A Multi-Objective Optimization Design Method for High-Aspect-Ratio Wing Structures Based on Mind Evolution Algorithm Backpropagation Surrogate Model

Abstract

The design of high-aspect-ratio wings enhances the flight efficiency of UAVs but also introduces significant aeroelasticity challenges. The efficient optimization of wing structures in complex environments has become critical. To address the current challenges in balancing wing strength with lightweight structural designs, this study proposed an intelligent solution method for optimizing wing dimensions and structural layout. Driven by mechanical simulation data, the method established a mapping relationship between the structural layout and dimensions of the wing and its bending stiffness. This approach was further enhanced by the mind evolution algorithm (MEA) to optimize the solution performance of the surrogate model. The wing structure optimization model was established using the multi-objective grey wolf optimizer (MOGWO) based on the surrogate model for search and optimization. This study focused on the composite material wing of a long-endurance unmanned aerial vehicle (UAV). The established MEA-BP surrogate model demonstrated high computational efficiency, with the prediction error standard deviation (STD) of wing deflection not exceeding 0.495 mm. The optimization model required 175 s to calculate the Pareto front solutions. The optimized structure resulted in a 28.32% increase in wing equivalent stiffness, and weight only increased by 6.67% compared to the original structure. These results showcased the effectiveness of the proposed method and validated the feasibility of integrating intelligent optimization algorithms and machine learning in the field of aircraft design.

1. Introduction

The development of long-endurance UAVs addresses the need for extended monitoring data collection in military reconnaissance, environmental monitoring, and emergency rescue operations [1,2,3]. To meet the requirements for long endurance, UAVs adopt high-aspect-ratio wing designs to improve cruise efficiency and fuel economy [4,5,6,7,8]. However, the reduced equivalent stiffness caused by the large wingspan makes the wing prone to uncontrollable deformation in complex flight environments, affecting flight safety [9,10,11]. The sensitivity of UAVs to flight weight further complicates the optimization of wing structures. Therefore, ensuring the mission characteristics and flight safety of long-endurance UAVs necessitates a lightweight structural design for high-aspect-ratio wings while maintaining bending stiffness [12,13].

The complex coupling effects between the structure and aerodynamic forces of high-aspect-ratio wings lead to prominent aeroelastic problems. As the aspect ratio of the wing increases, the wing is susceptible to significant bending and deformation due to aerodynamic forces during flight, which not only reduces the aerodynamic performance but may also cause flight instability [14,15]. Therefore, it is particularly important to ensure an efficient and optimized design of the wing.

The structural design of the wing represents a significant challenge in the field of aeronautical engineering. It is a complex, multidisciplinary, and multi-objective design problem that necessitates a comprehensive approach to mission requirements, flight performance, and environmental conditions in order to achieve the optimal design objectives [16,17]. Traditional optimization algorithms and intelligent optimization algorithms are currently the primary methods for addressing such multi-objective optimization problems in wing structural design [18,19]. Traditional optimization algorithms, such as the weighted method, ε-constraint method, and linear programming, simplify the solution process by transforming multi-objective problems into single-objective problems. For instance, Stewart et al. [20] studied the impact of wing planform shapes on the flight performance of flapping-wing aircraft, using wing area, peak power input, and motion configuration-based aerodynamic forces as objective functions, and employed the ε-constraint method to select aerodynamic force as the primary objective function for optimal wing shape design.

While traditional algorithms have demonstrated efficacy in engineering applications, they are not without limitations. The subjectivity inherent in weight selection and the lack of adaptability to nonlinear problems represent significant constraints on their broader applicability. Intelligent optimization algorithms, such as genetic algorithms and particle swarm optimization, address multi-objective problems directly, allowing for a more comprehensive identification of Pareto front solutions and providing designers with a wider range of choices. Wu et al. [21] proposed a controllable wing design method for underwater gliders based on multi-objective optimization algorithms. They established a relationship between controllable wing configuration parameters and the glider’s overall performance, using genetic algorithms to obtain the Pareto optimal set and verifying the design through a prototype. Each intelligent optimization algorithm possesses distinct advantages and requires careful design when applied to problem-solving tasks [22,23,24]. Compared with other optimization algorithms, the MOGWO algorithm achieved superior global search and local refinement capabilities through a roulette wheel selection strategy. This approach demonstrated outstanding performance in terms of optimization efficiency and solution quality, making it highly effective for rapid optimization design of high-aspect-ratio wings.

During the design process, it is essential to conduct a comprehensive and meticulous analysis to ensure the soundness and logical coherence of the design. Such analyses comprise aerodynamic performance analysis, structural strength analysis, and fatigue life analysis [25,26,27,28]. The extensive calculations required for wing performance analysis led to resource wastage and prolonged design cycles. To address this issue, researchers have developed surrogate models for the rapid computation of target performance levels for various structural designs. Common surrogate models include Kriging models, radial basis functions, and support vector regression models [29,30,31,32]. Shi et al. [33] investigated the aerodynamic configuration design of tandem wing UAVs, proposing the entropy rank and selection pooling (ESP) adaptive sequential sampling method to generate datasets for training radial basis function (RBF) surrogate models. The calculated Pareto front set for aerodynamic configuration achieved a performance improvement of over 6.44% compared to the original design. For more complex design models, neural networks, with their powerful nonlinear relationship fitting capabilities, offer an effective solution for high-dimensional nonlinear problems. Liu et al. [34] addressed the aerodynamic shape optimization of hypersonic vehicles over a wide Mach number range by establishing a neural network surrogate model with 18 wing design variables. They combined this with optimization algorithms to compute the optimization scenarios for different weight coefficient combinations.

While the application of surrogate models has accelerated wing structural design, they still fall short in directly solving complex multi-parameter design problems. Considering the rapid and accurate computation capabilities of surrogate models and the comprehensive optimization capabilities of multi-objective optimization algorithms, their combination can significantly enhance design efficiency [35,36,37]. Conventional wing design optimization often focuses on single objectives or local solutions, lacking the ability to handle multi-objective optimization and engineering complexities. Surrogate models and multi-objective algorithms effectively address efficiency and accuracy challenges in high-aspect-ratio wing stiffness and lightweight design. This paper proposes a lightweight optimization design method for high-aspect-ratio wings based on the MEA-BP neural network surrogate model. The MEA-BP neural network is employed to establish a surrogate model for wing structural design, enabling the rapid prediction of maximum wing deflection based on the structural layout and dimensions of ribs, spars, and skins. On this basis, with the goal of minimizing wing weight while maximizing equivalent stiffness, the MOGWO algorithm is used to compute the Pareto front solutions for the wing optimization problem. This paper conducts a comprehensive performance analysis of the surrogate model, discusses the advantages and disadvantages of different designs within the Pareto front solutions, selects the optimal design, and compares it with the original structure to validate the effectiveness of the design method.

2. Research Object

With the rapid advancement of UAV technology, high-aspect-ratio UAVs, characterized by their unique wing design and superior performance, have demonstrated significant potential in critical fields such as military reconnaissance, environmental monitoring, and logistics transportation. To meet the stringent requirements for strength, rigidity, and light weight in the wing structures of these UAVs, this paper proposed a novel optimization design method for wing structures. This method thoroughly investigated the complex loading environment faced by high-aspect-ratio UAV wings during flight. As illustrated in Figure 1, the overall wing structure and external loading conditions in this study referenced the UAV structural design scheme proposed by Liu et al. [34]. The research object was a high-aspect-ratio composite wing with a wingspan of 5.5 m, chord length of 0.6 m, and a NACA 4412 airfoil.

The wing under study was primarily subjected to distributed aerodynamic forces and concentrated loads from mounted equipment. Yan [38] simulated the aerodynamic load distribution of the wing using Profili 2.21 software, considering flight conditions at an altitude of 8,000 m, Reynolds number of 573,287, and an angle of attack of 6°. To simulate the loading conditions of mounted equipment, a vertical downward concentrated force of 200 N was applied at the middle of the wing. The external load distribution on the wing is shown in Figure 2. The aerodynamic load distribution of the wing was as follows: S₁: 3.68 × 10⁻⁴ MPa, S₂: −0.01x + 6.13 × 10⁻⁴ MPa (x: 0~610 mm), S₃: −1.12 × 10⁻⁴ MPa, S₄: −6.24 × 10⁻⁴ MPa. Based on the characteristics of the aerodynamic load distribution, this study considered the aerodynamic and mounted load effects on the wing during flight. By adjusting the layout and design of key structures such as wing ribs and spars, a comprehensive optimization of the wing structure was performed. The optimized wing aimed to exhibit a more uniform stress distribution under the same load conditions, with a significant reduction in maximum stress. It met the application conditions of a tensile limit of 655 MPa, a compressive limit of 621 MPa, and a deflection ≤ 5% of the wingspan, achieving the design objectives of minimal deformation and lightweight.

3. Research Contents

3.1. MEA-BP Surrogate Model

The backpropagation (BP) neural network is a multilayer feedforward neural network based on the error backpropagation algorithm (Figure 3a). It possesses strong nonlinear mapping capabilities and flexible network structures, making it widely used in fields such as signal processing, pattern recognition, and intelligent control [39]. However, traditional BP neural network surrogate models often face issues of insufficient fitting accuracy and low computational efficiency. Consequently, many researchers have adopted genetic algorithms (GAs) to optimize the initial weights and biases of BP neural networks [40,41]. Despite this, BP neural network models based on GA still suffer from premature convergence and slow convergence, limiting their optimization effectiveness.

The mind evolution algorithm (MEA) is a heuristic optimization algorithm that simulates the process of human cognitive evolution, solving problems by mimicking human learning, adaptation, and evolution processes [42]. MEA inherits concepts from GA, such as population, individual, environment, and evolution, and introduces new concepts like subpopulations, bulletin boards, convergence, and divergence to address GA’s optimization limitations. By dividing the population into superior and temporary subpopulations and employing operations like convergence, divergence, and information exchange via bulletin boards, MEA can more rapidly approach or find the global optimum. The algorithm structure is shown in Figure 3b.

The MEA-BP surrogate model combines the global search capability of MEA with the nonlinear mapping ability of BP neural networks, enhancing the training efficiency and prediction accuracy of neural networks [43]. The algorithm flow of the MEA-BP model is illustrated in Figure 4, with the main steps as follows:

• Data Preprocessing and Model Construction: Generate training and testing samples for the MEA-BP model and construct the topology of the BP neural network.
• MEA Parameter Setting: Define parameters such as the number of iterations, population size, and the size of the superior and temporary subpopulations.
• Initial Population Generation and Scoring: Randomly generate the initial population and score the individuals, with the highest-scoring individuals designated as superior and temporary individuals.
• Generation of Superior and Temporary Subpopulations: Generate superior and temporary subpopulations centered around the superior and temporary individuals.
• Convergence and Divergence: Perform convergence operations within subpopulations to obtain local optimal individuals, followed by divergence operations between subpopulations to retain the high-scoring subpopulations.
• Iterative Optimization: Repeatedly perform convergence and divergence operations until the maximum number of iterations is reached or the score of the optimal individual no longer changes.
• Assign Optimal Weights and Biases: Decode the optimal individual found by MEA according to encoding rules to obtain the initial weights and biases of the BP neural network.
• Train BP Neural Network: Perform iterative training until stopping criteria are met and output the final prediction results.

3.2. MOGWO Algorithm

The MEA-BP surrogate model combines the global search capability of the introduction of intelligent optimization algorithms represented by multi-objective genetic algorithm (MOGA) and multi-objective particle swarm optimization (MOPSO), which has improved the ability to solve multi-objective optimization problems. Different intelligent optimization algorithms have their own outstanding advantages in solving problems. The wing optimization design problem studied in this paper required the rapid optimization of the global space of the design domain. In comparison with algorithms represented by MOGA, the MOGA presents certain shortcomings, including a slow convergence speed and an ineffective search for global optimal solutions. Consequently, it is not an optimal choice for addressing this problem. In recent years, the multi-objective grey wolf optimizer (MOGWO) algorithm proposed by Mirjalili et al. [44] has been widely used. It is a meta-heuristic algorithm that simulates the social hierarchy and cluster hunting strategy of wild wolves. Compared to NSGA-II and MODEA, MOGWO demonstrated higher optimization efficiency and solution quality in 2E-COLRP and SHSC problems [45,46]. In hybrid energy system optimization, it outperformed MOGOA in reducing costs and improving reliability [47]. This study leveraged MOGWO to address the rapid lightweight design requirements of high-aspect-ratio wing structures, effectively enhancing optimization efficiency and solution quality.

MOGWO retains the fundamental concepts of GWO [48], viewing the parameters to be optimized as a wolf pack with a clear hierarchical structure, where α, β, δ, and ω wolves represent the best, second-best, third-best, and other solutions, respectively. The ω wolves hunt under the leadership of α, β, and δ wolves. Equation (1) simulated the encircling behavior during hunting.

$$\begin{array}{c} \overrightarrow{D}=\left|\overrightarrow{C} \cdot { \overrightarrow{X}}_p\left(t\right)- \overrightarrow{X}\left(t\right)\right|,\\ \overrightarrow{X}\left(t+1\right)={ \overrightarrow{X}}_p\left(t\right)- \overrightarrow{A} \cdot \overrightarrow{D} ,\\ \overrightarrow{A}=2 {\displaystyle }\overrightarrow{a} \cdot { \overrightarrow{r}}_1- {\displaystyle }\overrightarrow{a},\\ \overrightarrow{C}=2 \cdot { \overrightarrow{r}}_2,\end{array}$$

(1)

where $\overrightarrow{D}$ is the distance vector, $\overrightarrow{A}$ and $\overrightarrow{C}$ are coefficient vectors, $t$ is the iteration number, $\overrightarrow{X}$ is the position vector of a grey wolf, ${\overrightarrow{X}}_p$ is the position vector of the prey, $\overrightarrow{a}$ is the convergence factor, and ${ \overrightarrow{r}}_1$ and ${ \overrightarrow{r}}_2$ are random vectors in the range [0, 1]. In practice, due to the uncertainty of the prey’s position, the positions of the leading wolves α, β, and δ are used to estimate the potential location of the prey. This process is illustrated in Figure 5, and the position of other grey wolves is updated based on Equation (2), with the aim of gradually approaching the actual location of the prey.

$$\begin{array}{c}{\overrightarrow{D}}_i=\left|\overrightarrow{C} \cdot {\overrightarrow{X}}_i\left(t\right)-\overrightarrow{X}\left(t\right)\right|,(i=1,2,3),\\ \overrightarrow{X}\left(t+1\right)={\displaystyle \frac{1}{3}}{\displaystyle \sum _{i=1}^3\left({\overrightarrow{X}}_i\left(t\right)-\overrightarrow{A} \cdot {\overrightarrow{D}}_i\right)},\end{array}$$

(2)

where ${\overrightarrow{X}}_i$ represents the position vectors of the leading wolves α, β, and δ, and ${\overrightarrow{D}}_i$ represents the distance vectors between other wolves and the leading wolves.

Building on GWO, MOGWO introduces the concept of the Pareto optimal solution set, allowing the algorithm to accurately find a set of balanced solutions in a multi-objective optimization space, achieving a collaborative optimization of multiple objective functions. To effectively manage and utilize high-quality solutions obtained during iterations, MOGWO incorporates an external archive mechanism that stores and retrieves the most suitable non-dominated solutions, ensuring the diversity and quality of the solution set. Additionally, the algorithm employed a roulette wheel selection strategy for selecting the leaders, ensuring that the chosen leaders guide the pack towards the Pareto front. These advancements enabled MOGWO to solve multi-objective optimization problems more efficiently and accurately than existing multi-objective optimization algorithms. The algorithm flow of MOGWO is shown in Figure 6, with the main steps as follows:

• Initialization: Randomly generate an initial set of solutions as the grey wolf pack and initialize parameters a, A, and C.
• Evaluation and Archiving: Calculate the fitness values of each solution based on the objective functions and use an external archive to store non-dominated solutions.
• Leader Selection: Use the roulette wheel selection strategy to select the leaders (α, β, and δ wolves) based on fitness values.
• Position Update: Update the positions of other individuals in the pack using the leaders’ positions and calculate the fitness values of the updated individuals.
• Non-dominated Sorting and Archive Update: Perform non-dominated sorting on the new population and update the archive. If the archive is full, remove the least crowded solution.
• Iteration Termination: Check if the maximum number of iterations is reached. If so, terminate the iteration.
• Result Output: Output the Pareto optimal solution set from the archive as the optimization result.

4. Simulation Experiment Content

In order to realize the lightweight structural design of large aspect ratio wings, this paper employed an MEA-BP neural network as a surrogate model for wing design. In combination with the MOGWO algorithm, this approach addressed the optimization problem, resulting in the optimal structural design for the wing. The research process is illustrated in Figure 7. First, a series of simulation experiments were conducted on wings with various design structures using parametric modeling, establishing a data set of wing structural parameters and deformation results. The MEA-optimized BP neural network was then trained using simulation data, with the effectiveness of the surrogate model validated through comparative prediction data. Finally, a multi-objective optimization problem mathematical model was constructed based on UAV operating conditions. The MOGWO algorithm was utilized to perform target optimization based on the surrogate model’s calculation results. The correctness of the computed Pareto front solution set was validated, and the results were analyzed and discussed according to practical application requirements.

4.1. Acquisition of Training Data

The high-aspect-ratio wing structure consisted of wing ribs, beams, and skins in this paper. In view of the large wingspan and high aspect ratio of long-flight UAVs, composite materials with high strength, high stiffness, and strong designability must be used to meet the requirements of light weight, high stiffness, and high performance of long-flight UAVs. In this paper, the composite material structure of the high-aspect-ratio UAV wing uses Toray T300 plain weave bidirectional carbon fiber cloth and Nanjing Fiberglass Institute SW100A-100a high-strength bidirectional glass fiber cloth were used as reinforcement materials, and balsa wood was used as the wing rib matrix material. The density of balsa wood was 160 kg/m³, the elastic modulus was 4 GPa, and the Poisson’s ratio was 0.49.

Table 1. Mechanical property parameters.

#	T300	SW100A-100a
Density (kg/m3)	1.43 × 103	1.782 × 103
E11/GPa	62.4	38.36
E22/GPa	62.4	38.36
E33/GPa	8.10	8.10
G12/GPa	3.14	4.14
G13/GPa	3.14	4.14
G23/GPa	1.312	1.312
μ12	0.33	0.26
μ13	0.33	0.26
μ23	0.33	0.26

presents some mechanical properties parameters of the reinforced material. In order to reduce the weight of the wing and give full play to the performance of materials, the layup of each composite component of the wing was designed according to a specific scheme. The wing skin was laid in a combination of T300/SW100A-100a, with a layup scheme of 0°/90°/0°. The wing rib was constructed using a composite sandwich structure, comprising a balsa wood main body and two carbon fiber plies on either side, with a layup scheme of 0°/90°/0°/90°. The wing main beam was constructed from an anisotropic laminated structure comprising T300, with a layup scheme of 0°/90°/0°/90°.

For long-flight UAV wings, rib thickness, rib quantity, skin thickness, and longitudinal beam thickness play a vital role in improving the wing’s bending and shear resistance, torsional stiffness and durability, longitudinal load, and bending moment. Concurrently, enhancing the mechanical properties of the wings may also result in an increase in the mass of the UAV. Consequently, it is imperative to optimize the design of the number of wing ribs N and the thickness of the wing ribs d, the skin thickness h, and the wing beam thicknesses w₁ and w₂ to ensure the light weight of the UAV while ensuring its stable and reliable operation. This paper selected five parameters, as shown in

Table 2. Wing optimization design parameters and range.

Variables	Wing Rib Number	Rib Thickness (mm)	Beam 1 Thickness (mm)	Beam 2 Thickness (mm)	Skin Thickness (mm)
Lower Limit	10	10	2	2	2
Upper Limit	20	30	10	10	5

, as variables and calculated the mechanical simulation results of each variable within the specified range. The wing structure details are shown in Figure 8.

Simulating real flight environments through simulation experiments enables the rapid analysis of wings. This study simplified the high-aspect-ratio wing ribs into a complex solid structure composed of trusses and plates, covered with composite material skin. The internal ribs were evenly distributed along the wingspan and connected by two composite beams of different sizes, forming the main wing structure. During the simulation, the left side of the wing was constrained as a fixed end, with the aerodynamic load shown in Figure 2 applied to the skin surface. The skin was modeled using S4R elements, while the ribs, beams, and other components used C3D8R elements. With a mesh size of 5 mm, the computational model exhibited good mesh independence.

During the training process, neural networks learn from data to achieve good prediction results on new unknown data. Reasonable training data are conducive to the neural network model to fully learn the intrinsic physical knowledge of the data. To ensure the uniformity of experimental sampling, this paper adopted the Latin Hypercube Sampling (LHS) method for experimental design and generated 500 data sets with five input parameters. LHS is based on space filling technology, meets the projection characteristics, and can use fewer points to obtain better space filling initial sample points. In order to verify the rationality of the sampling space, the sampling space of the four parameters of the input parameters, namely, the rib thickness, the number of ribs, and the thickness of the two longitudinal beams, was plotted, as shown in Figure 9. The data space was evenly distributed and can cover the entire sample space more comprehensively. The sampling space was reasonable and reliable.

In order to quickly generate training data sets for training the surrogate model, this paper adopted a parametric modeling method and established a long-flight UAV wing model with different rib thickness, number of ribs, skin thickness, and longitudinal beam thickness in Abaqus2023. The 500 data sets with five input parameters generated by LHS are combined with Python 3.9 batch processing to calculate the 500 sets of data. The simulation model generated a 3D Stress hexahedral network by arranging seeds at a fixed interval of 5 mm. Python 3.9 software was used to read 500 sets of input data sets generated by MATLAB 2023, and each set of data was batch-substituted into the parametric modeling script for modeling and extracting the total mass of each wing model. The simulation calculation was submitted in batches using a Windows batch script. Finally, Python software was used to read the files and extract batch data from 500 sets of .odb files generated by simulation, and the maximum deflection and maximum stress of wings with different aspect ratios were obtained, forming 500 data sets containing the total mass, maximum deflection, and maximum stress of the model.

4.2. Training Method

During the neural network training process, the physical meanings of different parameters in the data set may lead to differences in the data dimensions, value ranges, etc., and affect the training results of the final model. In order to improve the accuracy and comparability of the data, we normalized the data, effectively eliminating the potential impact of singular sample data and maintaining the distribution characteristics of the data. This paper used the Min-max normalization method to normalize the training data to the range of −1 to 1. The training data set was divided into a training data set of 80% of the data, a validation set of 10% of the data, and a test set of 10% of the data.

This paper used BP neural network to construct a surrogate model, with five design parameters as model input and the maximum deflection Wmax, maximum stress, and total mass Mall of the wing as the three outputs of the model. In order to determine the optimal network structure, this study compared the performance of 12 groups of neural networks with different hidden layer structures and numbers, as shown in Figure 10. By comparing the error standard deviation (STD) of the three outputs across the training set, test set, and verification set with different hidden layer structures, it was evident that the 8-16-8 hidden layer structure yields better results. Additionally, this structure was relatively simple and easy to train. Therefore, the network structure of this paper was set as input layer (5), fully connected layer 1 (8), fully connected layer 2 (16), fully connected layer 3 (8), and output layer (3).

In order to further improve the training speed and prediction accuracy of the BP neural network, this paper employed the MEA to optimize the initial weights and biases of the established BP neural network. In this process, the inverse of the standard deviation of the model prediction error was established as the optimization objective function. This indicator provided a direct reflection of the model’s predictive capability, offering a basis for evaluating the quality of the sub-population. By considering the balance between the optimal sub-population score and the optimization time under each group of parameters shown in Figure 11, this paper selected the MEA initial parameter configuration of Popsize of 60, Bestsize of 10, and Tempsize of 10. The selected parameters resulted in a high score and the shortest optimization time. The objective of this optimization strategy was to rapidly identify the optimal initial weights and biases for the BP neural network, thereby achieving the most efficient model training convergence.

MEA was employed for the purpose of optimizing the BP neural network, and the optimization process is illustrated in Figure 12. Once the score of the temporary subpopulation surpassed that of the superior subpopulation, the alienation operation was initiated in accordance with the optimization strategy. This entailed the introduction of new subpopulations, which serve to enhance the diversity and performance of the superior subpopulation (Figure 12a,b). Following these adjustments, the superior subpopulation ultimately achieved a score transcendence, exhibiting a markedly superior performance compared to the temporary subpopulation. At this time, the global optimal solution was successfully extracted and output to the BP neural network (Figure 12c,d). The ADAM optimizer was used in the BP neural network training stage, and MATLAB 2023b software was used on the NVIDIA GeForce GTX 1650 platform to obtain the final model after 100 iterative calculations. During this process, the total training time was 51 s, with the MEA optimization time being 19 s and the neural network training time being 32 s.

4.3. Model Performance Evaluation

To ensure the smooth progress of the optimization process, after the model training was completed, it was necessary to comprehensively evaluate the performance of the model by determining the accuracy and calculation speed of the model. Common model performance evaluation indicators include standard deviation (STD), correlation coefficient (R), and determination coefficient (R²).

In order to evaluate the prediction performance of the constructed MEA-BP neural network model, this paper studied indicators such as the standard deviation and correlation coefficient of the prediction results of the surrogate model. The results are shown in Figure 13. The STD of the model on the training set was 0.495 mm, and R was 0.9998; the STD on the validation set was 0.631 mm, and R was 0.9998; the STD on the test set was 0.584 mm, and R was 0.9999. The results showed that the mean square error of the prediction model was extremely small, and the R values for the training set, validation set, and test set were all greater than 0.999. As shown in Figure 14, the error distributions of each data set were similar, proving that the model had the same fitting effect on different outputs. The model did not have overfitting or underfitting problems, had high prediction accuracy and good generalization ability, and met the needs of practical applications.

4.4. Optimization Design Method

The construction of the wing surrogate model realized the rapid calculation of the deformation and mass under different wing structural designs. To balance the requirements of wing performance in different application backgrounds, this paper solved the problem based on the surrogate model combined with the optimization algorithm. As a typical slender beam, the high-aspect-ratio wing produced large flexural deformation under the action of external loads, which changes the aerodynamic design of the entire aircraft and thus affects flight safety. Increasing the equivalent stiffness of the wing section to reduce the overall deformation is an effective method. The equivalent stiffness of the wing is related to the wing structure design, such as the number of ribs, rib size, and spar stiffness. The increased mass caused by the increased stiffness of the wing posed a challenge to the lightweight requirements of UAVs. In order to optimize the wing structure design to achieve less deformation and light weight, the five parameters shown in Table 2 were selected as decision variables, and the minimum total wing mass M under the condition of the minimum wingtip displacement W is taken as the optimization goal, and the solution equation was established as follows:

$$\begin{array}{l}\mathit{Find}: X=N,d,w_1,w_2,h\\ \mathit{Minimize}: F\left(X\right)=W\left(X\right),M\left(X\right)\\ s.t.:10\le N\le 20\\ \hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}10\le d\le 30\\ \hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}2\le w_1\le 10\\ \hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}2\le w_2\le 10\\ \hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}2\le h\le 5\end{array}$$

(3)

Intelligent optimization algorithms have outstanding advantages in solving multi-objective optimization problems. To solve the above problems, this paper adopted the multi-objective grey wolf optimization algorithm (MOGWO) to solve them. According to the optimization results, the number of grey wolves was 100, the maximum iteration was 100, and the archive size was 100. The Pareto frontier solution obtained after iterative calculation is shown in

Table 3. Pareto frontier solution.

	Decision Variable					Objective Function Value
	N	d (mm)	w1 (mm)	w2 (mm)	h (mm)	Wmax (mm)	Mall (mm)
1	10	10.00	2.00	2.00	2.35	193.58	35.88
2	10	10.00	2.11	2.11	2.94	151.77	42.42
…				…			…
11	19	27.29	5.13	3.43	5.00	82.49	77.09
12	10	10.00	2.10	2.00	2.66	169.00	39.25
13	10	10.00	2.02	2.00	2.02	222.61	31.48
14	10	10.55	2.05	2.06	2.21	205.19	34.76
15	10	11.07	2.34	2.12	3.94	108.95	53.38
16	12	10.88	3.25	2.15	4.84	88.79	64.82
…				…			…
100	16	21.41	4.53	2.91	4.93	84.75	72.69

. According to the distribution of the Pareto frontier solution shown in Figure 15, it can be seen that the objective function of the minimum wingtip displacement W_max is inversely proportional to the total mass M_all of the wing. When the minimum wingtip displacement W_max reaches 38.10 mm, further optimization leads to a significant increase in the total wing mass M_all, which cannot meet the lightweight requirements of the wing. Therefore, it is necessary to further discuss the Pareto frontier solution obtained and select a reasonable design to achieve a balance in target performance.

5. Results and Discussion

5.1. Optimization Effect Verification

In order to verify the convergence of MOGWO calculation, this paper calculated four groups of MOGWO algorithms with different parameters for optimization calculation. The four different parameter configurations are shown in

Table 4. MOGWO parameter configuration.

#	Grey Wolf Number	Maximum Iterations	Archive Size
1	200	200	200
2	200	50	50
3	50	50	200
4	50	200	50

. After optimization calculation, the Pareto frontier solution sets corresponding to four different parameter configurations were obtained, as shown in Figure 16. Compared with Figure 15, the Pareto frontier was relatively stable, and the optimization objectives and distribution of the frontier solution set did not change significantly. Therefore, it can be considered that the MOGWO calculation converged and the obtained optimization design results were effective.

5.2. Optimization Scheme Analysis

The combination of intelligent optimization algorithm and neural network effectively solved multi-objective optimization problems. MOGWO calculates a set of reliable Pareto frontier solutions. To gain further insight into the distinctions between various frontier solutions, this paper examined the optimization outcomes of the solution and identified a suitable design scheme that aligns with the practical application context. As shown in Table 3, the maximum deformation of the wing in the optimization result was 222.61 mm, corresponding to a total structural mass of 31.48 kg. The boundaries of the optimization target corresponded to the red triangle (minimum deformation) and red square (minimum mass) in Figure 15. When the wing deformation is the largest, the structure adopts the structural design of N = 10, d = 10 mm, w₁ = 2 mm, w₂ = 2 mm, h = 2 mm. At this time, the equivalent stiffness of the wing is the smallest, and the total mass of the structure is the smallest, which is in line with the deformation theory of material mechanics.

In order to meet the requirements of high-aspect-ratio wings for structural stiffness and lightness, the design schemes shown in Table 3 were compared, and the optimal design scheme was studied using the multi-criteria decision analysis (MCDA) method. In order to achieve a balance of target performance in a reasonable design, this paper adopted the superior–inferior solution distance (TOPSIS) method to select the optimal design. The core idea of this method is that the optimal solution should be closest to the ideal solution and farthest from the negative ideal solution. By analyzing the 100 selected design schemes, we found that the 16th group of schemes, that is, when the number of ribs N is 12, the rib thickness d is 10.88 mm, the skin thickness h is 3.25 mm, and the spar thickness w₁, w₂ are 2.15 mm and 4.84 mm, respectively, the wingtip displacement W_max is 87.49 mm, and the total wing mass M_all is 64.82 kg.

In order to verify the reliability of the optimization algorithm, this paper used finite element simulation based on the optimization results for verification, as shown in Figure 17. The experimental results showed that the optimization results W_{max_BP} and M_{all_BP} obtained from the BP neural network differ from the simulation results W_max and M_all by less than 5%, and the optimization design method achieved the lightweight design requirements for the wing structure. As shown in

Table 5. Comparison of different optimized structures with the original structure based on simulation results.

Variables	Optimal Model	Minimal Wmax Model	Minimal Mall Model	Original Model
N	12	19	10	16
d (mm)	10.88	27.29	10.00	25
w1 (mm)	3.25	5.13	2.02	6
w2 (mm)	2.15	3.43	2.00	6
h (mm)	4.84	5.00	2.02	3.5
σmax (MPa)	39.12	38.10	61.16	46.48
Wmax (mm)	87.49	82.21	226.24	122.06
Wmax_BP (mm)	88.79	82.49	222.61	/
|Wmax − Wmax_BP| (mm)	1.3	0.28	3.63	/
Mall (kg)	64.91	77.78	31.14	60.85
Mall_BP (kg)	64.82	77.09	31.28	/
|Mall − Mall_BP| (kg)	0.09	0.69	0.14	/

, compared with the original wing structure, the optimal model calculated by the TOPSIS method effectively improved the bending stiffness of the structure under the same load, reduced the deformation by 34.57 mm, and the total mass of the wing only increased by 4.06 kg, achieving the optimization and improvement of the wing performance.

6. Conclusions

To reconcile the apparent contradiction between the bending stiffness and lightweight nature of high-aspect-ratio wings, a lightweight mechanistic design is required. This paper introduced a novel wing structure design method based on the use of a surrogate model. The constructed surrogate model utilized the fitting ability of MEA-BP neural networks to enable rapid calculation of wing designs, while the MOGWO algorithm was used to solve the optimization problem of wing structure. By comparing the results from different Pareto frontier solution sets, the optimal design solution was reasonably selected. The results demonstrated that the calculation outcomes of the surrogate model and optimization method established in this paper are reliable. In the structural optimization of high-aspect-ratio wings, there is a strong coupling relationship between the structural layout and the design of ribs, spars, and skins. Compared with the original design, the selected optimal design significantly reduces the deformation and total mass of the wing, meeting the lightweight design requirements. The combination of neural network models and intelligent optimization algorithms not only provides an effective solution to multi-objective optimization problems but also expands the interdisciplinary application prospects of artificial intelligence. In the future, artificial intelligence-assisted design methods will promote technological progress and innovation across various fields and offer new perspectives and approaches for optimizing complex systems.