## 5 Optimizing the SMA-LRBs  
In the previous section, the approach for obtaining the hysteresis models for SMA-LRBs with three different configurations of wires was explained. As discussed, the influential parameters on the hysteresis behavior of the SMA-LRBs include the geometry of the isolator, wires arrangement, and the radius of the SMA wire (see Equation (10)). This research uses the SMA wire in three arrangements, as a supplementary element to the existed LRBs of the benchmark bridge, to enhance its seismic performance without changing its isolators. Therefore, the only unknown yet significant parameter is the radius of the wire. It should be determined carefully since it has a substantial effect on the force applied to the LRB isolator. By increasing the radius of the wire, the re-centering force generated in the SMA wire and applied to the LRB increases, and, as a result, the horizontal stiffness of the isolator increases. Consequently, the residual deformation declines, but the base shear force and the deck acceleration of the bridge rise. The main goal of this research is to minimize the residual displacements of the bridge after strong earthquakes; however, a detrimental increase in the base shear and acceleration is not acceptable. Therefore, it is decided to take advantage of a multi-objective optimization algorithm to simultaneously minimize displacement and acceleration and find the optimum radius of the wire. An evolutionary optimization algorithm named NSGA-II (non-dominated sorting genetic algorithm) is used to find the best radius of the SMA wire for each arrangement under the most severe earthquake. The NSGA-II is a multi-objective genetic algorithm that was proposed by [PERSON] et al. [55]. It is an extension and development of NSGA, which was proposed previously by [PERSON] and [PERSON] [56]. In the structure of the NSGA-II, in addition to genetic operators (i.e., crossover and mutation), two specified operators are defined and employed. The first one is non-dominated sorting, which sorts and partitions the population into fronts (F1, F2, etc. ), where F1 indicates the approximated Pareto front. The second one is crowding distance, which specifies a mechanism of ranking amongst members of a front, which are either dominated by or are dominating each other. These ranking mechanisms are utilized with genetic selection operators to generate the population of the next generation. The flowchart of NSGA-II is demonstrated in Figure 10[57]. Figure 10: Flowchart of non-dominated sorting genetic algorithm (NSGA)-II. In the NSGA-II algorithm of this project, there are two objective functions, namely, maximum mid-span displacement and maximum deck acceleration. For simplicity, these parameters are normalized with respect to the quantities of the uncontrolled bridge. The optimization should be carried out in the worst scenario of the earthquake, so the real record from the station El Centro Array #12 (RSN 8161) is used in the algorithm, as it has larger Arias Intensity and larger PGA value (Tables 1 and 2). The population includes 30 members, and the maximum number of iterations is equal to 30. The crossover and mutation rates are 0.7 and 0.4, in turn. Under these defined circumstances, the algorithm is run three times for the three various configurations of wires, and the Pareto front is achieved for each case. The Pareto front indicates the points that simultaneously and independently minimize the normalized maximum mid-span displacement and the normalized maximum deck acceleration. Figure 11 depicts the optimized point for the three wire configurations. Selecting the desired radius amongst the optimized points of a Pareto front depends on the designer's perspective. In this research, the main goal is to decrease displacements; hence, the final radius is determined based on this idea. The final optimized and rounded radii for the double-cross, straight, and cross arrangements are 5 mm, 6 mm and 7 mm, respectively.