Systematic topology optimization of solid-solid phononic crystals for multiple separate band-gaps with different polarizations.

Phononic crystals (PnCs) have attracted considerable interest due to their unique and outstanding band-gap characteristics. In many applications, it is desirable to have a unit cell with specific band-gaps. The distribution of elastic materials within a unit cell has significant effect on the band-gaps, which is extremely difficult to be determined without systematic synthesis method. In this paper, topology optimization techniques are utilized to obtain two-dimensional (2D) square lattice PnCs with maximized relative band-gaps between multiple consecutive bands. The optimization follows two-stage design process using Genetic algorithms (GAs) in combination with finite element method (FEM). Three numerical examples are given to optimize 2D steel/epoxy PnCs in one-eighth symmetry for coupled mode, shear mode and mixed mode respectively. The results show that the optimized PnCs with different band-gaps, which can easily be found by the developed method, have different materials layout, and the PnCs with the lowest order band-gap are simple lattice and have the highest value of application in noise reduction and vibration isolation. Some optimized PnCs with higher order band-gaps have the same lattice as those with the lowest order band-gap, and whose absolute band-gaps are inversely proportional to the minimum feature size of primitive cells.