A new uncertainty propagation method considering multimodal probability density functions

Abstract

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.

Appendix

Deduction of the ith non-standard central moment ${m}_i^{\prime }$:

$$ {m}_i^{\prime }=\int {\left(\frac{y-{\mu}_Y}{c}\right)}^i\rho (y) dy,i=0,1,...,l $$

(24)

where

$$ {\mu}_Y=\int y\rho (y) dy={m}_1 $$

(25)

Therefore, ${m}_i^{\prime }$ is rewritten as:

$$ {m}_i^{\prime }=\int {\left(\frac{y-{m}_1}{c}\right)}^i\rho (y) dy,i=0,1,...,l $$

(26)

According to the rules of integral operation:

$$ {\displaystyle \begin{array}{l}{m}_i^{\prime }=\frac{1}{c^i}\int {\left(y-{m}_1\right)}^i\rho (y) dy\\ {}\kern0.75em =\frac{1}{c^i}\int \left[\sum \limits_{j=0}^i{B}_i^{j+1}{y}^{i-j}{\left(-{m}_1\right)}^j\right]\rho (y) dy\\ {}\kern0.75em =\frac{1}{c^i}\sum \limits_{j=0}^i{B}_i^{j+1}{\left(-{m}_1\right)}^j\int {y}^{i-j}\rho (y) dy\\ {}\kern0.75em =\frac{1}{c^i}\sum \limits_{j=0}^i{B}_i^{j+1}{\left(-{m}_1\right)}^j{m}_{i-j}\end{array}} $$

(27)

${B}_i^{j+1}=\left(\begin{array}{c}i\\ {}j\end{array}\right)$ is the binomial coefficient. Let j = j + 1, then, we obtain:

$$ {m}_i^{\prime }=\frac{1}{c^i}\sum \limits_{j=1}^{i+1}{B}_i^j{\left(-1\right)}^{j-1}{m_1}^{j-1}{m}_{i-j+1} $$

(28)

where ${B}_i^j=\left(\begin{array}{c}i\\ {}j-1\end{array}\right)$.