## 3 Assembly of Substructure with Non-Matching Interfaces
### Assembly of IBS and Finite Element Substructures with Non-Matching Interfaces  
The IBS method needs to pre-compute the impulse response of a substructure. If a substructure is often changed during the design process, generating the impulse response is much more expensive than performing the Newmark method once. So, in this section, we propose a method of assembling the IBS and finite element substructure with non-matching interfaces. The substructure that changes frequently will be modeled as an FE model and others will be represented by an IBS. For a finite element substructure, the equations of motion can be described as  
\[\mathbf{M}^{(r)}\mathbf{u}^{\prime\prime}{}^{(r)}(t)+\mathbf{C}^{(r)}\mathbf{u}^{\prime(r)}(t)+ \mathbf{K}\mathbf{u}^{(r)}(t)=\mathbf{f}^{(r)}(t)+\mathbf{g}^{(r)}(t) \tag{24}\]  
where superscript \(r\) denotes the serial number of FE substructures. Separately, \(\mathbf{f}^{(r)}(t),\mathbf{g}^{(r)}(t)\) denote the external forces and interface forces acting on the FE substructure r.  
Similar to Equation (12), the displacement substructure can be described as  
\[\mathbf{u}^{(r)}=\left[\overline{\mathbf{u}}_{b}^{(r)T}\quad\mathbf{u}_{i}^{(r)T}\right]^{T}= \left[\mathbf{g}_{u}^{(r)}\left(\mathbf{u}_{b}^{(r)}\right)^{T}\quad\mathbf{u}_{i}^{(r)T} \right]^{T} \tag{25}\]  
where \(\mathbf{u}_{b}^{(r)}\) and \(\mathbf{u}_{i}^{(r)}\) are interface DOFs and internal DOFs of finite element substructures r. \(\mathbf{g}_{u}^{(r)}\) is the interpolation function relating interface DOFs \(\mathbf{u}_{b}^{(r)}\) to the intermediate surface with coincident intermediate surface DOFs \(\overline{\mathbf{u}}_{b}^{(r)}\). Then, the displacement compatibility of the IBS and FEM substructures is described as  
\[\sum_{s=1}^{N_{s}}\mathbf{B}^{(s)}\overline{\mathbf{u}}^{(s)}+\sum_{s=1}^{N_{r}}\mathbf{B}^ {(r)}\overline{\mathbf{u}}^{(r)}=0 \tag{26}\]  
Similar to Equation (14), the interface force \(\mathbf{g}^{(r)}(t)\) that directly interacts with substructure r can be described as  
\[\mathbf{g}^{(r)}=\mathbf{g}_{f}^{(r)}\Big{(}\mathbf{B}^{(r)T}\mathbf{\lambda}\Big{)} \tag{27}\]  
where \(\mathbf{g}_{f}^{(r)}\) is the force interpolation function, \(\mathbf{g}^{(r)}\) is the interface forces directly interacting with the substructure and \(\mathbf{\lambda}\) is the Lagrange multiplier vector, which stands for the intermediate surface forces. Substituting Equations (13), (14), (26) and (27) into Equations (2) and (24) yields the system-governing equation with non-matching interfaces  
\[\left\{ \begin{array}{l}\mathbf{u}^{(s)}(t)=\sum\limits_{i=0}^{n-1}\mathbf{H}_{n-i}^{ (s)}\Big{[}\mathbf{f}_{i}^{(s)}+\mathbf{f}_{i+1}^{(s)}+\mathbf{g}_{i}^{(s)}+\mathbf{g}_{i+1}^ {(s)}\Big{]}\frac{\mathrm{d}t}{2}\\ \mathbf{M}^{(r)}\mathbf{u}^{\prime\prime}(t)+\mathbf{C}^{(r)}\mathbf{u}^{\prime(r)}(t)+\mathbf{K}^ {(r)}\mathbf{u}^{(r)}(t)=\mathbf{f}^{(r)}(t)+\mathbf{g}^{(r)}(t)\\ \sum\limits_{s=1}^{N_{t}}\mathbf{B}^{(s)}\overline{\mathbf{u}}^{(s)}+\sum\limits_{r=1} ^{N_{r}}\mathbf{B}^{(r)}\overline{\mathbf{u}}^{(r)}=\mathbf{0}\end{array} \right. \tag{28}\]  
To solve this equation, using the same method mentioned in Section 3.1, we assume  
\[\mathbf{g}_{u}^{(r)}\Big{(}\mathbf{u}_{b}^{(r)}\Big{)}=\mathbf{G}_{u}^{(r)}\mathbf{u}_{b}^{(r)} \tag{29}\]  
\[\mathbf{g}_{f}^{(s)}\Big{(}\mathbf{B}^{(s)T}\mathbf{\lambda}\Big{)}=\mathbf{G}_{f}^{(s)}\mathbf{B} ^{(s)T}\mathbf{\lambda} \tag{30}\]  
where \(\mathbf{G}_{u}^{(r)}\) and \(\mathbf{G}_{f}^{(s)}\) are interpolation matrices, which transform displacements and forces from substructures to the intermediate surface, respectively. Substituting Equation (29) into Equation (25) yields  
\[\mathbf{u}^{(r)}=\left[\mathbf{G}_{u}^{(r)}\Big{(}\mathbf{u}_{b}^{(r)}\Big{)}^{T}\quad\bm {u}_{i}^{(r)T}\right]^{T}= \begin{bmatrix}\mathbf{G}_{u}^{(r)}&0\\ 0&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{b}^{(r)}\\ \mathbf{u}_{i}^{(r)}\end{bmatrix}