# Multi-Objective Optimization of Low Reynolds Number Airfoil Using Convolutional Neural Network and Non-Dominated Sorting Genetic Algorithm

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## Abstract

**:**

## 1. Introduction

^{5}are shown in Figure 1. There is a range of lift coefficients for which the drag coefficient remains almost constant or increases only slightly (points 1 and 2). In this range, the transition curve is steep, i.e., the transition moves towards the leading edge very slowly. Afterward, there is a sudden increase in the drag coefficient. This is because the associated transition locations move upstream abruptly.

## 2. Machine Learning Methods

## 3. Data Generation

^{5}. Experimental data can be found in [35]. CFD analysis requires grid study and solver settings. Ansys ICEM was used to generate a C-H grid around the airfoil. Four grids were studied to ensure that the selected grid was fine enough to capture the laminar separation bubble and associated flow physics. The grid study was performed for an alpha of 4°. The details of grids and the corresponding lift and drag coefficients are presented in Table 2. This analysis was carried out using Ansys Fluent pressure-based, steady, Transition SST, coupled pressure–velocity, and second-order upwind schemes for spatial discretization.

^{6}, representing flight conditions of 9 m/s at an altitude of 800 m with a chord length of 0.3 m.

## 4. CNN Structure

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Function | Linear | Sigmoid | Tanh | ReLU |
---|---|---|---|---|

Formula | $y=ax$ | $y=1/\left(1\text{}+\text{}{e}^{-x}\right)$ | $y=\frac{{e}^{x}\text{}-\text{}{e}^{-x}}{{e}^{x}\text{}+\text{}{e}^{-x}}$ | $y=\mathrm{max}\left(0,x\right)$ |

Nature | Linear | Non-Linear | Non-Linear | Non-Linear |

Range | [−inf inf] | [0 1] | [−1 1] | [0 inf] |

Response |

Grid | Size | ${\mathit{c}}_{\mathit{l}}$ | ${\mathit{c}}_{\mathit{d}}$ | Residuals |
---|---|---|---|---|

1 | 46,042 | 0.794 | 0.0131 | 1 × 10^{−4} |

2 | 69,192 | 0.794 | 0.0132 | 1 × 10^{−4} |

3 | 96,742 | 0.796 | 0.0132 | 1 × 10^{−5} |

4 | 128,692 | 0.796 | 0.0132 | 1 × 10^{−5} |

Layer | Filter Size | Pool Size | No. of Filters | Stride | Output Shape |
---|---|---|---|---|---|

Conv-1 | 5 × 5 | --- | 64 | 1 | (156, 156, 64) |

Pool-1 | --- | 2 × 2 | --- | 2 | (78, 78, 64) |

Conv-2 | 3 × 3 | --- | 128 | 1 | (76, 76, 128) |

Pool-2 | --- | 2 × 2 | --- | 2 | (38, 38, 128) |

Conv-3 | 3 × 3 | --- | 256 | 1 | (36, 36, 256) |

Pool-3 | --- | 2 × 2 | --- | 2 | (18, 18, 128) |

Conv-4 | 3 × 3 | --- | 256 | 1 | (16, 16, 256) |

Pool-4 | --- | 2 × 2 | --- | 2 | (8, 8, 256) |

Conv-5 | 3 × 3 | --- | 512 | 1 | (6, 6, 512) |

Pool-5 | --- | 2 × 2 | --- | 2 | (3, 3, 512) |

Learning Rate | Constant | Exp. Decay | Step Decay |
---|---|---|---|

Formula | $lr=cont$ | $lr=l{r}_{0}\ast exp\left(-k\ast eph\right)$ | $lr=l{r}_{0}\ast drop^\left(eph/ep{h}_{drp}\right)$ |

Parameters | $const=0.1$ | $l{r}_{0}=0.15,k=0.0025$ | $l{r}_{0}=0.1,drop=0.75,ep{h}_{drp}=80$ |

Loss Function |

Optimizer | AdaGrad | RMSprop | Adam |
---|---|---|---|

Parameters | $l{r}_{0}=0.1$ | $l{r}_{0}=0.0001$ | $l{r}_{0}=0.0001$ |

Loss Function |

${\mathit{f}}_{1}\left(\mathit{x}\right)$ | ${\mathit{f}}_{2}\left(\mathit{x}\right)$ | ${\mathit{c}}_{\mathit{l}}/{\mathit{c}}_{\mathit{d}}$ | $\mathit{t}/{\mathit{c}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{x}}_{\mathit{t}/{\mathit{c}}_{\mathit{m}\mathit{a}\mathit{x}}}$ | Airfoil Shape |
---|---|---|---|---|---|

0.515 | 0.0530 | 18.847 | 0.1212 | 0.435 | |

0.600 | 0.0305 | 32.724 | 0.1123 | 0.402 | |

0.698 | 0.0168 | 59.286 | 0.1036 | 0.37 | |

0.801 | 0.0140 | 70.983 | 0.0987 | 0.339 | |

0.900 | 0.0124 | 80.272 | 0.0982 | 0.309 | |

1.005 | 0.0120 | 83.310 | 0.0988 | 0.279 | |

1.103 | 0.0114 | 87.708 | 0.0970 | 0.309 | |

1.199 | 0.0109 | 91.4677 | 0.0915 | 0.309 |

Model | Operation | Time (Minutes) | Machine |
---|---|---|---|

NSGA-II with CNN | Data generation | 95 | Intel @ 2.9 GHz |

Training (1000 epochs) | 17 | NVIDIA GeForce | |

Optimization | 25 | Intel @ 2.9 GHz | |

Total | 137 | --- | |

NSGA-II with XFOIL | Optimization | 242 | Intel @ 2.9 GHz |

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**MDPI and ACS Style**

Bakar, A.; Li, K.; Liu, H.; Xu, Z.; Alessandrini, M.; Wen, D.
Multi-Objective Optimization of Low Reynolds Number Airfoil Using Convolutional Neural Network and Non-Dominated Sorting Genetic Algorithm. *Aerospace* **2022**, *9*, 35.
https://doi.org/10.3390/aerospace9010035

**AMA Style**

Bakar A, Li K, Liu H, Xu Z, Alessandrini M, Wen D.
Multi-Objective Optimization of Low Reynolds Number Airfoil Using Convolutional Neural Network and Non-Dominated Sorting Genetic Algorithm. *Aerospace*. 2022; 9(1):35.
https://doi.org/10.3390/aerospace9010035

**Chicago/Turabian Style**

Bakar, Abu, Ke Li, Haobo Liu, Ziqi Xu, Marco Alessandrini, and Dongsheng Wen.
2022. "Multi-Objective Optimization of Low Reynolds Number Airfoil Using Convolutional Neural Network and Non-Dominated Sorting Genetic Algorithm" *Aerospace* 9, no. 1: 35.
https://doi.org/10.3390/aerospace9010035