Maximization of operating frequency ranges of hyperbolic elastic metamaterials by topology optimization

Hyperbolic elastic metamaterials developed for sub-wavelength resolution allow wave propagation in the radial direction but prohibit wave propagation in the circumferential direction. Recently, a two-dimensional elastic metamaterial truly exhibiting the hyperbolic behavior has been realized and also experimented but there is a practically important design issue that its operating frequency range should be widened. Motivated by this need, the present investigation aims to set up a topology optimization formulation to maximize the operating frequency range. Because different wave physics are involved along the circumferential and radial directions, the topology optimization requires the extraction of the key physical phenomena along the two different directions. In doing so, the wave physics occurring in the hyperbolic elastic metamaterial is analyzed by using equivalent discrete models and the findings from the analysis are used to set up a topology optimization problem. The topology optimization that maximizes the operating frequency range of the hyperbolic elastic metamaterial is newly formulated by using the finite element method. After the metamaterial configuration maximizing the frequency range is found, the mechanics hidden in the optimized configuration is explained in some details by using analytic mass-spring model.