## 2 Concept of Impulse-Based Substructuring Method  
[PERSON] [4] introduced the IBS method in 2010. The basic idea of the IBS method is to describe the dynamic response of a substructure by the convolution of the IRFs' matrix \(H(t)\) and the applied force vector \(f(t)\) (i.e., [PERSON]'s integral) and use interface compatibility and equilibrium to assemble adjacent substructures.  
The displacement vector \(\mathbf{u}\) of a substructure at time \(t\) can be written as  
\[\mathbf{u}^{(s)}(t)=\int_{0}^{t}\mathbf{H}^{(s)}(t-\tau)\mathbf{f}^{(s)}(\tau)d\tau \tag{1}\]  
where superscript \(s\) denotes the serial number of substructures. Applying the trapezoidal rule to [PERSON]'s integral, the finite difference form of the displacement can be described approximately as  
\[\mathbf{u}^{(s)}_{n}=\sum_{i=1}^{n-1}\mathbf{H}^{(s)}_{n-i}(\mathbf{f}^{(s)}_{i}+\mathbf{f}^{( s)}_{i+1})\frac{dt}{2} \tag{2}\]  
where \(dt\) denotes the analysis time step and n denotes the \(N\)th analysis step. To assemble the substructures, interface displacement compatibility and interface force equilibrium are introduced. Displacement compatibility requires that the degrees of freedom on each side of the interface between any two connected substructures have to be equal. Displacement compatibility is described as  
\[\sum_{s=1}^{N_{s}}\mathbf{B}^{(s)}\mathbf{u}^{(s)}=0 \tag{3}\]  
where \(B\) is a signed Boolean matrix identifying the matching degrees of freedom (DOF) on the interfaces, and \(N_{s}\) is the number of substructures.  
The force equilibrium requires that interface forces on each side of the interface between any two connected substructures have to be an equilibrating force system. For a matching interface assembly, the interface forces should comply with [PERSON]'s third law. This can be described as  
\[g^{(s)}=\mathbf{B}^{(s)T}\mathbf{\lambda} \tag{4}\]  
where \(g^{(s)}\) denotes the full set of interface forces of substructure \(s\), and \(\lambda\) is known as the Lagrange multiplier vector.  
Combining displacement compatibility and substituting Equation (4) into Equation (2) yields the system-governing equation:  
\[\left\{\begin{array}{l}\mathbf{u}_{n}^{(s)}=\sum\limits_{i=1}^{n-1}\mathbf{H}_{n-i}^{( s)}[\mathbf{f}_{i}^{(s)}+\mathbf{f}_{i+1}^{(s)}+\mathbf{B}^{(s)T}(\mathbf{\lambda}_{i}+\mathbf{ \lambda}_{i+1})]\frac{dt}{2}\\ \sum\limits_{s=1}^{N_{s}}\mathbf{B}^{(s)}\mathbf{u}^{(s)}=0\end{array}\right. \tag{5}\]  
Rewrite the first equation of Equation (5), and we have  
\[\left\{\begin{array}{l}\mathbf{u}_{n}^{(s)}=\sum\limits_{i=1}^{n-2}\mathbf{H}_{n-i}^{ (s)}[\mathbf{f}_{i}^{(s)}+\mathbf{f}_{i+1}^{(s)}+\mathbf{B}^{(s)T}(\mathbf{\lambda}_{i}+\mathbf{ \lambda}_{i+1})]\frac{dt}{2}\\ +\mathbf{H}_{1}^{(s)}(\mathbf{f}_{n-1}^{(s)}+\mathbf{f}_{n}^{(s)}+\mathbf{B}^{(s)T}\mathbf{\lambda} _{n-1})+\mathbf{H}_{1}^{(s)}\mathbf{B}^{(s)T}\mathbf{\lambda}_{n}\\ \sum\limits_{s=1}^{N_{s}}\mathbf{B}^{(s)}\mathbf{u}^{(s)}=0\end{array}\right. \tag{6}\]  
Finally, by substituting the first line of Equation (6) into the second line of Equation (6), Lagrange multipliers \(\mathbf{\lambda}_{n}\) at time \(t_{n}\) can be found as  
\[\mathbf{\lambda}_{n}=-\left\{\sum\limits_{s=1}^{N_{s}}\mathbf{B}^{(s)}\mathbf{H}_{1}^{(s) }\mathbf{B}^{(s)^{T}}\right\}^{-1}\sum\limits_{s=1}^{N_{s}}\mathbf{B}^{(s)}\hat{\mathbf{n} }_{n-1}^{(s)} \tag{7}\]  
where  
\[\begin{array}{l}\hat{\mathbf{n}}_{n-1}^{(s)}=\sum\limits_{i=0}^{n-2}\mathbf{H}_{n-1 }^{(s)}\left[\mathbf{f}_{i}^{(s)}+\mathbf{f}_{i+1}^{(s)}+\mathbf{B}^{(s)^{T}}(\mathbf{\lambda} _{i}+\mathbf{\lambda}_{i+1})\right]\\ +\mathbf{H}_{1}^{(s)}\left(\mathbf{f}_{n-1}^{(s)}+\mathbf{f}_{n}^{(s)}+\mathbf{B}^{(s)^{T}}\bm {\lambda}_{n-1}\right)\end{array} \tag{8}\]  
Then, from Equation (7), we can see that Lagrange multipliers become the only direct unknown variables.