Research Papers

Sample-Based Synthesis of Functionally Graded Material Structures

[+] Author and Article Information
Xingchen Liu

Spatial Automation Laboratory,
University of Wisconsin-Madison,
Madison, WI 53706,
e-mail: xingchen@wisc.edu

Vadim Shapiro

Spatial Automation Laboratory,
University of Wisconsin-Madison,
Madison, WI 53706
e-mail: vshapiro@wisc.edu

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 25, 2016; final manuscript received April 12, 2017; published online May 19, 2017. Assoc. Editor: Jitesh H. Panchal.

J. Comput. Inf. Sci. Eng 17(3), 031012 (May 19, 2017) (10 pages) Paper No: JCISE-16-2054; doi: 10.1115/1.4036552 History: Received August 25, 2016; Revised April 12, 2017

Spatial variation of material structures is a principal mechanism for creating and controlling spatially varying material properties in nature and engineering. While the spatially varying homogenized properties can be represented by scalar and vector fields on the macroscopic scale, explicit microscopic structures of constituent phases are required to facilitate the visualization, analysis, and manufacturing of functionally graded material (FGM). The challenge of FGM structure modeling lies in the integration of these two scales. We propose to represent and control material properties of FGM at macroscale using the notion of material descriptors, which include common geometric, statistical, and topological measures, such as volume fraction, correlation functions, and Minkowski functionals. At microscale, the material structures are modeled as Markov random fields (MRFs): we formulate the problem of design and (re)construction of FGM structure as a process of selecting neighborhoods from a reference FGM, based on target material descriptors fields. The effectiveness of the proposed method in generating a spatially varying structure of FGM with target properties is demonstrated by two examples: design of a graded bone structure and generating functionally graded lattice structures with target volume fraction fields.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Birman, V. , and Byrd, L. W. , 2007, “ Modeling and Analysis of Functionally Graded Materials and Structures,” ASME Appl. Mech. Rev., 60(5), p. 195. [CrossRef]
Mahamood, R. M. , Member, E. T. A. , Shukla, M. , and Pityana, S. , 2012, “ Functionally Graded Material: An Overview,” World Congress on Engineering, London, UK, July 4–6, Vol. III, pp. 2–6.
Ivar, E. , 2004, “ Functionally Graded Materials,” Handbook of Advanced Materials, K. W. James , ed., pp. 465–486.
Zisis, T. , Kordolemis, A. , and Giannakopoulos, A. E. , 2010, “ Development of Strong Surfaces Using Functionally Graded Composites Inspired by Natural Teeth—Finite Element and Experimental Verification,” ASME J. Eng. Mater. Technol., 132(1), p. 011010. [CrossRef]
Mehboob, H. , and Chang, S.-H. , 2015, “ Optimal Design of a Functionally Graded Biodegradable Composite Bone Plate by Using the Taguchi Method and Finite Element Analysis,” Compos. Struct., 119, pp. 166–173. [CrossRef]
Thamaraiselvi, T. V. , and Rajeswari, S. , 2004, “ Biological Evaluation of Bioceramic Materials: A Review,” Trends Biomater. Artif. Organs, 18(1), pp. 9–17.
Biswas, A. , Shapiro, V. , and Tsukanov, I. , 2004, “ Heterogeneous Material Modeling With Distance Fields,” Comput. Aided Geom. Des., 21(3), pp. 215–242. [CrossRef]
Jaworska, L. , Rozmus, M. , Królicka, B. , and Twardowska, A. , 2006, “ Functionally Graded Cermets,” J. Achiev. Mater. Manuf. Eng., 17(1–2), pp. 73–76.
Silva, E. C. N. , Walters, M. C. , and Paulino, G. H. , 2006, “ Modeling Bamboo as a Functionally Graded Material: Lessons for the Analysis of Affordable Materials,” J. Mater. Sci., 41(21), pp. 6991–7004. [CrossRef]
Nogata, F. , and Takahashi, H. , 1995, “ Intelligent Functionally Graded Material: Bamboo,” Compos. Eng., 5(7), pp. 743–751. [CrossRef]
Park, S.-M. , Crawford, R. H. , and Beaman, J. J. , 2001, “ Volumetric Multi-Texturing for Functionally Gradient Material Representation,” Sixth ACM Symposium on Solid Modeling and Applications (SMA), Ann Arbor, MI, June 4–8, pp. 216–224.
Siu, Y. K. , and Tan, S. T. , 2002, “ ‘Source-Based’ Heterogeneous Solid Modeling,” Comput.-Aided Des., 34(1), pp. 41–55. [CrossRef]
Oxman, N. , Keating, S. , and Tsai, E. , 2011, “ Functionally Graded Rapid Prototyping,” Innovative Developments in Virtual and Physical Prototyping, CRC Press, Boca Raton, FL, pp. 483–489.
Vidimče, K. , Wang, S.-P. , Ragan-Kelley, J. , and Matusik, W. , 2013, “ OpenFab,” ACM Trans. Graphics, 32(4), p. 1. [CrossRef]
Doubrovski, E. , Tsai, E. , Dikovsky, D. , Geraedts, J. , Herr, H. , and Oxman, N. , 2014, “ Voxel-Based Fabrication Through Material Property Mapping: A Design Method for Bitmap Printing,” Comput.-Aided Des., 60, pp. 3–13. [CrossRef]
Seepersad, C. C. , Allen, J. K. , McDowell, D. L. , and Mistree, F. , 2006, “ Robust Design of Cellular Materials With Topological and Dimensional Imperfections,” ASME J. Mech. Des., 128(6), p. 1285. [CrossRef]
Corney, J. , and Torres-Sanchez, C. , 2008, “ Toward Functionally Graded Cellular Microstructures,” ASME J. Mech. Des., 131(9), p. 091011.
Gibson, L. J. , and Ashby, M. F. , 1997, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK.
Hashin, Z. , and Shtrikman, S. , 1963, “ A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials,” J. Mech. Phys. Solids, 11(2), pp. 127–140. [CrossRef]
Moulinec, H. , and Suquet, P. , 1998, “ A Numerical Method for Computing the Overall Response of Nonlinear Composites With Complex Microstructure,” Comput. Methods Appl. Mech. Eng., 157(1–2), pp. 69–94. [CrossRef]
Liu, X. , and Shapiro, V. , 2016, “ Homogenization of Material Properties in Additively Manufactured Structures,” Comput.-Aided Des., 78, pp. 71–82. [CrossRef]
Bendsøe, M. P. , and Sigmund, O. , 1999, “ Material Interpolation Schemes in Topology Optimization,” Arch. Appl. Mech., 69(9–10), pp. 635–654.
Bendsøe, M. P. , 1989, “ Optimal Shape Design as a Material Distribution Problem,” Struct. Optim., 1(4), pp. 193–202. [CrossRef]
Sigmund, O. , 1994, “ Materials With Prescribed Constitutive Parameters: An Inverse Homogenization Problem,” Int. J. Solids Struct., 31(17), pp. 2313–2329. [CrossRef]
Gibson, L. J. , and Ashby, M. F. , 1982, “ The Mechanics of Three-Dimensional Cellular Materials,” Proc. R. Soc. A, 382(1782), pp. 43–59. [CrossRef]
Liu, X. , and Shapiro, V. , 2015, “ Random Heterogeneous Materials Via Texture Synthesis,” Comput. Mater. Sci., 99, pp. 177–189. [CrossRef]
Liu, H. , Maekawa, T. , Patrikalakis, N. M. , Sachs, E. M. , and Cho, W. , 2004, “ Methods for Feature-Based Design of Heterogeneous Solids,” Comput.-Aided Des., 36(12), pp. 1141–1159. [CrossRef]
Huang, J. , Fadel, G. M. , Blouin, V. Y. , and Grujicic, M. , 2002, “ Bi-Objective Optimization Design of Functionally Gradient Materials,” Mater. Des., 23(7), pp. 657–666. [CrossRef]
Pasko, A. , Adzhiev, V. , Schmitt, B. , and Schlick, C. , 2001, “ Constructive Hypervolume Modeling,” Graphical Models, 63(6), pp. 413–442. [CrossRef]
Qian, X. , and Dutta, D. , 2003, “ Design of Heterogeneous Turbine Blade,” Comput.-Aided Des., 35(3), pp. 319–329. [CrossRef]
Kou, X. , and Tan, S. , 2007, “ Heterogeneous Object Modeling: A Review,” Comput.-Aided Des., 39(4), pp. 284–301. [CrossRef]
Hu, Y. , Blouin, V. Y. , and Fadel, G. M. , 2008, “ Design for Manufacturing of 3D Heterogeneous Objects With Processing Time Consideration,” ASME J. Mech. Des., 130(3), p. 031701. [CrossRef]
Cho, J. R. , and Ha, D. Y. , 2002, “ Optimal Tailoring of 2D Volume-Fraction Distributions for Heat-Resisting Functionally Graded Materials Using FDM,” Comput. Methods Appl. Mech. Eng., 191(29–30), pp. 3195–3211. [CrossRef]
Zhang, X. J. , Chen, K. Z. , and Feng, X. A. , 2004, “ Optimization of Material Properties Needed for Material Design of Components Made of Multi-Heterogeneous Materials,” Mater. Des., 25(5), pp. 369–378. [CrossRef]
Panchal, J. H. , Kalidindi, S. R. , and McDowell, D. L. , 2013, “ Key Computational Modeling Issues in Integrated Computational Materials Engineering,” Comput.-Aided Des., 45(1), pp. 4–25. [CrossRef]
Torquato, S. , 2002, Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Interdisciplinary Applied Mathematics: Mechanics and Materials, Springer, New York.
Latief, F. , Biswas, A. , Fauzi, U. , and Hilfer, R. , 2010, “ Continuum Reconstruction of the Pore Scale Microstructure for Fontainebleau Sandstone,” Physica A, 389(8), pp. 1607–1618. [CrossRef]
Redenbach, C., 2009, “ Microstructure Models for Cellular Materials,” Comput. Mater. Sci., 44(4), pp. 1397–1407.
Pasko, A. , Fryazinov, O. , Vilbrandt, T. , Fayolle, P.-A. , and Adzhiev, V. , 2011, “ Procedural Function-Based Modelling of Volumetric Microstructures,” Graphical Models, 73(5), pp. 165–181. [CrossRef]
Chen, Y. , 2007, “ 3D Texture Mapping for Rapid Manufacturing,” Comput.-Aided Des. Appl., 4(1–6), pp. 761–771. [CrossRef]
Kou, X. , and Tan, S. S. , 2010, “ Modeling Functionally Graded Porous Structures With Stochastic Voronoi Diagram and B-Spline Representations,” International Conference on Manufacturing Automation (ICMA), Hong Kong, China, Dec. 13–15, pp. 99–106.
Kou, X. , and Tan, S. , 2010, “ A Simple and Effective Geometric Representation for Irregular Porous Structure Modeling,” Comput.-Aided Des., 42(10), pp. 930–941. [CrossRef]
Yeong, C. , and Torquato, S. , 1998, “ Reconstructing Random Media,” Phys. Rev. E, 58(1), pp. 224–233. [CrossRef]
Jiao, Y. , Stillinger, F. , and Torquato, S. , 2007, “ Modeling Heterogeneous Materials Via Two-Point Correlation Functions: Basic Principles,” Phys. Rev. E, 76(3), pp. 1–15. [CrossRef]
Xu, H. , Li, Y. , Brinson, L. C. , and Chen, W. , 2014, “ A Descriptor-Based Design Methodology for Developing Heterogeneous Microstructural Materials System,” ASME J. Mech. Des., 136(5), p. 051007.
Graham-Brady, L. , and Xu, X. F. , 2008, “ Stochastic Morphological Modeling of Random Multiphase Materials,” ASME J. Appl. Mech., 75(6), p. 061001. [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]
Guo, E. Y. , Chawla, N. , Jing, T. , Torquato, S. , and Jiao, Y. , 2014, “ Accurate Modeling and Reconstruction of Three-Dimensional Percolating Filamentary Microstructures From Two-Dimensional Micrographs Via Dilation-Erosion Method,” Mater. Charact., 89, pp. 33–42. [CrossRef]
Zhou, S. , and Li, Q. , 2008, “ Design of Graded Two-Phase Microstructures for Tailored Elasticity Gradients,” J. Mater. Sci., 43(15), pp. 5157–5167. [CrossRef]
Bostanabad, R. , Chen, W. , and Apley, D. W. , 2016, “ Characterization and Reconstruction of 3D Stochastic Microstructures Via Supervised Learning,” J. Microsc., 264(3), pp. 282–297.
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]
Cang, R. , and Yi, R. , 2016, “ Deep Network-Based Feature Extraction and Reconstruction of Complex Material Microstructures,” ASME Paper No. DETC2016-59404.
Holdstein, Y. , Fischer, A. , Podshivalov, L. , and Bar-Yoseph, P. Z. , 2009, “ Volumetric Texture Synthesis of Bone Micro-Structure as a Base for Scaffold Design,” IEEE International Conference on Shape Modeling and Applications (SMI), Beijing, China, June 26–28, pp. 81–88.
Sundararaghavan, V. , 2014, “ Reconstruction of Three-Dimensional Anisotropic Microstructures From Two-Dimensional Micrographs Imaged on Orthogonal Planes,” Integr. Mater. Manuf. Innovation, 3(1), p. 19. [CrossRef]
Turner, D. M. , and Kalidindi, S. R. , 2016, “ Statistical Construction of 3-D Microstructures From 2-D Exemplars Collected on Oblique Sections,” Acta Mater., 102, pp. 136–148. [CrossRef]
Wei, L.-Y. , and Levoy, M. , 2000, “ Fast Texture Synthesis Using Tree-Structured Vector Quantization,” 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH), New Orleans, LA, July 23–28, pp. 479–488.
Levina, E. , and Bickel, P. J. , 2006, “ Texture Synthesis and Nonparametric Resampling of Random Fields,” Ann. Stat., 34(4), pp. 1751–1773. [CrossRef]
Ashikhmin, M. , 2001, “ Synthesizing Natural Textures,” Symposium on Interactive 3D Graphics, Research Triangle Park, NC, Mar. 19–21, pp. 217–226.
Efros, A. A. , and Freeman, W. T. , 2001, “ Image Quilting for Texture Synthesis and Transfer,” 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH), Los Angeles, CA, Aug. 12–17, Vol. 1, pp. 341–346.
Zhang, J. , Zhou, K. , Velho, L. , Guo, B. , and Shum, H.-Y. , 2003, “ Synthesis of Progressively-Variant Textures on Arbitrary Surfaces,” ACM Trans. Graphics, 22(3), p. 295. [CrossRef]
Zohdi, T. I. , 2013, “ Basic Microstructure-Macroproperty Calculations,” Effective Properties of Heterogeneous Materials SE-5 (Solid Mechanics and Its Applications), M. Kachanov and I. Sevostianov , eds., Vol. 193, Springer, Dordrecht, The Netherlands, pp. 365–389.
Ohser, J. , and Schladitz, K. , 2009, 3D Images of Materials Structures: Processing and Analysis, Wiley, Weinheim, Germany.
Hill, R. , 2002, “ The Elastic Behaviour of a Crystalline Aggregate,” Proc. Phys. Soc., Sect. A, 65(5), pp. 349–354. [CrossRef]
Kalidindi, S. R. , Binci, M. , Fullwood, D. , and Adams, B. L. , 2006, “ Elastic Properties Closures Using Second-Order Homogenization Theories: Case Studies in Composites of Two Isotropic Constituents,” Acta Mater., 54(11), pp. 3117–3126. [CrossRef]
Monetti, R. , Bauer, J. , Sidorenko, I. , Müller, D. , Rummeny, E. , Matsuura, M. , Eckstein, F. , Lochmüller, E.-M. , Zysset, P. , and Räth, C. , 2009, “ Assessment of the Human Trabecular Bone Structure Using Minkowski Functionals,” Medical Imaging 2009: Biomedical Applications in Molecular, Structural, and Functional Imaging (Society of Photo-Optical Instrumentation Engineers), X. P. Hu , and A. V. Clough , eds., Vol. 7262, p. 72620N.
Xu, H. , Dikin, D. , 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]
Kwatra, V. , Schodl, A. , Essa, I. , and Turk, G. , 2003, “ Graphcut Textures: Image and Video Synthesis Using Graph Cuts,” ACM Trans. Graphics, 22(3), pp. 277–286.
Barnes, C. , Shechtman, E. , Finkelstein, A. , and Goldman, D. B. , 2009, “ PatchMatch: A Randomized Correspondence Algorithm for Structural Image Editing,” ACM Trans. Graphics, 28(3), p. 1. [CrossRef]
Barnes, C. , Zhang, F.-L. , Lou, L. , Wu, X. , and Hu, S.-M. , 2015, “ PatchTable: Efficient Patch Queries for Large Datasets and Applications,” ACM Trans. Graphics, 34(4), pp. 97:1–97:10. [CrossRef]
Yuan, J. , Bae, E. , and Tai, X. C. , 2010, “ A Study on Continuous Max-Flow and Min-Cut Approaches,” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), San Francisco, CA, June 13–18, Vol. 1, pp. 2217–2224.
Jiang, C. , Giger, M. L. , Chinander, M. R. , Martell, J. M. , Kwak, S. , and Favus, M. J. , 1999, “ Characterization of Bone Quality Using Computer-Extracted Radiographic Features,” Med. Phys., 26(6), pp. 872–879. [CrossRef] [PubMed]
Hutmacher, D. W. , 2000, “ Scaffolds in Tissue Engineering Bone and Cartilage,” Biomaterials, 21(24), pp. 2529–2543. [CrossRef] [PubMed]
Bose, S. , Roy, M. , and Bandyopadhyay, A. , 2012, “ Recent Advances in Bone Tissue Engineering Scaffolds,” Trends Biotechnol., 30(10), pp. 546–554. [CrossRef] [PubMed]
Fryazinov, O. , Sanchez, M. , and Pasko, A. , 2015, “ Shape Conforming Volumetric Interpolation With Interior Distances,” Comput. Graphics, 46, pp. 149–155. [CrossRef]
Liu, K. , and Tovar, A. , 2014, “ An Efficient 3D Topology Optimization Code Written in Matlab,” Struct. Multidiscip. Optim., 50(6), pp. 1175–1196. [CrossRef]
Darabi, S. , Shechtman, E. , Barnes, C. , Goldman, D. B. , and Sen, P. , 2012, “ Image Melding: Combining Inconsistent Images Using Patch-Based Synthesis,” ACM Trans. Graphics, 31(4), pp. 1–10. [CrossRef]
Stoyan, D. , and Mecke, K. R. , 2005, “ The Boolean Model: From Matheron Till Today,” Space, Structure and Randomness, Springer, New York, pp. 151–181.


Grahic Jump Location
Fig. 1

Key problem: construct a material structure Y with target effective material properties distribution P(Y), given a material structure X with effective properties distributions P(X). X and Y are usually represented by some microscopic models. Map h evaluates the effective properties of X and Y. Map g represents gradation techniques to design the target properties fields on macroscopic scale. Map f symbolizes the integration of macroscopic and microscopic representations.

Grahic Jump Location
Fig. 2

Synthesis of graded material structures are reformulated in terms of neighborhoods

Grahic Jump Location
Fig. 3

Problem formulation via material descriptors

Grahic Jump Location
Fig. 4

Synthesis of Y neighborhood by neighborhood in steps of s–o. For this illustration, s = 7 is the neighborhood size, o = 2 is the size of overlap. Numbers in the center of the neighborhoods indicate the order of the synthesis. The piecewise linear curve illustrates the hypersurface that separates the overlap regions in two parts while minimizing the accumulated mismatches along the boundary. Two piecewise linear curves represent the hypersurfaces separating neighborhood 5 and 7 from synthesized material structure, respectively.

Grahic Jump Location
Fig. 5

Femur bone structure reconstruction (a) shows a cross section of a two-phase femur bone structure, (b)–(d) are Minkowski functionals fields of (a), (e) and (f) are reconstruction results without and with Minkowski functionals as descriptors fields. Masks are used to stop the algorithm from scanning black regions surrounding the bone. (a) Two-phase femur bone structure, (b) perimeter field, (c) area field, (d) Euler characteristic field, (e) reconstruction without Minkowski functionals fields, and (f) reconstruction with Minkowski functionals fields.

Grahic Jump Location
Fig. 6

Synthesis of missing femur bone structures (a) shows the Femur bone with missing bone structures, (b)–(d) are target Minkowski functional fields designed by inverse distance interpolation from the healthy bone regions, (e) and (f) show the synthesized bone structures different target descriptors fields. (a) Femur bone with the deteriorated region removed, (b) interpolated perimeter field, (c) interpolated area field, (d) interpolated Euler characteristic field, (e) synthesis of missing bone structure with Minkowski functionals field from original bone sample (Fig. 5(c)), and (f) synthesis of missing bone structure with Minkowski functionals field interpolated from existing structures.

Grahic Jump Location
Fig. 7

Synthesis of graded material structure that is functionally a cantilever beam with fixtures on the top and bottom left edges and downward loads on the bottom right edge. The target volume fraction field ranging from 0.3 to 0.85 is designed by SIMP with penalty factor equals 1. (a) Target volume fraction distribution from 0.3 to 0.85, 30 × 16 × 8, (b) reference material, 80 × 80 × 80, (c) FGM structure synthesized by the proposed framework, 300 × 160 × 80, and (d) FGM structure generated by procedural method, 300 × 160 × 80.

Grahic Jump Location
Fig. 8

Functionally graded lattice structures for a cantilever beam with different unit cells: (a) Type 1 unit cell, (b) type 2 unit cell, (c) FGM structure with type 1 unit cell, and (d) FGM structure with type 2 unit cell

Grahic Jump Location
Fig. 9

Synthesis of graded material structure that is functionally a wheel structure with fixtures on four lower corners and load on the center of bottom surface. The target volume fraction field ranging from 0.3 to 0.85 is designed by SIMP with penalty factor equals 1. (a) Volume fraction distribution, 20 × 20 × 10: view from top, (b) volume fraction distribution: view from bottom, (c) reference functionally graded lattice, 100 × 100 × 100, (d) FGM structure synthesized by the proposed framework: top, Resolution: 200 × 200 × 100, and (e) FGM structure synthesized by the proposed framework: bottom.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In