Available online at www.sciencedirect.com

www.elsevier.com/locate/actamat

Acta Materialia 59 (2011) 4595–4605

Formulation and calibration of higher-order elasticlocalization relationships using the MKS approach

Tony Fast a, Surya R. Kalidindi a,b,⇑

a Department of Materials Science and Engineering, Drexel University, Philadelphia, PA 19104, USAb Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA

Received 24 January 2011; received in revised form 3 April 2011; accepted 4 April 2011Available online 27 April 2011

Abstract

Localization (as opposed to homogenization) describes the spatial distribution of the response field of interest at the microscale for animposed loading condition at the macroscale. A novel approach called Materials Knowledge Systems (MKS) has recently been formu-lated to construct accurate, bidirectional, microstructure–property–processing linkages in hierarchical material systems to facilitate com-putationally efficient multiscale modeling and simulation. This approach is built on the statistical continuum theories developed byKroner that express the localization of the response field at the microscale using a series of highly complex convolution integrals, whichhave historically been evaluated analytically. The MKS approach dramatically improves the accuracy of these expressions by calibratingthe convolution kernels in these expressions to results from previously validated physics-based models. All of the prior work in the MKSframework has thus far focused on calibration and validation of the first-order terms in the localization relationships. In this paper, weexplore, for the first time, the calibration and validation of the higher-order terms in the localization relationships. In particular, it isdemonstrated that the higher-order terms in the localization relationships play an increasingly important role in the spatial distributionof elastic stress or strain fields at the microscale in composite systems with relatively high contrast.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Materials knowledge systems; Multi-scale modeling; Volterra kernels; Localization relationships; High contrast

1. Introduction

In recent papers [1–4], we have formulated, implementedand validated a novel approach called Materials Knowl-edge Systems (MKS) for accurate bidirectional exchangeof information between selected length scales in multiscalemodeling and simulation of a range of materials phenom-ena. This approach has the potential to facilitate concur-rent evolution of microstructure at multiple length scalesthroughout the sample, which is the equivalent of executinga large number of microscale numerical models (e.g. finite-element models or phase-field models) inside a macroscalenumerical model in a fully coupled manner. Central toMKS are the localization relationships that accurately

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.04.005

⇑ Corresponding author at: Department of Materials Science andEngineering, Drexel University, Philadelphia, PA 19104, USA.

E-mail address: skalidin@coe.drexel.edu (S.R. Kalidindi).

distribute the macroscale applied conditions (e.g. stress,strain or strain rate) to the microscale. In the MKS, theselocalization relationships are expressed as calibratedmeta-models that take the form of a simple algebraic serieswhose terms capture the individual contributions from ahierarchy of microstructure descriptors. Each term in thisseries expansion is expressed as a convolution of theappropriate local microstructure descriptor and its corre-sponding local influence at the microscale. Our past explo-rations with the MKS approach have all utilized only thefirst-order terms in the expansion [1–4]. It was generallyobserved that the first-order terms in the expansion wereadequate to capture accurately the microscale distributionof response fields of interest provided the contrast betweenthe possible local states in the microstructure was limited tomoderately low values. It was specifically noted that thereis a critical need to include the higher-order terms forimproving the accuracy of the localization relationships

rights reserved.

4596 T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605

in the MKS framework for material systems with moderateto high contrast. Strong contrast formulations have beenpreviously utilized to predict the effective properties of elec-trical conductivity [5,6] and elasticity [7,8]. These methodshave not yet been employed to predict the local responsefields necessary to perform fully coupled multiscalesimulations.

The main impediment to including the higher-orderterms has been the lack of a robust methodology to esti-mate the values of the coefficients (also called influencecoefficients) in the series expansion for the higher-orderterms. In prior work [1–4], we have relied heavily on thehighly efficient decoupling of the first-order influence coef-ficients achieved by the use of fast Fourier transforms(FFTs), which then made it trivial to calibrate these coeffi-cients to results from numerical models. The number ofinfluence coefficients in the higher-order terms of theexpansion of the localization relationship is unimaginablylarge. Although some decoupling is accomplished by theuse of FFTs, the numbers of even the second-order coeffi-cients for most problems are so large that it is not practicalsimply to extend and employ the same heuristics that wereused successfully for the first-order terms.

In this paper, we present the results from our first forayinto estimating the numerical values of a subset of the dom-inant higher-order influence coefficients for elastic defor-mations in moderate- to high-contrast composite materialsystems, and critically validate their contributions toimproving the accuracy of the localization relationshipsin these problems. It should be noted that this paper repre-sents the first ever calibration of the higher-order influencecoefficients in the localization relationships to data fromnumerical models. In order to accomplish this task, wehave reformulated the localization relationship and devel-oped a number of new computational protocols and heuris-tics to quantify the relative contributions of variousdominant higher-order influence coefficients. This papersummarizes these major advances in this novel scale-bridg-ing framework.

2. MKS framework

The MKS framework begins with a discrete representa-tion of the microstructure, analogous to a digital signal.The spatial domain, presumably a microscale volume ele-ment (MVE), is tessellated into S spatial cells indexed bys = 0, 1, 2, . . . , S � 1. The average structure in each spatialcell is quantified using as many microstructure variables asneeded (this could include phase identifiers, elemental com-positions, crystal lattice orientation); the combination of allrelevant local structure attributes is called the local state.The distinct space of all possible local states in a givenmaterial system constitutes the local state space for thatmaterial system. Similar to the spatial domain, the localstate space H is broken up into H total bins which areenumerated by h = 1, 2, . . . , H. To define the microstruc-ture as a digital signal, this work employs a discrete

approximation of the microstructure function proposedby Adams et al. [9] and denoted as mh

s . This variable reflectsthe volume fraction of each distinct local state h in the spa-tial cell s, and has the following properties:

XH

h¼1

mhs ¼ 1; mh

s � 0: ð1Þ

Let ps be the averaged local response in spatial cell s

defined on the same spatial domain as the microstructurefunction; it reflects a thoughtfully selected local physicalresponse central to the phenomenon being studied, andmay include stress, strain or microstructure evolution vari-ables [1–4]. In the MKS framework, the localization rela-tionship, extended from Kroner’s statistical continuumtheories [10,11], captures the local response field using aset of kernels and their convolution with higher-orderdescriptions of the local microstructure [12–18]. The local-ization relationship can be expressed as:

ps ¼XH

h1¼1

XS�1

t1¼0

ah1t1

mh1sþt1þXH

h1¼1

XH

h2¼1

XS�1

t1¼0

�XS�1

t2¼0

ah1h2t1t2

mh1sþt1

mh2sþt1þt2

þXH

h1¼1

XH

h2¼1

XH

h3¼1

XS�1

t1¼0

Xs�1

t2¼0

�XS�1

t3¼0

ah1h2h3t1t2t3

mh1sþt1

mh2sþt1þt2

mh2sþt1þt2þt3

ð2Þ

The kernels, or influence coefficients, are defined by theset ah1

t1 ; ah1h2t1t2 ; a

h1h2h3t1t2t3

; . . .� �

. ah1t1; ah1h2

t1t2and ah1h2h3

t1t2t3 are the first-,second- and third-order influence coefficients, respectively.The hierarchical higher-order descriptions of the localmicrostructure are expressed by the products mh1

sþt1mh2

sþt1þt2. . . mhN

sþt1þt2þ...þtN , where N denotes the order. The influencecoefficients in Eq. (2) provide a comprehensive and efficientframework to extract, store and recall microstructure–property–processing linkages. They capture quantitativelythe contributions of various microstructure arrangementsabout any spatial position s to the local response at thatposition. The first-order influence coefficients ah1

t1capture

the contribution of the local state h1 placed at a distancet1 away (the space of vectors is also tessellated using thesame scheme used for the spatial domain [9]). Likewise,the second-order influence coefficients ah1h2

t1t2 capture thecombined effect of placing local states h1 and h2 at distancest1 and t1 + t2 away, respectively.

Volterra series have been used extensively in the litera-ture to model the dynamic response of nonlinear systemsusing a set of kernel functions called Volterra kernels[19–24]. The higher-order localization relationship shownin Eq. (2) is a particular form of Volterra series andassumes that the system is spatially invariant, casual andhas finite memory. The influence coefficients in Eq. (2)are analogous to Volterra kernels. The higher-order termsin the series are designed to capture systematically the non-linearity in the system. In other words, the accuracy withwhich the nonlinearity is captured depends strongly on

T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605 4597

the specific higher-order terms retained in the localizationrelationship. The influence coefficients are expected to beindependent of the microstructure, since the system isassumed to be spatially invariant.

The localization relationship shown in Eq. (2) is to beapplied on each spatial cell of the MVE. The size of theMVE is generally selected based on the specific application.In applications involving homogenization, the MVE isessentially the representative volume element (RVE) or amember of an RVE set [25]. As mentioned earlier, in theMKS framework, we aim to calibrate the influence coeffi-cients in Eq. (2) against results from numerical models.Consequently, MVEs are also used here for generatingthe necessary numerical datasets for the calibration of theinfluence coefficients. Since the influence coefficients areexpected to decay to zero values for increasing values oft, the localization captured by Eq. (2) is associated with afinite interaction zone or finite memory. In order to capturethe spatial characteristics of localization accurately, we rec-ommend that the MVE size used for generating the calibra-tion datasets be at least twice the size of the interactionzone. Since the size of the interaction zone is not knowna priori, a few trials are typically needed to establish a suit-able MVE size for a given material system. In general, wenote that the MVE size increases as the contrast in the localproperties is increased.

Application of the concepts of Volterra series directly tomaterials phenomena poses a major challenge. Materialsphenomena require an extremely large number of inputchannels (because of the complexities associated withmicrostructure representation) and an extremely largenumber of output channels (because of the broad rangeof response fields of interest). However, the microstructurerepresentation presented in Eq. (1) allows certain simplifi-cations. The introduction of these simplifications allows areformulation of the localization relationship into a formthat is amenable for calibration by linear regression inthe FFT space.

3. Reformulation of the MKS framework

As discussed in the previous section, the interpretationof the higher-order terms in Eq. (2) is rather complex andelusive. In an effort to simply the problem, we restrictour attention in this paper to the class of eigen microstruc-tures [9], where each spatial cell is allowed to have only onelocal state. In other words, the microstructure variable mh

s

is allowed only one of two possible values, 0 or 1.In our attempts to understand the higher-order coeffi-

cients better, we made two important observations: (i) acloser look at Eq. (2) revealed that the lower-order influ-ence coefficients are always included in the higher-ordercoefficients for the class of microstructures studied here.As an example, it is easy to see that second-order terms cor-responding to t2 = 0 and h1 = h2 are identical to the first-order terms. (ii) It is advantageous to separate the set ofspatial correlation vectors ft1; t2; . . . ; tNg in Eq. (2) is such

a way that we use t1 for the convolution of microstructurewith the influence coefficients, but use the set of vectorsft2; . . . ; tNg to capture the details of the neighborhood. Thisidea is illustrated in Fig. 1a and b. Fig. 1a shows the previ-ous interpretation where each term is defined in a continu-ous serial chain of spatial vectors. Fig. 1b shows the newinterpretation introduced in this paper where the first vec-tor t1 is used to identify a neighbor of interest (with theintention of using this vector for spatial convolution aswe have previously done with the first-order terms), andthe rest of the vectors are defined to quantify the neighbor-hood in a concurrent manner. This new interpretationallows us to select and arrange the vectors ft2; . . . ; tNg suchthat we progress systematically from the nearest neighborsto the farthest. We will demonstrate in this paper the impli-cit advantage of this new interpretation; it allows us toidentify systematically the relevant subsets of the dominanthigher-order influence coefficients in the localization rela-tionship in the increasing order of their relativeimportance.

Using the above insights, Eq. (2) can be modified, with-out loss of generality, to include up to Nth-order influencecoefficients as follows:

ps ¼XH

h1¼1

. . .XH

hN¼1

XS�1

t1¼0

. . .XS�1

tN¼0

ah1h2...hNt1t2...tN

mh1sþt1

mh2sþt1þt2

. . . mhNsþt1þtN

ð3ÞAlthough Eq. (3) allows a simpler physical interpretationcompared to Eq. (2), it still has a very large number ofinfluence coefficients. In fact, the number of independentinfluence coefficients in Eqs. (2) and (3) remain exactlythe same. The main advantage of Eq. (3) is that it allowsus to organize the terms in a much simpler and intuitivemanner. As an example, Table 1 lists the choices for t2

for the second-order (N = 2) descriptors in the increasingorder of their likely importance (based on the length ofthe vectors) in quantifying the neighborhood. Likewise,Table 2 lists the choices for the combination of vectors{t2, t3, t4, t5, t6, t7} needed to quantify the seventh-order(N = 7) microstructure descriptors, once again in the orderof their likely importance for the localization relationships.Tables 1 and 2 also illustrate that the lower-order micro-structure descriptors are always included in the higher-order microstructure descriptors.

As noted earlier, the higher-order descriptors areexpected to play an important role in the higher-contrastcomposites, because they quantify accurately the localneighborhoods in the microstructure. It is well known thatVolterra series are only effective if the transfer functionsconverge with increasing complexity [19–21]. In the MKSapproach, this translates to the need to organize the termsin Eq. (3) in order of decreasing importance. The hierarchi-cal description of the microstructure variables as describedabove in Tables 1 and 2 allows us to accomplish this goal.

Previous explorations of the MKS have exploited thecomputational efficiency and properties of discrete FFTs.

Fig. 1. (a) Illustration of the previous interpretation of the higher-order terms in the localization relationship where each vector is selected independent ofall others. (b) Illustration of the interpretation of the higher-order terms in the reformulated localization relationship presented in this paper. In this newinterpretation, {t2, . . . , tN} are used concurrently to quantify the neighborhood of s + t1.

Table 1List of the choices for t2 to account up to fourth neighbors in the second-order influence coefficients. {e1, e2, e3} denote the three orthonormalpositive vectors, whose lengths reflect the spatial cell size in the tessellationof the MVE. The choices are grouped by the distance of the neighbors.There are a total of 6 first neighbors, 12 second neighbors, 8 thirdneighbors, and 6 fourth neighbors in the table.

Choices for t2 Comments

0 First-ordercoefficients

±e1, ±e2, ±e3 First neighbors±(e1 + e2), ±(e1 + e3), ±(e1 � e2), ±(e1 � e3),

±(e2 + e3), ±(e2 � e3)Second neighbors

±(e1 + e2 + e3), ±(e1 + e2 � e3), ±(e1 � e2 + e3),±(e1 � e2 � e3)

Third neighbors

±2e1, ±2e2, ±2e3 Fourth neighbors

Table 2Possible choices for the combination of vectors {t2, t3, t4, t5, t6, t7} neededto specify the seventh-order influence coefficients. For the lower-ordercoefficients, only the first- and second-order coefficients are included inthis table. By systematically increasing the number of non-zero ti, one canspecify the other lower-order coefficients up to the sixth-order coefficients.Note that there are a total of 64 seventh-order first neighbors.

Choices for{t2, t3, t4, t5, t6, t7}

Comments

All six ti = 0 First-order coefficientsFive ti = 0 one ti – 0 Second-order coefficients; if the non-zero ti are

equated to ej, we get the second-order coefficientsfor the first neighbors (same as those in Table 1)

All ti = ej Seventh-order coefficients for the first neighbors

4598 T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605

These heuristics have been developed based upon a multi-channel convolution of the microstructure and the influ-ence coefficients [26–28]. In this paper, we extend theseprotocols to establish the values of carefully selected subsetof the higher-order influence coefficients. For this purpose,the simplest way to interpret Eq. (3) is to see it as a simplesum of contributions from specific configurations of localstates at t1 away from the location of interest (seeFig. 1b). In order to visualize this better, let us define ahigher-order microstructure function as:

i ~m~hsþt1¼ mh1

sþt1mh2

sþt1þti2

. . . mhN

sþt1þtiNð4Þ

In Eq. (4), ~h denotes the local state identifier for the newmicrostructure function that enumerates all distinct or-dered sequences of the combination of local states(h1, . . . , hN). The corresponding local state space, eH, canbe expressed as the product space of the individual localstate variables. The number of distinct local state identifiersin eH is denoted by eH ¼ H N . The index i in Eq. (4) enumer-ates each choice of the set of vectors t2; . . . ; tNf g. It isimportant to note that each choice of t2; . . . ; tNf g and allof its permutations are captured by one microstructuredescriptor. With these simplifications in notation, the high-er-order localization relationship in Eq. (3) truncated toNth-order coefficients can be expressed as:

ps ¼XI

i¼1

XeH~h¼1

XS�1

t1¼0

ia~ht1

i ~m~hsþt1þ esðIÞ ð5Þ

where i~a~ht1

is a compact representation of the higher-orderinfluence coefficients associated with the local conforma-tion of the microstructure based upon the choice of theset of vectors indexed by i (see Tables 1 and 2 for exam-ples), and es(I) is the truncation error that depends criti-cally on the choice of the vectors sets.

Eq. (5) represents all terms in the expansion of the local-ization relationship up to order N. However, in this work,we found that it is advantageous to selectively pick combi-nations of terms from different orders. As shown in Table 2,the numbers of terms in the higher-order expansions areextremely large. For example, if one considers seventh-order terms, there are 64 different combinations of vectorsto account for first neighbors alone (see Table 2). The num-bers for seventh-order second neighbors are too large toinclude them in establishing the localization relationships,and therefore one may be forced to truncate the series toinclude only the first neighbors in the seventh-order expan-sions. However, it might be advantageous to include the 12second-order second neighbors (see Table 1) along withseventh-order first neighbors to improve on the accuracy(note that the second-order first neighbors are already

T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605 4599

included in the seventh-order first neighbors and thereforedo not need to be included again). The notation in Eq. (5)is cumbersome when it comes to capturing the idea of selec-tively picking terms from the different order expansions asthey are associated with different values of eH ¼ H N . Inorder to simplify the notation, Eq. (5) is rewritten as:

ps ¼XI

i¼1

XS�1

t¼0

iati ~msþt þ esðIÞ ð6Þ

where i now enumerates not only all the distinct combina-tions of the selected combinations of vectors (as shown inTables 1 and 2), but also the distinct combinations of the or-dered sequences of local states that could be potentially asso-ciated with each selected combination of vectors. Thenumber of distinct terms selected in this mixed higher-orderexpansion is denoted by I. It should also be noted that thatthe subscript on the vector t has been dropped for simplicity.

Representing the localization relationship as shown inEq. (6) allows us to recognize the fact that the heuristicsdeveloped earlier for the first-order influence coefficientscan now be applied to the higher-order influence coeffi-cients in a relatively straightforward manner. In this paper,we specifically explore the potential benefits to the localiza-tion relationships by considering a limited number of localmicrostructure conformations corresponding to specificselections of short vectors from Tables 1 and 2. However,in doing so, we also need to pay particular attention tothe redundancies in the microstructure descriptors (thesewill be discussed later) in order to achieve compactrepresentations.

Following our earlier approach, Eq. (6) can be effec-tively decoupled using FFTs as:

P k ¼XI

i¼1

iAki ~Mk þ EkðIÞ ¼ IP þ EkðIÞ

iAk ¼ Jkðiat1Þ; P k ¼ JkðpsÞ; i ~Mk ¼ Jkði ~msÞ;

EkðIÞ ¼ JkðesðIÞÞ

ð7Þ

where Jk() denotes the Fourier transform, and the super-script * denotes the complex conjugate; P k is the approxi-mate response from the MKS method for a distinctchoice of local conformations included in the series. Thedecoupling accomplished in Eq. (7) makes it much easierto calibrate the influence of coefficients against the resultsproduced by physics-based models (such as finite-elementmodels). This is one of the distinctive features in theMKS framework (as opposed to the previously attemptedapproaches based on Kroner’s statistical continuum theo-ries that use analytical approaches to estimate the valuesof the influence coefficients).

4. Calibrating higher-order influence coefficients

The next major challenge in the MKS framework isthe calibration of the selected higher-order influencecoefficients via linear regression. The approach currently

used in MKS for estimating the influence coefficients isordinary least-squares fitting (OLSF) on the numericaldatasets produced by the physics-based models by mini-mizing the norm of the truncation error for each frequencyexpressed as [26]:

£kðIÞ ¼ EkðIÞEkðIÞ� ¼ ðI P k � P kÞðI P k � P kÞ� ð8ÞThis method involves solving for iAk using the followingsystem of linear equations obtained as a solution for OLSF(minimizing the error defined in Eq. (8)) [29]:

XR

r¼1

rP kðri0 ~MkÞ� ¼XI

i¼1

ðiAkÞ�XR

r¼1

ri ~Mkðri0 ~MkÞ� ¼

XI

i¼1

ðiAkÞ�RMk

where

RMk ¼XR

r¼1

ri ~Mkðri0 ~MkÞ�;

ð9ÞIn Eq. (9), r indexes the MVEs simulated using the physics-based (often numerical) model. RMk is a condensed nota-tion for the calibration matrix computed in the linearregression analyses for the influence coefficients corre-sponding to the frequency k. Although other approacheshave been used in the literature to approximate higher-order Volterra kernels (e.g. cross-correlation methods[23,24], neural networks [22] and Bayesian methods[30,31]), these are not needed in the present application be-cause the final form of the localization relationship usedhere readily lends itself to OLSF methods. This is the pri-mary benefit of recasting the localization relationship inthe form shown in Eq. (6) eigen microstructures.

There exist several redundancies in the higher-ordermicrostructure descriptors defined earlier, causing the cali-bration matrix to be singular (and therefore not invertibleas needed to solve for the influence coefficients in Eq.(9)). In order to understand the redundancy implicit inthe higher-order microstructure descriptors introduced ear-lier, let us consider the set of second-order microstructuredescriptors in a highly simplified one-dimensional micro-structure. More specifically, consider combinations oftwo spatial cells to define one of the first-neighbor sec-ond-order microstructure descriptors. This produces a totalof four possible second-order local states as shown inFig. 2. Let us index these distinct local states as~h ¼ 1; . . . ; 4. For the example microstructure shown inFig. 2, it is easy to see that the specification of the localstate in spatial cells 3 and 5 automatically identifies thelocal state in spatial cell 4. Several additional redundanciesoccur when the higher-order descriptors defined earlier inthis paper are applied to three-dimensional microstruc-tures. Although it is possible to express all the inherentredundancies analytically for simple cases such as the oneshown in Fig. 2, it has not been possible to develop generalexpressions to take into account all of the redundancies inall of the higher-order microstructure descriptors in three-dimensional microstructure datasets.

Fig. 2. Illustration of the redundancy in the second-order microstructure descriptors for the vector t2 = +e1 For a local state space tessellated into twodiscrete local states (i.e. white and black) there are four possible arrangements of the local states in the higher-order microstructure description for theselected vector. All of these possible combinations are illustrated in the box on the left. On the right is an example 1-D microstructure, which contains 10spatial cells. The local state in each spatial cell depends upon the conformation of the local state in spatial cell at s in addition to the local state in thespatial cell to its right. As an example, the higher-order local state in spatial cell s = 4 is defined by the presence of the black local state in cell 4 and thepresence of the white local state in cell 5. This conformation matches the higher-order local state 3 as shown in the box on left s = 5.

4600 T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605

In order to circumvent the challenges associated withanalytically determining all the redundancies, we havedeveloped a numerical approach to resolve them usingthe reduced-row echelon form (RREF) of the calibrationmatrix, Gk ¼ rref ðRMkÞ. The RREF identifies all of theinherent redundancies of the higher-order microstructuredescriptors in the calibration matrix, and indexes the inde-pendent columns in Gk into a set vk. This is equivalent totransforming Eq. (7) to:

P k ¼Xi�vk

iBki ~Mk;

iBk ¼XI

i0¼1

ii0Gkði0AkÞ� ð10Þ

where iBk denote the FFTs of the reorganized influencecoefficients corresponding to the independent higher-ordermicrostructure descriptors. The new coefficients are ob-tained by a linear transformation of the original coefficientsas shown in Eq. (10). The relationship between the two setsof coefficients arises from the redundancy in the definitionsof the higher-order microstructure descriptors defined andused here. As such, the transformation described in Eq.(10) is independent of the physics of the phenomenon beingstudied. Note that this transformation is key to our abilityto estimate the numerical values for the reorganized coeffi-cients, iBk, using the linear regression methods already de-scribed in Eq. (9). It is required that there are at least asmany datasets in the calibration matrix as there are reorga-nized coefficients.

In prior work [1–4] with first-order influence coefficients,we noted that the influence coefficients decayed sharplywith increasing values of t away from s. An important con-sequence of the reformulated MKS framework presentedhere that it is expected to preserve this attribute for thehigher-order influence coefficients defined here in Eq. (6).The same feature is also expected to hold for the reorga-nized influence coefficients, whose FFTs are defined inEq. (10). In order words, an inverse FFT on iBk should pro-duce a function that decays sharply with increasing t. Asnoted in our prior work [1,4], this attribute of the influencecoefficients defined in the MKS framework is central to ourability to use the influence coefficients obtained from

smaller microstructure datasets on much larger microstruc-ture datasets. This concept will be illustrated later with acase study. Another advantage of the spectral form of thehigher-order localization relationship is that it is highlyparallelizable with regards to the calibration of the influ-ence coefficients and the evaluation of local response fields.In addition, graphics processing units (GPUs) may be usedto further expedite the computation of FFTs [32].

In closing this section, we note that our approach forestimating the influence coefficients deviates substantiallyfrom the approaches used in the literature based on Volter-ra kernels [19–24]. The approach described here explicitlyaccounts for all of the redundancies from the microstruc-ture signal in the Volterra series expansion, facilitates thesuccessful use of simple linear regression methods, and pro-duces influence functions with finite memory (with theexpected decay with increasing values of t).

5. Case study: elastic deformations in a two-phase composite

In this section, the new approach developed in thispaper for building MKS databases involving higher-orderlocalization relationships is applied to the elastic responseof a two-phase composite. Both phases are assumed toexhibit isotropic elastic properties. A large number of 3-D volumes of the composite material (MVEs) with very dif-ferent internal structures were created for this purpose.Each digital microstructure dataset was spatially tessellatedinto uniform cubes and enumerated as described earlier inSection 2. Each spatial cell in the microstructure datasetwas assigned to be occupied by one of the two phases Aor B, whose Young’s moduli are denoted as EA and EB,respectively. Furthermore, we have selected phase A tobe the stiffer phase, i.e. EA > EB. The ratio EA

EBreflects the

contrast in the composite material, as their Poisson ratiosare assumed to be the same. As noted earlier, we expectthe higher-order influence coefficients to play an importantrole as the contrast ratio EA

EBincreases. In prior work

[2,4,33], we have shown that the first-order influence coef-ficients produced excellent predictions for a contrast ratioof 1.5. In this case study, we critically evaluate the

Table 3Summary of the different selections of the combinations of influence coefficients evaluated in this study for k = 0.

Case Combination of coefficients selected Value of I

1 First-order coefficients 22 Second-order coefficients up to first neighbors 53 Second-order coefficients up to second neighbors 114 Second-order coefficients up to third neighbors 155 Second-order coefficients up to fourth neighbors 186 Second-order coefficients up to fifth neighbors 307 Second-order coefficients up to sixth neighbors 428 Seventh-order coefficients up to first neighbors 1139 Seventh-order coefficients up to first neighbors plus second-order coefficients from second neighbors to sixth neighbors 141

T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605 4601

contributions of various higher-order influence coefficientsfor composites with contrast ratios 5 and 10.

As mentioned earlier, in the MKS approach, the influ-ence coefficients are estimated by calibration to datasetsproduced by physics-based models for the phenomenonof interest. In the present case study, we have used the com-mercial finite-element code ABAQUS [34] to generate thedatasets needed for the calibration of the localization rela-tionships. Based on our prior experience in similar prob-lems [1,2,4], we have generated a large number ofexample 3-D microstructures made of cubical eight-nodedsolid elements (C3D8). The size of each microstructure inthis calibration dataset was selected to be 15 � 15 � 15 ele-ments. This was deemed large enough to capture the decayof the influence coefficients with increasing t. Either phaseA or phase B was assumed to occupy each cubical elementof the finite-element model, and the corresponding localproperties were assigned in the simulation. Building onour previous experience with the MKS framework, weimposed periodic boundary conditions on all [1,2,4,33]microstructures that corresponded to uniaxial extensione = 0.0005 in the e3 direction. As noted in our previouswork [4,33], it is adequate to consider a total of six different

Fig. 3. Mean and standard deviation of the mean absolute strain error(MASE) for the validation and calibration sets for each set of influencecoefficients in Table 3 for a contrast ratio of 5.

boundary conditions to build a comprehensive MKS data-base for elasticity problems to simulate an arbitrary uni-form boundary condition using the superpositionprinciple. In this paper, we focus on only one boundarycondition as our goal here is to critically validate the con-tributions of the higher-order influence coefficients toimproving the accuracy of the localization relationshipsfor moderate- to high-contrast composites.

In critically evaluating the contributions of the higher-order terms, we will increase the numbers and order ofthe influence coefficients in a systematic way so that wecan evaluate the relative contributions of the differentterms (i.e. different order and different level of neighborsat a given order). The different sets of influence coefficientsevaluated in this study are summarized in Table 3 alongwith the corresponding numbers of independent influencecoefficients in each set. In developing compact representa-tions for each set listed in Table 3, we exploited all of theredundancies present in each case.

Each set of the influence coefficients listed in Table 3 iscalibrated separately using linear regression methods (seeEq. (9)) for two different contrast ratios of 5 and 10, respec-tively. A total of 1100 MVEs (each with 15 � 15 � 15

Fig. 4. Mean and standard deviation of the mean absolute strain error(MASE) for the validation and calibration sets for each set of influencecoefficients in Table 3 for a contrast ratio of 10.

4602 T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605

cuboidal spatial cells) with varying volume fractions rangingfrom 1% to 99% were generated and used in this study. TheseMVEs were then randomly divided into two groups: (i) a setof 600 MVEs was selected as the calibration set, and (ii) theremaining 500 MVEs constituted the validation set.

Numerous approaches may be used to quantify theaccuracy of the influence coefficients. In this work, we havechosen to use the mean absolute strain error (MASE) E tocompare the strain fields obtained from the MKS approach(denoted by pMKS

s ) and the finite-element method (denotedby pMKS

s ). The MASE for the strain field in an MVE indexedby r is expressed as:

XS

s¼1

rpFEMs � rpMKS

s

�� ��S � e

¼ rE ð11Þ

where e denotes the average strain imposed on the MVE.This metric quantifies the error for a single microstructure.However, by applying Eq. (11) to all microstructures in aset, it would be possible to quantify the mean and standarddeviation of the MASE for that set. In this work, the meanand standard deviation of the MASE were established sep-arately for the calibration and validation sets.

Figs. 3 and 4 elucidate the robustness of each set ofinfluence coefficients calibrated in this study for contrastsof 5 and 10, respectively. The overlap in the means andstandard deviations of MASE for the calibration and vali-dation datasets for cases 1–7 indicates that we haveachieved an acceptable level of robustness for the influencecoefficients being estimated in these cases. This is becausethese plots indicate that the MASE for the validation setfor these cases is unlikely to improve further by addingmore microstructures to the calibration set. It is noted that

Fig. 5. MKS and FE predictions of the strain fields in an example 3-D 15 �section perpendicular to the loading direction and containing the highest loca

the number of microstructures needed to achieve robust-ness increased sharply with the inclusion of more influencecoefficients in the calibration (see Table 3). In fact, robust-ness was achieved for cases 1–7 with fewer than 200 micro-structures in the calibration set. However, cases 8 and 9required the addition of several new microstructures tothe calibration set. This is also reflected in the fact thatthe differences between the means of MASE for the calibra-tion and validation sets are highest for cases 8 and 9. Evenwith the inclusion of 600 microstructures in the calibrationset, Figs. 3 and 4 suggest that inclusion of additionalmicrostructures is likely to improve the calibration forcases 8 and 9.

Figs. 3 and 4 also reveal that there is a dramatic improve-ment in MASE from case 1 to case 2 with the inclusion ofthe second-order first neighbors. Comparing case 2 to cases3–7 reveals that adding more second-order neighbors(beyond the first neighbors) produced only a small improve-ment. In fact, the next significant improvement in MASEoccurred with the inclusion of seventh-order first neighbors,suggesting that it would be beneficial to go to the higher-order terms instead of increasing the number of neighborsat the lower order. This is particularly clear from comparingcase 8 (includes seventh-order first neighbors) with case 7(includes second-order sixth neighbors). This suggests thatthe dominant interactions are generally confined to thenearest neighbors. The systematic inclusion of the higher-order terms not only improved the mean value of MASE,but also lowered its standard deviation (see Figs. 3 and 4).

Figs. 5 and 6 present a comparison of the strain fieldsobtained from the MKS approach and the finite-elementsimulation in a mid-section of an example MVE forcontrast ratios of 5 and 10, respectively. In each figure,

15 � 15 MVE with a contrast ratio of 5. The strain fields are shown on al strain in the MVE in the FE predictions.

Fig. 6. MKS and FE predictions of the strain fields in an example 3-D 15 � 15 � 15 MVE with a contrast ratio of 10. The strain fields are shown on asection perpendicular to the loading direction and containing the highest local strain in the MVE in the FE predictions.

T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605 4603

we have presented the MKS predictions for three differentcombinations of influence coefficients listed in Table 3,along with the corresponding values of the MASE for theselected MVE. The mid-sections shown in Figs 5 and 6are selected perpendicular to the loading direction. It isimmediately evident from these figures that the first-orderinfluence coefficients (case 1) do not adequately capturethe strain fields. Furthermore, the error increases signifi-cantly with an increase in the contrast value (compareFig. 5 with Fig. 6). It also clear from these figures thatthe systematic inclusion of higher-order influence coeffi-cients steadily improves the MKS predictions for both con-trast ratios. In particular, note the considerableimprovement with the results shown from case 7 and case

Fig. 7. Frequency plots of the MKS and FE predicted strain distributions in anand dashed lines the compliant phase.

9. In fact, the reduction in MASE between case 7 and case9 is �45% and �80% for contrast ratios of 5 and 10, respec-tively. These results clearly indicate the important roleplayed by higher-order influence coefficients for materialswith higher contrast.

Figs. 7 and 8 compare the frequency distributions in thepredicted strain fields in each phase in an example MVE forcontrast ratios of 5 and 10, respectively. In these figures,the frequency distribution for the compliant phase (phaseA) is shown by dotted lines, while the frequency distribu-tion for the stiffer phase is shown by solid lines. The firstplot in each figure shows the contributions from the differ-ent neighbors to the second-order influence coefficients. Itis generally seen that the inclusion of first neighbors in

example MVE for a contrast ratio of 5. Solid lines indicate the stiff phase

Fig. 8. Frequency plots of the MKS and FE predicted strain distributions in an example MVE for a contrast ratio of 10. Solid lines indicate the stiff phaseand dashed lines the compliant phase.

4604 T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605

the second-rank coefficients resulted in a marked improve-ment in the predictions (by comparing all of the predictionsto finite-element predictions). However, there was only amodest improvement with inclusion of additional neighbors(from second neighbors to sixth neighbors). The second plot

Fig. 9. (a) MKS and FE predictions of the strain fields in an example 3-D 45 �section perpendicular to the loading direction and containing the highest localand FE predicted strain distributions of the same MVE. Solid lines indicate t

in Figs. 7 and 8 shows the contributions from only the firstneighbors of different orders of influence coefficients. It canbe clearly seen that there is substantial improvement in theaccuracy of the MKS predictions when we increase the orderof the influence coefficients. In order to further emphasize

45 � 45 MVE with a contrast ratio of 10. The strain fields are shown on astrain in the MVE in the FE predictions. (b) Frequency plots of the MKS

he stiff phase and dashed lines the compliant phase.

T. Fast, S.R. Kalidindi / Acta Materialia 59 (2011) 4595–4605 4605

this point, we have also added case 9 to Figs. 7b and 8b. Onceagain, we observe that adding higher-level neighbors pro-vides only a modest improvement in the accuracy of theMKS predictions. In other words, it might be much morebeneficial to increase the order of the influence coefficientsinstead of increasing the number of neighbors beyond theimmediately close neighbors. Comparing Figs. 7 and 8, wenote that the improvements are much more significant forthe higher-contrast composites.

Figs. 7 and 8 indicate that the higher-order MKS predic-tions capture the details of the tails of the distribution withremarkable accuracy. It is further noted that the neighborsbeyond the first neighbors do contribute to the tails of thedistributions shown in these figures. For example, it isclearly seen that the second neighbors made contributionsto the tails of the distributions shown for the compositewith a contrast of 10 in Fig. 8a.

It was previously shown that the first-order influence coef-ficients established on smaller spatial domains can also beapplied to significantly larger spatial domains [1,2]. In thiswork, we evaluate the viability of the same concept for thehigher-order influence coefficients. As noted earlier, weexpect an inverse FFT on iBk to produce a function thatdecays sharply with increasing t. This allows us to extendthe function to larger spatial domains by simply paddingthe functions with zeros for the larger values of t. Using thisconcept, we trivially extended the higher-order influencecoefficients extracted in this work on the 15 � 15 � 15domain from case 9 to a larger spatial domain of 45 �45 � 45 for a contrast ratio of 10 using the same method thatwas used in our earlier work [1,2]. These extended influencefunctions were then evaluated by comparing the finite-ele-ment results to the MKS results on an example MVE thatwas digitally generated using the same protocols that weredescribed earlier for the smaller MVEs. One such compari-son is presented in Fig. 9. It is clear that the trivially extendedinfluence coefficients accurately reproduce the microscalespatial distribution of the strain field on the larger MVEswith about the same accuracy that was realized earlier forthe smaller MVEs. The MASE on the 45 � 45 � 45 domainwas 6.12% which is within the second standard deviation ofthe case 9 influence coefficients (Fig. 4b). It should be notedthat the finite-element simulation required 45 min on asupercomputer, whereas the MKS predictions were obtainedwithin 15 s on a desktop computer (3.2 GHz and 3 GBRAM).

6. Conclusions

New heuristics were successfully developed for thehigher-order influence coefficients in the localizationexpressions in the MKS framework. This was accom-plished by a reformulation of the localization expression,which allowed a systematic identification of the relevantsubsets of the dominant higher-order influence coefficientsin the localization relationship in increasing order of theirrelative importance. The new methods developed in this

study were validated on a case study involving the elasticresponse of a two-phase composite with contrast ratios of5 and 10 in the respective elastic moduli of the constituentphases. The higher-order influence coefficients were foundto provide a drastic improvement over the first-order termsin the accuracy of the predictions of the microscale stressfields in these composites.

Acknowledgment

The authors acknowledge financial support for thiswork from the DARPA-ONR Dynamic 3-D Digital Struc-ture project, Award No. N000140510504.

References

[1] Fast T, Niezgoda SR, Kalidindi SR. Acta Mater 2011;59(2):699–707.[2] Kalidindi SR et al. Comput Mater Continua 2009;17(2):103–26.[3] Landi G, Kalidindi SR. Comput Mater Continua 2010;16:273–93.[4] Landi G, Niezgoda SR, Kalidindi SR. Acta Mater 2009;58(7):2716–25.[5] Torquato S. J Appl Phys 1985;58(10):3790–7.[6] Brown WF. J Chem Phys 1955;23(8):1514–7.[7] Fullwood DT, Adams BL, Kalidindi SR. J Mech Phys Solids

2008;56(6):2287–97.[8] Torquato S. J Mech Phys Solids 1997;45(9):1421–48.[9] Adams BL, Gao X, Kalidindi SR. Acta Mater 2005;53(13):3563–77.

[10] Kroner E. Statistical modelling. In: Gittus J, Zarka J, editors.Modelling small deformations of polycrystals. London: Elsevier;1986. p. 229–91.

[11] Kroner E. J Mech Phys Solids 1977;25:137–55.[12] Garmestani H et al. J Mech Phys Solids 2001;49(3):589–607.[13] Lin S, Garmestani H. Compos Part B: Eng 2000;31(1):39–46.[14] Mason TA, Adams BL. Metall Mater Trans A: Phys Metall Mater Sci

1999;30(4):969–79.[15] Garmestani H, Lin S, Adams BL. Int J Plast 1998;14(8):719–31.[16] Beran MJ et al. J Mech Phys Solids 1996;44(9):1543–63.[17] Beran MJ. Statistical continuum theories. New York: Interscience

Publishers; 1968.[18] Milton GW. The theory of composites. In: Ciarlet PG et al., editors.

Cambridge monographs on applied and computational mathemat-ics. Cambridge: Cambridge University Press; 2002.

[19] Cherry JA. Distortion analysis of weakly nonlinear filters usingvolterra series. Ottawa: National Library of Canada; 1995.

[20] Volterra V. Theory of functionals and integral and integro-differentialequations. New York: Dover; 1959.

[21] Wiener N. Nonlinear problems in random theory. Cambridge,MA: MIT Press; 1958.

[22] Wray J, Green G. Biological cybernetics 1994;71(3):187–95.[23] Korenberg M, HunterV I. Annals of biomedical engineering

1996;24(2):250–68.[24] Lee YW, Schetzen M. Int J Control 1965;2(3):237–54.[25] Niezgoda SR et al. Acta Mater 2010;58:4432–45.[26] Raz GM, Van Veen BD. IEEE Trans Signal Process

2000;48(1):192–200.[27] Lang T, Perreau S. Proc IEEE 1998;86(10):1951–68.[28] Fullwood DT et al. Comput Mater Continua 2009;9(1):25–39.[29] Hsia TC. System identification: least-squares methods. Lexington –

Mass. & Toronto: Heath; 1977.[30] Cockayne E, van de Walle A. Phys Rev B 2009;81(1).[31] Kerschen G, Golinval J-C, Hemez FM. J Vibrat Acoust

2003;125(3):389–97.[32] Nickolls J et al. Queue 2008;6(2):40–53.[33] Kalidindi SR, Landi G, Fullwood DT. Acta Mater

2008;56(15):3843–53.[34] ABAQUS, DSS Corp. Providence, RI; 2010.