This paper presents a comprehensive review of the literature and trends on the design of periodic microstructural composite materials including photonic crystals (PtCs), phononic crystals (PnCs) and metamaterials (MMs) by topology optimization. The properties of these materials rely highly on the constitutive materials and their spatial distributions and the resulting materials may exhibit various special properties, e.g., photonic/phononic band gaps, negative permittivity/permeability, negative effective modulus and negative refraction. Therefore, PtCs, PnCs and MMs can be viewed as structural materials with periodic unit cells and the design of which is a typical topology optimization problem for desired properties or functionalities. In recent years, a great amount of research has implemented topology optimization for designing these structural materials as well as the associated functional devices. This review summarises the most recent development in topology optimization of PtCs, PnCs and MMs and the possible directions for future research are recommended.

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This paper systematically investigated the topological design of cellular phononic crystals with a maximized gap size between two adjacent bands. Considering that the obtained structures may sustain a certain amount of static loadings, it is desirable to ensure the optimized designs to have a relatively high stiffness. To tackle this issue, we conducted a multiple objective optimization to maximize band gap size and bulk or shear modulus simultaneously with a prescribed volume fraction of solid material so that the resulting structures can be lightweight, as well. In particular, we first conducted the finite element analysis of the phononic band gap crystals and then adapted a very efficient optimization procedure to resolve this problem based on bi-directional evolutionary structure optimization (BESO) algorithm in conjunction with the homogenization method. A number of optimization results for maximizing band gaps with bulk and shear modulus constraints are presented for out-of-plane and in-plane modes. Numerical results showed that the optimized structures are similar to those obtained for composite case, except that additional slim connections are added in the cellular case to support the propagation of shear wave modes and meanwhile to satisfy the prescribed bulk or shear modulus constraints.

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