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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 79
Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 297

Application of the Mathematical Theory of Homogenization in Topology Optimization Problems

Y. Wang and D. Tran

School of Architectural, Civil and Mechanical Engineering, Victoria University of Technology, Melbourne, Australia

Full Bibliographic Reference for this paper
Y. Wang, D. Tran, "Application of the Mathematical Theory of Homogenization in Topology Optimization Problems", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 297, 2004. doi:10.4203/ccp.79.297
Keywords: structural optimization, topology optimization, homogenization theory, microstructure model, composite materials, topology optimization algorithm, mathematical theory of homogenization.

Broadly speaking, problems in Structural Optimization can be divided into three types: sizing, shape and topology optimization. With the exception of a few early landmark results of Maxwell in 1895 [1] and Michell in 1904 [2], the historical development has progressed from element stiffness design, through geometric and shape optimization to topology optimization design [3,4]. With the benefit of hindsight, a more logical sequence would be of conceptual design first using topological and shape designs and then finalize the structure by determining its geometry dimensions, as no amount of fine turning of the cross-sections and thickness of the structural members will compensate for a conceptual error in the topology of the structure [5]. In this paper, the application of mathematical homogenization theory (MHT) to the topology optimization problem is considered. This approach considers the structural body as made up of periodic microstructures, the homogenized properties of which can be obtained by mathematical homogenization theory (MHT) and the topology optimization problem is viewed as optimal distribution of materials of the components that make up the heterogeneous composite in a continuous manner, unlike the on-off distribution of homogeneous material adopted by other shape and topology optimization techniques. The topology optimization problem can be defined in such a way that the geometry parameters of the void, the soft and hard materials that define the composite become the design variables, thus converting the complex and apparently intractable topology optimization into a more manageable sizing optimization problem. The design variables would change during the optimization process, thus creating holes and redistributing materials so that the structural efficiency is improved, and converge to the optimum solution. The Optimality Criteria Method using Kuhn-Tucker conditions is used to derive the numerical algorithm to update the design variables. The algorithm also solves the homogenization problem within the microstructure cell and the topology optimization over the structure. Among various microstructures developed for structural topology optimization the cross shape one material and bi-material models are chosen to illustrate the technique by investigating a benchmark problem. It was shown that the method yields the expected optimal solution, the patterns of optimal layout are similar for different sizes of microstructures and the convergence is excellent even for very coarse meshing. Further examples are given in [6].

Maxwell, C., "Scientific Papers II", Cambridge University Press, pp. 175, 1895.
Michell, A.G.M., "The limit of economy of material in frame structures", Phil. Mag., Vol. 8, 589-597, 1904.
Haftka, R.T., Gurdal, Z., "Elements of Structural Optimization, 3rd revised edition", Kluwer Academic Publishers, Dordrecht, 1992.
Rozvany, G.I.N., "Aim, scope, method, history and unified terminology of computer-aided topology optimization in structural mechanics", Struct. Multidisc. Optim., Vol. 21, 90-108, 2001. doi:10.1007/s001580050174
Olhoff, N., Bendsoe, M.P., Rasmussen, J., "On CAD-integrated structural topology and design optimization", Comp. Meth. Appl. Mech. Engrg. 89, 259-279, 1991. doi:10.1016/0045-7825(91)90044-7
Wang, Y, "A Study on Microstructures of Homogenization for Topology Optimization", PhD Thesis, Victoria University, Melbourne, Australia.

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