Mathematical theory and numerical method for subwavelength resonances in multi-layer high contrast elastic media

Abstract

In this paper, we develop a rigorous mathematical framework and numerical method for analyzing and computing the subwavelength resonances of multi-layer structures in elastic system, respectively. The system considered is constituted of a finite alternance of high-contrast segments, called the “resonators”, and a background medium. Firstly, based on layer potential theory, we derive an integral equation explicitly involving the geometric and material configurations. By the Gohberg-Sigal theory, it is theoretically demonstrated that the number of resonance frequencies increases as the number of resonators increases. There are $6\cdot \lceil \frac{N}{2}\rceil$ resonance frequencies for $\lceil \frac{N}{2}\rceil$ resonators (high contrast domain) within the N-layer structure. In addition, we derive the quantitative expressions for the subwavelength resonance frequencies within concentric balls, i.e., coaxial resonators, calculated by solving the corresponding eigenvalue problem of an explicit matrix. Finally, some numerical experiments are also provided to collaborate with the theoretical results.