Research Article Open Access

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

Subramanian G.1 and P.V. Jeyakarthikeyan 1
  • 1 SRM University, India
American Journal of Applied Sciences
Volume 15 No. 4, 2018, 219-229

DOI: https://doi.org/10.3844/ajassp.2018.219.229

Submitted On: 20 May 2017 Published On: 27 September 2017

How to Cite: G., S. & Jeyakarthikeyan , P. (2018). Weighted Integration Route to Stiffness Matrix of Quadrilaterals for Speed, Accuracy and Functionally Graded Material Application. American Journal of Applied Sciences, 15(4), 219-229. https://doi.org/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. 

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Keywords

  • Weighted Integration
  • One-Point Sampling
  • Hourglass Control
  • Stiffness Matrix
  • Quadrilaterals/Parallelograms
  • Universal Matrices
  • Functionally Graded Materials
  • Super Element