Optimum design of phononic crystal perforated plate structures for widest bandgap of fundamental guided wave modes and maximized in-plane stiffness

This paper presents a topology optimization of single material phononic crystal plate (PhP) to be produced by perforation of a uniform background plate. The primary objective of this optimization study is to explore widest exclusive bandgaps of fundamental (first order) symmetric or asymmetric guided wave modes as well as widest complete bandgap of mixed wave modes (symmetric and asymmetric). However, in the case of single material porous phononic crystals the bandgap width essentially depends on the resultant structural integration introduced by achieved unitcell topology. Thinner connections of scattering segments (i.e. lower effective stiffness) generally lead to (i) wider bandgap due to enhanced interfacial reflections, and (ii) lower bandgap frequency range due to lower wave speed. In other words higher relative bandgap width (RBW) is produced by topology with lower effective stiffness. Hence in order to study the bandgap efficiency of PhP unitcell with respect to its structural worthiness, the in-plane stiffness is incorporated in optimization algorithm as an opposing objective to be maximized. Thick and relatively thin Polysilicon PhP unitcells with square symmetry are studied. Non-dominated sorting genetic algorithm NSGA-II is employed for this multi-objective optimization problem and modal band analysis of individual topologies is performed through finite element method. Specialized topology initiation, evaluation and filtering are applied to achieve refined feasible topologies without penalizing the randomness of genetic algorithm (GA) and diversity of search space. Selected Pareto topologies are presented and gradient of RBW and elastic properties in between the two Pareto front extremes are investigated. Chosen intermediate Pareto topology, even not extreme topology with widest bandgap, show superior bandgap efficiency compared with the results reported in other works on widest bandgap topology of asymmetric guided waves, available in the literature. Finally, steady state and transient frequency response of finite thin PhP structures of selected Pareto topologies are studied and validity of obtained bandgaps is confirmed.