Topological Design of Cellular Phononic Band Gap Crystals

Abstract

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.

Shape optimization of solid–air porous phononic crystal slabs with widest full 3D bandgap for in-plane acoustic waves

“…The question does not seem to have a unique answer: for instance, in the case of application of genetic algorithms, the Authors in [20] and [21] do not include the air portion in the model (which is limited to plane strain); conversely, the Authors in [23], who consider a full 3D model, prefer to take into account the void sections of porous PnC by introducing some very compliant material, but they finally point out some numerical drawbacks of that choice. In the application of BESO algorithm, see [26], the introduction of air in the model seems unaivoidable, if one wants to implement the standard optimization procedure that relies on the material switch in some specific elements. Nevertheless, the robustness and the computational burden are highly affected by the fictitious modeling of the void sections.…”

“…The Authors in [23] propose the application of a multi-objective optimization process, with the achievement of Pareto frontiers between the two conflicting objectives of maximum bandgap and maximum stiffness. Conversely, the Authors in [26] introduce additional constraints for imposing that the stiffness parameters are larger than some pre-defined thresholds: the objective function is then modified via the Lagrange multiplier technique and the optimization algorithm is modified in order to account for the additional unknowns.…”

“…Recently, genetic algorithms have been also used for topology optimization of solid-air PnCs, see [20], [21], [22] for the 2D case and [23] for the full 3D case. In the present work, we consider the application of Bidirectional Evolutionary Structural Optimization (BESO) that has been demostrated well suited for Photonic Crystals topology optimization [24], solid-solid PnCs topology optimization [25] and in sistematic 2D topology optimization of solid-air [26] PnCs. In those studies, it is shown how BESO algorithm can work well in bandgap optimization problems and can be faster than other algorithms in terms of number of iterations necessary to find the optimal topology.…”

Follow-up optimized phononic band structure design based on reserved inclusions

“…Zhang et al [31] performed the topology optimization for ABGs in PnCs with sixfold symmetric hexagonal lattice. Considering a relatively high stiffness of PnCs, Li et al [32] conducted optimization of cellular PnCs to simultaneously maximize band gap size, and bulk or shear modulus under prescribed filling fractions.…”

A polynomial‐based method for topology optimization of phononic crystals with unknown‐but‐bounded parameters

“…The bidirectional evolutionary structural optimization was developed to optimize the design of PnC with maximum BG width (BGW) under the volume constraint . Meanwhile, Li et al used the bidirectional evolutionary structural optimization in conjunction with the homogenization method for the topology optimization of cellular PnCs with a bulk or shear modulus constraint. Recently, Lu et al used the gradient‐based topology optimization (GTO) method to obtain 3D PnCs with ultrawide normalized all‐angle and all‐mode BGs.…”