Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework

TL;DR

This paper introduces a simple integration technique for XFEM that avoids element subdivision when modeling discontinuities, maintaining accuracy and ease of implementation in fracture mechanics and multi-material problems.

Contribution

The paper presents a novel integration method that eliminates the need for element subdivision in XFEM, simplifying implementation while preserving accuracy.

Findings

Accurate results in fracture mechanics benchmarks

Effective in multi-material problems

Easily integrable into existing codes

Abstract

Partition of unity methods, such as the extended finite element method (XFEM) allow discontinuities to be simulated independently of the mesh [1]. This eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re-meshing, as the discontinuity evolves. However, to compute the stiffness matrix of the elements intersected by the discontinuity, a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted. In this paper, we use a simple integration technique, proposed for polygonal domains [2] to suppress the need for element subdivision. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi-material problem, show that the proposed method yields accurate results. Owing to its simplicity, the proposed integration technique can be easily integrated in any existing code.