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Multimodal ellipsoid model for non-probabilistic structural uncertainty quantification and propagation

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Abstract

The traditional ellipsoid convex set is a kind of basic non-probabilistic model to measure uncertainties. However, it is difficult or inaccurate to quantify the uncertainties of variables with multimodal distributed samples. In this paper, a more generalized non-probabilistic ellipsoid model named multimodal ellipsoid model is proposed to effectively deal with the multimodal distributed samples. The samples with one or more similar properties are clustered together, and the principal directions of the samples and characteristic matrix are appropriately found through the Gaussian mixture model. Then, the multimodal ellipsoid model can be constructed by using the elliptical contour features of the Gaussian model to measure the uncertainties of variables. The proposed multimodal ellipsoid model can not only establish traditional ellipsoid model, but also establish multi-ellipsoid model for uncertain variables with multimodal samples. Furthermore, combining with the multimodal ellipsoid model and performance measure approach, the uncertain propagation results of system are obtained accurately. Three numerical examples and one engineering application are provided to demonstrate the effectiveness and accuracy of the proposed multimodal ellipsoid model.

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Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability

The code that used and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

This work is funded by the National Natural Science Foundation of China (Grant No. 51975199, 11811530285), and the independent research project of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University (Grant No. 71865010).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, People’s Republic of China

    Jie Liu, Zhongbo Yu, Hao Liu & Xu Han

  2. State Key Laboratory of Reliability and Intelligence of Electrical Equipment, School of Mechanical Engineering, Hebei University of Technology, Tianjin, 300401, People’s Republic of China

    Dequan Zhang & Xu Han

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  1. Jie Liu
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  2. Zhongbo Yu
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  3. Dequan Zhang
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  4. Hao Liu
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  5. Xu Han
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Contributions

JL Conceptualizaion, Methodology. ZY Methodology, Investigation, Writing-Reviewing and Editing. DZ Supervision, Validation. HL Writing- Original draft preparation. XH Conceptualizaion. All authors contributed to writing or correcting the article.

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Correspondence to Jie Liu.

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Appendix

Appendix

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Table 6 Samples on the uncertainty for numerical example 1
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Table 7 Samples on the uncertainty for numerical example 2
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Table 8 Samples on the uncertainty for numerical example 3
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Table 9 Samples on the uncertainty for numerical example 4
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Liu, J., Yu, Z., Zhang, D. et al. Multimodal ellipsoid model for non-probabilistic structural uncertainty quantification and propagation. Int J Mech Mater Des 17, 633–657 (2021). https://doi.org/10.1007/s10999-021-09551-z

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