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An efficient 3D homogenization-based topology optimization methodology. (English) Zbl 07360514
Summary: Homogenization theory forms the basis for solving the topology optimization problem (TOP) formulated for designing composite materials. Homogenization is proved to be an efficient approach to effectively determine the equivalent macroscopic properties of the composite material. It relies on the assumption that the composite material presents a periodic pattern on a microstructural level; the simplest repeating unit of the microstructure, that if isolated represents exactly the macroscopic behaviour of the material, is called the unit cell. Scope of homogenization is to determine the macroscopic (or else effective) properties of the non-homogeneous unit cell, that is, determine the properties of the unit cell as if it was composed by homogeneous material. In this study, a simple methodology is proposed where homogenization is implemented on a 3D lattice unit cell, with its radius being considered as the alternating parameter for the homogenization procedure; different values of the radius result to different unit cell configurations and hence, to different equivalent properties. A fitting process takes place in order to appropriately model the variations of the obtained effective properties w.r.t the design parameter. The corresponding, homogenization-based TOP is formed and the accuracy of the proposed methodology is assessed on several case studies.
MSC:
74-XX Mechanics of deformable solids
Software:
top88.m; SNOPT; top.m
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[1] Lagaros, ND, The environmental and economic impact of structural optimization, Struct Multidiscip Optim, 58, 4, 1751-1768 (2018)
[2] Kanellopoulos, I.; Sotiropoulos, S.; Kazakis, G.; Lagaros, ND, Topology optimization aided structural design: interpretation, computational aspects and 3d printing, Heliyon (2017)
[3] Sotiropoulos, S.; Kazakis, G.; Lagaros, ND, Conceptual design of structural systems based on topology optimization and prefabricated components, Comput Struct (2020)
[4] Christensen PW, Klarbring A (2009) An introduction to structural optimization, volume 153 of Solid mechanics and its applications. Springer, Netherlands. ISBN 978-1-4020-8665-6. doi:10.1007/978-1-4020-8666-3
[5] Bendsøe, MP; Lund, E.; Olhoff, N.; Sigmund, O., Topology optimization-broadening the areas of application, Control Cybern, 34, 7-35 (2005) · Zbl 1167.74529
[6] Sigmund O(2019) Topology optimization state-of-the-art and future perspective, (last accessed November ). https://goo.gl/PCqrgk
[7] Paulino GH (2013) Where are we in topology optimization? In: 10th World Congress on structural and multidisciplinary optimization. Orlando, FL, USA
[8] Rozvany, GIN; Olhoff, N., Topology optimization of structures and composite continua (2001), Dordrecht: Springer, Dordrecht
[9] Bendsøe, MP, Optimal shape design as a material distribution problem, Struct Multidiscip Optim, 1, 4, 193-202 (1989)
[10] Zhou, M.; Rozvany, GIN, The coc algorithm, part II: topological, geometrical and generalized shape optimization, Comput Methods Appl Mech Eng, 89, 1-3, 309-336 (1991)
[11] Mlejnek, HP, Some aspects of the genesis of structures, Struct Multidiscip Optim, 5, 1-2, 64-69 (1992)
[12] Wang, MY; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput Methods Appl Mech Eng, 192, 1-2, 227-246 (2003) · Zbl 1083.74573
[13] Allaire, G.; Jouve, F.; Toader, AM, Structural optimization using sensitivity analysis and level-set method, J Comput Phys, 194, 363-393 (2004) · Zbl 1136.74368
[14] Xie Y, Steven GP(1992) Shape and layout optimization via an evolutionary procedure. In: Proceedings of the international conference computational engineering science, volume 194. Hong Kong University, Hong Kong, 17-22 December, p 363
[15] Xie, Y.; Steven, GP, A simple evolutionary procedure for structural optimization, Comput Struct, 49, 5, 885-896 (1993)
[16] Querin, OM; Steven, GP; Xie, YM, Evolutionary structural optimization using a bidirectional algorithm, Eng Comput, 15, 8, 1031-1048 (1998) · Zbl 0938.74056
[17] Fujii, D.; Chen, BC; Kikuchi, N., Composite material design of two-dimensional structures using the homogenization design method, Int J Numer Meth Eng, 50, 2031-2051 (2001) · Zbl 0994.74055
[18] Liu, L.; Yan, J.; Cheng, G., Optimum structure with homogeneous optimum truss-like material, Comput Struct, 86, 1417-1425 (2008)
[19] Wang, Y.; Wang, MY; Chen, F., Structure-material integrated design by level sets, Struct Multidiscip Optim, 54, 1145-1156 (2016)
[20] Chen, W.; Tong, L.; Liu, S., Concurrent topology design of structure and material using a two-scale topology optimization, Comput Struct, 178, 119-128 (2017)
[21] Rodrigues, H.; Guedes, J.; Bendsøe, MP, Hierachical optimization of material and structure, Struct Multidiscip Optim, 24, 1, 1-10 (2002)
[22] Lazarov B(2013) Topology optimization using multiscale finite element method for high-contrast media. In: International conference on large-scale scientific computing-LSSC 2013, Sozopol, Bulgaria
[23] Alexandersen, J.; Lazarov, B., Topology optimization of manufacturable microstructural details without length scale separation using a spectral coarse basis precoditioner, Comput Methods Appl Mech Eng, 290, 156-182 (2015) · Zbl 1423.74738
[24] Pantz, O.; Trabelsi, K., A post-treatment of the homogenization method for shape optimization, SIAM J Control Optim, 47, 3, 1380-1398 (2008) · Zbl 1161.49042
[25] Pantz O, Trabelsi K(2010) Construction of minimization sequences for shape optimization. In: 15th International conference on methods and models in automation and robotics, Miedzyzdroje, Poland, pp 278-283. doi:10.1109/MMAR.2010.5587222
[26] Groen, JP; Sigmund, O., Homogenization-based topology optimization for high-resolution manufacturable microstructures, Int J Numer Methods Eng (2018)
[27] Westermann, N.; Sigmund, R.; Wu, O.; Aage, J., Infill optimization for additive manufacturing-approaching bone-like porous structures, IEE Trans Visual Comput Graph (2018)
[28] Sigmund, J.; Groen, O.; Wu, JP, Homogenization-based stiffness optimization and projection of 2d coated structures with orthotropic infill, Comput Methods Appl Mech Eng (2019) · Zbl 1441.74149
[29] Pantz, G.; Geoffroy-Bonders, O.; Allaire, P., 3-d topology optimization of modulated ad oriented periodic microstructures by the homogenization method, J Comput Phys (2020) · Zbl 1443.74246
[30] Cheng, X.; Yan, G.; Guo, J., Multi-scale concurrent material and structural design under mechanical and thermal loads, Comput Mech (2016) · Zbl 1382.74103
[31] Shukla, A.; Misra, A., Review of optimality criterion approach scope, limitation and development in topology optimization, Int J Adv Eng Technol, 6, 4, 1886-1889 (2013)
[32] Svanberg, K., The method of moving asymptotes - a new method for structural optimization, Int J Numer Meth Eng, 24, 2, 359-373 (1987) · Zbl 0602.73091
[33] Bendsøe MP, Sigmund O (2004) Topology optimization: theory, 2nd edn. Methods and applications. Springer, Berlin Heidelberg. ISBN 978-3-540-42992-0. doi:10.1007/978-3-662-05086-6 · Zbl 1059.74001
[34] Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures, vol 7, 1st edn. North Holland, New York. ISBN 978-0-8218-5324-5 · Zbl 0404.35001
[35] Bendsøe, MP; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput Methods Appl Mech Eng, 71, 197-224 (1988) · Zbl 0671.73065
[36] Hassani, B.; Hinton, E., A review of homogenization and topology optimization i-homogenization theory for media with periodic structure, Comput Struct, 69, 6, 707-717 (1998) · Zbl 0948.74048
[37] Hassani, B.; Hinton, E., A review of homogenization and topology optimization ii analytical and numerical solution of homogenization equations, Comput Struct, 69, 6, 719-738 (1998) · Zbl 0948.74048
[38] Hassani, B.; Hinton, E., A review of homogenization and topology optimization i-topology optimization using optimality criteria, Comput Struct, 69, 6, 739-756 (1998) · Zbl 0948.74048
[39] Groen J, Stutz F, Aage N , Bærentzen JA, Sigmund O. De-homogenization of optimal multi-scale 3d topologies. https://arxiv.org/abs/1910.13002 · Zbl 1442.74148
[40] Gao, J.; Luo, Z.; Xia, L.; Gao, L., Concurrent topology optimization of multiscale composite structures in matlab, Struct Multidiscip Optim, 60, 2621-2651 (2019)
[41] Allaire, G.; Geoffroy-Donders, P.; Pantz, O., Topology optimization of modulated and oriented periodic microstructures by the homogenization method, Comput Math Appl, 78, 7, 2197-2229 (2019) · Zbl 1443.74246
[42] Groen, JP; Wu, J.; Sigmund, O., Homogenization-based stiffness optimization and projection of 2d coated structures with orthotropic infill, Comput Methods Appl Mech Eng, 349, 722-742 (2019) · Zbl 1441.74149
[43] Monteiro A (2017) Topology optimization of microstructures with constraints on average stress and material properties. Master’s thesis, Instituto Superior Técnico, Universidade de Lisboa, Portugal
[44] Wu, J.; Wang, W.; Gao, X., Design and optimization of conforming lattice structures, IEEE Trans Visual Comput Graph (2019)
[45] Andreassen, E.; Andreasen, CS, How to determine composite material properties using numerical homogenization, Comput Mater Sci, 83, 488-495 (2014)
[46] Guoying, D.; Yunlong, T.; Yaoyao, FZ, A 149 line homogenization code for three-dimensional cellular materials written in matlab, J Eng Mater Technol, 141, 1, 488-495 (2018)
[47] Liu, K.; Tovar, A., An efficient 3d topology optimization code written in matlab, Struct Multidiscip Optim, 50, 6, 1175-1196 (2014)
[48] Gill, P.; Murray, W.; Saunders, M., SNOPT: an SQP algorithm for large-scale constrained optimization, Soc Ind Appl Math, 12, 4, 979-1006 (2002) · Zbl 1027.90111
[49] Bourdin, B., Filters in topology optimization, Int J Numer Meth Eng, 50, 9, 2143-2158 (2001) · Zbl 0971.74062
[50] Pedersen CG, Lund JJ, Damkilde L, Kristensen AS (2006) Topology optimization—improved checker-board filtering with sharp contours. In: Proceedings of the 19th nordic seminar on computational mechanics, Lund, Sweden, 20-21 October, p 182
[51] Wang, SY; Lim, KM; Khoo, BC; Wang, MY, A hybrid sensitivity filtering for topology optimization, Comput Model Eng Sci, 24, 1, 21-50 (2008) · Zbl 1232.74083
[52] Sigmund, O., A 99 line topology optimization code written in matlab, Struct Multidiscip Optim, 21, 2, 120-127 (2001)
[53] Andreassen, E.; Clausen, A.; Schevenels, M.; Lazarov, BS; Sigmund, O., Efficient topology optimization in matlab using 88 lines of code, Struct Multidiscip Optim, 43, 1, 1-16 (2011) · Zbl 1274.74310
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