Research Papers: Design Automation

Microstructure Representation and Reconstruction of Heterogeneous Materials Via Deep Belief Network for Computational Material Design

[+] Author and Article Information
Ruijin Cang, Yongming Liu, Max Yi Ren

Department of Mechanical Engineering,
Arizona State University,
Tempe, AZ 85287

Yaopengxiao Xu, Shaohua Chen, Yang Jiao

Department of Materials
Science and Engineering,
Arizona State University,
Tempe, AZ 85287

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 28, 2016; final manuscript received April 20, 2017; published online May 19, 2017. Assoc. Editor: Carolyn Seepersad.

J. Mech. Des 139(7), 071404 (May 19, 2017) (11 pages) Paper No: MD-16-1674; doi: 10.1115/1.4036649 History: Received September 28, 2016; Revised April 20, 2017

Integrated Computational Materials Engineering (ICME) aims to accelerate optimal design of complex material systems by integrating material science and design automation. For tractable ICME, it is required that (1) a structural feature space be identified to allow reconstruction of new designs, and (2) the reconstruction process be property-preserving. The majority of existing structural presentation schemes relies on the designer's understanding of specific material systems to identify geometric and statistical features, which could be biased and insufficient for reconstructing physically meaningful microstructures of complex material systems. In this paper, we develop a feature learning mechanism based on convolutional deep belief network (CDBN) to automate a two-way conversion between microstructures and their lower-dimensional feature representations, and to achieve a 1000-fold dimension reduction from the microstructure space. The proposed model is applied to a wide spectrum of heterogeneous material systems with distinct microstructural features including Ti–6Al–4V alloy, Pb63–Sn37 alloy, Fontainebleau sandstone, and spherical colloids, to produce material reconstructions that are close to the original samples with respect to two-point correlation functions and mean critical fracture strength. This capability is not achieved by existing synthesis methods that rely on the Markovian assumption of material microstructures.

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Sharma, V. , Wang, C. , Lorenzini, R. G. , Ma, R. , Zhu, Q. , Sinkovits, D. W. , Pilania, G. , Oganov, A. R. , Kumar, S. , and Sotzing, G. A. , 2014, “ Rational Design of All Organic Polymer Dielectrics,” Nat. Commun., 5, p. 4845.
Baldwin, A. F. , Huan, T. D. , Ma, R. , Mannodi-Kanakkithodi, A. , Tefferi, M. , Katz, N. , Cao, Y. , Ramprasad, R. , and Sotzing, G. A. , 2015, “ Rational Design of Organotin Polyesters,” Macromolecules, 48(8), pp. 2422–2428. [CrossRef]
Ma, R. , Sharma, V. , Baldwin, A. F. , Tefferi, M. , Offenbach, I. , Cakmak, M. , Weiss, R. , Cao, Y. , Ramprasad, R. , and Sotzing, G. A. , 2015, “ Rational Design and Synthesis of Polythioureas as Capacitor Dielectrics,” J. Mater. Chem. A, 3(28), pp. 14845–14852. [CrossRef]
Kalidindi, S. R. , and De Graef, M. , 2015, “ Materials Data Science: Current Status and Future Outlook,” Annu. Rev. Mater. Res., 45(1), pp. 171–193. [CrossRef]
Kaczmarowski, A. , Yang, S. , Szlufarska, I. , and Morgan, D. , 2015, “ Genetic Algorithm Optimization of Defect Clusters in Crystalline Materials,” Comput. Mater. Sci., 98, pp. 234–244. [CrossRef]
Kirklin, S. , Saal, J. E. , Hegde, V. I. , and Wolverton, C. , 2016, “ High-Throughput Computational Search for Strengthening Precipitates in Alloys,” Acta Mater., 102, pp. 125–135. [CrossRef]
Xu, H. , Dikin, D. A. , Burkhart, C. , and Chen, W. , 2014, “ Descriptor-Based Methodology for Statistical Characterization and 3D Reconstruction of Microstructural Materials,” Comput. Mater. Sci., 85, pp. 206–216. [CrossRef]
Xu, H. , Liu, R. , Choudhary, A. , and Chen, W. , 2015, “ A Machine Learning-Based Design Representation Method for Designing Heterogeneous Microstructures,” ASME J. Mech. Des., 137(5), p. 051403. [CrossRef]
Dosovitskiy, A. , and Brox, T. , 2015, “ Inverting Visual Representations With Convolutional Networks,” eprint arXiv:1506.02753.
Mahendran, A. , and Vedaldi, A. , 2015, “ Understanding Deep Image Representations by Inverting Them,” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, June 7–12, pp. 5188–5196.
Nguyen, A. , Yosinski, J. , and Clune, J. , 2015, “ Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images,” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, June 7–12, pp. 427–436.
Krizhevsky, A. , Sutskever, I. , and Hinton, G. E. , 2012, “ ImageNet Classification With Deep Convolutional Neural Networks,” Advances in Neural Information Processing Systems 25, F. Pereira , C. J. C. Burges , L. Bottou , and K. Q. Weinberger , eds., Curran Associates, Lake Tahoe, NV, pp. 1097–1105.
Simonyan, K. , and Zisserman, A. , 2014, “ Very Deep Convolutional Networks for Large-Scale Image Recognition,” eprint arXiv:1409.1556.
Yan, X. , Yang, J. , Sohn, K. , and Lee, H. , 2015, “ Attribute2Image: Conditional Image Generation From Visual Attributes,” eprint arXiv:1512.00570.
McDowell, D. L. , and Olson, G. B. , 2008, “ Concurrent Design of Hierarchical Materials and Structures,” Sci. Model. Simul., 15, pp. 207–240. [CrossRef]
Broderick, S. , Suh, C. , Nowers, J. , Vogel, B. , Mallapragada, S. , Narasimhan, B. , and Rajan, K. , 2008, “ Informatics for Combinatorial Materials Science,” JOM, 60(3), pp. 56–59. [CrossRef]
Ashby, M. F. , and Cebon, D. , 1993, “ Materials Selection in Mechanical Design,” J. Phys. IV, 3(C7), pp. C7-1–C7-9.
Karasek, L. , and Sumita, M. , 1996, “ Characterization of Dispersion State of Filler and Polymer-Filler Interactions in Rubber Carbon Black Composites,” Mater. Sci., 31(2), pp. 281–289. [CrossRef]
Rollett, A. D. , Lee, S.-B. , Campman, R. , and Rohrer, G. , 2007, “ Three-Dimensional Characterization of Microstructure by Electron Back-Scatter Diffraction,” Annu. Rev. Mater. Res., 37(1), pp. 627–658. [CrossRef]
Borbely, A. , Csikor, F. , Zabler, S. , Cloetens, P. , and Biermann, H. , 2004, “ Three-Dimensional Characterization of the Microstructure of a Metal–Matrix Composite by Holotomography,” Mater. Sci. Eng. A, 367(1), pp. 40–50. [CrossRef]
Tewari, A. , and Gokhale, A. , 2004, “ Nearest-Neighbor Distances Between Particles of Finite Size in Three-Dimensional Uniform Random Microstructures,” Mater. Sci. Eng. A, 385(1), pp. 332–341. [CrossRef]
Pytz, R. , 2004, “ Microstructure Description of Composites, Statistical Methods,” Mechanics of Microstructure Materials (Courses and Lectures), Springer, Vienna, Austria.
Steinzig, M. , and Harlow, F. , 1999, “ Probability Distribution Function Evolution for Binary Alloy Solidification,” Minerals, Metals, Materials Society Annual Meeting, Citeseer, San Diego, CA, pp. 197–206.
Scalon, J. , Fieller, N. , Stillman, E. , and Atkinson, H. , 2003, “ Spatial Pattern Analysis of Second-Phase Particles in Composite Materials,” Mater. Sci. Eng. A, 356(1), pp. 245–257. [CrossRef]
Torquato, S. , 2013, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Vol. 16, Springer Science & Business Media, New York.
Sundararaghavan, V. , and Zabaras, N. , 2005, “ Classification and Reconstruction of Three-Dimensional Microstructures Using Support Vector Machines,” Comput. Mater. Sci., 32(2), pp. 223–239. [CrossRef]
Basanta, D. , Miodownik, M. A. , Holm, E. A. , and Bentley, P. J. , 2005, “ Using Genetic Algorithms to Evolve Three-Dimensional Microstructures From Two-Dimensional Micrographs,” Metall. Mater. Trans. A, 36(7), pp. 1643–1652. [CrossRef]
Holotescu, S. , and Stoian, F. , 2011, “ Prediction of Particle Size Distribution Effects on Thermal Conductivity of Particulate Composites,” Materialwiss. Werkstofftech., 42(5), pp. 379–385. [CrossRef]
Klaysom, C. , Moon, S.-H. , Ladewig, B. P. , Lu, G. M. , and Wang, L. , 2011, “ The Effects of Aspect Ratio of Inorganic Fillers on the Structure and Property of Composite Ion-Exchange Membranes,” J. Colloid Interface Sci., 363(2), pp. 431–439. [CrossRef] [PubMed]
Gruber, J. , Rollett, A. , and Rohrer, G. , 2010, “ Misorientation Texture Development During Grain Growth—Part II: Theory,” Acta Mater., 58(1), pp. 14–19. [CrossRef]
Liu, Y. , Greene, M. S. , Chen, W. , Dikin, D. A. , and Liu, W. K. , 2013, “ Computational Microstructure Characterization and Reconstruction for Stochastic Multiscale Material Design,” Comput.-Aided Des., 45(1), pp. 65–76. [CrossRef]
Quiblier, J. A. , 1984, “ A New Three-Dimensional Modeling Technique for Studying Porous Media,” J. Colloid Interface Sci., 98(1), pp. 84–102. [CrossRef]
Jiang, Z. , Chen, W. , and Burkhart, C. , 2013, “ Efficient 3D Porous Microstructure Reconstruction Via Gaussian Random Field and Hybrid Optimization,” J. Microsc., 252(2), pp. 135–148. [CrossRef] [PubMed]
Grigoriu, M. , 2003, “ Random Field Models for Two-Phase Microstructures,” J. Appl. Phys., 94(6), pp. 3762–3770. [CrossRef]
Roberts, A. P. , 1997, “ Statistical Reconstruction of Three-Dimensional Porous Media From Two-Dimensional Images,” Phys. Rev. E, 56(3), p. 3203. [CrossRef]
Yeong, C. , and Torquato, S. , 1998, “ Reconstructing Random Media,” Phys. Rev. E, 57(1), p. 495. [CrossRef]
Jiao, Y. , Stillinger, F. , and Torquato, S. , 2008, “ Modeling Heterogeneous Materials Via Two-Point Correlation Functions—II: Algorithmic Details and Applications,” Phys. Rev. E, 77(3), p. 031135. [CrossRef]
Jiao, Y. , Stillinger, F. , and Torquato, S. , 2009, “ A Superior Descriptor of Random Textures and Its Predictive Capacity,” Proc. Natl. Acad. Sci., 106(42), pp. 17634–17639. [CrossRef]
Karsanina, M. V. , Gerke, K. M. , Skvortsova, E. B. , and Mallants, D. , 2015, “ Universal Spatial Correlation Functions for Describing and Reconstructing Soil Microstructure,” PloS One, 10(5), p. e0126515. [CrossRef] [PubMed]
Fullwood, D. , Kalidindi, S. , Niezgoda, S. , Fast, A. , and Hampson, N. , 2008, “ Gradient-Based Microstructure Reconstructions From Distributions Using Fast Fourier Transforms,” Mater. Sci. Eng. A, 494(1), pp. 68–72. [CrossRef]
Fullwood, D. T. , Niezgoda, S. R. , and Kalidindi, S. R. , 2008, “ Microstructure Reconstructions From 2-Point Statistics Using Phase-Recovery Algorithms,” Acta Mater., 56(5), pp. 942–948. [CrossRef]
Okabe, H. , and Blunt, M. J. , 2005, “ Pore Space Reconstruction Using Multiple-Point Statistics,” J. Pet. Sci. Eng., 46(1), pp. 121–137. [CrossRef]
Hajizadeh, A. , Safekordi, A. , and Farhadpour, F. A. , 2011, “ A Multiple-Point Statistics Algorithm for 3D Pore Space Reconstruction From 2D Images,” Adv. Water Resour., 34(10), pp. 1256–1267. [CrossRef]
Matthews, J. , Klatt, T. , Morris, C. , Seepersad, C. C. , Haberman, M. , and Shahan, D. , 2016, “ Hierarchical Design of Negative Stiffness Metamaterials Using a Bayesian Network Classifier,” ASME J. Mech. Des., 138(4), p. 041404. [CrossRef]
Tahmasebi, P. , and Sahimi, M. , 2013, “ Cross-Correlation Function for Accurate Reconstruction of Heterogeneous Media,” Phys. Rev. Lett., 110(7), p. 078002. [CrossRef] [PubMed]
Tahmasebi, P. , and Sahimi, M. , 2015, “ Reconstruction of Nonstationary Disordered Materials and Media: Watershed Transform and Cross-Correlation Function,” Phys. Rev. E, 91(3), p. 032401. [CrossRef]
Liu, X. , and Shapiro, V. , 2015, “ Random Heterogeneous Materials Via Texture Synthesis,” Comput. Mater. Sci., 99, pp. 177–189. [CrossRef]
Bostanabad, R. , Bui, A. T. , Xie, W. , Apley, D. W. , and Chen, W. , 2016, “ Stochastic Microstructure Characterization and Reconstruction Via Supervised Learning,” Acta Mater., 103, pp. 89–102. [CrossRef]
Hinton, G. , Deng, L. , Yu, D. , Dahl, G. E. , Mohamed, A.-R. , Jaitly, N. , Senior, A. , Vanhoucke, V. , Nguyen, P. , and Sainath, T. N. , 2012, “ Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups,” IEEE Signal Process. Mag., 29(6), pp. 82–97. [CrossRef]
Mnih, V. , Kavukcuoglu, K. , Silver, D. , Graves, A. , Antonoglou, I. , Wierstra, D. , and Riedmiller, M. , 2013, “ Playing Atari With Deep Reinforcement Learning,” eprint arXiv:1312.5602.
Schmidhuber, J. , 2015, “ Deep Learning in Neural Networks: An Overview,” Neural Networks, 61, pp. 85–117. [CrossRef] [PubMed]
Levine, S. , Finn, C. , Darrell, T. , and Abbeel, P. , 2015, “ End-to-End Training of Deep Visuomotor Policies,” eprint arXiv:1504.00702.
Reed, S. E. , Zhang, Y. , Zhang, Y. , and Lee, H. , 2015, “ Deep Visual Analogy-Making,” Advances in Neural Information Processing Systems, Curran Associates, Inc., Boston, MA, pp. 1252–1260.
Lee, H. , Grosse, R. , Ranganath, R. , and Ng, A. Y. , 2009, “ Convolutional Deep Belief Networks for Scalable Unsupervised Learning of Hierarchical Representations,” 26th Annual International Conference on Machine Learning (ICML), Montreal, QC, Canada, June 14–18, pp. 609–616.
Bousquet, O. , and Bottou, L. , 2008, “ The Tradeoffs of Large Scale Learning,” Proceedings of the 20th International Conference on Neural Information Processing Systems, Vancouver, BC, Canada, Dec. 3–6, pp. 161–168.
Rumelhart, D. E. , Hinton, G. E. , and Williams, R. J. , 1988, “ Learning Representations by Back-Propagating Errors,” Cognit. Model., 5(3), p. 1.
Hinton, G. E. , 2002, “ Training Products of Experts by Minimizing Contrastive Divergence,” Neural Comput., 14(8), pp. 1771–1800. [CrossRef] [PubMed]
Bengio, Y. , 2009, “ Learning Deep Architectures for AI,” Found. Trends Mach. Learn., 2(1), pp. 1–127. [CrossRef]
Yumer, M. E. , Asente, P. , Mech, R. , and Kara, L. B. , 2015, “ Procedural Modeling Using Autoencoder Networks,” 28th Annual ACM Symposium on User Interface Software and Technology (UIST), Charlotte, NC, Nov. 11–15, pp. 109–118.
Kingma, D. P. , and Welling, M. , 2013, “ Auto-Encoding Variational Bayes,” eprint arXiv:1312.6114.
Goodfellow, I. , Pouget-Abadie, J. , Mirza, M. , Xu, B. , Warde-Farley, D. , Ozair, S. , Courville, A. , and Bengio, Y. , 2014, “ Generative Adversarial Nets,” Advances in Neural Information Processing Systems, Curran Associates, Inc., Boston, MA, pp. 2672–2680. [PubMed] [PubMed]
Sohn, K. , and Lee, H. , 2012, “ Learning Invariant Representations With Local Transformations,” eprint arXiv:1206.6418.
van der Walt, S., Schönberger, J. L., Nunez-Iglesias, J., Boulogne, F., Warner, J. D., Yager, N., Gouillart, E., Yu, T., and The Scikit-Image Contributors, 2014, “ Scikit-Image: Image Processing in Python,” Peer J., 2, p. e453.
Jiao, Y. , Padilla, E. , and Chawla, N. , 2013, “ Modeling and Predicting Microstructure Evolution in Lead/Tin Alloy Via Correlation Functions and Stochastic Material Reconstruction,” Acta Mater., 61(9), pp. 3370–3377. [CrossRef]
Li, H. , Kaira, S. , Mertens, J. , Chawla, N. , and Jiao, Y. , 2016, “ Accurate Stochastic Reconstruction of Heterogeneous Microstructures by Limited X-ray Tomographic Projections,” J. Microsc., 264(3), pp. 339–350. [CrossRef] [PubMed]
Li, H. , Chawla, N. , and Jiao, Y. , 2014, “ Reconstruction of Heterogeneous Materials Via Stochastic Optimization of Limited-Angle X-ray Tomographic Projections,” Scr. Mater., 86, pp. 48–51. [CrossRef]
Chen, H. , Lin, E. , Jiao, Y. , and Liu, Y. , 2014, “ A Generalized 2D Non-Local Lattice Spring Model for Fracture Simulation,” Comput. Mech., 54(6), pp. 1541–1558. [CrossRef]


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Fig. 1

(a) Samples and random synthesis results of material systems that are assumed to be Markovian [48]. (b) and (c) Random synthesis of the Ti–6Al–4V alloy microstructure following the synthesis algorithm from Ref.[48]. The synthesis is based on (b) a single sample and (c) 100 samples from Fig. 14. Image courtesy of Dr.Ramin Bostanabad.

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Fig. 2

(a) Three layers of human face features extracted at increasing length scales [54] and (b) a schematic comparison between an RBM and an autoencoder. ai (bj) and vi (hj) are the bias and state values of visible (hidden) layer, respectively. Wij indicates the weights between the visible and hidden layers.

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Fig. 3

(a) CRBM and pooling layers and (b) forward and backward probabilistic max-pooling procedures with 2 × 2 blocks

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Fig. 4

Input images and extracted features. Visualizations of filters for the five layers: The first layer has two filters, each with 12 orientations (see Sec. 3.1). Due to limited space, only the first 144 and 25 filters from the third and fourth layers, respectively, are shown.

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Fig. 9

A comparison between random reconstructions (bottom) and the original samples (top) for four different material systems

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Fig. 8

(a) A threshold of 0.5 does not always guarantee valid reconstructions, see the second row (b) threshold can be fine-tuned based on a heuristic criterion: Here, we test a set of threshold values and pick the one that yields a volume fraction closest to the average volume fraction of reconstructions of the original samples (ρ = 0.51)

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Fig. 7

Postprocessing steps: Column (a) shows the original reconstruction and its third layer activations (288 channels) with enlarged sample channels; Column (b) shows the reconstruction after thresholding the third layer activations at 0.5; Column (c) shows the further improved reconstruction after skeletonization

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Fig. 6

Comparison between random reconstructions from (a) the fourth and (b) the fifth layers, sampled from the corresponding design spaces ({0, 1}1000 in (a) and {0, 1}30 in (b))

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Fig. 5

Activation on different filters. From left to right, reconstructions when only the first, only the second, and both nodes from the fifth layer are activated, respectively.

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Fig. 10

Comparison between the samples (top) and their reconstructions (bottom). Details of the two fail to match.

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Fig. 11

The two-point correlation functions for the four different materials

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Fig. 12

(a) Comparison on critical fracture strength among the four material systems between the original samples (black), their corresponding reconstructions (gray), and random reconstructions (white) and (b) comparison on critical fracture strength among individual original images and the related reconstruction images (see figure online for color)

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Fig. 13

(a)–(c) Sample reconstructions (without skeletonization) based on models trained from 10, 50, and 100 samples, respectively. The variances among random reconstructions for these three cases are 0.2644, 0.2863, and 0.4205, respectively. The variance is calculated from 100 reconstructions in each case.

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Fig. 14

Ti–6Al–4V alloy

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Fig. 15

Pb–Sn (lead–tin) alloy

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Fig. 16

Pore structure of Fontainebleau sandstone

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Fig. 17

Two-dimensional suspension of spherical colloids



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