## 3 Materials and Methods  
### Reliability Comprehensive Evaluation Index Considering Multisource Uncertainties  
When the aleatory uncertainties and epistemic uncertainties are fully considered, the limit state function of the multidisciplinary reliability design optimization model is transformed from the original one which only contains aleatory uncertain design variables and coupling state variables to one which includes both aleatory uncertain variables, epistemic uncertain variables and coupling state variables. Because of the existence of epistemic uncertain design variables, the value of limit state function is no longer a single value (i.e., no longer a single failure surface), but a series of values (i.e., there are a series of failure surface families) between the minimum and maximum values of the limit state function. At this time, the reliability value becomes an interval. In fact, this is also uncertainty, that is, the uncertainties brought by uncertainties. In order to ensure the high reliability of the design, we propose to use the minimum value of limit state function as the criterion to measure reliability, and the difference of limit state function value to describe the influence of epistemic uncertainties on reliability. The larger the interval difference, the greater the degree of epistemic uncertainties. In fact, the range of the limit state function also directly indicates that designers need to do more experiments on these epistemic variables to improve the reliability of design or obtain more data and knowledge through other channels, reduce the epistemic uncertainty of design parameters, reduce the impact on the limit state function, and then improve reliability of the complex engineering system design. As shown in Figure 2, the whole standard normal space (U space) is divided by \(\Omega\) into three parts: safety area (\(\Omega_{\text{s}}=\{\mathbf{u}\text{: min }\mathbf{G(u,v,y)>0\,|\,v\in E }\}\)), critical region (\(\Omega_{\text{c}}=\{\mathbf{u}\text{: min }\mathbf{G(u,v,y)=0\,|\,v\in E }\}\)), and failure region (\(\Omega_{\text{f}}=\{\mathbf{u}\text{: min }\mathbf{G(u,v,y)<0\,|\,v\in E }\}\)). Where \(\mathbf{u}\) is the standard normal random variables, \(\mathbf{v}\) is the normalized vector of epistemic variables, and \(\mathbf{y}\) is the vector of coupling state variable, \(\mathbf{G(u,v,y)}\) represent the limit state function after space transformation. \(\mathbf{G(u,v,y)>0}\) indicates that it meets the reliability requirements. The junction of safety area \(\Omega_{\text{s}}\) and critical area \(\Omega_{\text{c}}\) represents a unique spatial curve (surface) \(\widehat{\mathbf{G(u)}}=0\), which is called \"the most probable failure surface\". The comprehensive evaluation index of multidisciplinary reliability under multisource uncertainty is defined as: in the standard \(\mathbf{U}\) space, the shortest distance (i.e., the lower limit of interval) from \" the most probable failure surface \" to the coordinate origin is regarded  
Figure 2: Comprehensive reliability index in u space. as the reliability evaluation value, and is recorded as \(\beta_{C}^{L}\) (C is comprehensive, which means comprehensive reliability index, L is the lower limit of reliability interval), and the farther the most likely failure surface is from the origin, the smaller the failure probability of limit state function is. In fact, the solution of comprehensive reliability evaluation index can be given by Equation (10):  
\[\beta_{C}^{L}=\min\sqrt{\mathbf{u}^{\mathrm{T}}\mathbf{u}} \tag{10}\] \[\text{s.t. }\widehat{G}(\mathbf{u},\mathbf{v})=0\]  
where,  
\[\widehat{G}(\mathbf{u},\mathbf{v})=G(\mathbf{u},\mathbf{v}) \tag{11}\] \[\text{s.t. }\mathbf{v}_{i}^{\mathrm{T}}\mathbf{v}_{i}\leq 1\] \[\text{ }i=1,2,\cdots,n\]  
Equation (10) is used to search for the most probable point (\(\mathbf{u}^{MPP}\)) of aleatory uncertain variables, and Equation (11) is used to solve the worst point (\(\mathbf{v}^{WCP}\)) of epistemic uncertain variables.