## 3 Dynamic Response Topology Optimization with Condensed Localized Mortar Method
### Optimization Formulation with Condensed Localized Mortar Method  
The optimization formulation used to obtain an optimum topological layout in a linear dynamic system [13; 34; 35] is represented by Equation (15), where the condensed localizedmortar method described in the previous section is adopted by the matrices \(\hat{\mathbf{M}}\), \(\hat{\mathbf{K}}\), and vectors \(\hat{\mathbf{u}}(t)\), \(\hat{\mathbf{u}}(t)\) and \(\hat{\mathbf{F}}(t)\) to handle the impact problem with the non-matching interface. \[\underset{\eta}{\text{minimize}}\quad J_{d}\] (15) subject to \[\left\{ \begin{array}{c}\hat{\mathbf{M}}\ddot{\hat{\mathbf{u}}}(t)+\hat{ \mathbf{K}}\hat{\mathbf{u}}(t)=\hat{\mathbf{F}}(t)\text{,}t\in[0,t_{f}]\\ g(\eta)=V(\eta)-V_{rq}=\sum\limits_{e=1}^{N}\eta_{e}v_{e}-V_{req}\leq 0\\ 0<\eta_{\text{min}}\leq\eta\leq 1\end{array} \right.\]  
The optimization formulation (15) is organized to minimize the objective function \(J_{d}\) using the momentum equation and volume constraint, \(g(n)\)[13]. In this study, the objective function \(J_{d}\) is defined for mean dynamic compliance to find a stiff structure as follows:  
\[J_{d}=\frac{1}{t_{f}}\int\limits_{0}^{t_{f}}c(\hat{\mathbf{u}}(t),\mathbf{\eta })\text{dt},c(\hat{\mathbf{u}}(t),\mathbf{\eta})=\hat{\mathbf{F}}^{T}(t)\hat{ \mathbf{u}}(t)\text{ where }\frac{\partial c}{\partial\eta_{e}}=0\text{,}\frac{ \partial c}{\partial\hat{\mathbf{u}}}=\hat{\mathbf{F}}(t) \tag{16}\]  
Additionally, for the mean squared displacement of a target, DOF is given as:  
\[J_{d}=\frac{1}{t_{f}}\int\limits_{0}^{t_{f}}c(\hat{\mathbf{u}}(t),\mathbf{\eta })\text{dt},c(\hat{\mathbf{u}}(t),\mathbf{\eta})=(\mathbf{L}\hat{\mathbf{u}}( t))^{2}\text{ where }\frac{\partial c}{\partial\eta_{e}}=0\text{,}\frac{ \partial c}{\partial\hat{\mathbf{u}}}=2\hat{\mathbf{u}}(t)\mathbf{L} \tag{17}\]  
where \(\mathbf{L}\) represents the role in selecting the target DOF from the displacement \(\hat{\mathbf{u}}(t)\). The optimum topological layout that minimizes the objective function can be determined, which includes the mean dynamic compliance and mean squared displacement of a target degree-of-freedom while satisfying the constraints, by performing the optimization formulation defined in Equations (15)-(17). The sensitivity of the optimization formulation (15) must be calculated when a gradient-based optimization algorithm is used. Since the topology optimization presents several design variables by defining the relative density function \(\eta\), for each element, the adjoint method is more appropriate than the direct method [6,7,13]. The adjoint variables \(\lambda_{\text{adj}}\) must be introduced to simplify the objective function in order to use the adjoint method to calculate the sensitivities of the objective function. When the adjoint variable \(\lambda_{\text{adj}}\) is inserted with the additional adjoint equation \(\hat{\mathbf{M}}\ddot{\hat{\mathbf{u}}}+\hat{\mathbf{K}}\hat{\mathbf{u}}- \hat{\mathbf{F}}\) the objective function \(J_{d}\) can be rewritten as:  
\[J_{d}\cong J_{d}+\mathbf{\lambda}^{T}\int\limits_{0}^{t_{f}}(\hat{\mathbf{M} }\ddot{\hat{\mathbf{u}}}+\hat{\mathbf{K}}\hat{\mathbf{u}}-\hat{\mathbf{F}}) \text{dt} \tag{18}\]  
Since the additional adjoint equation \(\hat{\mathbf{M}}\ddot{\hat{\mathbf{u}}}+\hat{\mathbf{K}}\hat{\mathbf{u}}- \hat{\mathbf{F}}\) is always zero, as illustrated in Equation (11), it can be confirmed that the objective function \(J_{d}\) does not change even when the adjoint variable and the adjoint equation are added.