  The elastic wave propagation is investigated in the 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, a
nonlinear optimization problem is formulated. The maximum of the largest band
gap amplitudes in the low-frequency range is selected as suited objective
function. Proper inequality constraints are introduced to restrict the optimal
solutions within a compact set of mechanical and geometric parameters,
including only physically realistic properties of both the lattice and
resonators. The optimization problems related to full and partial band gaps are
solved independently, by means of a globally convergent version of the
numerical method of moving asymptotes, combined with a quasi-Monte Carlo
multi-start technique. The optimal solutions are 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.
