Generalized Gaussian quadrature rules on arbitrary polygons

In this paper, we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons. These quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required. We have used this algorithm for the construction of symmetric and non-symmetric quadrature rules over convex and concave polygons. While in the case of symmetric quadratures our results are comparable to available rules, the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions. To demonstrate the efficiency of the new quadrature rules, we have tested them for the integration of rational polygonal shape functions over a regular hexagon. For a relative error of 10(-8) in the computation of stiffness matrix entries, one needs at least 198 evaluation points when the region is partitioned, whereas 85 points suffice with our quadrature rule. Copyright (C) 2009 John Wiley & Sons, Ltd.