# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 2 Theoretical Development
This is due to the potential for large random fluctuations to amplify the spectral radius, thereby destabilizing the numerical solution process and adversely affecting its convergence and accuracy. As such, alternative numerical methods may necessitate exploration for problems exhibiting pronounced randomness in order to safeguard the stability and reliability of the results. In view of this, a stochastic finite element method based on a Krylov subspace (termed as SFE-KS) is developed next for static random analysis. For this purpose, both sides of Equation (4) are multiplied by \(K^{-1}\) to obtain  
\[K^{-1}K_{d}\cdot x_{d}=K^{-1}l \tag{10}\]  
The purpose of the above operation is to reduce the condition number of the coefficient matrix because matrix \(K^{-1}K_{d}\) is closer to the identity matrix than \(K_{d}\). From Equations (2) and (10), one has  
\[x=K^{-1}l \tag{2}\]0  
\[x=K^{-1}l \tag{2}\]1  
\[x=K^{-1}l \tag{2}\]2  
Note that the dimension of the matrix \(D\) is \(m\times m\). Usually, the dimension of a Krylov subspace takes a small value, such as \(m=2\) or \(3\). Therefore, Equation (17) can be easily solved by matrix inversion as:  
\[x=K^{-1}l \tag{2}\]3  
where \(\hat{y}\) denotes the solution of Equation (17). Finally, the approximate solution of \(x_{d}\) based on the \(m\)-dimensional Krylov subspace can be obtained as  
\[x=K^{-1}l \tag{2}\]4  
Overall, the proposed method avoids directly inverting the random stiffness matrix \(K_{d}\) through the preprocessed Krylov subspace, thus saving the calculation amount. Upon comparison, it can be observed that the main difference between the complete analysis, the NSFE method, and the proposed SFE-KS method lies in the algorithms used to solve for the displacement under random sampling. Specifically, the complete analysis employs Equation (5), the NSFE approach utilizes Equation (9), and the proposed method adopts Equation (21). As mentioned earlier, the complete analysis refers to the classical Monte Carlo finite element method, and its calculation results in the case studies presented will serve as the exact solution to validate the computational efficiencies and accuracies of the NSFE and SFE-KS methods. The steps of the SFE-KS method are summarized as follows: (1) Establish the FEM without considering randomness to obtain the original stiffness matrix \(K\). (2) For a static load \(l\), solve Equation (1) to obtain the vector \(x\) and the matrix \(K^{-1}\). (3) Perform random sampling according to Equation (3) to obtain a random stiffness matrix \(K_{d}\). (4) Calculate the matrix \(\Pi\) using Equations (12) and (14). (5) Compute the matrix \(D\) and the vector \(z\) using Equations (18) and (19). (6) Calculate the vector \(\hat{y}\) using Equation (20). (7) Calculate the displacement vector \(x_{d}\) using Equation (21). (8) Thousands of random samples (Steps 3-7) are obtained to obtain the statistical characteristics of the structural response and the failure probability of the structure. The formula for calculating the probability of structural failure is given as  
\[x=K^{-1}l \tag{2}\]5  
where \(q_{f}\) represents the number of times the vertex displacement exceeds the threshold, and \(q_{t}\) represents the total number of samples. Based on the principles of matrix computations [33], an approximate assessment of the computational expense associated with the complete analysis, the NSFE method, and the proposed SFE-KS method is conducted by quantifying the number of floating-point operations (flops) in their computation process. As mentioned previously, the primary distinction among these three methods lies in the differing algorithms employed to calculate displacements under random sampling. Consequently, the subsequent focus will be on estimating the number of floating-point operations corresponding to the distinct operations in each of these three methods.