Research Papers: Design Automation

Design and Optimization of Graded Cellular Structures With Triply Periodic Level Surface-Based Topological Shapes

[+] Author and Article Information
Dawei Li

College of Mechanical and Electrical Engineering,
Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China
e-mail: davidlee@nuaa.edu.cn

Ning Dai

College of Mechanical and Electrical Engineering,
Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China
e-mail: dai_ning@nuaa.edu.cn

Yunlong Tang

Department of Mechanical Engineering,
McGill University,
Montreal, Quebec, Canada H3A 0G4
e-mail: tang.yunlong@mail.mcgill.ca

Guoying Dong

Department of Mechanical Engineering,
McGill University,
Montreal, Quebec, Canada H3A 0G4
e-mail: guoying.dong@mail.mcgill.ca

Yaoyao Fiona Zhao

Department of Mechanical Engineering,
McGill University,
Montreal, Quebec, Canada H3A 0G4
e-mail: yaoyao.zhao@mcgill.ca

1Corresponding authors.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received June 27, 2018; final manuscript received January 14, 2019; published online March 13, 2019. Assoc. Editor: Xu Guo.

J. Mech. Des 141(7), 071402 (Mar 13, 2019) (13 pages) Paper No: MD-18-1491; doi: 10.1115/1.4042617 History: Received June 27, 2018; Accepted January 15, 2019

Periodic cellular structures with excellent mechanical properties widely exist in nature. A generative design and optimization method for triply periodic level surface (TPLS)-based functionally graded cellular structures is developed in this work. In the proposed method, by controlling the density distribution, the designed TPLS-based cellular structures can achieve better structural or thermal performances without increasing its weight. The proposed technique can be divided into four steps. First, the modified 3D implicit functions of the triply periodic minimal surfaces are developed to design different types of cellular structures parametrically and generate spatially graded cellular structures. Second, the numerical homogenization method is employed to calculate the elastic tensor and the thermal conductivity tensor of the cellular structures with different densities. Third, the optimal relative density distribution of the object is computed by the scaling laws of the TPLS-based cellular structures added optimization algorithm. Finally, the relative density of the numerical results of structure optimization is mapped into the modified parametric 3D implicit functions, which generates an optimum lightweight cellular structure. The optimized results are validated subjected to different design specifications. The effectiveness and robustness of the obtained structures is analyzed through finite element analysis and experiments. The results show that the functional gradient cellular structure is much stiffer and has better heat conductivity than the uniform cellular structure.

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Thompson, M. K., Moroni, G., Vaneker, T., Fadel, G., Campbell, R. I., Gibson, I., Bernard, A., Schulz, J., Graf, P., Ahuja, B., and Martina, F., 2016, “Design for Additive Manufacturing: Trends, Opportunities, Considerations, and Constraints,” CIRP Ann.-Manuf. Technol., 65(2), pp. 737–760. [CrossRef]
Orme, M. E., Gschweitl, M., Ferrari, M., Madera, I., and Mouriaux, F., 2017, “Designing for Additive Manufacturing: Lightweighting Through Topology Optimization Enables Lunar Spacecraft,” J. Mech. Des., 139(10), 100905. [CrossRef]
Dong, G., Tang, Y., and Zhao, Y. F., 2017, “A Survey of Modeling of Lattice Structures Fabricated by Additive Manufacturing,” J. Mech. Des., 139(10), 100906. [CrossRef]
Cheng, L., Liu, J., Liang, X., and To, A. C., 2018, “Coupling Lattice Structure Topology Optimization With Design-Dependent Feature Evolution for Additive Manufactured Heat Conduction Design,” Comput. Methods Appl. Mech. Eng., 332, pp. 408–439. [CrossRef]
Cheng, L., Liu, J., and To, A. C., 2018, “Concurrent Lattice Infill With Feature Evolution Optimization for Additive Manufactured Heat Conduction Design,” Struct. Multidiscip. Optim., 58, pp. 511–535. [CrossRef]
Bates, S. R. G., Farrow, I. R., and Trask, R. S., 2016, “3D Printed Elastic Honeycombs With Graded Density for Tailorable Energy Absorption,” Active and Passive Smart Structures and Integrated Systems 2016, International Society for Optics and Photonics, Vol. 9799, 979907.
Schaedler, T. A., and Carter, W. B., 2016, “Architected Cellular Materials,” Annu. Rev. Mater. Res., 46, pp. 187–210. [CrossRef]
Lu, L., Sharf, A., Zhao, H., Wei, Y., Fan, Q., Chen, X., Savoye, Y., Tu, C., Cohen-Or, D., and Chen, B., 2014, “Build-to-Last: Strength to Weight 3D Printed Objects,” ACM Trans. Graph. (TOG), 33(4), 97.
Li, D., Dai, N., Jiang, X., and Chen, X., 2016, “Interior Structural Optimization Based on the Density-Variable Shape Modeling of 3D Printed Objects,” Int. J. Adv. Manuf. Technol., 83(9–12), pp. 1627–1635. [CrossRef]
Brackett, D., Ashcroft, I., and Hague, R., 2011, “Topology Optimization for Additive Manufacturing,” Proceedings of the Solid Freeform Fabrication Symposium, Vol. 1, pp. 348–362.
Aremu, A. O., Maskery, I. A., Tuck, C. J., Ashcroft, I. A., Wildman, R. D., and Hague, R. J. M., 2016, “Effects of Net and Solid Skins on Self-Supporting Lattice Structures,” Challenges in Mechanics of Time Dependent Materials, Vol. 2, pp. 83–89.
Steven, G. P., 1997, “Homogenization of Multicomponent Composite Orthotropic Materials Using FEA,” Int. J. Numer. Methods Biomed. Eng., 13(7), pp. 517–531.
Xie, Y. M., Zuo, Z. H., Huang, X., and Yang, X., 2013, “Predicting the Effective Stiffness of Cellular and Composite Materials With Self-Similar Hierarchical Microstructures,” J. Mech. Mater. Struct., 8(5), pp. 341–357. [CrossRef]
Lord, E. A., and Mackay, A. L., 2003, “Periodic Minimal Surfaces of Cubic Symmetry,” Curr. Sci., 85, pp. 346–362.
Wu, X., Ma, D., Eisenlohr, P., Raabe, D., and Fabritius, H.-O., 2016, “From Insect Scales to Sensor Design: Modelling the Mechanochromic Properties of Bicontinuous Cubic Structures,” Bioinspir. Biomim., 11(4), 045001. [CrossRef] [PubMed]
Khaderi, S. N., Deshpande, V. S., and Fleck, N. A., 2014, “The Stiffness and Strength of the Gyroid Lattice,” Int. J. Solids Struct., 51(23–24), pp. 3866–3877. [CrossRef]
Michielsen, K., and Stavenga, D. G., 2008, “Gyroid Cuticular Structures in Butterfly Wing Scales: Biological Photonic Crystals,” J. R. Soc. Interface, 5(18), pp. 85–94. [CrossRef] [PubMed]
Gandy, P. J. F., Cvijović, D., Mackay, A. L., and Klinowski, J., 1999, “Exact Computation of the Triply Periodic D (‘Diamond’) Minimal Surface,” Chem. Phys. Lett., 314(5–6), pp. 543–551. [CrossRef]
Rajagopalan, S., and Robb, R. A., 2006, “Schwarz Meets Schwann: Design and Fabrication of Biomorphic and Durataxic Tissue Engineering Scaffolds,” Med. Image Anal., 10(5), pp. 693–712. [CrossRef] [PubMed]
Jung, Y., and Torquato, S., 2005, “Fluid Permeabilities of Triply Periodic Minimal Surfaces,” Phys. Rev. E, 72(5), 056319. [CrossRef]
Abueidda, D. W., Al-Rub, R. K. A., Dalaq, A. S., Lee, D.-W., Khan, K. A., and Jasiuk, I., 2016, “Effective Conductivities and Elastic Moduli of Novel Foams With Triply Periodic Minimal Surfaces,” Mech. Mater., 95, pp. 102–115. [CrossRef]
Mahmoud, D., and Elbestawi, M. A., 2017, “Lattice Structures and Functionally Graded Materials Applications in Additive Manufacturing of Orthopedic Implants: A Review,” J. Manuf. Mater. Process., 1(2), p. 13.
Torres-Sanchez, C., and Corney, J. R., 2009, “Toward Functionally Graded Cellular Microstructures,” J. Mech. Des., 131(9), 091011. [CrossRef]
Han, Y., and Lu, W. F., 2018, “A Novel Design Method for Nonuniform Lattice Structures Based on Topology Optimization,” J. Mech. Des., 140, 091403. [CrossRef]
Panesar, A., Abdi, M., Hickman, D., and Ashcroft, I., 2018, “Strategies for Functionally Graded Lattice Structures Derived using Topology Optimisation for Additive Manufacturing,” Addit. Manuf., 19, pp. 81–94. [CrossRef]
Liu, X., and Shapiro, V., 2016, “Sample-Based Design of Functionally Graded Material Structures,” ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, V02AT03A035.
Wu, J., Wang, C. C. L., Zhang, X., Zhang, X., and Westermann, R., 2016, “Self-Supporting Rhombic Infill Structures for Additive Manufacturing,” Comput. Aided Des., 80, pp. 32–42. [CrossRef]
Tang, Y., Kurtz, A., and Zhao, Y. F., 2015, “Bidirectional Evolutionary Structural Optimization (BESO) Based Design Method for Lattice Structure to Be Fabricated by Additive Manufacturing,” Comput. Aided Des., 69, pp. 91–101. [CrossRef]
Alzahrani, M., Choi, S. K., and Rosen, D. W., 2015, “Design of Truss-Like Cellular Structures Using Relative Density Mapping Method,” Mater. Des., 85, pp. 349–360. [CrossRef]
Daynes, S., Feih, S., Lu, W. F., and Wei, J., 2017, “Optimisation of Functionally Graded Lattice Structures Using Isostatic Lines,” Mater. Des., 127, 215–223. [CrossRef]
Cheng, L., Zhang, P., Bai, J., Robbins, J., and To, A., 2017, “Efficient Design Optimization of Variable-Density Cellular Structures for Additive Manufacturing: Theory and Experimental Validation,” Rapid Prototyping J., 23(4), pp. 660–677. [CrossRef]
Wang, X., Zhang, P., Ludwick, S., Belski, E., and To, A. C., 2018, “Natural Frequency Optimization of 3D Printed Variable-Density Honeycomb Structure via a Homogenization-Based Approach,” Addit. Manuf., 20, 189–198. [CrossRef]
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]
Bendsøe, M. P., 1989, “Optimal Shape Design as a Material Distribution Problem,” Struct. Optim., 1(4), pp. 193–202. [CrossRef]
Huang, X., and Xie, Y. M., 2007, “Convergent and Mesh-Independent Solutions for the Bi-Directional Evolutionary Structural Optimization Method,” Finite Elem. Anal. Des., 43(14), pp. 1039–1049. [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,” J. Appl. Mech., 81(8), 081009. [CrossRef]
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., Yang, W., Zhou, J., Li, D., and Guo, X., 2017, “Structural Topology Optimization Through Explicit Boundary Evolution,” J. Appl. Mech., 84(1), 011011. [CrossRef]
Zhu, Y., Li, S., Du, Z., Liu, C., Guo, X., and Zhang, W., 2019, “A Novel Asymptotic-Analysis-Based Homogenisation Approach Towards Fast Design of Infill Graded Microstructures,” J. Mech. Phys. Solids, 124, pp. 612–633. [CrossRef]
Xia, L., Xia, Q., Huang, X., and Xie, Y. M., 2016, “Bi-Directional Evolutionary Structural Optimization on Advanced Structures and Materials: A Comprehensive Review,” Arch. Comput. Methods Eng., 25, pp. 437–478. [CrossRef]
Sigmund, O., and Maute, K., 2013, “Topology Optimization Approaches,” Struct. Multidiscip. Optim., 48(6), pp. 1031–1055. [CrossRef]
Von Schnering, H. G., and Nesper, R., 1991, “Nodal Surfaces of Fourier Series: Fundamental Invariants of Structured Matter,” Z. Phys. B Condens. Matter, 83(3), pp. 407–412. [CrossRef]
Gandy, P. J. F., Bardhan, S., Mackay, A. L., and Klinowski, J., 2001, “Nodal Surface Approximations to the P, G, D and I-WP Triply Periodic Minimal Surfaces,” Chem. Phys. Lett., 336(3–4), pp. 187–195. [CrossRef]
Ashby, M. F., 2006, “The Properties of Foams and Lattices,” Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci., 364(1838), pp. 15–30. [CrossRef]
Bensoussan, A., Lions, J. L., and Papanicolaou, G., 2011, Asymptotic Analysis for Periodic Structures, American Mathematical Society, Providence, Rhode Island.
Andreassen, E., and Andreasen, C. S., 2014, “How to Determine Composite Material Properties Using Numerical Homogenization,” Comput. Mater. Sci., 83, pp. 488–495. [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]
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]
Liu, C., Du, Z., Zhang, W., Zhu, Y., and Guo, X., 2017, “Additive Manufacturing-Oriented Design of Graded Lattice Structures Through Explicit Topology Optimization,” J. Appl. Mech., 84(8), 081008. [CrossRef]


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Fig. 1

Procedures for TPLS-based functionally graded cellular structures design and optimization

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Fig. 2

Library of TPLS-based cellular structures: (a) G, D, P, and W level surfaces and (b) cellular structures with 3 × 3 × 3 unit cells

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Fig. 3

Scaling laws of the relative density of G-, D-, P-, and W-based cellular structures (R2: coefficient of determination)

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Fig. 4

Example of functionally graded cellular structures of G (a), D (b), P (c), and W (d) along the y-axis direction

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Fig. 5

Asymptotic homogenization concept of a cellular structure

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Fig. 6

Elastic tensor and thermal conductivity tensor scaling laws for TPLS-based cellular structures: (a) Gyroid, (b) diamond, (c) primitive, and (d) iWP

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Fig. 7

The procedure of the relative density mapping methodology

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Fig. 8

(a) A cubic block of 30 × 30 × 30 mm, (b) boundary conditions for stress deformation analysis, (c) boundary conditions for thermal analysis, and (d)–(l) the number of units along a single direction are 1, 2, 3, 4, 5, 6, 8, 10, and 12

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Fig. 9

(a) Maximum of the total deformation associated with unit cell number along one direction and (b) maximum temperature associated with unit cell number along one direction

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Fig. 10

(a) The design domain and boundary conditions, (b) and (c) an optimized cantilever beam and its cross section area, and (d)–(f) different filling densities, density intervals, and their results of compliance, respectively (ρ is the relative density and c is the compliance)

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Fig. 11

Comparison between the homogenization result and full-scale simulation of the optimized cantilever beam (kstiffness is the stiffness of the linear elastic region, Case 1 and Case 2 are the examples)

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Fig. 12

Illustration of local material damage: (a) standard topology optimization, the uniform gyroid-based cellular structure, and our proposed optimized cellular structures, respectively, (b) deficiencies in different areas (A, B, and C), and (c) displacement results with different design strategies and different deficiencies. (ρ is the relative density, c is the compliance).

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Fig. 13

Heat-sink optimization and full-scale simulation results: (a) design domain and boundary conditions, (b) the uniform gyroid structure with the volume fraction of 30%, (c) our optimized result with a density range of 0.1–0.8, and (d) our optimized result with a density range of 0.1–0.9



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