## 2 Linear Dynamic Analysis with Condensed Localized Mortar Method
### Application of Condensed Localized Mortar Method in Linear Dynamic Analysis  
Based on the basic concept and framework of the linear dynamic analysis considering frictionless contact conditions, this subsection presents a method to apply the condensed localized mortar method [19] in the linear dynamic analysis to derive the contact force vector \(\textbf{f}_{c}\) in Equation (4). The mortar method is the most widely used approach to solve the interface, contact, impact, and multi-physics problems by introducing Lagrange multipliers to impose the interface compatibility condition in a weak sense [24; 25; 26; 27]. The mortar methods can be divided into classical and localized versions depending on how the interface compatibility conditions and Lagrange multipliers are defined. The classical mortar method is the most widely used approach for treating the interface conditions by introducing Lagrange multipliers as the interface pressure, as shown in Figure 3b. A fictitious frame is introduced in the localized mortar method, and the interface compatibility condition is imposed through the frame domain, as shown in Figure 3c [19; 28; 29; 30; 31; 32]. This study employs the localized mortar method to impose the interface compatibility conditions because it is successfully utilized in various problems by uniquely defining the interface compatibility conditions through the frame domain [19; 28; 29; 30; 31; 32]. In the case of the localized mortar method, the interface compatibility condition between the newly introduced frame domain and the adjacent two parts, as shown in Figure 3c, is given by  
\[\mathbf{u}^{(1)}-\mathbf{u}^{(2)}\mathbf{=0}\ \Rightarrow\left\{ \begin{array}{l} \mathbf{u}^{(1)}-\mathbf{u}^{(f)}=0\\ \mathbf{u}^{(2)}-\mathbf{u}^{(f)}=0\end{array} \right. \tag{6}\]  
Subsequently, two sets of Lagrange multipliers, i.e., \(\boldsymbol{\lambda}^{(1)}\) and \(\boldsymbol{\lambda}^{(2)}\), are defined along the interface to impose the interface capability condition (6) in a weak sense. By substituting the interface condition (6) and two sets of Lagrange multipliers into the momentum equation of the mortar method (4) and organizing the matrix form, the total system equation of the localized mortar method in a linear dynamic condition is given by:  
\[\left[ \begin{array}{cccc}\mathbf{M}^{(1)}&0&0&0&0\\ 0&\mathbf{M}^{(2)}&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array} \right]\left[ \begin{array}{cccc}\tilde{\mathbf{u}}^{(1)} \\ \tilde{\mathbf{u}}^{(2)}\\ \tilde{\boldsymbol{\lambda}}^{(1)}\\ \tilde{\boldsymbol{\lambda}}^{(2)}\\ \tilde{\boldsymbol{\lambda}}^{(f)}\end{array} \right]+\left[ \begin{array}{cccc }\mathbf{K}^{(1)}&0&\mathbf{B}_{1}\mathbf{Q}_{1}&0&0\\ 0&\mathbf{K}^{(2)}&0&\mathbf{B}_{2}\mathbf{Q}_{2}&0\\ \mathbf{Q}_{1}^{T}\mathbf{B}_{1}^{T}&0&0&0&-\mathbf{W}_{1}\\ 0&\mathbf{Q}_{2}^{T}\mathbf{B}_{2}^{T}&0&0&-\mathbf{W}_{2}\\ 0&0&-\mathbf{W}_{1}^{T}&-\mathbf{W}_{2}^{T}&0\end{array} \right]\left[ \begin{array} []{c}\mathbf{u}^{(1)}\\ \mathbf{u}^{(2)}\\ \mathbf{\lambda}^{(1)}\\ \mathbf{u}^{(2)}\\ \mathbf{u}^{(f)}\end{array} \right]=\left[ \begin{array}{c}\mathbf{f}^{(1)} \\ \mathbf{f}^{(2)}\\ \mathbf{G}^{(1)}\\ \mathbf{G}^{(2)}\\ 0\end{array}