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

Abstract

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.

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

“…where A is a 24 × 24 complex matrix, function of the dimensionless angular frequencyω 1 and wave vector K, slenderness λ 1 and λ 2 , aspect ratio α and geometric ratio χ. Vector c collects the 24 complex constants, C pq and D pq , appearing in the displacement field, Eqs. (5). Introducing the following normalization…”

Machine-Learning Techniques for the Optimal Design of Acoustic Metamaterials

“…This computational cost-saving target can be achieved by introducing acceptable simplifications, including -for instance -the adoption of machine-learning techniques for function interpolation or approximation. It has to be remarked that the considerations motivating the employment of machine-learning techniques are quite general, and their validity holds also for other optimizable topologies of periodic metamaterials [15], [16], [17].…”

“…Differently from [15], [16], [17], [22], the SLP algorithm is used instead of the Globally Convergent version of the Method of Moving Asymptotes (GCMMA) [33], [34], since some preliminary simulations suggest that, at least for the specific optimization problem, the performance of SLP does not depend strongly on the interpolation quality. Technical details about the mesh-free interpolation are reported in Appendix C; d) an iterative optimization algorithm is applied for a finite number s N of times, by adopting a Quasi-Monte Carlo multi-start initialization approach (identical to that employed in [15], [16], [17], [22]), which produces s N different starting points. In addition, a subset of the same input training points used for the construction of the interpolant is selected also to perform the Quasi-Monte Carlo multi-start initialization, since on such points the surrogate objective function reproduces exactly the true objective function, committing a zero approximation error therein.…”

Multi-parametric sensitivity analysis of the band structure for tetrachiral acoustic metamaterials

“…where the O( 0 )-order combination parameter 2 s = 2 2 /χ 2 plays a key role. Indeed, it can be verified that the condition (18) is asymptotically equivalent to 2 s > 1 + 12 2 − δ 2 + 12 2 δ 2 (23) which essentially requires 2 s > 1, if higher orders are neglected. This low-order condition is fully consistent with the asymptotic approximation (19) obtained for the band gap amplitude.…”