## 3 Results  
### The Optimization Results  
The optimization object is a large wind tunnel nozzle section, and four types of Mach number profiles, with values of 1.3, 1.2, 1.1, and 1.0, were designed. The aerodynamic contours are illustrated in Figure 4a,b, wherein the Mach number 1.0 represents a contraction curve, while the remaining profiles depict contraction-expansion curves. The aerodynamic profile is divided into two regions, which are delineated by the throat and serve as the demarcation point. The inlet to the throat is designated as the contraction region, while the  
Figure 3: The optimization process of the multi-point flexible wall nozzle structure. The optimization process consists of four steps: (1) define the variables and select sampling points using the Latin hypercube method, (2) analyze the profile overlap by calculating elastic curves and comparing them with theoretical profiles, (3) build and validate the surrogate model using shape functions, and (4) optimize using the genetic algorithm to find the global solution. throat to the outlet is classified as the expansion region. Of particular note is the expansion region, which presents a significant challenge in terms of the elastic profile required for the flexible plate. This is a crucial aspect of the structural design. In order to ensure profile accuracy, the structure of the flexible wall nozzle is subjected to a rigorous calculation and analysis process. On this basis, the profile deviations, corresponding to the two cases of downstream setting of a single-group driving mechanism and a two-group driving mechanism, are compared in order to determine the final number and position of the driving points. (1) Single-group driving jacks  
Figure 4c gives the curve of the influence of the position of the single-group drive mechanism drive on the profile reclamping accuracy and the root mean square deviation,  
Figure 4: Optimization and analysis of the flexible nozzle design: (**a**) Theoretical profiles for the four different Mach numbers (1.3, 1.2, 1.1, and 1.0) showing the target shapes. (**b**) Schematic representation of the nozzle plate structure illustrating the key components and dimensions. (**c**) Analysis of the drive mechanism position impact on the profile accuracy, showing the root mean square deviation versus the drive point location. (**d**) Profile deviation analysis for all the four Mach numbers after optimal drive point placement. (**e**) The three-dimensional feasible solution space visualization generated using RPIM interpolation. (**f**) Comparative analysis of the profile surface deviations post-optimization for all the four Mach number configurations. from which it can be seen that the largest profile deviation had the same trend as the root mean square deviation, and the optimal design point was located near \(x=1600\). Once the location of the driving point was determined, the corresponding four Mach number faced deviations, as illustrated in Figure 4d. The largest profile deviation was observed in the vicinity of the coordinate origin at a distance of approximately 750, for the Mach 1.1 condition, with a magnitude of \(-0.7\) mm. The largest deviation for the Mach 1.1, 1.2, and 1.3 profiles was observed between the drive point and the fixed point. This was due to the fact that there was a slight change in the angle of the aerodynamic theoretical curve at this position, which was not accounted for by the structure's fixed connection. The profile exhibiting the greatest degree of overlap was that corresponding to Mach 1.0, with a maximum deviation of approximately 0.3 mm. However, the deviations observed in all four of the Mach profiles exceeded the \(\pm 0.2\) mm threshold specified in the design specifications. This finding suggests that a single drive point is insufficient to meet the design requirements.