## 3 Dynamic Force-Thermal Follow-Up Loading Method for Multi-Stage Telescopic Wings
### Method Comparison and Validation
#### 3.3.2 Initial Moment Analysis and Validation  
After obtaining the load distribution from both methods, a comparison analysis was conducted by simulating the deformation trend of the wingtip deflection under the equivalent loads and comparing the fitting degree with the aerodynamic load. The constraints for the initial moment analysis at 0 s are shown in Figure 23. The root of the equivalent wing was fixed, and simulations were carried out under the following three load conditions on the wing surface: 1 the evaluation aerodynamic load, 2 the equal-area distribution load, and 3 the dynamic multi-point allocation method equivalent load. The initial moment data of the three load cases were input into the simulation model to obtain the simulation results, as shown in Figure 24. The wingtip deformation trajectories were extracted from the three results to analyze the fitting effects of different equivalent methods and aerodynamic loads. Figure 25 shows the extracted wingtip deformation trajectories under the three loading conditions. To comprehensively evaluate the fitting performance of the proposed dynamic multi-point allocation method and the traditional equal-area allocation method in practical applications, this paper calculates two commonly used error metrics, namely root mean square error (RMSE) and the coefficient of determination (R\({}^{2}\)), for comparative analysis. RMSE is primarily used to measure the magnitude of fitting errors between the two methods, giving  
Figure 24: Simulation wingtip trajectory. Figure 23: Simulation constraints. higher penalties to larger errors. R2, on the other hand, reflects the correlation between the model's calculation results and actual values, indicating the model's ability to explain data variability. Through the joint analysis of these two metrics, the fitting performance of the model can be more comprehensively assessed. The root mean square error (RMSE) is a metric used to measure the difference between the fitted values and the true values. It is defined as the square root of the average of the squared differences between the fitted and actual values, as follows:  
\[\text{RMSE}=\sqrt{\frac{1}{\text{n}}\sum\limits_{\text{i=1}}^{\text{n}}{( \text{y}_{\text{pred,i}}-\text{y}_{\text{true,i}})}^{2}} \tag{15}\]  
where \(\text{y}_{\text{pred,i}}\) is the equivalent value for the i-th sample, \(\text{y}_{\text{true,i}}\) is the aerodynamic load value, and n is the number of samples in the dataset. A smaller RMSE value indicates that the difference between the model's fitting results and the actual results is smaller, meaning the fitting performance is better. The coefficient of determination R2 measures the correlation between the model's computed values and the actual values. Its formula is as follows:  
\[\text{R}^{2}=1-\frac{\sum\limits_{\text{i=1}}^{\text{n}}{(\text{y}_{\text{true, i}}-\text{y}_{\text{pres,i}})}^{2}}{\sum\limits_{\text{i=1}}^{\text{n}}{( \text{y}_{\text{true,i}}-\overline{\text{y}})}^{2}} \tag{16}\]  
where \(\overline{\text{y}}\) is the mean of the actual values. The R2 value ranges between 0 and 1, with a value closer to 1 indicating a stronger ability of the model to explain the data variability and a better fitting performance. This paper evaluates the superiority of the fitting performance of the two methods by comparing their performance based on RMSE and R2 metrics. For the error analysis of the two methods at the initial state, RMSE and R2 values were calculated using the wingtip deflection simulation trajectory data. The fitting error results for the dynamic multi-point allocation method and the equal-area distribution method at the initial moment are shown in Table 5.  
The results show that, at the initial moment, the dynamic multi-point allocation method proposed in this paper outperforms the equal-area distribution method in both metrics.