  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.
