computational continua

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2010; 84:774–802Published online 17 May 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.2918

Computational continua

Jacob Fish∗,† and Sergey Kuznetsov

Multiscale Science and Engineering Center, Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A.

SUMMARY

We develop a new coarse-scale continuum formulation hereafter referred to as computational continua.By this approach, the coarse-scale governing equations are stated on a so-called computational continuadomain consisting of disjoint union of computational unit cells, positions of which are determined toreproduce the weak form of the governing equations on the fine scale. The label ‘computational’ isconceived from both theoretical and computational considerations. From a theoretical point of view, thecomputational continua is endowed with fine-scale details; it introduces no scale separation; and makesno assumption about infinitesimality of the fine-scale structure. From computational point of view, thecomputational continua does not require higher-order continuity; introduces no new degrees-of-freedom;and is free of higher-order boundary conditions. The proposed continuum description features two buildingblocks: the nonlocal quadrature scheme and the coarse-scale stress function. The nonlocal quadraturescheme, which replaces the classical Gauss (local) quadrature, allows for nonlocal interactions to extendover finite neighborhoods and thus introduces nonlocality into the two-scale integrals employed in varioushomogenization theories. The coarse-scale stress function, which replaces the classical notion of coarse-scale stress being the average of fine-scale stresses, is constructed to express the governing equationsin terms of coarse-scale fields only. Perhaps the most interesting finding of the present manuscript isthat the coarse-scale continuum description that is consistent with an underlying fine-scale descriptiondepends on both the coarse-scale discretization and fine-scale details. As a prelude to introducing thecomputational continua framework, we unveil the relation between the generalized continua and higher-order mathematical homogenization theory and point out to their limitations. This serves as motivationto the main part of the manuscript, which is the computational continua formulation. Copyright � 2010John Wiley & Sons, Ltd.

Received 21 December 2009; Revised 15 March 2010; Accepted 17 March 2010

KEY WORDS: multiscale; mathematical homogenization; computational continua; unit cell; nonlocal;large deformation

∗Correspondence to: Jacob Fish, Multiscale Science and Engineering Center, Rensselaer Polytechnic Institute, Troy,NY 12180, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: Office of Naval Research; contract/grant number: N000140310396Contract/grant sponsor: National Science Foundation; contract/grant numbers: CMS-0310596, 0303902, 0408359Contract/grant sponsor: Rolls-Royce; contract/grant number: 0518502Contract/grant sponsor: Automotive Composite Consortium; contract/grant number: 606-03-063LContract/grant sponsor: AFRL/MNAC; contract/grant number: MNK-BAA-04-0001

Copyright � 2010 John Wiley & Sons, Ltd.

COMPUTATIONAL CONTINUA 775

1. INTRODUCTION

Development of new materials in the past decade spurred renewed interest in nonlocal and gener-alized continuum theories. These generalizations of the classical local continuum description wereconceived to account for subscale details in the continuum description and to deal with nonlocalnature of fracture and localization. Generalized and nonlocal continuum theories are equipped withmaterial microstructure information in the form of enhanced kinematics, balance and constitutiveequations or their nonlocal (integral) representation.

Generalized continuum models can be classified into two main categories [1]: higher-gradecontinua and higher-order continua. Higher-grade continuum is characterized by higher-orderspatial derivatives of the displacement field [2–8], whereas higher-order continuum is endowed withadditional degrees-of-freedom independent of the usual translational degrees-of-freedom rangingfrom 3 rotational degrees-of-freedom in the Cosserat or polar continuum [9] to 12 degrees-of-freedom in the micromorphic continuum [10] and more in the so-called multiscale micromorphiccontinuum [11]. For background information, we refer to review article of Green and Naghdi [12]and monograph of Eringen [13].

The link between the generalized continua, on one hand, and homogenization theories on theother, was shown to exist by several investigators [14–19]. The unit cell (UC) or the representativevolume element (RVE) in the first-order homogenization theories [20–28] experiences (or issubjected to) a constant macroscopic deformation gradient independent of its size. This is ananomaly that becomes unacceptable as the UC size and/or strain gradients become sufficientlylarge. Higher-order theories [29, 30], on other hand, are equipped with a mechanism of subjectingthe UC to ‘true’ macroscopic deformation. For example, in second-order theories, the macroscopicdeformation gradient is idealized to vary linearly over the UC domain. Yet, despite the noteworthyperformance in some localization problems [31–33], higher-order homogenization theories are notwithout shortcomings. From theoretical point of view, they hinge on the two-scale integrationscheme (see Equation (29)), which assumes infinitesimality of the UC although the coefficientsof the enriched coarse-scale continua depend on the size of the UC. Moreover, these theoriesrequire consideration of higher-order boundary conditions. While higher-order homogenizationand generalized continua theories are equipped with enriched kinematics, which approximate thefine-scale deformation, they remain local in nature. Trostel [34] and Forest and Sievert [1] referredto these type of continuum models as local, whereas Bazant and Jirasek [35] classified both thegeneralized continua and nonlocal gradient models as weakly nonlocal. From computational pointof view, the generalized continua models require consideration of additional degrees-of-freedomin combination with hybrid formulations, or alternatively, impose C1 continuity requirement.

Various nonlocal theories of either integral or gradient type include a nonlocal kernel functionwhose support provides an internal length scale. The integral formulation reduces to the gradienttype by truncating the series expansion of the nonlocality kernel [36]. By virtue of truncation,the nonlocal gradient models assume that nonlocal interactions are limited to close neighborhood.In the earlier works, nonlocality was introduced in the nonlocal approximations of fields andbalance equations [37, 38] with later works focusing on nonlocality in internal variables [39, 40],which is closely related to the gradient plasticity theories [41]. The nonlocal theories reduceto those of the generalized continua or higher-order homogenization theory when the coarse-scale problem size is much larger than the scale of heterogeneity. Selection of nonlocal kernelsand the magnitude of the internal length scale are still controversial issues. It is also unclearhow to construct nonlocal kernels that would have a sufficient degree of generality for a wide

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2010; 84:774–802DOI: 10.1002/nme

776 J. FISH AND S. KUZNETSOV

range of problems in heterogeneous media. For problems for which nonlocal interactions are wellunderstood, computational difficulties related to higher-order continuity and boundary conditionscan be partially alleviated by a reproducing kernel strain regularization of implicit gradient models[42]. Also, noteworthy are more recent variations of higher order and nonlocal continuum theoriesthat bring certain microstructural information to the macroscopic equation of motion [43–47].

The primary goal of the present manuscript is to develop a coarse-scale continuum description,which is consistent with an underlying fine-scale description, for heterogeneities of finite size.The so-called computational continua developed here possesses the fine-scale features, but withoutintroducing scale separation, which can be mathematically justified provided that the fine-scalestructural details are infinitesimally small. From computational point of view, the computationalcontinua does not require higher-order continuity, introduces no new degrees-of-freedom and isfree of higher-order boundary conditions. The proposed continuum description features (i) nonlocalquadrature scheme defined over computational continua domain consisting of disjoint union ofcomputational UCs, positions of which are determined to reproduce the weak form of the governingequations on the fine scale; and (ii) the coarse-scale stress function, which replaces the classicaldefinition of coarse-scale stress being the average of fine-scale stresses and thus allowing to restatethe governing equations of continua in terms of coarse-scale fields only. We distinguish betweenthe classical notion of the UC defined in the rescaled (stretched) coordinate system positionedat the coarse-scale element Gauss quadrature points and the computational UCs defined in thephysical domains whose locations are chosen to provide variational consistency with the fine-scalegoverning equations.

The outline of the manuscript is as follows. We consider a strong form of the boundary valueproblem on a composite domain ��

X with boundary ��X given as

�P�ij (F

�)

�X j+B�

i = 0 on ��X (1)

F�ik = �ik+

�u�i

�Xk(2)

P�ij N�

j = T ��i on ��t�

X (3)

u�i = ui on ��u�

X (4)

��t�X ∪��u�

X = ��X and ��t�

X ∩��u�X =0 (5)

where P� denotes the First Piola–Kirchhoff stress tensor; F� the deformation gradient; B� and T��

the body force and traction vectors, respectively; N� the unit normal to the boundary ��X . Lower

case subscripts i and j denote spatial dimensions except for subscripts X and x that refer to theinitial and deformed configurations, respectively. The superscript � denotes existence of fine-scalefeatures. Summation convention over repeated subscripts is employed except for the subscripts Xand x .

As a prelude to introducing the computational continua formulation, Section 2 establishes therelation between the generalized continua and the mathematical homogenization theory and pointsout to their limitations. To our knowledge, this is an original attempt to derive the governing equa-tions of the generalized continua for large deformation problems from asymptotic homogenization



method. The computational continua formulation is presented in Section 3, which is self contained.It includes the formulation of the coarse- and fine-scale problems, the nonlocal quadrature schemedefined over computational continua domain consisting of disjoint union of computational UCdomains, the derivation of the coarse-scale stress function and the finite element discretizationof the two-scale problem including treatment of weakly periodic boundary conditions. Numericalexamples conclude the article in Section 4. Conclusions and future research directions are discussedin Section 5.

2. HOMOGENIZATION, GENERALIZED CONTINUA AND THEIR LIMITATIONS

In the mathematical homogenization theory, various fields are assumed to depend on two coordi-nates: X, the coarse-scale coordinate in the initial domain �X , and Y, the fine-scale coordinate inthe initial UC domain �Y . The two coordinates are related by Y≡X/�� with 0<��1 and �=1in the classical theory. The initial composite domain ��

X is defined as a product space �X×�Y .Coordinates in the deformed (or current) configuration are x and y corresponding to the deformed

composite domain ��x , the coarse-scale domain �x and the UC domain �y . Dependence of various

fields on the two scale coordinates is denoted by P�=P(X ,Y),B�=B(X ,Y), T��=T�(X ,Y) andu�(X )=u(X ,Y).

In the mathematical homogenization theory, displacements are expanded as

u�i (X )=u0

i (X )+��u1i (X ,Y)+�2�u2

i (X ,Y)+O(�3�) (6)

where it is assumed that the size of the UC is infinitesimally small and therefore the leading-orderdisplacement u0

i (X ) is considered to be constant over the UC domain. For large UC distortions,u0

i (X ) is no longer constant over the UC domain and therefore Equation (6) has to be modifiedas described below.

We proceed by expanding u0i (X ) in Taylor’s series around the UC centroid X = X .

u0i (X )=u0

i (X )+ �u0i

�X j

∣∣∣∣∣X

(X j− X j )+ 1

2

�2u0i

�X j�Xk

∣∣∣∣∣X

(X j− X j )(Xk− Xk)+·· · (7)

In the classical theory, it is assumed that coarse-scale displacements u0i (X ) and its various order

derivatives are O(1) functions. Here we consider existence of high coarse-scale gradients

�n+1u0i

�X j . . .�Xk= O(�−n�)

l ′nfor n=0,1 and 0<��1 (8)

such that

�u0i

�X j

∣∣∣∣∣X

(X j− X j )= l ′O(��),�2u0

i

�X j�Xk

∣∣∣∣∣X

(X j− X j )(Xk− Xk)= l ′O(��) (9)



where l ′is dimensional characteristic parameter. For the second-order theory considered here, wewill assume that higher-order terms in (7) remain of order O(�2�) and higher. From (8) and (9) itfollows that X j− X j=��Y j and thus u0

i (X ) can be written as

u0i (X ,Y)=uC

i (X ,Y)+l ′O(�2�) (10)

where uCi (X ,Y) is termed as a coarse-scale displacement given by

uCi (X ,Y)=u0

i (X )+ �� u0i

�X j

∣∣∣∣∣X

Y j+ 1

2�2� �2u0

i

�X j�Xk

∣∣∣∣∣X

Y j Yk (11)

Similarly, higher-order terms uni (X ,Y), for n�1, are expanded around X = X as

uni (X ,Y)=un

i (X ,Y)+ �� uni

�X j

∣∣∣∣X

Y j+�2� 1

2

�2uni

�X j�Xk

∣∣∣∣∣X

Y j Yk+l ′O(�3�) (12)

Substituting (11) and (12) into (6) gives

u�i (X )=ui (X ,Y)=uC

i (X ,Y)+��u1i (X ,Y)+�2�

(u2

i (X ,Y)+ �u1i

�X j

∣∣∣∣∣X

Y j

)+l ′O(�3�) (13)

Spatial derivative of f (X ,Y) is given by

� f �

�Xi= 1

�� f (X ,Y)

�Yi(14)

The deformation gradient can be expressed as

F�ik=�ik+

�u�i

�Xk=�ik+ 1

��ui (X ,Y)

�Yk=F0

ik(X ,Y)+��F1ik(X ,Y)+O(�2�) (15)

where �ij is the Kronecker delta and

F0ik(X ,Y)= FC

ik (X ,Y)+F∗ik(X ,Y) (16)

FCik (X ,Y)= �ik+

�u0i

�Xk

∣∣∣∣∣X

+ �� 2u0i

�X j�Xk

∣∣∣∣∣X

Y j (17)

F∗ik(X ,Y)= �u1i (X ,Y)

�Yk(18)

F1ik(X ,Y)= �u2

i

�Yk+ �u1

i

�Xk

∣∣∣∣∣X

(19)



In the present manuscript, the First Piola–Kirchhoff stress Pij(F�) formulation is adopted dueto its conjugacy with the deformation gradient. Expanding Pij(F�) around the leading order defor-mation gradient F0

ik yields

Pij(F�)= Pij(F0)+ �� Pij

�F�mn

∣∣∣∣∣F0

F1mn+O(�2�)= P0

ij (X ,Y)+�� P1ij (X ,Y)+O(�2�) (20)

Further expanding Equation (20) in Taylor series around the centroid X = X yields

Pij(X ,Y)= P0ij (X ,Y)+

�P0ij

�Xk

∣∣∣∣∣X

(Xk− Xk)+�� P1ij (X ,Y)+��

�P1ij

�Xk

∣∣∣∣∣X

(Xk− Xk) · · ·

= P0ij (X ,Y)+��

(�P0

ij

�Xk

∣∣∣∣∣X

Yk+P1ij (X ,Y)

)+O(�2�) (21)

Substituting (21) into equilibrium equation (1) yields the two-scale residual equations

r1i (X ,Y) =

�P0ij (X ,Y)

�Y j=0

rCi (X ,Y) =

�P0ij

�X j

∣∣∣∣∣X

+�P1

ij (X ,Y)

�Y j+Bi=0

(22)

We now focus on the derivation of the weak form starting with a UC problem (22a). Let W�Y×�X

be the space of C0 continuous weakly Y -periodic weight functions on �Y defined as

W�Y×�X={w1(X ,Y) defined in ��×�X , C0(�Y ), weakly Y -periodic} (23)

Multiplying (22a) by w1i and integrating by parts yields the weak form of the UC problem

∫�Y

�w1i

�YkP0

ik(X ,Y)d�=0 ∀w1∈W�Y×�X(24)

where we assumed weak periodicity of P0ik defined as∫

��Y

w1i P0

ij (X ,Y)N�j d�=0 ∀w1∈W�Y×�X

(25)

We now turn to the derivation of the coarse-scale weak form. Let �(X ,Y) be a two-scale functiondefined by interpolation from the UC centroids

�(X ,Y)=∑I �(X I ,Y)�I (X ) (26)

where X I denotes the coordinates of the UC centroid I ; and �I (X )∈C0(�X ) possesses interpo-lation property �I (X J )=�I J .



We define a smooth coarse-scale residual rCi (X ,Y) as

rCi (X ,Y)=

�P0ij (X ,Y)

�X j+

�P1ij (X ,Y)

�Y j+Bi (X ,Y) (27)

where

P Sij (X ,Y)=∑

IP S

ij (X I ,Y)�I (X ) for s=0,1 (28)

The integral of the oscillatory two-scale function over composite domain ��X is defined in the

usual way as

lim��→0

∫��

X

�(X ,Y)d�= lim��→0

∫�X

(1

|�Y |∫

�Y

�(X ,Y)d�

)d� (29)

To construct the coarse-scale weak form, we define the space of C1(�X ) continuous functions as

W�X ={w0 defined in �X , w0∈C1(�X ), w0=0 on ��uX } (30)

and construct the coarse-scale test function wC(X ,Y) as

wCi (X ,Y)=w0

i (X )+�� w0i (X )

�X jY j (31)

The coarse-scale weak form is obtained by integrating the product of rCi (X ,Y) and wC

i (X ,Y) over

composite domain ��X , which yields

lim��→0

∫��

X

wCi (X ,Y)rC

i (X ,Y)d�

=∫

�X

(1

|�Y |∫

�Y

(w0

i (X )+�� w0i (X )

�XkYk

)�P0

ij (X ,Y)

�X jd�

)d�

+∫

�X

(1

|�Y |∫

�Y

(w0

i (X )+�� w0i (X )

�XkYk

)�P1

ij (X ,Y)

�Y jd�

)d�

+∫

�X

(1

|�Y |∫

�Y

(w0

i (X )+�� w0i (X )

�X jY j

)Bi (X ,Y)d�

)d� (32)

Denoting

Pij = 1

|�Y |∫

�Y

P0ij d�, Qikj= 1

|�Y |∫

�Y

P0ikY j d� (33)

Bi = 1

|�Y |∫

�Y

Bi d�, Ti= 1

|��X ∩��tX |∫

��X∩��tX

T �i d�, T �

i = P0ij N�

j (34)

Bij = 1

|�Y |∫

�Y

Bi Y j d�, Ti j= 1

|��X ∩��tX |∫

��X∩��tX

T �i Y j d� (35)



and integrating the coarse-scale spatial derivatives of Pij and Qikj by parts yields the leading-ordercoarse-scale weak form

∫�X

�w0i

�XkPik d�+��

∫�X

�2w0

i

�X j�XkQikj d�

=∫

��tX

w0i Ti d�+

∫�X

w0i Bi d�+��

∫��m

X

�w0i

�XkTij d�+��

∫�X

�w0i

�XkBij d� (36)

where ��mX is portion of the boundary where Tij is prescribed. In the above, we assumed weak

Y -periodicity conditions of P1ij and P1

ij Yk as

∫��Y

P1ij N�

j d�=0,

∫��Y

P1ij N�

j Yk d�=0 (37)

In summary, we seek for the trial function u�∈U�X×�Y , where

U�X×�Y ={

u� defined for (X ,Y)∈�X×�Y , weakly Y -periodic

C0(�Y ) and C1(�X ) in u�= u on ��u�X

}(38)

that satisfies the weak form of the coarse-scale problem (36) and the UC problem (24) subjectedto the weak periodicity conditions (25) and (37).

Equation (36) defines the weak form of the second-grade continuum boundary value problem[32, 48]. The strong form of the higher order/grade continua can be obtained by appropriateintegration by parts of the weak form (36). It involves higher-order derivatives and thus requiresC1 continuity. The second-order continuum can be constructed by defining coarse-scale displace-ment gradients as independent fields and then requiring the two to be equal in the weak sense.While such a formulation alleviates computational difficulties arising from C1 continuity require-ment, it introduces additional degrees-of-freedom and requires mixed or multi-field formulation[31]. At a more fundamental level, the two theories hinge on the two-scale integration scheme(29), which assumes infinitesimality of the UC.

3. SECOND-ORDER COMPUTATIONAL CONTINUA

In this section, we develop a second-order continuum formulation that is free of the theoreticaland the computational limitations discussed in the previous section. The so-called second-ordercomputational continua to be derived hereafter will make no assumption about infinitesimalityof the UC, will require C0 continuity only and will involve no additional degrees-of-freedom.Furthermore, we will make no assumption about scale decomposition, but will introduce a UC localcoordinate system v in the physical UC domain, which is related to the stretched coordinate byv=��Y. Both the test and the trial functions will be decomposed into oscillatory weakly periodicfunctions and smooth coarse-scale functions. In addition to various mathematical homogenizationtheories, such a decomposition has been used in various local enrichment methods includingenriched elements [49, 50], variational multiscale method [51], the s-version of the finite element



method [52] with application to strong [53] and weak [54] discontinuities, the multigrid-likemethods [55, 56] and the extended finite element method [57, 58].

3.1. Nonlocal quadrature

To construct the coarse-scale weak form, it is necessary to integrate functions on a compositedomain ��

X with finite size fine-scale details for which the two-scale integration scheme (29) is nolonger valid. The trivial solution is to replace (29) by a sum of integrals over UC domains. However,such an integration scheme is not practical for problems involving numerous UCs. Furthermore,it would give rise to cumbersome integration over coarse-scale finite element domains whoseboundaries do not coincide with UC boundaries. To circumvent these difficulties, we introducethe so-called nonlocal quadrature scheme by which the integration over composite domain ��

Xis replaced by integration over the so-called computational continua domain �C

X consisting of adisjoint union [59] (sometimes called direct sum or free union) of computational unit cell (CUC)domains � �

X Idenoted as

�CX =

�N∐

I=1� �

X I(39)

where�

X I denotes the coordinates of centroid of the CUC domain � �X I

. Note that if ��X I∩��

X J=0,

∀I �= J then the disjoint union reduces to a regular union.The nonlocal quadrature scheme is then defined as

∫��

X

�(X )d�=�N∑

I=1

∫��

X I

��(

�

X I ,v)�(�

X I ,v)d� (40)

where��(

�

X I ,v) is defined as

��(

�

X I ,v)= J e(�

X I ,v)�

W I

|� �X I| (41)

where�

W I denotes the nonlocal quadrature weight; |��X I| the volume of the CUC domain ��

X I;

J e(�

X I ,v) the Jacobian that maps a coarse-scale element into bi-unit cube (square, interval).�

W I

and�

X I are chosen to exactly evaluate integrals (40) on the composite domain with integrand �(X )

approximated by a polynomial of order m. The pair (�

W I ,�

X I ) depends on the CUC size relative tothe coarse-scale finite element size as will be subsequently discussed.

We first consider the nonlocal quadrature scheme for integrating smooth functions in one dimen-sion followed by the generalization to multidimensions in the remainder of this section. Discussionabout smoothness of �(X ) is given in Section 3.2 (see Remark 4).



Figure 1. Nonlocal quadrature: physical (top) and parent element (bottom) domains. Unit cell domains

are shown in the brackets.�

X I ,�I denote positions of quadrature points in the physical and parent elementdomains, respectively; � and �′ =�/J e are the size of the unit cell in the physical and parent element

domains, respectively; J e= (b−a)/2 is element Jacobian.

Consider a one-dimensional domain [a,b] mapped into a parent element domain [−1,1], asshown in Figure 1.

The goal is to find quadrature weights and sampling points that exactly integrate polynomials ofa given order. Applying the nonlocal quadrature scheme (40) to one-dimensional element domainyields

I e=∫ 1

−1J e�(�)d�=

�N e∑

I=1

WI

�

∫ �/2

−�/2J e�(

�

X I+�)d� (42)

where�

N e is number of quadrature points in the element domain. Applying element mapping tothe UC domain yields

I e=�N e∑

I=1

WI

(�/J e)

∫ (�/J e)/2

−(�/J e)/2J e�(�I+)d (43)

Further denoting �′(�)= J e�(�) and �′ =�/J e, Equation (42) can be rewritten as

I e=∫ 1

−1�′(�)d�=

�N e∑

I=1

WI

�′

∫ �′/2

−�′/2�′(�I+)d (44)

Let us now approximate the integrand �′(�) by a polynomial function

�′(�)=m∑

J=1�J �J−1 (45)

Substituting the polynomial expansion (45) into the quadrature scheme (42) yields

I e=�N e∑

I=1

WI

�′

∫ �′/2

−�′/2

m∑J=1

�J (�I+)J−1 d (46)



A closed-form integration of the above yields

I e =�N e∑

I=1WI

(�1+�2�I+�3

(�2

I+�′2

12

)+�4

(�3

I+�I �

′2

4

)+·· ·

)

= �1

�N e∑

I=1WI+�2

�N e∑

I=1WI �I+�3

⎛⎝�′2

12+

�N e∑

I=1WI �

2I

⎞⎠+�4

⎛⎝ �

N e∑I

WI

(�3

I+�I �

′2

4

)⎞⎠ · · · (47)

On the other hand, exact integration of (45) gives

I e=∫ 1

−1�′(�)d�=

∫ 1

−1

m∑I=1

�I �I−1=2�1+0�2+ 2

3�3+0�4+·· · (48)

Requiring the nonlocal quadrature to exactly integrate polynomials for arbitrary �I yields a non-linear system of equations from which the quadrature weights and positions of the CUCs can bedetermined. For instance, for two-point nonlocal quadrature we get⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1

�1 �2

�21+

�′2

12�2

2+�′2

12

�31+�1

�′2

4�3

2+�2�′2

4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

[W1

W2

]=

⎡⎢⎢⎢⎢⎣

2

0

2/3

0

⎤⎥⎥⎥⎥⎦ (49)

which yields

�1,2=±√

1

3−�′2

12, W1,2=1 (50)

Remark 1As expected in the limit as �′→0, the above reduces to the usual Gauss quadrature, hereafter tobe referred as local quadrature. When the CUC size is equal to one half of the element size, �′ =1,we get �1,2=±0.5. As the CUC size further increases the two sampling points move toward theorigin, and when the size of the CUC coincides with that of the element we get �1,2=0, whichis equivalent to having one quadrature point in the middle of the interval with quadrature weightequal to 2. Note that while nonlocal one-point quadrature element maintains full rank (providedthat the CUC is fully integrated), it should be used only if the CUC size is close to that of theelement. Also, it can be seen that �′�2, i.e. the CUC must be smaller than element.

Remark 2For three-point nonlocal quadrature, the weights and quadrature points are

W1,3 = 5(4−�′2)

3(12−7�′2), �1,3=±

√60−35�′2

10

W2 = 2−2W1, �2=0



Unit cell

1

2

1′Θ

2′Θ

Effective unit cell inparent domain

(a) (b)

Figure 2. Definition of the effective unit cell size in the parent element domain: (a) physical element andunit cell domains and (b) parent element on bi-unit square; shaded area denotes mapped unit cell; dashed

lines denote effective unit cell.

Note that when �′ =0 we have W1,3= 59 and �1,3=±0.774596692, which reduces to the local

three-point Gauss quadrature. For �′ =1, the three-point nonlocal quadrature reduces to two-pointnonlocal quadrature with W1,3=1, W2=0 and �1,3=±0.5. The three-point nonlocal quadratureshould be limited to �′�1 to avoid negative values of weights.

Remark 3The formula for nonlocal quadrature in multidimensions remains the same as defined byEquation (40), but the evaluation of sampling points and weights needs to be further addressed.For elements whose edges (faces in three dimensions) are parallel to those of the CUC, position of

quadrature points and weights in each space direction a of the parent element (�

W I a,�I a) can bedetermined by �′a=�a/J e

a where �a and J ea are the length of the CUC and one half of the element

length along the axis �a , respectively. When edges (faces in three dimensions) are not parallel, itis necessary to define an effective CUC domain �′a as described below. Consider a quadrilateralelement and a CUC in the physical (left) and parent (right) element domains as shown inFigure 2.

Note that a rectangular CUC in the physical domain becomes distorted following elementmapping. Thus, an effective rectangle (brick) CUC has to be defined whose edges (faces)are parallel to those of the parent element. Such an effective CUC can be determined tohave the same centroid, the same area (volume) and the same ratio between the moments ofinertia. An alternative to constructing quadrature scheme in multidimensions by tensor productof one-dimensional quadrature is to directly derive quadrature weights and sampling pointsin multidimensions. This can be accomplished by integrating the monomials in the Pascaltriangle (pyramid) exactly and by the nonlocal quadrature scheme (40) This forms a systemof non-linear equations for the unknown quadrature positions and weights that can be solvedfor each element in the preprocessing stage. We will elaborate on this variant in our futurework.



3.2. Coarse-scale problem and coarse-scale stress function

Consider the coarse-scale weak form∫��

X

wCi

(�

�XkP�

ik+B�i

)d�=0 ∀wC∈W C

�X(51)

where wC is C0 continuous test function on �X satisfying homogeneous boundary conditions on �uX

W C�X={wC defined in �X ,C0(�X ), wC=0 on ��u

X } (52)

Integrating by parts the divergence terms and applying the nonlocal quadrature scheme yields

�N∑

I=1

∫��

X I

��(

�

X I ,v)�wC

i

�X jP�

ij (�

X I ,v)d�=∫

��tX

wCi T ��

i d�+∫

�X

wCi B�

i d� (53)

We now define the so-called coarse-scale stress function to address the computational challengeof integrating over CUC domains � �

X Iwhen the size of the heterogeneity is much smaller than

the size of the CUC domain. We decompose P�ij (

�

X I ,v) into the coarse-scale stress PCij (

�

X I ,v) and

the fine-scale perturbation P∗ij (�

X I ,v) as

P�ij (

�

X I ,v)= PCij (

�

X I ,v)+P∗ij (�

X I ,v) (54)

where

PCij (

�

X I ,v)= Pij(�

X I )+Qijk(�

X I )�k (55)

The first term in (55) represents the constant part of the coarse-scale stress, whereas the second

term describes its linear variation. The coefficients, Pij(�

X I ) and Qijk(�

X I ), are constructed to satisfythe following conditions: ∫

��X I

��(

�

X I ,v)�wC

i

�X jP∗ij (

�

X I ,v)d�=0 (56)

with �wCi /�X j locally approximated by a linear field over ��

X I

�wCi

�X j=c0

ij+c1ijk�k ∀c0

ij,c1ijk (57)

The coefficients, Pij(�

X I ) and Qijk(�

X I ), can be evaluated by solving a system of linear equations

0(�

X I )Pij(�

X I )+1k(

�

X I )Qijk(�

X I ) =∫

��

��(

�

X I ,v)P�ij (

�

X I ,v)d�

1j (

�

X I )Pij(�

X I )+2jk(

�

X I )Qijk(�

X I ) =∫

��

��(

�

X I ,v)P�ij (

�

X I ,v)� j d�

(58)



where

0(�

X I ) =∫

��X I

��(

�

X I ,v)d�, 1i (

�

X I )=∫

��X I

��(

�

X I ,v)�i d�

2ij(

�

X I ) =∫

��X I

��(

�

X I ,v)�i� j d�

(59)

Furthermore, if the product�� wC

i /�X j is approximated by a linear field c0ij+c1

ijk�k , then thesolution of (56) reduces to

Pij(�

X I )= 1

|� �X I|∫

��X I

P�ij (

�

X I ,v)d� (60)

and

Qijk(�

X I )= 12

l2k |��

X I|∫

��X I

P�ij (

�

X I ,v)�k d� (61)

where lk is the CUC length in the k-direction and the underbar in (61) denotes no summation overthe repeated indices.

Substituting (56) into (53) yields the computational continua weak form

�N∑

I=1

∫��

X I

��(

�

X I ,v)�wC

i

�� jPC

ik (�

X I ,v)d�=∫

��tX

wCi T ��

i d�+∫

�X

wCi B�

i d� (62)

Remark 4The integrand on the left-hand side of Equation (62) is a smooth function defined over computationalcontinua domain, which is a disjoint union of CUC domains. Thus, the original integral (40) of anoscillatory fine-scale function over a composite domain ��

X has been replaced by the integrationof smooth coarse-scale function over computational continua domain. The nonlocal quadraturescheme smoothness requirement is therefore met by virtue of replacing the integration of theoscillatory function ��(X )∈��

X by integration of smooth coarse-scale function �C(X )∈�X .

3.3. Computational unit cell (CUC) problem

The CUC problem is constructed by defining a space of test functions over CUC domains as

W��X I

={w1(v,�

X I ) defined in � �X I

, C0(� �X I

), weakly v-periodic} (63)

and requiring the equilibrium equation (1) to be satisfied in the weak sense∫��

X I

w1i

(�

��kP�

ik+B�i

)d�=0 ∀w1∈W��

X I

(64)



Integration by parts of the above yields∫��

X I

�w1i

��kP�

ik d�=0 ∀w1∈W��X I

(65)

where the weak periodicity is defined as∫��

X I

w1i P�

ik N�k d� = 0

∫��

X I

w1i �p

i d� = 0 ∀p=0,1 . . .m(66)

Equation (66b) follows from the approximation of the body force B�i (v) by a polynomial function

of order m over the CUC domain. At a minimum we will require (66b) to be satisfied for p=0.The weak periodicity conditions (66) can be satisfied provided that w1

i (and u1i ), P�

ij are periodicfunctions. This apparently is not trivial for problems where the coarse-scale deformation gradient

FCjk(

�

X I ,v) varies in � �X I

. In this case u1i (

�

X I ,v) is no longer periodic although some investigators

[31, 33] assumed periodicity. This lack of periodicity can be attributed to the following. Consider

the usual decomposition u1i =Hijk(v)FC

jk(�

X I ,v) where Hijk(v) is v-periodic function. If FCjk is not

a function of v, then u1i is periodic. Otherwise u1

i is not periodic.The weak periodicity condition (66a) can be enforced using the Lagrange multiplier, penalty

or augmented Lagrange multiplier method. Here we will discuss how to approximately satisfy itindependent of the specific CUC.

We will require the constraint (66a) to hold for the stress field approximated by P�ik≈�ik+

�ik j� j+�ikjm �u1j/��m for arbitrary coefficients �ik,�ikj,�ikjm, which yields∫

��X I

u1i N�

k d� = 0

∫��

X I

u1i � j N�

k d� = 0

∫��

X I

�u1i

��mN�

k d� = 0

(67)

where w1i in (67a) and (67b) has been substituted by u1

i and in (67c) by a constant. Applying thedivergence theorem to (67) and combining with (66b) for p=0 leads to∫

��X I

�u1i

��md�=0,

∫��

X I

�u1i

��m�k d�=0,

∫��

X I

�2u1i

��m��kd�=0 (68)

The upshot of (68) is that the perturbation u1 should not affect the overall displacement, the overalldeformation gradient and the overall gradient of the deformation gradient.



Figure 3. Pad region around the unit cell domain.

Furthermore, combining (67a) with (67b) for k= j yields homogeneous constraints on each ofthe six bounding surfaces �� j

�X I

of the CUC∫�� j

�X I

u1i d�=0 ∀ j=1,2..6 (69)

where⋃6

j=1�� j�X I

=�� X I

and ��i�X I

⋂�� j

�X I

=0 ∀i �= j . Equation (69) can be interpreted as a

weak compatibility condition between adjacent CUCs. In [60], it has been shown that Equation(69a) is necessary to pass the patch test in a mesh consisting of finite elements enriched with afine-scale kinematics interfacing standard finite elements with homogenized material properties.Equation (69a) was also employed in [31, 32] in conjunction with periodic boundary conditions.

In approximating constraint equation (66b), we assumed that either the test function or the stressare oscillatory functions but not both. To account for the relaxed constraint, we will limit the valueof perturbation u1

i by inserting a thin pad region around the CUC domain as shown in Figure 3.

Remark 5Most of the investigators [16, 31, 33], who considered higher-order homogenization theories,enforced periodicity on u1, but not on its gradient. The fact that periodicity (or rather weakperiodicity) of u1 gradients is not enforced has interesting implications even in the context of homo-geneous materials. Consider a CUC made of a homogeneous material subjected to coarse-scaledisplacements uC

1 = X1 X2, uC2 =0. If we seek for a periodic correction u1

i without requiring peri-odicity of its gradients, then an equilibrated CUC solution uC

i +u1i will switch from the prescribed

hourglass mode to a pure bending mode. This is desirable when one wants to capture bending witha single solid element through the plate thickness. However, if an adjacent unit cell is subjectedto uniform field then the deformation between the two cells will be incompatible. In general,compatibility between the adjacent cells cannot be enforced if the deformation of each unit cell iscontrolled by what happens in the interior.

3.4. Discrete coarse-scale problem

We will consider a Galerkin approximation where the coarse-scale trial uCi and test wC

i functionsare discretized using the same C0(�X ) continuous coarse-scale shape functions N C

i�

uCi = N C

i�dC�

wCi = N C

i�cC�

(70)



where dC� and cC

� denote nodal values of trial and test functions. Greek subscripts are reserved fortensor components and summation convention over repeated indices is employed.

Inserting (70) into the coarse-scale weak form (62) yields the discrete coarse-scale problem,which states: Given n+1T ��

i , n+1 B�i and n+1d, find n+1�dC

� such that

n+1rC� (n+1�dC) ≡ n+1 f int

� −n+1 f ext� =0

n+1dC = n+1d on ��uX

n← n+1 Go to the next load increment

(71)

where n+1rC� and n+1�dC

� are the coarse-scale residual and displacement increment in the (n+1)thload increment, respectively, and

f int� =

�N∑

I=1

∫��

X I

��(

�

X I ,v)�N C

i�

�X jPC

i j (�

X I ,v)d� (72)

f ext� =

∫�X

N Ci�B�

i d�+∫

��tX

N Ci�T ��

i d� (73)

where f int� and f ext

� are the internal and the external forces, respectively.The integral over the CUC domain will be evaluated using local quadrature, which yields

f int� =

�N∑

I=1

N u∑M=1

J e(�

X I ,vM )�

W I

|� �X I| W u

M J u(�

X I , vM )�N C

i�

�� j

∣∣∣∣∣vM

PCi j (

�

X I , vM ) (74)

where the pair of coordinates (�

X I , vM ) denotes the position of local quadrature point vM in the

coordinate system of the CUC positioned at�

X I as shown in Figure 4; J u(�

X J , vM ) and W uM denote

the CUC Jacobian and the corresponding weight, respectively.

Figure 4. Mixed nonlocal–local quadrature scheme.



3.5. Discrete computational unit cell (CUC) problem

The unit cell displacements in the CUC can be expressed as a sum of the coarse-scale displacements

uCi (X ) and the fine-scale perturbation u1

i (�

X I ,v) :

u�i (

�

X I ,v)=uCi (X )+u1

i (�

X I ,v) (75)

where

u1i (

�

X I ,v)=N Fi(v)d1

(�

X I ) (76)

or by direct discretization

u�i (

�

X I ,v)=N Fi(v)d(

�

X I ) (77)

where N Fi(v) is a fine-scale shape function; d(

�

X I ) the total nodal displacement in the CUC and

d1(

�

X I ) the corresponding perturbation.

The CUC problem can be either solved directly for the total displacement d(�

X I ) or for the

perturbation d1(

�

X I ). The corresponding two approaches are referred to hereafter as the total [61]and the correction-based [61] approaches, respectively.

We consider the Galerkin approximation of the fine-scale test function on � �X I

w1i (

�

X I ,v)=N Fi(v)c1

(�

X I ) (78)

which yields the discrete CUC problem subjected to weak periodicity conditions. We will assume

that d1(

�

X I ) and c1(

�

X I ) satisfy homogeneous linear constraints. Let d(�

X I ) and c(�

X I ) be theindependent degrees-of-freedom, then linear transformation operator T can be defined as

d1 (

�

X I )=Td(�

X I ), c1(

�

X I )=Tc(�

X I ) (79)

In summary, the discrete CUC problem is formulated as follows. Given the coarse-scale deformationi+1n+1uC(X ) and prior solution nd(

�

X I ), find i+1n+1�d(

�

X I ) such that

i+1n+1r F

(i+1n+1�d(

�

X I ))=∫

��X I

T

�N Fi(v)

�� jP�

i j (�

X I ,v)d�=0 ∀ �

X I (80)

subjected to weak periodicity conditions. The left superscript in (80) denotes the iteration count

of the coarse-scale problem; nd(�

X I ) denotes previously (at the load step n) converged solution in

the CUC positioned at�

X I .For the correction-based approach, the leading-order deformation gradient can be written as a

sum of known coarse-scale deformation gradient FCik(X ) and the unknown perturbation

F�ik(

�

X I ,v)=FCik(X )+

�N Fi(v)

��kd1(

�

X I ) (81)

Consequently, the discrete CUC problem (80) can be solved for the unknown perturbation d1(�

X I ).



Figure 5. Computational continua: the algorithm and information flow.

The computational algorithm, consisting of evaluation of nonlocal quadrature points in thepreprocessing stage, solution of CUC problems, evaluation of the coarse-scale stress function,solution of the coarse-scale problem and the information flow between various stages, is summarizedin Figure 5.

4. NUMERICAL EXAMPLES

In this section, we consider several numerical examples to demonstrate the versatility of thecomputational continua (to be referred hereafter as C2) formulation to resolve the coarse-scalebehavior of interest for large CUCs subjected to considerable coarse-scale gradients. The resultsof the computational continua C2 are compared with the classical coarse-scale continua resultingfrom O(1)homogenization (to be referred here as O(1)) and the direct numerical simulation (DNS)capable of resolving fine-scale details. Here for simplicity, we restrict ourselves to two-dimensionalproblems. We first study a single coarse-scale element formulation consisting of multiple unit cellsfollowed by the consideration of multiple coarse-scale element domains.

4.1. A single coarse-scale element studies

Here we consider a two-dimensional coarse-scale domain discretized with a single four-nodequadrilateral element subjected to a hourglass deformation mode uC

1 = X1 X2, uC2 =0. The coarse-

scale quantities of interest are internal energy density and internal force vector and we reportthe normalized error in these quantities of interest as a function of the ratio between the size ofcoarse-scale domain L and the unit cell size l. We consider different material models including



2

3

LL 12

34

1 l

Figure 6. Problem domain consisting of four unit cells (left), coarse-scale element domain(center) and a unit domain (right).

Figure 7. Von Mise stress distribution: (left) DNS in the left upper portion of the coarse-scale domain—thepeak value of Mises stress is 4.696×108; (center) C2 formulation—the peak value of Mises stress is

4.177×108; (right) O(1) formulation—the peak value of Mises stress is 1.823×108.

elasticity, hyperelasticity, monotonic plasticity and cyclic plasticity. Figure 6 depicts an exampleof the coarse-scale problem domain and the CUC.

4.1.1. Linear elastic material. We consider a unit cell made of linear isotropic elastic material withE=4×1010 and �=0.3, having a circular hole in the center. We consider a square coarse-scaledomain with L=288 and subject its vertices to displacements d(1)= [1,0],d(2)= [−1,0],d(3)=[1,0],d(4)= [−1,0] with linear variation between the vertices.

Figure 7 compares the von Mises stress distributions obtained using DNS, C2 and O(1) formu-lations. It is not surprising that O(1) theory shows substantially different stress distributions as itis subjected to constant deformation gradient. Note that C2 formulation subjects the CUC to theboundary displacements similar to those obtained from DNS. Consequently, the peak stress valuesobtained by C2 are much better than those obtained by the O(1) formulation.

Internal force vectors in the coarse-scale element denoted by fintC2 and fint

O(1) were calculated for

the C2 and O(1) formulations, respectively, and were compared with the reaction force vectorrDNS obtained by DNS. The normalized errors, defined as f int

C2 =‖rDNS−fintC2‖/‖rDNS‖ and f int

O(1)=‖rDNS−fint

O(1)‖/‖rDNS‖, respectively, are depicted in Figure 8 (left) for various ratios of L/ l.Figure 7 (right) compares the relative error in the energy densities denoted by eO(1) and eC2 as a



Figure 8. The normalized error in the internal force (left) and in the energy density (right) asa function of unit cell size L/ l.

function of the unit cell size L/ l. It can be seen that unlike O(1) continua, C2 formulation capturesthe size dependence of the internal energy density. Figure 8 also gives the results of the formulation(denoted by C∗) when local Gauss quadrature points are used instead of nonlocal quadrature points.It can be seen that the resulting formulation is even less accurate then the classical O(1) theory,but the difference between the three formulations becomes insignificant as the unit cell becomesvery small.

It can be seen that for all CUC sizes considered, C2 has less than 3% error in the two quantitiesof interest; yet O(1) results in errors as high as 18% in particular when the CUC size is large. Asthe unit cell size reduces, the difference between the classical homogenization and computationalcontinua becomes insignificant and in the limit the two solutions coincide.

4.1.2. Hyperelastic material. A hyperelastic material with Young’s modulus E depending onequivalent strain εeq=ε2

11+ε222−ε11ε22+3ε2

12 and �=0.3 has been considered for the matrix phaseas depicted in the following table:

εeq 0 0.0025 0.005 0.0075 0.009 0.01

E (∗1010) 4 24 54 94 160 160

The inclusion phase is assumed to be elastic with E=20×1010,�=0.3.The normalized error in the internal force vector as a function of time and the CUC size is

depicted in Figure 9. The normalized error in the coarse-scale energy density as a function of timefor the CUC sizes of L/ l=2, L/ l=3 and L/ l=4 is depicted in Figure 10. Figure 11 summarizesthe dependence of the energy density on L/ l as obtained in the last load increment.



Figure 9. The normalized error in the internal force for hyperelastic material as a function of load forunit cell sizes of: L/ l=2 (left), L/ l=3 (center) and L/ l=4 (right).

Figure 10. The normalized error in the energy density for hyperelastic material as a function of load forunit cell sizes of: L/ l=2 (left), L/ l=3 (center) and L/ l=4 (right).

Figure 11. Energy density in the final load increment as a function of unit cell size L/ l.



Figure 12. The normalized error in the internal force for plasticity as a function of load for unit cell sizesof: L/ l=2 (left), L/ l=3 (center) and L/ l=4 (right).

Figure 13. The normalized error in the energy density for plasticity as a function of load for unit cellsizes of: L/ l=2 (left), L/ l=3 (center) and L/ l=4 (right).

As in the elasticity case, the C2 formulation gives rise to errors that do not exceed 4% in bothquantities of interest throughout the entire loading history. The O(1) formulation, on the otherhand, results in errors that are up to four times higher.

4.1.3. Monotonic plasticity. We consider von Mises plasticity for the matrix phase and linearelasticity for the inclusion phase with E=12×1010,�=0.3. The yield stress in the matrix phaseis assumed to depend on the equivalent plastic strain as shown in the following table:

εeq 0.0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Y,107 3 3.2 3.6 4.6 6.2 9.4 15.8 28.6 54.2 105.4 206.6

The normalized error in the internal force vector and in the energy density as a function of timeand the CUC size are depicted in Figures 12 and 13, respectively.

4.1.4. Cyclic plasticity. The cyclic load history prescribed by controlling nodal displacements isshown in Table I.

The error in the internal force is depicted in Figure 14 for L/ l=2. The error in the internalforce was not normalized since the denominator might be close to zero due to force removal.



Table I. Prescribed cycling load history.

Step

Node 1 2 3 4 5 6 7 8

1 [1, 0] [0, 0] [−1, 0] [0, 0] [1, 0] [0, 0] [−1, 0] [0, 0]2 [−1, 0] [0, 0] [1, 0] [0, 0] [−1, 0] [0, 0] [1, 0] [0, 0]3 [1, 0] [0, 0] [−1, 0] [0, 0] [1, 0] [0, 0] [−1, 0] [0, 0]4 [−1, 0] [0, 0] [1, 0] [0, 0] [−1, 0] [0, 0] [1, 0] [0, 0]

Figure 14. Cyclic plasticity: the error in the internal force and the norm of the reaction force vectorobtained by DNS as a function of time for L/ l=2.

4.2. Cantilever beam problem

We consider a cantilever beam problem consisting of eight unit cells subjected to uniform pressureload as shown in Figure 15(a). The left end of the cantilever is fixed. For the C2 and O(1) homog-enization, the coarse-scale domain is modeled by 8 eight-node quadrilateral elements. Quadraticelements are required for the C2 formulation to resolve the linear coarse-scale stress field withinthe element. The CUC has a square domain with a very weak circular inclusion of a diameterequal to half of the unit cell size as shown in Figure 15(c). We consider a CUC size equal tothe coarse-scale element size. Linear elastic material properties are assumed. The nondimensionalunits are as follows: Ematrix=4.0e+10, �matrix=0.3, Einclusion=4.0, �inclusion=0.3, pressure loadequal to 1.5e+6 and coarse-scale element size equal to 288.

The reference solution is obtained by DNS using a very fine finite element mesh as shown inFigure 16, which depicts the deformed mesh and the von Mises stress distribution.

The maximum deflection (point A in Figure 15) as obtained by C2 is compared with the referencesolution (DNS) and the following three O(1) formulations:

O(1)-4: Classical O(1) formulation where the CUC is subjected to constant deformation gradientand periodicity constraint; reduced integration (four quadrature points) is used to integrate thecoarse-scale element.



A

(a)

(b) (c)

Figure 15. (a) Geometry, load and boundary conditions; (b) coarse-scale finite element mesh consistingof 8 eight-node elements; and (c) a unit cell equal to the size of the coarse-scale element.

Figure 16. Deformation and von Mises stress distribution as obtained by direct numerical simulation.

Table II. Maximum displacement (Point A in Figure 15) as obtained by DNS,C2 and three O(1) formulations.

DNS C2 O(1)-4 O(1)-4-Full O(1)-9

uy(X A) −67.7839 −65.5284 −97.5683 −97.4983 −97.503uy(X A) −5.39595 −5.29128 −8.00685 −8.0064 −8.00444

O(1)-9: Classical O(1) formulation where the CUC is subjected to constant deformation gradientand periodicity constraint; full integration (nine quadrature points) is used to integrate the coarse-scale element.

O(1)-4-Full: Modified O(1) formulation where the CUC is subjected to complete deformationfield extracted from the coarse-scale element and periodicity constraint; reduced quadrature is usedto integrate the coarse-scale element.

The results of the maximum displacements at point A are given in Table II.The error in the displacement norm at point A is 3.99% for the C2 formulation and over 43%

for the three O(1) formulations—more than a factor of 10 in error reduction. Obviously as thecoarse-scale fields in the CUC become more uniform, improvements offered by C2 become moremodest and in the limit the two formulations coincide. For instance, for the same beam size butwith two rows of unit cells, the C2 provides reduction of error by a factor of 1.7.



DNS C2

O(1)-9O(1)-4O(1)-4-Full

Figure 17. von Mises stress distribution in the unit cell adjacent to the clamped end as obtained by directnumerical simulation (DNS), Computational Continua (C2) and three variants of the O(1) homogenization.

Table III. Maximum von Mises stress in the unit cell adjacent to the clamped end as obtained by DNS,C2 and three O(1) formulations.

DNS C2 O(1)-4 O(1)-4-Full O(1)-9

Maximum von Mises stress 3.186e+08 2.982e+08 4.280e+07 3.720e+08 4.296+07

Figure 17 compares the von Mises stress distribution in the unit cell adjacent to the clampedend. The maximum values are compared in Table III. The stress distribution in the CUC is obtainedin the post-processing stage by subjecting the CUC to the coarse-scale deformation.

5. CONCLUSIONS AND DISCUSSION

Unlike the generalized continua, which assumes scale separation and is limited to infinitesimalunit cells, and the nonlocal continua for which construction of general purpose nonlocal kernelsfor heterogeneous media is questionable, the computational continua possess the generality andthe versatility not found in the existing methods. The combination of nonlocal quadrature andthe coarse-scale function reproduces the exact coarse-scale variational statement provided thatthe coarse-scale fields are smooth. The unit cell is permitted to be as large as a coarse-scale elementand the resulting formulation requires C0 continuity only, involves no additional degrees-of-freedom



and no higher-order boundary conditions. The versatility of the formulation was confirmed onlimited numerical examples.

Several important characteristics of the method, such as

1. Can the method be applied to unit cell larger than coarse-scale elements?2. Does the method serve as a localization limiter?3. Can the method be used to propagate discontinuities?

have not been investigated. In the following, we will comment on the above issues, but in depthstudy is required. The answer to the first question is ‘yes’. However, if the unit cell is largerthan coarse-scale element, it is necessary to carry out the integration over the unit cell domainrather than over the elements. This of course complicates the integration as it requires additionaltriangulation or increasing number of quadrature points [52].

The answer to the second question is ‘no’. In other words, if the fine-scale equations are notwell posed (strain softening, mesh dependence, etc.) the computational continua will inherit thesame deficiencies. The nonlocal character of the computational continua formulation is intendedto reproduce the overall fine-scale behavior rather than to modify it as is often the case in nonlocaltheories. The responsibility of constructing a well-posed problem falls squarely on the governingequations on the fine scale rather than on the coarse graining process.

The answer to the third question is mixed. The discontinuities contained in the interior of theunit cell are computationally resolved and then transferred to the coarse-scale in the form of linearcoarse-scale stress function. Larger discontinuities are reflected by directionally softer coarse-scaleelement behavior, but in the absence of coarse-scale kinematical enrichment (such as multiscaleXFEM [45, 62]) or remeshing, computational continua without enrichment is not well equipped topropagate large discontinuities.

ACKNOWLEDGEMENTS

This work was supported by the Office of Naval Research grant N000140310396, National Science Foun-dation grants CMS-0310596, 0303902, 0408359, Rolls-Royce contract 0518502, Automotive CompositeConsortium contract 606-03-063L and AFRL/MNAC contract MNK-BAA-04-0001.

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