Research Papers: Design Automation

Connecting Microstructures for Multiscale Topology Optimization With Connectivity Index Constraints

[+] Author and Article Information
Zongliang Du

Structural Engineering Department,
University of California, San Diego,
San Diego, CA 92093
e-mail: zodu@eng.ucsd.edu

Xiao-Yi Zhou

School of Engineering,
Cardiff University,
Cardiff CF24 3AA, UK
e-mail: zhoux19@cf.ac.uk

Renato Picelli

School of Engineering,
Cardiff University,
Cardiff CF24 3AA, UK
e-mail: PicelliR@cardiff.ac.uk

H. Alicia Kim

Structural Engineering Department,
University of California, San Diego,
San Diego, CA 92093;
School of Engineering,
Cardiff University,
Cardiff CF24 3AA, UK
e-mail: alicia@ucsd.edu

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 1, 2018; final manuscript received August 6, 2018; published online October 1, 2018. Assoc. Editor: James K. Guest.

J. Mech. Des 140(11), 111417 (Oct 01, 2018) (12 pages) Paper No: MD-18-1272; doi: 10.1115/1.4041176 History: Received April 01, 2018; Revised August 06, 2018

With the rapid developments of advanced manufacturing and its ability to manufacture microscale features, architected materials are receiving ever increasing attention in many physics fields. Such a design problem can be treated in topology optimization as architected material with repeated unit cells using the homogenization theory with the periodic boundary condition. When multiple architected materials with spatial variations in a structure are considered, a challenge arises in topological solutions, which may not be connected between adjacent material architecture. This paper introduces a new measure, connectivity index (CI), to quantify the topological connectivity, and adds it as a constraint in multiscale topology optimization to achieve connected architected materials. Numerical investigations reveal that the additional constraints lead to microstructural topologies, which are well connected and do not substantially compromise their optimalities.

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Corni, I. , Harvey, T. J. , Wharton, J. A. , Stokes, K. R. , Walsh, F. C. , and Wood, R. J. K. , 2012, “ A Review of Experimental Techniques to Produce a Nacre-Like Structure,” Bioinspiration Biomimetics, 7(3), p. 031001. [CrossRef] [PubMed]
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]
Wickham, S. , Large, M. C. , Poladian, L. , and Jermiin, L. S. , 2006, “ Exaggeration and Suppression of Iridescence: The Evolution of Two-Dimensional Butterfly Structural Colours,” J. R. Soc. Interface, 3(6), pp. 99–109. [CrossRef] [PubMed]
Shalaev, V. M. , 2007, “ Optical Negative-Index Metamaterials,” Nat. Photonics, 1(1), pp. 41–48. [CrossRef]
Zheng, X. , Lee, H. , Weisgraber, T. H. , Shusteff, M. , DeOtte, J. , Duoss, E. B. , Kuntz, J. D. , Biener, M. M. , Ge, Q. , and Jackson, J. A. , 2014, “ Ultralight, Ultrastiff Mechanical Metamaterials,” Science, 344(6190), pp. 1373–1377. [CrossRef] [PubMed]
Andreassen, E. , Lazarov, B. S. , and Sigmund, O. , 2014, “ Design of Manufacturable 3D Extremal Elastic Microstructure,” Mech. Mater., 69(1), pp. 1–10. [CrossRef]
Takezawa, A. , Kobashi, M. , and Kitamura, M. , 2015, “ Porous Composite With Negative Thermal Expansion Obtained by Photopolymer Additive Manufacturing,” APL Mater, 3(7), p. 076103. [CrossRef]
Berger, J. B. , Wadley, H. N. G. , and McMeeking, R. M. , 2017, “ Mechanical Metamaterials at the Theoretical Limit of Isotropic Elastic Stiffness,” Nature, 543(7646), pp. 533–537. [CrossRef] [PubMed]
Sigmund, O. , 1994, “ Materials With Prescribed Constitutive Parameters: An Inverse Homogenization Problem,” Int. J. Solids Struct., 31(17), pp. 2313–2329. [CrossRef]
Sigmund, O. , and Torquato, S. , 1997, “ Design of Materials With Extreme Thermal Expansion Using a Three-Phase Topology Optimization Method,” J. Mech. Phys. Solids, 45(6), pp. 1037–1067. [CrossRef]
Rodrigues, H. , Guedes, J. M. , and Bendsøe, M. P. , 2002, “ Hierarchical Optimization of Material and Structure,” Struct. Multidiscip. Optim., 24(1), pp. 1–10. [CrossRef]
Coelho, P. G. , Fernandes, P. R. , Guedes, J. M. , and Rodrigues, H. C. , 2008, “ A Hierarchical Model for Concurrent Material and Topology Optimisation of Three-Dimensional Structures,” Struct. Multidiscip. Optim., 35(2), pp. 107–115. [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]
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]
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]
Guedes, J. , and Kikuchi, N. , 1990, “ Preprocessing and Postprocessing for Materials Based on the Homogenization Method With Adaptive Finite Element Methods,” Comput. Methods Appl. Mech. Eng., 83(2), pp. 143–198. [CrossRef]
Alexandersen, J. , and Lazarov, B. S. , 2015, “ Topology Optimisation of Manufacturable Microstructural Details Without Length Scale Separation Using a Spectral Coarse Basis Preconditioner,” Comput. Methods Appl. Mech. Eng., 290, pp. 156–182. [CrossRef]
Xie, Y. M. , Zuo, Z. H. , Huang, X. , and Rong, J. H. , 2012, “ Convergence of Topological Patterns of Optimal Periodic Structures Under Multiple Scales,” Struct. Multidiscip. Optim., 46(1), pp. 41–50. [CrossRef]
Coelho, P. G. , Amiano, L. D. , Guedes, J. M. , and Rodrigues, H. C. , 2016, “ Scale-Size Effects Analysis of Optimal Periodic Material Microstructures Designed by the Inverse Homogenization Method,” Comput. Struct., 174, pp. 21–32. [CrossRef]
Dumontet, H. , 1985, “ Boundary Layers Stresses in Elastic Composites,” Studies in Applied Mechanics, Vol. 12, Elsevier, Amsterdam, The Netherlands, pp. 215–232.
Liu, C. , Zhang, W. , Du, Z. , and Guo, X. , “ Multidomain Topology Optimization of Manufacturable Microstructures Using Homogenization Method,” submitted.
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]
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]
Deng, J. , and Chen, W. , 2017, “ Concurrent Topology Optimization of Multiscale Structures With Multiple Porous Materials Under Random Field Loading Uncertainty,” Struct. Multidiscip. Optim., 56(1), pp. 1–19. [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]
Dunning, P. D. , and Kim, H. A. , 2015, “ Introducing the Sequential Linear Programming Level-Set Method for Topology Optimization,” Struct. Multidiscip. Optim., 51(3), pp. 631–643. [CrossRef]
Johnson, S. G. , 2014, “ The NLopt Nonlinear-Optimization Package,” epub, accessed Aug. 27, 2018, http://ab-initio.mit.edu/nlopt
Xia, L. , and Breitkopf, P. , 2014, “ Concurrent Topology Optimization Design of Material and Structure Within Nonlinear Multiscale Analysis Framework,” Comput. Methods Appl. Mech. Eng., 278, pp. 524–542. [CrossRef]
Bendsøe, M. P. , and Sigmund, O. , 2004, Topology Optimization: Theory, Methods and Applications, Springer, Berlin.
Xie, Y. M. , and Steven, G. P. , 1993, “ A Simple Evolutionary Procedure for Structural Optimization,” Comput. Struct., 49(5), pp. 885–896. [CrossRef]
Querin, O. M. , Steven, G. P. , and Xie, Y. M. , 1998, “ Evolutionary Structural Optimisation (ESO) Using a Bidirectional Algorithm,” Eng. Comput., 15(8), pp. 1031–1048. [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]
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. , Yuan, J. , Zhang, J. , and Guo, X. , 2016, “ 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. , Yang, W. , Zhou, J. , Li, D. , and Guo, X. , 2017, “ Structural Topology Optimization Through Explicit Boundary Evolution,” ASME J. Appl. Mech., 84(1), p. 011011. [CrossRef]


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Fig. 1

An illustration of multiscale topology optimization without connectivity

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Fig. 2

An L-beam example with (a) the design domain, (b) the initial macroscale structure, and (c) the initial microstructure

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Fig. 3

Multiscale optimization solution of the L beam example (a) optimized macroscale structure and (b) optimized microstructures illustrating the connectivity challenges

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Fig. 4

An illustration of the proposed connectivity measure, where yellow structure on the left represents one-unit cell and the green structure on the right represents the adjacent unit cell, and the light red strips represent the interface regions used to quantify the connectivity between the two cells

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Fig. 5

Two test cases obtained by microstructural optimization via homogenization, (a) maximum shear modulus and bulk modulus solutions and (b) maximum D11H and D22H solutions

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Fig. 6

(a) Initial design with five circular holes, (b) optimized unit cell with maximum shear modulus, (c) optimized unit cell with maximum bulk modulus, and (d) optimized unit cell with maximum D22H

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Fig. 7

The maximum shear and bulk microstructure design

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Fig. 8

(a) Illustration of the self-connectivity index and (b) maximum shear and bulk modulus microstructure designs with self-connectivity constraints

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Fig. 10

Illustrations of the connected cellular materials of the cantilevered beams (a) design domain 1:3 and (b) design domain 2:3

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Fig. 11

Optimization history of cantilevered beam 2:3

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Fig. 12

Optimized solution for case 1 (a) macrostructure and (b) microstructure

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Fig. 13

The optimized multiscale structure of case (3) with (a) the optimized macrostructure and (b) the optimized microstructures

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Fig. 14

Optimization history of L-beam case 2

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Fig. 15

Optimized solution for case 3 (a) macrostructure and (b) microstructure and (c) an illustration of the optimized structure–material system

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Fig. 16

Optimization history of L-beam case 3



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