Abstract
Periodic topology optimization has been suggested as an effective means to design efficient structures which address a range of practical constraints, such as manufacturability, transportability, replaceability and ease of assembly. This study proposes a new approach for design of finite periodic structures by allowing variable orientation states of individual unit-cells. In some design instances of periodic structures, the unit-cell may exhibit certain geometric features allowing multiple possible assembly orientations (e.g. facing up or down). For the first time, this work incorporates such assembly flexibility within the periodic topology optimization, which enables to greatly expand the conventional periodic design space and take more advantage of structural periodicity. Given its broad applications, a methodology for the design of more efficient periodic structures while maintaining the same degree of periodic constraint may be of significant benefit to engineering practice. In this study, several numerical examples are presented to demonstrate the effectiveness of this new approach for both static and vibratory criteria. Brute force analysis is also utilized to compare all possible assembly configurations for several periodic structures with a small number of unit-cells. A heuristic approach is suggested for selecting more beneficially oriented configurations in periodic structures with a large number of unit-cells for which an exhaustive search may be computationally infeasible. It is found that in all the presented cases, the oriented periodic structures outperform the conventional non-oriented (or namely translational) periodic counterparts. Finally, an educational MATLAB code is provided for replication of the design results in this paper.
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The result data are available in the reference data set (Thomas 2021).
References
Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in matlab using 88 lines of code. Struct Multidisc Optim 43(1):1–16. https://doi.org/10.1007/s00158-010-0594-7
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654. https://doi.org/10.1007/s004190050248
Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26):3443–3459. https://doi.org/10.1016/S0045-7825(00)00278-4
Cadman JE, Zhou S, Chen Y, Li Q (2013) On design of multi-functional microstructural materials. J Mater Sci 48(1):51–66. https://doi.org/10.1007/s10853-012-6643-4
Chen Y, Zhou S, Li Q (2010) Multiobjective topology optimization for finite periodic structures. Comput Struct 88(11):806–811. https://doi.org/10.1016/j.compstruc.2009.10.003
Da D, Yvonnet J, Xia L, Le MV, Li G (2018) Topology optimization of periodic lattice structures taking into account strain gradient. Comput Struct 210:28–40. https://doi.org/10.1016/j.compstruc.2018.09.003
Fu J, Xia L, Gao L, Xiao M, Li H (2019) Topology optimization of periodic structures with substructuring. J Mech Des 141(7):071403. https://doi.org/10.1115/1.4042616
He G, Huang X, Wang H, Li G (2016) Topology optimization of periodic structures using beso based on unstructured design points. Struct Multidisc Optim 53(2):271–275. https://doi.org/10.1007/s00158-015-1342-9
Huang X, Xie Y (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elements Anal Des 43(14):1039–1049. https://doi.org/10.1016/j.finel.2007.06.006
Huang X, Xie YM (2008) Optimal design of periodic structures using evolutionary topology optimization. Struct Multidisc Optim 36(6):597–606. https://doi.org/10.1007/s00158-007-0196-1
Huang X, Xie M (2010a) Evolutionary topology optimization of continuum structures: methods and applications. Wiley, New York
Huang X, Xie YM (2010b) A further review of eso type methods for topology optimization. Struct Multidisc Optim 41(5):671–683. https://doi.org/10.1007/s00158-010-0487-9
Lazarov BS, Wang F, Sigmund O (2016) Length scale and manufacturability in density-based topology optimization. Arch Appl Mech 86(1):189–218. https://doi.org/10.1007/s00419-015-1106-4
Li Q, Steven GP, Xie YM (2001) A simple checkerboard suppression algorithm for evolutionary structural optimization. Struct Multidisc Optim 22(3):230–239. https://doi.org/10.1007/s001580100140
Liu J, Ma Y (2016) A survey of manufacturing oriented topology optimization methods. Adv Eng Softw 100:161–175. https://doi.org/10.1016/j.advengsoft.2016.07.017
Lund E, Stegmann J (2005) On structural optimization of composite shell structures using a discrete constitutive parametrization. Wind Energy 8(1):109–124. https://doi.org/10.1002/we.132
Osanov M, Guest JK (2016) Topology optimization for architected materials design. Annu Rev Mater Res 46(1):211–233. https://doi.org/10.1146/annurev-matsci-070115-031826
Otomori M, Yamada T, Izui K, Nishiwaki S (2015) Matlab code for a level set-based topology optimization method using a reaction diffusion equation. Struct Multidisc Optim 51(5):1159–1172. https://doi.org/10.1007/s00158-014-1190-z
Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidisc Optim 21(2):120–127. https://doi.org/10.1007/s001580050176
Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidisc Optim 48(6):1031–1055. https://doi.org/10.1007/s00158-013-0978-6
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75. https://doi.org/10.1007/BF01214002
Thomas S (2021) Finite periodic topology optimization with oriented unit-cells data set. https://doi.org/10.17632/wj6s2d8xbg.2
Thomas S, Li Q, Steven G (2020) Topology optimization for periodic multi-component structures with stiffness and frequency criteria. Struct Multidisc Optim 61(6):2271–2289. https://doi.org/10.1007/s00158-019-02481-7
van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidisc Optim 48(3):437–472. https://doi.org/10.1007/s00158-013-0912-y
Wei P, Li Z, Li X, Wang MY (2018) An 88-line matlab code for the parameterized level set method based topology optimization using radial basis functions. Struct Multidisc Optim 58(2):831–849. https://doi.org/10.1007/s00158-018-1904-8
Wu J, Sigmund O, Groen JP (2021) Topology optimization of multi-scale structures: a review. Struct Multidisc Optim 63(3):1455–1480. https://doi.org/10.1007/s00158-021-02881-8
Xia L, Xia Q, Huang X, Xie YM (2018) Bi-directional evolutionary structural optimization on advanced structures and materials: a comprehensive review. Arch Comput Methods Eng 25(2):437–478. https://doi.org/10.1007/s11831-016-9203-2
Xie YM, Zuo ZH, Huang X, Rong JH (2012) Convergence of topological patterns of optimal periodic structures under multiple scales. Struct Multidisc Optim 46(1):41–50. https://doi.org/10.1007/s00158-011-0750-8
Zhang S (in press) A comprehensive review of educational articles on structural and multidisciplinary optimization. https://doi.org/10.1007/s00158-021-03050-7
Zhang W, Sun S (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Methods Eng 68(9):993–1011. https://doi.org/10.1002/nme.1743
Zuo Z (2009) Topology optimization of periodic structures. RMIT University, Melbourne PhD thesis
Zuo ZH, Xie YM, Huang X (2011) Optimal topological design of periodic structures for natural frequencies. J Struct Eng 137(10):1229–1240. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000347
Zuo ZH, Huang X, Yang X, Rong JH, Xie YM (2013) Comparing optimal material microstructures with optimal periodic structures. Comput Mater Sci 69:137–147. https://doi.org/10.1016/j.commatsci.2012.12.006
Acknowledgements
The funding support from Australian Research Council (ARC) through Discovery scheme (DP190103752) is acknowledged. The first author is a recipient of Australian Government Research Training Program (RTP) Scholarship.
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ST: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing—Original Draft, Writing—Review & Editing, and Visualization. QL: Conceptualization, Resources, Writing—Review & Editing, Supervision, and Funding acquisition. GS: Resources, Writing—Review & Editing, and Supervision.
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Replication of results code is available in the reference data set (Thomas 2021).
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Thomas, S., Li, Q. & Steven, G. Finite periodic topology optimization with oriented unit-cells. Struct Multidisc Optim 64, 1765–1779 (2021). https://doi.org/10.1007/s00158-021-03045-4
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DOI: https://doi.org/10.1007/s00158-021-03045-4