## 5 Wing Surface Structure Application  
In this section, the proposed two-step bionic curved stiffener layout generation method is implemented for the wing surface structural design. As illustrated in Figure 19, a simplified model with boundary and loading conditions is presented. The optimization problem is listed in Equation (8), and the optimization options are shown in Table 4. With the GS data, a wing box is set to simulate the bionic grid wing's structural properties. \[\left\{ \begin{array}{l}\text{Find }:\mathbf{w}=\left[w^{1}{}_{1},w^{1}{}_{2}, \ldots,w^{1}{}_{n},w^{2}{}_{1},w^{2}{}_{2},\ldots,w^{2}{}_{m}\right]^{T}\\ \text{min: }\text{M}_{\text{bic}}\\ \text{s.t.}:\theta_{L}\leq\theta_{twist}\leq\theta_{U}\\ \text{U}_{tip}\leq\text{U}_{U}\\ S_{\text{max}}\leq S_{U}\end{array} \right. \tag{8}\]  
Figure 17: Buckling modes of both the initial and optimal structures using detailed finite element analysis under uniaxial compression. (**a**) Initial straight stiffeners. (**b**) Optimal mass stiffeners. (**c**) Optimal RBL stiffeners. Figure 18: Buckling modes of both the initial and optimal structures using detailed finite element analysis under biaxial compression. (**a**) Initial straight stiffeners. (**b**) Optimal RBL stiffeners. Here, \(\theta_{twist}\) is the wing's maximum twist angle, U\({}_{tip}\) is the displacement of the wing tip, and S\({}_{max}\) is the maximum stress value of the stiffeners. The optimized process is presented in Figure 20. In this figure, the color represents the tip displacement that corresponds to the bending stiffness; blue indicates a low level and red indicates a high level. The detailed FEM result are shown in Figure 21. For the upper and bottom surfaces' optimal results in Figure 21b,c, the difference is caused by the opposite load condition to the bottom surface suffered by the tension force, while the upper surface resists the compression force. The layout of the upper surface revealed a large set of stiffeners towards the wingspan, and the bottom surface revealed a chord-wise like layout, which is a twist that resists performance to improve torsional rigidity. \begin{table}
\begin{tabular}{c c} \hline \hline
**Optimization Technique Options** & **Value** \\ \hline Number of designs per generation & 40 \\ Number of generations & 25 \\ Number of islands & 5 \\ Mutation rate & 0.01 \\ Cross-over rate & 1.0 \\ Mutation rate & 0.03 \\ Maximum stress limit & 900 MPa \\ Maximum displacement limit & 150 mm \\ Twist angle limit & \([-3^{\circ},\,3^{\circ}]\) \\ Mass & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Multi-island genetic algorithm setting and optimization results for the wing structure. Figure 19: Boundary condition (BC) and load conditions with the ground structure (GS) method and the topology result of the wing surface structure. (**a**) Model setting of the GS method. (**b**) Wing topology result of the upper and bottom surface. A comparison between the bionic structure and traditional structure is listed in Table 5. The three types are under the same load conditions, and the grid structure uses the same cross section. The bionic grid structure has a 29.1% weight loss compared with the traditional structure and a 22.8% weight loss compared with the initial straight structure. In the table, this surface grid structure shows great advantages in weight loss. As the traditional structure is a beam and slab structure, this weight loss  
Figure 21: The detailed FE result for the bionic grid wing surface structure. (**a**) Bionic grid wing structure static FE result. (**b**) Bionic grid of the upper wing surface. (**c**) Bionic grid of the bottom wing surface. Figure 20: All designs explored during the optimization process. is mainly relayed to the materials' improvement. These remarkable reductions are caused by the different uses of the structure's layout, and the novel bionic design of the stiffener can further reduce the weight. However, we must discuss the twist angle along the wing span (Figure 22).