Under the condition of maintaining the material properties and working conditions of the wing box from Locatelli’s literature, a straight beam rib wing box with a spanwise length of 400 mm, a chordwise length of 800 mm, upper and lower skins of 2 mm, beam rib thickness of 5 mm, and a section thickness of 20 mm is established. A frame with the same thickness as the skins is constructed around the perimeter of the wing box. The internal structure of the model is shown in Figure 1 .

The wing box is made of A12124-T851, a material characterized by high hardness, corrosion resistance, and significant heat treatment effects. It maintains excellent performance even under high temperatures. The material’s performance parameters are listed in Table 1 .

The stress characteristics and constraints of the rectangular wing box, as described in the literature, indicate that it is subjected to a non-uniformly distributed load. The load condition is such that at the leading edge (x = 0 mm), the pressure load is 0.31 MPa, while at the trailing edge (x = 800 mm), the load is 0.142 MPa. This can be approximated as a linearly varying load distributed along the chordwise direction. A schematic diagram of the load distribution is shown in Figure 2 .

The constructed wing box model is imported into ANSYS software for static simulation analysis. Since the model is a three-dimensional solid model, the default solid element type, namely Solid186, is selected in the Workbench static module. This solid element type is more intuitive and capable of calculating local stress concentrations.

In finite element analysis, mesh generation is a crucial step that directly affects the accuracy of subsequent calculation results [6] . The density of the mesh and the refinement zones significantly influence the finite element calculation results. However, more and denser meshes do not necessarily lead to more accurate results. Ideally, the mesh should be selected based on the point where the calculation results no longer exhibit significant fluctuations with further refinement. The Automatic adaptive meshing method is employed, with quadrilateral elements as the unit type. The mesh is divided into sizes of 2.2 mm, 2.5 mm, 2.8 mm, 3 mm, 3.5 mm, 4 mm, 4.5 mm, 5 mm, 7 mm, and 9 mm, and the equivalent stress and deformation are calculated for each. The specific values are shown in Table 2 .

As shown in the table data, a mesh size of 3 mm is selected. The mesh density is 0.86 and the results of the static analysis are illustrated in Figure 3 .

As shown in Figure 3 , the deformation of the wing box is smallest at the fixed surface and gradually increases along the spanwise direction, with more noticeable deformation at the corners away from the fixed end. The maximum equivalent stress of the wing box is higher at the fixed surface and gradually decreases along the spanwise direction, approaching zero at the outermost edge. Due to the wing box bearing a non-uniform distributed load, both the structural deformation and the maximum equivalent stress values increase gradually from the leading edge to the trailing edge. The mass of the wing box is 4.62 kg, the maximum deformation is 3.2567 mm, and the maximum equivalent stress is 106.55 MPa.

When modeling the curved beam rib structure, it is necessary to consider how to optimize the curve in subsequent steps. A three-point control method is adopted, where the initial point, intermediate control point, and endpoint are used as curve control points. A B-spline curve is used to connect these three points, and then the wing box model is established. The mass of the curved beam rib wing box is kept the same as that of the straight beam rib wing box. The model is imported into ANSYS software for simulation analysis. The curved wing box has a mass of 4.62 kg, a maximum deformation of 3.2345 mm, and a maximum equivalent stress of 99.837 MPa. The structural diagram and simulation contour of the curved wing box are shown in Figure 4 .

As shown in Figure 4 , under the condition of equal mass, the maximum deformation of the curved wing box is reduced by 0.68%, and the maximum equivalent stress is reduced by 6.3%. Compared to the straight beam rib wing box, this solution demonstrates superior performance.

The BP neural network [9] is one of the most widely used models in artificial neural networks (ANN). It is inspired by the neurons in the biological brain and mimics the human brain’s response to external stimuli, establishing a brain-like perception system. Through the forward propagation of information and the backward propagation of errors, it repeatedly learns and trains, resulting in an intelligent network. The advantages of the BP neural network are as follows:

1) Strong Approximation Capability

The BP neural network has powerful mapping and prediction abilities for solving complex problems. With reasonable sample data, a three-layer neural network structure can predict accurate solutions with high precision, making the BP neural network particularly suitable for problems with complex internal mechanisms.

2) Fault Tolerance

The BP neural network allows for minor errors during training. Even if some neurons are compromised, the overall training results are not significantly affected, demonstrating a certain level of fault tolerance.

3) Self-Learning and Adaptability

Once parameters are set, the BP neural network can autonomously conduct training and learning, storing the learned information in the network’s weights. It iteratively trains until the error falls within an acceptable range.

The BP neural network is one of the foundational models of deep learning.

The BP neural network consists of an input layer, hidden layers, and an output layer. The hidden layers can be single or multiple, with each layer containing a certain number of nodes. Nodes between layers are interconnected and represented by weights. Generally, a three-layer neural network can effectively solve problems. The number of nodes in the hidden layer of the neural network can be determined using an empirical formula [10] , as shown in Equation (1).

$p=\sqrt{n+l}+\partial$ (1)

here, n represents the number of input layer nodes, determined by the number of input variables, and l represents the number of output layer nodes, determined by the output variables. α is a constant between 1 and 10. In this chapter, the design variables are the parameters of all control points in the wing box and the thickness values of the beam ribs. A single hidden layer neural network is used, and the number of hidden layer nodes is automatically calculated by MATLAB based on the formula. Before training the neural network, Latin Hypercube Sampling is first performed on all design variables of the curved wing box, and the structural performance of each sample is calculated. From the sample dataset, 50 data points are selected to check the quality of the network. MATLAB calculations determine that the optimal number of hidden layer nodes is 16. The prediction regression plot, mean squared error plot, and validation point error plot are shown in Figures 5-7 , respectively.

As shown in Figure 5 , the neural network prediction regression R-value has high accuracy. From Figure 6 , it can be seen that the mean squared error (MSE) of the BP neural network is small, meeting the program’s error requirements. Figure 7 shows that the maximum stress error of randomly selected validation points ranges between 0.08% and 3.26%, with most validation point errors below 2%. The maximum deformation prediction error ranges between 0.06% and 1.24%, and the mass prediction error ranges between 0.07% and 0.28%. Combined with

the regression R-value in Figure 5 and the prediction variance MSE in Figure 6 , it indicates that the neural network has high prediction accuracy.

The Fast Non-Dominated Sorting Genetic Algorithm with Elitist Strategy (NSGA-II) [11] is a classic multi-objective genetic algorithm. It balances multiple objectives through non-dominated sorting and crowding distance calculation. When generating offspring, non-dominated sorting is performed, where solutions with higher rankings are retained, while those with lower rankings undergo crossover and mutation. The optimal solutions computed by the NSGA-II algorithm are evenly distributed, and its convergence and robustness are superior to other multi-objective optimization algorithms [12] . It has significant advantages in the field of aircraft structural design, as detailed below:

1) The multi-objective genetic algorithm can simultaneously optimize multiple conflicting objectives (such as structural mass, deformation, and stress) without simplifying the problem into a single-objective optimization.

2) The genetic algorithm, based on population evolution, can search for the solution space globally, avoiding getting trapped in local optima.

3) Through operations such as crossover and mutation, the diversity of solutions is increased, enhancing the probability of finding the global optimal solution.

4) The multi-objective genetic algorithm provides designers with multiple feasible optimization solutions, facilitating the trade-off between different objectives. In aircraft design, the most suitable structural design solution can be selected based on performance requirements.

The mathematical model of the multi-objective genetic algorithm in this paper is shown in Equation (2):

$\begin{array}{l}\text{Min}F\left(x\right)=\left(f_{\text{mass}}\left(X\right),f_{\text{deformation}}\left(X\right),f_{\text{stress}}\left(X\right)\right)\\ s.t.\{\begin{array}{l}\beta <3.23\text{mm}\\ \sigma <100.16\text{MPa}\\ X\in \left(M_i,N_i\right)\end{array}\end{array}$ (2)

here, f_mass(X) is the mathematical model for mass, f_deformation(X) is the mathematical model for maximum deformation, and f_stress(X) is the mathematical model for maximum equivalent stress. B represents the maximum deformation of the structure, σ represents the maximum equivalent stress of the structure, X represents the structural design variables, and Miand Nidenote the upper and lower limits of the variables.