# Optimal Design of Acoustic Metamaterial of Multiple Parallel Hexagonal Helmholtz Resonators by Combination of Finite Element Simulation and Cuckoo Search Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Initial Structural Design

#### 2.1. Design Objective

#### 2.2. Geometric Parameters Design

## 3. Optimization of Acoustic Metamaterial

#### 3.1. Establishment of Finite Element Simulation Model

#### 3.2. Optimization Algorithm

- (1)
- Set the initial parameters for the algorithm. In the current optimization process, the number of host nest populations was set as $N=20$, the maximum number of iterations was $N\_iterTotal=100$, and the maximum probability of discovery was ${p}_{a}=0.25$.
- (2)
- Calculate the fitness function of the population individual. In this research, sound absorption performance within the target frequency range of the acoustic metamaterial of multiple parallel hexagonal Helmholtz resonators was the research object and the depth of neck of each resonator was the optimization parameter. To ensure that the resulted absorption curve could maintain a high absorption coefficient in the target frequency range, the fitness value function $\mathrm{max}(\alpha )$ was chosen as the total amount of sound energy absorbed at the target frequency range, as shown in Equation (1).

- (3)
- Update all nests according to Equation (3).

- (4)
- For all nests ${x}_{i}(i=1,\cdots ,N)$, a random number ${r}_{i}\in [0,1]$ is generated and compared to the probability ${p}_{a}$. If ${r}_{i}>{p}_{a}$, then ${x}_{i}^{t+1}$ is randomly changed, otherwise there will be no change. The adaptation of the nests before and after the random change was evaluated, retaining the nest with better adaptation as the final ${x}_{i}^{t+1}$. Afterwards, the return step (2) was iterated.
- (5)
- When all the absorption coefficients at the target frequency range were above 0.85, or it had reached the maximum number of iterations, the iteration ended, and the current optimal individual was output.

#### 3.3. Interactive Operations with MATLAB and COMSOL

## 4. Results and Discussion

#### 4.1. Optimization Results

#### 4.2. Theoretical Analysis

#### 4.3. Experimental Validation

#### 4.3.1. Methodology

#### 4.3.2. Experimental Results

#### 4.4. Applicability of the Optimal Design

## 5. Conclusions

- (1)
- For the problem of low frequency broadband noise in the workshop and the narrow absorption bandwidth of a single Helmholtz resonator, the acoustic metamaterial of multiple parallel hexagonal Helmholtz resonators was designed and optimized. Its effective absorption capacity was verified by experimental validation. Therefore, the proposed acoustic metamaterial could attenuate the noise with a broad frequency range, which was also beneficial in practical engineering applications.
- (2)
- A joint simulation method incorporating the finite element simulation and cuckoo search algorithm was employed for optimization of the proposed acoustic metamaterial to achieve broadband sound absorption effect at the low frequency range in this research. Compared with the optimization method based on theoretical models, the design method proposed in this research could obtain more accurate optimization results, which satisfied the requirements of certain practical conditions.
- (3)
- The optimal design method proposed in this research could be applied to the absorption needs for different conditions, which also proved its feasibility and practicality. It presented a novel method for the development and application of acoustic metamaterial, which would be favorable to promote its practical applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The designed acoustic metamaterial of multiple parallel hexagonal Helmholtz resonators. (

**a**) Schematic diagram of the whole structure; (

**b**) Structure of a single hexagonal Helmholtz resonator; (

**c**) Top view of the single hexagonal Helmholtz resonator; (

**d**) Main view of the single hexagonal Helmholtz resonator.

**Figure 2.**The single Helmholtz resonator. (

**a**) Schematic diagram of the cylindrical representative hexagonal structure; (

**b**) Schematic diagram of the 2D rotationally symmetric model for the finite element analysis.

**Figure 3.**Sound absorption coefficient of the initial structure gained by finite element simulation. The gray area represents the frequency range that meets the design requirements.

**Figure 4.**Finite element simulation model. (

**a**) Finite element model of the acoustic metamaterial of multiple parallel hexagonal Helmholtz resonators. (

**b**) Finite element mesh division of the proposed acoustic metamaterial.

**Figure 5.**The optimization process of cuckoo search algorithm for the acoustic metamaterial of multiple parallel hexagonal Helmholtz resonators.

**Figure 7.**The optimization results. (

**a**) The optimized sound absorption coefficients obtained in simulation for the acoustic metamaterial of multiple parallel hexagonal Helmholtz resonators with its corresponding nineteen single structures. (

**b**) Instantaneous local velocity of corresponding absorption peaks.

**Figure 8.**Fabrication and detection of the acoustic metamaterial. (

**a**) The AWA6290T transfer function sound absorption coefficient measurement system; (

**b**) Schematic diagram of the transfer function tube measurement; (

**c**) The low force stereolithography (LFS) 3D printer of Form3; (

**d**) The 3D printed sample of acoustic metamaterial with the diameter of 100 mm and the thickness of 40 mm.

**Figure 9.**Comparisons of theoretical, simulation and experimental sound absorption coefficients of the proposed acoustic metamaterial of multiple parallel hexagonal Helmholtz resonators.

**Figure 10.**The optimization results of Condition-1. (

**a**) Schematic diagram of the 3D printed sample with the diameter of 100 mm and the thickness of 20 mm. (

**b**) The corresponding actual sound absorption coefficient.

**Figure 11.**The optimization results of Condition-2. (

**a**) Schematic diagram of the 3D printed sample with the diameter of 100 mm and the thickness of 30 mm. (

**b**) The corresponding actual sound absorption coefficient.

Group | Serial Number | Thickness/mm | Diameter of Micropore/mm | Depth of Neck/mm |
---|---|---|---|---|

I | 1 | 40 | 4.47 | 13.0 |

2 | 12.1 | |||

3 | 11.2 | |||

II | 4 | 3.54 | 14.7 | |

5 | 13.4 | |||

6 | 12.3 | |||

7 | 11.3 | |||

III | 8 | 3.75 | 12.1 | |

9 | 11.2 | |||

10 | 10.3 | |||

11 | 9.5 | |||

12 | 8.8 | |||

IV | 13 | 4.74 | 12.3 | |

14 | 11.4 | |||

15 | 10.7 | |||

16 | 10.0 | |||

V | 17 | 5.16 | 11.8 | |

18 | 11.0 | |||

19 | 10.3 |

Group | Serial Number | Thickness/mm | Diameter of Micropore/mm | Depth of Neck/mm |
---|---|---|---|---|

I | 1 | 40 | 4.47 | 15.7 |

2 | 15.2 | |||

3 | 14.4 | |||

II | 4 | 3.54 | 15.9 | |

5 | 15.2 | |||

6 | 14.7 | |||

7 | 14.3 | |||

III | 8 | 3.75 | 16.7 | |

9 | 15.3 | |||

10 | 14.1 | |||

11 | 13.1 | |||

12 | 11.8 | |||

IV | 13 | 4.74 | 16.2 | |

14 | 14.5 | |||

15 | 12.6 | |||

16 | 10.9 | |||

V | 17 | 5.16 | 12.0 | |

18 | 10.3 | |||

19 | 9.5 |

Group | Serial Number | Thickness/mm | Diameter of Micropore/mm | Depth of Neck/mm |
---|---|---|---|---|

I | 1 | 20 | 4.47 | 8.8 |

2 | 8.2 | |||

3 | 7.3 | |||

II | 4 | 3.54 | 8.1 | |

5 | 7.6 | |||

6 | 6.8 | |||

7 | 6.1 | |||

III | 8 | 3.75 | 9.1 | |

9 | 8.2 | |||

10 | 7.1 | |||

11 | 6.3 | |||

12 | 5.1 | |||

IV | 13 | 4.74 | 9.0 | |

14 | 7.6 | |||

15 | 6.4 | |||

16 | 5.3 | |||

V | 17 | 5.16 | 6.2 | |

18 | 5.0 | |||

19 | 4.5 |

Group | Serial Number | Thickness/mm | Diameter of Micropore/mm | Depth of Neck/mm |
---|---|---|---|---|

I | 1 | 30 | 4.47 | 10.3 |

2 | 9.1 | |||

3 | 8.0 | |||

II | 4 | 3.54 | 9.8 | |

5 | 9.0 | |||

6 | 8.4 | |||

7 | 7.7 | |||

III | 8 | 3.75 | 10.8 | |

9 | 9.9 | |||

10 | 8.6 | |||

11 | 7.6 | |||

12 | 6.5 | |||

IV | 13 | 4.74 | 9.5 | |

14 | 8.2 | |||

15 | 7.0 | |||

16 | 5.9 | |||

V | 17 | 5.16 | 6.4 | |

18 | 5.4 | |||

19 | 4.8 |

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## Share and Cite

**MDPI and ACS Style**

Yang, F.; Wang, E.; Shen, X.; Zhang, X.; Yin, Q.; Wang, X.; Yang, X.; Shen, C.; Peng, W.
Optimal Design of Acoustic Metamaterial of Multiple Parallel Hexagonal Helmholtz Resonators by Combination of Finite Element Simulation and Cuckoo Search Algorithm. *Materials* **2022**, *15*, 6450.
https://doi.org/10.3390/ma15186450

**AMA Style**

Yang F, Wang E, Shen X, Zhang X, Yin Q, Wang X, Yang X, Shen C, Peng W.
Optimal Design of Acoustic Metamaterial of Multiple Parallel Hexagonal Helmholtz Resonators by Combination of Finite Element Simulation and Cuckoo Search Algorithm. *Materials*. 2022; 15(18):6450.
https://doi.org/10.3390/ma15186450

**Chicago/Turabian Style**

Yang, Fei, Enshuai Wang, Xinmin Shen, Xiaonan Zhang, Qin Yin, Xinqing Wang, Xiaocui Yang, Cheng Shen, and Wenqiang Peng.
2022. "Optimal Design of Acoustic Metamaterial of Multiple Parallel Hexagonal Helmholtz Resonators by Combination of Finite Element Simulation and Cuckoo Search Algorithm" *Materials* 15, no. 18: 6450.
https://doi.org/10.3390/ma15186450