Derivative recovery and a posteriori error estimate for extended finite elements

www.elsevier.com/locate/cma

Comput. Methods Appl. Mech. Engrg. 196 (2007) 3381–3399

Derivative recovery and a posteriori error estimatefor extended finite elements

Stephane Bordas a,*, Marc Duflot b

a University of Glasgow, Civil Engineering, Rankine Building, Glasgow G12 8LT, United Kingdomb CENAERO, Avenue Jean Mermoz, 30 B-6041 Gosselies, Belgium

Received 6 November 2006; received in revised form 9 March 2007; accepted 12 March 2007Available online 19 April 2007

Abstract

This paper is the first attempt at error estimation for extended finite elements. The goal of this work is to devise a simple and effectivelocal a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM).In each element, the local estimator is the L2 norm of the difference between the raw XFEM strain field and an enhanced strain field com-puted by extended moving least squares (XMLS) derivative recovery obtained from the raw nodal XFEM displacements. The XMLS con-struction is tailored to the nature of the solution. The technique is applied to linear elastic fracture mechanics, in which near-tip asymptoticfunctions are added to the MLS basis. The XMLS shape functions are constructed from weight functions following the diffraction crite-rion to represent the discontinuity. The result is a very smooth enhanced strain solution including the singularity at the crack tip. Resultsare shown for two- and three-dimensional linear elastic fracture mechanics problems in mode I and mixed mode. The effectivity index ofthe estimator is close to 1 and improves upon mesh refinement for the studied near-tip problem. It is also shown that for the linear elasticfracture mechanics problems treated, the proposed estimator outperforms one of the superconvergent patch recovery technique ofZienkiewicz and Zhu, which is only C0. Parametric studies of the general performance of the estimator are also carried out.� 2007 Elsevier B.V. All rights reserved.

Keywords: Extended finite element method; Derivative recovery; A posteriori error estimation; Extended moving least squares; Fracture mechanics;Enrichment; Partition of unity; Three-dimensional cracks

1. Introduction

Determining the accuracy of a numerical solution isessential, both to gain a better understanding of the numer-ical method used, and to measure its reliability in the con-text of real-world problems. This paper extends thepreliminary ideas presented by the authors in Ref. [17]and offers an alternate method to the extended globalrecovery method that we proposed in Ref. [23] presents

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.03.011

* Corresponding author. Tel.: +44 (0) 141 330 4075; fax: +44 (0) 141 3304557.

E-mail addresses: [email protected] (S. Bor-das), [email protected] (M. Duflot).

URL: http://www.civil.gla.ac.uk/~bordas (S. Bordas).

an error estimator for partition of unity enriched finite ele-ment methods such as the extended finite element method(XFEM) based on derivative recovery with intrinsicallyenriched extended moving least squares (XMLS) approxi-mants and diffraction. The adequacy of the proposed esti-mator on two- and three-dimensional linear elastic fracturemechanics problems treated with an extended finite elementformulation is shown through 2D and 3D numerical exam-ples. MLS derivative recovery in finite elements was firstproposed in Ref. [50] of which this work is a generalization.

The extended finite element method (XFEM) is amongpartition of unity methods [33,7], and has been extensivelyput to work for problems with discontinuities, since 1999[10]. Typically, XFEM and related methods add, locally,special functions to the FE approximation, in order to bet-ter approximate particular features of a problem. For

mailto:[email protected]

mailto:[email protected]

http://www.civil.gla.ac.uk/~bordas

3382 S. Bordas, M. Duflot / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3381–3399

problems with discontinuities in the primal dependentvariable (cracks [35,34,14], fluid–structure interaction[32]), discontinuous functions are added. To handle discon-tinuities in the dual unknown (multi-materials [47], solidifi-cation [20], biofilms [41], multi-phase flow [19], Stokes flow[53]), functions with discontinuous derivatives enhance thestandard basis. These additional functions allow to modelthe discontinuous features without recourse to a conform-ing mesh, while retaining an exact representation of the dis-continuities, to which the mesh need not conform. Inaddition to discontinuities, special functions can be addedto tell the approximation about a known characteristic ofthe expected solution, around crack tips or boundary layersfor instance. This increases accuracy for coarse meshes,and improves the convergence rate. The extended finite ele-ment was coupled with the level set method, and the latestresults are given in Ref. [21]. Hansbo and Hansbo came upwith yet another technique to describe strong discontinu-ities within finite elements [28] whose kinematics wasproved equivalent to that of XFEM [5].

Three types of procedures are currently available forderiving error estimators. They are either residual based,recovery based, or constitutive behaviour based. The residual

based error estimators were first introduced by Babuskaand Rheinboldt in 1978 [6] and have been since used veryeffectively and further developed by many others. Ains-worth and Oden made substantial progress with the intro-duction of so-called residual equilibration in 1992-1993[3,1]. On the other hand, the recovery based estimates havebeen first introduced by Zienkiewicz and Zhu in 1987 [55].Again, these were extensively improved by the introductionof new recovery processes. These authors have introducedthe so-called superconvergent patch recovery (SPR) in1992 [56,57]. This method has produced a very significantimprovement of performance of recovery based methods.Finally, the French team of Professor Ladeveze devisedthe error on constitutive behaviour [30]. The book [4] isan excellent reference on error estimates, as is the reviewpaper [2].

Although enriched finite element methods have beenvery successful in solving problems with singularities anddiscontinuities, to the authors’ knowledge, work on a pos-teriori error estimates for such methods is by the group ofProfessor Strouboulis in the context of the generalizedfinite element method (GFEM) [43–45], and in work ofProfessor Karihaloo [54] concerning the XFEM.

In general, the exact solution to a problem is unknown.In order to approximate the approximation error of anumerical technique, an enhanced solution is required.How should one go about constructing improved (recov-ered) fields from a raw XFEM solution? The use of SPRtechniques would be possible, provided the existence, andlocation of superconvergent points [8] was known. Addi-tionally, such recovery techniques perform poorly inregions of high gradients, which does not make it attractiveto solve fracture problems. The next possibility that comesto mind is the use of smoothing techniques. One of the sim-

plest and versatile fitting method is the moving leastsquares approximation. A major point in favor of usingMLS approximants is their ability to include special func-tions such as jumps, kinks, and any other analytic functionsuch as those commonly used in XFE approximations. Wecall this enhanced recovery extended moving least squaresrecovery (XMLS recovery).

The method proposed in this paper is rooted in recoverybased methods and extends the work of Ref. [50] to parti-tion of unity enriched finite elements. We compute therecovered (smoothed) field and its spatial derivatives usingthe nodal values of the primary dependent variable to con-struct local, enriched, MLS approximations. In a given ele-ment, the error estimate is the L2 norm of the differencebetween the raw (XFEM) strain field and an enhancedstrain field. The enhanced strain field is computed usingthe raw nodal XFEM displacements through extended(enriched) MLS derivative recovery (XMLS), where theweight function is computed through the diffraction crite-rion, and the basis incorporates near-tip fields spanningthe linear elastic fracture mechanics solution space.

The main improvements from the work of [50] lie in thefact that (i) this work is suited to enriched FEM approxi-mations (ii) we introduce the Westergaard near-tip solutioncompletely up to first order (iii) we present three-dimen-sional examples. The crack may be arbitrarily oriented withrespect to the mesh, because we use enrichment for the rawsolution.

This method is general, because it does not require theexistence of superconvergent points, can handle arbitraryenrichment schemes through intrinsic enrichment of theMLS basis and leads to very smooth recovered derivativesthanks to the MLS approximation. It is simple, since thesmoothed displacement (strain) field is constructed locallyby a simple MLS approximation. It is accurate and the effi-ciency index of the obtained estimator is close to unity, asnumerical studies show. Additionally, it outperforms thesuperconvergent patch recovery technique of Ref. [55] forthe fracture mechanics examples studied, which is no sur-prise since the latter estimator yields only a C0 smoothingof the strain. One drawback of the approach is its compu-tational cost, since the enriched MLS basis comprisingseven terms must be employed everywhere in the domain.This could be remedied by only enriching around the cracktip, and transitioning to a standard approximation farfrom the crack.

An outline of the remainder of this paper is as follows.The basics of the extended finite element method and par-tition of unity methods are recalled in Section 2. In Section3, the construction of moving least squares approximantson a cloud of points is shown, the derivative recovery isexplained, and error measures defined. Section 4 solvestwo-dimensional fracture mechanics problems with theextended finite element method, and computes the associ-ated approximation error distribution through extendedmoving least squares recovery. First, a mode I problem isstudied. Structured meshes are first considered, for which

Fig. 1. Selection of enriched nodes. Circled nodes (set Ncr) are enriched by the step function whereas the squared nodes (set Ntip) are enriched by thecrack tip functions. (a) Structured mesh of quadrangles. (b) Unstructured mesh of triangles.

S. Bordas, M. Duflot / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3381–3399 3383

the influence of the XFEM and MLS enrichment radii arestudied. The adequacy of the proposed error estimator isalso compared to one of the standard superconvergentpatch recovery techniques of Zienkiewicz and Zhu. Then,the performance of the estimator is studied on unstructuredmeshes to show the versatility of the method. A mixed-mode crack problem is studied next, for both structuredand unstructured discretizations. Section 5 studies the per-formance of our method on three-dimensional crack prob-lems in mode I and mixed mode. Finally, in Section 6,conclusions are drawn from the work, and extensionsproposed.

1 As opposed to topological.2 For instance, quadratic shape functions NIðxÞ but linear shape

functions eN J ðxÞ were used for six-noded triangular elements [42].

2. Extended finite element method

The extended finite element method introduced by Bely-tschko and Black [10] and improved in Ref. [35] relies onpartition of unity enrichment [7,33]. Other partition ofunity enriched finite element methods are the generalizedfinite element method (GFEM) [43,44] and Nitsche’smethod [28], although the latter’s kinematics is identicalto that of the XFEM [5]. The key idea is to locally add arbi-trary special functions to the finite element basis. Thesefunctions can make the approximated field or its derivativediscontinuous inside an element, thereby releasing therequirement for the mesh to conform to discontinuitiesand making the method quite suitable to treat fracturemechanics problems. Additionally, the functions addedcan help capture a known local character in the soughtsolution. For instance, near-tip crack tip fields of linearelastic fracture mechanics can be added to the approxima-tion basis. This gives the method the ability to achieve goodaccuracy with relatively coarse meshes. Note that partitionof unity has also been applied in a so-called meshfree con-text, through extrinsic enrichment [25,52], intrinsic enrich-ment [49] or weight function enrichment [22]. One of themost recent examples for fracture of quasi-brittle materials

using meshfree methods with extrinsic enrichment is shownhere: [39,18].

Much work has been done since the seminal work ofProfessor Belytschko’s team on the XFEM. Particularly,the notion of geometrical1 enrichment was introduced[29,9], in which the enriched region becomes of fixed mea-sure upon mesh refinement, as opposed to shrinking to zeroas was so in the original version of the method. This nov-elty confers the modified XFEM with optimal convergencerates for linear elastic fracture mechanics. This geometricenrichment method will be used consistently in this paper,except otherwise stated.

Define N as the set of all nodes in the domain. Nodeswhose support is bisected by a crack form set Ncr. Fortopological enrichment (Fig. 1), nodes whose support con-tains the tips are grouped in set Ntip. For geometric enrich-ment (Fig. 2), set Ntip contains all nodes in a disk centeredon the crack tip, with a radius fixed in advance, but prob-lem dependent.

The finite element shape functions are noted N IðxÞ andeN J ðxÞ and uI , aJ and baK are the displacement and enrich-ment nodal variables, respectively. The partition of unityassociated with the enrichment, eN J ðxÞ may differ fromthe N IðxÞ used for the standard part of the displacementapproximation.2 In this paper, both sets of shape functionsare identical and are the standard shape functions of theunderlying finite element approximation: three-noded tri-angles in the 2D examples, four-noded tetrahedra in the3D examples.

Let H(x) be the modified Heaviside function which takesthe value +1 above the crack and �1 below the crack.Finally, note Ba(x) a basis that spans the near-tip asymp-totic fields:

Fig. 2. Another scheme for selection of enriched nodes for 2D crack problem. Circled nodes (set of nodes Ncr) are enriched by the step function whereasthe squared nodes (set of nodes Ntip) are enriched by the crack tip functions. The enrichment radius rXFEM

enr is here equal to R.

Fig. 3. Points added at all intersections of the crack with the XFEM mesh.The union of these points with the XFEM nodes forms set NXMLS,containing all the nodes used to support the XMLS approximation.Adding nodes along the crack improves its representation by thediffraction method and decreases the size of the domains of influence(smoothing length) required for analysis.

3 The documents are accessible online at http://www.civil.gla.ac.uk/~bordas.


B � ½B1;B2;B3;B4�

¼ffiffirp

sinh2;ffiffirp

cosh2;ffiffirp

sinh2

cos h;ffiffirp

cosh2

cos h

� �:

ð1ÞWith these definitions, the XFEM approximation for acrack in linear elastic fracture mechanics reads

uhðxÞ ¼XI2N

N IðxÞuI þX

J2Ncr

eN J ðxÞðHðxÞ � HðxJ ÞÞaJ

þX

K2Ntip

eN KðxÞX4

a¼1

ðBaðxÞ � BaðxKÞÞbaK : ð2Þ

From this enriched approximation (2), the Bubnov–Galer-kin procedure yields the standard discrete system of equa-tions Kd = f. The numerical integration for elements cut by

the discontinuity or containing the crack tip requires subdi-vision, except for very special cases [51]. Interested readerscan refer to [38,16] among other sources for details on theextended finite element method.3

3. Diffracted extended moving least squares recovery

3.1. Moving least squares approximations

Moving least square approximations (MLS) were origi-nally created by mathematicians to fit and construct sur-faces [31]. This fitting method has been used to constructshape functions for meshfree methods [12]. As introducedabove, the idea is to use the raw XFEM displacements tocompute an enhanced strain field. The enhanced strain fieldis taken as the derivative of the MLS smoothing of the rawXFEM displacements. The latter shall be indicated with asuperscript h, the enhanced by a superscript s (for smooth).

Let us denote by NXMLS the set of points composed bythe XFEM nodes and the points on either sides of all inter-sections of the crack with the XFEM mesh. These addi-tional points are shown in Fig. 3. Note that in [50], theauthors use only the FEM nodes as MLS support points;in their method, the mesh then conforms with the crackfaces.

The XFEM displacement approximation writes as inEq. (2) with eN ¼ N . At a node xI, we note the nodalXFEM displacement vector uh

I ¼ uhðxIÞ. The strain fieldis deduced by the strain–displacement relations. We definethe symmetric gradient partial differential operator Dð�Þ

eh ¼ 1

2ð$þ $TÞ � uh ¼ D� uh: ð3Þ

In the following, quantities in square brackets [Æ] andcurly brackets {Æ} are matrix operators and vectors ofnodal values, respectively. Let x be an arbitrary point in



Table 1Some possible basis functions used for MLS approximations

1D 2D 3D

Constant [1] [1] [1]Linear ½1; x� ½1; x; y� ½1; x; y; z�Quadratic ½1; x; x2� ½1; x; y; x2; y2; xy� ½1; x; y; z; x2; y2; z2; xy; xz; yz�Linear + tip – ½1; x; y; ½B1ðr; hÞ;B2ðr; hÞ;B3ðr; hÞ;B4ðr; hÞ�� ½1; x; y; z; ½B1ðr; hÞ;B2ðr; hÞ;B3ðr; hÞ;B4ðr; hÞ��The branch functions ðBiÞ16i64 are given by Eq. (1).

Fig. 4. Diffraction method for introducing discontinuities in moving leastsquares approximations.


X, point x belongs to the domain of influence of nx pointsof set NXMLS defined in Fig. 3. With these notations, thesmoothed displacement and strain fields write

usðxÞ ¼Xnx

I

UIðxÞuhI ¼ ½U�

TðxÞfuhg; ð4Þ

esðxÞ ¼Xnx

I

DðUIÞðxÞuhI ¼ ½D�½U�

TðxÞfuhg; ð5Þ

UI(x) is the value of the MLS shape function associatedwith node I at point x. {uh} and [U] are nx � 1 vectors,½D� is a 3� nx matrix in 2D and a 6� nx matrix in 3D.

The MLS shape functions of the nx points whosedomains of influence contain point x write0

½U1ðxÞ;U2ðxÞ; . . . ;UnxðxÞ� ¼ UTðxÞ¼ pTðxÞA�1ðxÞBðxÞ: ð6Þ

• p is the vector of m basis functions defined in Table 1.Unless otherwise stated, we use the enriched linear basisgiven in the last row of the table. The approximationwrites uh ¼

Pmj¼1ajpj � pTa.

• A is an m� m matrix that needs to be inverted at allpoints where the MLS functions are required:AðxÞ ¼

Pnx

I¼1wIðxÞpðxIÞpTðxIÞ. A is invertible if nx > m

i.e., there are more points whose domains of influencecontain x than basis functions in p, but this is not a suf-ficient condition.

• B is an m� nx matrix defined by BðxÞ ¼ ½w1ðxÞpðx1Þ;w2ðxÞpðx2Þ; . . . ;wnx

ðxÞpðxnxÞ�.

• wI is the weight function associated with point xI. Toaccount for the presence of the discontinuity, the diffrac-tion criterion is used to construct the weight function, asexplained below.

For each supporting point xI, a weight functionwI : X! R is defined. In this work, the fourth order splineweight function is chosen. In 1D, if s is the normalized dis-tance (s P 0) between node xI and point x in the computa-tional domain:

s ¼ x� xI

dI

�� ; ð7Þ

where dI is the smoothing length of the support domain ofnode xI, also known as domain of influence. The quarticspline function writes

f4ðsÞ ¼1� 6s2 þ 8s3 � 3s4 if jsj 6 1;

0 if jsj > 1:

�ð8Þ

In the 2D (3D) examples, we use circular (spherical) do-mains of influence, such that

wIðxÞ ¼ f4

kx� xIkdI

� �: ð9Þ

To account for the presence of the crack, the computa-tion of s is altered following the diffraction method intro-duced in [11,37] to represent discontinuities in the contextof the element free Galerkin method. The basic idea ofthe diffraction technique is depicted in Fig. 4. The standardweight function (discontinuities not present) associatedwith a node evaluated at a point increases when this pointbecomes closer to the node. When discontinuities exist, thedistance calculation is modified to account for them. If acrack hides a point, the weight of a node at this point isdiminished. When the line segment [IX] bissected by acrack, the normalized distance of Eq. (7) in the definitionof the weight function is replaced by the normalized lengthof the shortest path from I to B passing by a point of thefront (path IXB in Fig. 4):

s ¼ kx� xck þ kxc � xIkdI

: ð10Þ

The weight function of node I is then continuous exceptacross the crack. We choose to use additional nodes oneach side of the crack [22], as shown in Fig. 3. This is byno means mandatory to obtain an accurate representation


of the discontinuity, but reduces the support size (smooth-ing length) required for the calculations, and we make useof this technique in all calculations presented in this paper.

The derivatives of the shape functions may be obtainedby differentiating (6). This process involves the evaluation

Fig. 5. Results for a structured, moderately fine mesh: (a) XFEM mesh. (b) Rarepresented. (d) XMLS recovered strain field. (e) Distribution of ehs

Xqof Eq. (1

of additional matrices and carrying out several matrixproducts. This is why the computation of the derivativesof MLS shape functions is a costly operation. The readeris referred to [12] for details on the computation of the suc-cessive derivatives of MLS shape functions.

w strain field. (c) Displacement of the nodes in NXMLS. Only the nodes are1) on the mesh.

0.01

0.1

0.02 0.04 0.06 0.08 0.1

Stra

in d

iffe

renc

e no

rm

Element size

only tip elementrenr=0.1renr=0.2renr=0.3renr=0.4

Fig. 7. Mode I structured mesh: strain difference convergence with meshrefinement. See Table 3 for the values of the convergence rates. The largerthe enrichment radius, the lower the error and the more monotonic theconvergence. If only the tip element is enriched (geometrical enrichment),the convergence rate is 0.46022, for an enrichment radius of 0.4 it reaches0.91052, close to optimum.

1.1only tip element


For structured meshes, the smoothing length is fixed inadvance to three times the characteristic element size,which ensures that at least seven nodes lie within eachdomain of influence. This is mandatory for matrix A tobe invertible. For unstructured triangular meshes, thesmoothing length is taken to be three times the averagecharacteristic length of the elements sharing the node.For the mode I problem, we also study the influence ofthe smoothing length.

MLS approximations yield very regular fields on thewhole domain. Additionally, the degree of consistency ofthe approximation is arbitrary (p refinement, shown inTable 1) and can be increased straightforwardly. Moreimportantly for the object of this paper, special, non-poly-nomial functions can intrinsically enrich the MLS approx-imation basis, which, similarly to partition of unityenriched methods, makes it possible to closely reproducea given local behaviour (last row of Table 1). The majordifference is that in XFEM, enrichment is local, while inMLS, intrinsic enrichment must be performed on the wholedomain, even if the region where it is actually needed isonly a small subset of this domain. This increases the com-putational cost. To avoid this, it is possible to transitionfrom an enriched to a standard MLS approximation, atthe cost of additional implementation complexity. An alter-native is to enrich the basis extrinsically, leading to parti-tion of unity enriched MLS approximations.

3.2. Measure of the adequacy of the estimator

On element Xq, the approximate error between the rawand the smoothed solution is measured by the norm

ehsXq¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZXq

kehðxÞ � esðxÞk2 dx

s: ð11Þ

The global approximate error is measured by the sum ofthe element-wise errors on the nelt elements of the mesh

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1 -0.5 0 0.5 1

ε yy

y

εh

εs

εexact

Fig. 6. �yyðx ¼ 0Þ near-tip problem, structured mesh.

ehs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnelt

q¼1

ehsXq

vuut 2

: ð12Þ

Define eheXq

as the following error norm between the rawXFEM solution and the exact solution on element Xq.We call this error the exact error:

eheXq¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZXq

kehðxÞ � eexactðxÞk2 dx

s: ð13Þ

0.9

0.95

1

1.05

0.02 0.04 0.06 0.08 0.1

Eff

ectiv

ity in

dex

Element size

renr=0.1renr=0.2renr=0.3renr=0.4

Fig. 8. Mode I problem, structured mesh: effectivity index for variousXFEM enrichment radii. Enriching only the tip element causes anineffective error indicator non-convergent to 1. For coarse meshes, highervalues of the enrichment radius increases the effectivity of the indicator.Upon mesh refinement, whatever the enrichment radius, the effectivitycurves meet in the close vicinity of unity.


The global exact error is

ehe ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnelt

q¼1

ehe2

Xq

vuut : ð14Þ

The global error effectivity index of the estimator isdefined as the ratio of the approximate error to the exacterror

h ¼ ehs

ehe: ð15Þ

A good estimator has an effectivity close to unity, whichmeans that the measured error is close to the exact error. Inother words, the enhanced solution is close to the analyticalsolution.

3.3. Recovery algorithm

The algorithm is particularly simple. The goal is to com-pute a recovered strain field at each XFEM Gauss point in

0.01

0.1

0.02 0.04 0.06 0.08 0.1

Stra

in d

iffe

renc

e no

rm

Element size

rs=3rs=4rs=5

0.9

0.95

1

1.05

1.1

0.02 0.04 0.06 0.08 0.1

Eff

ectiv

ity in

dex

Element size

rs=3rs=4rs=5

Fig. 9. Influence of the MLS radius on convergence of the strain differenceand the effectivity index. The convergence curve is unaltered whenincreasing the XMLS recovery radius, while the effectivity index improvesslightly. (a) Convergence of the norm of the difference between the rawXFEM and recovered strain field with mesh refinement. (b) Convergenceof the effectivity index with mesh refinement.

the mesh. Once the nodal values of the XFEM displace-ment field are computed at all points in NXMLS, a loopon XFEM Gauss points in the mesh is performed. TheXMLS shape functions are calculated based on the loca-tions of the points in NXMLS. For each Gauss point, thevalues of the XMLS shape functions derivatives are nowknown. The smoothed strain can be computed using theXFEM displacements of points in NXMLS and the XMLSshape functions derivatives.

4. Two-dimensional numerical examples

In all examples, linear elasticity is assumed and thematerial parameters are Young’s modulus E = 1.0 andPoisson’s ratio m = 0.3. The two-dimensional resultsassume plane strain conditions.

4.1. Introduction

This section shows the performance of the error estima-tor for two problems. The domain is a biunit square (bot-tom left coordinate [�1,1] top right coordinate [1,1]). Thezone enriched with near-tip fields is a disk centered on thecrack tip of radius 0.1, unless otherwise noted. A horizon-tal edge crack of unit length is located along the line y = 0and the crack tip is at (0,0). Two loading conditions areassessed (mode I and mixed mode conditions), both for astructured and unstructured mesh of linear triangular ele-ments. For each mesh type, a convergence study is per-formed, and the effectivity index of the proposed errorestimator is analysed for a mode I problem, where the ana-lytical solution is known. In particular, the influence of theenrichment radius on the effectivity index is evaluated, and,

Table 2Mode I problem, structured mesh: convergence for various XMLSenrichment radii

MLS smoothing length rXMLS Convergence rate

3h 0.812714h 0.811865h 0.81231

h is the characteristic element size around the tip element.

Table 3Mode I problem, convergence for various XFEM enrichment radii

Enrichment radius: rXFEMenr Convergence rate

Only tip element 0.460220.1 0.812710.2 0.867280.3 0.906440.4 0.91052

See also Fig. 7. Close to optimal convergence is attained for a fixed areaenrichment. The results tend to show that the rate of 1. would be reachedif the whole component were enriched.

0.01

0.1

0.1 0.08 0.06 0.04 0.02St

rain

dif

fere

nce

norm

Enriched basisLinear basis

SPR


for fixed enrichment radius, it is verified that the effectivityis close to one.

For each test case, six figures are shown, labeled (a)through (e).

(a) Mesh used for the XFEM calculation.(b) Raw deformation field computed with XFEM: eh.(c) Displacements of the nodes in NXMLS obtained by

XFEM: uhI ¼ uhðxIÞ.

(d) Smoothed deformation obtained by the recoveryalgorithm: es.

(e) The distribution of the error norm ehsXq

of Eq. (11) inthe domain. In a remeshing algorithm, in each ele-ment, this error norm would be compared to a limiterror, and would determine whether an elementshould be enlarged, or shrunk.

In a separate figure, values �yyðx ¼ 0Þ for both the rawXFEM deformation field and the smoothed deformationfield obtained through MLS derivative recovery. The exactstrain field is also plotted, when available.

Element size

2

3

4

5

Eff

ectiv

ity in

dex

Enriched basisLinear basis

SPR

4.2. Edge crack under mode I loading

This section concerns the case of the edge crack in modeI loading subjected to uniform tension applied at infinityand described in Section 4.1. Traction boundary conditionsrepresenting the exact stress state due to the far field areapplied on the boundary, while the bottom left corner isrestrained in both directions and the bottom right cornerin the vertical (y) direction. The MLS basis can representexactly the exact solution to the problem.

0

1

0.1 0.08 0.06 0.04 0.02

4.2.1. Structured mesh

First, the problem is solved on a moderately fine, regularstructured mesh. Two cases are compared to show the

Fig. 10. Mode I structured mesh, case of a linear MLS basis used forrecovery: smooth (recovered) strain field.

necessity for the addition of the near-tip fields to theMLS basis. In the first, the four additional functions asso-ciated with the asymptotic fields are added to the MLSbasis upon computation of the recovered derivative. Inthe second, only a linear MLS basis is used.

Table 4Mode I problem, convergence of the L2 norm of the raw/smooth straindifference, for the linear and enriched (extended) MLS bases, comparison

with SPR

Recovery method Convergence rate

MLS linear 0.478468XMLS linear + near-tip 0.81271SPR 0.53445

Element size

Fig. 11. Mode I problem: linear versus full XMLS recovery basis:convergence of the strain difference and effectivity index with meshrefinement. Notice that the convergence rate is 0.478468 for the linearbasis: a value of 1/2 is expected because of the

ffiffirp

singularity. SPR yields aconvergence rate of 0.53445, which is of order 1/2, as expected. Theenriched basis yields a convergence rate of 0.81271 and the overall error ismuch less than with a linear basis. The deficiency of the linear basis is evenmore striking when looking at the effectivity index, which does notconverge to unity as the mesh size goes to zero. (a) Convergence of thenorm of the difference of the raw strain with the recovered strain. (b)Effectivity index versus mesh size.


With near-tip fields in the MLS basis: The results aregiven in Fig. 5. It is clear from Figs. 5b and 6e that theraw strain field from XFEM is piecewise continuous whilethe recovered strain field, computed from the XFEM nodaldisplacements of Fig. 5c are smooth, and very close to theexact solution, even on this moderately fine mesh.

Fig. 7 shows that the norm of the difference between theraw XFEM strain field and the XMLS recovered strainfield converges to zero with a rate close to the optimal rateof 1.0 as the mesh size tends to zero, as long as a fixed areais enriched around the crack tip during mesh refinement. Ifonly the crack tip element is enriched, the convergence rateis 0.46022, close to 1/2, which is the strength of the cracktip stress singularity. This corroborates the findings ofRefs. [29,9]. Five convergence curves are shown, for fourvalues of the XFEM enrichment radius and one for topo-logical enrichment (only the tip element is enriched). Notethat the higher the enrichment radius, the lower the error,the straighter the convergence line, and the closer to opti-mal the convergence rate is, as summarized in Table 3.

Fig. 8 shows the evolution of the effectivity index withmesh refinement for four values of the XFEM enrichmentradius. It is interesting to notice that the effectivity e versusmesh size h curves become closer to the horizontal linee = 1 with increasing enrichment radius. Additionally, theeffectivity of the error indicator clearly becomes closer tounity as the mesh size goes to zero, as is expected. If onlythe tip element is enriched, the effectivity is not optimal(oscillates about 95%) and does not converge to 1. withmesh refinement. For coarse meshes, large enrichment radiiare necessary to obtain an effective indicator. This isexpected, enrichment takes over the standard FEMapproximation, which in itself is poorly suited to representthe near-tip behaviour. As the mesh is refined, the standardFEM part of the approximation becomes more important,and the enriched part less important, which explains the

Fig. 12. Comparison of the superconvergent patch recovery (SPR) smooth straindicator. (a) SPR, (b) XMLS.

effectivity curves all meeting in the vicinity of 1. for smallh’s.

Influence of the MLS enrichment radius: In this section,the influence of the MLS radius on the error estimate isinvestigated. The mode I problem presented in Section4.2 is revisited with three values of the XMLS recoveryradius of influence. If h is the characteristic tip-element,the MLS smoothing length is chosen as 3h (smallest valuepossible to compute the MLS shape functions, and usedpreviously), 4h and 5h. Fig. 9 shows that the convergencerate of the norm of the difference between the raw XFEMstrain field and the recovered strain field remains the samewhen increasing the recovery radius, as does the overallvalue of this error norm. On the contrary, the effectivityindex of the error estimate becomes closer to 1. It is theopinion of the authors that this relative increase is toosmall to justify the increased computational cost incurredwhen using larger interpolation radii. Table 2 summarizesthe convergence rates obtained for various XMLS enrich-ment radii.

Without near-tip fields in the MLS basis: To motivate theneed for the near-tip fields to be included in the MLS basisused for the derivative recovery, the mode I problem is nowsolved without adding those fields to the MLS basis. Whenexamining Fig. 10 and comparing to Fig. 5e, it is clear thatthe recovered strain field is not as accurate as before. Thedeformation is more spread out, and the correct strainfringes are not reproduced. Additionally, Fig. 11 shows(a) the convergence of the norm of the difference betweenthe raw and the recovered strain field and (b) the conver-gence of the effectivity index as the mesh size tends to zero.In the figures, both the results for the linear and the full(linear plus near-tip enrichment) MLS basis are presented.It is evident that the convergence rate for the linear basis isasymptotically lower and the error is much higher. Simi-larly, the error estimate is not effective, since upon mesh

in field [55] and the proposed extended moving least squares XMLS error

Fig. 13. Results for the mode I problem and a moderately fine unstructured mesh. (a) XFEM mesh. (b) Raw strain field. (c) Displacement of the nodes inNXMLS. Only the nodes are represented. (d) XMLS recovered strain field. (e) Distribution of ehs

Xqof Eq. (11) on the mesh.

4 Which we shall name XMLS.


refinement, its effectivity diverges from the optimal value of1. To conclude, Table 4 compares the convergence rates forthe linear and full enrichment bases.

Comparison of the recovered solution with SPR To fur-ther motivate the need for extended moving least squaresrecovery, this section is dedicated to a comparison of the

proposed recovered solution4 with the now standard SPRof Ref. [55]. Table 4 and Fig. 11 show that SPR convergesat a suboptimal rate of 0.53445, comparable to the rate of


0.478468 attained by MLS recovery without tip-enrich-ment, while XMLS reaches the almost-optimal rate of0.81271, as expected. Fig. 12 compares the smoothed strainresults using SPR (a) and XMLS (b) smoothing. The supe-riority of the XMLS smoothing method is clear.

4.2.2. Unstructured meshNow that the effectivity of the error indicator was pro-

ven on structured mesh, the case of unstructured XFEMmeshes is treated, to show the versatility of the method.In an unstructured mesh, the position of the tip in the tipelement can vary considerably from one discretization toanother, creating additional scatter in the solution.. Themesh is generated using the open source software Gmsh[40]. Results are shown in Fig. 13. Again, the raw strainfield obtained from XFEM suffers from the low degreeapproximation (jumps in Fig. 14), where the unstructurednature of the mesh is revealed in the uneven strain steps.On the other hand, the XMLS recovered strain field is very

0.01

0.1

0.02 0.04 0.06 0.08 0.1

Stra

in d

iffe

renc

e no

rm

Element size

ValuesLeast square linear fit

Fig. 15. Mode I problem unstructured mesh: strain difference L2 norm,convergence slope = 0.76200, linear triangular elements.

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1 -0.5 0 0.5 1

ε yy

y

εh

εs

εexact

Fig. 14. �yyðx ¼ 0Þ, near-tip unstructured mesh.

regular, and follows nicely the analytical solution. Thesmoothness of the recovered field is evident fromFig. 13d. The map shown in Fig. 13e, showing the differ-ence between the raw and recovered strain field, indicatesthe region where the error indicator advises to refine themesh.

Now, it is interesting to note that the XFEM results onan unstructured mesh are poorer than their structuredmesh counterpart. In particular, the convergence is nonmonotonic, as shown in Fig. 15. Remember that theMLS approximation is non-polynomial (even for a linearbasis) and are not exactly integrable using Gauss–Lob-atto–Legendre quadrature. One could wonder if the con-vergence oscillations depicted in Fig. 15 are caused by aninaccurate integration. This is not the case, since the resultsshown are converged down to 10�4 precision; additionalGauss points do not substantially decrease the error. Theauthors speculate that this non-monotonic convergenceemanates from the fact that upon unstructured mesh refine-ment, the position of the tip with respect to the nodes of thetip element varies. It is likely that, for some mesh configu-ration, the tip falls very close to a node, leading to badlyconditioned matrices, and oscillations.

In Fig. 16, one can see that the effectivity index is oscil-latory, but still remains close to unity as the mesh size goesto zero, as in the structured mesh case. However, we noticemore ample oscillations in the effectivity than for the struc-tured mesh. This is certainly due to badly conditionedmatrices caused by the tip being close to one of the tip ele-ment’s nodes as the mesh is unstructurally refined. Theseresults are a good indication that our method performs wellfor unstructured meshes.

4.3. Edge crack under mixed-mode loading

In this mixed-mode loading case, the x and y displace-ments are fixed on the bottom edge of the square plate,

0.9

0.95

1

1.05

1.1

0.02 0.04 0.06 0.08 0.1

Eff

ectiv

ity in

dex

Element size

Fig. 16. Mode I problem unstructured mesh: effectivity index for lineartriangular meshes.


and a horizontal traction tx = 1 is imposed on the top edge.The MLS basis is unable to represent the solution to thisproblem exactly, as opposed to the mode I case treatedpreviously.

Fig. 17. Results for the mixed-mode case and a structured mesh. Notice that bethe lower right corner, leading to a higher predicted error in this zone. (a) XFEMthe nodes are represented. (d) XMLS recovered strain field. (e) Distribution o

4.3.1. Structured mesh

The mixed-mode problem is first examined using a struc-tured mesh. From Fig. 17, several points are to be noticed.Figs. 17b and 18 show even more clearly than in the mode I

cause of the restraints on the bottom edge, strain concentration appears atmesh. (b) Raw strain field. (c) Displacement of the nodes in NXMLS. Only

f ehsXq

of Eq. (11) on the mesh.


case, the irregularity of the raw XFEM strain field. Becauseof the restraints on the lower edge, the bottom right cornerof the domain sees a strain concentration, which leads to ahigher error. The error estimate on the whole domainshown in Fig. 17e shows that the heaviest refinement isto occur in a ball of radius 0.2 centered on the crack tipclose to the crack tip. We also note that a mesh adaptionstep based on this indicator would also advise to refinethe mesh at the bottom corners, which is adequate. Sinceno close form solution is known for this particularmixed-mode problem, Fig. 18 only shows the raw and therecovered strain fields, where, again, the smoothness ofthe recovered field is to be remarked.

It is seen from Fig. 19 that the convergence of the L2

norm of the strain difference with mesh refinement is quiteuniform, with a rate of 0.89677, close to the optimal rate of1. The error committed here is comparable to that reportedin the pure mode I case of the earlier sections.

1

0.1 0.08 0.06 0.04 0.02

Stra

in d

iffe

renc

e no

rm

Element size


Fig. 19. Mixed mode, structured case, strain difference L2 norm:slope = 0.89677.

0

5

10

15

20

-1 -0.5 0 0.5 1

ε yy

y

εh

εs

Fig. 18. �yyðx ¼ 0Þ, mixed mode, structured mesh.

4.3.2. Unstructured mesh

The unstructured mesh calculations are shown in Figs.20 and 22 for a moderately fine mesh. The predicted refine-ment zone is very similar to that of the structured mesh cal-culations. Studying the convergence of the norm of thedifference between the raw XFEM strain and the XMLSrecovered strain shows that the optimal rate is nearlyattained. This corroborates the results found for the modeI case, and indicates that our method is suitable for mixed-mode failure as well as simple mode I loading conditions(see Fig. 21).

5. Three-dimensional cracks

This section presents two three-dimensional crack exam-ples, still in the context of linear elastic fracture mechanics.The domain is ½�1; 1� � ½�1; 1� � ½�0:5; 0:5� (x� y � z).The crack is defined by the equation y ¼ 0; x 6 0, its frontis along the z axis. We show both the raw and enhancedstrain field on the deformed configuration, so that the val-ues on the faces of the crack, and along its front may bebetter identified.

5.1. 3D mode I edge crack

The geometry is defined as in the introduction to Sec-tion 5, and the domain is subjected to uniform uniaxialtension, fixture conditions being added to prevent rigidbody modes. Fig. 24 compares the deformation field fromthe XFEM (a) to the smoothed (recovered) field (b) andshows clearly the superiority of the XMLS recovered solu-tion. Fig. 23 shows the strain eyy along line x = 0. Intui-tively, one would think that using the recovered stress,strain and displacement fields to compute the stress inten-sity factors through domain integral methods woulddecrease the amount of oscillations apparent in previous3D fracture computations with XFEM [36,27,14,46,48,13]. Another alternative, which we did not try in thiswork, would be to compute the stress intensity factorsdirectly from the stress and displacement fields, withoutrecourse to domain integrals. Since the local fields aroundthe crack tip are now more accurate than the raw fields,this method may yield sufficiently accurate results withoutthe burden of domain integral evaluation. The computa-tion of the stress intensity factors may possibly be doneby an evaluation of a contour integral, where the inte-grand would be based on the smooth recovered fields.Indeed, contour integrals are transformed to domain inte-grals mainly for two reasons (i) it is usually easier given afinite element solution to compute integrals over the ele-ment interiors (ii) the strain/stress fields are usually non-smooth through element boundaries, which negativelyaffects the accuracy of contour integration. The first pointis not so true in modern codes where neighbourhood andedge/interior information are readily accessible [24,15].The accuracy difficulty could be lifted by the use of accu-rate recovered fields [54].

Fig. 20. Mixed mode problem, unstructured mesh. (a) XFEM mesh. (b) Raw strain field. (c) Displacement of the nodes in NXMLS. Only the nodes arerepresented. (d) XMLS recovered strain field. (e) Distribution of ehs

Xqof Eq. (11) on the mesh.


5.2. 3D edge crack under combined tension and torsion

loading

The geometry is as defined in the introduction of Section5. On face y = 1, tractions are imposed as follows:

ty ¼ 1;

tx ¼ 4zð1� x2Þ;

tz ¼ 4xð0:25� z2Þ:

ð16Þ

0

5

10

15

20

-1 -0.5 0 0.5 1

ε yy

y

εh

εs

Fig. 21. �yyðx ¼ 0Þ, mixed mode, unstructured mesh.

1

0.1 0.08 0.06 0.04 0.02

Stra

in d

iffe

renc

e no

rm

Element size


Fig. 22. Mixed mode problem, unstructured mesh: strain difference L2

norm, convergence at average rate 0.87611, close to the optimal rate of 1.0.

0

5

10

15

20

-1 -0.5 0 0.5 1

ε yy

y

εh

εs

Fig. 23. 3D edge crack under mode I loading: plot of eyy along line x = 0.

Fig. 24. Deformed configuration and strain field for the mode I crack, note trough solution (a) compared to the recovered fields (b).


On face y = �1, tractions are imposed as follows:

ty ¼ �1;

tx ¼ �4zð1� x2Þ;tz ¼ �4xð0:25� z2Þ:

ð17Þ

Additionally, six nodal displacements are fixed so as toavoid rigid body modes. Fig. 25 compares the deformationfield obtained with the XFEM (a) to the recovered(smoothed) deformation field obtained through MLSderivative recovery. Again, the results are quite satisfying.Fig. 26 shows eyy along line x = 0.

6. Conclusions and extensions

Through applications to two- and three-dimensionalfracture, this paper shows how the diffraction method cou-pled with intrinsically enriched moving least squaresapproximations provides for a simple and accurate deri-vative recovery technique for enriched (extended) finite

hat the linear tetrahedra used for the XFEM approximation yield a very

Fig. 25. Deformed configuration and strain field for the combined tension/torsion loading case of the 3D crack. Here again, the smoothing fulfills its rolenicely. (a) Raw XFEM deformation field. (b) Smoothed (recovered) deformation field.

0

5

10

15

20

-1 -0.5 0 0.5 1

ε yy

y

εh

εs

Fig. 26. 3D edge crack under mixed-mode loading: plot of eyy along linex = 0.


element solutions. In turn, this recovery procedure is usedto compute an error estimate whose effectivity index isstudied. A numerical convergence study proves that it isnecessary to use near-tip enrichment to get optimal conver-gence rates and effectivity indices that are close to unity.Additionally, the proposed estimator outperforms theSPR-based estimator of Ref. [55] for the problems studied.

The next steps to further develop and improve the pro-posed methodology would be, in the authors’ minds, thefollowing:

• Treat more complex cases, with complex geometries andcurved cracks, where the use of level sets may bebeneficial.

• Extend to other enrichment types such as discontinuousderivative enrichments and test the method on multi-material problems.

• Come up with a suitable adaptive strategy for crackproblems, where a combination of h, p and r adaptivity

would be considered. The challenge resides in decidingwhat adaptive step should be carried out, and in whichorder. A coupling with moving meshes is under consid-eration and could be beneficial to accuracy.

• Handle the case of multiple tips, which requires definingthe polar coordinates adequately for the computation ofthe near-tip fields associated with both crack tips.

• Now that an error estimator is available, it is natural tothink about local remeshing. Several remeshing stepscould be imagined at each crack propagation step, inorder to limit the global error. Additionally, it will beinteresting to see the influence of an adapted mesh oncrack path.

• In fracture mechanics, the goal of the computation is theobtention of fracture parameters (J or interaction inte-grals, stress intensity factors). The authors are develop-ing a posteriori goal-oriented error estimates for theextended finite element method in fracture mechanics.

• To improve computational efficiency of the approach, itwould be desirable to limit the enriched area (in theMLS approximation) to regions where enrichment isuseful, as opposed to using an enriched basis for allpoints in the domain. The question then translates intothe management of the transition area between enrichedand non-enriched MLS approximations.

• Another alternative to compute the error estimatorwould be to use XMLS close to the tip, and SPRelsewhere.

• It could also be beneficial to construct the XMLS shapefunctions with weight functions w defined using theXFEM mesh as opposed to circles or spheres. This issimilar in principle to the work of the intrinsic XFEM[26].

• A study on the computation of stress intensity factorswould be interesting. For instance a comparison ofdirect calculation from the raw (smooth) stress fieldsto domain integral calculations using the raw (smooth)


fields, could be helpful in an effort to understand thesource of the oscillations reported in the SIFs computedon XFEM solutions.

Acknowledgments

The support of the Walloon Region and FEDER Euro-pean funds under contract No. EP1A122030000102 for M.Duflot is gratefully acknowledged.

The first author thanks Dr. Thomas Zimmermann andthe whole LSC team for welcoming him at EPFL from2003 to 2006. A special thank to Dr. Raphael Bonnaz forvery helpful discussions on error estimates during theircommon time at EPFL, Lausanne, Switzerland. Mr. Ngu-yen Vinh Phu is thanked for his dedication and contribu-tion to some of the figures, borrowed from his M.Sc.dissertation. The authors are grateful to Dr. Marcus Ruterand Dr. Sergey Korotov for their input on goal-oriented er-ror estimation and to Arnaud Francois from CENAERO,Belgium, for his help with the SPR comparison.

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Derivative recovery and a posteriori error estimate for extended finite elements