## 6 Conclusions and Future Works  
This paper extended the application of the cubic normal transformation method from interval variables to multiple types of epistemic uncertainties and developed a unified probabilistic representation framework for uncertainty propagation analysis of multiple types of epistemic uncertainties. A unified probabilistic representation approach was proposed. The uncertain intervals of four statistical moments were calculated for epistemic uncertainties represented by using the interval approach, triangular fuzzy approach, and evidence theory. A construction algorithm of continuous cubic normal distribution function sets was developed based on the uncertain intervals of four statistical moments, and these epistemic uncertainties were represented using continuous probabilistic belief and plausibility measure functions. Compared with the traditional MCS method, the proposed methodology can acquire accurate uncertainty bounds of performance functions with higher computational efficiency. Especially, the computational efficiency in the uncertainty analysis of the total heat transfer rate of the 24-stream plate-fin heat exchanger was improved by 85.7%.  
Although some improvements were made for the uncertainty propagation analysis of multiple types of epistemic uncertainty, the proposed algorithms have some limitations, which can be further researched in the future:  
(1). The uncertainty representation results could be integrated into probabilistic reliability optimization methodologies. The lower and upper bounds of the probability density function of the performance function could be applied into first-order or second-order reliability methods. Reliability optimization algorithms could be developed to solve the reliability problem considering multiple types of epistemic uncertainties.  
(2). The proposed uncertainty representation method could be integrated with an adaptive approximation method to solve the reliability problem of complex engineering products. Though the proposed algorithm can decrease the evaluation number required to calculate the performance function in the uncertainty propagation analysis, the performance function needs to be calculated several times in the reliability optimization process. Therefore, determining how to construct an adaptive  
Figure 10: Uncertainty analysis results of the heat exchanger.  
approximation model for probabilistic reliability optimization under multiple types of epistemic uncertainties is an issue worth researching in the future.  
(3). The probabilistic uncertainty representation framework can be extended to integrate more epistemic uncertainty types, such as \(p\)-box, convex model, insufficient data, and variables with uncertainty distribution parameters.  
Conceptualization, X.P.; Methodology, X.P. and Q.G.; Supervision, Z.L. and S.J.; Writing--original draft, X.P. and Q.G.; Writing--review and editing, J.L. and B.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key Research and Development Program of China [under grant number 2017 YFB0603704], the National Natural Science Foundation of China under grant [number: 51875525, U1610112, U1608256], the Zhejiang Provincial Natural Science Foundation of China under grant [number: LY20E050020], and Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems [number: GZKF-201916]. No potential conflict of interest was reported by the authors.