## 1 Introduction  
There are various sources of uncertainties in real-world engineering conditions, such as loading uncertainties, material uncertainties, geometric and boundary uncertainties due to manufacturing tolerances, variation in operating environments, and differences in technical levels. A small amount of uncertainties may have a severe influence on the overall performance of a system owing to the nonlinear coupling uncertainty propagation of multiple uncertain variables in the complex engineering problems. Therefore, uncertainties should be appropriately quantified and analyzed to improve the performance of engineering products in actual situations. The uncertainties are assumed to be aleatory uncertainties, which have determinate and complete uncertain information, and are represented using accurate probability density functions (PDFs). Many probabilistic uncertain analysis and optimization methodologies have been proposed to decrease the deteriorative effects of uncertainties and improve the reliability of engineering products, such as the first-order reliability design method [1; 2; 3], the second-order reliability design method [4; 5; 6], the time-dependent reliability design method [7], and the reliability-based robust design method [8]. However, it is difficult to acquire an accurate PDF of uncertainties in some actual engineering applications. Only insufficient information can be acquired, and the economic cost or computational complexity is too high to acquire further uncertainty information. New uncertainty problems are produced when using aleatory uncertainties to address these uncertainty problems, the reliability design becomes the calculation of the uncertainty of uncertain variables, and the computational complexity increases exponentially with the number of uncertain variables, which makes it impossible to calculate in complex engineering problems. To address the problems of incomplete information or insufficient data for uncertain variables, the use of epistemic uncertainty is proposed. This has been widely applied to the uncertainty analysis and reliability design of engineering products [9; 10; 11]. Some non-probabilistic uncertainty analysis methods have been proposed to address epistemic uncertainties, and summarized by [PERSON] [12], which includes the interval analysis approach [13; 14], the convex model [15; 16], the fuzzy set [17; 18], and the evidence theory [19; 20; 21; 22]. These non-probabilistic analysis methods have different application situations based on the available uncertainty information. The interval analysis method is used when the uncertainty information is lacking, only lower and upper bounds of uncertain variables can be acquired. The convex model discerns the boundaries of uncertain variables instead of their precise probability distributions, which can be represented with a few sample information. The interval model and ellipsoid convex model are two common convex models [23]. When some quantitative uncertainty information is available, fuzzy set theory can be used, and the uncertainty is represented using membership functions. Evidence theory uses belief assignment as its basis to describe uncertainty information, and then it represents the variable uncertainty with the belief and plausibility measure function. With these non-probabilistic uncertainty representation methods, epistemic uncertainty is quantified, propagated, and optimized in the reliability design of engineering products [24; 25]. However, there are some challenges in the application of these non-probabilistic uncertainty representation methods for actual engineering problems. One of the challenges is the high computational cost [26], which is caused by the discretized quantification of epistemic uncertainties. The calculation complexity increases with the number of epistemic variables and the discrete interval numbers of all epistemic variables. Another challenge is the unified calculation of aleatory uncertainty and epistemic uncertainty.