An Efficient and Rapid Numerical Quadrature to generate element matrices for quadrilateral and hexahedral elements in Functionally Graded Materials (FGMs)

Abstract

This research paper introduces a robust, efficient and stabilized midpoint quadrature to generate element stiffness matrices for quadrilateral and hexahedral elements in finite element method (FEM). This new quadrature is established based on Passive Richardson extrapolation which has the inherent ability to stabilize and improve the accuracy of the extrapolated results. The new quadrature includes a simple artificial stabilizing function with Gauss one-point quadrature (1GP) as an anti-hourglass effect. For the first approximation, the 1GP (sampling point to be considered at origin (0,0)) is done to evaluate element matrices (thermal conductivity matrices, capacitance matrices etc.,) in the 4-square element ( [-2, 2] x [-2, 2] ) in the (ξ, η) plane. As a stabilizing function, sampling points are considered to be either at the center of the four element edges or center of the four quadrants in the reference square element with appropriate constant weighting functions. For hexahedral elements also, the element stiffness coefficients are calculated only in the center of the element and center of all the octants of the reference cubic element ([-2, 2] x [-2, 2] x [-2, 2]), the proposed method deals mostly with integer variables and sampling at (0,0,0), whereas Gauss quadrature uses lengthy float variables corresponding to irrational numbers in terms of decimals in the stiffness matrix calculations, thus consuming time. A number of standard example problems are demonstrated to assess the efficiency of the proposed method. The results of all the numerical examples are found to be in good agreement with the exact results and conventional Gauss quadrature results.