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Reduced Order Model for Broadband Superabsorption of Waves by Metascreens

Habib Ammari, Yu Gao, Lara Vrabac

Abstract

This work presents a new design for broadband absorption of low-frequency acoustic waves using a thin coating made of subwavelength acoustic resonators arranged periodically on a reflective surface. We first study the associated scattering problem and the corresponding subwavelength resonance problem, and then derive analytical approximations for the resonant frequencies and the reflection coefficient in terms of the periodic capacitance matrix in a half-space with a Dirichlet boundary condition. These approximations yield an effective macroscopic description of the coating via an impedance boundary condition and clarify the mechanism of superabsorption through an approximate coupling condition. Moreover, they lead to a reduced order model that enables efficient evaluation of the scattered waves over a frequency band and accelerates broadband absorption design. Building on this reduced order model, we develop a gradient based shape optimization method using shape derivatives of the resonant quantities to achieve broadband absorption. Numerical experiments demonstrate the broadband performance and the effectiveness of the proposed design procedure.

Reduced Order Model for Broadband Superabsorption of Waves by Metascreens

Abstract

This work presents a new design for broadband absorption of low-frequency acoustic waves using a thin coating made of subwavelength acoustic resonators arranged periodically on a reflective surface. We first study the associated scattering problem and the corresponding subwavelength resonance problem, and then derive analytical approximations for the resonant frequencies and the reflection coefficient in terms of the periodic capacitance matrix in a half-space with a Dirichlet boundary condition. These approximations yield an effective macroscopic description of the coating via an impedance boundary condition and clarify the mechanism of superabsorption through an approximate coupling condition. Moreover, they lead to a reduced order model that enables efficient evaluation of the scattered waves over a frequency band and accelerates broadband absorption design. Building on this reduced order model, we develop a gradient based shape optimization method using shape derivatives of the resonant quantities to achieve broadband absorption. Numerical experiments demonstrate the broadband performance and the effectiveness of the proposed design procedure.
Paper Structure (21 sections, 13 theorems, 166 equations, 9 figures, 1 algorithm)

This paper contains 21 sections, 13 theorems, 166 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement and background
  3. Main results and outline
  4. Preliminaries
  5. Quasi-periodic Green's function
  6. Layer potential techniques
  7. Periodic capacitance matrix
  8. Approximation of the total field
  9. Integral formulation
  10. Subwavelength resonances
  11. Approximation of the scattering problem
  12. Reduced order model for broadband absorption
  13. Shape sensitivity analysis
  14. Optimal design algorithm
  15. Numerical experiments
  16. ...and 6 more sections

Key Result

Lemma 2.1

For any $\psi \in L^2(\partial D)$, the following identities hold:

Figures (9)

  • Figure 1: Illustration of a metascreen above a reflective surface.
  • Figure 2: Geometry (top) and absorptance spectra (bottom) for single and three resonator configurations (per unit cell), comparing the exact solution and the reduced order model.
  • Figure 3: Geometry (top) and absorptance spectra (bottom) for three and nine resonator configurations (per unit cell), comparing the exact solution and the reduced order model.
  • Figure 4: Shape optimization results for a single resonator configuration ($1\times 1$). The left column corresponds to $J^{\mathrm{res}}$ and the right column to $J^{\mathrm{ref}}$.
  • Figure 5: Shape optimization results for a three resonators configuration arranged horizontally ($3\times 1$). The left column corresponds to $J^{\mathrm{res}}$ and the right column to $J^{\mathrm{ref}}$.
  • ...and 4 more figures

Theorems & Definitions (32)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.1
  • Remark 2.1
  • Lemma 2.3
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 22 more