Reduced order derivation of the two-dimensional band structure of a mixed-mode resonator array

Abstract

In this paper, the 2D band structure of a mixed-mode metamaterial resonator array for in-plane waves is investigated. The band structure in the interior and on the boundary of the irreducible Brillouin zone as well as 1D dispersion diagrams for different propagation angles are calculated numerically and presented. Additionally, a reduced order analytical method is established to compare and approximate the band structure. The studied metamaterial, with a T-shaped cantilever beam as resonator in its square array repeating unit cells, exhibits branches with mixed P and SV waves except at exactly one angle of propagation. This paper also reports on the occurrence of avoided level crossings, which are related to the existence of exceptional points in the complex domain. A reduced order analytical approach is used that can generate partial (low branches) band structure with relatively little computational effort. The reduced order model agrees well with the numerical results for these low branches and can provide support in mode identification and band sorting. With proper adjustments in parameters, this analytical method will be applicable for other metamaterials that have similar unit cell structure.

Reduced Order Modeling of Dynamic Mechanical Metamaterials for Analysis of Infinite and Finite Systems

“…The same is true for continuum FE models, which may lack symmetry due to their meshes. Another example can be found in 22 , where the direct FEM result suggests a real crossing between two branches, but the ROM and a further refined FEM study indicate otherwise. The ROM approach is more efficient for band sorting purposes (and can easily be leveraged in calculation of topological invariants based on path integrals), and it provides benefits in understanding the difference between an ideal model and a realistic one, as well as in distinguishing normal and accidental degeneracy.…”

“…The materials are assumed to have no loss or gain mechanisms, but the inclusion of linear viscoelastic response is considered as a natural future expansion. Detailed numerical and experimental studies on some similar micro-structures can be found in previous work 21,22 .…”

“…To accurately capture such dynamics, the elastic spring elements cannot be simple 2-DOF elements. In our previous work 22 , the elastic spring elements are physically represented as beams. The reduced stiffness and mass matrices can then be derived using simple strength of materials analysis.…”