# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 3 Numerical Verification
### A Plate Structure  
As shown in Figure 9, a plate structure with a geometric size of 120 mm \(\times\) 40 mm \(\times\) 20 mm serves as a demonstration for applying the introduced methodology. This structure is digitally replicated utilizing solid elements, each with dimensions of 10 mm \(\times\) 10 mm \(\times\) 10 mm. One extremity of the plate is securely anchored, while the opposing end experiences multiple concentrated forces, each individually exerting a force of 5 kN. Assuming that the elastic modulus of the material follows a normal distribution with a mean value of 206 GPa, two scenarios are considered for the standard deviation: 0.05 and 0.2 times the mean value. Without loss of generality, 3000 random samples are taken from the structure to analyze the random response and failure probability of the plate structure. Structural failure is defined as the vertical displacement at the free end of the structure exceeding a certain threshold. This threshold is taken as 1.05 times the vertical displacement of the free end of the structure without considering randomness. The plate structure possesses 540 degrees of freedom. Applying the flops estimation formula, it was found that the total flops required for a comprehensive analysis, the NSFE method, and the proposed method are 157,755,600, 2,916,000, and 1,751,772, respectively. Notably, the proposed method significantly reduces the flops count, demonstrating its efficiency and superiority in terms of computational requirements. The standard deviation of the elastic modulus being 0.05 times the mean value is used to simulate situations with a very low probability of failure. Table 11 presents the computation time for the three methods under this scenario. Figure 10 shows the vertical displacements at the free end of the plate obtained using the three methods. Table 12 provides the statistical measures of the vertical displacement at the free end. As shown in Table 11, the time required for the NSFE method is approximately 43% of the complete analysis, while the SFE-KS method requires approximately 22% of the complete analysis time and 51% of the NSFE algorithm time. Clearly, the proposed SFE-KS method exhibits the highest computational efficiency. From Table 12, one can find that the failure probabilities calculated using the three methods are all zero. This shows that when the standard deviation of the elastic modulus is small, the failure probability of the structure is also very small and close to 0. In addition, the ranges, means, and standard deviations of the displacements at the free end of the plate structure obtained using the three methods are almost equal, indicating that both the NSFE and SFE-KS rapid methods can achieve highly accurate calculation results when the standard deviation of the elastic modulus is very small. Taking into account both the computation time and accuracy, it has been demonstrated that the proposed method performs the best. Figure 9: A plate structure. Table 13 presents the computation times for the three methods under the scenario of the standard deviation of the elastic modulus being 0.2 times the mean value. Figure 11 shows the vertical displacements at the free end of the plate obtained using the three methods. Table 14 provides the statistical measures of the vertical displacement at the free end. As shown in Table 13, the time required for the NSFE method is approximately 49% of the full analysis, while the SFE-KS method requires approximately 25% of the full analysis time and 50% of the NSFE algorithm time. Clearly, the proposed SFE-KS method exhibits the highest computational efficiency. From Table 14, one can find that the calculation results of the proposed method are closer to the exact solution obtained using the complete analysis than those of the NSFE method. This indicates that when the standard deviation of the elastic modulus is relatively large, the calculation accuracy of the NSFE method begins to significantly lag behind that of the SFE-KS method.