Topology optimization design of multi-modal resonators for metamaterial panels with maximized broadband vibroacoustic attenuation

Resonant metamaterial panels can achieve exceptional vibroacoustic attenuation by subwavelength local resonators attached to a host structure. While single-mode resonators provide improvements only in a narrow band, recent advancements propose multi-modal resonators for broadband attenuation. However, existing designs utilize only a limited number of modes, and effective methodologies are lacking to systematically design resonator layouts that achieve an appropriate amount of resonances across the target frequency range, with an adequate participating mass. In this study, we tackle this challenge by developing a dedicated topology optimization method for multi-modal resonator design in metamaterial panels. The method leverages effective thin plate modelling, i.e. a homogenized metamaterial representation through an effective mass density, which provides accurate and very efficient vibroacoustic predictions. The optimization objective is to maximize the broadband diffuse field sound transmission loss (STL) of the metamaterial panel while constraining mass. The optimization problem is solved by gradient-based mathematical programming, for which the necessary (adjoint) sensitivities of objective and constraints are derived. The efficacy of the proposed topology optimization approach is demonstrated by targeting the suppression of coincidence dips in orthotropic host plates. For narrowband coincidence dips, the optimized resonator exhibits maximized mass participation in one single mode of interest. For broadband coincidence dips, the method effectively generates multi-modal resonators with up to 6 resonances distributed across the target frequency range. It is finally demonstrated that the topology-optimized designs surpass the performance of simpler, parametrically optimized multi-modal resonator layouts, as well as of conventional treatments by a damping layer of rubber.