inexact schwarz-algebraic multigrid preconditioners for crack problems modeled by extended finite...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2012; 90:311–328Published online 2 December 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3318

Inexact Schwarz-algebraic multigrid preconditioners for crackproblems modeled by extended finite element methods

Luc Berger-Vergiat1, Haim Waisman1,*,†, Badri Hiriyur1, Ray Tuminaro2 andDavid Keyes3

1Department of Civil Engineering & Engineering Mechanics, Columbia University, New York, NY 10027, U.S.A.2Sandia National Laboratories, Scalable Algorithms, P.O. Box 969, MS 9159, Livermore, CA 94551, U.S.A.

3Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027, U.S.A.

SUMMARY

Traditional algebraic multigrid (AMG) preconditioners are not well suited for crack problems modeled byextended finite element methods (XFEM). This is mainly because of the unique XFEM formulations, whichembed discontinuous fields in the linear system by addition of special degrees of freedom. These degreesof freedom are not properly handled by the AMG coarsening process and lead to slow convergence. In thispaper, we proposed a simple domain decomposition approach that retains the AMG advantages on well-behaved domains by avoiding the coarsening of enriched degrees of freedom. The idea was to employa multiplicative Schwarz preconditioner where the physical domain was partitioned into “healthy” (orunfractured) and “cracked” subdomains. First, the “healthy” subdomain containing only standard degrees offreedom, was solved approximately by one AMG V-cycle, followed by concurrent direct solves of “cracked”subdomains. This strategy alleviated the need to redesign special AMG coarsening strategies that can handleXFEM discretizations. Numerical examples on various crack problems clearly illustrated the superior per-formance of this approach over a brute force AMG preconditioner applied to the linear system. Copyright© 2011 John Wiley & Sons, Ltd.

Received 4 February 2011; Revised 16 June 2011; Accepted 26 August 2011

KEY WORDS: XFEM; extended finite elements; domain decomposition; Schwarz preconditioner; fractureanalysis; algebraic multigrid; smoothed aggregation multigrid

1. INTRODUCTION

Fracture mechanics is an established field of the engineering sciences, which has enormously ben-efited from the development of advanced computational techniques. The classical approaches tomodeling cracks using Galerkin finite element methods [1,2] start with discretizations that conformto crack interfaces and additionally use mesh refinement coupled with special elements (such ashaving additional quarter-point nodes) near the crack tip. The special elements are necessitated bythe fact that generic isoparametric elements fail to capture the singularities in stresses (proportionalto r�1=2, where r is the local radial coordinate) at the crack tip.

Obtaining a discretization that conforms to the crack geometry is a nontrivial task, especially incases where cracks propagate as a result of quasi-static or fatigue loadings (requiring remeshing) orwhen multiple random micro-cracks are considered (involving complex topologies). To overcomesuch difficulties, Belytschko et al. [3, 4] introduced the extended finite element method (XFEM),and around the same time, Babuška and co-authors [5, 6] introduced the generalized finite elementmethod. Both methods have similar features, and in this paper, we will focus on XFEM. The key idea

*Correspondence to: Haim Waisman, Department of Civil Engineering & Engineering Mechanics, Columbia University,New York, NY 10027, USA.

†E-mail: [email protected]

Copyright © 2011 John Wiley & Sons, Ltd.

312 L. BERGER-VERGIAT ET AL.

of XFEM, when applied to problems in linear elastic fracture mechanics, is to resolve the complex-ities associated with modeling of cracks by using a mesh that is independent of the crack geometry.The discontinuities along the crack interface and singularities near the crack tip are instead capturedthrough an “enriched” space of basis functions that model the underlying physics of the fractureproblem. These additional enrichment functions have local support near a crack, satisfy a partitionof unity, and add to the number of degrees of freedom at the nodes near the crack. The use of anenriched space of basis functions alleviates the need for remeshing the domain in the case of prop-agating cracks. The crack tip enrichment functions also model singularities without the need forspecial elements or mesh refinement. This technique has gained a wide acceptance in the scientificcommunity during the past decade.

Concurrent with the development of computational modeling tools, numerical solvers too haveevolved considerably during the past few decades. Currently, the most advanced solvers for sparselinear systems, such as those obtained from finite element discretizations, are based on building abasis for the Krylov subspace associated with the linear system and finding an approximate solu-tion iteratively in this span. There are various preconditioning schemes available to accelerate theKrylov subspace solvers, and well-known among them is the multigrid method [7–10]. This methodis based on generating a hierarchy of discretizations for the problem in order to resolve and smooththe error at different modes, thereby improving the convergence properties of the solver. Whereasgeometric multigrid methods have been in use for a long time [11, 12], the state-of-the-art in thisfield are the algebraic multigrid methods (AMG), which construct coarse grids directly from thelinear system without a need for geometric information. A particularly promising AMG techniqueis the one based on the concept of smoothed aggregation [13, 14]. In this approach, grid transferoperators are obtained directly from the system matrix in two steps: construction of a tentative pro-longator by aggregation of nodes and smoothing of the tentative prolongator with a damped Jacobimethod. The smoothed aggregation multigrid method has been shown to work well on a variety ofproblems [15] and in parallel architectures [14].

In this article, we propose an approach that combines the use of XFEM modeling of fractureproblems with AMG preconditioning of a generalized minimum residual (GMRES) solver [16].The coupling of these methods has proven to be challenging because the coarsening by AMG needsto preserve the representation of discontinuities across a hierarchy of discretizations. In particular,a dedicated AMG method would require special awareness of cracks, so that aggregates do notcross cracks in the coarsening process. In addition, it will require accurate approximation of smoothmodes such as constants and rigid-body modes on coarse levels and address the variability in thenumber of degrees of freedom between enriched and standard nodes. Whereas a specially designedAMG may lead to more optimal results, addressing the aforementioned issues is not straightfor-ward. On the other hand, brute force application of AMG to XFEM problems may lead to poorconvergence (as will be shown later in this paper).

Our contribution overcomes these difficulties by using domain-decompostion techniques to obtainsubdomains that contain the elements with enriched degrees of freedom separated from the rest ofthe problem. The subdomains containing the enriched degrees of freedom are relatively small andcan be easily handled by direct solvers, whereas AMG is used to solve approximately the other,so-called “healthy”, subdomains. This strategy is attractive because it alleviates the need to designspecial AMG methods for XFEM discretizations.

An interesting approach, which is quite different from the approach in this paper, that employsmultigrid and XFEM was proposed by Rannou et al. [17] and later extended in [18, 19]. Thesemethods employ geometric multigrid in localized regions to achieve an accurate and fast modelingof small and local defects such as cracks or inclusions in large structures. Another method, whichattempts to resolve local features by superimposing refined patches is the so-called s-version of thefinite element method and was developed by Fish et al. [20, 21]. This method has recently beenextended to account for material failure in arbitrary directions [22].

Alternatively, to resolve local or global features, one can also consider self correcting multigridmethods, for example, the generalized global basis method, developed by Waisman et al.[23, 24].The method adds a two-level filtering cycle to any multigrid scheme (geometric or algebraic) toresolve nonconverging error modes on an auxiliary coarse grid. However, computing the additional

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2012; 90:311–328DOI: 10.1002/nme

XFEM_DD 313

prolongation operator may be expensive for such problems as it requires solution of large eigenvalueproblems.

Another interesting study that combines domain decomposition and XFEM has just been pub-lished by Menk and Bordas [25]. In this approach, domain decomposition is employed in order toobtain stiffness matrices whose condition number is closely related to the one of the finite elementmatrices without any enrichments. A similar domain decomposition concept has been applied tofluid flow problems where localized phenomena such as shock waves had to be resolved exactly toenable quick convergence of Newton–Krylov–Schwarz [26] schemes. Other close strategies havebeen previously used by Wyart et al. [27, 28]. Whereas [28] used a finite element tearing and inter-connecting (FETI) method by Farhat et al. [29, 30] for the separation phase, we use a multiplicativeSchwarz algorithm to perform the domain decomposition in order to implement it as a precondi-tioner for the GMRES solver. A Schwarz solver algorithm is more adapted to the inexact AMGsubdomain solves that are considered in this paper, because a FETI-like approach would requirehigh accuracy subdomain solutions, which are fairly expensive to obtain and therefore has a negativeimpact on the efficiency of the preconditioner.

The outline of this paper is as follows: the XFEM model for the fracture problem being con-sidered is explained in Section 2, including a description of how level set functions can be usedto generate subdomains enclosing the cracks. This is followed in Section 3 by an overview of thedomain decomposition technique used to efficiently precondition the global iterative solvers. Theproposed approach of preconditioning using multiplicative Schwarz algorithm and AMG for sub-domains without cracks is explained in detail in Section 3, and finally some numerical convergenceresults are presented in Section 4.

2. A BRIEF OVERVIEW OF THE EXTENDED FINITE ELEMENT METHOD FORCRACK PROBLEMS

In this work, we focus on 2D crack problems modeled using XFEM. The XFEM is implementedwith level set functions to detect the locations of the nodes around the crack and its tips. Typically,in XFEM, two types of enrichment functions are used:

� Nodes along the crack line, excluding those at the tips, are enriched by the Heaviside stepfunction to incorporate a strong discontinuity. This enrichment adds two degrees of freedomper node, one associated with each spatial dimension and models the opening of the crack:

H.x/D

²C1 above crack line�1 below crack line

(1)

� Nodes of the elements containing crack tips are enriched with a set of four functions, so-called branch functions that model the near-tip analytical solution and therefore incorporatetip singularities:

Fj .r.x/, �.x//D

�pr sin

�

2,pr cos

�

2,pr sin

�

2sin � ,

pr cos

�

2sin �

�jD1:::4

(2)

To preserve a partition of unity and to ensure local support near the crack, the enrichment functionsare enveloped by the shape functions associated with the enriched nodes. The general expression forthe discretized trial function for the XFEM weak form is then written as:

uh.x/D

nXiD1

Ni .x/ui C

nCXiD1

Ni .x/H.x/ai C

nTXiD1

0@Ni .x/

4XjD1

Fj .x/bj i

1A (3)

where Ni .x/ are the standard finite element shape functions associated with degrees of freedom ui .ai and bj i are the degrees of freedom associated with the enriched nodes, n is the number of nodes,nC is the number of nodes enriched with Heaviside functions along the crack, and nT is the numberof nodes enriched with tip functions. The enrichment strategy is illustrated in Figure 1. The purpleline represents the crack, the red squares highlight the nodes enriched with tip functions, and the



Figure 1. Modeling a crack with the extended finite element method. Heaviside enrichment is representedby blue circles and tip functions by red squares.

light blue circles show the nodes enriched with Heaviside functions. For more information on themodeling of cracks with the XFEM, the reader is referred to [4].

3. PROPOSED INEXACT SCHWARZ-ALGEBRAIC MULTIGRID PRECONDITIONER

Domain decomposition methods, such as the Schwarz alternating method [31], solve boundary valueproblems by subdividing the computational domain into subdomains. Each subdomain is solvedindependently using data from neighboring subdomains to define boundary conditions. This pro-cess is iterated until a converged solution is reached [32, 33]. In addition, these techniques havebeen found to give fast convergence when employed as preconditioners to Krylov-type solvers[34]. Domain decomposition techniques have been widely used in computational mathematics andmechanics because of two main reasons: (1) suitability for massively parallel computing when solv-ing extremely large problems [35]; and (2) their ability to isolate regions that pose difficulties tothe solution of the problem and are better handled separately, for example, contact problems incontinuum mechanics or shock waves in fluid mechanics [36, 37].

In the present work, we employ domain decomposition concepts, based on the multiplicativeSchwarz method, to reformulate the problem so that AMG would retain its convergence featureswhen applied to the bulk of the domain that contains no cracks. Hence, we partition the problemin a way that separates the enriched degrees of freedom from the nodes that are not enriched. Theformulation and the proposed preconditioner are outlined in the following subsections.

3.1. Domain decomposition formulation for cracks: a multiplicative Schwarz approach

The proposed multiplicative Schwarz preconditioner begins with a special domain decomposition.As shown in Figure 2, we consider two possible partitioning strategies: (1) a single subdomain con-taining all cracks (see Figure 2(a)); and (2) multiple crack subdomains (see Figure 2(b)), whereeach crack owns its own subdomain. In many cases, the physics of the problem will determine thepartitioning scheme. For example, clusters of cracks and microcracks, formed as a result of a local-ized impact or indentation loads, can be aggregated into a single subdomain, whereas more isolatedcracks, nucleating at far distances from each other, for example, formed as a result of fatigue loadsapplied to the whole structure, may be too far apart and will require their own subdomains. In anycase, the two strategies lead to the same type of systems.

In Figure 2, we refer to the subdomain �h1 that does not contain any enriched nodes (or cracks)as a “healthy” subdomain (for simplicity of the presentation, we will only consider one healthysubdomain). The other subdomains containing at least one crack, and hence all enriched degreesof freedom associated with that crack, are referred to as “cracked” subdomains and denoted by�ci . Cracked subdomains may also contain one or several layers of elements around cracks thatconstitute the overlap with the healthy subdomain.


XFEM_DD 315

/////////////////

//

Γ1

Γ

c

Γ

///////////////

/

Γ

(a)

1h

1h

(b)

1c

1c

1h

2c

3c

Γ2c

Γ3c

Γ1h

Γ1c

Figure 2. Schematic representation of the “healthy” and “cracked” subdomains in the formulation of domaindecomposition. (a) Multiple cracks share a single cracked subdomain, (b) each crack is assigned to a different

cracked subdomain.

The general formulation leads to a coupled set of linear systems associated with the healthy andcracked subdomains. The coupling occurs through the boundary conditions and overlapping ele-ments. Note that, in this formulation, cracked subdomains are decoupled from each other and areonly coupled to the healthy subdomain. This property is not an essential feature of the proposedapproach but is adopted in the current work to keep the description of the method simple, withoutany loss of generality.

Mathematically, the decomposition into subdomains is written in the following way:

8ˆ<ˆˆ:

Kh1uh1 D fh1 in �h1 with uh1 D uE on �

and with uh1 D uc1 on �cjKc1uc1 D fc1 in �c1 with uc1 D uh1 on �c1

Kc2uc2 D fc2 in �c2 with uc2 D uh1 on �c2

...Kcnucn D fcn in �cn with ucn D uh1 on �cn

(4)

The superscript h is employed to denote an operator or variable defined on a healthy sub-domain, and the superscript c denotes operators or variables defined on a cracked subdo-main. Note that the healthy subdomain is solved first, followed by cracked subdomains.

�D�h1 ˚�c1˚ : : :˚�

cncD�h1

ncMiD1

�ci is the finite element space associated with the problem,

nc is the number of cracked subdomains, uE are the values of the displacement field where essen-tial boundary is imposed, � is the boundary of �, and �i is the boundary of subdomain �i . Thesolution over the full problem domain (henceforth referred to as global) is formed by assemblingthe solutions from the subdomain problems. Once the problem is discretized by finite elements, wedenote, in each subdomain, the stiffness matrix, force vector, and the unknown displacement vectorby K,f, and u, respectively.

A simple procedure to generate a cracked subdomain employs the level set formulation and isdescribed as follows. Start by forming a set, which is the union of all elements having specified over-lap for a given crack with the healthy subdomain. This is easily generated by selecting all the nodesthat have level set values smaller than no�he ,where he is the characteristic dimension of an element.If no is chosen equal to 1, only the elements containing the crack are selected. Choosing no D 2

selects all elements containing at least one enriched node. This would create a non-overlapping par-tition of the domain between cracked subdomains and the healthy subdomain, and could be wellsuited for application of a FETI algorithm. In the numerical tests presented here, we choose no D 3



to have an additional layer of overlapping elements. An illustration of the domain decompositionand the overlapping elements for XFEM is shown in Figure 3.

The Schwarz algorithm may be written in the following way. The linear system on subdomaini (healthy or cracked) is extracted from the global system by employing a restriction operator

Ri W �N�N ! �N id�N

i , where N denotes the overall number of degrees of freedom and N id

the number of degrees of freedom associated with subdomain i . The restriction operator is also thetranspose of the prolongation operator Ri D .Pi /T , which will be used in subsequent notation. Theprolongator matrix Pi is a Boolean matrix constructed such that each row corresponds to a globaldegree of freedom, and each column corresponds to a degree of freedom belonging to the particularsubdomain �i . The entries of Pi are as follows:

Pi .j , Ok/D ıjk & k 2�i (5)

where ıjk denotes the Kronecker symbol, j being the current global degree of freedom, and k isthe global degree of freedom in subdomain �i that corresponds to local column index Ok.

The application of the prolongation operator to the inverse of the restricted stiffness matrixPTi KPi is defined as:

Bi D Pi�PTi KPi

��1PTi (6)

For notation simplicity but without the loss of generality, we assume a decomposition into twosubdomains, one healthy and one cracked subdomain, as illustrated in Figure 3. For this domaindecomposition, the Schwarz preconditioning iterate may be written by

��

input rnuhn D Bhrnucn D Bc

�rn �KuhE

�unC1 D uhnC ucnoutput unC1

(7)

where rn is the residual at iteration n given as rn D f � Kun, and uhE is the essential boundaryconditions applied to the cracked subdomain, which are obtained by solving (approximately) thehealthy subdomain (note that these values are changing with every solver iteration as opposed touE , which are the given boundary conditions of the problem). uhn and ucn are the updated terms

Overlapping elements Overlapping elements

Figure 3. Two overlapping domains employed in the Schwarz method. The following color legend is used:the black squares represent Schwarz essential boundary conditions, the black triangles represent clampednodes, the red circles represent pulled nodes, the green zone represents the elements belonging to the same

subdomain, the blue zone represents the elements that are part of the overlapping layer.


XFEM_DD 317

of the solution vectors corresponding to healthy and cracked subdomains, respectively. In a morecompact notation, Equation (7) becomes:

unC1 D�

BhCBc �BcKBh�

rn DMrn (8)

Where M is the preconditioning operator. Note that M defined in Equation (8) is not symmetric. Onecould symmetrize the preconditioner by adding an additional solve of cracked subdomains beforethe healthy part, as follows: ��

input rnuc1n D Bcrnuhn D Bh

�rn �Kuc1E

�uc2n D Bc

�rn �KuhE

�unC1 D uc1n C uhnC uc2noutput unC1

(9)

In a compact matrix form, it reads:

Msym D Bc CBh �BhKBc �BcKBhCBcKBhKBc (10)

D Bc C .I�BcK/Bh .I�KBc/ (11)

Normally, a symmetric form of the preconditioner allows for a bigger choice of iterative solvers.However, it also must be noted that it requires the application of the operator Bc twice, which ismore expensive, and hence will not be pursued in this paper.

Another way of creating a symmetric system is to use an additive Schwarz algorithm instead ofa multiplicative one. The idea of the additive Schwarz algorithm is to solve all subdomains concur-rently. In contrast with the multiplicative Schwarz approach, the nonsymmetric term in Equation 8is dropped in this formulation. The additive Schwarz algorithm has the following form:��

input rnuhn D Bhrnucn D BcrnunC1 D uhnC ucnoutput unC1

(12)

or in a compact matrix form:

unC1 D�

BhCBc�

rn DMrn (13)

One obvious advantage of this approach is the ease of parallelization of this algorithm. In addition,it is also much simpler to symmetrize the system of equations in this case without introducing newmatrix operations. In the additive Schwarz approach, because the residual is not updated after eachsubdomain solve, convergence is slower compared with the multiplicative Schwarz approach. In thenumerical examples presented in Section 5, we use a GMRES solver with the preconditioner schemegiven in Equation 8 and employ only a single smoothed aggregation AMG cycle during the solutionphase. A brief overview of the smoothed aggregation method is given in the following subsection.

3.2. The smoothed aggregation multigrid method

The multigrid method is well known to be an efficient tool for solving large sparse systems arisingfrom the discretization of partial differential equations [12,38,39]. It is frequently used as a precon-ditioner to a Krylov-subspace-based iterative solver. The key idea is to attenuate both high-energyoscillatory error modes, as well as low energy, smooth error modes, by resolving them using a hier-archy of discretizations. The high-energy modes obtained at fine discretizations are tackled througha simple smoothing procedure, whereas the smooth error is resolved at coarser discretizations whereeventually, at the coarsest grid, a direct solve is relatively inexpensive to perform. Whereas standard



multigrid uses a geometric notion of coarsening to obtain the grid hierarchy and the correspondingtransfer operators, this is accomplished in AMG directly from the matrix data through the use ofgraph algorithms. One promising AMG scheme is the one based on smoothed aggregations, wherenearly optimal grid-transfer operators are available.

Smoothed aggregation is an approach for automatically generating a grid transfer matrix,Q [13, 40]. It begins by associating a graph with the matrix system being solved. Graph verticescorrespond to matrix rows for scalar PDEs, whereas for PDE systems, it is natural to associate onevertex with each nodal block of unknowns, for example, x and y displacements at a particular gridpoint. A graph edge exists between vertex i and j if there is a nonzero in the block matrix, whichcouples i’s rows with j ’s columns or j ’s rows with i’s columns. In some situations, it may beadvantageous to omit edges if all entries within the coupling block are small [41]. This graph isthen automatically coarsened by aggregating neighboring vertices together. Each aggregate definesa vertex on the next coarser mesh. For details on aggregation, readers may refer to [13] and [42]. Thegoal is to create ideal aggregates, which consist of a single central vertex and all of its immediateneighbors. Whereas it is not usually possible to coarsen completely with ideal aggregates, a largefraction of the computed aggregates are typically ideal.

A matrix B with m columns corresponding to m vectors is then defined. These vectors are therigid body modes. In two dimensions, m D 3 and the vectors are representations of a constant xdisplacement, a constant y displacement, and a constant rotation. That is,

B D

266664

1 0 Oy � y10 1 x1 � Ox1 0 Oy � y20 1 x2 � Ox...

......

377775

where . Ox, Oy/ are coordinates of an arbitrary reference point, .xj ,yj / is the coordinate location of thej th node, and an ordering is assumed so that all degrees of freedom at a node appear in consecutiverows. With aggregates in hand, a tentative (or initial prolongator) can now be constructed accordingto

Qtent D

26664Q.1/ 0 : : :

0 Q.2/ 0 : : :. . .

: : : 0 Q. On/

37775 , BC D

26664R.1/R.2/

...R. On/

37775 , (14)

where first B.i/ is defined as a submatrix of B obtained by taking only rows associated with degreesof freedom within nodes assigned to the i th aggregate and then B.i/ D Q.i/R.i/ is constructedvia a QR factorization. It follows that B D QtentBC (i.e., Qtent perfectly interpolates rigid bodymodes). In a two-level method, BC is not needed. However, if one wishes to recursively generatemore levels, BC assumes the role of B in the recursive generation of the next prolongator operator.

Whereas Qtent could be used as a prolongator, it is improved by a smoothing process given by

QD .I �!D�1K/Qtent

where ! is a damping parameter, andD is the diagonal ofK. This corresponds to applying one stepof a Jacobi iteration procedure to Qtent . This process smooths columns (or basis functions) of the


XFEM_DD 319

Figure 4. Generic two-level multigrid cycle to solve K1u1 D f1.

prolongator while preserving the quality of the interpolation of rigid body modes. Further detailscan be found in [13, 40].

The key steps of a simple two-level AMG cycle are listed in Figure 4. In this algorithm, subscript“1” denotes fine-level quantities whereas subscript “0” indicates coarse-level quantities. The cyclestarts with a specified (�1/ number of presmoothing steps through the operator D (e.g., Jacobi,Gauss–Seidel or Chebyshev sweeps) at the fine level to reduce the the high-energy oscillatorymodes. The residual obtained after the presmoothing step is resolved at a coarser scale througha restriction operator QT W Rn! Rm where n and m denote the number of fine and coarse degreesof freedom, respectively. Next, the solution of the coarse scale is interpolated back to the fine scaleby a prolongation operator Q W Rm ! Rn and added as a correction to the fine-level solution. Byusing the complementary nature of the smoothing and the prolongator operators, fast convergenceis ensured. The final step in the AMG cycle is to perform a specified (�2/ number of post-smoothingsteps to attenuate the errors that are not reduced in the previous steps.

Replacing K�10 by a recursive invocation of the multigrid procedure will lead to a multilevelversion of the multigrid algorithm presented in Figure 4.

3.3. Preconditioner setup and algorithm flow

Our proposed approach uses a domain decomposition algorithm with an inexact multiplicativeSchwarz method as the preconditioner for the residual obtained at each iteration of a global GMRESsolver. This partitioning is only performed once and is reused over successive iterations. The healthysubdomain is approximately solved using one AMG V-cycle (details in Section 3.2) and crackedsubdomains are solved concurrently with a direct solver. The two solutions are then assembled backto be returned to the global GMRES solver. A schematic representation of the algorithm is illustratedin Figure 5 and shown in a condensed form in Figure 6.

Whereas one could argue that the use of a direct solver on the cracked subdomains may be com-putationally too expensive, we point out that cracked subdomains are relatively small comparedwith the overall size of the problem, and therefore, this step is fairly inexpensive. Moreover, linearsystems associated with cracked subdomains are factored before starting GMRES iterations, andreused in successive iterations of the preconditioner. Nonetheless, in the case of propagating cracks,a new partition may need to be built at every crack propagation step. Hence, the linear systemsassociated with cracked subdomains must also be refactored after each crack propagation step.

One of the great interests of this process is that it is utterly simple to implement and provides away to apply the AMG method to the bulk of the domain, retaining its convergence properties forcrack problems modeled by XFEM.



Exact Solves on cracked domains

Inexact AMG

solve on

GMRES

Figure 5. Schematic description of the inexact Schwarz-AMG preconditioner.

Figure 6. GMRES preconditioned by an inexact Schwarz-AMG preconditioner.

4. NUMERICAL EXAMPLES

Numerical results comparing the inexact Schwarz-AMG preconditioner versus a brute force AMGpreconditioner within GMRES [16] are provided in this section. A brute force AMG method isdefined as an AMG method that is directly applied to the entire domain without redesigning theAMG to take into account enriched degrees of freedom. The AMG method employed in our studiesis the one based on smoothed aggregation concepts [13] and has been implemented in the MueMatpackage at Sandia Labs.

For all examples in this section, the following material properties are set: Young’s modulus istaken as E D 107 ŒPa�, Poisson’s ratio is set to � D 0.3, and a plane strain formulation is assumed.The inexact Schwarz-AMG preconditioner is used in a nonsymmetrized form (see Section 3). A sin-gle AMG V-cycle with two grid levels is applied to the healthy subdomain with one presmoothingand one post-smoothing sweep of a Gauss–Seidel smoother. We conducted several different numer-ical tests to study the convergence of the proposed preconditioner. The examples in this sectioninclude crack problems that investigate: (1) the length of cracks; (2) the number of cracks and theeffects of mesh refinement; (3) a three-point bending problem with multiple arbitrarily distributedcracks; and (4) analysis of multiple cracks in an unstructured mesh. For each of these exampleproblems, the stopping criteria were set to a residual tolerance of 10�10 in the k.k2 norm. Thepreconditioners were implemented in MATLAB.


XFEM_DD 321

wa

w

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.730

40

50

60

70

Nu

mb

er o

f it

erat

ion

s

Length of the crack

AMG brute force

Inexact Schwarz

(b)

Figure 7. Convergence of the preconditioners on the length growth problem. (a) Geometric definition of theproblem, (b) number of iterations to converge as function of crack length.

4.1. Crack growth under pure mode I loading

The first example compares the convergence of the AMG brute force preconditioner to the oneobtained by the inexact Schwarz-AMG preconditioner as function of increasing crack size. Theinexact Schwarz-AMG preconditioner is a multiplicative Schwarz method in which the healthy sub-domain is approximated with a single AMG cycle, and the cracked subdomain is solved exactly.Note that, in this problem, the cracked subdomain grows and follows the crack path as it propa-gates in the domain. The problem considered is a square plate with dimensions of 1 � 1 Œm�, and acrack propagating in mode I from one end to the other end of the plate as shown in Figure 7(a). Auniformly distributed tensile load of 1000 ŒN �m�1� is applied to its top and bottom edges. Crackpropagation is performed quasi-statically from an initial crack of length ainitial D 0.12 Œm� to a finallength of af inal D 0.71 Œm�. A mesh of 30 � 30 elements is chosen for this problem. Figure 7(b)shows the number of iterations required by each method as a function of the crack length. Thisnumerical experiment reveals that the inexact Schwarz-AMG preconditioner exhibits a faster con-vergence than that of the AMG brute force preconditioner, but when the size of the crack increases,the difference between the number of iterations needed by the two methods decreases.

4.2. Mesh refinement and the effect of multiple cracks

4.2.1. Multiple cracks in a plate. We study the convergence of the preconditioners when the num-ber of cracks is increased, and the mesh is refined. The geometric description of the domain, its load,and boundary conditions are identical to the previous example shown in Figure 7. In this case, thenumber of cracks is varied from one to four, all the cracks being horizontal and of the same length:a D 0.9m. The cracks are equally spaced with the physical distance between them being kept thesame even as the mesh is refined. In this problem, each crack owns its own subdomain, that is, up tofour cracked subdomains are considered as shown in Figure 8.

The convergence results comparing brute force AMG, inexact multiplicative Schwarz-AMG, andexact Schwarz preconditioners with GMRES, are reported in Table I. An exact Schwarz precondi-tioner is obtained when direct solves are performed on both the healthy and cracked subdomains.We consider the following three mesh refinements: 30� 30, 50� 50, and 80� 80 elements.

Table I shows three trends. The difference in the number of iterations needed to converge betweenthe following:

(i) AMG brute force and inexact Schwarz-AMG preconditioners are decreasing as the mesh isrefined,

(ii) AMG brute force and inexact Schwarz-AMG methods are increasing with the number ofcracks present in the problem,

(iii) inexact Schwarz-AMG and exact Schwarz preconditioners are fairly constant, and(iv) inexact additive and multiplicative Schwarz-AMG is about 40% of the latter.



a

w

w

(a)

(b) (c)

Figure 8. XFEM discretization of a plate four cracks. (a) Problem description, (b) mesh refinement with30 � 30 elements, and (c) mesh refinement with 50 � 50 elements. Note the “dilution” effect, where the

cracks are separated by more elements.

Table I. Effect of the number of tracks and the mesh refinement on the convergence of the preconditioners.

Mesh AMG Additive Multiplicative Exactsize brute force Schwarz-AMG Schwarz-AMG Schwarz

30� 30 73 82 59 57One crack 50� 50 83 108 75 70

80� 80 101 142 97 9330� 30 118 129 66 61

Two cracks 50� 50 145 169 121 11680� 80 174 221 156 15030� 30 161 171 123 118

Three cracks 50� 50 195 225 161 15580� 80 231 293 208 20030� 30 208 204 154 150

Four cracks 50� 50 255 275 205 19780� 80 302 359 264 255

The first trend is interesting: the cracks are fixed at the same physical locations, but more ele-ments are introduced in between the cracks as a result of the h-refinement. This causes a “dilution”effect of the enriched nodes in the stiffness matrix, that is, the enriched nodes in the matrix aregetting more separated and decoupled from each other. As a result, the AMG brute force methodovercomes some of its coarsening difficulties, and the difference between the Schwarz and AMGmethods decreases.


XFEM_DD 323

The second trend is expected. Additional cracks introduce more enriched degrees of freedom,and hence the performance of the brute force AMG degrades much faster than that of the Schwarzmethods.

Finally, the third trend shows that the inexact Schwarz approach performs only slightly worsethan the exact method, suggesting that the “inexact” approach provides a good approximation.

4.2.2. A three-point bending test with multiple cracks. Another series of experiments is conductedto study the effect of mesh refinement and multiple cracks. However, in this example, the densityof enrichments (ratio of enriched nodes versus overall nodes) is kept constant. The example cho-sen is that of a beam under three-point bending test, as shown in Figure 9(a), with W D 1 Œm�

and H D 0.2 Œm�. The beam is supported by a hinge and a simple support, and a concentratedpoint load of P D 103 ŒN � is applied at the top middle of the beam. Hence, the bottom part ofthe beam is subject to tension, and all cracks open up. This example test is important to civil engi-neers, and the multiple-crack formation has been studied by Ba Lant [43–45]. In his work, Ba Lantdetails how almost collinear cracks appear on smooth surfaces such as concrete roads or dry lakesurfaces. In this study, because of the close proximity of the cracks, only one cracked subdomain isconsidered to properly capture the coupled crack interactions. We choose an enrichment density of�cracks D ¹0.05, 0.075º, and the cracked subdomain is kept with the same physical dimensions, thatis, the cracks are localized within this subdomain. We considered five different mesh refinements.The convergence results are shown in Figure 10. First, it is clear that the Schwarz preconditionersare less sensitive to the mesh refinement than that of the AMG brute force preconditioners. More

W

H

P

(a)

(b)

(c)

Figure 9. Three-point bending test. (a) Geometric definition of the problem, (b) enrichment and crackedsubdomain, and (c) deformed shape.



0 1000 2000 3000 4000 5000 6000 7000 8000 90000

50

100

150

200

250

300

350

Number of nodes on the mesh

Nu

mb

er o

f it

erat

ion

s

AMG brute force =0.05AMG brute force =0.075Additive Schwarz AMG =0.05Additive Schwarz AMG =0.075Multiplicative Schwarz AMG =0.05Multiplicative Schwarz AMG =0.075Exact Schwarz AMG =0.05Exact Schwarz AMG =0.075

Figure 10. Convergence of brute force AMG, inexact Schwarz-AMG, and exact Schwarz preconditionersin the three-point bending test. Crack densities of �cracks D 0.05 and �cracks D 0.075 and different mesh

refinements have been considered.

importantly, notice that the Schwarz preconditioners are insensitive to variation of the crack den-sity whereas the AMG brute force preconditioners converge slower when the density of enrichmentincreases. This results imply that, for a given partition of the domain into healthy and cracked sub-domains, the Schwarz preconditioners will converge in the same number of iterations no matterwhat the density of cracks are. The inexact Schwarz-AMG method gives satisfactory performancecompared with the exact Schwarz approach, which is quite encouraging. Finally, the penalty for notupdating the Schwarz projections multiplicatively is within the small factor that one expects fromthe classic elliptic case, offering a trade of extra concurrency for the convergence rate [46].

4.3. Multiple cracks with different lengths and orientations

In this example, we investigate the convergence of the preconditioners on a plate containing threecracks of different lengths and orientations. Two strategies, illustrated in Figure 2, are used topartition the domain into healthy and cracked subdomains. In the first case, the cracked subdomainowns all the cracks (a single cracked subdomain) whereas, in the second approach, each crack ownsits own local subdomain (multiple cracked subdomains). In addition, an additive Schwarz method isinvestigated and compared with the other methods [47]. The plate dimensions, boundary conditions,and loading are given in Figure 7(a). The mesh and the partitioned domains are shown in Figure 11,the convergence results are plotted in Figure 12 and summarized in Table II.

It is clear that both domain decomposition strategies give excellent results compared with theAMG brute force preconditioner. The AMG performance is poor, which is mainly attributed to thecracks having two sets of tip functions inside the domain, and in close proximity to each other.Moreover, the different orientation of the cracks makes it significantly harder for the AMG to gener-ate appropriate aggregates, and the coarsening of these special functions, significantly deterioratesits performance.

As expected, the multiplicative inexact Schwarz method with a single cracked subdomain givesslightly better performance than its counterpart with multiple cracked subdomains. The singlecracked subdomain has converged in 50 iterations, whereas it converged in 63 iterations when mul-tiple subdomains are introduced. This behavior is caused by the fact that all the cracks are solvedconcurrently in the single cracked subdomain case, whereas solving them in a sequential mannerintroduces a small delay in the coupling of these cracks. The convergence of the additive Schwarzmethod is slightly worse than that of the multiplicative Schwarz method as has been noticed before.


XFEM_DD 325

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Figure 11. Domain decomposition and mesh of a plate with three cracks with different lengths and orien-tations. (a) Decomposition with multiple cracked subdomains, and (b) decomposition with a single cracked

subdomain.

0 20 40 60 80 100 120 140 160 180 20010 16

10 14

10 12

10 10

10 8

10 6

10 4

10 2

100

Number of iterations

No

rm o

f th

e re

sid

ual

AMG brute forceAdditive Schwarz AMG multipleAdditive Schwarz AMG singleMultiplicative Schwarz AMG multipleMultiplicative Schwarz AMG singleExact Schwarz AMG multipleExact Schwarz AMG single

Figure 12. Comparison of the convergence rate for the decomposition strategies shown in Figure 11.

Table II. Summary of the convergence results for the problem considered in Figure 11.

AMG Additive Multiplicative Exactbrute force Schwarz-AMG Schwarz-AMG Schwarz-AMG

Single “crack” subdomain 190 68 50 48Multiple “crack” subdomains 190 74 63 59

4.4. Multiple cracks on an unstructured mesh

Finally, we present here a model of a fractured rail, typically observed in high-speed railways. Giventhe geometry of the cross section of the rail, it is not easy to mesh the domain with a structured mesh;therefore, we employ an unstructured mesh consisting of 3000 quads. In the analysis, we considerthe following cracking scenarios:

� Case 1 (one crack): crack #1� Case 2 (three cracks): cracks #1–#3



� Case 3 (five cracks): cracks #1–#5� Case 4 (seven cracks): cracks #1–#7

A uniform tension is applied at the top of the rail while its base is clamped as shown in Figure 13.For each analysis, all the preconditioners (additive, multiplicative, exact, and brute force AMG) aretested.

The convergence behavior for each of the cases studied is reported in Table III. It can clearly beobserved that the convergence rate of AMG brute force preconditioner is very poor as comparedwith the other methods. This is probably caused by the fact that the crack tip enrichment is greatlydependent on the size of the elements. Because the mesh is unstructured, the values of the crack tipenrichment are more diverse and become harder to handle properly when they are aggregated by the

(a) (b)

(c)

8000

6000

4000

2000

0

2000

4000

6000

(d)

10000

1 23

45

6

7

Figure 13. Unstructured mesh of a rail (a) geometry and boundary conditions, (b) mesh and domaindecomposition, (c)enrichments along the cracks, and (d) �yy stresses on undeformed mesh.

Table III. Summary of the convergence results for the problemconsidered in Figure 13.

AMG Additive Multiplicative ExactBrute force Schwarz-AMG Schwarz-AMG Schwarz-AMG

Case 1 62 44 37 22Case 2 123 58 45 34Case 3 309 92 70 64Case 4 358 110 82 77


XFEM_DD 327

multigrid preconditioner. This example also shows the versatility of the proposed preconditionersbecause it is not really affected by the use of an unstructured mesh.

5. CONCLUSION

The paper presents an inexact Schwarz-AMG preconditioning approach for crack problems mod-eled by XFEM in order to retain the convergence properties of smoothed aggregation AMG forelastostatic problems. The preconditioner is based on the domained partition into a “healthy” (orunfractured) part and a “cracked” part that include all the XFEM-enriched degrees of freedom.Whereas the solution in the “healthy” subdomain is approximated inexactly by a single multigrid V-cycle, the “cracked” subdomains (single or multiple) are solved exactly. This alleviates the need toredesign the AMG coarsening algorithms to handle enriched degrees of freedom. Numerical exam-ples on various crack problems clearly illustrate the superior performance of this approach over abrute force AMG preconditioner, in particular when multiple random cracks are fully embeddedin the structure. This approach allows one to apply AMG schemes to XFEM problems with noadjustment of the original AMG method.

ACKNOWLEDGEMENTS

The authors are grateful to the funding support provided by the Department of Energy under grantDE-SC0002137.

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s NationalNuclear Security Administration under contract DE-AC04-94AL85000, and his co-authors are grateful forthe support provided by Sandia National Laboratories.

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