## 1 Introduction  
Advances in material science and intelligent systems have driven the increasing use of large-deflection flexible aerodynamic structures in modern aerospace equipment. These structures--including flexible wall wind tunnel nozzles, intelligent variant wings, and variable aeroengine nozzles--can modify their shape in response to changing mission requirements and operating conditions, achieving optimal performance beyond what traditional rigid structures can offer. The flexible wall nozzle is particularly crucial in aerodynamics research as wind tunnels are essential tools for investigating fundamental aerodynamic principles. This research both validates and advances relevant theories while contributing directly to flight vehicle development [1, 2, 3, 4]. The precision of nozzle shape control directly affects flow field quality and Mach number simulation accuracy in trans-supersonic wind tunnels--a fundamental aspect of wind tunnel technology [5, 6, 7]. In typical applications, nozzles accelerate the gas from inlet to outlet to create a uniform flow field at a specified velocity. However, the initial acceleration creates an inhomogeneous expansion wave at the flexible plate interface. To establish a homogeneous flow field, the downstream flexible plate must generate an opposing compression wave. The nozzle's profile plays a critical role in determining flow field quality [8; 9]. Therefore, research into flexible profile control and structural optimization methods is vital for enhancing flow field quality. In flexible nozzle structures with large deflection deformation, there is significant interdependence between support point positions and the elastic deformation of the flexible plate. This makes it difficult to determine support point positions using the direct cubic curve approximation method, regardless of driving rod elongation [10]. To solve this problem, researchers have proposed a variational decomposition method [11] for analyzing the large-deflection beam model of a flexible wall nozzle under multi-point dynamic constraint boundaries. They analyzed how key parameters--including the elastic beam deformation curve, rotation angle, and force load--affect the solution accuracy. While this method efficiently analyzes flexible wall nozzles, optimizing their shape control remains challenging due to numerous design and state variables, as well as due to the complex coupling between the driving mechanism and flexible plate. Recent advances in multidisciplinary design optimization, especially surrogate-based modeling [12], offer ways to improve optimization efficiency while reducing process complexity. Common surrogate models include polynomial response surfaces, radial basis functions, and Kriging models [13]. The Kriging model, introduced by [PERSON] in 1951, provides unbiased estimation with minimum variance--making it ideal for this application. [PERSON] et al. [14] successfully applied finite element and Kriging surrogate models to solve drum brake noise problems, significantly improving brake stability optimization efficiency while reducing noise. Similarly, [PERSON] et al. [15] developed a high-dimensional multi-objective optimization method for rotor blade airfoils with multiple design points and objectives. Their method comprehensively considers high- and low-speed lift-resistance characteristics, torque characteristics, and drag dispersion using a high-precision Kriging model to enhance optimization efficiency. This paper presents a mathematical model of large-deflection deformation for a multi-pivot flexible nozzle, which is a typical flexible aerodynamic structure. Based on the principle of variational division, the model enables efficient deformation control and an optimal design of wind tunnel nozzles. We established a flexible wall nozzle structure optimization surrogate model using the radial basis point interpolation method to optimize the multi-pivot flexible wall nozzle structure.