Research Papers

Additive Manufacturing-Oriented Design of Graded Lattice Structures Through Explicit Topology Optimization

[+] Author and Article Information
Chang Liu, Zongliang Du, Weisheng Zhang, Yichao Zhu

State Key Laboratory of Structural Analysis for
Industrial Equipment,
Department of Engineering Mechanics,
International Research Center for
Computational Mechanics,
Dalian University of Technology,
Dalian 116023, China

Xu Guo

State Key Laboratory of Structural Analysis for
Industrial Equipment, Department of
Engineering Mechanics,
International Research Center for
Computational Mechanics,
Dalian University of Technology,
Dalian 116023, China
e-mail: guoxu@dlut.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 24, 2017; final manuscript received May 29, 2017; published online June 16, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(8), 081008 (Jun 16, 2017) (12 pages) Paper No: JAM-17-1271; doi: 10.1115/1.4036941 History: Received May 24, 2017; Revised May 29, 2017

In the present work, a new approach for designing graded lattice structures is developed under the moving morphable components/voids (MMC/MMV) topology optimization framework. The essential idea is to make a coordinate perturbation to the topology description functions (TDF) that are employed for the description of component/void geometries in the design domain. Then, the optimal graded structure design can be obtained by optimizing the coefficients in the perturbed basis functions. Our numerical examples show that the proposed approach enables a concurrent optimization of both the primitive cell and the graded material distribution in a straightforward and computationally effective way. Moreover, the proposed approach also shows its potential in finding the optimal configuration of complex graded lattice structures with a very small number of design variables employed under various loading conditions and coordinate systems.

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


Lakes, R. , 1993, “ Materials With Structural Hierarchy,” Nature, 361(6412), pp. 511–515. [CrossRef]
Sigmund, O. , and Torquato, S. , 1996, “ Composites With Extremal Thermal Expansion Coefficients,” Appl. Phys. Lett., 69(21), pp. 3203–3205. [CrossRef]
Knight, J. C. , 2003, “ Photonic Crystal Fibres,” Nature, 424(6950), pp. 847–851. [CrossRef] [PubMed]
Kushwaha, M. S. , Halevi, P. , Dobrzynski, L. , and Djafari-Rouhani, B. , 1993, “ Acoustic Band Structure of Periodic Elastic Composites,” Phys. Rev. Lett., 71(13), pp. 2022–2025. [CrossRef] [PubMed]
Liu, C. , Du, Z. L. , Sun, Z. , Gao, H. J. , and Guo, X. , 2015, “ Frequency-Preserved Acoustic Diode Model With High Forward-Power-Transmission Rate,” Phys. Rev. Appl., 3(6), p. 064014. [CrossRef]
Zheng, X. , Lee, H. , Weisgraber, T. H. , Shusteff, M. , DeOtte, J. , Duoss, E. B. , Kuntz, J. D. , Biener, M. M. , Ge, Q. , Jackson, J. A. , Kucheyev, S. O. , Fang, N. X. , and Spadaccini, C. M. , 2014, “ Ultralight, Ultrastiff Mechanical Metamaterials,” Science, 344(6190), pp. 1373–1377. [CrossRef] [PubMed]
Kim, T. , Hodson, H. P. , and Lu, T. J. , 2004, “ Fluid-Flow and Endwall Heat-Transfer Characteristics of an Ultralight Lattice-Frame Material,” Int. J. Heat Mass Transfer, 47(6–7), pp. 1129–1140. [CrossRef]
Zheng, J. , Zhao, L. , and Fan, H. , 2012, “ Energy Absorption Mechanisms of Hierarchical Woven Lattice Composites,” Compos. Part B: Eng., 43(3), pp. 1516–1522. [CrossRef]
Vasiliev, V. V. , and Razin, A. F. , 2006, “ Anisogrid Composite Lattice Structures for Spacecraft and Aircraft Applications,” Compos. Struct., 76(1–2), pp. 182–189. [CrossRef]
Zheludev, N. I. , and Kivshar, Y. S. , 2012, “ From Metamaterials to Metadevices,” Nat. Mater., 11(11), pp. 917–924. [CrossRef] [PubMed]
Bendsøe, M. P. , and Kikuchi, N. , 1988, “ Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 197–224. [CrossRef]
Rodrigues, H. , Guedes, J. M. , and Bendsøe, M. P. , 2002, “ Hierarchical Optimization of Material and Structure,” Struct. Multidisciplinary Optim., 24(1), pp. 1–10. [CrossRef]
Coelho, P. G. , Fernandes, P. R. , Guedes, J. M. , and Rodrigues, H. C. , 2007, “ A Hierarchical Model for Concurrent Material and Topology Optimisation of Three-Dimensional Structures,” Struct. Multidiscip. Optim., 35(2), pp. 107–115. [CrossRef]
Coelho, P. G. , Guedes, J. M. , and Rodrigues, H. C. , 2015, “ Multiscale Topology Optimization of Bi-Material Laminated Composite Structures,” Compos. Struct., 132, pp. 495–505. [CrossRef]
Liu, L. , Yan, J. , and Cheng, G. , 2008, “ Optimum Structure With Homogeneous Optimum Truss-Like Material,” Comput. Struct., 86(13–14), pp. 1417–1425. [CrossRef]
Niu, B. , Yan, J. , and Cheng, G. , 2009, “ Optimum Structure With Homogeneous Optimum Cellular Material for Maximum Fundamental Frequency,” Struct. Multidiscip. Optim., 39(2), pp. 115–132. [CrossRef]
Deng, J. , Yan, J. , and Cheng, G. , 2012, “ Multi-Objective Concurrent Topology Optimization of Thermoelastic Structures Composed of Homogeneous Porous Material,” Struct. Multidiscip. Optim., 47(4), pp. 583–597. [CrossRef]
Yan, J. , Guo, X. , and Cheng, G. , 2016, “ Multi-Scale Concurrent Material and Structural Design Under Mechanical and Thermal Loads,” Comput. Mech., 57(3), pp. 437–446. [CrossRef]
Guo, X. , Zhao, X. , Zhang, W. , Yan, J. , and Sun, G. , 2015, “ Multi-Scale Robust Design and Optimization Considering Load Uncertainties,” Comput. Methods Appl Mech. Eng., 283, pp. 994–1009. [CrossRef]
Sivapuram, R. , Dunning, P. D. , and Kim, H. A. , 2016, “ Simultaneous Material and Structural Optimization by Multiscale Topology Optimization,” Struct. Multidiscip. Optim., 54(5), pp. 1267–1281. [CrossRef]
Tan, T. , Rahbar, N. , Allameh, S. M. , Kwofie, S. , Dissmore, D. , Ghavami, K. , and Soboyejo, W. O. , 2011, “ Mechanical Properties of Functionally Graded Hierarchical Bamboo Structures,” Acta Biomater., 7(10), pp. 3796–3803. [CrossRef] [PubMed]
Rho, J.-Y. , Kuhn-Spearing, L. , and Zioupos, P. , 1998, “ Mechanical Properties and the Hierarchical Structure of Bone,” Med. Eng. Phys., 20(2), pp. 92–102. [CrossRef] [PubMed]
Gibson, L. J. , 2012, “ The Hierarchical Structure and Mechanics of Plant Materials,” J. R. Soc. Interface, 9(76), pp. 2749–2766. [CrossRef] [PubMed]
Norris, A. N. , 2008, “ Acoustic Cloaking Theory,” Proc. R. Soc. A: Math. Phys. Eng. Sci., 464(2097), pp. 2411–2434. [CrossRef]
Maldovan, M. , 2013, “ Sound and Heat Revolutions in Phononics,” Nature, 503(7475), pp. 209–217. [CrossRef] [PubMed]
Conway, J. H., Burgiel, H., and Goodman-Strauss, C., 2016, The Symmetries of Things, CRC Press, Boca Raton, FL, p. 218.
M. C. Escher, “M. C. Escher,” The M. C. Escher Company B.V., Baarn, The Netherlands, accessed June 13, 2017, http://www.mcescher.com/
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]
Lin, D. , Li, Q. , Li, W. , Zhou, S. , and Swain, M. V. , 2009, “ Design Optimization of Functionally Graded Dental Implant for Bone Remodeling,” Compos. Part B: Eng., 40(7), pp. 668–675. [CrossRef]
Radman, A. , Huang, X. , and Xie, Y. M. , 2013, “ Topology Optimization of Functionally Graded Cellular Materials,” J. Mater. Sci., 48(4), pp. 1503–1510. [CrossRef]
Radman, A. , Huang, X. , and Xie, Y. M. , 2014, “ Maximizing Stiffness of Functionally Graded Materials With Prescribed Variation of Thermal Conductivity,” Comput. Mater. Sci., 82, pp. 457–463. [CrossRef]
Wang, Y. , Chen, F. , and Wang, M. Y. , 2017, “ Concurrent Design With Connectable Graded Microstructures,” Comput. Methods Appl. Mech. Eng., 317, pp. 84–101. [CrossRef]
Bendsøe, M. P. , 1989, “ Optimal Shape Design as a Material Distribution Problem,” Struct. Optim., 1(4), pp. 193–202. [CrossRef]
Bendsøe, M. P. , Guedes, J. M. , Haber, R. B. , Pedersen, P. , and Taylor, J. E. , 1994, “ An Analytical Model to Predict Optimal Material Properties in the Context of Optimal Structural Design,” ASME J. Appl. Mech., 61(4), pp. 930–937. [CrossRef]
Mlejnek, H. P. , 1992, “ Some Aspects of the Genesis of Structures,” Struct. Optim., 5(1–2), pp. 64–69. [CrossRef]
Zhou, M. , and Rozvany, G. I. N. , 1991, “ The COC Algorithm, Part II: Topological, Geometrical and Generalized Shape Optimization,” Comput. Methods Appl. Mech. Eng., 89(1–3), pp. 309–336. [CrossRef]
Allaire, G. , Jouve, F. , and Toader, A.-M. , 2004, “ Structural Optimization Using Sensitivity Analysis and a Level-Set Method,” J. Comput. Phys., 194(1), pp. 363–393. [CrossRef]
Wang, M. Y. , Wang, X. , and Guo, D. , 2003, “ A Level Set Method for Structural Topology Optimization,” Comput. Methods Appl. Mech. Eng., 192(1–2), pp. 227–246. [CrossRef]
Guo, X. , Zhang, W. , and Zhong, W. , 2014, “ Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework,” ASME J. Appl. Mech., 81(8), p. 081009. [CrossRef]
Zhang, W. , Yang, W. , Zhou, J. , Li, D. , and Guo, X. , 2016, “ Structural Topology Optimization Through Explicit Boundary Evolution,” ASME J. Appl. Mech., 84(1) p. 011011.
Guo, X. , Zhang, W. , Zhang, J. , and Yuan, J. , 2016, “ Explicit Structural Topology Optimization Based on Moving Morphable Components (MMC) With Curved Skeletons,” Comput. Methods Appl. Mech. Eng., 310, pp. 711–748. [CrossRef]
Zhang, W. , Li, D. , Yuan, J. , Song, J. , and Guo, X. , 2017, “ A New Three-Dimensional Topology Optimization Method Based on Moving Morphable Components (MMCs),” Comput. Mech., 59(4), pp. 647–665. [CrossRef]
Zhang, W. , Yuan, J. , Zhang, J. , and Guo, X. , 2015, “ A New Topology Optimization Approach Based on Moving Morphable Components (MMC) and the Ersatz Material Model,” Struct. Multidiscip. Optim., 53(6), pp. 1243–1260. [CrossRef]
Zhang, W. , Zhang, J. , and Guo, X. , 2016, “ Lagrangian Description Based Topology Optimization—A Revival of Shape Optimization,” ASME J. Appl. Mech., 83(4), p. 041010. [CrossRef]
Zhang, W. , Li, D. , Zhang, J. , and Guo, X. , 2016, “ Minimum Length Scale Control in Structural Topology Optimization Based on the Moving Morphable Components (MMC) Approach,” Comput. Methods Appl. Mech. Eng., 311, pp. 327–355. [CrossRef]
Guo, X. , Zhou, J. , Zhang, W. , Du, Z. , Liu, C. , and Liu, Y. , 2017, “ Self-Supporting Structure Design in Additive Manufacturing Through Explicit Topology Optimization,” Comput. Methods Appl. Mech. Eng., epub.
Zhang, W. , Chen, J. , Zhu, X. , Zhou, J. , Xue, D. , Lei, X. , and Guo, X. , 2017, “ Explicit Three Dimensional Topology Optimization Via Moving Morphable Void (MMV) Approach,” Comput. Methods Appl. Mech. Eng., 322, pp. 590–614. [CrossRef]
Svanberg, K. , 1987, “ The Method of Moving Asymptotes—A New Method for Structural Optimization,” Int. J. Numer. Methods Eng., 24(2), pp. 359–373. [CrossRef]
Wu, J. , Dick, C. , and Westermann, R. , 2016, “ A System for High-Resolution Topology Optimization,” IEEE Trans. Visualization Comput. Graphics, 22(3), pp. 1195–1208. [CrossRef]


Grahic Jump Location
Fig. 1

A print with graded patterns [26]

Grahic Jump Location
Fig. 2

Generating a periodic lattice structure through a primitive cell

Grahic Jump Location
Fig. 3

Generating graded lattice structures using different approaches

Grahic Jump Location
Fig. 4

Generating graded effect through coordinate transformation

Grahic Jump Location
Fig. 5

The simply supported beam example

Grahic Jump Location
Fig. 6

Numerical results of the simply supported beam example: (a) and (f) initial designs for the two cases, respectively; (b)–(d) and (g)–(i) some intermediate results for the two cases, respectively; (e) and (j) the optimized structures

Grahic Jump Location
Fig. 7

Convergence history of the values of the objective function: (a) case 1 and (b) case 2

Grahic Jump Location
Fig. 8

A circle region under distributed pressure

Grahic Jump Location
Fig. 9

Numerical results of the pressured circle region example: (a) initial design; (b)–(e) some intermediate results; (f) optimized structure

Grahic Jump Location
Fig. 10

Convergence history of the values of the objective function for the pressured circle region example

Grahic Jump Location
Fig. 11

A torque arm example

Grahic Jump Location
Fig. 12

Initial designs for the torque arm example

Grahic Jump Location
Fig. 13

Optimized graded lattice structure of the torque arm



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