Fast numerical algorithms with universal matrices for finding element matrices of quadrilateral and hexahedral elements

Abstract

In this paper, a new closed-form formulation with universal matrices strongly recommends that Weighted Richardson Extrapolation (WRE) with robust and hourglass controlled one-point quadrature can absolutely replace the conventional Gauss Quadrature in terms of efficiency, accuracy, and speed to find element stiffness matrices of quadrilateral and hexahedral elements. Linearizing the geometric transformation and averaging the material property over an element, using sampling point at the origin (0,0) of a standard mapped 2-square (ξ,η) plane for two-dimensional analysis and origin (0,0,0) of a standard mapped 2-cube (ξ,η,ζ) system in the hexahedral element (mid-point rule), helps to obtain element stiffness matrix quickly and explicitly through constant universal matrices. This technique is used independently to each of the eight sub-cubes of the mapped 2-cube (ξ,η,ζ) of the element for three-dimensional problems and four sub-squares of the mapped 2-square of the element for two-dimensional problems. These matrices are assembled appropriately for a second and better approximation. A weighted addition of the two approximations produces a stiffness matrix as accurate as from conventional Gauss Quadrature. However, finding the stiffness matrix in this way, due to explicit integrations, demands only a third of the time needed for Gauss Quadrature.