Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method

Abstract

New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method.For discontinuous functions, we construct Gauss-like quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the quadratures, which ensures that the final quadratures have minimal number of Gauss points. For weakly singular integrands, we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately. Numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques.

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

“…For some of the existing methods for integrating discontinuous functions, see Refs. [32,[37][38][39].…”

“…The difference between our method and the technique presented in Ref. [32] is two-fold: (1) we evaluate the lhs of (14) using homogeneous quadratures presented in this paper, which results in fast and efficient evaluation of the integrals; and (2) we fix the locations of the integration points and solve a linear system of equations to obtain the corresponding weights. The number of integration points numx is proportional to the number of basis functions num f present in the moment equations (numx ∝ num f ), and is not affected by the shape of the domain or the configuration of the discontinuity.…”

“…They also showed that the method is not restricted to regular domains-moderate degree close-to-minimal quadrature rules were produced over arbitrary convex and concave polygons. Node elimination algorithm was also used for the integration of discontinuous functions by replacing the weight function with a step function [32]. A nice feature of the node elimination algorithm is that the Newton iterations are only for the optimization of the quadrature rule and the output of each iteration is a solution of (14) by construction.…”

“…For this purpose, similar to Ref. [32], we solve the moment equations (14) over the entire domain after replacing the weight function with a discontinuous function. The difference between our method and the technique presented in Ref.…”

“…For example, in the node elimination algorithm [29,30,32], the basis functions can be selected as monomials that are homogeneous functions (see Sect. 3).…”