# First-order reliability method to problems involving multimodal distributions

Structural and Multidisciplinary Optimization（2023）

摘要

In practical reliability problems, input variables may follow multimodal distributions. Traditional reliability methods developed for unimodal variables may not work well for multimodal ones. Therefore, in this study, an efficient reliability method is proposed to solve problems involving multimodal distributions. First, the multimodal problem is decomposed into multiple unimodal subproblems. We theoretically prove that the failure probability of a multimodal problem equals to the weighted sum of failure probability of its unimodal subproblems. Second, the unimodal subproblems are ranked according to their potential contributions to the failure probability of the original multimodal problem. Third, the first-order reliability method (FORM) is used to solve the ranked subproblems one by one. A dependent stopping criterion is proposed to improve the efficiency of FORM by adaptively allocating computational effort so that subproblems ranked in the front are solved with stricter accuracy requirement. Finally, the failure probability of the multimodal problem is obtained as the weighted sum of failure probabilities of the subproblems. Three examples are used to illustrate the effectiveness of the proposed method.

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