The eXtended Finite Element Method (X-FEM)

Nicolas Moes

Ecole Centrale de NantesInstitut GeM, UMR CNRS 8183

1 Rue de la Noe, 44321 Nantes, France

Abstract

The purpose of this document is to provide a state of the art in the use of theX-FEM method to model discontinuities.

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Contents

1 Introduction 3

2 Background on discretization methods 32.1 Problem description and notations . . . . . . . . . . . . . . . . . . . . . . 32.2 Rayleigh-Ritz approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Meshless methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Discontinuity modeling 103.1 A simple 1D problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Case a : crack located at a node . . . . . . . . . . . . . . . . . . . . 103.1.2 Case b: crack located in between two nodes . . . . . . . . . . . . . 12

3.2 Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Extension to 2D and 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Cracks located by level sets . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Mathematical and technical aspects 244.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Topological and geometrical enrichment strategies . . . . . . . . . . . . . . 244.3 Condition number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Stress intensity factor evaluation 335.1 2D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Eshelby tensor and J integral . . . . . . . . . . . . . . . . . . . . . . . . . 335.3 Interaction integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 Extension to body forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.5 Extension to thermal loadings . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Level sets propagation 406.1 Level set methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 Fast marching method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.4 Propagation of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.5 Specific methods for crack growth . . . . . . . . . . . . . . . . . . . . . . . 41

7 Applications currently treated with X-FEM 44

A the auxiliary field in the 2D setting 59

B The auxiliary fields in the 3D setting 59

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1 Introduction

In spite of its decades of existence, the finite element method coupled with meshing toolsdoes not yet manage to simulate efficiently the propagation of 3D cracks for geometriesrelevant to engineers in industry. This fact was the motivation behind the design of theeXtended Finite Element Method (X-FEM).

In spite of constant progress of meshers, initial creation of the mesh and modification ofthis mesh during the propagation of a crack, remain extremely heavy and lack robustness.

Even if this operation were easy, the question of the projection of fields from one meshto the next one would still be raised for history dependent problems (plasticity, dynamics,. . . ). The possibility offered to preserve the mesh through the simulation is undoubtlyappealing.

The basic idea is to introduce inside the elements the proper discontinuities so asto relax the need for the mesh to conform to them. This introduction is done via thetechnique of the partition of unitu (Melenk and Babuska 1996; Babuska and Melenk1997). It should be noted that X-FEM is not the only method based on the partition ofthe unity (as in painting several schools exist). The GFEM approach (generalized finiteelement method) and PUFEM (partition of unity finite element) are also based on thepartition of unity. We will of course detail in this document the work carried out underthese different names.

The constant ambition which distinguishes the X-FEM approach since its beginningsis to use the partition of unity to release the mesh from the constraints to conform tosurfaces of discontinuity, while keeping the same performance as traditional finite element(optimality of convergence). Quickly also the X-FEM was coupled to the level set methodto locate and evolve the position of surfaces of discontinuities.

2 Background on discretization methods

2.1 Problem description and notations

The solid studied is depicted in figure 1. It occupies a domain Ω whose boundary isdenoted by Γ. This boundary is composed of the crack faces Γc+ and Γc+ assumed tractionfree, as well as a part Γu on which displacement u are imposed and, finally, a part Γt onwhich tractions t are imposed.

Stresses, strains and displacements are denoted by σ, ε et u, respectively. For now,small strains and displacements are assumed. In the absence of volumic forces, equilibriumequations read

∇ · σ = 0 sur Ω (1)

σ · n = t sur Γt (2)

σ · n = 0 sur Γc+ (3)

σ · n = 0 sur Γc− (4)

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Γc

uΓ

Γtt

Ω

ex

ey

ez

Figure 1: Reference problem.

where n is the outward normal. Kinematics equations read

ε = ε(u) = ∇su sur Ω (5)

u = u on Γu (6)

where ∇s is the symmetrical part of the gradient operator. Finally, the constitutive lawis assumed elastic: σ = C : ε where C is Hooke’s tensor. The space of admissibledisplacement field is denoted U . whereas the space of admissible displacements to zero(i.e. zero imposed displacements) is denoted U0:

U = v ∈ V : v = u on Γu (7)

U0 = v ∈ V : v = 0 on Γu (8)

where space V is connected to the regularity of the solution and is detailed in (Babuskaand Rosenzweig 1972) and (Grisvard 1985). This space contains discontinuous fields ofdisplacement across the crack faces Γc. The weak form of the equilibrium equations iswritten ∫

Ω

σ : ε(v) dΩ =

∫Γt

t · v dΓ ∀v ∈ U0 (9)

Let us note that the border Γc does not contribute to the weak form because it is trac-tion free. Combining (9) with the constitutive law and the kinematics equations, the

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displacement variational principle is obtained: find u ∈ U such that∫Ω

ε(u) : C : ε(v) dΩ =

∫Γt

t · v dΓ ∀v ∈ U0 (10)

2.2 Rayleigh-Ritz approximation

We point out the approach of Rayleigh-Ritz because the partition of the unity may be seenas a support wise use of this approach. In the method of Rayleigh-Ritz, the approximationis written as a linear combination of displacement modes φi(x), i = 1, . . . , N defined onthe domain of interest:

u(x) =N∑i

aiφi(x) (11)

These modes must satisfy a priori the essential boundary conditions (imposed displace-ments are considered null to simplify the presentation). The introduction of this approx-imation into the variational principle (10) leads to the following system of equations

Kijaj = fi, j = 1, . . . , N (12)

with the summation rule applied to index j.

Kij =

∫Ω

ε(φi) : C : ε(φj) dΩ (13)

fi =

∫Γt

t · φi dΓ (14)

The method of Rayleigh-Ritz offers a great freedom in the choice of the modes. Thesemodes can for example be selected so as to satisfy the interior equations. This methodhowever has the disadvantage of leading to a linear system with dense matrix.

2.3 The finite element method

In the finite element method, the domain of interest, Ω, is broken up into geometricalsubdomains of simple shape Ωe, e = 1, . . . , Ne called elements:

Ω = ∪Nee=1Ωe (15)

the set of elements constitutes the mesh. On each element, the unknown field is approx-imated using simple approximation functions, of polynomial type, as well as unknowncoefficients called degrees of freedom. Degrees of freedom have a simple mechanical sig-nificance in general. For first degree elements, the degrees of freedom are simply thedisplacement of the nodes along x and y′ directions. Let us indicate by uαi the displace-ment of node i in direction α (α = x or y) and by φαi the corresponding approximationfunction. The finite element approximation on element Ωe is written

u(x) |Ωe=∑i∈Nn

∑α

aαi φαi (x) (16)

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where Nn denotes the set of nodes of element Ωe. For instance, for a triangle, they aresix approximation functions

φαi = φ1ex, φ2ex, φ3ex, φ1ey, φ2ey, φ3ey (17)

where φ1, φ2 and φ3 are scalar linear functions over the element with value of 0 or 1 at thenodes. Approximation (16) allows one to model any rigid mode or constant strain over theelement. This condition must be fulfilled by the approximation for any type of elements.Continuity of the approximation over the domain is obtained by the use of nodal degreesof freedom shared by all elements connected to the node. The stiffness matrix, Ke

ij, andload vector, f ei , are given for a finite element by

Keiα,jβ =

∫Ωe

ε(φαi ) : C : ε(φβj ) dΩ (18)

f eiα =

∫Γt∩∂Ωe

t · φαi dΓ (19)

The global system of equations is obtained by assembling the elementary matricesand forces in a global stiffness and force vector. In the assembly process, the equationsassociated with the degrees of freedom fixed by Dirichlet boundary conditions are notbuilt.

Contrary to the approximation of Rayleigh-Ritz, the local character of the finite ele-ment approximation leads to sparse matrices. Moreover, the finite element has a strongmechanical interpretation: kinematics is described by nodal displacements which are as-sociated by duality to nodal forces. The behavior of the element is characterized by theelementary stiffness matrix which connects the nodal forces and displacements. The globalsystem to solve enforces the equilibrium of the structure: the sum of the nodal forces ateach node must be zero. Lastly, the finite element method did demonstrate a high level ofrobustness in industry which makes it a very appreciated approach for most applications.

However, the use of the finite element method for problems with complex geometryor evolution of internal surfaces is currently obstructed by meshing issues. This did yielda motivation to design the so called meshless methods.

2.4 Meshless methods

We give some insights on meshless methods because they are important to understandthe history of the concept of enrichment. Within the framework of meshless methods, thesupport of the approximation function is more important than the elements (which actu-ally do not exist any more). On these supports, enrichment functions may be introducedfor example to model a crack tip (work carried out for example by (Fleming, Chu, Moran,and Belytschko 1997)).

Years of active research on meshless mesthods did show the importance of the support.Many researches were undertaken in the nineties (always active to date for certain

applications) to develop methods in which the approximation does not rest on a meshbut rather on a set of points. Various methods exist to date: diffuse elements (Nayroles,

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Touzot, and Villon 1992), Element Free Galerkin method (EFG) (Belytschko, Lu, and Gu1994), Reproducing Kernel Particle Method (RKPM) (Liu, Adee, Jun, and Belytschko1993), h−p cloud method (Duarte and Oden 1996). Each point has a domain of influence(support) with a simple shape (circle or rectangle for example in 2D) on which approxi-mations are built. These functions are zero on the boundary and outside the domain ofinfluence. This is why the field of influence is generally called the support. Abusively, wewill speak about the support i for the support associated with node i. The approximationfunctions defined on the support i are denoted φαi , α = 1, . . . , Nf (i) where Nf (i) is thenumber of functions defined over support i. The corresponding degrees of freedom aredenoted aαi . The approximation at a given point x is written

u(x) =∑

i∈Ns(x)

Nf (i)∑α=1

aαi φαi (x) (20)

where Ns(x) is the set of points whose support contains point x. Figure 2 shows forexample a point x covered by three supports. The approximations functions are builtso that the approximation (20) can represent all rigid modes and constant strain modeson the domain. These conditions are necessary to prove the convergence of the method.Various approaches (diffuse element, EFG, RKPM, . . . ) are distinguished, among otherthings, by the techniques used for the construction of these approximation functions.

Figure 2: Three supports covering node x.

Once a set of approximation functions has been built, it is possible to add some by en-richment. Various manners of enriching exist and we will describe an enrichment describedas external by Belytschko and Fleming (1999). . The enrichment of the approximationmakes it possible to represent a given displacement mode, for example F (x)ex on a sub-domain denoted as ΩF ⊂ Ω. Let us note NF the set of supports which have an non emptyintersection with the Ωf under-field. The enriched approximation is written

u(x) =∑

i∈Ns(x)

Nf (i)∑α=1

aαi φαi (x) +

∑i∈Ns(x)∩NF

Nf (i)∑α=1

bαi φαi (x)F (x) (21)

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where the new degrees of freedom, bαi , multiply the enriched approximation functionsφαi (x)F (x). Let us show that the function F (x)ex may be represented on the Ωf subdo-main. By setting to zero all degrees of freedom aαi and taking the function F (x) out ofthe sum, the approximation in a point x ∈ ΩF is written

u(x) =

∑i∈Ns(x)∩NF

Nf (i)∑α=1

bαi φαi (x)

F (x) (22)

The degrees of freedom bαi can be selected so that the factor in front of F (x) isthe rigid mode ex. That is possible since functions φαi are able to represent any rigidmode. In conclusion, the approximation (21) can represent F (x)ex on ΩF . Enrichmentmade it possible within the framework of the Element Free Galerkin Method to solveproblems of propagation of cracks in two and three dimensions without remeshing (Krysland Belytschko 1999): the crack is propagated through a set of points and is modelledby enrichment of the approximation with discontinuous functions F (x) on the crack orrepresenting the singularity on the crack front. Great flexibility in the writing of theapproximation and its enrichment as well as the possibility of creating very regular fieldsof approximation are two important assets of meshless methods and the EFG approach inparticular. The use of meshless methods however presents a certain number of difficultiescompared to the finite element method:

• Within the finite element method the assembly of the stiffness matrix can be doneby assembling the contributions of each element. In meshless methods, the assemblyis done rather by covering the domain by points of integration and by adding thecontribution of each one of them. The choice of the position and the number ofintegration points is tedious for an arbitrary set of approximation points;

• the boundary conditions of the Dirichlet type are delicate to impose;

• the approximation funtions are to be built and are not explicit;

• the size of the domain of influence is a parameter in the method which the usermust choose carefully.

2.5 Partition of unity

Melenk et Babuska (1996) did show that the traditional finite element approximationcould be enriched so as to represent a specified function on a given domain. Their pointof view can be summarized as follows. Let first us recall that the finite element approxi-mation is written on an element as

u(x) |Ωe=∑i∈Nn

∑α

aαi φαi (x) (23)

As the degrees of freedom defined at a node have the same value for all the elementsconnected to it. The approximations on each element can be “ assembled’ ’ to give a valid

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approximation in any point x of the domain:

u(x) =∑

i∈Nn(x)

∑α

aαi φαi (x) (24)

where Nn(x) is the set of nodes belonging to the elements containing point x. The domainof influence (support) of the approximation function φαi is the set of elements connectedto node i. The set Nn(x) is thus also the set of the nodes whose support covers point x.The finite element approximation (24) can thus be interpreted as a particularization ofthe approximation (20) used in meshless methods:

• The set of points is the set of nodes in the mesh;

• The domain of influence of each node is the set of elements connected to it.

It is thus possible to enrich the finite element approximation by the same techniquesas those used in meshless methods. Here is the enriched approximation which makes itpossible to represent function F (x)ex on domain ΩF :

u(x) =∑

i∈Nn(x)

∑α

φαi aαi +

∑i∈Nn(x)∩NF

∑α

bαi φαi (x)F (x) (25)

where NF is the set of nodes whose support has an intersection with domain ΩF . Theproof is obtained by setting to zero coefficients aαi and by taking into account the factthat the finite element shape functions are able to represent all rigid modes and thus theex mode. We move now to the concrete use of the partition of unity for the modeling ofdiscontinuities.

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3 Discontinuity modeling

We start from the 1D case.

3.1 A simple 1D problem

We consider a bar shown in figure 3. Two cases are considered : a crack is located at anode or between two nodes.

case bcase a

Figure 3: A bar with a crack located at a node (case a) or between two nodes (case b).

3.1.1 Case a : crack located at a node

Using classical finite elements, this case is treated by introducing double nodes. Node 2is replaced by nodes 2− and 2+ sharing the same location but bearing different unknownsas shown in figure 4. The approximation reads

u = u1N1 + u−2 N−2 + u+

2 N+2 + u3N3 (26)

where Ni indicates the approximation functions and ui the corresponding degrees of free-dom. Defining the displacement average, < u >, and (half) jump, [u], at node 2

< u >=u−2 + u+

2

2[u] =

u−2 − u+2

2(27)

the approximation may be rewritten as

u = u1N1+ < u > N2 + [u]N2H(x) + u3N3 (28)

whereN2 = N−2 +N+

2 (29)

The generalised Heaviside function H (generalized because the original Heaviside func-tion goes from 0 to 1) is represented in figure 5. Abusively, we shall however call it Heav-iside function. In the approximation (28), one distinguishes the continuous part modeledby functions N1, N2 and N3 to which is added a discontinuous part given by the productof N2 by the Heaviside function. Node 2 is called an enriched nodes because an additionaldegree of freedom is given to it.

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1 2 3- +

N+2 N3N−

2N1

Figure 4: Double node to model a discontinuity located at a node.

1 2 3- +

H = −1

H = +1

1 2 3- +

HN2

Figure 5: The generalised Heaviside function (left) as well as its product with functionN2 (right).

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3.1.2 Case b: crack located in between two nodes

Let us study now case b in which the crack is located between two nodes. As in case a,we wish to write the approximation as the sum of a continuous and a discontinuous part.Evolving on case a, we propose

u = u1N1 + u2N2 + u3N3 + u4N4 + a2N2H + a3N3H (30)

Nodes 2 and 3 are enriched by the Heaviside function. This enrichment was presentedthe first time (in 2D) by (Moes, Dolbow, and Belytschko 1999). If the crack is locatedat a node, there is only one enriched node. In case b, two nodes are enriched becausethe support of nodes 2 and 3 are cut by the crack. A node is enriched by the Heavisidefunction if its support is cut into two by the crack. One can show that the approximation(30) makes it possible to represent two rigid modes (on the left of the crack and the otheron the right). The fact that two (and not one) additional degree are necessary may besurprising. Indeed, a crack implies a jump in displacement but can also imply a jumpin strain. By linear combination of the various functions implied in (30), one noticesthat enrichment brings two functions on the element joining nodes 2 and 3. These twofunctions are shown in figure 7. Note that for case a, only one additional degree of freedomis needed because finite element natively exhibits strain jumps across element boundaries.

2 3

N2H

N3H

H = +1

H = −1

1 2 43

Figure 6: A crack located between two nodes. The Heaviside function (left) and enrich-ment functions (right).

Finally, it should be noted that proposed enrichment yields the same approximationspace as if the cracked element is replaced by two elements and a double node. Thisobservation is limited to 1D and will not carry over to 2D and 3D.

3.2 Alternatives

Consider a cracked element going from node 1 to node 2. The X-FEM basisN1, N2, HN1, HN2

is depicted in figure 8. It spans the same approximation space as the one introduced in(Hansbo and Hansbo 2002), shown in figure 9. The “Hansbo” alternative was shownidentical to the X-FEM enrichment in Areias and Belytschko (2006a). We also note thatthe way a crack is modeled in the “Hansbo” alternative is similar to the way a hole ismodeled with X-FEM (see (Daux, Moes, Dolbow, Sukumar, and Belytschko 2000)) exceptthat the operation is done twice (as if the crack was creating two holes, one on each sideof the crack).

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2 3

Figure 7: Two functions modeled by the X-FEM enrichment. We observe a continu-ous function with discontinuous slope (dashed line) and a discontinuous function withcontinuous slope (solid line).

The relation between the X-FEM and Hansbo bases is given by

N1 = NI +N ′I (31)

N ′1 = NI −N ′I (32)

N2 = NII +N ′II (33)

N ′2 = −NII +N ′II (34)

Owing to

u = u1N1 + u2N2 + u′1N′1 + u′2N

′2 (35)

= uINI + uIINII + u′IN′I + u′IIN

′II (36)

we also get uIu′IuIIu′II

=

1 1 0 01 −1 0 00 0 1 −10 0 1 1

u1

u′1u2

u′2

(37)

The Hansbo alternative may also be interpreted as the superposition of two elements.The first one is used to define functions NI and N ′II . On this element, the right node iscalled virtual. The second element is used to define functions N ′I and NII . For this secondelement, it is the left node which is virtual. The virtual qualifier has been introduced by(Molino, Bao, and Fedkiw 2004). The phantom qualifier may also be found in (Song,Areais, and Belytschko 2006).

Another alternative consists in replacing the enriched functions N1H(x) and N2H(x)by the functions known as shifted N1(H(x)−H(x1)) and N2(H(x)−H(x2)) where x1 andx2 indicate the positions of nodes 1 and 2, respectively. This alternative was proposedby Zi and Belytschko (2003). It is show in figure 10. It is important to note that whenthe crack falls exactly on a node the Heaviside function is not defined there and a signmust be selected to shift the base. There is thus a certain number of equivalent basesallowing to model the discontinuity. The choice is guided in general by the simplicity of

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N1 N2 N ′1 = N1H N ′

2 = N2H

Figure 8: Classical and X-FEM functions defined on a cracked element.

NI NII N ′II N ′

I

Figure 9: Hansbo and Hansbo alternative to model a discontinuity.

N1 N2 N1(H(x)−H(x1)) N2(H(x)−H(x2))

Figure 10: Shifted base alternative.

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implementation according to the target code. Also note that even if these various baseswill lead to the same solution, the generated matrices will not be identical (and will nothave the same condition number, see section 4.3). These alternatives were presented inthe 1D case but also exist in 2D and 3D. Lastly, the interested reader will be able to findin the following reference other alternatives or additional information on the alternativespresented above: (Svahn, Ekevid, and Runesson 2007).

3.3 Extension to 2D and 3D

We consider now 2D and 3D meshes cut by a crack. Just like in the 1D case, we beginwith the case of a crack inserted with double nodes.

Figure 11: Finite element meshnear a crack tip, the circled num-bers are element numbers

Figure 12: Regular mesh withouta crack.

Figure 11 is taken from (Moes, Dolbow, and Belytschko 1999) and shows a four elementmesh in which a crack has been introduced through double nodes (nodes 9 and 10). Thefinite element approximation associated with the mesh in figure 11 is

u =10∑i=1

uiNi (38)

where ui is the (vectorial) displacement at node i and φi is the bilinear shape functionassociated with node i. Each shape function φi has a compact support ωi given by theunion of the elements connected to node i.

Let us rewrite (38) in such a way that we recover an approximation without crackcorresponding to figure 12 and a discontinuous additional displacement. Defining theaverage displacement a and the displacement jump b on the crack faces as

a =u9 + u10

2b =

u9 − u10

2(39)

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we can express u9 and u10 in terms of a and b

u9 = a+ b u10 = a− b (40)

Then replacing u9 and u10 in terms of a and b in (38) yields

u =8∑i=1

uiNi + a(N9 +N10) + b(N9 +N10)H(x) (41)

where H(x) is referred to here as a discontinuous, or ‘jump’ function. This is defined inthe local crack coordinate system as

H(x, y) =

+1 for y > 0

−1 for y < 0(42)

If we now consider the mesh in figure 12, φ9 + φ10 can be replaced by φ11, and a byu11. The finite element approximation now reads

u =8∑i=0

uiNi + u11N11 + bN11H(x) (43)

The first two terms on the right hand side represent the classical finite element approx-imation, whereas the last one represents the addition of a discontinuous enrichment. Inother words, when a crack is modeled by a mesh as in figure 11, we may interpret thefinite element space as the sum of one which does not model the crack (such as figure 12)and a discontinuous enrichment. The third term may be interpreted as an enrichment ofthe finite element functio by a partition of unity technique.

Derivation that we have just carried out on a small grid of four elements can bereiterated on any 1D, 2D or 3D grid containing a discontinuity modeled by double nodes.This derivation will yield to the same conclusion: the modeling of a discontinuity by doublenodes is equivalent to traditional finite element modeling if one adds terms correspondingto an enrichment by the partition of unity of the nodes located on the path of discontinuity.Let us note that the nodes which are enriched are characterized by the fact that theirsupport is cut into two by the discontinuity.

Let us suppose now that one wishes to model a discontinuity which does not follow theedge of the elements. We propose to enrich all the nodes whose support is (completely)cut into two by the discontinuity (Moes, Dolbow, and Belytschko 1999). At these nodes,we add a degree of freedom (vectorial if the field is vectorial) acting on the traditionalshape function at the node multiplied by a discontinuous function H(x) being 1 on aside of the crack and -1 on the other. For example, in figures 13 and 14, circled nodesare enriched. A node whose support is not completely cut by discontinuity must not beenriched by function H because that would result in enlarging the crack artificially. Forexample, for the mesh shown in figure 14, if nodes C and D are enriched, the crack willbe active up to the point q (the displacement field will be discontinuous up to point q).

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However, if only nodes A and B are enriched by the discontinuity, the displacement fieldis discontinuous only up to point p and the crack appears unfortunately shorter.

In order to represent the crack on its proper length, nodes whose support contains thecrack tip (squared nodes shown in figure 14) are enriched with discontinuous functions upto the point t but not beyond. Such functions are provided by the asymptotic modes ofdisplacement (elastic if calculation is elastic) at the crack tip. This enrichment, alreadyused by (Belytschko and Black 1999) and (Strouboulis, Babuska, and Copps 2000) allowsmoreover precise calculations since the asymptotic characteristics of the displacement fieldare built-in. Let us note that if the solution is not singular at the crack tip (for exampleby the presence of a cohesive zone), other functions of enrichment can be selected (Moesand Belytschko 2002; Zi and Belytschko 2003).

Figure 13: Crack not aligned witha mesh, the circled nodes areenriched with the discontinuousfunction H(x).

Figure 14: Crack not alignedwith a mesh, the circled nodesare enriched with the discontin-uous function and the squarednodes with the tip enrichmentfunctions.

We are now able to detail the complete modeling of a crack with X-FEM locatedarbitrarily on a mesh, figure 15. The enriched finite element approximation is written:

uh(x) =∑i∈I

uiNi(x) +∑i∈L

aiNi(x)H(x) (44)

+∑i∈K1

Ni(x)(4∑l=1

bli,1Fl1(x)) +

∑i∈K2

Ni(x)(4∑l=1

bli,2Fl2(x))

where:

• I is the set of nodes in the mesh;

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Figure 15: Crack located on a structured (left) and unstructured mesh (right). Circlednodes are enriched with the Heaviside function while squared nodes are enriched by tipfunctions.

• ui is the classical (vectorial) degree of freedom at node i;

• Ni is the scalar shape function associated to node i;

• L ⊂ I is the subset of nodes enriched by the Heaviside function. The corresponding(vectorial) degress of freedom are denoted ai. A node belongs to L is its support iscut in two by the crack and does not contain the crack tip. Those nodes are circledon figure 15;

• K1 ⊂ I et K2 ⊂ I are the set of nodes to enrich to model crack tips numbered 1 and2, respectively. The corresponding degrees of freedom are bli,1 and bli,2, l = 1, . . . , 4.A node belongs to K1 (resp. K2) if its support contains the first (resp. second)crack tip. Those nodes are squared in figure 15.

Functions F l1(x), l = 1, . . . , 4 modeling the crack tip are given in elasticity by:

F l1(x) ≡

√rsin( θ

2),√rcos( θ

2),√rsin( θ

2)sin(θ),

√rcos( θ

2)sin(θ)

(45)

where (r, θ) are the polar coordinates in local axis at the crack tip (“tip” 1 on figure 16).It must be noted that the first function is discontinuous across the crack. Similarly,functions F l

2(x) are also given by (45); the local system of coordinates being now locatearound the second crack tip.

The jump function H(x) is defined as follows. It is discontinuous on the crack and con-stant on each side of the crack (+1 or -1). The sign is defined in the following way (Moes,Dolbow, and Belytschko 1999). The crack is considered to be a curve parametrized by thecurvilinear coordinate s, as in figure 17. The origin of the curve is taken to coincide withone of the crack tips. Given a point x in the domain, we denote by x∗ the closest pointon the crack to x. At x∗, we construct the tangential and normal vector to the curve, esand en, with the orientation of en taken such that es ∧ en = ez. The function H(x) isthen given by the sign of the scalar product (x − x∗) · en. In the case of a kinked crack

18

Figure 16: Local axis at the crack tips.

as shown in figure 17b, where no unique normal but a cone of normals is defined at x∗,H(x) = 1 if the vector (x− x∗) belongs to the cone of normals at x∗ and -1 otherwise.

Figure 17: Illustration of normal and tangential coordinates for a smooth crack (a) andfor a crack with a kink (b). x∗ is the closest point to x on the crack. In both of the abovecases, the jump function H(x) = −1.

This manner of calculating the function H(x) is rather general but cumbersome toimplement. One will see thereafter within the level set framework an approach much moreeffective. The use of the level sets also allows a fast evaluation of the distance r and angleθ necessary in the evaluation of the enrichment functions.

Description that we made above of enrichment for a crack is the original version pre-sented in (Moes, Dolbow, and Belytschko 1999). Since then, improvements were proposed.In particular, nothing prevents from enriching at a larger distance from the crack tip thanjust for the nodes whose support touches the crack tip.

The extension to the three-dimensional case of the modeling of cracks by X-FEM

19

was carried out in (Sukumar, Moes, Belytschko, and Moran 2000). Just like in the two-dimensional case, the fact that a node is enriched or not and the type of enrichmentdepend on the relative position of the support associated with the node compared to thecrack location. The support of a node is a volume, the crack front is a curve (or severaldisjoined curves) and the crack itself is a surface. Enrichment functions for the crackfront remain given by (45). A node is enriched if its support touches the crack front. Theevaluation of r and θ can be done by finding the nearest point on the crack front, thenby establishing a local base there. The use of level sets dealt with in the following sectionmakes it possible to avoid this operation.

3.4 Cracks located by level sets

To locate a curve in 2D, one can indicate all points located on this curve for exampleusing a parametric equation. One can qualify this representation as explicit. Anothermanner, implicit, to represent the curve is to know around the curve to distance of allpoints distance to this curve. This distance will positively be counted if one is inside thecurve and negatively in the contrary case (the curve is supposed to separate the spacein two zones). On a finite element mesh the level set is interpolated between the nodesby traditional finite element shape functions. In short, the location of a surface in 3D(curve in 2D) is given by a finite element field defined near the surface (curve in 2D). Theknowledge of the signed distance is indeed needed in a narrow band around the surface.

For instance, in the figure 19 one can see the value of a level set locating a circle on agrid (negative inside the circle and positive outside). The iso-zero of the level set functionindicates the position of the circle. The level set is defined likewise in 3D. Figure 20 givesfor example the iso-zero for a level set defined on a fine grid. The level set locates thematerial interface between strands and matrix in a so-called 4D composite.

The implicit representation is particularly interesting when the curve (surface) evolves.Indeed, contrary to the explicit representation which does not make it possible to managetopological changes easily. These changes are taken into account very naturally in theimplicit level set representation. By topological changes, one understands for example thefact that two bubbles meet to form a unique bubble or the fact that a drop can separatein two drops. Another example of topological change is the case of a crack initially insidea cube (figure 18 up) which after some propagation cuts the four faces of the cube. Thecrack front initially circular is split into four independent curves (figure 18 bottom).

The article (Osher and Sethian 1988) was one of the first to present robust algorithmsfor level sets propagation. The use of the level sets for computational science then veryquickly developed as attested by a sequence of three books (Sethian 1996; Sethian 1999;Osher and Fedkiw 2002). These algorithms of propagation were initially mainly developedwithin the framework of finite differences since they were coupled to the computationphysical fields by finite differences. Indeed, level sets were initially used for fluid mechanicsapplications : for instance to follow free interfaces or interfaces between various phases.Some articles however did develop algorithms appropriate for unstructured finite elementmeshes as (Barth and Sethian 1998).

As we indicated above, a level set separates space in two zones, a positive zone and

20

Figure 18: A lens shaped crack (up) propagating in a cube and finally cutting four cubefaces (bottom).

21

Figure 19: A level set locating a circle.

Figure 20: The iso-zero of a level set function locatingthe interface between strands and a matrix in a 4D com-posite.

22

a negative zone. A crack does not separate a domain into two (unless it is broken!). Aunique level set is thus not enough to locate a crack. One needs two of them. The firstone denoted lsn separates space into two by considering a tangent extension from thecrack whereas the function lst makes it possible to locate the front. These two level setsare represented in figure 21. The set of points characterized by lsn = 0 and lst ≤ 0defines the position of the crack whereas points for which lsn = lst = 0 defines thefront. The representation of a crack by two level set functions was for the first timesuggested by (Stolarska, Chopp, Moes, and Belytschko 2001) in 2D and (Moes, Gravouil,and Belytschko 2002) in 3D. Figure 22 gives the iso-zero of both level sets for a crack in3D whereas figure 23 gives an idea of the location of the iso-zero of both level sets in 3D.

lsn

lst

Figure 21: Two level set functions locating a crack on a2D mesh.

Coordinates r and θ appearing in the enrichment functions (45) are computed fromthe equations given in (Stolarska, Chopp, Moes, and Belytschko 2001)

r = (ls2n + ls2

t )1/2 θ = arctan(lsn

lst

) (46)

23

Surface exterieure du solide

Front de fissure

lst=0

lsn=0 : surface de la fissure

Figure 22: Iso-zero of lsn and lst locating a crack in 3D.

4 Mathematical and technical aspects

4.1 Integration

Integration on the elements cut by the crack is made seperately on each side of the crack.The lsn level set cuts a triangular (tetrahedral) element along a line (a plane). Thepossible cuts are indicated on figures 24 and 25. For elements close to the crack tip, use ofnonpolynomial enrichment functions requires special care (Bechet, Minnebo, Moes, andBurgardt 2005).

4.2 Topological and geometrical enrichment strategies

The initial enrichment strategy for the crack tip consisted in enriching a set of nodesaround the tip. A node is enriched if its support touches the crack tip (Moes, Dolbow,and Belytschko 1999). In 3D, nodes for which the support touches the crack front areenriched (Sukumar, Moes, Belytschko, and Moran 2000).

This type of enrichment may be called topological because it corresponds becauseit does not involve the distance from the node to the tip. As a matter of fact, thetopological enrichment is active over an area which vanishes to zero as the mesh sizegoes to zero. Another enrichment, developped independently in (Bechet, Minnebo, Moes,and Burgardt 2005) and (Laborde, Pommier, Renard, and Salaun 2005) may be calledgeometrical because it consists in enriching all nodes located within a given distance tothe crack tip. Both enrichment strategies are compared in figure 26.

In order to study the influence of the enrichment on the convergence rate, a planestrain benchmark problem is set up. A square domain in plane strain is subjected to a

24

Figure 23: Sign of the lst level set (blue negative, redpositive) on the lsn iso-zero plane.

25

Figure 24: Two scenarii of the level set cut of a triangle.

pure mode I. The square boundary is subjected to the exact tractions corresponding tothe mode I of infinite problem given in appendix A. Rigid modes are prevented. Theexaggerated deformed shape of the benchmark problem is shown in figure 27. To beprecise the domaine size is Ω = [0, 1]× [0, 1] and tractions applied correspond to K1 = 1and K2 = 0. The crack tip is located at the center of the square. Young modulus is 1and Poisson ratio 0. A convergence analysis is performed for a uniform grid shown infigure 29. The energy norm error, ε, measuring the distance between the exact, σ,u, andapproximated field σh,uh

ε =

(∫Ω

(σh − σ) : C−1 : (σh − σ) dΩ∫Ωσ : C−1 : σ dΩ

)1/2

=

(∫Ωε(uh − u) : C : ε(uh − u) dΩ∫

Ωε(u) : C : ε(u) dΩ

)1/2

(47)is plotted in figure 28. For the topologicval enrichment only the nodes whose supportis touching the crack tip are enriched. In the case of the geometrical enrichment, nodeswithin a distance of re = 0.05 from the crack tip are enriched. It can be observed that theconvergence rate is 0.5 when the toplogical or no enrichment is present. The topologicalenrichment yielding however a smaller error. On the contrary, the geometrical enrichmentproduces a order of 1 convergence. In order to analyze these convergence rates, we mustrecall the convergence rate result of the classical finite element method (see for instance(Bathe 1996))

ε = O(hmin(r−m,p+1−m)) (48)

The regularity of the solution is indicated by r (u ∈ Hr(Ω)) whereas p is the degree of thefinite element interpolation and m is the error norm used. For our benchmark, r = 3/2,p = 1 and m = 1, so we indeed get a convergence rate of 0.5 The topological enrichment

26

Figure 25: Four scenarii of the level set cut of a tetra-hedron.

27

* A new Enrichment

Computing the stress

Figure 26: Extent of the enrichment zone as the mesh size decreases. Geometrical enrich-ment (top) and topological (bottom) enrichments are compared.

yields a lower error than a pure FEM analysis because the X-FEM approximation spans alarger space than the FEM one. However, it does not affect the convergence rate since theenrichment area goes to zero as the mesh size goes to zero. In the case of the geometricalenrichment, the enrichment is able to represent exactly (even as h to zero) the roughpart of the solution. The classical part of the approximation is thus only in charge of thesmooth part of the solution yielding the optimal order of 1 convergence rate. This wasproved by Laborde, Pommier, Renard, and Salaun (2005). It was also shown in this paperthat (for the benchmark problem) if the polynomial approximation is raised, higher (stilloptimal) convergence rates are obtained. The polynomial degree needs only to be raisedin the classical part and Heaviside parts of the approximation (first and second term termin the right hand side of (45).

4.3 Condition number

When solving a linear system of equation Kx = f , an important number to take intoaccount is the condition number defined as the ratio between the maximum and minimumeigenvalue of the K matrix.

κ =λmax

λmin

(49)

This condition number has a direct impact on the convergence rate for an iterativesolver and on the propagation of roundoffs for a direct solver. For instance, for theconjugate gradient ietartive solver, the error at iteration m reads (Saad 2000):

‖x−xm‖K ≤ 2

[√κ− 1√κ+ 1

]m‖x−x0‖K (50)

28

Figure 27: Mode I benchmark problem : Exaggerated deformed shape of the square undermode one loading.

29

0.01

0.1

1

1 10 100 1000

err

or

1/h

w /o e n ric h m e n t (s lo p e ~ 0.5 )to p o lo g ic a l e n ric h m e n t (s lo p e ~ 0.5 )

g e o m e tric a l e n ric h m e n t (s lo p e ~ 1.0)

Figure 28: Relative error in the energy norm for the mode I benchmark problem.

where x0 is the initial guess, xm the iterate m and

‖a‖K =√aTKa (51)

Thus the higher the condition number, the slower is the convergence. To be precise, thebound (50) is in general pessimistic. Indeed, first of all κ can be calculated on the basisof the eigenvalues for which the corresponding eigenvector projected on the right handside is not zero. Then, the κ can be readjusted progressively with the iterations whilebeing based only on the eigenvectors remaining active through the iterations (see detailin (Saad 2000)).

The conditioning of X-FEM was studied in (Bechet, Minnebo, Moes, and Burgardt2005) and (Laborde, Pommier, Renard, and Salaun 2005) for the two types of enrichment:topological and geometrical. The evolution of the condition number according to the sizeof elements of the grid is given in figure 30 for the stiffness and mass matrices. They areplotted for the benchmark problem already discussed in section 4.2.

We note that for the geometrical enrichment, the condition number grows dramaticallywith the mesh size. A specific preconditioner was designed in (Bechet, Minnebo, Moes, andBurgardt 2005) circumvent the increase. The effect of the preconditioner is also given infigure 30. This preconditioner could be called pre-preconditioner X-FEM preconditionerbecause it takes care of the specificity of X-FEM. After its application, regular FEMpreconditioner may be used. The idea behind the X-FEM precodntioner is quite simple.On enriched nodes, the enriched shapes fonctions are orthogonalized with respect to theclassical shape function. The matrices related to a given node are thus diagonal. ....

30

Figure 29: Mesh type used for the convergence analysis.

31

10

100

1000

10000

100000

1e + 06

1 10 100

Co

nd

itio

n N

um

be

r

1/h

M -c la s sK -c la s s

M -e n ric h e dK -e n ric h e dM -p re c o n dK -p re c o n d

1

10 0

10 0 0 0

1e + 0 6

1e + 0 8

1e + 10

1e + 12

1e + 14

1 10 10 0

Conditio

n N

um

ber

1/h

M -c la s sK -c la s s

M -e n ric h e dK -e n ric h e dM -p re c o n dK -p re c o n d

Figure 30: Condition number as a function of the mesh size for the mass and stiffnessmatrices. Topological and geometrical enrichments are considered as well as the influenceof the preconditioner.

32

K11 K12 K13 K14

K22 K23 K24

K13 K23 K33 K34

K14 K24 K34 K44

K12

0 0 0

K11

K11

K11

0

0

0

00

0

0

0

0

K11

Figure 31: Preconditioning strategy.

5 Stress intensity factor evaluation

This section summarizes the use of integrals for the extraction of the stress intensityfactors along a crack front. The integral J is recalled as well as interaction integralswhich make it possible to separately extract each stress intensity factor. The originalityof the presentation lies in the use of a representation by level sets of the crack. Thisrepresentation makes it possible in each point of the domain around the crack to define acurvilinear basis in which the auxiliary solution can be expressed. This curvilinear basemay be used even for non planar crack or plane crack with curved front. The influence ofthe presence of volumic force or thermal loading is also discussed.

5.1 2D case

The domain integral is defined by :

J = −∫S

qi,jPijdS +

∫C+∪C−

qiPijnjdC (52)

J =(1− ν2)

E(K2

1 +K22) (53)

5.2 Eshelby tensor and J integral

In sections 5.2 and 5.3, one considers a linear elastic solid containing a crack. No bodyforces or thermal loads act on the body. Also, the crack faces are considered traction free.

Eshelby tensor is a second order tensor generaly non symmetric defined by

Pij =1

2σklεklδij − σkjuk,i (54)

33

Figure 32: Notations for the domain integral.

Assuming the stress σ, strain ε and displacement u fields satisfy elasticity equations(equilibrium, compatibility and behavior) :

σij,j = 0 εij =1

2(ui,j + uj,i) σij = Eijklεkl (55)

the Eshelby tensor is divergence free

Pij,j = 0 (56)

We consider in figure 32 a domain of integration V . This domain contains a part ofthe crack. The upper and lower faces are denoted S+ et S−, respectively. The domainalso contains a part of the front denoted by Γ.

The J domain integral is defined by:

J = −∫V

qi,jPijdV +

∫S+∪S−

qiPijnjdS (57)

The n vector is the outward normal to each crack face. The integral (domain form ofthe Rice integral) was first introduced in 2D by Destuynder, Djaoua, and Lescure (1983).The physical meaning of the integral is the power dissipated as the crack front advancesunder the q velocity:

J =

∫Γ

G · qdΓ (58)

where G is the vectorial energy release rate at the crack tip. It’s tangential component(written as G ) is related to the stress intensity factors by the following formula :

G =(1− ν2)

E(K2

1 +K22) +

1

2µK2

3 (59)

34

Integral (57) is independent from the choice of volume V provided that

• the vector field q is zero on the outer boundary of V ;

• volume V interesects the same front piece Γ;

• there exist a domain V 0 around Γ in which function q has a given value independentof the choice of V .

Figure 33: Notations used to prove the independence of J with respect to the domain V .

Let us prove the independence. Consider two domains V and V ′ shown in figure 33 (thefigure depicts the 2D case but the 3D case is similar). Suppose V ⊂ V ′ (this restrictionwill be waived later). Let us write the difference of J computed on these two domains.

J(V ′)− J(V ) = (60)

−∫V ′\V 0

(qiPij),jdV −∫∂V 0

qiPijnjdS +

∫(S′+\S0

+)∪(S′−\S0

−)

qiPijnjdS (61)

+

∫V \V 0

(qiPij),jdV +

∫∂V 0

qiPijnjdS −∫

(S+\S0+)∪(S−\S0

−)

qiPijnjdS (62)

35

where ∂V 0 is the external boundary of V 0 (crack faces are excluded). We have also usedthe following:

qi,jPij = (qiPij),j (63)

due to the fact that the Eshelby tensor is divergence free. Gauss theorem may be used ondomains V \ V 0 and V ′ \ V 0 since these domains do not contain singularities. We deducethat lines (61) and (62) are zero leading to

J(V ′) = J(V ) (64)

If domain V is not included in V ′, one can define domain V included in the intersectionof V and V ′; and write

J(V ′) = J(V ) = J(V ) (65)

If the crack faces are traction free and if the vector q is tangent to the crack, integralson the crack faces in (57) vanishes. We assume traction free crack faces in the following.

Also let us note that if tensor P is not divergence free (it will not be divergence freefor instance in the presence of body forces), it is the expression for J below that it isnecessary to consider

J = −∫V

(qiPij),jdV (66)

5.3 Interaction integrals

Let us consider that the fields ui, εij, σij involved in the Eshelby tensor are the sum of twofields, denoted by indices 1 and 2, satisfying all the elastic interior equations:

ui = u1i + u2

i εij = ε1ij + ε2ij σij = σ1ij + σ2

ij (67)

Inserting these expression in integral (66), we obtain

J = J (1) + J (2) + I(1,2) (68)

The term I is called an interaction integral

I(1,2) = −∫V

qi,j(σ1klε

2klδij − σ1

kju2k,i − σ2

kju1k,i)dV (69)

Since integrals J , J (1) and J (2) are contour independent, so is I

I(1,2) =

∫Γ

G(1,2) · qdΓ (70)

Tangential component of G(1,2) is

G(1,2) =2(1− ν2)

E(K1

1K21 +K1

2K22) +

1

µK1

3K23 (71)

36

Exponents 1 and 2 denote stress intensity factors related to fields 1 and 2, respectively.Field 1 is chosen as the finite element solution and field 2 is called an auxiliary field chosenas pure mode I, II or III. These pures modes are recalled in appendix 7. Auxiliary fieldssatisfy

σ2ij,j = 0 ε2ij =

1

2(u2

i,j + u2j,i) σ2

ij = Eijklε2kl (72)

σ2ijnj = 0 sur S+ ∪ S− (73)

where nj is normal (outward) to each crack face. Equation (73) indicates that the auxiliaryfield is traction free on the crack faces.

Let us now consider the case of a plane crack with a curved front or a non planarcrack. We suggest to use the curvilinear basis given by the pair of level sets to define theauxiliary fields (Moes, Gravouil, and Belytschko 2002). The curvilinear basis is definedby

e1 = ∇lst e2 = ∇lsn e3 = e1 × e2 (74)

This basis moves from one point to the next but is assumed orthonormal (it is areasonable assumption since one forces the orthogonality and the unit standard of thegradients of both level sets in the resolution of level set equations). The basis is illustratedat the crack tips in figure 16 (although this basis is drawn at a point on the crack face, weinsist that it can be defined anywhere the levekl sets are defined thrigh the relations (28),see section 6.5 ). Auxiliary fields now read:

σ2 = σ2αβeα ⊗ eβ (75)

ε2 = ε2αβeα ⊗ eβ (76)

u2 = u2αeα (77)

The curvilinear basis is also used to define the vectorial field q:

q = qe1 (78)

Indices α and β of the local basis should not be confused with the indices i and j of theglobal basis. The global basis is denoted using the vectors E1, E2, E3.

Scalar q is worth 1 in the vicinity of the crack front falls to zero on the outer boundaryof V . The q field is illustrated on figure 34(b) and is compared with a field q that wouldresult from global axis

For curved front or non planar cracks, relations (72-73) are no longer valid and shouldbe replaced by:

σ2ij,j 6= 0 (79)

ε2ij 6=1

2(u2

i,j + u2j,i) (80)

σ2ij = Eijklε

2kl (81)

σ2ijnj = 0 sur S+ ∪ S− (82)

37

Figure 34: A virtual velocity field with fixed orientation (a) and respecting the crackgeometry (b).

Equation (81) remains valid because we decide to calculate strains using the constitu-tive law. Relation (82) remains also valid with the use of the curvilinear basis.

Note that to obtain the expression of the interaction integral (69), we assumed thatthe auxiliary satisfy all internal equations. It will no longer be the case with a curvedfront or non planar crack.

5.4 Extension to body forces

In the presence of body forces, f , Eshelby tensor reads

P volij = (

1

2σklεkl − fkuk)δij − σkjuk,i (83)

It is no longer divergence freeP volij,j = −fk,iuk (84)

except in the case of uniform body forces. J integral is no longer given by (57) and isnow given by

Jvol = −∫V

(qi,jPvolij + qifk,iuk)dV (85)

Regarding the interaction integral, we need to adapt the expression (69). It becomes

38

I(1,2)vol = −

∫V

qi,j((σ1klε

2kl − fku2

k)δij − σ1kju

2k,i − σ2

kju1k,i)dV −

∫V

qifk,iu2kdV (86)

Note that the auxilary fielf statisfy the interal equations without body forces (as before).In the particular case of centrifugal forces, body forces are written

f = ρω2rer (87)

where ρ is density, ω angular speed, r the distance to the rotation axis and, finally, er isa vector linking the rotation axis to the considered point. In a cartesian basis, if x axis isthe rotation axis, the body forces are

f = ρω2(yey + zez) (88)

Note that they are linear with respect to the coordinates. The gradient is constant

[∇f ](ex,ey ,ez) = ρω2

0 0 00 1 00 0 1

(89)

5.5 Extension to thermal loadings

We consider a thermoelastic constitutive law and a given temperatuire field

σij = Eijkl(εkl − εthkl), εthkl = α(T − To)δkl (90)

where α is the thermal expansion coefficient and To the reference temperature. Eshelbytensor reads

P thij = ψδij − σkjuk,i (91)

where

ψ =1

2Eijkl(εij − εthij )(εkl − εthkl) (92)

The divergence of Eshelby tensor is non zero

P thij,j = −σklεthkl,i (93)

If α is uniform, we getP thij,j = −σklεthkl,TT,i (94)

J integral is now given by

Jth = −∫V

(qi,jPthij + qiσklε

thkl,TT,i)dV (95)

and the interaction integral by

I(1,2)th = −

∫V

qi,j(σ1klε

2klδij − σ1

kju2k,i − σ2

kju1k,i)dV (96)

−∫V

qiσklεthkl,TT,idV (97)

Auxiliary fields satisfy the classical elastic equations with no thermal effects

39

6 Level sets propagation

We have seen in a previous section how convenient is the use of level set to locate materialinterfaces, voids or crack on a mesh. The level set is defined on the mesh as any otherphysical field. (In fact the level set does not need to be defined at all nodes on themesh but only close to the surface. It is common to call this zone the narrow band.)Another important gain in locating surfaces with level sets is the possibility to use efficientalgorithms to propagate the surfaces. Surfaces may evolve under a physical velocity : crackgrowth, void growth, phase tranformation, ... They can also evolve under a “numerical”speed for instance in the case of topology optimization.

The litterature describing algorithms to evolve level sets is now quite rich. An overviewof the algorithms and applications may be found in the sequel of the following three books: (Sethian 1996) (Sethian 1999) (Osher and Fedkiw 2002).

Algorithms developped to evolve interfaces located by level sets may be placed inat least three categories: level set mesthods, fast marching methods and other (newer)methods : finite element and discontinuous Galerkin methods.

6.1 Level set methods

These methods tackle the evolution equation given by Osher and Sethian (1988).

∂φ

∂t+ F‖∇φ‖ = 0 given φ(x, t = 0) (98)

in which φ is the level and F the normal velocity to the front. It is discretized using finitedifferences over a uniform grid. Great care is taken in developping the stencil formulato make sure that the the level set update at a node is based on information getting tothe node and not leaving it (so called upwinding). Equation (98) is indeed an hyperbolicconservation law.

Although intially developped for uniform grids, the level sets methods has been ex-tended for non-uniform finite element type meshes for instance in (Barth and Sethian1998). It is called the triangular version of the level set method.

6.2 Fast marching method

The fast marching method is restricted to the case for which the front velocity F is alwayspositive. It tackles the so-called Eikonal equation

‖∇φ‖F = 1 (99)

Unilke for the level set equation, the location of the front at time t is given by φ = t andnot φ = 0. For a grid of N the fast marching method will take no more than N log(N)operations to find the φ value at each grid point.

40

6.3 Other methods

The discontinuous Galerkin method was used in (Marchandise, Remacle, and Chevaugeon2006). In the approach of Mourad, Dolbow, and Garikipati (2005), the level set equationis transformed by assuming that the level set remains a signed distance function (i.e. thegradient factor is dropped in (98).

6.4 Propagation of curves

The method described above tackle the problem of the evolution of surfaces in R3 or curvesin R2. These are so-called codimension 1 problem. Problems of codimension 2 are also ofinterest, i.e. the evolution of a curve in R3. A specific level set approach was developpedfor this more complex problem in (Burchard, Cheng, Merriman, and Osher 2001; Cheng,Burchard, Merriman, and Osher 2002) and more recently in (Carlini, Falcone, and Ferretti2007).

6.5 Specific methods for crack growth

As seen in section 3.4, two level sets are needed to locate a crack on a mesh. The levelset update as the crack evolves is thus a bit more tricky than in the case of a single levelset. Note that the use of two level sets to locate the crack was inspired from the use oftwo level sets used to locate a curve (Burchard, Cheng, Merriman, and Osher 2001). Animportant difference however between a curve and a crack front is that the path followedby the crack front must be stored.

For planar cracks a single level set function may be considered and a crack growthalgorithms based on the fast marching method was proposed by (Sukumar, Chopp, andMoran 2003). Using the ease with which level sets may handle topological changes,merging of co-planar cracks was also considered by Chopp and Sukumar (2003).

Going back now to cases for which two level sets are needed, the first paper dealingwith the update of 2D cracks located by two level sets was written by Stolarska, Chopp,Moes, and Belytschko (2001). Geometrical formulas are used to update the level sets inthis work. The basic idea is depicted in figure 35. The tangential level set at time stepn lsnt is first rotated to be orthognal to the crack speed. Then it is shifted by an amountF∆t corresponding to the crack advance over a time step ∆t. Regarding the normal levelset, lsn, is simply updated in the non-shaded area (given by ˆlst > 0) as the sign distanceto the line oriented in the F direction.

Note that it was recently shown by (Duflot 2007) that the level set update describedabove yields non smooth variation of the r and θ fields (computed from (46).

For 2D cracks update, a another geometrical type update is introduced in (Ventura,Budyn, and Belytschko 2003). It is based on a so-called vector level set representation ofthe crack.

Concerning 3D cracks, the paper (Gravouil, Moes, and Belytschko 2002) is not using ageometrical update but rather a sequence of level set methods. The scheme is summarizedin the table below. It was applied to the growth of cracks in complex structural parts by

41

lsnt = 0 lsn+1

t = 0lst = 0

F∆t

Figure 35: Sketch of the geometrical update for the tangential level as the crack grows inthe direction F .

Bordas and Moran (2006). The scheme was studied by Duflot (2007) and the 2D case andagain it was shown that the r and θ fields were not smooth. It was also stated that it is ingeneral impossible to reach orthogonality of the levels as well normalized distance at thesame time (that is impossibility to have both ∇lst · ∇lsn = 0 and ‖∇lsn‖ = ‖∇lst‖ = 1except for straight cracks). Finally, although focused mainly on 2D crack growth, thepaper Duflot (2007) also suggest a new algorithm for 3D crack growth in which theexplicit definition of the front is needed.

All methods described above, solve the level set equations on the same grid as theone used to solve the mechanical fields; Some authors prefer to solve the level set updateon a different grid as Prabel, Combescure, Gravouil, and Marie (2007) using the levelset method and and Sukumar, Chopp, Bechet, and Moes (2008) using the fast marchingmethod.

The advantage of using non-matching meshes is the capability to use regular structcuredgrid for which algorithm are faster and simpler to code. In particular, the level set evo-lution equation involving a virtual time as in (100) converge much faster on structuredgrid.

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1. Extend Vt to the domain

∂Vt

∂τ+ sign(lst)

∇lst

‖∇lst‖· ∇Vt = 0

∂Vt

∂τ+ sign(lsn)

∇lsn

‖∇lsn‖· ∇Vt = 0 (100)

2. Extend Vn to the domain

∂Vn

∂τ+ sign(lsn)

∇lsn

‖∇lsn‖· ∇Vn = 0

∂Vn

∂τ+ sign(lst)

∇lst

‖∇lst‖· ∇Vn = 0 (101)

3. Adjustment to prevent modification of the previous crack surface

V n = H(lst)lstVn

∆tVt

(102)

where H is the Heaviside function.

4. Update and reinitialize lsn

∂lsn

∂t+ V n‖∇lsn‖ = 0

∂lsn

∂τ+ sign(lsn)(‖∇lsn‖ − 1) = 0 (103)

5. Update lst∂lst

∂t+ Vt‖∇lst‖ = 0 (104)

6. Othogonalize and reinitialize lst

∂lst

∂τ+ sign(lsn)

∇lsn

‖∇lsn‖· ∇lst = 0

∂lst

∂τ+ sign(lst)(‖∇lst‖ − 1) = 0 (105)

Table 1: Scheme for the level set update introduced in Gravouil, Moes and Belytschko2002

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7 Applications currently treated with X-FEM

The new possibilities offered by the modeling of discontinuities by X-FEM or its alter-natives were quickly used in many applications which we try below to account for in themost exhaustive possible way. The initial paper (Moes, Dolbow, and Belytschko 1999)is now quoted more than 250 times. A state of the art in the use of X-FEM in fracturemechanics was written by (Karihaloo and Xiao 2003).

• Powder Compaction: (Khoei, Anahid, and Shahim 2007; Khoei, Shamloo, andAzami 2006; Khoei, Shamloo, Anahid, and Shahim 2006)

• Interface modeling between two plastic phases: (Shamloo, Azami, and Khoei 2005)

• Biomechanics : modeling organ deformations associated with surgical cuts (Vi-gneron, Verly, and Warfield 2004a; Vigneron, Verly, and Warfield 2004c; Vigneron,Verly, and Warfield 2004b; Vigneron, Robe, Warfield, and Verly 2006)

• Homogenization of complex microstructures: (Moes, Cloirec, Cartraud, and Remacle2003)

• Homogenization and Imaging : Homogeneisation directe sur base d’image de bi-phases (Ionescu, Moes, Cartraud, and Beringhier 2006)

• Image correlation to analyze crack paths: (Ferrie, Buffiere, Ludwig, Gravouil, andEdwards 2006; Rethore, Roux, and Hild 2007; Rethore, Hild, and Roux 2007a)

• Digital image correlation for shear bands: (Rethore, Hild, and Roux 2007b)

• Solidification and phase transformation: (Merle and Dolbow 2002; Ji, Chopp, andDolbow 2002; Chessa, Smolinski, and Belytschko 2002)

• Modelling of tectonic plates subduction: (Zlotnik, Diez, Fernandez, and Verges2007)

• Contact in multi-material arbitrary Lagrangian-Eulerian calculations: (Vitali andBenson 2006)

• Modeling of thermal barrier coatings: (Michlik and Berndt 2006)

• Uncertainties and perturbations: (Grasa, Bea, Rodriguez, and Doblare 2006; Nouy,Schoefs, and Moes 2007)

• Contact and friction using a penalty approach : (Khoei and Nikbakht 2006)

• Contact et friction using Lagrange Multipliers and LATIN method: (Dolbow, Moes,and Belytschko 2001)

44

• Stiff interfaces (Dirichlet, contact, ...) using Lagrange multipliers and treatmentof the locking issue (Ji and Dolbow 2004; Bechet, Minnebo, Moes, and Burgardt2005; Geniaut, Massin, and Moes 2007; Kim, Dolbow, and Laursen 2007; Mourad,Dolbow, and Harari 2007)

• Calculation of dislocations : (Gracie, Ventura, and Belytschko 2007)

• Modeling thermal fatigue cracking in integrated circuits: (Stolarska and Chopp2003)

• Shear bands modeling: (Song, Areais, and Belytschko 2006; Areias and Belytschko2006b; Areias and Belytschko 2007)

• Modeling strong gradients and transition from damage to fracture : (Patzak andJirasek 2003; Areias and Belytschko 2005b; Comi, Mariani, and Perego 2007)

• Appropriate enrichment to model incompressibility (Dolbow and Devan 2004; Legrain,Moes, and Huerta )

• Multi-scale modelling and simulation of textile reinforced materials: (Haasemann,Kastner, and Ulbricht 2006)

• Numerical strategy for investigating the kinetic response of stimulus-responsive hy-drogels: (Dolbow, Fried, and Ji 2004; Dolbow, Fried, and Ji 2005)

• Shape optimization: (Belytschko, Xiao, and Parimi 2003; Miegroet and Duysinx2007)

• Modeling polycrystals with discontinuous grain boundaries: (Simone, Duarte, andGiessen 2006)

• Channel-cracking of thin films: (Huang, Prevost, and Suo 2002; Huang, Prevost,Huang, and Suo 2003; Liang, Huang, Prevost, and Suo 2003b; Liang, Huang, Pre-vost, and Suo 2003a)

• Crack modeling in elasto-plastic media: (Elguedj, Gravouil, and Combescure 2006;Prabel, Combescure, Gravouil, and Marie 2007)

• Modeling cracks in quasi-brittle materials using cohesive zones: (Wells and Sluys2001; Moes and Belytschko 2002; Zi and Belytschko 2003; Simone 2004; Gasser andHolzapfel 2005; Mergheim, Kuhl, and Steinmann 2005; Asferg, Poulsen, and Nielsen2007; Xiao, Karihaloo, and Liu 2007; Meschke and Dumstorff 2007)

• The use of X-FEM for the cracking of composites was first suggested by (Cox andYang 2006). It was implemented in (Hettich and Ramm 2006; Hettich, Hund, andRamm 2007) (micro-cracking of the matrix in the presence of fibers) and in (Wells,de Borst, and Sluys 2002; de Borst and Remmers 2006) (delamination)

45

• Multiple cracks: (Budyn, Zi, N.Moes, and Belytschko 2004)

• Fracture dynamics: : (Duarte, Hamzeh, Liszka, and Tworzydlo 2001; Rethore,Gravouil, and Combescure 2005b; Zi, Chen, Xu, and Belytschko 2005; Rethore,Gravouil, and Combescure 2005a) and also (Nistor, Pantale, and Caperaa 2006;Menouillard, Rethore, Combescure, and Bung 2006; Menouillard, Rethore, Moes,Combescure, and Bung 2007; Prabel, Combescure, Gravouil, and Marie 2007). Morerecently, comparison with experiments were given in (Gregoire, Maigre, Rethore,and Combescure 2007).

• Interface cracks: (Nagashima, Omoto, and Tani 2003; Sukumar, Huang, Prevost,and Suo 2004; Liu, Xiao, and Karihaloo 2004) and cracks in orthotropic media(Asadpoure, Mohammadi, and Vafai 2006a; Asadpoure, Mohammadi, and Vafai2006b; Asadpoure and Mohammadi 2007)

• Mixed mode cracks in graded materials (Menouillard, Elguedj, and Combescure2006)

• Crack modeling in heterogeneous media: (Dolbow and Nadeau 2002)

• Cracks in body under large deformations: (Legrain, Moes, and Verron 2005)

• Modeling of material interfaces under large deformation: (Khoei, Biabanaki, andAnahid 2007)

• Configurational forces: (Larsson and Fagerstrom 2005)

• Cracks in plates and shells: (Dolbow, Moes, and Belytschko 2000; Areias, Song, andBelytschko 2006; Areias and Belytschko 2005a)

• Space-time discontinuity: (Chessa and Belytschko 2004; Rethore, Gravouil, andCombescure 2005a)

• Network of discontinuities and tangential discontinuities: : (Belytschko, Moes, Usui,and Parimi 2001)

• Fluid-structure interactions: (Wagner, Moes, Liu, and Belytschko 2001; Wagner,Ghosal, and Liu 2003; Legay, Chessa, and Belytschko 2006)

• Two-phase fluids: (Chessa and Belytschko 2003a; Chessa and Belytschko 2003b)

• Propagation algorithms for 3D cracks modeled by X-FEM: (Gravouil, Moes, andBelytschko 2002; Sukumar, Chopp, and Moran 2003; Chopp and Sukumar 2003;Duflot 2007)

• Damage tolerance assessment of complex structures : (Bordas and Moran 2006)

46

• A posteriori error estimation for partition of unity based approaches: (Strouboulis,Zhang, Wang, and Babuska 2006; Bordas and Duflot 2007; Strouboulis, Zhang, andBabuska 2007)

• Partition of untiy based multi-scale strategy: (Strouboulis, Zhang, and Babuska2004; Fish and Yuan 2005; Guidault, Allix, Champaney, and Cornuault 2007)

• Numerical construction of the enrichment: (Duarte and Kim 2007)

• X-FEM on polygonal and quadtree meshes: (Tabarraei and Sukumar 2007)

• Coupling between “volume of fluid” representation of the interface and X-FEM:(Dolbow 2007)

• Modeling impact on non matching meshes: (Rozycki, Moes, Bechet, and Dubois2007)

Acknowledgment

This document is a summary of the work produced by many researchers which are quoted,I hope in a suitable manner. Moreover, I wish to thank Eric Bechet for the quality ofmany figures which were drawn as he was a post-doctoral fellow at the Ecole Centrale ofNantes.

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A the auxiliary field in the 2D setting

Stresses are given by

σaux11 =

1√2πr

Kaux

1 cosθ

2[1− sin

θ

2sin

3θ

2]−Kaux

2 sinθ

2[2 + cos

θ

2cos

3θ

2]

(106)

σaux22 =

1√2πr

Kaux

1 cosθ

2[1 + sin

θ

2sin

3θ

2] +Kaux

2 sinθ

2cos

θ

2cos

3θ

2

(107)

σaux12 =

1√2πr

Kaux

1 sinθ

2cos

θ

2cos

3θ

2+Kaux

2 cosθ

2[1− sin

θ

2sin

3θ

2]

(108)

and displacements by

uaux1 =

1

2µ

√r

2π

Kaux

1 cosθ

2(κ− cos θ) +Kaux

2 sinθ

2(κ+ 2 + cos θ)

(109)

uaux2 =

1

2µ

√r

2π

Kaux

1 sinθ

2(κ− cos θ) +Kaux

2 cosθ

2(κ− 2 + cos θ)

(110)

where µ = E2(1+ν)

. For plane strain, κ = 3− 4ν and for plane stress κ = 3−ν1+ν

.

B The auxiliary fields in the 3D setting

Stresses are given by

σaux11 =

1√2πr

Kaux

1 cosθ

2[1− sin

θ

2sin

3θ

2]−Kaux

2 sinθ

2[2 + cos

θ

2cos

3θ

2]

(111)

σaux22 =

1√2πr

Kaux

1 cosθ

2[1 + sin

θ

2sin

3θ

2] +Kaux

2 sinθ

2cos

θ

2cos

3θ

2

(112)

σaux12 =

1√2πr

Kaux

1 sinθ

2cos

θ

2cos

3θ

2+Kaux

2 cosθ

2[1− sin

θ

2sin

3θ

2]

(113)

σaux23 =

Kaux3√2πr

cosθ

2, σaux

13 = − Kaux3√2πr

sinθ

2, σaux

33 = ν(σaux11 + σaux

22 ) (114)

and displacements by

uaux1 =

1

2µ

√r

2π

Kaux

1 cosθ

2(κ− cos θ) +Kaux

2 sinθ

2(κ+ 2 + cos θ)

(115)

uaux2 =

1

2µ

√r

2π

Kaux

1 sinθ

2(κ− cos θ) +Kaux

2 cosθ

2(κ− 2 + cos θ)

(116)

uaux3 =

2

µ

√r

2πKaux

3 sinθ

2(117)

where µ = E2(1+ν)

et κ = 3− 4ν.

59

Figure 36: Coordinate system for the auxiliary fields.

60