# Metaheuristic Approaches to Solve a Complex Aircraft Performance Optimization Problem

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## Abstract

**:**

## 1. Introduction

## 2. Brief Review of Aircraft Multidisciplinary Analysis

#### 2.1. Aerodynamics

#### 2.2. Flight Stability

#### 2.3. Aeroelasticity

## 3. Genetic Algorithm (GA) Based Optimization Strategy

- Step 1: Generate the initial population using genetic parameters and each individual represents a specific aircraft model.
- Step 2: Perform the aforementioned multidisciplinary analysis to evaluate the responses of the interests for the current population.
- Step 3: Apply genetic operators to create the next generation of population and repeat Step 2 until the number of generations is met.

## 4. Genetic Programming Aided Optimization of Conceptual Aircraft Design

- Design of experiments
- Metamodel building by genetic programming (GP)
- Parameterized finite element model
- Optimization formulation, design variables and constraints

#### 4.1. Design of Experiments

#### 4.2. Metamodel Building by Genetic Programming (GP)

_{1}× x

_{2}− x

_{3})

^{1/2}, is shown in Figure 3.

^{2}) was the other fit criterion used to control the predictive performance of the metamodels:

#### 4.3. Parameterized Finite Element Model

#### 4.4. Optimization Formulation, Design Variables and Constraints

## 5. Case Study

^{−2}and R

^{2}= 0.995 for the predicted wingtip torsion response, indicate the metamodels by GP are quite accurate to estimate the nonlinear responses of the interests. Furthermore, design variables regarding horizontal bending stiffness and torsional stiffness have slight effects on the wingtip torsion, and this can be proved by sensitivity analysis performed in Section 5.2. To validate the accuracy of generated metamodels, a validation data set of 100 sampling points was also used to check the overall performance capacity of the metamodels. In this paper, the relative error was used to measure the discrepancy between the actual response and the predicted model,

#### 5.1. Optimal Designs by GP and GA

#### 5.2. Sensitivity Analysis

## 6. Summary

#### 6.1. Research Significance

#### 6.2. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Minimum distances between points in 250-point optimal Latin hypercube DoE (Design of Experiments).

**Figure 5.**(

**a**) A parametric wing structure; (

**b**) lifting surface in the aerodynamic model; (

**c**) model for structural analysis.

**Figure 9.**Stiffness comparison between the GA and GP methods. (

**a**) Vertical bending; (

**b**) Horizontal bending; (

**c**) Torsional.

**Figure 10.**Characteristic transient: (

**a**) perturbation pitch rate; (

**b**) perturbation angle of attack; (

**c**) perturbation velocity; (

**d**) perturbation pitch angle.

Design Variables | Lower Bound | Upper Bound |
---|---|---|

X1, Inboard taper ratio | 1.457 | 1.781 |

X2, Outboard taper ratio | 3.658 | 4.471 |

X3, Inboard aspect ratio | 0.937 | 1.146 |

X4, Outboard aspect ratio | 2.509 | 3.067 |

X5, Sweep angle (degree) | 24.651 | 30.130 |

X6, Inboard dihedral angle (degree) | 4.346 | 5.312 |

X7, Outboard dihedral angle (degree) | 0.645 | 0.789 |

X8, Axis position (root) | 0.306 | 0.374 |

X9, Axis percent (tip) | 0.338 | 0.413 |

X10, Axis percent (kink) | 0.389 | 0.476 |

X11, Vertical bending (a) | 0.600 | 2.000 |

X12, Vertical bending (b) | 0.600 | 2.000 |

X13, Vertical bending (c) | 0.600 | 2.000 |

X14, Horizontal bending (a) | 0.600 | 2.000 |

X15, Horizontal bending (b) | 0.600 | 2.000 |

X16, Horizontal bending (c) | 0.600 | 2.000 |

X17, Torsional (a) | 0.800 | 1.200 |

X18, Torsional (b) | 0.800 | 1.200 |

X19, Torsional (c) | 0.800 | 1.200 |

Parameters | Values |
---|---|

Population size | 500 |

Generations | 100 |

Number of fitness cases | 1997 |

Initialisation | 50% full + 50% grow |

Maximum depth initialised trees | 4 |

Minimum depth initialised trees | 2 |

Maximal depth for a tree | 50 |

Reproduction rate | 20% |

Crossover rate | 40% |

Mutation rate | 40% |

Termination of the evolution if RMSE is less than | 1.0 × 10^{−7} |

Responses | GP | Validation | GA | Constraint |

Stiffness | 0.533 | 0.535 | 0.806 | - |

Displacement | 5.61% | 5.65% | 6.16% | <7% |

Torsion (deg) | 1.39 | 1.44 | 2.05 | <2.14 |

Efficiency | 0.677 | 0.680 | 0.655 | >0.6 |

Flutter (m/s) | >320 | >320 | >320 m/s | >320 |

Drag (N) | 65,158.2 | 63,090.8 | 64,661.2 | <65,438 |

Geometric Design Variables | GP | GA |
---|---|---|

Inboard taper ratio | 1.74 | 1.60 |

Outboard taper ratio | 4.25 | 3.52 |

Inboard aspect ratio | 0.96 | 0.91 |

Outboard aspect ratio | 2.90 | 2.83 |

Sweep angle (deg) | 25.63 | 29.90 |

Inboard dihedral angle (deg) | 5.18 | 4.50 |

Outboard dihedral angle (deg) | 0.69 | 0.82 |

Axis position (root) | 0.31 | 0.33 |

Axis percent (tip) | 0.41 | 0.45 |

Axis percent (kink) | 0.46 | 0.44 |

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**MDPI and ACS Style**

Dong, G.; Wang, X.; Liu, D. Metaheuristic Approaches to Solve a Complex Aircraft Performance Optimization Problem. *Appl. Sci.* **2019**, *9*, 2979.
https://doi.org/10.3390/app9152979

**AMA Style**

Dong G, Wang X, Liu D. Metaheuristic Approaches to Solve a Complex Aircraft Performance Optimization Problem. *Applied Sciences*. 2019; 9(15):2979.
https://doi.org/10.3390/app9152979

**Chicago/Turabian Style**

Dong, Guirong, Xiaozhe Wang, and Dianzi Liu. 2019. "Metaheuristic Approaches to Solve a Complex Aircraft Performance Optimization Problem" *Applied Sciences* 9, no. 15: 2979.
https://doi.org/10.3390/app9152979