# Multiple Aerodynamic Coefficient Prediction of Airfoils Using a Convolutional Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Airfoil Data Processing

#### 2.2. Structure Design

#### 2.3. Parameter Selection

#### 2.4. Training Process

## 3. Results and Discussion

#### 3.1. Data Preparation

#### 3.2. Model Training

#### 3.3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ANN | artificial neural network |

BP | back propagation |

CAI | composite airfoil image |

CFD | computational fluid dynamics |

CNN | convolutional neural network |

DAG | directed acyclic graph |

MLP | multilayered perceptron |

MSE | mean square error |

NACA | National Advisory Committee for Aeronautics |

ReLU | rectified linear unit |

RMSE | root mean squared error |

SGDM | stochastic gradient descent with momentum |

SVM | support vector machine |

SVR | support vector regression |

TAI | transformed airfoil image |

## References

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**Figure 7.**Transformed airfoil images (TAIs) generated by convolving angle of attack and Mach number with airfoil image.

**Figure 9.**Comparisons between actual and predicted aerodynamic coefficients of random airfoil. (

**a**) Pitch-moment coefficient; (

**b**) drag coefficient; (

**c**) lift coefficient.

**Figure 10.**Linear regressions of actual relative to predicted aerodynamic coefficients. (

**a**) Pitch-moment coefficient; (

**b**) drag coefficient; (

**c**) lift coefficient.

**Table 1.**Computational-fluid-dynamics (CFD) results of ${C}_{D}$ and ${C}_{L}$ ($\alpha $ = 2, Ma = 0.4, Re = $6.5\times {10}^{6}$).

Grid | ${\mathit{C}}_{\mathit{m}\mathit{z}}$ | $\mathit{\u03f5}(\%)$ | ${\mathit{R}}_{\mathit{G}}$ | ${\mathit{C}}_{\mathit{m}\mathit{z}}$ | $\mathit{\u03f5}(\%)$ | ${\mathit{R}}_{\mathit{G}}$ |
---|---|---|---|---|---|---|

gird1 ($210\times 70$) | 0.01088 | 9.46 | 0.34306 | 1.2 | ||

gird1 ($300\times 100$) | 0.01028 | 3.42 | 0.42 | 0.34443 | 0.8 | 0.53 |

gird1 ($420\times 140$) | 0.01003 | 0.9 | 0.34516 | 0.6 | ||

gird1 ($600\times 200$) | 0.00994 | - | - | 0.34740 | - | - |

Methods of Input Image Preparation | RMSE | ||
---|---|---|---|

${\mathit{C}}_{\mathit{m}\mathit{z}}$ | ${\mathit{C}}_{\mathit{D}}$ | ${\mathit{C}}_{\mathit{L}}$ | |

Method used in this paper (convolution of flow conditions and airfoil images to generate TAIs). | 0.0027 | 0.0035 | 0.0273 |

Construct 2 $85\times 85$ constant two-dimensional input matrices using angle of attack and Mach number. | 0.0030 | 0.0044 | 0.0341 |

Construct $85\times 85$ two-dimensional input matrix by using combination of angle of attack and Mach number. Upper part of the matrix is angles of attack, and the lower part is Mach number. | 0.0052 | 0.0057 | 0.0492 |

Method in [10]. | 0.0042 | 0.0042 | 0.0296 |

Prediction Network | Resolution of Airfoil Image | Training Time (min) | RMSE | ||
---|---|---|---|---|---|

${\mathit{C}}_{\mathit{m}\mathit{z}}$ | ${\mathit{C}}_{\mathit{D}}$ | ${\mathit{C}}_{\mathit{L}}$ | |||

CNN | $85\times 85$ | 35 | 0.0027 | 0.0035 | 0.0273 |

DAG | $85\times 85$ | 35 | 0.0137 | 0.0413 | 0.2834 |

MLP | $16\times 16$ | 35 | 0.0028 | 0.0040 | 0.0332 |

Case | Training Time (s) | RMSE | Mean Relative Error | Maximum Relative Error |
---|---|---|---|---|

Case 1 (300 samples) | 192 | 0.0029 | $0.59\%$ | $2.44\%$ |

Case 2 (4200 samples) | 85,095 | 0.0433 | 1.94% | 80.20% |

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## Share and Cite

**MDPI and ACS Style**

Chen, H.; He, L.; Qian, W.; Wang, S.
Multiple Aerodynamic Coefficient Prediction of Airfoils Using a Convolutional Neural Network. *Symmetry* **2020**, *12*, 544.
https://doi.org/10.3390/sym12040544

**AMA Style**

Chen H, He L, Qian W, Wang S.
Multiple Aerodynamic Coefficient Prediction of Airfoils Using a Convolutional Neural Network. *Symmetry*. 2020; 12(4):544.
https://doi.org/10.3390/sym12040544

**Chicago/Turabian Style**

Chen, Hai, Lei He, Weiqi Qian, and Song Wang.
2020. "Multiple Aerodynamic Coefficient Prediction of Airfoils Using a Convolutional Neural Network" *Symmetry* 12, no. 4: 544.
https://doi.org/10.3390/sym12040544