## 2 Linear Dynamic Analysis with Condensed Localized Mortar Method
### Application of Condensed Localized Mortar Method in Linear Dynamic Analysis
\right], \tag{7}\]  
\[\mathbf{Q}_{1}=\int_{\Gamma_{I}}\mathbf{N}_{u_{\Gamma 1}}^{T}\mathbf{N}_{\lambda 1}d\Gamma,\quad \mathbf{Q}_{2}=\int_{\Gamma_{I}}\mathbf{N}_{u_{\Gamma 2}}^{T}\mathbf{N}_{\lambda 2}d\Gamma,\quad\mathbf{W}_{1}=\int_{ \Gamma_{I}}\mathbf{N}_{\lambda 1}^{T}\mathbf{N}_{uf}d\Gamma,\quad\mathbf{W}_{2}=\int_{ \Gamma_{I}}\mathbf{N}_{\lambda 2}^{T}\mathbf{N}_{uf}d\Gamma\] \[\mathbf{G}^{(1)}=\int_{\Gamma_{I}}\mathbf{N}_{\lambda 1}^{T} \mathbf{S}_{l}d\Gamma,\quad\mathbf{G}^{(2)}=\int_{\Gamma_{I}}\mathbf{N}_{ \lambda 2}^{T}\mathbf{S}_{2f}d\Gamma\]  
where \(\mathbf{B}_{1}\) and \(\mathbf{B}_{2}\) are the Boolean matrices representing the boundary displacement components, and \(\mathbf{Q}_{1}\), \(\mathbf{Q}_{2}\), \(\mathbf{W}_{1}\), and \(\mathbf{W}_{2}\) are the projection matrices, which represent the interface condition between the interface displacements and frame displacement. Additionally, \(\mathbf{G}^{(1)}\) and \(\mathbf{G}^{(2)}\) indicate the non-penetration condition caused by the KKT condition (2) and the interface compatibility condition (6). The condensed localized mortar method, which was developed to be easily applied in topology optimization algorithms even in the presence of a non-matching interface, is employed [19]. This can be obtained by condensing the Lagrange multipliers and overlapping interface displacements using the localized mortar method (7). When the total system equation is condensed, the numerical efficiency is increased drastically, and the total system equation can be used universally because it can be considered equivalent to the framework of the general structural finite element analysis [19]. Before proceeding with the condensed localized mortar method, the displacement \(\mathbf{u}^{(n)}\) at each part must be divided into the internal component \(\mathbf{u}_{i}^{(n)}\) and the interface component  
Figure 3: Simple patch test problem: (**a**) a continuum structure composed of two subdomains, and each subdomain is connected using the (**b**) classical and (**c**) localized version. \(\mathbf{u}_{b}^{(n)}\) as shown in Figure 4. When the displacements are decomposed, the total system Equation (7) for the localized mortar method is rewritten as:  
\[\left[ \begin{array}{cccccc}\mathbf{M}_{u}^{(1)}&\mathbf{M}_{u}^{(1)}&0&0&0&0&0 \\ \mathbf{M}_{u}^{(1)}&\mathbf{M}_{u}^{(1)}&0&0&0&0&0\\ 0&0&\mathbf{M}_{u}^{(2)}&\mathbf{M}_{u}^{(2)}&0&0&0\\ 0&0&\mathbf{M}_{u}^{(2)}&\mathbf{M}_{u}^{(2)}&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ \end{array} \right]\left[ \begin{array}{cccccc}\mathbf{M}_{u}^{(1)}&\mathbf{M}_ {u}^{(1)}&0&0&0&0&0\\ \mathbf{M}_{u}^{(1)}&\mathbf{M}_{u}^{(1)}&0&0&0&\mathbf{Q}^{(1)}&0&0\\ 0&\mathbf{Q}_{u}^{(2)}&\mathbf{K}_{u}^{(2)}&0&0&0\\ 0&0&\mathbf{K}_{u}^{(2)}&\mathbf{K}_{u}^{(2)}&0&0&\mathbf{Q}^{(2)}&0\\ 0&\mathbf{Q}_{u}^{(1)T}&0&0&0&0&-\mathbf{W}^{(1)}\\ 0&0&0&\mathbf{Q}^{(2)T}&0&0&-\mathbf{W}^{(1)T}&0\\ \end{array} \right]\left[ \begin{array}{cccccc}\mathbf{u}_{u}^{(1)}\\ \mathbf{u}_{u}^{(1)}\\ \mathbf{u}_{u}^{(2)}\\ \mathbf{u}_{u}^{(1)}\\ \mathbf{u}_{u}^{(2)}\\ \mathbf{u}^{(1)}\\ \mathbf{u}^{(2)}\end{array} \right]=\left[ \begin{array}{cccccc}\mathbf{f}_{ \xi}^{(1)}\\ \mathbf{f}_{\xi}^{(1)}\\ \mathbf{f}_{\xi}^{(1)}\\ \mathbf{f}_{\xi}^{(2)}\\ \mathbf{f}_{\xi}^{(2)}\\ \mathbf{q}^{(2)}\end{array} \right] \tag{8}\]  
The following two steps must be performed to derive the condensed localized mortar method. Firstly, two relations must be obtained through the fifth and sixth rows of the total system Equation (8), to condense the boundary displacements \(\mathbf{u}_{b}^{(1)}\), \(\mathbf{u}_{b}^{(2)}\) into the frame displacement \(\mathbf{u}^{(f)}\). \[