Efficient wave analysis in multi-layered locally resonant metamaterials: A semi-analytical approach integrating dynamic homogenization

This work proposes a novel framework that combines dynamic computational homogenization with an extended transfer matrix method (TMM) to efficiently model wave propagation in locally resonant metamaterials (LRMs) with arbitrary microstructures. Unlike other methods in the literature, which assume specific symmetries and normal incidences, the presented approach addresses general multi-layered LRM setups for 2D and 3D wave propagation, including oblique incidences. First, the dynamic computational homogenization is applied to an LRM to extract the effective homogenized inertial and mechanical characteristics, yielding a macro-scale homogenized enriched continuum description. The enriched continuum description provides frequency-dependent properties, such as the effective dynamic impedance tensor, revealing wave attenuation behaviors near resonance frequencies. Wave propagation is then analyzed in multi-layered LRM setups with acoustic and/or elastic incoming media. A constrained dispersion equation is solved numerically to accurately model interface interactions without relying on analytical simplifications. The framework is validated against direct numerical simulations (DNS) through several representative case studies, demonstrating its versatility and significant computational efficiency. This novel approach paves the way for efficient wave impedance control and transmission analyses, providing new insights into the design and functionality of LRMs for advanced acoustic devices, such as acoustic filters and waveguides.