# Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation Method and Discrete Adjoint Equation

#### 2.1. Numerical Simulation Method

#### 2.2. Validation of Simulation Method

#### 2.3. Discrete Adjoint Equation

#### 2.4. Gradient Verification

## 3. Optimization Design Framework for Supersonic Wing

#### 3.1. FFD Geometry Parameterization

#### 3.2. IDW Mesh Warping

#### 3.3. SNOPT Optimization Algorithm

#### 3.4. Optimization Framework

- 1
- Constructing the initial model and the corresponding mesh, and giving the initial design variable ${\mathit{X}}^{0}$;
- 2
- The surface mesh ${\mathit{x}}_{\mathrm{s}}$ is parameterized using the FFD method;
- 3
- Based on the deformed surface mesh, the deformed volume mesh ${\mathit{x}}_{\mathrm{v}}$ is obtained through IDW mesh warping technology;
- 4
- Using ADflow solver to solve the RANS equation to obtain the convergent flow solution vector $\mathit{Q}$ and the aerodynamic objective function value $\mathit{W}$;
- 5
- Based on the flow field results, ADflow constructs the adjoint equation and solves it through the GMRES algorithm. Finally, the gradient of the aerodynamic objective function with respect to the design variables is obtained;
- 6
- Combining the calculation results of the flow field in step 4 and the gradient information in step 5, the SNOPT optimization algorithm is used to update the design variables;
- 7
- Repeating steps 2 to 6 until the optimization design converges.

## 4. Aerodynamic Optimization Design of Subsonic Leading Edge Configuration

#### 4.1. Optimization Problem

#### 4.2. Optimization Results

## 5. Aerodynamic Optimization Design of Supersonic Leading Edge Configuration

#### 5.1. Optimization Problem

#### 5.2. Optimization Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 11.**Different deformation forms of the FFD frame. (Orange points are after deformation and blue points are before deformation).

**Figure 12.**Comparison of the distribution of spanwise lift and twist angle before and after optimization for subsonic leading edge wing (‘Twist’ represents the geometric angle of the chord of the wing profile relative to the horizontal axis of the fuselage).

**Figure 13.**Comparison of spanwise drag distribution before and after optimization for subsonic leading edge wing.

**Figure 14.**Comparison of pressure distribution of the upper surface before and after optimization for subsonic leading edge wing.

**Figure 15.**Comparison of pressure coefficients and airfoil before and after optimization for subsonic leading edge wing ($Y=2y/b$ is the nondimensional spanwise percentage position, where y is spanwise coordinates and b is the span).

**Figure 17.**Comparison of pressure distribution and Mach number spatial distribution at 40% of the wingspan before and after optimization ($Y=2y/b$ is the nondimensional spanwise percentage position, where y is spanwise coordinates and b is the span).

**Figure 18.**Derivatives of the upper (

**left**) and lower (

**right**) lift/drag coefficients with respect to the Z-axis displacement of the wing surface grid points for subsonic leading edge configuration.

**Figure 19.**The optimization history of the subsonic leading edge configuration. Five typical steps were chosen to visualize the changing trend of pressure distribution during the optimization. The pressure distribution with different colors comes from the dot which has the same color in the function reduction line.

**Figure 27.**Comparison of the distribution of spanwise lift and twist angle before and after optimization for supersonic leading edge wing.

**Figure 28.**Comparison of spanwise drag distribution before and after optimization for supersonic leading edge wing.

**Figure 29.**Comparison of pressure distribution of the upper surface before and after optimization for supersonic leading edge wing.

**Figure 30.**Comparison of airfoil and pressure coefficients before and after optimization for supersonic leading edge wing ($Y=2y/b$ is the nondimensional spanwise percentage position, where y is spanwise coordinates and b is the span).

**Figure 31.**Comparison of pressure distribution of the upper surface before and after optimization ($Y=2y/b$ is the nondimensional spanwise percentage position, where y is spanwise coordinates and b is the span).

**Figure 32.**Derivatives of the upper (

**left**) and lower (

**right**) lift/drag coefficients with respect to the Z-axis displacement of the wing surface grid points for supersonic leading edge configuration.

**Figure 33.**The optimization history of the supersonic leading edge configuration. Four typical steps were chosen to visualize the changing trend of pressure distribution during the optimization. The pressure distribution with different colors comes from the dot which has the same color in the function reduction line.

**Table 1.**Gradient verification of lift coefficients with respect to design variables. (Underlines show the different significant numbers betweent the result from finite difference and adjoint method).

Var | Finite Difference | Adjoint Equation | Rel. Error | ${\mathit{h}}_{\mathit{opt}}$ |
---|---|---|---|---|

4 | 0.0024191310 | 0.0024183648 | $3.17\times {10}^{-4}$ | ${10}^{-6}$ |

6 | 0.0018757146 | 0.0018757950 | $4.29\times {10}^{-5}$ | ${10}^{-5}$ |

9 | 0.0008184027 | 0.0008184097 | $8.65\times {10}^{-6}$ | ${10}^{-5}$ |

16 | −0.0007589824 | −0.0007589494 | $4.34\times {10}^{-5}$ | ${10}^{-3}$ |

20 | −0.0018793665 | −0.0018790061 | $1.92\times {10}^{-4}$ | ${10}^{-3}$ |

**Table 2.**Gradient verification of drag coefficients with respect to design variables. (Underlines show the different significant numbers betweent the result from finite difference and adjoint method).

Var | Finite Difference | Adjoint Equation | Rel. Error | ${\mathit{h}}_{\mathit{opt}}$ |
---|---|---|---|---|

4 | −0.0466292505 | −0.0466005468 | $6.16\times {10}^{-4}$ | ${10}^{-6}$ |

6 | −0.0361944061 | −0.0361960611 | $4.57\times {10}^{-5}$ | ${10}^{-5}$ |

9 | −0.0157922613 | −0.0157923472 | $5.44\times {10}^{-6}$ | ${10}^{-5}$ |

16 | 0.0146452139 | 0.0146445737 | $4.37\times {10}^{-5}$ | ${10}^{-3}$ |

20 | 0.0362649685 | 0.0362579965 | $1.92\times {10}^{-4}$ | ${10}^{-3}$ |

Parameters | Value |
---|---|

Length of fuselage (m) | 70.1 |

Semi-span (m) | 12.8 |

Reference semi-area (${\mathrm{m}}^{2}$) | 153.2 |

Average aerodynamic chord length (m) | 21.0 |

Sweep angle of inner wing section (${}^{\circ}$) | 78 |

Sweep angle of outer wing section (${}^{\circ}$) | 69 |

Grid Level | Grid Size (Million) | ${\mathit{C}}_{\mathit{d}}$ (Counts) | GCI/% | Order p |
---|---|---|---|---|

L2 | 1.3 | 160.88 | / | |

L1 | 5.3 | 152.87 | 1.488 | |

L0.5 | 22.6 | 151.39 | 0.266 | 3.57 |

Category | Name | Quantity |
---|---|---|

Objective | min${C}_{D}$ | 1 |

Design Variables | Shape | 240 |

Twist | 5 | |

AoA | 1 | |

Constraints | ${C}_{L}$ = 0.142 | 1 |

$V\ge {V}_{\mathrm{initial}}$ | 1 | |

$t\ge 0.97{t}_{\mathrm{initial}}$ | 1 | |

${C}_{{m}_{y}}\ge 0$ | 1 |

**Table 6.**Comparison of aerodynamic coefficients between the optimized configuration and the initial configuration for subsonic leading edge wing.

Configuration | ${\mathit{C}}_{\mathit{l}}$ | ${\mathit{C}}_{\mathit{d}}$ | $\mathbf{\Delta}{\mathit{C}}_{\mathit{d}}$ (Counts) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{dp}}$ (Counts) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{dv}}$ (Counts) | AoA (Degree) | ${\mathit{C}}_{\mathit{my}}$ |
---|---|---|---|---|---|---|---|

Initial | 0.142 | 152.87 | / | 86.88 | 66.00 | 2.15 | 0.0039 |

Twist + Shape | 0.142 | 147.09 | −5.78 | 81.32 | 65.77 | 1.85 | 0.0040 |

Shape | 0.142 | 149.00 | −3.87 | 83.06 | 65.94 | 2.04 | 0.0040 |

Parameters | Value |
---|---|

Length of fuselage (m) | 48.7 |

Semi-span(m) | 10.5 |

Reference semi-area (${\mathrm{m}}^{2}$) | 71.4 |

Aspect ratio | 2.6 |

Average aerodynamic chord length (m) | 7.8 |

Chord length of wing root | 10.4 |

Chord length of wing tip | 4.4 |

Sweep angle of leading edge (${}^{\circ}$) | 19.7 |

Grid Level | Grid Size (Million) | ${\mathit{C}}_{\mathit{d}}$ (Counts) | GCI/% | Order p |
---|---|---|---|---|

L2 | 3.0 | 289.56 | / | |

L1 | 6.5 | 288.01 | 0.870 | |

L0.5 | 14.0 | 287.13 | 0.487 | 2.27 |

Category | Name | Quantity |
---|---|---|

Objective | min${C}_{D}$ | 1 |

Design Variables | Shape | 120 |

Twist | 3 | |

AoA | 1 | |

Constraints | ${C}_{L}$ = 0.195 | 1 |

$V\ge {V}_{\mathrm{initial}}$ | 1 | |

$t\ge {t}_{\mathrm{min}}$ | 1 | |

${C}_{{m}_{y}}\ge 0$ | 1 |

**Table 10.**Comparison of aerodynamic coefficients between the optimized configuration and the initial configuration for supersonic leading edge wing.

Configuration | ${\mathit{C}}_{\mathit{l}}$ | ${\mathit{C}}_{\mathit{d}}$ | $\mathbf{\Delta}{\mathit{C}}_{\mathit{d}}$ (Counts) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{dp}}$ (Counts) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{dv}}$ (Counts) | AoA (Degree) | ${\mathit{C}}_{\mathit{my}}$ |
---|---|---|---|---|---|---|---|

Initial | 0.195 | 288.01 | / | 211.46 | 76.55 | 2.97 | 0.03 |

Opt_0.7t | 0.195 | 274.97 | −13.04 | 198.30 | 76.67 | 3.00 | 0.02 |

Opt_0.9t | 0.195 | 277.91 | −10.10 | 201.26 | 76.65 | 2.98 | 0.02 |

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## Share and Cite

**MDPI and ACS Style**

Rao, H.; Shi, Y.; Bai, J.; Chen, Y.; Yang, T.; Li, J.
Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint. *Aerospace* **2023**, *10*, 420.
https://doi.org/10.3390/aerospace10050420

**AMA Style**

Rao H, Shi Y, Bai J, Chen Y, Yang T, Li J.
Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint. *Aerospace*. 2023; 10(5):420.
https://doi.org/10.3390/aerospace10050420

**Chicago/Turabian Style**

Rao, Hanyue, Yayun Shi, Junqiang Bai, Yifu Chen, Tihao Yang, and Junfu Li.
2023. "Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint" *Aerospace* 10, no. 5: 420.
https://doi.org/10.3390/aerospace10050420