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

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Abstract

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.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 51775193 and 52175267), Science and Technology Planning Project of Guangzhou City of China (No. 202007020007) and Science and Technology Planning Project of Guangdong Province of China (No. 2015B010137002).

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Authors and Affiliations

  1. School of Mechanical & Automotive Engineering, South China University of Technology, Guangzhou, 510640, China

    Junfeng Wang, Jiqing Chen, Yunjiao Zhou, Fengchong Lan & Qingshan Liu

  2. Guangdong Provincial Key Laboratory of Automotive Engineering, South China University of Technology, Guangzhou, 510640, China

    Junfeng Wang, Jiqing Chen, Yunjiao Zhou, Fengchong Lan & Qingshan Liu

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  1. Junfeng Wang
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Correspondence to Yunjiao Zhou.

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Replication of results

The “Replication of results” is now modified. This revision provides data results and corresponding visualization codes for all computable figures in the paper. The rest of the figures are schematics and do not need to be replicated. Because of the large amount of data and codes used in figures, the author uploaded the codes to Github. Readers can visit https://github.com/Wang920133787/Matlab-figures.git to download. The steps to run the code are very simple, please download the folder and put it in MATLAB environment, open the corresponding “m” file and run directly, Some Figures need arrow.p file to call the arrow function, so the “arrow.p” file should be in the same folder. The engineering example involves a complex process (finite element modeling and time-consuming simulation), some parameters of the battery pack and the innovative core algorithm are still under a confidentiality agreement. Therefore, we have removed the tedious process here and all illustrative examples provide only the results of the simulation. But the pseudo-code of the core algorithm is provided in Sect. 5 for readers’ reference. We do not know whether this revision can meet the requirements. If there are still any deficiencies, please give us feedback. We will try our best to meet the requirements. Thank you.

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Wang, J., Chen, J., Zhou, Y. et al. Vector-angle geometric mapping-based directional importance sampling method for reliability analysis. Struct Multidisc Optim 65, 154 (2022). https://doi.org/10.1007/s00158-022-03217-w

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