# Kriging-Based Framework Applied to a Multi-Point, Multi-Objective Engine Air-Intake Duct Aerodynamic Optimization Problem

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Optimization Framework

#### 2.1.1. Kriging Surrogate

- Linearity—linear combination of random functions

- Unbiasedness—absence of systematic bias

- Minimal prediction variance—minimizing the mean squared error

#### 2.1.2. RBF-Based Mesh Morphing

**$\mathcal{P}$**containing a unity column and all control points’ positions (Equation (20)).

#### 2.1.3. Achievement Scalarizing Function

#### 2.2. Optimization Problem

#### 2.3. Evaluation of Objectives

#### 2.3.1. Flow Solver Governing Equations

- The continuity equation:

- The momentum conservation equation:

#### 2.3.2. Validation of Turbulence Modeling Technique

#### 2.3.3. Details of the Flow Field Modeling Approach

^{−5}.

## 3. Results and Discussion

#### 3.1. Metamodel Fit Validation

#### 3.2. Sensitivity Analysis

#### 3.3. Assessment of Deformed Mesh Quality

#### 3.4. Optimization Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Leave-One-Out Cross-Validation

## Appendix B. Functional Analysis of Variance (FANOVA)

## References

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**Figure 3.**Initial mesh with an indication of flow stations and mesh morpher control points’ locations.

**Figure 4.**Illustration of $\underset{\theta \in \left[0,2\pi \right]}{\mathrm{min}}{P}_{t}^{AIP}\left(\theta -\frac{\pi}{6},\theta +\frac{\pi}{6}\right)$ evaluation process at the AIP. The blue segment represents the pressure averaging region.

**Figure 5.**Prediction of normalized longitudinal (U) and circumferential (V) velocity components for various turbulence models in a 180° curved duct with a circular cross-section. Station: (

**a**,

**b**) at 90° bend; (

**c**,

**d**) two diameters downstream of the curved section. Adapted from [37].

**Figure 7.**Results of the duct grid independence study. L = duct centerline length. Numerical values indicate the number of mesh elements.

**Figure 8.**LOO−CV plots for various objective function transformations. Blue lines indicate locations of perfect predictions.

**Figure 9.**Plots of standardized cross-validated residuals for various objective function transformations. Red dashed lines indicate interval of +/− 3 standard deviations.

**Figure 10.**Q−Q plot for various objective function transformations. Blue lines indicate target normal distribution.

**Figure 11.**The effect of FANOVA analysis on model considering 12 variables. Bar height indicates the total effect Sobol’ index value; black color represents the main effect index, and gray color denotes all higher-order interactions.

**Figure 12.**Assessment of deformed mesh quality: (

**a**) mesh orthogonality angle distribution at the duct symmetry plane for a selected solution; (

**b**) histogram of minimum value of mesh elements’ orthogonality angle distribution in all optimization cases. The dashed line indicates the orthogonality angle value for the reference case.

**Figure 14.**Cumulated minimum value of the ASF (

**a**) and convergence criterion evolution (

**b**) in the course of optimization.

**Figure 15.**Optimization solutions in the objective function space for the three design points. Markers indicate optimization target (target), evaluation of initial geometry (reference), and best obtained solution (best).

**Figure 16.**Details of the distortion coefficient improvements: (

**a**) radially averaged total pressure distribution at AIP. Solid and dashed lines indicate total pressure levels averaged over the AIP and worst 60° sector, respectively; (

**b**) total pressure at AIP normalized by the average dynamic head.

**Figure 17.**Details of the pressure loss coefficient improvements: (

**a**) maps of local pressure loss coefficient; (

**b**) flow streamlines colored by the vertical velocity component at cross−section plane downstream of the AIP.

Design Variable | Range of Displacement |
---|---|

${x}_{1},\dots ,{x}_{4}$ | $\pm 100\mathrm{mm}$ |

${x}_{5},{x}_{7},{x}_{8}$ | $\pm 50\mathrm{mm}$ |

${x}_{6}$ | $\pm 75\mathrm{mm}$ |

${x}_{9},\dots ,{x}_{12}$ | $\pm 15\mathrm{mm}$ |

Design Point | Altitude (m) | A/C Velocity (m/s) | Ambient Pressure (Pa) | Ambient Temperature (K) | Ambient Density (kg/m ^{3}) | Engine Mass Flow Rate (kg/s) |
---|---|---|---|---|---|---|

DP1: Nominal cruise | 3000 | 65 | 69,700 | 268.6 | 0.909 | 1.5 |

DP2: Low-altitude climb | 100 | 46 | 100,129 | 287.5 | 1.213 | 1.9 |

DP3: High-altitude cruise | 3700 | 71 | 64,089 | 264.1 | 0.845 | 1.3 |

Design Point | $\mathit{d}\mathit{P}$ | $\mathit{D}{\mathit{C}}_{60}$ |
---|---|---|

DP1: Nominal cruise | target: 10% nadir: 200% | target: 10% nadir: 200% |

DP2: Low-altitude climb | target: 5% nadir: 200% | target: 5% nadir: 200% |

DP3: High-altitude cruise | target: 5% nadir: 200% | target: 5% nadir: 200% |

Transforming Function | Definition |
---|---|

Natural logarithm | $\mathrm{ln}y$ |

Square root | $\sqrt{y}$ |

Negative inverse square root | $\frac{-1}{\sqrt{y}}$ |

Negative inverse | $\frac{-1}{y}$ |

Cube root | $\sqrt[3]{y}$ |

Reference | Optimized Solution | |||||
---|---|---|---|---|---|---|

$\mathit{d}\mathit{P}$ | $\mathit{D}{\mathit{C}}_{60}$ | $\mathit{d}\mathit{P}$ | $\mathit{D}{\mathit{C}}_{60}$ | |||

Design Point | Absolute | Relative | Absolute | Relative | ||

DP1: Nominal cruise | 0.00512 | 0.09640 | 0.00493 | −3.71% | 0.09190 | −4.67% |

DP2: Low-altitude climb | 0.00459 | 0.09369 | 0.00443 | −3.49% | 0.08943 | −4.55% |

DP3: High-altitude cruise | 0.00471 | 0.09679 | 0.00454 | −3.61% | 0.09252 | −4.41% |

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Drężek, P.S.; Kubacki, S.; Żółtak, J.
Kriging-Based Framework Applied to a Multi-Point, Multi-Objective Engine Air-Intake Duct Aerodynamic Optimization Problem. *Aerospace* **2023**, *10*, 266.
https://doi.org/10.3390/aerospace10030266

**AMA Style**

Drężek PS, Kubacki S, Żółtak J.
Kriging-Based Framework Applied to a Multi-Point, Multi-Objective Engine Air-Intake Duct Aerodynamic Optimization Problem. *Aerospace*. 2023; 10(3):266.
https://doi.org/10.3390/aerospace10030266

**Chicago/Turabian Style**

Drężek, Przemysław S., Sławomir Kubacki, and Jerzy Żółtak.
2023. "Kriging-Based Framework Applied to a Multi-Point, Multi-Objective Engine Air-Intake Duct Aerodynamic Optimization Problem" *Aerospace* 10, no. 3: 266.
https://doi.org/10.3390/aerospace10030266