## 2 Large-Scale MDO by Using PODA and MMRT
### Inherent Rigid-Flex Coupling Relation of HSRS
\begin{array}{c}\overline{M}_{ii}=\mathbf{\Phi}^{T}M_{ii}\mathbf{\Phi}=\mathbf{I}_{P \times P}\\ \overline{M}_{io}=\overline{M}_{oi}^{T}=\mathbf{\Phi}^{T}M_{ii}\mathbf{\Psi}+\mathbf{\Phi}^ {T}M_{io}\\ \overline{M}_{oo}=\mathbf{\Psi}^{T}M_{ii}\mathbf{\Psi}+\mathbf{\Psi}^{T}M_{io}+M_{oi}\mathbf{ \Psi}+M_{oo}\\ \overline{K}_{ii}=\mathbf{\Phi}^{T}K_{ii}\mathbf{\Phi}=\mathbf{A}_{ii}\\ \overline{K}_{io}=K_{oi}^{T}=\mathbf{\Phi}^{T}K_{ii}\mathbf{\Psi}+\mathbf{\Phi}^{T}K_{io}= \mathbf{\theta}_{P\times Q}\\ \overline{K}_{oo}=\mathbf{\Psi}^{T}K_{ii}\mathbf{\Psi}+\mathbf{\Psi}^{T}K_{io}+K_{oi}\mathbf{ \Psi}+K_{oo}=K_{ii}\mathbf{\Psi}+K_{oo}\\ \overline{f}_{o}=f_{o}\end{array} \]  
Suppose that \(\Lambda_{ii}=diag(\omega_{1}^{2},\quad\cdots\quad\omega_{k}^{2},\quad\cdots \quad\omega_{P}^{2})\), \(\omega_{k}^{2}\) is the \(k\)-th inherent vibration frequency of service system under the condition that all DoFs on the interface are constrained. Introducing the modal damping ratio \(\zeta_{k}\), \(k=1,2,\cdots P\), the first line of Equation (15) can be expanded into \(P\) independent equations  
\[\ddot{\eta}_{k}+2\zeta_{k}\omega_{k}\dot{\eta}+\omega_{k}^{2}\eta_{k}=-L_{k} \ddot{\delta}_{k} \tag{16}\]  
where the modal participation factor under multi-axial excitation  
\[L_{k}=\mathbf{\Phi}_{k}^{T}M_{ii}\mathbf{\Psi}+\mathbf{\Phi}_{k}^{T}M_{io} \tag{17}\]  
Substituting \(\eta_{k}=\Pi_{k}e^{i\omega t}\) and \(\ddot{\delta}_{k}=-\omega^{2}\Delta_{k}e^{i\omega t}\) into Equation (16), the result can be obtained after sorting  
\[\Pi_{k}=\frac{1}{\left(\frac{\omega_{k}}{\dot{\omega}^{\prime}}\right)^{2}+2j \zeta_{k}\frac{\omega_{k}}{\dot{\omega}^{\prime}}-1}L_{k}\Delta_{k}=H_{k}L_{k }\Delta_{k} \tag{18}\]According to Equation (18), perform the MSR correctly based on node force and displacement in global coordinates and draw von Misses dynamic stress cloud. Through effective modal excitation \(\ddot{\delta}_{k}\), the seamless integration with Dong's structure stress recovery and collaborative operation with weldline fatigue assessment are realized under the technical support of relevant analysis softwares [35; 36]. The relationship between loading excitation and displacement/acceleration response of the interface can be obtained from the second line of Equation (15):  
\[\overline{\mathbf{K}}_{oo}\mathbf{X}_{o}+\left(\sum_{k=1}^{P}H_{k}L_{k}^{T}L_{k} \right)\ddot{\mathbf{X}}+\overline{\mathbf{M}}_{oo}\ddot{\mathbf{X}}_{o}=\mathbf{F}_{o} \tag{19}\]  
in which, the stiffness matrix of the interface after reduction is weakened by the definition of interface DoFs, i.e.,  
\[\overline{\mathbf{K}}_{oo}=\mathbf{K}_{oo}-\mathbf{K}_{io} \tag{20}\]  
where \(\mathbf{M}_{oo}\) is Cuyan mass matrix, in a specific response surface direction \(l_{r}\), the mass or moment of inertia of rigid body \(\mathbf{m}_{T}=l_{r}^{T}\mathbf{M}_{oo}l_{r}\), and the mass matrix of the interface after reduction is enhanced by the definition of interface DoFs,  
\[l_{r}^{T}\overline{\mathbf{M}}_{oo}l_{r}>l_{r}^{T}\mathbf{M}_{oo}l_{r} \tag{21}\]  
If the weak coupling interfaces of the service system have only one node (no constrained DoF), the constrained mode is composed of six rigid body motion modes, and Equation (19) can be converted into the loading excitation and acceleration response of the interface as follows  
\[\left(\sum_{k=1}^{P}H_{k}L_{k}^{T}L_{k}+\overline{\mathbf{M}}_{oo}\right)\ddot{ \mathbf{X}}_{o}=\left(\sum_{k=1}^{P}H_{k}M_{k}^{\textit{eff}}+\overline{\mathbf{M}}_{oo }\right)\ddot{\mathbf{X}}_{o}=F_{\text{omin}} \tag{22}\]  
where \(L_{k}^{T}L_{k}\) is the effective mass of the \(k\)-th inherent mode of service system, it can be noted as \(M_{k}^{\textit{eff}}\); and \(F_{\text{omin}}\) is referred as the minimum amplitude of excitation load acting on the interface DoFs.