Computational & Technology Resources
an online resource for computational,
engineering & technology publications
Civil-Comp Proceedings
ISSN 1759-3433
CCP: 77
Edited by: B.H.V. Topping
Paper 131

Topology Optimization using Homogenization

Y. Wang, M. Xie and D. Tran

Faculty of Science, Engineering and Technology, Victoria University of Technology, Melbourne, Australia

Full Bibliographic Reference for this paper
Y. Wang, M. Xie, D. Tran, "Topology Optimization using Homogenization", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 131, 2003. doi:10.4203/ccp.77.131
Keywords: structural optimization, topology optimization, homogenization theory, microstructure model, composite materials, topology optimization algorithm.

Among structural topology optimization methods, the homogenization method proves to be one of the most general approach to shape and topology optimization. This method is based on the mathematical homogenization theory (MHT) and by considering a structural body as consisting of microstructures or base cells which are periodically distributed, each cell is made of materials with holes or voids. Thereby the complex nature of the topological optimization problem can be converted to a sizing problem. In fact, the idea of using a cellular body with a periodic microstructure moves the on-off nature of the topology optimization problem of distributing materials from the macroscopic scale to the microscopic scale. In mathematical terms the introduction of microstructures corresponds to a relaxation of the variation problem that can be established for the optimization problem. So the heart of homogenization method for topology optimization is formulating a microstructure model for a design domain. Although there has been considerable work done in the fields of the structural optimization, at present the studies on the microstructures of the homogenization method is comparatively modest and limited, extensive research has been carried out on one-material microstructures, few examples exist in the literature on bi-material microstructures and their applications to topology optimization.

In this paper, new one-material microstructures and bi-material microstructures are proposed. The topology optimization problem can be defined in such a way that the geometry parameters of the void, the soft and hard materials become the design variables. The dimensions the void and the percentages of the soft and hard materials (volume fractions) will change during the optimization process, thus creating holes and distributing the two materials so that the structural efficiency is improved. First a brief review of MHT and its application in topology optimization is presented, followed by the development of new microstructures, of both the one-material and bi-material models. Numerical algorithms to exploit both presently available and newly developed microstructures models in topology optimization are presented. The checkerboard problem in the layout of structure due to numerical instability is overcome by a filtering scheme applying to Lagrangian functions of the optimization algorithm. A number of case studies of topology optimization of two-dimensional structures are investigated using various microstructures. It was shown that new microstructures developed are capable of producing stable solution with improved convergence. Further works are needed to extend the homogenization method to three-dimensional problems.

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. and 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
Bennett, J.A. and Botkin, M.E. (Eds.), "The Optimum Shape, Automated Structural Design", Plenum, New York, 1986.
Mattheck, C., "Design in Nature: Learning from Trees" Springer-Verlag, Berlin, 1997.
Xie, Y.M. and Steven, G.P., "A simple evolutionary procedure for structural Optimization", Computers and Structures, Vol. 49, 885-896, 1993. doi:10.1016/0045-7949(93)90035-C
Babuska, I., "Solution of interface problems by homogenization I", Journal of Mathematical Analysis, Vol. 8, 603-634,1976. doi:10.1137/0507048
Bendsoe, M.P. and Kikuchi, N., "Generating optimal topologies in structural design using a homogenization method", Computer Methods in Applied Mechanics and Engineering, Vol. 71, 197-224, 1998. doi:10.1016/0045-7825(88)90086-2
Hassani, B. and Hinton, E., "Homogenization and Structural Topology Optimization", Springer-Verlag, Berlin, 1998.
Bendsoe, M.P., "Optimization of Structural Topology, Shape, and Material", Springer-Verlag, Berlin, 1995.
Sigmund, O., "Systematic Design of Mechanical Systems using Topology Optimization", in "Engineering Design Optimization", Sienz, J. (Editor), 5-12, 2000.
Bendsoe, M.P. and Sigmund, O., "Topology Optimization, Theory, Methods and Applications", Springer, Berlin, 2003.
Thomsen, J., "Topology optimisation of structures composed of one or two materials", Structural Optimization, Vol. 5, 108-115, 1992. doi:10.1007/BF01744703
Sigmund, O., "A new class of extremal composites", Journal of the Mechanics and Solids, 48, 397-428, 2000. doi:10.1016/S0022-5096(99)00034-4

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £123 +P&P)