## 3 Assembly of Substructure with Non-Matching Interfaces
### Assembly of IBS and Finite Element Substructures with Non-Matching Interfaces
=\overline{\mathbf{G}}_{u}^{(r)}\mathbf{u}^{(r)} \tag{31}\]  
where \(\mathbf{I}\) is identity matrix and \(\overline{\mathbf{G}}_{u}^{(r)}\) is defined as  
\[\overline{\mathbf{G}}_{u}^{(r)}= \begin{bmatrix}\mathbf{G}_{u}^{(r)}&0\\ 0&\mathbf{I}\end{bmatrix}  \tag{32}\]  
The CB method is based on a consistent Rayleigh-Ritz coordinate transformation  
\[\mathbf{u}=\mathbf{T}\eta \tag{42}\]  
where \(\mathbf{T}\) is a coordinate transformation matrix and \(\eta\) is the generalized coordinates. Dividing the original \(\mathbf{u}\) into the boundary and internal DOFs \(\mathbf{u}_{b}\) and \(\mathbf{u}_{i}\), the modal substructure formulation can be partitioned as  
\[ \begin{bmatrix}\mathbf{M}_{ij}^{(m)}&\mathbf{M}_{ib}^{(m)}\\ \mathbf{M}_{bi}^{(m)}&\mathbf{M}_{bb}^{(m)}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{i}^{ \prime\;(m)}\\ \mathbf{u}_{b}^{\prime\;(m)}\end{bmatrix} + \begin{bmatrix}\mathbf{C}_{ij}^{(m)}&\mathbf{C} _{ip}^{(m)}\\ \mathbf{C}_{bi}^{(m)}&\mathbf{C}_{bb}^{(m)}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{i}^{ \prime\;(m)}\\ \mathbf{u}_{b}^{\prime\;(m)}\end{bmatrix} + \begin{bmatrix}\mathbf{K}_{ii}^{(m)}&\mathbf{K} _{ib}^{(m)}\\ \mathbf{K}_{bi}^{(m)}&\mathbf{K}_{bb}^{(m)}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{i}^{(m) }\\ \mathbf{u}_{b}^{\prime\;(m)}\end{bmatrix} = \begin{bmatrix}\mathbf{f}_{i}^{(m)}\\ \mathbf{g}_{b}^{(m)}\end{bmatrix}  \tag{43}\]  
where the superscript m denotes the serial number of modal substructures. According to the CB method [1], the constraint modes are defined as  
\[\Phi_{c}^{(m)}= \begin{bmatrix}-\left(K_{ii}^{(m)}\right)^{-1}K_{ib}^{(m)}\\ \mathbf{I}\end{bmatrix}  \tag{44}\]  
To obtain the fixed-interface modes, the substructure is assumed to be fixed at its boundary DOF \(\mathbf{u}_{b}^{m}\) and, from Equation (43), the eigen equation of the fixed substructure is  
\[\mathbf{M}_{ii}^{(m)}\mathbf{u}_{i}^{\prime\;(m)}+\mathbf{K}_{ii}^{(m)}\mathbf{u}_{i}^{(m)}=0 \tag{45}\]  
The fixed-interface modes \(\Phi_{b}^{(m)}\) are mass normalized and are  
\[\Phi_{B}^{(m)^{\mathsf{T}}}\mathbf{M}_{ii}^{(m)}\Phi_{B}^{(m)}=\mathbf{I},\Phi_{B}^{(m )^{\mathsf{T}}}\mathbf{K}_{ii}^{(m)}\Phi_{B}^{(m)}=\mathbf{\Lambda}^{(m)} \tag{46}\]  
where \(\mathbf{\Lambda}^{(m)}\) is a diagonal matrix whose elements are the squares of the corresponding eigenfrequencies of the fixed substructure. In the CB method,  
\[\mathbf{u}_{i}^{(m)}=\Phi_{B}^{(m)}\mathbf{\eta}_{B}^{(m)}+\[PERSON]_{Ci}^{(m)}\mathbf{u}_{b}^ {(m)} \tag{47}\]  
where \(\mathbf{\eta}_{B}^{(m)}\) is the fixed-interface generalized coordinates, and \(\[PERSON]_{Ci}^{(m)}=-\left(K_{ii}^{(m)}\right)^{-1}K_{ib}^{(m)}\) is the partition of constraint modes associated with the internal DOFs. Then, the coordinate transformation can be described as  
\[u^{(m)}= \begin{bmatrix}\[PERSON]_{B}^{(m)}&\[PERSON]_{Ci}^{(m)}\\ 0&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{\eta}_{B}^{(m)}\\ \mathbf{u}_{b}^{(m)}\end{bmatrix} =\mathbf{T}^{(m)}\mathbf{\eta}^{(m)} \tag{48}\]  
where \(\mathbf{T}^{(m)}\) is the reduction matrix of the Craig-Bampton method. In Equation (48), all boundary DOFs are retained. Usually, not all eigenvectors are retained for \(\Phi_{B}^{(m)}\) and that will result in a reduction of the substructure. According to Equation (48), the modal substructures are given as  
\[\overline{\mathbf{M}}^{(m)}\mathbf{\eta}^{\prime\;(m)}(t)+\overline{\mathbf{C}}^{(m)}\mathbf{ \eta}^{\prime\;(m)}(t)+\overline{\mathbf{K}}^{(m)}\mathbf{\eta}^{(m)}(t)=\vec{\mathbf{f}}^ {(m)}(t)+\mathbf{T}^{(m)\mathsf{T}}\mathbf{g}^{(m)}(t) \tag{49}\]  
where \(\overline{\mathbf{M}}^{(m)}=\mathbf{T}^{(m)\mathsf{T}}\mathbf{M}^{(m)}\mathbf{T}^{(m)}\), \(\overline{\mathbf{C}}^{(m)}=\mathbf{T}^{(m)\mathsf{T}}\mathbf{C}^{(m)}\mathbf{T}^{(m)}\), \(\overline{\mathbf{K}}^{(m)}=\mathbf{T}^{(m)\mathsf{T}}\mathbf{K}^{(m)}\mathbf{T}^{(m)}\), and \(\vec{\mathbf{f}}^{(m)}=\mathbf{T}^{(m)}\mathbf{f}^{(m)}\). Then, the formulation of the assembled system considering non-matching interfaces can be described as  
\[\left\{