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 2
  • 1 Department of Aerospace Engineering, SRM University, India
  • 2 Department of Mechanical Engineering, SRM University, India

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. 

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

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Keywords

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