## 2 Materials and Methods  
### Optimization of Seismic Isolators  
The main purpose of seismic isolation design is to ensure the stability of the superstructure and isolation system during earthquake ground motions. In order to prevent possible damage to the superstructure, the aim is to minimize the roof floor acceleration (PRA) and inter-story drift ratio (ISDR). The horizontal displacement (\(D\)) should be kept to a minimum to avoid harming the base isolation system. Many studies and seismic codes have similar purposes in base isolation modeling [23]. This study uses energy components to target base isolation and superstructure stability. To demonstrate the effectiveness of energy components in base isolation optimization, four different objective functions are proposed below. * **Case 1: Input Energy. ** The most basic goal of energy-based seismic design is to determine the energy transmitted to the structure by the earthquake. This energy, which must be completely dissipated by the structure, is called input energy. As the input energy decreases, the potential for earthquake-induced structural damage decreases. Consequently, the objective function was developed as input energy minimization. * **Case 2: Hysteretic Energy. ** The repeated cyclic movement of base isolation systems produces hysteretic energy. The seismic isolation mechanism dissipates most of the input energy as hysteretic energy. The part of the input energy that cannot be dissipated reaches the superstructure and causes damage. The energy consumed in the seismic isolation system should be as large as possible so that the amount of energy reaching the superstructure is reduced. For this reason, the maximization of hysteretic energy is selected as the target function in this study. * **Case 3: Energy Ratio. ** The optimization of seismic isolation parameters should aim to decrease the input energy and increase the hysteretic energy. In order to achieve this goal, the ratio of the difference in the input energy from the hysteretic energy to the input energy has been used in previous studies [40]. In this study, the relevant ratio was used as the objective function. Therefore, the objective functions in the first two situations were combined to generate the study's main suggestion. * **Case 4: PRA/PGA. ** PRA/PGA has been used as an objective function in many structural optimization studies to ensure the earthquake resistance of structures. In this study, in which energy components were suggested as the objective function, PRA/PGA was used to verify the results. As explained in the previous section, seismic isolation modeling requires the \(T_{0}\) and \(F_{Q}/W\) parameters. Since these two parameters are sufficient for seismic isolation modeling, they are taken as variable parameters. The objective is to find the nonlinear isolation parameters of a seismic-isolated structure:  
\[\overset{\rightarrow}{X}=\left(T_{0},\ \ F_{Q}/W\right) \tag{7}\]  
where \(T_{0}\) and \(F_{Q}/W\), which are calculated by Equations (1)-(4), represent the isolation period and the ratio of characteristic strength to weight, respectively. Seismic isolators have some limitations due to their physical properties. The objective functions of this study are presented with Equations (8) and (9) regarding these physical constraints. \[fasc_{n}= \begin{cases}E_{\text{Imp.En'}}f_{penalty},&n=1\\ E_{\text{Hys.En'}}f_{penalty},&n=2\\ E_{\text{R.En'}}f_{penalty},&n=3\\ E_{\text{PRA/PGA'}}f_{penalty},&n=4\end{cases}  \tag{8}\]\[f_{penalty}=\left\{ \begin{array}{ll}\left(1+\kappa_{1}\cdot\frac{D}{D_{limit}} \right)^{\kappa_{2}},&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad if\ D>D_{limit}\\ \left(1+\kappa_{1}\cdot\frac{\beta_{eff}}{\beta_{eff_{limit}}}\right)^{ \kappa_{2}},&\quad\quad\quad if\ \beta_{eff}>\beta_{eff_{limit}}\\ \left(1+\kappa_{1}\cdot\frac{ISDR}{ISDR_{limit}}\right)^{\kappa_{2}},&\quad if \ ISDR>ISDR_{limit}\end{array} \right.