Abstract
Based on hybrid cellular automata (HCA), we present a two-scale optimization model for heterogeneous structures with non-uniform porous cells at the microscopic scale. The method uses the K-means clustering algorithm to achieve locally nonperiodicity through easily obtained elemental strain energy. This energy is used again for a two-scale topological optimization procedure without sensitivity analysis, avoiding drastically the computational complexity. Both the experimental tests and numerical results illustrate a significant increase in the resulting structural stiffness with locally nonperiodicity, as compared to using uniform periodic cells. The effects of parameters such as clustering number and adopted method versus classical Optimality Criteria (OC) are discussed. Finally, the proposed methodology is extended to 3D two-scale heterogeneous structure design.
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References
ASTM International A (2015) Standard test method for flexural properties of polymer matrix composite materials. ASTM International,
Baughman RH, Zakhidov AA, de Heer WA (2002) Carbon nanotubes—the route toward applications. Science 297:787–792. https://doi.org/10.1126/science.1060928
Coelho PG, Fernandes PR, Guedes JM, Rodrigues HC (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidiscip O 35:107–115. https://doi.org/10.1007/s00158-007-0141-3
Da DC (2019) Topology optimization design of heterogeneous materials and structures. Wiley, pp 29–55
Da DC, Chen JH, Cui XY, Li GY (2017) Design of materials using hybrid cellular automata. Struct Multidiscip O 56:131–137. https://doi.org/10.1007/s00158-017-1652-1
Deng JD, Yan J, Cheng GD (2013) Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Struct Multidiscip O 47:583–597. https://doi.org/10.1007/s00158-012-0849-6
Evans AG, Hutchinson JW, Fleck NA, Ashby MF, Wadley HNG (2001) The topological design of multifunctional cellular metals. Prog Mater Sci 46:309–327. https://doi.org/10.1016/S0079-6425(00)00016-5
Feyel F, Chaboche JL (2000) FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Method Appl M 183:309–330. https://doi.org/10.1016/S0045-7825(99)00224-8
Gao LZ, Li H, Gao L (2019a) Topology optimization for multiscale design of porous composites with multi-domain microstructures. Comput Method Appl M 344:451–476. https://doi.org/10.1016/j.cma.2018.10.017
Gao LZ, Li H, Li P, Gao L (2019b) Dynamic multiscale topology optimization for multi-regional micro-structured cellular composites. Compos Struct 211:401–417. https://doi.org/10.1016/j.compstruct.2018.12.031
Jie G, Hao L, Liang G, Mi X (2018) Topological shape optimization of 3D micro-structured materials using energy-based homogenization method. Adv Eng Softw 116:89–102. https://doi.org/10.1016/j.advengsoft.2017.12.002
Lakes R (1993) Materials with structural hierarchy. Nature 361:511–515. https://doi.org/10.1038/361511a0
Li H, Luo Z, Gao L, Qin QH (2018) Topology optimization for concurrent design of structures with multi-patch microstructures by level sets. Comput Method Appl M 331:536–561. https://doi.org/10.1016/j.cma.2017.11.033
Liang X, Du JB (2019) Concurrent multi-scale and multi-material topological optimization of vibro-acoustic structures. Comput Method Appl M 349:117–148. https://doi.org/10.1016/j.cma.2019.02.010
Liu L, Yan J, Cheng GD (2008) Optimum structure with homogeneous optimum truss-like material. Comput Struct 86:1417–1425. https://doi.org/10.1016/j.compstruc.2007.04.030
Niu B, Yan J, Cheng GD (2009) Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidiscip O 39:115–132. https://doi.org/10.1007/s00158-008-0334-4
Patel NM, Kang BS, Renaud JE, Tovar A (2009) Crashworthiness design using topology optimization. J Mech Design 131 Artn:061013. https://doi.org/10.1115/1.3116256
Rodrigues H, Guedes JM, Bendsoe MP (2002) Hierarchical optimization of material and structure. Struct Multidiscip O 24:1–10. https://doi.org/10.1007/s00158-002-0209-z
Schaedler TA et al (2011) Ultralight metallic microlattices. Science 334:962–965. https://doi.org/10.1126/science.1211649
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip O 33:401–424. https://doi.org/10.1007/s00158-006-0087-x
Sigmund O, Maute K (2013) Topology optimization approaches: a comparative review. Struct Multidiscip O 48:1031–1055. https://doi.org/10.1007/s00158-013-0978-6
Sivapuram R, Dunning PD, Kim HA (2016) Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidiscip O 54:1267–1281. https://doi.org/10.1007/s00158-016-1519-x
Tan P-N, Steinbach M, Kumar V (2006) Introduction to data mining, 1st edn. Pearson Addison Wesley, Boston
Tovar A, Patel NM, Niebur GL, Sen M, Renaud JE (2006) Topology optimization using a hybrid cellular automaton method with local control rules. J Mech Des 128:1205–1216. https://doi.org/10.1115/1.2336251
Wang C, Zhu JH, Zhang WH, Li SY, Kong J (2018) Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures. Struct Multidiscip O 58:35–50. https://doi.org/10.1007/s00158-018-2009-0
Wang QM, Jackson JA, Ge Q, Hopkins JB, Spadaccini CM, Fang NX (2016) Lightweight mechanical metamaterials with tunable negative thermal expansion. Physical Review Letters 117:175901. https://doi.org/10.1103/PhysRevLett.117.175901
Xia L, Breitkopf P (2014) Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework. Comput Method Appl M 278:524–542. https://doi.org/10.1016/j.cma.2014.05.022
Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in Matlab. Struct Multidiscip O 52:1229–1241. https://doi.org/10.1007/s00158-015-1294-0
Xie YM, Zuo ZH, Huang XD, Rong JH (2012) Convergence of topological patterns of optimal periodic structures under multiple scales. Struct Multidiscip O 46:41–50. https://doi.org/10.1007/s00158-011-0750-8
Xu B, Huang X, Zhou SW, Xie YM (2016) Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase material for maximum stiffness. Compos Struct 150:84–102. https://doi.org/10.1016/j.compstruct.2016.04.038
Xu B, Xie YM (2015) Concurrent design of composite macrostructure and cellular microstructure under random excitations. Compos Struct 123:65–77. https://doi.org/10.1016/j.compstruct.2014.10.037
Xu J et al (2018) Compressive properties of hollow lattice truss reinforced honeycombs (Honeytubes) by additive manufacturing: patterning and tube alignment effects. Mater Design 156:446–457. https://doi.org/10.1016/j.matdes.2018.07.019
Xu J, Wu YB, Gao X, Wu HP, Nutt S, Yin S (2019) Design of composite lattice materials combined with fabrication approaches. J Compos Mater 53:393–404. https://doi.org/10.1177/0021998318785710
Yan X, Huang X, Zha Y, Xie YM (2014) Concurrent topology optimization of structures and their composite microstructures. Comput Struct 133:103–110. https://doi.org/10.1016/j.compstruc.2013.12.001
Yan XL, Huang XD, Sun GY, Xie YM (2015) Two-scale optimal design of structures with thermal insulation materials. Compos Struct 120:358–365. https://doi.org/10.1016/j.compstruct.2014.10.013
Yao HB, Fang HY, Wang XH, Yu SH (2011) Hierarchical assembly of micro-/nano-building blocks: bio-inspired rigid structural functional materials. Chem Soc Rev 40:3764–3785. https://doi.org/10.1039/c0cs00121j
Yin S, Li JN, Liu BH, Meng KP, Huan Y, Nutt SR, Xu J (2017) Honeytubes: hollow lattice truss reinforced honeycombs for crushing protection. Compos Struct 160:1147–1154. https://doi.org/10.1016/j.compstruct.2016.11.007
Zhang W, Yin S, Yu TX, Xu J (2019) Crushing resistance and energy absorption of pomelo peel inspired hierarchical honeycomb. Int J Impact Eng 125:163–172. https://doi.org/10.1016/j.ijimpeng.2018.11.014
Zhang WH, Sun SP (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Methods Eng 68:993–1011. https://doi.org/10.1002/nme.1743
Zhang ZQ, Zhang YW, Gao HJ (2011) On optimal hierarchy of load-bearing biological materials. Proceedings of the Royal Society B-Biological Sciences 278:519–525. https://doi.org/10.1098/rspb.2010.1093
Zhou SW, Li Q (2008) Design of graded two-phase microstructures for tailored elasticity gradients. J Mater Sci 43:5157–5167. https://doi.org/10.1007/s10853-008-2722-y
Funding
This work was financially supported by the Natural Science Foundation of China (NSFC) (No. 11902015, 41804134), National Key Research and Development Program of China (2017YFB0103703), and the Fundamental Research Funds for the Central Universities, Beihang University, Young Elite Scientist Sponsorship Program by CAST.
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Jia, J., Da, D., Loh, CL. et al. Multiscale topology optimization for non-uniform microstructures with hybrid cellular automata. Struct Multidisc Optim 62, 757–770 (2020). https://doi.org/10.1007/s00158-020-02533-3
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DOI: https://doi.org/10.1007/s00158-020-02533-3