## 3 Assembly of Substructure with Non-Matching Interfaces  
### Assembly of Impulse-Based Substructures with Non-Matching Interfaces  
Substructures rarely have a perfect node-to-node interface when established by different design groups. As shown in Figure 1, two substructures, \(\Omega^{(1)}\) and \(\Omega^{(2)}\), are coupled with non-matching interfaces. The nodes in each substructure's interface are not coincident. In this case, it is necessary to reconsider the compatibility of the interface displacement and the equilibrium of the interface force. In this section, based on the work of [PERSON] [16; 17], who proposed an assembly method for matching interfaces, a method of assembling impulse-based substructures with non-matching interfaces is proposed. As illustrated in Figure 1, consider a structure with \(N_{s}\) substructures. Substructure \(\Omega^{(s)}\), where superscript \(s\) denotes the serial number of substructures, is coupled with other substructures with non-matching interfaces. Vector \(\mathbf{u}^{(s)}\) denotes the displacement of the substructure, \(\mathbf{u}^{(s)}\) is a constitutive of interface DOFs \(\mathbf{u}_{b}^{(s)}\) and internal DOFs \(\mathbf{u}_{i}^{(s)}\), giving  
\[\mathbf{u}^{(s)}=\left[\mathbf{u}_{b}^{(s)T}\quad\mathbf{u}_{i}^{(s)T}\right]^{T} \tag{9}\]Define a Boolean matrix \(\mathbf{B}_{b}^{(s)}\) that projects vector \(\mathbf{u}^{(s)}\) to vector \(\mathbf{u}_{b}^{(s)}\)  
\[\mathbf{B}_{b}^{(s)}\mathbf{u}^{(s)}=\mathbf{u}_{b}^{(s)} \tag{10}\]  
As the nodes in each interface of the substructure do not coincide, we consider the interface displacement compatibility and force equilibrium on an intermediate surface. Thus,  
\[\mathbf{\overline{u}}_{b}^{(s)}=\mathbf{g}_{u}^{(s)}\mathbf{u}_{b}^{(s)} \tag{11}\]  
where \(\mathbf{g}_{u}^{(s)}\) is the interpolation function relating the interface DOFs \(\mathbf{u}_{b}^{(s)}\) of substructure s to the intermediate surface DOFs \(\mathbf{\overline{u}}_{b}^{(s)}\). Then, the displacement of substructure \(s\) can be described as  
\[\mathbf{\overline{u}}_{b}^{(s)}=\left[\mathbf{\overline{u}}_{b}^{(s)T}\quad\mathbf{u}_{i }^{(s)T}\right]^{T}=\left[\mathbf{g}_{u}^{(s)}\mathbf{u}_{b}^{(s)T}\quad\mathbf{u}_{b}^{( s)T}\right]^{T} \tag{12}\]  
The displacement compatibility is described as  
\[\sum_{s=1}^{N_{s}}\mathbf{B}^{(s)}\mathbf{\overline{u}}^{(s)}=0 \tag{13}\]  
\(g^{(s)}\) is the interface force directly interacting with substructure \(s\) and \(\mathbf{\lambda}\) is the Lagrange multiplier vector, which stands for the intermediate surface forces. The force equilibrium can be described as  
\[\mathbf{g}^{(s)}=\mathbf{g}_{f}^{(s)}\Big{(}\mathbf{B}^{(s)T}\mathbf{\lambda}\Big{)} \tag{14}\]  
where \(\mathbf{g}_{f}^{(s)}\) is the force interpolation function. Substituting Equations (13) and (14) into Equation (2) yields the system-governing equation with the non-matching interfaces  
\[\left\{ \begin{array}{l}\mathbf{u}_{n}^{(s)}=\sum\limits_{i=1}^{n-1}\mathbf{H}_{n-i}^ {(s)}[\mathbf{f}_{i}^{(s)}+\mathbf{f}_{i+1}^{(s)}+\mathbf{B}^{(s)T}(\mathbf{\lambda}_{i}+\mathbf{ \lambda}_{i+1})]\frac{dt}{2}\\ \sum\limits_{s=1}^{N_{k}}\mathbf{B}^{(s)}\mathbf{\overline{u}}^{(s)}=0\end{array} \right. \tag{15}\]  
To solve this equation, the two critical interpolation functions \(\mathbf{g}_{u}^{(s)}\) and \(\mathbf{g}_{f}^{(s)}\) should be considered. Then, we assume  
\[\mathbf{g}_{u}^{(s)}\Big{(}\mathbf{u}_{b}^{(s)}\Big{)}=\mathbf{G}_{u}^{(s)}\mathbf{u}_{b}^{(s)} \tag{16}\]  
\[\mathbf{g}_{f}^{(s)}\Big{(}\mathbf{B}^{(s)T}\mathbf{\lambda}\Big{)}=\mathbf{G}_{f}^{(s)}\mathbf{B} ^{(s)T}\mathbf{\lambda} \tag{17}\]  
Figure 1: A two-substructure system with non-matching interfaces. where \(\mathbf{G}_{u}^{(s)}\) and \(\mathbf{G}_{f}^{(s)}\) are interpolation matrices, which transform displacements and forces from substructures to intermediate surfaces, respectively. Substituting Equation (16) into Equation (12) yields  
\[\overline{\mathbf{u}}^{(s)}= \begin{bmatrix}\mathbf{G}_{u}^{(s)}\mathbf{u}_{b}^{(s)}\\ \mathbf{u}_{i}^{(s)}\end{bmatrix} =