A topology optimization of open acoustic waveguides based on a scattering matrix method

TL;DR

This paper introduces a topology optimization method for open acoustic waveguides that employs a scattering matrix approach and level-set method to achieve bound states in the continuum, demonstrated through numerical experiments.

Contribution

It develops a novel topology optimization scheme using a scattering matrix and topological derivative to realize bound states in open acoustic waveguides.

Findings

Successfully designed waveguides with bound states in the continuum

Demonstrated the effectiveness of the optimization scheme through numerical examples

Achieved control over resonant frequencies and wavenumbers

Abstract

This study presents a topology optimization scheme for realizing a bound state in the continuum along an open acoustic waveguide comprising a periodic array of elastic materials. First, we formulate the periodic problem as a system of linear algebraic equations using a scattering matrix associated with a single unit structure of the waveguide. The scattering matrix is numerically constructed using the boundary element method. Subsequently, we employ the Sakurai--Sugiura method to determine resonant frequencies and the Floquet wavenumbers by solving a nonlinear eigenvalue problem for the linear system. We design the shape and topology of the unit elastic material such that the periodic structure has a real resonant wavenumber at a given frequency by minimizing the imaginary part of the resonant wavenumber. The proposed topology optimization scheme is based on a level-set method with a novel topological derivative. We demonstrate a numerical example of the proposed topology optimization and show that it realizes a bound state in the continuum through some numerical experiments.