An adaptive interface-enriched generalized FEM for the ... · neous adhesives. A detailed...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2015; 102:1352–1370Published online 9 January 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4860

An adaptive interface-enriched generalized FEM for the treatmentof problems with curved interfaces

Soheil Soghrati1,2,*,† , C. Armando Duarte3 and Philippe H. Geubelle4

1Department of Mechanical and Aerospace Engineering, The Ohio State University, 201 W. 19th Avenue,Columbus, OH 43210, USA

2Department of Materials Science and Engineering, The Ohio State University, 2041 College Rd.,Columbus, OH 43210, USA

3Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign,205 North Mathews Avenue, Urbana, IL 61801, USA

4Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign,405 North Mathews Avenue, Urbana, IL 61801, USA

SUMMARY

An adaptive refinement scheme is presented to reduce the geometry discretization error and provide higher-order enrichment functions for the interface-enriched generalized FEM. The proposed method relies on theh-adaptive and p-adaptive refinement techniques to reduce the discrepancy between the exact and discretizedgeometries of curved material interfaces. A thorough discussion is provided on identifying the appropriatelevel of the refinement for curved interfaces based on the size of the elements of the background mesh.Varied techniques are then studied for selecting the quasi-optimal location of interface nodes to obtain amore accurate approximation of the interface geometry. We also discuss different approaches for creating theintegration sub-elements and evaluating the corresponding enrichment functions together with their impacton the performance and computational cost of higher-order enrichments. Several examples are presentedto demonstrate the application of the adaptive interface-enriched generalized FEM for modeling thermo-mechanical problems with intricate geometries. The accuracy and convergence rate of the method are alsostudied in these example problems. Copyright © 2015 John Wiley & Sons, Ltd.

Received 21 October 2013; Revised 19 October 2014; Accepted 24 November 2014

KEY WORDS: IGFEM; GFEM/XFEM; h-adaptivity; p-adaptivity; mesh refinement; high-order enrich-ment; heat transfer; linear elasticity

1. INTRODUCTION

Since the introduction of the FEM, the development of robust techniques for the generation ofmeshes that conform to complex geometrical features has been a growing area of research [1, 2].However, to this day, creating conforming meshes with proper element aspect ratios for problemswith complex geometries is a major challenge [3–5]. Moreover, to simulate phenomena with evolv-ing geometries such as crack propagation and moving front problems, one needs to continuouslyre-generate or adaptively refine the finite element (FE) mesh [6–10]. A similar issue is encounteredin evaluating the statistical physical response of problems with geometric uncertainties and hetero-geneous microstructures, where multiple numerical simulations, each requiring a new conformingfinite element mesh, must be performed [11].

To address the concerns related to the mesh quality and difficulties associated with the adaptivemesh refinement schemes, more advanced techniques have been proposed to decouple the numerical

*Correspondence to: Soheil Soghrati, Department of Mechanical and Aerospace Engineering and Department ofMaterials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA.

†E-mail: [email protected]

Copyright © 2015 John Wiley & Sons, Ltd.

ADAPTIVE INTERFACE-ENRICHED GENERALIZED FEM 1353

approximation from the finite element mesh. Meshfree methods tackle this issue by either limitingthe discretization to the domain boundaries [12, 13] or replacing the surface/volume mesh with adomain (cloud) of influence [14, 15]. In the realm of mesh-independent FEM (methods that elimi-nate the requirement of using conforming meshes), the generalized/extended FEM (GFEM/XFEM)[16–20] is one of the most successful techniques. This method employs a partition of unity com-bined with appropriate enrichment functions to capture the weak and/or strong discontinuities in adomain discretized with non-conforming FE meshes [21].

Recently, Soghrati et al. [22, 23] have developed an interface-enriched generalized FEM(IGFEM) for the mesh-independent treatment of interface problems. In this method, the non-conforming elements intersected by the material/phase interface are divided into smaller integrationsub-elements, which also serve as means for constructing enrichment functions. Unlike the con-ventional GFEM/XFEM, the method relies on enriching the nodes created at the intersection of theinterface with the non-conforming elements edges and does not use a partition of unity. The IGFEMyields the same precision and convergence rate as the standard FEM with conforming meshes, whileproviding advantages such as a lower computational cost and straightforward imposing of Dirichletboundary conditions compared with the GFEM/XFEM [22]. In [24, 25], the IGFEM was appliedto the simulation of actively cooled microvascular materials and the multi-scale failure of heteroge-neous adhesives. A detailed comparison of the IGFEM with standard FEM and its advantages overGFEM/XFEM are presented in [22, 26].

In this manuscript, we study the application of h-adaptive and p-adaptive refinement techniquesfor discretizing the material interfaces and evaluating the corresponding enrichment functions in theIGFEM framework. In the remainder of the paper, we refer to these approaches as the h-IGFEM andp-IGFEM, respectively. In the h-IGFEM, the objective is to improve the geometric approximationof curved interfaces by reducing the size and thus increasing the number of integration sub-elementsin a non-conforming element. In the p-IGFEM, an iso-parametric interpolation of the geometryusing higher-order integration elements is implemented to reduce the geometry discretization erroralong the interface. In addition to geometry, the higher-order enrichment functions associated withthe h-IGFEM and p-IGFEM can improve the field approximation in the vicinity of the interface.The selection of the optimal locations for the interface nodes and the creation of the integrationelements in the hp-IGFEM are discussed in Section 2.

In the context of GFEM/XFEM, Cheng et al. used the higher-order mapping of level sets [27]with transition integration elements, that is, elements with one curved edge, to improve the implicitrepresentation of the interface [28]. In [29, 30], the non-conforming elements of the backgroundmesh are hierarchically divided into smaller sub-elements to obtain a more accurate approximationof the interface. Legrain et al. [31] developed a higher-order XFEM based on a similar idea, wherethe mesh used to represent the domain geometry is decoupled from the approximation mesh. Theh-adaptive and p-adaptive refinement schemes using elements with hanging nodes are combinedwith the XFEM to improve the field approximation near the interface in [32, 33]. It must be notedthat the hp-IGFEM studied hereafter differs from the conventional hp-refinement technique, asthe original (background) mesh remains unchanged and the refinement is conducted at the level ofintegration sub-elements.

The manuscript is organized as follows: In Section 2, we describe the strong and weak formsof the governing equations of interest and present their standard IGFEM approximation. The algo-rithms adopted for creating the integration elements and evaluating the corresponding higher-orderenrichment functions in the hp-IGFEM are presented in Section 3. Several implementation issuesincluding the optimal location of interface nodes and the adaptive refinement criteria are also dis-cussed in that section. Three example problems with complex interface geometries are studied inSection 4, including an investigation of the accuracy and convergence rate of the adaptive IGFEM.

2. GOVERNING EQUATIONS AND IGFEM FORMULATION

In this paper, we propose and study the application of the h-IGFEM and p-IGFEM for model-ing conductive heat transfer and linear elasticity problems in multi-material domains. Consider a

Copyright © 2015 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2015; 102:1352–1370DOI: 10.1002/nme

1354 S. SOGHRATI, C. A. DUARTE AND P. H. GEUBELLE

bimaterial open domain � D �1 [ �2 � R2 such that N�1 \ N�2 D ; and with the bound-ary � D N�n� and outward unit normal vector n. Also, assume Q W � ! R is a heat source,�1 W N�1 ! R2 � R2 and �2 W N�2 ! R2 � R2 are the conductivity tensors in each materialphase, and �u and �q are portions of the boundary with the prescribed temperature Nt W �u ! R andapplied heat flux q W �q ! R, respectively. Decomposing the temperature field u W N� ! R intou D u0 C ut , where ut W N� ! R with ut j�u

D Nu, the weak form of he steady-state heat transferequations is expressed as follows: Find u0 2 V WD ®

u0 W N� ! R; u0j�uD 0

¯such thatZ

�1

ru0 � �1rv d�CZ

�2

ru0 � �2rv d�CZ

�

vQ d�

CZ

�1

rut � �rv d�1 CZ

�2

rut � �rv d�2 CZ

�q

vq d� D 0 8v 2 V :(1)

To express the weak form of linear elasticity equations, assume b W � ! R2 is the body forcevector, t W �t ! R2 is the applied tractions vector, and Nu W �u ! R2 is the prescribed displacementvector along the boundary. The weak form of the governing equations is then written as: Find u0 2W WD ®

u0 W N� ! R2; u0j�u D 0¯

such thatZ�1

Lu0 � C1LT wT d�CZ

�2

Lu0 � C2LT wT d�CZ

�

wb d�

CZ

�1

Lud � C1LT wT d�CZ

�2

Lud � C2LT wT d�CZ

�t

wt d� D 0 8w 2 W;

(2)

where ud W N� ! R2 with ud j�u D Nu, and the operator L is defined as

L D

264

@@x

0

0 @@y

@@x

@@y

375 : (3)

In (2), C1 and C2 denote the fourth-order isotropic stiffness tensors in each material phase.The Galerkin approximation of (1) and (2) is obtained by replacing V and W with appropriate

finite dimensional spaces Vh � V and Wh � W , respectively. To revisit the standard IGFEMformulation described in [22], assume the domain� Š �h is discretized intom finite elements thatdo not conform to the materials interface. The IGFEM approximation of the temperature field uh in(1) can then be written as

uh .x/ DnX

iD1

Niui CnenXj D1

sj j˛j ; (4)

where ¹Ni ºniD1 2 Vh is a set of n Lagrangian shape function,

® j

¯nen

j D12 Vh is a set of nen

enrichment functions, and ui and ˛j are nodal values of the temperature and generalized DOFs,respectively. The scaling factor sj avoids the ill-conditioning of the stiffness matrix in the case ofan interface passing very close to nodes of a non-conforming element [26]. The second term in (4)is added to capture weak discontinuities caused by the mismatch between material properties acrossthe interface in non-conforming (parent) elements of the mesh. The enrichment function associatedwith the j -th interface node is calculated as

j Dn

.j /ceX

kD1

N .k/p ; (5)

where n.j /ce is the number of integration (children) elements sharing node j and N k

p is the p-th Lagrangian shape function of the k-th children element associated with that node (Figure 1).



Figure 1. Discretization of a non-conforming element into triangular and rectangular children elements,and visualization of the interface-enriched generalized FEM enrichment function 2 associated with theinterface node 2. The distances x.1/

1and x.1/

2are used for computing the scaling factor sj in (6) for interface

node 1.

Because of the local support of the IGFEM enrichment functions, no enrichment is needed inblending elements, that is, those adjacent to non-conforming elements, to achieve the optimalconvergence rate.

The scaling factor sj in (4) is evaluated as

sj D

vuuutmin�x

.j /1 ; x

.j /2

�x

.j /1 C x

.j /2

; (6)

where x.j /1 and x.j /

2 are the distances between the interface node j and the two end nodes of theparent element edge cut by the materials interface, as shown in Figure 1. This scaling factor avoidsill-conditioning in the stiffness matrix in the case of an interface passing very close to nodes of anon-conforming element.

A comparison between the performance of the IGFEM and the standard FEM is presented inthe Appendix, where standard FEM model is discretized using a mesh composed of conformingelements with bad aspect ratios identical to those created as children elements in the IGFEM approx-imation. This study shows the improved accuracy obtained via IGFEM by avoiding the constructionof sliver and degenerate elements, which also improves the condition number of the stiffnessmatrix [34].

3. ADAPTIVE IGFEM FORMULATION

As shown in [22, 23], the standard IGFEM yields the same precision and convergence rate asthe standard FEM but without the restriction of using finite element meshes that conform tothe problem morphology. However, for problems with curved interfaces similar to that shown inFigure 2(a), the geometry discretization error associated with a coarse background mesh can con-siderably deteriorate the IGFEM precision. As apparent in that figure, the linear approximation ofthe interface between the two interface nodes does not accurately capture of the interface geom-etry, which causes error in evaluating material properties in the shaded region. While refining thebackground non-conforming mesh alleviates this source of the error (Figure 2(b)), this approachleads to an unnecessary increase in the computational cost if the interface interacts with a smallregion of the non-conforming mesh. Moreover, as shown in the shaded region in Figure 2(b),refining the background mesh cannot effectively reduce the geometry discretization error in allthe parent elements. An adaptive mesh refinement scheme can be used to locally refine noncon-forming elements, although this approach requires an special treatment of elements with hangingnodes [32].

As an alternative approach to provide a more accurate approximation of curved interfaces, weimplement an adaptive refinement scheme using the IGFEM, which is restricted to the level ofchildren elements. Two different techniques are used for this purpose: (i) increasing the number of



(a) (b)

Figure 2. (a) Standard interface-enriched generalized FEM approximation of a curved interface using acoarse parent element, where the shaded region corresponds to the mismatch between the approximate andthe actual interface geometries; (b) geometry discretization error associated with a more refined background

mesh, where the size of parent elements is half of that shown in Figure (a).

(a)

(b)

Figure 3. Creation of the integration children elements and enrichment functions for (a) second-levelh-interface-enriched generalized FEM and (b) first-level p-interface-enriched generalized FEM. The figureson the left illustrate the interface nodes numbering, and those on the right show the local node numbering.

children elements along the interface, a method referred to hereafter as h-IGFEM (Figure 3(a)), and(ii) using higher-order iso-parametric children elements or the p-IGFEM (Figure 3(b)).

In the k-th level h-IGFEM, we first create k dummy nodes at equally distanced locations on one ofthe edges of the parent element that does not intersect with the interface. Unlike interface nodes, nogeneralized DOF is assigned to the dummy nodes, and therefore, no enrichment is attached to thosenodes. The next step is to create k new interface nodes in addition to those already existing on theparent element edges to construct a combination of three-node triangular and four-node tetrahedralchildren elements along the interface, as shown in Figure 3(a). Different approaches for determining



the location of these internal interface nodes are discussed in Section 3.4. In the p-IGFEM, theseadditional interface nodes are implemented to create a combination of triangular and quadrilateraltransition children elements (Figure 3(b)). A higher-order polynomial approximation of the interfacegeometry can then be obtained using an isoparametric mapping of the children elements. Alternativeapproaches for creating the children elements and their impact on the precision of the h-IGFEM andp-IGFEM are discussed in Section 3.2.

3.1. Scaling factor

The only modification needed in the formulation of the h-IGFEM and p-IGFEM compared withthat of the standard IGFEM is in evaluating the scaling factor sj (Equation (4)) for internalinterface nodes. To facilitate our discussion on this topic, four possible case scenarios for thelocation of interface nodes and the corresponding configuration of h-IGFEM children elementsare depicted in Figure 4. To ease the discussion, we assume that the parent elements are refinedenough such that the material interface cuts each element edge only once, although cases in whicheach edge is cut multiple times can be treated similarly by modifying the arrangement of inte-gration elements [22]. Equation (6) can still be implemented to evaluate sj for edge interfacenodes, that is, the enriched nodes located on the parent element edges. As shown in Figure 4(b),when an edge interface node approaches one of the nodes of the background mesh, the aspectratio of the corresponding children element goes to infinity. According to (6), a small scal-ing factor is then assigned to that interface node to avoid sharp gradients in the associated en-richment function.

The scaling factor sj associated with an internal interface node is independent of the value of sjfor the edge interface nodes. For example, the values of sj computed from (6) for the two edge inter-face nodes illustrated in Figure 4(a) and (b) are close to the unity and zero, respectively. However,because there are no children elements with high aspect ratios connected to the internal interfacenode 2 in those figures, the scaling factor associated with this node must be considerably largerthan zero in both cases. On the other hand, the high aspect ratio of the children elements sharingthe internal interface nodes 2 and 3 in Figure 4(c) and (d), respectively, leads to constructing

(a) (b)

(c) (d)

Figure 4. Four different case scenarios of the location of interface nodes, and the configuration of childrenelements configuration in the h-interface-enriched generalized FEM. The distances dD and dM , shown in

Figure (c), are used in (8) to evaluate sj for the internal interface node 2.



enrichment functions with very sharp gradients. A small scaling factor must then be assigned tothese nodes to ensure a well-conditioned stiffness matrix. The scaling parameter sj for an inter-nal interface node can be defined as a function of the distances dD and dM from that node to theattached dummy and mesh nodes, respectively (Figure 4(c)). As either of dD or dM becomes toosmall, the aspect ratio of the corresponding children elements goes to infinity, which implies theuse of an infinitesimally small scaling factor. A similar discussion would be valid for higher-orderenrichment functions associated with the internal interface nodes in the p-IGFEM. To capture thisbehavior, our numerical experiments show that appropriate scaling factor associated with an internalinterface node can be evaluated as

sj Ds

min .dD; dM /

max .dD; dM /; (7)

provided that the Jacobian of at least one of the children elements connected to that node tendstowards zero. Otherwise, the scaling factor is given by

sj D min .dD; dM /

max .dD; dM /: (8)

3.2. Creating p-IGFEM children elements

The primary objective of the approach adopted for constructing the children elements in Figure 3was to obtain the least number of generalized DOFs for both the h-IGFEM and p-IGFEM. Itis evident that alternative approaches can be used to create children elements for the same levelof refinement, which will affect how the corresponding enrichment functions are constructed. InFigure 3(b), a combination of triangular and quadrilateral transition elements were employed to cre-ate the children elements for the p-IGFEM. An alternative approach is illustrated in Figure 5(a),

(a)

(b)

Figure 5. Two alternative schemes for creating children elements in the p-interface-enriched generalizedFEM using (a) triangular transition elements and (b) a combination of six-node triangular and eight-node

quadrilateral elements. Note the negative Jacobian associated with the children element 2 in Figure (a).



which relies on using only triangular transition elements to discretize the parent element. How-ever, as shown there, this approach can become problematic when children elements with negativeJacobians are created (children element 2).

The second alternative approach for creating children elements in the p-IGFEM is depicted inFigure 5(b), where a combination of the six-node triangular and eight-node serendipity quadrilateralelements are adopted as children elements. This scheme requires creating additional non-interfaceenriched nodes over the edges of the parent element, for which the enrichment functions and scalingfactors are still evaluated from (5) and (6), respectively. Note that despite the larger number ofgeneralized DOFs in this approach than that with transition elements (Figure 3(b)), the isoparametricmapping of the children elements geometry yields the same level of precision for approximatingthe interface geometry. Moreover, because of the presence of non-interface enriched nodes on theparent element edges in the current approach, the order of the refinement must be kept constant inall non-conforming elements. However, the higher-order enrichment functions associated with thechildren elements shown in Figure 5(b) can contribute to improving the field approximation nearthe interface. In Section 4, we study the impact of the two approaches illustrated in Figures 3(b) and5(b) on the accuracy and convergence rate of the p-IGFEM.

3.3. Refinement criteria

To adaptively refine the parent elements along the interface in the h-IGFEM and p-IGFEM,we must define the criteria used to identify the required level of the refinement. There are twogoals in using an adaptive refinement scheme: (i) to improve the approximation of curved inter-faces (geometric-based adaptivity) and (ii) to reduce the error in areas with sharp field gradients(error-based adaptivity). The latter scheme is often combined with a posteriori error analysisto identify regions of the domain that require adaptive refinement. Unlike the geometric basedadaptivity, an error-based adaptive refinement of the mesh might be required in the vicinity ofa straight interface with large mismatch between material properties of each phase. The focusof the h-IGFEM and p-IGFEM presented in this work is to reduce the geometry discretiza-tion error associated with curved interfaces by using a geometric-based adaptive refinement ofthe background mesh, and the error-based adaptive refinement is out of the scope of the cur-rent manuscript.

In the h-adaptive and p-adaptive IGFEM, the level of the refinement in parent elements is deter-mined based on the interface curvature. If the interface geometry and its first derivative are givenin the form of analytical functions, the variations of the slopes �1 and �2 at the edge interfacenodes can be used to specify the required refinement level (Figure 6). A k-th order h-refinement orp-refinement is thus required when

jtan �1 � tan �2j > tolk ; (9)

where tolk is a user-defined tolerance. As apparent in Figure 6(a), large values of �1 and �2 indicatea large mismatch between the interface geometry and its linear approximation L12. Equation (9)

(a) (b)

Figure 6. Variations of the interface slope at edge interface nodes for two different configurations of thematerial interface.



provides a computationally inexpensive approach to determine the appropriate level of the refine-ment, as it only requires the evaluation of the interface slope at the edge interface nodes. However,this approach fails to correctly identify the required number of internal interface nodes for specialcases with sharp variations of the interface curvature inside the parent element, such as that shownin Figure 6(b). Moreover, for problems involved creating physical models of multi-phase materialsfrom tomographic images or modeling evolving topologies, the interface slope at the edge inter-face nodes is not given explicitly. A first-order finite difference approximation can be employed toevaluate tan �1 and tan �2 for these cases, but it requires more rigorous computations to identify therequired order of the refinement.

As an alternative approach, one can determine the required order of the refinement in the parentelement as

PmiD1 di

m cos �> tolk ; (10)

where di is the vertical distance from the interface to L12 atm equally distanced points and � is theangle between L12 and the horizontal coordinate axis (Figure 7). If � > �=4, di is considered asthe horizontal distance to obtain a better estimate of the interface curvature. Note that if there are noabrupt changes in the interface curvature inside the parent element (Figure 7(a)), evaluating (10) atonly one point is often sufficient to determine the order of the refinement. However, for a materialinterface similar to that shown in Figure 7(b), more interfacial nodes are needed to accurately capturethe offset between the curved interface and L12.

3.4. Location of internal interface nodes

After determining the number of internal interface nodes, the next step is to find the location of thesenodes, which plays an important role in accuracy of the interface geometry approximation in theh-IGFEM and p-IGFEM. Unlike FEM-based adaptive mesh refinement schemes, we are no longerconcerned with the aspect ratio of children elements in the IGFEM. We can take advantage of thisunique feature to arbitrarily select the location of internal interface nodes. Figure 8 presents six dif-ferent case scenarios for determining the internal interface nodes locations and the correspondingpiece-wise linear approximation of the interface geometry in the h-IGFEM. To simplify the discus-sion on the appropriate location of the internal interface nodes in this section, it is assumed that thephase interface does not intersect with the parent element edges more than once. In Figure 8(a), thehorizontal distance L between the edge interface nodes is divided into three equal sections usingthe vertical dashed lines drawn at x1 and x2. The intersection point of these vertical lines with thecurved interface is then adopted as the location of the internal interface nodes 3 and 4. If the interfacegeometry is given explicitly, computing the location of these nodes simply translates to comput-ing the values of an analytical function at x1 and x2. Similar to (10), the vertical dashed lines inFigure 8(a) are replaced with horizontal lines when � > �=4 to obtain a more uniform distributionof interface nodes.

(a) (b)

Figure 7. Vertical distances from the interface to the line segment between edge interface nodes at threeequally distanced points used for determining the appropriate level of the refinement.



(a) (b) (c)

(d) (e) (f)

Figure 8. Six different schemes for determining the location of the internal interface nodes and thecorresponding discretization of the interface geometry in the h-interface-enriched generalized FEM.

As shown in Figure 8(a), the approach used for selecting the location of internal interface nodesin the previous paragraph does not yield a satisfactory approximation of the interface geometry inall children elements. One can alleviate this issue by recursively revisiting the refinement criteriain each discretized sub-section, that is, between nodes 1–3, 3–4, and 4–2 in the first iteration, toadd new interface nodes where necessary. This approach suggests that the interface node 5 must beadded to reduce the geometry discretization error in the sub-section 1–3, as shown in Figure 8(b).We can then continue this algorithm recursively to obtain a proper discretization of the interfacegeometry in the parent element. However, despite the simplicity and low computational cost of thisapproach, no information regarding the interface curvature is incorporated, which results in a largenumber of internal interface nodes to accurately capture the interface curvature.

The optimal location of internal interface nodes can be determined such that they minimizeˇ̌A12 �Ah

12

ˇ̌, where A12 is the area between the material interface and L12 and Ah

12 is the areabetween its h-IGFEM or p-IGFEM approximation and L12 (Figure 8(b)). If the interface curva-ture can be reasonably approximated to be constant in the parent element, the scheme used inFigure 8(c) yields close-to-optimal locations of internal interface nodes. As shown there, the linesegment L12 is first divided into three segments with same lengths using nodes x1 and x2. Theintersections of the lines perpendicular to L12 and passing x1 and x2 is then adopted as the loca-tion of the internal interface nodes 3 and 4. Note that this approach does not yield a satisfactoryapproximation of the interface geometry when there are sharp variations of the interface curvatureinside the parent element (Figure 8(d)). A recursively adaptive approach similar to that explainedin the previous paragraph must then be used to ensure an accurate representation of the inter-face geometry for such special cases. A node 5 is then added in Figure 8(e) at the intersectionof the interface and the line passing node x3 and perpendicular to L24 to reduce the geometrydiscretization error.

As an alternative approach, we use the scheme shown in Figure 8(f) to select the location ofinternal interface nodes. It is worth mentioning that evaluating the optimal location of k internalinterface nodes via minimizing

ˇ̌A12 � Ah

12

ˇ̌leads to a complex nonlinear optimization prob-

lem when k > 1. Clearly, solving such a computationally demanding problem is not justifiableduring the adaptive refinement of the interface. However, for the particular case of k D 1 andan interface with non-changing curvature sign inside the parent element, the optimal location ofthe internal interface node can simply be obtained by maximizing Ah

12, that is, the area of thetriangle 4123 shown in Figure 8(f). The optimal location of the interface node 3 is determinedsuch that the tangent vector to the interface at that node be parallel to the line segment L12, that



is, f0

.x1/ D tan �12. The refinement criteria are then recursively revisited in each sub-sectionto identify the requirement for adding the internal interface node 4 in the sub-section 2–3 suchthat f

0

.x2/ D tan �23.

4. NUMERICAL EXAMPLES

Three example problems are presented in this section to demonstrate the application of the h-IGFEMand p-IGFEM for simulating multi-material thermal and structural problems. To study the accuracyand convergence rate of the higher-order IGFEM in these examples, we monitor the variation of theL2-norms andH 1-norms of the error, expressed as

��u � uh��

L2.�/D

qR�

�u � uh

�2d�;��u � uh

��H 1.�/

DqR

�

�u � uh

�2 C R�

��ru � ruh��2d�:

(11)

4.1. First example problem: rectangular domain with elliptic inclusions

In this example, we model the conductive heat transfer in a rectangular domain with elliptic inclu-sions, as shown in Figure 9(a). The boundary conditions consist in the prescribed temperatureNu D 20 ıC and the uniform heat flux q D 800 kW/m along the bottom and top edges, respectively,while the side edges are insulated. The thermal conductivity � of ellipsoid inclusions is ten timeslarger than that of the surrounding material, which results in a weak temperature discontinuity alongtheir interface. The temperature field evaluated using the standard IGFEM over a fine 560 � 240

structured mesh is adopted as the reference solution for this problem (Figure 9(b)). As shown in [22],the IGFEM yields a comparable level of precision as the standard FEM with conforming mesheswhen the size of elements are identical in both methods.

To study the variation of norms of the error associated with the h-IGFEM and p-IGFEM in thisproblem, three structured meshes of three-node triangular elements built on 14 � 6, 28 � 12, and56�24 grids are adopted to approximate the temperature field. The refinement criterion given in (10)is used to adaptively discretize the interface in this model, with the tolerance values tolk D 10%, 5%,and 1% of the element size corresponding to the first-level, second-level, and third-level refinements,respectively. The discretized domains using the standard, first-level, third-level, and the adaptiveh-IGFEM for the 14 � 6 background mesh are depicted in Figure 10. As shown in Figure 10(a),using the standard IGFEM with such a coarse mesh leads to a poor approximation of the inclu-sions geometry. The first-level and third-level h-IGFEM can be employed to reduce the geometrydiscretization error, as shown in Figure 10(b) and (c), respectively. Figure 10(d) illustrates the dis-cretized domain for the adaptive h-IGFEM, which yields a comparable precision to the third-levelh-IGFEM for approximating the inclusions geometry while requiring fewer integration elementsand enriched nodes.

(a) (b)

Figure 9. First example problem: (a) applied thermal loads, boundary conditions, and thermal properties ofthe rectangular domain with elliptic inclusions and (b) interface-enriched generalized FEM approximation

of the temperature field using a fine 560 � 240 structured finite element mesh.



(a) (b)

(c) (d)

Figure 10. Discretized domain and configuration of children elements in the first example problem using (a)standard interface-enhanced generalized FEM (IGFEM), (b) h-IGFEM with the first-level of refinement, (c)

h-IGFEM with the second-level of refinement, and (d) adaptive h-IGFEM.

(a) (b)

Figure 11. (a) L2-norms and (b) H1-norms of the error versus the element size (h) associated with thetemperature field evaluated for the first example problem (Figure 9) using the standard, h-interface-enhancedgeneralized FEM (IGFEM), p-IGFEM, and adaptive h-IGFEM. The p-IGFEM results marked as (tr) and(st) correspond to using transition and standard higher-order children elements for evaluating the enrichment

functions, respectively.

Figure 11 illustrates the variation of the L2-norms andH 1-norms of the error versus the elementsize h for the first example problem (Figure 9). The two types of the quadratic p-IGFEM solu-tions presented there correspond to using standard and transition children elements for constructingthe enrichment functions. As shown in Figure 11(a), the impact of the h-IGFEM and p-IGFEMon reducing the L2-norm of the error in this problem is not considerable. However, according toFigure 11(b), the H 1-norms of the error associated with the h-IGFEM and p-IGFEM are con-siderably lower than the error evaluated for the standard IGFEM. As expected, the h-IGFEM andp-IGFEM have a more pronounced impact on reducing the error when the coarsest backgroundmesh is used to discretize the domain. Further refinement of the background mesh eventually fadesaway the difference between the standard and h-IGFEM and p-IGFEM results, as the smaller



size of parent elements in the refined meshes can readily capture the inclusions geometry with agood precision.

4.2. Second example problem: perforated plate

In this example, the h-IGFEM and p-IGFEM are implemented to evaluate the linear elastic responseof a perforated plate subjected to a uniform traction t in the y-direction, as shown in Figure 12(a).The ratio of the normal stress field �yy to the applied traction t over a quarter of the domain isdepicted in Figure 12(b). This plane stress reference solution is obtained using the standard IGFEMwith a 200 � 200 structured mesh, with symmetric boundary conditions are imposed along theleft and bottom edges of the quarter plate. Two structured triangular meshes built on 10 � 10 and20 � 20 grids are then used to evaluate the norms of the error associated with the h-IGFEM and p-IGFEM. The same refinement criterion and tolerance values as the previous example problem areused in the adaptive h-IGFEM solutions, for which the resulting discretized domains are depictedin Figure 12(c) and (d).

The variation of the L2-norms and H 1-norms of the error versus the element size (h) and totalnumber of degrees of freedom (N ) for the second example problem solved with the standard, h-IGFEM, p-IGFEM, and adaptive h-IGFEM are presented in Figure 13. As shown in Figure 13(a)and (b), implementing the h-IGFEM and p-IGFEM can effectively reduce both the L2-norms and

(a) (b)

(c) (d)

Figure 12. Second example problem: (a) dimensions, elastic properties, and applied tractions to theperforated plate of interest; (b) ratio of the normal stress in the y-direction to the boundary traction eval-uated with the standard interface-enriched generalized FEM over a 200 � 200 structured mesh, which isalso adopted as the reference solution. The discretized domains of the perforated plate in the adaptiveh-interface-enriched generalized FEM using 10�10 and 20�20 structured meshes are shown in Figures (c)

and (d), respectively.



(a) (b)

(c) (d)

Figure 13. L2-norms andH1-norms of the error versus the element size (h) for the second example problem(Figure 12) using the standard, h-interface-enriched generalized FEM (IGFEM), p-IGFEM, and adaptive h-IGFEM for approximating the displacement field. The p-IGFEM results marked as (tr) and (st) correspondto using transition and standard higher-order children elements for evaluating enrichment functions, respec-tively. The insets of figures (c) and (d) clearly show that the adaptive h-IGFEM yields a higher accuracythan the third-level h-IGFEM for similar values of N , while the p-IGFEM provides a considerably more

accurate approximation than both methods.

the H 1-norms of the error compared with the standard IGFEM results. In addition to the improve-ment in the L2-norm of the error, the key difference between the results of the previous heat transferproblem (shown in Figure 11) and those of the current example lies in the behavior of the p-IGFEMwith the transition and standard children elements. While both approaches yielded a similar levelof accuracy in the first example problem, the computed error for the p-IGFEM with standard chil-dren elements in this example is lower than that with the transition elements, and even lower thanthe error associated with the third-level h-IGFEM. This is due to the presence of sharp displace-ment gradients (stress concentration) near the circular holes in the perforated plate, which can bemore accurately approximated using higher-order enrichment functions built on standard childrenelements of the p-IGFEM.

Figure 13(c) and (d) illustrates the variation of the errors associated with the h-IGFEM andp-IGFEM versus the total number of DOFs, N . As shown there, for coarser background meshes,the h-IGFEM and p-IGFEM yield a lower error compared with the standard IGFEM for thesame values of N . The most outstanding feature in these figures is the significantly lower normsof the error evaluated for the p-IGFEM with the standard children elements than the otherIGFEM schemes. As explained in the previous paragraph, this can attributed to the impact of thehigher order enrichments implemented in this method on capturing the stress concentrations nearthe inclusions.



4.3. Third example problem: plane strain multi-layered material

As final example problem, we demonstrate the application of the h-IGFEM and p-IGFEM forevaluating the structural responses of a multi-layered material subjected to the mechanical loadsshown in Figure 14(a). A three-node triangular structured mesh built on a 120 � 60 grid is usedto discretize the domain, for which the standard IGFEM is not able to fully capture the complexgeometry of the materials interfaces. Therefore, the quadratic p-IGFEM with standard childrenelements is implemented to approximate the plane strain elastic response of this layered material.According to the previous example, the higher-order enrichment functions used in the p-IGFEM canconsiderably improve the field approximation near the materials interfaces, where the stress concen-tration is expected. Figure 15 illustrates the evaluated normal strain and normal stress fields in they-direction together with the deformed configuration of the domain (scaled by a factor of 20) for this

(a)

(b) (c) (d)

Figure 14. Third example problem: (a) domain dimensions, material properties, and applied thermo-mechanical loads to a multi-layered material; (b–d) portions of discretized domain using the adaptive

h-interface-enhanced generalized FEM corresponding to the regions labeled similarly in Figure (a).

(a) (b)

Figure 15. (a) Normal strain and (b) normal stress field in the y-direction evaluated with the p-IGFEM forthe third example problem (Figure 14(a)). The deformed configuration of the domain is magnified by a scale

factor of 20.



problem. As apparent in Figure 15(a), the p-IGFEM has nicely captured the discontinuous strainfield and the high strains near the peaks of the second material layer caused by their drilling impacton the softer upper layer. It must be noted that enforcing the Dirichlet boundary condition in enrichedelements along the bottom edge of the domains similar to the standard IGFEM, which avoids theimplementation of the penalty method, Lagrange multipliers, or Nitsche methods commonly usedin GFEM/XFEM [22].

5. CONCLUSION

An adaptive IGFEM combined with h-refinement and p-refinement techniques was presented formodeling problems with complex microstructures. The main objective of the proposed method is toreduce the geometry discretization error associated with curved material interfaces without refiningthe non-conforming background mesh. Varied approaches were discussed to determine the locationof the interface nodes in the h-IGFEM and p-IGFEM to attain the best possible approximation ofthe problem geometry. Moreover, varied techniques for creating integration children elements andtheir impact on the accuracy and robustness of the numerical solution were studied. An adaptivescheme was then developed to identify the appropriate level of the refinement in the non-conformingelements based on the interface curvature. The h-IGFEM and p-IGFEM were employed to simu-late several thermo-mechanical problems, through which the accuracy and the computational costof these methods were also examined. Both methods were proved to be effective in reducing thenumerical error via more accurate approximation of the interface geometry and providing appropri-ate enrichment functions to capture the corresponding discontinuous gradient field. Furthermore, thehigher-order enrichment functions used in the p-IGFEM were shown to be superior to those of the h-IGFEM for simulating structural problems, as they more accurately capture the stress concentrationin the vicinity of the interface.

APPENDIX

Figure A.1 presents the results of a comparison study between the performance of the IGFEM andthe standard FEM for simulating a plane stress linear elasticity problem. The internal geometryand boundary conditions of this 1 � 1 domain are depicted in Figure A.1(a), which is composedof two material phases with E1 D 20 GPa, E2 D 200 GPa, �1 D 0:28, and �2 D 0:3. Thetwo phases are separated by sinusoidal-shape materials interfaces with a wavelength of D 13

cm and a small amplitude of A D 1:56 mm. Figure A.1(b) shows the reference solution for thedisplacement field in this problem evaluated using a 100 � 100 mesh of six-node triangular ele-ments. The IGFEM approximation of the displacement field in the x-direction using a 16 � 16

nonconforming structured mesh is depicted in Figure A.1(b), which results in an L2-norm of theerror of kekIGFEM

L2.�/ D 0:00516. This figure also illustrates children elements used for the numericalquadrature and evaluation of the IGFEM enrichment functions in nonconforming elements cut bymaterials interfaces.

The standard FEM approximation of the displacement field in the x-direction is depicted inFigure A.1(c). In this simulation, the conforming elements created along the materials inter-faces are identical to those used as the IGFEM children elements. Due to the location ofmaterials interfaces with respect to edges of finite elements in the original 16 � 16 struc-tured mesh, the conforming elements created along materials interfaces have poor aspect ratiosthat lead to an ill-conditioned stiffness matrix (Cn D 4:7 � 109) and expected to deterio-rate the accuracy of the FEM approximation. This hypothesis is confirmed by the standardFEM solution, which results in a larger error than the corresponding IGFEM approximation(kekFEM

L2.�/ D 0:0129). We have also presented the IGFEM and standard FEM simulations ofthe shear stress field in Figures A.1(e) and A.1(f), respectively, where the IGFEM predictionof the maximum shear stress is approximately three times more accurate than the correspond-ing standard FEM solution. The H 1-norm of the error associated with the IGFEM simulation



(a) (b)

(c) (d)

(e) (f)

Figure A.1. Comparison study between the performance of the interface-enriched generalized FEM(IGFEM) and standard FEM: (a) problem description; (b) reference solution using a 100 � 100 mesh ofsix-node triangular elements; (c, d) IGFEM and FEM approximations of the displacement field in thex-direction; (e, f) IGFEM and FEM approximations of the shear stress field. The conforming mesh usedfor the standard FEM simulation is composed of elements identical to the IGFEM children elements along

materials interfaces.

is kekIGFEMH 1.�/

D 0:0138, which is also more accurate than that of the standard FEM solution(kekFEM

H 1.�/D 0:0472).

ACKNOWLEDGEMENTS

This work has been supported by the Air Force Office of Scientific Research Multidisciplinary Univer-sity Research Initiative (MURI-18) on Functionally Graded Hybrid Materials under the contract numberFA9550-09-1-0686.



REFERENCES

1. Ho-Le K. Finite element mesh generation methods: a review and classifications. Computer-Aided Design 1988;20(1):27–38.

2. Topping BHV, Muylle J, Ivanyi P, Putanowicz R, Cheng B. Finite Element Mesh Generation. Saxe-CobourgPublications: Kippen, Strlingshire, UK, 2002.

3. Ortiz M, Quigley JJ. Adaptive mesh refinement in strain localization problems. Computer Methods in AppliedMechanics and Engineering 1991; 90(1-3):781–804.

4. Young PG, Bresford-West TBH, Coward SRL, Notarberardino B, Walker B, Abdud-Aziz A. An efficient approachto converting three-dimensional image data into highly accurate computational model. Philosophical Transactions ofthe Royal Society A 2008; 355:3155–3173.

5. Geuzaine C, Remacle JF. Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities.International Journal for Numerical Methods in Engineering 2009; 79(11):1309–1331.

6. Lo SH. Automatic mesh generation and adaptation by using contours. International Journal for Numerical Methodsin Engineering 1991; 31:689–707.

7. Plewa T, Linde TJ, Weirs VG. Adaptive mesh refinement-theory and applications: Proceedings of the ChicagoWorkshop on Adaptive Mesh Refinement Methods. Lecture Notes in Computer Science and Engineering 41, 2005.

8. Phongthanapanich S, Dechaumphai P. Adaptive Delaunay triangulation with object-oriented programming for crackpropagation analysis. Finite Elements in Analysis and Design 2004; 40:1753–1771.

9. Khoei AR, Azadi H, Moslemi H. Modeling of crack propagation via an automatic adaptive mesh refinement basedon modified superconvergent patch recovery technique. Engineering Fracture Mechanics 2008; 75:2921–2945.

10. Schrefler BA, Secchi S, Simoni L. On adaptive refinement techniques in multi-field problems including cohesivefracture. Computer Methods in Applied Mechanics and Engineering 2006; 195:444–461.

11. Kulkarni MG, Matous K, Geubelle PH. Coupled multi-scale cohesive modeling of failure in heterogeneous adhesives.International Journal for Numerical Methods in Engineering 2010; 84:916–964.

12. Boroomand B, Soghrati S, Movahedian M. Exponential basis functions in solution of static and time harmonicproblems in a meshless style. International Journal for Numerical Methods in Engineering 2010; 81(8):971–1018.

13. Movahedian B, Boroomand B, Soghrati S. A Trefftz method in space and time using exponential basis functions:application to direct and inverse heat conduction problems. Engineering Analysis with Boundary Elements 2013;37(5):868–883.

14. Li S, Liu WK. Meshfree and particle methods and their applications. Applied Mechanics Reviews 2002; 55(1):1–34.15. Nguyen NP, Rabczuk T, Bordes S, Duflot M. Meshless methods: a review and computer implementation aspects.

Mathematics and Computers in Simulation 2008; 79:763–813.16. Duarte CA, Oden TJ. H-p clouds – an h-p meshless method. Numerical Methods for Partial Differential Equations

1996; 12(6):673–705.17. Oden TJ, Duarte CA, Zienkiewicz OC. A new cloud-based hp finite element method. Computer Methods in Applied

Mechanics and Engineering 1998; 153(1-2):117–126.18. Melnek JM, Babuska I. The partition of unity finite element method: basic theory and applications. Computer

Methods in Applied Mechanics and Engineering 1996; 139(1-4):289–314.19. Babuska I, Melnek JM. The partition of unity method. International Journal for Numerical Methods in Engineering

1997; 40(4):727–758.20. Babuška I, Strouboulis T. The Finite Element Method and Its Reliability. Oxford University Press: New York, NY,

USA, 2001.21. Belytschko T, Gracie R, Ventura G. A review of extended/generalized finite element methods for material modeling.

Modeling and Simulation in Material Science and Engineering 2009; 17:043001.22. Soghrati S, Aragón AM, Duarte CA, Geubelle PH. An interface-enriched generalized finite element method for

problems with discontinuous gradient fields. International Journal for Numerical Methods in Engineering 2012;89(8):991–1008.

23. Soghrati S, Geubelle PH. A 3D interface-enriched generalized finite element method for weakly discontinuousproblems with complex internal geometries. Computer Methods in Applied Mechanics and Engineering 2012;217-220:46–57.

24. Soghrati S, Thakre PR, White SR, Sottos NR, Geubelle PH. Computational modeling and design of actively-cooledmicrovascular materials. International Journal for Heat and Mass Transfer, 2012; 55(19-20):5309–5321.

25. Aragón AM, Soghrati S, Geubelle PH. Effect of in-plane deformation on the cohesive failure of heterogeneousadhesives. Journal of Mechanics and Physics of Solids 2013; 61:1600–1611.

26. Soghrati S. Hierarchical interface-enriched finite element method: an automated technique for mesh-independentsimulations. Journal of Computational Physics 2014; 275:41–52.

27. Osher S, Sethian JA. Fronts propagations with curvature dependent speed: algorithms based on Hamilton–Jacobiformulations. Journal of Computational Physics 1988; 79:12–49.

28. Cheng KW, Fries TP. Higher-order XFEM for curved and strong discontinuities. International Journal for NumericalMethods in Engineering 2009; 82:564–590.

29. Dreau K, Chevaugeon N, Moës N. Studied X-FEM enrichment to handle material interfaces with higher order finiteelement. Computer Methods in Applied Mechanics and Engineering 2010; 199(29-32):1922–1936.

30. Fries TP, Byfut A, Alizada A, Cheng KW, Schroöder A. Hanging nodes and XFEM. International Journal forNumerical Methods in Engineering 2011; 86(4-5):404–430.



31. Legrain G, Chevaugeon N, Dreau K. High order X-FEM and levelsets for complex microstructures: uncouplinggeometry and approximation. Computer Methods in Applied Mechanics and Engineering 2012; 241-244:172–189.

32. Byfut A, Schroöder A. hp-adaptive extended finite element method. International Journal for Numerical Methods inEngineering 2012; 89(11):1392–1418.

33. Strouboulis T, Zhang L, Babuška I. Generalized finite element method using mesh-based handbooks: applica-tion to problems in domains with many voids. Computer Methods in Applied Mechanics and Engineering 2003;192(28):3109–3161.

34. Fried I. Condition of finite element matrices generated from nonuniform meshes. AIAA Journal 1972; 10(2):219–221.


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