# Theoretical and Numerical Modeling of Acoustic Metamaterials for Aeroacoustic Applications

## Abstract

**:**

## 1. Introduction

## 2. Integral Formulation of the Problem

## 3. Numerical Solution

## 4. Effect of Motion on the Metamaterial Design

## 5. Results and Discussion

#### 5.1. Effect of Convective Cloak

#### 5.2. Boundary Impedance Estimate

#### 5.3. Computational Efficiency

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix: The Integral Coefficients

## References

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**Figure 2.**Effect of coordinate transformation on straight lines (red, dashed) and wave fronts (black, continuous lines) from the virtual domain $\overline{\mathrm{\Omega}}$ to the physical domain Ω. Object and source at rest.

**Figure 3.**Effect of coordinate transformation on straight lines (red, dashed) and wave fronts (gray, continuous lines) from the virtual domain $\overline{\mathrm{\Omega}}$ to the physical domain Ω. Object and source moving with velocity ${\mathbf{v}}_{b}$.

**Figure 4.**Convergence of ${\mathrm{\sigma}}_{cs}$ with the number of field elements in ${\mathrm{\Omega}}_{c}$, ${N}_{el}$, for four values of $ka$. Equation (3). Mach number ${M}_{b}=0.0,0.1,0.2$ (

**A**,

**B**, and

**C**, respectively). The red line indicates the ${h}^{2}$ rate.

**Figure 5.**Convergence of ${\mathrm{\sigma}}_{cs}$ with the number of field elements in ${\mathrm{\Omega}}_{c}$, ${N}_{el}$, for four values of $ka$. Equation (5). Mach number ${M}_{b}=0.0,0.1,0.2$ (

**A**,

**B**, and

**C**, respectively). The red line indicates the ${h}^{2}$ rate.

**Figure 7.**Scattering cross-section ${\mathrm{\sigma}}_{cs}$ as a function of the reduced frequency $ka$ at ${M}_{b}=0.0,0.1,0.2$. Black dashed lines for a static cloak; red continuous lines for a convective cloak.

**Figure 8.**Directivity pattern at $r=5{R}_{2}$ of the insertion loss ${I}_{L}$ for $ka=1$ (Equation (5)). Mach number ${M}_{b}=0.0,0.1,0.2$.

**Figure 9.**Directivity pattern at $r=5{R}_{2}$ of the insertion loss ${I}_{L}$ for $ka=3.4$ (Equation (5)). Mach number ${M}_{b}=0.0,0.1,0.2$.

**Figure 10.**Directivity pattern at $r=5{R}_{2}$ of the insertion loss ${I}_{L}$ for $ka=6$ (Equation (5)). Mach number ${M}_{b}=0.0,0.1,0.2$.

**Figure 11.**Relative cloaking recovery performance of the convective cloak, $|{\mathrm{\sigma}}_{cs}^{SC}-{\mathrm{\sigma}}_{cs}^{CC}|/|{\mathrm{\sigma}}_{cs}^{SC}|$. Black dots $M=0.1$; red squares ${M}_{b}=0.2$.

**Figure 12.**Field of total pressure in ${\mathrm{\Omega}}_{c}\cup {\mathrm{\Omega}}_{h}$. Comparison of results obtained with static (SC) and convected (CC) cloaks at $ka=3.4$.

**Figure 13.**Field of total pressure in ${\mathrm{\Omega}}_{c}\cup {\mathrm{\Omega}}_{h}$. Comparison of results obtained with static (SC) and convected (CC) cloaks at $ka=6$.

**Figure 15.**Directivity pattern at $r=5{R}_{2}$ of the local impedance $\mathcal{Z}=\mathcal{R}+j\phantom{\rule{0.166667em}{0ex}}\mathcal{X}$ at $ka=3.4$ obtained using Equation (5) for ${M}_{b}=0.1$ and ${M}_{b}=0.2$.

**Table 1.**Average Wall Clock Time (WTC) in seconds for a complete simulation with and without the evaluation of the acoustic field in ${\mathrm{\Omega}}_{h}$.

CPU (Year) | Op.System | WCT ${\mathrm{\Omega}}_{c}$ | WCT ${\mathrm{\Omega}}_{c}\cup {\mathrm{\Omega}}_{h}$ |
---|---|---|---|

Intel Xeon E5405 2.0 GHz (2007) | GNU/Linux kernel 2.6.18 | 230 | 275 |

Intel Xeon E5520 2.2 GHz (2009) | OSX 10.11.2 | 150 | 188 |

Intel i7 4 GHz (2015) | OSX 10.11.3 | 68 | 90 |

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**MDPI and ACS Style**

Iemma, U.
Theoretical and Numerical Modeling of Acoustic Metamaterials for Aeroacoustic Applications. *Aerospace* **2016**, *3*, 15.
https://doi.org/10.3390/aerospace3020015

**AMA Style**

Iemma U.
Theoretical and Numerical Modeling of Acoustic Metamaterials for Aeroacoustic Applications. *Aerospace*. 2016; 3(2):15.
https://doi.org/10.3390/aerospace3020015

**Chicago/Turabian Style**

Iemma, Umberto.
2016. "Theoretical and Numerical Modeling of Acoustic Metamaterials for Aeroacoustic Applications" *Aerospace* 3, no. 2: 15.
https://doi.org/10.3390/aerospace3020015