## 2 Large-Scale MDO by Using PODA and MMRT
### Inherent Rigid-Flex Coupling Relation of HSRS  
The improved transaction strategy of the flexible body to MBS interface by using MMAT has enriched the intension of the surrogate model. Specifically, by applying the CMS (Component Mode Synthesis) method, both interface displacement condensation treatment and CC (Characteristic Constraint) mode make the surrogate model of a flexible body neater to deal with the nonlinear influence factors on relevant interfaces [31; 32], so as to implement the robust design of lightweight structures [33]. As shown in Equations (4) and (5), the hypothesis of classic structural dynamics is changed from the linear elastic deformation under the service condition of small displacement to the one under the large displacement service system, and the modal analysis of the rigid-flex coupling system includes two subsets of motion and elasticity. Under the changed assumption of linear elastic deformation under large displacement service systems, large-scale nonlinear systems will have the inherent rigid-flex coupling relationship due to the singularities or complex constraints. For a linear elastic system with total \(n\) DoFs (Degree of freedom), \(Q\) is taken as the interface DoFs, and \(P\) is the internal DoFs, \(n=Q+P\), which are represented by subscriptsand \(i\), respectively. The displacement \(x_{o}=X_{o}e^{i\omega t}\) on the interface DoFs, the displacement \(x_{i}=X_{i}e^{i\omega t}\) on the internal DoFs, and the excitation load \(f_{o}=F_{o}e^{i\omega t}\) acting on the interface DoFs. Without considering damping for the time being, the motion equation of the service system of a lightweight structure can be written in the following block form [34]:  
\[ \begin{bmatrix}M_{ii}&M_{io}\\ M_{oi}&M_{oo}\end{bmatrix}\begin{Bmatrix}\ddot{x}_{i}\\ \ddot{x}_{o}\end{Bmatrix} + \begin{bmatrix}K_{ii}&K_{io}\\ K_{oi}&K_{oo}\end{bmatrix}\begin{Bmatrix}x_{i}\\ x_{o}\end{Bmatrix} = \begin{Bmatrix}0_{P}\\ f_{o}\end{Bmatrix}  \tag{13}\]  
where \(\mathbf{M},\mathbf{K},\ddot{\mathbf{x}}\) and \(\mathbf{x}\) are the mass matrix, stiffness matrix, acceleration and displacement response of service system respectively, 0 is the zero vector matrix, and the subscripts represent the number of rows and columns. The displacement \(\mathbf{x}\) of service system can be extended to the linear combination of inherent modes and constrained modes for the fixed interface with mass normalization. Inherent modes \(\mathbf{\Phi}=[\Phi_{1}\quad\cdots\quad\Phi_{P}]\). Constrained modes \( \begin{bmatrix}\mathbf{\Psi}&\mathbf{I}_{Q\times Q}\end{bmatrix} ^{T}\) and \(\mathbf{\Psi}=-K_{ii}^{-1}K_{io}= \begin{bmatrix}\mathbf{\Psi}_{1}\quad\cdots\quad\Psi _{Q}\end{bmatrix} \). The relationship between generalized displacement \(\begin{Bmatrix}\eta_{i}&x_{o}\end{bmatrix}^{T}\) and physical displacement \( \begin{Bmatrix}x_{i}&x_{o}\end{Bmatrix} ^{T}\) is as follows:  
\[ \begin{Bmatrix}x_{i}\\ x_{o}\end{Bmatrix} = \begin{bmatrix}\mathbf{\Phi}&\mathbf{\Psi}\\ \mathbf{0}_{Q\times P}&\mathbf{I}_{Q\times Q}\end{bmatrix}\begin{Bmatrix}\eta_{i}\\ x_{o}\end{Bmatrix} =\mathbf{T} \begin{Bmatrix}\eta_{i}\\ x_{o}\end{Bmatrix}  \tag{14}\]  
Substitute Equation (14) into Equation (13), left multiple by \(\mathbf{T}^{T}\), and transform the system motion equation into a description form based on generalized displacement  
\[ \begin{bmatrix}\overline{M}_{ii}&\overline{M}_{io}\\ \overline{M}_{oi}&\overline{M}_{oo}\end{bmatrix}\begin{Bmatrix}\ddot{\eta}_{i} \\ \ddot{x}_{o}\end{Bmatrix} + \begin{bmatrix}\overline{K}_{ii}&\overline{K}_{io}\\ \overline{K}_{oi}&\overline{K}_{oo}\end{bmatrix}\begin{Bmatrix}\eta_{i}\\ x_{o}\end{Bmatrix} = \begin{Bmatrix}0_{P}\\ \overline{f}_{o}\end{Bmatrix}  \tag{15}\]  
where  
\[