Guilhem FERTÉ - lamsid.cnrs-bellevue.fr · XFEM, which is achieved in quasi-statics with the use of XFEM-suited multiplier spaces in a consistent formulation, blockwise diagonal

Guilhem FERTÉ

Mémoire présenté en vue de l’obtention du grade de Docteur de l’Ecole Centrale de Nantes

sous le label de L’Université Nantes Angers Le Mans École doctorale : Sciences Pour l’Ingénieur, Géosciences, Architecture (SPIGA) Discipline : Mécanique des solides, des matériaux, des structures et des surfaces Unité de recherche : Institut de recherche en génie civil et mécanique (GeM) Soutenue le 7 novembre 2014

Développement de l’approche X-FEM cohésive pour la modélisation de fissures et d’interfaces avec le

logiciel libre EDF R&D Code_Aster.

JURY Président : Jean-Jacques MARIGO, Professeur des universités, Ecole Polytechnique, Palaiseau. Rapporteurs : Patrick HILD, Professeur des universités, Université Paul Sabatier – Toulouse 3.

Anthony GRAVOUIL, Professeur des universités, INSA LYON. Examinateur : Barbara WOHLMUTH, Professeur, Technische Universität München Allemagne Directeur de Thèse : Nicolas MOËS, Professeur des universités, Ecole Centrale de Nantes Co-directeur de Thèse : Patrick MASSIN, HDR, EDF R & D, Clamart.

Numerical simulation of cracks and interfaces with cohesive zone models in the extended finite element method, with EDF R&D software Code_Aster.

Résumé Afin d’évaluer la nocivité de défauts détectés dans certaines de ses centrales, EDF est amené à utiliser des outils de simulation avancés. Les phénomènes visés sont la propagation de fissures 3D sur trajet inconnu, mais aussi les transitoires dynamiques durant les phases de propagation instable.

Cette thèse se propose d’associer la méthode des éléments finis étendus (XFEM) et les modèles de zones cohésives dans ce but. Celles-ci sont définies sur des surfaces potentielles de fissuration étendues. Ainsi, la loi cohésive séparera naturellement les domaines adhérents et ouverts. Une actualisation implicite du front de propagation peut donc être menée, ce qui fait l’originalité de cette méthode. Ceci demande une insertion robuste de lois d’interface non-régulières dans la XFEM. En statique, l’utilisation d’espaces de multiplicateurs de Lagrange dédiés dans une formulation consistante, des opérateurs diagonaux par blocs à l’interface et une écriture de la loi cohésive dans le formalisme du lagrangien augmenté permettent d’y parvenir. A partir de là, et avec un critère directionnel écrit sur les champs cohésifs uniquement, une procédure de propagation sur trajet inconnu est proposée et confrontée à des résultats expérimentaux de la littérature. En dynamique, une loi cohésive initialement adhérente est traitée implicitement au sein d’un schéma originellement explicite, ce qui permet une détermination analytique des contraintes cohésives si une discrétisation appropriée est adoptée. La formulation est validée sur un essai de type éprouvette DCB conique. En perspective, une application de ces méthodes sur des études industrielles est envisagée.

Nous étudions ensuite l’extension aux éléments quadratiques. Pour les fissures libres de contraintes, une découpe en cellules d’intégration quadratiques s’avère nécessaire pour avoir une formulation optimale. Pour les interfaces adhérentes, un nouvel espace de multiplicateurs est proposé, qui est à la fois stable et précis puisqu’il produit une convergence à l’ordre 2 lorqu’utilisé avec des cellules d’intégration quadratiques.

Mots clés Zone cohésive, XFEM, fracture fragile, contact, éléments quadratiques, dynamique rapide.

Abstract In order to assess the harmfulness of detected defects in some nuclear power plants, EDF Group is led to develop advanced simulation tools. Among the targeted mechanisms are 3D non-planar quasi-static crack propagation, but also dynamic transients during unstable phases.

In the present thesis, quasi-brittle crack growth is simulated based on the combination of the XFEM and cohesive zone models. These are inserted over large potential crack surfaces, so that the cohesive law will naturally separate adherent and debonding zones, resulting in an implicit update of the crack front, which makes the originality of the approach. This requires a robust insertion of non-smooth interface laws in the XFEM, which is achieved in quasi-statics with the use of XFEM-suited multiplier spaces in a consistent formulation, blockwise diagonal interface operators and an augmented lagrangian formalism to write the cohesive law. Based on this concept and a novel directional criterion appealing to cohesive integrals, a propagation procedure over non-planar crack paths is proposed and compared with literature benchmarks. As for dynamics, an initially perfectly adherent cohesive law is implicitely treated within an explicit time-stepping scheme, resulting in an analytical determination of interface tractions if appropriate discrete spaces are used. Implementation is validated on a tapered DCB test.

Extension to quadratic elements is then investigated. For stress-free cracks, it was found that a subdivision into quadratic subcells is needed for optimality. Theory expects enriched quadrature to be necessary for distorted subcells, but this could not be observed in practice. For adherent interfaces, a novel discrete multiplier space was proposed which has both numerical stability and produces quadratic convergence if used along with quadratic subcells.

Key Words Cohesive zone model (CZM), XFEM, brittle fracture, contact, quadratic elements, explicit dynamics.

L4u L’Université Nantes Angers Le Mans

Développement de l’approche X-FEM cohésive pour la modélisation de fissures et d’interfaces avec le logiciel libre EDF R&D Code_Aster.

Guilhem FERTÉ

Acknowledgements

Je tiens à remercier mon directeur de thèse, Nicolas Moës, pour son aide et ses conseils.

Je lui suis très reconnaissant de sa disponibilité. Malgré un emploi du temps chargé, il est

resté accessible et a toujours su trouver du temps pour me conseiller et me transmettre

un peu de son expérience. Cela a été une chance et un honneur de travailler avec lui. Il

m'a beaucoup appris scientiquement et professionnellement, par son esprit de synthèse

et son souci de clarté. Je remercie également mon co-directeur Patrick Massin, qui m'a

donné goût à la recherche scientique. Je le remercie de la liberté et la conance qu'il

m'a accordée dans la conduite de mon travail. Sa disponibilité au quotidien pour discuter

des problèmes rencontrés m'a permis de progresser à un bon rythme. Merci également à

Gilles Debryune, Jérôme Laverne et Samuel Géniaut pour l'éclairage qu'ils m'ont apporté

sur la compréhension des problématiques industrielles auxquelles ma thèse était liée.

Je tiens également à remercier MM. Anthony Gravouil et Patrick Hild d'avoir accepté

la tâche de relire mon manuscrit. Je leur suis reconnaissant du temps important qu'ils

ont bien voulu me consacrer en corrigeant mon rapport, et en assistant à la soutenance.

Je remercie également Madame Barbara Wohlmuth et M. Jean-Jacques Marigo qui a

accepté de présider le jury.

Je tiens à témoigner ma gratitude à mes collègues et amis du LaMSID, pour tous les bon

moments partagés, au laboratoire et à l'extérieur, et leur soutien pendant les galères.

Merci à André (dont on ne saura jamais la date de naissance), à Sam (fraîchement

Papa, comme le temps passe vite), à Jérôme et nos longs débats footballistiques, à Dzifa,

Fabien, Emricka, Najib, Alexandre Bérard, Alexandre Martin, Marie (Aveyron forever!),

Marcel, Nazir, Christelle, Dina, Bertand, Abdallah, Hippolyte, Maximilien, Alex et à

toutes les personnes que j'y ai croisées. Nous aurons l'occasion de nous revoir autour

d'un verre, ou pour un foot à Orsay. Merci également à tous mes collègues et amis

du groupe T64, qui m'on accueilli pour ma dernière année de thèse, en particulier mes

collègues du bureau des thèsards Eric et Emricka. Je vous suis reconnaissant du soutien

que vous m'avez apporté pendant la phase de rédaction. Au plaisir de vous revoir (autour

d'un goûter?...)

Cette thèse n'aurait pu se faire sans le soutien actif de la tribu Ferté, ma famille. Merci

à mes parents et mes frères et soeurs Clarisse, Marie-Liesse, Alexis et Marie-Aimée pour

leur aide, ainsi qu'à Camille. Je remercie tous les proches qui m'ont encouragé dans cette

entreprise, en particulier Frédéric, Leila et Ahmedou.

Enn, je dédie cette thèse à la mémoire d'Alexis. Tu es parti trop vite, nous sommes

avec toi par la pensée, tu restes notre ami pour toujours et tu nous manques à tous.

i

Contents

Acknowledgements i

Contents ii

List of Figures vi

List of Tables x

1 Introduction and industrial context 1

1.1 Cracking of graphite bricks in advanced gas-cooled reactors . . . . . . . . 2

1.2 Microcracks under inox coating in pressurized water reactors . . . . . . . . 4

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 3D crack propagation with cohesive elements in the extended niteelement method 8

2.1 Literature survey: linear elastic fracture mechanics . . . . . . . . . . . . . 10

2.1.1 Linear elastic fracture mechanics (LEFM): basics . . . . . . . . . . 10

2.1.1.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1.2 The energy release rate as derivative with respect to thedomain: G-integral . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1.3 Relation to J-integral by Rice [1] . . . . . . . . . . . . . . 13

2.1.1.4 Asymptotic elds . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1.5 Stress intensity factors with interaction integrals and Ir-win formula . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1.6 Determining the crack increment . . . . . . . . . . . . . . 16

2.1.1.7 Crack bifurcation angle . . . . . . . . . . . . . . . . . . . 17

2.1.1.8 Crack advance . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2 Level-set description of a propagating crack . . . . . . . . . . . . . 18

2.1.2.1 Representation of the crack by means of level-set functions 18

2.1.2.2 Level-set update to model the propagation of a crack . . 19

2.1.3 The X-FEM to model singular cracks . . . . . . . . . . . . . . . . . 21

2.1.4 Limitations of LEFM . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Literature survey: cohesive zone models . . . . . . . . . . . . . . . . . . . 22

2.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Equilibrium of cracked bodies with cohesive forces . . . . . . . . . 23

2.2.3 Invariant integrals [2, 3] . . . . . . . . . . . . . . . . . . . . . . . . 24

ii

Contents iii

2.2.4 Asymptotic elds and equivalent stress intensity factors . . . . . . 25

2.2.5 Determining a crack increment : crack advance and angle . . . . . 25

2.2.5.1 Bifurcation angle: criteria from equivalent stress intensityfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.5.2 Bifurcation angle : the meso-local approach . . . . . . . . 27

2.2.5.3 Crack advance . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.6 Determining the crack path as a whole: crack tracking algorithms . 28

2.2.6.1 Local crack tracking algorithm . . . . . . . . . . . . . . . 29

2.2.6.2 Non-local crack tracking algorithm . . . . . . . . . . . . . 31

2.2.6.3 Global crack tracking algorithm . . . . . . . . . . . . . . 33

2.2.7 2D implicit crack advance . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.8 Enrichment strategies for cohesive zone models . . . . . . . . . . . 34

2.2.8.1 Stopping enrichment on crack edges . . . . . . . . . . . . 34

2.2.8.2 With dedicated asymptotic functions around the tip . . . 34

2.2.8.3 With a corrected Heaviside enrichment . . . . . . . . . . 35

2.3 XFEM cohesive zone models with large adherent zone . . . . . . . . . . . 36

2.3.1 Discrete displacement . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.2 Why a penalized law is not ecient to describe a large adherentzone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.3 The use of Lagrange multipliers in the X-FEM . . . . . . . . . . . 40

2.3.4 A mixed law, in the augmented Lagrangian formalism . . . . . . . 41

2.3.5 Stable mortar formulation for inserting an interfacial law inXFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.6 Blockwise diagonal discrete operators at the interface . . . . . . . . 44

2.3.7 Numerical validation with the inclusion debonding test . . . . . . . 45

2.4 3D cohesive crack propagation with implicit crack advance . . . . . . . . . 47

2.4.1 Overview of the procedure . . . . . . . . . . . . . . . . . . . . . . . 47

2.4.2 Update of the crack front . . . . . . . . . . . . . . . . . . . . . . . 47

2.4.3 Bifurcation angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4.4 Update of the potential crack surface . . . . . . . . . . . . . . . . . 51

2.4.5 Extension of the multiplier space and initial internal variables . . . 52

2.5 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.5.1 An extruded test : the L-shaped panel . . . . . . . . . . . . . . . . 55

2.5.2 Three-point bending test with an initial skew crack . . . . . . . . . 57

2.5.3 Brokenshire's torsion test . . . . . . . . . . . . . . . . . . . . . . . 59

2.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.6.1 Toward the use of quadratic elements with the two-eld formula-tion by Lorentz [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Dynamic cohesive crack growth in the extended nite element method 67

3.1 Literature survey: traction-free cracks . . . . . . . . . . . . . . . . . . . . 68

3.1.1 Dynamic equilibrium of cracked bodies . . . . . . . . . . . . . . . . 68

3.1.2 Dynamic energy release rate and invariant integrals [5, 6] . . . . . 69

3.1.2.1 Path-independant integrals under the steady state as-sumption . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.1.2.2 A general path-independent integral . . . . . . . . . . . . 71

3.1.3 Asymptotic elds for a dynamic analysis . . . . . . . . . . . . . . . 71

Contents iv

3.1.4 Irwin's relation and interaction integrals . . . . . . . . . . . . . . . 72

3.1.5 Discretization with the X-FEM . . . . . . . . . . . . . . . . . . . . 73

3.1.6 Explicit time-stepping with central nite dierences . . . . . . . . 73

3.1.7 X-FEM mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.1.8 Crack increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2 Literature survey: cohesive zone models . . . . . . . . . . . . . . . . . . . 76

3.2.1 Cohesive zone models to model rate eects . . . . . . . . . . . . . 76

3.2.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Semi-explicit initially rigid cohesive law in the X-FEM . . . . . . . . . . . 77

3.3.1 The initially ridig cohesive law . . . . . . . . . . . . . . . . . . . . 77

3.3.2 Semi-discretization in space . . . . . . . . . . . . . . . . . . . . . . 79

3.3.2.1 Toward dening a consistent jump . . . . . . . . . . . . . 79

3.3.2.2 First lumping step: turning H into a diagonal H . . . . . 80

3.3.2.3 Second lumping step: turning B into a diagonal B . . . . 80

3.3.2.4 Adopted lumped mass matrix . . . . . . . . . . . . . . . . 80

3.3.2.5 Semi-discrete problem in space . . . . . . . . . . . . . . . 81

3.3.3 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.3.4 Detail of the analytical determination of cohesive stresses . . . . . 82

3.4 Numerical test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.5 Determining the critical load with a path-following method by Lorentz [7] 86

3.6 Outlook: toward non-planar dynamic crack propagation . . . . . . . . . . 89

4 Convergence analysis of linear or quadratic X-FEM for curved freeboundaries 92

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Formulation of the continuous problem . . . . . . . . . . . . . . . . . . . . 95

4.3 The discrete problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3.1 The approximation of the geometry . . . . . . . . . . . . . . . . . . 96

4.3.2 The interpolation of the eld of unknowns . . . . . . . . . . . . . . 99

4.4 The accuracy of the geometry description . . . . . . . . . . . . . . . . . . 100

4.4.1 Interpolating the level-set function . . . . . . . . . . . . . . . . . . 100

4.4.2 Constructing subcells . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5 A priori error estimates depending on geometry description and quadrature105

4.5.1 Applying the rst Strang lemma . . . . . . . . . . . . . . . . . . . 105

4.5.2 About the quadrature rules in the subcells . . . . . . . . . . . . . . 112

4.5.3 Geometry inuence on the error . . . . . . . . . . . . . . . . . . . . 116

4.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.6.1 Plate with a hole under tension . . . . . . . . . . . . . . . . . . . . 119

4.6.2 Plate with an elliptic hole . . . . . . . . . . . . . . . . . . . . . . . 121

4.6.3 A practical study of problematic cases . . . . . . . . . . . . . . . . 122

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5 Interface problems with quadratic X-FEM: design of a stable multiplierspace and error analysis 125

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2 Formulation of the continuous problem . . . . . . . . . . . . . . . . . . . . 129

Contents v

5.3 Discretization of primal and dual spaces . . . . . . . . . . . . . . . . . . . 131

5.3.1 Discretization of the eld of unknowns . . . . . . . . . . . . . . . . 131

5.3.2 Discretization of the multipliers . . . . . . . . . . . . . . . . . . . . 131

5.3.3 Design of multiplier space P1* . . . . . . . . . . . . . . . . . . . . 132

5.3.4 Non redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.3.5 Proof of the discrete inf-sup condition . . . . . . . . . . . . . . . . 135

5.3.6 Interpolation properties of the discrete multiplier spaces . . . . . . 139

5.4 Changes due to a curved geometry . . . . . . . . . . . . . . . . . . . . . . 142

5.4.1 Changes to the operators . . . . . . . . . . . . . . . . . . . . . . . 142

5.4.2 Transnite elements . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.4.3 Conforming discretization spaces over transnite elements . . . . . 144

5.5 Convergence analysis with a weak discontinuity . . . . . . . . . . . . . . . 145

5.5.1 Convergence analysis on transnite elements . . . . . . . . . . . . . 145

5.5.2 Convergence analysis on the actual elements . . . . . . . . . . . . . 148

5.5.3 About the optimal convergence orders . . . . . . . . . . . . . . . . 153

5.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.6.1 Cracked block under cubic pressure. . . . . . . . . . . . . . . . . . 155

5.6.2 Circular inclusion under compressive loads . . . . . . . . . . . . . . 156

5.6.3 Bimaterial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.6.4 A practical study of the problematic case of slanted triangles . . . 159

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6 Conclusion and outlook 162

A Constructing level-sets from implicit or parametric representation 164

B Transnite maps and their errors 167

C Elements of proof for lengthways intersected elements 171

D Appendix Proof of theorem 5.9 176

E Cohesive crack propagation: some implementation details 180

E.1 Implemention points about the geometrical update of level-sets . . . . . . 180

E.1.1 The inclined crack surface with respect to the boundary of thestructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

E.1.2 A curved crack front or a curved structure boundary . . . . . . . . 181

E.2 Implemention details about the computation of equivalent stress intensityfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

E.2.1 Discrete energy release rate G and discrete stress intensity factors(Keq

I ,KeqII ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

E.2.2 Direction of θ near the extremities of the crack front . . . . . . . . 183

E.2.3 Choice of the basis to split modes . . . . . . . . . . . . . . . . . . . 184

Bibliography 187

List of Figures

1.1 Schematic diagram of nuclear power generation. Source : InternationalAtomic Energy Agency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Schematic diagram of an advanced gas-cooled reactor (AGR). . . . . . . . 2

1.3 Graphite keying structure. Source IAEA. . . . . . . . . . . . . . . . . . . 3

1.4 Upper view of graphite keying structure. Source: NKS (nordic nuclearsafety research) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Schematic diagram of a pressurized water reactor (PWR). Source Wikimedia. 5

1.6 Microcracks appearing under inox coating. Source IRSN. . . . . . . . . . . 6

2.1 Linear Elastic Fracture Mechanics: problem denition. . . . . . . . . . . . 11

2.2 Virtual crack extension and covariant basis along the crack front . . . . . 12

2.3 Denition of the J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Modes of fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Bifurcation angle and crack advance . . . . . . . . . . . . . . . . . . . . . 17

2.6 The inuence of mode III on the bifurcation angle. . . . . . . . . . . . . . 18

2.7 3D description of the crack and the covariant basis attached to the crackfront. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 2D level-set description and polar basis around the front. . . . . . . . . . . 19

2.9 XFEM enrichment strategy for stress free cracks. . . . . . . . . . . . . . . 21

2.10 Cohesive zone model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.11 Notations of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.12 Dening invariant integrals in the presence of cohesive forces. . . . . . . . 24

2.13 Far elds reported by Planas and Elices [3]. . . . . . . . . . . . . . . . . . 26

2.14 Principle of explicit crack increment and crack tracking algorithms. . . . . 26

2.15 Computation domain of the smeared stress for the meso-local approach . . 27

2.16 Common architecture for local or non-local crack tracking. . . . . . . . . . 30

2.17 Constructing the crack surface with a local tracking algorithm. . . . . . . 31

2.18 Local crack tracking algorithm: extension with an element. . . . . . . . . 31

2.19 Non-local crack tracking algorithm. . . . . . . . . . . . . . . . . . . . . . . 32

2.20 2D implicit crack advance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.21 Enrichment stopping at a crack edge. . . . . . . . . . . . . . . . . . . . . . 35

2.22 Enrichment with asymptotic function. . . . . . . . . . . . . . . . . . . . . 35

2.23 Alternative to Heaviside function in a triangular element 123 completelycut by the crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.24 Discontinuous part of the displacement eld for the triangular element 123containing the tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.25 Potential crack surface not matching the mesh edges, and subsequent en-riched nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vi

List of Figures vii

2.26 Penalized linear-softening mode-coupling cohesive law. . . . . . . . . . . . 38

2.27 Inclusion debonding test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.28 Normal opening with the inclusion debonding test. . . . . . . . . . . . . . 39

2.29 Normal tractions with the inclusion debonding test. . . . . . . . . . . . . . 39

2.30 Mesh not matching the crack surface and reduced multiplier space. . . . . 41

2.31 Mode-coupling mixed cohesive law. . . . . . . . . . . . . . . . . . . . . . . 42

2.32 Total number of Newton iterations to solve the problem in 3 time steps. . 46

2.33 Cohesive stress along the interface for intermediate formulations. . . . . . 47

2.34 Overview of the procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.35 Computation of a rough crack front . . . . . . . . . . . . . . . . . . . . . . 49

2.36 Reconstruction of a smooth tangential level-set . . . . . . . . . . . . . . . 50

2.37 Dening invariant integrals in the presence of cohesive forces. . . . . . . . 51

2.38 Geometric update algorithm, as introduced by Colombo [8]. . . . . . . . . 52

2.39 Extension of the restricted multiplier space and update of the internalvariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.40 Bilinear-softening law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.41 L-shaped panel: geometry and loading. . . . . . . . . . . . . . . . . . . . . 55

2.42 Comparison of computed and experimental crack paths for the L-shapedpanel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.43 Computed and experimental load-deection curves. . . . . . . . . . . . . . 56

2.44 L-shaped panel: deformed shape and stress modulus. . . . . . . . . . . . . 57

2.45 L-shaped panel: prole of the normal cohesive traction on the crack surface. 57

2.46 Three-point-bending test with an initial skew crack. . . . . . . . . . . . . 58

2.47 Perspective view of the crack surface and eld of cohesive tractions. . . . . 58

2.48 Top view of the crack surface. . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.49 Experimental crack path from [9, 10]. . . . . . . . . . . . . . . . . . . . . . 59

2.50 3-point bending test with an initial skew crack: deformed shape and mapof stress modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.51 crack path in the cut plane (X=-5). . . . . . . . . . . . . . . . . . . . . . . 60

2.52 Geometry and loading for Brokenshire's torsion test. . . . . . . . . . . . . 61

2.53 Load deection curves for Brokenshire's torsion test. . . . . . . . . . . . . 61

2.54 Shape of the computed crack path and map of normal cohesive tractionsfor Brokenshire's test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.55 Deformed shape and map of stress modulus for Brokenshire's test. . . . . 62

2.56 Experimental crack path from [11]. . . . . . . . . . . . . . . . . . . . . . . 62

2.57 Local constitutive law giving w, in the case of pure mode I. . . . . . . . . 64

2.58 Inclusion test with quadratic elements: deformed shape and displacementalong x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.59 Inclusion test with quadratic elements: deformed shape and displacementalong y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.60 Inclusion test with linear elements: deformed shape and displacementalong x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.1 : Notations for dening a dynamic energy release rate. . . . . . . . . . . . 70

3.2 Enrichment basis of Hansbo and Hansbo [12]. . . . . . . . . . . . . . . . . 75

3.3 Cut triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.4 Set-valued linear-softening mixed-mode cohesive law. . . . . . . . . . . . . 78

List of Figures viii

3.5 Reduced space for the denition of discrete tractions. . . . . . . . . . . . . 79

3.6 Analytical detemination of cohesive tractions. . . . . . . . . . . . . . . . . 82

3.7 Numerical test: geometry and loading. . . . . . . . . . . . . . . . . . . . . 84

3.8 Stress component σyy and deformed shape at dierent times. . . . . . . . 85

3.9 Top displacement as a function of time, for dierent time steps. . . . . . . 86

3.10 Zoom of the intersected mesh and stress component σyy at t=4µs. . . . . 86

3.11 Load-deection curve obtained with a path-following method and crackarrest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.12 The path-following method by Lorentz [7]. . . . . . . . . . . . . . . . . . . 89

3.13 Deformed shaped and stress component σyy. . . . . . . . . . . . . . . . . . 90

3.14 Deformed shaped and stress modulus. . . . . . . . . . . . . . . . . . . . . 90

3.15 Deformed shaped and stress component σyz. . . . . . . . . . . . . . . . . . 91

4.1 Denition of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 Denition principle and properties of the level-set function. . . . . . . . . 97

4.3 Subdivision of cut elements, in the case g = 2. . . . . . . . . . . . . . . . . 98

4.4 Approximation process of the geometry. . . . . . . . . . . . . . . . . . . . 99

4.5 Interface, mesh not matching the interface and enriched nodes, in the casep=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.6 Classical and transnite subcells, in the case where g=2. . . . . . . . . . . 101

4.7 Decomposition of the errors. . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.8 Intersected triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.9 Various congurations of intersected triangles. . . . . . . . . . . . . . . . . 104

4.10 Typical convergence rates (e.g. for an exponential shape). . . . . . . . . . 105

4.11 A superconvergent case : the circle. . . . . . . . . . . . . . . . . . . . . . . 105

4.12 Regular, slender and crescent-shaped subcells. . . . . . . . . . . . . . . . . 109

4.13 Situation responsible for local high errors. . . . . . . . . . . . . . . . . . . 112

4.14 Intersected triangle and measures of interest. . . . . . . . . . . . . . . . . 117

4.15 Numerical test: model and loading. . . . . . . . . . . . . . . . . . . . . . . 120

4.16 Numerical results for a problem with a strong discontinuity. . . . . . . . . 121

4.17 Plate with an elliptic hole: geometry and loading. . . . . . . . . . . . . . . 121

4.18 Plate with an elliptic hole: results. . . . . . . . . . . . . . . . . . . . . . . 122

4.19 Quadrature error: crescent-shaped subcells. . . . . . . . . . . . . . . . . . 123

4.20 Quadrature error: slender subcells. . . . . . . . . . . . . . . . . . . . . . . 123

4.21 The eect of lengthways intersected elements: test 1. . . . . . . . . . . . . 124

4.22 The eect of lengthways intersected elements : test 2. . . . . . . . . . . . . 124

5.1 Some topological denitions around the interface. . . . . . . . . . . . . . . 131

5.2 Restriction algorithm P1−, as dened by Béchet and al [13]. . . . . . . . 132

5.3 Closely intersected edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 Intersected triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5 Building algorithm P1* with closely intersected edges. . . . . . . . . . . . 134

5.6 Minimal restriction algorithm without closely intersected edges. . . . . . . 134

5.7 Subdivision of the interface. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.8 Intersected element K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.9 Support of a node n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.10 Nodes classication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

List of Figures ix

5.11 Node of type 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.12 Cut element K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.13 Illustration of the interpolation of a multiplier by algorithm (P1*). . . . . 142

5.14 Transnite elements and transnite map. . . . . . . . . . . . . . . . . . . . 144

5.15 Exact and approximate domains. . . . . . . . . . . . . . . . . . . . . . . . 152

5.16 Block submitted to a cubic pressure. . . . . . . . . . . . . . . . . . . . . . 156

5.17 Convergence curves on energy and multipliers for the block under pressure.157

5.18 Circular inclusion under compressive loads: multiplier convergence. . . . . 157

5.19 Circular inclusion under compressive loads: energy error. . . . . . . . . . . 158

5.20 Circular inclusion under compressive loads. . . . . . . . . . . . . . . . . . 158

5.21 Bimaterial ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.22 Convergence curves for a bimaterial ring. . . . . . . . . . . . . . . . . . . . 160

5.23 Cracked block: local pressure errors. . . . . . . . . . . . . . . . . . . . . . 160

B.1 Classical and transnite subcells, and local system of coordinates. . . . . . 167

C.1 Lengthways intersected triangle. . . . . . . . . . . . . . . . . . . . . . . . . 172

C.2 Lengthways intersected triangles. . . . . . . . . . . . . . . . . . . . . . . . 174

D.1 Transnite elements and transnite map. . . . . . . . . . . . . . . . . . . . 176

D.2 Transnite map: rst step. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

D.3 Transnite map: second step. . . . . . . . . . . . . . . . . . . . . . . . . . 178

E.1 Assumption about the projection point onto the front for the geometricalupdate algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

E.2 Inclined crack surface with respect to the boundary of the structure . . . . 181

E.3 The curved crack front case . . . . . . . . . . . . . . . . . . . . . . . . . . 181

E.4 The curved crack boundary case . . . . . . . . . . . . . . . . . . . . . . . 182

E.5 Shape functions for discrete G and theta . . . . . . . . . . . . . . . . . . . 183

E.6 Various ways of dening eld theta . . . . . . . . . . . . . . . . . . . . . . 184

E.7 Possible basis to split nodes . . . . . . . . . . . . . . . . . . . . . . . . . . 184

E.8 The arc-shaped crack test, computed with a small cohesive zone . . . . . . 185

List of Tables

2.1 Tested formulations and Newton iterations needed to solve the problemin 3 time steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2 Material data for the plain concrete of the L-shaped panel. . . . . . . . . . 55

2.3 Material data for the PMMA sample . . . . . . . . . . . . . . . . . . . . . 58

2.4 Material data for the plain concrete of Brokenshire's test. . . . . . . . . . 60

3.1 Material data for the TDCB test. . . . . . . . . . . . . . . . . . . . . . . . 84

4.1 Expected rates of convergence as a function of the displacement, geometryand quadrature according to (4.59) . . . . . . . . . . . . . . . . . . . . . . 119

5.1 Expected convergence orders for the displacement in the energy norm. . . 155

5.2 Expected convergence orders for the multiplier. . . . . . . . . . . . . . . . 155

E.1 Material data for arc-shaped crack test. . . . . . . . . . . . . . . . . . . . 186

E.2 Relative errors for the arc-shaped crack test. . . . . . . . . . . . . . . . . . 186

x

À la mémoire d'Alexis

xi

Chapter 1

Introduction and industrial context

Résumé en français

Dans une centrale nucléaire, une réaction de ssion en chaîne est amorcée et contrôlée.La chaleur produite est utilisée par un uide caloporteur pour générer de la vapeur,qui en passant à travers des turbines, entraîne un alternateur (voir g.1.1). La réactionne peut être auto-entretenue que si les neutrons émis par le combustible sont ralentispar un modérateur, de sorte qu'ils puissent interagir avec d'autres noyaux. Les réacteursavancés au gaz (AGR) en service en Grande-Bretagne (voir g.1.2) ont un coeur graphitequi remplit la fonction de modérateur tout en assurant le maintien en position des autreséléments nécessaires à la réaction (crayons combustibles, conduits de passage du gaz,barres de contrôle). Le coeur graphite est composé de briques assemblées par rainures etclavettes (voir g.1.3). En opération, le graphite est soumis à un ux neutronique et unchamp de températures variables en espace, qui, combinés à l'oxydation par radiolyse duCO2, rendent le matériau hétérogène et produisent des contraintes résiduelles dans lesbriques. Tard dans la vie du matériau, des ssures peuvent s'initier à partir des fonds derainure. An d'avoir un appui au contrôle sur site, EDF Energy développe des outils demodélisation pour prédire dans quelles conditions précises ces ssures sont susceptiblesd'apparaître. Dans ce but, il est nécessaire de disposer d'une procédure de modélisationde propagation de ssures sur trajet inconnu, en trois dimensions, et pouvant prendreen compte les phases de propagation instables par une modélisation dynamique.

In nuclear power plants, a nuclear ssion chain reaction is initiated and controlled. Thelarge amount of heat that it produces is moved away from the reactor by a coolant andused to turn water into steam, which ows to turbines, which, in turn, drive an electricalgenerator. The generated electrical current is sent onto the grid (see g.1.1). For thechain reaction to be sustained, fast ssion neutrons released from the fuel need to beslowed down in order to interact with other ssion neutrons, a process that is known asmoderation. The reaction is stimulated or held down as neutron-absorbant control rodsare raised or dropped (respectively), or in some cases, by controlling the concentration ofa neutron absorbant in the coolant (see g.1.1). Additional secondary shutdown systemsexist to stop the reaction in the event of an emergency.

1

Chapter 1. Introduction and industrial context 2

Figure 1.1: Schematic diagram of nuclear power generation. Source : Interna-tional Atomic Energy Agency

1.1 Cracking of graphite bricks in advanced gas-cooled re-actors

Advanced gas-cooled reactors (AGR) were designed and built in the UK in the mid-60sand commissioned in the early 80's. They use carbone dioxide as a coolant and graphitebricks as a moderator (see g.1.2). Secondary emergency shutdown systems comprisesthe addition of methane to the cooling gas and the injection of boron beads into thereactor. EDF Energy operates 7 such plants in the UK, each made up of two reactors.

Figure 1.2: Schematic diagram of an advanced gas-cooled reactor (AGR).


The reactor core is made of interconnected graphite bricks of two types (see gs.1.3-1.4):

circular bricks which let fuel assemblies through ;

square interstitial bricks that contain either control rods, or secondary shutdownsystems, or cooling gas channels.

Circular bricks are held together by loose keys, while the key connecting interstitial andcircular bricks is an integral part of the interstitial brick (see gs.1.3-1.4).

Figure 1.3: Graphite keying structure. Source IAEA.

Figure 1.4: Upper view of graphite keying structure. Source: NKS (nordic nu-clear safety research)


While in operation, graphite is subjected to thermal loads and irradiation. The eect ofradiation is twofold:

as neutron ux varies over space, material properties are heterogeneously aected,which, combined with the heterogeneous temperature eld and anisotropic proper-ties of extruded graphite bricks, leads to heterogeneous shrinkage of the material.The phenomenon is known as dimensional change and produces internal stressesin the brick;

radiolytic oxidation occurs when CO2 is radiolysed, allowing an atom of oxygen toreact with carbon from graphite to release CO. This mainly happens in graphitepores, and leads to graphite loss of weight over time.

Both phenomena imply that the internal stresses at stake change with the ageing ofgraphite. Late in the core operating life, cracks may develop from keyway roots. Crackgrowth monitoring is undertaken, especially during statutory outages every three years.EDF Energy launched a vast research program about the modelling of crack growth, inorder to:

provide an enhanced description of the conditions leading to keyway root cracking;

provide simulation-based support to crack monitoring and decision-making assis-tance about the investments to be undertaken and possible lifetime extensions.

In addition to establishing the load conditions leading to cracking, it is of critical interestto determine the morphology of generated cracks. Indeed, a major concern is to preventgraphite blocks from falling apart and hindering the good operation of the core. Thedetermination of crack path is also important to ensure that the structural function ofa brick is not aected by an existing crack. A modelisation technique meeting the EDFEnergy's requirements should include the description of:

3D quasi-static stable crack propagation ;

unstable crack propagation in a dynamic framework.

With this second requirement, we aim at evaluating the potential for secondary crackpropagation initiating from other keyway roots, partly due to inertial eects.

1.2 Microcracks under inox coating in pressurized water re-actors

In pressurized water reactors (PWR) operated by EDF in France, pressurized light wateris used both as a coolant and, by owing through fuel assemblies, as a moderator (seeg.1.5). In addition to control rods, the injection of boron to the water allows to holdthe reaction down. Fuel assemblies rest on a lower core plate and are maintained inposition by intermediate grids and an upper support plate. The reactor core is housedin a pressurized steel envelope called reactor vessel. Its inner wall is lined with inox


coating, which is a thermal shield and protects it against corrosion (see g.1.6). On afew identied vessels manufactured before 1979, some microcracks have been detectedunder the coating by extensive ultrasonic inspection during periodical shutdowns. Theyare due to unappropriate thermal conditioning at the time when coating was welded.These defects have been proven not to further propagate, according to the specicationsof the safety case, even when the most severe accidental cases were assumed.

Figure 1.5: Schematic diagram of a pressurized water reactor (PWR). SourceWikimedia.

The most severe accidental case that the reactor vessel should be able to withstand isthe loss-of-coolant accident (LOCA). If the coolant ow is reduced, or lost altogether,emergency shutdown systems are operated. However, because of natural radioactivedecay, residual heat still has to be removed from the core a while after the ssion reactionhas stopped. If not, a temperature increase in the reactor can result in very severe damageto the core, such as the melting of fuel. Residual heat removal is ensured by emergencycore cooling systems (ECCM). In PWR, a backup system is activated that injects coldwater into the reactor. Due to the sudden depressurization and massive injection of coldwater, the nuclear vessel is subjected to extreme mechanical and thermal loads.

Originally made of ductile steel, irradiation causes the steel to get brittle near the innerwall, while steel stays ductile near the outer wall. Assuming large operating life exten-sions, further studies are undertaken considering propagating defects, in order to predictthe direction and arrest (because of tenacity gradient, see g.1.6) of such cracks. Thesestudies consider plane predenite crack path until now, and there is a need for methodsallowing for curved non-predenite crack paths. This problem also requires methods ac-counting for quasi-static crack propagation and sudden switch to unstable propagationin a dynamic framework.


Figure 1.6: Microcracks appearing under inox coating. Source IRSN.

1.3 Outline of the thesis

The present thesis intends to model both industrial cases by a combined use of:

the extended nite element method (XFEM) for its ability to describe complexcrack paths in allowing discontinuity surfaces within elements;

and cohesive zone models (CZM), which model crack growth by vanishing interfa-cial forces as a crack opens. CZM have advantages of their own, to be detailed inchapter 2.

All newly proposed methods in this thesis have been implemented in the EDF R&Dsoftware Code_Aster. Some literature overview is provided in section 2.1 about crackpropagation strategies relying on classical linear elastic fracture mechanics (LEFM). Sec-tion 2.2 oers a literature survey about existing method coupling the XFEM and cohesivezones. In section 2.3 a robust way of inserting an initially perfect adherent law in theXFEM is presented as a prerequisite to the description of the propagation procedure insection 2.4. The originality of the procedure lies in the a posteriori computation of thecrack advance from the cohesive state, while in most existing methods thus far it waspostulated from the stress state ahead of the front. Numerical tests and validation areproposed in section 2.5.

In chapter 3, a short review is presented of some existing numerical methods to simulatedynamic crack propagation, either in the frame of the LEFM (section 3.1) or cohesivezones (section 3.2). A robust way of inserting an initially perfectly adherent cohesivelaw in the XFEM in explicit dynamics is presented in section 3.3. A numerical test iscarried out in section 3.4 to validate the implementation, and some details are providedin section 3.5 to determine the critical load triggering unstable crack propagation.


For complex crack shapes and large structures, it might be interesting to use quadraticelements for accuracy. Besides, as the method is intended to be implemented in anindustrial code (Code_Aster), extension to quadratic elements is worth investigating inorder to keep as much compatibility as possible with other features of the code. Forinstance, one might want to use the XFEM with quadratic elements for incompressibleplasticity in the medium term. In that respect, chapter 4 investigates the optimalityof the XFEM when computing stress free cracks with quadratic elements, depending onthe quadrature scheme and description of the geometry. Aiming at a generalization tocohesive zone models, adherent interfaces are considered in chapter 5, and optimality isdiscussed as a function of the discrete multiplier space in addition to the above. A novelmultiplier is designed that is suitable for use with quadratic displacements, and exhibitsquadratic convergence.

Chapter 2

3D crack propagation with cohesive

elements in the extended nite

element method


Parmi les approches généralement admises en mécanique de la rupture fragile pour mod-éliser la dégradation d'un matériau, deux concentrent cette dégradation sur une surfacede discontinuité. L'une, la mécanique linéaire de la rupture, repose sur l'écriture de loisglobales de conservation de l'énergie, associées à des grandeurs caractéristiques seuilsactivant la propagation. L'autre introduit des zones cohésives, c'est-à-dire des forces decohésion tc s'exerçant entre les lèvres de la ssure, fonction du saut de déplacement [u]selon une loi adoucissante à partir d'un seuil déduit du champ local de contraintes (voirg.2.10). Dans ce contexte, l'approche X-FEM est une technique numérique particulière-ment exible car elle permet l'insertion dans un modèle de surfaces de discontinuité in-dépendamment du maillage, ce qui permet d'étudier la propagation sur un trajet inconnuà l'avance. La ssure est localisée de façon implicite par des fonctions dites level-sets(voir g.2.7-2.8). Numériquement, propager la ssure consiste à actualiser ces level-sets.

Ce chapitre commence par une étude bibliographique détaillée des procédures numériquesde modélisation de la propagation de ssures, utilisant X-FEM soit dans le cadre de lamécanique linéaire de la rupture (section 2.1), soit avec les modèles de zones cohésives(section 2.2). Pour ces dernières, dans la plupart des stratégies, l'avancée de la zonecohésive est donnée par un critère explicite, qui se base sur le champ de contraintes enamont du front (g.2.14). Nous proposons une procédure alternative, dans laquelle uneloi cohésive est dénie sur une large zone potentielle de ssuration . L'avancée dufront de ssure se fait de façon implicite, par un post-traitement du résultat cohésif.On laisse ainsi l'équilibre décider naturellement de la frontière entre zones adhérentes etouvertes.

Dans ce but, nous illustrons par une série d'exemples numériques qu'une loi avec rigiditéinitiale nie tc([u]) échoue à décrire ces larges zones d'adhérence dans X-FEM. Si leparamètre de pénalisation est trop faible, la solution est fausse: on observe une ouverture

8

Chapter 2. Quasi-static cohesive crack propagation 9

signicative dans les zones censées être adhérentes (g.2.28). S'il est trop important, onobserve des oscillations d'origine numérique dues à la non-conformité de l'interface parrapport au maillage (g.2.29). Pour pallier le problème, nous proposons:

l'introduction d'un espace de multiplicateurs de Lagrange λ dédié à X-FEM, etune formulation de type mortier pour utiliser cet espace de façon consistante;

l'écriture de la loi cohésive avec rigidité initiale innie tc([u],λ) dans le formalismedu Lagrangien augmenté;

une diagonalisation de certains opérateurs d'interface.

Sur un test d'arrachement, nous montrons que chacune de ces modications permet,en plus de résoudre le problème d'oscillations numériques, de gagner en robustesse, endiminuant le nombre d'itérations de Newton nécessaires pour résoudre le problème (voirtable 2.1).

En se basant sur cette capacité à résoudre de façon exacte la zone cohésive en introduisantune loi sur une large surface potentielle, la procédure d'étude de propagation sur trajetinconnu se décompose comme suit (voir g.2.34):

étant donnée une surface potentielle de ssuration, on calcule l'équilibre de lastructure,

on actualise le front de propagation par un post-traitement des grandeurs cohésivesdu calcul,

on détermine les angles de propagation le long du front,

on actualise la zone potentielle de ssuration en fonction,

on prépare l'espace de multiplicateurs et les variables internes initiales pour lenouveau pas de propagation.

Un premier front de propagation brut est déterminé en déterminant l'iso-zéro de lavariable interne de la loi cohésive, et en calculant l'intersection de cet iso-zéro avecles faces du maillage (voir g.2.35). A partir de ce nuage de points, on reconstruita posteriori l'avancée de ssure qu'il aurait fallu appliquer depuis l'ancien front pourpropager jusqu'à cet endroit. On lisse cette fonction d'avancée le long du front, puis unalgorithme d'actualisation des level-sets est appliqué, ce qui produit un nouveau frontplus régulier (voir g.2.36).

En ce qui concerne la détermination de l'angle de propagation, on se base sur le critère decontrainte circonférentielle maximale d'Erdogan et Sih [14], traduit en termes de facteursd'intensité des contraintes (2.69). En présence d'une zone cohésive, on peut prouver queles champs loins de la zone cohésive ont la même allure que des champs d'une ssurelibre (voir g.2.13), et dénir des facteurs d'intensité des contraintes équivalents. Nousmontrons que ces facteurs d'intensité des contraintes équivalents peuvent être déduitsdirectement du saut de déplacement et de la contrainte cohésive par des intégrales surl'interface.


L'actualisation de la surface potentielle de ssuration se fait par une méthodegéométrique d'actualisation des level-sets [8].

La procédure proposée est validée à l'aide de trois tests issus de la littérature (section2.5). Le premier consiste en l'étude de la ssuration d'une éprouvette de béton en formed'équerre, issu de [15]. Les résultats obtenus montrent un bon accord avec l'expérience,tant en terme de trajet de ssuration que de courbe force-déplacement. Le résultatobtenu pour cette dernière pourrait cependant être amélioré en utilisant une loi expo-nentielle plutôt qu'une loi linéaire ou bilinéaire. Le second test est un test de exiontrois points sur une éprouvette de Plexiglas avec une pré-ssure inclinée par rapport àl'axe d'application du chargement. Pour ce test, seul le trajet de ssuration est comparéavec les expériences, et montre une bonne correspondance qualitative. Le troisième testest un test de torsion d'un parallélépipède de béton entaillé sur la diagonale. Le trajetde ssuration est complexe, et exhibe une forme en S. La comparaison est satisfaisantetant du point de vue du trajet de ssuration que de la réponse globale de la structure,avec la même remarque que précédemment pour cette dernière.

A model is presented that accurately describes brittle failure in the presence of cohesiveforces, with a particular focus on the prediction of non planar crack paths. In comparisonwith earlier literature, the originality of the procedure lies in the a posterori computationof the crack advance from the equilibrium, instead of a most common determination be-forehand from the stress state ahead of the front. To this aim, a robust way of introducingbrittle non-smooth cohesive laws in the X-FEM is presented. Then the a posteriori updatealgorithm of the crack front is detailed. A novel bifurcation criterion is used that relieson cohesive quantities exclusively, by introducing equivalent stress intensity factors. Theprocedure shows good accordance with experiments from the literature.

2.1 Literature survey: linear elastic fracture mechanics

Four families of models are generally admitted to compute the brittle failure of com-ponents or structures : non local approaches relying rstly on regularization techniques[16, 17] with limits described in [18], rst order [1922] and higher order [23, 24] gradientapproaches sharing similarities with damage phase-eld models [25, 26], cohesive zonemodels (CZM) [27] and the linear elastic fracture mechanics (LEFM) [28, 29]. While therst two types of models introduce continuous deterioration of the mechanical propertiesin the bulk, strain localization is modelled by surfaces of discontinuity in the others, inter-preted as cracks propagating in the structure in the case of brittle failure. Recently, thethick level-set model [30] brought together the two approaches into a united formalism:surfaces of discontinuity automatically appear in some continuous damaged zones.

2.1.1 Linear elastic fracture mechanics (LEFM): basics

2.1.1.1 Principle

This consists in nding the displacement u and stress σ at equilibrium for a crackeddomain Ω subjected to volume forces f and whose boundary is composed of:


the free crack surface Γ (both lips);

a part Γg of the external boundary over which surface load g is applied;

a part Γu over which displacement is prescribed (see g.2.1).

u g

Initial configuration Deformed configuration

x+u- (x )

x+u+ ( x )

⟦u⟧

ΩΓx

n

g

f

Figure 2.1: Linear Elastic Fracture Mechanics: problem denition.

Equilibrium is then given by :

divσ + f = 0 on Ω (2.1)

σ · n = 0 on Γ (2.2)

σ · n = g on Γg (2.3)

u = 0 on Γu (2.4)

where n is the outward normal to Ω. The behaviour law is :

σ = A : ε on Ω (2.5)

where A is the fourth order elasticity tensor.

Under the small strain assumption, the compatibility condition is :

ε =1

2

(∇u+∇uT

)(2.6)

When advancing the crack, some stored energy in the structure is released and dissipatedin the cracking process. The energy release rate is then dened as the opposed derivative

of the potential energy W (u(Ω),Ω) =

∫Ω

1

2σ(u) : ε(u) dΩ −

∫Ωf · udΩ −

∫Γg

g · u dΓ

of the structure with respect to the cracked computation domain Ω, by G = −dWdΩ

(see

[3133]), where the solution u depends on the computation domain Ω. After Grithcriterion [34], an energy Gc has to be provided to break up a unit surface of new crack,with the propagation criterion reads:


G < Gc → no propagationG = Gc → propagation

(2.7)

For a cracked structure, propagation is stable if the increase of the crack length A tendsto decrease G, the load being kept constant. This is translated as:

∂G

∂A< 0→ stable propagation

∂G

∂A> 0→ unstable propagation

(2.8)

2.1.1.2 The energy release rate as derivative with respect to the domain:G-integral

Now, domain Ω evolves through crack extension from a preexisting crack surface, whilepreserving both the old crack and the external boundary. Hence, we are led to dene atransformation F (A) : x → x+ Aθ (x) where θ is the virtual crack extension. In orderto dene a unit crack extension, it holds |θ| = 1 in a torus Din containing the cracktip T (see g.2.2a.) So that the external boundary be left as is, |θ| smoothly decreasesfrom 1 to 0 in Dex := D \ Din, so that θ = 0 outside D (see g.2.2a). It holds then

G = −dW (u(A),Ω(A))

dA|A=0. It is a Lagrangian derivative.

Γ

Dex

C

Din

D

values of θΓ

n

t

P

a. virtual crack extension b. covariant basis along the crack tip T

b

T

Figure 2.2: Virtual crack extension and covariant basis along the crack front

Invoking Reynolds transport theorem, the denition of a Lagrangian derivative, and theweak form of equilibrium, the authors of [31, 33, 35] nd an expression G (θ) or G-integral reproduced in (2.10), for the energy release rate. Dening G along the crackfront T , parametrized by its curvilinear abcsissa s, and with outward pointing normalvector t (s) (see g.2.2b), it holds:

G (θ) =

∫TG (s)θ (s) · t (s) ds (2.9)


with:

G (θ) =

∫ΩE : ∇θ dΩ +

∫Ω

[u · ∇f · θ + (f · u) div(θ)] dΩ

+

∫Γg

[u · ∇g · θ + (g · u) (div(θ)− n · ∇θ · n)] dΓ(2.10)

where E is Eshelby's tensor (see e.g. [32]):

E = ∇uT · σ − 1

2(σ : ε) 1 (2.11)

These expressions hold under the assumptions that:

(H1) the material is homogeneous and isotropic;

(H2) linear isothermal elasticity holds;

(H3) the virtual crack extension eld θ is tangent to the cracked surface Γ, so∀x ∈ Γ,θ (x) · n (x) = 0 (see g.2.2), where n (x) is the normal to the cracksurface;

(H4) the virtual crack extension θ is tangent to the external boundary ∂Ω\Γ atany point ∀x ∈ ∂Ω,θ (x) ·n (x) = 0, where n(x) is here the normal to the extenalboundary.

Validity. Note that this expression for G remains valid for 3D problems near boundaries,for curved crack surfaces, and for crack extension elds θ not necessarily directed towardt (s) on T .

Constructing θ satisfying (H4). While constructing θ with Code_Aster, θ (x) is mod-ied to be tangent to the boundary if x is located on a side face, that is, a face whichbelongs to a single nite element.

2.1.1.3 Relation to J-integral by Rice [1]

Let us consider the additional assumptions that:

(H5) a plane problem is analysed;

(H6) the crack is rectilinear;

(H7) there are no volume or surface forces applied in the vicinity of the tip.

In this case, for any contour C fully owned by Din and connecting crack lips with each

other (see g.2.2a), G is given by Rice's J-integral as G = J := −∫Cθ · E · nda and

that integral is independent of contour C (see [1]).


CDCC

ϵ

Dϵ

D in

Γ

ϵr cθc

Figure 2.3: Denition of the J-integral

Proof (see e.g. [33]). Let us denote by DC the enclosed domain by C and Γ (see g.2.3).Note that DC is not Lipschitz since it has a tip. Hence, the divergence theorem cannotbe applied, so a vanishing disk Dε with a radius ε → 0 is removed (see g.2.3). We

denote Jε := −∫Cε

θ ·E · nda. Applying the divergence theorem in domain DC\Dε, it

comes Jε − J =

∫DC\Dε

div (θ ·E) dΩ.

Yet, we have div (θ ·E) = θ · div (E) +∇θ : E. With lack of volume forces, let us recallthat div (E) = 0. Moreover, with a straight crack, θ is uniform over Din ⊃ DC , where wethen have ∇θ = 0. It follows that Jε − J = 0. It may then be deduced that J := lim

ε→0Jε,

and this for any chosen contour C, hence the independence with respect to the contour.

Notice that this limit is well dened, and it is not zero: σ and ∇u exhibit ε−1/2 singular-ity, as will be seen in section 2.1.1.4 coming next, so E behaves as ε−1, which, integratedover a circle with a radius ε, yields a nite value.

Moreover, writing the divergence theorem over D\DC , it comes:

J =

∫D\DC

∇θ : EdΩ (2.12)

The same arguments about the uniformity of θ in Din allow to write that:

J =

∫D∇θ : EdΩ = G (θ) = G (2.13)

Curved cracks. For curved cracks, G = limε→0

Jε still applies. However, G 6= J in general.

In other words, G still quanties the intensity of the singularity at the tip, but addi-tional curvature-related terms are necessary for this intensity to be measured by J , sinceG = J +

∫DC∇θ : EdΩ.


Proof. Due to the crack curvature, θ is no longer uniform, which leads

to Jε − J =

∫DC\Dε

∇θ : EdΩ 6= 0. Adding this to (2.12), we

get Jε =∫D\Dε ∇θ : EdΩ. Let rc ∈ [0, ε], owing to the singu-

larity r−1c of E, and to dΩ = rcdr cdθc (see gs.2.3 or 2.8b), it holds

limε→0

∫Dε

∇θ : EdΩ = 0. Hence limε→0

Jε = G (θ) = G.

2.1.1.4 Asymptotic elds

An asymptotic analysis of the displacement eld near the crack tip yields [33, 36]:

u = uR +KIuSI +KIIu

SII +KIIIu

SIII (2.14)

where uR remains nite in the vicinity of the tip, and uSi are singular elds expressedas:

uSi (rc, θc) =

√rc

2πµgi (θc) (2.15)

where µ is the shear modulus and (rc, θc) are the polar coordinates related to the cracktip (see gs.2.3 or 2.8). Expressions for gi are available in [33, 36].

These expressions were established under assumption that :



(H7) there are no volume forces;

(H8) in 3D, crack tip points far from boundaries are considered.

Expression (2.27) reveals three modes of fracture, as represented on g.2.4.

nt

b

Figure 2.4: Modes of fracture


3D crack tip points near boundaries. The singular displacement does not necessarilybehave as

√rc. Instead, the extremity point of the crack front should be considered

an intersection with a free surface, and studied as a corner singularity. The cornerasymptotic behaviour is a topic on its own, and depends upon POisson's ratio (on thissubject, see [3739]).

2.1.1.5 Stress intensity factors with interaction integrals and Irwin formula

Under assumptions (H1-H4) and (H7-H8), with G-integral (2.10), it can be noticed thatG =

∫ΩE : ∇θ dΩ is expressed as g (u,u) where g is a symmetric bilinear form. It can

be shown that (see [33]):

g(uR,u

SI

)= g

(uR,u

SII

)= g

(uR,u

SIII

)= 0 (2.16)

and:

g(uSI ,u

SII

)= g

(uSI ,u

SIII

)= g

(uSII ,u

SIII

)= 0 (2.17)

Invoking bilinearity, g (u,u) = K2I g(uSI ,u

SI

)+K2

II g(uSII ,u

SII

)+K2

III g(uSIII ,u

SIII

). As-

suming a plane strain state, it comes (see [33]) g(uSI ,u

SI

)= g

(uSII ,u

SII

)=

1− ν2

E, and

g(uSIII ,u

SIII

)=

1

2µ. 3D Irwin's formula is deduced as:

G =1− ν2

E

(K2I +K2

II

)+

1

2µK2

III (2.18)

Besides g(u,uSI

)= KIg

(uSI ,u

SI

), so KI , as well as the other stress intensity factors,

may be computed from the bilinear form, as:

KI =E

1− ν2g(u,uSI

)(2.19)

3D crack tip points near boundaries. Strictly speaking, stress intensity factors may notbe computed as they may not exist.

J-integral. It is possible to consider for g the bilinear form from J-integral, but it has tobe under the additional assumptions (H5-H7).

Curved cracks. When computing the gradient ∇uSI for the auxiliary eld, the variationof the change of basis due to crack curvature has be taken into account.

2.1.1.6 Determining the crack increment

The question is to deduce from the previous LEFM quantities, for all points P along thecrack front, the bifurcation angle βP between the crack increment and the previous crack(see g.2.5) and the crack advance AP , which is the length of the crack increment (seeg.2.5).


ϕn=0P

Q

t P

nP

βPA P

tQnQ

Figure 2.5: Bifurcation angle and crack advance

A description and comparison of various criteria can be found in Meschke et al [40, 41].

2.1.1.7 Crack bifurcation angle

Common criteria in the literature are :

The maximal hoop stress criterion of Erdogan and Sih [14]. It consists in maximiz-ing σββ , and is translated in terms of stress intensity factors:

βP = 2 arctan

[1

4

(KI/KII − sign(KII )

√(KI/KII )2 + 8

)](2.20)

The minimum strain energy density criterion [42]. It's about nding the angle thatminimizes S = 1

2σ : ε around an arc of given radius.

The maximum energy release rate criterion by [43, 44]. It consists in maximizingG(β). Neglecting KIII , it is expressed as:

KI sin(βP ) +KII (3 cosβP − 1) = 0 (2.21)

Taking KIII into account, SIF are parametrized into (see [44])KI

|K|= cos(γ) cos(ψ),

KII

|K|= sin(γ) cos(ψ) and

KIII

|K|= sin(ψ), with |K| :=

(K2I +K2

II +K2III

)1/2. Bi-

furcation angle is then given by a curved surface, reproduced on g.2.6.

The maximum KI or minimum KII criterion (see [45]). Amestoy and Leblond [46]reports a maximal dierence of 7° between (2.20),(2.21) and these criteria.

2.1.1.8 Crack advance

In 2D, the advance is simply prescribed to a xed value A0 at each step of the pro-cedure. The issue is more serious in 3D. In most cases, a rule of thumb determinesexplicit advances for points along the crack front. For instance in [47], the crack is


moved forward proportionally to a quantity that is regarded as driving the cracking pro-cess: for the maximum energy release rate citerion, points along the front are moved by

A(s) = A0G(s)

maxs∈T

G(s).

Figure 2.6: The inuence of mode III on the bifurcation angle.

2.1.2 Level-set description of a propagating crack

2.1.2.1 Representation of the crack by means of level-set functions

Most of the time, cracks are implicitely represented by so-called level-set functions (see[4850]). A normal level-set φn denes the position of the cracked surface and a po-tential extension of it, through the isosurface φn = 0, which will be called potentialcrack surface in this thesis. A tangential level-set φt allows to discriminate between thecracked surface and the adherent zone located ahead of the tip (see g.2.9a), so that:

Γ = x, φn(x) = 0 and φt(x) ≤ 0 (2.22)

The crack front is then represented by T : (φn = 0) ∩ (φt = 0)(see g.2.7 and g.2.8a).This allows to dene the covariant basis along the front (see g.2.7) as n := ∇φn andt := ∇φt.


ϕn=0

ϕt=0crack frontnP

t P

P

bP

Figure 2.7: 3D description of the crack and the covariant basis attached to thecrack front.

Curved cracks. Equation (2.22) stays valid but the crack should always be at an angleθc = π with its tangential extension to the crack tip. The denition of θc should then beadapted accordingly. With this description, the correction of the polar basis for curved

cracks is written θc = arc tan

[sign (φt)φn√

r2c − φ2

n

](see Stazi and Belytschko [51] or Ventura et

al. [52]), which is illustrated on g.2.8b. Other authors [8, 53] rather take rc =√φ2n + φ2

t

and θc = arc tanφnφt

, as illustrated on g.2.8c.

ϕn

r c

√r c2−ϕn2

ϕtϕn

ϕn=0

ϕt=0

θc

ϕn

θc

r cϕt

a. crack representation by level-set functions

b. polar basis as in [Stazi et al, 2003] and [Ventura et al, 2005]

c. polar basis as in [Colombo et al, 2011]

tn

tn

tn

Figure 2.8: 2D level-set description and polar basis around the front.

2.1.2.2 Level-set update to model the propagation of a crack

Given a prescribed crack increment A(s) along the front, propagating the crack meansextending the crack surface and nding the new crack tip. Since the crack has implicitdescription, this consists in updating level-sets φn and φt.

Level-set update algorithms. Level-set update algorithms may be grouped into :

standard moving interfaces algorithms;


geometrical moving front algorithms.

Standard moving interfaces algorithms. They are generally reduced to a set of Hamilton-Jacobi equations, solved abiding by the following procedure (see [48]):

1. Extension of the interface velocity V φ = ∆A∆t on the nodes of the mesh.

2. Level-set update by solving Hamilton-Jacobi equation ∂φ∂t +∇φ ·V φ = 0 (see Duot

[54]).

In the case of crack propagation, additional steps are designed to keep signeddistance properties for φn and φt and to ensure that ∇φn ·∇φt = 0 (see Osher andSethian [48]):

3. Adjustment of the normal level-set. It is about enforcing |∇φn| = 1.

4. Orthogonalization of the tangent level-set, the normal one being xed. Given thenew normal level-set, it's about enforcing ∇φn · ∇φt = 0.

5. Adjustment of the tangential level-set to have |∇φt| = 1.

These three last steps are also resolved by Hamilton-Jacobi equations of the form∂φ∂t +F (φ,x) |∇φ| = f(x), where t is a ctitious time that evolves until a stationarystate is reached.

Numerical schemes considered by the literature until now are:

Implicit nite dierences [48] : it is rather slow and has been reported to besubjected to unstabilities [53, 55].

Explicit nite dierences [5658] : for a random unstructured mesh, the as-sociated stability (CFL) condition requires very small time steps, so that a hugenumber of iterations would be necessary. Consequently, this scheme is more e-ciently used on an auxiliary grid [58, 59]. An alternative method exists withoutauxiliary grid for simplicial meshes, according to [60].

Fast marching methods [55, 60, 61] consist in solving a polynomial equation oneach node, proceeding iteratively from the closest nodes to the interface to remotenodes.

Geometrical moving front algorithms. However, since the existing crack surface is pre-served, the level-set update is more readily done by simple geometrical considerations ina fully explicit way, by taking advantadge of the existence of a moving crack front fromwhich distances can be measured. It has been implemented:

with classical level-sets by [8, 54, 62];

with vector level-sets by [52];

with an implicit-explicit crack description by [47].


2.1.3 The X-FEM to model singular cracks

So that possibly non-planar surfaces of discontinity may be inserted in a model regardlessof the underlying mesh, an extended nite element method (X-FEM) was developed by[63] that accounts for discontinuities within elements by means of a local enrichmentwith discontinuous functions, based on the partition of unity method [64].

To increase accuracy, the asymptotic solution (2.15) is inserted into the nite elementbasis. Classically, the nodes H of the element containing the crack tip are enriched withasymptotic functions, and all remaining nodes K belonging to crack-through elementsare enriched with a Heaviside-like function representing the jump (see g.2.9), so thatthe discrete displacement space is:

V h :=

∑i∈N

aiNi (x) +∑i∈KbiNi (x)H (x) +

∑i∈H

Ni (x)4∑l=1

cilFl (x) ,

ai ∈ Rd, bi ∈ Rd, cil ∈ Rd (2.23)

where N represents the set of nodes of the mesh, Ni are the linear shape functions and:

H (x) :=

+1 if φn(x) ≥ 0−1 if φn(x) < 0

(2.24)

[Fl] =

[√rc sin(

θc2

),√rc cos(

θc2

),√rc sin(

θc2

) sin(θc),√rc cos(

θc2

) cos(θc)

](2.25)

node in K

node in H

crack

Figure 2.9: XFEM enrichment strategy for stress free cracks.

2.1.4 Limitations of LEFM

Though the LEFM is probably the most widespread engineering tool to study crackpropagation, it implies fairly strong assumptions and therefore suers from limitations:


If the size of the process zone approaches any relevant lengthscale of the model(structure typical size, initial crack length, distance between crack tips), the re-sponse of the stucture is subjected to a size eect, that LEFM fails to predict.

In 3D, Grith (implicit) criterion that crack propagates until G < Gc is very hardto verify along a front. A rule of thumb is generally used to compute an explicitcrack advance instead (see [47] for discussion).

The crack may not initiate from a sound structure: a pre-crack has to be put in.Thus, the sensitivity of the response to the shape and orientation of the pre-crackhas to be studied, and the failure load cannot always be accurately reproduced.

The calculation of stress intensity factors (SIF) requires the construction of ade-quate contours surrounding the front [31, 6567], or specic elements to achieveoptimal quality of the results [68], both techniques being generally coupled, whichrequires additional monitoring.

The notion of stress intensity factor may not even exist in somes cases, as hetero-geneous or anisotropic materials or non-linear materials (except for very speciccases). In others, the notion of SIF may be extended, but at the cost of a morecomplicated computation: von Mises plasticity under monotonic radial loads, geo-metrical non-linearities, some rate-dependent eects...

2.2 Literature survey: cohesive zone models

2.2.1 Principle

In order to adress some of these issues, cohesive zone models were originally proposedby Barenblatt [27]. They consist of restraining forces which vanish as the crack opens,thus obeying a softening traction-displacement relation (see g.2.10).

−t c

Zone ouverte

Zone fermée

Zone cohésive

t c

l c

c

l c

t c

−t c

w c

crack front

[u ]

Cohesive zone Adherent zoneTraction-free zone

Open or cracked zone

Figure 2.10: Cohesive zone model


The potential crack surface φn = 0 is then made up of (see g.2.10):

an adherent zone,

a cohesive process zone, where crack lips are open while subjected to restrainingforces, with characteristic length lc and characteristic opening wc,

a traction-free zone.

The crack front or crack tip is dened as the boundary between the cohesive andadherent zone: it is located at the onset of debonding(see g.2.10), where thestress reaches a critical value σc. The cohesive characteristic length is then given

by lc = MGcE

σ2c (1− ν2)

where M is a multiplicative factor depending on the shape

of the traction-opening curve (see [69]).

2.2.2 Equilibrium of cracked bodies with cohesive forces

This time, let Ω ⊂ Rd be the uncracked domain of interest. We denote by Γ , not apreexisting crack in the domain, but a potential crack surface. As a reminder, the externalboundary ∂Ω is still decomposed into non-overlapping parts Γu and Γg, with prescribeddisplacements and forces g, respectively (see g.2.11). If ui are the displacement eldson each of the crack lip Γi (Γ+ is the upper crack lip, and Γ- the lower crack lip), thedisplacement jump is [u] (x) = u+ (x) − u- (x). Let n be the outward normal vectorto Γ - and tc be the restraining (cohesive) force that Γ + applies over Γ- (g.2.11). Inaddition to (2.1) and (2.3-2.6), we have −σ+ ·n+ = −σ- ·n+ = σ+ ·n = σ- ·n = tc onΓ.

potential crack surface Γ

traction-free zone

u g

Reference configuration Deformed configuration

n

x

x+u+ (x )

x+u- (x )

tc

[u]Γ

-Γ+

g (t )

−tc

adherent zonecohesive zone

ug(t) g

Ω+

Ω-

Figure 2.11: Notations of the problem.


ΓR+

ΓR-

ΓΓ+

Γ-

C

crack front

cohesive zone

DC

a. contour surrounding the cohesive zone

ΓR+∖ΓC

+

ΓC+

ΓC-

C

ΓR-∖ΓC

-

b. contour not surrounding the cohesive zone

θvector field

Figure 2.12: Dening invariant integrals in the presence of cohesive forces.

2.2.3 Invariant integrals [2, 3]

Let us consider assumptions (H1) to (H7). This time, the virtual crack extension eldθ of the crack is taken such that |θ| = 1 everywhere. Again, we consider a contourC surrounding the tip, and denote by Γ+

C and Γ-C the crack lip segments between the

extremities of C and the tip (see g.2.12b), so that the whole denes a domain DC (seeg.2.12a). A generalized Rice integral is dened as:

J := −∫∂DC

θ ·E · ndΓ (2.26)

where n is the outward normal to the contour. Due to the fact that asymptotic cohesivenear-elds are not singular, we have J = 0.

Proof. Let us take on proof 2.1.1.3 again. We had applied a divergence theorem inDC\Dε. Then, given the lack of volume forces and the uniformity of θ, we had obtainedJ := lim

ε→0Jε. This time, stresses and displacement gradients are bounded in the vicinity

of the tip, so E is bounded, which implies limε→0−∫Cεθ ·E · ndΓ = 0 and thus J = 0.

By ∂DC = C∪Γ+C∪Γ-

C , J may be split up into J = Jcoh+Jext, with Jext := −∫C θ·E·ndΓ

and Jcoh =∫

ΓCθ · [E] · ndΓ (see Moës and Belytschko [2]) with [E] = E+ −E-. Given

the denition for E, the fact that θ · n = 0 and the fact that σ · n = tc on Γ, it comes:

Jcoh =

∫ΓC

tc · ∇[u] · θ dΓ (2.27)

It is noticeable that the integrand of (2.27) vanishes outside the cohesive zone, sincetc = 0 on the traction-free zone and [u] = 0 on the adherent zone. Hence Jcoh is


independent of contour C as long as it surrounds the cohezive zone (see g.2.12a), andmay be written as:

Jcoh =

∫Γtc · ∇[u] · θ dΓ (2.28)

Consequently, Jext does not depend upon C for such contours. It is worth remarking thatJext has the same denition as the J-integral from LEFM. It may then be interpreted asan intensity measure for an equivalent singularity, when looking far elds.

Other interpretation. For any contour C, a corrected J-integral JCext may be dened thatis always independent of the contour, and quanties the apparent singularity far awayfrom the process zone. Correction consists in including cohesive forces over the part Γ\ΓCof the crack not covered by C (see g.2.12b):

JCext = Jext −∫

Γ\ΓCtc · ∇[u] · θ dΓ (2.29)

If C surrounds the cohesive zone, JCext is simply Jext, and for a vanishing contour Caround the tip, we nd −Jcoh.

Curved cracks. In this case, Jcoh rather than Jext is reasonable to quantify the equivalentsingularity, on condition that C surrounds the cohesive zone (see g.12a). Indeed, let usconsider the proof of 2.1.1.3 again. For a curved crack, the divergence theorem over DC

yields Jext + Jcoh = −∫DC∇θ : E dΩ 6= 0 . Similarly, given that C owns the cohesive

zone, the divergence theorem over D\DC gives Jext =∫D\DC ∇θ : E dΩ (see 2.1.1.3),

hence −Jcoh =∫D∇θ : E dΩ = G (θ) .

2.2.4 Asymptotic elds and equivalent stress intensity factors

Asymptotic elds. Away from the process zone, far elds are similar to asymptotic eldsof a free crack located at distance A∞ from the crack tip (see g.2.13 from [3]). For a

linear-softening law, [70] reports A∞ =π

8lc.

Hence, a mode I far eld u is similar at order 0 to a KI,equSI , where u

SI is the asymptotic

eld of LEFM and thus KI,eq an equivalent stress intensity factor. Hence, KI,eq may stillbe computed with the bilinear form:

gext(u,uSI

)= KI,eqg

ext(uS1 ,u

S1

)=

1− ν2

EKI,eq (2.30)

2.2.5 Determining a crack increment : crack advance and angle

In a rst family of approaches (g.2.14), the crack path is fully extended before computingthe equilibrium. While conserving the crack surface, the load increment is applied. The


-10 -8 -6 -4 -2 0 2

0

2

4

6

8

10

12

exact solution

order 0 approximation

order 1 approximation

normalized coordinate along the crack surface

norm

aliz

ed o

peni

ng

Figure 2.13: Far elds reported by Planas and Elices [3].

Load increase and equilibrium, no propagation

Extension of the crack surface

Equilibrium with the extended crack surface

1. 2.

3.2.

σ > σ c

Legend: Traction-free zone

Cohesive zone

Extended crack surface

Figure 2.14: Principle of explicit crack increment and crack tracking algorithms.

extension of the crack path is then determined from the stress state ahead of the front,where elements are going to exceed a critical stress. The cohesive calculation is performedwith this new path, and the procedure is repeated iteratively (g.2.14). The crack pathmay be extended by segments or faces through the dermination of a bifurcation angleand crack advance, as in [41, 7173].


crack front

radius R

crack Gauss point

radius R

volume V

Figure 2.15: Computation domain of the smeared stress for the meso-local ap-proach

2.2.5.1 Bifurcation angle: criteria from equivalent stress intensity factors

The use of SIF-based criteria in cohesive models has been suggested by [74, 75]. Authorsexplain that this is a reasonable assumption for concrete, since experiments showed thatthe crack path, contrary to the load-deection curve, is almost insensitive to the size ofthe cohesive zone, to such a point that the crack path may still reasonably be predictedwith LEFM assumptions.

Criteria from section 2.1.1.7 may be computed from the equivalent stress intensity factorsdeduced straight from the cohesive result, as was done by Moës and Belytschko [2] or Ziand Belytschko [76]. On the contrary, Meschke et al [40] or Zamani et al [77] carry outa dedicated computation with a free crack.

Advantages and drawbacks. Relying on global energetic quantities rather than local elds,this criterion is reported as giving smooth paths by [77]. However, it can only be usedfor mode I dominated problems on brittle or quasi-brittle materials.

2.2.5.2 Bifurcation angle : the meso-local approach

In this approach, the crack increment along the front is deduced from a smeared stress,computed in a half disk (in 2D) or a half ball (in 3D) ahead of the front (see g.2.15).The smeared stress is then taken as the weighted average over that domain V , either

with Gaussian weights as σ =

∫V ω(r)σdV∫V ω(r)dV

with ω(r) = exp(− r2

2R2

)(see Wells [72] or

Meschke [78]) or with a mere average σ = 1|V |∫V σdV (see [41]).

Rankine criterion. The principal components of this smeared stress are computed as

σ =d∑i=1λσi n

σi ⊗ nσi with λσ1 ≥ λσ2 ≥ λσ3 . For brittle failure, the propagation direction is

considered as being normal to the direction nσ1 of the maximal principal stress (Rankine'sdirection). This was converted into an angle expression in 2D and 3D by [79, 80]. Forinstance in 2D (with (n, t) the covariant basis at the crack tip):


β =1

2arc tan

(2σnt

σnn − σtt

)(2.31)

Advantadges and drawbacks. This criterion is robust and easy to implement. In somecases (see the L-shaped panel test in [41]), a perturbed crack path is reported. Zamaniand al [77] suggest to drop axial load σtt in (2.31) when computing the criterion. Toaccount for the switch to shear band propagation in the brittle-ductile transition re-gion, more elaborate criteria have been designed by Haboussa [79] by combination withmaximal von Mises stress direction.

2.2.5.3 Crack advance

Elementwise local extension. In 2D, the crack may be propagated iteratively, as in [72, 73],in all cut elements, as long as λσ1 > σc in the element.

Prescribed length. Alternatively, the crack may be extended over several elements atonce, with a prescribed length A0 [41, 71] - which is chosen depending on the expectedcurvature of the crack, and covers at least a few mesh elements - under condition thatλσ1 > σc be reached by Gauss points lying (see [71]):

in a close direction to that given by the bifurcation angle,

at a certain distance from the existing tip, so as to avoid locking eects that couldappear if the crack were extended with too small a load increment. An empiricalvalue 0.25A0 is reported in [71].

SIF-based criterion. Instead, in [41], the prescribed length A0 is applied as long as thisresults in a decrease E(A0) < E(0) of the total mechanical energy E := W+

∫Γ Π([u])dΓ,

where Π([u]) =∫ [u]

0 tc · d[u] is the surface cohesive energy. If this is not the case, theload is increased without extending the crack, which avoids the aforementioned lockingeects.

Variational method. In [78], bifurcation angle and crack advance are included as newunknows in the energy of the structure to be minimized, and the related derivatives arenumerically computed by nite dierences. Load increment is monitored, so that:

if the computed advance is to small, load is increased without even extending thecrack,

if it is too large, the load step size is decreased.

2.2.6 Determining the crack path as a whole: crack tracking algo-rithms

Hovewer, the crack path is often determined as a whole, by so-called crack trackingalgorithms : Jäger [81] oers a nice review of these methods. Let us recall the context(see g.2.14).


1. Having a result at time step ti−1, or starting from an initially sound structure, loadis increased until time step ti, without extending the crack (see g.2.14).

2. In each element, the stress on a central integration point, or a smeared stress, is

decomposed into principal components σ =3∑i=1λσi n

σi ⊗ nσi with λσ1 ≥ λσ2 ≥ λσ3 . In

some elements, the critical stress will be exceeded λσ1 ≥ σc.

A crack tracking algorithm then consists in extending the surface of internal discontinuityfrom (see g.2.17):

a eld of normal directions nσ1 ,

the preexisting crack surface.

Specications for a good algorithm are:

the crack surface should be smooth and continuous,

the previous crack surface should not be moved.

2.2.6.1 Local crack tracking algorithm

It is an iterative process by Areias and Belytschko [82], in which elements are treatediteratively as (see g.2.17):

Initialization. A series of edges or crack roots are given as inputs, which locate crackinitiation. They gererally corresponds to sharp edges of the structure (L-shaped panel,Nooru-Muhamed test, notched sample).

For each time step. The elements where critical stress is reached are stored. A planesurface of internal discontinuity is then inserted, element by element. Each element istreated if it has a cracked neighbor or is adjacent to a crack root (see g.2.17). Theprocedure stops when there are no elements left to handle.

Building the crack surface on an element. In addition to Rankine's direction, the planecrack surface takes cracked neighbour elements into account, by enforcing continuity forthe crack surface (see [81, 82]). Hence, if the current element shares a single point withthe previous crack surface, Rankine's direction is the normal to the inserted discontinuitysurface. If it shares a line, it is the projection of Rankine's direction on the normal planeto the line. If it is two lines or more, the normal is fully determined by the plane denedby the two neighbours (see gs. 2.17-2.18).


Cut ← FALSE

Extension of the crack by adding a plane internal

discontinuity surfacewithin element i

Element i is maked as processed

i = nf ?

Remove processed elements from the listUpdate nf

Cut = TRUE?

START

crack roots

Cut ← TRUEi ← 0

Initialization of the crack surface: crack roots

i ← i + 1

Is element i intersected by the current

crack surface?

First time step ?

Previous crack surface

Initialization of the crack surface: previous surface

This step is different in local or non-local crack tracking

END

NO

YES

YES

NO

YES

NO

NO

YES

List the nf elements wherecritical stress is reached

Figure 2.16: Common architecture for local or non-local crack tracking.


ne=n1σ

C1

ne=(1−C1C 2⊗C1C 2

∣C1C 2∣2 )⋅n1

σ

C1

C2 C1

C2

C3

ne=C 1C 2∧C1C3

a. one point in commun with the previous crack surface

b. one line in common c. two lines or more

Figure 2.17: Constructing the crack surface with a local tracking algorithm.

START LOCAL TRACKING

Element iIntersection with the previous crack surface

The intersection is a point ?

The intersection isA single line ?

Normal given by the stressfirst eigenvector

Normal given by the stressand the common line

with the previous crack surface

Normal exclusively given by the previous crack surface

A plane internal discontinuity surface is added which goes through an intersection point

and is orthogonal to the normal determined above

END LOCAL TRACKING

Figure 2.18: Local crack tracking algorithm: extension with an element.

2.2.6.2 Non-local crack tracking algorithm

Gasser and Holzapfel [83, 84] keep the same overall procedure, but non-local quantitiesare used to extend the crack surface. Hence, the surface of discontinuity is obtained byinterpolation of a cloud of points, consisting in the intersection points for the currentelement as given by Rankine's direction, and points of the cracked surface in a neighbor-hood. The overall crack surface is not continuous strictly speaking, but it is not a realproblem in the PU-FEM formulation used by the authors, as long as nodes of the parentmesh may be attributed one ot the other side of the cracked surface (g.2.19).


START NON-LOCAL TRACKING

Element iIntersection points with the previous crack surface

Rankine direction at an integration point in the element

Predicting surface of discontinuitygoing through the average P of intersection points

and normal to Rankine direction

Cloud of points of the previous crack surfaceand the predicted surfacefar from P by less than R

Least square approximation of the cloud of pointsby a plane

Corrected normal within the element :still going through P but orthogonal

to the plane determined above

END NON-LOCAL TRACKING

n1σ

P

P

R

P

n1σ

n1σ

P

Figure 2.19: Non-local crack tracking algorithm.


2.2.6.3 Global crack tracking algorithm

Principle. In this approach, suggested by Oliver et al [85, 86], the crack surface isextended globally. For each cut mesh element, vector nσ1 should be as close as possibleto the normal to the crack surface. Consequently, the crack surface is searched amongthe envelop of this vector elds, with the form of iso-contours of a scalar function φn,which represents potential crack paths. As for us, a single crack surface is tracked, asΓ : φn = 0.

Vector elds envelops should verify:

On each element ∇φn should be colinear to nσ1 . Introducing K = 1−nσ1 ⊗nσ1 , wethen have j := K · ∇φn = 0.

Let T be the previous crack front, or the crack roots. So as to ensure continuitywith the previous crack surface, we wish to have φn = 0 on T .

We would like to preclude constant-φn solutions, which do not provide iso-contours.

We solve the following problem over the computation domain Ω:

∇ · j := 0 over Ω (2.32)

j = K · ∇φn over Ω (2.33)

j · n = 0 over ∂Ω (2.34)

φn = 0 over T (2.35)

φn = 1 on A1, a point on ∂Ω (2.36)

It is then easy to verify that j = 0 is the solution. It is unique and satises φn 6= cte.

Compatibility of boundary conditions. Point A1 should be put at a place where the cracksurface will not go.

Extension of the crack surface. Once φn = 0 is known, a discontinuity surface is itera-tively added in the intersected elements for which λσ1 > σc.

Well posed problem. To have a well-posed problem, K should be invertible. As a result,K = (1 + ε) 1− nσ1 ⊗ nσ1 is used, with ε 1.


2.2.7 2D implicit crack advance

In a second family of approaches (g.2.20), the crack advance is controlled at each step:from a previous crack path, a bifurcation angle is determined. The crack is extended bya controlled length in this direction, an additional equation Keq

I (σN ) = KIc giving theintensity σN of load to be applied to have this advance in practice (see [2, 76, 77]).

Bifurcation angle

Equilibrium with a prescribed advance, load factor adapted

accordingly

1.

2.

1.

Legend:

Traction-free zone

Cohesive zone

Extended crack surface

Figure 2.20: 2D implicit crack advance.

2.2.8 Enrichment strategies for cohesive zone models

For all the above techniques, the surface of discontinuity has to stop within the structure.When using X-FEM enrichment strategies [72, 73] or PU-FEM methods [84, 87, 88],surfaces of disconinuity may stop either at the edges (2D) or faces (3D) of the mesh.Alternatively, a crack tip may be located within an element, which would be enrichedwith regular functions as in [2, 41, 71, 89]. An alternative is proposed by [76] to locatea crack tip within an element with no other enrichment than Heaviside's.

2.2.8.1 Stopping enrichment on crack edges

In Wells et al [72], the crack ends at the intersection with a mesh edge (see g.2.21).Only Heaviside enrichment is applied, on all nodes of cut elements, except the verticesof the considered intersected edge (see g.2.21).

2.2.8.2 With dedicated asymptotic functions around the tip

The above description would result in a rough crack front in 3D. To tackle the issue, Moësand Belytschko [2] suggest a description with a cohesive tip ending within elements, withhelp of a dedicated asymptotic enrichment on the nodes of the element containing thetip, similarly to LEFM.


Heaviside enriched node

crack

Figure 2.21: Enrichment stopping at a crack edge.

Cohesive enrichment function are not singular at the tip, though, but proposed from theasymptotic analysis by Planas and Elices [3]. Near-eld terms are taken on, in the polar

basis of the crack tip (see g.2.22) rc sin

(θc2

)and r

3/2c sin

(θc2

).

Heaviside enriched node

crack

asymptotic enrichment

Figure 2.22: Enrichment with asymptotic function.

2.2.8.3 With a corrected Heaviside enrichment

Zi and Belytschko [76], work out the Heaviside-like enrichment function. Instead ofH (x), they use node specic enrichment functions, as Hi (x) = sign (φ(x)) − sign (φi)(see g.2.23), so that the discontinuous part of the discrete displacement isudisch (x) =

∑i∈K

Ni (x) Hi (x) bi.

Advantadges. Such udisch is zero on common edges with non-intersected elements, whichsimplies the implementation.

To extend to a discontinuity ending within an element, this property should be respected.Therefore, for the represented conguration on g.2.24, the enriched part udisc should bezero on edges 12 and 13. Consequently, only node 3 will be enriched. Besides, it shouldbe continuous over 23 with the adjacent element, and should be zero over segment 2P ,over which the crack tip is located (see g.2.24).

Consequently, on this element the discontinuous eld is udisc = N3

(ξ*)H3

(ξ*)b3, so

with linear shape functions udisc = ξ*3H3(ξ*)b3 where ξ* =(ξ*P , ξ

*2 , ξ

*3

)is the reference

coordinate over triangle P23 .


Enriched nodes

crack

Value of H 3(x )

3

1

2

Figure 2.23: Alternative to Heaviside function in a triangular element 123 com-pletely cut by the crack.

Only enriched node

crack

Value of udisc∝ H 3(ξ*)ξ3

*

3

1

2PCT

Figure 2.24: Discontinuous part of the displacement eld for the triangular ele-ment 123 containing the tip.

2.3 XFEM cohesive zone models with large adherent zone

In this thesis, we aim at having 3D implicit crack advance: the new crack front willbe determined afterhand based on cohesive elds. The cohesive law is then inserted onthe whole potential crack surface φn = 0 and the equilibrium will naturally separateit into the cracked surface and the adherent zone. Hence, a robust implementation ofnon-smooth cohesive laws in the X-FEM is presented in this section, that is capable ofhandling perfect adherence over large areas. This turns out to be necessary when thecrack advance is not known beforehand.

2.3.1 Discrete displacement

In this strategy, the elements crossed by the potential crack are all enriched with aHeaviside-like function H, so that the discrete displacement space is:

V h :=

∑i∈N

aiNi (x) +∑i∈KbiNi (x)H (x) ,ai ∈ Rd, bi ∈ Rd

(2.37)

where N represents the set of nodes of the mesh, K is the set of enriched nodes (seeg.2.25), Ni are the linear shape functions and:


enriched nodesK

intersected elementsE

potential crack surface Γ

Figure 2.25: Potential crack surface not matching the mesh edges, and subsequentenriched nodes.

H (x) :=

+1 if φn(x) ≥ 0−1 if φn(x) < 0

(2.38)

2.3.2 Why a penalized law is not ecient to describe a large adherentzone

As a trial run, the cohesive traction tc is written as an explicit function of [u] (see

g.2.26). Hence, an equivalent jump is dened as [u]eq =√〈[u]n〉2+ + β2[u]2s, where:

[u]n = [u] · n is the normal jump;

〈[u]n〉+ = max [u]n, 0 and 〈[u]n〉- = min [u]n, 0 are its positive and negativeparts;

[u]s = (1− n⊗ n) · [u] is the tengential jump;

β is a material parameter that quanties the ratio between tensile and shearstrengths.

A dimensionless internal variable α may then be dened as:

[u]eqwc− α ≤ 0 (2.39)

α ≥ 0 (2.40)([u]eqwc− α

)α = 0 (2.41)

where wc corresponds to the rst opening for which cohesive forces are reduced to zero.The internal variable has non-zero initial value α0: α = α0 indicates adherence with theinitial stiness, α ∈]α0, 1[ describes a damaged material, and α = 1 characterizes fulldebonding. For loading conditions (that is, if α = [u]eq/wc ), the surface energy reads :


σc

wc

t c⋅n

⟦u⟧n

Opening mode

βσc

β−1wc

−β−1w c

t c , s

⟦u⟧s

Shear mode

−βσc

α0wc αwc

w c :=2G c

σc

Figure 2.26: Penalized linear-softening mode-coupling cohesive law.

Π ([u]) = 2Gcα(

1− α

2

)+

σc2α0wc

〈[u]n〉2- (2.42)

The cohesive traction is deduced from (2.42) by tc ([u]) =∂Π

∂[u]. Its expression may be

synthetized as follows: dening an equivalent traction as tc,eq =√〈tc,n〉2+ + β−2t2c,s, it

holds:

tc,eq = σc (1− α)[u]eqαwc

(2.43)

The normal traction is then dened by tc,n =tc,eq[u]eq

〈[u]n〉+ + σc〈[u]n〉-α0wc

and the shear

component is tc,s = β2 tc,eq[u]eq

[u]s.

To see how such a formulation performs with large adherent zones, an inclusion debondingtest is carried out, whose geometry and loading conditions are represented on g.2.27(all dimensions are millimetters). It consists of a plate in plane strain under tension, forwhich linear isotropic elasticity is assumed with coecients E = 36.56GPa and ν = 0.2.A circular inclusion is inserted into the plate, which is prone to debonding and thereforesubjected to the above cohesive law, with tensile strength σc = 2.7MPa, fracture energyGc = 0.095N .mm−1 and mixed-mode parameter β = 1. With strong mode-coupling, acurved crack surface and some contact at the upper and lower extremities of the inclusion,this problem is a good challenger for the accuracy and robustness of a formulation.


ux=0ux=u0

500

25075

x

y

θ

(E ,ν )

(σc ,Gc )

Figure 2.27: Inclusion debonding test.

deplacement

Page 1

-3,5 -2,5 -1,5 -0,5 0,5 1,5 2,5 3,5-1,0E-4

1,0E-4

3,0E-4

5,0E-4

7,0E-4

9,0E-4

1,1E-3

1,3E-3 α0 = 1.e-2

α0 = 1.e-3

α0 = 1.e-4

α0 = 1.e-6

reference solution conforming mesh α0 = 1.e-6

Angle along the interface (rad)

Nor

mal

ope

ning

(m

m)

1

Figure 2.28: Normal opening with the inclusion debonding test.

-3,5 -2,5 -1,5 -0,5 0,5 1,5 2,5 3,5

-0,5

0

0,5

1

1,5

2

2,5

3

reference solution : matching mesh α0=1.e-6

non-matching mesh α0=1.e-6

Angle along the interface (rad)

Nor

mal

trac

tion

(MPa

)

2

3

Figure 2.29: Normal tractions with the inclusion debonding test.


Normal opening and tractions these being computed straight from the displacementby (2.43) have been plotted on gs.2.28-2.29 at load u0 = 0.04mm, for crack-surface-matching and non matching meshes, and with various penalization parameters α−1

0 . Itcan be seen on gs.2.28-2.29 that such classical penalized cohesive laws tc ([u]) raise threeissues:

1. non physical opening or interpenetration is observed, impairing the accuracy of theresults, when penalization parameter α−1

0 is too low (tag 1 on g.2.28): α−10 = 104

is the minimal value which accurately resolves the adherent phase;

2. spurious oscillations of the traction are observed in the (almost) adherent zone,when α−1

0 is on the contrary too high (tag 2 on g.2.29). This numerical issue hasbeen widely reported and studied by X-FEM literature (see [13, 90, 91]). In oneword, it comes from the fact that the discretization space for traction gets too richin comparison with that of the displacement, when the stiness of the interfaciallaw gets so high that it is akin to enforcing a Dirichlet condition;

3. misevaluation of the regime (adherence or debonding) may occur as a collateraldamage of these oscillations, with some Gauss points in the adherent zone beingmistakenly considered as dissipative (tag 3 on g.2.29).

To conclude, classical penalized laws fail to achieve good resolution of large adherentzones, since a too low α−1

0 implies large opening in the adherent zone leading to a wrongsolution in terms of accuracy, while a higher value of α−1

0 arises spurious oscillations dueto stability issues, as soon as the penalization is about to get suciently sti to correctlydescribe adherence.

Changes to the formulation are proposed as remedial actions to each of the aforemen-tioned drawback:

1. amixed law with initial perfect adherence is proposed to avoid non physical opening;

2. a stable mortar formulation is introduced to handle perfect adherence and thesudden switch to a Neumann-like condition when debonding starts;

3. a discretization with blockwise diagonal operators is suggested to avoid regimemisevaluations.

2.3.3 The use of Lagrange multipliers in the X-FEM

The ability to prescribe Dirichlet conditions in the X-FEM is a prerequisite to introducingmixed laws. Literature on the topic can be divided into stable methods, which worksout discrete spaces by enriching the displacement space [92] or reducing the traction multiplier space [13, 91, 93], and stabilized methods, which consists of adding stabilizingterms to the formulation, either by having the stress trace on the interface play the roleof Lagrange multiplier, as in Nitsche's methods [94], or with stabilization term on themultiplier/stress discrepancy [95]. In this paper, the reduced multiplier space by Géniaut[93] is used.


Group of nodes sharing the same Lagrange degree of freedom

Support of the non-local shape function ψ3

Potential crack surface

Intersected edge supporting an equality relation

Other intersected edge

Figure 2.30: Mesh not matching the crack surface and reduced multiplier space.

In this space, multipliers components are supported by the nodes in K (see g.2.25).This initial set is reduced into a fewer number Nλ of degrees of freedom by prescrib-ing equality relations between the components, supported by some intersected edges V:a truly independent Lagrange degree of freedom I is shared by a group of nodes i inK (see g.2.30), hence making a non-local shape function ψI :=

∑i∈INi (see g.2.30).

A eld is then interpolated over cut elements and the discrete multiplier is nallythe trace of this eld on the crack surface, so that the discrete multiplier space is

Mh :=

∑I

µIψI |Γ,µI ∈ Rd.

2.3.4 A mixed law, in the augmented Lagrangian formalism

For the sake of conciseness, let us call w the displacement jump. In the augmentedLagrangian formalism, a general expression of the surface density of energy for othotropicpotential laws is:

Π(w,λ) = φ(λn + rwn,λs + rsws)−λ2n

2r− λs · λs

2rs(2.44)

where φ is a derivable function, r and rs are the normal and tangential augmentationparameters. Here, tangential augmentation is taken as rs = β2r.

While the cohesive traction is still given by tc =∂Π

∂w, dual equations are needed to

know where Dirichlet conditions take place, and enforce them there. This additional

interfacial law is determined by stating that∂Π

∂λ= 0 in the augmented Lagrangian

formalism, which ensures that there is no need to verify any further inequality constraintthroughout computation: all constraints are embedded in that (equality) interfacial law,which makes it eligible for resolution with a mere Newton-Raphson procedure.

The cohesive traction thus reads :

tc,n (λn + rwn,λs + rsws) =∂Π

∂wn= r

∂φ

∂(λn + rwn)(2.45)


σc

wc

t c ,n

wn

Opening mode

βσc

β−1w c

−β−1w c

t c , s

w s

Shear mode

−βσc

Figure 2.31: Mode-coupling mixed cohesive law.

tc,s(λn + rwn,λs + rβ2ws

)=

∂Π

∂ws= rβ2 ∂φ

∂(λs + rβ2ws)(2.46)

The dependence of tc on λn + rwn and λs + rsws will be omitted from here onward toalleviate notations. The interfacial law is simply λ = tc.

Let (λ+ rw)eq :=√〈λn + rwn〉2+ + β−2 (λs + β2rws)

2 be an equivalent augmented trac-

tion. A threshold function ϕ is then introduced as ϕ(

(λ+ rw)eq

):=

(λ+ rw)eq − σcrwc − σc

,

so that a scalar dimensionless internal variable α is dened as verifying:

ϕ(

(λ+ rw)eq

)− α ≤ 0 (2.47)

α ≥ 0 (2.48)

α[ϕ(

(λ+ rw)eq

)− α

]= 0 (2.49)

It holds α ≤ 0 for an uncracked material, and α ≥ 1 for a fully cracked material. For

loading conditions, that is to say if α = ϕ(

(λ+ rw)eq

), function φ is dened by:

φ (λn + rwn,λs + rsws) = 2Gc

(1− σc

rwc

)α(

1− α

2

)+

1

2r〈λn + rwn〉2- (2.50)

For contact-free situations, and not considering the related term in 2.50, the surface

energy 2.44 depends upon α and λeq only, through Π(α, λeq) = φ(α) −λ2eq

2r. When

dissipation starts, it holds α = 0 and λeq = σc. When it ends, it holds α = 1 andλeq = 0. Then, we have π(α = 1, λeq = 0) − π(α = 0, λeq = σc) = Gc since φ(α =

1) − φ(α = 0) = Gc −σ2c

2r, which ensures that an energy Gc shall be provided to fully

open a unit surface of crack.


The resulting traction deduced from (2.50) by (2.45-2.46) may be synthetized as follows:

still dening equivalent tractions as tc,eq =√〈tc,n〉2+ + β−2t2c,s, the traction-separation

law is expressed in terms of equivalent quantities as tc,eq = (1− Td) (λ+ rw)eq , whereTd is the damage tensor. Its expression for linear softening is:

Td =α(

1− σcrwc

)α+ σc

rwc

(2.51)

We may readily check that Td = 0 if α = 0 (perfect adherence), and Td = 1 ifα = 1 (full debonding). The vector traction-separation law is then dened in terms ofnormal component tc,n = (1− Td) 〈λn+rwn〉++〈λn+rwn〉-(the latter term being addedto account for unilateral contact) and shear component tc,s = (1− Td)

(λs + rβ2ws

).

2.3.5 Stable mortar formulation for inserting an interfacial law inXFEM

As mentioned before, interface quantities (tractions tc and energy density Π) shouldbe dened over the reduced space Mh to avoid spurious oscillations during adherencephases. This still applies to the displacement jumpw, because it actually comes into playduring adherence phases as well (through the augmentation). Hence, w is introduced asa new unknown of the problem, to be discretized over a dierent space from that of [u]:Mh. The total energy of the problem reads:

E(u,λ,w) =1

2

∫Ωε(u) : C : ε(u)dΩ−

∫Γg

g · udΓg +

∫Γ

Π (w,λ) dΓ (2.52)

Now, nding the solution of the continuous problem would imply to nd (u,w,λ) =

argminw*=[u*]

E(u*,λ*,w*

), hence the Lagrangian of the problem:

L(u,w,λ,µ) =1

2


∫Γg

g ·udΓg+

∫Γ

Π(w,λ)dΓ+

∫Γµ·([u]−w) dΓ

(2.53)The optimality conditions of Lagrangian (2.53) give the following discrete weak form:

∀u* ∈ V h,

∫Ωσ(u) : ε(u*)dΩ−

∫Γg

g · u*dΓg +

∫Γµ · [u*]dΓ = 0 (2.54)

∀µ* ∈Mh,

∫Γ

([u]−w) · µ*dΓ = 0 (2.55)

∀w* ∈Mh,−∫

Γ[µ− tc] ·w*dΓ = 0 (2.56)


∀λ*n ∈Mh,−∫

Γ

[λn − tc,n]

r· λ*ndΓ = 0 (2.57)

∀λ*s ∈Mh,−∫

Γ

[λs − tc,s]rβ2

· λ*sdΓ = 0 (2.58)

2.3.6 Blockwise diagonal discrete operators at the interface

Brittle cohesive laws exhibit sudden changes of tangent behaviour matrix, and as suchbelong to the family of non-smooth interface laws (as unilateral contact for exemple). Itis thus better, to prevent cyclic ip-op behaviour of a Newton-Raphson algorithm, tolimit the number of points where these changes may occur to the number Nλ of Lagrangedegrees of freedom, which leads to the discretization adopted below.

The components of unknown vectors u and µ are dened in a xed basis (eX , eY , eZ),while the components of unknown vectors w and λ are dened in the covariant basis(n, t, b) to the crack surface Γ at each point x ∈ Γ(see g.2.7), so that:

w(x) =

Nλ∑i=1

ψI(x) (wI,nn(x) + wI,tt(x) + wI,bb(x)) (2.59)

The analog of (2.59) holds for λ. At a Lagrange degree of freedom I ∈ 1..Nλ, it ispossible to compute cohesive components tIc,n, t

Ic,t, t

Ic,b from components (wI,n, wI,t, wI,b)

and (λI,n, λI,t, λI,b) with the aforementioned cohesive law. These cohesive componentsare not meant to be associated with particular directions around degree of freedom I ,but they are meant to be used in a weak sense. The discretisation procedure is as follows:

for (2.54), classical Gaussian integration is applied;

for (2.55), classical Gaussian integration is performed in the xed basis∀I ∈ 1..Nλ,

∫Γ ([uX ](x)−w(x) · eX)ψI(x) dΓ = 0, and similarly for directions

eY , eZ ;

for (2.56), a lump is applied to the cohesive traction only, so that, (2.56) becomesin the covariant basis ∀I ∈ 1..Nλ, tIc,n

∫Γ ψI(x)dΓ −

∫Γµ(x) · n(x)ψI(x)dΓ = 0

and similarly for directions t, b;

as for (2.57-2.58), which represents the interfacial law, it is lumped everywhere, so

that ∀I ∈ 1..Nλ,∫

Γ ψI(x)dΓ

r

(tIc,n − λI,n

)= 0 and similarly for directions t, b.

In this expression, the multiplying factor has been conserved to keep the tangentstiness matrix symmetric.


This highly non-linear set of equations is solved with a Newton-Raphson algorithm. Thetangent stiness matrix of the problem reads:

K :=

kuu (kµu)T 0 0

kµu 0 (−kwµ)T 0

0 −kwµ dww(dλw

)T0 0 dλw dλλ

(2.60)

where:

kuu is the bulk stiness matrix;

kµu and kwµ are classical mass matrices stemming from the discretisation of mortar operators, the latter also handling the change from xed to covariantbasis;

matrices d are all blockwise diagonal: for I and J distinct Lagrange degrees offreedom, it holds dIJ = 0.

Owing to (2.45-2.46) we have∂tc,s

∂ (λn + rwn)= β2

(∂tc,n

∂ (λs + rβ2ws)

)Tso that a symmet-

ric matrix derived from the cohesive law is introduced as:

kc :=

∂tc,n

∂ (λn + rwn)β2 ∂tc,n∂ (λs + rβ2ws)

∂tc,s∂ (λn + rwn)

β2 ∂tc,s∂ (λs + rβ2ws)

=

[kcnn kcnskcsn kcss

](2.61)

Diagonal blocs are then determined by:

dwwII = rkc (2.62)

dλwII =

[kcnn kcns

β−2kcns β−2kcss

](2.63)

dλλII =

kcnn − 1

rsym.

kcnsrβ2

β−2kcss − 1

rβ2

(2.64)

2.3.7 Numerical validation with the inclusion debonding test

The above three changes (mixed law, stable mortar formulation, blockwise diagonaloperators) may be applied independently from one another. To assess their individual


numerical eect, they were tested incrementally in intermediate formulations, which areall summarized on table 2.1.

The inclusion debonding is considered (g.2.27), with a load u0 = 0.2 mm being originallyapplied in 3 time steps. At a given step, if the Newton-Raphson algorithm fails toconverge, computation restarts with a load increment that is half its previous value.As an indicator for robustness, we take the total number of iterations needed to applythe full load: it is obtained by summing up the Newton iterations of all converged loadincrements, and reported on table 2.1 and g.2.32. For penalized laws, the smallerpenalization parameter which still ensures adequate adherence (non physical openingwas not observed) is α−1

0 = 104. This value was chosen in table 2.1, and only highervalues are tested on g.2.32. The total number of Newton iterations is then observed todecrease signicantly for each of the changes, illustrating the ability of each of them tobring additional robustness (see table 2.1 and g.2.32).

Classical formulationStable mortar formulationConsistent operators Blockwise diago-

nal operators

Penalized law tc([u]) tc(w) tc(w)

Newton iterations 729 311 147

Mixed law tc([u],λ) tc(w,λ) tc(w,λ)

Newton iterations 63 43 15

Table 2.1: Tested formulations and Newton iterations needed to solve the problem in3 time steps.

1E+3 1E+4 1E+5 1E+610

100

1000

10000penalized law tc([u])

stable penalized law tc(w) non-diagonal operators

stable penalized law tc(w) blockwise diagonal operators

mixted law tc([u],λ)

stable mixted law tc(w,λ) non-diagonal operators

stable mixted law tc(w,λ) blockwise diagonal operators

Penalization parameter (1/α0)

Tot

al n

umbe

r of

New

ton

itera

tions

Lower limit for an adequate resolution of adherence zones

Figure 2.32: Total number of Newton iterations to solve the problem in 3 timesteps.

Moreover, normal cohesive tractions have been plotted along the inclusion on g.2.33,for u0 = 0.04 mm. It can be deduced from this plot that:


the stable multiplier space, used along with a mixed law allows to remedy theaforementioned spurious oscillations issue;

the implementation of mixed cohesive law is checked, since its results coincide withthose of the linear-softening penalized law;

neither the stable mortar formulation, nor blockwise diagonal operators alterthe results, since they coincide with other laws.

-3,5 -2,5 -1,5 -0,5 0,5 1,5 2,5 3,5-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0Penalized law tc([u])

Stable penalized law tc(w) classical operators

Stable penalized law tc(w) diagonal operators

Mixed law tc(λ,[u])

Stable mixed law tc(λ,w) classical operators

Stable mixed law tc(λ,w) diagonal operators

Angle (rad)

No

rma

l co

he

siv

e tr

act

ion

(M

pa

)

Figure 2.33: Cohesive stress along the interface for intermediate formulations.

2.4 3D cohesive crack propagation with implicit crack ad-vance

2.4.1 Overview of the procedure

The procedure that we propose to study crack propagation is as follows (g.2.34) :

given a potential crack surface, the equilibrium state is computed (g.2.34b);

update of the crack front based on the computed cohesive state (g.2.34c);

determination of bifurcation angles along the front (g.2.34d);

update of the potential crack surface accordingly (g.2.34e);

correction of the cohesive internal variables (g.2.34a).

2.4.2 Update of the crack front

This is the main novelty of our propagation procedure. When used on undenite paths,the cohesive crack advance is generally determined beforehand. On the contrary, whenused on predenite paths, cohesive laws do not require a priori knowledge of the crack


Load step and

equilibrium

Update of the crack

front

Bifurcation angle

Update of the

potential crack

surface

Correction of the

cohesive state

Legend :

Traction-free zone

Cohesive zone

Adherent zone

Extended potential crack surface

a. b. c.

d.e.a.

Crack front

Figure 2.34: Overview of the procedure.

front: that information is naturally embbeded in the model, which is an advantage ofcohesive laws in comparison with LEFM. Hence, we would like to benet from this featurefor an undenite path as well.

We shall then set up a detection phase, in which the tangential level-set and conse-quently the crack front should be updated a posteriori, based on the new computedcohesive state and on the crack front from the previous load increment. A requirementfor that detection is that it should produce a smooth φt: it has to be suitable for fur-ther use by level-set update algorithms during the next propagation steps. To do so, arst rough front detection is performed using the cohesive internal variable state only(g.2.35): that eld is similar to a tangential level-set, but it is too rough for a straightuse in the level-set update procedure. A smooth crack advance eld will therefore berecreated a posteriori, from the former crack front (g.2.36a-b), and put into a level-setupdate algorithm which will produce a smoother tangential level-set (g.2.36c-d).

The detection process is decomposed as follows:

1. Rough crack front detection (g.2.35)

(a) The nodal eld of internal variables α is computed, which is allowed to benegative for the adherent part (g.2.35a).

(b) Elements which are intersected by the isozero of that eld are computed(g.2.35b).

(c) Intersection points of the rough crack front with the faces of the mesh arecomputed, as in [53], creating a cloud of points (g.2.35c).


b. computation of intersected elements

a. nodal values of the internal variables, negative values for undamaged material

c. intersection points with the mesh faces

Upcoming views : representation of the crack

surface intersecting the mesh

α≤0

α>0

Figure 2.35: Computation of a rough crack front

Reconstruction of a smooth tangential level-set (g.2.36)

2. (a) Construction of a crack advance eld at each point P of the previous front, asthe distance from P to the projection of the cloud of points in the propagationplane (P,nP , tP ) (g.2.36a).

(b) Smoothing of that advance eld as a scalar function (g.2.36b).

(c) Application of a level-set update algorithm (to the tangential level-set only)with that advance eld and the bifurcation angle computed at earlier steps(g.2.36c-d).

2.4.3 Bifurcation angle

To recall the results of section 2.2.3 briey, it has been proven that a J-integral Jext maystill be dened in the presence of cohesive forces. Provided its contour surrounds thecohesive zone (g.2.37), it is still independent of the contour. When used along with


t P

nPAP=∑

i

ω(d i) Ai

A

s

P

d i

Pt P

nP

APβP

a. reconstruction of a crack advance distance at each point of the previous front

b. smoothing of the crack front advance

c. deduction of the vector crack advance, bifurcation angle βP being available from the propagation stage of the previous regularized crack front

d. tangential level-set update (with the geometrical algorithm of [Colombo 2012])

Q

ϕn=0

ϕt =

0

P

M

Q

t P

nPtQ

nQ

βPA P

iAi

previous crack front

rough crack frontpoint cloud

Curvilinear abscissa along the previous front

Figure 2.36: Reconstruction of a smooth tangential level-set

LEFM asymptotic elds, it may be used to compute equivalent stress intensity factors,as in [2, 76]. These equivalent SIF quantify the apparent singularity of the far elds.

If a virtual crack extension eld θ is taken tangent to the cohesive zone, directed towardt on the front and so that |θ| = 1 (g.2.37), then it holds:

Jext = −Jcoh = −∫

Γtc ·∇[u] · θ dΓ (2.65)

From Irwin's formula, it comes that Jext = −Jcoh =1− ν2

E

(K2I,eq +K2

II ,eq

)+

1

2µK2

III ,eq .

Besides, the cohesive traction is decomposed as tc = tc,nn+ tc,tt+ tc,bb. Making use of

the notation [∇u] · θ =∂[u]n∂θ

n+∂[u]t∂θ

t+∂[u]b∂θ

b, the equivalent SIF may alternatively

be computed by:

K2I,eq = − E

1− ν2

∫Γ

∂[u]n∂θ

tc,ndΓ (2.66)

K2II ,eq = − E

1− ν2

∫Γ

∂[ut]

∂θtc,tdΓ (2.67)

K2III ,eq = −2µ

∫Γ

∂[ub]

∂θtc,bdΓ (2.68)


Γ

contourC

values of θ

cohesive zone

Figure 2.37: Dening invariant integrals in the presence of cohesive forces.

Physically, expressions (2.66-2.68) quantify the dissipated energy in the three fracturemode for a self-similar propagation of the crack in the direction θ, provided the cohesiveprocess zone is small compared to the sample size. There is therefore no need to constructcontours surronding the front and no need to compute auxiliary elds with this method.

The adopted bifurcation angle is then the classical maximum hoop stress criterion denedby Erdogan and Sih [14]:

β = 2 arctan

[1

4

(KI,eq/KII,eq − sign(KII,eq)

√(KI,eq/KII,eq)

2 + 8

)](2.69)

Further implementation details are available in AppendixD.

2.4.4 Update of the potential crack surface

At this stage, the potential crack surface should be extended from the crack front afterthe bifurcation angle determined at the previous step. In other terms, the cracked zone which contains the cohesive zone and the traction-free zone is kept unchanged whilethe adherent zone is forgotten and repositioned according to the bifurcation angle.In practice, this means applying a level-set update algorithm to the normal level-set (thetangential level-set having already been updated during the detection phase of section2.4.2).

General procedures rely on numerical shemes to solve Hamilton-Jacobi equations [48].However, simpler geometrical algorithms have emerged in the literature that take advan-tadge of the fact that the previous cracked surface is frozen, to such a point that theuse of such procedures is now prevailing for crack propagation problems. One of themwas adopted here, named geometrical level-set update by [8]. New level-set values arecomputed from the previous crack front and advance vector eld as follows:

For every node M of the mesh:


ϕn=0P

M

t P

nP

ϕn=0P

M

Q

t P

nP

βPA P

ϕn=0 ϕt =0

P

M

Q

tQnQ

a. b.

ϕn

ϕt

c.

Figure 2.38: Geometric update algorithm, as introduced by Colombo [8].

node M is projected onto the previous crack front (see g.2.38a) ;

its projection Q onto the new front is deduced from the advance vector eld (seeg.2.38b) ;

the updated normal (for this stage) or tangential level-sets (for the crack frontdetection of section 2.4.2) is directly computed, respectively as the normal andtangential components of vector QM . In the case where we had φt (M) ≤ 0before the update of the crack front, φn (M) is not updated, so that the previouscracked surface is kept unchanged (see g.2.38c).

Further implementation details are available in AppendixD.

2.4.5 Extension of the multiplier space and initial internal variables

As illustrated by g.2.39a-b, the update of the crack surface implies that it has turnedahead of the front. Consequently, the edges which are intersected by the new cracksurface are not the same as those which were intersected by the former crack surface,especially ahead of the crack front (see g.2.39a-b).

In g.2.39d, Q denotes the set of intersected edges by the new crack surface, V thesubset of edges with an equality relation, and K denotes the set of enriched nodes. Wedenote Q0, V0 and K0 these (dierent) sets if the former crack surface congurationis considered (g.2.39a). Hence, the construction of the set of equality relations V dening the new reduced multiplier space Mh non longer relies upon Q0: it has to beformed again starting from Q.

Now, a new V started from scratch would possibly lead to completely dierent com-binations of nodes sharing a Lagrange degree of freedom. As the internal variableswere dened on the older Mh , they would have to be projected onto a quite dierentspace, which would involve serious energetic issues. It is therefore wiser to start the newspace abiding by preexisting combinations behind the crack front, and extend it withnew groups on the formerly non intersected area, located ahead of the crack front (seeg.2.39a-c).

In practice, edges of Q0\V0 whose both vertices belong to K are removed from the set Qprovided as an input to the restriction algorithm (see g.2.39b). The restriction algorithm


by Géniaut et al [93] is then performed, thus naturally resulting in a V which preservespreexisting combinations (see g.2.39c).

The new restricted space involves nodes in K\K0 which had no attributed value forthe internal variables (white nodes on g.2.39c). For such a node n, an initial value isdetermined as follows (see g.2.39c-d):

if there exists an edge in V connecting n to a node m ∈ K0, the value of m isattributed to n;

otherwise, the initial internal variable is set to 0 (uncracked material).

There are also a few nodes in K0 whose value must still be changed, since the relatedconnecting edge has changed between V and V0 (grey nodes on g.2.39c). Once more iffor such a node, there exists an edge in V connecting it to a node m ∈ K0, the value ofm is attributed to n.


α1

α1

α2

α2

α2

α3

α3α4

α4

α5

α5≃0

α5

α6

α6≃0

α1

α1

α2

α2

α2

α3

α3α4

α4

(α5)

α5

α6

α1

α1

α2

α2

α2

α3

α3α4

α4

α4

α5

α6

α5

α6

0

0

a. previous crack configuration

intersected edges V 0 with an equality relation

other intersected edges Q 0∖V 0enriched nodes K 0

previous crack surface

crack front

new crack surface

crack front

set of intersected edges Qinitially provided to the restriction algorithm

enriched nodes K

intersected edges V with an equality relation

other intersected edges Q ∖V

enriched nodes K

new crack surface

nodes in K 0 with a value

nodes in K 0 whose value needs correction

nodes in K ∖K 0 without value

b. update of the crack surface

c. update of the reduced space d. update of the internal variables

Figure 2.39: Extension of the restricted multiplier space and update of the internalvariables.

2.5 Numerical tests

Several tests of increased complexity were carried out, which all have been numericallyand experimentally investigated in ealier literature. Whenever possible, these bench-marks have been carried out with the same data.


σc

wc

t c , eq

weq

wc3

σc3 Gc

Figure 2.40: Bilinear-softening law.

2.5.1 An extruded test : the L-shaped panel

Based on the experimental results by Wrinkler [15] on plain concrete, this L-shapedpanel test was numerically reproduced by [71, 77, 78]. The geometry and loading aresummarized on g.2.41. The documented material data of the experimented concrete aregiven in table 2.2, as well as the values reported by [71] as best representing the results,also used here for computation. Instead of linear softening, the bilinear-softening law ofg.2.40 is used, which is closer to the actual fracture properties of concrete.

Experimented concrete Used for computation (see [71])

Modulus of elasticity E = 25.85 GPa E = 25 GPa

Poisson's ratio ν = 0.18 ibid.

Tensile strength σc = 2.7 MPa σc = 2.5 MPa

Fracture energy Gc = 0.095 N.mm−1 Gc = 0.13 N.mm−1

Table 2.2: Material data for the plain concrete of the L-shaped panel.

f

u=0

250 250

250

250

100

30

POY

Z

X

Figure 2.41: L-shaped panel: geometry and loading.


As shown by g.2.42, the computed crack path appears to be within the experimentalrange. As for the load-deection curve, represented on g.2.43, the peak is rather accu-rately reproduced, but the computed post-peak behaviour exhibits a steeper slope thanthe experiments. This is probably due to the use of bilinear instead of exponential soft-ening (see [67]), the latter being closer to the actual softening behaviour of concrete [84].As expected from the values of table 2.2, the process zone is of the same order than thecharacteristic dimension of the specimen, as can be seen on the map of normal cohesivetractions on the crack surface (see g.2.45). The stress modulus map in represented ong.2.44.

Figure 2.42: Comparison of computed and experimental crack paths for the L-shaped panel.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,90

1

2

3

4

5

6

7

8

computed load deflection curve

experimental range

deflection (mm)

ap

plie

d fo

rce

(kN

)

Figure 2.43: Computed and experimental load-deection curves.


Figure 2.44: L-shaped panel: deformed shape and stress modulus.

Figure 2.45: L-shaped panel: prole of the normal cohesive traction on the cracksurface.

2.5.2 Three-point bending test with an initial skew crack

This test consists of an experimental and numerical study of the propagation of a fatiguecrack in PMMA (Plexiglas©) by [9, 10], and was used as a numerical benchmark by[8]. Geometry and loading conditions are summarized on g.2.46. The actual materialparameters are given in Table 2.3. Here, the applied load is monotonic instead of cyclic,and focus is made on an accurate prediction of the crack path. The typical length of

the cohesive zone is then expected to be lc =EGc

(1− ν2)σ2c

= 0.3mm. As such the process

zone is much smaller than a characteristic dimension of the sample. The response ofthe structure should tend to that given by LEFM, and a larger size of the process zonemay even be used for computation without signicant change of results (see table 2.3)as long as it remains small compared to the specimen size. Consequently, as the crackpath is controlled by the stress intensity factors ratio, it is expected to be identical formonotonic and cyclic loading: this is why our computed crack path in the monotoniccase should compare well to fatigue experiments for this specic case.


f

120 120

60

10

10

2045 °

xy

z

Figure 2.46: Three-point-bending test with an initial skew crack.

Physical values Used for computation

Modulus of elasticity E = 2.8 GPa ibid.

Poisson's ratio ν = 0.38 ibid.

Fracture energy Gc = 0.5 N.mm−1 ibid.

Tensile strength σc = 40 MPa σc = 15 MPa

Table 2.3: Material data for the PMMA sample

The computed crack path is in good accordance with the experiment, with an initialtwist of the crack path to recover mode I loading conditions (see gs.2.47-2.48): thequalitative evolution of the crack front is similar to that of the experiments. Fig.2.51shows the comparison of the positions of the endpoint of the crack in the cut planex = −5mm.

Figure 2.47: Perspective view of the crack surface and eld of cohesive tractions.


Figure 2.48: Top view of the crack surface.

Figure 2.49: Experimental crack path from [9, 10].

2.5.3 Brokenshire's torsion test

Again, a prismatic specimen with an initial skew crack is considered, and submitted totorsion (see g.2.52). Experiments (see gs.2.56 and 2.53) were carried out by Broken-shire [11], and the test was numerically reproduced by [47, 83], as a benchmark to testthe ability of several crack propagation algorithms to capture non-planar crack paths.Geometry and loading are represented on g.2.52. The material parameters of plainconcrete of table 2.4 and the bilinear-softening law of g.2.40 are adopted.

The experimental S-shaped crack path is accurately reproduced for this test, ascan be seen on g.2.54. Plotting the applied load as a function of the crack mouthopening displacement (CMOD) at the center top of the notch (see g.2.52), the peakis accurately reproduced while the post-peak behaviour appears to be too sti again(see g.2.53), which is still due to the use of bilinear instead of exponential softening (see[67]). Again, the size of the process zone is large compared to the size of the specimen, ascan be seen on the prole of normal cohesive tractions on the crack surface (see g.2.54).The stress modulus map is represented on g.2.55.


Figure 2.50: 3-point bending test with an initial skew crack: deformed shape andmap of stress modulus.

-6 -5 -4 -3 -2 -1 0 10

2

4

6

8

10

12

14

16

18

computed crack path

crack path from [Citarella, Buchholz, 2008]

Y coordinate (mm)

Z c

oo

rdin

ate

(m

m)

Figure 2.51: crack path in the cut plane (X=-5).

Modulus of elasticity E = 35 GPa

Poisson's ratio ν = 0.2

Fracture energy Gc = 0.0825 N.mm−1

Tensile strength σc = 2.3 MPa

Table 2.4: Material data for the plain concrete of Brokenshire's test.


frigid

rigid

100

25

25

50

100

100100

250

5

Figure 2.52: Geometry and loading for Brokenshire's torsion test.

0 0,1 0,2 0,3 0,4 0,5 0,60

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

computed load deflection curve

experimental curve 1

experimental curve 2

CMOD (mm)

ap

plie

d fo

rce

(kN

)

Figure 2.53: Load deection curves for Brokenshire's torsion test.


Figure 2.54: Shape of the computed crack path and map of normal cohesivetractions for Brokenshire's test.

Figure 2.55: Deformed shape and map of stress modulus for Brokenshire's test.

Figure 2.56: Experimental crack path from [11].


2.6 Outlook

This chapter was focused on stable crack propagation, in the quasi-static framework.If the required load to further propagate a crack decreases as crack grows, which is

energetically translated by∂G

∂A≥ 0, sudden propagation occurs. During this stage,

inertial eects have to be taken into consideration: the dynamic modelling of theseunstable crack growth stages will be the topic of chapter 3.

2.6.1 Toward the use of quadratic elements with the two-eld formu-lation by Lorentz [4]

In spite of their robustness and ease of implementation, the linear elements that wereused thus far have some shortcomings:

elds are quite rough: stresses are for instance constant elementwise;

due to the properties of the discrete multiplier space, the multiplier eld is quiterough: it has large piecewise constant zones;

for curved interface problems, such as the inclusion debonding test, the curvedinterface is approximated by a broken line.

Hence, improving the quality of the solution would mean either to rene the mesh, butthis is not always possible for the large structures, or to use higher order elements. Hence,we have considered the use of quadratic elements to improve the quality of the results.However, the use of quadratic elements in the X-FEM brings about new issues:

how should the curved geometry be approximated to make sure that quadraticelements are optimally used? This will be the topic of chapter 4;

is there a multiplier space, intended for use with quadratics exclusively, whichprovides a better description of interface elds, with no piecewise constant zonesfor instance? This issue is adressed in chapter 5.

There is also the question of the cohesive formulation. In spite of its good robustness,the formulation with linear elements of sections 2.3.5-2.3.6 has some drawbacks:

it is rather heavy, with three additional interface elds;

the interface law is quite poorly integrated: it is written once per group of nodesmaking up a multiplier DOF (see sections 2.3.3 and 2.3.6).

For quadratic elements, lighter robust formulations exist in the literature. Let us considerthe total energy (2.52), but this time, the surface energy Π(w) is a semi-dierentiablefunction of the consistant jump w:

E(u,w) =1

2


∫Γg

g · udΓg +

∫Γ

Π (w) dΓ (2.70)


As in section 2.3.5, (2.70) is minimized under the condition that w be equal to [u] in aweak sense. Hence, the Lagrangian of the problem is written and augmented by a term∫

Γr2 ([u]−w)2 dΓ over the discrepancy between the primal and consistent jumps:

L(u,w,λ) =1

2

∫Ω

σ(u) : ε(u)dΩ−∫

Γg

g · udΓg +

∫Γ

Π(w)dΓ +

∫Γ

λ · ([u]−w) dΓ +

∫Γ

r

2([u]−w)2 dΓ

(2.71)

The conditions for a stable discretization of this problem, as exposed in [96], are:

numerical stability of the discrete multiplier space Mh in the sense of the inf-supcondition,

that the trace application from V h into the denition space for w be surjective.

As discussed by [96], for linear elements this would imply using complex bubble functions,but for quadratic elements, there is an easy set of discrete space for the problem [4, 96]:

the displacement is quadratic;

the multiplier is piecewise linear;

the consistent jump w is collocated over the Gauss points of the interface.

This set is prone to extension to the X-FEM: it relies on P2/P1 elements for standardFEM (displacement is P2, multiplier P1), and this concept has a very natural extensionin the X-FEM, as will be detailed in chapter 5, with the denition of a multiplier space(P1*) with almost no piecewise constant parts.

For each Gauss point g along the interface, denoting xg its location, the optimality

condition∂L∂w

= 0 of Lagrangian (2.71) gives:

λ(xg) + r[u(xg)]− rw(xg) ∈ ∂Π (w(xg)) (2.72)

t c , n(wn)=∂π(wn)

wn

λn+r⟦u⟧n−r wn

w n(λn+r ⟦u⟧n)

λn+r⟦u⟧n

Figure 2.57: Local constitutive law giving w, in the case of pure mode I.

As shown by [4], in fact (2.72) amounts to a local constitutive law w(λ + r[u]) that iswritten over each integration point, and is deduced from an analytical determination of


the intersection point of the non-smooth cohesive law with a straight line, as representedon g.2.57. With this static condensation of w, the problem to be solved is simply:

∀u* ∈ V h,


∫Γg

g ·u*dΓg+

∫Γ

(λ+ r ([u]−w(λ+ r[u]))) · [u*]dΓ = 0

(2.73)

∀λ* ∈Mh,

∫Γ

([u]−w(λ+ r[u])) · λ*dΓ = 0 (2.74)

The suggested extension of this formulation to the X-FEM was implemented. For theinclusion test of g.2.27, the displacement elds along axis x and y are represented ongures 2.58-2.59 for quadratic elements, and compared to gure 2.60 for linear elements.The quality of the results is indeed improved through:

quadratic displacements (see g.2.58) and the quadratic description of the curvedinterface (to be detailed in chapter 4);

a richer integration of the interface law, and a richer multiplier space (to be detailedin chapter 5).

Figure 2.58: Inclusion test with quadratic elements: deformed shape and displace-ment along x.


Figure 2.59: Inclusion test with quadratic elements: deformed shape and displace-ment along y.

Figure 2.60: Inclusion test with linear elements: deformed shape and displacementalong x.

Chapter 3

Dynamic cohesive crack growth in

the extended nite element method


Avec ce chapitre, nous essayons d'étendre les méthodes et résultats du chapitre 2, quiconcernaient la propagation stable en modélisation quasi-statique, à la description des casde propagation instable, nécessitant une modélisation dynamique. En fait, la propagationbrutale est même considérée comme un phénomène de dynamique rapide, et est étudiéedans le cadre de la dynamique explicite. Nous commençons également par une étudebibliographique des méthodes existantes pour traiter de la propagation dynamique dessures avec X-FEM, soit dans le cadre des approches traditionnelles avec ssures libresde contraintes (section 3.1), soit dans celui des zones cohésives (section 3.2).

Par analogie avec le chapitre 2, nous cherchons à avoir une formulation qui permettel'introduction de la loi cohésive sur une large surface de ssuration potentielle: cetteformulation devra pouvoir décrire correctement de larges zones adhérentes et une tran-sition marquée avec la zone ouverte. Dans ce but, nous considérons une loi initialementrigide, qui est introduite de façon implicite dans le schéma des diérences centrées (sec-tion 3.3). Cette méthode a été proposée par Doyen [97] pour des interfaces conformes aumaillage; nous en proposons une extension à X-FEM. En utilisant alors un espace dédiépour discrétiser la contrainte cohésive qui n'est autre que l'espace de multiplicateurs duchapitre 2 et en diagonalisant certains opérateurs d'interface, on parvient à ramener ladétermination de la contrainte cohésive à la résolution analytique d'une série d'équationsscalaires (section 3.3).

L'implémentation est validée sur un test de type DCB trapézoïdale (section 3.4). Dans cetest, une ssure s'amorce et se propage brutalement avant de s'arrêter dans la structure(g.3.8): la longueur d'arrêt est la quantité d'intérêt. En réalisant, comme d'usage endynamique explicite, le calcul avec diérentes valeurs du pas de temps, on observe desrésultats sensiblement similaires (voir g.3.9). On en déduit que la méthode présentedes propriétés énergétiques acceptables, quoique perfectibles, et permet d'évaluer cor-rectement la distance d'arrêt. La détermination de la charge critique déclenchant lapropagation instable se fait par une méthode de pilotage du chargement proposée par

67

Chapter 3. Dynamic crack propagation with cohesive elements in the XFEM 68

Lorentz [7], qui est utilisée dans le cadre XFEM (section 3.5). A cette occasion, onpeut constater que la prédiction de l'arrêt de ssure nécessite une méthode dynamique:un calcul quasi-statique avec pilotage la sous-estimerait (g.3.11). Il s'agit d'une méth-ode permettant de suivre des instabilités lors d'un calcul quasi-statique, en déterminantl'intensité du chargement comme une nouvelle inconnue du problème, et en prescrivantune ouverture maximale comme une nouvelle équation. En d'autres termes, on pre-scrit une certaine dissipation maximale, et on en déduit l'intensité de chargement qui laproduit.

En perspective, la procédure d'étude de propagation sur trajet inconnu du chapitre 2 estadaptée pour la dynamique. Tandis que les premiers résultats pour décrire une propa-gation instable sont encourageants, la méthodologie semble plus dicilement utilisablepour décrire les chargements transitoires (chocs): le front apparaît moins marqué (zoned'élaboration diuse) ce qui rend plus dicile son actualisation. Par ailleurs, la taillede la zone cohésive ne peut plus être estimée à priori comme en quasi-statique, ce quicomplique le choix des bons paramètres de calcul.

3.1 Literature survey: traction-free cracks

In quasi-statics, the energy release rate was dened from the change of potential energywhen the cracked domain was changed by advancing the crack. By advancing thecrack , we mean that a transformation F (A) : x → x + Aθ (x) is applied to thecracked domain Ω, where θ is the virtual crack extension (see g.3.1), and we had then

Gstat = −dW (u(A),Ω(A))

dA|A=0.

Unstable crack propagation of a crack appears when the quasi-static energy release rate

is such that∂Gstat

∂A> 0. During this phase, the applied load is sucient for a sudden

propagation of a crack, until it arrests or breaks through the structure. Inertial eectsplay a key role during this transient stage, in particular through the interaction of thestress waves with the crack tip. Again, the most widespread tools for dynamic analysisof crack propagation is the linear elastic fracture mechanics (LEFM).

3.1.1 Dynamic equilibrium of cracked bodies

We consider an elastic body with density ρ, occupying a domain Ω ⊂ Rd whose boundaryis decomposed into Γu and Γg, where displacements are set to zero on Γu and surfaceforces g are applied on Γg (see g.3.1).

In dynamic analysis, displacement u and stresses σ depend on the physical time t inaddition to the crack advance A. Let us call:

Ekin =∫

Ω ekin dΩ the kinematic energy, with ekin = 1/2ρu2;

Wel :=∫

Ωwel dΩ the elastic energy, with wel = 1/2σ : ∇u;

Wext :=∫

Γgwext dΓ the work of external forces, with wext = u · g = u · σ · n.


During the crack propagation, the mechanical energy E of the system, not including thefracture energy, is written as:

E (t, A(t)) = Wel (t, A(t)) + Ekin (t, A(t))−Wext (t, A(t)) (3.1)

Indeed, the cracked domain Ω depends upon time as its external boundary ∂Ω\Γ moves,and through the crack advance A(t), which we have denoted as Ω(t, A(t)). As the solutionu of the problem involves Ω through the resolution of (3.2), it depends upon the samequantities. The equilibrium of the domain is characterized by div(σ) = ρu in addition to(2.2-2.6). Denoting by V :=

v ∈ H1(Ω),v|Γu = 0

, a weak formulation of the problem

is deduced by integrating by parts, as:

∀u* ∈ V ,∫

Ωρu · u*dΩ +


∫Γg

g · u*dΓg = 0 (3.2)

The material is assumed isotropic and homogeneous, and is loaded in the linear elasticelastic range. Waves then propagate at constant speed in the solid. Among the vawesthat it experiences are:

dilatational waves, with velocity cd =

√E(1− ν)

ρ(1 + ν)(1− 2ν)were E is Young's mod-

ulus and ν Poisson's ratio;

shear waves, with velocity cs =

√E

2ρ(1 + ν).

3.1.2 Dynamic energy release rate and invariant integrals [5, 6]

By choosing u as a virtual displacement in (3.2), we have (see [5])

∫Ω(t)

(wel + ekin) dΩ =∫Γg

wext dΓ. Hence, it comes:

∂Wel

∂t+∂Ekin

∂t= Pext (3.3)

This may be interpreted as a stationary condition of (3.1) with respect to time: whenthe crack surface does not evolve, the mechanical energy of the system is conserved.

Owing to the fact thatdWel

dt=∂Wel

∂AA+

∂Wel

∂t, a dissipation rate D naturally appears

with a total dierential of (3.1), as [5]:

D = −∂ (Wel + Ekin −Wext)

∂AA = Pext −

d (Wel + Ekin)

dt(3.4)

In order to calculate those total derivatives, the cracked domain is split into a domainΩC which travels as the crack propagates, and the remaining domain Ω\ΩC (see g.3.1).


C

values of θ

n

ΩC

Ω

Γu

Γg

g ( t)

Figure 3.1: : Notations for dening a dynamic energy release rate.

Reynold's transport theorem then states that :

d

dt

(∫Ω\ΩC

(wel + ekin) dΩ

)=

∫Ω\ΩC

(wel + ekin) dΩ−∫C

(wel + ekin) An · θ dΓ (3.5)

As for Pext , it holds:

Pext =

∫Γg

wext dΓ =

∫∂(Ω\ΩC)

wext dΓ +

∫Cwext dΓ (3.6)

By applying the divergence theorem and remarking that div (σ · u) = wel + ekin (see [5]),it nally comes:

Pext =

∫Ω\ΩC

(wel + ekin) dΩ +

∫Cn · σ · udΓ (3.7)

Reporting (3.5) and (3.7) into (3.4), we get:

D =

∫C

[(wel + ekin) An · θ + u · σ · n

]dΓ− d

dt

(∫ΩC

(wel + ekin) dΩ

)(3.8)

Due to its construction, D is independent of contour C. The rst term in (3.8) is naturallyinterpreted as an energy ux through the moving contour as the crack propagates (see[6]). As for the second term, it vanishes as C → 0. Hence, the dynamic energy releaserate is dened as (see [6]):


G = limC→0

1

A

∫C

[(wel + ekin) An · θ + u · σ · n

]dΓ (3.9)

3.1.2.1 Path-independant integrals under the steady state assumption

Let us introduce a steady state assumption as:

(H9) Near elds are supposed to depend upon t only through x− At = x ·θ− At.In particular, for close points to the crack tip, it holds u = − (∇u · θ) A.

If (H9) is satised, the second term in (3.8) cancels out by denition, so that the dynamicenergy release rate is:

G = Jstea =

∫C

[(wel + ecin)n · θ − θ · ∇uT · σ · n

]dΓ (3.10)

and that integral is independent of contour C. From (3.10), it is clear that G is expressedby a bilinear form, as G = g(u,u).

3.1.2.2 A general path-independent integral

Attigui et al [98] and Réthoré [5] (for the 3D case) also propose a path-independentintegral, valid even in the case of non-steady crack propagation (H9 is not veried):

J = Jstea −∫C

(2ekin + ρA u · ∇u · θ

)(n · θ) dΓ +

d

dt

(∫ΩC

ρu · ∇u · θ dΩ

)(3.11)

It is easily checked that in the steady state case, (3.11) comes down to (3.10).

3.1.3 Asymptotic elds for a dynamic analysis

Let us consider the following assumptions:



(H7) there are no volume or surface forces applied in the vicinity of the tip;

(H8) in 3D, crack tip points far from boundaries are considered.

Near eld symptotic analyses by Bui ([99], Section 4.5) for plane and antiplane conditionsshows that displacements and stresses exhibit dinstinct stress intensity factors, that weshall denote byKcin

I andKdynI . Hence, the intensity of the singularity of the displacement

jump is:


KcinI = lim

rc→0

µ

k + 1

√2π

rc[un(θc = π)] (3.12)

KcinII = lim

rc→0

µ

k + 1

√2π

rc[ut(θc = π)] (3.13)

KcinIII = lim

rc→0

µ

4

√2π

rc[ub(θc = π)] (3.14)

where k = 3 − 4ν is Kolosov's constant (in plane strains), µ is the shear modulus and(rc, θc) are the polar coordinates related to the crack tip (see gs.2.3 or 2.8). The asymp-totic discontinuity of the stress eld veries:

KdynI = lim

rc→0

√2πrcσnn(θc = 0) (3.15)

KdynII = lim

rc→0

√2πrcσnt(θc = 0) (3.16)

KdynIII = lim

rc→0

√2πrcσnb(θc = 0) (3.17)

Let us denote β1 :=

1−

(A

cd

)21/2

and β2 :=

1−

(A

cs

)21/2

. The dynamic and

kimematic stress intensity factors are related by universal functions:

KcinI = fI(A)Kdyn

I ,KcinII = fII(A)Kdyn

II ,KcinIII = fIII(A)Kdyn

III (3.18)

fI(A) =4β1(1− β2

2)

(k + 1)(

4β1β2 −(1 + β2

2

)2) (3.19)

fII (A) =4β2(1− β2

1)

(k + 1)(

4β1β2 −(1 + β2

2

)2) (3.20)

fIII (A) =1

β2(3.21)

Note, as illustrated by Freund [6], that the singular part of the asymptotic elds is stillvalid if (H9) is not satised. Indeed, transient terms due to a non-steady state only aect

higher-order expansions (rc, r32c ...).

3.1.4 Irwin's relation and interaction integrals

Under assumptions (H1-H2), (H7-H8), and by inserting asymptotic expressions (3.12-3.17) into (3.10) and considering the limit as C → 0, Réthoré [5] proves an analoguousexpression to Irwin's relation. In a 3D case, and far from boundaries, it holds:

G =1− ν2

E

(KcinI Kdyn

I +KcinII Kdyn

II

)+

1

2µKcin

III KdynIII (3.22)


Similarly to the quasi-static case, SIF are computed by interaction integrals using thebilinear form emanating from (3.10), as:

g(u,uSI ) =(1− ν2)

EfI(A)Kdyn

I (3.23)

3.1.5 Discretization with the X-FEM

The discrete displacement uh is classically enriched with a discontinuous Heaviside-likefunction across the crack surface, as:

uh(x) =∑i

U cli Ni(x)︸︷︷︸

classical part

+∑i

U enri Ni(x)H(x)︸︷︷︸enriched part

(3.24)

where:

H (x) =

−1 if φn(x) < 01 if φn(x) ≥ 0

(3.25)

The column vector collecting the components of uh in the nite element basis is denotedby U . As is discussed in the rst part of this thesis, the crack tip may be accounted byenrichment ending at crack edges [88, 100102], or singular enrichment for the elementcontaining the tip [59, 103, 104], or Heaviside enrichment ending within an element [105].As was proven by Réthoré et al. [106, 107], stability for the time-integration scheme whenpropagating the crack and thus modifying enrichment is ensured by initializing Un+1

as Un+1 =

Un

−000

. In other terms, all enriched degrees of freedom (DOF) of time

tn are kept at time tn+1 and the initial values for the new enriched DOF is set to 0[106, 107]. For asymptotic enrichment, this means keeping all asymptotic DOF fromthe start throughout computation, which could turn out cumbersome to implement, andexplains why several references still use Heaviside's enrichment alone.

3.1.6 Explicit time-stepping with central nite dierences

As crack velocities may have the same order or magnitude as waves in the solid, a largenumber of small time steps is needed for computation. Then, it seems natural to useexplicit time-integration schemes. Indeed, provided a diagonal approximation M of themass matrixM is used, they do not require the resolution of linear systems, which makesthe computational cost per time step much lower than for implicit schemes. In return,explicit schemes are subjected to a CFL stability condition ∆t ≤ ∆tc: the time stepshould be smaller than a critical value. However, for quick phenomena such as ours, thisstill makes the total number of time steps reasonable.


The time interval is subdivided into segments of equal length ∆t, and we denote tn :=n∆t. The time discretization of U(t) at time tn is denoted by Un. The central nitedierences scheme is adopted, which consists of dening time-discretized velocity and

acceleration by Un =Un+1 − Un−1

2∆tand Un =

Un+1 − 2Un + Un−1

∆t2.

For free cracks, at each step of the central nite dierence scheme,(Un−1, Un

)being

considered as input data, Un+1 is computed by:

MUn+1 + Un−1 − 2Un

∆t2= Fext (tn)−KUn (3.26)

where K is the stiness matrix.

The critical time step for this scheme is ∆tc = 2ω−1max where ωmax is the largest eigen-

value of the system, that is ωmax = maxω : det

(K − ω2M

)= 0. On classical (non-

enriched) nite elements, ∆tFEMc =hmin

cdis taken in practice, where hmin is the length

of the smallest edge of the mesh. However, in the X-FEM, ∆tc strongly depends uponthe choice that is made for M .

3.1.7 X-FEM mass matrices

In the rst use of X-FEM by [105], authors report trouble nding an ecient M in theX-FEM, which led them, in a communication [100] released shortly after, to use theconsistent mass matrix instead, as:

M cl ,cli,j =

∫Ω ρNi(x)Nj(x)dΩ

M cl ,enri,j =

∫Ω ρH(x)Ni(x)Nj(x)dΩ

M enr ,enri,j =

∫Ω ρH

2(x)Ni(x)Nj(x)dΩ

(3.27)

Indeed, trying to use a straight row-sum lump of (3.27), Belytschko et al. [105] observed adramatic decrease of the critical time step when the crack cuts the mesh in the vicinity ofa node. The eect is also pointed out by [88, 101]. In addition to the use of the consistentmass matrix in [88, 100], the issue was rst adressed with implicit time-integration forenriched elements [105], or an algorithm by which mesh nodes avoid getting close to thecohesive surface [88, 101].

By denoting Ω = ∪eΩe the decomposition of the domain into nite elements, the mass

matrix is assembled from elementary contributions as M =∑eMe. Menouillard et al.

[102] proposed an alternative mass matrix as:

(Me)enr ,enri,i =

ρ

ne

∫Ωe

H2dΩ (3.28)

for each node i belonging to Ωe, and with ne the number of such nodes and ρ the mass

density of the solid. Authors proved a 1D theoretical critical time step ∆tc =∆tFEMc√

2


node i

crack

Value of Value ofH i H i

Figure 3.2: Enrichment basis of Hansbo and Hansbo [12].

and noticed on 2D and 3D computation that ∆tc =∆tFEMc

2would always yield stable

results. Gravouil et al. [103] proposed another lumping strategy and tested it to otherenrichments H than Heaviside's with:

(Me)enr ,enri,i =

1ne∑j=1

H2(xj)

∫Ωe

ρH2dΩ (3.29)

with xj the position of node j. By studying comprehensive intersection conguration,they bring analytical evidence that in any case, the critical time step is bounded by halfthe FEM value:

∆tXFEMc =

∆tFEMc

2(3.30)

In Menouillard et al. [62], the authors adopted an enrichment strategy by Hansbo andHansbo [12]:

uh =∑i

U cli Ni(x)Hi(x)︸︷︷︸

classical part

+∑i

U enri Ni(x)Hi(x)︸︷︷︸enriched part

(3.31)

where functions Hi(x) = sign(φn(x)) + sign(φi) and Hi(x) = sign(φn(x))− sign(φi) arerepresented on g.3.2.

Let us consider a triangle with cut ratio ε (see g.3.3). Menouillard et al. [62] proposeda diagonal mass matrix in Hansbo and Hansbo basis (3.31), with classical and enrichedpart given by:

(Me)cl ,cl(1,2,3) =

ρ|Ωe|ne

ε 0 00 1− ε 00 0 1− ε

(3.32)

(Me)enr ,enr(1,2,3) =

ρ|Ωe|ne

1− ε 0 00 ε 00 0 ε

(3.33)


ϵ∣Ωe∣

(1−ϵ)∣Ωe∣

1

2

3

Figure 3.3: Cut triangle.

Authors proved that the critical time step is ∆tc = ∆tFEMc for constant strain elements,and that ∆tc = 0.76∆tFEMc is a lower bound that ensures stability for other elements.

3.1.8 Crack increment

A theoretical speed limit for mode I crack propagation is the Rayleigh wave speed cR (thevelocity of surface waves). Kanninen and Popelar [108] thus bring theoretical evidencethat the crack speed may be related to stress intensity factors. For instance in pure modeI (analoguous expressions exist for other modes), the analytical formulas that relate themare well approximated by:

A =

(1− KIc

KdI

)cR (3.34)

Should tenacity KIc be assumed a constant value, this expression may stand up toqualitative comparison with experimental data only at low speeds. Indeed, Ravi-Chandarand Knauss [109, 110] have shown experimental limitation to (3.34) on a series of testson Homalite-100 sheets. The observed speed hardly ever reaches 0.6cR in practice [110].It was observed, for such fast propagating cracks, that a large number of microcracksappear in a diuse zone ahead of the tip, thus slowing down the macro-cracking process [109]. In some cases, this may eventually lead to crack slowing down by branchinginto two macro-cracks [109]. In almost all brittle materials, dynamic cracking is alwaysaccompagnied by diuse cracking patterns and wavy crack proles [6]. Thus, (3.34) hasto be understood with an apparent tenacity KIc which depends upon A, to such a pointthat crack velocities exceeding half of the Rayleigh wave speed are rarely observed inpractice.

3.2 Literature survey: cohesive zone models

3.2.1 Cohesive zone models to model rate eects

The rst use of cohesive zone models in a dynamic framework by [111, 112] is actuallyintended to numerically reproduce the eect of such diuse micro-cracks on the overall


response. Cohesive surfaces are introduced at some element boundaries. If an additionallengthscale is provided to the model, as the typical distance between two cohesive surfaces[113], results happen to be insensitive to the mesh size. Multi-microcracking [111, 113] and the subsequent limited speed as well as crack branching situations [112, 113] couldbe reproduced. However, [113] points out very dierent results for initially elastic [112]or rigid laws [111]. Moreover, it explains, with a simple dimensional analysis, that withinitially elastic laws it is not possible to resolve the cohesive zone without aecting thewave speeds of the solid .

Rather than introducing cohesive surfaces at the meso-scale, [114, 115] have suggested toaccount for rate eects with a more phenomenological opening-rate-dependent cohesivelaw, introduced on the main crack path only.

3.2.2 Numerical implementation

To handle initially rigid laws, Camacho and Ortiz [111] proposed an implementationby doubling the nodes on cohesive surfaces. Initial adherence is accounted for in thefollowing way:

opposite nodes are initially tied, and split up according to the cohesive law whenreaching a critical traction: initial tractions are computed from nodal forces, andcohesive tractions are then determined by a return mapping algorithm;

contact, when it occurs, is enforced afterhand, by assuming inelastic collision be-tween interpenetrating nodes;

a regularized tangential law is used once the cohesive surface has opened in traction.

Similar methodologies have been applied to the X-FEM by Beltytschko et al. [105] orde Borst et al. [88, 101]: cohesive zones are inserted iteratively as hyperbolicity is lost[105] or a critical traction is reached [88, 101]. Cohesive tractions are still computed by areturn mapping algorithm: if it produces a negative jump, the cohesive surface at stakeis set inactive but its position is kept and interfacial stresses are computed by theequations of motion.

A more general way of inserting any set-valued interface law into explicit time-integrationschemes is brought by Doyen et al. [97] in a standard procedure that adresses all threeaforementioned issues at once. Interface forces are implicit and analytically determined,while nodes outside cohesive surfaces are explicitely treated. While [97] concentrates onpredenite crack paths, this chapter aims at adapting that work to unpredicted crackpaths, by extending it to the X-FEM.

3.3 Semi-explicit initially rigid cohesive law in the X-FEM

3.3.1 The initially ridig cohesive law

Let us call λ the cohesive traction at the interface. The set-valued linear-softeningcohesive law from g.3.4 is considered.


σc

wc

λn

⟦u⟧n

Opening mode

σc

wc

−w c

λs

⟦u⟧s

Shear mode−σc

αwc

Figure 3.4: Set-valued linear-softening mixed-mode cohesive law.

Again using the cohesive law of section 2.3.4, and with a strength ratio β = 1, mode-

coupling in ensured by an equivalent jump [u]eq :=√〈[u]n〉2+ + [us]2, itself dening a

dimensionless scalar internal variable α, as:

ϕ ([u]eq , α) :=[u]eqwc− α ≤ 0 (3.35)

α ≥ 0 (3.36)

αϕ ([u]eq , α) = 0 (3.37)

It holds α ∈ [0, 1] where α = 0 is perfect adherence and α = 1 is full debonding. Interms of primal potential, for loading conditions (that is, if α = [u]eq/wc) the associatedenergy may be written:

Π ([u]) =

∞ if [u]n < 02Gcα

(1− α

2

)if [u]n ≥ 0 and [u]eq ≤ wc

Gc if [u]n ≥ 0 and [u]eq > wc

(3.38)

Cohesive tractions are then represented as a set valued map (see g.3.4), as λ ∈K whereK := ∂Π ([u]) is the subdierential of Π at point [u]. Denoting by x a singleton xfor simplicity, this states that the normal traction belongs to (see the opening mode ofg.3.4):

λn ∈

]−∞, σc] if α = 0

σc (1− α)[u]nαwc

if α > 0 and [u]n > 0

]−∞, 0] if α > 0 and [u]n = 0

(3.39)

In 2D, the shear component belongs to (see the shear mode of g.3.4):

λs ∈

[−σc, σc] if α = 0

σc (1− α)[u]sαwc

if α > 0(3.40)

In 3D, it belongs to:


λs ∈

D(0, σc) if α = 0

σc (1− α)[u]sαwc

if α > 0(3.41)

where D(0, σc) is the disk of radius σc centered on the origin.

3.3.2 Semi-discretization in space

The discrete interface traction is denoted by λh. As it is a set-valued map of uh, it is implicit it plays the role of a Lagrange multiplier in the adherent parts of the law.As such, numerical stability requires λh to be dened over a reduced space Mh, as wasextensively discussed in section 2.3.3. Briey, multiplier components are initially denedon the enriched nodes, and the space is then reduced by prescribing equality relationsbetween those components, so that a Lagrange degree of freedom I (always denoted bycapital letters) is eventually shared by several nodes i ∈ I (see g.3.5), thus producinga non-local shape function ψI :=

∑i∈INi (see g.3.5). Let us call Λ the associated column

vector to λh in that reduced space nite element basis.

Group of nodes sharing the same Lagrange degree of freedom

Support of the non-local shape function ψ3

Potential crack surface

Intersected edge supporting an equality relation

Other intersected edge

Figure 3.5: Reduced space for the denition of discrete tractions.

3.3.2.1 Toward dening a consistent jump

The discrete interfacial traction then obeys λh ∈ Kh where Kh ⊂ Mh is the discreteset for K. Dening it componentwise would be natural: inclusion tests λh ∈Kh wouldthen amount to a set of independent inequations, which is more easily included into aresolution procedure. However, while K is expressed as a function of [u], and uh hascomponents in V h, the components of λh are dened over Mh.

A componentwise denition of Kh thus requires that a consistent jump wh ∈Mh

be built from [uh]. With W the column vector of components of wh, the discrete convexset Kh would then be dened componentwise by:

Kh := λh, ∀I, ΛI ∈ ∂Π(WI) (3.42)

Let us then take as wh the projection of [uh] onto Mh, which is most natural. Bydenition, it is dened in a unique manner by wh ∈Mh and [uh] −wh ∈M⊥

h , that isto say ∀µh ∈Mh,

∫Γµh · ([uh]−wh) dΓ = 0. This is matricially translated by:

BU −HW = 0 (3.43)


where BenrI,j =

∫Γ 2ψINjdΓ, Bcl

I,j = 0 and HI,J =∫

Γ ψIψJdΓ.

3.3.2.2 First lumping step: turning H into a diagonal H

Now, keeping in mind that we are going to use explicit time-stepping, an explicit expres-sion should be available for W , hence the need for a lumped, easily invertible H:

HI,J = 0 if J 6= I

hI := HI,I =∫

Γ ψIdΓ(3.44)

Expressing W componentwise, at this intermediate stage, (3.43-3.44) gives:

∀I, WI =

∫Γ[uh]ψIdΓ∫

Γ ψIdΓ(3.45)

3.3.2.3 Second lumping step: turning B into a diagonal B

As will be apparent in section 3.3.3, an explicit expression of Un+1 also requires a block-wise diagonal B, expressed as:

BenrI,j = 0 if j /∈ I

BenrI,j =

∫Γ 2NjdΓ if j ∈ I

(3.46)

Owing to these expressions, W is nally expressed componentwise as a weighted averageWI = 2

∑i∈IαiU

enri of the enriched components of U belonging to group I, where:

αi =

∫ΓNidΓ∫Γ ψIdΓ

(3.47)

which given that ψI =∑i∈INi ensures that

∑i∈Iαi = 1.

3.3.2.4 Adopted lumped mass matrix

As is standard, classical components of the lumped mass matrix are computed by:

M cl ,cli,i =

∫ΩρNidΩ (3.48)

As for the enriched part, the lumped matrix (3.28) by Menouillard [102] is adopted. In

the case of simplicial linear elements, since we have ∀i,∫

ΩeNidΩ =

|Ωe|ne

, the lumped

mass matrix reduces to:


mi := M enr ,enri,i =

∫ΩρNiH

2dΩ (3.49)

In our case, since H2 = 1, we simply have mi = M enr ,enri,i = M cl ,cl

i,i .

3.3.2.5 Semi-discrete problem in space

The semi-discrete problem in space then reads:

MU(t) +KU(t) + BTΛ(t) = Fext(t) (3.50)

with ∀I, ΛI ∈ ∂Π(WI) where WI is given by (3.47) through WI = 2∑i∈IαiU

enri .

3.3.3 Time discretization

As this cohesive law enforces perfect initial adherence, it is an essential condition onthe eld of unknowns. As such, it can only be applied implicitely, and the central nitedierence scheme becomes in the presence of cohesive forces:

MUn+1 + Un−1 − 2Un

∆t2+KUn + BTΛn+1 = Fext(tn) (3.51)

with ∀I, Λn+1I ∈ ∂Π(Wn+1

I ).

For each enriched node i, this becomes:

mi

∆t2(Un+1)enri + (Fn)enri + (BT )enri,I Λn+1

I = 0 (3.52)

where Fn := MUn−1 − 2Un

∆t2+KUn − Fext(tn).

Multiplying (3.52) by 2αim−1i , and given that (BT )enri,I = 2hIαi after (3.44, 3.46-3.47),

one gets:

2αi∆t2

(Un+1)enri + 2αim−1i (Fn)enri + 4hIα

2i m−1i Λn+1

I = 0 (3.53)

A sum over i ∈ I gives1

∆t2Wn+1I + 2

∑i∈Iαim

−1i (Fn)enri + 4hI

(∑i∈Iα2i m−1i

)Λn+1I = 0.

Hence, the consistent jump and interfacial tractions may be determined analytically,solving the equation:

Λn+1I = ∂Π(Wn+1

I ) = −AWn+1I +B (3.54)


with A :=1

4∆t2hI


) and B := −

∑i∈Iαim

−1i (Fn)enri

2hI


) .

Indeed, solving (3.54) consists of computing the intersection of the cohesive law with aline, as shown on g.3.6.

c

wc

λn

wn

opening mode

−Awn+B

∂Π(wn)

W I , n

Λ I , n

Figure 3.6: Analytical detemination of cohesive tractions.

3.3.4 Detail of the analytical determination of cohesive stresses

Denoting by subscripts n and s normal and tangential components of interface quantities,from (3.39), it is clear that 〈wn〉- = 0. Combining (3.54) with the expression (3.39) forthe cohesive stress, considering cases when α > 0, and taking the positive part of it, itcomes:

σc(1− α)wnwcα

= −Awn + 〈Bn〉+ (3.55)

Note that if α = 0, then wn = 0 after (3.35). Likewise, combining (3.54) with (3.40), itholds:

σc(1− α)ws

wcα= −Aws +Bs (3.56)

With (3.55,3.56), and classically introducing Beq :=√〈Bn〉2+ +B2

s, we may deduce that:

[σc(1− α)

wcα+A

]weq = Beq (3.57)


Loading conditions apply when the predicted value ofweqwc

exceeds the value of α at theprevious time step. In this case, α is updated. If this condition is written replacing weqby its expression (3.57), one gets:

ifBeq − σcAwc − σc

> α, then α is updated asBeq − σcAwc − σc

← α;

otherwise, α is not updated.

Note that if a strength ratio β 6= 1 had been chosen at this stage, this would impairedfactorization (3.57) and made the determination of the internal variable quite more com-plex. Assuming α to have been updated if necessary from now on, we have from (3.55)that:

wn =〈Bn〉+

σc(1−α)wcα

+A(3.58)

ws =Bs

σc(1−α)wcα

+A(3.59)

Combining (3.54) with (3.39) and taking the negative part of it, we have 〈λn〉− = 〈Bn〉−.Using this and inserting (3.58) into (3.39), it comes:

λn =σc(1− α)〈Bn〉+σc(1− α) +Awcα

+ 〈Bn〉− (3.60)

Likewise, inserting (3.59) into (3.40), it comes:

λs =σc(1− α)Bs

σc(1− α) +Awcα(3.61)

For loading conditions, (3.58-3.61) may be simplied. Owing to (3.57), (3.58-3.59) be-come:

wn =〈Bn〉+Beq

αwc (3.62)

ws =Bs

Beqαwc (3.63)

Still due to (3.57), for loading conditions, (3.60-3.61) become:

λn = σc(1− α)〈Bn〉+Beq

+ 〈Bn〉− (3.64)

λs = σc(1− α)Bs

Beq(3.65)


3.4 Numerical test

In this test, the ability of the procedure to describe unstable crack growth and arrestunder a quasi-static load is tested. Double cantilever beam (DCB) samples are oftenused for experiments on metals [116]. When loaded in force, a classical DCB exhibitsunstable crack growth (see [117]), while tapered trapezoidal DCB (TDCB) mayhave continuous stable crack growth [117]. Here, an intermediate geometry is tested,with a rectilinear then trapezoidal sample (see g.3.7), so that sudden crack growth willoccur from the pre-crack and be followed by arrest (see g.3.11). The critical load whichtriggers unstable crack growth is determined by a quasi-static analysis to be developed insection 3.5 (see g.3.11). That load is applied and assumed constant during the dynamicpropagation stage. The quasi-static result for this load is provided as an initial state todynamic computation, all initial velocities being set to zero. Material properties from[97] are adopted (see table 3.1).

Modulus of elasticity E = 200 GPa

Poisson's ratio ν = 0.3

Density ρ = 7800 Kg.m−3

Tensile strength σc = 1000. MPa

Fracture energy Gc = 16 N.mm−1

Table 3.1: Material data for the TDCB test.

ux=0

F

−F

15

10

10

15

100

90

P

x

y

2

Figure 3.7: Numerical test: geometry and loading.

Displacement along y at point P (g.3.7) has been represented on g.3.9 as a function oftime: sudden crack growth followed by arrest is indeed observed, with small vibrationsof the structure after arrest. Now, for classical central dierences, the optimal timeincrement ∆t in terms of accuracy is the critical value ∆tc: energy losses occur forsmaller time steps. How our central-dierences-based scheme performs energetically maytherefore be assessed by comparing solutions between a time step close to ∆tc (which is2.10−8s for this test) and smaller steps. Hence, three dierent time steps were tested (seeg.3.9): while energy losses are observed for smaller steps, they remain small enough tokeep a very acceptable prediction of crack arrest. Stress component σyy is represented


for various time steps on g.3.8, and g.3.10 shows a zoom at an early stage of crackpropagation, where it can be seen that the interface does not cut the mesh along edges.

Figure 3.8: Stress component σyy and deformed shape at dierent times.


0,0E+0 5,0E-5 1,0E-4 1,5E-4 2,0E-4 2,5E-40,0E+0

2,0E-1

4,0E-1

6,0E-1

8,0E-1

1,0E+0

1,2E+0

1,4E+0

1,6E+0

1,8E+0

2,0E+0Δt = 2.E-8 s

Δt = 1.5E-8 s

Δt = 1.E-8 s

Time (s)

Dis

plac

emen

t alo

ng y

at p

oint

P (m

m)

Figure 3.9: Top displacement as a function of time, for dierent time steps.

Figure 3.10: Zoom of the intersected mesh and stress component σyy at t=4µs.

3.5 Determining the critical load with a path-following methodby Lorentz [7]

Following the previous dynamic analysis, two questions araise:

Could the dynamic arrest prediction be obtained by a quasi-static analysis?

How to determine the critical load in practice?

About the rst question, the load-deection curve of the TDCB test, obtained with thequasi-static method to be presented next, was computed on g.3.11. It can be seen thatthe crack arrest is severely misevaluated in the quasi-static approach (g.3.11), whichmakes dynamic computation compulsory: it appears that a physical prediction of thecrack arrest requires inertial eects to be taken into account.


0,0E+0 2,0E-1 4,0E-1 6,0E-1 8,0E-1 1,0E+01,2E+01,4E+01,6E+01,8E+02,0E+00

200

400

600

800

1000

1200

Displacement along y at point P (mm)

Forc

e F

(N)

Quasi-static prediction for crack arrest

Dynamic prediction for crack arrest

Figure 3.11: Load-deection curve obtained with a path-following method and crackarrest.

About the second question, one could argue that the critical load may be obtained byloading the structure with a prescribed displacement instead of a force. However, this isnot always possible, in particular when:

a distributed force is applied, as is the case here;

the response to a prescribed displacement is still unstable. In this case, the responseis said to exhibit snap-backs. An example with X-FEM can be found in [118].

Hence, we appeal to path-following methods, for which structure unstabilities can befollowed in the quasi-static framework (see Riks [119]). With a radial externalload, that is to say such that Fext(t) = σN (t)F , it consists in imposing that a scalardimentionless quantity f(U), which we take as representative for crack growth, be equalto the virtual time increment ∆τ . Then, we deduce the load intensity σN that actuallyproduces this prescribed crack advance. Practically, a new unknown σN and equationf(U) = ∆τ have been added to the system, so that instead of solving Fint(U)−Fext = 0 with Fint(U) = −KU −BTΛ(U) we solve:

Fint(U) = σN F (3.66)

and:f (U(σN )) = ∆τ (3.67)

A robust choice for f should follow the crack front by prescribing an opening incrementin the current process zone. As a result, function f should be dened as a maximalopening increment rather than an average one in a xed zone. The elastic predictorpath-following method by Lorentz [7] thus controls the maximal overtaking of ϕ, thedamage function dened by (3.35), for the current load increment. Indexing Gauss


points along the interface by g, and calling xg their positions, function f for computingthe virtual time τn is dened as:

f(U) = maxgϕ([un(σN ,xg)]eq , α

n−1)

(3.68)

With Fint a non-linear function of U due to cohesive forces, equilibrium (3.66) is computedby means of a Newton Raphson procedure. With KT the global tangent stiness matrix,the displacement is incremented at each iteration by δU = σNK

−1T F − K−1

T Fint(U).Then, it is clear that the displacement is an ane function of σN , as U

n(σN ) = Un−1 +∆U + δUint + σNδUext where:

Un−1 is the displacement at the previous time step;

∆U is the increment predicted by previous Newton-Raphson iterations;

δUint := −K−1T Fint(U

n−1 + ∆U);

δUext := K−1T F .

Owing to (3.35), for a given Gauss point it is clear that σN → ϕ([un(σN ,xg)]eq , α

n−1)is a

quadratic function. A qualitative representation of such functions ϕ(g) for dierent Gausspoints is plotted on g.3.12, as well as the solutions of the global maximization problem(3.67-3.68). Then, (3.67-3.68) amounts to nding the intersection points of the maximumfunction of a set of parabolas with a xed value (see g.3.12). It's then apparent that

these points can equivalently be obtained by a piecewise ane problem maxgA

(g)1 σN +

A(g)0 = ∆τ where A

(g)1 σN + A

(g)0 is the tangent curve to ϕ(g) at its intersection points

σ(g)N with ∆τ (see g.3.12). The solutions of (3.67-3.68) are easily found as the maximal

σ(g)N with a negative A

(g)1 and the minimal σ

(g)N with nonnegative A

(g)1 (see g.3.12). The

chosen solution is the one which minimizes the Newton-Raphson residual.


Figure 3.12: The path-following method by Lorentz [7].

φ(g)

σN

g1

g2

g3

g4

Δτ

solutions of the global problem

σN(2) , A1

(2) < 0

max σN(g) , A1

(g) < 0

g

min σN(g) , A1

(g) > 0

g

max φ(g)

g

φ(1)

φ(3)

φ(4)

φ(2)

σN(2) , A1

(2) > 0

3.6 Outlook: toward non-planar dynamic crack propagation

To conclude with this chapter, we are now able to insert an initially rigid cohesive lawover large zones. A natural prospect, and the ultimate goal of this chapter, is to pro-pose a procedure for modelling dynamic crack propagation over non-predenite crackpaths. Of course, the procedure from section 2.4 may straight be adapted to the dy-namic framework, by achieving the "computation of equilibrium" stage with the explicitdynamics formulation of section 3.3 (and this is pretty much the only thing that needsto be changed). We did implement a mock-up of such a program, and began to run itfor unstable crack growth problem, both with quasi-static loads (see gs.3.13-3.15) andtransient loads (shocks). Sadly, as we go to press, it is not so far advanced as to produceextensive numerical results standing comparison with experiments on common bench-marks. Nevertheless, it seems interesting to share some early feedback and meaningfulidentied trends. First, the procedure seems to behaves rather properly for quasi-staticloads. Figures 3.13 and 3.15 thus show stress components σzz and σzy, for an asymetricDCB test: while pulling on both arms with equal opposite forces, an arm being widerthan the other causes the crack to tilt. The only hitch is the size of the cohesive zone,which is sometimes observed to dier signicantly from its predicted value by:

lc = MGcE

σ2c (1− ν2)

(3.69)


Indeed, (3.69) is known to be valid for quasi-statics only [69]. This means that additionalmonitoring of the mesh size may be necessary from the user. Otherwise, the cohesive zoneis not accurately resolved, which results in a misevuation of the stress intensity factors(2.66-2.68) in the event of a small cohesive zone, and thus leads to a wrong directionalcriterion (2.69).

Figure 3.13: Deformed shaped and stress component σyy.

Figure 3.14: Deformed shaped and stress modulus.

For transient loads, things get harder, as the issue of the cohesive-zone size is made evenworse. Besides, over large zones, the pattern of the cohesive internal variable does notalways exhibit a sharp transition to be identied as the updated crack front: as statedbefore (section 3.1.8), the crack pattern is known to be diuse in the elaboration zone.Hence, we are sceptical about the applicability of the "a-posteriori crack front update"strategy (section 2.4) for fast transient load problems. A solution would be to switch toKanninen's criterion instead [108], and to introduce a surface of discontinuity only overthe area that is thus expected to open. Another prospect would be considering the stressstate ahead of the front instead of a criterion relying on SIFs, as the latter suers fromtoo small process zone sizes.


Figure 3.15: Deformed shaped and stress component σyz.

Chapter 4

Convergence analysis of linear or

quadratic X-FEM for curved free

boundaries

This chapter has been published in Computer Methods in Applied Mechanics and Engi-neering [120]


Dans ce chapitre, nous étudions la convergence de la méthode X-FEM pour décrire desinterfaces libres (ssures, trous), en particulier l'inuence de la quadrature. Ceci a étéétudié numériquement par de nombreux auteurs [51, 121123] pour diérentes méthodesde quadrature, mais jamais théoriquement. Les éléments intersectés sont divisés encellules d'intégration selon la position de l'interface (voir g.4.3). L'interface est alorsapproximée par des bords de ces cellules. Cette procédure de découpe est détailléed'après ce qui est fait par [121], et l'approximation de l'interface qui en résulte est décrite(section 4.4). L'erreur géométrique commise par ce procédé est dénie par une mesureε qui quantie l'écart maximal entre interfaces exactes et approximées, puis évaluée parune analyse de convergence. Pour une description de la géométrie d'ordre g (divisionen triangles linéaires pour g = 1 et quadratiques pour g = 2), l'erreur géométriqueest ε ≤ Chg+1. Ce résultat théorique est validé par des expériences numériques pourdiérentes formes d'interfaces.

Une analyse de convergence du problème est ensuite menée (section 4.5). Nous montrons(proposition 4.7) que l'erreur sur la solution se décompose en une erreur de consistancefonction de la description de la géométrie (donc de ε), une erreur de consistance fonctiondu choix du schéma d'intégration dans les cellules, et de l'erreur d'interpolation duchamp de déplacement. An d'évaluer le schéma d'intégration nécessaire pour avoirl'optimalité, nous classons les cellules d'intégration suivant leur régularité (voir g.4.12).L'analyse montre alors (proposition 4.14) que le schéma d'intégration doit être enrichi,de façon à inclure des quantités géométriques dues à la courbure, à mesure que la celluledevient plus distordue . Pour dire les choses de façon inverse, pour un certain ordred'intégration élevé on intègre exactement; cependant plus la cellule d'intégration a une

92

Chapter 4. X-FEM for curved free boundaries 93

forme régulière , plus on peut négliger certains eets de géométrie et par là appauvrirle schéma d'intégration sans aecter l'optimalité de la procédure. En ce qui concernel'erreur de consistance due à l'approximation de la géométrie, nous montrons (théorème4.17) qu'elle s'écrit comme εh−1/2.

Ces estimations théoriques sont eectivement observées sur des tests numériques (section4.6), à savoir une plaque sous tension percée d'un cercle ou d'une ellipse, ce qui valide leuroptimalité. En particulier, on peut voir qu'une description quadratique de la géométrieest nécessaire à l'optimalité si une interpolation quadratique est utilisée.

The aim of this paper is to provide a-priori error estimates for problems involving curvedinterfaces and solved with the linear or quadratic extended nite-element method (X-FEM), with particular emphasis on the inuence of the geometry representation and thequadrature. We focus on strong discontinuity problems, which covers the case of holesin a material or cracks not subjected to contact as the main applications. The well-known approximation of the curved geometry based on the interpolated level-set functionand straight linear or curved quadratic subcells is used, whose accuracy is quantied bymeans of an appropriate error measure. A priori error estimates are then derived, whichdepend upon the interpolation order of the displacement, and foremost upon the aboveerror measure and the quadrature scheme in the subcells. The theoretical predictions aresuccessfully compared with numerical experiments.

4.1 Introduction

The extended nite element method (X-FEM) is an extension to the classical nite el-ement method introduced by Moës, Dolbow and Belytschko in [63] which allows easyhandling of problems with jumps or singularities. Adequate resolution of such problemsusually widely relies on the quality of the mesh, which should conform to the interfacegeometry. In particular, problems with evolving interfaces should be adressed with elab-orated remeshing tools (see [124126]). The X-FEM circumvents the diculty by a localenrichment of the polynomial interpolation space with non-polynomial functions, basedon the partition of unity method [64]. This enables a full independence of the mesh withrespect to the interface location.

The extension of the X-FEM to higher order elements was rst considered by Stazi, Be-lytschko and al in [51] with the aim of describing cracks. Relying on the work of Chessa[127] on the partition of unity, they proposed a quadratic interpolation for the classi-cal part of the displacement but a linear one for the enriched Heaviside and crack-tippart. Shortly after, Zi and Belytschko [76] considered the use of higher-order elementsto describe cohesive cracks. To this purpose, they removed crack-tip enrichments butused a corrected Heaviside enrichment instead at the cohesive tip to account for dis-continuities ending at the interior of an element. This absence of crack tip enrichmentallowed the authors to use quadratic interpolation for the enriched Heaviside part aswell (see [76]). Later on, Laborde and coworkers [128] extensively discussed crack-tipenrichment strategies that can be used with higher-order interpolation for the Heavisideenrichment. Despite the singularity, they recover the optimal convergence by enrichingwith the crack-tip functions on a xed area. As this makes the conditioning number


to soar, they proposed corrections with degree-of-freedom gathering (the rate of con-vergence being suboptimal, short of a 0.5 exponent) and pointwise matching (optimal).Alternative strategies were also proposed to restaure acceptable conditioning withoutdamaging the convergence, by means of enrichment with cuto functions [129], vectorialenrichments [130] or both combined [131], or enrichment over a xed area with an ap-propriate preconditioner [132]. This paper is restricted to Heaviside enrichment, but theabove bibliography highlights existing tools to extend it to crack-tips while preservingboth optimal convergence and acceptable conditioning.

In early papers about higher-order X-FEM, the authors highlighted the opportunity tobetter describe crack curvature. Stazi and Belytschko [51] interpolated the level set func-tion using the same quadratic shape functions as those of the displacement. Based onthe isozero of this interpolant, they would construct a quadrature scheme based on alinear subdivision. Later on, Legay, Wang and Belytschko [121] would again considerthe isozero of the level-set functions, but use quadratic curved subcells for the quadra-ture. The authors observed suboptimal rates for curved interfaces and optimal rates forstraight ones, suggesting that a poor representation of the geometry would hinder theconvergence. More recently, and almost simultanously, Cheng and Fries in [122], Dréauand al in [133, 134], Moumnassi and coworkers [123] also stressed out experimentallysuboptimal rates for higher-order formulations when used along with a linear descriptionof the geometry, provided the same grid is used as a starting point for the quadrature andthe bulk elds. All three papers proposed sucessful remedies to the problems, namely:

1. the description of the crack geometry on a ner subgrid, in [123, 133, 134], so thatpiecewise-linear cuts over relatively small subcells drops the quadrature error to beas small as the interpolation error,

2. curved subcells with one curved 4-node edge in [122].

Despite these experiments, to our knowledge the link between the description of thegeometry on the one hand, the quadrature rule in the subcells on the other and the errorof the problem was never theoretically quantied to a predictive rule with X-FEM. Thisis the task that we propose to examine in this paper.

In this paper, the approach of Legay and coworkers [121] is adopted for the interfaceresolution: starting from the interpolated level set on the nite-element mesh, classicalsubdivision of cut elements into straight triangles or curved quadratic 6-node trianglesis carried out. Each such subcell is then considered as belonging to one or the other sideof the interface.

An error measure is provided in part 4 to assess the accuracy of this geometrical rep-resentation. Corresponding theoretical orders of convergence are provided and checkedagainst numerical experiments. We link in part 5 geometrical representation, quadra-ture and convergence in X-FEM, based on the work of [135] and [136] in FEM. To thisaim, we derive a priori estimates as functions of the above geometrical error measure,the quadrature scheme and the interpolation order of the displacement. In part 6, theestimates are validated by numerical experiments.

The main result of this article is that, in practice, it suces to perform a subdivision afterthe same order than the interpolation and use the related typical quadrature scheme toget optimality. However, the theory allows to nd out problematic cases that have to behandled with thorough consideration.


4.2 Formulation of the continuous problem

We consider an elastic body occupying an domain Ω in R2, which is mathematically abounded open set. This body is cut through by a curved interface Γ, which can typicallybe a crack or a hole. In this way, Ω is separated into two open sets Ωi, i ∈ 1, 2, so thatΩ = Ω1 ∪ Ω2 ∪ Γ (see g.4.1). The restriction of any eld v to Ωi is denoted by vi. Foreach body, the remaining of the boundary ∂Ωi\Γ is composed of non-overlapping partsΓiu and Γig where conditions are prescribed on the displacement ui and the surface forcedistribution gi, respectively. We assume that ∀i ∈ 1, 2,Γiu has non zero measure, so asto prevent rigid body motion. Interface Γ is assumed to be traction free, which meansthat both parts constitute non-interacting solids. Holes in a material and cracks are themain applications of this case of study.

Figure 4.1: Denition of the problem.

The body is subjected to volume forces f i in addition to the surface loads. Smalldisplacements and strains are assumed. The equations in Ωi read:

∇ · σi + f i = 0 in Ωi (4.1)

σi · ni = gi on Γig (4.2)

ui = 0 on Γiu (4.3)

σi · ni = 0 on Γ (4.4)

where σi is the Cauchy stress tensor, dened from the elasticity fourth-order tensor Ai

and the strain tensor εi = 12

(∇ui +∇uTi

), as σi = Ai · εi.

In what follows, we adopt classical notations for the functional spaces: Wm,p(Ω) denotesthe Sobolev space of functions v for which all derivatives up to order m lie in Lp(Ω).In other words, for any multi-index α := (α1, α2) whose size |α| = α1 + α2 is less than

m, it holds ∂αv := ∂|α|v∂xα11 ∂x

α22

∈ Lp(Ω). The associated semi-norm is denoted |.|m,p,Ω, andthe associated norm ‖.‖m,p,Ω. Classically, index p is omitted when it is 2 so that wenote Hm(Ω) := Wm,2(Ω) and ‖v‖m,Ω := ‖v‖m,2,Ω. We also introduce product spacesHm (Ω1)×Hm (Ω2), which are endowed with the broken norm ‖v‖2m,Ω1∪Ω2

:= ‖v‖2m,Ω1+

‖v‖2m,Ω2. In this paper, C will denote a generic non-negative constant, and c a strictly

positive constant.


The components of the solution belong to V := v ∈ H1(Ω1)×H1(Ω2), v | Γiu= 0,

i ∈ 1, 2. A bilinear form a ∈ L(V 2 × V 2;R) and linear form b ∈ L(V 2;R) are denedas:

For (u,v) ∈ V 2 × V 2 , a (u,v) :=

2∑i=1

∫Ωi

σi(ui) : εi(vi)dx (4.5)

For v ∈ V 2 , l (v) :=2∑i=1

∫Ωi

f i · vidx +

∫Γig

gi · vids (4.6)

The resulting weak formulation consists in nding u ∈ V 2, such that ∀v ∈ V 2:

a (u,v) = l (v) (4.7)

Equation (4.7) constitute a vectorial second-order problem. For the sake of conciseness,and as is very common in nite-element analysis, we shall in the mathematical devel-opements of this paper rather work with a scalar second-order elliptic problem: thiscorresponds to a heat diusion problem through a cracked structure. All theorems ofthis paper are given in the scalar setting, but their vectorized counterparts could bederived in a similar way. We set a ∈ L(V × V ;R) as:

a (u, v) :=2∑i=1

∫Ωi

∇ui ·Ai · ∇vidx (4.8)

Here is Ai the conductivity second-order tensor, whose components are assumed to haveregularity properties except across the interface: for i ∈ 1, 2 and (k, l) ∈ 1, 22,(Ai)kl ∈ L∞(Ωi). This obviously ensures the continuity of a. Tensor Ai is also supposedto verify an ellipticity property, that is to say there exists c > 0, ∀i ∈ 1, 2,∀x ∈ Ωi, ∀ξ ∈R2,

∣∣ξT ·Ai (x) · ξ∣∣ ≥ c|ξ|2. This makes a coercive. The source terms are fi ∈ L2(Ωi)

and we introduce l ∈ L(V ;R) as:

l (v) :=

2∑i=1

∫Ωi

fividx (4.9)

Again, the problem may be written as (4.7).

4.3 The discrete problem

4.3.1 The approximation of the geometry

Since the X-FEM is widely used to describe propagating interfaces, the interface is usuallyimplicitely represented, by means of a signed distance function on Ω, called level-setfunction (see e.g. [137]).


Assumption (H10). The existence of a level-set function φ is assumed in a 2δ-xed-width strip Sδ centered on the interface, whose isozero coincides with the interface:

Γ = x, φ(x) = 0 (4.10)

This assumption is fully discussed in appendix A, where a rigourous denition is givenunder regularity assumptions on the interface, the semi-width δ being related to theminimal radius of curvature. In a nutshell, for any point x ∈ Sδ, a unique projectionpoint x onto the interface may be dened (see g.4.2), as:

x := x− φ(x)∇φ(x) (4.11)

Hence, φ is the signed distance to the interface and ∇φ, which veries |∇φ| = 1, givesthe normal direction to the interface at the projection point (see g.4.2).

x

∇ϕ

x

ϕ

Figure 4.2: Denition principle and properties of the level-set function.

We suppose that the whole domain Ω is meshed with a family Th of triangular anemeshes, regardless of the location of the interface. We now aim at subdividing cut ele-ments into quadrature subcells whose boundary would dene an approximated interfacelocation Γh.

Stage 1 of the geometry approximation.

It consists of an interpolation of the level-set function, as (see g.4.3b):

φh(x) =∑j∈Nh

Nj(x)φj (4.12)

where Nh denotes the nodes of the mesh, Nj the shape function of order g associated withnode j, and φj the values of the level-set function at this node (see g.4.3a). Exponent gis thus the representation order of the geometry. If g = 1, this amounts to the well-knownlinear subdivision.

The need for a second stage when g = 2

Now, if g = 1, the zero isobar Γhφ : φh = 0 is a straight line elementwise, so it may readilyserve as boundary for linear conforming subcells. On the contrary, if g = 2, as in [121],iso-zero Γhφ is constituted by conic sections elementwise. Now, subcells that would havesuch a curve as boundary are unusual (for instance, a quadratic-subcell edge is not a


conic section strictly speaking, and would be only if the inverse jacobian of the subcellwas approximated by its second-order Taylor expansion). Hence we would like to t Γhφto the border of a quadratic subcell Γh. This will be the purpose of our second stage.

Second stage of the geometry approximation.

If g = 2, in addition to both intersection points between Γhφ and the edges of K, a middle

point on on Γhφ is determined using the perpendicular bissector to the segment betweenthe previous points (see g.4.3c). The cut element is then subdivided into quadratictriangular subcells (see g.4.3d), an edge of which interpolates the three points on theisozero curve. This edge is nally the approximation Γh of the interface (see g.4.3d):any subcell E then fully belongs to one or the other side of the interface. Approximatedbodies Ωh

i may then be dened as on g.4.3d.

About Γhφ and Γh being dierent curves when g = 2.

When g = 1, Γhφ and Γh are two straight lines, and necessarily coincide since they have

two points in common. On the contrary, when g = 2, Γhφ is a conic section (the 2Dcase of quadric surface), and as such is determined in a unique manner not by three, butby ve points. Consequently, having three points in common does not ensure that thequadratic-subcell edge Γh coincides with Γhφ.

a. Computation of nodal level sets b. Quadratic interpolation from nodal values

c. Determination of some points on the interpolated level-set isozero

d. Subdivision with corresponding quadratic triangles and approximated domains

:=0

Γϕh

Γϕh :ϕh=0

:=0

h

Subcell

ΓϕhΩ1

h Ω2h

Figure 4.3: Subdivision of cut elements, in the case g = 2.

Figure 4.4 below recaps the approximation process:


Γ Exact interface, ϕ = 0

ΓhΦ

Conic-section approximation, ϕh = 0

ΓhQuadratic-edge approximation

Stage 1

Stage 2

Figure 4.4: Approximation process of the geometry.

4.3.2 The interpolation of the eld of unknowns

Again, Th denotes a family of ane triangular meshes of Ω. Since in the X-FEM the posi-tion of the interface is independent of the mesh, this does not prevent the representationof arbitrary curved interfaces. Family Th is assumed to be quasi-uniform and regular,that is to say for K ∈ Th, denoting hK the radius of the smallest circle containing K andρK the radius of the largest circle included in K and setting the characteristic mesh sizeh := max

KhK , we have:

Assumption (H11): There exists a constant C independant of h and K such thathρK≤ C. This ensures that the angles of triangle K are uniformly bounded away

from zero.

In order to account for the eld jump across the interface, the classical nite elementapproximation is enriched with a Heaviside-like function (see [63]):

Vh :=

∑i∈Nh

aiNi(x) +∑j∈Kh

bjNj(x)H(x), ai ∈ R, bi ∈ R

(4.13)

In this expression Ni is the shape function of order p ∈ 1, 2 at node i, Kh is the set ofenriched nodes - the nodes whose shape function is not identically zero on the interface,as pictured on g.4.5 - and H is the Heaviside-like function used to represent the jump:

H(x) =

−1 if x ∈ Ωh

1

+1 if x ∈ Ωh2

(4.14)


enriched nodesK h

intersected elementsE h

interface

Figure 4.5: Interface, mesh not matching the interface and enriched nodes, in thecase p=2.

Note that the interpolation order p may be dierent from the representation order g ofthe geometry. We denote Eh the set of cut elements.

4.4 The accuracy of the geometry description

We shall now be equipped with an error measure for assessing the accuracy of the geom-etry description:

Denition 4.1. The interface resolution is introduced as ε := maxx∈Γ

(minx′∈Γh

|x− x′|). Of

course, ε depends upon h, but this dependence is omitted in the notation for the sake ofconciseness. The dependence of ε upon h indicates a convergence in L∞-norm.

This measure is an extension to X-FEM of denition 3.1 in [138] for classical FEM.Geometrically, this means that for any subdivision, we have Γh ⊂ Sε, where Sε is the2ε-xed-width strip centered on Γ: the interface is thus said to be ε-resolved by thesubdivision.

4.4.1 Interpolating the level-set function

The rst stage of our description consists in approximating the level-sets. For K ∈ Th,let Pg(K) be the space of polynomials of total order g on K. Let Ig be the standardnodal interpolation operator. We have:

Lemma 4.2. Let g ∈ 1, 2, δ′ < δ and assume the level-set φ to satisfy φ ∈ Cg+1 (Sδ).Then for h ≤ h0 := δ−δ′ the interpolation estimate ‖φ−Igφ‖0,∞,Sδ′ 6 Ch

g+1‖φ‖g+1,∞,Sδholds.

Proof. We apply [135], theorem 3.1.6 (interpolation with ane meshes). Since h ≤ h0 :=δ − δ′, φ is well-dened on all elements K ∈ Th verifying K ∩ Sδ′ 6= ∅.


4.4.2 Constructing subcells

In order to analyse the error that we have made approximating Γhφ by Γh (second stage),we should introduce, for the purpose of the demonstration exclusively, subcells whichexactly resolve a curved interface (see [139]), referred to as transnite subcells. Theirmappings are no longer polynomial but simply analytical, and constructed by transniteinterpolation from the analytical expression of the curved edge to be matched (see g.4.6).In order to evaluate the error mentioned above, we have constructed a mapping FE fortransnite subcells and compared it with the polynomial mapping FE of classical ones.The analysis is detailed in Appendix B. Its main result reads as follows (lemma B.1):under the assumption that Γhφ admits a local Cg+1-parametrization fφ from a segment

I, which is the case since Γhφ is a conic-section, we have:

‖FE − FE‖g+1,∞,K ≤ Chg+1|fφ|g+1,∞,I (4.15)

a1

a2

x1E

x2E

x3E

x

y

x13E

F E

F E

Reference triangle K

Γh

Classical subcell

Transfinite subcell

x2

x1x13x3

E

E

K

x13E Γ

Figure 4.6: Classical and transnite subcells, in the case where g=2.

We are now in a position to give an estimate for the whole process:

Theorem 4.3. Let g ∈ 1, 2 be the representation order of the geometry. Assumeinterface Γ to have a geometric continuity Gg+2. Then the interface is O(hg+1) resolvedby the subdivision, that is to say ε ≤ Chg+1.

Appendix A may be consulted for a comprehensive denition of geometric continuity.

Proof. Since Γ has a geometric continuity Gg+2, after lemma A.1 we have φ ∈ Cg+1(Sδ).We may therefore apply lemma 4.2, which for x ∈ Γh gives (see g.4.7):

|φ(x)| ≤ minx′∈Γhφ

(|φ(x′)− φh(x′)|+ |x′ − x|)

≤ Chg+1|φ|g+1,∞,Sδ + minx′∈Γhφ

(‖x′ − x‖) (4.16)


x

Γϕh :ϕh=0x

hx '

x '

Figure 4.7: Decomposition of the errors.

We may then apply lemma B.1. to any subcell E with a curved edge for each cut parentelement:

|φ(x)| ≤ Chg+1(|φ|g+1,∞,Sδ + |fφ|g+1,∞,I

)(4.17)

Let

I → Rs → fφ(s) = (x(s), y(s))

be a parametrization of Γφ. Then we have ∀s ∈ I ,

f ′φ (s) · ∇φh (x(s), y(s)) = 0 (4.18)

It may be deduced that parametrization fφ may be chosen such that y′(s) =∂φh∂x (x(s), y(s)) and x′(s) = −∂φh

∂y (x(s), y(s)). So dierentiating with the chain ruleand making use of the previous expressions yields (dropping the dependence to s toalleviate notations):

y′′(s) = −∂2φh∂x2

∂φh∂y

+∂2φh∂x∂y

∂φh∂x

(4.19)

and:

y(3)(s) = −∂φh∂x

[∂2φh∂x2

∂2φh∂y2

−(∂2φh∂x∂y

)2]

(4.20)

We derive similar expressions for the successive derivatives of x, and deduce

|fφ|g+1,∞,I ≤ C‖φh‖g+1g+1,∞,Sδ . Hence |φ(x)| ≤ Chg+1‖φ‖g+1,∞,Sδ (1 + ‖φ‖gg+1,∞,Sδ

)so

ε ≤ Chg+1 which means that the interface is O(hg+1) resolved by the subdivision.

Remark 4.4. However, a locally better resolution of the interface would be appreciablein some situations, for instance cut triangles which are cut close to a node. To remaingeneral, let us consider any K ∈ Eh. Two of its edges are intersected and the subcellcontaining their common node is called triangular subcell. This common node is mappedonto the origin of the reference triangle K (see g.4.8). Such a map is moreover chosensuch that the rst reference coordinate correspond to the largest coordinate aM amongthe intersection points, while the second reference coordinate is associated with thesmallest am.


The rst stage of the approximation may then be consequently improved in|φ − Igφ|0,∞,Γh ≤ aMh|φ − Igφ|1,∞,E ≤ CaMh

g+1. The second stage occurs with atriangular subcell E of a characteristic size aMh(see g.4.8), so in K the interface islocally O(aMh

g+1)-resolved by the subdivision.

x3K

x2K

x1K

h

F K−1 (affine)

a1

a 2

aM

am

KEKE

F K (affine)

x3 x1

x2

Figure 4.8: Intersected triangle.

With this local property of an improved resolution of the interface, problematic casesare those for which one side has a small area and the interface does not remain in thevicinity of a node, so that no better resolution is available. The characteristic feature ofthose intersected elements is that the area of the one side is small when compared to thelength of the cut interface, hence the terminology developed below:

Denition 4.5. Let 0 < κ 1 be a xed parameter. A triangle K will be calledlengthways intersected if:

∃i ∈ 1, 2, meas(Ωi ∩K)

meas(Γ ∩K)2≤ κ (4.21)

For the sake of the demonstration, we use alternative notations from g.4.8: K is length-ways intersected if:

min(am, 1− am)

aM≤ 2κ (4.22)

To be more practical, for these triangles there is an almost coincident edge with theinterface (see g.4.9a and b). To be more mathematical, problems will arise from trian-gulations (family of meshes) for which ratios (4.21) or (4.22) are not bounded away from0 as h→ 0. Strictly speaking, we should call lengthways intersected a triangulation, nota triangle.


Almost coincident edge

a. lengthways intersected triangle whose associated edge is crossed

b. lengthways intersected triangle whose associated edge is not crossed

c. small section but not a lengthways intersected triangle

d. not a lengthways intersected triangle

Figure 4.9: Various congurations of intersected triangles.

4.4.3 Numerical experiments

The resolution of the interface in the sense of our error measure (denition 4.1) wascomputed for various shapes, namely polynomial, exponential, sinusoidal and circularcurves. For the sake of comparison with the literature, the classical area error measureis also assessed, which consists of comparing the approximate area on one side of theinterface with its analytical value:

area error =

∣∣meas(Ω1)−meas(Ωh1)∣∣

meas(Ω1)(4.23)

It is clear from this denition that both error measures should have the same order ofconvergence, unless a compensation phenomenon occurs on the area error which makesits rate of convergence higher. In accordance with the theory, the observed convergenceorder for both error measures was always 2 with a linear subdivision. For all shapes butthe circle, the interface resolution was observed to converge at order 3 with a quadraticsubdivision, as predicted by the theory, while the area error was always 4. The resultsfor the exponential curve are displayed on g.4.10 as an example. It is still unclear tous what systematic compensation eect is at stake. A super-convergent resolution of theinterface was observed for the circle (see g.4.11), as was already noticed by [122].


0,010,05

1E-12

1E-10

1E-8

1E-6

1E-4

1E-2

1E+0

1E-12

1E-10

1E-8

1E-6

1E-4

1E-2

1E+0Interface resolution. Linear subdivision Slope 1.97

Area error. Linear subdivision Slope 2.0

Interface resolution. Quadratic subdivision Slope 2.97

Area error. Quadratic subdivision Slope 3.99

mesh size (m)

inte

rfa

ce r

es

olu

tion

are

a e

rro

r

Figure 4.10: Typical convergence rates (e.g. for an exponential shape).

0,010,05

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

approximation of the geometry

the circle

Area error. Linear subdivision Slope 2.0

Interface resolution. Linear subdivision Slope 2.0

Interface resolution. Quadratic subdivision Slope 3.64

Area error. Quadratic subdivision Slope 4.04

mesh size

inte

rfa

ce r

es

olu

tion

are

a e

rro

r

Figure 4.11: A superconvergent case : the circle.

4.5 A priori error estimates depending on geometry descrip-tion and quadrature

As already pointed out in [51, 76], a piecewise linear subdivision yields suboptimal con-vergence rate with higher-order eld interpolation, when the same mesh is used forinterpolation and for quadrature. So optimal convergence for curved interfaces reliespartly on the representation of the interface geometry, and we shall now quantify thisdependence for the strong discontinuity problem presented in section 2.

4.5.1 Applying the rst Strang lemma

Given that u ∈ V is discontinuous across Γ and its interpolant uh ∈ Vh is discontinuousacross Γh, ∇(u − uh) would not be dened on both of them, thus making dicult thevery denition of a H1-norm. Hence, we chose to dene the error as ‖u − uh‖21,Ωh1∪Ωh2where u is an extension to u to be dened with a discontinuity across Γh instead of Γ.

Mathematical denition of u.


To dene such u, we introduce bounded sets Ωi which contain all discretizations Ωhi

from a certain mesh size: ∀h ≤ h0, Ωi ⊂(Ωi ∪ Ωh

i

). Let p ∈ 1, 2, we assume that

u ∈ Hp+1(Ω1) × Hp+1(Ω2) and denote ui its restriction to Ωi. We may extend ui toui ∈ Hp+1

(Ωi

)in a stable way (see Stein [140]). We may then dene u ∈ Hp+1(Ωh

1) ×Hp+1(Ωh

2) such as u|Ωhi:= ui. The error on the solution is dened as ‖u− uh‖21,Ωh1∪Ωh2

=∑i=1,2‖ui − uh‖21,Ωhi .

Conversely, for a discrete wh ∈ Vh, we will sometimes be led to use extensions wh beingdiscontinuous across Γ instead of Γh. To dene them, we note wih := wh|Ωhi

. Since it is

polynomial on each intersected triangle, we denote wih its natural polynomial extension,and build wh by combining the restrictions to Ωi.

Domain approximation stage

Approximating the operators, the rst step is to consider exact integrals but over the ap-proximated domain, which leads to intermediate bilinear and linear forms overH1(Ωh

1) ×H1(Ωh2): ah (v, w) :=

∑i=1,2

∫Ωhi

(Ai · ∇v

)· ∇wdx and lh (v) :=

∑i=1,2

∫Ωhifivdx ,

where for i ∈ 1, 2, Ai and fi are the stable extensions to Ai and fi related to theSobolev spaces they respectively belong to.

Quadrature approximation stage

The second step is the quadrature. On a cut element K, the quadrature scheme isdened over each subcell E. The reference triangle K is endowed with an integrationscheme whose weights are (wl)l=1..L and integration points are located at (bl)l=1..L. De-noting DFE the Jacobian matrix of FE and JE := det(DFE), this yields an integrationscheme on each subcell with weights wl,E = wlJE(bl) and integration points locations

bl,E = FE(bl). Let Vh be the interpolation space (4.14), the discrete linear and bilinearforms are dened by ∀ (uh, vh) ∈ Vh:

ah(uh, vh) :=

2∑i=1

∑E∈Ωhi

L∑l=1

wl,E([Ai · ∇uh

]· ∇vh

)(bl,E) (4.24)

lh(vh) :=

2∑i=1

∑E∈Ωhi

L∑l=1

wl,E(fivh

)(bl,E) (4.25)

The discrete problem then amounts to nding uh ∈ Vh such that:

∀vh ∈ Vh, ah (uh, vh) = lh (vh) (4.26)

We recall:

Denition 4.6. The bilinear forms ah are said to be uniformly Vh-elliptic if there existsc > 0,∀vh ∈ Vh, c‖vh‖21,Ωh1∪Ωh2

≤ ah (vh, vh).

A general abstract error estimate then reads:


Proposition 4.7. Extension to X-FEM of the theorem 4.4.1 in [135]). Assume that thediscrete bilinear forms ah are uniformly Vh-elliptic, there exists a constant C independentof h such that:

‖u− uh‖1,Ωh1∪Ωh2≤ C inf

vh∈Vh

‖u− vh‖1,Ωh1∪Ωh2

(a)

+ supwh∈Vh

|ah (vh, wh)− ah (vh, wh) |‖wh‖1,Ωh1∪Ωh2

+ C sup

wh∈Vh

|lh (wh)− lh(wh)|‖wh‖1,Ωh1∪Ωh2

(b)

+C supwh∈Vh

|ah (u, wh)− a (u, wh) |‖wh‖1,Ωh1∪Ωh2

+ C supwh∈Vh

|lh(wh)− l(wh)|‖wh‖1,Ωh1∪Ωh2

(c)

(4.27)

Proof. ∀(v, w) ∈ H1(Ωh1 ∪ Ωh

2), by Cauchy-Schwarz and concavity of the square root :

|ah (v, w) | ≤ |A|0,∞,Ω1∪Ω2

(‖∇v‖Ωh1 ‖∇w‖Ωh1 + ‖∇v‖Ωh2 ‖∇w‖Ωh2

)≤ |A|0,∞,Ω1∪Ω2

( ∑i=1,2

∑j=1,2‖∇v‖Ωhi ‖∇w‖Ωhj

)≤ 2|A|0,∞,Ω1∪Ω2‖v‖Ωh1∪Ωh2

‖w‖Ωh1∪Ωh2

Then with a proof strictly identical to [135], 4.4.1, we get:

‖u− uh‖1,Ωh1∪Ωh2≤ C inf

vh∈Vh

‖u− vh‖1,Ωh1∪Ωh2

+ supwh∈Vh

|ah (vh, wh)− ah (vh, wh) |‖wh‖1,Ωh1∪Ωh2

+C sup

wh∈Vh

|ah (u, wh)− lh(wh)|‖wh‖1,Ωh1∪Ωh2

We rst decompose:

ah (u, wh) = [ah (u, wh)− a (u, wh)] + a (u, wh) (4.28)

And since a (u, wh) = l (wh), we deduce from (4.28) that:

ah (u, wh)− lh (wh) = [ah (u, wh)− a (u, wh)]

+[l (wh)− lh (wh)

]+(lh − lh

)(wh)

(4.29)

which leads to the result.

Three kinds of errors appear in estimate (4.27) of proposition 4.7. The rst one (a) is arather well-documented interpolation error. The second (b) is the consistency error that


we have made by using a quadrature scheme to compute the integrals on the approxi-mated geometry. The third (c) is the consistency error that arises by using operators onthe approximated geometry rather than the exact one, the integrals being assumed tobe computed exactly.

Because we will need to prove uniform ellipticity for ah before going any further, let usfocus on the integration quantities involved in its expression (4.24). To this purpose, werecall:

Remark 4.8. Estimates |DFK | ≤ Ch and |JK | ≤ Ch2 hold for any parent element.For m ∈ 1, 2, |DmFE |0,∞,K ≤ Chm and for l ∈ 0, 1, 2, |JE |l,∞,K ≤ Chl+2 holdfor any subcell (see [135], theorem 4.3.3 and its proof). However, while the regularityassumption (H11) of the parent mesh ensures the inverse properties |DF−1

K | ≤ Ch−1 and|J−1K | ≤ Ch−2, the inverse properties |DF−1

E |0,∞,E ≤ Ch−1 and |J−1E |0,∞,E ≤ Ch−2 do

not hold for all subcells. To have additional properties for some of them, they should beclassied after their shape properties.

Denition 4.9. From the more demanding to the more permissive, we will call (seeg.4.12):

Regular, a subcell for which assumption (H11) holds. For those subcells the inverseproperties |DF−1

E |0,∞,E ≤ Ch−1 and |J−1E |0,∞,E ≤ Ch−2 hold.

Slender, a non regular subcell E, which, given a constant C independent of h anddenoting J−1

E := det(DF−1E ), nevertheless veries:

|J−1E |0,∞,E |JE |0,∞,K ≤ C (4.30)

Any subcell with no curved edge is either regular or slender.

Crescent-shaped, a neither regular nor slender subcell.

Again, asymptotically speaking, we should call regular, slender or crescent-shaped fami-lies of subcells, but not individuals. Hence, strictly speaking, we call regular a family ofsubcells of the triangulation for which the ratio in (H11) remains bounded as h→ 0, slen-der a family for which (4.30) remains bounded, and crescent-shaped subcells belongingto neither families.

Remark 4.10. In practice, the slenderness property excludes curved subcells which leadto a degenerescence of the angle at the vertex with respect to the similar ane subcellangle (see the third triangle on g.4.12).

Let us set out the following assumption about quadrature schemes for k ∈ N:

Assumption (H12, k): The quadrature scheme (wl, bl) is exact for Pk(K). Furtherin this paper, we will note (H12, k=3) if the quadrature scheme is exact for P3(K)for instance.

We know give a condition under which the approximate bilinear forms are uniformlyVh-elliptic:


h

h

h

Regular subcell Slender subcell Crescent-shaped subcell

Figure 4.12: Regular, slender and crescent-shaped subcells.

Proposition 4.11. (Extension to X-FEM of [135], theorem 4.4.2). Let p = 2 and g = 2.Under assumptions (H11), (H12, k=4) for regular and slender subcells and (H12, k=6)for crescent-shaped subcells, the bilinear forms ah are uniformly Vh-elliptic.

For p = 1 and g = 1, uniform ellipticity holds under the assumptions (H11) and (H12,k=0), and for p = 2 and g = 1 it holds under (H11) and (H12, k=2), the easier proofbeing left to the reader.

Proof. Step 1. Let K ∈ Eh and for each side i ∈ 1, 2, let E ⊂(K ∩ Ωh

i

)be a subcell.

Let vh ∈ Vh , and call p := vh|K∩Ωhi∈ P2 (K) . The ellipticity condition on A on the

Gauss point number l of E yields:

[(A · ∇p) · ∇p] (bl,E) ≥ c|∇p|2 (bl,E) (4.31)

The pull-back of p onto the reference parent element will be noted p := p FK . Thus,we have for x ∈

(K ∩ Ωh

i

), Dp (x) = Dp

(F−1K (x)

)·DF−1

K so since FK is ane:

|Dp(x)|2 ≥ 1

|DFK |2|Dp

(F−1K (x)

)|2 (4.32)

After (4.31) and (4.32), it holds:

L∑l=1

wl,E [(A · ∇p) · ∇p] (bl,E) ≥ c 1

|DFK |2L∑l=1

(|Dp|2 F−1

K FE(bl))wlJE(bl) (4.33)

Step 2. Proof for slender subcells. We can estimate JE(bl) = 1J−1E (bl,E)

≥∣∣J−1E

∣∣−1

0,∞,E .

Hence:


L∑l=1

wl,E [(A · ∇p) · ∇p] (bl,E) ≥ c

∣∣J−1E

∣∣−1

0,∞,E|DFK |2

L∑l=1

(|Dp|2 F−1

K FE(bl))wl (4.34)

We have |Dp|2 F−1K FE ∈ P4(K) which thanks to (H2, k=4) allows the conversion of

right-side member into an exact integral that may be estimated in the following way:

L∑l=1

wl|Dp|2(F−1K FE(bl)

)=∫K

(|Dp|2 F−1

K FE(x))dx

≥ |JE |−10,∞,EJK

∫E |Dp|

2dx

(4.35)

where we have noted E := F−1K (E) ⊂ K (see g.4.8). Combining (4.34) and (4.35), the

denition (4.30) of slender subcells, and the classical estimates |p|21,E≥ cJ−1

K |DFK |2|p|21,E(see [135], theorem 4.3.2), we may conclude:

L∑l=1

wl,E [(A · ∇p) · ∇p] (bl,E) ≥ c|p|21,E (4.36)

Step 3. Proof for crescent-shaped subcells. Since FK is ane so is F−1K , and FE ∈ P2(K)

which implies:

JE(|Dp|2 F−1

K FE)∈ P6(K) (4.37)

Thanks to (4.37) and (H2, k=6), the right-side member of (4.33) may be converted intoan exact integral, which after a change of integration domain gives :

L∑l=1

wl,E [(A · ∇p) · ∇p] (bl,E) ≥ c 1

|DFK |2JK |p|21,E (4.38)

By the same classical estimates |p|21,E≥ cJ−1

K |DFK |2|p|21,E , we may conclude:

L∑l=1

wl,E [(A · ∇p) · ∇p] (bl,E) ≥ c|p|21,E (4.39)

Step 4. Summing up (4.36) over E yields:

∃c > 0,∀vh ∈ Vh, ah (vh, vh) ≥ c|vh|21,Ωh1∪Ωh2(4.40)

We may now prove the same property with the norm instead of the semi-norm followingthe proof of [135], theorem 4.4.2. Applying the Poincaré inequality over Ωi to Stein


extensions of vh, we may nd a constant such that ∀v ∈ Vh, ‖v‖1,Ωh1∪Ωh2≤ C|v|1,Ωh1∪Ωh2

which yields the nal result ∃c > 0, ∀vh ∈ Vh, ah (vh, vh) ≥ c‖vh‖21,Ωh1∪Ωh2.

Having established a condition which ensures that the assumption of the abstract errorestimate (4.27) holds, we take to estimating its dierent terms. Let us start with theinterpolation error (a) in (4.27). As it is well-documented in the literature, we stick tothe essential.

Denition 4.12. ([95, 129, 131]). Let p ∈ 1, 2 be the interpolation order, let u ∈Hp+1 (Ω1 ∪ Ω2), we introduce an interpolation operator Πh onto Vh byΠhu :=

∑l∈Nh

alNl +∑l∈Kh

blNlH, where ai, bi are given by (xi being the position of the

node associated with Ni):

if l ∈ Nh\Kh then al := u(xl),

if l ∈ Kh then:

al := 1

2 (ui(xl) + uj(xl))bl := 1

2(−1)i (ui(xl)− uj(xl))where xl ∈ Ωh

i and j := 3− i.

Note that Denition 4.12 implies that if Ip is the nodal interpolation operator, thenΠhu|Ωhi

= Ipui. The denition is consistent since H2 (Ω1 ∪ Ω2) ⊂ C0 (Ω1 ∪ Ω2).

Proposition 4.13. Let p ∈ 1, 2, let u ∈ Hp+1 (Ω1 ∪ Ω2), the following interpolationestimate holds:

‖u−Πhu‖1,Ωh1∪Ωh2≤ Chp‖u‖p+1,Ω1∪Ω2 (4.41)

Proof. The reader is referred to [129, 131] for the case where p = 1. For p = 2, theregularity of the parent elements and the stability of the extension give:

‖ui − I2ui‖1,Ωhi ≤ Chp‖ui‖p+1,Ωi

≤ Chp‖ui‖p+1,Ωi (4.42)

Summing up the results yields the desired property by denition of Πh.

Note that the properties above say nothing about punctual values of the gradient (thatin so say in the sense of the norm of W 1,∞ (Ωi)): let us rst illustrate how a clas-sical (naive) estimate for this norm indeed fails to prove local convergence for smallsections. Take as the exact solution a polynomial, but with an order q > p too highto be spanned by the basis. Let K ∈ Eh be a triangle cut on a small section, so

thatmeas (K ∩ Ωi)

meas (K)= κ 1. The equivalence of all norms on Pq (K ∩ Ωi) yields


‖u − Πhu‖1,∞,Ωi∩K ≤ Cκ−1/2h−1‖u − Πhu‖1,Ωi∩K . We may then use the classical esti-mate from [129, 131, 141] ‖u− Πhu‖1,K∩Ωi ≤ Chp‖ui‖p+1,K . The equivalence of normsover Pq (K) nally yields:

‖u−Πhu‖1,∞,Ωi∩K ≤ Cκ−1/2hp+1‖ui‖p+1,K (4.43)

Using quasi-interpolation operators improves the classical estimates of [131] into‖u − Πhu‖1,K∩Ωi ≤ Chp‖ui‖p+1,K∩Ωi except for an alternance of small and large sec-tions, such as that represented on g.4.13. The analysis has not been reported here sincequasi-interpolation operators are beyond the scope of this paper.

Ωi

Γ

Figure 4.13: Situation responsible for local high errors.

In practice, the phenomenon is familiar to X-FEM users, where incorrect values are oftennoticed at small sections. However, the very fact that they occur only on small areasensures that the (theoretically proven) H1-convergence is not aected. So the wise userwill either have engineering quantities of interest dened by integrals, or drop those smallsections when looking for maximal stress or strains.

4.5.2 About the quadrature rules in the subcells

We shall handle consistency errors (b) in (4.27), arising from using a quadrature to evalu-ate the intergrals. The point of this section is to prove that they do not yield suboptimalrates of convergence. In a nutshell, the paradigm of the next two propositions is to rely onthe fact that interpolation occurs on ane parent elements whose mapping, jacobian andtheir inverse are well-known, and the fact that the subcells have polynomial geometricalquantities (mappings and jacobian). The idea is then to increase the quadrature order toinclude the description of those geometrical quantities as the shape of the subcells getsworse.

To explain it the other way round, for a uniform material and the case p = 2, a schemecorrectly integrating P6 would be exact as it corresponds to the maximal order obtainedtaking into account geometrical and interpolation eects. As the shape of the subcellsgets more regular, some contributions due to geometrical eects become very small, andmay be neglected while keeping the integration error reasonably low.

Let ϕ be a continuous function over K, we dene a quadrature error by

∆(ϕ) :=∫K ϕ dx −

L∑l=1

wlϕ(bl). In the same way, the quadrature error on a subcell


E ∈ Sh of a continuous function over E is ∆E(ϕ) :=∫E ϕ dx −

L∑l=1

wl,Eϕ(bl,E). It is

estimated by:

Proposition 4.14. (Extension to X-FEM of [135], Theorem 4.4.4). Let p = 2 andg = 2, E ∈ Sh, K ∈ Eh such that E ⊂ K. Under assumptions (H11), (H12, k=7) andA ∈ W 2,∞(E) for crescent-shaped subcells, (H12, k=5) and A ∈ W 4,∞(E) for slendersubcells, (H12, k=3) and A ∈W 2,∞(E) for regular subcells; we have ∀ (p, p′) ∈ (P2(E))2

|∆E ((A · ∇p′) · ∇p)| ≤ Ch2‖A‖2,∞,E‖p′‖2,E |p|1,E.

In case p = 1 and g = 1, assumptions (H11), (H12,k=0) and A ∈ W 1,∞(E) en-sure a quadrature consistency error (whose easier demonstration is left to the reader):∀ (p, p′) ∈ (P1(E))2, |∆E ((A · ∇p′) · ∇p)| ≤ Ch‖A‖1,∞,E‖p′‖1,E |p|1,E .

Proof. Step 1. Let (e1, e2) be an orthonormal basis of the plane, let ∂iv := Dv · ei, for(i, j) ∈ 1, 22, we have:

∆E

(Aij∂ip

′∂jp)

= ∆[(Aij FE)

(∂ip′ FE

)(∂jp FE) JE

]Introducing the pullback with respect to the parent element p := p FK yields

∂jp (x) = Dp(F−1K (x)

)· DF−1

K · ej . Let us call ej' :=DF−1

K ·ej|DF−1

K ·ej |, then this may be

reformulated into ∂jp (x) =∣∣DF−1

K · ej∣∣ ∂j′ p (F−1

K (x)). Since

∣∣DF−1K · ej

∣∣ ≤ |DF−1K |, we

have:

∆E (Aij∂ip′∂jp) ≤ |DF−1

K |2∆[(Aij FE)

(∂i′ p

′ F−1K FE

) (∂j′ p F−1

K FE)JE] (4.44)

Step 2: Proof for crescent-shaped subcells. To alleviate notations, we denotew := (∂i′ p

′) F−1K FE

(∂j′ p

) F−1

K FE JE , a := a FE := Aij FE and notice

that w ∈ P6(K). Hence with (H12, k=7) the continuous linear form ϕ→ ∆ (ϕw) assess-ing the quadrature error vanishes over the space P1(K), so the Bramble-Hilbert Lemma([135], theorem 4.1.3) asserts that:

∃C, ∆ (aw) ≤ C|a|2,∞,K |w|0,K (4.45)

The second term is evaluated by |w|20,K≤ |JE |0,∞,KJK |∂i′ p

′∂j′ p|20,E . By virtue of the

Cauchy-Schwarz inequality and |p|1,E ≤ J−1/2K |DFK ||p|1,E , we assess:

|w|0,K ≤ |JE |1/2

0,∞,KJ−1/2K |DFK |2|p|1,E |p′|1,E (4.46)

Moreover D2a = D2 (a FE) = DF TE · D2a · DFE + Da · D2FE , which given that|DFE | ≤ Ch and |D2FE | ≤ Ch2 yields:


|a|2,∞,K ≤ Ch2‖a‖2,∞,E (4.47)

Assumption (H11) implies that |DFK |2J−1/2K |DF−1

K |2 ≤ Ch−1, which combined with

|JE |1/20,∞,K≤ Ch and equations (4.45), (4.46) and (4.47) yields:

∣∣∆E

(a∂ip

′∂jp)∣∣ ≤ Ch2‖a‖2,∞,E |p|1,E |p′|1,E (4.48)

Step 3: Proof for slender subcells. Starting from equation (4.44) we setv := (∂i′ p

′) F−1K FE ∈ P2(K), u :=

(∂j′ p

)F−1

K FE ∈ P2(K) and b := (a FE) JE .

Given (H12, k=5), the continuous linear form ϕ → ∆ (ϕu) vanishes over P3(K). Usingthe Bramble Hilbert lemma and the Leibniz rule for derivating a product yields:

∣∣∣∆(bvu)∣∣∣ ≤ C 2∑j=0

|b|4−j,∞,K |v|j,K

|u|0,K (4.49)

For j ∈ 0, 1, we have:

|v|j,K ≤ J1/2K |DFE |

j

0,∞,K|J−1E |

1/2

0,∞,K|DF−1

K |j |p′|1+j,E

≤ |DFE |j0,∞,K |J−1E |

1/2

0,∞,K|DF−1

K |j |DFK |1+j‖p′‖1+j,E

(4.50)

Since FK is ane and p′ ∈ P2(E):

|v|2,K ≤ |J−1E |

1/2

0,∞,K|D2FE |0,∞,K |DFK |

2 |DF−1K | ‖p′‖2,E (4.51)

Moreover: |b|4−j,∞,K ≤ 6min4−j,2∑

l=0

|a|4−j−l,∞,K |JE |l,∞,K . Given that FE ∈ P2(K) and

|DlFE | ≤ Chl for l ∈ 0, 1, 2, one could easily prove |a|4−j−l,∞,K ≤ Ch4−j−l‖a‖4,∞,E .Since JE ∈ P2(K), we get |JE |l,∞,K ≤ C|JE |0,∞,K by virtue of the equivalence of all

norms over P2(K), which leads to:

|b|4−j,∞,K ≤ C|JE |0,∞,Kh2−j‖a‖4,∞,E (4.52)

Combining (4.49), (4.50), (4.51) and (4.52), using the estimates of remark (4.8) and thedenition (4.30) of a slender subcell, it follows that:

∣∣∣∆(bvu)∣∣∣ ≤ Ch4‖p′‖2,E |p|1,E‖a‖4,∞,E (4.53)

Given (4.44) and (4.53), we may conclude:

∆E

(a∂ip

′∂jp)≤ Ch2‖p′‖2,E |p|1,E‖a‖4,∞,E (4.54)


Step 4: Proof for regular subcells. The proof follows the same line than in Step 3 but forlinear forms vanishing over P1(K) instead of P3(K). Equation (4.49) becomes:

∣∣∣∆(bvu)∣∣∣ ≤ C 2∑j=0

|b|2−j,∞,K |v|j,K

|u|0,K (4.55)

Again, the derivatives of b may be estimated by a Leibniz formula, but this time aclassical estimate |JE |l,∞,K ≤ Chl+2 is used to estimate the derivative of the Jacobian.Along with remark 4.8, it holds:

|b|2−j,∞,K ≤ Ch4−j‖a‖2,∞,E (4.56)

This time the inverse properties of denition 4.9 may be used to evaluate (4.50) and(4.51) in |v|j,∞,K ≤ Ch

j‖p′‖2,E , which leads to (4.53) and (4.54).

Remark 4.15. Note that the property |JE |l,∞,K ≤ Chl|JE |0,∞,K would have improvedestimate (4.52) by two orders and decreased by as much the required quadrature scheme,but we failed to prove it for all slender subcells.

Having evaluated the quadrature error in the integral dening the stiness matrix, weshall now estimate the one dening the volume forces:

Proposition 4.16. (generalization to X-FEM of [19], Theorem 4.4.5). Let p = 2 andg = 2, E ∈ Sh, K ∈ Th such that E ⊂ K and assume that f ∈ W 1,∞(E). Underassumptions (H11), (H12, k=6) for crescent-shaped or slender subcells and (H12, k=4)for regular subcells; we have ∀p ∈ P2(E): |∆E (fp)| ≤ Ch2‖p‖0,E‖f‖1,∞,E.

In case p = 1 and g = 1, under the assumption (H11), (H12, k=0) and f ∈ L∞(E), wehave ∀p ∈ P1(E), |∆E (fp)| ≤ Ch‖p‖0,E‖f‖0,∞,E . The proof is left to the reader.

Proof. Step 1: proof for crescent-shaped or slender subcells.

The proof is analogous to that of proposition 4.14, Step 2. We have ∆E (fp) = ∆(f v)

with f := f FE and v := (p FE) JE . Since v ∈ P6(K), by (H12, k=6) the continuouslinear form ϕ→ ∆ (ϕv) vanishes over P0(K), so after the Bramble-Hilbert Lemma:

∣∣∣∆(f v)∣∣∣ ≤ C|f |1,∞,K |v|0,K≤ C|DFE |0,∞,K |JE |

1/2

0,∞,K|p|0,E‖f‖1,∞,E

≤ Ch2|p|0,E‖f‖1,∞,E

(4.57)


Step 2: proof for regular subcells. We have ∆E (fp) = ∆(f v) but with f := (f FE) JEand v := pFE . Since v ∈ P4(K) and (H12, k=4), the continuous linear form ϕ→ ∆ (ϕv)

vanishes over P0(K). The Bramble Hilbert lemma gives∣∣∣∆(f v)

∣∣∣ ≤ C|f |1,∞,K |v|0,K .Then |f |1,∞,K ≤ |DFE ||f |1,∞,E |JE |0,∞,K + |f |0,∞,E |JE |1,∞,K and |v|0,K ≤|J−1E |

1/2

0,∞,K|p|0,E , and we get the result after the properties of denition 4.9.

4.5.3 Geometry inuence on the error

We now focus on consistency error (c) in (4.27), which is the most interesting, and arisesby approximating the geometry. Excluding lengthways intersected elements from theproof, we show that (c) is expressed as εh−1/2, ε being the geometrical error measureof section 4. However, the numerical experiments in section 6 and the extended proofin Appendix C suggest that the estimates holds even for those elements in practice. Arecap of all results then yields the nal estimates:

Theorem 4.17. (Inspired from [136], section 4.4). Assume (H11) and a quadraturescheme adapted to the regularity properties of the subcells (see propositions 4.11, 4.14 and4.16 for full details). The interface is supposed to be ε-resolved by the subdivision, withno lengthways intersected triangles. Suppose A ∈ W 2,∞ (Ω1 ∪ Ω2), f ∈ W 1,∞ (Ω1 ∪ Ω2)and u ∈ Hp+1 (Ω1 ∪ Ω2), then the following error estimate holds:

‖u− uh‖1,Ωh1∪Ωh2≤ Chp [‖u‖p+1,Ω (1 + ‖A‖p,∞,Ω) + ‖f‖p−1,∞,Ω]

+ Cε

h1/2(‖f‖0,∞,Ω + ‖u‖1,∞,Ω‖A‖0,∞,Ω)

(4.58)

Proof. Step 1: Recap the results. Due to propositions 4.11, the approximate bilinearforms ah are uniformly Vh-elliptic, so the abstract error estimate (4.27) of proposition4.7 may be used. By letting vh = Πhu in this estimate, we may estimate the interpolationerror (4.27a) by proposition 4.13. Besides, all requirements are met to use the consistencyestimate of proposition 4.14, which summing up the results over all subcells and usingthe stability properties of Πh gives (in case p = 2):

|ah(Πhu,wh)− ah(Πhu,wh)|‖wh‖1,Ωh1∪Ωh2

≤ Ch2‖u‖2,Ω1∪Ω2‖A‖2,∞,Ω1∪Ω2 (4.59)

By the same token, the second term in (4.27b) is estimated by theorem 4.16.

Step 2: The error due to the change of domain. Let us estimate terms (4.27c), whichare due to the use of a non-conforming nite element method because of the change indomain. Appealing to the section 4.4 of [136], the last two terms may be written asintegrals over the skin (the part of Ωi which does not belong to Ωh

i and conversely):


ah (u, wh)− a (u, wh) =2∑i=1

(∫Ωhi \Ωi

A∇ui∇whdx −∫

Ωi\ΩhiA∇ui∇whdx

)(4.60)

The Cauchy-Schwarz inequality on the above equation yields (as a reminder, Sε is the2ε-wide strip centered on Γ, see g.4.14):

|ah (u, wh)− a (u, wh)| ≤ meas(Sε)1/2‖A‖0,∞,Ω

(‖u‖1,∞,Ω1∪Ω2 |wh|1,Sε∩(Ω1∪Ω2)

+ ‖u‖1,∞,Ωh1∪Ωh2|wh|1,Sε∩(Ωh1∪Ωh2)

) (4.61)

It is obvious that meas(Sε) = O(ε), and we may use Berger's lemma (see [136], Lemma2.2), which states that the integral of a polynomial function over a subdomain of itsoriginal denition domain is controlled by the ratio of the areas multiplied by the integralover the whole domain:

‖wh‖21,Sε∩Ωi∩K ≤ Cmeas(Sε ∩ Ωi ∩K)

meas(Ωi ∩K)‖wh‖21,Ωi∩K (4.62)

x3K

x 2K

x1K

i∩K∩S

S

∼(1−am)h

x3K

x 2K

x1K

i∩K∩S

S

∼am h

∼ aM h

~1

~h

a. side of the top subcell b. other side

~1 i∩K

i∩K

ϵ a M locallyϵ aM locally

Figure 4.14: Intersected triangle and measures of interest.

In order to estimate the ratio, equivalent measures to the quantities of interest arerepresented on g.4.14:

with assumption (H11) that the mesh is regular, the angles of the parent triangleK are bounded away from zero,

the interface Γ is globally ε-resolved by the subdivision, but as we have seen inremark 4.4, we have a better precision when considering the intersection with Konly: it is locally εaM -resolved by the subdivision,


the length of this intersection Γ ∩K is equivalent to aMh.

We may now evaluate the area of the dark grey strip on g.4.14, as we haveestimated its width and length respectively from the last two points. It readsmeas(Sε∩Ωi∩K) ≤ C(εaM )aMh. If i is the side of the triangular subcell (see g.4.14a),then it comes meas(Ωi ∩ K) ≥ c(aMh)amh. Otherwise (see g.4.14b), we may assertmeas(Ωi ∩K) ≥ c(1− am)h2 ≥ c(1− am)aMh

2. Hence, it comes:

meas(Sε ∩ Ωi ∩K)

meas(Ωi ∩K)≤ C ε

h

aMminam, 1− am

(4.63)

At this stage, we come across the denition 4.5 of lengthways intersected triangles.Indeed, for triangles which are not lengthways intersected, denition 4.5 states that

aMminam,1−am is bounded, so that by combining (4.62) with (4.63):

‖wh‖21,Sε∩Ωi∩K ≤ Cε

h‖wh‖21,Ωi∩K (4.64)

As a remark, we may still derive this property on the side of the intersection withthe larger area (g.4.14b) for lengthways intersected triangles. On the contrary, onthe smaller side (g.4.14a), no such property can be established. Then, decomposing‖wh‖21,Ωi∩K ≤ ‖wh‖

21,(Ωi\Sε)∩K + ‖wh‖21,Ωi∩Sε∩K , and making use of (4.64), we have

‖wh‖21,Ωi∩K ≤ ‖wh‖21,Ωhi ∩K

+C εh‖wh‖

21,Ωi∩K . Given theorem 4.3, ε is at least a O(h2), so

for h small enough, it comes:

‖wh‖1,Ωi∩K ≤ C‖wh‖1,Ωhi ∩K (4.65)

Combining (4.65) and (4.64), summing up over all intersected elements, and replacingthe result in (4.61), we obtain:

|ah (u, wh)− a (u, wh)|‖wh‖1,Ωh1∪Ωh2

≤ C ε

h1/2‖A‖0,∞,Ω‖u‖1,∞,Ω1∪Ω2 (4.66)

Following the very same procedure, one shows that:

∣∣l(wh)− lh(wh)∣∣

‖wh‖1,Ωh1∪Ωh2

≤ C ε

h1/2‖f‖0,∞,Ω (4.67)

which ends the proof.


4.6 Numerical experiments

The ability of the X-FEM analysis to catch the analytical solution is assessed throughthe relative energy error, namely:

energy error =

√∫Ωh1ε (u1 − uh) : A : ε (u1 − uh)√∫

Ω1ε (u1) : A : ε (u1)

(4.68)

This is equivalent to theH1(Ωh

1 ∪ Ωh2

)-norm considered in section 5: as in [135, 136], uh is

compared with an extension u rather than with u. Comparing thus solutions domainwise,we study in practice the impact of an approximated geometry on the solution over adomain of interest: the matter, as opposed to the holes. This corresponds to the actualengineering concern. The expected convergence rates are:

DisplacementRepresentationorder g of thegeometry

Assumedaccuracy of

thequadraturescheme

Convergenceorder ofthe

interfaceresolution ε

Consistencygeometry:order ofεh−1/2

Displacementinterpolationerror rate p

Energy

error rate

rate

P1 1 1 2 1,5 1 1

P2 1 2 2 1,5 2 1,5

P2 2 2 3 2,5 2 2

Table 4.1: Expected rates of convergence as a function of the displacement, geometryand quadrature according to (4.59)

4.6.1 Plate with a hole under tension

In this test, tested with an X-FEM formulation in [49], an innite plate with a centralhole is subjected to a uniform traction σ along the x-axis. In plane stress, the analyticalsolution of the problem, as available in [49], is:

σxx (r, θ) = σ

(1− a2

r2

[3

2cos (2θ) + cos (4θ)

]+

3

2

a4

r4cos (4θ)

)(4.69)

σyy (r, θ) = σ

(−a

2

r2

[1

2cos (2θ)− cos (4θ)

]− 3

2

a4

r4cos (4θ)

)(4.70)

σxy (r, θ) = σ

(−a

2

r2

[1

2sin (2θ) + sin (4θ)

]+

3

2

a4

r4sin (4θ)

)(4.71)

In practice, a square plate is considered of length 2 m with a hole radius a = 0.4 m.A contour traction is prescribed on its edges, immediately deduced from the analyticalstress (see [49, 123]). To prevent any rigid-body motion, the displacement is preventedalong the x-axis at the middle of the upper and lower edges, and along the y-axis atthe middle of the lateral edges (see g.4.15). The material is isotropic and elastic, withparameters E = 105 Pa and ν = 0.3.


The orders of convergence for P2 or P1 interpolation and subdivision were close to thetheoretical values (see g.4.16). However, and in accordance with the results in [133, 134],the case of a P2 displacement with a P1 subdivision yielded a superconvergent rate 1.87,closer to 2 than to the theoretical value 1.5, though values themselves are signicantlyhigher. This superconvergence probably occurs due to symmetry eects related to thecircular surface. To highlight this fact, the geometry description is perturbed by arandomly distributed quantity O

(h2), corresponding to the geometry error determined

in section 4 for a linear subdivision. We then observed a recovered theoretical rate 1.5(see g.4.16).

xx

− xy

xy

yy

− xx

xy

x

yr

Figure 4.15: Numerical test: model and loading.

Moreover, several quadrature rules were tested over the subcells, from 3-point to 12-pointGauss schemes. Very little inuence could be observed, as can be seen on g.4.16. Giventhat 3-point Gauss rule exactly integrates elements of P2(K), it seems like an exactintegration of elements in P2(K) suces to yield optimal orders of convergence, whichis precisely the case in FEM (see [135]).


0,010,05

1E-4

1E-3

1E-2

1E-1

1E+0

Numerical test

Plate with a hole in traction

Linear displacement and subdivision. Slope 1,04

Quadratic displacement – linear perturbed subd. Slope 1,54

Quadratic displacement – linear subdivision. Slope 1,87

Quadratic displacement and subdivision. 3 Gauss points Slope 1,88

Quadratic displacement and subdivision. 12 Gauss points Slope 1,88

mesh size (m)

en

erg

y e

rro

r

Figure 4.16: Numerical results for a problem with a strong discontinuity.

4.6.2 Plate with an elliptic hole

This time, an elliptic hole is inserted into the plate (see g.4.17), which is subjected tosurfaces forces g = σ · n where σ obeys (4.69-4.71) and n is the normal vector to theellipse. In this way, the analytical solution (4.69-4.71) still applies.

The observed convergence rate are close to their predicted values (see g.4.18), thusensuring the optimality of the estimates. In particular, the distinction between linearand quadratic subdivision is more obvious, with respective convergence rates 1.54 and1.94 close to their theoretical values.

xx

− xy

xy

yy

− xx

xy

0,8

y

0,5

g

x

Figure 4.17: Plate with an elliptic hole: geometry and loading.


0,010,051,0E-5

1,0E-4

1,0E-3

1,0E-2

1,0E-1

1,0E+0Linear displacement and subdivision. Slope 1,01

Quadratic displacement – linear subdivision. Slope 1,54

Quadratic displacement and subdivision. Slope 1,94

mesh size (m)

ener

gy e

rror

Figure 4.18: Plate with an elliptic hole: results.

4.6.3 A practical study of problematic cases

Let us now discuss practical aspects, to see if the demonstration shortcomings reallyinduce problems or are just theoretical subtleties. Let us rst investigate wether a poorintegration scheme actually induces outstanding quadrature errors in badly shaped sub-cells, that is to say slender or crescent-shaped subcells, according to denition 4.9. Wehave plotted on gures 4.19 and 4.20 the local relative integration errors for all curvedsubcells, respectively with respect to the aspect ratio hE/ρE of the subcell and ratio ofthe jacobians |J−1

E |0,∞,E |JE |0,∞,K : the rst is high for a slender subcell (g.4.19) andthe other is high for a crescent-shaped subcell. Two conclusions emerge from gs.4.19and 4.20.

1. The quadrature error in slender subcells appears to be no larger than that of reg-ular subcells, when tested with the rather poor quadrature scheme recommendedfor the latter. Hence, the quadrature in slender subcells seems to require no par-ticular treatment. The reader is referred to remark 4.15 for an explanation of thephenomenon.

2. The quadrature error for crescent-shaped subcells appears rather higher than theother values, though there are not enough points to assert it with certainty. For ourstrategy, there are indeed very few of them, too few to have a signicant impact onthe error. However, for a subdivision strategy that would involve a lot of crescent-shaped subcells, suboptimal rates may be obtained if the quadrature scheme is notincreased, as has been observed in [123].


1E+00 1E+01 1E+02 1E+03 1E+04

1E-14

1E-12

1E-10

1E-08

1E-06

1E-04

1E-02

1E+00

aspect ratio of the subcella high ratio indicates a slender subcell

rela

tive

inte

grat

ion

erro

r fo

r 3

Gau

ss p

oint

s

Figure 4.19: Quadrature error:crescent-shaped subcells.

1E+0 1E+1 1E+21E-14

1E-12

1E-10

1E-08

1E-06

1E-04

1E-02

1E+00

ratio of the Jacobiansa high ratio indicates a crescent-shaped subcell

rela

tive

inte

grat

ion

erro

r f

or 3

Gau

ss p

oint

s

Figure 4.20: Quadrature error: slen-der subcells.

Secondly, we recall that we have proven convergence properties with respect to thegeometry error in theorem 4.17 excluding the case of lengthways intersected triangles(see denition 4.5). Denoting N the number of intersected triangles K, let ∆EK =∫

Ωh1∩Kε (u1 − uh) : A : ε (u1 − uh) dΩ, EK =

∫Ωh1∩K

ε (u1) : A : ε (u1) dΩ and E =

N∑K=1

EK/N . We have plotted on g.4.21 the local relative energy errors ∆EK/E for the

above problem of a plate with a hole, as a function of the characteristic ratio of theintersection (4.22), which is high for lengthways intersected triangles. The normalizationhas been done with the average local energy on tested elements, so as to prevent thespurious eect of higher local interpolation errors on small sections mentioned in section5.

On g.4.22, we have considered another test of a full plate (no hole) under uniaxialtension, whose boundary has been perturbed according to the O

(h2)error measure

determined in section 4 for linear subdivision. As the exact displacement is spannedby the basis, there is no interpolation in this case, so we have directly plotted the localrelative error given by ∆EK/EK . Both on gures 4.21 and 4.22, we see that errors donot increase for lengthways intersected triangles, so this suggests that the exclusion oflengthways intersected triangles from the theoretical estimate is not a practical issue.We propose in Appendix C a proof of convergence for lengthways intersected trianglesin a simplied case: this analysis shows that the only problematic case will be detected


beforehand by the subdivision process, so that there will be no negative impact over theconvergence.

1E+0 1E+1 1E+2

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

characteristc ratio of the intersectionA high ratio indicates a lengthways intersected triangle

loca

l re

lativ

e e

ne

rgy

err

or

Figure 4.21: The eect of lengthwaysintersected elements: test 1.

1E+0 1E+1 1E+2 1E+31E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

characteristic ratio of the intersectionA high ratio indicates a lengthways intersected triangle

Lo

cal r

ela

tive

err

or

Figure 4.22: The eect of lengthwaysintersected elements : test 2.

4.7 Conclusion

The practical conclusion of these special cases of study is that a subdivision with subcellswith the same order than the nite-element mesh, used along with the classical relatedquadrature scheme, is successful to yield optimal orders of convergence. As for thetheory, theoretical a-priori error estimates were derived in this paper for the X-FEMmethod which take into account quadrature eects and the description of the geometry,and can anticipate the convergence rate of a formulation, and predict whether it will beoptimal or not. Our estimates hold for linear or quadratic elements, but we believe thatthe methodology can easily be extended to higher orders.

Chapter 5

Interface problems with quadratic

X-FEM: design of a stable

multiplier space and error analysis

This chapter has been accepted for publication in International Journal for NumericalMethods in Engineering [142].


Ce chapitre se propose d'étudier la convergence de la méthode X-FEM, en particulier pourdes éléments quadratiques, mais pour des interfaces adhérentes, par exemple séparantdeux matériaux (inclusion...). Pour de tels problèmes, le champ d'intérêt est continuau travers de l'interface, mais son gradient est discontinu, ce qui rend en réalité larésolution plus complexe que les problèmes à discontinuité forte du chapitre 4. Unepremière stratégie consiste à utiliser une fonction d'enrichissement dont la dérivée estdiscontinue, une seconde stratégie à utiliser des méthodes de stabilisation (Nitsche...), etnous donnons une courte bibliographie sur le sujet.

Nous choisissons une troisième solution: l'utilisation de multiplicateurs de Lagrange. Lescomposantes d'un multiplicateur sont dénies aux n÷uds extrémités des éléments inter-sectés, et le champ de multiplicateurs est déduit par interpolation sur ces éléments ettrace sur l'interface (voir g.5.1). L'espace de multiplicateurs de Lagrange est soumisà une condition numérique de stabilité, dite condition inf-sup. Sa vérication nécés-site d'appauvrir l'espace de multiplicateurs en attribuant le même DDL de Lagrange àplusieurs n÷uds: en d'autres termes, des relations d'égalité sont imposées entre les com-posantes nodales (voir g.5.2). Pour les éléments linéaires, un espace (P1-) a été conçupar [13] en ce sens. Un nouvel espace (P1*) de multiplicateurs dédié à un usage avec deséléments quadratiques est déni en section 5.3, et la stabilité numérique de cette asso-ciation (condition inf-sup) est démontrée (théorème 5.5). Les propriétés d'interpolationde cet espace sont établies: en norme L2, un champ est quasiment approximé avec uneprécision d'ordre h2. Pour construire cet espace, on dénit le minimum de relationsd'égalité qui évite les traces nulles ou quasiment nulles: en pratique, on dénit des re-lations d'égalité sur les arêtes intersectées à proximité d'un n÷ud (voir g.5.5), et on

125

Chapter 5. X-FEM for curved interface problems 126

met une relation d'égalité par composante du graphe (des arêtes intersectées) qui necomporte pas déjà de telles arêtes (voir g.5.6). De cette façon, on s'est ramené à unespace semblable à une formulations P2/P1 d'éléments nis classiques, et on a prouvéque ceci était en fait susant pour assurer la stabilité numérique.

Des estimations d'erreur a priori sont ensuite établies (section 5.5), sous l'hypothèse destabilité inf-sup, en fonction des propriétés d'interpolation du multiplicateur de Lagrangeet du déplacement, mais également de la description de la géométrie courbe. Ces estima-tions sont résumées en (5.64-5.65) et dans les tableaux 5.1 et 5.2. Quand on les confronteà des résultats numériques, certaines de ces estimations se révèlent sous-optimales, maisles facteurs manquants sont documentés par la littérature. Le premier test numériqueest un bloc avec interface droite sous pression cubique. Il permet de valider les pro-priétés d'interpolation des deux espaces, et surtout leur inuence sur la convergence duproblème. On voit qu'avec des éléments quadratiques, l'espace (P1-) produit une conver-gence sous-optimale en h pour la norme L2 du multiplicateur et en h3/2 pour la normeH1 du déplacement, quand l'espace (P1*) permet de recouvrer l'optimalité avec uneconvergence en h2 pour ces deux grandeurs (voir g.5.17). Le second test consiste en uneinclusion circulaire sous un chargement biaxial: pour ce test, le déplacement analytiqueest dans la base d'interpolation, ce qui permet d'isoler les termes liés au choix de l'espacede multiplicateurs à à la géométrie courbe, et de valider les estimations correspondantes.En plus de conrmer l'eet du choix de l'espace, il montre qu'une découpe linéaire en-traine une convergence sous-optimale, toujours en h pour la norme L2 du multiplicateuret en h3/2 pour la norme H1 du déplacement (voir gs.5.18 et 5.19). Le troisième test estun test de compression de deux anneaux emboîtés de matériaux diérents. Il permet devalider les estimations sur un vrai bimatériau, et conrme les observations précédentes:avec des éléments quadratiques, l'obtention d'une convergence en h2 nécessite à la foisune découpe quadratique et l'utilisation de l'espace (P1*) (voir g.5.22).

The aim of this paper is to propose a procedure to accurately compute curved inter-faces problems within the extended nite-element method (X-FEM) and with quadraticelements. It is dedicated to gradient discontinuous problems, which covers the case ofbimaterials as the main application. We focus on the use of Lagrange multipliers to en-force adherence at the interface, which makes this strategy applicable to cohesive laws orunilateral contact. Convergence then occurs under the condition that a discrete inf-supcondition is passed. A dedicated P1 multiplier space intended for use with P2 displace-ments is introduced. Analytical proof that it passes the inf-sup condition is presentedin the two-dimensional case. Under the assumption that this inf-sup condition holds, apriori error estimates are derived for linear or quadratic elements, as functions of thecurved interface resolution and of the interpolation properties of the discrete Lagrangemultipliers space. The estimates are successfully checked against several numerical ex-periments: disparities, when they occur, are explained in the literature. Besides, thenew multiplier space is able to produce quadratic convergence from P2 displacements andquadratic geometry resolution.


5.1 Introduction

Solving interface problems with the classical nite element method requires a considerablemeshing eort when interfaces have complex geometries. Hence, to reduce the burdenof meshing, Moës, Dolbow and Belytschko [63] proposed an extended nite elementmethod (X-FEM) allowing surfaces of discontinuities within elements. The discontinuityis described by a local enrichment of the basis functions on crossed-through elementswith discontinuous functions across the interface, abiding by the rules of the partition ofunity method in [64].

In weak discontinuity problems, such as heat conduction or linear elasticity in bimate-rials, a continuous eld of interest is searched but with a discontinuous gradient acrossthe interface. Those problems are actually more tricky to handle than strong disconti-nuities: while fully uncoupling elds on either sides basically consists in doubling somedegrees of freedom around the interface, and is in principle easy to implement; weak dis-continuity problems bring about additional issues by demanding eld continuity acrossthe interface, which could be viewed as some partial uncoupling. There are basicallythree approaches for enforcing that continuity. The rst consists in using continuous butgradient-discontinuous functions across the interface. The other two consist in enforc-ing the continuity in a weak sense either with Nitsche's stabilization method, or withLagrange multipliers. We shall focus on the latter approach in this paper, after givingsome background details about the others.

Sukumar and coworkers [49] were the rst to consider using X-FEM to account forinclusions. They would then take the absolute value of the level-set function as theenrichment function. With this enrichment, they would observe suboptimal convergencerates, due to poor properties on the elements adjacent to those intersected . Theywould then modify the enrichment to even the enrichment function on those elementsand then almost get optimal rates. Moës and coworkers [143] then proposed a modiedabs-enrichment, named ridge enrichment, that does not require any treatment for such blending elements and therefore yields optimal rates. Aiming at a general methodthat would be applicable to other enrichment than the ridge, Fries and al [144] proposed acorrected XFEM formulation, which works out the set of enriched nodes and associatedshape functions rather than a specic enrichment function. Meanwhile, Legay and al[121] used enhanced strain techniques to tackle the issue, inspired from assumed strainstrategies in [127, 145].

In the same article [121], the authors rst used higher-order X-FEM for weak discon-tinuity problems without assumed strain, or any other specic handling of blendingelements, but by multiplying the enrichment with shape functions one degree lower thanthose of the classical part. Finally, Cheng and Fries [122] adapted standard and mod-ied abs-enrichment, as well as corrected XFEM for higher orders, compared the threeand concluded that at higher orders, the corrected XFEM would behave better than thestandard one, the latter converging faster than the modied abs-enrichment.

These gradient-discontinuous displacement enrichment strategies have the advantage ofproducing discrete solutions which exhibit zero jump across the interface exactly, by con-struction of the enrichment functions. On the contrary, the multiplier approach enforcesthis condition weakly, so that the discrete jump is not necessarily exactly zero across theinterface. However, designing a gradient-discontinuous enrichment that preserves opti-mality in blending elements even at higher orders and for curved interfaces is somehow


cumbersome. This consideration motivated the rst use of Lagrange multipliers to ac-count for material interfaces in [146], which was recently further developed in [147]. Themultiplier method is moreover prone to extensions to other interface laws (e.g. unilateralcontact or cohesive law).

Ji and Dolbow [90] were the rst to introduce Lagrange multipliers in the X-FEM. Theyreported that a naive discretization triggers oscillations of the multipliers and a loss ofconvergence, and so does high-stiness penalization. The critical point lies in a discreteinf-sup condition which determines the stability of the formulation. In X-FEM, thiscondition is even more technical to verify due to the non-conformity of the mesh, and isviolated for a naive discretization. There are several ways to restore it in the literature,namely:

1. an enrichment of the displacement basis with bubble function on the interface, in[92, 148, 149],

2. the denition of a reduced Lagrange multiplier space, which was improved in asequence of three papers by Moës and al [150], Géniaut and al [93], Béchet andal [13]. These spaces are designed for use with linear elements, and we discussextension to quadratic elements in this paper.

These are called stable formulations, in that they work out the discrete spaces directly,as opposed to stabilized formulations which circumvent the inf-sup condition with amodication of the variational form by:

1. an adaptation of Nitsche's approach [151] to the X-FEM by Hansbo [152], furtherdeveloped by [94, 149] and [153]. In this approach no Lagrange multiplier is intro-duced in the formulation: the normal ux directly plays its role, and a stabilizationterm on the jump is added to avoid oscillations ;

2. the addition of a stabilization term on the multiplier/ux discrepancy into theweak formulation by [95, 154], based on Barbosa and Hughes approach [155]. Aconnection was made by Stenberg [156] between this approach and Nitsche's one[151] ;

3. a three eld approach with ellipticity-enhancing terms by Gravouil and al [157].

Dealing more specically with curved interfaces, several authors reported a loss of opti-mality of the convergence rate with higher-order X-FEM when approximating the inter-face geometry by linear segments (see [121, 122, 133, 134]). Ranging from a geometricaldescription on a ner submesh ([133, 134] to curved subcells ([121, 122]), successfulremedies to the problem were proposed. To say things briey, we adopted the geom-etry description suggested by Legay and coworkers [121] with a higher-order level-setinterpolation and curved quadratic triangles as subcells.

Studying interface problems with X-FEM and Lagrange multipliers, possibly at higherorders and/or with curved interfaces, our task is threefold:

1. design a both optimal and stable reduced multiplier space, suited for use togetherwith quadratic displacements,


2. derive theoretical a priori estimates for stable formulations of X-FEM with La-grange multipliers to our knowledge, this has solely been done for stabilizedformulations up to now,

3. include the inuence of the geometry representation in those estimates.

In section 3, the new P2-dedicated discrete multiplier space is introduced. Analyticalproof that it passes the stability inf-sup condition is presented in the two-dimensionalcase. In section 4, the description of the geometry is presented and some results arerecalled about assessing its accuracy. In section 5, the a priori estimates are derived, andhereby the inuence on the convergence rate of the aforementioned geometrical descrip-tion and the interpolation properties of the multiplier space is investigated. Numericaltests are carried out in section 6, and numerical orders are compared with their theoret-ical counterparts. Based on these tests, some theoretical predictions appear suboptimal,down h1/2, but the reason for this appears documented by [120, 158]. We show, for aquadratic displacement, that the combination of a quadratic geometrical description ofthe interface with the new restriction algorithm yields optimal convergence in practice,with an order 2.

5.2 Formulation of the continuous problem

The denition is the same as in section 4.2, but this time, the interface Γ typicallycorresponds to a change in material properties: in addition to equilibrium equations(4.1-4.3) in Ωi, the following continuity and transmission conditions across the interfacehold:

[u] = 0 on Γ (5.1)

[σ · n] = 0 on Γ (5.2)

where for x ∈ Γ, [u] (x) = u2(x)− u1(x) is the displacement jump.

In this chapter, C will denote a generic nonnegative constant, and c a positive constant.

The components of the solution to problem (4.1-4.3, 5.1-5.2) are assumed to belong

to the functional space V :=v ∈ H1(Ω1)×H1(Ω2), v|Γiu = 0, i ∈ 1, 2, [v]|∂Γ = 0

,

where ∂Γ denotes the extremities of Γ. Closed curves are included in this formalism bythe convention that ∂Γ = ∅ for them. We call H1/2 (Γ) the trace Sobolev-Slobodeckijspace, which is endowed with the Slobodeckij norm (see e.g. [139] or [159]):

‖w‖21/2,Γ := ‖w‖20,Γ +

∫Γ

∫Γ

|w(x)− w(y)|2

|x− y|2dxdy (5.3)

For open curves, H1/200 (Γ) ⊂ H1/2 (Γ) is introduced as the subspace of functions which

can be extended by 0 as they approach the extremity points ∂Γ. It is endowed with thenorm (see e.g. [139], section 1.1):


‖w‖H

1/200 (Γ)

:= ‖w‖21/2,Γ +

∫Γ

1

dist (x, ∂Γ)|w(x)|2dx (5.4)

For closed curves, we set the convention H1/200 (Γ) := H1/2(Γ). For any v ∈ V , owing to

the fact that [v]|∂Γ = 0, we have [v] ∈ H1/200 (Γ). Let λ := σ ·n be the surface force that

Ω2 applies to Ω1 at the interface. Its components then belong to H−1/2 (Γ) which is the

topological dual space of H1/200 (Γ), thus equipped with the operator norm:

‖λ‖−1/2,Γ := supw∈H1/2

00 (Γ)

∫Γ λwds

‖w‖H

1/200 (Γ)

(5.5)

We then have an equivalence of norms, in the sense that for any v ∈ V ,

‖[v]‖H

1/200 (Γ)

≤ C‖v‖1,Ω1∩Ω2 and for any w ∈ H1/200 (Γ), there exists a v ∈ V such that

[v] = w and ‖v‖1,Ω1∩Ω2 ≤ C‖w‖H1/200 (Γ)

(see [160], Appendix A). In this way an equivalent

denition of H−1/2-norm is:

‖λ‖−1/2,Γ := supw∈V

∫Γ λ[w]ds

‖w‖1,Ω1∪Ω2

(5.6)

To obtain (5.6) while releasing the assumption that [v]|∂Γ = 0, the appendix A of [160]may be consulted. To give problem (4.1-4.3, 5.1-5.2) a weak formulation, in addition tobulk bilinear forms bilinear and linear forms (4.5-4.6), a surface bilinear form is intro-duced, which will be useful to enforce constraint (5.1), as:

For v ∈ V 2 and µ ∈M2, b (v,µ) :=

∫Γ[u] · µds (5.7)

where M := H−1/2 (Γ). The weak formulation of the problem then reads: nd u ∈ V 2,λ ∈M2 such that:

∀v ∈ V 2, a (u,v) + b (v,λ) = l (v) (5.8)

∀µ ∈M2, b (u,µ) = 0 (5.9)

For the sake of conciseness, we shall in the mathematical developments of this paperrather work with a scalar second-order elliptic problem this corresponds to a heatdiusion problem in a bimaterial. All theorems will be presented in their scalar version,but their vectorized counterparts could be derived in a similar way. In addition to bulkoperators (4.8-4.9), the scalar surface operator b ∈ L(V,M) is:

b (v, µ) :=

∫Γ[u]µds (5.10)

The multiplier λ := (A · ∇u)·n then corresponds to the normal heat ux on the interface.


5.3 Discretization of primal and dual spaces

In this rst phase, we focus exclusively on the use of Lagrange multipliers. So theinterface is assumed to be straight for now, and thus exactly described. We still assumethat Ω is meshed with a quasi-uniform and regular family Th of ane triangular meshes(assumption H11).

5.3.1 Discretization of the eld of unknowns

We call Eh the set of elements of Th cut by Γ, and Vh the corresponding intersectededges. Let Nh ⊂ N be the nodes of the mesh, each node being identied by a number.Nodes of elements in Eh will be denoted Kh ⊂ Nh , and the subset of vertex nodes byMh ⊂ Kh (see g.5.1). For an intersected edge q ∈ Vh, xq represents the position ofthe associated intersection point. Moreover, we denote qm ∈ Mh and qM ∈ Mh thenearest and farthest connected vertex nodes, respectively, and xmq , x

Mq their positions

(see g.5.1).

nodes in M h

q∈V h

K∈E h

Intersection points

xq

xqm

xqM

j∈K h

Figure 5.1: Some topological denitions around the interface.

Classically, the discrete eld space is then (see [63]):

Vh :=

∑i∈Nh

aiNi(x) +∑j∈Kh

bjNj(x)H(x), ai ∈ R, bi ∈ R

(5.11)

where H is the Heaviside-like function used to represent the jump:

H(x) =

−1 if x ∈ Ω1

1 if x ∈ Ω2(5.12)

5.3.2 Discretization of the multipliers

For the approximation to be independent of the crack location, the multiplier componentsare dened on the parent nodesMh, as in [13, 93, 95]. Then:

Mh :=

x ∈ Eh → ∑i∈Mh

µiNlini (x), µi ∈ R

(5.13)


where N lini is the linear shape function associated with vertex node i. The approximation

space Mh for the multipliers is then:

Mh = µ |Γ, µ ∈ Mh (5.14)

As mentioned in the introduction, the interpolation of the pressure multipliers shouldadditionnaly abide by a so-called discrete inf-sup condition to yield stable results (see[13, 150]):

∃c > 0 independent of h ,∀µh ∈Mh, supvh∈V h

∫Γ µh[vh]dΓ

‖vh‖1,Ω1∪Ω2

≥ ch1/2‖µh‖0,Γ (5.15)

This condition basically states that the discrete multiplier space should not be too richin comparison with the discrete displacement one. Otherwise, spurious oscillations ofthe pressure multipliers are observed (see [90]). Hence, equality relations have to beprescribed between some multiplier components onMh, thus reducing Mh, as in [13, 93].Two restriction algorithms have been considered (the rst from [13], the second is newlyproposed):

1. the algorithm of [13] with continuous piecewise linear or constant multipliers, thatwe will denote by the acronym P1. It is illustrated on g.5.2, and is intended tobe used along with a linear interpolation of the displacement. Such a formulationwill be denoted by P1/P1;

2. a new algorithm of P1 multipliers with less constraints, referred to as P1∗ fromhere onward. It veries P1 ⊂ P1∗ and is intended for a combined use along witha piecewise quadratic displacement and will be called P2/P1∗.

1 23

equality relations

nodes sharing a degree of freedom

Figure 5.2: Restriction algorithm P1−, as dened by Béchet and al [13].

5.3.3 Design of multiplier space P1*

As a higher-order interpolation of the displacement makes a given pressure interpolationmore likely to satisfy the inf-sup condition, the idea behind the constuction of algorithm(P1∗) was to release some constraints from (P1). Let us set a parameter κ 1 andintroduce the following preliminary denitions:

Denition 5.1. Closely intersected edge. Let q ∈ Vh be an intersected edge. It is mappedonto a reference segment q with an ane map, in such a way that qm be mapped onto 0,so that the reference coordinate aq of the intersection point veries aq ≤ 1/2, as shownon g.5.3. The edge is called closely intersected if aq ≤ κ.


Almost coincident node. n ∈ Mh is an almost coincident node if there exists a closelyintersected edge q ∈ Vh such that n = qm.

xqM

xqm

F q−1 (affine)

F q (affine)

Γ

aq edge q

node qm

node qM

xqm

0 1

xqM

q

Figure 5.3: Closely intersected edge.

Denition 5.2. Slanted triangle. Let us consider any K ∈ Eh. Two of its edges areintersected and the subcell containing their common node is called triangular subcell.This common node is mapped onto the origin of the reference triangle K (see g.5.4).Such a map is moreover chosen such that the rst reference coordinate correspondsto the largest coordinate aM among the intersection points, while the second referencecoordinate is associated with the smallest am (see g.5.4). We then call slanted a triangle

with an elongated triangular subcell, such thatamaM≤ κ.

x3K

x2K

x1K

Γ

F K−1 (affine)

a1

a 2

aM

am

KEKE

F K (affine)

x3 x1

x2

Figure 5.4: Intersected triangle.

Asymptotically speaking, we should rather call slanted family of meshes, for which ratioamaM

decreases with mesh renement and is not bounded away from zero as h→ 0.

Algorithm P1∗ will be built so as to prevent some basic aws high conditioning andredundancy in a rst stage, and in a second stage we will analytically prove that theresulting construction is actually sucient to pass the inf-sup condition.

Denition 5.3. Restricted multiplier space P1∗ dening Mh. The minimal restrictionalgorithm is as follows:

1. Initialization of a working set of edges Ch = Vh

2. For each almost coincident node n, for each closely intersected edge q emanatingfrom n (i.e. n = qm), an equality relation is assigned linking n to the oppositevertex qM , provided qM is not already involved in another relation.


3. All edges thus supporting an equality relation are removed from Ch .

4. Components of Ch, in the sense of the graph theory, are listed.

5. For each such component, if a vertex is already linked, no additional equality rela-tion is assigned (see g.5.5). If not, a single additional equality relation is assignedfor the whole component (see g.5.6), so as to preclude redundant combinations,as is explained in theorem 5.4.

Fig.5.5 and 5.6 provide examples of this algorithm with and without almost coincidentnodes. Note that no further relation is needed after dealing with almost coincident nodeson g.5.5.

edges supporting an equality relation

1st closely intersected node 2nd closely intersected node 3rd closely intersected node &final relations

intersected edges

other edges components of the graph of remaining edges

Figure 5.5: Building algorithm P1* with closely intersected edges.

edges finally supporting an equality relation

connected components on the graph of remaining edges

Figure 5.6: Minimal restriction algorithm without closely intersected edges.

5.3.4 Non redundancy

Any well-dened algorithm restriction requires the linear independance of the traces onΓ of the functions in Mh, otherwise the problem becomes singular. Singular combi-nations could be detected through the computation of the kernel of the mass matrix∫

ΓNlink N lin

l dΓ, as in [95]. Here, aiming at a topological construction of Mh, we willrather show that:

Theorem 5.4. Let µh ∈ Mh, if µh|Γ = 0 then µh = 0.


Proof. For L a Lagrange degree of freedom and L ⊂ Mh the set of nodes sharing it,we set ψL :=

∑l∈LN linl . Let us suppose a combination µh =

∑L

µLψL having zero trace.

The sum can be split into the connected components of the graph of Vh. For eachcomponent, after the minimal restriction algorithm dening P1∗, there exists an edgeq supporting an equality relation. Since µq := µh(xq) = 0, µh(xmq ) = µh(xMq ) = 0.It follows that for an edge q′ emanating from qm or qM , since µq′ = 0, we haveµh(xmq′ ) = µh(xMq′ ) = 0. The coecients are shown to be zero on all nodes of the con-nected component repeating the procedure iteratively. In other terms, we have proventhat for each connected component of the graph, without restriction algorithm, therewould be exactly one degenerate mode, which corresponds to the case where the multi-plier components are proportional to the signed distance to the interface.

5.3.5 Proof of the discrete inf-sup condition

We are now able to prove the inf-sup condition analytically:

Theorem 5.5. Assume Γ to be a straight line, and assume that none of the intersectedtriangle is slanted. Then, discretization (P2/P1∗) fullls the discrete inf-sup condition(5.15).

Proof. Step 1. As in [13], a subdivision of Γ is introduced for the sake of the proof: a singlerepresentative edge is selected among closely intersected edges emanating from a commonalmost coincident node, while non closely intersected edges are all selected. Segmentse with length he between the thus selected intersection points make a subdivision for Γ(see g.5.7).

he

e

Intersected edges

Edges emanating from a closely intersected node

Intersection points / edges in C h

Subdivision of

Figure 5.7: Subdivision of the interface.

Let µh ∈ Mh. We dene mesh-dependent norms as ‖µh‖2−1/2,Γh :=∑e∈Γ

he‖µh‖20,e and

‖µh‖21/2,Γh :=∑e∈Γ

1

he‖µh‖20,e.

Step 2. Let us prove the existence of c, C such that ch ≤ he ≤ Ch. Let e be an elementof the subdivision, K be the triangle such that e ∩ K 6= ∅ and such that λM > κ (seeg.5.8): reductio ad absurdum, it is obvious that such a triangle exists after the denitionof the subdivision.


x3

x1

x2e

K

hK

ρK

K∣x1−x3∣aM

Figure 5.8: Intersected element K.

Then he ≥ aM |x1 − x3| sin(θK) ≥ 2ρKκ sin(θK) (see g.5.8). As stated previously,regularity assumption (H1) ensures that θK is uniformly bounded away from zero, whichimplies that he ≥ ch. This also implies that a limited number N of triangles maybe crossed by element e, hence he ≤ 2Nh. An immediate consequence is that norms‖.‖−1/2,Γh and h1/2‖.‖0,Γ are equivalent.

Step 3. As has been done in [13], we look for vh ∈ Vh such that [vh] = µh, which giventhe displacement approximation (5.11) amounts to seeking the enriched DOF bn suchthat:

µh = 2∑n∈Kh

bnNn|Γ (5.16)

and such that those coecients be uniformly bounded by the neighboring pressure:

∃C, ∀n, |bn| ≤ C|µh|0,∞,Γ∩Sn (5.17)

where Sn denotes the support of n (see g.5.9).

nS n

∩S n

Figure 5.9: Support of a node n.

Let us classify the nodes in Kh (see g.5.10) :

Type 1: n is located on an edge supporting an equality relation, or is a node locatedexactly on the interface.


Type 2: n is a non type 1 vertex node.

Type 3: n is a non type 1 middle node, whose associated edge is intersected andboth ends of which are subjected to equality relations.

Type 4: n is a non type 3 middle node on an intersected edge.

Type 5: n is a middle node on a non intersected edge.

Type 1

Type 2

Type 3

Type 4

Type 5

Edge supporting a relation

Figure 5.10: Nodes classication.

Coecients bn were chosen such that:

Type 1: node n belongs to an edge q supporting an equality relation. We denote Lthe Lagrange degree of freedom thus shared by the whole edge: we have µq = µL.Let us set bn = µL/2, it implies [u](xq) = µq and (5.17) holds.

Type 2: we set bn = 0.

Type 3: Let q be the intersected edge n belongs to (see g.5.11). Let K and L bethe Mh degrees of freedom qM and qm are associated with (see g.5.11). We set

bn =µK + µL

4, which obviously satises (5.17): given that µq = (1− aq)µL+aqµK ,

we may check that [u](xq) = 2∑

l∈n,qM ,qmblNl(aq) = µq.

Type 1

Type 3

Edge supporting a relation

n

qm

qM μL

Edge q

μL Common Lagrange degrees of freedom

μL

μLμK

μK

μKμKμK

Figure 5.11: Node of type 3.

Type 4: Once again, q denotes the intersected edge n belongs to. bn is then

taken as 2bn =1

Nn(aq)

(µq − 2

∑l∈qM ,qm

blNl(aq)

). An easy calculation then yields

[u](xq) = µq. We know that q is not closely intersected (otherwise, n would be a


type 1 or 3). As a result, it holds aq ≥ κ and Nn(aq) ≥ 4κ(1− κ) since aq ≤ 1/2.Any vertex nodes at the end of q is either a type 2, in which case it bears azero as enriched degree of freedom, or a type 1, in which case it bears a Mh

degree of freedom µL. It implies (5.17) since 2|bn| ≤1

2κ(1− κ)maxµq, µL ≤

1

2κ(1− κ)|µh|0,∞,Γ∩Sn .

Type 5: At this stage, the displacement jump equals µh on all intersection points.So the remaining work to obtain (5.16) consists in making this jump linear in be-tween. Let K ∈ Eh be the intersected triangle n belongs to (see g.5.12). Denotinge := Γ ∩K, we propose some evidence that there can be found bn such that:

∑l∈Kh∩K

blNl|e ∈ P1(e) (5.18)

As the interface is straight and the mapping ane, e is described by

e : a2 − am +amaM

a1 = 0 (see g.5.12). Therefore, we may rewrite:

∑l∈Kh∩K

blNl|e =∑

l∈Kh∩KblNl

(a1,am

[1− a1

aM

])(5.19)

This expression is a second order polynomial of a1, and as such is linear if and onlyif the second-order coecient is zero. Calling bKi the enriched degree of freedomof the node located at xKi (see g.5.12) and replacing the shape functions by theirexpressions the corresponding equation turns into:

bn = bK12 =bK1 + r2bK2 + (1− r)2bK3 + 2r(1− r)bK23 − 2(1− r)bK13

2r(5.20)

where r := am/aM veries 0 < κ ≤ r ≤ 1 since slanted triangles have been

excluded. It follows that |bn| ≤3

κmax

i∈1,2,3,23,13|bKi | ≤ C|µh|0,∞,Γ∩Sn so (5.17) holds.

x3K

x 2K

x1K

ΓF K

−1 (affine)

a1

a2

aM

am

KK

F K (affine)

x13K

x12Kx 23

K

x3

x 23

x 2

x12

x1

x13

node n

e

Figure 5.12: Cut element K.

The displacement thus generated veries (5.17). Since it coincides with the pressure atthe intersection points and is piecewise linear in between, (5.16) holds. The demonstra-tion follows [13] from now on. It is briey recalled for the sake of self-suciency.


Step 4. Prove that ‖µh‖−1/2,Γh‖[wh]‖1/2,Γh ≤ C∫

Γ µh[wh]dΓ.

Since [wh] = µh|Γ, we have∫

Γ µh[wh]ds = ‖µh‖20,Γ. This implies that ‖µh‖−1/2,Γh‖[wh]‖1/2,Γh =(∑e,e′

heh′e‖µh‖20,e‖µh‖20,e′

)1/2

. Sinceheh′e

is bounded, as has been proven in step 2, it may

be concluded that ‖µh‖−1/2,Γh‖[wh]‖1/2,Γh ≤ C‖µh‖20,Γ = C∫

Γ µh[wh]dΓ.

Step 5. Prove that∑q∈Vh

µ2q ≤ C‖[wh]‖21/2,Γh.

Using the pullback onto the reference unit segment e for the mesh-dependent norms,it holds ‖µh‖21/2,Γh =

∑eh−1e ‖µh‖20,e =

∑e‖µh‖20,e . The equivalence of all-norms on the

nite dimensional space of linear functions on e yields ‖µh‖0,∞,e = ‖µh‖0,∞,e ≤ C‖µh‖0,e.Moreover an element e crosses a limited number of edges, given the regularity of the mesh,as has been pointed out in step 2, so that

∑q∈Vh

µ2q ≤ C

∑e‖µh‖20,∞,e, which gives the result

since [wh] = µh|Γ .

Step 6. Prove that ‖wh‖21,Ω1∪Ω2≤ C

∑q∈Vh

µ2q

It can easily be shown with a transformation onto the reference triangle that‖wh‖20,M ≤ Ch2

∑K∈Th

‖wh‖20,K . Since ‖wh‖20,K ≤ 4∑

i=1..6b2i ‖Ni‖20,K and the triangulation

is regular, we may assert that ‖wh‖20,Ω ≤ Ch2∑n∈Kh

b2n. An inverse approximation yields

‖wh‖1,Ω1∪Ω2 ≤ Ch−1‖wh‖0,Ω, so ‖wh‖21,Ω1∪Ω2≤ C

∑n∈Kh

b2n. Using step 3, this implies that

‖wh‖21,Ω1∪Ω2≤ C

∑n∈Kh

‖µh‖20,∞,Γ∩Sn . Each intersected edge belongs to a limited number

of supports Sn (with quadratic triangles, 9 is an upper bound for a shared edge), so∑n∈Kh

‖µh‖20,∞,Γ∩Sn ≤ 9∑q∈Vh

µ2q , which yields the result.

Step 7. Conclusion.

Steps 5 and 6 yield ‖wh‖1,Ω1∪Ω2 ≤ C‖[wh]‖1/2,Γh. Hence we may assert that:

supvh∈V h

∫Γ µh[vh]dΓ

‖vh‖1,Ω1∪Ω2

≥∫

Γ µh[wh]dΓ

‖wh‖1,Ω1∪Ω2

≥ c∫

Γ µh[wh]dΓ

‖[wh]‖1/2,Γh(5.21)

And due to step 4,

∫Γ µh[wh]dΓ

‖[wh]‖1/2,Γh≥ c‖µh‖−1/2,Γh. With the equivalence proven in step 2

between ‖.‖−1/2,Γh and h1/2‖.‖0,Γ, we nally have

∫Γ µh[wh]dΓ

‖wh‖1,Ω1∪Ω2

≥ ch1/2‖µh‖0,Γ.

5.3.6 Interpolation properties of the discrete multiplier spaces

An important point to the error analysis is to estimate the ability of the discrete multiplierspaces to approach a continuous function. Hence, we shall dene an interpolant πh of λonto Mh:


Denition 5.6. For L a degree of freedom of the reduced space Mh, ψL its shape func-

tion, and SL the support of ψL, we set λL :=

∫Γ λψLdΓ∫Γ ψLdΓ

=

∫Γ∩SL λψLdΓ∫Γ∩SL ψLdΓ

. Taking a cue on

the quasi-interpolation operators of Clément [161], we buildπhλ as πhλ :=

∑L

λLψL, which may as well be regarded as an element of Mh.

Let us start with stability properties for πh:

Lemma 5.7. Operator πh is H1- and L2-stable, that is to say ‖πhλ‖1,Γ ≤ C‖λ‖1,Γ and‖πhλ‖0,Γ ≤ C‖λ‖0,Γ. Moreover, |πhλ|21,Eh ≤ Ch|λ|

21,Γ.

Proof. We have ‖πhλ‖20,Γ ≤∑L

λ2L‖ψL‖20,Γ. Since ψL ≥ 0, applying Schwarz inequal-

ity to the denition of λL gives λ2L ≤

‖λ‖20,Γ∩SL‖ψL‖21,0,Γ

‖ψL‖20,Γ. Shape function ψL veries

‖ψL‖0,Γ ≤ Ch12 and the inverse property ‖ψL‖0,Γ ≤ Ch−

12 ‖ψL‖0,1,Γ hence ‖πhλ‖20,Γ ≤

C∑L

‖λ‖20,Γ∩SL ≤ C‖λ‖20,Γ so the operator is L2-stable.

As for the H1-stability, on a support SL, let us call I and J the adjacent degrees offreedom. Since ψI+ψJ+ψL = 1 over SL, we have∇(πhλ) = (λJ−λI)∇ψJ+(λL−λI)∇ψLso that:

|πhλ|21,SL∩Γ ≤ Ch−1|λJ − λI |2 (5.22)

Let F := h Id be a dilatation mapping from an adimensional reference space ontothe real one. Let ψ := ψF , λ := λF , Γ := F−1(Γ) and SL := F−1(SL). An equivalent

denition for λL is then λL :=

∫Γ∩SL λψLdΓ∫Γ∩SL ψLdΓ

. With this denition, we introduce the linear

form l(λ) = λL − λI . Schwarz inequality and the equivalence of all norms for ψL yield|l(λ)| ≤ C‖λ‖0,SL∩Γ ≤ C‖λ‖1,SL∩Γ hence l is continuous and vanishes for constants, so we

may apply Bramble-Hilbert lemma ([135], theorem 4.1.3): |l(λ)| ≤ C|λ|1,SL∩Γ. Pulling

the right-side member onto the real space, we have |l(λ)|2 ≤ Ch|λ|21,SL∩Γ, which given

(5.22) yields |πhλ|21,K∩Γ ≤ C|λ|21,SL∩Γ. Summing up over the intersected elements yields

|πhλ|21,Γ ≤ C|λ|21,Γ. Replacing (5.22) with |πhλ|21,K ≤ C|λJ − λI |2, it comes |πhλ|21,Eh ≤Ch|λ|21,Γ.

Let us know determine common interpolation properties to P1∗ and P1−, as:

Lemma 5.8. For multiplier spaces P1∗ and P1−, it holds:

∀λ ∈ H1(Γ), ‖πhλ− λ‖0,Γ ≤ Ch‖λ‖1,Γ (5.23)

and:∀λ ∈ H1/2(Γ), ‖πhλ− λ‖0,Γ ≤ Ch1/2‖λ‖1/2,Γ (5.24)


Proof. Adopting similar denitions as in the proof of lemma 5.7, over SL, it readsπhλ = λL + (λJ − λL)ψJ + (λI − λL)ψI . Then, we have:

‖πhλ− λ‖0,Γ∩SL ≤ ‖λL − λ‖0,Γ∩SL + |λI − λL|‖ψI‖0,Γ∩SL + |λJ − λL|‖ψJ‖0,Γ∩SL

. From Poincaré-Wirtinger inequality (see [162], Corrolary 5.4.1 and its proof), it comes‖λL − λ‖0,Γ∩SL ≤ C|λ|1,Γ∩SL . Since ‖ψI‖0,Γ∩SL is bounded, and |λI − λL| ≤ C|λ|1,Γ∩SLafter the proof of lemma 5.7, it comes ‖πhλ− λ‖0,Γ∩SL ≤ C|λ|1,Γ∩SL . Pulling this resultonto the physical space and summing up over L yields (5.23). As for (5.24), it followsfrom (5.23) and lemma 5.7 appealing to the interpolation theory: the argument is rathertechnical and classically used for quasi-interpolation operators (see for instance Chenand Nochetto [163], lemma 3.2).

Establishing an improved accuracy for P1∗ would imply to improve (5.23) by usingvanishing operators for all ane functions. In fact, this would solely be possible ifthe interface cuts the mesh far from nodes: otherwise, the multiplier stays piecewiseconstant on small intersections of the mesh and the interface. Nevertheless, to havesome understanding of this case without going too far into technicalities, let us considera nodal interpolation operator instead of πh. In this simplied analysis, it is assumedthat meas (Γ) = 1 and that it is uniformly subdivided into segments of length h (seeg.5.13). If a ∈ [0, 1] is the extremity of such a segment, let us assume that λ and πhλcoincide at a and a + h. In the case where a corresponds to a almost coincident node,πhλ is taken to be constant on [a, a + κh] and linear on [a + κh, a + h] (see g.5.13).Otherwise, πhλ is linear on [a, a+ h] (see g.5.13).

In the rst case (almost coincident node), some trapezoid-rule analogous L1-error esti-mate yields:

‖λ− πhλ‖0,1,[a,a+h] ≤ h2κ

2|λ|1,∞,Γ +

h3

12|λ|2,∞,Γ (5.25)

We recall that κ represents the intersection ratio under which an edge is consideredclosely intersected, which implies the imposition of some equality relations making theP0-segment to appear. Assuming a random conguration of intersection which shouldhappen if the mesh is built regardless of the interface such P0-segments embeddingelements therefore occur with a probability κ on Γ. The rest of them are fully P1.Therefore an L1-error estimate is:

‖λ− πhλ‖0,1,Γ ≤ hκ2

2|λ|1,∞,Γ +

h2

12|λ|2,∞,Γ (5.26)

The same Taylor series-based estimations may be carried out for the L2 error. Reportingonly dominant terms in κ or h, we have :

‖λ− πhλ‖20,[a,a+h] ≤ |λ|21,∞,Γκ

2h3

3+ |λ′λ''|0,∞,Γh4 κ

12+ |λ|22,∞,Γ

h5

120(5.27)


Assuming the same probability of occurrence, the error estimates reads :

‖λ− πhλ‖20,Γ ≤ |λ|21,∞,Γκ3h2

3+ |λ′λ''|0,∞,Γh3κ

2

12+ |λ|22,∞,Γh4 1

120(5.28)

0

1

h

πhλ

a

a+κ h

a+h

Figure 5.13: Illustration of the interpolation of a multiplier by algorithm (P1*).

It can be inferred from estimate (5.28) that even if algorithm P1∗ is not fully P1 strictlyspeaking, it decreases the size and occurrence of P0 zones so drastically that the cor-responding suboptimal error components the rst two terms in (5.28) have a verylow coecient, and thus the optimal component the third term in (5.28) becomesdominant unless extremely high renement are considered. Therefore an optimal rate ofconvergence of 2 for ‖λ−πhλ‖0,Γ is expected for usual renements. On the contrary, theoccurrence of large P0 zones in the P1 algorithm suggests a predicted convergence rateof 1.

Finally, the choice of parameter κ in algorithm P1∗ has to be a compromise betweenthe precision that implies a decrease in the coecient of the suboptimal component in(5.28) and the necessity to keep the conditioning number reasonable for almost coincidentnodes. The conditioning number for a linear triangular element intersected with ratioκ 1 at both edges is around κ−2, as shown in [164] or [165]. In our calculationsκ = 1.10−3 was adopted.

5.4 Changes due to a curved geometry

We now consider problems where the interface may have any shape. The approximationof a curved geometry has been described in details in section 4.3.1 and the accuracy ofthis description has been assessed in section 4.4. Here, the changes that they imply tothe problem are detailed.

5.4.1 Changes to the operators

Approximating domains change denition (5.12) of the Heaviside function to

H(x) =

−1 if x ∈ Ωh

1

1 if x ∈ Ωh2. Given expression (5.11) of the discrete space Vh, this implies

that an interpolant vh ∈ Vh is discontinuous across Γh instead of Γ. As in section 4.5.1,


the discrete solution uh should then not be compared with u but with an extension u ofu with a discontinuity across Γh instead of Γ (see section 4.5.1 for the denition of thatu).

Discrete bulk operators ah and lh are also dened with integrals over approximated do-mains, as in section 4.5.1, which may be consulted for their denitions. Finally, bilinearforms to enforce the constraint should be distinguished between integrals over Γ or Γh,as:

∀ (u, µ) ∈ V ×M, b (u, µ) :=

∫Γµ[u]dΓ (5.29)

and:

∀ (uh, µh) ∈ Vh ×Mh, bh (u, µ) :=

∫Γh

µh[uh]dΓ (5.30)

This distinction is transmitted to the denition of the discrete multiplier space Mh,dened as the trace over Γh of functions in Mh:

Mh :=λh|Γh, λh ∈ Mh

(5.31)

5.4.2 Transnite elements

Based on the geometry approximation from section 4.3.1, we may build transnite ele-ments, which dier from transnite subcells in the sense that K will this time be mappedonto a triangle K close to K (see g.5.14) rather than onto E. Triangles K shouldconstitute a mesh where the interface is exactly resolved (see g.5.14). This turns out tobe useful in the upcoming analysis, allowing the separation between interpolation andconsistency errors.

So let K be a cut parent element. We are looking for a transnite map FK which would

map K onto a transnite triangle K , in such a way that the ane pull-back Γ of Γh bemapped onto Γ (see g.5.14). This condition is stated by the following theorem:

Theorem 5.9. For each non slanted element K ∈ Eh, a transnite map FK : K → Kmay be established, which observes the following properties:

1.(K)K∈Th

is an admissible mesh of Ω. In other terms, the intersection of two

transnite triangles K ∩ K ′ is either empty, or equal to K, or reduces to an edgeor a vertex (see g.5.14),

2. RK := FK − FK veries ‖RK‖g+1,∞,K ≤ Chg+1 ,

3. F−1K

(Γ ∩ K

)= F−1

K (Γh ∩K) (see g.5.14).

The proof of this theorem is rather technical and proposed in appendix D.


F K (affine)

maps Γ onto Γh∩K

F K transfinite

maps onto ∩ K

K K K ' K '

Γ

K

a2

a1

ΓΓh

am

aM

Figure 5.14: Transnite elements and transnite map.

5.4.3 Conforming discretization spaces over transnite elements

For the sake of the error analysis, we will be led to dene intermediate discretizationspaces over transnite elements. In this context, a discrete space of conforming elds isintroduced as in [139]:

Vh :=v ∈ V, v|K∩Ωi

FK ∈ Pp(K)

(5.32)

Still following the guideline of [139], a map is then established between the elements of

Vh and Vh, by letting S :

Vh → Vhvh → vh GK

and C = S−1 :

Vh → Vhvh → vh G−1

K

where

GK := FK F−1K

. In a similar way, the intermediate multiplier space is:

Mh := JtSµh, µh ∈Mh (5.33)

where Jt is the dilatation induced by GK in the tangential direction t to Γ:

Jt :=(tT ·DGTK ·DGK · t

)1/2(5.34)

It is then worth noticing that for vh ∈ Vh and µh ∈Mh:

bh(vh, µh) =∫

Γh[vh]µhds

=∫

Γ ([vh] GK) (µh GK) Jtds= b (S[vh], JtSµh)

(5.35)

Let us provide evaluations for the involved quantities. Since FK = FK + RK , it followsthat G−1

K = Id +RK F−1K . It has been proven in theorem 5.9 that |RK |0,∞,K ≤ Ch

g+1,

which implies |G−1K − Id |0,∞,K ≤ Chg+1. In the same way, the jacobian matrix of

G−1K reads DG−1

K = I +[DRK F−1

K

]· DF−1

K . Since |DF−1K |0,∞,K ≤ Ch−1, it satises

|DG−1K − I|0,∞,K ≤ Chg. As was proven by Ciarlet (see [135], theorem 4.3.3), when h is

suciently small, the inverse GK of G−1K satises:


|GK − Id |0,∞,K ≤ Chg+1 (5.36)

|DGK − I|0,∞,K ≤ Chg (5.37)

5.5 Convergence analysis with a weak discontinuity

Let us now analyse the convergence of the interface problem. To study the convergencewith Lagrange multipliers, our key starting assumption is the stability of the discretemultiplier space. In X-FEM, this is classically translated by inf-sup condition (5.15),that becomes for us:

∃c > 0, ∀µh ∈ Mh, supvh∈Vh

b(vh, µh)

‖vh‖1,Ω1∪Ω2

≥ ch1/2‖µh‖0,Γ (5.38)

Let us point out that this condition is dierent from those usually admitted for mor-tar methods or standard nite element method, where h1/2‖µh‖0,Γ is replaced with‖µh‖−1/2,Γ ≥ ch1/2‖µh‖0,Γ. So we have to gure out what to keep and what to rewritefrom classical nite-element proofs. In the end, the remarkable result is that it may in-deed be proven that this inf-sup condition is sucient to yield appropriate convergence,though it is a bit less stringent than those classically assumed.

5.5.1 Convergence analysis on transnite elements

The rst step of the convergence study consists in analysing an intermediate interfaceproblem relying on transnite elements where the interface is, geometrically speaking,exactly resolved. This step is intended to produce interpolation-related errors, whilepreventing any domain-related consistency error from appearing.

The intermediate discrete problem that we suggest to analyse reads: nd(uh, λh) ∈ Vh × Mh such as:

∀vh ∈ Vh, a (uh, vh) + b(vh, λh) = l (vh) (5.39)

∀µh ∈ Mh, b (uh, µh) = 0 (5.40)

Due to condition that v|Γiu = 0 in the denition of V , a veries a coercivity assumption on

V . Introducing the constrained space V ⊃ Kh :=vh ∈ Vh,∀µh ∈ Mh, b (vh, µh) = 0

,

the following error estimates holds for the intermediate discrete problem:

Theorem 5.10. Extension to X-FEM of [139], Proposition 2.1. Let u be the solution ofthe exact problem, and λ be the corresponding normal ux, let uh be the solution of theintermediate discrete problem, then:

‖u− uh‖1,Ω1∪Ω2 ≤ C infvh∈Kh

‖u− vh‖1,Ω1∪Ω2 + C supwh∈Kh

b (wh, λ)

‖wh‖1,Ω1∪Ω2

(5.41)


Moreover, if λ ∈ L2 (Γ) then:

‖u− uh‖1,Ω1∪Ω2 ≤ C infvh∈Kh

‖u− vh‖1,Ω1∪Ω2 + Ch1/2 infµh∈Mh

‖λ− µh‖0,Γ (5.42)

Proof. The proof relies on the coercivity and continuity of a on V , on the expression ofthe intermediate and exact problem and on the denition of Kh. It is therefore strictlyidentical to proposition and remark 2.6 in [166] for (5.41), from an original proof in[167]. The proof to get (5.42) from (5.41) comes from [139] and relies on the fact that∀ωh ∈ Vh, ‖[ωh − πhωh]‖0,Γ ≤ Ch1/2‖ωh‖1,Ω1∪Ω2 . While this is a classical property ofmultiplier spaces with standard nite element method, in our X-FEM case it had to beexplicitely proven in (5.24) by the dedicated lemma 5.8.

We now aim at estimating the rst term of (5.42), making use of the nodal interpolationoperators in a rst stage (lemma 5.12) and extending the results to the minimum overthe restricted space in a second stage (theorem 5.14), by taking advantage of the inf-supcondition. Let us introduce:

Denition 5.11. Let u ∈ Hp+1(Ω1 ∪ Ω2). We dene a nodal interpolation operator byIp := Ip F−1

Kon K. We may then dene an interpolation operator onto Vh as in [95],

[131] or [129] by Πhu|Ωi = Ipui.

Lemma 5.12. Extension to X-FEM of [139], Lemma 2.3. Let u ∈ Hp+1(Ω1)×Hp+1(Ω2),then‖u− Πhu‖1,Ω1∪Ω2 ≤ Chp‖u‖p+1,Ω1∪Ω2.

Proof. Extension to X-FEM of the proof in [139]. By letting ûi := ui|K FK , we obtain:

‖ûi − Ip ûi‖1,K ≤ C|ûi|p+1,K ≤ Chp‖ui‖p+1,K (5.43)

Transforming the left hand side of (5.43) back to K and using the concavity of the squareroot, we deduce h−1|ui− Ipui|0,K + |ui− Ipui|1,K ≤ Ch

p‖ui‖p+1,K , and the result follows

by summation, denition of K and stability of the extensions ui.

In the upcoming developments, an estimate of the trace of this interpolation also comesout to be necessary. Hence the statement of the following lemma:

Lemma 5.13. The trace of the interpolation error abides by the following inequalitiy.For v ∈ V :

‖[v − Πhv]‖0,Γ ≤ Chp+1/2‖v‖p+1,Ω1∪Ω2 (5.44)


Proof. With the notations of g.5.14, and denoting vh := Πhv for the sake of convenience,

we have ‖[v − vh]‖0,Γ∩K =

(meas(Γ ∩ K)

meas(Γ)

)1/2

‖[v − ˆvh]‖0,Γ, andmeas(Γ ∩ K)

meas(Γ)≤ Ch.

Let us now consider the linear form Θ :

Hp(K)→ R

ϕ →∫

Γ

([v − ˆvh]

)(ϕ− Ipϕ

)ds

.

By virtue of the Schwarz inequality and the trace theorem, Θ is continuous, and theassociated operator norm ‖Θ‖* satises ‖Θ‖* ≤ C‖[v − ˆvh]‖0,Γ. In this estimate, theconstant in the trace theorem solely depends upon the Lipschitz constant of the domainformed by Γ and the boundaries of K. Hence, since there is no slanted triangle, C isindependent of K.

Let us now apply the Bramble-Hilbert lemma (see section 3.1 of [135]), which due to thefact that Θ is continuous and vanishes over Pp(K) states that |Θ(ϕ)| ≤ C‖Θ‖*|ϕ|p+1,K .

Applying this result with the specic choice ϕ = ˆv1 − ˆv2, whose trace on Γ veriesϕ|Γ = [v], and simplifying by ‖[v − ˆvh]‖0,Γ it comes ‖[v − ˆvh]‖0,Γ ≤ C|ˆv1 − ˆv2|p+1,K .

Pulling these quantities back onto the physical space, summing up the re-sults and appealing to the stability properties of the extensions, it comes thath−1/2‖[v − vh]‖0,Γ ≤ Chp‖v1 − v2‖p+1,Ω1∪Ω2 . Applying this result to p = 0 gives‖[v − vh]‖0,Γ ≤ Ch1/2‖v1 − v2‖1,Ω1∪Ω2 . The stability of the extensions nally yieldsthe result.

The interpolation estimates may now be extended to the inmum over the constrainedspace:

Theorem 5.14. Extension to X-FEM of [139], Proposition 2.2. The estimate for thelower bound over Vh given in lemma 5.13 also holds over the constrained space Kh,namely:

infvh∈Kh

‖u− vh‖1,Ω1∪Ω2 ≤ Chp‖u‖p+1,Ω1∪Ω2 (5.45)

Proof. In this proof we aim at correcting Πhu to cancel out its jump, while keeping asimilar estimate. We call z the L2(Γ)-projection of [Πhu] onto Mh, that is to say the

multiplier z ∈ Mh such that ∀µ ∈ Mh,∫

Γ

(z − [Πhu]

)µds = 0. Let then wh ∈ Vh be the

displacement introduced in the proof of theorem 5.5, which fullls the inf-sup conditionb(wh, z)

‖wh‖1,Ω1∪Ω2

≥ ch1/2‖z‖0,Γ and z = [wh]. These properties and the denition of z lead

to:

‖wh‖1,Ω1∪Ω2 ≤ Ch−1/2 b(Πhu, z)

‖z‖0,Γ(5.46)


The Schwarz inequality reads b(Πhu, z) ≤ ‖[Πhu]‖0,Γ‖z‖0,Γ, so that ‖wh‖1,Ω1∪Ω2 ≤Ch−1/2‖[Πhu]‖0,Γ. Since [u] = 0, we may write [Πhu] = [Πhu − u]. By virtue oflemma 5.13, ‖[Πhu − u]‖0,Γ ≤ Chp+1/2‖u‖p+1,Ω1∪Ω2 , which leads to ‖wh‖1,Ω1∪Ω2 ≤Chp‖u‖p+1,Ω1∪Ω2 . Considering the particular choice vh = Πhu − wh ∈ Kh in the in-ferior bound, a triangle inequality yields inf

vh∈Kh‖u − vh‖1,Ω1∪Ω2 ≤ ‖u − Πhu‖1,Ω1∪Ω2 +

‖wh‖1,Ω1∪Ω2 . The conclusion then follows immediately by virtue of Lemma 5.12.

As for the error on the Lagrange multiplier, the following error estimate holds:

Lemma 5.15. Provided λ ∈ L2(Γ), there exists C such that:

‖λ− λh‖−1/2,Γ ≤ C

(h1/2 inf

µh∈Mh

‖λ− µh‖0,Γ + ‖u− uh‖1,Ω1∪Ω2

)(5.47)

Proof. Provided the inf-sup condition (5.38) and (5.44), we apply the theorem 4.8 of[159] to prove (5.47).

5.5.2 Convergence analysis on the actual elements

Carrying out the analysis on actual elements, the idea is to come down to the estimatesof the previous conforming analysis, to which only geometry-related consistency errorsshould be added.

By letting then u′h := Suh , λ′h := JtSλh and a′h (wh, vh) := ah (Cwh, Cvh), the discreteproblem may be reformulated in:

∀v ∈ Vh, a′h(u′h, v

)+ b

(v, Jtλ

′h

)= lh (Cv) (5.48)

∀µ ∈ Mh, b(u′h, Jtµ

)= 0 (5.49)

We have the general estimate:

Lemma 5.16. Provided λ ∈ L2(Γ), the following estimates holds:

‖u− u′h‖1,Ω1∪Ω2 ≤ C infvh∈Kh

(‖u− vh‖1,Ω1∪Ω2 + sup

wh∈Kh

a(vh, wh)− a′h(vh, wh)

‖wh‖1,Ω1∪Ω2

)+C inf

µh∈Mh

h1/2‖λ− µh‖0,Γ + supwh∈Kh

l(wh)− lh(Cwh)

‖wh‖1,Ω1∪Ω2

(5.50)


Proof. Very classical proof which consists in using the rst Strang lemma together withtheorem 5.10. It is for instance identical to proposition 2.16 of [166].

The rst term in the right-hand side in (5.50) has already been assessed in theorem 5.14by (5.49). The third term is related to the interpolation error of the multiplier and willbe estimated by lemma 5.19. Regarding the second and fourth terms in (5.50), theseconsistency errors are estimated by:

Lemma 5.17. (From [139], propositions 2.4 and 2.5). Let us recall that g ∈ 1, 2 is theorder of description of the interface, the second and fourth terms in (5.50) are respectivelybounded by O(hg) and O(hg+1).

Proof. The proof is a straightforward extension to the case g = 2 of [139]. It is notrecalled for the sake of conciseness.

The error in the Lagrange multiplier is now assessed for this discrete problem:

Lemma 5.18. Extension to X-FEM of [139], Proposition 2.4. The error of the Lagrangemultiplier is given by:

‖λ− λ′h‖0,Γ ≤ Cinf

µh∈Mh

‖λ− µh‖0,Γ + h−1/2 (‖u− u′h‖1,Ω1∪Ω2 + hg (1 + ‖u‖1,Ω1∪Ω2) + hp‖u‖p+1,Ω1∪Ω2)

(5.51)

Proof. Once again, it is analogous to that in [139] except for subtleties due to usinginf-sup condition (5.38) instead of a classical H−1/2(Γ)-norm uniform inf-sup conditionas in [139].

Step 1. With the inf-sup condition, it holds ‖λh−λ′h‖0,Γ ≤ Ch−1/2 supwh∈Vh

b(wh, λh − λ′h)

‖wh‖1,Ω1∪Ω2

,

whose numerator is split into b(wh, λh − λ′h) = b(wh, λh − λ) + b(wh, λ− λ′h).

Step 2. Estimating supwh∈Vh

b(wh, λh − λ)

‖wh‖1,Ω1∪Ω2

.

We have supwh∈Vh

b(wh, λh − λ)

‖wh‖1,Ω1∪Ω2

≤ supw∈V

b(w, λh − λ)

‖w‖1,Ω1∪Ω2

= ‖λh − λ‖−1/2,Γ. Lemma 5.15 then

states that ‖λ − λh‖−1/2,Γ ≤ C

(h1/2 inf

µh∈Mh

‖λ− µh‖0,Γ + ‖u− uh‖1,Ω1∪Ω2

). As for


‖u − uh‖1,Ω1∪Ω2 , it holds ‖u − uh‖1,Ω1∪Ω2 ≤ C infvh∈Kh

‖u − vh‖1,Ω1∪Ω2 + Ch1/2 infµh∈Mh

‖λ −

µh‖0,Γ after theorem 5.10, the rst term of which is nally estimated by theorem 5.14,as inf

vh∈Kh‖u− vh‖1,Ω1∪Ω2 ≤ Chp‖u‖p+1,Ω1∪Ω2 .

Step 3. Estimating supwh∈Vh

b(wh, λ− λ′h)

‖wh‖1,Ω1∪Ω2

.

Subtracting the discrete problem (5.48) to the continuous one (5.8), one getsb(wh, λ − λ′h) = l(wh) − lh(Cwh) + a′h(u′h, wh) − a(u,wh). We nally decomposea′h(u′h, wh) − a(u,wh) = a(u′h − u,wh) + (a′h − a)(u′h, wh). Appealing to the continu-ity of a and the estimates of lemma 5.17, it may be concluded that:

supwh∈Vh

b(wh, λ− λ′h)

‖wh‖1,Ω1∪Ω2

≤ Chg(1 + ‖u′h‖1,Ω1∪Ω2) + C‖u− u′h‖1,Ω1∪Ω2

By writing that ‖u′h‖1,Ω1∪Ω2 ≤ ‖u‖1,Ω1∪Ω2 +‖u−u′h‖1,Ω1∪Ω2 , for h small enough we have:

supwh∈Vh

b(wh, λ− λ′h)

‖wh‖1,Ω1∪Ω2

≤ Chg(1 + ‖u‖1,Ω1∪Ω2) + C‖u− u′h‖1,Ω1∪Ω2

Step 4. The expected result follows from combining the results of step 1 to 3, andremarking that ‖λh − λ′h‖0,Γ ≥ ‖λ− λ′h‖0,Γ − ‖λh − λ‖0,Γ.

We may then evaluate:

Lemma 5.19. Suppose λ ∈ H1(Γ), then the interpolation error over the Lagrange mul-tiplier is:

infµ∈Mh

‖λ− µ‖0,Γ ≤ ‖λ− πhλ‖0,Γ + Chg‖λ‖1,Γ (5.52)

The regularity of λ is related to that of u given that λ := (A · ∇u) · n.

Proof. Since S(πhλ) ∈ Mh, infµ∈Mh

‖λ−µ‖0,Γ ≤ ‖λ−S(πhλ)‖0,Γ, which may be decomposed

into:

‖λ− S(πhλ)‖0,Γ ≤ ‖λ− πhλ‖0,Γ + ‖πhλ− S(πhλ)‖0,Γ + ‖1− Jt‖0,∞,Γ‖πhλ‖0,Γ (5.53)

The second term may be split into ‖πhλ − S(πhλ)‖20,Γ =∑K

‖πhλ − S(πhλ)‖20,Γ∩K . We

may then assert that ‖πhλ−S(πhλ)‖0,Γ∩K ≤ h‖πhλ−S(πhλ)‖0,∞,Γ∩K and make use of


the Taylor inequality to evaluate ‖πhλ − S(πhλ)‖0,∞,Γ∩K ≤ |πhλ|1,∞,K‖GK − Id‖0,∞,Γ.The equivalence of all norms on the nite dimensional space of ane functions over thereference element yields |πhλ|1,∞,K ≤ Ch

−1|πhλ|1,K . Combining those results yields:

‖πhλ− S(πhλ)‖0,Γ ≤ C|πhλ|1,Eh‖GK − Id‖0,∞,Γ (5.54)

After lemma 5.7, we have |πhλ|1,Eh ≤ Ch1/2‖λ‖1,Γ. Since ‖GK − Id‖0,∞,K ≤ Chg+1, itholds:

‖πhλ− S(πhλ)‖0,Γ ≤ Chg+3/2‖λ‖1,Γ (5.55)

Given the expression (5.35) that Jt :=(tT ·DGTK ·DGK · t

)1/2and estimate (5.36) that

|DGK − 1|0,∞,K ≤ Chg, and the Taylor expansion of the square root, it follows immedi-ately that:

|Jt − 1|0,∞,K ≤ Chg (5.56)

Moreover, after lemma 5.7 (L2-stability), we have:

‖πhλ‖0,Γ ≤ C‖λ‖0,Γ (5.57)

Then, estimate (5.52) to be proven is deduced from (5.53), by replacing its second termwith (5.55) and its third term with (5.56) and (5.57).

We are now in position to conclude with the nal usual estimates:

Theorem 5.20. Assume u ∈ W 1,∞(Ω1 ∪ Ω2), the following a priori estimate holds forthe displacement error:

‖u−uh‖1,Ωh1∪Ωh

2≤ C

hp‖u‖p+1,Ω1∪Ω2

+ h1/2‖λ− πhλ‖0,Γ + hg (1 + ‖u‖1,Ω1∪Ω2) + hg+1/2‖λ‖1,Γ

(5.58)

Proof. Step 1. Given estimate (5.37) of GK − Id , it holds:

c‖u− uh‖1,Ωh1∪Ωh2≤ ‖Su− u′h‖1,Ω1∪Ω2 ≤ C‖u− uh‖1,Ωh1∪Ωh2

(5.59)

This can be broken down into:

‖Su− u′h‖1,Ω1∪Ω2 ≤ ‖Su− u‖1,Ω1∪Ω2 + ‖u− u′h‖1,Ω1∪Ω2 (5.60)


Step 2.The second term ‖u − u′h‖1,Ω1∪Ω2 of (5.60) is assessed with lemma 5.16 byestimation (5.50), whose rst term inf

vh∈Kh‖u − vh‖1,Ω1∪Ω2 is bounded in theorem

5.14 by Chp‖u‖p+1,Ω1∪Ω2 , second and fourth terms are evaluated by lemma 5.17 as

supv∈Vh

supw∈Vh

a(v, w)− a′h(v, w)

‖w‖1,Ω1∪Ω2‖v‖1,Ω1∪Ω2

≤ Chg and supwh∈Kh

l(wh)− lh(Cwh)

‖wh‖1,Ω1∪Ω2

≤ Chg+1 and third

term h1/2 infµ∈Mh

‖λ−µ‖0,Γ is bounded in lemma 5.19 by h1/2‖λ−πhλ‖0,Γ +Chg+1/2‖λ‖1,Γ.

Step 3. The rst term of (5.60) is exclusively due to the approximation of the ge-ometry. Let us call Sh the part of the domain which is in (Ω1/Ω

h1) ∪ (Ωh

1\Ω1): it isbasically the domain enclosed by Γh ∪ Γ (see the colored domain on g.5.15). It holds|Su − u|21,Ω1∪Ω2

= |Su − u|21,(Ω1∪Ω2)\Sh + |Su − u|21,Sh . Since u = u on Ω\Sh, we have

|Su − u|21,(Ω1∪Ω2)\Sh =

∫(Ω1∪Ω2)\Sh

∇(Su − u) · ∇(Su − u)dΩ. Moreover,

|Su − u|21,Sh =

∫Sh

∇(Su − u) · ∇(Su − u)dΩ +

∫Sh

∇(Su − u) · ∇(u − u)dΩ. Let us

sum up these results:

|Su− u|21,Ω1∪Ω2≤∫

Ω1∪Ω2

∇(Su− u) · ∇(Su− u)dΩ +

∫Sh

∇(Su− u)∇(u− u)dΩ (5.61)

Γ

Γh

S hCh g+1

Ω1∖S h

Ω2∖ S h

Figure 5.15: Exact and approximate domains.

The rst term at the right hand side of (5.61) may be bounded by Chg|Su − u|1,Ω1∪Ω2

with Schwarz inequality and the very same argument about GK − Id as in step 1. As forthe second term of (5.61), it may be bounded by Chg+1‖u‖1,∞,Ω1∪Ω2 |Su−u|1,Ω1∪Ω2 sincethe width of Sh is O(hg+1) after theorem 5.9 and meas(Sh) ≤ Chg+1meas(Γ) (g.5.15).

Then |Su − u|1,Ω1∪Ω2 ≤ Chg(1 + h‖u‖1,∞,Ω1∪Ω2), or for h suciently small,|Su− u|1,Ω1∪Ω2 ≤ Chg. Repeating the proof for the L2 norm and scalar product brings‖Su− u‖0,Ω1∪Ω2 ≤ Chg hence:

‖Su− u‖1,Ω1∪Ω2 ≤ Chg (5.62)


Theorem 5.21. The following a priori estimates holds for the multiplier error:

‖λ− λh‖0,Γ ≤ C ‖λ− πhλ‖0,Γ + hg‖λ‖1,Γ+ h−1/2

(‖u− uh‖1,Ωh1∪Ωh2

+ hg (1 + ‖u‖1,Ω1∪Ω2) + hp‖u‖p+1,Ω1∪Ω2

)(5.63)

Proof. The estimate is split into ‖λ−λ′h‖0,Γ and ‖λ′h−λh‖0,Γ. By virtue of lemmas 5.18and 5.19, the rst is estimated by:

‖λ− λ′h‖0,Γ ≤ C‖λ− πhλ‖0,Γ+ h−1/2 ‖u− u′h‖1,Ω1∪Ω2 + hg(1 + ‖u‖1,Ω1∪Ω2) + hg‖λ‖1,Γ + hp‖u‖p+1,Ω1∪Ω2

We may then split ‖u − u′h‖1,Ω1∪Ω2 ≤ ‖u − Su‖1,Ω1∪Ω2 + ‖Su − u′h‖1,Ω1∪Ω2 , for which‖Su − u′h‖1,Ω1∪Ω2 ≤ C‖u − uh‖1,Ωh1∪Ωh2

(see (5.59)) and ‖Su − u‖1,Ω1∪Ω2 ≤ Chg after

(5.62). As for ‖λ′h − λh‖0,Γ, it holds ‖λ′h − λh‖0,Γ ≤ Chg+3/2‖λ‖1,Γ arguing as in lemma5.19, given that λ′h = JtSλh.

5.5.3 About the optimal convergence orders

Before we check those results against numerical experiments, let us mention some pointswhich are likely to improve our theoretical prediction:

As was pointed out by Melenk and Wohlmuth in [158], in the theoretical estimatesfor the dual variable, the displacement-related terms were found to be suboptimalwhen compared to experiments in numerous papers, down a factor h1/2. Theauthors extensively discuss the issue in [158], and oer a proof that the missingfactor may almost be theoretically recovered (in fact a factor h1/2 log(h) may berecovered). This would allow to remove the highlighted factor h−1/2 from (5.63).

Secondly, when comparing our domain consistency error hg from lemma 5.17, ob-tained with transnite elements, to the domain consistency error hg+1/2 from [120],obtained with a direct analysis, we may be led to think that our estimates is subop-timal, down h1/2. Such a suboptimality was actually observed on all our numericalexperiments. This would allow to replace term hg (1 + ‖u‖1,Ω1∪Ω2) in (5.58) and(5.63) by hg+1/2 (1 + ‖u‖1,Ω1∪Ω2).

To sum it up, taking into account the aforementioned facts leads us to the following ex-pected convergence orders, denoting O (.) Landau's notation for an asymptotic boundingfunction:


‖u− uh‖1,Ωh1∪Ωh2= O(hp + hg+

12 ) +O(h

12 ‖λ− πhλ‖0,Γ) (5.64)

‖λ− λh‖0,Γ = O (‖λ− πhλ‖0,Γ) +O(hg) +O(‖u− uh‖1,Ωh1∪Ωh2

)(5.65)

In the upcoming numerical experiments, keeping the origin of these terms in mind iscrucial. It is especially useful to know in what particular cases their related errorsvanish, in which case the term disappears from the estimate:

Term O(hp) disappears when the interpolation basis owns the exact displacement.

Term O (‖λ− πhλ‖0,Γ) disappears in case λ ∈ Mh, which holds true if the exactpressure is spanned by the basis. It is the case for a constant pressure for instance.

Term O(hg+1/2) originating from term hg+1/2 (1 + ‖u‖1,Ω1∪Ω2) in (5.58) and termO(hg) originating from terms hg (1 + ‖u‖1,Ω1∪Ω2) in (5.63) are called bulk consis-tency error. They vanish if the interface is exactly resolved.

Terms O(hg+1/2) and O(hg), respectively originating from term hg+1/2‖λ‖1,Γ in(5.58) and hg‖λ‖1,Γ in (5.63) are called surface consistency error. They are inducedby the presence of Lagrange multipliers and vanish if the interface is exactly resoved,or if λ ∈ Mh.

Let us now recapitulate the expected displacement and multiplier convergence rates forsome combinations (the mention exact means that we consider a particular casewhose exact solution belongs to the interpolation space):


DisplacementMultiplierspace

Subdivision(geometry

representationorder g)

Multiplierinterpolation

relatederror: orderof(πhλ-λ)+0,5

Displacementinterpolationerror rate p

Geometry-related

consistency:order g + 0, 5

Energy er-

ror rate

P1 P1-- P1 1,5 (1 + 0,5) 1 1,5 (1 + 0,5) 1

P1 P1* (any)No convergence (inf-sup condition is vio-lated)

P2 P1* P1 2,5 (2 + 0,5) 2 1,5 (1 + 0,5) 1,5

P2 P1-- P2 1,5 (1 + 0,5) 2 2,5 (2 + 0,5) 1,5

P2 P1* P2 2,5 (2 + 0,5) 2 2,5 (2 + 0,5) 2

P1-exact P1-- P1 1,5 (1 + 0,5) - 1,5 (1 + 0,5) 1,5

P2-exact P1* P2 2,5 (2 + 0,5) - 2,5 (2 + 0,5) 2,5

P2 P1-- - exact P2 - 2 2,5 (2 + 0,5) 2

P1 P1-- - exact P1 - 1 1,5 (1 + 0,5) 1

Table 5.1: Expected convergence orders for the displacement in the energy norm.

As for the multiplier, the expected convergence rates are:

DisplacementMultiplierspace

Subdivision(geometry

representationorder g)

Multiplierinterpolationerror rate

Displacementrelated error

rate

Geometry-related

consistencyerror rate

Multiplier

L2-error

rate

P1 P1-- P1 1 1 1 1

P1 P1* (any)No convergence (inf-sup condition is vio-lated)

P2 P1* P1 2 1,5 1 1

P2 P1-- P2 1 1,5 2 1

P2 P1* P2 2 2 2 2

P1-exact P1-- P1 1 1,5 1 1

P2-exact P1* P2 2 2,5 2 2

P2 P1-- - exact P2 - 2 2 2

P1 P1-- - exact P1 - 1 1 1

Table 5.2: Expected convergence orders for the multiplier.

5.6 Numerical results

5.6.1 Cracked block under cubic pressure.

In this test, a square elastic block of side 1m is clamped on its lower boundary. Third-order polynomial boundary conditions are prescribed, namely a displacement ua on theupper boundary, and surface loads gl and gr on the side boundaries (see g.5.16):

ua =−1

Ep3x

3 + p2x2 + p0y (5.66)

gl = −p2y(y − 1)x+ p3y2(y − 3

2) + 5

p3

16y (5.67)

gr = (3p3 + p2) y(y − 1)x+ −3p3

4− p2

2− p3y

2(y − 3

2)− 5

p3

16y (5.68)

where p3 = 3.104Pa/m3; p2 = 3.104Pa/m2; p0 = 1.104Pa . This block is cut through bya straight 20-degree leaning adherent interface (see g.5.16). The coecients of elasticity


areE = 1010Pa and ν = 0, so as to obtain an analytical solution easily. It reads:

σxx = 3p3xy(y − 1) + p2y(y − 1) (5.69)

σyy = −p3x3 − p2x

2 − p0 + 3p3x

2

(y − 1

2

)+p2

2

(y − 1

2

)(5.70)

σxy = −3p3x2

4− p2

x

2− p3y

3 + 3p3y2

2− 5

p3

16(5.71)

Imposed displacement

Surface load

ua

g l gr

x

y

Figure 5.16: Block submitted to a cubic pressure.

Obviously the interface is exactly resolved here, so the terms with g are removed from(5.64) and (5.65). The interpolation errors are the only remaining, so the test is wellsuited to study the ability of the P1 and P1∗ algorithms to produce optimal orders ofconvergence. For linear interpolation of the displacement, and therefore P1 algorithmfor the multipliers, well-known optimal orders around 1 are observed for the energy andmultiplier errors. Using algorithm P1 together with a quadratic displacement yieldssuboptimal orders of convergence for the multipliers and the energy, close to 1 and 1.5respectively (see g.5.17), as predicted by the theory. Indeed, this suggests that thesuboptimal term dominates in (5.28), due to large P0 zones in algorithm P1. Optingfor algorithm P1∗ instead allows us to recover the optimal orders of accuracy 2 both inenergy and multipliers, as illustrated by g.5.17.

5.6.2 Circular inclusion under compressive loads

.

We consider an elastic circular inclusion of radius 0, 4m embedded in a square block ofside 1m made of the same material. Coecients of elasticity are E = 100Pa and ν = 0.A uniform pressure p1 = 2Pa is applied on the upper and lower boundary, and a smallerpressure p2 = 1Pa is applied on the side boundaries (see g.5.20). Lateral motionis prevented on the middle nodes of those boundaries, so as to prevent rigid motion.Since the interface is adherent, the analytical solution is the one without inclusion σ =−p1y⊗y−p2x⊗x. We point out that the exact displacement belongs to the interpolationbasis in this test, so term p is removed from (5.64).

For (P1/P1) interpolation the convergence order is found out to be 1.2 for the multiplierand 1.54 for the energy when neglecting the coarser value (see gs.5.18 and .5.19). The


5E-35E-2

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

P1 / P1-- : energy Slope 1.02

P1 / P1-- : multiplier Slope 1.12

P2 / P1-- : multiplier Slope 1.03

P2 / P1-- : energy Slope 1.55

P2 / P1* : multiplier Slope 2.03

P2 / P1* : energy Slope 1.99

mesh size (m)

rela

tive

en

erg

y e

rro

r

rela

tive

mu

ltip

lier

err

or

Figure 5.17: Convergence curves on energy and multipliers for the block under pres-sure.

energy order of convergence is close to the theoretical value 1.5 and above the theoreticalvalue of 1. for the multiplier, possibly an eect of P1 -zones in the (P1) algorithm whichstill tends to increase the observed order of convergence at low levels of renement.More important, the results for (P2 / P1*) interpolation with a linear description of thegeometry conrms the predicted surface consistency error, with observed suboptimalslopes 0.99 and 1.52 for the multiplier and energy respectively close to the theoreticalpredictions 1. and 1.5 (see gs. 5.18 and 5.19).

A quadratic description of the geometry allows instead to recover the optimal orders ofconvergence. Once again, the observed values are in accordance with their theoreticalcounterparts 2 for the multiplier and 2.5 for the energy. Finally, the results for (P2/P1)interpolation (with a quadratic description of the geometry) oers a new illustrationof suboptimal rates with this set of discretizations, with an interpolation order of themultipliers at 1.19 slightly above 1, which triggers a roughly 0.5 higher order 1.72 for theenergy, when 1.5 was expected.

5,00E-0035,00E-002

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Multipliers error

P2 / P1* / subdivision P1 Slope 0.99

P2 / P1- / subdivision P2 Slope 1.19

P1 / P1- / subdivision P1 Slope 1.2


mesh size (m)

rela

tive

mu

ltip

lier

err

or

Figure 5.18: Circular inclusion under compressive loads: multiplier convergence.


0,010,05

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

Convergence of the energy

P2 / P1-- / subdivision P2 Slope 1.72

P1 / P1-- / subdivision P1 Slope 1.54 (1st point neglected)



mesh size (m)

rela

tive

en

erg

y e

rro

r

Figure 5.19: Circular inclusion under compressive loads: energy error.

p1

p2

Figure 5.20: Circular inclusion under compressive loads.

5.6.3 Bimaterial ring

This numerical experiment aims at testing the ability of the Lagrange multipliers ap-proach to deal with an actual interface problem. The reference model, which is availablein [49] or [138], consists of an outer ring 1 that is made of a material E1 = 109Pa andν1 = 0.3 and geometrically delimited by radii R1 = 1m and R2 = 0, 6m. It enclosesan inner ring 2 of material properties E2 = 108Pa and ν2 = 0.2, which has an innerradius R3 = 0, 2m (see g.5.21). The interface bonding the rings is adherent. A uniformexternal pressure σ = 100Pa is applied on the outer boundary r = R1 while the innerboundary r = R3 is free.

An analytical solution in plane stress is available for this problem, for which the contactpressure on the interface r = R2 is found to be:

λ =2σR2

1

R21(1 + ν1) +R2

2(1− ν1) +E1

E2

R21 −R2

2

R22 −R2

3

(R2

2(1− ν2) +R23(1 + ν2)

) (5.72)


Introducing A1 =λR2

2 − σR21

R21 −R2

2

, B1 = (λ− σ)R2

1R22

R21 −R2

2

, A2 = −λ R22

R22 −R2

3

and

B2 = −λ R22R

23

R22 −R2

3

, the analytical displacement on ring i is:

u = ui(r)er =1

Ei

(1− νi)Air + (1 + νi)

Bir

er (5.73)

In the adaptation to X-FEM, a square with a side of 2m is meshed. So as to account forexternal pressure, the analytical displacement is prescribed on the side of this mesh as aboundary condition. The material interface and inner hole are represented by respectivelyadherent and free X-FEM interfaces (see g.5.21).

Prescribed analytical displacementR3

r

R1

Reference problem X-FEM modelisation

Adherent interface

Free interface

Material 1

Material 2R2

Figure 5.21: Bimaterial ring.

In this test the exact pressure is constant along the interface. As a consequence theinterpolation error of the Lagrange multiplier may be removed from (5.64) and (5.65).

The results on multipliers and energy for the bimaterial test show optimal rates for P1displacement and P1 subdivision, with convergence orders slightly above the theoreti-cal value 1, as well as for P2 displacement and P2 subdivision with experimental ratesmatching the theoretical value 2 almost exactly (see g.5.22). This illustrates the abil-ity of the method to optimally solve interface problems, including when higher orderinterpolations are used.

The test was also run for a P2 displacement and P1 subdivision. The pressure convergesat the suboptimal 0.98 rate, close to the prediction of 1 given by (5.65). However, theenergy exhibits a superconvergent rate 1.88, where we would expect 1.5 (see g.5.22).As was done in [120], we highlight that the eect is partly due to symetry compensationsby randomly perturbing the geometry description within the O(h2) range predicted intheorem 5.9. The expected rate 1.5 is then observed, illustrating the correctness of theprediction for the bulk consistency error.

5.6.4 A practical study of the problematic case of slanted triangles

We shall now discuss the case of slanted triangles, which were excluded from the demon-stration of the inf-sup (theorem 5.5) and examine whether it actually causes trouble orif it is only a technical issue of the demonstration. So we would like to see whetherthe constants of the convergence analysis especially that of the inf-sup condition


0,010,1

1E-4

1E-3

1E-2

1E-1

1E+0

1E-5

1E-4

1E-3

1E-2

1E-1P2 / subdivision P1. multiplier error Slope 0.98P1 / subdivision P1. multiplier error Slope 1.21P1 / subdivision P1. energy error Slope 1.28P2 / subdivision P1. energy error Slope 1.88P2 / sub. pert. P1 energy error Slope 1.52P2 / subdivision P2. multiplier error Slope 2.0P2 / subdivision P2. energy error Slope 2.0

mesh size (m)

rela

tive

mu

ltip

lier

err

or

rela

tive

en

erg

y e

rro

r

Figure 5.22: Convergence curves for a bimaterial ring.

are aected or not in the borderline case where ratio aM/am gets big, which indicatesthe presence of slanted triangles. We then consider the cracked block test of section 6.1again, and give it a small but nonzero leaning angle, so that slanted triangles will appear.The relative pressure error or local multiplier error is then plotted against ratio aM/amfor the dierent triangles on g.5.23. No dierence could be observed between slantedand other triangles, suggesting that the constants in the convergence analysis are notaected, and the problem is more a shortcoming of the demonstration than a practicalissue.

1E+0 1E+1 1E+21E-12

1E-10

1E-8

1E-6

1E-4

1E-2

1E+0

intersection ratio of the elementa high ratio indicates a slanted element

rela

tive

pres

sure

err

or

Figure 5.23: Cracked block: local pressure errors.

5.7 Conclusion

A Lagrange multiplier approach was adopted to deal with weak discontinuity problems.A new restriction algorithm (P1*)was proposed for the Lagrange multipliers in part 3,which is dedicated to a combined use with P2 -displacements. A mathematical proofthat it would then pass the inf-sup condition was presented. In part 4, an a-priorierror estimate was established for weak discontinuity problems for the H1-norm of thedisplacement and L2-norm of the Lagrange multiplier, as a function of the resolution ofthe interface and the interpolation error of the Lagrange and displacement spaces. Itwas checked against numerical experiments in part 5: when observed rates were abovetheoretical ones, this has been explained in light of what earlier literature says. We


show, among other things, that the (P1*) algorithm and quadratic subdivision achievequadratic convergence with a P2 displacement.

Chapter 6

Conclusion and outlook

In order to help decision-making about the late operating life of some nuclear powerplants, EDF is led to develop advanced modelling tools. In particular, the load conditionscausing cracks to occur in the graphite bricks of some British AGR require advancedfracture mechanics to be fully determined, as a support to onsite monitoring. Besides,assessing the nocivity of microcracks under the inox lining of a few French PWR demandsadvanced modelisation, when the worst accidental load cases and an extended plantlifetime are assumed. In this thesis, novel numerical procedures to model quasi-brittlecrack growth have been implemented in the industrial FEM software Code_Aster, andcompared with experiments from literature benchmarks including complex crack paths.

They rely on the coupling of the XFEM and cohesive zone models. The guideline ofthis thesis consists in assigning a cohesive behaviour to large potential crack surfaces, sothat the cohesive law will naturally discriminate between adherent and debonding zones,thus implicitely dening an updated crack front as the boundary between these two.In return, this strategy demands that large adherent phases be accurately accountedfor, as well as the sudden transition from adherence to nite stiness at the onset ofdebonding. With this switch from Dirichlet to Neumann-like conditions when somecriterion is met, this problem belongs to non-smooth boundary conditions, also knownas variational inequalities, along with unilateral contact for instance. When classicalpenalized laws proved unsuitable for this purpose, the issue was tackled by the use ofXFEM-suited multiplier spaces in a dedicated consistent formulation, blockwise diagonalinterfacial operators and the use of the augmented Lagrangian formalism to write aninitially perfect adherent cohesive law. A numerical test was run, and showed thateach three remedial actions would bring additional robustness in allowing to cut thenumber of Newton iterations needed for resolution, whithout impairing the accuracy ofthe results, eventually making the procedure applicable to industrial cases. While thishas solely been implemented and tested for a linear-softening cohesive law in this thesis,an important outlook is to consider extension to frictionless and frictional contact. Itwould be interesting to see how the procedure can be adapted to these cases, and wetherit reduces Newton iterations to such great extent.

With this ability of an implicit update of the crack front, a crack propagation proce-dure was set up, relying on classical level-set update algrithms and a directional anglecomputed from cohesive quatities exclusively for robustness. Numerical results werecompared with experiments from the literature on plain concrete, which is the most ex-perimented material with available data for cohesive parameters. The results showed

162

Chapter 6. Conclusion and outlook 163

rather good agreement as for the crack path, critical load and early post-peak behaviour,but we failed to reproduce the late post-peak behaviour. This is due to large processzones in concrete, and requires the use of exponential instead of linear softening to bexed. However, the eect is expected to be less pronounced for graphite and steel, giventhe smaller process zones. An major prospect would be to consider extension to het-erogeneous materials: our directional criterion then no longer holds, and a stress-basedcriterion should be adopted instead.

In chapter 3, unstable crack growth has been considered. A set-valued cohesive law isstill used. It is implicitely treated, but inserted into an explicit time-stepping scheme.With an appropriate choice of the discrete space for discrete cohesive tractions, this isfound to come down to an analytical resolution of a set of scalar equations. The switchfrom stable to unstable crack propagation is done by determining the critical load bymeans of path-following methods, and feeding the quasi-static result as an initial stateto the dynamic computation. A numerical example on a tapered DCB test shows rathergood energetic behaviour for this scheme and bears out the need for a dynamic model foran accurate prediction of crack arrest. In spite of these acceptable results, a rst outlookto this work would be to test some improvements by other authors to see if energy lossescan be reduced. Another important prospect is to include this dynamic implementationin the propagation procedure mentioned before. We did mock up a model of such aprogram, but we ran out of time for it to produce convincing results bearing comparisonwith established experimental data. In that regard, it remains to determine whether animplicit update of the crack front is still a successful strategy for dynamic crack growth.Indeed, as the process zone is found to be diuse for fast crack growth, this strategy mayfail for a range of applications. It would be interesting to see how it compares with anexplicit description of crack advance, for instance with Kanninen law.

Chapter 4 has investigated extension to quadratic nite elements, with a view to using theXFEM with specic features on the mid-term. For free curved surfaces, it was establishedtheoretically and veried numerically that a subdivision of cut elements into quadraticsubcells is required to have optimality with quadratic elements. While the theory predictsthat quadrature should be increased in distorted subcells to get optimality, the eect hasnot been observed in practice. Important prospects are the reduction of the conditioningnumber and 3D subcell construction for quadratics. These aspects are undertaken byMarcel Ndeo in his thesis to be defended next year, with very promising results.

Chapter 5 investigates the use of quadratics for adherent interfaces. A-priori error esti-mates have been given as a function of the subcell construction and interpolation proper-ties for both the primal variable and multiplier space. When checked against numericaltests, some proved suboptimal, but the missing factors could be explained in light ofwhat literature says. In particular, a novel multiplier space has been designed that issuitable for use with quadratic displacements: it has numerical stability, and producesquadratic convergence when used with quadratic subcells.

Appendix A

Constructing level-sets from implicit

or parametric representation

We need to dene regularity properties for the curve Γ to be represented. Here, it isassumed to have a geometric continuity Gk with k ≥ 2 : in general this means that Γis part of the boundary of an open set Ω of class Ck, which is translated by equations(A.2-A.4) for an implicit representation, and means that Γ admits a parametrization ofclass Ck.

Then, we derive level-sets in the upcoming lemma. Note that we always need the as-sumption of a regular closed curve to do so. Hence if Γ is open, it has to be regularlyextended beyond its extremities to get a regular level-set. This phenomenon has alreadybeen highlighted by Fries [122].

Lemma A.1. Let Ω be a bounded open set of class Ck with k ≥ 2, whose boundaryis denoted ∂Ω. Then there exists δ > 0 such that the following mapping be a Ck−1-dieomorphism:

Ξ :∂Ω×]− δ, δ[ → Sδ ⊂ R2

(x, t) → x+ tn(x)(A.1)

Its inverse will be denoted Ξ−1 : y → (π(y), φ(y)). π(y) is then called the orthogonalprojection onto ∂Ω. In the case where Ω is convex, it coincides for points outside D withthe classical denition of the orthogonal projection onto a convex set. φ(y) is our objectof interest, it is called the signed distance function to ∂Ω and used as a denition of theexact level-set.

Proof. By denition of the regularity of a bounded open set, there exists a functionϕ ∈ Ck(R2) such that :

Ω = x ∈ R2, ϕ(x) < 0 (A.2)

∂Ω = x ∈ R2, ϕ(x) = 0 (A.3)

164

Appendix A. Mathematical denition of level-sets 165

∃c > 0,∀x ∈ ∂Ω, |∇ϕ|(x) ≥ c (A.4)

So ∂Ω is an implicitly dened curve, whose points are all regular. So we may dene unitnormal and tangent vectors to this curve:

n (ϕ) :=∇ϕ|∇ϕ|

(A.5)

and:

t(ϕ) := Rot −π2

(n(ϕ)) (A.6)

With the help of the implicit functions theorem, there is a regular parametrisation of∂Ω in the vicinity of each of its points. Let x ∈ ∂Ω, there exists an open portion of thecurve containing x which admits a regular parametrization, that we can Ck-equivalentlyreparametrize with a normal parametrization s → fx(s). Then, the denition of thedirectional derivative on the dierential manifold ∂Ω yields for the derivative relative tothe rst variable:

∂Ξ

∂x= t(s) + t

dn(s)

ds= (1− tγ(s)) t(s) (A.7)

where we have made use of the Fréchet formula and called the curvature γ.

The derivative relative to the second variable is simply:

∂Ξ

∂t= n(s) (A.8)

Hence the rst derivative DΞ is:

DΞ = (1− tγ(ϕ)) t(ϕ)⊗ t(ϕ) + n(ϕ)⊗ n(ϕ) (A.9)

where the curvature is bounded. Indeed, since ϕ is of class C2 on a compact set on the

rst hand, its gradient is bounded away from zero in the other, and γ(ϕ) = t(ϕ)T ·D2ϕ·t(ϕ)|∇ϕ|

(see [137]), we may deduce that there exists R0 > 0 such that |γ(ϕ)| < 1R0

. Let ξ ∈ R2,

denoting |.| any norm on R2, we have |DΞ(ξ)| ≥ |ξ|(

1− |t|R0

). Let us set δ < R0, then

infξ

|DΞ(ξ)||ξ| ≥ 1− δ

R0> 0. Hence DΞ is invertible, and |DΞ−1| ≤ 1

1−δ/R0.

We have shown that Ξ has a nonzero Jacobian everywhere. Let us prove that it isinjective. By contradiction, let x,x′ be distinct points on ∂Ω such that there exists(t, t′) ∈] − δ, δ[2 verifying x + tn(x) = x′ + t′n(x′). As a consequence |x − x′| ≤δ|n(x)−n(x′)| and since the curvature is bounded |n(x)−n(x′)| ≤ R−1

0 |(x)− (x′)|, so

Appendix A. Mathematical denition of level-sets 166

|x − x′| ≤ δ/R0|x − x′| which is impossible since δ/R0 < 1. After the global inversiontheorem, we conclude that Ξ is a Ck−1 dieomorphism.

Lemma A.2. Let Γ be a closed regular simply-connected parametric curve, represented

by a parametrisation f :I ⊂ R → R2

s → f(s)of class Ck with k ≥ 2. Then there exists δ > 0

such that the following mapping be a Ck−1-dieomorphism:

Ξ :I×]− δ, δ[ → Sδ ⊂ R2

(s, t) → f(s) + tn(s)(A.10)

This lemma is similar to lemma A.1, but for parametrically-represented curves of geo-metrical continuity Gk.

Proof. The assumptions on f imply that it is Ck-equivalent to the normal parametriza-tion. Without loss of generality, the parametrisation may therefore be considered normal.The curvature is bounded since it is expressed as γ(s) = −t(s)T ·Dn(s) · t(s) and f is ofclass C2. The Jacobian of Ξ is then found to be bounded away from zero proceeding as inLemma A.1 and the injectivity of Ξ is proven in the same way, knowing that f(s′) = f(s)yields s′ = s since the curve is simple.

Remark A.3. Suppose that assumptions of either lemma A.1 or lemma A.2 hold. Thenthe signed distance function is of class Ck−1, by denition of a Ck−1-dieomorphism.

Appendix B

Transnite maps and their errors

Lemma B.1. Let Γ be a curved interface admitting the normal parametrization in

the local basis of g.B.1

I → Rs → f(s) = (x(s), y(s))

. Let [s1, s2] ⊂ I such that f ∈

Cg+1([s1, s2]). Let E be a transnite subcell whose boundary matches Γ exactly on [s1, s2].Then denoting FE the transnite map, FE the corresponding isoparamatric map (see

g.B.1) and RE := FE − FE, we have ‖RE‖g+1,∞,K ≤ Chg+1|f |g+1,∞,[s1,s2] where C isindependent of h and f .

Actually, with the notations of g.B.1, the characteristic size of the subcell is aMh, sowe have ‖RE‖g+1,∞,K ≤ Ch

g+1ag+1M |f |g+1,∞,[s1,s2].

Proof. We refer the reader to the appendix A of [139] for a comprehensive proof in thecase g = 1. We shall detail the proof in the case where g = 2.

a1

a2

x1E

x2E

x3E

x

y

x13E

F E

F E

Reference triangle K

Γh

Classical subcell

Transfinite subcell

x2

x1x13x3

E

E

K

x13E Γ

Figure B.1: Classical and transnite subcells, and local system of coordinates.

Step 1 : Introduce an explicit denition of Γ on [s1, s2] and show that h = O(|xE3 − xE1 |

).

Using the appropriate local system of coordinates (see g.B.1), we observe that

167

Appendix B. Transnite maps and their errors 168

y(s1) = y(s2) = 0. Hence with Rolle's theorem ∃sx ∈ [s1,s2], y′(sx) = 0. Let us seth := |s2 − s1| and h0 < |f |−1

2,∞,I , for h < h0 , the Taylor inequality on interval [sx, s]

yields x′(s)2 = 1−y′(s)2 ≥ 1−(h/h0)2 > 0. So x is C3 and strictly monotonic on [s1, s2],

hence it is invertible and the inverse s :

[0, |xE1 − xE3 |] → [s1,s2]

x → s(x)is C3.

Moreover h ≤∫ s2s1x′(s)ds +

∫ s2s1|y′(s)|ds ≤ |xE3 − xE1 | + (h/h0)h, hence |xE3 − xE1 | ≥

(1 − h/h0)h. So when the mesh is suciently small, h and |xE3 − xE1 | are equivalentquantities. From here onward we will then rather work with h := |xE3 − xE1 |. Wemay dene the part of Γ that the subcell approximates explicitely Γ : y = ν(x) where

ν :

[0, h] → R

x → y (s(x)). The regularity assumptions about f and the regularity of s

yield ν ∈ C3([0, h]).

Step 2 : Introduce the polynomial and transnite maps and prove |RE |0,∞,K ≤ Ch3|f |3,∞,[s1,s2].

Calling (ai)i∈1..3 the reference barycentric coordinates (x(a1, a2) and a3 = 1− a1− a2),and N13 the quadratic shape function of the node located at x13, the expressions of thepolynomial and transnite maps are (see the appendix of [139]):

FE(x) =3∑i=1

aixEi +

(xE13 − xE13

)N13 =

3∑i=1

aixEi + 4y

(xE13

)a1a3y (B.1)

FE(x) =3∑i=1

aixEi +

1− a1 − a2

1− a1ν(a1h)y (B.2)

Then RE := FE − FE has the expression RE (x) = a3a3+a2

ν(a1h) − 4ν(h2 )a1a3. Let uspose ν(a1) := ν(a1h), the error would then take the appropriate form:

RE (x) =a3

a2 + a3RE (a1) (B.3)

where RE(a1) := ν(a1)− 4ν(12)a1(1− a1) is the residual along the lower edge [x3, x1] of

the reference triangle on g.B.1. Hence |RE (x) | ≤ |RE(a1)| with equality if x belongsto that edge. We may then consider this case without loss of generality. Expanding ν toTaylor series at 0, 1 and 1/2 gives:

ν(a1) = a1ν′(0) + a2

1ν′′(0)

2 +R2(0; a1)

= (a1 − 1)ν ′(1) + (a1 − 1)2 ν′′(1)2 +R2(1; a1)

= ν(

12

)+ (a1 − 1

2)ν ′(

12

)+ (a1 − 1

2)2 ν′′( 1

2)2 +R2(1

2 ; a1)

(B.4)

where the Laplace residual has been noted Rk (a; b) :=∫ baν(3)(t)k! (b− t)k dt for k ∈

0, 1, 2.


We may then multiply each of the above Taylor series with its corresponding shapefunction along the segment: the series at 0, 1 and 1/2 are respectively multiplied byN3(a1) = (1− 2a1)(1− a1), N1(a1) = a1(2a1 − 1) and N13(a1) = 4a1(1− a1). Owing tothe fact that N1(a1) +N13(a1) +N3(a1) = 1, the addition yields:

RE(a1)a1(1−a1)(1−2a1) = [ν ′(0) + ν ′(1)− 2ν ′(1/2)]

+12 [a1ν

′′(0) + (a1 − 1)ν ′′(1) + (1− 2a1)ν ′′(1/2)]

+R2(0; a1)

a1− R2(1; a1)

1− a1+ 4

R2(1/2; a1)

1− 2a1

(B.5)

Further expansion into Taylor series of ν ′(0), ν ′(1), ν ′′(0) and ν ′′(1) around 1/2 allowsto cancel out rst and second order terms in the above expression:

ν ′(0) + ν ′(1)− 2ν ′(1/2) = R1(1/2; 0) +R1(1/2; 1) (B.6)

a1ν′′(0) + (a1 − 1)ν ′′(1) + (1− 2a1)ν ′′(

1

2) = a1R0(

1

2; 0) + (a1 − 1)R0(

1

2; 1) (B.7)

Hence:

RE(a1)a1(1−a1)(1−2a1) = R1(1/2; 0) +R1(1/2; 1)

+12 [a1R0(1/2; 0) + (a1 − 1)R0(1/2; 1)]

+R2(0; a1)

a1− R2(1; a1)

1− a1+ 4

R2(1/2; a1)

1− 2a1

(B.8)

From this expression it is obvious that :

|RE |0,∞,K ≤ C|ν|3,∞,[0,1] ≤ Ch3|ν|3,∞,[0,h] ≤ Ch3|f |3,∞,[s1,s2] (B.9)

Step 3 : Prove the same properties for the derivatives with respect to x(a1, a2)

Combining (B.8) with (B.3), we have:

RE(x)a1a3(1−2a1) = R1(1/2; 0) +R1(1/2; 1)

+12 [a1R0(1/2; 0) + (a1 − 1)R0(1/2; 1)]

+R2(0; a1)

a1− R2(1; a1)

1− a1+ 4

R2(1/2; a1)

1− 2a1

(B.10)

Let µ ∈ [0, 1]. We have ∂R0(µ;a1)∂a1

= ν(3)(a1). Let l ∈ 1, 2, it comes∂Rl(µ;a1)

∂a1= Rl−1(µ; a1). Hence, for m ∈ 0..3 , a straightforward recursion yields∣∣∣∂mR2(µ;a1)

∂am1

∣∣∣ ≤ |ν|3,∞,[0,1]|a1−µ|3−m. The only term of (B.10) that could be problematic

to prove the result is a3R2(1;a1)1−a1

. Its k-order derivative with respect to x, k ∈ 0..3,


involves terms of the form Rk;m,l := 1(1−a1)l+1

(∂m(a3)∂ami

∂k−m−lR2(1;a1)

∂ak−m−l1

)where m ∈ 0, 1 ,

i ∈ 1, 2 and l is an integer verifying m+ l ≤ k. In case m = 0, we have:

|Rk;m,l| ≤a3

1− a1

∣∣Dk−lR2(1, a1)∣∣

(1− a1)l≤ |ν|3,∞,[0,1]|1− a1|3−k (B.11)

Else, if m = 1 then |Rk;m,l| ≤|Dk−(l+1)R2(1,a1)|

|1−a1|l+1 ≤ |ν|3,∞,[0,1]|1 − a1|3−k, so we get to

bound all derivatives of RE with |ν|3,∞,[0,1], which given (B.9) yields the results that weintended to prove ‖RE‖3,∞,K ≤ Ch3|f |3,∞,[s1,s2].

Appendix C

Elements of proof for lengthways

intersected elements

In this appendix, we derive an estimate of the error due to the change in domain inthe case of lengthways intersected triangle, under the simplifying assumptions that (toalleviate expressions):

the interpolation is linear,

the subdivision is linear,

there is no source term: f = 0 ,

the conductivity is isotropic and uniform: A = 1 .

To this aim, we would like to rene estimate (c) in (4.27) by getting rid of the superior

bound in the consistency estimates supwh∈Vh

|ah (u, wh)− a (u, wh)|‖wh‖1,Ωh1∪Ωh2

. Considering again the

proof of theorem 4.7, and the proof of theorem 4.1.1 in [135], it can be seen that itactually suces to consider wh = uh − Πhu. Abiding by the same principle, Strangand Fix [136] also proposed an analysis of the error due to the change of domain byintroducing uh that would be the solution over the discrete space if the operators werecomputed on the exact domain:

∀vh ∈ Vh, a (uh, vh) = 0 (C.1)

The authors ([136], section 4.4) would then study E := ‖uh − uh‖1,Ωh1∪Ωh2as the consis-

tency error and come across the estimates:

E2 ≤ C |ah (u− uh, uh − uh)|+∣∣ah (u, uh − uh)− a

(u, uh − uh

)∣∣≤ ChpE + ε1/2|uh − uh|1,Sε

(C.2)

171

Appendix C. Elements of proof for lengthways intersected elements 172

h

I

KK ''

K 'e : almost intersected edge

am h

aM h

n

i

Figure C.1: Lengthways intersected triangle.

We shall adopt the same paradigm and look for an estimation of the second term. Aswe have seen in (4.63), in a lengthways intersected element K (see g.C.1), the Bergerlemma would only give:

|uh − uh|21,Sε∩K∩Ωhi≤ C aM

am

ε

h|uh − uh|21,K∩Ωhi

(C.3)

So now we aim at nding a better approximation for |uh− uh|21,K∩Ωhithat would remove

theaMam

factor. The gradient is split into a tangential and normal part to the almost

coincident edge e, according to the notations of g.C.1 :

∇ (uh − uh) = ∇τ (uh − uh) +∇n (uh − uh) (C.4)

Given that the gradients are constant over K since the interpolation is linear and thatmeas

(K ∩ Ωh

i

)≤ amhmeas

(e ∩ Ωh

i

)(see g.C.1), integrating over K ∩ Ωh

i yields:

|uh − uh|21,K∩Ωhi≤ amh|∇τ (uh − uh) |2

0,e∩Ωhi+ |∇n (uh − uh) |2

0,K∩Ωhi(C.5)

We denote K ′ the adjacent element to K such that K ′ ∩ K = e (see g.C.1). Ele-

ment K′obviously veries

meas(K′ ∩ Ωh

i

)meas (K ′)

≥ c, so that meas(e ∩ Ωh

i

)≤ Cmeas(K′)

h ≤

Cmeas(K′∩Ωhi )

h . Still keeping in mind that gradients are constant, we have then:

|∇τ (uh − uh) |20,e∩Ωhi

≤ C

h|∇τ (uh − uh) |2

0,K′∩Ωhi(C.6)


As for the second term, the discrete traduction of the free surface boundary conditionmakes ∇uh nearly othogonal to the normal of Γh, itself being a close direction to nKsince the element is lengthways intersected. To give it a mathematical traduction, wewrite the weak formulation of the problem:

∫Ωi

∇uh · ∇vhdΩ = 0 (C.7)

We then choose as a test function the shape function of the opposite node I to the almostcoincident edge. On g.C.1, we would take vh = NI for instance. Calling K

′′ the elementbelonging to the support of I along with K, it holds:

∫K∩Ωi

∇uh · ∇NIdΓ = −∫K′′∩Ωi

∇uh · ∇NIdΓ (C.8)

By the same token, we have:

∫K∩Ωhi

∇uh · ∇NIdΓ = −∫K′′∩Ωhi

∇uh · ∇NIdΓ (C.9)

Since ∇NI |K is directed toward nK , it holds ∇NI |K =C

hnK . Given that ∇uh and ∇uh

are constant elementwise, it may be deduced from (C.8) that

∇uh|K · nK = −meas(Ωi∩K′′)meas(Ωi∩K) ∇uh|K′′ · nK′′ and from (C.9) that

∇uh|K · nK = −meas(Ωhi ∩K′′)meas(Ωhi ∩K)

∇uh|K′′ · nK′′ . Substracting these expressions yields:

|∇ (uh − uh) · nK |K ≤ Cmeas(Ωhi ∩K′′)meas(Ωhi ∩K)

|∇ (uh − uh)|K′′

+C

∣∣∣∣meas(Ωi∩K′′)meas(Ωi∩K) −

meas(Ωhi ∩K′′)meas(Ωhi ∩K)

∣∣∣∣ |∇uh|K′′ (C.10)

As meas(Ωhi ∩K ′′

)∼ a2

mh2 and meas

(Ωhi ∩K

)∼ amaMh2, it holds:

meas(Ωhi ∩K ′′

)meas

(Ωhi ∩K

) ≤ C amaM

(C.11)

Moreover, we may decompose:

meas(Ωhi ∩K′′)meas(Ωhi ∩K)

− meas(Ωi∩K′′)meas(Ωi∩K) =

meas(Ωhi ∩K′′)−meas(Ωi∩K′′)meas(Ωhi ∩K)

−meas(Ωi∩K′′)meas(Ωhi ∩K)

meas(Ωhi ∩K)−meas(Ωi∩K)

meas(Ωi∩K)

(C.12)


We have |meas(Ωhi ∩K ′′

)− meas (Ωi ∩K ′′) | ≤ a2

mεh. The grey area on g.C.2a maybe estimated by |meas

(Ωhi ∩K

)−meas (Ωi ∩K) | ≤ a2

M εh, from which we immediatelydeduce meas (Ωi ∩K) ≥ aMh (amh− aM ε). Note that if aM ε ≥ amh then the edgeis intersected twice, as represented on g.C.2b. In our program, this conguration isdetected by the subdivision process and an error is issued since it cannot be handled. Sowe will assume from now on that aM ε < amh. With these estimates and (C.11), (C.12)gives:

∣∣∣∣meas(Ωhi ∩K′′)meas(Ωhi ∩K)

− meas(Ωi∩K′′)meas(Ωi∩K)

∣∣∣∣ ≤ amaM

ε

h

(1 + C

am/aM−ε/h

)≤ C ε

h

(1− aM

amεh

)−1(C.13)

Since the gradients are constant by elements, recalling the areas of meas(Ωhi ∩K ′′

)and

meas(Ωhi ∩K

), given (C.11) and (C.13), and assuming that ∇uh is bounded we have:

∫K∩Ωhi

|∇n (uh − uh) |2dΩ ≤ C(amaM

∫K′′∩Ωhi

|∇ (uh − uh) |2

+ε2amaM

(1− aM

am

ε

h

)−2)

(C.14)

h

K

am h

aM h

aM ϵ

h

K

am h aM ϵ

a. accepted configuration b. rejected configuration

Figure C.2: Lengthways intersected triangles.

Combining the results of (C.14) and (C.6) in (C.5), reporting in (C.3) yields:

|uh − uh|21,K∩Sε∩Ωhi≤ C ε

h

(|uh − uh|21,K′∩Ωhi

+ |uh − uh|21,K′′∩Ωhi

)+ C ε3

h a2M

(1− aM

amεh

)−2 (C.15)

Summing (C.15) over all lengthways intersected triangles (there are at most meas(Γ)h of

them) yields:

|uh − uh|21,Sε ≤ Cε

hE2 + C

ε3

h2a2M

(1− aM

am

ε

h

)−2

(C.16)


Reporting the expression in (C.2), we have:

E2 − CE(hp + εh−1/2

)− C

(εh−1/2

)2(

1− aMam

ε

h

)−1

≤ 0 (C.17)

Hence E lies between the roots of this second-order polynomial, so giving a superiorbound to the abs-value of the roots, we conclude about the consistency error:

E ≤ Chp + Cεh−1/2

[(1− aM

am

ε

h

)−1/2

+ 1

](C.18)

To conclude about this proof, for lengthways intersected triangles, a similar error esti-mates than theorem 4.17 may still be derived. If the intersection is convex, as on g.C.1,the estimate cannot be degenerated. On the contrary, it can if it is too concave, as ong.C.2b, but in such cases the subdivision algorithm alerts us before the estimates getscritical.

Appendix D

Appendix Proof of theorem 5.9

The statement of the theorem is recalled hereafter :

Theorem D.1. For each non slanted element K ∈ Eh, a transnite map FK : K → Kmay be established, which observes the following properties:

1.(K)K∈Th

is an admissible mesh of Ω. In other terms, the intersection of two

transnite triangles K ∩ K ′ is either empty, or equal to K, or reduced to an edgeor a vertex (see g.D.1),

2. RK := FK − FK veries ‖RK‖g+1,∞,K ≤ Chg+1,

3. F−1K

(Γ ∩ K

)= F−1

K (Γh ∩K) (see g.D.1).

F K (affine)

maps Γ onto Γh∩K

F K transfinite

maps onto ∩ K

K K K ' K '

Γ

K

a2

a1

ΓΓh

am

aM

Figure D.1: Transnite elements and transnite map.

Proof. The process of geometry approximation involves two stages. It is extensivelydiscussed in section 4.1, and summarized on g.4.3. In one word, the rst stage deducesa rst approximation Γhφ as the iso-zero of the discrete level-set function. In the secondstage, each cut element is subdivided into conforming quadratic subcells, an edge ofwhich ts three points on Γhφ. The set of all such edges constitutes Γh .

First stage of the construction.

176

Appendix D. Proof of theorem 5.9 177

A map F 1K is constructed in such a way thatG1

K := F 1KF

−1K maps Γhφ onto Γ. An element

K ∈ Eh is not invariant by G1K , but it is mapped onto a triangle with curved boundaries

K (see g.D.1). Since Γhφ has a poor regularity across the elements (as is the case for

Γh on g.D.1), K will be constructed such that Γhφ ∩K be mapped onto Γ∩ K, so as to

ensure sucient regularity properties for G1K . Let x ∈ Γhφ, then φh(x) = 0 (see g.D.2).

After (4.11), it holds x − φ(x)∇φ(x) ∈ Γ , so that x − (φ(x)− φh(x))∇φ(x) ∈ Γ (seeg.4.11). Hence, G1

K := (Id + [φh − φ]∇φ) is adopted, so that F 1K is dened by means

of:

R1K := F 1

K − FK := ([φh − φ]∇φ) FK (D.1)

x

Γϕh

∇ x

x− x ∇ x

Figure D.2: Transnite map: rst step.

The dierentiation rule for composed functions is then used to dierentiate (D.1). Pro-vided |DFK | ≤ Ch and the higher-order derivatives of FK are zero; assuming that φ andits derivatives are regular; and given that for j ∈ 0..g + 1 we have, after lemma 4.2,‖φ− φh‖j,∞,K ≤ Chg+1−j ; it comes ‖R1

K‖g+1,∞,K ≤ Chg+1.

Second stage of the construction.

In this second stage, F 2K is created in such a way that G2

K := F 2K F

−1K maps Γh onto Γhφ,

while the intersected edges of K remain invariant by this transformation (see g.D.3).

The map for points located on the interface, when g=1. In the case where g = 1, thepullback Γ := F−1

K (Γh) is merely the segment e linking the pullbacks of the intersection

points (see g.D.3). To any point on Γ , we may associate a coordinate a = a1/aM rangingfrom 0 to 1 along e. For such points, F 2

K is dened such that R2K,1 := F 2

K−FK := RE(a)y,

where RE(a) := FE(a, 0)− FE(a, 0) is the error between Γh and Γhφ (g.4.6).

The map for any point, when g=1. First, the map R2K,g=1 must be extended to the whole

triangle in such a way that it be zero on the pullbacks of the intersected edges. Hence,denition a = a1/aM may no longer be used, since it is not uniformly 1 along the edge

a2 = 0. Instead, we set a :=ra1

a2 + ra1with r := am/aM . Firstly, this reduces to

a1/aM on e, since an equation of e is e : a2 + ra1 = am. Secondly, we do have a = 0along the edge a1 = 0 and a = 1 along the edge a2 = 0 this time, implying thatRE(a) = 0 along these edges. We then set F 2

K such that:

R2K,1 := F 2

K − FK :=

(a2 + ra1

am

)g+1

RE(a)y (D.2)


In the previous expression, factor

(a2 + ra1

am

)g+1

is meant to recover the regularity

properties demanded by b) as will be highlighted in a next step. This coecient valueis obviously 1 on Γ .

The map for any point, when g=2. A preliminary step has to be added which maps Γonto e. Let us denote y and Y the second local axis of K and K , and yc and Yc therelated coordinates at the summit of parabola Γh and Γ (see g.D.3), we may dene

F (x) = x+4a1a2

amaMYcY . It satises F (e) = Γ and we shall construct R2

K,2 = R2K,1 F−1

where the expression for R2K,1 is given by (D.2). We may evaluate yc by applying (4.15)

with g = 1, as yc = O(a2Mh

2) , so its pull back satises Yc = O(a2Mh), and we can deduce

that |F − Id |0,∞,K ≤C

rh ≤ Ch since there are no slanted elements. Hence, when h is

suciently small, F is invertible and the successive derivatives of F−1 are bounded, aswas shown by Ciarlet (see [135], theorem 4.3.3). We may therefore denote R2

K = R2K,1

from here onward, and study this map without loss of generality.

x3K

F K affine

maps onto h∩K

F K2

( transfinite)

maps Γ onto Γϕh∩Kam

aM a1

a2

x1K

x 2K

Kh

e

x1x3

x 2

xy

Y c

yc

Y

Figure D.3: Transnite map: second step.

Proof that R2K and its derivatives satisfy c). Equation (4.15) states that

‖RE‖g+1,∞,K ≤ Chg+1ag+1M . Replacing am = raM , and since r is bounded away from 0

for non slanted triangles, we have

(a2 + ra1

am

)g+1

≤ C

ag+1M

. The two previous results lead

to‖R2

K‖0,∞,K ≤ Chg+1. We aim at proving the same properties for the derivatives.Denoting D the derivation operator with respect to x (a1, a2) and considering j ∈

1..g+1, derivativeDjR2K involves in (D.2) sums

dlREdal

1

ag+1m

Dj−l(

[a2 + ra1]g+1)⊗Dla

where l ∈ 0..j. The involved terms are estimated by ‖RE‖g+1,∞,K ≤ Chg+1ag+1M ,∣∣∣Dj−l

([a2 + ra1]g+1

)∣∣∣ ≤ |a2 + ra1|g+1−j+l and the dierenciation∣∣Dla

∣∣ ≤ r(l!)

(a2 + ra1)l

of the denition of λ. Hence |DjR2K | ≤ Chg+1 1

rg, so if the element is not slanted,

‖R2K‖g+1,∞,K ≤ Ch

g+1 is obtained.

Pulling together the results.


We shall now draw the conclusions of the theorem. Let us remind that we are lookingfor FK such that property c) F−1

K(Γ ∩ K) = F−1

K (Γh ∩K) be fullled. Applying FK to

both sides, this amounts to seeking G−1K := FK F

−1K such that Γ ∩ K = G−1

K (Γh ∩K).

Since G2K maps Γh ∩K onto Γφ ∩K and G1

K maps Γφ ∩K onto Γ ∩ K, we set:

G−1K = G1

K G2K = F 1

K F−1K F 2

K F−1K (D.3)

Hence the expression of FK :

FK = F 1K F−1

K F 2K (D.4)

So property c) of the theorem holds by construction. Besides, let (K ′,K) ∈ (Eh)2

and x ∈ K ∩ K ′ , then G2K(x) = x. Moreover, G1

K(x) is composed of continuousfunctions depending upon the location of x, but not upon K, so that the mapping iscontinuous across the elements boundaries: G1

K(x) = G1K′(x). Hence G−1

K (x) = G−1K′ (x).

Reductio ad absurdum, this proves property a): should there be any overlap or gapbetween neighboring transnite triangles K and K ′ , there would be a point x ∈ K ∩K ′whose image by G−1

K would be strictly inside or outside of K ′, that is to say such thatG−1K (x) 6= G−1

K′ (x), which is impossible.

Let us prove b). The residual is RK =(FK +R1

K

) F−1

K (FK +R2

K

)− FK , which can

nally be reduced to:

RK = R1K +R1

K F−1K R2

K +R2K (D.5)

Then the dierenciation of (D.5) yields ‖RK‖g+1,∞,K ≤ Chg+1 according to the results

of steps 1 and 2, and the estimation |DF−1K | ≤ Ch−1.

Appendix E

Cohesive crack propagation: some

implementation details

E.1 Implemention points about the geometrical update oflevel-sets

About the geometrical update algorithm from section 2.4.4, let M be the point wherelevel-set functions are to be updated, and P its projection point onto the crack front (seegs.2.38 and E.1). An important assumption in the geometrical update algorithm is thatpoint M should belong to plane (P,np, tp), as on g.E.1. If it does not, basis (np, tp)has to be corrected to ensure it, before the update algorithm itself can be applied.

P

nP

t P

bP

Mt

n

P

Mt

n

a. basis at the projection point on the front

P

M

tn

b. local basis c. polar basis related to the projection point on the front

crack frontcohesive zone

Figure E.1: Assumption about the projection point onto the front for the geometricalupdate algorithm

There are three scenarios for which M /∈ (P,np, tp). They share the fact that theprojection point P then lies at an extremity of the crack front. These cases and thecorresponding corrections to be performed, in order of time, are listed below.

180

Appendix E. Details about cohesive crack propagation 181

E.1.1 The inclined crack surface with respect to the boundary of thestructure

In this case, nP has to be replaced by a corrected vector n*P , so that M ∈ (P,n*

P , tP ),as is shown on g.E.2. That correction reads:

n*P ← (1− tP ⊗ tP ) ·PM (E.1)

followed by:

n*P ←

n*P

‖n*P ‖

(E.2)

t P

PnP

M

nP*

Figure E.2: Inclined crack surface with respect to the boundary of the structure

The normal vector will be considered corrected accordingly in what follows: we use n*P

instead of nP from here onward.

E.1.2 A curved crack front or a curved structure boundary

In these cases, it is necessary to replace tP by a corrected t*P , so that M ∈ (P,n*P , t

*P ),

as is shown on gures E.3 and E.4. That correction reads:

t*P ←(1− n*

P ⊗ n*P

)·PM (E.3)

followed by:

t*P ←t*P‖t*P ‖

(E.4)

t Pt P*

P

M

nP*

Figure E.3: The curved crack front case


t PP

M

t P*

nP*

Figure E.4: The curved crack boundary case

E.2 Implemention details about the computation of equiv-alent stress intensity factors

As was pointed out in section 2.1.1.2, the energy release rate G is dened along the crackfront T , parametrized by its curvilinear abscissa s, by (2.9), recalled thereafter:

∀crack virtual extensionθ,

∫TG(s)θ(s) · t(s)ds = G(θ) (E.5)

In the presence of cohesive forces, it has been shown in section 2.2.3 that the expressionof G(θ) comes down to a surface integral (see (2.28), recalled thereafter):

G(θ) =

∫Γtc · ∇[u] · θdΓ (E.6)

Equivalent stress intensity factors (SIF) are deduced from similar computations of G butalong specic directions, by (2.66-2.67 from section 2.4.3), recalled thereafter:

K2I,eq = − E

1− ν2

∫Γ

∂[u]n∂θ

tc,ndΓ (E.7)

K2II ,eq = − E

1− ν2

∫Γ

∂[ut]

∂θtc,tdΓ (E.8)

E.2.1 Discrete energy release rate G and discrete stress intensity fac-tors (Keq

I , KeqII )

Hence, to compute the value ofG along the front, it is necessary to dene a discrete G anda nite number of test functions θi, representing a range of crack extension modes alongthe front. In this thesis, as in Code_Aster, a xed number Ng of uniformly discributedlinear shape functions Ψi is used (see g.E.5), so that:

G(s) =

Ng∑i=1

Ψi(s)Gi (E.9)


with, for i ∈ 1..Ng:

Ψi(s) =

2(Ng − 1)

s− si−1

|T |if s ∈ [si−1, si]

2(Ng − 1)si+1 − s|T |

if s ∈ [si, si+1]

0 otherwise

(E.10)

where we have dened si =i− 1

Ng − 1|T |, |T | being the length of the crack front.

0 ∣T∣sisi−1 si+1

Ψi

1Ψi−1 Ψi+1

shape function

curvilinear abscissa along the front

Figure E.5: Shape functions for discrete G and theta

LetM ∈ Γ, P be its projection point onto the crack front and sP the curvilinear abscissaat P . We dene Ng test functions, by:

θi(M) = Ψi(sP )tP for i ∈ 1..Ng (E.11)

In this way, G being represented by an unknown vector with a size Ng, determining itthrough (E.6) comes down to solving a linear system.

E.2.2 Direction of θ near the extremities of the crack front

As was mentioned in section 2.4.3, the equivalent stress intensity factors quantify thedissipated energy in the three fracture mode for a self-similar propagation of the cohesivecrack in the direction θ. Hence, θ is expected to be a smooth eld whose lines connectthe zero-stress front the separation line between the cohesive and traction-free zones to the crack front (see g.E.6c). In particular, it implies that θ should satisfy θ · ν = 0on the boundaries of the structure, with ν the normal to the boundary.

Now, using the uncorrected tangent vector tP in (E.11) does not allow to satisfy this(see g.E.6a). However, this has still been used in this thesis. The fact that θ · ν 6= 0has been handled by discarding the values of G at the extremities, and replacing themby an interpolation from the closest inner values.

However, as a prospect for future work, we could think of several ways to solve the issue:

1. Equation (E.11) could be replaced by θi(M) = Ψi(sP )t*P , with t*P the corrected

value from (E.3-E.4). This xes the problem, but does not produce a very smooth


eld for θ (see g.E.6b), resulting in strong dierences between extremities andinner values.

2. A smoother construction of θ could be imagined by drawing lines, each of whichconnects a point on the crack front to the point on the zero-stress front with thesame s/|T | ratio (see g.E.6c)

cohesive zone

a. Uncorrected tangent vector

crack front

zero-stress front

b. Corrected tangent vector c. Construction from the zero-stress front

νnormal vectorto the boundary

Figure E.6: Various ways of dening eld theta

E.2.3 Choice of the basis to split modes

The cohesive zone may be spread over a curved zone. The choice of the basis (n, t, b)to be used in (E.7-E.8) to split modes and compute (KI ,KII ,KIII ) is debatable. Onemight choose:

the basis at the crack front (g.E.7a);

the basis at the considered point (g.E.7b);

the basis related to the polar coordinates of the point, relatively to the crack front(g.E.7c).

P

nP

t P

bP

Mt

n

P

Mt

n

a. basis at the projection point on the front

P

M

tn

b. local basis c. polar basis related to the projection point on the front

crack frontcohesive zone

Figure E.7: Possible basis to split nodes

To choose the best option, let us consider a test with an arc-shaped crack under tensionfrom [51], represented on g.E.8, for which reference analytical solutions (E.12-E.13) areavailable for the stress intensity factors:

KI =σ

2

√πR sin(α)

[[1− sin2(α/2) cos2(α/2)

]cos(α/2)

1 + sin2(α/2)+ cos(3α/2)

](E.12)


KII =σ

2

√πR sin(α)

[[1− sin2(α/2) cos2(α/2)

]sin(α/2)

1 + sin2(α/2)+ sin(3α/2)

](E.13)

A small arc-shaped interface is inserted that extends the pre-crack and is prone to co-hesive debonding (see g.E.8). The chosen material parameters are summarized in tableE.1. They have been chosen so that:

no unsteady large crack propagation will occur: the applied tensile load is less thanthe tensile strength and the corresponding analytical G is less than the fractureenergy Gc;

a cohesive zone will develop ahead of the pre-crack tip, with a typical length lc thatremains small enough in comparison with the pre-crack (represented in blue ong.E.8), so that the analytical solution is still acceptable for the cohesive problem.

R= 4.25 40α=28°10°

σ

-σ

Cohesive zone

x

y

ux = 0

lc

Figure E.8: The arc-shaped crack test, computed with a small cohesive zone

The mesh size is chosen to have a good resolution of the cohesive zone and a convergednumerical solution: the errors with respect to the analytical solution reported on tableE.2 would no longer evolve with further mesh renement.


Modulus of elasticity E = 1000MPa

Poisson ratio ν = 0

Fracture energy Gc = 0.002N.mm−1

Tensile strength σc = 0.8MPa

Augmentation parameter r = 10

Applied tensile load σ = 0.4MPa

Mesh size h = 0.06mm

Table E.1: Material data for arc-shaped crack test.

The dierence between the computed value and its analytical reference is reported ontable E.2, for the three options. In this thesis, equivalents SIF are only used to computethe crack bifurcation angle β through (2.69). Thus, the adopted option is the one withsmaller relative error on β: the third option.

Relative error KI KIIKI

KIIcrack tilt angle β

Solution a 7.03 % 2.53 % 8.94 % 4.39 %

Solution b 3.38 % 9.62 % 6.03 % 2.96 %

Solution c 5.2 % 3.79 % 1.32 % 0.64 %

Table E.2: Relative errors for the arc-shaped crack test.

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Guilhem FERTÉ - lamsid.cnrs-bellevue.fr · XFEM, which is achieved in quasi-statics with the use of XFEM-suited multiplier spaces in a consistent formulation, blockwise diagonal