A new uncertainty propagation method considering multimodal probability density functions

  • Z. Zhang
  • J. Wang
  • C. JiangEmail author
  • Z. L. Huang
Research Paper


In practical engineering applications, random variables may follow multimodal distributions with multiple modes in the probability density functions, such as the structural fatigue stress of a steel bridge carrying both highway and railway traffic and the vibratory load of a blade subject to stochastic dynamic excitations, etc. Traditional uncertainty propagation methods are mainly used to treat random variables with only unimodal probability density functions, which, therefore, tend to result in large computational errors when multimodal probability density functions are involved. In this paper, an uncertainty propagation method is developed for problems in which multimodal probability density functions are involved. Firstly, the multimodal probability density functions of input random variables are established using the Gaussian mixture model. Secondly, the uncertainties of the input random variables are propagated to the response function through an integration of the sparse grid numerical method and maximum entropy method. Finally, the convergence mechanism is developed to improve the uncertainty propagation accuracy step by step. Two numerical examples and one engineering application are studied to demonstrate the effectiveness of the proposed method.


Uncertainty propagation Multimodal probability density function Convergence mechanism Sparse grid numerical integration Maximum entropy method 


Funding information

This work is supported by the National Natural Science Foundation of China (Grant No. 51805157, Grant No. 51725502 and Grant No. 51490662), Hunan Natural Science Foundation (Grant No. 2019JJ40015).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

158_2019_2301_MOESM1_ESM.rar (7 kb)
ESM 1 (RAR 6 kb)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle EngineeringHunan UniversityChangsha CityPeople’s Republic of China
  2. 2.School of Mechanical and Electrical EngineeringHunan City UniversityYiyang CityPeople’s Republic of China

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