## 3 Materials and Methods
### Decoupling Strategy for Multidisciplinary Reliability Approach  
As shown in Figure 4, when both aleatory and epistemic uncertainties are involved, the procedure of MRA will become a nested three-layer loop. The MPRA loop lies in the outer loop, which searches the MPP through calling convex analysis and MDA repeatedly in the standard normal space. The second loop is the MCRA, which aims at finding the minimum of limit state function through calling MDA. The MDA in the inner loop is always invoked repeatedly by MPRA and MCRA, providing the value of coupled limit state function. Obviously, the computational effort of the three-layered procedure may be prohibitive especially for large-scale and coupled multidisciplinary systems. Therefore, a decoupling strategy for multidisciplinary reliability is proposed to decouple the nested MRA into a sequence of cycles of three modules that are multidisciplinary probabilistic reliability analysis (MPRA), multidisciplinary convex reliability analysis (MCRA) and multidisciplinary analysis (MDA). As a result, a great number of MDA and reliability analyses can be eliminated. In the MDA module, the values of the coupling state variables are obtained through the coordination relationship among the sub disciplines, and the values of the coupling state variables are used by MCRA and MPRA. So that all the reliability constraints only contain the design variables without coupling state variables. In convex reliability analysis, Lagrange multiplier method [19] is used to convert constrained optimization problems to unconstrained problems. All epistemic uncertain design variables and Lagrange multiplicator \(\lambda\) are taken derivative based on the differential principle. KKT conditions is used to replace extremum search algorithm. In this way, the calculation efficiency of multidisciplinary convex reliability analysis can be improved. For multidisciplinary probabilistic reliability analysis. Global sensitivity equations (GSE) [20] based on CSSO strategy are used to provide the prerequisite for parallel subspace sensitivity analysis and provides gradient information for updating the random design variables. Modified advanced mean value (MAMV) [21] method is used to search MPP. The characteristics of the proposed method are as follows: (1) using PMA to improve the efficiency of reliability analysis from the model itself; (2) decoupling the three-layer nested analysis process; (3) adopting CSSO strategy and executing MDA and global sensitivity analysis to provide reliability analysis with the value and sensitivity information of limit-state functions; (4) adopting KKT conditions to replace the expensive extremum analysis. When there are only aleatory uncertainties, the method proposed in this paper became a sequential multidisciplinary probabilistic reliability analysis which integrates the CSSO and PMA, as a matter of convenience, we call it SMPRA. When aleatory uncertainties and epistemic uncertainties exist simultaneously, the proposed method is a decoupling based multidisciplinary reliability analysis; we call it MU-DBMRA. The flowchart of the proposed decoupling strategy is shown in Figure 5. Five steps are involved in this strategy. Step 1: Set initial values \(\mathbf{u}_{s}\), \(\mathbf{u}_{i}\), \(\mathbf{v}_{s}\), \(\mathbf{v}_{i}\), \(k=1\). Step 2: Multidisciplinary probabilistic reliability analysis. Step 2.1: Fix epistemic uncertain design variables (characterized by their mean values in first loop), and set loop number \(k=1\). Step 2.2: Execute multidisciplinary analysis. Evaluate the value of state variable \(\mathbf{y}_{k}\) and limit-state functions \(g(\mathbf{x}_{k})\). Step 2.3: Execute system sensitivity analysis. Use GSE method to obtain the gradient of limit-state function \(\
abla_{\mathbf{x}}g(\mathbf{x}_{k})\). Figure 5: The decoupling strategy for multidisciplinary reliability analysis approach.