It is known that composite materials with improved properties can be achieved through modifications to the topology of their microstructures. Structural topology optimization approaches can be utilized as a systematic way for finding the best spatial distribution of constituent phases within the microstructures of materials/composites. This study presents a new approach for designing material’s microstructures based on the Bi-directional Evolutionary Structural Optimization (BESO) methodology. It is assumed that the materials/composites are composed of repeating microstructures known as periodic base cells (PBC). The goal is to find the best spatial distribution of constituent phases within the PBC, in such a way that materials with desired or improved functional properties are achieved. To this end, the Homogenization theory is applied to establish a relationship between properties of materials microstructure and their macroscopic characteristics. As the first step of this study, the optimization problem is formulated to find microstructures for materials with maximum stiffness, in the form of bulk or shear modulus, or thermal conductivity. Cellular materials, which are composed of one solid phase and one void phase, are considered at this stage. By conducting finite element analysis of the PBC, and applying the Homogenization theory, elemental sensitivity numbers are derived. By gradual removing and adding elements in an iterative process, the optimal topology of the solid phase within the PBC is found. In the next stage of this study, the aim is to combine additional performance constraint to the above procedure. Maximization of bulk or shear modulus is selected as the objective of the material design, subject to the constraint on the isotropy of material and volume constraint. The methodology is extended into topology optimization of microstructures for composites of two or more non-zero constituent phases. For design of material with maximum stiffness or thermal conductivity, the constituent phases are divided into groups and sensitivity analysis is performed between different groups. The developed methodology is also applied in designing functionally graded material (FGM), in which the mechanical property of material gradually changes. It is assumed that the microstructure of the FGM is composed of a series of cellular base cells in the direction of gradation and self-repeated in other directions. Finally, an approach is proposed for the topological design of FGMs with two non-zero constituent phases and multi graded properties. The objective of optimization is defined to find the stiffest materials with prescribed gradation of thermal conductivity. Similar to the approach used for cellular FGMs, the connectivity of base cells is maintained by considering three base cells at each stage. The effectiveness and computational efficiency of the proposed approaches are numerically tested, through designing a range of 2D and 3D microstructures for materials. A series of new and interesting microstructures of materials are presented. The results clearly indicate the advantages of BESO utilization in terms of computational costs and convergence speed, quality of generated microstructures, and ease of implementation as a post processing algorithm
