# A quasi-optimal convergence result for fracture mechanics with XFEM

Abstract: The aim of this Note is to give 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. The difficulty is caused by the discontinuity of the displacement field across the crack, but we prove that a quasi-optimal convergence rate holds in spite of the presence of elements cut by the crack. The global linear convergence rate is obtained by using an enriched linear finite element method. To cite this article: E. Chahine et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

## Summary (1 min read)

### 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].

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