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.

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