Purpose This work investigates the design of graded locally resonant acoustic metamaterials for bandgap widening in a finite one-dimensional chain. Unlike conventional approaches based on dispersion analysis of an infinite periodic medium, the attenuation performance is evaluated directly from the finite-chain transmissibility. The study aims to identify resonator grading profiles that maximize the width of a single continuous attenuation region and to compare the effectiveness of sinusoidal and power-law grading parameterizations. Methods The metamaterial is modeled as a mass–spring–damper system in which each primary mass is coupled to a local resonator. The optimization objective is defined from the largest connected frequency interval below a prescribed transmissibility threshold and quantified through a capped log-area measure. Bayesian optimization is used to identify effective resonator grading profiles under two design parameterizations. The first uses a sinusoidal parameterization to grade the resonator frequencies, whereas the second uses a two-parameter power-law parameterization. The influence of the target resonant frequency and the robustness of the optimized designs with respect to the transmissibility threshold, parameter bounds, and resonator damping are also investigated. Results For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega _0 = 1~\mathrm {rad/s}\)</EquationSource> </InlineEquation>, the sinusoidal parameterization increases the average bandgap width from 0.041 to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(0.086~\mathrm {rad/s}\)</EquationSource> </InlineEquation>, an improvement of about <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(110.3\%\)</EquationSource> </InlineEquation> relative to the uniform chain. The power-law design performs better, increasing the average bandgap width to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(0.087~\mathrm {rad/s}\)</EquationSource> </InlineEquation>, or about <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(112.0\%\)</EquationSource> </InlineEquation> relative to the same baseline. As the target resonant frequency increases, the gains from both parameterizations increase, with the power-law design becoming more effective, yielding average improvements of about <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(255.9\%\)</EquationSource> </InlineEquation> at <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\omega _0 = 2~\mathrm {rad/s}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(373.1\%\)</EquationSource> </InlineEquation> at <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\omega _0 = 3~\mathrm {rad/s}\)</EquationSource> </InlineEquation>. The relative advantage of the power-law parameterization over the sinusoidal one also grows with the target frequency. The optimized designs consistently exhibit a smooth oscillatory structure under the sinusoidal parameterization. The optimal profile transitions from an approximately half-cycle sinusoidal form at <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\omega _0 = 1~\mathrm {rad/s}\)</EquationSource> </InlineEquation> to an approximately one-cycle form at <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\omega _0 = 2\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(3~\mathrm {rad/s}\)</EquationSource> </InlineEquation>, accompanied by a larger optimized grading amplitude. Under the power-law parameterization, the optimal grading profile remains approximately linearly increasing across all target frequencies. Conclusions Bandgap widening in the present finite structure is governed by a coherent redistribution of local resonant frequencies that promotes overlap between neighboring attenuation regions. The proposed Bayesian optimization framework provides an efficient and robust approach for designing graded resonator configurations that substantially outperform uniform designs while requiring only a small number of design parameters.

Bayesian Optimization of Resonator Grading for Bandgap Widening in Finite Locally Resonant Acoustic Metamaterials

期刊论文
发表日期：

2026年7月8日

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Ali Abdulai,
Mustafa Alshaqaq

摘要

Purpose

This work investigates the design of graded locally resonant acoustic metamaterials for bandgap widening in a finite one-dimensional chain. Unlike conventional approaches based on dispersion analysis of an infinite periodic medium, the attenuation performance is evaluated directly from the finite-chain transmissibility. The study aims to identify resonator grading profiles that maximize the width of a single continuous attenuation region and to compare the effectiveness of sinusoidal and power-law grading parameterizations.

Methods

The metamaterial is modeled as a mass–spring–damper system in which each primary mass is coupled to a local resonator. The optimization objective is defined from the largest connected frequency interval below a prescribed transmissibility threshold and quantified through a capped log-area measure. Bayesian optimization is used to identify effective resonator grading profiles under two design parameterizations. The first uses a sinusoidal parameterization to grade the resonator frequencies, whereas the second uses a two-parameter power-law parameterization. The influence of the target resonant frequency and the robustness of the optimized designs with respect to the transmissibility threshold, parameter bounds, and resonator damping are also investigated.

Results

For \(\omega _0 = 1~\mathrm {rad/s}\) , the sinusoidal parameterization increases the average bandgap width from 0.041 to \(0.086~\mathrm {rad/s}\) , an improvement of about \(110.3\%\) relative to the uniform chain. The power-law design performs better, increasing the average bandgap width to \(0.087~\mathrm {rad/s}\) , or about \(112.0\%\) relative to the same baseline. As the target resonant frequency increases, the gains from both parameterizations increase, with the power-law design becoming more effective, yielding average improvements of about \(255.9\%\) at \(\omega _0 = 2~\mathrm {rad/s}\) and \(373.1\%\) at \(\omega _0 = 3~\mathrm {rad/s}\) . The relative advantage of the power-law parameterization over the sinusoidal one also grows with the target frequency. The optimized designs consistently exhibit a smooth oscillatory structure under the sinusoidal parameterization. The optimal profile transitions from an approximately half-cycle sinusoidal form at \(\omega _0 = 1~\mathrm {rad/s}\) to an approximately one-cycle form at \(\omega _0 = 2\) and \(3~\mathrm {rad/s}\) , accompanied by a larger optimized grading amplitude. Under the power-law parameterization, the optimal grading profile remains approximately linearly increasing across all target frequencies.

Conclusions

Bandgap widening in the present finite structure is governed by a coherent redistribution of local resonant frequencies that promotes overlap between neighboring attenuation regions. The proposed Bayesian optimization framework provides an efficient and robust approach for designing graded resonator configurations that substantially outperform uniform designs while requiring only a small number of design parameters.