Structural topology optimization approaches help engineers to find the best layout or configuration of members in structural systems. However, these approaches differ in terms of computational costs and efficiency, quality of generated topologies, robustness, and the level of effort for implantation as a computational post-processing procedure. On the other hand, one common approach for saving resources is the application of porous or composite materials that have extreme or tailored properties. It is known that composite materials with improved properties can be designed by modifications into the topology of their microstructures. A systematic way for improving the properties of these types of materials consists of application of a structural topology optimization approach to find the best spatial distribution of materials within the microstructures of composites. This study presents new approaches for design of microstructures for materials based on the bidirectional evolutionary structural optimization (BESO) methodology. It is assumed that the materials are composed of repeating microstructures known as periodic base cells (PBC). The goal is to apply the BESO topology optimization 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 material properties in microstructural and macrostructural length scales. In the first stage of this study, the optimization problem is formulated to find microstructures for composites of prescribed volume constrains with maximum effective Young's moduli. It is assumed that the base materials are composed of two materials with different Poisson’s ratios. By performing finite element analysis on the PBC and applying the homogenization theory, an elemental sensitivity analysis is conducted. Following by removing and adding elements gradually in an iterative process according to their sensitivity ranking, the optimal topology for the PBC can be generated. The effectiveness and computational efficiency of the proposed approach is numerically exemplified through a range of 3D topology optimization problems. In the next stage of this study, the optimization problem is formulated to find microstructures for composites of prescribed volume constrains with maximum stiffness in the form of bulk or shear modulus. Here the composites possess two constituent phases. Compared with cellular materials whose microstructures are made of a solid phase and a void phase, composites of two different material phases are more advantageous since they can provide a wider range of performance characteristics. Maximization of bulk or shear modulus subject to a volumetric constraint is selected as the objective of the material design. Adding and removing of elements is performed based on the ranking of sensitivity numbers and imposed volumetric constraint between different base materials. The proposed procedure demonstrates very stable convergence without any numerical difficulty. The computational efficiency of the proposed approach has been demonstrated by numerical examples. A series of new and interesting microstructures of two base materials are presented. The other major advantage of the BESO in design of composites of two base materials is the distinctive interfaces between constituent phases in the generated microstructures, which make the manufacturing of microstructures viable. The methodology has the capability to be extended for material optimization with other objective or constraint functions.
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