scispace - formally typeset

Journal ArticleDOI

A quasi-optimal convergence result for fracture mechanics with XFEM

01 Apr 2006-Comptes Rendus Mathematique (Elsevier Masson)-Vol. 342, Iss: 7, pp 527-532

AbstractThe 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].

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

HAL Id: hal-00690581
https://hal.archives-ouvertes.fr/hal-00690581
Submitted on 7 Feb 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A quasi-optimal convergence result for fracture
mechanics with XFEM
Elie Chahine, Patrick Laborde, Yves Renard
To cite this version:
Elie Chahine, Patrick Laborde, Yves Renard. A quasi-optimal convergence result for fracture mechan-
ics with XFEM. Comptes rendus de l’Académie des sciences. Série I, Mathématique, Elsevier, 2006,
342, pp.527-532. �10.1016/j.crma.2006.02.002�. �hal-00690581�

A quasi-optimal convergence result for fracture mechanics with
XFEM
Elie Chahine
a
, Patrick Laborde
b
, Yves Renard
a
a
MIP, CNRS UMR 5640, INSAT, 135, avenue de Rangueil, 31077 Toulouse cedex 4, France
b
MIP, CNRS UMR 5640, UPS Toulouse 3, 118 route de Narbonne, 31062 Toulouse cedex 4, France
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.
Résultat de convergence quasi-optimal en mécanique de la rupture avec XFEM. Le but de cette Note
est de donner un résul-
tat de convergence pour une variante de la méthode XFEM (eXtended Finite Element Method) sur un domaine fissuré en utilisant
une fonction cut-off pour localiser l’enrichissement par les fonctions singulières. La difficulté est causée par la discontinuité du
champ de déplacement à travers la fissure, mais on montre une convergence quasi-optimale malgré la présence d’éléments coupés
par la fissure. Le résultat de convergence globale linéaire est obtenu en utilisant une méthode d’éléments finis affines enrichis.
Version française abrégée
Les méthodes d’éléments finis classiques utilisées pour la modélisation des fissures, pour être efficaces, doivent
respecter la géométrie de la fissure étudiée. Cette approche n’est pas très avantageuse dans la mesure elle nécessite
un grand nombre de degrés de liberté pour le raffinement en fond de fissure, ainsi qu’un remaillage lors de la pro-
pagation de la fissure. C’est la raison qui a motivé l’introduction de la méthode des éléments finis enrichis (XFEM :
eXtended Finite Element Method) par Moës, Dolbow et Belytschko en 1999 (voir [11]). Celle-ci consiste à ajouter
E-mail addresses: elie.chahine@insa-toulouse.fr (E. Chahine), laborde@mip.ups-tlse.fr (P. Laborde), yves.renard@insa-toulouse.fr
(Y. Renard).
1

à la base éléments finis classique des fonctions singulières au voisinage de la pointe de la fissure, et une fonction
discontinue tout au long de la fissure.
Le but de cette Note est de donner un résultat de convergence pour une variante de la méthode XFEM. Des résultats
numériques ont montré que la méthode XFEM classique converge en
h,oùh est le pas du maillage. Ceci ne présente
pas une amélioration de l’ordre de convergence de la solution de la méthode des éléments finis classique (voir [8,13]).
Cette dernière permet même de réaliser une meilleure convergence pour le calcul des facteurs d’intensité de contraintes
(voir [3]). Nous établissons, dans ce qui suit, que la variante proposée converge en h pour des éléments finis affines. On
considère, dans la Section 2, le problème d’élasticité linéaire (1) sur un domaine bidimensionnel borné et fissuré ,
et on suppose que le déplacement u solution est décomposé comme en (2). Dans la section suivante, on définit un
espace d’éléments finis affines enrichi par des fonctions singulières et une fonction discontinue sur une triangulation
de
(voir (4) et Fig. 1(b)). Dans la Section 4, le résultat de convergence est donné par le Théorème 4.1, suivi des
étapes principales de la preuve. Le domaine est divisé en deux suivant la fissure et son prolongement (Fig. 1(a)),
ce qui permet de définir un opérateur d’interpolation (voir (11) et (12)) en utilisant un opérateur d’extension (10).
Puis, on calcule les erreurs d’interpolation locales sur les élements totalement enrichis par la fonction discontinue H
(Lemme 4.2), sur l’élément contenant le fond de la fissure (Lemme 4.3), et sur les éléments partiellement enrichis
par H (Lemme 4.4).
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 allows us to use a fixed mesh independent of the crack geometry. A better approximated solution,
than the classical finite element one, can be obtained by choosing the appropriate singular enrichment (see [5]). For a
two-dimensional crack, the singular enrichment functions are based on the exact solution given in [9,10].
The aim of this Note is to give a convergence result for a variant of the XFEM. Numerical results showed that
the convergence error of the XFEM is of order
h, where h is the mesh parameter. 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]). We will prove, in what follows,
that the convergence error of the proposed variant is of order h for a linear finite element method. We consider in
Section 2, the elasticity problem over a cracked bi-dimensional domain and we suppose that the displacement field
u can be written as a sum of a singular part belonging to H
3/2
and a ‘regular discontinuous’ part (see [6]). In
Section 3, we define an affine finite element space enriched by singular functions and by a discontinuous function
over a triangulation of
. In Section 4, the main convergence result is given in Theorem 4.1 followed by the essential
steps of the proof. Indeed, the domain is divided into two subdomains which allow us to define an interpolation
operator based on an extension operator. Then we compute, in separate lemmas, the local interpolation errors over
the elements totally enriched by the discontinuous function H , over the element containing the crack tip and over the
elements partially enriched by H .
2. Model problem
Let be a bounded cracked domain in R
2
and let Γ
C
be the crack which is assumed to be straight. We consider
the linear elasticity problem on this domain for an isotropic material. The boundary of , denoted ∂Ω, is partitioned
into Γ
D
where a Dirichlet condition is applied, Γ
N
where a Neumann condition is applied, and Γ
C
(the crack) where
we assume having a traction free condition.
Let ϑ ={v H
1
(Ω);v = 0onΓ
D
} be the space of admissible displacements and
a(u,v) =
σ(u): ε(u) dx, l(v) =
f ·v dx +
Γ
N
g ·v dΓ, σ(u) =λ tr ε(u)I +2µε(u),
2

with σ(u) denoting the stress tensor, ε(u) the linearized strain tensor, g and f some given densities on Γ
N
and
respectively, λ>0 and µ>0 the Lamé coefficients. The problem can be written
find u ϑ such that a(u,v) = l(v) v ϑ. (1)
We suppose that f and g are sufficiently smooth (f H
(Ω), and g H
1/2+
N
) for some ∈]0;1/2[) such that
the solution u to the elasticity problem can be written as a sum of a singular part u
s
and a ‘regular discontinuous’ part
satisfying (see [6]):
u r
1/2
(c
1
u
I
+c
2
u
II
) = u u
s
H
2+
(Ω), (2)
where r denotes the distance to the crack tip. The normal (respectively tangential) component of the function u
I
(respectively u
II
) 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]). The asymptotic displacement at the crack tip for these two modes is given in
polar coordinates by
u
I
=
K
I
E
r
2π
(1 +ν)
cos
θ
2
(3 4ν cos θ)
sin
θ
2
(3 4ν cos θ)
,u
II
=
K
II
E
r
2π
(1 +ν)
sin
θ
2
(C
1
+2 +cos θ)
sin
θ
2
(C
1
2 +cos θ)
, (3)
where K
I
, K
II
denote the stress intensity factors, ν the Poisson ratio and C
1
= 3 4ν in the plane stress problem.
Note that u
I
and u
II
belong to H
3/2
(Ω) for any >0 (see [6]).
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]). We
consider a linear Lagrange finite element method defined on a regular triangulation of the non-cracked domain
.
The piecewise P
1
basis functions are denoted ϕ
1
,...,ϕ
N
(P
1
is the set of degree one polynomials). The enriched
space of the XFEM variant we propose is then
ϑ
h
=
v
h
=
iI
a
i
ϕ
i
+
iI
H
b
i
i
+
4
j=1
c
j
F
j
γ ; a
i
R
2
,b
i
R
2
,c
j
R
2
, (4)
where I is the set of the indices of the classical finite element nodes and I
H
is the set of the indices of the nodes
enriched by the following Heaviside function:
H(x)=
1if(x x
) ·n 0,
1 elsewhere,
(5)
x
denoting the crack tip and n is a normal unit vector to the crack. Moreover, (F
j
,j=1,...,4) are singular functions
given in polar coordinates by
F
j
(x)
r sin
θ
2
,
r cos
θ
2
,
r sin
θ
2
sin),
r cos
θ
2
sin)
, (6)
and γ is a C
3
cut-off function such that (0 <r
0
<r
1
)
γ(x)=1ifx<r
0
, 0 (x)<1ifr
0
<x<r
1
(x)= 0ifr
1
<x. (7)
The discrete problem can be written as follows
Find u
h
ϑ
h
such that a
u
h
,v
h
= l
v
h
v
h
ϑ
h
. (8)
4. Error estimate
Theorem 4.1. Let the displacement field u, solution to problem (1), satisfy the condition (2). Then, the following
estimate holds
u u
h
1,Ω
Chu γu
s
2+,Ω
, (9)
where u
h
is the solution to problem (8) and C>0 is a constant only depending on .
3

Fig. 1. The domain decomposition and the enrichment strategy.
Essential steps of the proof. The linear finite element method we consider is defined on a regular triangulation T
h
of
the non-cracked bi-dimensional domain
, where h = max
KT
h
(h
K
) with h
K
= diam(K) = max
x
1
,x
2
K
x
1
x
2
(K T
h
). Let ρ
K
={sup(diam(B));B ball of R
2
,B K}. In order to define an interpolation operator, and since the
displacement field is discontinuous across the crack, we divide into two sub-domains
1
and
2
according to the
crack and a straight extension of this crack (Fig. 1(a)). The displacement fields over
1
and
2
are denoted u
1
and
u
2
respectively. Let u
rd
be the ‘regular discontinuous’ displacement defined by u
rd
= u γu
s
, thus u
1
and u
2
are
the restrictions of u
rd
over
1
and
2
respectively. Let u
1
and u
2
be the extensions of u
1
over
2
and u
2
over
1
respectively such that (see [1]):
u
1
2+,
C
1
u
1
2+,Ω
1
and
u
2
2+,
C
2
u
2
2+,Ω
2
. (10)
Every triangle K cut by the crack is divided into K
1
= K
1
(where u
1
lives) and K
2
= K
2
(where u
2
lives).
In order to interpolate on K
1
(respectively on K
2
) one needs three interpolation points, that is why the nodal values of
the extension of u
1
(respectively u
2
) over K will be used. Thus every triangle totally enriched by H will have twelve
degrees of freedom. 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.
We define now the interpolation operator for a function u satisfying (2) as follows
Π
h
u =
iI
a
i
ϕ
i
+
iI
H
b
i
i
+
4
i=1
c
i
F
i
γ, (11)
such that (x
i
denoting the node of index i):
if i I \I
H
then a
i
= u(x
i
),
if i I
H
and if x
i
1
then a
i
+b
i
= u
1
(x
i
) and a
i
b
i
= u
2
(x
i
), (12)
if i I
H
and if x
i
2
then a
i
b
i
= u
2
(x
i
) and a
i
+b
i
= u
1
(x
i
).
Finally c
i
, i = 1,...,4, are defined such that
i
c
i
F
i
= u
s
.
In order to find the global interpolation error, we will proceed by computing local error estimates over the triangles
totally enriched by H , the triangle containing the crack tip, the triangles partially enriched by H and the non-enriched
triangles (Fig. 1(b)).
Lemma 4.2. Let T
H
h
be the set of triangles totally enriched by H (Fig. 1(b)) and σ
K
=
h
K
ρ
K
. There exists a constant
C>0 independent of h such that, for all K in T
H
h
and for all u satisfying (2), we have
u Π
h
u
1,K
1
Ch
K
σ
K
u
1
2+,K
and
u Π
h
u
1,K
2
Ch
K
σ
K
u
1
2+,K
. (13)
In fact, as we said before, the triangles totally enriched by H are cut in two parts. Using the extensions of u
1
and u
2
,
we associate three interpolation points to every part. Thus the interpolation operator we 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]). Lemma 4.2 is a direct consequence of this theorem.
4

Citations
More filters

Journal ArticleDOI
Abstract: The extended and generalized finite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture, (2) dislocations, (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with level set methods, as for example in the simulation growing cracks or moving phase interfaces. The state of the art for these problems is described along with the history of developments.

630 citations


Book
01 Jan 2008
Abstract: Dedication. Preface . Nomenclature . Chapter 1 Introduction. 1.1 ANALYSIS OF STRUCTURES. 1.2 ANALYSIS OF DISCONTINUITIES. 1.3 FRACTURE MECHANICS. 1.4 CRACK MODELLING. 1.4.1 Local and non-local models. 1.4.2 Smeared crack model. 1.4.3 Discrete inter-element crack. 1.4.4 Discrete cracked element. 1.4.5 Singular elements. 1.4.6 Enriched elements. 1.5 ALTERNATIVE TECHNIQUES. 1.6 A REVIEW OF XFEM APPLICATIONS. 1.6.1 General aspects of XFEM. 1.6.2 Localisation and fracture. 1.6.3 Composites. 1.6.4 Contact. 1.6.5 Dynamics. 1.6.6 Large deformation/shells. 1.6.7 Multiscale. 1.6.8 Multiphase/solidification. 1.7 SCOPE OF THE BOOK. Chapter 2 Fracture Mechanics, a Review. 2.1 INTRODUCTION. 2.2 BASICS OF ELASTICITY. 2.2.1 Stress-strain relations. 2.2.2 Airy stress function. 2.2.3 Complex stress functions. 2.3 BASICS OF LEFM. 2.3.1 Fracture mechanics. 2.3.2 Circular hole. 2.3.3 Elliptical hole. 2.3.4 Westergaard analysis of a sharp crack. 2.4 STRESS INTENSITY FACTOR, K . 2.4.1 Definition of the stress intensity factor. 2.4.2 Examples of stress intensity factors for LEFM. 2.4.3 Griffith theories of strength and energy. 2.4.4 Brittle material. 2.4.5 Quasi-brittle material. 2.4.6 Crack stability. 2.4.7 Fixed grip versus fixed load. 2.4.8 Mixed mode crack propagation. 2.5 SOLUTION PROCEDURES FOR K AND G . 2.5.1 Displacement extrapolation/correlation method. 2.5.2 Mode I energy release rate. 2.5.3 Mode I stiffness derivative/virtual crack model. 2.5.4 Two virtual crack extensions for mixed mode cases. 2.5.5 Single virtual crack extension based on displacement decomposition. 2.5.6 Quarter point singular elements. 2.6 ELASTOPLASTIC FRACTURE MECHANICS (EPFM). 2.6.1 Plastic zone. 2.6.2 Crack tip opening displacements (CTOD). 2.6.3 J integral. 2.6.4 Plastic crack tip fields. 2.6.5 Generalisation of J . 2.7 NUMERICAL METHODS BASED ON THE J INTEGRAL. 2.7.1 Nodal solution. 2.7.2 General finite element solution. 2.7.3 Equivalent domain integral (EDI) method. 2.7.4 Interaction integral method. Chapter 3 Extended Finite Element Method for Isotropic Problems. 3.1 INTRODUCTION. 3.2 A REVIEW OF XFEM DEVELOPMENT. 3.3 BASICS OF FEM. 3.3.1 Isoparametric finite elements, a short review. 3.3.2 Finite element solutions for fracture mechanics. 3.4 PARTITION OF UNITY. 3.5 ENRICHMENT. 3.5.1 Intrinsic enrichment. 3.5.2 Extrinsic enrichment. 3.5.3 Partition of unity finite element method. 3.5.4 Generalised finite element method. 3.5.5 Extended finite element method. 3.5.6 Hp-clouds enrichment. 3.5.7 Generalisation of the PU enrichment. 3.5.8 Transition from standard to enriched approximation. 3.6 ISOTROPIC XFEM. 3.6.1 Basic XFEM approximation. 3.6.2 Signed distance function. 3.6.3 Modelling strong discontinuous fields. 3.6.4 Modelling weak discontinuous fields. 3.6.5 Plastic enrichment. 3.6.6 Selection of nodes for discontinuity enrichment. 3.6.7 Modelling the crack. 3.7 DISCRETIZATION AND INTEGRATION. 3.7.1 Governing equation. 3.7.2 XFEM discretization. 3.7.3 Element partitioning and numerical integration. 3.7.4 Crack intersection. 3.8 TRACKING MOVING BOUNDARIES. 3.8.1 Level set method. 3.8.2 Fast marching method. 3.8.3 Ordered upwind method. 3.9 NUMERICAL SIMULATIONS. 3.9.1 A tensile plate with a central crack. 3.9.2 Double edge cracks. 3.9.3 Double internal collinear cracks. 3.9.4 A central crack in an infinite plate. 3.9.5 An edge crack in a finite plate. Chapter 4 XFEM for Orthotropic Problems. 4.1 INTRODUCTION. 4.2 ANISOTROPIC ELASTICITY. 4.2.1 Elasticity solution. 4.2.2 Anisotropic stress functions. 4.2.3 Orthotropic mixed mode problems. 4.2.4 Energy release rate and stress intensity factor for anisotropic. materials. 4.2.5 Anisotropic singular elements. 4.3 ANALYTICAL SOLUTIONS FOR NEAR CRACK TIP. 4.3.1 Near crack tip displacement field (class I). 4.3.2 Near crack tip displacement field (class II). 4.3.3 Unified near crack tip displacement field (both classes). 4.4 ANISOTROPIC XFEM. 4.4.1 Governing equation. 4.4.2 XFEM discretization. 4.4.3 SIF calculations. 4.5 NUMERICAL SIMULATIONS. 4.5.1 Plate with a crack parallel to material axis of orthotropy. 4.5.2 Edge crack with several orientations of the axes of orthotropy. 4.5.3 Single edge notched tensile specimen with crack inclination. 4.5.4 Central slanted crack. 4.5.5 An inclined centre crack in a disk subjected to point loads. 4.5.6 A crack between orthotropic and isotropic materials subjected to. tensile tractions. Chapter 5 XFEM for Cohesive Cracks. 5.1 INTRODUCTION. 5.2 COHESIVE CRACKS. 5.2.1 Cohesive crack models. 5.2.2 Numerical models for cohesive cracks. 5.2.3 Crack propagation criteria. 5.2.4 Snap-back behaviour. 5.2.5 Griffith criterion for cohesive crack. 5.2.6 Cohesive crack model. 5.3 XFEM FOR COHESIVE CRACKS. 5.3.1 Enrichment functions. 5.3.2 Governing equations. 5.3.3 XFEM discretization. 5.4 NUMERICAL SIMULATIONS. 5.4.1 Mixed mode bending beam. 5.4.2 Four point bending beam. 5.4.3 Double cantilever beam. Chapter 6 New Frontiers. 6.1 INTRODUCTION. 6.2 INTERFACE CRACKS. 6.2.1 Elasticity solution for isotropic bimaterial interface. 6.2.2 Stability of interface cracks. 6.2.3 XFEM approximation for interface cracks. 6.3 CONTACT. 6.3.1 Numerical models for a contact problem. 6.3.2 XFEM modelling of a contact problem. 6.4 DYNAMIC FRACTURE. 6.4.1 Dynamic crack propagation by XFEM. 6.4.2 Dynamic LEFM. 6.4.3 Dynamic orthotropic LEFM. 6.4.4 Basic formulation of dynamic XFEM. 6.4.5 XFEM discretization. 6.4.6 Time integration. 6.4.7 Time finite element method. 6.4.8 Time extended finite element method. 6.5 MULTISCALE XFEM. 6.5.1 Basic formulation. 6.5.2 The zoom technique. 6.5.3 Homogenisation based techniques. 6.5.4 XFEM discretization. 6.6 MULTIPHASE XFEM. 6.6.1 Basic formulation. 6.6.2 XFEM approximation. 6.6.3 Two-phase fluid flow. 6.6.4 XFEM approximation. Chapter 7 XFEM Flow. 7.1 INTRODUCTION. 7.2 AVAILABLE OPEN-SOURCE XFEM. 7.3. FINITE ELEMENT ANALYSIS. 7.3.1 Defining the model. 7.3.2 Creating the finite element mesh. 7.3.3 Linear elastic analysis. 7.3.4 Large deformation. 7.3.5 Nonlinear (elastoplastic) analysis. 7.3.6 Material constitutive matrix. 7.4 XFEM. 7.4.1 Front tracking. 7.4.2 Enrichment detection. 7.4.3 Enrichment functions. 7.4.4 Ramp (transition) functions. 7.4.5 Evaluation of the B matrix. 7.5 NUMERICAL INTEGRATION. 7.5.1 Sub-quads. 7.5.2 Sub-triangles. 7.6 SOLVER. 7.6.1 XFEM degrees of freedom. 7.6.2 Time integration. 7.6.3 Simultaneous equations solver. 7.6.4 Crack length control. 7.7 POST-PROCESSING. 7.7.1 Stress intensity factor. 7.7.2 Crack growth. 7.7.3 Other applications. 7.8 CONFIGURATION UPDATE. References . Index

305 citations


Journal ArticleDOI
Abstract: This article presents an overview and recent progress of the extended finite element method X-FEM in the analysis of crack growth modeling. It summarizes the important milestones achieved by the finite element community in the arena of computational fracture mechanics. The methodology of X-FEM, different from that of the classical finite element method, presents a very particular interest since it does not force the discontinuities to be in conformity with the borders. It makes possible the accurate solution of engineering problems in complex domains, which may be practically impossible to solve using the classical finite element method.

186 citations


Journal ArticleDOI
Abstract: This paper presents a review of the extended finite element method X-FEM for computational fracture mechanics. The work is dedicated to discussing the basic ideas and formulation for the newly developed X-FEM method. The advantage of the method is that the element topology need not conform to the surfaces of the cracks. Moreover, X-FEM coupled with LSM makes possible the accurate solution of engineering problems in complex domains, which may be practically impossible to solve using the standard finite element method.

112 citations


Journal ArticleDOI
Abstract: We consider a variant of the eXtended Finite Element Method (XFEM) in which a cutoff function is used to localize the singular enrichment surface. The goal of this variant is to obtain numerically an optimal convergence rate while reducing the computational cost of the classical XFEM with a fixed enrichment area. We give a mathematical result of quasi-optimal error estimate. One of the key points of this paper is to prove the optimality of the coupling between the singular and the discontinuous enrichments. Finally, we present some numerical computations validating the theoretical result. These computations are compared with those of the classical XFEM and a non-enriched method. Copyright © 2008 John Wiley & Sons, Ltd.

96 citations


References
More filters

Book
01 Jan 1989
Abstract: Keywords: methodes : numeriques ; fonction de forme Reference Record created on 2005-11-18, modified on 2016-08-08

17,323 citations


Book
01 Jan 1978
Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.

8,401 citations


Book
01 Apr 2002
Abstract: From the Publisher: This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained. About the Author Philippe G. Ciarlet is a Professor at the Laboratoire d'Analyse Numerique at the Universite Pierre et Marie Curie in Paris. He is also a member of the French Academy of Sciences. He is the author of more than a dozen books on a variety of topics and is a frequent invited lecturer at meetings and universities throughout the world. Professor Ciarlet has served approximately 75 visiting professorships since 1973, and he is a member of the editorial boards of more than 20 journals.

7,599 citations


Journal ArticleDOI
Abstract: SUMMARY An improvement of a new technique for modelling cracks in the nite element framework is presented. A standard displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Copyright ? 1999 John Wiley & Sons, Ltd.

5,156 citations


Book
01 Jan 1973

3,652 citations