Vector-angle geometric mapping-based directional importance sampling method for reliability analysis.

In reliability analysis, the probability density function (PDF) of the directional importance sampling method is based on a multi-dimensional vector (i.e., multivariate), thus it is inefficient to obtain the important directional vectors (IDVs) by sampling each dimensional component randomly. In this paper, an efficient solution approach of vector-angle geometric mapping is proposed. Firstly, the angles between IDVs and the design point position vector are set as the important direction angles (IDAs) in the standard Gaussian space. By exploring the geometric relationship between IDVs and IDAs, the PDF of multi-dimensional IDV can be converted into the PDF of one-dimensional IDA, following which, the cumulative distribution function of IDA is derived by integration. Further, the cumulative distribution is sampled uniformly using the Latin hypercube technique, and then the uniform IDAs are generated by inversion. Finally, the IDVs are shown by geometric mapping of the IDAs. The research results show that the PDF of IDA is jointly determined by the two parameters, dimensionality and reliability index. Therefore, the distribution characteristics of IDA can be explored and diagrammatically represented, and the obtained IDVs can be used repeatedly for other reliability analysis with the same mentioned parameters to improve the computational efficiency. The applicability, accuracy, and robustness of the proposed approach are proved on illustrative examples, battery pack and truss structure engineering applications. [ABSTRACT FROM AUTHOR]