# Optimization Environment Definition for Beam Steering Reflectarray Antenna Design

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## Abstract

**:**

## 1. Introduction

## 2. Antenna and Optimization Environment Description

#### 2.1. Antenna Geometry

#### 2.2. Performance Parameters

#### 2.3. Optimization Environment

## 3. Evolutionary Algorithms

#### 3.1. Differential Evolution

#### 3.2. Genetic Algorithm

#### 3.3. Biogeography Based Optimization

#### 3.4. Particle Swarm Optimization

#### 3.5. Social Network Optimization

## 4. Antenna Optimization Results and Discussion

#### 4.1. Feasibility Function

#### 4.2. Cost Function Parameter Definition

#### 4.3. Algorithm Comparison

#### 4.4. Analysis of the Final Solution

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Geometry of the planar reflectarray antenna: feeder (upper dark red horn antenna) and the reflector with in red the all patches.

**Figure 2.**Reflection characteristics of the selected patch: the upper diagram shows the reflection amplitude as function of the patch size, while the bottom part shows the angle variation induced by the reflection.

**Figure 4.**Optimization environment: it is composed by all the elements involved in the optimization process and it guarantees the quality of the final solution.

**Figure 5.**Comparison among the different convergence curves obtained with the four box boundary conditions. The curves are represented in semilogarithmic scale.

**Figure 6.**Analysis of scalarization factor $\lambda $ for the four scan angles: for each plot, the left axis shows the average values of ${\mathrm{RE}}_{\mathrm{s}}$ (solid line) and the confidence interval of 90% (light area). Similarly, the right axis represents the same values for the $\Delta {\theta}_{s}$. (

**a**) Results for ${\theta}_{s}={10}^{\circ}$; (

**b**) Results for ${\theta}_{s}={20}^{\circ}$; (

**c**) Results for ${\theta}_{s}={30}^{\circ}$; (

**d**) Results for ${\theta}_{s}={40}^{\circ}$.

**Figure 8.**Convergence curves of mBBO, nBBO, DE, and SNO: each thin line corresponds to a single optimization trial, while the thick red line is the average one. (

**a**) mBBO; (

**b**) nBBO; (

**c**) DE; (

**d**) SNO.

**Figure 10.**Comparison between radiation patterns of the aperture field method and of the full-wave. Each subfigure represents a different scan angle; the right diagram is the radiation pattern on the E-plane and the left one in the H-planes. (

**a**) ${\theta}_{scan}={10}^{\circ}$; (

**b**) ${\theta}_{scan}={20}^{\circ}$; (

**c**) ${\theta}_{scan}={30}^{\circ}$; (

**d**) ${\theta}_{scan}={40}^{\circ}$.

**Table 1.**Comparison of the results with the four feasibility function tested. The results are obtained with 12 independent trials and with 50,000 objective function calls. In bold the best values.

Box Condition | Mean | Standard Deviation | Best Result |
---|---|---|---|

Impenetrable wall | 191.41 | 19.75 | 164.14 |

Elastic wall | 201.08 | 52.76 | 159.1 |

Eliminating wall | 2566.66 | 527.95 | 1452.67 |

Closed space | 291.09 | 62.12 | 219.96 |

Algorithm | DE | GA | SGA | mBBO | nBBO | PSO | SNO |
---|---|---|---|---|---|---|---|

Population size | 25 | 50 | 25 | 25 | 25 | 25 | 100 |

Algorithm | Mean | Standard Deviation | Best Result |
---|---|---|---|

DE | 196.57 | 67.49 | 125.13 |

GA | 6124.97 | 1329.65 | 4224.91 |

SGA | 873.38 | 1778.9 | 234.97 |

mBBO | 539.64 | 106.72 | 335.32 |

nBBO | 402.39 | 106.84 | 269.46 |

PSO | 26,823.4 | 6314.18 | 13,603.59 |

SNO | 195.95 | 27.65 | 154.14 |

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**MDPI and ACS Style**

Niccolai, A.; Grimaccia, F.; Mussetta, M.; Zich, R.; Gandelli, A.
Optimization Environment Definition for Beam Steering Reflectarray Antenna Design. *Mathematics* **2022**, *10*, 33.
https://doi.org/10.3390/math10010033

**AMA Style**

Niccolai A, Grimaccia F, Mussetta M, Zich R, Gandelli A.
Optimization Environment Definition for Beam Steering Reflectarray Antenna Design. *Mathematics*. 2022; 10(1):33.
https://doi.org/10.3390/math10010033

**Chicago/Turabian Style**

Niccolai, Alessandro, Francesco Grimaccia, Marco Mussetta, Riccardo Zich, and Alessandro Gandelli.
2022. "Optimization Environment Definition for Beam Steering Reflectarray Antenna Design" *Mathematics* 10, no. 1: 33.
https://doi.org/10.3390/math10010033