Frontiers of Mechanical Engineering

, Volume 14, Issue 1, pp 33–46 | Cite as

Uncertainty propagation analysis by an extended sparse grid technique

  • X. Y. Jia
  • C. Jiang
  • C. M. Fu
  • B. Y. Ni
  • C. S. Wang
  • M. H. Ping
Open Access
Research Article
Part of the following topical collections:
  1. Innovative Design and Intelligent Design


In this paper, an uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy of the probability density function (PDF) of the system response. The proposed method incorporates the extended Gauss integration into the uncertainty propagation analysis. Moreover, assisted by the Rosenblatt transformation, the various types of extended integration points are transformed into the extended Gauss-Hermite integration points, which makes the method suitable for any type of continuous distribution. Subsequently, within the sparse grid numerical integration framework, the statistical moments of the system response are obtained based on the transformed points. Furthermore, based on the maximum entropy principle, the obtained first four-order statistical moments are used to fit the PDF of the system response. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed method, which includes two mathematical problems with explicit expressions and an engineering application with a black-box model.


uncertainty propagation analysis extended sparse grid maximum entropy principle extended Gauss integration Rosenblatt transformation high-order moments analysis 



This work was supported by the National Science Fund for Distinguished Young Scholars (Grant No. 51725502), the major program of the National Natural Science Foundation of China (Grant No. 51490662), and the National Key Research and Development Project of China (Grant No. 2016YFD0701105).


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Copyright information

© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the appropriate credit is given to the original author(s) and the source, and a link is provided to the Creative Commons license, indicating if changes were made.

Authors and Affiliations

  • X. Y. Jia
    • 1
  • C. Jiang
    • 1
  • C. M. Fu
    • 1
  • B. Y. Ni
    • 1
  • C. S. Wang
    • 2
  • M. H. Ping
    • 1
  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle EngineeringHunan UniversityChangshaChina
  2. 2.Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, School of Electro-Mechanical EngineeringXidian UniversityXi’anChina

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