Effective mass density based topology optimization of locally resonant acoustic metamaterials for bandgap maximization

Because effective material properties are essential concepts in the analyses of wave phenomena in metamaterials, they may also be utilized in the optimal design of metamaterials. In this work, we propose a topology optimization method directly using the Effective Mass Density (EMD) concept to maximize the first bandgaps of two-dimensional solid Locally Resonant Acoustic Metamaterials (LRAMs). When the first bandgap is characterized by the negative EMD, the bandgap maximization can be formulated efficiently as a topology optimization problem to broaden the frequency zone of the negative EMD values. In this work, EMD is calculated by considering the macroscopic isotropy of LRAMs in the long wavelength limit. To facilitate the analytical sensitivity analysis, we propose an elaborate calculation scheme of EMD. A sensitivity averaging technique is also suggested to guarantee the macroscopically isotropic behavior of the LRAMs. In the present study, the coating layer interfacing the core and the matrix of a ternary LRAM is chosen as the design region because it significantly influences the bandgap. By considering several numerical examples, the validity of this method is verified, and the effects of the mass constraint ratios on the optimized results are also investigated.