Efficient Formation of Element Stiffness Matrices on Personal Computers, using Finite-Element Methods

Abstract

This paper compares one-point quadrature with analytical integration as efficient alternative integration techniques to standard two-point Gauss-Legendre quadrature for finite-element codes that adopt bi-linear quadrilateral elements. The accuracy of the solutions obtained by the two alternative schemes for the heat-conduction problem on a square domain was compared with that obtained by standard two-point quadrature. The results show that one-point quadrature saves 75.0% of computer time compared to the two-point quadrature scheme while analytical integration saves 37.5% of computer time. Although, one-point quadrature generally requires an “hour-glass” correction, the paper shows that such a correction is not necessary when a Dirichlet boundary condition is applied over a large part of the solution domain. Solution of the conduction problem on a skewed domain proves that one-point quadrature remains acceptably accurate even for highly distorted elements.