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CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY 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 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", CivilComp 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.
Summary
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 crosssections 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 onoff 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 KuhnTucker 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 bimaterial
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].
References
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