The current techniques for topology optimization of material microstructure are typically based on infinitely small and periodically repeating base cells. These base cells have no actual size. It is uncertain whether the topology of the microstructure obtained from such a material design approach could be translated into real structures of macroscale. In this work we have carried out a first systematic study on the convergence of topological patterns of optimal periodic structures, the extreme case of which is a material microstructure with infinitesimal base cells. In a series of numerical experiments, periodic structures under various loading and boundary conditions are optimized for stiffness and frequency. By increasing the number of unit cells, we have found that the topologies of the unit cells converge rapidly to certain patterns. It is envisaged that if we continue to increase the number of unit cells and thus reduce the size of each unit cell until it becomes the infinitesimal material base cell, the optimal topology of the unit cell would remain the same. The finding from this work is of significant practical importance and theoretical implication because the same topological pattern designed for given loading and boundary conditions could be used as the optimal solution for the periodic structure of vastly different scales, from a structure with a few (e.g. 20) repetitive modules to a material microstructure with an infinite number of base cells.
Topology optimization Stiffness optimization Frequency optimization Topological pattern
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This work is supported by the Australian Research Council under its Discovery Projects funding scheme (Project No. DP1094401).
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, method and application. Springer, BerlinGoogle Scholar
Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–390CrossRefGoogle Scholar
Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43:1039–1049CrossRefGoogle Scholar
Huang X, Xie YM (2008) Optimal design of periodic structures using evolutionary topology optimization. Struct Multidisc Optim 36:597–606CrossRefGoogle Scholar
Huang X, Xie YM (2010) Evolutionary topology optimization of continuum structures: methods and applications. Wiley, ChichesterzbMATHCrossRefGoogle Scholar
Huang X, Zuo ZH, Xie YM (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88:357–364CrossRefGoogle Scholar
Liu L, Yan J, Cheng G (2008) Optimum structure with homogeneous optimum truss-like material. Comput Struct 86:1417–1425CrossRefGoogle Scholar
Moses E, Fuchs MB, Ryvkin M (2003) Topological design of modular structures under arbitrary loading. Struct Multidiscipl Optim 24:407–417CrossRefGoogle Scholar
Neves MM, Rodrigues H, Guedes JM (2000) Optimal design of periodic linear elastic microstructures. Comput Struct 76(1–3):421–429CrossRefGoogle Scholar
Sigmund O (1995) Tailoring materials with prescribed elastic properties. Mech Mater 20:351–368CrossRefGoogle Scholar
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33:401–424CrossRefGoogle Scholar
Sigmund O, Peterson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75CrossRefGoogle Scholar
Wang MY, Wang, X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246zbMATHCrossRefGoogle Scholar
Zhang W, Sun S (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Methods Eng 68:993–1011zbMATHCrossRefGoogle Scholar
Zhou M, Rozvany GIN (1992) DCOC: an optimality criteria method for large systems. Part I: theory. Struct Optim 5:12–25CrossRefGoogle Scholar