal rate of convergence. • The modified mapping coordinates lead to constant strain displacement matrix in every sampling. • Two stage sampling with RE suppress the hourglass effects. • Patch test, Displacement, and Energy norms are verified. • Though one point has been extensively used, the proposed method works for Functionally graded Materials.