# Novel Design Scheme for Structural Fundamental Frequency of Porous Acoustic Metamaterials

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## Abstract

**:**

## 1. Introduction

## 2. Research Problems on FMAM

## 3. Topology Optimization for Fundamental Frequency of Porous Acoustic Metamaterial Structure

#### 3.1. Modelling for Topology Optimization

#### 3.2. Topology Optimization

## 4. Parametric Optimization Based on Surrogate Model

#### 4.1. The Theoretical Background of Surrogate Model

#### 4.2. The Process of Parametric Optimization

#### 4.3. Results Analysis of Parametric Optimization

#### 4.4. Secondary Optimization

## 5. Experimental Analysis of Frequency Response Based on Closed Acoustic Box

#### 5.1. Prototype Manufacturing and Scheme Design

#### 5.2. Modal Experiment

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FIP | Full-cycle interactive progressive |

ASI systems | Acoustic-structure interaction systems |

FMAM | Frequency modulation acoustic metamaterial |

SM | Surrogate model |

SIMP method | Solid isotropic microstructures with penalization method |

SPL | Sound pressure level |

TO | Topology optimization |

EGO | Effective global optimization |

MSG exploration | Multi-start search with geometric global exploration |

EI | Expected improvement |

PO | Parametric optimization |

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**Figure 2.**Iterative process of topology optimization, which also shows the structural topologies of the 1st iteration, 5th iteration, 18th iteration, 25th iteration and 100th iteration.

**Figure 3.**The nephograms of modal simulation for unit cell of FMAM: (

**a**) represents the initial structure; (

**b**) represents the optimal structure.

**Figure 4.**Nephograms of modal simulation for 10 × 10 multi-cell of FMAM before topology optimization: (

**a**–

**f**) represent the first six modes of the FMAM in the ASI system respectively.

**Figure 5.**Nephograms of modal simulation for 10 × 10 multi-cell of FMAM after topology optimization: (

**a**–

**f**) represent the first six modes of the FMAM in the ASI system, respectively.

**Figure 6.**Composition of three-dimensional honeycomb porous FMAM: (

**a**) the remodeled two-dimensional configuration obtained through the topology optimization; (

**b**) the cross-sectional shape of the three-dimensional cellular porous FMAM; (

**c**) the three-dimensional unit cell; (

**d**) the 10 × 10 array three-dimensional honeycomb porous structure.

**Figure 7.**Parametric model of three-dimensional unit cell of FMAM: (

**a**) the cross-section of the three-dimensional honeycomb porous; (

**b**) the internal section of the three-dimensional honeycomb porous.

**Figure 8.**Change trend of input (structural) and output parameters in SM: (

**a**) the different parameters fluctuate in different states during the convergence process; (

**b**) the fundamental frequency of the FMAM.

**Figure 9.**Nephograms of modal simulation for 10 × 10 multi-cell of FMAM after parametric optimization: (

**a**–

**f**) represent the first six modes of the FMAM in the ASI system, respectively.

**Figure 10.**Change trend of SPL and $f$ during the parametric optimization: (

**a**) represents SPL; (

**b**) represents $f$.

**Figure 12.**Change trend of input (structural) and output parameters in the secondary optimization: (

**a**) the different parameters fluctuate in different states during the convergence process; (

**b**) the fundamental frequency of the FMAM.

**Figure 13.**Additive manufacturing of FMAM specimens: (

**a**) is the basic preprocessing model; (

**b**) is the specimen manufacturing and post-processing model; (

**c**) is the partial physical diagram of FMAM.

**Figure 14.**Frequency response experiments based on the closed acoustic box: (

**a**) Physical layout of the frequency response experiment; (

**b**) Flowchart of the frequency response experiment.

**Figure 15.**Site of frequency response experiments: (

**a**) is a magnified view of (

**c**); (

**b**) is the complete layout site of the experimental device, (

**c**) is the closed ASI interaction space connected to the vibrating device.

**Figure 16.**Main equipment for frequency response experiments: (

**a**–

**f**) represent the multichannel data acquisition instruments, power amplifiers, acoustic-vibration integrated sensors, signal generators, vibration exciters, and acoustic speakers, respectively.

**Figure 17.**Amplitude experimental device for FMAM: (

**a**) is a vibration controller used to adjust the intensity of the collected signal and (

**b**) is an interferometer used to emit polarized light to measure the structural vibration.

**Figure 18.**Comparison of experiment and simulation results of FMAM: (

**a**) represents the natural frequency; (

**b**) represents the surficial SPL.

**Table 1.**Results of former six modal analysis for porous acoustic metamaterials before and after topology optimization (TO).

Mode | Natural Frequency/Hz | Change Rate/% | Maximum Amplitude/mm | Change Rate/% | ||
---|---|---|---|---|---|---|

Before TO | After TO | Before TO | After TO | |||

First order | 677.15 | 1026.5 | 51.6 | 2.9615 | 2.2895 | −22.69 |

Second order | 1185.4 | 1833.4 | 54.67 | 2.8006 | 2.1134 | −24.54 |

Third order | 1185.4 | 1834.9 | 54.79 | 2.8006 | 2.1175 | −24.39 |

Forth order | 1554.1 | 2175.1 | 39.96 | 2.705 | 2.039 | −24.62 |

Fifth order | 1554.1 | 2178.9 | 40.2 | 2.705 | 2.0386 | −24.64 |

Six order | 1593.7 | 2446.4 | 53.5 | 2.6863 | 1.6228 | −39.59 |

Solid | Young’s Modulus | Density | Poisson’s Ratio |

2.6 GPa | 1.12 g/cm^{3} | 0.35 | |

Acoustic (20 °C) | Speed | Density | |

344 m/s | 1.29 kg/m^{3} |

Design Parameters | L1 (mm) | L2 (mm) | L3 (mm) | L4 (mm) | L5 (mm) |
---|---|---|---|---|---|

Value range | (100, 250) | (80, 250) | (80, 250) | (3, 6) | (3, 6) |

Design parameters | L6 (mm) | L7 (mm) | L8 (mm) | L9 (mm) | L10 (mm) |

Value range | (2.7, 5) | (2.7, 5) | (2.5, 4.5) | (2.5, 4.5) | (2, 4) |

Design parameters | L11 (mm) | L12 (mm) | L13 (mm) | Azimuth angle α1 | Slope angle α2 |

Value range | (2, 4) | (0.5, 1.5) | (0.5, 1.5) | (0°, 45°) | (0°, 30°) |

Design Parameters | L1 (mm) | L2 (mm) | L3 (mm) | L4 (mm) | L5 (mm) |
---|---|---|---|---|---|

Value range | 212.68 | 178.91 | 196.46 | 4.65 | 5.23 |

Design parameters | L6 (mm) | L7 (mm) | L8 (mm) | L9 (mm) | L10 (mm) |

Value range | 4.95 | 4.69 | 4.49 | 4.37 | 3.07 |

Design parameters | L11 (mm) | L12 (mm) | L13 (mm) | Azimuth angle α1 | Slope angle α2 |

Value range | 2.92 | 1.21 | 1.21 | 43.47 | 22.36 |

**Table 5.**Results of former six modal analyses for porous acoustic metamaterials before and after parametric optimization (PO).

Mode | Natural Frequency/Hz | Change Rate/% | Maximum Amplitude/mm | Change Rate/% | ||
---|---|---|---|---|---|---|

Before PO | After PO | Before PO | After PO | |||

First order | 1026.5 | 1662.4 | 61.95 | 2.2895 | 1.6459 | −28.11 |

Second order | 1833.4 | 2550.7 | 39.12 | 2.1134 | 1.3537 | −35.95 |

Third order | 1834.9 | 2560.1 | 39.52 | 2.1175 | 1.3793 | −34.86 |

Forth order | 2175.1 | 2868.1 | 31.86 | 2.039 | 1.9087 | −6.39 |

Fifth order | 2178.9 | 2889.5 | 32.61 | 2.0386 | 1.8456 | −9.47 |

Six order | 2446.4 | 3128.2 | 27.87 | 1.6228 | 1.3659 | −15.83 |

Design Parameters | L1 (mm) | L2 (mm) | L3 (mm) | L6 (mm) | L7 (mm) |
---|---|---|---|---|---|

Value range | (197, 228) | (161, 196) | (179, 214) | (4.7, 5) | (4.4, 5) |

Optimal value | 207.14 | 185.23 | 199.75 | 4.86 | 4.74 |

Design parameters | L8 (mm) | L9 (mm) | L10 (mm) | Slope angle α2 | |

Value range | (4.2, 4.5) | (4.1, 4.5) | (2.8, 3.3) | (19.3°, 25.4°) | |

Optimal value | 4.31 | 4.18 | 3.23 | 24.11 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, Y.; Li, H.; Ye, M.; Shi, Y.; Gao, L.
Novel Design Scheme for Structural Fundamental Frequency of Porous Acoustic Metamaterials. *Materials* **2022**, *15*, 6569.
https://doi.org/10.3390/ma15196569

**AMA Style**

Zhou Y, Li H, Ye M, Shi Y, Gao L.
Novel Design Scheme for Structural Fundamental Frequency of Porous Acoustic Metamaterials. *Materials*. 2022; 15(19):6569.
https://doi.org/10.3390/ma15196569

**Chicago/Turabian Style**

Zhou, Ying, Hao Li, Mengli Ye, Yun Shi, and Liang Gao.
2022. "Novel Design Scheme for Structural Fundamental Frequency of Porous Acoustic Metamaterials" *Materials* 15, no. 19: 6569.
https://doi.org/10.3390/ma15196569