Available online at www.sciencedirect.com

www.elsevier.com/locate/actamat

Acta Materialia 59 (2011) 699–707

A new framework for computationally efficientstructure–structure evolution linkages to facilitate high-fidelity

scale bridging in multi-scale materials models

Tony Fast a, Stephen R. Niezgoda a, Surya R. Kalidindi a,b,⇑

a Department of Materials Engineering, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USAb Department of Mechanical Engineering, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

Received 20 July 2010; received in revised form 1 October 2010; accepted 2 October 2010

Abstract

A novel mathematical framework called materials knowledge systems (MKS) was recently formulated to extract, store and recall com-putationally efficient hierarchical linkages that are at the core of multi-scale modeling of materials phenomena. A salient feature of thisnew framework is that it facilitates flow of high-fidelity information in both directions between the constituent length scales, and therebyoffers a new strategy for concurrent multi-scale modeling. The viability of this new framework has thus far been largely explored forcapturing the mechanical response of composite material systems. This paper extends the MKS framework to applications involvingmicrostructure evolution, where the local states are typically defined in a continuous local state space. In particular, it will be shown thatit is possible to obtain an efficient discretization of the local state space to produce a sufficiently accurate description of the linearizedstructure–structure evolution linkages for modeling spinodal decomposition. Furthermore, it will be shown that these linkages can beused successfully to accurately predict the continuous evolution of microstructure over the long time periods involved in such problems.� 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Spinodal decomposition; Phase field models; Statistical mechanics; Simulation

1. Introduction

The recently developed framework for materialsknowledge systems (MKS) [1,2] facilitates establishmentof computationally efficient hierarchical structure–perfor-mance–structure evolution linkages for multi-scale model-ing of materials phenomena. At the core of this newframework are computationally efficient localizationrelationships that capture accurately the microscale spatialdistribution of any local field of interest (e.g. a performancevariable such as stress or strain, or a structure evolutionvariable such as the time derivative of concentration) fora selected macroscale boundary condition, while taking

1359-6454/$36.00 � 2010 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2010.10.008

⇑ Corresponding author at: Department of Mechanical Engineering,Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA.Tel.: +1 215 895 1311; fax: +1 215 895 1478.

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

into account rigorously the topological details of themicrostructure [3–6]. These localization relationships areexpressed as a polynomial series sum, where each succes-sive term captures the contribution from a higher-orderlocal spatial correlation in the microstructure. This frame-work has its theoretical roots in the statistical continuummechanics theories of Kroner [7,8]. In the approach origi-nally described by Kroner, the values of the physics-relatedcoefficients (called influence coefficients) in the series wereestablished by selecting a reference medium and numeri-cally evaluating a complex series of nested convolutionintegrals. This approach leads to the principal value prob-lem (the integrand exhibits a singularity in the domain ofintegration) and exhibits high sensitivity to the selectionof the reference medium [9]. In the MKS framework[1,2], the same series expressions were recast into muchmore computationally efficient representations using dis-crete Fourier transforms (DFTs) [10]. The main advantage

rights reserved.

700 T. Fast et al. / Acta Materialia 59 (2011) 699–707

of using the DFT representations is that it allows easy cal-ibration of the localization relationships to results fromestablished numerical approaches. This novel approach cir-cumvents all of the main obstacles associated with the prin-cipal value problem, precludes the need to select a referencemedium and produces much more accurate predictions.

This paper focuses on extending the MKS framework tomicrostructure evolution problems. In particular, we focusour attention on modeling microstructure evolution duringspinodal decomposition [11]. Spinodal decomposition is aphase transformation where a homogenous mixture sepa-rates into regions of distinct chemical composition or phaseby a diffusion clustering mechanism, assisted by a negativeenergy gradient in free energy and modulated by the energycost of creating an interface between the phase-separatedregions. Phase separation by spinodal decomposition of abinary alloy is usually expressed as a set of highly nonlineardifferential field equations (these will be presented later). Inprior work [1], we reported the successful application of theMKS framework in capturing the underlying physics in thespinodal decomposition of a binary alloy. However, thiswas accomplished by choosing to represent the local statein the microstructure using two state variables: atomic frac-tion (c) and the gradient of free energy with concentration

dfdc

� �. It was further assumed that the nonlinear interdepen-

dence of these two local state variables was known a priori.These assumptions reduced the numerical complexity ofthe problem substantially by transforming the governingequations into a set of linear differential equations in theselected local state variables. In the present work, weaddress the same problem without making such simplifica-tions. Specifically, we represent the local state using a singlestate variable (i.e. atomic fraction, c). The nonlinearity ishandled by adequately binning the local state space. Thisnew approach is much more generally applicable to abroad range of microstructure evolution phenomena. Themain challenges involved in accomplishing this taskinclude: (i) an efficient discretization of the continuouslocal state space while accomplishing a sufficiently accuratedescription of the linearized structure–structure evolutionlinkages; (ii) a suitable time-integration scheme to accu-rately predict the continuous evolution of microstructureover long time periods involved; and (iii) a systematicapproach to selecting adequate datasets for calibrationand validation of the MKS. All of these aspects will beaddressed in detail in the following sections.

2. The MKS framework

The MKS framework was recently presented and suc-cessfully applied to model the mechanical response ofmulti-phase composite materials [1,2]. For the convenienceof the reader, the main elements of the framework are pre-sented here. The spatial domain of the material internalstructure is assumed to be binned into a uniform grid ofspatial cells (or voxels) that are enumerated by s = 1,2, . . . , S. The microstructure datasets identify the amount

of each local state present in each spatial cell. The set ofall distinct local states that are possible in a given materialsystem is referred to as the local state space, which is typ-ically a multi-dimensional space wherein each independentlocal state descriptor is represented on a set of orthogonalaxes. The local state space of interest is also tessellated intoindividual bins and enumerated by h = 1, 2, . . . , H. Let thevariable mh

s define the volume fraction of local state h in thespatial cell s. Based on this definition of the discretizedmicrostructure variable, it is easy to establish the followingproperties [12]:

XH

h¼1

mhs ¼ 1; mh

s � 0;1

S

XS

S¼1

mhs ¼ V h ð1Þ

where Vh denotes the volume fraction of local state h in theentire microstructure dataset. The microstructure functiondescribed above is essentially a discrete approximation of acontinuous probability distribution associated with thespatial placement of local states in the internal structureof the material.

Fig. 1 clarifies the definition of the discrete microstruc-ture function defined in Eq. (1) for an example binaryalloy, where the local state in each spatial cell is identifiedby the atomic fraction, c, of one of the elements. Each dis-crete spatial cell in the micrograph is associated with anatomic fraction, which represents the average value overthat spatial cell. The values of the atomic fraction are usu-ally bounded for a given binary alloy. For the alloy systemshown in Fig. 1, c 2 ½0:15; 0:23� constitutes the local statespace. This local state space is tessellated into 11 discretebins (i.e. H = 11), with the first and last bins centred on0.15 and 0.23, respectively. As an example, consider thespatial cell indexed by s = 3000 with an atomic fractionc = 0.219. This atomic fraction lies between discretizedlocal states h = 9 and h = 10, corresponding to atomic frac-tions 0.214 and 0.222, respectively. Consequently, themicrostructure variable takes the values M9

3000 ¼ 0:34 andM10

3000 ¼ 0:66, with all other Mh3000 ¼ 0 (for h not equal to

9 or 10).Let ps denote the local response variable in the spatial

bin of interest, s. For the microstructure evolution prob-lems of interest in this paper, ps could represent any localmicrostructure evolution field variable (e.g. the time-deri-vate of the local atomic fraction). Drawing on analoguesin systems theory [13–15], the response field in the micro-structure can be expressed as a series of higher-order con-volutions involving the microstructure signal as

ps ¼XH

h¼1

XS

t¼1

aht mh

sþt þXH

h¼1

XH

h0¼1

XS

t¼1

XS

t0¼1

ahh0

tt0 mhsþtm

h0

sþtþt0

!

ð2Þ

Eq. (2) expresses the local interactions as a series sum [1,2].In Eq. (2), ah

t and ahh0tt0 are referred to as the first-order and

second-order influence coefficients, respectively, and consti-tute the MKS mentioned earlier. The values of these

Fig. 1. Illustration of the definition of the microstructure variable used in establishing the materials knowledge system described in this work for spinodaldecomposition. The color scale identifies the atomic fractions that are tessellated into 11 bins. The values of the microstructure variables for the spatialbins indexed by s as 2578, 3000 and 3376 are shown. The figure also illustrates the physical meaning of the influence coefficients ah

t and ahh0tt0 (For

interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

T. Fast et al. / Acta Materialia 59 (2011) 699–707 701

coefficients are assumed to be independent of the micro-structure coefficients mh

s � aht captures the influence of the

placement of the local state h in a spatial location that ist away from the spatial cell of interest denoted by s. In thisnotation, t enumerates the bins in the vector space [12],which has been tessellated using the exact same scheme thatwas used for the spatial domain of the material internalstructure, i.e. t = 1, 2, . . . , S. Fig. 1 further clarifies thephysical interpretation of the influence coefficients definedin Eq. (2). It should be noted that the influence coefficientsah

t are closely related to the Green’s function kernels usedin other problems [16]. In the rest of this paper, we will lim-it our attention to the first-order influence coefficients. Theestimation of the second-order influence coefficients is sig-nificantly more complicated and is the goal of ongoinginvestigations.

The main challenge with Eq. (2) is the estimation of thenumerical values of the influence coefficients ah

t . In recentwork [1,2], we have demonstrated that it is possible toestablish highly accurate localization relationships for themechanical response of composite material systems by cal-

ibrating the series expansions of Eq. (2) to results obtainedfrom finite element models. In these papers, we also recog-nized and exploited the fact that Eq. (2) takes a much sim-pler form when transformed into the discrete Fouriertransform (DFT) space. Eq. (2), truncated to included onlythe first-order terms, can be conveniently expressed as

P k ¼XH

h¼1

bhk

� ��Mh

k

!" #; bh

k ¼ F k aht

� �;

P k ¼ F kðpsÞ; Mhk ¼ F k mh

s

� �ð3Þ

where F kðÞ denotes the DFT operation with respect to thespatial variables s or t, and the superscript star denotes thecomplex conjugate. Note that the number of coupled first-order coefficients in Eq. (3) is only H, although the totalnumber of first-order coefficients still remains as S*H. Be-cause of this dramatic uncoupling of first-order coefficientsinto smaller sets, the problem is cast as an embarrassinglyparallelized problem for estimating the values of influencecoefficients bh

k by calibrating them to results from estab-lished numerical models for the selected problem.

702 T. Fast et al. / Acta Materialia 59 (2011) 699–707

It is emphasized here that establishing bhk is a one-time

computational task for a selected composite material sys-tem because these coefficients are expected to be indepen-dent of the morphology of the microstructure (defined bymh

s ). As such, they offer a compact representation of theunderlying knowledge regarding the localization of theselected response variable for all possible topologies thatcould be defined in the given composite material system.The simplicity of Eq. (3) also presents a computationallyefficient procedure for computing the spatial distributionof the selected response variable for any microstructuredataset, after the corresponding influence coefficients areestablished and stored. However, it should be noted thatthe influence coefficients are expected to be strongly depen-dent on the imposed boundary conditions, i.e. they need tobe established separately for different boundary conditionsof interest.

3. Application of the MKS framework to spinodal

decomposition

3.1. The physics-based Cahn–Hilliard model

Spinodal decomposition occurs when a homogeneousmixture separates into regions of distinct chemical compo-sitions (or phases). Phase separation of a binary alloy intotwo phases, a and b, can be described by the well-knownCahn–Hilliard equation [17,18] as

_cðxÞ ¼ r2Ddf ðcðxÞÞ

dc�r2KcðxÞ

� �ð4Þ

where x is the spatial variable, c is a conserved orderparameter that defines the atomic fraction of phase a, Dis the diffusion coefficient, f is the free energy expressedas a function of c, and K gives information about the inter-face between the two phases. The free energy, f, is classi-cally assumed to be a double-well potential and is oftenapproximated as a fourth-order polynomial as [19]

f ðcðxÞÞ ¼ 4FðcðxÞ2 � 1Þ2 ð5Þwhere F is the height of the energy barrier between the min-ima in the potential. Eq. (5) implicitly identifies the atomicfractions where spinodal decomposition will occur. For thepresent study, the various parameters in Eqs. (4) and (5)were assigned the following values: D = .1 lm2/s andK = .3 lm2. These values are assigned on a spatial grid ofDx = 1 lm. For these parameters, the compositions canbe shown to lie in the range, c e [0.15, 0.23].

3.2. Local state variables and response variables

The first step in formulating MKS is the proper identifi-cation of the local state variables (along with the corre-sponding local state space) and the response variables. Asmentioned earlier, the proper local state variable for thespinodal decomposition phenomenon is the atomic frac-tion, c. The corresponding local state space depends on

the parameters used to describe the free energy function.As also mentioned earlier, for the parameters chosen forthe present study, the local state space is identified asc(x) e [0.15, 0.23].

In order to capture accurately the underlying nonlinearphysics in the spinodal decomposition phenomenon in theMKS framework, it is necessary to adequately discretizethe local state space. This approach is tantamount to line-arizing the underlying physics over small intervals in thelocal state space. This discretization is shown schematicallyin Fig. 1. As the figure shows, it is fairly easy to describe themapping between the average atomic fraction in a spatialbin, cs, and the microstructure function, mh

s .The response variable of interest in the spinodal decom-

position problem described above is the time derivative ofatomic fraction, _cðxÞ. In the discretized representationsused in the MKS framework, the response variable wouldbe the time derivative of the averaged atomic fraction ina spatial bin, _cS . The MKS framework described earlieraims to establish the correlations between _cs and mh

s . Thesecorrelations will be cast in the simple algebraic formsexpressed earlier in Eqs. (2) and (3) (with ps replaced by_cs). It is important to note that we have cast the problemin such a way that we are capturing the correlationsbetween _cs and mh

s at a given instant of time. In otherwords, the correlations by themselves do not push the timeforward. Once _cs is established, it is necessary to evolve themicrostructure using an appropriate time-integration pro-cedure. It is envisioned that the MKS framework will beapplied recursively in microstructure evolution problemsin order to march forward in time, thereby capturing thepath-dependence of the phenomenon studied.

In the present work, for consistency and simplicity, anEuler forward method has been implemented for time-inte-gration of both the physics-based model and the MKSapproach. It is, however, noted that more sophisticatedtime-integration schemes can be implemented as needed.Since most typically used time-integration schemes canaccumulate errors over a large number of time steps, it isimportant to evaluate critically the accumulation of errorsat various intermediate steps in the overall simulation ofthe phenomenon.

3.3. Calibration of the influence coefficients

The next step in the formulation of the MKS frameworkis the calibration of the first-order influence coefficients. Asnoted earlier, we intend to establish them by calibration toselected results from the numerical implementation of thephysics-based models described above. In prior work onmechanical response of composite systems [2], we have cho-sen to calibrate the influence coefficients to the finite ele-ment results on eigen microstructures. However, thisapproach is not directly applicable to the spinodal decom-position problem studied here, because of the very largenumber of possible eigen microstructures that could bedefined and the very limited information carried in each

T. Fast et al. / Acta Materialia 59 (2011) 699–707 703

such dataset. Instead, we use here a collection of micro-structures to build a calibration dataset to estimate theinfluence coefficients.

In this work, we have generated 20 different two-dimen-sional (2-D) initial microstructures of that are spatiallyresolved into 20 � 20 spatial bins and reflect a random per-turbation in average atomic fraction that lies within thespinodal points identified above. Eqs. (4) and (5) werenumerically integrated using a phase field model, whereineach selected initial microstructure was evolved using theEuler forward method with a time step of Dt = 0.1 s [20].This time step, for the selected model parameters, assuredthat the numerical integration was well within the stabilitycriterion for the finite difference method and showed excel-lent convergence [21]. Periodic boundary conditions weremaintained by imposing the minimum image conventionon the system [22]. Several intermediate datasets, eachincluding both the spatial distribution of the atomic frac-tion and its time derivative in the entire microstructure,were extracted. A total of about 2000 datasets wereextracted from the simulations on 20 different initial micro-structures described earlier. From each simulation, datasetswere collected at several intermediate stages of microstruc-ture evolution. These datasets were later grouped into twogroups: a calibration group, R, and a validation group, Q,with R \Q ¼ ;.

The constraints implicit in the definition of the micro-structure function (see Eq. (1)) translate to the followingrequirements in DFT space:

XH

h¼1

Mh0 ¼ S;

XH

h¼1

Mhk–0 ¼ 0: ð6Þ

Introducing Eq. (6) into Eq. (3) removes the redundan-cies implicit in Eq. (3) and yields the following expression(considering only the first-order terms):

P k–0 ¼XH�1

h¼1

bhk

� ��Mh

k

!" #; P 0 ¼ 0; bh

k ¼ bhk � bH

k

� �:

ð7ÞThe expression for P0 in Eq. (7) imposes the requirement

that the average atomic fraction in the microstructure isconserved (i.e. the volume averaged value of the time deriv-ative of the atomic fraction in the microstructure is identi-cally equal to zero).

The bhk coefficients in Eq. (7) constitute the MKS for

spinodal decomposition as they connect the microstructureand the response variables of interest. The datasets in thecalibration group were used to establish the values of bh

k ,by performing a standard linear regression analyses [23].The linear regression may be formulated as

XR

r¼1

rP k rMh0

k

� ��h i¼XH�1

h¼1

bhk

� ��XR

r¼1

rMhk rMh0

k

� ��h i" #

¼XH

h¼1

Ahk

� ��RMk

� ð8Þ

where r enumerates the datasets in the calibration group,R. Since there are H � 1 unknown influence coefficients,we need at least H � 1 independent datasets in R to suc-cessfully invert the RMk matrix in Eq. (8). However, in or-der to establish accurate values of bh

k , we will need toinclude many more datasets; this will be discussed later inmore detail. One major advantage of the formulation inEq. (8) is the fact that the calibration matrix RMk is inde-pendent of the values of the response. Therefore, if a givenset of microstructures are to be used in the calibration data-set for multiple response variables (e.g. for different bound-ary conditions), we need to assemble and invert the RMk

matrix only once.The factors that affect the accuracy of the influence coef-

ficients established by the procedure described aboveinclude: (i) the discretization of the local state space; and(ii) the number of datasets included in the calibrationgroup R. These will be discussed next.

3.4. Accuracy of the MKS

A volume-averaged root mean squared error metric hasbeen used here to quantify the accuracy of the MKS frame-work in capturing the underlying correlations in the spin-odal decomposition phenomenon. For a dataset in thevalidation group Q enumerated by q, the error is defined as

EHq ¼

1

S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXs

s¼1

MKSpHs� PBMpH

s

� �2

sð9Þ

where the superscript H reminds us about the sensitivity ofthe MKS predictions to the discretization of the local statespace. The superscripts MKS and PBM refer to the factthat the predictions were made using the MKS methodand the physics-based model, respectively.

Parametric studies were performed to understand theeffect of the discretization of the local state space (i.e. H)and the selection of the number of datasets included inthe calibration group R. More specifically, we created sev-eral different calibration groups with different numbers ofdatasets in each group. For each calibration group, we firstcomputed the values of the influence coefficients and usedthose to compute the error for each dataset according toEq. (9). The computed errors were then used to evaluatethe respective mean and standard deviation for the calibra-tion and the validation groups. For each calibration group,this process was then repeated for several values of H.Fig. 2a and b depicts typical results obtained from ourparametric study. The calibration groups used for Fig. 2aand b had 200 and 300 datasets, respectively.

Fig. 2a and b confirms our expectation that the accuracyof MKS improves substantially with increasing discretiza-tion of the local state space (i.e. the value of H). It is alsointeresting to note that the improvement in accuracy slowsdown with the very high values of H. This indicates thatwe have obtained a sufficient discretization of the localstate space at the high values of H that the underlying

Fig. 2. Influence of the discretization of the local state space and the number of datasets used in the calibration group on the accuracy of the MKS: (a) 200datasets in the calibration group; (b) 300 datasets in the calibration group. The mean and the standard deviation of the errors are shown in these plots forboth the calibration group (i.e. the datasets used in the regression analyses) and the validation group (i.e. the datasets that were not used in the regressionanalyses).

704 T. Fast et al. / Acta Materialia 59 (2011) 699–707

correlations can be expressed to the best possible accuracyin the form of the simplified linearized expressions formu-lated in Eq. (7). In other words, beyond these levels of dis-cretization of the local state space, the gains in accuracy areexpected to be minimal. Further gains in accuracy have tocome from the inclusion of the higher-order terms (see Eq.(2)).

The number of datasets included in the calibrationgroup also has a tremendous influence on the accuracy ofthe MKS. With fewer datasets in the MKS, although theerror in the calibration group is low, the error in the vali-dation group is substantially higher as seen in Fig. 2a. Fur-thermore, there is significant separation between the meansof the errors in the two groups that is significantly largerthan their respective standard deviations. These observa-tions point to the fact that the influence coefficientsobtained from the relatively small calibration group arenot sufficiently accurate and do not represent the bestMKS for the spinodal decomposition. On the other hand,the inclusion of a larger number of datasets in the calibra-tion group for Fig. 2b resulted in very close mean errors forthe calibration and validation groups with the difference inthe errors being significantly smaller than their respectivestandard deviations. It is therefore critical to have suffi-ciently rich datasets to establish high-fidelity MKS, andthe overall approach described above can be used to quan-tify its accuracy.

Fig. 3 shows an example initial microstructure, the cor-responding predicted response (i.e. the time derivative ofthe atomic fraction), and the error in the predictedresponse (i.e. the absolute difference between the predictedresponses by the MKS and the direct computation usingthe physics-based model). The prediction was obtainedusing H = 125 and the calibration group used forFig. 2b. It is seen that the MKS predicts the response fieldwith very high accuracy.

3.5. Potential benefits of MKS

A major advantage of MKS lies in its scalability to largemicrostructure datasets. The Green’s functions based influ-ence coefficients, ah

t , are expected to decay to zero values ast takes on large values. In the present example, it was notedthat the influence coefficients generally approached zerovalues for spatial cells that were removed by about 4 or 5cells from the spatial cell of interest. In other words, theinfluence coefficients obtained from relatively small models(e.g. the models with 20 � 20 spatial cells described above)can be applied to extremely large microstructure datasetswithout any need for recalibration. This concept was previ-ously validated in the application of MKS approach tomechanical response of composites [2]. In order to demon-strate the validity of this concept to the spinodal decompo-sition problem studied here, we applied the influencecoefficients obtained from the smaller 20 � 20 models toa new microstructure described on a much larger modelwith 100 � 100 spatial cells. Fig. 4 shows an example initialmicrostructure, the corresponding predicted response (i.e.the time derivative of the atomic fraction) and the errorin the predicted response (i.e. the absolute differencebetween the predicted responses by the MKS and the directcomputation using the physics-based model). For a valida-tion dataset of 100 RVEs, the mean squared error and itsstandard deviation for 20 � 20 and 100 � 100 spatial cellsare 4:89� 10�13 � 2:52� 10�13 and 5:27� 10�13 � 2:79�10�13, respectively. Meanwhile, the maximum meansquared errors are 5:30� 10�12 � 2:90� 10�12 and 8:17�10�12 � 3:96� 10�12, respectively. It is clearly seen thatthe MKS approach provides excellent predictions evenfor the larger models.

The use of FFT methods in the MKS approach affordsseveral computational advantages. (i) It is well known thatthe computational cost of FFTs scales as O(Nlog(N)),

Fig. 3. The predicted time derivatives of the atomic fraction by the MKS approach developed in this study and the error in this prediction (in comparisonto values computed directly from the physics-based model) for an example microstructure for a RVE size of 20 � 20. MSE refers to the mean squarederror. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 4. The predicted time derivatives of the atomic fraction by the MKS approach developed in this study and the error in this prediction (in comparisonto values computed directly from the physics-based model) for an example microstructure for a RVE size of 100 � 100. (For interpretation of thereferences to color in this figure, the reader is referred to the web version of this article.)

T. Fast et al. / Acta Materialia 59 (2011) 699–707 705

whereas most other numerical approaches scale at least asO(N2) [24]. (ii) The decoupling achieved by the use of FFTs(see Eqs. (2) and (3)) allows easy implementation of parall-elized computations. (iii) The recent advent of graphicsprocessing units promises to dramatically enhance thespeed of FFTs computations [25].

3.6. Evolution of microstructure-processing linkages

The predicted response field (i.e. the time derivative ofthe atomic fraction in each spatial cell) was used to updatethe microstructure using a simple Euler forward time-inte-gration scheme with a time step of Dt = 0.1 s. This time-integration procedure is exactly the same as that used inthe direct physics-based model computations described ear-lier. The MKS produced with the calibration group shownin Fig. 2b corresponding to H = 125 was used, as it was themost accurate one produced in this study. It is important torecognize that the MKS linkages describe the correlationsbetween microstructure variables and response fields atan instant of time, and therefore are not tied to a particular

time-integration scheme. For example, they can be utilizedwith an implicit scheme. However, for this first example,for consistency and simplicity, we opted to utilize the sameEuler forward time-integration scheme that was used in thephysics-based model.

Starting with a selected initial microstructure, Fig. 5shows the MKS predicted evolution of the microstructureat certain intermediate time steps and the correspondingerrors (in comparison with direct computations using thephysics-based model). It is clear that the MKS approachprovided excellent predictions of the evolution of themicrostructure. Fig. 6 shows the variation of the accumu-lated mean squared error as the simulation progresses intime. It is seen that the mean squared error accumulatesup to a certain time in the simulation and then it remainsmore or less constant, indicating that the error in these sim-ulations does not grow in an unbounded manner. It is alsoimportant to note that the accumulated error in the MKSpredictions of the final evolved microstructure is very low.

With the relatively small spatial domains and for therelatively simple problems described in this paper, the

Fig. 5. MKS prediction of the microstructure evolution for an example initial microstructure (first column) along with the corresponding predictions ofthe time derivative of atomic fractions (second column) and the associated error (in comparison with results from the direct physics-based modelcomputations). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 6. The accumulation of the mean squared error in the MKSprediction of microstructure evolution shown in Fig. 5.

706 T. Fast et al. / Acta Materialia 59 (2011) 699–707

computational cost advantage of the MKS approach overthe physics-based model was observed to be a factor of abouttwo. However, it is emphasized that the MKS approachallows for a true multi-scale simulation of the physical phe-nomenon with the information flowing in both directionsbetween the constituent length scales. As an example, con-sider the simulation of a complex processing operation wheredifferent macroscale spatial locations in the sample experi-ence different thermal histories (often an unavoidable

consequence of the boundary conditions imposed at themacroscale). Consequently, strong variations in the micro-structure should be expected at different macroscale loca-tions in the sample. In other words, it is not enough totrack the evolution of a single representative microstructurefor the entire sample. Furthermore, the development ofmicrostructure heterogeneities can be expected to have astrong influence on the macroscale simulation by alteringthe local effective properties at different locations in the sam-ple. In such a situation, it is necessary to track evolution ofseveral representative microstructures at various macroscalelocations in the sample and effectively pass information inboth directions between the microscale simulations and themacroscale simulation. This is extremely difficult, if notimpossible, using any of the currently employed techniques.The MKS approach described in this work offers a viableapproach for such problems. For a given problem, it shouldbe possible to set up the necessary MKS linkages as describedin this paper (this constitutes a one-time computationalexpense). After the MKS linkages are set up, they can beretrieved with minimal computational expense in a multi-scale simulation to not only track accurately the evolutionof the microstructure at each spatial location of interest inthe macroscale simulation, but also to pass their associatedupdated effective properties influencing the macroscalesimulation.

T. Fast et al. / Acta Materialia 59 (2011) 699–707 707

4. Conclusions

The recently developed mathematical framework calledmaterials knowledge systems (MKS) has been successfullyextended to address multi-scale materials modeling prob-lems involving microstructure evolution. It was shown thatit is possible to obtain an efficient discretization of the con-tinuous local state space to produce a sufficiently accuratedescription of the linearized structure–structure evolutionlinkages. The viability and the computational advantagesof the MKS methodology have been demonstrated formodeling the spinodal decomposition in an example binaryalloy. It was shown that the MKS linkages developed inthis study can be used to accurately predict the continuousevolution of microstructure in highly nonlinear phenome-non such as spinodal decomposition.

Acknowledgements

The authors acknowledge financial support for thiswork from the DARPA-ONR Dynamic 3D Digital Struc-ture project, Award No. N000140510504 (Program Man-ager: Dr. Julie Christodoulou).

References

[1] Kalidindi SR, Niezgoda SR, Landi G, Vachhani S, Fast A. ComputMater Continua 2010;395:1.

[2] Landi G, Niezgoda SR, Kalidindi SR. Acta Mater 2010;58:2716.

[3] Duvvuru HK, Wu X, Kalidindi SR. Comput Mater Sci 2007;41:138.

[4] Binci M, Fullwood D, Kalidindi SR. Acta Mater 2008;56:2272.[5] Fullwood DT, Kalidindi SR, Adams BL, Ahmadi S. Comput Mater

Continua 2009;9:25.[6] Kalidindi SR, Landi G, Fullwood DT. Acta Mater 2008;56:3843.[7] Kroner E. In: Gittus J, Zarka J, editors. Modelling small deformations

of polycrystals. London: Elsevier Science Publishers; 1986. p. 229.[8] Kroner E. J Mech Phys Solids 1977;25:137.[9] Kalidindi SR, Binci M, Fullwood D, Adams BL. Acta Mater

2006;54:3117.[10] Cooley JW, Tukey JW. Math Comput 1965;19:297.[11] Cahn JW. Acta Metall 1961;9:795.[12] Adams BL, Gao X, Kalidindi SR. Acta Mater 2005;53:3563.[13] Boyd S, Chua L. IEEE Trans Circ Syst 1985;32:1150.[14] Nikias C, Petropulu A. Higher-order spectra analysis: a nonlinear

signal processing framework. Englewood Cliffs, NJ: PTR PrenticeHall; 1993.

[15] Tong L, Xu G, Kailath T. IEEE Trans Inform Theory 1995;41.[16] Khojasteh A, Rahimian M, Pak RYS. Int J Solids Struct 2008;45:

4952.[17] Cahn JW, Hilliard JE. J Chem Phys 1958;28:258.[18] Novick-Cohen A, Segel L. Phys D: Nonlinear Phenom 1984;10:277.[19] Sander E, Wanner T. J Stat Phys 1999;95:925.[20] Wang Y, Chen LQ, Khachaturyan AG. Acta Metall Mater 1993;41:

279.[21] Incropera F, DeWitt D. Fundamentals of heat and mass transfer. New

York: Wiley; 2001.[22] Hloucha M, Deiters UK. Mol Simulat 1998;20:239.[23] Montgomery D, Peck E, Vining G. Introduction to linear regression

analysis. Hoboken, NJ: Wiley-Interscience; 2006.[24] Hoffman AJ, Martin MS, Rose DJ. SIAM J Numer Anal 1973;10:

364.[25] Nickolls J, Buck I, Garland M, Skadron K. Queue 2008;6:40.