Weighted Integration Route to Stiffness Matrix of Quadrilaterals for Speed, Accuracy and Functionally Graded Material Application

Subramanian G,P.V Jeyakarthikeyan

doi:10.3844/ajassp.2018.219.229

Abstract

A weighted integration route with robust one-point integration (hourglass-controlled) is proposed as efficient, time saving alternative to Gauss quadrature for stiffness matrix of bilinear quadrilaterals. One-point rule relies on sampling at the center of the element to linearize the geometric transformation and average the material property over it. This enables, for a given element, explicit integration of stiffness matrix yielding a first approximation. For a second and better approximation, this procedure is applied independently to each of the four sub-squares of the mapped 2-square of the element and the matrices are assembled. A weighted addition of the two approximations produces a stiffness matrix as accurate as from 3-point Gauss-quadrature (G9P). Whereas, due to explicit integrations, obtaining stiffness matrix in this way demands less than a third of the time needed for 2-point Gauss-quadrature (G4P). On both counts (speed and accuracy) this approach outperforms Gauss-quadrature. Sampling (material and geometry) at 5-points makes this element superior to G4P for Functionally Graded Material (FGM) applications. Bench mark examples by this approach are validated with Gauss quadrature and analytical solutions.

Highlights

Even to this day developing stiffness matrices for plane quadrilateral elements keeps enticing researchers (Jeyakarthikeyan et al, 2017)
Following a procedure similar to that for triangles the stiffness matrix is conveniently generated in terms of numerical universal matrices for the matching parallelogram
Despite the apparently elaborate exercise to find the stiffness matrix, the entire process still requires a third of the time needed for G4P

Summary

Introduction

Even to this day developing stiffness matrices for plane quadrilateral elements keeps enticing researchers (Jeyakarthikeyan et al, 2017). For a second approximation a cluster of four stable one-point quadrilaterals are carved out of the given quadrilateral and assembled into an (18×18) matrix (in principle) and reduced to (8×8) super element by enforcing the bilinear nature of the displacement description over the element (This procedure is not a static condensation, a procedure associated usually with standard super element formation, but is akin to obtaining transition elements from higher order elements: Zienkiewicz et al (2005; Jeyachandrabose and Kirkhope, 1984)) These two approximations (with h-squared error) are weighted and extrapolated to give a vastly improved stiffness matrix that is as good as employing G9P. Following a procedure similar to that for triangles the stiffness matrix is conveniently generated in terms of numerical universal matrices for the matching parallelogram (as the mid-point rule for a quadrilateral renders the geometric transformation linear). Actual calculations would follow the procedure outlined here to optimize computing effort

Procedure

Conclusion