# Metaheuristic Approaches to Solve a Complex Aircraft Performance Optimization Problem

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## 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

- Elsayed, M.S.A.; Sedaghati, R.; Abdo, M. Accurate Stick Model Development for Static Analysis of Complex Aircraft Wing-Box Structures. AIAA J.
**2009**, 47, 2063–2075. [Google Scholar] [CrossRef] - Wang, C. Insights from Developing a Multidisciplinary Design and Analysis Environment. Comput. Ind.
**2014**, 65, 786–795. [Google Scholar] [CrossRef] - Kenway, G.K.W.; Martins, J.R.R.A. Multipoint Aerodynamic Shape Optimization Investigations of the Common Research Model Wing. AIAA J.
**2016**, 54, 113–128. [Google Scholar] [CrossRef][Green Version] - Zhang, K.S.; Han, Z.H.; Li, W.J.; Song, W.P. Coupled Aerodynamic/ Structural Optimization of a Subsonic Transport Wing Using a Surrogate Model. J. Aircr.
**2008**, 45, 2167–2170. [Google Scholar] [CrossRef] - Yang, G.W.; Chen, D.W.; Cui, K. Response Surface Technique for Static Aeroelastic Optimization on a High-Aspect-Ratio Wing. J. Aircr.
**2009**, 46, 1444–1450. [Google Scholar] [CrossRef] - Mastroddi, F.; Gemma, S. Analysis of Pareto Frontiers for Multidisciplinary Design Optimization of Aircraft. Aerosp. Sci. Technol.
**2013**, 28, 40–55. [Google Scholar] [CrossRef] - Travaglini, L.; Ricci, S.; Bindolino, G. PyPAD: A Multidisciplinary Framework for Preliminary Airframe Design. Aircr. Eng. Aerosp. Technol.
**2016**, 88, 649–664. [Google Scholar] [CrossRef] - Brooks, T.R.; Kenway, G.K.W.; Martins, J.R.R.A. Benchmark Aerostructural Models for the Study of Transonic Aircraft Wings. AIAA J.
**2018**, 56, 2840–2855. [Google Scholar] [CrossRef] - Gray, J.S.; Hwang, J.T.; Martins, J.R.R.A.; Moore, K.T.; Naylor, B.A. Open MDAO: An Open-source Framework for Multidisciplinary Design, Analysis, and Optimization. Struct. Multidisc. Optim.
**2019**, 59, 1075–1104. [Google Scholar] [CrossRef] - McAllister, C.D.; Simpson, T.W. Multidisciplinary Robust Design Optimization of an Internal Combustion Engine. J. Mech. Des.
**2003**, 125, 124–130. [Google Scholar] [CrossRef] - Wang, K.; Zheng, Y. A New Particle Swarm Optimization Algorithm for Fuzzy Optimization of Armored Vehicle Scheme Design. Appl. Intell.
**2012**, 37, 520–526. [Google Scholar] [CrossRef] - Van Dijk, M.T.; van Wingerden, J.W.; Ashuri, T.; Li, Y.Y. Wind Farm Multi-Objective Wake Redirection for Optimizing Power Production and Loads. Energy
**2017**, 121, 561–569. [Google Scholar] [CrossRef] - Tabatabaei, S.K.; Behbahani, S.; de Silva, C.W. Self-Adjusting Multidisciplinary Design of Hydraulic Engine Mount Using Bond Graphs and Inductive Genetic Programming. Eng. Appl. Artif. Intel.
**2016**, 48, 32–39. [Google Scholar] [CrossRef] - Sekar, W.K.; Laschka, B. Calculation of the Transonic Dip of Airfoils Using Viscous-Inviscid Aerodynamic interaction method. Aerosp. Sci. Technol.
**2005**, 9, 661–671. [Google Scholar] [CrossRef] - Michalewicz, Z. Genetic Algorithms + Data Structures = Evolution Programs; Springer: Berlin, Germany, 2013. [Google Scholar]
- Manan, A.; Vio, G.A.; Harmin, M.Y.; Cooper, J.E. Optimization of Aeroelastic Composite Structures Using Evolutionary Algorithms. Eng. Optim.
**2010**, 42, 171–184. [Google Scholar] [CrossRef] - Wang, G.G.; Shan, S. Review of Metamodeling Techniques in Support of Engineering Design Optimization. J. Mech. Des.
**2006**, 129, 370–380. [Google Scholar] [CrossRef] - Forrester, A.I.J.; Sobester, A.; Keane, A.J. Engineering Design via Surrogate Modelling: A Practical Guide; John Wiley & Sons: New York, NY, USA, 2008. [Google Scholar]
- Timme, S.; Marques, S.; Badcock, K.J. Transonic Aeroelastic Stability Analysis Using a Kriging-Based Schur Complement Formulation. AIAA J.
**2011**, 49, 1202–1213. [Google Scholar] [CrossRef] - Raghavan, B.; Hamdaoui, M.; Xiao, M.; Breitkopf, P.; Villon, P. A Bi-level Meta-Modeling Approach for Structural Optimization Using Modified POD Bases and Diffuse Approximation. Comput. Struct.
**2013**, 127, 19–28. [Google Scholar] [CrossRef] - Liu, D.; Lohse-Busch, H.; Toropov, V.; Huhne, C.; Armani, U. Detailed Design of a Lattice Composite Fuselage Structure by a Mixed Optimization Method. Eng. Optim.
**2016**, 48, 1707–1720. [Google Scholar] [CrossRef] - Portelette, L.; Roux, J.; Robin, V.; Feulvarch, E. A Gaussian surrogate model for residual stresses induced by orbital multi-pass TIG welding. Comput. Struct.
**2017**, 183, 27–37. [Google Scholar] [CrossRef] - Amouzgar, K.; Bandaru, S.; Ng, A.H.C. Radial Basis Functions with a Priori Bias as Surrogate Models: A Comparative Study. Eng. Appl. Artif. Intell.
**2018**, 71, 28–44. [Google Scholar] [CrossRef] - Koza, J.R. Genetic Programming: On the Programming of Computers by Means of Natural Selection; MIT Press: Cambridge, MA, USA, 1992. [Google Scholar]
- Armani, U. Development of a Hybrid Genetic Programming Technique for Computationally Expensive Optimisation Problems. Ph.D. Thesis, University of Leeds, Leeds, UK, 2014. [Google Scholar]
- Fajfar, I.; Bűrmen, Ä.; Puhan, J. Grammatical evolution as a hyper-heuristic to evolve deterministic real-valued optimization algorithms. In Genetic Programming and Evolvable Machines; Spector, L., Ed.; Springer: Berlin, Germany, 2018; Volume 19. [Google Scholar]
- Faradonbeh, R.S.; Monjezi, M.; Armaghani, D.J. Genetic Programing and Non-linear Multiple Regression Techniques to Predict Backbreak in Blasting Operation. Eng. Comput.
**2016**, 32, 123–133. [Google Scholar] [CrossRef] - Pawiński, G.; Sapiecha, K. Speeding up Global Optimization with the Help of Intelligent Supervisors. Appl. Intell.
**2016**, 45, 777–792. [Google Scholar] [CrossRef] - Rostami, M.F.; Sadrossadat, E.; Ghorbani, B.; Kazemi, S.M. New Empirical Formulations for Indirect Estimation of Peak-confined Compressive Strength and Strain of Circular RC Columns Using LGP Method. Eng. Comput.
**2018**, 34, 865–880. [Google Scholar] [CrossRef] - Van Dam, C.P.; Nikfetrat, K.; Wong, K.; Vijgen, P.M.H.W. Drag Prediction at Subsonic and Transonic Speeds Using Euler Methods. J. Aircr.
**1995**, 32, 839–845. [Google Scholar] [CrossRef] - Fang, Z.P.; Chen, W.C.; Zhang, S.G. Flight Dynamics of Aircraft; Beihang University Press: Beijing, China, 2005. [Google Scholar]
- MSC Software. 2014. MSC Nastran Version 68: Aeroelastic Analysis User’s Guide; MSC Software: Newport Beach, CA, USA, 1992. [Google Scholar]
- Harder, R.L.; Desmarais, R.N. Interpolation Using Surface Splines. J. Aircr.
**1972**, 9, 189–191. [Google Scholar] [CrossRef] - Holland, J.H. Adaptation in Natural and Artificial Systems, 2nd ed.; MIT Press: Cambridge, MA, USA, 1992. [Google Scholar]
- Carr, J. An Introduction to Genetic Algorithms. Senior Project
**2014**, 40, 7. [Google Scholar] - Lin, H.; Lin, C. Optimization of Printed Circuit Board Component Placement Using an Efficient Hybrid Genetic Algorithm. Appl. Intell.
**2016**, 45, 622–637. [Google Scholar] [CrossRef] - Box, J.F.; Fisher, R.A. Design of Experiments. Am. Stat.
**1980**, 34, 1–7. [Google Scholar] - Simpson, T.; Peplinski, J.; Koch, P.; Allen, J. On the Use of Statistics in Design and the Implications for Deterministic Computer Experiments. Des. Theory Method. DTM
**1997**, 97, 14–17. [Google Scholar] - Audze, P.; Eglais, V. New Approach for Planning out of Experiments. Probl. Dyn. Strengths
**1977**, 35, 104–107. [Google Scholar] - Zhong, J.; Feng, L.; Ong, Y.-S. Gene Expression Programming: A Survey. IEEE Comput. Intell. Mag.
**2017**, 12, 54–72. [Google Scholar] [CrossRef] - Fajfar, I.; Tuma, T. Creation of Numerical Constants in Robust Gene Expression Programming. Entropy
**2018**, 20, 756. [Google Scholar] [CrossRef] - Sette, S.; Boullart, L. Genetic Programming: Principles and Applications. Eng. Appl. Artif. Intell.
**2001**, 14, 727–736. [Google Scholar] [CrossRef] - Wang, H.; Yao, X. Corner Sort for Pareto-Based Many-Objective Optimization. IEEE Trans. Cybern.
**2014**, 44, 90–102. [Google Scholar] - Wan, Z.Q.; Wang, X.; Yang, C. Integrated Aerodynamics/Structure/StabilitOptimization of Large Aaircraft in Conceptual Design. Proc. IMechE Part G J. Aerosp. Eng.
**2018**, 232, 745–756. [Google Scholar] [CrossRef]

**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 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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