Optimal design of low-frequency band gaps in anti-tetrachiral lattice meta-materials

The elastic wave propagation is investigated in a beam lattice material characterized by a square periodic cell with anti-tetrachiral microstructure. With reference to the Floquet-Bloch spectrum, focus is made on the band structure enrichments and modifications which can be achieved by equipping the cellular microstructure with tunable local resonators. By virtue of its composite mechanical nature, the so-built inertial meta-material gains enhanced capacities of passive frequency-band filtering. Indeed the number, placement and properties of the inertial resonators can be designed to open, shift and enlarge the band gaps between one or more pairs of consecutive branches in the frequency spectrum. In order to improve the meta-material performance, several nonlinear optimization problems are formulated. The largest among the band gap amplitudes in the low-frequency range is selected as suited objective function. Proper inequality constraints are introduced to restrict the admissible solutions within a compact set of mechanical and geometric parameters, including only physically realistic properties of both the lattice and the resonators. The optimization problems related to full and partial band gaps are solved by using a globally convergent version of the numerical method of moving asymptotes, combined with a quasi-Monte Carlo multi-start technique. The optimal solutions are numerically computed, discussed and compared from the qualitative and quantitative viewpoints, bringing to light the limits and potential of the meta-material performance. The clearest trends emerging from the numerical analyses are pointed out and interpreted from the physical viewpoint. Finally, some specific recommendations about the microstructural design of the meta-material are synthesized.