Symmetric and asymmetric Gauss and Gauss–Lobatto quadrature rules for triangles and their applications to high-order finite element analyses

Abstract

Although symmetric Gauss quadrature rules have been studied extensively, the investigations of asymmetric Gauss quadrature rules and symmetric Gauss–Lobatto (GL) quadrature rules are limited. Moreover, GL quadrature rules are essential for high-order finite element methods like the quadrature element method (QEM) and the spectral element method (SEM) in strong and weak form, and asymmetric Gauss quadrature rules are even more efficient than the symmetric ones. This work first presents new optimization methods for solving symmetric and asymmetric Gauss quadrature rules and then generalizes the techniques to solve GL and hierarchical Gauss–Lobatto (HGL) quadrature rules. Most of the asymmetric Gauss quadrature rules with precision degree range from 1 to 50, and symmetric GL and HGL quadrature rules presented in this work are new. They are attached to this article as MATLAB m-files for future usage. The GL and HGL quadrature rules are (1) used for constructing weak form QEM and HQEM with diagonal-mass-matrix (DMM) that is promising in dynamic analyses, and (2) used as interpolation nodes in high-order finite element analyses like Fekete points. Applying the present asymmetric Gauss quadrature rules and symmetric GL and HGL quadrature rules to in-plane vibration shows that both cases of usage of the GL and HGL quadrature rules can provide results with high accuracy and convergence rate.