## 2 Linear Dynamic Analysis with Condensed Localized Mortar Method
### Application of Condensed Localized Mortar Method in Linear Dynamic Analysis
\begin{array}{ll}5\text{th}-\text{ row}:\mathbf{Q}^{(1)T}\mathbf{u}_{b}^{(1)} -\mathbf{W}^{(1)}\mathbf{u}^{(f)}=0\rightarrow\mathbf{u}_{b}^{(1)}=\Big{(} \mathbf{Q}^{(1)T}\Big{)}^{-1}\mathbf{W}^{(1)}\mathbf{u}^{(f)}=\mathbf{C}_{1f} \mathbf{u}^{(f)}\\ 6\text{th}-\text{ row}:\mathbf{Q}^{(2)T}\mathbf{u}_{b}^{(2)}-\mathbf{W}^{(2)} \mathbf{u}^{(f)}=0\rightarrow\mathbf{u}_{b}^{(2)}=\Big{(}\mathbf{Q}^{(2)T} \Big{)}^{-1}\mathbf{W}^{(2)}\mathbf{u}^{(f)}=\mathbf{C}_{2f}\mathbf{u}^{(f)} \end{array}  \tag{9}\]  
Secondly, the second and fourth rows of the total system Equation (8) are used as follows to eliminate the Lagrange multipliers, \(\mathbf{\Lambda}^{(1)}\) and \(\mathbf{\Lambda}^{(2)}\):  
\[ \begin{array}{ll}2\text{nd}-\text{ row}:\mathbf{M}_{bi}^{(1)}\tilde{ \mathbf{u}}_{i}^{(1)}+\mathbf{K}_{bi}^{(1)}\mathbf{u}_{i}^{(1)}+\mathbf{M}_{bb }^{(1)}\tilde{\mathbf{u}}_{b}^{(1)}+\mathbf{K}_{bb}^{(1)}\mathbf{u}_{b}^{(1)}+ \mathbf{Q}^{(1)}\mathbf{\Lambda}^{(1)}=\mathbf{f}_{b}^{(1)}\\ \rightarrow\mathbf{\Lambda}^{(1)}=\Big{(}\mathbf{Q}^{(1)}\Big{)}^{-1}\Big{(} \mathbf{f}_{b}^{(1)}-\mathbf{M}_{bi}^{(1)}\tilde{\mathbf{u}}_{i}^{(1)}-\mathbf{ K}_{bi}^{(1)}\mathbf{u}_{i}^{(1)}-\mathbf{M}_{bb}^{(1)}\tilde{\mathbf{u}}_{b}^{(1)}- \mathbf{K}_{bb}^{(1)}\mathbf{C}_{1f}\mathbf{u}^{(f)}\Big{)}\\ 4\text{th}-\text{ row}:\mathbf{M}_{bi}^{(2)}\tilde{\mathbf{u}}_{i}^{(2)}+ \mathbf{K}_{bi}^{(2)}\mathbf{u}_{i}^{(2)}+\mathbf{M}_{bb}^{(2)}\tilde{\mathbf{u}} _{b}^{(2)}+\mathbf{K}_{bb}^{(2)}\mathbf{u}_{b}^{(2)}+\mathbf{Q}^{(2)}\mathbf{ \Lambda}^{(2)}=\mathbf{f}_{b}^{(2)}\\ \rightarrow\mathbf{\Lambda}^{(2)}=\Big{(}\mathbf{Q}^{(2)}\Big{)}^{-1}\Big{(} \mathbf{f}_{b}^{(2)}-\mathbf{M}_{bi}^{(2)}\tilde{\mathbf{u}}_{i}^{(2)}-\mathbf{ K}_{bi}^{(2)}\mathbf{u}_{i}^{(2)}-\mathbf{M}_{bb}^{(2)}\tilde{\mathbf{u}}_{b}^{(2)}- \mathbf{K}_{bb}^{(2)}\mathbf{C}_{2f}\mathbf{u}^{(f)}\Big{)}\end{array}  \tag{10}\]  
When the relations (9) and (10) are substituted into the total system Equation (8), the Lagrange multipliers \(\mathbf{\lambda}^{(1)}\) and \(\mathbf{\Lambda}^{(2)}\) and the overlapped boundary displacements \(\mathbf{u}_{b}^{(1)}\) and \(\mathbf{u}_{b}^{(2)}\) are condensed by the internal displacements \(\mathbf{u}_{i}^{(1)}\) and \(\mathbf{u}_{i}^{(2)}\) and the frame displacement \(\mathbf{u}^{(f)}\). Lastly, the matrix form of the total system equation can be obtained as follows by summarizing and re-organizing Equation (8):  
Figure 4: Division of the total displacements into the internal and interface components. \[\hat{\mathbf{M}}\tilde{\mathbf{u}}+\hat{\mathbf{K}}\tilde{\mathbf{u}}=\hat{ \mathbf{F}}\text{ where }\quad\hat{\mathbf{M}}=\left[ \begin{array}{ccc}\mathbf{M}_{ii}^{(1)}&0& \mathbf{M}_{ib}^{(1)}\mathbf{C}_{1f}\\ 0&\mathbf{M}_{ii}^{(2)}&\mathbf{M}_{ib}^{(2)}\mathbf{C}_{2f}\\ \mathbf{C}_{1f}^{T}\mathbf{M}_{ki}^{(1)}&\mathbf{C}_{2f}^{T}\mathbf{M}_{ki}^{ (2)}&\mathbf{C}_{1f}^{T}\mathbf{M}_{bb}^{(1)}\mathbf{C}_{1f}+\mathbf{C}_{2f}^ {T}\mathbf{M}_{bb}^{(2)}\mathbf{C}_{2f}\end{array} \right],\hat{\mathbf{a}}= \left[ \begin{array}{c}\mathbf{u}_{i}^{(1)}\\ \mathbf{u}_{i}^{(2)}\\ \mathbf{u}_{i}^{(f)}\end{array} \right] \tag{11}\] \[\hat{\mathbf{K}}=\left[ \begin{array}{ccc}\mathbf{K}_{ii}^{(1)}&0& \mathbf{K}_{ib}^{(1)}\mathbf{C}_{1f}\\ 0&\mathbf{K}_{ii}^{(2)}&\mathbf{K}_{ib}^{(2)}\mathbf{C}_{2f}\\ \mathbf{C}_{1f}^{T}\mathbf{K}_{ki}^{(1)}&\mathbf{C}_{2f}^{T}\mathbf{K}_{bi}^{ (2)}&\mathbf{C}_{1f}^{T}\mathbf{K}_{bb}^{(1)}\mathbf{C}_{1f}+\mathbf{C}_{2f}^ {T}\mathbf{K}_{bb}^{(2)}\mathbf{C}_{2f}\end{array} \right],\hat{\mathbf{F}}= \left[ \begin{array}{c}\mathbf{f}_{i}^{(1)}\\ \mathbf{f}_{i}^{(2)}\\ \mathbf{C}_{1f}^{T}\mathbf{f}_{b}^{(1)}+\mathbf{C}_{2f}^{T}\mathbf{f}_{b}^{(2 )}\end{array} \right]\]  
The internal displacements \(\mathbf{u}_{i}^{(1)}\) and \(\mathbf{u}_{i}^{(2)}\) and the frame displacement \(\mathbf{u}^{(f)}\) can be obtained directly by solving the total system equation of the condensed localized mortar method (11).