implementation of 2d xfem in vast - defence...

Defence R&D Canada – Atlantic

Copy No. _____

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

Implementation of 2D XFEM in VAST

DEFENCE DÉFENSE&

Contract Report

DRDC Atlantic CR 2010-098

July 2010

The scientific or technical validity of this Contract Report is entirely the responsibility of the contractor and the contents do not necessarily have the approval or endorsement of Defence R&D Canada.

Lei Jiang

Martec Limited

Martec Limited

1888 Brunswick St., Suite 400

Halifax, Nova Scotia B3J 3J8

Project Manager: Lei Jiang, 902-425-5101 x228

Contract Number: W7707-088100/001/HAL CU2

Contract Scientific Authority: Dr. Dave Stredulinsky, 902-426-3100 x352

This page intentionally left blank.

�


Lei Jiang Martec Limited Prepared By: Martec Limited 400-1800 Brunswick Street Halifax, Nova Scotia B3J 3J8 Canada

Contract Project Manager: Lei Jiang, 902-425-5101 Ext 228 Contract Number: W7707-088100/001/HAL CU2 CSA: Dr. Dave Stredulinsky, 902-426-3100 Ext 352 The scientific or technical validity of this Contract Report is entirely the responsibility of the Contractor and the contents do not necessarily have the approval or endorsement of Defence R&D Canada.

Defence R&D Canada – Atlantic

Contract Report


July 2010

Principal Author

Original signed by Lei Jiang

Lei Jiang

Senior Research Engineer

Approved by

Original signed by Neil Pegg

Neil Pegg

Head/Warship Performance

Approved for release by

Original signed by Ron Kuwahara for

Calvin Hyatt

Chair/Document Review Panel

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2010

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale,

2010

DRDC Atlantic CR 2010-098 i

Abstract ……..

This report is concerned with a recent implementation of the extended finite element method

(XFEM) in the VAST finite element program for 2D fracture mechanics analyses. The XFEM is a

new finite element formulation recently developed based on the method of partition of unity, in

which the classical finite element approximation is enriched by a discontinuous function and the

asymptotic displacement functions around crack tips. It allows the crack to be in the interior of

elements, so eliminates the need for explicitly modelling cracks in the finite element mesh and the

need for remeshing for crack propagation. In this report, the theoretical background of the XFEM

formulation and various issues related to its implementation in VAST are discussed in detail. The

results from an extensive numerical verification are also presented. The effects of various factors

on the accuracy of XFEM predicted mixed mode stress intensity factors, including the mesh size,

the area of crack tip enrichment, the minimum orders of numerical integration and the extent of

constraints, are investigated and guidelines for properly using the XFEM capability are proposed.

The results presented in this report demonstrated the potential and unique advantages of XFEM in

analysing crack propagations in ship structures and a number of possible future developments are

suggested.

Résumé ….....

Le présent rapport porte sur une mise en œuvre récente de la méthode à éléments finis élargie

(XFEM) dans le programme à éléments finis VAST pour les analyses en deux dimensions de la

mécanique des fissures. La XFEM est une nouvelle méthode à éléments finis récemment élaborée

à partir de la méthode de partition d’unité, dans laquelle l’approximation à éléments finis

classique est assortie d’une fonction discontinue et des fonctions de déplacement asymptotique

autour des extrémités de fissure. Cela permet à la fissure d’être à l’intérieur des éléments, ce qui

élimine le besoin de modéliser de façon explicite les fissures dans le modèle à éléments finis,

ainsi que le besoin de remailler pour la propagation des fissures. Dans le présent rapport, les

notions théoriques de la méthode XFEM et les différentes questions liées à sa mise en œuvre dans

le programme VAST sont étudiées en détails. Les résultats d’une vérification numérique étendue

sont aussi présentés. Les effets de divers facteurs sur la précision de facteurs d’intensité de

contrainte en mode mixte prévus par le programme XFEM, y compris la taille du modèle, la zone

d’enrichissement d’extrémité de fissure, les ordres minimaux d’intégration numérique et la portée

des contraintes, font l’objet d’une analyse, et des lignes directrices permettant d’utiliser

adéquatement les capacités de la méthode XFEM sont proposées. Les résultats présentés dans le

présent rapport ont démontré le potentiel de la méthode XFEM et les avantages uniques qu’elle

présente dans l’analyse de la propagation des fissures dans la structure des navires; de plus, on

suggère un certain nombre de développements à venir possibles.

ii DRDC Atlantic CR 2010-098


DRDC Atlantic CR 2010-098 iii

Executive summary


L. Jiang; DRDC Atlantic CR 2010-098; Defence R&D Canada – Atlantic; July 2010.

Introduction: The modeling of fracture and material damage has been a problem of significant

interest in solid mechanics for a long time. This is because crack initiation and propagation are

important factors that need to be considered in design and maintenance of practical engineering

systems, such as fatigue crack propagation in ship structures subjected to cyclic loading. Many

finite element formulations have been proposed for fracture mechanics analyses over the years.

However, all the classical finite element approaches require the crack be explicitly modeled in the

finite element mesh. As a result, for crack propagation, continuous remeshing has to be performed

that requires repeated mapping of the field variables between meshes which may affect both the

efficiency and the accuracy of the numerical solutions. In order to minimize the requirement of

remeshing during crack propagation analysis, a new finite element formulation, named the

extended finite element method (XFEM), has been developed. In this method, the classical finite

element approximation is enriched by a discontinuous function and the asymptotic displacement

field around the crack tips, so cracks are permitted in the interior of elements.

Results: In the present work, the extended finite element method (XFEM) is implemented in the

VAST finite element program for 2D fracture mechanics analyses. The theoretical background of

the XFEM formulation is presented in this report and the various issues related to the present

implementation of XFEM in VAST are discussed. Compared to the standard XFEM formulation

presented in the literature, a number of simplifying modifications were adopted, such as the use of

the shifted enrichment field that simplifies the procedure for displaying the deformed shape in

post-processing and the direct use of the asymptotic displacement functions that eliminated the

need of interaction integral for mixed mode fracture mechanics problems. This new 2D XFEM

capability has been verified by test example problems including plates with horizontal and slant

edge cracks and a center crack of arbitrary orientations. The VAST predicted mixed mode stress

intensity factors are in good agreement with the published analytical solutions.

Significance: DRDC and Martec have been involved in development of structural analysis

software aimed at providing DND with tools to make maintenance management decisions

regarding damaged ship structures in a timely manner. For example, if a crack is discovered in the

ship structure, can the ship undertake a certain operation in the damaged condition or should it be

taken out of service until repairs can be made. The numerical results obtained in this work

demonstrated the XFEM’s potential and unique advantage in modelling crack propagation in ship

structures and thus should help improve the efficiency and accuracy of crack modelling tool.

Future plans: Before the XFEM capability in VAST is applied to practical engineering analyses,

a substantial effort is still required to further verify its performance for curved and kinked cracks,

to improve the robustness of the pre- and post-processor, to compare the XFEM results obtained

from the interaction integral direct approach and to extend the present 2D XFEM capability to

3D. Some of these issues will be addressed in the next phase of the development.

iv DRDC Atlantic CR 2010-098

Sommaire .....

Implementation of 2D XFEM in VAST (Mise en œuvre de XFEM 2D dans le programme VAST)

L. Jiang; DRDC Atlantic CR 2010-098; R & D pour la défense Canada – Atlantique; juillet 2010.

Introduction ou contexte : La modélisation des fissures et des dommages causés aux matériaux

est un problème qui suscite de l’intérêt depuis longtemps dans le domaine de la mécanique des

solides. C’est parce que la formation et la propagation des fissures sont des facteurs importants à

considérer dans la conception et l’entretien des systèmes techniques, par exemple dans le cas de

la propagation des fissures causées par la fatigue dans les navires assujettis à des charges

cycliques. De nombreuses méthodes à éléments finis ont été proposées pour expliquer les

mécanismes de fissuration au fil des ans. Cependant, dans toutes les méthodes à éléments finis

classiques, il faut modéliser explicitement la fissure dans le modèle à éléments finis. Par

conséquent, il faut procéder continuellement à des remaillages nécessitant une cartographie

répétée de la variable de champ entre les mailles, ce qui peut avoir une incidence sur l’efficacité

et l’exactitude des solutions numériques. Afin de réduire au minimum la nécessité de « remailler

» durant l’analyse de la propagation des fissures, une nouvelle méthode à éléments finis, appelée

« méthode à éléments finis élargie (XFEM) », a été mise au point. Dans cette méthode,

l’approximation à éléments finis classique est assortie d’une fonction discontinue et d’un champ

de déplacement asymptotique autour des extrémités de fissure, ce qui fait en sorte que des fissures

sont admises à l’intérieur des éléments.

Résultats : Dans les travaux actuellement en cours, la méthode à éléments finis élargie (XFEM)

est appliquée au programme à éléments finis VAST dans le cadre d’analyses 2D de la mécanique

des fissures. Les notions théoriques appliquées dans la méthode XFEM sont examinées dans le

présent rapport et les différentes questions liées à la mise en œuvre de XFEM dans le programme

VAST sont étudiées. Comparativement à la méthode XFEM standard présentée dans la

documentation, un certain nombre de modifications ayant pour but de simplifier la méthode ont

été adoptées, notamment l’utilisation d’un champ d’enrichissement décalé qui simplifie la

procédure pour afficher les déformations observées dans le post traitement et l’utilisation directe

de fonctions de déplacement asymptotique, qui ont éliminé le recours à une intégrale d’interaction

dans le cas des problèmes de mécanique des fissures en mode mixte. Les capacités de la nouvelle

méthode 2D XFEM ont été vérifiées en solutionnant des exemples de problèmes, par exemple la

formation de fissures à arête oblique, ou la formation d’une fissure centrale présentant des

orientations arbitraires. Le programme VAST a prévu des facteurs d’intensité de contrainte en

mode mixte qui concordent avec les solutions d’analyse publiées.

Importance : RDDC et Martec ont participé à la mise au point d’un logiciel d’analyse structurale

afin de donner au MDN des outils servant à prendre des décisions éclairées en gestion de la

maintenance des structures de navires endommagées en temps opportun. Par exemple, si l’on

découvre qu’une fissure s’est formée dans la structure d’un navire, il convient de se demander si

le navire devrait ou non effectuer certaines opérations dans l’état endommagé ou s’il devrait être

mis hors service jusqu’à ce que les réparations soient effectuées. Les résultats numériques

obtenus dans le cadre de ces travaux ont démontré le potentiel de la méthode XFEM et les

DRDC Atlantic CR 2010-098 v

avantages uniques qu’elle présente dans la modélisation de la propagation des fissures qui se

forment dans la structure des navires, et cela devrait permettre au MDN d’améliorer l’efficacité et

l’exactitude des outils de modélisation des fissures.

Perspectives : Avant que la méthode XFEM intégrée au programme VAST soit appliquée à des

analyses techniques pratiques, des efforts importants sont encore requis pour vérifier son

rendement dans le cas des fissures courbées et déformées, afin d’améliorer la robustesse du

prétraitement ou du post traitement, de comparer les différents résultats XFEM obtenus à l’aide

de la méthode de l’intégrale d’interaction directe et d’appliquer la capacité 2D XFEM actuelle à

une méthode 3D. Certaines de ces questions seront traitées lors de la prochaine étape de la mise

au point.

vi DRDC Atlantic CR 2010-098


DRDC Atlantic CR 2010-098 vii

Table of contents

Abstract …….. ................................................................................................................................ i

Résumé …..... .................................................................................................................................. i

Executive summary ...................................................................................................................... iii

Sommaire ..... ................................................................................................................................ iv

Table of contents .......................................................................................................................... vii

List of figures ............................................................................................................................. viii

List of tables ................................................................................................................................... x

1 Introduction ............................................................................................................................. 1

2 Theoretical Formulation of the Extended Finite Element Method (XFEM) ............................. 2

2.1 Purpose of XFEM Development ................................................................................... 2

2.2 Foundation of XFEM: Partition of Unity Method ......................................................... 3

2.3 Formulation of the XFEM for Crack Modeling ............................................................ 4

2.4 Evaluation of Stress Intensity Factors ........................................................................... 7

2.5 Comparison of XFEM and the Existing Fracture Elements in VAST ........................... 9

3 Implementation of XFEM in VAST....................................................................................... 11

3.1 Computer Implementation of XFEM .......................................................................... 11

3.2 Development of a Pre-Processor for Mesh-Crack Interaction ..................................... 12

3.3 Development of a Inverse Map Capability ................................................................. 16

3.4 Evaluation of Higher-Order Numerical Integration Rules for Triangles ..................... 16

3.5 Interpretation of the Heaviside Function ..................................................................... 20

3.6 Selection of the Crack Tip Enrichment Functions ...................................................... 23

3.7 Use of a Shifted Displacement Field ........................................................................... 25

4 Verification of the 2D XFEM Capability in VAST................................................................ 27

4.1 Plate with an Edge Crack ............................................................................................ 27

4.2 Plate with a 45o Slant Crack ........................................................................................ 28

4.3 Plate with Angled Centre Crack.................................................................................. 29

5 Conclusions ........................................................................................................................... 42

References ..... .............................................................................................................................. 43

Annex A .. Input Data for 4-Noded 2D XFEM Fracture Element (IEC = 68) ............................... 45

viii DRDC Atlantic CR 2010-098

List of figures

Figure 1: A curved crack problem solved by the enriched fracture element in VAST. ................... 2

Figure 2: A dynamic crack propagation problem solved by XFEM (Reproduced from [20]). ........ 3

Figure 3: Different types of enrichments in the XFEM formulation. .............................................. 5

Figure 4: Conventions at crack tip. Domain A is enclosed by , C+, C�, and C0. Unit normal mj

= nj on C+, C�, and C0 and mj = ��j on � (Reproduced from [7]). ................................. 8

Figure 5: Elements selected about the crack tip for calculation of the interaction integral

(Reproduced from [7]). ................................................................................................. 9

Figure 6: Subdivision of elements in 2D. (a) Element e1 that is intersected by the crack (dark

line), (b) Element e2 that contains the crack tip (Reproduced from [7]). ..................... 12

Figure 7: Interaction of a straight crack with a regular mesh ........................................................ 14

Figure 8: Interaction of a curved crack with a regular mesh ......................................................... 14

Figure 9: Interaction of a curved crack with an irregular mesh ..................................................... 15

Figure 10: Interaction of a curved crack with a refined irregular mesh ........................................ 15

Figure 11: Domain divisions used to test numerical integration rules for triangles ...................... 18

Figure 12: A patch of conventional finite elements near a crack tip in which the dots indicate

nodes and the circled numbers are element numbers (Reproduced from [7]). ............ 22

Figure 13: A patch of extended finite element near a crack tip (Reproduced from [7]). ............... 22

Figure 14: Global and local coordinate systems at a crack tip. ..................................................... 24

Figure 15: A finite plate with an edge crack subjected to uniform tensile stress. .......................... 32

Figure 16: XFEM model of a plate with edge crack ..................................................................... 33

Figure 17: Details around the crack with different areas for assignment of crack tip

enrichment functions. ................................................................................................. 33

Figure 18: Deformed configuration obtained using R=0.3. .......................................................... 34

Figure 19: Deformed configuration obtained using R=1.0. .......................................................... 34

Figure 20: Problem geometry, external load and boundary condition for a plate with a 450

slanted edge crack under uni-axial tension. ................................................................ 35

Figure 21: XFEM model of a plate with a 45O slanted crack. ...................................................... 36

Figure 22: Details around the crack with different areas for assignment of crack tip

enrichment functions. ................................................................................................. 36

Figure 23: Deformed configuration of the plate with a 45O slanted crack obtained using

R=0.5, (10×10, 12) integration and global MPC. ....................................................... 37

Figure 24: Deformed configuration of the plate with a 45O slanted crack obtained using

R=0.5, (20×20, 19) integration and 6 MPC equations. ............................................... 37

DRDC Atlantic CR 2010-098 ix

Figure 25: Plate with angled center crack. .................................................................................... 38

Figure 26: XFEM model for the plate with an angled center crack (�=15O) superimposed on a

40×40 uniform mesh. ................................................................................................. 39

Figure 27: Details around the crack in the coarse mesh showing nodes with different types of

enrichments. ............................................................................................................... 39


200×200 uniform mesh. ............................................................................................. 40

Figure 29: Details around the crack in the fine mesh. ................................................................... 40

Figure 30: Comparison of analytical and XFEM predicted stress intensity factors for different

� values using the coarse and fine underlying finite element meshes. ........................ 41

x DRDC Atlantic CR 2010-098

List of tables

Table 1: Integration over domain (a) evaluated using different numerical integration rules. ........ 18

Table 2: Integration over domain (b) evaluated using different numerical integration rules. ........ 19

Table 3: Integration over domain (c) evaluated using different numerical integration rules. ........ 19

Table 4: Integration over domain (d) evaluated using different numerical integration rules. ........ 20

Table 5: Comparison of analytical and XFEM results for plate with an edge crack ..................... 30

Table 6: Comparison of Analytical and XFEM Results for Plate with 45O Slant Crack ............... 31

DRDC Atlantic CR 2010-098 1

1 Introduction

The modeling of fracture and material damage has been a problem of significant interest in solid

mechanics for a long time. This is because crack initiation and propagation are important factors

that need to be considered in design and maintenance of practical engineering systems. One

example is the accurate prediction of fatigue crack propagation in ship structures subjected to

cyclic loading. Many finite element formulations have been proposed for fracture mechanics

analyses over the years. However, all the classical finite element approaches have a common

disadvantage. They require the crack be explicitly modeled in the finite element mesh, which can

be very challenging for complex engineering structures with curved crack geometry. In addition,

to simulate crack propagation, continuous remeshing has to be performed and repeated mapping

of the field variables, such as stresses and strains, are required between the old and new meshes

which may raise concerns on accuracy of the numerical solutions.

In order to minimize the requirement of remeshing during crack propagation analysis, a new finite

element formulation, named the extended finite element method (XFEM), has been developed. In

this method, the standard displacement field in the finite element method is enriched by applying

a discontinuous displacement function along the crack line and the asymptotic displacement field

around the crack tips based on a recently developed mathematical formulation named partition of

unity. XFEM has been applied to a wide range of fracture mechanics problems, including

arbitrary branching and interaction of multiple cracks, dynamic and fatigue crack propagation and

arbitrary crack evolution in shells undergoing large displacements. In addition, the XFEM has

been extended to non-planar 3D crack growth simulations. A literature review of the XFEM

formulation was completed by Martec under a previous call-up tasking [1].

Comparing with earlier numerical methods for fracture mechanics, XFEM has a number of

advantages, including (a) it does not require the cracks be explicitly modeled, so no remeshing or

minimal remeshing is needed for crack propagation; (b) it is a finite element method, so it can be

implemented in existing general-purpose finite element programs, such as VAST; (c) in contrast

to boundary elements, it is readily applicable to non-linear problems; and (d) in contrast to finite

elements with remeshing, it does not require as many projections between different meshes.

In the present contract, the extended finite element method (XFEM) is implemented in the VAST

program to solve two-dimensional fracture mechanics problems. In the next chapter, the

theoretical background of the XFEM formulation is presented in detail and the various issues

related to the present implementation of the XFEM capability in VAST are discussed in Chapter

3. The results from an extensive numerical verification are provided in Chapter 4. Finally,

conclusions from the present work are given in Chapter 5, where some guidelines for using the

XFEM capability are provided and a number of possible future developments are proposed.

2 DRDC Atlantic CR 2010-098

2 Theoretical Formulation of the Extended Finite Element Method (XFEM)

2.1 Purpose of XFEM Development

As mentioned before, fracture mechanics is of particular interest in prediction of crack growth in

ship structures. Many theories of crack propagation are based on stress intensity factors, which

need to be evaluated using linear elastic fracture mechanics theory. Over the past few decades,

significant developments have been made to solve two- and three-dimensional linear fracture

mechanics problems using the finite element method. One widely used method is to utilize the

quarter-point isoparametric elements, which contain a singular stress field at the crack tip and

then, the stress intensity factors are calculated from either the stress or the displacement field or

by energy based procedures, such as stiffness derivatives and the J-integral [2]. An alternative

finite element method is the enriched isoparametric solid element formulation which has been

implemented in VAST [3, 4]. A standard test example for the 4-noded quad enriched fracture

element in VAST is shown in Figure 1, which involves mixed-mode deformation of a curved

crack under a constant stress field in the horizontal direction.

Figure 1: A curved crack problem solved by the enriched fracture element in VAST.

These classical finite element approaches for fracture mechanics have all been proved to be very

effective in prediction of the stress intensity factors in engineering structures under practical

loading conditions. However, they also have a common disadvantage. They require the crack be

explicitly modelled in the finite element mesh, which creates challenges for current meshing tools

when dealing with cracked structural components with highly complex geometry. In addition, in

simulation of crack propagation, continuous remeshing has to be performed which requires

repeated mapping of the field variables, such as stresses and strains, between different meshes.

These mappings of solution variables not only reduce computational efficiency, but may also

compromise the accuracy of the numerical results.


In order to minimize the requirement for remeshing during crack propagation analyses, a new

finite element formulation, named extended finite element method (XFEM) was developed in the

late 1990s [5]. Compared with the earlier finite element methods for fracture mechanics, XFEM

has a number of advantages, including:

It does not require the cracks be explicitly modeled, so no remeshing or minimal remeshing is

needed for crack propagation.

It is a finite element method, so it can be implemented in existing general-purpose finite

element programs such as VAST.

In contrast to boundary elements, it is readily applicable to non-linear problems.

In contrast to finite elements with remeshing, it does not require as many projections between

different meshes.

The XFEM has now been applied to a wide range of fracture mechanics problems, including

arbitrary branching and interaction of multiple cracks, dynamic and fatigue crack propagation and

arbitrary crack evolution in shells undergoing large displacements. In addition, the XFEM has

been successfully extended to solve problems involving non-planar 3D crack growth. An example

for application of XFEM for simulation of dynamic crack propagation is given in Figure 2 [20],

where a fixed finite element mesh is utilized for the entire simulation. A detailed review of the

XFEM formulation and its applications has been provided in [1].

Figure 2: A dynamic crack propagation problem solved by XFEM (Reproduced from [20]).

2.2 Foundation of XFEM: Partition of Unity Method

The extended finite element method (XFEM) is built on a new finite element formulation, named

the Partition of Unity Method (PUM). The method is motivated by the need for new techniques

for solving problems where the classical FEM fails or is prohibitively expensive, such as the

problem of crack propagation [6]. The basic concept of this method is to construct a conforming

finite element space with local properties of the partial differential equations being solved. The

(c) t=103.01�s

(b) t=99.44�s (d) t=105.64�s

(a) t=97.54�s (c) t=103.01�s

(b) t=99.44�s (d) t=105.64�s

(a) t=97.54�s


approach taken in the PUM is to start from a variational formulation and then design the trial (and

test) spaces in view of the problem under consideration.

The general form of approximation of a vector-valued function with the PUM is

� � � � � �1 1

N Mh

I I

I

N ��

�

�

� �

� �� x x x au (1)

where NI(x) are finite element shape functions, which form a partition of unity, such that

� � 1IIN � x . (2)

In Equation (1) given above, �� are enrichment functions and I

�a are the nodal unknowns.

Comparing this equation with the assumed displacement field of a classical finite element, it is

readily recognized that the classical finite element space � �� 1 1; 0 1�� is a special case of

the partition of unity method.

2.3 Formulation of the XFEM for Crack Modeling

The XFEM for crack modeling is a special application of the Partition of Unity method (PUM). In

XFEM, an enrichment of the finite element partition of unity near the crack tip is added to the

classical finite element space as [5, 7]

� � � � � � � �4

1 1

Nh

I I I I

I

N H ��

�

� � � � ��

� �� u x x u x a x b (3)

where uI denote the normal nodal degrees-of-freedom in a displacement-based finite element

formulation, such as displacements along the axes of a global coordinate system. H(x) is the

generalized discontinuous Heaviside function to model the interior of a crack defined as [7]

� �*1 ( ) 0

1

ifH

otherwise

��

�

x x nx (4)

where x is a sample (Gauss) point, x* is a point which lies on the crack and closest to x, n is the

unit outward normal to the crack at x*, and aI is the nodal enriched degree of freedom vector

associated with the Heaviside function.

In order to model the crack and also to improve the representation of crack-tip fields in fracture

computations accurately, crack-tip enrichment functions, � �x�� , are used in the finite elements

containing the crack-tip. The crack-tip enrichment functions are derived from the 2D asymptotic

crack-tip displacement field. The use of crack-tip functions serves two purposes:

When a crack terminates in the interior of an element, use of the Heaviside function will result

in inaccurate solutions because in this case, the crack is virtually extended to the intersection


with an element edge. The use of crack-tip functions ensures that the crack terminates at the

correction location.

The use of the linear elastic asymptotic crack-tip fields as the crack-tip functions ensures not

only correct representation of the near-tip behaviour, but also better accuracy for relatively

coarse finite element meshes in both 2D and 3D analyses.

The crack-tip enrichment functions for isotropic elasticity are [5, 9-10]

� �! " ! "##$$

$$

$$$

�� ,,2

cossin2

sinsin2

cos2

sin4...1, �%��

��

�� rrrrx

(5)

where r and $ are polar coordinates in the local crack-tip coordinate system as illustrated in

Figure 3. Note that the first term in the right hand side of the above equation is discontinuous

across the crack ($ = &180o). In Equation (3), I

�b indicates the nodal enriched degree of freedom

vector associated with the crack-tip functions.

Figure 3: Different types of enrichments in the XFEM formulation.

As illustrated in Figure 3, based on the method of partition of unity, the enrichment functions are

applied nodewise. As a result, different nodes in the same element can be enriched by different

functions. In the implementation of XFEM, the nodes in a finite element model are divided into

three distinct sets. The first set includes nodes whose shape function support contains the crack-

tip. The second set includes nodes whose shape function support is cut by the interior of the

crack. Finally, the third set contains nodes which do not belong to the first two sets mentioned

above. This definition of node sets is shown in Figure 3, where the nodes in the first and second

Enriched by Heaviside

function H(x)

Enriched by crack

tip functions

x’y’

r$

Enriched by Heaviside

function H(x)

Enriched by crack

tip functions

x’y’

r$


sets are indicated as red and blue circles, respectively. During the computation of the element

stiffness matrices, as will be discussed later, the normal finite element field is applied to all nodes,

but the crack-tip functions and the Heaviside discontinuous function are only applied to the nodes

in the first and second sets, respectively. It should be noted that the Heaviside function and the

crack tip enrichment functions are never applied to the same nodes.

For nodes in the second set, the support of the nodal shape function is fully cut into two disjoint

pieces by the crack. If for a certain node, one of the two pieces is very small compared to the

other, the generalized Heaviside function used for the enrichment is almost a constant over the

support, leading to an ill-conditioned stiffness matrix [7]. In this case, this node will need to be

removed from this set.

For computational efficiency considerations, the nodal enrichments should be localized to the

sub-domain where the enrichments are beneficial. However, Chessa et al. [8] demonstrated that

the use of a blending area, in which the enriched elements blend to the un-enriched elements, is

often crucial for good performance of local partition of unity enrichments. For polynomial

enrichments, the accuracy and convergence can also be improved by proper choice of the finite

element shape functions and the partition of unity shape functions. The improvement of several

enriched finite element schemes for correctly constructed blending elements were illustrated [8].

Once the displacement approximation for XFEM is established as given in Equation (3), the finite

element discretization can be readily accomplished using the principle of virtual work as stated

below

:h h h

h hd d d' ' '( ( �

( � (� � �) ) )b u t u** + (6)

where ** and + denote the stress and strain tensors, respectively, the colon between them indicates

tensor product. b and t are the body force vector and boundary tractions. Substituting the

displacement field in Equation (6) results in the discretized equilibrium equation as

Kd f (7)

where

uu ua ub

ij ij ij

e au aa ab

ij ij ij ij

bu ba bb

ij ij ij

� ��

� ��

k k k

K k k k

k k k

and

, -1 2 3 4T

u a b b b b

i i i i i if f f f f f f

are the stiffness matrix and equivalent force vector, respectively, and d contains generalized nodal

degrees of freedom, including nodal displacements as in the classical finite elements u, the nodal

unknowns associated with the generalized discontinuous Heaviside function a, and the

coefficients of the crack-tip enrichment functions b. The subscripts i, j denote nodes. Detailed


mathematical expressions for element stiffness matrix and equivalent nodal force vector are given

in [9].

Due to the existence of discontinuous functions in the enriched displacement field, some special

treatments are required to ensure accuracy of the numerical integration and thus to prevent the

rank deficiency of the computed stiffness matrix. This has been achieved by subdividing the

elements involving the enrichment functions into triangular domains. As will be discussed in the

next chapter, this subdivision of elements requires extensive numerical operations.

2.4 Evaluation of Stress Intensity Factors

In the XFEM formulation for fracture mechanics, stress intensity factors are normally computed

using domain forms of the interaction integrals and an excellent overview of this procedure was

presented in [7]. The coordinates are taken to be the local crack tip co-ordinates with the x1-axis

parallel to the crack faces. For general mixed-mode problems we have the following relationship

between the value of the J-integral and the stress intensity factors

E

K

E

KJ III

22

� (8)

where E is the Young’s modulus. Consider two states of a cracked body. State 1, ( )1()1()1( ,, ijiji u+* )

corresponds to the present state and State 2, ( )2()2()2( ,, ijiji u+* ), is an auxiliary state which will be

chosen as the asymptotic fields for Modes I or II. The J-integral for the sum of the two states is

� ��

��

�

��

�

.

�.�� )�

� dnx

uuJ j

ii

jijijjijijiji

1

)2()1(

)2()1(

1

)2()1()2()1()21(

2

1**'++** (9)

Expanding and rearranging terms gives

)2,1()2()1()21( IJJJ �� (10)

where I(1,2) is called the interaction integral for States 1 and 2 and defined as

��

��

�

.

.�

.

.� )

�

dnx

u

x

uWI j

iji

ijij

1

)1()2(

1

)2()1(

1

)2,1()2,1( **' (11)

where W(1;2) is the interaction strain energy

)1()2()2()1()2,1(

jijijijiW +*+* (12)

Writing Equation (8) for the combined the states and rearranging terms, we have

� �)2()1()2()1()2()1()21( 2IIIIII KKKK

EJJJ �� (13)


Equating (9) with (13) leads to the following relationship:

� �)2()1()2()1()2,1( 2IIIIII KKKK

EI � . (14)

Making the judicious choice of State 2 as the pure Mode I asymptotic fields with )2(

IK =1 gives

Mode I stress intensity factor for State 1 in terms of the interaction integral

),1()2( 2 IMode

I IE

K . (15)

Mode II stress intensity factor can be determined in a similar fashion.

Figure 4: Conventions at crack tip. Domain A is enclosed by , C+, C�, and C0. Unit normal mj =

nj on C+, C�, and C0 and mj = ��j on � (Reproduced from [7]).

The contour integral (11) is not in a form best suited for finite element calculations. We therefore

recast the integral into an equivalent domain form by multiplying the integrand by a sufficiently

smooth weighting function q(x) which takes a value of unity on an open set containing the crack

tip and vanishes on a prescribed outer contour C0 as shown in Figure 4. Then for each contour �,

assuming the crack faces are traction free and straight in the region A bounded by the contour C0,

the interaction integral may be written as

��

��

�

.

.�

.

.� )

�

dmqx

u

x

uWI j

i

ji

i

jij

1

)1(

)2(

1

)2(

)1(

1

)2,1()2,1( **' (16)

where the contour C=�+C++C�+C0 and m is the unit outward normal to the contour C. Now using

the divergence theorem and assuming that the inner contour � is shrunk to the crack tip, gives the

following equation for the interaction integral in domain form


dAx

qW

x

u

x

uI

jA

ji

jii

ji ..

��

��

��

..

�..

) 1

)2,1(

1

)1()2(

1

)2()1()2,1( '** (17)

where the relations mj = �nj on � and mj = nj on C0, C+ and C� have been utilized.

Figure 5: Elements selected about the crack tip for calculation of the interaction integral

(Reproduced from [7]).

For the numerical evaluation of the above integral, the domain A is set from the collection of

elements about the crack tip as illustrated in Figure 5. This selection of elements for computation

of the domain interaction integral requires the characteristic length of elements touched by the

crack tip, hlocal, to be first determined. In 2D cases, this quantity can be calculated as the square

root of the element area. The domain A was then set to be all elements which have a node within a

sphere of radius rd about the crack tip. Figure 5 gives an example where a set of elements for the

domain A with the domain radius rd taken to be twice the length hlocal. The q(x) function is taken

to have a value of unity for all nodes within the sphere rd, and zero on nodes outside the sphere.

The function is then easily interpolated within the elements using the nodal shape functions. It

should be noted that because only the derivative of function q(x) appears in the domain integral

given in (17), only the elements cut by the spherical surface, in which the value of q(x) varies, are

actually involved in the calculation.

2.5 Comparison of XFEM and the Existing Fracture Elements in VAST

In order to better understand the advantages and disadvantages of the XFEM, it is beneficial to

compare it with the existing fracture elements in the VAST program. These fracture elements in

VAST, such as the 4-noded and 8-noded quadrilateral elements for 2D cracks and 20-noded brick

element for 3D cracks, are developed based on an enriched solid element formulation [11], in

which the standard isoparametric finite element displacement field is enriched by the asymptotic

crack tip displacement field obtained from 2D linear elastic fracture mechanics as


I I II II

i i I i i II i i

i i i

N K N K N � �

� � � ��

� � �u u u u u u (18)

where uI and uII are the asymptotic displacement fields for Modes I and II, respectively. The

coefficients of the enrichment functions are the stress intensity factors, which are unique for each

crack-tip in the problem and are solved together with the nodal displacements, so no post-

processing, such as the J-integral, is required to calculate the stress intensity factors. Because the

additional degrees of freedom, the stress intensity factors, are shared by all the fracture elements

surrounding a crack-tip, they can be accommodated by introducing a new node at each crack-tip

point.

This treatment of the enrichment field is very different from that in the XFEM formulation. In

XFEM, the enrichment field is expressed in terms of additional nodal degrees of freedom,

resulting in a larger finite element system which requires more computational time to solve. In

addition, the solutions of the linear algebraic equations generated by the XFEM include nodal

displacements and nodal parameters associated with the discontinuous Heaviside and crack-tip

functions, but not the stress intensity factors. A post-processing step, such as the J-integral, is

normally required as described in Section 2.4.

However, the enriched displacement field in Equation (18) requires that the crack-tip be at one of

the corner nodes of the fracture element. In addition, no discontinuity is permitted in the interior

of the element. These restrictions require that the cracks must be explicitly modelled in a finite

element mesh, and in simulations of crack propagation, repeated remeshing must be performed

whenever the crack geometry altered. Furthermore, in order to ensure convergence, displacement

compatibility must be enforced at the interface of the enriched and non-enriched areas.

On the other hand, the XFEM permits discontinuities and crack-tip singularities be imposed onto

the standard finite element field at arbitrary locations, so the finite element mesh does not have to

be restricted to have the cracks aligned with element boundaries. This is extremely beneficial for

simulating crack propagations in geometrically complicated structural components because the

need for repeated remeshing is eliminated. As a result, the XFEM provides a better hope for

developing a fully automated software tool for 3D crack propagation.


3 Implementation of XFEM in VAST

3.1 Computer Implementation of XFEM

Sukumar and Prevost [9] provided a detailed description of the five main steps involved in the

implementation of the XFEM formulation into an existing general-purpose finite element code

based on their experience on incorporating the XFEM capabilities in Dynaflow.

The first step involves modification of the input data to the finite element program to include

descriptions of crack geometry. In 2D analyses, cracks are normally approximated by multiple

line segments. The modified input data also include other XFEM parameters, such as the

definition of the enrichment functions. Differing from the previous implementations of XFEM, in

the present VAST implementation, the asymptotic crack tip displacement functions were directly

utilized instead of the expansions of the asymptotic displacement functions given in Equation (5).

Details will be presented later in this chapter.

The second step requires expansion of the nodal degrees of freedom in the classical finite element

formulation to accommodate the additional nodal unknowns associated with the enrichment

functions, such as the generalized Heaviside function to describe the discontinuity and the crack-

tip functions to obtain an accurate representation of the local field in the vicinity of the crack tip.

The third step is to treat mesh-crack geometry interaction, including crack-element intersection

and element partitioning as indicated in Figure 6. The element partitioning is required to insure

equivalence between the strong and weak forms of the governing system of equations and to

preserve accuracy of numerical integration in order to eliminate potential rank deficiency of the

stiffness matrix. The element partitioning, in which the elements enriched by the discontinuous

and the crack-tip functions are subdivided into sub-triangles, requires extensive operations of

computational geometry to determine the intersections of the crack line with the element edges.

However, it should be realized that there are fundamental differences between element

partitioning and remeshing, for the following reasons:

Element partitioning is done solely for the purpose of numerical integration, so no new degrees

of freedom are introduced.

No restrictions are placed on the shape of the partitioned elements as the finite element field is

defined by the parent element.

During an analysis involving crack propagation, the crack geometry is continuously updated

using some sort of crack growth criteria, such as the Paris law. Once updated crack geometry is

available, the calculations for mesh-crack geometry interaction, outlined above, must be repeated.

In the present work, a pre-processor was developed to deal with mesh-crack interactions as will

be detailed in the next section.


Figure 6: Subdivision of elements in 2D. (a) Element e1 that is intersected by the crack (dark line), (b) Element e2 that contains the crack tip (Reproduced from [7]).

The fourth step is to generalize the assembly procedure and the matrix solver in the finite element

program to permit the additional nodal degrees of freedom associated with the enrichments. At

the present time, only the original skyline-based linear algebraic equation solver allows more than

six degrees-of-freedom per node, so all the numerical results given in this report were obtained

using this skyline-based solver. An investigation is underway to extend the far more efficient

sparse direct matrix solver to accommodate the additional nodal degrees-of-freedom resulting

from the enrichment functions.

The fifth and final step is to develop post-processing capabilities to support XFEM, including the

evaluation of stress intensity factors and modification of the graphical capabilities for displaying

displacements and stresses to include the contributions of the enrichment field. In the present

implementation of XFEM, the direct use of the asymptotic crack tip displacement field eliminated

the requirement for the domain integral because the mixed mode stress intensity factors were

obtained along with the nodal displacements, as in the existing enriched fracture elements in

VAST. As will also be described later in this chapter, the use of a shifted enriched displacement

field in this work resulted in a simplified procedure for graphical display of the deformed

configuration. However, due to the subdivision of the elements intersected by crack and the use of

higher order numerical integrations in elements involving enrichment functions, special

considerations are still required for proper modification of the graphical capabilities for proper

display of the field solutions, such as stresses and strains.

3.2 Development of a Pre-Processor for Mesh-Crack Interaction

As mentioned above, in the XFEM application for two-dimensional linear elastic fracture

mechanics, the nodes around the crack tips are enriched by the asymptotic displacement fields,

whereas the nodes along the crack line, excluding the crack tips are enriched by the discontinuous


Heaviside function. In order to ensure that appropriate enrichment functions are assigned to each

node in the finite element model, a pre-processor must be developed to analyze the interaction

between the crack and the underlying finite element mesh to identify the elements that are cut by

the crack and to calculate the intersections between the element edges and the crack line.

Because both the Heaviside function and the asymptotic displacement functions are discontinuous

across the crack, the numerical integration in enriched elements must be carried out separately

over domains above and below the crack line to ensure accuracy of the computed stiffness

matrices. This requires that the domains above and below the crack are automatically subdivided

into triangles. However, it should be noted that this subdivision process is fundamentally different

from mesh refinement because it is purely for the purpose of numerical integration and does not

introduce any additional degrees of freedom to the finite element system.

A pre-processor for treating mesh-crack interactions was developed in the present work. The

input data to this pre-processor included a standard finite element mesh not containing any cracks,

and a crack line defined in terms of a set of straight line segments. The output from the pre-

processor includes:

the nodes that need to be enriched and the associated type of enrichment functions,

the orientations of the enriched nodes (above or below the crack);

the coordinates of the intersecting points of the element edges and the crack,

the triangular domains in all elements that require subdivision.

the coordinates of the crack tips and the direction cosines defining the local coordinate system

at the crack tips.

A GOM file containing all this information is generated by the pre-processor for the new XFEM

element types IEC68.

The present version of pre-processor has been tested using a number of example problems. The

first example involved interaction of a plate discretized using a regular finite element mesh with

an inclined straight crack as shown in Figure 7. For this example, the pre-processor identified all

the elements cut by the crack correctly and subdivided the domains above and below the crack

into sub-triangles as required. In order to verify the capability of the pre-processor for dealing

with curved crack geometry, we then considered a circular crack in the same plate as shown in

Figure 8. The circular crack was defined using 8 line segments and all the intersections between

these line segments and the element edges were computed correctly. In order to further test the

robustness of the pre-processor, we considered interaction between the same circular crack with a

irregular mesh as shown in Figure 9. As can be seen from the figure, the pre-processor worked

well for irregular meshes. The last example was used to benchmark the computational efficiency

of the pre-processor for dealing with large finite element models. In this example, the same plate

and crack geometries were considered again. However, the circular crack was represented by 80

linear segments and the plate was discretized into a much finer irregular mesh containing

approximately 8000 nodes and elements as shown in Figure 10. On a typical laptop computer, the

pre-processor took less than one second to complete, indicating that the present algorithm of pre-

processing is very efficient for practical problems.


Figure 7: Interaction of a straight crack with a regular mesh

Figure 8: Interaction of a curved crack with a regular mesh


Figure 9: Interaction of a curved crack with an irregular mesh

Figure 10: Interaction of a curved crack with a refined irregular mesh


3.3 Development of a Inverse Map Capability

As mentioned before, the subdivision of the quadrilateral elements into triangles is purely for the

purpose of numerical integration. The standard and enriched displacement approximations are

still defined in the original quadrilateral element in terms of the parametric coordinates, and the

displacement-strain matrices must still be formulated in the original quadrilateral element as well.

During the numerical integration, the global coordinates of the numerical integration points in the

triangular domains are first computed from the global coordinates of the vertices and the area

coordinates at the integration points. However, in order to evaluate the integrand at these points,

an inverse map is required to solve for the corresponding parametric coordinates in the original

quadrilateral element.

In an isoparametric quadrilateral element, the physical coordinates, X,Y, and the parametric

coordinates, /,0, are related through the shape functions

� � � �� 1

23

��

123

��

123

��

� 0/

0/0/

,

,,

4

1 Y

X

Y

XN

Y

X

i

i

i

i (19)

where i indicates the node number and N, the standard bi-linear shape functions. The inverse map

algorithm can be formulated using the Newton method given below

123

��4

4

123

��4

4

��

�

�

��

�

�

.

...

.

...

Y

X

YX

YX

0/

0/

0/ . (20)

The right hand side of this equation is the difference between the specified physical coordinates

and those computed from the isoparametric expression (19) using the most updated parametric

coordinates. This equation is applied repeatedly until this difference becomes sufficiently small.

This inverse map algorithm was programmed and extensively tested using quadrilateral elements

of various geometries. The test results indicated that this inverse map program was both robust

and efficient. After testing, the subroutine for the inverse mapping was incorporated in the VAST

code.

3.4 Evaluation of Higher-Order Numerical Integration Rules for Triangles

Because the evaluation of stiffness matrices of elements enriched by the asymptotic displacement

field requires numerical integration of singular functions, higher-order numerical quadrature rules

for triangles must be used in these elements to ensure that sufficient accuracy is achieved. To

identify appropriate numerical integration schemes for the present implementation, a literature

search was performed and a number of high-order numerical integration rules for triangular

domains were identified [12-15].


To assess the relative accuracy of these numerical integration rules, two FORTRAN programs

were created to evaluate the following three integrations over a number of different quadrilateral

domains as indicated in Figure 5 (a-d):

)) dXdYXYI1

� � dXdYrdXdYYXI )))) � 222

2 (21)

� � dXdYr

dXdYYXI )))) �� 1422

3

where 22 YXr � denotes the distance of a point (X,Y) to the origin. It should be noted that

the third integration, I3, replicated the singularity produced by the asymptotic displacement field

around a crack tip.

In addition to the numerical quadrature rules identified in the literature, the integration schemes

currently utilized in VAST for the triangular plate element was implemented in these programs.

In order to provide a basis for comparison, high order numerical integration rules for rectangular

domains based on the Gauss and Simpson methods were also considered. In the first program, a

square domain was divided into two equal triangles as shown in Figure 11(a) and the inverse map

algorithm was not implemented as it was not required for rectangular element shape. In the

second program, the domain of the quadrilateral element was subdivided into four triangles as

shown in Figure 11 (b, c, d) and many of the standard computations in finite element method,

such as evaluation of the shape functions, computation of the Jacobian matrix and the inverse

map outlined above, were all implemented. For this reason, the skewed quadrilateral element

could be conveniently considered using the second program.

The results of these numerical integrations over different domains are presented in Tables 1-4.

Among them, the results obtained using the 40×40 Gaussian quadrature should be regarded as the

most accurate and used as reference solutions. Surprisingly, the present tests indicated that none

of the integration rules proposed by Sunder and Cookson [12] provided correct results, even for

the cases with non-singular integrand functions. The 7-point rule implemented in VAST yielded

results identical to the 7-point rule of Dunavant [13] which should normally be sufficient for

XFEM applications. If a higher accuracy is required, either the 12-point rule by Cowper [14] or

the 19-point rule by Dunavant [13] can be utilized.

In the present implementation of XFEM in VAST, 10×10 Gaussian quadrature rule is always

used for elements that contain enrichment functions, but not intersected by the crack. This order

of numerical integration is identical to that utilized in the original 4-noded fracture element in

VAST. For elements that are cut by the crack, but only involve the discontinuous Heaviside

function, either 4 or 7-point integration rule for triangles is employed. For elements intersected by

the crack and involved singularity-producing crack tip enrichment functions, three higher-order

numerical integration rules for triangles can be utilized which contain 12, 19 and 25 integration

points, respectively. In VAST analyses, the numerical integration orders are controlled by users

through the GOM file.


Figure 11: Domain divisions used to test numerical integration rules for triangles

Table 1: Integration over domain (a) evaluated using different numerical integration rules.

Integration Rule I1 I2 I3 SIMPSON RULE 0.2500000000E+00 0.6666666667E+00 0.1249986331E+01

20X20 GAUSSIAN 0.2499999981E+00 0.6666666652E+00 0.1249990128E+01

40X40 GAUSSIAN 0.2500000000E+00 0.6666666667E+00 0.1249986844E+01

4-POINT P=3 RULE OF

COOKSON

0.2478634836E+00 0.6752127321E+00 0.1278789841E+01

6-POINT P=4 RULE OF

COOKSON

0.2479822811E+00 0.6747375422E+00 0.1293223753E+01

7-POINT P=5 RULE OF

COOKSON

0.2476439881E+00 0.6745907143E+00 0.1302180608E+01

7-POINT P=5 RULE OF

DUNAVANT

0.2500000000E+00 0.6666666667E+00 0.1251037442E+01

19-POINT P=9 RULE OF

DUNAVANT

0.2500000000E+00 0.6666666667E+00 0.1250176573E+01


DUNAVANT

0.2500000000E+00 0.6666666667E+00 0.1247655168E+01

9-POINT P=5 RULE OF COWPER 0.2499999889E+00 0.6666666412E+00 0.1239527020E+01


4 POINT RULE IN VAST 0.2500000000E+00 0.6666666667E+00 0.1240831799E+01


(1,0)(0,0)

(1,1)(0,1)

X

Y

(1,0)(0,0)

(1,1)(0,1)

X

Y

(1,0)(0,0)

(2,1)(0,1)

X

Y

(1,0)(0,0)

(2,2)

(0,1)

X

Y

a

b

c

d

(1,0)(0,0)

(1,1)(0,1)

X

Y

(1,0)(0,0)

(1,1)(0,1)

X

Y

(1,0)(0,0)

(1,1)(0,1)

X

Y

(1,0)(0,0)

(1,1)(0,1)

X

Y

(1,0)(0,0)

(2,1)(0,1)

X

Y

(1,0)(0,0)

(2,1)(0,1)

X

Y

(1,0)(0,0)

(2,2)

(0,1)

X

Y

(1,0)(0,0)

(2,2)

(0,1)

X

Y

a

b

c

d


Table 2: Integration over domain (b) evaluated using different numerical integration rules.

Integration Rule I1 I2 I3 SIMPSON RULE 0.2500000000E+00 0.6666666667E+00 0.1249986331E+01

20X20 GAUSSIAN 0.2499999981E+00 0.6666666652E+00 0.1249990128E+01

40X40 GAUSSIAN 0.2500000000E+00 0.6666666667E+00 0.1249986844E+01

4-POINT P=3 RULE OF

COOKSON

0.2500000000E+00 0.6709396994E+00 0.1260338240E+01

6-POINT P=4 RULE OF

COOKSON

0.2500000000E+00 0.6707021044E+00 0.1274462339E+01

7-POINT P=5 RULE OF

COOKSON

0.2497750000E+00 0.6703286905E+00 0.1282783016E+01

7-POINT P=5 RULE OF

DUNAVANT

0.2500000000E+00 0.6666666667E+00 0.1244225461E+01


DUNAVANT

0.2500000000E+00 0.6666666667E+00 0.1248435366E+01


DUNAVANT

0.2500000000E+00 0.6666666667E+00 0.1248428016E+01





Table 3: Integration over domain (c) evaluated using different numerical integration rules.

Integration Rule I1 I2 I3

20X20 GAUSSIAN 0.7083333311E+00 0.1833333333E+01 0.1662752873E+01

40X40 GAUSSIAN 0.7083333333E+00 0.1833333333E+01 0.1662749650E+01

4-POINT P=3 RULE OF

COOKSON

0.7099374158E+00 0.1846916240E+01 0.1677163592E+01

6-POINT P=4 RULE OF

COOKSON

0.7100111955E+00 0.1846346265E+01 0.1693743100E+01

7-POINT P=5 RULE OF

COOKSON

0.7096768235E+00 0.1846194625E+01 0.1703396940E+01

7-POINT P=5 RULE OF

DUNAVANT

0.7083344673E+00 0.1833308193E+01 0.1656826048E+01


DUNAVANT

0.7084584735E+00 0.1833619843E+01 0.1661088156E+01


DUNAVANT

0.7084028353E+00 0.1833733771E+01 0.1660687877E+01






Table 4: Integration over domain (d) evaluated using different numerical integration rules.

Integration Rule I1 I2 I3

20X20 GAUSSIAN 0.1666666666E+01 0.3666666671E+01 0.2037693499E+01

40X40 GAUSSIAN 0.1666666667E+01 0.3666666667E+01 0.2037690331E+01

4-POINT P=3 RULE OF

COOKSON

0.1676280991E+01 0.3694441380E+01 0.2052715549E+01

6-POINT P=4 RULE OF

COOKSON

0.1675746402E+01 0.3692897012E+01 0.2071589919E+01

7-POINT P=5 RULE OF

COOKSON

0.1674756221E+01 0.3691069822E+01 0.2082707767E+01

7-POINT P=5 RULE OF

DUNAVANT

0.1666666667E+01 0.3666666668E+01 0.2030611715E+01


DUNAVANT

0.1666666667E+01 0.3666666667E+01 0.2035764803E+01


DUNAVANT

0.1666666666E+01 0.3666666666E+01 0.2035475539E+01





3.5 Interpretation of the Heaviside Function

One of the most important developments in XFEM is the use of the discontinuous Heaviside

function to represent the discontinuous displacement fields across a crack. The mathematical

expression of the Heaviside function is given in Equation (4) in the previous chapter. In this

section, we will demonstrate that the finite element approximation for a mesh with an explicitly

modelled crack is equivalent to the summation of an approximation for a mesh without a crack

and a discontinuous enrichment along the crack line. This example provides insights into the

Heaviside enrichment function.

Consider a patch of conventional 4-noded quad finite elements around a crack tip as shown in

Figure 12. The displacement at an arbitrary point inside these elements can be expressed in terms

of the shape functions and nodal displacements of the 10 nodes as

�

10

1i

iiN uu (22)

We then introduce two new vectors, a and b, which indicate the average displacements at the

nodes on either side of the crack face and the displacement differences between these nodes as

2,

2

109109 uub

uua

�

� (23)


Rearranging terms, we have

baubau �� 109 , (24)

Substituting (24) back into (22), we have

� � � � � �xxbauu HNNNNNi

ii 109109

8

1

��

(25)

where H(x) is referred to here as a discontinuous, or ‘jump’ function. This is defined in the local

crack coordinate system as

� � ��

5�

6�

0,1

0,1,

y

yyxH (26)

such that H(x) = 1 on element 1 and ��.

We now consider a finite element mesh shown in Figure 13, in which the crack is not explicitly

modelled, so N9+N10 can be replaced by N11 and a by u11. The finite element approximation given

in (25) for the mesh shown in Figure 12 can now be expressed as

� �xxbuuu HNNNi

ii 111111

8

1

��

(27)

It should be noted that the first two terms in the right hand side of the above equation represent

the standard finite element approximation, whereas the last term indicates a discontinuous

enrichment in terms of the Heaviside function, H(x). In other words, when a crack is modeled by

a mesh as in Figure 12, we may interpret the finite element space as the sum of one which does

not model the crack (such as Figure 13) and a discontinuous enrichment function.

The previous derivation provides insight into the extension of the technique for the case when the

crack does not align with the mesh. The key issues are the selection of the appropriate nodes to

enrich, and the form of the associated enrichment functions. In terms of enrichment with the jump

function, we adopt the convention that a node is enriched if its support is cut by the crack into two

disjoint pieces. This rule is seen to be consistent with the previous example, in which only node

11 was enriched.


Figure 12: A patch of conventional finite elements near a crack tip in which the dots indicate nodes and the circled numbers are element numbers (Reproduced from [7]).

Figure 13: A patch of extended finite element near a crack tip (Reproduced from [7]).


3.6 Selection of the Crack Tip Enrichment Functions

In more standard implementation of XFEM for fracture mechanics applications, the crack tip

enrichment functions were chosen to be the expansion of the asymptotic displacement field near a

crack tip obtained from linear elastic fracture mechanics. These enrichment functions are given in

Equation (5) in the previous chapter and are repeated below

� �! " ! "##$$

$$

$$$

�� ,,2

cossin2

sinsin2

cos2

sin4...1, �%��

��

�� rrrrx

(5)

The corresponding enriched displacements resulted from these enrichment functions are

123

��

��

��

�

��

��

123

��

y

x

Cv

u

b

b

4321

4321

0000

0000 (28)

where the vectors of nodal enriched displacements, bx and by, result in a total of eight additional

degrees of freedom at each node enriched by the crack tip functions. Although these enrichment

functions provide more flexibility in the finite element system and a better chance for obtaining

more accurate numerical solutions, the stress intensity factors cannot be directly solved along

with the nodal unknown and a post-processing step, such as the domain interaction integral, must

be performed.

In the present work, we have attempted to combine the advantages of the XFEM formulation and

the original fracture elements based on an enriched isoparametric solid element formulation. In

particular, instead of using the expansions of the asymptotic displacement field given in Equation

(5), the asymptotic displacement field is directly utilized, so the additional enriched nodal degrees

of freedom are the mixed mode stress intensity factors as

123

��

��

��

�

123

��

II

I

III

III

CK

K

vv

uu

v

u (29)

where u and v are displacement along the global X and Y-axes, as shown in Figure 14, and the

superscripts I and II denote asymptotic fields for Mode I and Mode II deformations, respectively.

The asymptotic displacements in the global coordinate system can be related to displacements in

local coordinates through coordinate transformation as

),('cos),('sin

),('sin),('cos

$�$�

$�$�

rvruv

rvruu

III

III

�

� (30)

where u’ and v’ indicate displacements in the crack tip coordinate system, x’-y’, and � is the

angle between the global and local systems as shown in Figure 14. For an arbitrary point near the

crack tip having polar coordinates (r,$) with respect to the local crack tip coordinate system, the

asymptotic displacement field can be expressed as [16]


� �

� � ��

��

� ��

��

��

� ��

2

3sin

2sin12

24

1'

2

3cos

2cos12

24

1'

$$7

#

$$7

#

r

Gv

r

Gu

I

I

(31a)

� �

� � ��

��

� ��

��

��

� ��

2

3cos

2cos12

24

1'

2

3sin

2sin32

24

1'

$$7

#

$$7

#

r

Gv

r

Gu

II

II

(31b)

for Mode I and Mode II deformations. For plane stress, we have

88

7��

1

3 (32)

where 8 denotes Poisson’s ratio.

Figure 14: Global and local coordinate systems at a crack tip.

The formulation of the strain-displacement matrix, B, requires evaluation of derivative of the

approximate displacement field with respect to the parametric coordinate of the quadrilateral

finite element, / and 0. Consider derivative of u’I with respect to / as an example, we have

crack

X

Y

x’

y’

$

�

r

u

v

u’

v’

crack

X

Y

x’

y’

$

�

r

u

v

u’

v’


� �

� �/$$$

7#

/$$

7#/

.

.��

��

� ��

�.

.��

��

� ��..

2

3sin3

2sin12

28

1

2

3cos

2cos12

24

1'

r

G

r

rG

u I

(33)

In order to obtain derivatives of the polar coordinates, we considered the following relations

� � �

��

� �

x

yyxr 12/122 tan, $ (34)

Differentiating the equations in (34) and rearranging the terms lead to

�

��

..

�..

..

�

��

..

�..

..

///$

///x

yy

xr

yy

xx

r

r2

1,

1 (35)

where derivatives of the global coordinates, x and y, with respect to the parametric coordinates

can be evaluated through the Jacobian matrix at the particular numerical integration point.

Comparing with the crack tip enrichment functions given in Equation (5), the present selection of

the crack tip enrichment functions has at least two advantages. First of all, it permits direct

calculation of mixed mode stress intensity factors, so the requirement of the domain interaction

integral is eliminated. Secondly, it only introduces two additional nodal degrees of freedom, KI

and KII. Comparing with the normally commonly used crack tip enrichment field given in (5)

which introduced eight additional nodal unknowns, the present approach results in a significantly

smaller finite element system and thus an improved computational efficiency. A similar idea for

treating crack tip enrichments has been proposed in [17] where the use of multi-point constraints

on the stress intensity factor degrees of freedom around the crack tip was suggested.

3.7 Use of a Shifted Displacement Field

Based on the presentations in the preceding sections in this report, we realize that the complete

displacement approximation used in XFEM formulation contains three parts: the standard finite

element displacement filed, the discontinuous Heaviside function along the crack but excluding

the crack tips and the crack tip enrichment functions around the crack tips. This displacement

approximation can be expressed below

� � � � � � � �! "III

N

I

I HN bxaxuxxu ��1

. (36)

However, if this expression is directly used in the implementation of XFEM in a finite element

program, the evaluation of the nodal displacements requires computation of the nodal values of

the enrichment functions in the post-processing stage as


� � � � � � � �! "IIIII

N

I

I HN bxaxuxxu ��1

. (37)

In order to avoid this complication, a technique of using a shifted displacement field has been

proposed [18] in which the original displacement approximation (36) is modified as

� � � � � � � �� ! "IIIII

N

I

I HHN bxxaxxuxxu �� 1

(38)

Because the shifted enrichment functions vanish at the nodes, the original degrees of freedom, uI,

contain the total nodal displacements, so the deformed configurations, including crack opening,

can be displayed using the existing graphics capabilities. The shifted displacement field was used

in the present implementation. As will be demonstrated in the next chapter, the deformed shapes

resulted from XFEM calculations can be displayed correctly using the current version of Trident.

In addition, it is worth mentioning that because the Heaviside and crack tip functions will never

be applied to the same nodes, we do not need to reserve spaces for both of them in the vector of

nodal degrees of freedom, as was done in some of the published work such as [18]. In VAST, we

expanded the nodal degrees of freedom from six to nine. The first six still represented translations

along the global axes and rotations about the global axes. For nodes enriched by the Heaviside

function, degrees of freedom 7 to 9 contained global vector values of aI. For nodes enriched by

the crack tip functions, degrees of freedom 7 and 8 contained the stress intensity factors, KI and

KII, and the 9th degree of freedom was constrained in the element stiffness matrix.


4 Verification of the 2D XFEM Capability in VAST

Since the completion of XFEM implementation in VAST for two-dimensional fracture mechanics

applications, numerical verification has been carried out to confirm the correctness of the

implementation and to evaluate its performance on predicting stress intensity factors for mixed

mode fracture mechanics problems. This numerical investigation and verification has focused on

the effects of the numerical integration order, the effects of constraint equations on the enriched

degrees of freedom and the effects on the underlying mesh on the accuracy of the XFEM results.

The numerical results from this study are presented in this chapter.

4.1 Plate with an Edge Crack

The first test case involved Mode I fracture of a finite plate with a straight edge crack subjected to

an uniform tensile stress as shown in Figure 15. A small crack with dimensions a=1, W=10 was

considered. For this test problem, the analytical solution of the stress intensity factor is available

in standard textbooks on fracture mechanics, such as [19] and can be expressed as

aK I #*12.1 (39)

The present XFEM model for this fracture mechanics problem is given in Figure 16, where an

edge crack is superimposed onto an underlying 40×40 uniform finite element mesh. The quad

elements cut by the crack line were subdivided into triangles to maintain accuracy of numerical

integration as detailed in Figure 17. In XFEM, the crack tip enrichments can be applied to nodes

within an arbitrarily selected area around the crack tip. In the present study, four different circular

areas were considered corresponding to radius R=0.3, 0.5, 0.75 and 1.0, respectively. The nodes

next to the crack line but outside the circular areas were enriched by the discontinuous Heaviside

function.

As detailed in the preceding chapter, in the present VAST implementation of XFEM formulation,

the nodes around the crack tip were enriched by the asymptotic displacement fields obtained from

linear elastic fracture mechanics. Thus, the enriched degrees of freedom at these nodes are the

stress intensity factors which can be solved directly along with the regular nodal displacements.

As a result, no post-processing, such as the interaction integral, is required for evaluating the

stress intensity factors. However, because the displacement field enrichments are applied node

wise, the resulting stress intensity factors vary from node to node. In order to generate unique K

values for a given crack tip, multi-point constraint equations must be utilized to enforce the

compatibility between the nodal variables of stress intensity factors at the nodes surrounding the

crack tip. Because applications of the constraint equations reduced the flexibility in the finite

element system, the numerical solutions would be influenced by the extent of this constraint.

In this numerical example, we focused on the effects of three factors on the accuracy of XFEM

predicted stress intensity factor in a pure Mode I fracture mechanics problem. These factors

included the order of numerical integration, the size of the circular area for crack tip enrichment

and the number of nodes included in the multi-point constraint equations. Three sets of stress

intensity factors obtained using XFEM with different areas of crack tip enrichment are compared

with the analytical solution in Table 5. In the first two cases, multi-point constraints were applied


to all the nodes with crack tip enrichments, so the compatibility between the crack tip fields

superimposed to all the enriched nodes was enforced. In case (a), lower orders of numerical

integration were employed where 2×2 Gaussian integration was used for elements not intersected

by the crack line, whereas 12 integration points were employed in each of the triangles resulting

from element subdivision for elements cut by the crack and containing the crack tip. In case (b),

higher numerical integration orders, 10×10, were utilized. The results in Table 5 indicates that,

although the use of more accurate numerical integration rules did not improve overall accuracy,

they did make the computed stress intensity factors increase monotonically with the increase of

the area of the crack tip enrichment.

As mentioned before, in the above two cases, the multipoint constraints were applied to all nodes

with crack tip enrichment, thus these cases were referred to as Global MPC. In the third case (c),

the multipoint constraints were only applied to the six nodes that were nearest to the crack tip.

These were the nodes within the inner circle in Figure 17. The reduction of constraint equations

provided more flexibility to the XFEM system and results in much improved numerical results. It

should be noted that the differences between the analytical solution and the improved numerical

results obtained with sufficiently large areas of crack tip enrichment, i.e. R=0.5 and 1.0, are less

than 1%. This is in the order of the lowest difference that can be achieved by using the traditional

fracture element in VAST as indicated in a previous numerical study [4].

The deformed configurations obtained using different areas of crack tip enrichment in case (a) are

shown in Figures 18 and 19. These plots of deformed configurations could be generated by

Trident because of the adoption of the shifted displacement filed in the current XFEM

implementation. These deformed shapes confirmed that the application of crack tip and Heaviside

enrichment functions along the crack line did lead to the same crack opening displacement.

4.2 Plate with a 45o Slanted Crack

In this numerical example, a plate with a 45O slanted edge crack in a uniform tensile stress field

was considered. The problem geometry, external load and boundary conditions are displayed in

Figure 20. This example was selected because it was one of the very few mixed mode fracture

mechanics problems having an analytical solution [3]. A finite element discretization required by

XFEM for this problem is depicted in Figure 21, where a slanted crack line with a length of 1.0

was superimposed onto an underlying 50×100 uniform mesh. All the elements cut by the crack

were subdivided into triangles as indicated in Figure 22. Similar to the previous numerical

example, four different circular areas were considered for application of the crack tip enrichment

functions. The radius of these areas were taken as R=0.15, 025, 0.5 and 1.0, respectively.

The Mode I and Mode II stress intensity factors obtained by using XFEM with different solution

parameters are compared with the analytical solutions in Table 6. These numerical parameters

included the size of the crack tip enrichment area (R), the order of the numerical integration rules

and the number of nodes involved in the multipoint constraint equations. Similar to the previous

test example, the XFEM solutions for this mixed mode fracture mechanics problem also

suggested that this finite element formulation is in favour of larger areas of crack tip enrichment

and higher orders of numerical integration, but the multipoint constraints should only be applied

to the nodes surrounding the crack tip. In particular, the best overall results were obtained using

an enrichment area R=0.5, integration orders (10×10, 12) and 6 (3 for each of KI and KII)


constraint equations involving only the four nodes of the element that included the crack tip. It is

quite interesting to note that the increase of the area of crack tip enrichment does not necessarily

result in an improvement of the numerical results. In particular, the solutions obtained using the

largest crack tip enrichment area, R=1.0, are less accurate than the case R=0.5. This is probably

because when R=1.0, a part of the circular area is outside the finite element model, so the

asymptotic crack tip field that is actually included in the solution is not symmetrical about the

slant crack line.

The deformed configurations obtained for cases (a) and (e) in Table 6 with R=0.5 are displayed in

Figures 23 and 24. These plots indicated that unlike the stress intensity factor, the deformed shape

predicted by XFEM is less sensitive to the numerical parameters, such as the order of numerical

integration and the extent of the multipoint constraint.

4.3 Plate with Angled Centre Crack

The third test example involved a plate with an angled centre crack subjected to uniform stress as

shown in Figure 25. The in-plane dimension of the square plate was W=1.0 and the half crack

length a was taken as 0.5. In this example, we attempted to obtain the KI and KII stress intensity

factors as a function of the angle � by using two fixed uniform underlying meshes including a

40×40 coarse mesh and a 200×200 fine mesh. A XFEM model based on the coarse mesh for

�=15O is given in Figure 26 and the local details around the crack, including element subdivisions

and nodal enrichments, are shown in Figure 27. The XFEM model based on the fine uniform

mesh with �=30O are given in Figures 28 and 29.

For the given loading condition and crack geometry, analytical solutions of stress intensity factors

for an infinite plate are available as [7]

� �� #*

�#*

sincos

cos2

aK

aK

II

I

(40)

The present XFEM results obtained using different underlying meshes are compared with the

analytical solution in Figure 30. In the solutions using the coarse mesh, all the nodes within a

radius R=0.3 were enriched with the crack tip functions as shown in Figure 27, and multipoint

constraints were applied to all the nodes with the crack tip enrichment. On the other hand, to

obtain solutions using the fine mesh, the guidelines generated from the solution of the previous

numerical examples were applied. They included: an appropriate area for crack tip enrichments

with a radius that equals one half of the characteristic crack length (R=a/2=0.25), moderately high

orders of numerical integration (10×10, 12) and multipoint constraints that involved only nodes of

the elements that contained the crack tips. It is interesting to noted that the XFEM was able to

correctly predict the trend of variations of the stress intensity factors with angle � even using the

extremely coarse mesh, whereas the fine mesh resulted in a very close agreement between the

analytical and numerical solutions for the entire range of �.


Table 5: Comparison of analytical and XFEM results for plate with an edge crack

Radius KI Diff (%)

Analytical 7.9406

(a) (2×2, 12) integration, Global MPC

R=0.30 7.2122 -9.1731

R=0.50 7.2127 -9.1668

R=0.75 7.4207 -6.5474

R=1.00 7.4040 -6.7577

(b) (10×10, 12) integration, Global MPC

R=0.30 7.1638 -9.7826

R=0.50 7.1832 -9.5383

R=0.75 7.3801 -7.0587

R=1.00 7.3855 -6.9907

(c) (10×10, 12) integration, 10 MPC

R=0.30 7.1638 -9.7826

R=0.50 7.6342 -3.8587

R=0.75 7.9570 0.2065

R=1.00 8.0127 0.9076


Table 6: Comparison of analytical and XFEM results for plate with 45O slanted crack

Radius KI Diff (%) KII Diff (%)

Analytical 1.86 0.88

(a) (10×10, 12) integration, Global MPC

R=0.15 1.7904 -3.7419 0.7451 -15.3250

R=0.25 1.8791 1.0269 0.7907 -10.1466

R=0.50 1.9939 7.1989 0.7579 -13.8795

R=1.00 2.0742 11.5161 0.7558 -14.1159

(b) (10×10, 12) integration, 56 MPC

R=0.15 1.7904 -3.7419 0.7451 -15.3250

R=0.25 1.8972 2.0000 0.8345 -5.1670

R=0.50 1.9367 4.1237 0.8611 -2.1511

R=1.00 1.9546 5.0860 0.8705 -1.0852

(c) (10×10, 12) integration, 30 MPC

R=0.15 1.8237 -1.9516 0.8224 -6.5455

R=0.25 1.8836 1.2688 0.8573 -2.5818

R=0.50 1.9168 3.0538 0.8789 -0.1284

R=1.00 1.9339 3.9731 0.8882 0.9261

(d) (10×10, 12) integration, 6 MPC

R=0.15 1.7721 -4.7258 0.8405 -4.4886

R=0.25 1.8240 -1.9355 0.8677 -1.4000

R=0.50 1.8623 0.1237 0.8925 1.4182

R=1.00 1.8840 1.2903 0.9070 3.0659

(e) (20×20, 19) integration, 6 MPC

R=0.15 1.8243 -1.9194 0.8358 -5.0205

R=0.25 1.8789 1.0161 0.8623 -2.0170

R=0.50 1.9191 3.1774 0.8865 0.7375

R=1.00 1.9419 4.4032 0.9006 2.3420

(f) (40×40, 25) integration, 6 MPC

R=0.15 1.8327 -1.4677 0.8368 -4.9045

R=0.25 1.8878 1.4946 0.8633 -1.8977

R=0.50 1.9284 3.6774 0.8875 0.8568

R=1.00 1.9514 4.9140 0.9017 2.4625


Figure 15: A finite plate with an edge crack subjected to uniform tensile stress.

*

W

a


Figure 16: XFEM model of a plate with edge crack

Figure 17: Details around the crack with different areas for assignment of crack tip enrichment functions.


Figure 18: Deformed configuration obtained using R=0.3.

Figure 19: Deformed configuration obtained using R=1.0.


Figure 20: Problem geometry, external load and boundary condition for a plate with a 450 slanted edge crack under uni-axial tension.


Figure 21: XFEM model of a plate with a 45O slanted crack.

Figure 22: Details around the crack with different areas for assignment of crack tip enrichment functions.


Figure 23: Deformed configuration of the plate with a 45O slanted crack obtained using R=0.5, (10×10, 12) integration and global MPC.

Figure 24: Deformed configuration of the plate with a 45O slanted crack obtained using R=0.5,

(20×20, 19) integration and 6 MPC equations.


Figure 25: Plate with angled center crack.

W

W

�2a

*

W

W

�2a

*



40×40 uniform mesh.

Figure 27: Details around the crack in the coarse mesh showing nodes with different types of enrichments.

crack tip 1 enrichment


Heaviside enrichment









200×200 uniform mesh.

Figure 29: Details around the crack in the fine mesh.


Figure 30: Comparison of analytical and XFEM predicted stress intensity factors for different �

values using the coarse and fine underlying finite element meshes.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 10 20 30 40 50 60 70 80 90

Angle (Degree)

Str

ess In

ten

sit

y F

acto

rsAnalytical, K_I

Analytical, K_II

XFEM, Coarse Mesh, K_I

XFEM, Coarse Mesh, K_II

XFEM, Fine Mesh, K_I

XFEM, Fine Mesh, K_II


5 Conclusions

The recent implementation of the extended finite element method (XFEM) in VAST has been

described in detail in this report. The theoretical background of the XFEM formulation and

various issues related to computer implementation were presented. The numerical results

presented in this report confirmed that XFEM is a promising technology and has unique

advantages for modelling crack growth in ship structures. In particular, XFEM eliminated the

need of explicit modelling of cracks in finite element models for fracture mechanics applications,

and thus voided repeated remeshing and mapping of field variables, such as stresses and strains,

in crack propagation analyses. Based on the numerical studies performed so far, some general

guidelines have been proposed to ensure accuracy of the mixed mode stress intensity factors:

1. The element size of the underlying uniform mesh should not exceed one tenth of the crack

length, so the crack is represented by at least 10 elements.

2. The radius of the circular area of crack tip enrichment should be approximately half of the

characteristic crack length.

3. For numerical integration, 10×10 Gaussian quadrature should be utilized for elements which

are not intersected by the crack, but contain enriched nodes. For each triangular domain

resulting from subdivision of elements cut by the crack, the 12-point Gaussian integration

rule is recommended.

4. The multipoint constraint equations should only be applied to the nodes of the elements that

contain the crack tip(s). This step can be automated in the future.

However, before the XFEM capability in VAST can be applied to practical engineering analyses,

a substantial amount of effort is still required:

To further verify the capability using additional example problems, especially those involving

curved or kinked cracks, to ensure the reliability of the present implementation,

To improve the robustness of the pre-processor which is responsible for dealing with crack-

mesh interaction and is a very important part of the XFEM algorithm,

To implement the pre-processor into a GUI program, such as Trident, to automatically deal

with the mesh-crack interactions and generate XFEM modes with little user input,

To develop and implement appropriate post-processing capability for XFEM to display the

results, especially the stress output from XFEM,

To implement the domain interaction J-integral algorithm and compare the results from the J-

integral with the current results which are obtained using a direct approach,

To extend the sparse solver to work with XFEM which involves more than six degrees of

freedom per node,

To test the XFEM capability on practical crack propagation problems in ship structures,

To extend the present 2D XFEM capability to 3D.


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368, 2009.

[19] T.L. Anderson, Fracture Mechanics: Fundamentals and Applications. CRC Press, Boca

Raton, Florida, 1991.

[20] G. Zi, H. Chen, J. Xu and T. Belytschko, “The Extended Finite Element Method for

Dynamic Fractures”, Shock and Vibration, Vol. 12, 00.9-23, 2005.


Annex A Input Data for 4-Noded 2D XFEM Fracture Element (IEC = 68)

Provide Cards Ea to define material properties for this element group

Card Ea (3E10.3, 7I5)

E = Young's modulus (force/length2) of isotropic material (see Figure

C2-68a).

PR = Poisson's ratio of isotropic material.

DEN = Material mass density, i.e. weight per unit volume/g (force-time2/

length4).

Note: Thermal stress calculation is not available for this element.

NIP1 = number of numerical integration point along the / direction in

the local parametric coordinate system for elements not

intersected by the crack. (Default equals to 2 for regular element

and 10 always used for enriched element).

NIP2 = number of numerical integration point along the 0 direction in

the local parametric coordinate system for elements not

intersected by the crack. (Default equals to 2 for regular element

and 10 always used for enriched element).

NIP3 = number of numerical integration point along the thickness

direction of the shell element. (Default equals to 2).

NIP1T = number of numerical integration point for subdivided triangles in

elements which are not enriched by asymptotic displacement

fields. (Default equals to 4. If NIP1T>4, a 7-point quadrature

rule is used)

NIP2T = number of numerical integration point for subdivided triangles in

elements which are enriched by the singular asymptotic


displacement fields. (Default equals to 12. If 12<NIP1T<25, a

19-point quadrature rule is used. If NIP1T>25, a 25-point

quadrature rule is used.)

IESF = 0, the incompatible “bubble” displacement modes will be used if

the element is not enriched

= 1, the incompatible “bubble” displacement modes will not be

used.

Note: The incompatible “bubble” displacement modes will always be turned

off if an element is enriched by either the discontinuous function or the

asymptotic displacement field, regardless the value of IESF in the input

data.

IPAN = 0, crack propagation angle is calculated using a criterion based

on the maximum tangential stress.

= 1, crack propagation angle is calculated using a criterion based

on the maximum strain energy release rate.

= 2, crack propagation angle is calculated using a criterion based

on the minimum strain energy density.

Supply Card Eb for each of the NELM elements.

*Card Eb.1 (4I5, 4E10.3, 4I2, I2, 2I5)

N19N4 = Element nodes.

TK (1) = Thickness at element nodes (length). (If the element is of

: constant thickness supply only TK(1).)

TK (4)

IENR19 IENR4= Flags to indicate the type of enrichment to be assigned to each

node in the element. IENRI=0 indicates that node I is not

enriched. IENRI=1 indicates that node I is enriched with the

discontinuous displacement function. IENRI=2 indicates that

node I is enriched with the crack tip asymptotic displacement

function.


Note: The following cards are controlled by the types of enrichments defined

above.

KODE = 0, element stiffness, geometric stiffeners and mass matrices are

calculated for the current element.

= 1, geometry, orientation and initial stresses of the current

element are same as those of the previous element and the

element matrices of the previous element are used for the current

element.

Note: If KODE = 1, TK(1) - TK(4) may be set to zero and values from the

previous element will then be used.

NEXG = number of extra elements generated through incrementation of

node numbers N1 to N4 from element connectivity array.

INCR = integer by which node numbers from previous element

connectivity array are incremented to generate a new element.

Card Eb.2 (4I5) (Provide this card if at least one of the IENR values is non-zero.)

IESEC1 = Flags to indicate whether element edges are intersected with the

: crack. The element edges are numbered counter clockwise

IESEC4 starting from the edge connecting node 1 and 2. IESECI=0 indicates that edge I is not intersected by the crack. IESECI=1 indicate that edge I is intersected by the crack.

Card Eb.3 (3F10.3) (Provide one card for each edge intersected by the crack IESECI=1)

XI, YI, ZI = Global coordinates of the intersection of crack and the I-th edge

of the element.

Card Eb.4 (4I5) (Provide this card if at least one of the IENR values is non-zero.)

IONT1 = Flags to indicate orientations of each node in the element with


: respect to the crack. IONTI=1 indicates node I is above the crack

IONT4 line, whereas IONTI=-1 indicates that node I is below the crack. This definition of node orientation is critical for correct evaluation of the discontinuous enrichment function.

Card Eb.5 (6F10.5) (Provide this card if at least one of the IENR values equal to 2)

XC, YC, ZC = Global coordinates of the crack tip to which the crack tip

enrichment functions are associated.

DC1, DC2, DC3 = Direction cosines defining crack orientation, i.e. direction along

crack and away from crack-tip as shown in Figure C2-68b. If

direction cosines do not correspond with element plane, the

projection on the element plane will be used.


! "

;;;

1

;;;

2

3

;;;

;;;

�

�

;;;

1

;;;

2

3

;;;

;;;

�

�

EZX

EYZ

EXY

EYY

EXX

D

SZX

SYZ

SXY

SYY

SXX

SXX, SYY, SXY, SYZ, SZX - local stress components

EXX, EYY, EXY, EYZ, EZX - local strain components

[D] - elasticity matrix

Isotropic Material

! "

��

�

�

��

�

�

�

�

��

kPR

kPR

PR

PR

PR

PR

ED

2/)1(0000

02/)1(000

002/)1(00

0001

0001

1 2

where k = 1.2 indicates the shear correction factor.

Anisotropic Material

! "

D11 D12 D13 D14 D15

D22 D23 D24 D25

D = D33 D34 D35

(sym.) D44 D45

D55

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FIGURE C2-68a: Stress-Strain Relationships for 4-Noded

2D XFEM Fracture Element


FIGURE C2-68b: 4-Noded 2D XFEM Fracture Element


Distribution list

DRDC Atlantic TM 2010-098

Internal distribution 4 Scientific Authority (2 hard copies, 2 CDs) 1 Author 3 DRDC Atlantic Library (1 hard copy, 2 CDs) Total internal copies: 8

External distribution (within Canada by DRDKIM)

1 Library and Archives Canada, Atten: Military Archivist, Government

Records Branch 1 NDHQ/DRDKIM 3 1 NDHQ/DMSS 2

Louis St. Laurent Bldg. 555 boul de la Carriere Ottawa, Hull K1A OK2

Total external copies: 3 TOTAL COPIES: 11

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Implementation of 2D XFEM in VAST:

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L. Jiang

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JULY 2010

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11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by security classification.)

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12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspond to the

Document Availability (11). However, where further distribution (beyond the audience specified in (11) is possible, a wider announcement

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13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable

that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification

of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include

here abstracts in both official languages unless the text is bilingual.)

This report is concerned with a recent implementation of the extended finite element method

(XFEM) in the VAST finite element program for 2D fracture mechanics analyses. The XFEM is

a new finite element formulation recently developed based on the method of partition of unity,

in which the classical finite element approximation is enriched by a discontinuous function and

the asymptotic displacement functions around crack tips. It allows the crack to be in the interior

of elements, so eliminates the need for explicitly modelling cracks in the finite element mesh

and the need for remeshing for crack propagation. In this report, the theoretical background of

the XFEM formulation and various issues related to its implementation in VAST are discussed

in detail. The results from an extensive numerical verification are also presented. The effects of

various factors on the accuracy of XFEM predicted mixed mode stress intensity factors,

including the mesh size, the area of crack tip enrichment, the minimum orders of numerical

integration and the extent of constraints, are investigated and guidelines for properly using the

XFEM capability are proposed. The results presented in this report demonstrated the potential

and unique advantages of XFEM in analysing crack propagations in ship structures and a

number of possible future developments are suggested

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be

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Extended Finite Element Method, XFEM, Fracture Mechanics, Mixed Mode, VAST


�

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