Abstract
This paper presents a multiscale level set topology optimization method for the design of graded cellular structures. The micro-scale level set-based prototype microstructure is periodically repeated to constitute a basic structure, which is cut by the macro-scale level set surface to generate the progressively-changing cellular structure. In the proposed method, the level set functions evolve in both scales, and then the configuration-optimized cellular microstructures are distributed in a well-arranged graded pattern. Due to the high-order continuity of parameterized level set function, the adjacent cellular microstructures connect to each other smoothly. For bridging different scales, the effective properties of microstructures are calculated by the numerical homogenization method. By studying the relation of microstructural elasticity tensors to the level set functions of both scales, the mechanical property and sensitivity analysis for cellular microstructure at an arbitrary position can be estimated effectively. Numerical examples demonstrate that the proposed method can customize both the configurations and distributions for graded cellular structures with favourable continuity.
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References
Alexandersen J, Lazarov B (2015) Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner. Comput Methods Appl Mech Eng 290:156–182. https://doi.org/10.1016/j.cma.2015.02.028
Alzahrani M, Choi SK, Rosen D (2015) Design of truss-like cellular structures using relative density mapping method. Mater Des 85:349–360. https://doi.org/10.1016/j.matdes.2015.06.180
Arabnejad S, Pasini D (2012) Multiscale design and multiobjective optimization of orthopedic hip implants with functionally graded cellular material. J Biomech Eng 134:031004. https://doi.org/10.1115/1.4006115
Bendsøe M, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bendsøe M, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654. https://doi.org/10.1007/s004190050248
Chu S, Gao L, Xiao M, Zhang Y (2020) Multiscale topology optimization for coated structures with multifarious-microstructural infill. Struct Multidisc Optim 61:1473–1494. https://doi.org/10.1007/s00158-019-02428-y
Coelho P, Fernandes P, Guedes J, Rodrigues H (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidisc Optim 35:107–115. https://doi.org/10.1007/s00158-007-0141-3
Cramer A, Challis V, Roberts A (2016) Microstructure interpolation for macroscopic design. Struct Multidisc Optim 53:489–500. https://doi.org/10.1007/s00158-015-1344-7
Dunning P (2017) Design parameterization for topology optimization by intersection of an implicit function. Comput Methods Appl Mech Eng 317:993–1011. https://doi.org/10.1016/j.cma.2017.01.008
Eschenauer H, Kobelev V, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51. https://doi.org/10.1007/BF01742933
Gao J, Xiao M, Gao L, Yan J, Yan W (2020) Isogeometric topology optimization for computational design of re-entrant and chiral auxetic composites. Comput Methods Appl Mech Eng 362:112876. https://doi.org/10.1016/j.cma.2020.112876
Gibson LJ, Ashby MF (1997) Cellular solids: structure and properties, 2 edn. Cambridge Solid State Science Series. Cambridge University Press. https://doi.org/10.1017/CBO9781139878326
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically-a new moving morphable components based framework. J Appl Mech 81:081009. https://doi.org/10.1115/1.4027609
Huang X, Zhou S, Xie Y, Li Q (2013) Topology optimization of microstructures of cellular materials and composites for macrostructures. Comput Mater Sci 67:397–407. https://doi.org/10.1016/j.commatsci.2012.09.018
Li H, Luo Z, Zhang N, Gao L, Brown T (2016) Integrated design of cellular composites using a level-set topology optimization method. Comput Methods Appl Mech Eng 309:453–475. https://doi.org/10.1016/j.cma.2016.06.012
Liang S, Gao L, Zheng Y, Li H (2020) A transitional connection method for the design of functionally graded cellular materials. Appl Sci 10:7449. https://doi.org/10.3390/app10217449
Liu H, Hongming Z, Tielin S, Xia Q (2020) M-vcut level set method for optimizing cellular structures. Comput Methods Appl Mech Eng 367:113154. https://doi.org/10.1016/j.cma.2020.113154
Liu H, Wang Y, Zong H, Wang M (2018) Efficient structure topology optimization by using the multiscale finite element method. Struct Multidisc Optim 58:1411–1430. https://doi.org/10.1007/s00158-018-1972-9
Liu Y, Li Z, Wei P, Wang W (2018) Parameterized level-set topology optimization method considering symmetry and pattern repetition constraints. Comput Methods Appl Mech Eng 340:1079–1101. https://doi.org/10.1016/j.cma.2018.04.034
Long K, Han D, Gu X (2017) Concurrent topology optimization of composite macrostructure and microstructure constructed by constituent phases of distinct poisson’s ratios for maximum frequency. Comput Mater Sci 129:194–201. https://doi.org/10.1016/j.commatsci.2016.12.013
Luo Z, Tong L, Wang M, Wang S (2007) Shape and topology optimization of compliant mechanisms using a parameterization level set method. J Comput Phys 227:680–705. https://doi.org/10.1016/j.jcp.2007.08.011
Luo Z, Wang M, Wang S, Wei P (2008) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Meth Eng 76:1–26. https://doi.org/10.1002/nme.2092
Nakshatrala P, Tortorelli D, Nakshatrala K (2013) Nonlinear structural design using multiscale topology optimization. part i: static formulation. Comput Methods Appl Mech Eng 261–262:167–176. https://doi.org/10.1016/j.cma.2012.12.018
Radman A, Huang X, Xie Y (2013) Topology optimization of functionally graded cellular materials. J Mater Sci 48:1503–1510. https://doi.org/10.1007/s10853-012-6905-1
Rodrigues H, Guedes J, Bendsøe M (2002) Hierarchical optimization of material and structure. Struct Multidisc Optim 24:1–10. https://doi.org/10.1007/s00158-002-0209-z
Sethian J, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163:489–528. https://doi.org/10.1006/jcph.2000.6581
Shah K, Pinkerton A, Haq I, Khan A, Khan M, Li L (2014) Parametric study of development of inconel-steel functionally graded materials by laser direct metal deposition. Mater Design 54:531–538. https://doi.org/10.1016/j.matdes.2013.08.079
Sigmund O (1994) Materials with prescribed constitutive parameters: an inverse homogenization problem. Int J Solids Struct 31:2313–2329. https://doi.org/10.1016/0020-7683(94)90154-6
Sokolowski J, Zolésio JP (1992) Introduction to shape optimization: shape sensitivity analysis, vol. 10. Springer series in computational mathematics. https://doi.org/10.1007/978-3-642-58106-9
Song Y, Ma Q, he y, Zhou M, Wang M, (2020) Stress-based shape and topology optimization with cellular level set in b-splines. Struct Multidisc Optim 62:2391–2407. https://doi.org/10.1007/s00158-020-02610-7
Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Meth Eng 24:359–373. https://doi.org/10.1002/nme.1620240207
Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12:555–573. https://doi.org/10.1137/S1052623499362822
Wadley H (2006) Multifunctional periodic cellular metals. Philos Transact A Math Phys Eng Sci 364:31–68. https://doi.org/10.1098/rsta.2005.1697
Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246. https://doi.org/10.1016/S0045-7825(02)00559-5
Wang M, Zong H, Ma Q, Tian Y, Zhou M (2019) Cellular level set in b-splines (clibs): a method for modeling and topology optimization of cellular structures. Comput Methods Appl Mech Eng 349:378–404. https://doi.org/10.1016/j.cma.2019.02.026
Wang Y, Chen F, Wang M (2017) Concurrent design with connectable graded microstructures. Comput Methods Appl Mech Eng 317:84–101. https://doi.org/10.1016/j.cma.2016.12.007
Wang Y, Kang Z (2019) Concurrent two-scale topological design of multiple unit cells and structure using combined velocity field level set and density model. Comput Methods Appl Mech Eng 347:340–364. https://doi.org/10.1016/j.cma.2018.12.018
Wang Y, Liao Z, Shi S, Wang Z, Poh LH (2020) Data-driven structural design optimization for petal-shaped auxetics using isogeometric analysis. Comput Model Eng Sci 122:433–458. https://doi.org/10.32604/cmes.2020.08680
Wang Y, Liao Z, Ye M, Zhang Y, Li W, Zhaohui X (2020) An efficient isogeometric topology optimization using multilevel mesh, mgcg and local-update strategy. Adv Eng Softw 139:102733. https://doi.org/10.1016/j.advengsoft.2019.102733
Wang Y, Luo Z, Kang Z, Zhang N (2015) A multi-material level set-based topology and shape optimization method. Comput Methods Appl Mech Eng 283:1570–1586. https://doi.org/10.1016/j.cma.2014.11.002
Wang Y, Wang M, Chen F (2016) Structure-material integrated design by level sets. Struct Multidisc Optim 54:1145–1156. https://doi.org/10.1007/s00158-016-1430-5
Wang Y, Xu H, Pasini D (2017) Multiscale isogeometric topology optimization for lattice materials. Comput Methods Appl Mech Eng 316:568–585. https://doi.org/10.1016/j.cma.2016.08.015
Wei P, Li Z, Li X, Wang M (2018) An 88-line matlab code for the parameterized level set method based topology optimization using radial basis functions. Struct Multidisc Optim 58:831–849. https://doi.org/10.1007/s00158-018-1904-8
Wei P, Wang W, Yang Y, Wang M (2020) Level set band method: A combination of density-based and level set methods for the topology optimization of continuums. Front Mech Eng 15:390–405. https://doi.org/10.1007/s11465-020-0588-0
Wendland H (2006) Computational aspects of radial basis function approximation. Stud Comput Math 12:231–256. https://doi.org/10.1016/S1570-579X(06)80010-8
Wu Z, Xia L, Wang S, Shi T (2019) Topology optimization of hierarchical lattice structures with substructuring. Comput Methods Appl Mech Eng 345:602–617. https://doi.org/10.1016/j.cma.2018.11.003
Xia Q, Zong H, Shi T, Liu H (2020) Optimizing cellular structures through the m-vcut level set method with microstructure mapping and high order cutting. Comp Struct 261:113298. https://doi.org/10.1016/j.compstruct.2020.113298
Xie Y, Steven G (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896. https://doi.org/10.1016/0045-7949(93)90035-C
Yu C, Wang Q, Mei C, Xia Z (2020) Multiscale isogeometric topology optimization with unified structural skeleton. Comput Model Eng Sci 122:779–803. https://doi.org/10.32604/cmes.2020.09363
Zhang H, Wang Y, Kang Z (2019) Topology optimization for concurrent design of layer-wise graded lattice materials and structures. Int J Eng Sci 138:26–49. https://doi.org/10.1016/j.ijengsci.2019.01.006
Zhang Y, Li H, Xiao M, Gao L, Chu S, Zhang J (2019) Concurrent topology optimization for cellular structures with nonuniform microstructures based on the kriging metamodel. Struct Multidisc Optim 59:1273–1299. https://doi.org/10.1007/s00158-018-2130-0
Zheng X, Lee H, Weisgraber T, Shusteff M, DeOtte J, Duoss E, Kuntz J, Biener M, Ge Q, Jackson J, Kucheyev S, Fang N, Spadaccini C (2014) Ultralight, ultrastiff mechanical metamaterials. Science 344:1373–1377. https://doi.org/10.1126/science.1252291
Zhou M, Rozvany G (1991) The coc algorithm, part ii: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336. https://doi.org/10.1016/0045-7825(91)90046-9
Zong H, Liu H, Ma Q, Tian Y, Zhou M, Wang M (2019) Vcut level set method for topology optimization of functionally graded cellular structures. Comput Methods Appl Mech Eng 354:487–505. https://doi.org/10.1016/j.cma.2019.05.029
Funding
This work was supported by the National Key R&D Program of China (2018YFB1700803, 2018YFB1700804, 2020YFB1708300), the National Natural Science Foundation of China (52005192) and the Fundamental Research Funds for the Central Universities (HUST: 500310 0089).
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This work is built on parameterized level set topology optimization and inverse homogenization method. The opening source codes are presented in the related references. Therefore, the results can be replicated by implementing the formulas in this study.
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Yu, C., Wang, Q., Xia, Z. et al. Multiscale topology optimization for graded cellular structures based on level set surface cutting. Struct Multidisc Optim 65, 32 (2022). https://doi.org/10.1007/s00158-021-03097-6
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DOI: https://doi.org/10.1007/s00158-021-03097-6