Platonic quasi-normal modes expansion

TL;DR

This paper introduces a robust quasi-normal modes framework for elastic wave control in thin plates, enabling efficient modeling, optimization, and design of resonator arrays for advanced wave manipulation.

Contribution

It develops a dispersive QNM expansion for elastic plates, deriving sensitivities and applying gradient-based optimization for designing resonant states and tailored wave responses.

Findings

Validated the QNM approach with scattering simulations

Demonstrated design of quasi-bound states in the continuum

Enabled precise eigenfrequency positioning in complex structures

Abstract

Elastic wave manipulation using large arrays of resonators is driving the need for advanced simulation and optimization methods. To address this we introduce and explore a robust framework for wave control: Quasi-normal modes (QNMs). Specifically we consider the problem for thin elastic plates, where the Green's function formalism is well known and readily exploited to solve multiple scattering problems. By studying the associated nonlinear eigenvalue problem we derive a dispersive QNM expansion, providing a reduced-order model for efficient forced response computations which reveals physical insight into the resonant mode excitation. Furthermore, we derive eigenvalue sensitivities with respect to resonator parameters and apply a gradient-based optimization to design quasi-bound states in the continuum and position eigenfrequencies precisely in the complex plane. Scattering simulations validate our approach in structures such as graded line arrays and quasi-crystals. Drawing on QNM concepts from electromagnetism we demonstrate significant advances in elastic metamaterials, highlighting their potential for tailored wave manipulation.