Machine learning approaches for elastic localization linkages in highcontrast composite materials
 Ruoqian Liu^{1},
 Yuksel C. Yabansu^{2},
 Ankit Agrawal^{1}Email author,
 Surya R. Kalidindi^{2, 3} and
 Alok N. Choudhary^{1}
DOI: 10.1186/s401920150042z
© Liu et al. 2015
Received: 13 August 2015
Accepted: 16 November 2015
Published: 4 December 2015
Abstract
There has been a growing recognition of the opportunities afforded by advanced data science and informatics approaches in addressing the computational demands of modeling and simulation of multiscale materials science phenomena. More specifically, the mining of microstructure–property relationships by various methods in machine learning and data mining opens exciting new opportunities that can potentially result in a fast and efficient material design. This work explores and presents multiple viable approaches for computationally efficient predictions of the microscale elastic strain fields in a threedimensional (3D) voxelbased microstructure volume element (MVE). Advanced concepts in machine learning and data mining, including feature extraction, feature ranking and selection, and regression modeling, are explored as data experiments. Improvements are demonstrated in a gradually escalated fashion achieved by (1) feature descriptors introduced to represent voxel neighborhood characteristics, (2) a reduced set of descriptors with top importance, and (3) an ensemblebased regression technique.
Keywords
Materials informatics Data mining Elastic localization linkages Structure feature selection Structure feature ranking Ensemblebased regressionBackground
Material data sciences and informatics [1–12] are emerging as foundational disciplines in the realization of the vision set forth in various highprofile national strategic documents [13, 14]. The novel tools developed in these emerging fields focus mainly on transforming large amounts of collected data (from both experiments and computer simulations) into higher value knowledge that can also be easily disseminated to the broader research community. More specifically, various emerging concepts and tools in machine learning and data mining methods are applied to represent, parse, store, manage, and analyze material data. The higher value knowledge extracted using these tools can be used to dramatically accelerate material development efforts for a range of advanced technologies. One of the central tasks in the analyses of materials data is the identification and extraction of robust and reliable structure–property relationships [15–33].
The internal structure of a material system exhibits multiple hierarchical length scales that play a pivotal role in the behavior and performance characteristics of the material. Consequently, multiscale modeling is an integral component of any effort aimed at rational material design. Almost all multiscale models currently employed in materials design involve oneway coupling, where the information is passed mainly from a lower to a higher length scale (also called homogenization). Communication of highvalue information in the opposite direction (also called localization) is usually very limited. For the purpose of achieving efficient scale bridging, datadriven approaches for establishing localization structure–property relationships as lowcomputationalcost linkages (i.e., surrogate models or metamodels) are of great interest.
Physicsbased multiscale material models provide tools needed to explore the role of material structure in optimizing the overall (effective) properties of interest. This is generally accomplished by solving governing field equations numerically (e.g., finite element models), while satisfying the appropriate (lower length scale) material constitutive laws and the imposed boundary and initial conditions. However, the computational resource requirements of such multiscale materials models are usually very high, rendering these tools impractical for the needs of rational material design and optimization. Besides the high computational requirements, there is not enough attention paid to systematic learning from these simulations. In other words, in any typical design and optimization effort, solutions of the governing field equations are generally obtained for multiple trials of the material structures. However, most solutions that do not produce the desired property or performance are routinely discarded without distilling transferable knowledge from them. It is extremely important to recognize that even when the trial did not produce the desired solution, there is a great deal of information in the solution obtained. Since a significant computational cost was expended in arriving at the solution, it only behooves us to learn as much as we can from the solution obtained. Machine learning techniques and datadriven methods are ideally suited for this task and can lead to dramatic savings in both time and effort, when implemented properly into the material development efforts. In the present study, we demonstrate the implementation of one such strategy for capturing the elastic localization in highcontrast composite material systems in a lowcost surrogate model that is applicable to a very broad set of potential material internal structures.
Materials informatics is an emerging discipline that leverages information technology and data science to uncover the essential process–structure–property (PSP) relationships central to accelerated discovery and design of new/improved materials. A large part of materials informatics involves the use of data mining and machine learning techniques to exploit materials databases and discover trends and mathematical relations for material design [34]. Datacentered methods, as opposed to ab initio methods, are generally expressed as heuristic models, statistically learned from large amounts of historically accumulated observations. Bearing sound generality, they are also able to adapt quickly to new observations. The capability of establishing models from a pure statistical or “machinelike” standpoint avoids human interference and thus enhances the chance of finding the embedded highvalue information in an objective manner, especially when this knowledge is not easily expressed through simple equations.
The rich complexity of the material internal structure typically demands a highdimensional representation [3, 5, 10, 20, 35, 36]. In general, it is actually preferable to start with a more than sufficient list of potential descriptive features (interpreted here as measures of material internal structure) prior to building the models. In this phase of model building, it is fully acknowledged that the salient features are only expected to naturally lie in a much lower dimensional space. An important step of machine learning is the identification of these salient features using suitable feature selection techniques or a transformation of features from a higher dimensional space to a lower dimensional space, known as feature extraction. Both selection and extraction can be either supervised or unsupervised. If the response of the material structure (e.g., the elastic response associated with the material structure in the present case study) needs to be predicted, supervised learning provides more insights in the selection process.

Features identifying the local neighborhood of a voxel to different degrees of adequacy are explored systematically with carefully defined neighbor levels.

Multiple strategies are explored for ranking the large number of potential features that could be used to quantify the neighborhood of the voxel of interest.

Different strategies for formulating regression models are critically evaluated and contrasted for their computational efficacy and accuracy for the selected task. Ensemble methods, which aggregate a number of weak regressors each specializing in a subdomain of the original task, have shown substantial promise.
Methods
Problem statement
Localization, as opposed to homogenization, describes the spatial distribution of the response at the microscale for an imposed loading condition (e.g., averaged strain) at macroscale. Localization is critically important in correlating various failurerelated macroscale properties of the material with the specific local microstructure conformations responsible for the (local) damage initiation in the material. In this work, these two scales are to be connected through linkages extracted by datadriven processes used in machine learning systems.
More specifically, we focus our effort in this study on extracting localization relationship for elastic deformation in a twophase composite [15, 16, 18, 19]. The input into such a linkage typically includes the material microstructure (defined in a threedimensional (3D) microscale volume element (MVE)) and the applied macroscale loading condition (typically expressed as the averaged elastic strain imposed on the MVE). The output from the linkage is the microscale elastic strain field throughout the MVE.
The first step in the application of data science methods is the collection and organization of appropriate data from which the linkages can be mined efficiently. At the present time, suitable datasets for this purpose can only be obtained using numerical models. The experimental protocols for measuring 3D stress (or strain fields) are still very much in developmental stages [37–39]. Therefore, we proceed here with datasets created by numerical physicsbased models (e.g., finite element (FE) models). In other words, we consider the predictions obtained by the FE models as the “ground truth” and we want to establish the localization linkages as a surrogate model for the actual FE model. Our expectation is that the surrogate model will provide a much faster answer compared to the FE model with only a modest loss in accuracy.
Design of data experiments
A total of 2500 MVEs with varying volume fractions were included in this study. They are evenly distributed in 100 variations of volume fraction values, from 1.0 to 99.4 %. Therefore, 25 MVEs are present in each variation, within which, 15 are used as calibration (for feature extraction, model training), and the remaining 10 are used for validation.
where p _{imposed} denotes the average strain imposed on the MVE, and p _{ s } and \(\hat p_{s}\) denote the values of the strain in the voxel s from the FE model and the surrogate model developed in this work, respectively. This metric quantifies the average error for a single MVE microstructure. In the data experiments presented here, we show both individual e for each MVE as well as averaged MASE, \(\bar e\), over the entire set of 1000 validation MVEs.
In constructing training and test data for predictive modeling, each voxel in the MVE is examined, represented, and transformed into a data instance consisting of “inputs” and “outputs”. Each MVE generates 9261 data samples (this is the number of voxels in each MVE). The complete calibration set hence contains 13,891,500 samples and validation contains 9,261,000 samples.
We term the voxel under examination as the “focal voxel”, whose response (average elastic strain in the voxel) is to be predicted. Each voxel in the MVE gets to be the focal voxel once, and when it does, other voxels in its local environment are taken to construct input features for it. By doing this, we are assuming that the response of a focal voxel is strongly influenced by some shortrange interactions with neighboring voxels in its local environment. This concept is highly consistent with the concepts of Green’s functions utilized extensively in composite theories [42–47].
Following the symbolic definitions in [15–19, 22], we let the microstructure variables \({m_{s}^{0}}\) and \({m_{s}^{1}}\) denote the volume fraction of each local state in each voxel of the composite MVE, where 0<s≤S indexes the voxels; S=9261 is the total number of voxels in an MVE. Since \({m_{s}^{1}}+{m_{s}^{0}} = 1\) and we employ eigenmicrostructures (each voxel is assigned exclusively to one of the two phases allowed) in the present case study, we further simplify the notation and use m _{ s } to simply denote \({m_{s}^{1}}\) in some of the case studies presented here.

Level of neighbors, l. Neighbors generally refer to voxels adjoining a given voxel. Here, we extend the definition and serialize neighbors based on their scalar distances from the voxel of interest. Figure 3 shows a 3D voxel of interest in pink, surrounded by its different levels of neighbor voxels. The level of a neighbor, l, is used in this study to identify all of the voxels that are at a distance of \(\sqrt {l}\) from the voxel of interest. In Fig. 3, l=1, 2, 3, 4 from the upper left to the lower right. In this work, where MVEs are of dimension 21×21×21, a voxel can have up to 300 levels of neighbors, although, at some of these levels, there do not exist any neighbor members (for example, l=7 and l=15 do not have any members invalid as their squared values cannot be represented by a sum of squares of 3 whole numbers).

Individual voxel t in a neighbor layer. In each layer of the same neighbor level, there can be none or a number T _{ l } of neighbor member voxels. As shown in Fig. 3, there are T _{1}=6 firstlevel neighbors, T _{2}=12 secondlevel neighbors, T _{3}=8 thirdlevel neighbors, and T _{4}=6 fourthlevel neighbors. To address each of them, we assign an index variable t=0,…,T _{ l }−1. For example, all voxels at neighbor level 1 of s can be indexed as (s,1,0), (s,1,1), (s,1,2), (s,1,3),(s,1,4), and (s,1,5), following the notation introduced earlier.
Following this nomenclature, m _{ s,0,0} is the (binary) microstructure variable at s, i.e., the focal voxel. Its neighboring voxels, m _{ s,l,t }, along with other extracted feature variables are included in the input feature vector when modeling p _{ s }.
Three data exercises are designed and conducted here to study the important subprocesses involved in building a datacentered learning system for localization: (i) neighbor inclusion—how large a spatial neighborhood of voxels should be considered in formulating the statistical model for the response at the focal voxel; (ii) feature extraction—what salient features should be considered in building simplified geometrical constructs among the neighborhood voxels; and (iii) regressors—what learning algorithm should be used for connecting the microstructure and the desired local response.
Design of exercise 1
In this first exercise, namely, Ex 1, we focus on identifying the amount of information needed in forming an accurate representation of a focal voxel, with its local neighborhood. By only using the structure information given by m _{ s,l,t }, we explore how much of a l is necessary in order to represent adequately the neighborhood of m _{ s,0,0} for the elastic localization linkages of interest. As we increase l, the number of input variables used in the modeling p _{ s } will also increase.
Six variations are designed, varying the number of inputs by adjusting the extent to which level neighbors are to be included. Only input features are varied, and the prediction target (p _{ s }) and regression scheme are fixed. A M5 model tree, which is a type of decision tree with linear regression functions at the leaves, is used as the regression model for the data experiments in this case study. The M5 model tree is based on the M5 scheme described by Quinlan [48] and implemented by Wang and Witten [49]. This set of experiments is aimed at answering the question: Will using more information about the neighbors’ help improve the prediction model for the elastic response at the focal voxel?
Design of exercise 2

m _{ s,l,t } is what has been used in Ex 1, the microstructure value of voxels in the neighborhood of s. We use up to the 12th level, and the total number of neighbor voxels are 1+6+12+8+…=179.Table 1
Definition of the set of features constructed in Ex 2, with regard to the representation of a focal voxel at s
Symbol
Meaning
Count
Scope
m _{ s,l,t }
Microstructure value of voxels at a neighbor level l, with index t, of a focal voxel at s
179
Binary, {0,1}
l=1,…,12
\(\text {pr}_{l}^{h}\)
Fraction of voxels with microstructure phase h at neighbor level l
24
Real, [0,1]
\(\text {pr}_{l}^{h}\)
Fraction of voxels with microstructure phase h up to neighbor level l
24
Real, [0,1]
\(I_{\text {norm}}^{h}\)
The normalized impact of all 12 levels of neighbors of phase h
2
Real
\( I_{\text {norm}}^{h} = \sum _{i=1}^{12} T_{l}\cdot \text {pr}_{l}^{h}/\sqrt {l} + T_{0}\cdot \text {pr}_{0}^{h}/0.5\)
S _{3}
3plane symmetry index
1
Real
S _{9}
9plane symmetry index
1
Real

\(\text {pr}_{l}^{h}\) is the volume fraction of phase h in neighborhood level l.

\(\text {pr}_{l}^{h}\) is the accumulated volume fraction of phase h up to neighborhood level l.

\(I_{\text {norm}}^{h}\) is defined as the aggregated “impact” to a focal voxel of all its neighbors up to a specified level (in this exercise, we include up to the 12th level). For this purpose, we first quantify the impact of each voxel in neighbor level l to be given by \(1/\sqrt {l}\); as expected, closer neighbors have higher impact values. For all voxels at l (l>0), the overall impact is computed as \({I_{l}^{h}}=T_{l}\cdot \text {pr}_{l}^{h}/\sqrt {l}\). For l=0, the impact value is assigned as \({I_{0}^{h}}=2\). \(I_{\text {norm}}^{h}\) is then calculated as a sum of impacts from all levels (up to 12), \(I_{\textit {norm}}^{h}=\sum _{i=0}^{12}{I_{i}^{h}}\). It is easy to see that the sum of \(I_{\text {norm}}^{0}\) and \(I_{\text {norm}}^{1}\) is always a constant value (\(=2.0+T_{1}/\sqrt {1}+T_{2}/\sqrt {2}+T_{3}/\sqrt {3}+\dots \)) where T _{1}=6,T _{2}=12,T _{3}=8,….

S _{3} and S _{9} stand for two symmetry descriptors looking at a 3D local microstructure, including up to the 12 neighbor levels, centered at the focal voxel. Symmetry is defined as the degree of similarity between the two halves of the 3D structure when bisected by a specified plane. S _{3} considers three dividing planes passing through the center focal voxel, and S _{9} uses nine, adding six diagonal ones. Planes are illustrated in Fig. 4, where the focal voxel is placed at the center of the structure. Note that the MVE structure in the figure is only for illustration. In actual calculation, planes cut through an irregular but symmetrical structure where a focal voxel is in the center and all of its neighbors up to the 12th level (in total, 178 neighbor voxels) scatter around it. For every dividing plane, we assess how similar the resulting two halfstructures are to each other, by computing a voxeltovoxel exclusive nor (XNOR, giving one when two voxels are the same) of the two halfstructures and then taking a distancenormalized sum. In this way, nonconformity farther away from the focal voxel has a smaller effect.
where P(X=h) is the prior probability that the microstructure X takes value h and X _{ h } is a binary attribute that takes the value 1 when X=h and 0 otherwise.
Features ranked by the correlation with the response. Top 30 are shown
Rank  Feature 

1  m _{ s,0,0} 
2–7  m _{ s,1,2}, m _{ s,1,3}, m _{ s,1,1}, m _{ s,1,0}, m _{ s,1,4}, m _{ s,1,5} 
8–13  m _{ s,2,2}, m _{ s,2,3}, m _{ s,2,0}, m _{ s,2,1}, m _{ s,4,4}, m _{ s,2,4} 
14–16  \(\text {pr}_{1}^{1}\), m _{ s,2,4}, \(\text {pr}_{1}^{0}\) 
17  \(I_{\text {norm}}^{0}\) 
18  \(\text {pr}_{1}^{1}\) 
19–20  S _{9}, S _{3} 
21–23  m _{ s,2,8}, m _{ s,2,5}, m _{ s,3,3} 
24  \(\text {pr}_{1}^{0}\) 
25–30  m _{ s,2,6}, m _{ s,5,6}, m _{ s,5,10}, m _{ s,2,9}, m _{ s,8,28}, m _{ s,5,11} 
Design of exercise 3

Ex 3a As an extension to M5 regression tree, a random forest (RF) [50] regressor that forms an ensemble of many tree estimators is explored. The concept of ensemble learning or using a number of estimators and aggregating their results is expected to give a better generalization towards unseen data. The number of member estimators in RF is set to be 50.

Ex 3b As a classic kernelbased learning model, support vector machine [51] finds an optimized hyperplane in feature space to separate classes. To deal with continuous class outputs, Support Vector Regression (SVR) [52] is used.
Results and discussion
The following subsections present performances of each designed data experiment in terms of average prediction errors. Another measure of performance is the computational time. FEM simulations for each MVE took 23 s with two processors in a supercomputer, whereas with data models, once the model parameters are fixed, the prediction only takes a few milliseconds per MVE.
Data exercise 1: neighbor inclusion
The first exercise studies the feature space constructed by neighbor voxels only. Since our goal at this point is to explore potential features for building the prediction model subsequently, it is not essential to use the entire dataset. In order to save computational cost, the six models (described earlier) are built and tested on a small subset that contained 50 MVEs with volume fractions of 48–53 %, which are regarded as the most difficult MVEs, because the response field exhibits the highest level of heterogeneity. Tenfold crossvalidation is conducted where in each fold, 45 MVEs are used for training the model, and the remaining 5 MVEs are used for testing.
The results are summarized in Fig. 5, showing six variations in the inclusion of neighbor voxels m _{ s,l,t } in building a relationship between m _{ s } (or m _{ s,0,0}) and p _{ s }. l varies from 0 up to 3, 5, 6, 8, 9, and 10, from left to right in Fig. 5. This corresponds to a number of inputs of 27, 57, 81, 93, 123, and 147, respectively.
The results indicate that using more neighbors does not necessarily continue to enhance the accuracy. The model with an inclusion of neighbor level l up to 8 gives the best (least) test error. The speed of the learning of model trees is influenced linearly by feature dimensions.
Figure 5 also indicates that most of the experiments have a very high test error of over 50 %. The shortcoming of this series of modeling lies in the inadequacy of microstructure representation coming solely from individual components of the m _{ s,l,t }. In Ex 2, we aim to identify a set of engineered microstructure features in addition to m _{ s,l,t } to represent more effectively the salient neighborhood features of the focal voxel.
Data exercise 2: feature extraction
With the set (see Table 1) containing 231 feature variables devised, a series of exercises (labeled Ex 2) are conducted using different combinations of the feature variables based on a rank generated by correlation measures, while keeping the regression model the same. With regard to the rank of importance (partially shown in Table 2), we take various numbers of top features with the best relevance quality in constructing prediction models and thus designed Ex 2a, Ex 2b, Ex 2c, and Ex 2d. The top 27, 57, and 93 features are selected to match the number of inputs used in Ex 1, and the last exercise uses all 231 features in the set.
The example shown in this figure corresponded to a volume fraction of 50.22 % (this is one of the cases with the highest error). The improvements in the accuracy of Ex 2b over Ex 1d is clearly evident. In particular, it should be noted that Ex 2b is doing a very reasonable job in predicting the locations and distributions of the hot spots (voxels with the highest local elastic strain).
Data exercise 3: regressors
Conclusions
In this paper, we explored multiple data mining experiments and strategies for establishing statistical models for capturing elastic localization relationships in high contrast composites. More specifically, our focus was on a composite with a contrast of 10. The efficacy of different approaches for feature selection and regression were studied systematically. We demonstrated that a set comprised of basic feature descriptors combined with engineered (constructed) features is able to boost the prediction performance. Moreover, a reduced set of descriptors generated by feature ranking methods offers even better results. In terms of regression techniques, ensemble methods such as random forests show superiority when both accuracy and time consumption are taken into account.
Declarations
Acknowledgements
All authors gratefully acknowledge primary funding support from AFOSR award FA95501210458 for this work. RL, AA, and AC also acknowledge partial support from NIST award 70NANB14H012 and DARPA award N6600115C4036.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Rajan K (2005) Materials informatics. Mater Today 8(10): 38–45.View ArticleGoogle Scholar
 Panchal JH, Kalidindi SR, McDowell DL (2013) Key computational modeling issues in integrated computational materials engineering. Comput Aided Des 45(1): 4–25.View ArticleGoogle Scholar
 Kalidindi SR (2015) Data science and cyberinfrastructure: critical enablers for accelerated development of hierarchical materials. Int Mark Rev 60(3): 150–168.Google Scholar
 Niezgoda SR, Turner DM, Fullwood DT, Kalidindi SR (2010) Optimized structure based representative volume element sets reflecting the ensembleaveraged 2point statistics. Acta Mater 58(13): 4432–4445.View ArticleGoogle Scholar
 Niezgoda SR, Yabansu YC, Kalidindi SR (2011) Understanding and visualizing microstructure and microstructure variance as a stochastic process. Acta Mater 59(16): 6387–6400.View ArticleGoogle Scholar
 Ferris KF, Peurrung LM, Marder JM (2007) Materials informatics: fast track to new materials. Adv Mater Process1: 50–51. 165(PNNLSA52427).Google Scholar
 Rodgers JR, Cebon D (2006) Materials informatics. MRS Bull 31(12): 975–80.View ArticleGoogle Scholar
 Rajan K, Suh C, Mendez PF (2009) Principal component analysis and dimensional analysis as materials informatics tools to reduce dimensionality in materials science and engineering. Stat Anal Data Mining: ASA Data Sci J 1(6): 361–371.View ArticleGoogle Scholar
 McDowell DL, Olson GB (2008) Concurrent design of hierarchical materials and structures. Sci Model Simul SMNS 15(13): 207–240. doi:10.1007/s1082000891006.View ArticleGoogle Scholar
 Niezgoda SR, Kanjarla AK, Kalidindi SR (2013) Novel microstructure quantification framework for databasing, visualization, and analysis of microstructure data. Integrating Mater Manuf Innov 2(1): 1–27.View ArticleGoogle Scholar
 Kalidindi SR, Gomberg JA, Trautt ZT, Becker CA (2015) Application of data science tools to quantify and distinguish between structures and models in molecular dynamics datasets. Nanotechnology 26(34): 344006.View ArticleGoogle Scholar
 Steinmetz P, Yabansu YC, Hotzer J, Jainta M, Nestler B, Kalidindi SR (2016) Analytics for microstructure datasets produced by phasefield simulations. Acta Mater 103: 192–203.View ArticleGoogle Scholar
 Allison J, Backman D, Christodoulou L (2006) Integrated computational materials engineering: A new paradigm for the global materials profession. JOM 58(11): 25–27.View ArticleGoogle Scholar
 Warren J (2012) Materials genome initiative In: AIP Conference Proceedings.. American Institute of Physics, Ste. 1 NO 1 Melville NY 117474502 United States.Google Scholar
 Landi G, Niezgoda SR, Kalidindi SR (2010) Multiscale modeling of elastic response of threedimensional voxelbased microstructure datasets using novel dftbased knowledge systems. Acta Mater 58(7): 2716–2725.View ArticleGoogle Scholar
 Landi G, Kalidindi SR (2010) Thermoelastic localization relationships for multiphase composites. Comput Mater Continua 16(3): 273–293.Google Scholar
 Fast T, Niezgoda SR, Kalidindi SR (2011) A new framework for computationally efficient structure–structure evolution linkages to facilitate highfidelity scale bridging in multiscale materials models. Acta Mater 59(2): 699–707.View ArticleGoogle Scholar
 Fast T, Kalidindi SR (2011) Formulation and calibration of higherorder elastic localization relationships using the MKS approach. Acta Mater 59(11): 4595–4605.View ArticleGoogle Scholar
 Kalidindi SR, Niezgoda SR, Landi G, Vachhani S, Fast T (2010) A novel framework for building materials knowledge systems. Comput Mater Continua 17(2): 103–125.Google Scholar
 Çeçen A, Fast T, Kumbur E, Kalidindi S (2014) A datadriven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells. J Power Sources 245: 144–153.View ArticleGoogle Scholar
 AlHarbi HF, Landi G, Kalidindi SR (2012) Multiscale modeling of the elastic response of a structural component made from a composite material using the materials knowledge system. Model Simul Mater Sci Eng 20(5): 055001.View ArticleGoogle Scholar
 Yabansu YC, Patel DK, Kalidindi SR (2014) Calibrated localization relationships for elastic response of polycrystalline aggregates. Acta Mater 81: 151–160.View ArticleGoogle Scholar
 Adams BL, Lyon M, Henrie B (2004) Microstructures by design: linear problems in elastic–plastic design. Int J Plast 20(8): 1577–1602.View ArticleGoogle Scholar
 Belvin A, Burrell R, Gokhale A, Thadhani N, Garmestani H (2009) Application of twopoint probability distribution functions to predict properties of heterogeneous twophase materials. Mater Charact 60(9): 1055–1062.View ArticleGoogle Scholar
 Adams BL, Kalidindi SR, Fullwood DT (2012) Microstructure sensitive design for performance optimization. ButterworthHeinemann, Boston.Google Scholar
 Böhlke T, Lobos M (2014) Representation of Hashin–Shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with application in materials design. Acta Mater 67: 324–334.View ArticleGoogle Scholar
 Proust G, Kalidindi SR (2006) Procedures for construction of anisotropic elastic–plastic property closures for facecentered cubic polycrystals using firstorder bounding relations. J Mech Phys Solids 54(8): 1744–1762.View ArticleGoogle Scholar
 Kalidindi SR, Binci M, Fullwood D, Adams BL (2006) Elastic properties closures using secondorder homogenization theories: case studies in composites of two isotropic constituents. Acta Materialia 54(11): 3117–3126.View ArticleGoogle Scholar
 Kalidindi SR, Niezgoda SR, Salem AA (2011) Microstructure informatics using higherorder statistics and efficient datamining protocols. Jom 63(4): 34–41.View ArticleGoogle Scholar
 Knezevic M, Kalidindi SR (2007) Fast computation of firstorder elastic–plastic closures for polycrystalline cubicorthorhombic microstructures. Comput Mater Sci 39(3): 643–648.View ArticleGoogle Scholar
 Fromm BS, Chang K, McDowell DL, Chen LQ, Garmestani H (2012) Linking phasefield and finiteelement modeling for process–structure–property relations of a nibase superalloy. Acta Mater 60(17): 5984–5999.View ArticleGoogle Scholar
 Binci M, Fullwood D, Kalidindi SR (2008) A new spectral framework for establishing localization relationships for elastic behavior of composites and their calibration to finiteelement models. Acta Mater 56(10): 2272–2282.View ArticleGoogle Scholar
 Yabansu YC, Kalidindi SR (2015) Representation and calibration of elastic localization kernels for a broad class of cubic polycrystals. Acta Mater 94: 26–35.View ArticleGoogle Scholar
 Suh C, Rajan K (2009) Invited review: data mining and informatics for crystal chemistry: establishing measurement techniques for mapping structure–property relationships. Mater Sci Technol 25(4): 466–471.View ArticleGoogle Scholar
 Li Z, Wen B, Zabaras N (2010) Computing mechanical response variability of polycrystalline microstructures through dimensionality reduction techniques. Comput Mater Sci 49(3): 568–581.View ArticleGoogle Scholar
 Torquato S (2002) Random heterogeneous materials: microstructure and macroscopic properties vol. 16. Springer, New York.View ArticleGoogle Scholar
 Schmidt T, Tyson J, Galanulis K (2003) Fullfield dynamic displacement and strain measurement using advanced 3D image correlation photogrammetry: part 1. Exp Tech 27(3): 47–50.View ArticleGoogle Scholar
 Germaneau A, Doumalin P, Dupré JC (2008) Comparison between xray microcomputed tomography and optical scanning tomography for full 3D strain measurement by digital volume correlation. NDT & E Intl 41(6): 407–415.View ArticleGoogle Scholar
 Tezaki A, Mineta T, Egawa H, Noguchi T (1990) Measurement of three dimensional stress and modeling of stress induced migration failure in aluminium interconnects In: Reliability Physics Symposium, 1990. 28th Annual Proceedings., International, 221–229.. IEEE.
 ABAQUS (2000) ABAQUS/standard user’s manual vol. 1. Hibbitt, Karlsson & Sorensen, Pawtucket, RI.Google Scholar
 scikitlearn: machine learning in Python. http://scikitlearn.github.io/. [Online; accessed August 2015].
 Garmestani H, Lin S, Adams B, Ahzi S (2001) Statistical continuum theory for large plastic deformation of polycrystalline materials. J Mech Phys Solids 49(3): 589–607.View ArticleGoogle Scholar
 Saheli G, Garmestani H, Adams B (2004) Microstructure design of a two phase composite using twopoint correlation functions. J Computeraided Mater Des 11(23): 103–115.View ArticleGoogle Scholar
 Fullwood DT, Adams BL, Kalidindi SR (2008) A strong contrast homogenization formulation for multiphase anisotropic materials. J Mech Phys Solids 56(6): 2287–2297.View ArticleGoogle Scholar
 Adams BL, Canova GR, Molinari A (1989) A statistical formulation of viscoplastic behavior in heterogeneous polycrystals. Textures Microstruct 11: 57–71.View ArticleGoogle Scholar
 Kröner E (1977) Bounds for effective elastic moduli of disordered materials. J Mech Phys Solids 25(2): 137–155.View ArticleGoogle Scholar
 Kröner E (1986) Statistical modelling In: Modelling small deformations of polycrystals, 229–291.. Springer, Netherlands.View ArticleGoogle Scholar
 Quinlan JR (1992) Learning with continuous classes In: Proceedings of the 5th Australian Joint Conference on Artificial Intelligence, 343–348.. World Scientific, Singapore.Google Scholar
 Wang Y, Witten IH (1996) Induction of model trees for predicting continuous classes. (Working paper 96/23), Hamilton, New Zealand. University of Waikato, Department of Computer Science.
 Breiman L (2001) Random forests. Mach Learn 45(1): 5–32.View ArticleGoogle Scholar
 Vapnik V (2000) The nature of statistical learning theory. Springer, New York.View ArticleGoogle Scholar
 Smola AJ, Schölkopf B (2004) A tutorial on support vector regression. Stat Comput 14(3): 199–222.View ArticleGoogle Scholar