Abstract Failure probability function (FPF) can reflect quantitative effects of random input distribution parameter (DP) on failure probability, and it is significant for decoupling reliability‐based design optimization (RBDO). But the FPF estimation is time‐consuming since it generally requires repeated reliability analyses at different DPs. For efficiently estimating FPF, an augmented directional sampling (A‐DS) is proposed in this paper. By using the property that the limit state surface (LSS) in physical input space is independent of DP, the A‐DS establishes transformation of LSS samples in standard normal spaces corresponding to different DPs. By the established transformation in different standard normal spaces, the LSS samples obtained by DS at a given DP can be transformed to those at other DPs. After simple interpolation post‐processing on those transformed samples, the failure probability at other DPs can be estimated by DS simultaneously. The main novelty of A‐DS is that a strategy of sharing DS samples is designed for estimating the failure probability at different DPs. The A‐DS avoids repeated reliability analyses and inherits merit of DS suitable for solving problems with multiple failure modes and small failure probability. Compared with other FPF estimation methods, the examples sufficiently verify the accuracy and efficiency of A‐DS.
For reliability analysis of complex structures with time-consuming implicit performance functions, the computational cost required by direct sampling methods is usually unaffordable for engineering applications. However, the adaptive metamodel embedded in sampling methods can significantly improve reliability analysis efficiency. Therefore, a new metamodel-based directional importance sampling method (Meta-DIS-AK) is proposed for reliability analysis in this article. The main novelty of Meta-DIS-AK is constructing the quasi-optimal DIS density (DIS-D) by the Kriging model and giving a simple rejection sampling algorithm for accurately extracting DIS-D samples. By the constructed quasi-optimal DIS-D, the failure probability is transformed into a two-step estimation of augmented failure probability and a correction item. The updating of the first step is designed to ensure that the constructed DIS-D is well approaching to theoretical optimal DIS-D and obtain augmented failure probability estimation, and the second step updating can guarantee the accuracy of correction item estimation. Meta-DIS-AK not only overcomes the difficulties of constructing DIS-D and extracting importance direction vector samples in the existing DIS but also inherits the advantages of DIS in dealing with high-dimensional and small failure probability problems. Compared with the existing efficient sampling methods combined with AK, the results of the examples fully verify the accuracy and efficiency of the proposed Meta-DIS-AK.
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