Shortened quadrature rules for finite elements

Abstract

A numerical method to calculate Rieman's integrals for complete polynomials on R n space is presented. The method is based on the direct elimination of some polynomial tenns, whose contribution to the value of the integral is zero for particular positions of the sampling points. Using the proposed scheme to carry out the integrations occurring in the finite element formulations leads to a remarkable reduction in the computing time, with respect to the employment of the ordinary Gauss-Legendre quadrature rule. Moreover, if this rule is available at least as an alternative, those indeterminations can be removed, that depend on local singularities arising in the transformations performed before the integrations themselves. Tables are reported giving the normalised coordinates and the weighting factors of the sampling points, for various quadrature orders and polynomial degrees. Examples are produced from which an estimate can be drawn of the computing savings obtained by the method.