A quasi-optimal convergence result for fracture mechanics with XFEM

TL;DR: Chahine et al. as discussed by the authors gave a convergence result for a variant of the eXtended Finite Element Method (XFEM) on cracked domains using a cut-off function to localize the singular enrichment area.

About: This article is published in Comptes Rendus Mathematique.The article was published on 2006-04-01 and is currently open access. It has received 33 citations till now. The article focuses on the topics: Extended finite element method & Rate of convergence.

Summary

1. Introduction

• Classical finite element methods used for modeling crack propagation are subjected to several constraints: the mesh should match the crack geometry, should always evolve with the crack growth and should be refined near the crack tip.
• This motivated Moës, Dolbow and Belytschko to introduce an approach called XFEM (eXtended Finite Element Method) in 1999 (see [11]).
• The idea is to add singular functions to the finite element basis taking into account the singular behavior around the crack tip, and a step function modeling the discontinuity of the displacement field across the crack.
• This is not an improvement of the convergence order of the classical finite element method solution (see [8,13]).
• This later method can even realize better convergence results for the computation of the stress intensity factors (see [3]).

2. Model problem

• The authors consider the linear elasticity problem on this domain for an isotropic material.
• The normal (respectively tangential) component of the function uI (respectively uII) is discontinuous along the crack.
• They both correspond to the well known I and II opening modes for a bi-dimensional crack (see [9,10]).

3. XFEM: description and discretized problem

• The idea of XFEM is to use a classical finite element space enriched by some additional functions.
• These functions result from the product of global enrichment functions and some classical finite element functions (see [11]).

4. Error estimate

• Let the displacement field u, solution to problem (1), satisfy the condition (2).
• Note that a similar work has been done in [7], but for a domain totally cut by the crack, which means that the domain does not contain a crack tip.
• Thus the interpolation operator the authors defined allows us to make a classical interpolation over each part of the triangle, and to have the same optimal rate of convergence obtained in the classical global interpolation theorem (see [2,4,12]).

5. Concluding remarks

• (ii) Let us note that the work presented in [12] is applied to a mesh respecting the crack geometry.
• Thus it does not involve the problem presented here of the triangles partially enriched by the Heaviside function.
• On the other hand, this note offers an improvement for the ‘classical’ XFEM method where the convergence rate remains of order √ h for some reasons detailed in [8].

Figures

Fig. 1. The domain decomposition and the enrichment strategy.