Sample Article

On the role of enrichment andstatical admissibility of recovered

fields in a posteriori errorestimation for enriched finite

element methodsOctavio Andres Gonzalez-Estrada, Juan Jose Rodenas,

Stephane Pierre Alain Bordas, Marc Duflot, Pierre Kerfriden andEugenio Giner

(Authors’ affiliations are shown at the end of the article.)

Abstract

Purpose – The purpose of this paper is to assess the effect of the statical admissibility of therecovered solution and the ability of the recovered solution to represent the singular solution; also theaccuracy, local and global effectivity of recovery-based error estimators for enriched finite elementmethods (e.g. the extended finite element method, XFEM).

Design/methodology/approach – The authors study the performance of two recovery techniques.The first is a recently developed superconvergent patch recovery procedure with equilibration andenrichment (SPR-CX). The second is known as the extended moving least squares recovery (XMLS),which enriches the recovered solutions but does not enforce equilibrium constraints. Both are extendedrecovery techniques as the polynomial basis used in the recovery process is enriched with singularterms for a better description of the singular nature of the solution.

Findings – Numerical results comparing the convergence and the effectivity index of both techniqueswith those obtained without the enrichment enhancement clearly show the need for the use of extendedrecovery techniques in Zienkiewicz-Zhu type error estimators for this class of problems. The results alsoreveal significant improvements in the effectivities yielded by statically admissible recovered solutions.

Originality/value – The paper shows that both extended recovery procedures and staticaladmissibility are key to an accurate assessment of the quality of enriched finite element approximations.

Keywords Finite element analysis, Error analysis, Extended finite element method, Error estimation,Linear elastic fracture mechanics, Statical admissibility, Extended recovery

Paper type Research paper

1. IntroductionEngineering structures, in particular in aerospace engineering, are intended to operatewith flawless components, especially for safety critical parts. However, there is alwaysa possibility that cracking will occur during operation, risking catastrophic failure andassociated casualties.

The mission of damage tolerance assessment (DTA) is to assess the influence of thesedefects, cracks and damage on the ability of a structure to perform safely and reliablyduring its service life. An important goal of DTA is to estimate the fatigue life of astructure, i.e. the time during which it remains safe given a pre-existing flaw.

DTA relies on the ability to accurately predict crack paths and growth rates incomplex structures. Since the simulation of three-dimensional crack growth is either

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814

Received 18 August 2011Revised 5 December 2011Accepted 8 December 2011

Engineering Computations:International Journal forComputer-Aided Engineering andSoftwareVol. 29 No. 8, 2012pp. 814-841q Emerald Group Publishing Limited0264-4401DOI 10.1108/02644401211271609

not supported by commercial software, or requires significant effort and time for theanalysts and is generally not coupled to robust error indicators, reliable,quality-controlled, industrial DTA is still a major challenge in engineering practice.

The extended finite element method (XFEM) (Moes et al., 1999) is now one of manysuccessful numerical methods to solve fracture mechanics problems. The particularadvantage of the XFEM, which relies on the partition of unity (PU) property (Melenk andBabuska, 1996) of finite element shape functions, is its ability to model cracks without themesh conforming to their geometry. This allows the crack to split the background mesharbitrarily, thereby leading to significantly increased freedom in simulating crack growth.

This feat is achieved by adding new degrees of freedom (dof) to:. Describe the discontinuity of the displacement field across the crack faces, within

a given element.. Reproduce the asymptotic fields around the crack tip. Thanks to the advances

made in the XFEM during the last years, the method is now considered to be arobust and accurate means of analysing fracture problems, has beenimplemented in commercial codes (ABAQUS, 2009; CENAERO, 2011) and isindustrially in use for DTA of complex three dimensional structures (Bordas andMoran, 2006; Wyart et al., 2007; Duflot, 2007).

While XFEM allows to model cracks without (re)meshing the crack faces as theyevolve and yields exceptionally accurate stress intensity factors (SIF) for 2Dproblems (Bordas and Moran, 2006; Wyart et al., 2007; Duflot, 2007) show that forrealistic 3D structures, a “veryfine” mesh is required to accurately capture the complexthree-dimensional stress field and obtain satisfactory SIFs, the main drivers of linearelastic crack propagation.

Consequently, based on the mesh used for the stress analysis, a new mesh offeringsufficient refinement throughout the whole potential path of the crack must beconstructed a priori, i.e. before the crack path is known. Practically, this is done byrunning preliminary analyses on coarse meshes to obtain an approximative crack pathand heuristically refining the mesh around this path to increase accuracy.

Typically, this refinement does not rely on sound error measures, thus theheuristically chosen mesh is in general inadequately suited and can cause largeinaccuracies in the crack growth path, especially around holes, which can lead tonon-conservative estimates of the safe life of the structure.

Thus, it is clear that although XFEM simplifies the treatment of cracks compared tothe standard FEM by lifting the burden of a geometry conforming mesh, it stillrequires iterations and associated user intervention and it employs heuristics which aredetrimental to the robustness and accuracy of the simulation process.

It would be desirable to minimise the changes to the mesh topology, thus userintervention, while ensuring the stress fields and SIFs are accurately computed at eachcrack growth step. This paper is one more step in attempting to control the discretizationerror committed by enriched FE approximations, decrease human intervention in DTAof complex industrial structures and enhance confidence in the results by providingenriched FEMs with sound error estimators which will guarantee a predeterminedaccuracy level and suppress recourse to manual iterations and heuristics.

The error assessment procedures used in finite element analysis are well knownand can be usually classified into different families (Ainsworth and Oden, 2000):

Admissibility ofrecovered fields

815

residual based error estimators, rcovery based error estimators, dual techniques, etc.Residual based error estimators were substantially improved with the introduction ofthe residual equilibration by Ainsworth and Oden (2000). These error estimators havea strong mathematical basis and have been frequently used to obtain lower and upperbounds of the error (Ainsworth and Oden, 2000; Dıez et al., 2004). Recovery based errorestimates were first introduced by Zienkiewicz and Zhu (1987) and are often preferredby practitioners because they are robust and simple to use and provide an enhancedsolution. Further improvements were made with the introduction of new recoveryprocesses such as the superconvergent patch recovery (SPR) technique proposed byZienkiewicz and Zhu (1992a, b). Dual techniques based on the evaluation of twodifferent fields, one compatible in displacements and another equilibrated in stresses,have also been used by Pereira et al. (1999) to obtain bounds of the error. Herein we aregoing to focus on recovery based techniques which follow the ideas of theZienkiewicz-Zhu (ZZ) error estimator proposed by Zienkiewicz and Zhu (1987).

The literature on error estimation techniques for mesh based PU methods is stillscarce. One of the first steps in that direction was made in the context of the generalizedfinite element method (GFEM) by Strouboulis et al. (2001). The authors proposed arecovery-based error estimator which provides good results for h-adapted meshes.In a later work two new a posteriori error estimators for GFEM were presented(Strouboulis et al., 2006). The first one was based on patch residual indicators andprovided an accurate theoretical upper bound estimate, but its computed versionseverely underestimated the exact error. The second one was an error estimator based ona recovered displacement field and its performance was closely related to the quality ofthe GFEM solution. This recovery technique constructs a recovered solution on patchesusing a basis enriched with handbook functions.

In order to obtain a recovered stress field that improves the accuracy of the stressesobtained by XFEM (Xiao and Karihaloo, 2006) proposed a moving least squares (MLS)fitting adapted to the XFEM framework which considers the use of staticallyadmissible basis functions. Nevertheless, the recovered stress field was not used toobtain an error indicator.

Pannachet et al. (2009) worked on error estimation for mesh adaptivity in XFEM forcohesive crack problems. Two error estimates were used, one based on the error in theenergy norm and another that considers the error in a local quantity of interest. Theerror estimation was based on solving a series of local problems with prescribedhomogeneous boundary conditions.

Panetier et al. (2010) presented an extension to enriched approximations of theconstitutive relation error (CRE) technique already available to evaluate error boundsin FEM (Ladeveze and Pelle, 2005). This procedure has been used to obtain local errorbounds on quantities of interest for XFEM problems.

Bordas and Duflot (2007) and Bordas et al. (2008) proposed a recovery based errorestimator for 2D and 3D XFEM approximations for cracks known as the extendedmoving least squares (XMLS). This method intrinsically enriched an MLS formulation toinclude information about the singular fields near the crack tip. Additionally, it used adiffraction method to introduce the discontinuity in the recovered field. This errorestimator provided accurate results with effectivity indices close to unity (optimal value)for 2D and 3D fracture mechanics problems. Later, Duflot and Bordas (2008) proposeda global derivative recovery formulation extended to XFEM problems. The recovered

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solution was sought in a space spanned by the near tip strain fields obtained fromdifferentiating the Westergaard asymptotic expansion. Although the results provided bythis technique were not as accurate as those in Bordas and Duflot (2007) and Bordas et al.(2008), they were deemed by the authors to require less computational power.

Rodenas et al. (2008) presented a modification of the SPR technique tailored to theXFEM framework called SPRXFEM. This technique was based on three key ingredients:

(1) the use of a singular þ smooth stress field decomposition procedure around thecrack tip similar to that described by Rodenas et al. (2006) for FEM;

(2) direct calculation of recovered stresses at integration points using the PU; and

(3) use of different stress interpolation polynomials at each side of the crack when acrack intersects a patch.

In order to obtain an equilibrated smooth recovered stress field, a simplified version ofthe SPR-C technique presented by Rodenas et al. (2007) was used. This simplifiedSPR-C imposed the fulfilment of the boundary equilibrium equation at boundary nodesbut did not impose the satisfaction of the internal equilibrium equations.

The numerical results presented by Bordas and Duflot (2007), Bordas et al. (2008) andRodenas et al. (2008) showed promising accuracy for both the XMLS and the SPR XFEM

techniques. It is apparent in those papers that SPR XFEM led to effectivity indicesremarkably close to unity. Yet, these papers do not shed any significant light onthe respective roles played by two important ingredients in those error estimators,namely:

(1) the statical admissibility of the recovered solution; and

(2) the enrichment of the recovered solution.

Moreover, the differences in the test cases analysed and in the quality measuresconsidered in each of the papers makes it difficult to objectively compare the merits ofboth methods.

The aim of this paper is to assess the role of statical admissibility and enrichment ofthe recovered solution in recovery based error estimation of enriched finite elementapproximation for linear elastic fracture. To do so, we perform a systematic study ofthe results obtained when considering the different features of two error estimationtechniques: XMLS and SPR-CX. SPR-CX is an enhanced version of the SPR XFEM

presented by Rodenas et al. (2008) for 2D problems. Advantages and disadvantages ofeach method are also provided.

The outline of the paper is as follows. In Section 2, the XFEM is briefly presented.Section 3 deals with error estimation and quality assessment of the solution.In Sections 4 and 5 we introduce the error estimators used in XFEM approximations:the SPR-CX and the XMLS, respectively. Some numerical examples analysing bothtechniques and the effect of the enrichment functions in the recovery process arepresented in Section 6. Finally some concluding remarks are provided in Section 7.

2. Reference problem and XFEM solutionLet us consider a 2D linear elastic fracture mechanics (LEFM) problem on a boundeddomain V , R2. The unknown displacement field u is the solution of the boundaryvalue problem:


817

7 ·s ðuÞ þ b ¼ 0 in V ð1Þ

s ðuÞ ·n ¼ t on GN ð2Þ

s ðuÞ ·n ¼ 0 on GC ð3Þ

u ¼ u on GD ð4Þ

whereGN andGD are the Neumann and Dirichlet boundaries andGC represents the crackfaces, with ›V ¼ GN < GD < GC and GN > GD > GC ¼ B. b are the body forces perunit volume, t the tractions applied on GN (being n the normal vector to the boundary)and u the prescribed displacements on GD. The weak form of the problem reads: findu [ V such that:

;v [ V aðu; vÞ ¼ lðvÞ; ð5Þ

where V is the standard test space for elasticity problems such that:

V ¼ vjv [ H 1ðVÞ; vjGDðxÞ ¼ 0; v discontinuous on GC;

and:

aðu; vÞ :¼

ZV

s ðuÞ : 1ðvÞdV ¼

ZV

s ðuÞ : S : s ðvÞdV ð6Þ

lðvÞ :¼

ZV

b · vdVþ

ZGN

t · vdG; ð7Þ

where S is the compliance tensor, s and 1 represent the stress and strain operators.LEFM problems are denoted by the singularity present at the crack tip. The

following expressions represent the first term of the asymptotic expansion whichdescribes the displacements and stresses for combined loading modes I and II in 2D.These expressions can be found in the literature (Szabo and Babuska, 1991;Rodenas et al., 2008) and are reproduced here for completeness:

u1ðr;fÞ ¼KI

2m

ffiffiffiffiffiffir

2p

rcos

f

2ðk2 cosfÞ þ

KII

2m

ffiffiffiffiffiffir

2p

rsin

f

2ð2 þ kþ cosfÞ

u2ðr;fÞ ¼KI

2m

ffiffiffiffiffiffir

2p

rsin

f

2ðk2 cosfÞ þ

KII

2m

ffiffiffiffiffiffir

2p

rcos

f

2ð2 2 k2 cosfÞ

ð8Þ

s 11ðr;fÞ ¼KIffiffiffiffiffiffiffiffi2pr

p cosf

21 2 sin

f

2sin

3f

2

2

KIIffiffiffiffiffiffiffiffi2pr

p sinf

22 þ cos

f

2cos

3f

2


p cosf

21 þ sin

f

2sin

3f

2

þ

KIIffiffiffiffiffiffiffiffi2pr

p sinf

2cos

f

2cos

3f

2


p sinf

2cos

f

2cos

3f

2þ

K IIffiffiffiffiffiffiffiffi2pr

p cosf

21 2 sin

f

2sin

3f

2

ð9Þ

where r and f are the crack tip polar coordinates, KI and KII are the SIFs for modes Iand II, m is the shear modulus, and k the Kolosov’s constant, defined in terms of

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theparameters of material E (Young’s modulus) and y (Poisson’s ratio), according tothe expressions:

m ¼E

2ð1 þ y Þ; k ¼

3 2 4y plane strain

32y1þy

plane stress

8<:

This type of problem is difficult to model using a standard FEM approximation as themesh needs to explicitly conform to the crack geometry. With the XFEM thediscontinuity of the displacement field along the crack faces is introduced by addingdof to the nodes of the elements intersected by the crack. This tackles the problem ofadjusting the mesh to the geometry of the crack (Moes et al., 1999; Stolarska et al.,2001). Additionally, to describe the solution around the crack tip, the numerical modelintroduces a basis that spans the near tip asymptotic fields. The following expressionis generally used to interpolate the displacements at a point of coordinates xaccounting for the presence of a crack tip in a 2D XFEM approximation:

uhðxÞ ¼i[I

XNiðxÞai þ

j[J

XNjðxÞH ðxÞbj þ

m[M

XNmðxÞ

X4

I¼1

FI ðxÞcIm

!ð10Þ

where Ni are the shape functions associated with node i, ai represent the conventionalnodal dof, bj are the coefficients associated with the discontinuous enrichmentfunctions, and cm those associated with the functions spanning the asymptotic field. Inthe above equation, I is the set of all the nodes in the mesh, M is the subset of nodesenriched with crack tip functions, and J is the subset of nodes enriched with thediscontinuous enrichment (Figure 1). In equation (10), the Heaviside function H, withunitary modulus and a change of sign on the crack face, describes the displacementdiscontinuity if the finite element is intersected by the crack. The FI are the set ofbranch functions used to represent the asymptotic expansion of the displacement fieldaround the crack tip seen in equation (8). For the 2D case, the following functions areused (Belytschko and Black, 1999):

Figure 1.Classification ofnodes in XFEM

Fl enrichment

H enrichment

H+Fl enrichment

x1

y1

r

re

φ

crack

Note: Fixed enrichment area of radius re


819

FI ðr;fÞf g ;ffiffiffir

psin

f

2; cos

f

2; sin

f

2sinf; cos

f

2sinf

ð11Þ

The main features of the XFEM implementation considered to evaluate the numericalresults is described in detail in Rodenas et al. (2008) and can be summarized as follows:

. Use of bilinear quadrilaterals.

. Decomposition of elements intersected by the crack into integration subdomainsthat do not contain the crack. Alternatives which do not required this subdivisionare proposed by Ventura (2006) and Natarajan et al. (2010).

. Use of a quasi-polar integration with a 5 £ 5 quadrature rule in triangularsubdomains for elements containing the crack tip.

. No correction for blending elements. Methods to address blending errorsare proposed by Chessa et al. (2003), Gracie et al. (2008), Fries (2008) andTarancon et al. (2009).

2.1 Evaluation of SIFThe SIFs in LEFM represent the amplitude of the singular stress fields and are keyquantities of interest to simulate crack growth in LEFM. Several post - processingmethods, following local or global (energy) approaches, are commonly used to extractSIFs (Banks-Sills, 1991) or to calculate the energy release rate G. Energy or globalmethods are considered to be the most accurate and efficient methods (Banks-Sills, 1991;Li et al., 1985). Global methods based on the equivalent domain integral (EDI methods)are specially well-suited for FEM and XFEM analyses as they are easy to implement andcan use information far from the singularity. In this paper, the interaction integral asdescribed in Shih and Asaro (1988) and Yau et al. (1980) has been used to extractthe SIFs.This technique provides KI and KII for problems under mixed mode loading conditionsusing auxiliary fields. Details on the implementation of the interaction integral can befound for example in Moes et al. (1999), Rodenas et al. (2008) and Giner et al. (2005).

3. Error estimation in the energy normThe approximate nature of the FEM and XFEM approximations implies a discretizationerror which can be quantified using the error in the energy norm for the solutionkek ¼ ku2 uhk. To obtain an estimate of the discretization error keesk, in the context ofelasticity problems solved using the FEM, the expression for the ZZ estimator is definedas (Zienkiewicz and Zhu, 1987) (in matrix form):

kek2 < keesk

2¼

ZV

ðs * 2 s hÞTD21ðs * 2 s hÞdV ð12Þ

or alternatively for the strains:

keesk2¼

ZV

ð1 * 2 1hÞTDð1 * 2 1hÞdV;

where the domain V refers to the full domain of the problem or a local subdomain(element),s h represents the stress field provided by the FEM,s * is the recovered stressfield, which is a better approximation to the exact solution than s h and D is theelasticity matrix of the constitutive relation s ¼ D1.

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820

The recovered stress field s * is usually interpolated in each element using theshape functions N of the underlying FE approximation and the values of the recoveredstress field calculated at the nodes s

*:

s *ðxÞ ¼Xne

i¼1

NiðxÞs*

i ðxiÞ; ð13Þ

where ne is the number of nodes in the element under consideration and s*

i ðxiÞ are thestresses provided by the least squares technique at node i. The components of s

*

i areobtained using a polynomial expansion, si;j ¼ pa (with j ¼ xx; yy; xy), defined over aset of contiguous elements connected to node i called patch, where p is the polynomialbasis and a are the unknown coefficients.

The ZZ error estimator is asymptotically exact if the recovered solution used in theerror estimation converges at a higher rate than the finite element solution (Zienkiewiczand Zhu, 1992b).

In this paper, we are interested in the role played by statical admissibility andenrichment of the recovered solution in estimating the error committed by XFEM. Todo so, we study the performance of two recovery-based error estimators, which exhibitdifferent features, that have been recently developed for XFEM:

(1) the SPR-CX derived from the error estimator developed by Rodenas et al. (2008)and summarized in Section 4 (two other versions of the SPR-CX technique havebeen also considered); and

(2) the XMLS proposed by Bordas and Duflot (2007) and summarized in Section 5.

Both estimators provide a recovered stress field in order to evaluate the estimated errorin the energy norm by means of the expression shown in equation (12).

4. SPR-CX error estimatorThe SPR-CX error estimator is an enhancement of the error estimator first introducedby Rodenas et al. (2008), which incorporates the ideas proposed in Rodenas et al. (2007)to guarantee the exact satisfaction of the equilibrium locally on patches. InRodenas et al. (2008) a set of key ideas are proposed to modify the standard SPR byZienkiewicz and Zhu (1992a), allowing its use for singular problems.

The recovered stresses s * are directly evaluated at an integration point x through theuse of a PU procedure, properly weighting the stress interpolation polynomials obtainedfrom the different patches formed at the vertex nodes of the element containing x:

s *ðxÞ ¼Xnv

i¼1

NiðxÞs*i ðxÞ; ð14Þ

where Ni are the shape functions associated to the vertex nodes nv.One major modification is the introduction of a splitting procedure to perform the

recovery. For singular problems the exact stress field s is decomposed into two stressfields, a smooth field s smo and a singular field s sing :

s ¼ s smo þ s sing: ð15Þ

Then, the recovered field s * required to compute the error estimate given inequation (12) can be expressed as the contribution of two recovered stress fields, onesmooth s*

smo and one singular s*sing :


821

s * ¼ s*smo þ s*

sing: ð16Þ

For the recovery of the singular part, the expressions in equation (9) which describe theasymptotic fields near the crack tip are used. To evaluate s*

sing from equation (9) wefirst obtain estimated values of the SIFs K I and KII using the interaction integral asindicated in Section 1. The recovered part s*

sing is an equilibrated field as it satisfies theinternal equilibrium equations.

Once the field s*sing has been evaluated, an FE approximation to the smooth part

s hsmo can be obtained subtracting s*

sing from the raw FE field:

s hsmo ¼ s h 2 s*

sing: ð17Þ

Then, the field s*smo is evaluated applying an SPR-C recovery procedure over the field

s hsmo.For patches intersected by the crack, the recovery technique uses different stress

interpolation polynomials on each side of the crack. This way it can represent thediscontinuity of the recovered stress field along the crack faces, which is not the casefor SPR that smoothes out the discontinuity (Bordas and Duflot, 2007).

In order to obtain an equilibrated recovered stress field s*smo, the SPR-CX enforces

the fulfilment of the equilibrium equations locally on each patch. The constraintequations are introduced via Lagrange multipliers into the linear system used to solvefor the coefficients of the polynomial expansion of the recovered stresses on each patch.These include the satisfaction of the:

. Internal equilibrium equations.

. Boundary equilibrium equation. A point collocation approach is used to imposethe satisfaction of a second order approximation to the tractions along theNeumann boundary.

. Compatibility equation. This additional constraint is also imposed to furtherincrease the accuracy of the recovered stress field.

To evaluate the recovered field, quadratic polynomials have been used in the patchesalong the boundary and crack faces, and linear polynomials for the remaining patches.As more information about the solution is available along the boundary, polynomialsone degree higher are useful to improve the quality of the recovered stress field.

The enforcement of equilibrium equations provides an equilibrated recovered stressfield locally on patches. However, the process used to obtain a continuous fields * shownin equation (14) introduces a small lack of equilibrium as explained in Rodenas et al.(2010a). The reader is referred to Rodenas et al. (2010a) and Dıez et al. (2007) for details.

5. XMLS error estimatorIn the XMLS the solution is recovered through the use of the MLS technique, developedby mathematicians to build and fit surfaces. The XMLS technique extends the work ofTabbara et al. (1994) for FEM to enriched approximations. The general idea of theXMLS is to use the displacement solution provided by XFEM to obtain a recoveredstrain field (Bordas and Duflot, 2007; Bordas et al., 2008). The smoothed strains arerecovered from the derivative of the MLS-smoothed XFEM displacement field:

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822

u*ðxÞ ¼i[N x

XciðxÞu

hi ð18Þ

1 *ðxÞ ¼i[N x

X7sðciðxÞu

hi Þ; ð19Þ

where the uhi are the raw nodal XFEM displacements and ciðxÞ are the MLS shape

function values associated with a node i at a point x. N x is the set of nx nodes inthe domain of influence of point x, 7s is the symmetric gradient operator, u* and 1* arethe recovered displacement and strain fields, respectively. At each point xi the MLSshape functions ci are evaluated using weighting functionsvi and an enriched basispðx iÞ. The total nx non-zero MLS shape functions at point x are evaluated as:

ðciðxÞÞ1#i#nx¼ ðA21ðxÞpðxÞÞTpðxiÞviðxÞ ð20Þ

where:

AðxÞ ¼Xnx

i¼1

viðxÞpðxiÞpðxiÞT

is a matrix to be inverted at every point x (Bordas and Duflot, 2007; Bordas et al., 2008)for further details).

For each supporting point xi , the weighting function vi is defined such that:

viðsÞ ¼ f 4ðsÞ ¼1 2 6s 2 þ 8s 3 2 3s 4 ifjsj # 1

0 ifjsj . 1

(ð21Þ

where s is the normalized distance between the supporting point xi and a point x in thecomputational domain. In order to describe the discontinuity, the distance s in theweight function defined for each supporting point is modified using the “diffractioncriterion” (Belytschko et al., 1996). The basic idea of this criterion is shown in Figure 2.The weight function is continuous except across the crack faces since the points at theother side of the crack are not considered as part of the support. Near the crack tip theweight of a node i over a point of coordinates x diminishes as the crack hides the point.When the point x is hidden by the crack the following expression is used:

s ¼kx2 xCk þ kxC 2 x ik

di

ð22Þ

where di is the radius of the support.

Figure 2.Diffraction criteria

to introduce thediscontinuity in the

XMLS approximation

xi

xC

x


823

The MLS shape functions can reproduce any function in their basis. The basis p usedis a linear basis enriched with the functions that describe the first order asymptoticexpansion at the crack tip as indicated in equation (11):

p ¼ ½1; x; y; ½F1ðr;fÞ;F2ðr;fÞ;F3ðr;fÞ;F4ðr;fÞ ð23Þ

Note that although the enriched basis can reproduce the singular behaviour of thesolution around the crack tip, the resulting recovered field not necessarily satisfies theequilibrium equations.

6. Numerical resultsIn this section, numerical experiments are performed to verify the behaviour of bothXFEM recovery based error estimators considered in this paper.

Babuska et al. (1994a, b, 1997) proposed a robustness patch test for qualityassessment of error estimators. However, the use of this test is not within the scope ofthis paper and furthermore, to the authors’ knowledge, it has not been used in thecontext of XFEM. Therefore, the more traditional approach of using benchmarkproblems is considered here to analyse the response of the different error estimators.

The accuracy of the error estimators is evaluated both locally and globally. Thisevaluation has been based on the effectivity of the error in the energy norm, which isquantified using the effectivity index u defined as:

u ¼keesk

kek: ð24Þ

When the enhanced or recovered solution is close to the analytical solution theeffectivity approaches the theoretical value of 1, which indicates that it is a good errorestimator, i.e. the approximate error is close to the exact error.

To assess the quality of the estimator at a local level, the local effectivity D, inspiredon the robustness index found in Babuska et al. (1994a), is used. For each element e,D represents the variation of the effectivity index in this element, u e, with respect to thetheoretical value (the error estimator can be considered to be of good quality if it yieldsD values close to zero). D is defined according to the following expression, wheresuperscripts e indicate the element e:

D ¼ u e 2 1 if u e $ 1

D ¼ 1 2 1u e if u e , 1

with u e ¼jee

es

jjeej jj

: ð25Þ

To evaluate the overall quality of the error estimator we use the global effectivityindex u, the mean value mðjDjÞ and the standard deviation s ðDÞ of the local effectivity.A good quality error estimator yields values of u close to one and values of mðjDjÞ ands ðDÞ close to zero.

The techniques can be used in practical applications. However, in order to properlycompare their performance we have used an academic problem with exact solution. Inthe analysis we solve the Westergaard problem (Gdoutos, 1993) as it is one of the fewproblems in LEFM under mixed mode with an analytical solution. In Giner et al. (2005)and Rodenas et al. (2008) we can find explicit expressions for the stress fields interms of the spatial coordinates. In the next subsection we show a description of the

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Westergaard problem and XFEM model, taken from Rodenas et al. (2008) andreproduced here for completeness.

6.1 Westergaard problem and XFEM modelThe Westergaard problem corresponds to an infinite plate loaded at infinity withbiaxial tractions s x1 ¼ s y1 ¼ s1 and shear traction t1, presenting a crack of length2a as shown in Figure 3. Combining the externally applied loads we can obtaindifferent loading conditions: pure modes I, II or mixed mode.

In the numerical model only a finite portion of the domain (a ¼ 1 and b ¼ 4 inFigure 3) is considered. The projection of the stress distribution corresponding to theanalytical Westergaard solution for modes I and II, given by the expressions below, isapplied to its boundary:

s Ixðx; yÞ ¼

s1ffiffiffiffiffitj j

p xcosf

22 ysin

f

2

þ y

a2

tj j2

msinf

22 ncos

f

2

s Iyðx; yÞ ¼

s1ffiffiffiffiffitj j

p xcosf

22 ysin

f

2

2 y

a 2

tj j2

msinf

22 ncos

f

2

t Ixyðx; yÞ ¼ y

a 2s1

tj j2ffiffiffiffiffitj j

p mcosf

2þ nsin

f

2

ð26Þ

s IIx ðx; yÞ ¼

t1ffiffiffiffiffitj j

p 2 ycosf

2þ xsin

f

2

2 y

a 2

tj j2

mcosf

2þ nsin

f

2

s IIy ðx; yÞ ¼ y

a 2t1

tj j2ffiffiffiffiffitj j

p mcosf

2þ nsin

f

2

t IIxyðx; yÞ ¼

t1ffiffiffiffiffitj j

p xcosf

22 ysin

f

2

þ y

a 2

tj j2

msinf

22 ncos

f

2

ð27Þ

Figure 3.Westergaard problem

2ba

y

x

Ω0

b

σ∞

τ∞

τ∞

σ∞

σ∞ σ∞

Notes: Infinite plate with a crack of length 2aunder uniform tractions σ∞ (biaxial) and τ∞;finite portion of the domain Ω0, modelled withFE


825

where the stress fields are expressed as a function of x and y, with origin at the centre ofthe crack. The parameters t, m, n and f are defined as:

t ¼ ðx þ iyÞ2 2 a 2 >¼ ðx 2 2 y 2 2 a2Þ þ ið2xyÞ ¼ m þ in

m ¼ ReðtÞ ¼ Reðz 2 2 a 2Þ ¼ x 2 2 y 2 2 a 2

n ¼ ImðtÞ >¼ ðz2 2 a2Þ ¼ 2xy

f ¼ ArgðtÞ ¼ Argðm 2 inÞ with f [ ½2p;p; i 2 ¼ 21

ð28Þ

For the problem analysed, the exact value of the SIF is given as:

KI;ex ¼ sffiffiffiffiffiffipa

pKII;ex ¼ t

ffiffiffiffiffiffipa

pð29Þ

Three different loading configurations corresponding to the pure mode I(s1 ¼ 100; t1 ¼ 0), pure mode II (s1 ¼ 0; t1 ¼ 100), and mixed mode(s1 ¼ 30; t1 ¼ 90) cases of the Westergaard problem are considered. The geometricmodels and boundary conditions are shown in Figure 4.

Bilinear elements are used in the models, with a singular þ smooth decomposition areaof radius r ¼ 0:5 equal to the radius re used for the fixed enrichment area (note that thesplitting radius r should be greater or equal to the enrichment radius re (Rodenas et al.,2008)). The radius for the Plateau function used to extract the SIF is rq ¼ 0:9. Young’smodulus is E ¼ 107 and Poisson’s ratio y ¼ 0:333. For the numerical integration ofstandard elements we use a 2 £ 2 Gaussian quadrature rule. For split elements we useseven Gauss points in each triangular subdomain, and a 5 £ 5 quasipolar integration(Bechet et al., 2005) in the subdomains of the element containing the crack tip.

Regarding global error estimation, the evolution of global parameters in sequences ofuniformly refined structured (Figure 5) and unstructured (Figure 6) meshes is studied. Inthe first case, the mesh sequence is defined so that the crack tip always coincided witha node. For a more general scenario, this condition has not been applied for theunstructured meshes.

Figure 4.Model for an infiniteplate with a cracksubjected to biaxialtractions s1; t1in the infinite

ba

a = 1b = 4

σy

σx

τyx

τyx

τxy

σy

x x

yy

σx 2b

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826

6.2 Mode I and structured meshesThe first set of results presented in this study are for the Westergaard problem undermode I load conditions. Figure 7 shows the values for the local effectivity indexD for thefirst mesh in the sequence analysed. In the figure, the size of the enrichment area isdenoted by a circle. It can be observed that far from the enrichment area both techniquesyield similar results, however, the XMLS technique exhibits a higher overestimation ofthe error close to the singularity.

The same behaviour is also observed for more refined meshes in the sequence.Figure 8 shows a zoom in the enriched area of a finer mesh where, as before, a higheroverestimation of the error can be observed for the results obtained with the XMLSerror estimator. In this example and in order to study the evolution of the accuracy ofthe recovered stress field when considering different features in the recovery process,we consider two additional versions:

(1) The first considers a recovery procedure which enforces both internal andboundary equilibrium, but does not include the singular þ smooth splittingtechnique (SPR-C). This approach is very similar in form to conventionalSPR-based recovery techniques widely used in FEM. It is well-known that thistype of recovery process will produce unreliable results when the stress fieldcontains a singularity because the polynomial representation of the recoveredstresses is not able to describe the singular field (Bordas and Duflot, 2007).

Figure 6.Sequence of unstructured

meshesMesh 1273 elem.

Mesh 2525 elem.

Mesh 3818 elem.

Mesh 43,272 elem.

Mesh 513,088 elem.

Figure 5.Sequence of

structured meshesMesh 1288 elem.

Mesh 2512 elem.

Mesh 3800 elem.

Mesh 43,200 elem.

Mesh 512,800 elem.


827

(2) The second version of the error estimator performs the splitting but does notequilibrate the recovered field (SPR-X). As previously commented, s*

sing is anequilibrated field but s*

smo is not equilibrated. The aim ofthis second version isto assess the influence of enforcing the equilibrium constraints.

Table I summarizes the main features of the different recovery procedures weconsidered.

The performance of SPR-X in Figure 8 is poor, especially along the Neumannboundaries, where the equilibrium equations are not enforced, as they are in the SPR-CX.Particularly interesting are the results for the SPR-C technique, where considerableoverestimation of the exact error is observed “in the whole enriched region”. This is dueto the inability of polynomial functions to reproduce the near-tip fields, even relativelyfar from the crack tip.

The above compared the relative performance of the methods locally. It is also useful tohave a measure of the global accuracy of the different error estimators, Figure 9 (left)shows the convergence of the estimated error in the energy norm keesk (equation (12))evaluated with the proposed techniques, the convergence of the exact error kek[1] isshown for comparison. It can be observed that SPR-CX leads to the best approximation tothe exact error in the energy norm. The best approach is that which captures as closely aspossible the exact error. Non-singular formulations such as SPR-C greatly overestimatethe error, especially close to the singular point. Only SPR-C suffers from the characteristicsuboptimal convergence rate (factor 1=2, associated with the strength of the singularity).

In order to evaluate the quality of the recovered field s * obtained with each of therecovery techniques, the convergence in energy norm of the approximate error on therecovered field ke*k ¼ ku2 u*k is compared for the error estimators in Figure 9(right). It can be seen that the XMLS, the SPR-CX and the SPR-X provide convergencerates higher than the convergence rate for the exact error kek, which is an indicator of

Figure 7.Mode I: structured mesh 1

0

1

2

–1

–2

(a) SPR-CX (b) XMLS

Notes: Local effectivity index D (ideal valueD = 0); the circles denote the enriched zonesaround the crack tips; the superiority of thestatically admissible recovery (SPR-CX) comparedto the standard XMLS is clear

EC29,8

828

the asymptotic exactness of the error estimators based on these recovery techniques(Zienkiewicz and Zhu, 1992b).

Figure 10 shows the evolution of the parameters u, mðjDjÞ y s ðDÞ with respect to thenumber of dof for the error estimators. It can be verified that although the SPR-Ctechnique includes the satisfaction of the equilibrium equations, the effectivity of theerror estimator does not converge to the theoretical value (u ¼ 1). This is due to theabsence of the near-tip fields in the recovered solution. The influence of introducing

Singular functions Equilibrated singular functions Locally equilibrated field

SPR-X Yes Yes NoSPR-C No Not applicable YesSPR-CX Yes Yes YesXMLS Yes No No

Table I.Comparison of features

for the different recoveryprocedures

Figure 8.Mode I: structured mesh 4

(a) SPR-CX (b) XMLS

(d) SPR-C(c) SPR-X

3

0

–3

Notes: Local effectivity index D (ideal value D = 0); the circles denote the enrichedzones around the crack tips; the results show the need for the inclusion of the near-tipfields in the recovery process (SPR-X is far superior to SPR-C); it is also clear thatenforcing statical admissibility (SPR-CX) greatly improves the accuracy along thecrack faces and the Neumann boundaries; also notice that XMLS leads to slightlyimproved effectivities compared to SPR-CX around the crack faces, between theenriched zone and the left boundary; SPR-C is clearly not able to reproduce the singularfields, which is shown by large values of D inside the enriched region


829

singular functions in the recovery process can be observed in the results provided bySPR-X. In this case, the convergence is obtained both locally and globally. This showsthe importance of introducing a description of the singular field in the recovery process,i.e. of using extended recovery techniques in XFEM.

Regarding the enforcement of the equilibrium equations, we can see that anequilibrated formulation (SPR-CX) leads to better effectivities compared to othernon-equilibrated configurations (SPR-X, XMLS), which is an indication of the advantagesassociated with equilibrated recoveries in this context.

Still in Figure 10, the results for the SPR-CX and XMLS show that the values ofthe global effectivity index are close to (and less than) unity for the SPR-CX estimator,whilst the XMLS is a lot less effective and shows effectivities larger than 1.

The next step in our analysis is the verification of the convergence of the mean valueand standard deviation of the local effectivity index D. From a practical perspective, onewould want the local effectivity to be equally good, on average, everywhere inside thedomain; one would also want this property to improve with mesh refinement. In otherwords, the average local effectivity m and its standard deviation s should decrease withmesh refinement: m and s measure the average behaviour of the method and how far theresults deviate from the mean, i.e. how spatially consistent they are. The idea is to spotareas where there may be compensation between overestimated and underestimated areasthat could produce an apparently “accurate” global estimation. It can be confirmed that,although the curves for mðjDjÞ and s ðDÞ for both estimators tend to zero with meshrefinement, the SPR-CX is clearly superior to the XMLS. These results can be explained bythe fact that the SPR-CX enforces the fulfilment of the equilibrium equations in patches,and evaluates the singular part of the recovered field using the equilibrated expression

Figure 9.Mode I: structured meshes:convergence of theestimated error in theenergy norm je esj (left)

103 104 105

10–1

10–2

10–3

10–4

dof

||ees||

SPR-CX s = 0.48

XMLS s = 0.57

SPR-X s = 0.50

SPR-C s = 0.26

||e|| s = 0.48

10–1

10–2

10–3

10–4

10–5

103 104 105

dof

||e*es||

SPR-CX s = 0.73

XMLS s = 0.72

SPR-X s = 0.77

SPR-C s = 0.26

||e|| s = 0.48

Notes: All methods which include the near-tip fields do converge with the optimalconvergence rate of 1/2;; the best method is SPR-CX; XMLS performs worse than SPR-X forcoarse meshes, but it becomes equivalent to SPR-CX for fine meshes whilst the SPR-X errorincreases slightly; convergence of the error for the recovered field ke ees* esk (right); for therecovered field, SPR-CX is the best method closely followed by SPR-X; XMLS errors are halfan order of magnitude larger than SPR-CX/SPR-X, but superior to the non-enriched recoverymethod SPR-C; for all methods, except SPR-C, the error on the recovered

EC29,8

830

that represents the first term of the asymptotic expansion, whereas the XMLS enrichmentuses only a set of singular functions that would be able to reproduce this term but are notnecessarily equilibrated.

Although the results for the SPR-CX proved to be more accurate, the XMLS-typetechniques will prove useful to obtain error bounds. There is an increasing interest inevaluating upper and lower bounds of the error for XFEM approximations. Some workhas been already done to obtain upper bounds using recovery techniques as indicated inRodenas et al. (2010a), where the authors showed that an upper bound can be obtained ifthe recovered field is continuous and equilibrated. However, they demonstrated that due tothe use of the conjoint polynomials process to enforce the continuity ofs *, some residualsin the equilibrium equations appear, and the recovered field is only nearly equilibrated.Then, correction terms have to be evaluated to obtain the upper bound. An equilibratedversion of the XMLS could provide a recovered stress field which would be continuous andequilibrated, thus facilitating the evaluation of the upper bound. Initial results for thisclass of techniques can be found in Rodenas et al. (2009, 2010b).

Figure 10.Global indicators u,

m jDj

and s Dð Þ for modeI and structured meshes

103 104 105

2

4

6

8

10

dof

q

SPR-CXXMLSSPRXSPRC

103 104 105

10–1

10–2

dof

m(|D|)

SPR-CXXMLSSPRXSPRC

103 104 105

10–1

10–2

10–3

100

dof

s (D)

SPR-CXXMLSSPRXSPRC

103 104 1050.8

1

1.2

1.4

dof

q (zoomed y-axis)

SPR-CXXMLSSPRX


831

In this first example we have shown the effect of using the SPR-C and SPR-Xrecovery techniques in the error estimation. Considering that these two techniques arespecial cases of the SPR-CX with inferior results, for further examples we will focusonly on the SPR-CX and XMLS techniques.

6.3 Mode II and structured meshesFigure 11 shows the results considering mode II loading conditions for the localeffectivity index D on the fourth mesh of the sequence. Similarly to the results formode I, the XMLS estimator presents a higher overestimation of the error near theenrichment area. This same behaviour is observed for the whole set of structuredmeshes analysed under pure mode II.

As for the mode I case, the evolution of global accuracy parameters in mode IIexhibits the same behaviour seen for mode I loading conditions. Figure 12 shows theresults for the convergence of the error in the energy norm. Figure 13 shows theevolution of global indicators u, mðjDjÞ and s ðDÞ. Again, SPR-CX provides the bestresults for the Westergaard problem under pure mode II, using structured meshes.

6.4 Mixed mode and unstructured meshesConsidering a more general problem, Figure 14 shows the local effectivity index D forthe fourth mesh in a sequence of unstructured meshes, having a number of dof similarto the mesh represented previously for load modes I and II. The XMLS recovered fieldpresents higher overestimation of the error around the crack tip which corroborates theresults found in previous load cases.

Figures 15 and 16 show the evolution of global parameters for unstructured meshesand mixed mode. Once more, the best results for the error estimates are obtained usingthe SPR-CX technique.

7. ConclusionsThe aim of this paper was to assess the accuracy gains provided by:

Figure 11.Mode II: structured mesh 4

0

4

–4(a) SPR-CX (b) XMLS

2

–2

Notes: Local effectivity index D (ideal value D = 0); SPR-CXleads to much better local effectivities than XMLS; it is alsoremarkable that the effectivities are clearly worse in theenriched region (circle) for both the XMLS and the SPR-CX,and that this effect is more pronounced in the former method;note the slightly worse results obtained by SPR-CX around thecrack faces between the boundary of the enriched region and theleft boundary, as in mode I

EC29,8

832

. including relevant enrichment functions in the recovery process; and

. enforcing statical admissibility of the recovered solution.

We focused on two recovery-based error estimators already available for LEFM problemsusing the XFEM. The first technique called SPR-CX is an enhancement of the SPR-basederror estimator presented by Rodenas et al. (2008), where the stress field is split into twoparts (singular and smooth) and equilibrium equations are enforced locally on patches.The second technique is the XMLS proposed by Bordas and Duflot (2007) and Bordas et al.(2008) which enriches the basis of MLS shape functions, and uses a diffraction criterion, inorder to capture the discontinuity along the crack faces and the singularity at the crack tip.

To analyse the behaviour of the ZZ error estimator using both techniques and toassess the quality of the recovered stress field, we have evaluated the effectivity indexconsidering problems with an exact solution. Convergence of the estimated error in theenergy norm and other local error indicators are also evaluated. To analyse theinfluence of the special features introduced in the recovery process we have alsoconsidered two additional configurations: SPR-C and SPR-X.

The results indicate that both techniques, SPR-CX and XMLS, provide error estimatesthat converge to the exact value and can be considered as asymptotically exact.Both techniques could be effectively used to estimate the error in XFEMapproximations, while other conventional recovery procedures which do not includethe enrichment functions in the recovery process have proved not to converge to theexact error. This shows (albeit only numerically) the need for the use of extendedrecovery techniques for accuracy assessment in the XFEM context.

Figure 12.Mode II: structured

meshes: convergence ofthe estimated error in the

energy norm je esj

103 104 105

10–1

10–2

10–3

dof

||ees||

SPR-CX s = 0.48XMLS s = 0.54||e|| s = 0.48

10–1

10–2

10–3

10–4

103 104 105

dof

||e*es||

SPR-CX s = 0.59XMLS s = 0.71||e|| s = 0.48

Notes: The convergence rates are very similar to those obtained in the mode I case and,SPR-CX is still the most accurate method, i.e. that for which the estimated error is closestto the exact error; convergence of the error in the recovered field ees (right); it can benoticed that the SPR-CX error still converges faster than the exact error, but with a lowerconvergence rate (0.59 versus 0.73) than in the mode I case; there is no such difference for theXMLS results, which converge at practically the same rate (0.71 versus 0.72). The error leveldifference between SPR-CX and XMLS is about half an order of


833

Figure 14.Mixed mode:unstructured mesh 4

0

1

–1

(a) SPR-CX (b) XMLS

0.5

–0.5

Notes: Local effectivity index D (ideal value D = 0); note theimproved results compared to the structured case with D valuesranging from –1.5 to 1.5 as opposed to –4 to 4

Figure 13.Global indicators u, mðjDjÞand sðDÞ for mode II andstructured meshes

10–1

10–2

m(|D|)

SPR-CXXMLS

103 104 105

dof

10–1

10–2

100

s (D)

SPR-CXXMLS

103 104 105

dof

1

SPR-CXXMLS

103 104 105

dof

0.8

1.2

1.4q

10–1

10–2

m(|D|)

SPR-CXXMLS

103 104 105

dof

10–1

10–2

100

s (D)

SPR-CXXMLS

103 104 105

dof

1

SPR-CXXMLS

103 104 105

dof

0.8

1.2

1.4q

Note: The results are qualitatively and quantitatively similar to those obtained in mode I

EC29,8

834

Better results are systematically obtained when using the SPR-CX to recover the stressfield, specially in the areas close to the singular point. For all the different test casesanalysed, the XMLS produced higher values of the effectivity index in the enrichedarea, where the SPR-CX proved to be more accurate. This can be ascribed to the factthat the SPR-CX technique recovers the singular part of the solution using the knownequilibrated exact expressions for the asymptotic fields around the crack tip andenforces the fulfilment of the equilibrium equations on patches. Further work currentlyin progress will include the development of an equilibrated XMLS formulation, whichcould provide a continuous and globally equilibrated recovered stress field which canthen be used to obtain upper bounds of the error in energy norm.

The aim of our project is to tackle practical engineering problems such as thosepresented in Bordas and Moran (2006) and Wyart et al. (2007) where the accuracy of thestress intensity factor is the target. The superiority of SPR-CX may then be particularlyrelevant. To minimise the error on the SIFs we will target this error directly, throughgoal-oriented error estimates. This will be reported in a forthcoming publication.However, although the SPR-CX results are superior in general, an enhanced version ofthe XMLS technique presented in this paper where we enforce equilibrium conditionscould result useful to evaluate upper error bounds when considering goal-orientederror estimates, as it directly produces a continuous equilibrated recovered stress field.

Our next step will be the comparison of available error estimators in threedimensional settings in terms of accuracy versus computational cost to minimise theerror on the crack path and damage tolerance of the structure.

AcknowledgementsStephane Bordas would like to thank the Royal Academy of Engineering and of theLeverhulme Trust for supporting his Senior Research Fellowship entitled “Towardsthe next generation surgical simulators” as well as the EPSRC for support under grant

Figure 15.Mixed mode: unstructured

meshes: convergence ofthe estimated error in the

energy norm jeesj (left)

103 104 105

10–2

10–3

10–4

dof

||ees||

SPR-CX s = 0.54XMLS s = 0.57||e|| s = 0.48

SPR-CX s = 0.77XMLS s = 0.70||e|| s = 0.48

10–2

10–3

10–4

103 104 105

dof

||e*es||

Notes: Convergence of the error for the recovered field ees* (right); the results are almostidentical to the structured mesh case, except for the faster convergence obtained for SPR-CX inthe unstructured compared to the structured case (0.77 versus 0.59), keeping in mind thatoptimal convergence rates are only formally obtained for structured meshes


835

EP/G042705/1 Increased Reliability for Industrially Relevant Automatic Crack GrowthSimulation with the eXtended Finite Element Method.

This work has been carried out within the framework of the research projectsDPI2007-66773-C02-01, DPI2010-20542 and DPI2010-20990 of the Ministerio de Cienciae Innovacion (Spain). Funding from Feder, Universitat Politecnica de Valencia andGeneralitat Valenciana is also acknowledged.

Note

1. The exact error measures the error of the raw XFEM compared to the exact solution.The theoretical convergence rate is OðhpÞ, h being the element size and p the polynomialdegree (1 for linear elements,. . .) or Oð2dofp=2Þ if we consider the number of degrees offreedom.

Figure 16.Global indicators u,m jDj

and s Dð Þ formixed mode andunstructured meshes

SPR-CXXMLS

10–1

10–2

m(|D|)

103 104 105

dof

SPR-CXXMLS

10–1

10–2

100

s (D)

103 104 105

dof

SPR-CXXMLS

1

103 104 105

dof

0.8

1.2

1.4q

Note: Notice that the XMLS performs more closely to SPR-CX using those indicators, forunstructured than for structured meshes, especially when measuring the mean value of theeffectivities m( D )

EC29,8

836

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Author’s affiliationOctavio Andres Gonzalez-Estrada, Institute of Modelling and Simulation in Mechanics andMaterials, Cardiff School of Engineering, Cardiff University, Cardiff, UK.

Juan Jose Rodenas, Research Center for Vehicle Technology, Department of Mechanical andMaterials Engineering, Universitat Politecnica de Valencia, Valencia, Spain.

Stephane Pierre Alain Bordas, Institute of Modelling and Simulation in Mechanics andMaterials, Cardiff School of Engineering, Cardiff University, Cardiff, UK.

Marc Duflot Multiscale Modeling of Materials and Structures, Cenaero, Gosselies, Belgium.Pierre Kerfriden, Institute of Modelling and Simulation in Mechanics and Materials, Cardiff

School of Engineering, Cardiff University, Cardiff, UK.Eugenio Giner, Research Center for Vehicle Technology, Department of Mechanical and

Materials Engineering, Universitat Politecnica de Valencia, Valencia, Spain.

About the authorsOctavio Andres Gonzalez-Estrada joined the Theoretical, Applied and Computational Mechanicsteam at the School of Engineering at Cardiff University on 1 May 2010, as a Postdoctoral ResearchAssociate. Before this, he was working at the Research Centre for Vehicle Technology (CITV) atthe Universidad Politecnica de Valencia as a Research Associate. He received his PhD inMechanical and Materials Engineering (2010) and MSc in Mechanical and Materials Engineering(2008) from the Universidad Politecnica de Valencia, and his Bachelor in Mechanical andManufacturing Engineering from the Universidad Autonoma de Manizales, Colombia, in 2001,with First Class Honours. His area of expertise and research directions include computationalsolid mechanics and numerical methods, with an emphasis in fracture mechanics, structuralreliability, damage tolerant analysis, evolving discontinuities, partition of unity methods, theextended finite element method and level set method, accuracy and convergence of the solutionin FEM/XFEM approximations, error estimation and error bounding for FEM/XFEM.Octavio Andres Gonzalez-Estrada is the corresponding author and can be contacted at:[email protected]

Juan Jose Rodenas obtained a BEng Honours degree in Combined Engineering Studies fromCoventry Polytechnic (UK) in 1990, a degree in Mechanical Engineering from UniversidadPolitecnica de Valencia (Spain) in 1993 and an MSc in Design of Rotating Machines fromCranfield Institute of Technology (UK) also in 1993. He received his PhD from the UniversidadPolitecnica de Valencia (Spain) with the thesis Discretization error in shape sensitivity analysisby means of the Finite Element Method. He joined the Department of Mechanical and MaterialsEngineering at the Universidad Politecnica de Valencia (Spain) in 1993 and the Research Centerof Vehicle Technology in 2003. His areas of expertise include error estimation and errorbounding in FEM and XFEM based on the use of recovered solutions, shape sensitivity analysis,shape optimization and Cartesian grids for FE analysis with meshes independent of thegeometry of the domain.

Stephane Pierre Alain Bordas joined the Theoretical Applied and Computational Mechanicsteam at Cardiff University on 1 September 2009, as a Professor. Before this, he was a Lecturerin Glasgow University Civil Engineering Department for three years (2006-2009). Between 2003and 2006, he was at the Laboratory of Structural and Continuum Mechanics at the Swiss FederalInstitute of Technology in Lausanne, Switzerland, working under the support of Professor

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Thomas Zimmermann on meshfree point collocation methods and partition of unity enrichment(extended finite elements) with applications to geomechanics. In 2003, he graduated in Theoreticaland Applied Mechanics with a PhD from Northwestern University, under the guidance ofProfessor Brian Moran. His thesis, funded by the Federal Aviation Administration, concentratedon applications of the extended finite element method (XFEM) to damage tolerance analysis ofcomplex structures, casting design and biofilm growth processes. In 1999, through a joint graduateprogramme of the French Institute of Technology (Ecole Speciale des Travaux Publics) and theAmerican Northwestern University he completed a dual MSc after a thesis work on Time domainreflectometry simulation to assess ground movements, with Professor Charles H. Dowding.His areas of expertise are: computational mechanics with an emphasis on moving discontinuities(mechanics of fracture, biofilm and tumour growth, etc.); method development (enriched/extendedfinite elements, meshfree methods, smooth strain finite elements); evolving discontinuities(level set methods, partition of unity enrichment; academic research/industrial applications:bridging the gap (porting novel methods to industrial codes, real-world applications ofcomputational mechanics and novel method development). More recently developed areas ofinterest include high performance computing, surgical simulation, biomechanics,microstructurally-faithful material modelling, multiscale simulation, and model reductiontechniques.

Marc Duflot obtained his Bachelor and Master degrees from the University of Liege (his masterthesis entitled The non-conforming boundary element method). He worked at University of Liegeas a research fellow, obtaining his PhD in 2004 with the thesis Application of meshless methods infracture mechanics. Later, he moved to industry to work as a research engineer at Cenaero, wherehe is one of the lead developers of the software Morfeo. Research interests include numericalsimulation in fracture mechanics, extended finite element method (XFEM), meshfree methods,level set method, fast marching method, error estimation and adaptivity.

Pierre Kerfriden obtained his PhD degree from ENS Cachan, France, in 2008 with the thesisA three-scale domain decomposition method for the analysis of the debounding in laminates.He was a temporary Assistant Professor at LMT Cachan from 2008 to 2009. In 2010 he joinedCardiff School of Engineering as a Lecturer. His research interests include damage prediction incomposites using multiscale methods, model order reduction, XFEM and alternative enrichmentmethods, domain decomposition and parallel computing for intensive nonlinear computations.

Eugenio Giner obtained a BEng Honours degree in Combined Engineering Studies fromCoventry Polytechnic (UK) in 1990 and a degree in Mechanical Engineering from UniversidadPolitecnica de Valencia (Spain) in 1993. He received his PhD from the Universidad Politecnica deValencia (Spain) with the thesis Discretization error for the analysis of the stress intensity factorby means of finite elements. He joined the Department of Mechanical and Materials Engineeringat the Universidad Politecnica de Valencia (Spain) in 1994 and the Research Center of VehicleTechnology in 2003. His areas of expertise include computational fracture mechanics with FEMand X-FEM; postprocessing of numerical solutions to extract SIFs and generalized SIFs insingular domains and estimation of the discretization error; 3D fracture mechanics; crack contactmodelling; applications to fatigue and fretting-fatigue; sliding contacts, etc.

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