Development of a new reliability extrapolation method for the reliability estimation of mechanical systems

Asymptotic sampling (AS) is an efficient simulation-based technique for estimating the small failure probabilities of structures. AS utilizes the asymptotic behavior of the reliability index with respect to the standard deviations of random variables. In this method, the standard deviations of random variables are progressively inflated using a scale parameter to obtain a set of scaled reliability indices. The collection of the standard deviation scale parameters and corresponding scaled reliability indices are called support points. Then, least squares regression is performed using these support points to establish a relationship between the scale parameter and scaled reliability indices. Finally, extrapolation is performed to estimate the actual reliability index. The accuracy and performance of the AS method are affected by various factors including the sampling method used, the values of the scale parameters, the number of support points, and the formulation of extrapolation models. One of the purpose of this study is to make a critical evaluation of the performance of the asymptotic sampling method for highly safe structures, and to provide some guidelines to improve the performance of the AS method. It is found that generating the random variables by Sobol sequences and using the 6-model mean extrapolation formulation give slightly more accurate results rather than Latin hypercube sampling (LHS) and 10-model mean extrapolation formulation. Besides, the optimum initial scale parameter is approximately around 0.4, and the optimum number of support points is typically 4 for all problems. As the reliability level increases, the optimum initial scale parameter value decreases, and the optimum number of support points increases. Another factor affecting the performance of the method is the regression type. In the previous studies, the relationship between reliability indices and support points has been established using nonlinear regression. In this study, we explored the use of more advanced machine learning (e.g., Gaussian process, support vector regression) and surrogate modeling (e.g., Kriging) techniques, and compared the accuracies of these techniques to that of the nonlinear regression on various example problems. It is found that using nonlinear regression yields more accurate results than machine learning and surrogate modeling techniques evaluated within the scope of this study. Various extrapolation models have been used in AS to improve accuracy. Moreover, a mean extrapolation formulation using the average value of different extrapolation models was proposed to further improve its accuracy. Although the mean extrapolation formulation protects against using the wrong extrapolation model, it did not guarantee a reliability estimation better than that of the best available extrapolation model. In this study, we propose a weighted average AS formulation in which the weight factors are optimized to minimize the variance of the reliability index estimation through the bootstrapping method. In the weight factor determination, both convex and affine formulations are considered and the results are compared. The performance of the proposed method is evaluated using various example problems. It is found that the proposed weighted average formulation has higher accuracy than the mean extrapolation formulation. For weight factor optimization, the affine formulation yields more accurate results than the convex formulation in most cases.