**1. Introduction**

Aircraft shape optimization is one of the key problems in aerodynamic configuration design. The traditional aerodynamic optimization design methods mainly rely on experience and trial-and-error methods, which require a lot of human, material and financial resources, and not only take a long time but also require a lot of computational resources [1]. In recent years, with the rapid development of computational fluid dynamics (CFD) technology, the combination of numerical methods and optimization algorithms for the aerodynamic shape optimization of aircraft can significantly shorten the development cycle and reduce the design cost [2]. Therefore, it is important to carry out research on efficient aerodynamic optimization design methods based on the combination of CFD technology and optimization algorithms for the development of aerodynamic optimization design.

Among numerous aerodynamic optimization studies, gradient-based methods and heuristic algorithms are two of the most widely used methods. Gradient-based methods are particularly attractive due to their ability to significantly improve the efficiency of high-dimensional optimization problems. The adjoint method proposed by Jameson [3] is

**Citation:** Zhao, X.; Tang, Z.; Cao, F.; Zhu, C.; Periaux, J. An Efficient Hybrid Evolutionary Optimization Method Coupling Cultural Algorithm with Genetic Algorithms and Its Application to Aerodynamic Shape Design. *Appl. Sci.* **2022**, *12*, 3482. https://doi.org/10.3390/ app12073482

Academic Editor: Vincent A. Cicirello

Received: 23 February 2022 Accepted: 23 March 2022 Published: 29 March 2022

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an effective sensitivity analysis method that evaluates sensitivity information by solving the adjoint problem regardless of the number of design variables. Therefore, the computational time of sensitivity analysis can be significantly reduced. By combining the adjoint method with the gradient method, the optimization efficiency can be greatly improved. In recent years, this technique has been widely used in aerodynamic optimization [4,5]. However, two reasons make this technique less attractive: one is its difficulty in dealing with constrained/multi-objective problems, and the other is that it is easy for it to fall into local optima.

Heuristic algorithms do not need to rely on information about a specific problem and have good global performance in finding the optima; they are thus particularly suitable for solving problems with complex multiple local optima. Among them, genetic algorithms (GAs), differential evolution (DE) algorithm and particle swarm optimization (PSO) algorithm are the most popular methods in the field of aerodynamic optimization, and they have all been successfully applied in aerodynamic optimization [6–9]. However, their evolutionary procedures require multiple calls to the CFD analysers, which significantly increases the computational cost. Therefore, it is necessary to improve the optimal efficiency and therefore to develop optimization algorithms in particular allowing for balanced exploitation and exploration capabilities [10].

Many engineering problems are complex high-dimensional multimodal problems, so that most algorithms converge slowly, easily fall into local optima and are inefficient in dealing with such problems. Aerodynamic optimization is a highly complex nonlinear problem with multi-parameter, high-dimensional and multimodal characteristics. In order to solve aerodynamic optimization problems effectively, it is undoubtedly necessary to develop new intelligent and knowledge-based algorithms with satisfactory performance. The genetic algorithm has good robustness and global search capability [11–15], and can be well adapted to solve various types of problems. The cultural algorithm is a knowledgebased super-heuristic algorithm, and its unique two-layer evolutionary mechanism can improve the evolutionary efficiency very well. The hybrid of genetic algorithms and cultural algorithm can combine the advantages of both, and then solve aerodynamic optimization problems efficiently.

Cultural algorithm (CA) [16] is an evolutionary algorithm based on the simulation of a two-layer evolutionary mechanism of human society, proposed by R.G. Reynolds in 1994. It was inspired by and developed from human sociology and aimed to model the evolution of the cultural component of evolutionary systems over time [17]. CA simulates the development of society and culture, which can be divided into two parts, the population space and the belief space, which are independent from each other but interconnected through communication protocols. CA extracts the implicit information carried by the population evolution process, such as the location of the optimal individuals or the range of the best individuals, into the belief space and stores it in knowledge sources. CA provides a new framework and mechanism for evolutionary models or swarm intelligence systems [18], such as genetic algorithms [19], ant colony algorithms [20], particle swarm algorithms [21] and differential evolution [22], etc. The two-layer evolutionary mechanism of CA improves the efficiency of the algorithm. Compared with other evolutionary algorithms, CA has stronger global optimization capability and higher optimization precision, and it has been successfully applied to optimization problems such as clustering analysis [23], sensor localization [24], multi-objective optimization [25] and vehicle routing [26]. Although the cultural algorithm can use knowledge sources to improve evolutionary efficiency, its global convergence and evolutionary efficiency are deficient due to its single mutation operator [27]. Therefore, the cultural algorithm needs to be improved for better performance of the optimization.

In this paper, an efficient hybrid evolutionary optimization method coupling CA with GAs (HCGA) is introduced with a validation background of the application of evolutionary algorithms to aerodynamic optimization design. Considering the features of CA and GAs, the proposed algorithm reconstructs the framework of cultural algorithms, which uses

GAs as a population space evolutionary model of the cultural framework, with the three types of knowledge, namely situational knowledge, normative knowledge and historical knowledge; these kinds of knowledge construct the knowledge sources of the belief space. In addition, HCGA introduces population variance and population entropy to determine population diversity, and it develops a new knowledge-guided *t*-mutation operator to dynamically adjust the mutation step based on the change of population diversity during the evolutionary process. It further introduces the *t*-mutation operator into the influence function to balance the exploration and exploitation ability of the algorithm and improve its optimization efficiency.

The rest of the paper is organized as follows. A brief introduction to the basic principles and framework of the cultural and genetic algorithms is given in Section 2. The proposed algorithm HCGA is introduced in Section 3. Numerical results and comparisons are presented and discussed in Section 4.2. The HCGA is applied in Section 5 to the aerodynamic optimization design of the wing cruise factor. Conclusions and perspectives are discussed in Section 6.

### **2. Brief Description of GAs and CA**
