multiscale enrichment based on partition of unity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2005; 62:1341–1359Published online 8 December 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1230

Multiscale enrichment based on partition of unity

Jacob Fish∗,† and Zheng Yuan

Departments of Civil, Mechanical and Aerospace Engineering, Rensselaer Polytechnic Institute,Troy, NY 12180, U.S.A.

SUMMARY

A new multiscale enrichment method based on the partition of unity (MEPU) method is presented. Itis a synthesis of mathematical homogenization theory and the partition of unity method. Its primaryobjective is to extend the range of applicability of mathematical homogenization theory to problemswhere scale separation may not be possible. MEPU is perfectly suited for enriching the coarse scalecontinuum descriptions (PDEs) with fine scale features and the quasi-continuum formulations withrelevant atomistic data. Numerical results show that it provides a considerable improvement overclassical mathematical homogenization theory and quasi-continuum formulations. Copyright 2005John Wiley & Sons, Ltd.

KEY WORDS: multiscale; homogenization; partition of unity; quasi-continuum; finite elements

1. INTRODUCTION

This manuscript is concerned with developing a systematic approach aimed at enriching coarsescale mathematical models with relevant fine scale features. The fine scale enrichment could beeither discrete (atomistic) or continuous. The discrete variant of the enrichment is designed toenrich quasi-continuum formulations, whereas the continuum enrichment is targeted for coarsescale models governed by PDEs.

The method advocated in this paper belongs to the category of methods employing hierarchi-cal decomposition of the approximation space in the form of u = uc +uf , where uc and uf arecoarse and fine scale solutions, respectively. The literature on this type of methods is rapidlyexpanding, in particular due to the renewed interest in scale bridging methods spurred by theemergence of nanotechnology.

∗Correspondence to: J. Fish, Departments of Civil, Mechanical and Aerospace Engineering, Rensselaer PolytechnicInstitute, Troy, NY 12180, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: National Science Foundation; contract/grant number: CMS 0303902, CMS 0310596

Received 28 June 2004Revised 16 July 2004

Copyright 2005 John Wiley & Sons, Ltd. Accepted 1 September 2004

1342 J. FISH AND Z. YUAN

There seems to be no consensus on the origin of this decomposition, but quite often theglobal–local paper of Mote [1] that appeared in 1971 is mentioned in this regard in finiteelements. For review articles on the early attempts in mid and late 1970s see Noor [2] andDong [3]. In most of these early attempts global functions describing closed form analyticalsolutions (uf ), such as crack tip asymptotic fields [4] or discontinuities in shear bands [5], wereused to enrich finite element approximations (uc). These so-called global enrichment methods(GEM) are very efficient in enhancing finite element discretizations with very few fine scalefunctions known to approximately describe the asymptotic behaviour of the solution, but onthe negative side, they often result in dense matrices.

These considerations spurred the development of local (element level) enrichment methods(LEM) where the fine scale function(s) can be condensed out on the element level. Localenrichment functions have been used to embed discontinuous strains [6, 7], curvatures [8]and displacements [9, 10]. Variational multiscale method (VMS) [11] falls into this category ofmethods. VMS was partially motivated by identifying it with classical stabilized methods whereuf takes the form of a residual-free bubble [12, 13]. These bubbles are functions required tovanish on element boundaries for second order problems. VMS has been subsequently appliedto embed micromechanical fields in solid mechanics by Garikipati and Hughes [14, 15].

Mathematical homogenization [16–18] can be interpreted as an LEM. By this approachuc = u0(x) and uf = u1(x, y) + O(2), where x and y = x/ are coarse and fine scaleposition vectors, respectively; 0 < = l/L>1, and l, L denote the characteristic size of thefine and coarse scales, respectively. Note that the leading order fine scale solution is O(),whereas the leading order solution gradient (or strain), 0 = ∇s

xu0+∇s

yu1, has O(1) contribution

from both the fine and coarse scales, where ∇sx , ∇s

y are symmetric gradient with respect to thecoarse and fine scale coordinates, respectively, and 0 is the leading order strain. Moreover,since the fine scale is related to the coarse scale by ∇s

yu1 = (y)∇s

xu0(x), it can be eliminated

at the level of the coarse scale material point (constitutive equation) by means of homogenizedor effective material properties.

An attractive feature of the LEM is that fine scale features can be eliminated either on thecoarse scale element level (static condensation) or at the coarse scale material point (effectiveproperties) resulting in embedded fine scale physics at the coarse scale without significantlyincreasing the cost of coarse scale computations. On the negative side, fine scale features areonly approximated, and their gross response is injected into the coarse scale. This type of scalebridging is often coined as sequential, serial, information-passing, or parameter-passing.

The need to control fine scale approximations without compromising the sparsity of discreteapproximations produced several enrichment schemes to be referred here as sparse globalenrichment methods (SGEM). Among the noteworthy SGEMs are the s-version of the finiteelement method [19–22] with application to strong [23, 24] and weak [25–28] discontinuities,various multigrid-like scale bridging methods [29–32], the extended finite element method(XFEM) [33–35] and the generalized finite element method (GFEM) [36, 37] both based onthe partition of unity (PU) framework [38, 39] and the discontinuous Galerkin (DG) [40, 41]method. Multiscale methods [19–41] based on the concurrent resolution of multiple scales areoften called as embedded, concurrent, integrated or hand-shaking multiscale methods.

In the s-version, the fine grid is superimposed in subdomain(s) where fine scale resolution isneeded. It offers the flexibility of retaining fixed coarse grid, while adapting the fine grid basedon where and when is required. The downside is the need for remeshing the fine grid andsomewhat cumbersome quadrature required to integrate coupling terms between the meshes.

Copyright 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 62:1341–1359

MULTISCALE ENRICHMENT METHOD 1343

Unstructured multigrid-like methods require remeshing to capture evolving fine scale features,but the need for integrating coupling terms is circumvented. Multigrid-like approaches aimed atbridging between discrete and continuum scales have been reported in References [32, 42, 43].

The attractiveness of PU-based methods stems from the fact that special functions knownto approximate the solution can be embedded into the coarse scale by a simple multiplicationof these functions with standard finite element shape functions without introducing finer grids.The PUM-based methods are based on the theorem [38], which states that if there exists sucha enrichment function known to approximate the exact solution uex on a patch , suchthat

∑i ‖uex − ‖2

U() 2, then the global error is bounded by∣∣∣∣

∣∣∣∣uex −∑

N

∣∣∣∣∣∣∣∣()

where N and are the standard finite element shape functions and special functions knownto approximate the solution on a patch.

Special functions can be also introduced locally using Lagrange multipliers [44] or bymeans of the DG method [40], but the question of stability cannot be easily resolved, and theimplementation of such methods requires major changes to existing finite element codes.

In XFEM, which use local partition of unity [45], the enrichment functions are usedto describe spatial features, such as asymptotic crack fields [33] or local flow fields [46];in addition they provide means to model arbitrary discontinuities. For this discontinuousenrichment functions, the problem of linear dependency does not arise, but the issue ofensuring the integration errors to be significantly smaller than the approximation errorsrequires special attention [35]. In GFEM, which uses special handbook function [36, 37], theresulting space is not linearly independent and the computational overhead (needed for eitherreorthogonalization or for using an indefinite solver) might be quite significant. Moreover,integration of coupling terms involving special handbook functions is challenging at best,and therefore it is not surprising that the method has been implemented for two-dimensionalproblems only.

The method presented in this manuscript is a synthesis of the mathematical homogenizationtheory [16, 17] and the PU method. It is intended for class of problems involving multiplespatial scales. One of the main objectives of the so-called multiscale enrichment based onpartition of unity (MEPU) method is to extend the range of applicability of the mathematicalhomogenization theory to problems where scale separation may not exist, i.e. beyond thetraditional range of = l/L>1. MEPU is applicable to both enriching coarse scale continuumdescription with fine scale (computationally unresolvable) features as well quasi-continuumformulations [47, 48] with relevant atomistic data. The latter can be viewed as an atomisticallyinformed (and therefore nearly optimal) generalization of the classical Born rule.

MEPU takes advantage of the simplicity of the information transfer between the scalesinherent to the mathematical homogenization theory and the elegance of enforcing C0 continuityof solution without compromising on sparsity provided by the PU framework. MEPU is free ofsome of the drawbacks inherent in each of its two constituents, namely: (i) the discontinuity ofthe fine scale enrichment function u1(x, y) resulting from the mathematical homogenizationtheory and (ii) some challenges involved in integrating coupling scale terms appearing in PUbased methods. Moreover, MEPU is a hierarchical method in the sense that it converges to thefine scale description as the space of enrichment functions is expanded and the size of the coarse



mesh is decreased. Therefore, it could serve as a dual-purpose concurrent-information-passingmethod.

The outline of this paper is as follows. Section 2 describes the basic formulation of themethod. Analysis of the accuracy of the integration scheme is given in Section 3. The issueof interfacing MEPU and standard elements is studied in Section 4. Validation studies for con-tinuum and quasi-continuum problems are given in Section 5. Conclusions and future researchdirections are outlined in Section 6. Attention is restricted to linear constitutive equations (orharmonic potentials) and steady state problems.

2. FORMULATION OF MEPU

We start by stating the key result from the mathematical homogenization theory for periodicelastic heterogeneous media. Consider a two-term double-scale asymptotic expansion of thesolution, ui = u0

i (x) + u1i (x, y), where x and y = x/ are coarse and fine scale position

vectors, respectively; 0 < = l/L>1, and l, L denote characteristic size of the fine and coarsescale, respectively. Following References [16–18], the fine scale solution is given as

u1i (x, y) = ikl(y)0

kl(x) (1)

where lower case roman subscripts are reserved for spatial dimensions; 0ij the symmetric

gradient of the coarse scale solution given by 0ij ≡ u0

(i,j) = (u0i,j + u0

j,i)/2; comma denotesthe spatial derivative; regular brackets around two subscripts denote symmetric gradient; andikl(y) is the influence function (symmetric with respect to kl indices) obtained from the unitcell solution, the continuum version of which is given by

yj

[Lijkl((i,j)kl + Iijkl)] = 0 on

ikl(y) = ikl(y + y) on

ikl(y) = 0 on vert

(2)

where is a unit cell domain, is its boundary and vert are the vertices of the unitcell; Lijkl is the linear elasticity constitutive tensor, Iklmn = (mknl + nkml)/2; y is thebasic period of the unit cell; Equation (2) is often replaced by the normalization condition∫ ikl d = 0. The unit cell problem is typically solved using finite element method.

One of the salient features of the mathematical homogenization theory is that the fine scalesolution is completely described by the coarse scale and that the influence functions can beprecomputed at a material point. The microstructure in the unit cell could be periodic orrandom, but the solution has to be locally periodic, i.e. at the neighbouring points in the coarsemesh, homologous by periodicity (same y), the value of a response function is the same, butat points corresponding to different points in the coarse scale (different x), the value of thefunction can be different.

For these assumptions to be valid a unit cell has to ‘experience’ a constant variation of 0ij

over its domain . This assumption breaks down in the high gradient regions, such as in thevicinity of cracks or cutouts or where the characteristic size of the unit cell is comparable to



Ω

1

1

-1

ΘI

Ωe

Figure 1. Homogenization-like integration scheme.

the coarse scale features. To allow for variation of the coarse scale solution over the unit celldomain, MEPU removes the dependency of the fine scale functions on the coarse scale solutionby replacing 0

ij and ikl(y) in Equation (1) with an independent set of degrees-of-freedomaij and the influence functions defined over the local supports ikl(x)N(x), respectively. Itis convenient to replace a pair of subscripts kl in ikl and akl denoting the enrichment modeby a single upper case Roman subscripts, A, i.e. ikl → iA and akl → aA. The resultingsolution approximation states

ui = N(x)di︸︷︷︸u0

i

+iA(x)N(x)aA (3)

where summation convention is employed for the repeated indices; Greek subscripts denotefinite element nodes.

The discrete system of equations is obtained using a standard Galerkin method. MEPUexploits the special structure of the influence functions by utilizing homogenization-like inte-gration (HLI) scheme. The accuracy of the integration scheme is studied in Section 3. LetI = ∫

e f d be a typical integral to be evaluated over the element domain e, then the HLIscheme states

I =∫

Jf d =

ngauss∑I=1

WIJIfI =ngauss∑I=1

WIJI1

||∫

I

f d︸︷︷︸fI

(4)

where is the biunit parent domain, J the Jacobean, and ngauss the number of quadraturepoints.

The HLI scheme schematically depicted in Figure 1, centres a unit cell at each of the coarsescale Gauss points. The value of the integrand at a coarse scale Gauss point is replaced by itsintegral over the unit cell domain normalized by the volume of the unit cell. For coarse scaleelements, encompassing numerous unit cells there is a significant cost savings compared to thedirect numerical integration over all unit cells contained in the coarse scale element. Moreover,



direct numerical integration difficulties caused by intersecting coarse scale element boundarieswith unit cell boundaries are automatically resolved by HLI.

Compared to the classical homogenization theory, there is an overhead associated with:(i) additional degrees-of-freedom, aA, which cannot be condensed out on the element level,and (ii) the need for integration over all the coarse scale Gauss points as opposed to a singleintegration over the unit cell domain required by the classical homogenization theory for linearproblems. Ideally, MEPU should be used in the critical regions only, whereas the classicalhomogenization should be employed elsewhere. The issue of transitioning between the tworegions is discussed in Section 4.

Remark 1MEPU can be generalized to quasi-continuum as follows. The starting point is the asymptoticexpansion ui = u0

i (x)+ u1i (x, y) and u1

i (x, y) = iA(y)0A(x) where y is a discrete variable de-

noting position of atoms. For more details and extension to multiple temporal scales we refer toReference [49]. The discrete influence function can be obtained by minimizing potential energyin a unit cell, (u0

i (x) + iA(y)0A(x)), with respect to the discrete influence function iA(y)

subjected to periodic boundary conditions iA(y) = iA(y+ y) on while keeping the coarsescale fields fixed over the unit cell domain. Once the influence functions are computed, theenriched quasi-continuum interpolation is given by Equation (3), and the formulation proceedsalong the lines described in References [47, 48].

3. ANALYSIS OF THE INTEGRATION SCHEME

Let e = [0, 1] be the domain of coarse scale elements in 1D such that nuc = 1 where nucrepresents the number of unit cells in e and is the size of the unit cell. Consider a typicalintegrand of the form (c0 + c1x + c2x

2)g(x) where g(x) is a -periodic function such that∫

g(x) dx = =

nuc(5)

In this section we show that the integration error resulting from the HLI scheme is of theorder O(1/nuc).

Consider a typical integral

=∫ 1

0(c0 + c1x + c2x

2)g(x) dx =nuc∑i=1

∫i

(c0 + c1x + c2x2)g(x) dx (6)

To evaluate the exact integral, we utilize the mean value theorem for each unit cell domain,which gives

ex =nuc∑i=1

[(c0 + c1i + c2

2i )

∫i

g(x) dx

]=

nuc

nuc∑i=1

(c0 + c1i + c22i )

= c0 + c11

nuc

nuc∑i=1

i + c21

nuc

nuc∑i=1

2i (7)



where

i ∈ [(i − 1), i] =[(i − 1)

nuc,

i

nuc

]

For numerical integration, we use two Gauss points

nu =2∑

I=1WIJI

1

||∫

I

(c0 + c1x + c2x2)g(x) dx (8)

where

W1 = W2 = 12 , J1 = J2 = 1

1 =[

1

2

(1 − 1√

3

)− 1

2nuc,

1

2

(1 − 1√

3

)+ 1

2nuc

]

2 =[

1

2

(1 + 1√

3

)− 1

2nuc,

1

2

(1 + 1√

3

)+ 1

2nuc

]

As before, to evaluate the two integrals in (8) we use the mean value theorem

nu = 1

2||[(c0 + c11 + c2

21)

∫1

g(x) dx + (c0 + c12 + c222)

∫2

g(x) dx

]

= c0 + c112 (1 + 2) + c2

12 (2

1 + 22) (9)

where

1 ∈[

1

2

(1 − 1√

3

)− 1

2nuc,

1

2

(1 − 1√

3

)+ 1

2nuc

]

2 ∈[

1

2

(1 + 1√

3

)− 1

2nuc,

1

2

(1 + 1√

3

)+ 1

2nuc

] (10)

The integration error is given by

e = |ex − uc|

=∣∣∣∣c1

(1

nuc

nuc∑i=1

i − 1

2(1 + 2)

)+ c2

(1

nuc

nuc∑i=1

2i − 1

2(2

1 + 22)

)∣∣∣∣ |c1| ·

∣∣∣∣ 1

nuc

nuc∑i=1

i − 1

2(1 + 2)

∣∣∣∣︸︷︷︸A

+|c2| ·∣∣∣∣ 1

nuc

nuc∑i=1

2i − 1

2(2

1 + 22)

∣∣∣∣︸︷︷︸B

(11)



The bounds for terms A and B can be estimated in the closed form

Amax = max

(∣∣∣∣∣(

1

nuc

nuc∑i=1

i

)max

−(

1

2(1 + 2)

)min

∣∣∣∣∣ ,∣∣∣∣∣(

1

2(1 + 2)

)max

−(

1

nuc

nuc∑i=1

i

)min

∣∣∣∣∣)

= max

(1

nuc,

1

nuc

)= 1

nuc

Bmax = max

(∣∣∣∣∣(

1

nuc

nuc∑i=1

2i

)max

−(

1

2

(2

1 + 22

))min

∣∣∣∣∣ ,∣∣∣∣∣(

1

2(2

1 + 22)

)max

−(

1

nuc

nuc∑i=1

2i

)min

∣∣∣∣∣)

= max

(1

nuc− 1

12n2uc

,1

nuc+ 1

12n2uc

)= 1

nuc+ 1

12n2uc

(12)

The integration error can be bounded by

e |c1| · Amax + |c2| · Bmax

= |c1| · 1

nuc+ |c2| ·

(1

nuc+ 1

12n2uc

)∼ O

(1

nuc

)(13)

It can be seen that the integration error of the homogenization-like integration scheme is oforder O(1/nuc), and thus e → 0 as nuc → ∞.

4. FINE-TO-COARSE SCALE TRANSITION

In this section, we show that MEPU elements degenerate to the standard finite elements withhomogenized properties in case of a constant (coarse scale) strain field. We then show thatcontinuity of average tractions and coarse scale displacements between MEPU and standardfinite elements is automatically maintained for the locally constant coarse scale strain field.We then devise a patch to test the validity of the fine-to-coarse scale transitioning. We discussthe formulation in the context of continuum enrichment, but the same procedure applies to thequasi-continuum enrichment (see numerical experiments in Section 5).

Consider the strain field obtained from Equation (3)

B = 0B + (BA(x)N(x) + BBi(x)iA(x))aA (14)

where AB is symmetric gradient of iA

ijA = (iA,j + jA,i)/2 → BA (15)

and BBm the coarse scale strain–displacement matrix defined as

Bijm = 1

2

(N

xj

im + N

xi

jm

)= N

xk

Ikmji → BBm (16)



Given the constant coarse scale strain field, 0B , over the unit cell domain, find aA, such

that ∫

(BAN + BBiiA)DBC[0C + (CDN + BCjjD)aD] d = 0 (17)

The system has a unique solution since∫(BAN + BBiiA)DBC(CDN + BCjjD) d

is symmetric positive definite and the right hand side is non-zero.Taking summation over the element nodes in Equation (17) yields∫

BAB +∑

(BBiiA)DBC[0

C + (CDN + BCjjD)aD]

d = 0 (18)

Due to the PU property,∑

BBi = 0, we get∑

(BBiiA) = 0 (19)

which yields ∫BADBC[0

C + (CDN + BCjjD)aD] d = 0 (20)

Since jD is a weak form solution of Equation (2) satisfying,∫ BADBC(ICD+CD) d =

0, Equation (20) can be written as

∫

BADBCICD0D + BADBCCD0

D︸︷︷︸Unit Cell equation

−BADBCCD0D + BADBC(CDN + BCjjD)aD

d = 0 (21)

which can be further simplified as∫

[BADBCCD(NaD − 0D) + BADBCBCjjDaD] d = 0 (22)

It can be seen that if aD ≡ aD = 0D ∀ then NaD = ∑

NaD = aD = 0D and

the first term in Equation (22) vanishes. Moreover, BCjjDaD = ∑ BCjjDaD = 0, and

thus aD = 0D ∀ is the solution of Equation (22). The resulting stress is given by B =

DBC(ICD + CD)0D . The non-local or average stress becomes

B =∫

DBC(ICD + CD) d︸︷︷︸

DBC

0D ≡ DBC0

D

which is identical to the stress obtained by standard finite element with homogenized properties.The upshot of this finding is that at the interface between the MEPU and standard finite elementswith homogenized properties continuity of average tractions is maintained. Moreover, continuityof coarse scale displacements N(x)di is automatically enforced at the interface, whereas theenrichment iA(x)N(x)aA is allowed to be discontinuous. We will refer to such continuity



Reference Mesh Structured Mesh Unstructured Mesh

Homogenized

MEPU

Homogenized

MEPU

Figure 2. The patch test for interfaces.

Figure 3. Displacements u2 obtained from 011: (a) coarse scale continuity; and (b) fine scale continuity.

condition, which enforces continuity of average tractions and coarse scale displacements, ascoarse scale continuity.

We now consider a patch of elements where a single MEPU element is surrounded bystandard finite elements with homogenized material properties. Both structured and unstructuredmeshes are considered as shown in Figure 2.

For comparison, we also consider the formulation where total displacements across theinterface are enforced to be continuous. We will refer to such continuity formulation as finescale continuity. Figure 3 shows the displacement field u2 obtained by subjecting the patchof elements to Dirichlet boundary conditions corresponding to the constant coarse scale strainfield 0

11.The relative error |eref − eFEM|/eref in strain energy density, e = A0

A/2, is plotted inFigure 4. It can be seen that the formulation enforcing fine scale displacement continuity givesrise to large errors, whereas in case of coarse scale continuity the solution of the strain energydensity is exact.



-20%

50%

-30%

(a) (b)

60%

Figure 4. Errors in relative strain energy density e = A0A/2: (a) coarse scale continuity;

and (b) fine scale continuity.

x

y

Figure 5. Configuration for the two-dimensional continuum fracture problem.

5. NUMERICAL EXAMPLES

In this section, MEPU is tested for various continuum and quasi-continuum fracture problems.We start with a two-dimensional coarse scale configuration shown in Figure 5. The uniform



Figure 6. Reference mesh for the two-dimensional continuum fracture problem: (a) undeformedconfiguration; and (b) deformed configuration.

Table I. Mode I stress intensity factor for the two-dimensional continuum fracture problem.

Structured mesh Unstructured mesh

Methods K Error (%) K Error (%)

HOMO 5.4377E + 0 3.75 5.2848E + 0 6.45MEPU_HOMO 5.5872E + 0 1.10 5.3471E + 0 5.35MEPU 5.6290E + 0 0.36 5.3956E + 0 4.49REF 5.6493E + 0 — 5.6493E + 0 —

displacement boundary condition is applied along the top and bottom edges of the plate. Dueto symmetry, only the upper half of the plate is analysed.

We consider a unit cell with a square inclusion as shown in Figure 6. The phase propertiesof the fine scale constituents are as follows: inclusion: Young’s modulus = 60 GPa, Poisson’sratio = 0.2. Matrix: Young’s modulus = 2 GPa, Poisson’s ratio = 0.2. The coarse scale domaincontains 16 × 16 unit cells. The reference mesh is shown in Figure 6.

For comparison, three methods are investigated: (i) homogenization (HOMO), which usesstandard finite element with homogenized properties, (ii) MEPU, and (iii) combination ofHOMO and MEPU, where MEPU is used around the crack tip and HOMO elsewhere. Bothstructured and unstructured meshes are considered. All the meshes are shown in Figure 7. Theenrichment functions, A, are obtained using finite element solution of the corresponding unitcell problem (2).

The results of mode-I stress intensity factor are summarized in Table I. The stress intensityfactors were evaluated using virtual crack closure integral method [50].

For rate of convergence studies, three meshes shown in Figure 8 were considered. Table IIgives the relative error in the stress intensity factor obtained by using three different methods.The logarithm of error is plotted in Figure 9.



Figure 7. Homogenized and MEPU meshes for the two-dimensional continuum fracture problem: (a)HOMO (MEPU) structured mesh; (b) MEPU_HOMO structured mesh; (c) HOMO (MEPU) unstructured

mesh; and (d) MEPU_HOMO unstructured mesh.

Figure 8. Mesh refinement for the two-dimensional continuum fracture problem: (a) coarse mesh;(b) fine mesh; and (c) very fine mesh.



Table II. The relative error in mode-I stress intensity factor fortwo-dimensional continuum fracture problem.

Methods\mesh Coarse (%) Fine (%) Very fine (%)

MEPU 76.11 32.74 0.36MEPU_HOMO 74.48 30.86 1.10HOMO 98.28 34.72 3.75

-9

-8

-7

-6

-5

-4

-3

-2

-1

0coarse fine very fine

MEPU

HOMO_MEPU

HOMO

Log

2 o

f St

ress

Int

ensi

ty f

acto

r

Figure 9. Convergence of mode-I stress intensity factor.

In the next set of numerical examples, we consider a three-dimensional fracture problem withtwo different unit cells. The former is a fibrous composite with the following phase properties.Fiber: Young’s modulus = 379 GPa, Poisson’s ratio = 0.21. Matrix: Young’s modulus = 69 GPa,Poisson’s ratio = 0.33. The problem configuration and the corresponding unit cell are shownin Figure 10. The latter is a satin weave [51]. The phase properties of the weave constituentsare as follows. Blackglas matrix: Young’s modulus = 69 GPa, Poisson’s ratio = 0.33. Bundlephase: Young’s modulus = 679 GPa, Poisson’s ratio = 0.21. The unit cell shown in Figure 11is discretized with 1566 elements.

In general, it is not feasible to use a discretization scheme with grid spacing comparable tothe scale of spatial heterogeneities. To obtain the reference solution the satin weave problemconsidered in Figure 12 was discretized with 626 400 standard elements. On the hand, MEPUrequires, only 400 elements, but each element has additional 6 degrees-of-freedom per node.

The results of the strain energy release rate for the two unit cells are summarized in Table III.In the final numerical example, we consider a discrete (atomistic) model problem in two

dimensions. The problem configuration is shown in Figure 13. Atoms are assumed to interactonly with their nearest neighbours in horizontal, vertical and diagonal directions. The interatomicinteractions are assumed to be stronger for the four atoms positioned at the centre of theunit cell (black–black connections) and weaker for the remaining atomic pairs (grey–grey and



Figure 10. Three-dimensional continuum fracture problem for fibrous composite and the correspondingunit cell: (a) reference mesh; and (b) MEPU (HOMO) mesh.

Figure 11. Woven unit cell [51]: (a) geometric model; and (b) finite element mesh.

Figure 12. Three-dimensional continuum fracture problem for woven composites.



Table III. Strain energy release rate G of three-dimensional continuum fracture problem.

Fibrous composites Woven composites

Methods G Error (%) G Error (%)

HOMO 2.0564 4.61 28.0564 4.16MEPU 1.9991 1.69 27.4670 1.97REF 1.9659 — 26.9371 —

Figure 13. Two-dimensional quasi-continuum fracture problem and the corresponding unit cell.

Table IV. Potential energy E and potential energy release rate G oftwo-dimensional atomic fracture problem.

Potential energy Potential energy release rate

Methods E Error (%) G Error (%)

QC 1.75 90.42 0.041460 89.88MEPU 0.925 0.65 0.021139 3.18REF 0.919 — 0.021835 —

grey–black connections) as shown in Figure 13. The quadratic approximation of the Lennard–Jones potential

Ea(r) = −1 + 12 a(r − r1

0 )2, Eb(r) = −2 + 12 b(r − r2

0 )2 (23)

has been employed, where rk0 (k = 1, 2) are the interatomic distances at the equilibrium; a and

b are constants defined by

a = 36 3√

41

21

and b = 36 3√

42

22

where 1 = 3.405 × 10−10, 1 = 1.6572 × 10−21; 2 = 3.405 × 10−10, 2 = 1.6572 × 10−20.The results of the potential energy and the energy release rate obtained using molecular

mechanics (the reference solution), MEPU and quasi-continuum method based on the Bornrule are summarized in Table IV.



It can be seen that MEPU provides a very accurate solution for both the potential energyand the energy release rate, whereas quasi-continuum assuming Born rule behaves poorly forproblems involving heterogeneous potentials.

6. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

Multiscale enrichment based on PU presented in this manuscript provides a considerable im-provement over classical mathematical homogenization theory and quasi-continuum for contin-uum and discrete systems, respectively. One of its main attractions stems from the ease ofimplementation into most of the commercial software packages. Any commercial code thatallows for adding user-defined elements with arbitrary number of degrees-of-freedom per nodecan be used for implementation. MEPU can be easily implemented into the quasi-continuumframework by replacing the Born rule with appropriate enrichment functions. Homogenizationerror estimators [52, 53] can be utilized to identify the location where enrichment is needed.

So far, the method has been studied assuming small deformations, harmonic potentials andsteady state conditions. It remains to be seen whether the coarse scale continuity conditions setfourth in this paper are sufficient to avoid spurious wave reflections at the atomistic/continuuminterface or whether some sort of overlapping as described in Reference [54] is needed.

Wave dispersion is another issue, which needs a careful examination. Existing methodologies(see for example [55–57]) are based on continuum (or quasi-continuum) enrichment. It remainto be seen if higher order enrichment functions Pijmn (18 functions in 3D corresponding tothe gradients of the coarse scale strain field), which play a central role in the formulation ofexisting dispersive models, have to be added to enrich the element kinematics. With increasein the order of enrichment functions, the polynomial order of the coarse scale fields may haveto be appropriately increased to let the unit cell ‘experience’ higher order coarse fields over itsdomain. Thus, it may be advantageous to use different polynomial order of the shape (support)functions in the first and second term of Equation (3).

These are just few of the issues that will be addressed in our future research work.

ACKNOWLEDGEMENTS

This work was supported by the National Science Foundation under grants CMS 0303902 and CMS0310596.

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