## 1 Introduction
Optimisation-based methods estimate the range of uncertainty in the structural response by iteratively searching the best combination of parameters that minimise or maximise the structural response. [PERSON] et al. [20] reduce the computational cost of structural analyses with interval uncertainty using gradient descent. [PERSON] et al. [21] proposed a Newton iteration interval analysis method to quantify the uncertainty of nonlinear structural response. [PERSON] et al. [22] applied the lightning attachment procedure optimization to the interval finite element analysis. However, these methods often require repetitive solving of the system model, resulting in high computational costs. In addition, machine-learning-based methods [23; 24; 25] have been widely applied in uncertainty propagation analysis. These methods utilize machine learning algorithms to construct models that infer the propagation of uncertainty by learning the mapping relationship between input parameters and output responses. By learning and training on extensive datasets, these methods can provide a relatively high predictive accuracy. However, these methods have strict requirements for the quality and quantity of data, and they can be affected by errors in the dataset. The above non-intrusive methods typically involve sampling and statistical analysis of the model's output to establish a statistical model and quantify uncertainty. This approach does not require modification of the original model or system, making it easier to apply to existing complex systems. However, non-intrusive methods often come with high computational costs and may overlook certain characteristics of the model or system. In contrast, intrusive methods can directly output quantified information about uncertainty by modifying the existing model or system. This approach often requires improvement or expansion of the model to accurately capture uncertainty. Perturbation methods are an important class of intrusive methods in the field of uncertainty quantification. These methods introduce small perturbations into the system or model to investigate the resulting output uncertainty. [PERSON] et al. [26] proposed a matrix perturbation method that represents the deviation of an interval matrix as a perturbation about the nominal value of the interval matrix using interval arithmetic. Subsequently, [PERSON] et al. [27] developed an interval perturbation finite element method for structures with uncertain-but-bounded parameters. [PERSON] et al. [28] proposed an interval perturbation finite element-boundary element method for structural-acoustic systems based on first-order Taylor series expansion and perturbation theory. In these methods, the interval stiffness matrix is expanded using the first-order Taylor series, and its inverse matrix is approximated using the first-order Neumann series. To enhance the accuracy, researchers have conducted further investigations based on interval perturbation methods. [PERSON] et al. [29] proposed modified interval perturbation finite element methods to address the issue of neglecting higher-order Neumann series terms in traditional interval perturbation finite element methods. However, methods based on classical interval analysis (CIA) [30] are inevitably affected by dependency. To overcome this problem, [PERSON] et al. [31] proposed an enhanced interval finite element method based on the affine algorithm. The affine algorithm keeps track of variables during the computation process, but it can be challenging to implement when dealing with a large number of variables and complex relationships. [PERSON] et al. [32; 33] incorporated the physical characteristics of uncertain parameters into the finite element method by introducing an extra unitary interval (EUI) for each uncertain parameter during the assembly and solution stages. [PERSON] et al. [34] developed a second-order Taylor expansion-based method (STM) for estimating the natural frequencies of uncertain structures.