## 4 Evaluation and Discussion
### Numerical Example  
As shown in Figure 6, the numerical example [22] consists of three subsystems and five design variables. Where \(x_{1}\), \(x_{2}\), \(x_{3}\) belongs to subsystem 1, subsystem 2 contains three design variables: \(x_{1}\), \(x_{4}\), \(x_{5}\). \(y_{12}\) and \(y_{21}\) are coupling-state variables; \(g_{1}\) and \(g_{2}\) are reliability constraints of subsystem 1 and subsystem 2 respectively. The functional relationship between the two subsystems is as below:  
(1) Subsystem 1:  
\[ \begin{array}{l}\mathbf{x}_{1}=\{x_{1},x_{2},x_{3},x_{4},x_{5}\},\mathbf{y}_ {1}=\{y_{12}\},\mathbf{g}_{1}=\{g_{1}\}\\ y_{12}=x_{1}^{2}+2x_{2}-x_{3}+2\sqrt{y_{21}}+x_{4}+x_{5}\\ g_{1}=5-\left(x_{1}^{2}+2x_{2}+2x_{3}x_{4}+x_{2}e^{-y_{21}}\right)-1.1x_{5} \end{array}  \tag{24}\]  
(2) Subsystem 2:  
\[ \begin{array}{l}\mathbf{x}_{2}=\{x_{1},x_{4},x_{5}\},\mathbf{y}_{2}=\{y_{21 }\},\mathbf{g}_{2}=\{g_{2}\}\\ y_{21}=x_{1}x_{4}+x_{4}^{2}+x_{5}+y_{12}\\ g_{2}=\sqrt{x_{1}}+x_{4}+x_{5}(0.4x_{1})+y_{2}\end{array}  \tag{25}\]  
In order to verify the effectiveness and computational efficiency of the proposed method, two cases are selected:  
**Case 1**: Suppose that aleatory design variables \(\mathbf{x}=(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\), \(x_{5})\)-N(\(\mu_{\mathbf{x}}\), \(\sigma_{\mathbf{x}}\)), and the mean value \(\mu_{\mathbf{x}}=(1\), \(1\), \(1\), \(1\)), \(\sigma_{\mathbf{x}}=(0.1\), \(0.1\), \(0.1\), \(0.1\)). **Case 2**: Suppose that \(x_{1}\), \(x_{2}\) and \(x_{3}\) are design variables with aleatory uncertainty, the mean value \(\mu_{\mathbf{x}}\) of \(\mathbf{x}=(x_{1}\), \(x_{2}\), \(x_{3})\) equals (1, 1, 1) and the variance \(\sigma_{\mathbf{x}}\) equals (0.1,0.1,0.1). Similarly, suppose that \(x_{4}\) and \(x_{5}\) are design variables with epistemic uncertainty. Furthermore, in order to further verify the influence of variation range of design parameters with epistemic uncertainty on limit-state functions, three test points are set:  
**Test point 1:** The variation range can be described as below: \(\mathbf{x}\in E=\left\{\mathbf{x}\Big{|}(\mathbf{x}-\overline{\mathbf{x}})^{T }W(\mathbf{x}-\overline{\mathbf{x}})\leq 0.04^{2}\right\}\); the calibration value is \(\overline{\mathbf{x}}=[\overline{x}_{4},\overline{x}_{5}]^{T}=[1,1]^{T}\) and the characteristic matrix is \(\mathbf{W_{x}}=\left[ \begin{array}{cc}64&0\\ 0&16\end{array} \right]\). Therefore, the conversion relationship of epistemic uncertainties in x-space and in \(\mathbf{v}\) space is \(x_{4}=\frac{v_{4}}{20}+1\), \(x_{5}=\frac{v_{5}}{100}+1\). **Test point 2:** Take the variation \(\varepsilon\) as \(0.02\) and the characteristic matrix \(\mathrm{w_{11}}\) as \(64\), \(\mathrm{w_{12}}\) as \(0\), \(\mathrm{w_{21}}\) as \(0\) and \(\mathrm{w_{22}}\) as \(16\), then the conversion relationship of epistemic uncertainties in \(\mathbf{x}\) space and in \(\mathbf{v}\) space is: \(x_{4}=v_{4}/400+1\), \(x_{5}=v_{5}/200+1\). **Test point 3:** Take the variation \(\varepsilon\) as \(0.04\) and the characteristic matrix \(\mathrm{w_{11}}\) as \(16\), \(\mathrm{w_{12}}\) as \(0\), \(\mathrm{w_{21}}\) as \(0\) and \(\mathrm{w_{22}}\) as \(64\), then the conversion relationship of epistemic uncertainties in \(\mathbf{x}\) space and in \(\mathbf{v}\) space is: \(x_{4}=v_{4}/100+1\), \(x_{5}=v_{5}/400+1\). Following multidisciplinary reliability analysis with SMPRA and MU-DBMRA (three test points) methods for the limit-state function \(g_{1}\) respectively, the obtained MPPs, their values and iteration times are shown in Table 1. In order to prove the effectiveness of the algorithm, we chose the method proposed in reference [7] to compare the results in test point 1. The solver of PMA in the contrasting method is sequential quadratic programming (SQP). Figure 6: The coupling relationship diagram of the numerical example.