Comput MechDOI 10.1007/s00466-016-1366-z

ORIGINAL PAPER

A conforming to interface structured adaptive meshrefinement technique for modeling fracture problems

Soheil Soghrati1,2 · Fei Xiao1 · Anand Nagarajan1

Received: 24 September 2016 / Accepted: 9 December 2016© Springer-Verlag Berlin Heidelberg 2016

Abstract A Conforming to Interface Structured AdaptiveMesh Refinement (CISAMR) technique is introduced for theautomated transformation of a structured grid into a con-forming mesh with appropriate element aspect ratios. TheCISAMR algorithm is composed of three main phases: (i)Structured Adaptive Mesh Refinement (SAMR) of the back-ground grid; (ii) r -adaptivity of the nodes of elements cut bythe crack; (iii) sub-triangulation of the elements deformedduring the r -adaptivity process and those with hanging nodesgenerated during the SAMR process. The required consid-erations for the treatment of crack tips and branching cracksare also discussed in this manuscript. Regardless of the com-plexity of the problem geometry and without using iterativesmoothing or optimization techniques, CISAMRensures thataspect ratios of conforming elements are lower than three.Multiple numerical examples are presented to demonstratethe application of CISAMR for modeling linear elastic frac-ture problems with intricate morphologies.

Keywords CISAMR · Mesh generation · Linear elasticfracture · Branching crack · r -adaptivity

1 Introduction

The finite element method (FEM) is one of the most popu-lar numerical techniques for simulating fracture mechanics

B Soheil Soghratisoghrati.1@osu.edu

1 Department of Mechanical and Aerospace Engineering, TheOhio State University, 201 West 19th Avenue, Columbus,OH, USA

2 Department of Materials Science and Engineering, The OhioState University, 2041 N. College Road, Columbus, OH, USA

problems.However, creating appropriate conformingmeshesformodeling such problems, specially in the presence ofmor-phological complexities such as crack branching, could be achallenging and time consuming task. To avoid this issue,one can implement more simplified techniques such as theinter-element separationmethod [1], which obviates the needfor the adaptation of the mesh to the crack geometry. In thisapproach, cracks are only allowed to form and propagatealong the element edges by removing the node connectiv-ity, although this constraint can undermine the fidelity if thebackground mesh is not highly refined.

Significant research effort has been dedicated towarddeveloping robust algorithms for creating appropriate con-forming meshes for modeling problems with embeddedcracks (strong discontinuities). For example, a nodal relax-ation and remeshing technique is introduced in [2,3] to ensurethat the crack path overlaps with the element edge. One canalso use moving meshes relying on the Delaunay triangula-tion scheme to simulate crack growth [4]. Azócar et al. [5]introduced a Lepp–Delaunay based technique for generatingan initial 2D mesh and updating that for modeling propa-gating cracks without reconstructing the whole mesh. Khoeiet al. [6] developed an adaptive mesh refinement algorithmrelying on the super convergent patch recovery technique todetermine the optimal element size. Meyer et al. [7] usedan iterative numerical approach relying on hierarchical pre-conditioners for the efficient simulation of 2D crack growthproblems by preserving the initial hierarchical data structure.In the method proposed by Askes et al. [8], an iterative h/r -adaptive scheme is employed to optimize the discretizationfor modeling such problems. Areias et al. [9,10] proposed anedge rotation technique for efficient local refinement andgen-erating conforming meshes for crack growth and nucleationproblems [11–13].More recently, the universal meshmethod[14] is introduced, which employs an iterative relaxation

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algorithm to relocate the vertices of background triangularelements within a distance from the crack to generate a highquality conforming mesh.

Creating appropriate conforming elements for simulatingsingular stresses at the crack tip is crucial to the accuracyof linear elastic fracture finite element simulations. Sev-eral techniques are used to alleviate the error in this region,including the collapsed quadrilateral quarter-point elementsintroduced in [15,16]. In this approach, the non-vertex nodesof element edges connected to a crack tip are located at aquarter of the length of each edge towards the tip to simulatethe stress singularity. Rashid [17] introduced the ArbitraryLocal Mesh Replacement (ALMR) method, in which a mov-ing mesh patch is employed to approximate the field inthe vicinity of a growing crack tip. A weak statement ofcompatibility is then used between this patch of elementsand the non-matching background mesh by introducing aset of force-type unknowns. Intrinsic and extrinsic cohesivezone models [1,18] are also widely used for approximatingthe mechanical behavior of cracks. Intrinsic cohesive zonemodels generate interface elements before the start of thesimulation [19,20], whereas extrinsic cohesive zone modelsrequire creating interface elements as the simulation pro-ceeds [21]. Thus, the implementation of the latter techniquerequires the adaptive refinement and coarsening of the meshthroughout the solution process [22,23].

While the mesh generation techniques mentioned abovecan successfully create appropriate standard FE models forsimulating a variety of fracture problems, the computationalcost and complexity associatedwith this process could still besignificant. Further, creating acceptable conforming meshesfor problems with complex morphologies and in particularthose involving branching cracks would be a challengingtask. In order to resolve this issue, one can employ mesh-free methods to fully eliminate the need for the constructionof conforming meshes [24]. For example, Belytschko et al.[25] introduced the Element Free Galerkin (EFG) method,which employs a set of moving least-square interpolants toapproximate the weak form of the problem. In this method,the effect of a crack can easily be incorporated in the solu-tion by removing the nodes located on one side of that crackfrom the domain of influence of the nodes on the oppositeside [26,27]. Rabczuk and Belytschko [28] presented an h-adaptive refinement scheme for simulating fracture problemsusing the EFG method, which is also expanded to modelingcrack propagation and shear bands [29]. Among other meth-ods in this category we can mention the Reproducing KernelParticle Method (RKPM) [30] and the EFG-Particle method[31]. Also, the Peridynamic method [32–34] is a non-localmeshfree method for approximating damage and fractureproblems, in which spatial derivatives are replaced with anintegral equation. The domain is then discretized using a setof material points, where each two points within a specified

distance are connected to one another using a virtual springwith a critical stretch parameter to replicate the discontinuitycaused by the development of a crack [35,36].

The idea of making the approximate field independentfrom the mesh structure can also be incorporated in the FEformulation to allow the use of nonconforming meshes fordiscretizing the problem. In the phase field method [37–39],the sharp interface model often representing strong discon-tinuities in the FEM is replaced with a continuously varyingdiffuse interface model. A phase field variable is also addedto the governing equations for simulating the crack growth,which can properly simulate the evolution of cracks withcomplex morphologies [40,41]. However, introducing thephase field variable, together with the need for excessiverefinement of elements in the vicinity of the crack to accu-rately capture sharp gradients of this variable within thediffuse interface [42], leads to a considerably higher com-putational cost than an FE simulation relying on the sharpinterface model.

Another class of advanced FEMs, often referred to asmesh-independent FEMs, use an a priori knowledge of thesolution field in the form of enrichment functions to simu-late the strong discontinuity in a nonconforming element cutby a crack. In the embedded discontinuity method [43], thisis achieved by incorporating discontinuous shape functions,together with additional degrees of freedom (DOFs) alongthe crack, in the FE approximation. In another approach, thefield and its gradients are enhanced with embedded disconti-nuities to accurately capture displacement jumps and relativerotations accross the crack path [44–46].

The eXtended/Generalized FEM (X/GFEM) is anothermesh-independent method [47–50], which is one of the mostsuccessful techniques in this category for simulating prob-lems with strong and weak (gradient) discontinuities. TheX/GFEM creates additional generalized DOFs at the nodesof elements intersecting with a crack, as well as neighboringelements (blending elements) in some approaches. Appro-priate enrichment functions computed using the partition ofunitymethod (PUM) [51] are then interpolated at these gener-alized DOFs to reconstruct the discontinuous field along thecrack [52,53], which yield an optimal rate of convergence.Special enrichment functions are often used in the vicinityof a crack tip or at a branching point to accurately simulatethe stress singularity [54–56]. The X/GFEM has also beencombined with adaptive hp-refinement techniques to reducethe approximation error without significant increase in thenumber of DOFs [57,58].

While mesh-independent methods such as X/GFEMenable the use of nonconforming meshes for modeling prob-lems with embedded cracks, the background elements cut bya crack are often subdivided into smaller conforming sub-elements (children elements) to allow accurate numericalquadrature [59,60]. However, the computational cost asso-

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ciated with this process is considerably lower than that ofmesh generation algorithms, as no further treatment (e.g.,the iterative relaxation scheme used in the universal meshmethod [14]) is required to improve the aspect ratios ofchildren elements. Moreover, additional effort is required inthe X/GFEM for computing the enrichment functions andinverse mapping of quadrature points between children andparent elements for accurate numerical integration. The com-putational cost associated with this process is justified by thespecial enrichment functions applied to the elements locatedin the vicinity of the crack tip, which improve both the accu-racy and convergence rate of X/GFEM [61,62]. Note thatperforming a successful X/GFEM simulation often requiresadditional treatments to resolve issues such as enforcingDirichlet boundary conditions (e.g., using the penaltymethod[63]) and ill-conditioning of the stiffness matrix (e.g., via thepre-conditioning scheme introduced in [64]).

The remainder of this manuscript is structured as fol-lows: For completeness, in Sect. 2 we briefly present thestrong and weak forms of the governing equations for mod-eling linear elastic fracture mechanics problems. An efficientalgorithm for identifying the elements of a background struc-tured mesh that intersect with a crack is presented in Sect. 3.The CISAMR algorithm for transforming this backgroundgrid into a conforming mesh is introduced in Sect. 4, whichalso addresses the treatment of branching cracks and spe-cial cases such as cracks that are in close proximity of thedomain boundaries. Five numerical examples are provided inSect. 5 to verify the appropriateness of conforming meshesgenerated using CISAMR for modeling fracture problemsand demonstrate its application for modeling problems withintricate geometries. Final concluding remarks are presentedin Sect. 6.

2 Governing equations

Consider an open domainΩ ⊂ R2 with the outward unit nor-

mal vector n on its boundary, a pre-existing crack denotedby Γc, and boundaries ∂Ω = Γu ∪ Γt corresponding to theDirichlet and Neumann conditions, respectively. The strongform of governing equations for a linear elastic fracture prob-lem can be expressed as: Find the displacement field u suchthat

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

∇ · σ + b = 0 in Ω Equilibrium eqn.σ = C : ε in Ω Constitutive eqn.ε = 1

2

(∇u + ∇uT)

in Ω Kinematic eqn.u = u on Γu Dirichlet BCσ · n = t on Γt Neumann BCt = 0 on Γc Crack condition,

(1)

where ε is the infinitesimal strain tensor, σ is the Cauchystress tensor, b is the body force, t is the traction vector, u is

the prescribed displacement along the Dirichlet boundary, tis the applied traction along the Neumann boundary, and C

is the fourth-order elasticity (Hooke’s) tensor.To derive the weak form of (1), u must be selected from a

set of admissible function space U such that

u ∈ U :={v : Ω → R

2 ∈ H3/2−ε(Ω) ∀ε > 0, u|Γu = 0}

,

(2)

where H3/2 is theHilbertianSobolev space of order 32 and ε is

an infinitesimally small scalar parameter. The smoothness ofthis fractional space is intermediate between integer Sobolevspaces of order 1 and 2, i.e., if v ∈ H3/2 then v ∈ H1 butv /∈ H2, which allows the discontinuity of the displacementfield across Γc. The weak form of the problem can then bewritten as: Find u ∈ U such that ∀v ∈ U

∫

Ω

LTuT · CLvdΩ +∫

Ω

vb dΩ +∫

Γt

vt dΓ = 0, (3)

where the differential operator L is given by

L =[ ∂

∂x 0 ∂∂x

0 ∂∂y

∂∂y

]

. (4)

TheGalerkinFEMapproximation of (3),uh , can be evaluatedby using a subset of the allowable function space Uh ⊂ U

composed of Lagrangian shape functions. This requires dis-cretizing Ω ∼= Ωh ≡ ∪m

i=1Ω i into m finite elements thatconform to the crack geometry. Note that the fidelity of theresulting FE simulation depends on the quality of this con-forming mesh, i.e., minimizing the geometric discretizationerrors and creating elements with small aspect ratios, spe-cially in the vicinity of the crack tip.

3 Crack/grid interaction

Before describing the CISAMR technique, in this sectionwe introduce an efficient algorithm for identifying relativelocations of the nodes and edges of a structured mesh withrespect to a pre-existing crack. In the context of theX/GFEM,this task is often accomplished using a level-set function [65],which yields an implicit representation of the crack geometry.Implementing a level set approach in the X/GFEM has theadditional advantage that it can also be used for evaluatingthe enrichment functions.

In this work, we employ an explicit description ofcracks using Non-Uniform Rational B-splines (NURBS)[66], which are parametric functionswidely used in theCom-puter Aided Design (CAD). Complex geometries such asbranching cracks can easily be modeled using NURBS by

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decomposing the geometry into multiple NURBS curves,each representing a single branch of the crack. A NURBScurve C(u) is a function of parametric coordinate u, whichis composed of n B-splines Mp

i (u)|ni=1 of order p. C(u) isevaluated by interpolating the B-spline functions over a setof n control points with physical coordinates xi and weightswi as

C(u) =n∑

i=1

xiwi Mpi (u)

∑nj=1 w j M

pj (u)

. (5)

A set of n+ p+1 breakpointsU = {u1, u2, u3, . . . , un+p+1}(knot vector) is employed to divide the parametric spaceu into smaller intervals. It must be noted that C(u) is aC0-continuous function, which is not differentiable at thebreakpoints.

As noted previously, the aim of CISAMR is to transforma structured grid composed of quadrilateral elements withaspect ratios of one into a hybrid conforming mesh for mod-eling fracture problems. Thus, a prelude to the modelingprocess is to locate the background elements that intersectwith one or more branches of a crack. Further, modelingthe strong discontinuity along a cracks requires creating twonodes with the same coordinates but different connectivities(equation numbers) at each side of the crack. Hence, it is alsoessential to identify relative locations of nodes of the noncon-forming background elements with respect to a pre-existingcrack and its branches.

While NURBS provide an exact representation of thecrack geometry, determining the relative location of aNURBS curve with respect to all nodes of a backgroundmesh and evaluating its intersection points with the ele-ment edges would be a computationally demanding task.This computational burden could be unfeasibly high whenusing a refined background grid for modeling a problemwith multiple micro-cracks or branching cracks, as mil-lions of nonlinear equations must be solved throughout thisprocess. Note that in such problems, each NURBS curvecorresponding to a crack might only intersects with a hand-ful of background elements; therefore determining whetherevery element of the mesh is cut by every crack and itsbranches would significantly undermine the efficiency. Thus,using a simple, robust, and yet efficient algorithm for thisinitial phase of the modeling process is crucial to minimiz-ing the overall complexity and computational cost associatedwith the implementation of CISAMR. In this manuscript,we employ the following three-step algorithm to extract thepreliminary information required for transforming the back-ground grid into a hybrid conforming mesh:

• Step 1 For each crack, we first employ the quad-treesearch algorithm [67] to identify background elementsholding crack tip(s) and/or branching point(s). In thissearch algorithm, the domain is first divided into four

Second crack tip

First crack tip

Crack path

Element holdingthe crack tip

Edges intersectingwith crack path

Nodes belongingto Side 1

(a)

(b)

Fig. 1 a Implementation of the quad-tree search algorithm to identifybackground elements holding the crack tips;b identifying the remainingelements cut by the crack and the relative locations of their nodes withrespect to the crack using a sequential algorithm

quadrants, followed by selecting the quadrant that con-tains one of the crack tips (branching points). Asschematically shown in Fig. 1a, this process is recur-sively continued until the element containing the cracktip (branching point) is identified. The quad-tree searchis a highly efficient algorithm that operates in O(logN )

time; thus locating the elements holding the crack tipsand branching points via this technique would be com-putationally inexpensive. It is worth mentioning that inthe NURBS representation of a crack morphology, coor-dinates of crack tips (branching points) are given by x1and xn , i.e., the coordinates of the first and the last controlpoints.

• Step 2 Pick one of the elements containing the crack tip(branching point) and identify which edge (or node) ofthat element intersects with the crack. The neighboring

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Fig. 2 Refined mesh afterapplying the proposed SAMRalgorithm to the originalbackground grid

Crack

Hangingnode

element that shares that edge would be the second ele-ment cut by the crack. In the special case that a crackpasses through one of the element nodes, the search mustbe expanded to all three neighboring elements sharingthat node to identify which one is located on the crackpath. As shown in Fig. 1b, this process is continueduntil reaching the element holding the other crack tip(branching point). This sequential algorithm can effi-ciently identify all the elements cut by the crack withminimum number of calculations.

• Step3Asimilar approach as that described in the previousstep is employed to identify the nodes of nonconformingbackground elements that are on each side of the crack(Fig. 1b). Starting with one of the elements holding thecrack tip (branching point), we identify the nodes of thiselement located on one side of the crack (marked as side1) and those on the opposite side (marked as side 2).We then move to the adjacent element cut by the crack,which is already identified in Step 2, to determine therelative locations of nodes of this elements with respectto the crack. Assume that nodes Ni and N j are the nodesdefining edge E of this element, and Ni is assigned to side1 after visiting the previous element. Node N j belongs toside 1 if E does not intersect with the crack and otherwiseit belongs to side 2. We repeat this sequential process bymarching toward the element holding the other crack tip(branching point) to determine the side corresponding toall nodes of the nonconforming background elements.

4 CISAMR modeling of cracks

After identifying the background elements intersecting withthe crack and the relative locations of their nodeswith respect

to that using the algorithm described in Sect. 3, the CISAMRis employed to transform this grid into a conforming mesh.The CISAMR employs a non-iterative algorithm to gener-ate this conforming mesh, which ensures that aspect ratiosof resulting elements are lower than three and the geomet-ric discretization error in negligible. As noted previously,this method does not require the use of computationallydemanding iterative smoothing or optimization schemes toimprove the quality of elements, which highly facilitatesits implementation for modeling geometrically elaborateproblems. The CISAMR algorithm is composed of threemajor steps, involving the h-adaptivity, r -adaptivity, and sub-triangulation of background elements. In the remainder ofthis section, we describe in more detail each step of thisalgorithm, together with the required considerations for thetreatment of crack tips and branching cracks.

4.1 h-adaptivity

As the first step, a customized Structured Adaptive MeshRefinement (SAMR) algorithm is employed to achieve thedesired level of refinement along the crack and in the vicin-ity of its tip(s), as shown in Fig. 2. While this step is anoptional phase of the CISAMR algorithm, applying that isessential for minimizing the geometric discretization error,as well as the accurate approximation of stress concentra-tions, without using an exceedingly fine background grid.During this process, at each level of refinement we subdividethe elements cut by the crack and their selected neighboringelements into four sub-quadrangles. A neighboring elementis selected for refinement if it shares at least one node withone of the nonconforming elements, such that the distanced between this nodes and the intersection point of the crackwith one of the edges connected to that is d < 0.5h, where

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h

d <0.5hi Cracktip

(a) (b)

Fig. 3 Identifying the neighboring elements that must be refined during the SAMR phase for elements a cut by the crack and b holding the cracktip

h is the length of that edge (Fig. 3a). In other words, if oneof the midpoints of the edges connected to a mesh node islocated on the opposite side of the crack compared to thatnode, all the neighboring elements sharing that node are sub-jected to SAMR. Exception is applied to elements holding acrack tip, where all neighboring elements sharing at least oneof the nodes of such elements must be refined (Fig. 3b). Also,as shown in Fig. 2, we recursively apply additional levels ofSAMR to each element holding a crack tip and its neighbor-ing elements to achieve a more accurate approximation ofthe stress field. It is worth mentioning that the SAMR of thebackground mesh leads to the generation of hanging nodeson the edges of adjacent elements (Fig. 2), which will betreated in the last step of the CISAMR algorithm describedin Sect. 4.3.

One of the main objectives of the SAMR phase is reduc-ing the geometric discretization error and providing a moreaccurate approximation of the field near the crack tip. How-ever, in certain conditions, execution of this step might beessential for the construction of conforming triangular ele-ments with appropriate aspect ratios in the final step of theCISAMR algorithm. In order to handle such cases, the fol-lowing constraints must be satisfied during the SAMRphase:(i) the refinement must be continued until none of the back-ground elements havemore than one hanging node on each oftheir edges, although they can have multiple hanging nodeson multiple edges; (ii) none of the edges of an element canintersect with a crack more than once; (iii) if two cracksare located in close proximity, the SAMR must be recur-sively continued until none of the elements cut by one of thecracks or its neighboring elements intersect with the othercrack. The only exception to the latter case would be perti-nent to elements holding a crack branching point, which willbe discussed in detail in Sect. 4.2. Additionally, more levelsof SAMR might be required when a crack is located in the

close proximity of the domain boundaries, which will alsobe addressed in the following section.

4.2 r-adaptivity

A non-iterative r -adaptivity algorithm is then implementedto move selected nodes of the nonconforming backgroundelements to the crack in the direction of element edges,as depicted in Fig. 4. To accomplish this, we first calcu-late the coordinates of intersection points of the crack withthe edges of such elements. Assuming that d is the dis-tance between the intersection point and one of the endnodes of the corresponding edge with length h, the new loca-tion of this node is determined according to the followingalgorithm:

1. If none of the edges connected to that node intersect withthe crack: the mesh node maintains its current location.

2. If only one of the edges connected to that node intersectswith the crack (Fig. 5a):

(a) If d ≥ 0.5h: the node maintains its current location.(b) If d < 0.5h: the node moves to the crack/edge inter-

section point.

3. If two perpendicular edges connected to that mesh nodeintersect with the crack (Fig. 5b):

(a) If d1 ≥ 0.5h1 and d2 ≥ 0.5h2: the node maintains itscurrent location.

(b) If d1 < 0.5h1 and d2 ≥ 0.5h2: the node moves to theintersection point at distance d1.

(c) If d1 < 0.5h1 and d2 < 0.5h2 and d1 ≤ d2: thenode moves to the intersection point at distance d1and the other intersection point at distance d2 isdiscarded.

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Fig. 4 Deformed meshobtained after applying ther -adaptivity to the refinedstructured mesh shown in Fig. 2.The arrows show the directionsthe nodes of the backgroundelements are relocated in thisprocess

Collapse all nodeson the crack tip

Snap to the closestintersection point

4. If the node belongs to an element that holds the crack tip:

(a) If the crack tip has the same coordinates as one of thenodes of the element: no action is required.

(b) If the crack tip is on one of the edges of the element:the nodes defining that edge are snapped to the cracktip.

(c) If the crack tip is inside the element: annihilate theelement by snapping all its nodes to the crack tip, asshown in Fig. 4.

The constraints imposed on Step 1 (h-adaptivity) of theCISAMR algorithm ensure that cases 1, 2, and 3 in the algo-rithm above do not contradict one another and no other casescenario would be feasible. Thus, the proposed non-iterativer -adaptivity algorithm can be employed to determine thenew location of each mesh node independently, i.e., with-out accounting for the proximity of adjacent nodes to thecrack/edge intersection points. As discussed next, additionalconsiderations would be required for the treatment of twospecial cases that might be encountered during this process.

• Special case 1 Applying the proposed r -adaptivity algo-rithm to nonconforming elements located along thedomain boundaries might result in moving their nodesaway from the boundary. Clearly, such cases mustbe avoided not to introduce a significant geometricdiscretization error resulting from compromising theintegrity of domain morphology. Figure 6 illustrates fourdifferent case scenarios encountered in elements cut by acrack along the domain boundary, where applying the r -adaptivity is only allowed in the element shown in Fig. 6a.In this case, the relocated node moves along the bound-ary and not away from that, which maintains the originalshape of the domain. If dp and da are the distances

between a node on the boundary and crack/edge inter-section points corresponding to the edges perpendicularto and along that boundary, respectively, the r -adaptivityin elements sharing that node is unallowable if:

1. dp < 0.5h, and either da > dp or the element edgealigned with the boundary does not intersect with thecrack (Fig. 6b, c).

2. The element located on the boundary holds a cracktip (Fig. 6d).

As shown in Fig. 7, the unallowable case scenarios men-tioned above can easily be resolved by applying more levelsof SAMR before performing the r -adaptivity. This can beconsidered as a fourth constraint on the SAMR phase (seeSect. 4.1), which must be applied to applicable elementsrecursively to avoid moving the mesh nodes away from theboundary. Note that the extra computational cost associ-ated with this task must not be considered as a limitationof CISAMR, as it is also essential to reduce the error inapproximating the field in such regions of the domain.

• Special case 2 In order to handle the crack branching phe-nomenon (Fig. 8a), we apply the r -adaptivity algorithmintroduced for elements located along the crack path witha minor modification. In this approach, we first visit thenodes of the element holding a branching point to relocatethem according to the regular r -adaptivity algorithm, i.e.,based on the proximity of each node to the crack/edgeintersection points, as described earlier in this section.Next, we identify the node in the resulting deformed ele-ment with the shortest distance to the branching pointand relocate that node to the branching point, as shownin Fig. 8b.

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Mesh nodemaintains its

locationCase 2(a)

d > 0.5h

Case 2(b)

d < 0.5h

Node snaps tointersection point

Crack

Intersectionpoint

d >0.5h

Case 3(b) Case 3(c)

1 1

d > 0.5h2 2

d <d1

Discard 2 intersection

point

2

Case 3(a)Mesh node

maintains itslocation

Node snaps tointersection point

at distance d 1

d <0.5h1 1

d > 0.5h2 2

d <0.5h1 1

d < 0.5h2 2

Node snaps tointersection point

at distance d 1

nd

(a)

(b)

Fig. 5 Different case scenarios for relocating a mesh node during the r -adaptivity phase based on its relative distance to the intersection points ofthe crack with the element edges connected to that node

Domainboundary

Crack

d <0.5ha

d >0.5hp

d <0.5hp2

d >0.5hp1

d <0.5hp2

d <0.5hp1

Crack tip

(a) (b) (c) (d)

Fig. 6 Applying the r -adaptivity to nonconforming elements located along the domain boundary: a allowable case; b, c, d unallowable cases,where mesh nodes move away from the boundary

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Hangingnode

App

ly m

ore

SAM

R le

vels

App

ly r

-ada

ptiv

ity

Cracktip

(a) (b) (c)

Fig. 7 Resolving the unallowable cases shown in Fig. 6b–d by applying additional levels of SAMR before performing the r -adaptivity

Branchingpoint

Regularr-adaptivity

Moving closestnode to the

branching point

(a) (b)

Fig. 8 Applying the modified r -adaptivity algorithm to a background element holding a crack branching point. The arrows in figure b show thedirections in which background nodes are relocated during this process

Note that one of the constraints of the h-adaptivity phaseof the CISAMR is that the refinement must be recursivelycontinued until none of the element edges intersect withmore than one branch of the crack. For an element holdingthe branching point, it is also not desirable that two distinctbranches of the crack intersect with two perpendicular edgeswith distances d1 and d2 to the node shared between themsuch that that d1 < 0.5h and d2 < 0.5h. This case scenario,although extremely rare, could result in a conflict during ther -adaptivity of such nodes. Hence, we add another constraintto the SAMR phase to resolve this conflict by recursivelyrefining the elements holding a branching point if this specialcase is encountered. Nevertheless, the treatment of the crack

branching in CISAMR is straightforward and still does notrequire an iterative smoothing process to improve the aspectratios of resulting elements.

4.3 Sub-triangulation

As the last step, we sub-triangulate all the elements deformedduring the r -adaptivity process, as well as elements withhanging nodes on their edges to create the final conform-ing mesh (Fig. 9). Note that after applying the r -adaptivity,which is the main difference between CISAMR and the pro-cess of creating integration sub-elements in X/GFEM, theremaining nonconforming elements are diagonally cut by the

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Fig. 9 Creating the finalconforming mesh bysub-triangulating the deformedbackground mesh shown inFig. 4

Double-diagonalsub-triangulation

Quarter-point cracktip elements

Largest angle Removing hanging node by sub-triangulation

Hangingnode

Crackdouble-diagonal

Removinghanging

node

(a) (b)

Fig. 10 Eliminating the hanging nodes by the sub-triangulation of background elements: a regular (undeformed) element with two hanging nodes;b special case, where the background element with a hanging node is deformed during the r -adaptivity phase and cut by the crack

crack. Thus, despite the presence of crack tips and branch-ing points, a simple diagonal subdivision algorithm can beemployed to create a set of conforming triangular elements.To ensure that aspect ratios of these sub-elements are lowerthan three, whenever possible, we cut the background quad-rangle along the diagonal emanating from its largest angle.Following this simple rule results in up to four times loweraspect ratios for the resulting conforming elements comparedto cutting the background element along the other diagonal.If the element is already cut by the crack along the diago-nal corresponding to its smallest angle, we can implementa double-diagonal sub-triangulation scheme, as shown inFig. 9. This simple approach avoids the construction of sub-elements with poor aspect ratios by cutting the backgroundelement along both its diagonals, which results in four con-forming sub-triangles. Note that this scheme would not benecessary when the smallest angle of the deformed back-ground element is larger than 60◦, as in this case using thediagonal emanating from this angle still yields sub-elementswith acceptable aspect ratios (i.e.,<3). Note that, as shown inFig. 9, quarter point sub-elements can be used in the vicinityof the crack tip to more accurately approximate the corre-sponding singular stress field.

In addition to the elements deformed during the r -adaptivity process, we expand the sub-triangulation to ele-ments with hanging nodes to obtain a hybrid conformingmesh (Fig. 9). Other techniques such as constraining theDOFs associated with hanging nodes [68] or modifyingthe basis functions of corresponding elements by addingC0-continuous shape functions can also be used to handlehanging nodes [58]. As shown in Fig. 10a, in this work, wesimply eliminate the hanging nodes by sub-triangulating thecorresponding background elements. It must be noted thatif during the SAMR phase a higher level of refinement isused in the vicinity of the crack tip than along the crackpath, some of the elements with hanging nodes might alsointersect with the crack and being deformed after applyingthe r -adaptivity. A similar rare case scenario could occur ifadditional levels of SAMR is required in some regions of themesh due to the close proximity of a crack to the domainboundary or crack branching. As depicted in Fig. 10b, thesingle-diagonal and double-diagonal rules described in thepreceding paragraph must be applied before performing thesub-triangulation for eliminating the hanging nodes to ensurethat the resulting sub-elements have appropriate aspectratios.

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t

t

2a

β

b

0.5 t

0.5t

1Element Aspect Ratio

1.96

Fig. 11 First example problem: domain geometry and boundary conditions. The inset shows a small portion of the conforming elements createdusing CISAMR and the aspect ratios of resulting elements when discretizing the domain using a 200 × 200 background grid (β = 30◦)

0 15 30 45 60 75 90

Analytical

1.6E+05

1.4E+05

1.2E+05

1.0E+05

8.0E+04

6.0E+04

4.0E+04

2.0E+04

β (◦)

Ene

rgy

Rel

ease

Rat

e (J

/m )2 50 50

200 200

Fig. 12 First example problem: CISAMRapproximation of the energyrelease rate for two background meshes with 50 × 50 and 200 × 200resolutions

5 Numerical results

Five numerical examples are provided in this section. Thefirst two examples aim to verify the accuracy for FE simula-tions relying on CISAMRmeshes and the last three exampledemonstrate the ability of this method for solving two-dimensional linear elastic fracture problems with complexgeometries.

5.1 Domain with a slanted crack

In this example, we aim to verify the suitability of the FEmodels constructed using CISAMR for the accurate predic-tion of the fracture energy release rate (G) in a square domainwith a slanted crack, modeled as a straight line, as shown

Fig. 13 Second example problem: deformed shape and stress fieldobtained from CISAMR simulation of the mode I linear elastic fractureproblem using a 50 × 50 background grid for discretizing the domain

in Fig. 11. The domain has a length of b = 10mm, elas-tic moduli E = 10GPa and ν = 0.3, and is subjected tobiaxial normal tractions tv = t = 1N/m and th = 0.5talong its vertical (y-direction) and horizontal (x-direction)edges, respectively. The slanted crack shown in Fig. 11 hasa length of 2a = 0.9mm and passes through the centroidof the domain, forming an angle of β with the x-axis. Theanalytical values of the stress intensity factor (SIF) for modeI and mode II fractures in this problem are given by [69]

KI = t√

πa

2(1 + cos2 β), KII = t

√πa

2sin β cosβ. (6)

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0%

1%

10%

50×50 100×100 200×200

Rel

ativ

e E

rror

in G

Mesh size

0

0.01

0.1

10×10 20×20 50×50 100×100

Err

or

Mesh size

H -norm (slope: 0.53)L -norm (slope: 1.17)

1

2

(a) (b)

Fig. 14 Second example problem: variation of the error associated with predicting a the fracture energy release rate and b L2 and H1 norms of theerror with respect to size of the background grid for the FE simulations of mode I fracture using CISAMR for generating the conforming meshes

400

μm

t = 1 kN/m

t = 1 kN/m(a) (b)

Fig. 15 Third example problem: a domain geometry and boundary conditions; b small portion of the conformingmesh constructed using CISAMRcorresponding to the box shown in figure a

The SIF values can then be employed to evaluate the energyrelease rate as

G = K 2I + K 2

II

E. (7)

To compare the FE results with the analytical values of Ggiven in (7), the approximate energy release rate can be com-puted using the J -integral as

G =∫

Γ

(

Wdx2 − t∂u∂x1

ds

)

, (8)

where Γ is an arbitrary curve with length s, which beginsfrom one side of the crack and ends at a point with the samecoordinate on the opposite side. Also, W is the strain energydensity, and x1 and x2 are local coordinates parallel and per-pendicular to the crack face, respectively.

In order to create FE models for this problem usingCISAMR, we employ two 50× 50 and 200× 200 structuredmeshes for discretizing the domain. Referring to the numberof SAMR levels along the crack path as Nc and the additionallevels of refinement applied to the elements holding the cracktips as Nt , we use Nc = 1 and Nt = 2 for both backgroundmeshes. The CISAMR is then employed to create conform-ing meshes for seven domains with β = 0◦, 15◦, 30◦, 45◦,60◦, 75◦, and 90◦. The inset of Fig. 11 illustrates a small por-tion of the conforming mesh corresponding to the 200×200background grid and β = 30◦, together with the aspect ratiosof resulting elements. It is worth mentioning that in all 14conforming meshes created for this example, the maximumaspect ratio of the conforming elements does not exceed 2.4.The predicted values of G corresponding to different valuesof β, as well as their comparison with analytical results arepresented in Fig. 12. This example shows that the energy

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Fig. 16 Third example problem: FE simulations of a normal stress and b shear stress fields in the unit cell of brittle material with multiplemicro-cracks (Fig. 15a)

100μm

70 μm

t=

1kN

/m

t=

1kN

/m

(b)

(d)

(c)

(a) (b)

(c) (d)

Fig. 17 Fourth example problem: a domain geometry and boundary conditions; b–d portions of the conforming mesh generated using CISAMRcorresponding to the boxes labeled similarly in figure a

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Fig. 18 Fourth example problem: a deformed shape and displacement field; b, normal and shear stress fields corresponding to the boxes labeledsimilarly in figure a

release rate can be approximated with acceptable accuracyusing the high quality conforming meshes constructed usingthe CISAMR for discretizing the domain.

5.2 Mode I fracture: convergence rate

In this example, we study the the error associated withapproximating G, as well as the convergence rates of the L2

and H1 norms of the error for modeling a classical mode Ifracture problem using CISAMR. The values of these normsof the error are computed as

EL2 =√∫

Ω

∥∥u − uh

∥∥2 dΩ, (9)

EH1 =√∫

Ω

(∥∥u − uh

∥∥2 + ∥

∥∇u − ∇uh∥∥2

)dΩ. (10)

The simulations are conducted on a 5 m × 5 m domain withE = 10 GPa and ν = 0.3, and prescribed boundary condi-tions evaluated based on the analytical solution given in [53].Figure 13 shows the deformed shape and approximate stressfield in this problem using a 50×50 background grid for dis-cretizing the domain. The variations of the error associated

with G, L2, and H1 norms of the error for different meshsizes are depicted in Fig. 14. Only one level of refinementis used for creating the conforming mesh near the crack. Asexpected for such a geometrically simple problem, the con-vergence rates associated with FE simulations relying on themeshes generated using CISAMR are compatible with theanalytical values. However, the main advantage of CISAMRis realized when it is employed for modeling fracture prob-lems with complex geometries, as shown in the followingexamples (Fig. 15).

5.3 Domain with multiple micro-cracks

As the third example problem, we aim to simulate themechanical behavior of a 400µm × 400µm unit cell of abrittle material (E = 300GPa and ν = 0.29) with mul-tiple pre-existing micro-cracks subject to uniform tractiont = 1N/m in the y-direction, as shown in Fig. 15a. To createthe conforming mesh using CISAMR, the domain is dis-cretized using a 400 × 400 background grid, while Nc = 2and Nt = 1 is used during the SAMR phase. Figure 15billustrates a small portion of the resulting conforming meshcorresponding to the box shown in Fig. 15a. Despite the com-plexity of the material microstructure in this problem, the

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400 μm

800 μm

ux

=0

ux

=2

μm

(a) (b)

Fig. 19 Fifth example problem: a domain geometry and boundary conditions; b a small portion of the conforming mesh generated using CISAMRcorresponding to the box shown in figure a

Fig. 20 Fifth exampleproblem: deformed shape andapproximate normal stress fieldσx

CISAMR can successfully build conforming elements withappropriate aspect ratios along each micro-crack (maximumaspect ratio: 2.74). The FE approximation of the deformedshape of the unit cell (with magnified displacement vectors)in this example is depicted in Fig. 16, which also shows theresulting normal stress (σx ) and shear stress (σxy) fields. Itis worth mentioning that no special recovery scheme (e.g.,the super convergent patch recovery [70]) has been used forcreating the contour plots shown in this figure. Instead, toevaluate the stress value at a node, we have used the stan-dard approach of averaging the stresses corresponding to allelements sharing that node.

5.4 Domain with a branching crack

In this example, we implement CISAMR to create the FEmodel of a 100µm × 70µm aluminum domain (E =68.9GPa and ν = 0.33) with a pre-existing branching cracksubject to traction t = 1N/m in the x-direction, as depictedin Fig. 17a. The crack shown in this figure resembles the frac-

ture patterns often caused by the stress corrosion crackingphenomenon [71]. In order to generate a conforming meshfor discretizing the domain using CISAMR, we employ a420 × 600 background grid, together with Nc = 2 andNt = 1. Figure 17b–d illustrate three insets of the result-ing conforming mesh, which show appropriate aspect ratiosof elements in the vicinity of the branching points and cracktips. Themaximumaspect ratio of elements in this problem is2.63. The resulting FE simulation of the deformed shape andthe displacement field are depicted in Fig. 18a. The normalstress (σx ) and shear stress (σxy) fields in two sub-domainsof this figure are shown in Fig. 18b.

5.5 Fractured porous material

In the final example problem, we demonstrate the abil-ity of CISAMR to simulate the linear elastic response ofa 800µm × 400µm perforated plate (E = 14GPa andν = 0.27) with multiple pre-existing cracks subject to a pre-scribed displacement ux = 2µmalong its left edge, as shown

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in Fig. 19a. A 160×80 background grid is transformed into aconformingmesh usingCISAMR formodeling this problem.Two levels of SAMR is used along and the curved bound-aries, crack paths, and crack tips. Figure 19b illustrates asmall portion of the resulting conforming mesh correspond-ing to the box shown in Fig. 19a. As shown in these figures,CISAMRgenerates elements with proper aspect ratios (max-imum: 2.23) and can easily handle the intersection betweencracks and pre-existing pores without the use of an iterativealgorithm. It is worth mentioning that a key advantage ofCISAMR for modeling this problem is that each geometricentity (crack and inclusion) can independently be added to thebackground grid. In this approach, we first add the inclusionsand perform SAMR and r -adaptivity without considering thepresence of cracks. The cracks are then added to themesh anda similar process is carried out for each crack to locally mod-ify the background mesh. The intersection points of cracksand inclusions is treated similar to a branching point. Afterperforming non-iterative SAMRand r -adaptivity steps for allinclusions and cracks, the sub-triangulation step is executedto create the final conforming mesh. Figure 20 illustrates theresultingFE simulation of the deformed shape and the normalstress field σx .

6 Conclusion

A new non-iterative algorithm was introduced for creat-ing high quality conforming meshes for modeling fracturemechanics problems. The proposed method, named Con-forming to Interface Structured Adaptive Mesh Refinement(CISAMR), enables the automated transformation of a sim-ple structured gird into a conforming mesh with low elementaspect ratios and negligible discretization error via threesynergistic steps: (i) customized Structured Adaptive MeshRefinement (SAMR) of the background mesh in the vicin-ity of a crack; (ii) r -adaptivity of the elements intersectingwith the crack by moving their selected nodes to the crack;(iii) sub-triangulation of the elements deformed during ther -adaptivity phase, as well as the elements with hangingnodes. Special cases such as cracks that are in close proximityof domain boundaries and the crack branching phenomenonwere also addressed in this manuscript. We also presentedan efficient algorithm for identifying the elements of abackground grid cut by an arbitrary-shaped crack, which isexplicitly described in terms of NURBS functions. The accu-racy of the FE solutions using conforming meshes generatedby the CISAMR was verified by comparing the results withanalytical values of the fracture energy release rate in a biax-ially loaded square domain with varying angular orientationsof a pre-existing crack.We also demonstrated the applicationof CISAMR for simulating three linear elastic fracture prob-lems with complex geometries, including a brittle material

with multiple embedded micro-cracks, an aluminum domainwith a branching crack, and a perforated plate with pre-existing cracks.

Acknowledgements This article is based upon work supported by theNational Science Foundation under Grant No. 1608058.

References

1. Camacho GT, Ortiz M (1996) Computational modelling of impactdamage in brittle materials. Int J Solids Struct 33(20):2899–2938

2. Bouchard PO, Bay F, Chastel Y, Tovena I (2000) Crack propaga-tion modelling using an advanced remeshing technique. ComputMethods Appl Mech Eng 189(3):723–742

3. Colombo D, Giglio M (2006) A methodology for automatic crackpropagation modelling in planar and shell FE models. Eng FractMech 73(4):490–504

4. Nishioka T, Tokudome H, Kinoshita M (2001) Dynamic fracture-path prediction in impact fracture phenomena using moving finiteelement method based on Delaunay automatic mesh generation.Int J Solids Struct 38(30):5273–5301

5. Azócar D, Elgueta M, Rivara MC (2010) Automatic LEFM crackpropagation method based on local Lepp–Delaunay mesh refine-ment. Adv Eng Softw 41(2):111–119

6. Khoei AR, Azadi H, Moslemi H (2008) Modeling of crack prop-agation via an automatic adaptive mesh refinement based onmodified superconvergent patch recovery technique. Eng FractMech 75(10):2921–2945

7. Meyer A, Rabold F, Scherzer M (2006) Efficient finite elementsimulation of crack propagation using adaptive iterative solvers.Commun Numer Methods Eng 22(2):93–108

8. Askes H, Sluys LJ, de Jong BBC (2000) Remeshing techniquesfor r-adaptive and combined h/r-adaptive analysis with applicationto 2D/3D crack propagation. In: European Congress on Computa-tional Methods in Applied Sciences and Engineering, ECCOMAS

9. Areias P, Rabczuk T (2013) Finite strain fracture of plates andshells with configurational forces and edge rotations. Int J NumerMethods Eng 94(12):1099–1122

10. Areias P, Rabczuk T, Camanho PP (2013) Initially rigid cohe-sive laws and fracture based on edge rotations. Comput Mech52(4):931–947

11. Areias P,Reinoso J, CamanhoPP,RabczukT (2015)A constitutive-based element-by-element crack propagation algorithm with localmesh refinement. Comput Mech 56(2):291–315

12. Areias P,MsekhMA,Rabczuk T (2016) Damage and fracture algo-rithmusing the screened poisson equation and local remeshing. EngFract Mech 158:116–143

13. Areias P, Rabczuk T, César de Sá J (2016) A novel two-stage dis-crete crack method based on the screened Poisson equation andlocal mesh refinement. Comput Mech 58(6):1003–1018

14. Rangarajan R, Chiaramonte MM, Hunsweck MJ, Shen Y, Lew AJ(2015) Simulating curvilinear crack propagation in two dimensionswith universal meshes. Int J Numer Methods Eng 102(3–4):632–670

15. Henshell RD, ShawKG (1975) Crack tip finite elements are unnec-essary. Int J Numer Methods Eng 9(3):495–507

16. Barsoum RS (1976) On the use of isoparametric finite elements inlinear fracture mechanics. Int J Numer Methods Eng 10(1):25–37

17. Rashid MM (1998) The arbitrary local mesh replacement method:an alternative to remeshing for crack propagation analysis. ComputMethods Appl Mech Eng 154(1):133–150

18. Xu XP, Needleman A (1994) Numerical simulations of fast crackgrowth in brittle solids. J Mech Phys Solids 42(9):1397–1434

123

Comput Mech

19. Zhang Z, Paulino GH, Celes W (2008) Cohesive modeling ofdynamic crack growth in homogeneous and functionally gradedmaterials. Multiscale Funct Grad Mater 973:562–567

20. Jin ZH, PaulinoGH,Dodds RH (2002) Finite element investigationof quasi-static crack growth in functionally graded materials usinga novel cohesive zone fracture model. J Appl Mech 69(3):370–379

21. Zhang ZJ, Paulino GH, Celes W (2007) Extrinsic cohesive mod-elling of dynamic fracture and microbranching instability in brittlematerials. Int J Numer Methods Eng 72(8):893–923

22. Park K, Paulino GH, Celes W, Espinha R (2012) Adaptive meshrefinement and coarsening for cohesive zone modeling of dynamicfracture. Int J Numer Methods Eng 92(1):1–35

23. Nguyen VP (2014) An open source program to generate zero-thickness cohesive interface elements. Adv Eng Softw 74:27–39

24. Li S, Liu WK (2002) Meshfree and particle methods and theirapplications. Appl Mech Rev 55(1):1–34

25. BelytschkoT, LuYY,GuL (1994) Element-freeGalerkinmethods.Int J Numer Methods Eng 37(2):229–256

26. Belytschko T, Gu L, Lu YY (1994) Fracture and crack growthby element free Galerkin methods. Model Simul Mater Sci Eng2(3A):519

27. Belytschko T, Lu YY, Gu L, Tabbara M (1995) Element-freeGalerkinmethods for static and dynamic fracture. Int J Solids Struct32(17):2547–2570

28. Rabczuk T, Belytschko T (2005) Adaptivity for structured mesh-free particle methods in 2D and 3D. Int J Numer Methods Eng63(11):1559–1582

29. Rabczuk T, Samaniego E (2008) Discontinuous modelling of shearbands using adaptive meshfree methods. Comput Methods ApplMech Eng 197(6):641–658

30. Liu WK, Hao S, Belytschko T, Li S, Chang CT (1999) Multi-ple scale meshfree methods for damage fracture and localization.Comput Mater Sci 16(1):197–205

31. Rabczuk T, Belytschko T (2004) Cracking particles: a simplifiedmeshfree method for arbitrary evolving cracks. Int J Numer Meth-ods Eng 61(13):2316–2343

32. Silling SA, Askari E (2005) A meshfree method based on the peri-dynamic model of solid mechanics. Comput Struct 83(17):1526–1535

33. Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridy-namic states and constitutive modeling. J Elast 88(2):151–184

34. Hu W, Ha YD, Bobaru F (2012) Peridynamic model for dynamicfracture in unidirectional fiber-reinforced composites. ComputMethods Appl Mech Eng 217:247–261

35. HaYD, Bobaru F (2011) Characteristics of dynamic brittle fracturecaptured with peridynamics. Eng Fract Mech 78(6):1156–1168

36. Breitenfeld MS, Geubelle PH, Weckner O, Silling SA (2014)Non-ordinary state-based peridynamic analysis of stationary crackproblems. Comput Methods Appl Mech Eng 272:233–250

37. KarmaA, Kessler DA, Levine H (2001) Phase-field model of modeIII dynamic fracture. Phys Rev Lett 87(4):045501

38. Miehe C, Welschinger F, Hofacker M (2010) Thermodynamicallyconsistent phase-field models of fracture: variational principlesand multi-field FE implementations. Int J Numer Methods Eng83(10):1273–1311

39. Hofacker M, Miehe C (2013) A phase field model of dynamicfracture: Robust field updates for the analysis of complex crackpatterns. Int J Numer Methods Eng 93(3):276–301

40. Borden MJ, Verhoosel CV, Scott MA, Hughes TJ, Landis CM(2012) A phase-field description of dynamic brittle fracture. Com-put Methods Appl Mech Eng 217:77–95

41. Welschinger F, Hofacker M, Miehe C (2010) Configurational-force-based adaptive FE solver for a phase field model of fracture.PAMM 10(1):689–692

42. Heister T, Wheeler MF, Wick T (2015) A primal-dual active setmethod and predictor-corrector mesh adaptivity for computing

fracture propagation using a phase-field approach. Comput Meth-ods Appl Mech Eng 290:466–495

43. Wells GN, Sluys LJ (2001) A new method for modelling cohe-sive cracks using finite elements. Int J Numer Methods Eng50(12):2667–2682

44. Armero F, Linder C (2008) New finite elements with embed-ded strong discontinuities in the finite deformation range. ComputMethods Appl Mech Eng 197(33):3138–3170

45. Armero F, Linder C (2009) Numerical simulation of dynamic frac-ture using finite elementswith embedded discontinuities. Int J Fract160(2):119–141

46. Linder C, Armero F (2007) Finite elements with embedded strongdiscontinuities for the modeling of failure in solids. Int J NumerMethods Eng 72(12):1391–1433

47. Babuska I, Melenk JM (1997) The partition of unity method. Int JNumer Methods Eng 40(4):727–758

48. Simone A, Duarte CA, Van der Giessen E (2006) A generalizedfinite element method for polycrystals with discontinuous grainboundaries. Int J Numer Methods Eng 67(8):1122–1145

49. Duddu R, Bordas S, Chopp D, Moran B (2008) A combinedextended finite element and level set method for biofilm growth.Int J Numer Methods Eng 74(5):848–870

50. DolbowJ,MoësN,BelytschkoT (2001)Anextendedfinite elementmethod for modeling crack growth with frictional contact. ComputMethods Appl Mech Eng 190(51):6825–6846

51. Melenk JM, Babuska I (1996) The partition of unity finite ele-mentmethod: basic theory and applications.ComputMethodsApplMech Eng 139(1–4):289–314

52. Chessa J, Wang H, Belytschko T (2003) On the construction ofblending elements for local partition of unity enriched finite ele-ments. Int J Numer Methods Eng 57:1015–1038

53. Fries TP (2008) A corrected XFEM approximation without prob-lems in blending elements. Int J Numer Methods Eng 75(5):503–532

54. Zi G, Belytschko T (2003) New crack-tip elements for XFEMand applications to cohesive cracks. Int J Numer Methods Eng57(15):2221–2240

55. Chahine E, Laborde P, Renard Y (2008) Crack tip enrichmentin the XFEM using a cutoff function. Int J Numer Methods Eng75(6):629–646

56. Song JH, Wang H, Belytschko T (2008) A comparative studyon finite element methods for dynamic fracture. Comput Mech42(2):239–250

57. Byfut A, Schroöder A (2012) hp-adaptive extended finite elementmethod. Int J Numer Methods Eng 89(11):1392–1418

58. Fries TP, Byfut A, Alizada A, Cheng KW, Schroöder A (2011)Hanging nodes and XFEM. Int J Numer Methods Eng 86(4–5):404–430

59. Oden TJ, Duarte CA, Zienkiewicz OC (1998) A new cloud-basedhpfinite elementmethod.ComputMethodsApplMechEng 153(1–2):117–126

60. Moës N, Belytschko T (2002) Extended finite element method forcohesive crack growth. Eng Fract Mech 69:813–833

61. Karoui A, Mansouri K, Renard Y, Arfaoui M (2014) The extendedfinite element method for cracked hyperelastic materials: a conver-gence study. Int J Numer Methods Eng 100(3):222–242

62. Nicaise S, Renard Y, Chahine E (2011) Optimal convergence anal-ysis for the extended finite element method. Int J Numer MethodsEng 86(4–5):528–548

63. Belytschko T, Gracie R, Ventura G (2009) A review ofextended/generalized finite element methods for material model-ing. Model Simul Mater Sci Eng 17(4):043001

64. Lang C,Makhija D, Doostan A,Maute K (2014) A simple and effi-cient preconditioning scheme for heaviside enriched xfem.ComputMech 54(5):1357–1374

123

Comput Mech

65. Chessa J, Belytschko T (2003) An enriched finite element methodand level sets for axisymmetric two-phase flow with surface ten-sion. Int J Numer Methods Eng 58:2041–2064

66. Piegl L, Tiller W (2012) The NURBS book. Springer, Berlin67. Baase S (2009) Computer algorithms: introduction to design and

analysis. Pearson Education, India68. Demkowicz L (2007) Computing with hp-adaptive finite elements.

Vol. 1: One- and two-dimensional elliptic and Maxwell problems.CRC Applied Mathematics and Nonlinear Science Series. Chap-man and Hall

69. Anderson TL, Anderson TL (2005) Fracture mechanics: funda-mentals and applications. CRC Press, Boca Raton

70. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite elementmethod: its basis and fundamentals. Elsevier, Amsterdam

71. Dietzel W, Turnbull A (2007) Stress corrosion cracking. GKSS-Forschungszentrum, Bibliothek

123