Improve the accuracy of asymptotic approximation in reliability problems involving multimodal distributions

Asymptotic approximations such as the first- and second-order reliability method (FORM, SORM) are sometimes preferred in reliability analysis due to their efficiency. Bayesian updating for practical engineering problems can produce unnormalized and multimodal distributions, which cannot be directly used in FORM. The Laplace method can approximate such distributions using a weighted multivariate normal distribution. The weighting scheme for different modes must be carefully made in order to achieve accurate results. In this paper, a new weighting scheme and an optimal weighting scheme are proposed to improve the accuracy of Laplace approximation for multimodal distributions. The proposed weighting schemes are compared with existing weighting schemes in detail, and it is shown that the proposed weighting schemes can produce more accurate results in some cases. Based on that, an asymptotic approximation method for reliability estimation and system response predictions involving unnormalized multimodal distributions is proposed. The method integrates the Laplace method, FORM, and inverse FORM to formulate a completely asymptotic analysis cycle. The overall method is applied to two engineering cases: A simplified bridge model and a steam turbine rotor Bayesian analysis with real ultrasonic inspection data. Results are compared with existing simulation-based methods to investigate the efficiency and accuracy of the method.