American Journal of Applied Sciences

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

Subramanian G. and P.V. Jeyakarthikeyan

DOI : 10.3844/ajassp.2018.219.229

American Journal of Applied Sciences

Volume 15, Issue 4

Pages 219-229


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. 


© 2018 Subramanian G. and P.V. Jeyakarthikeyan . This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.