# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 1 Introduction  
In engineering structures, the physical properties of materials are usually random, such as the elastic modulus of concrete and residual stress of steel, which can be regarded as random variables. On the other hand, there will also be random dimensional deviations in the process of making and installing components. The random changes of these physical quantities will affect the bearing capacity and deformation of the whole structure. Therefore, in structural design, reliability analysis is usually necessary [1, 2]. The Monte Carlo simulation is one of the most commonly used reliability analysis algorithms [3, 4]. Its basic idea is to assume that the physical quantity is a random variable that obeys a certain random distribution and then randomly selects samples from the original structure thousands of times. By calculating the response of each sampling and comparing it with the predetermined threshold, the failure times in these samples that exceed the response threshold can be obtained so as to calculate the failure probability of the structure. In the above operations, the response data of different samples need to be calculated repeatedly. For large-scale structures with a huge number of degrees of freedom, such static or dynamic response analysis requires a huge amount of calculation, which will make it difficult to carry out reliability analysis. The stochastic finite element (SFE) method is a tool to quickly calculate the random response of structures. With the aid of the stochastic finite element method, the calculation of the Monte Carlo simulation in reliability analysis can be greatly reduced. [PERSON] and [PERSON] [5] proposed the iterative Krylov subspace method, coupled with diverse preconditioning strategies, to solve nonlinear computational problems. For a parallelimplementation, domain decomposition methods are employed, with each processor working on a local grid to compute a partial stiffness matrix, thereby accelerating the computational speed. [PERSON] et al. [6] reviewed the current state of reliability analysis methods based on the SFE method. Emphasis was placed on four prevalent SFE techniques: direct Monte Carlo, Taylor expansion, perturbation, and Neumann expansion. [PERSON] et al. [7] combined a non-invasive SFE method with a direct probability integration technique to generate precise probability density functions for each individual seismic response moment. [PERSON] and [PERSON] [8] refined the SFE method to conduct an in-depth investigation into the random vibrations exhibited by functionally graded material plates when subjected to moving loads. [PERSON] et al. [9] employed the SFE method to explore the capacity for accounting for spatial variability in concrete tensile strength. [PERSON] et al. [10] employed the SFE method to investigate the natural vibration of thin plates on viscoelastic supports with time-fractional orders. [PERSON] et al. [11] used the SFE method to analyze the mean and standard deviation of the dynamic response of the train bridge system. [PERSON] [12] introduced a deep learning algorithm grounded on convolutional neural networks and trained it with a probabilistic failure analysis dataset derived from an SFE model. [PERSON] et al. [13;14] proposed a new method for structural reliability analysis using the SFE approach, as well as solving partial differential equations defined on random domains. [PERSON] et al. [15] used the SFE method to ascertain the spectral response and statistical parameters of the structural natural frequencies. [PERSON] et al. [16] proposed a novel SFE method that leverages the Karhunen-Loeve expansion for discretizing the stochastic field, coupled with the point estimate method for accurately calculating the random response of a structure. [PERSON] et al. [17] presented a triangular extended SFE model for simulating random void problems. [PERSON] et al. [18] used the SFE method to study the random response of offshore strip foundations on spatially variable soils considering a combination of horizontal and rotational loads.