## 4 Modeling the SMA-LRB Isolator
### Optimizing the SMA-LRB Isolator  
[PERSON] and [PERSON] [7] proved that the hysteresis behavior of SMA was related to various parameters, including the base isolator's size, radius of the wire, and temperature. This research used the SMA wire as a supplementary element to the base isolators of the benchmark bridge to improve its seismic performance without changing its LRBs. Moreover, it was assumed that the temperature was constant and equal to 20 \({}^{\circ}\)C, and only the radius of the wire was unknown. This parameter has a profound effect on the force applied to the LRB isolator. Increasing the wire's radius led to an increase in the re-centering-induced force from the wire to the LRB and consequently an increase in the horizontal stiffness of the isolators. As a result, by increasing the radius of the wire, the residual deformation is reduced, but the base shear force and the deck acceleration of the bridge increase. Figure 6: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Yield Displacement(m)** & **Post-Yield Stiffness (kN/mm)** & **Initial Stiffness (kN/mm)** \\ \hline \hline  & 0.015 & 0.6 & 4.8 \\  & 0.015 & 1.2 & 9.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: LRB characteristics of the benchmark bridge. In this work, an evolutionary optimization algorithm named NSGA-II (non-dominated sorting genetic algorithm) was utilized to find the optimum radius of the wires for each earthquake record. The non-dominated sorting genetic algorithm type II is a multi-objective genetic algorithm, proposed by [PERSON] et al. [36]. It is an extension and improvement of NSGA, which was proposed earlier by [PERSON] and [PERSON] [37]. In the structure of NSGA-II, in addition to genetic operators (crossover and mutation), two specialized operators are defined and utilized. The first operator is called non-dominated sorting, in which the population is sorted and partitioned into fronts (F1, F2, etc. ), where F1 indicates the approximated Pareto front. The second one is the crowding distance, which indicates a mechanism of ranking among members of a front, which are dominating or dominated by each other. These ranking mechanisms are used with genetic selection operators to create the population of the next generation. Figure 7 depicts the flow chart of NSGA-II ([PERSON] et al. [38]). Figure 6: Hysteresis curve plotted by this paper’s written code (PCODE), and with [PERSON] Dezfuli and Alam (MCODE) [7]. It is worthwhile to mention that the used algorithm in this project includes two objective functions, which are parameters J1 and J3. As can be seen in Table 2, these parameters are the maximum base shear force of the controlled bridge normalized with respect to the maximum base shear force of the uncontrolled structure and the maximum mid-span displacement of the controlled bridge normalized with respect to the corresponding displacement of the uncontrolled structure, respectively. For each earthquake, the population included 40 members, and it was assumed that the maximum number of iterations was equal to 40. The mutation and crossover rates were 0.4 and 0.7, respectively. Using these parameters and the algorithm mentioned previously, the Pareto front was achieved, which contained the points simultaneously and independently minimized J1 and J3 for an earthquake record. For the far-field North Palm Springs earthquake, Figure 8 presents the optimized points. This algorithm can find more counts of optimum points by increasing the population size and the maximum iteration. However, the runtime also increases. Choosing the desired radius among these points depends on the designer's purpose. In this study, the main aim was to alleviate displacements, so among the optimum points with the least J3, the point with minimum J1 was picked. Consequently, the optimum radius was 6 mm for the North Palm Springs earthquake. This process was repeated for other earthquakes.