Generalized Gaussian Quadrature Rules in Enriched Finite Element Methods

In this paper, we present new Gaussian integration schemes for the efficient and accurate evaluation of weak form integrals that arise in enriched finite element methods. For discontinuous functions we present an algorithm for the construction of Gauss-like quadrature rules over arbitrarily-shaped elements without partitioning. In case of singular integrands, we introduce a new polar transformation that eliminates the singularity so that the integration can be performed with fewer integration points while maintaining high accuracy. We combine the quadrature construction technique with a point elimination algorithm, which ensures that the final quadratures have as few number of Gauss points as possible without sacrificing accuracy.