Shape optimization of phononic band gap structures using the homogenization approach

TL;DR

This paper presents a homogenization-based shape optimization method to maximize low-frequency phononic band gaps in elastic composites, enabling improved design of acoustic devices through sensitivity analysis and numerical validation.

Contribution

It introduces a novel shape optimization framework for phononic band gaps using homogenization and sensitivity analysis, specifically targeting low-frequency gaps in elastic composites.

Findings

Optimized inclusion shapes increase low-frequency band gaps.

Derived analytic formulas for dispersion effects of inclusion resizing.

Numerical examples demonstrate effective optimization under stiffness constraints.

Abstract

The paper deals with optimization of the acoustic band gaps computed using the homogenized model of strongly heterogeneous elastic composite which is constituted by soft inclusions periodically distributed in stiff elastic matrix. We employ the homogenized model of such medium to compute intervals --- band gaps --- of the incident wave frequencies for which acoustic waves cannot propagate. It was demonstrated that the band gaps distribution can be influenced by changing the shape of inclusions. Therefore, we deal with the shape optimization problem to maximize low-frequency band gaps; their bounds are determined by analysing the effective mass tensor of the homogenized medium. Analytic transformation formulas are derived which describe dispersion effects of resizing the inclusions. The core of the problem lies in sensitivity of the eigenvalue problem associated with the microstructure. Computational sensitivity analysis is developed, which allows for efficient usage of the gradient based optimization methods. Numerical examples with 2D structures are reported to illustrate the effects of optimization with stiffness constraint. This study is aimed to develop modelling tools which can be used in optimal design of new acoustic devices for "smart systems".