Efficient and robust quadratures for isogeometric analysis: Reduced Gauss and Gauss–Greville rules

Abstract

This work proposes two efficient quadrature rules, reduced Gauss quadrature and Gauss–Greville quadrature, for isogeometric analysis. The rules are constructed to exactly integrate one-dimensional B-spline basis functions of degree p, and continuity class Cp−k, where k is the highest order of derivatives appearing in the Galerkin formulation of the problem under consideration. This is the same idea we utilized in Zou et al. (2021), but the rules therein produced negative weights for certain non-uniform meshes. The present work improves upon Zou et al. (2021) in that the weights are guaranteed to be positive for all meshes. The reduced Gauss quadrature rule is built element-wise according to the element basis degree and smoothness. The Gauss–Greville quadrature rule combines the proposed reduced Gauss quadrature and Greville quadrature Zou et al. (2021). Both quadrature rules involve many fewer quadrature points than the full Gauss quadrature rule and avoid negative quadrature weights for arbitrary knot vectors. The proposed quadrature rules are stable and accurate, and they can be constructed without solving nonlinear equations, therefore providing efficient and easy-to-use alternatives to full Gauss quadrature. Various numerical examples, including curved shells, demonstrate that they achieve good accuracy, and for p=5 and 6 eliminate locking.