## 6 Single and Dual Optimization Formulations
For instance, [PERSON] and [PERSON] [56] stated that eccentricities related to different isolator parameters can affect the response of the system. In addition, the placement of the isolators--in conjunction with structural irregularities--can affect the torsional response of the superstructure [57]. These important issues are taken into account in the proposed optimization formulations via the minimization of the isolators' eccentricity, as described in the sequence. Figure 6: Flowchart of the conventional ELF design process for single (SFPB1) and triple (TFPB1) isolators. In the current study, a realistic range of values is used for the design variables in the optimization calculations, which are treated as continuous parameters. More specifically, the values of the friction coefficient and the radius of curvature range between 0.03 and 0.08 and 1 to 5 m, respectively, for SFPB [58] as well as the outer sliding surfaces of TFPB. With respect to the inner surfaces of TFPB isolators, the following values are considered, e.g., \(R_{1A}\) = 3-\(R_{2,3}\) and \(\mu_{1,4}\) = 3-\(\mu_{2,3}\), as recommended by [PERSON] and [PERSON] [59]. Subscripts correspond to the sliding surface of the bearings, as shown in Figure 1. Certainly, in engineering projects, additional technical/economical constraints could be imposed on these parameters due to budget limitations, availability and ease of implementation (e.g., grouping of isolators with small variations, using discrete values for design variables, etc.). As previously mentioned, when isolators' parameters are different, adverse torsional effects can affect the superstructure. In order to minimize the torsional response of the tank, the objective function is set in both formulations as follows:  
\[e_{min}=\sqrt{\left(x_{cw}-x_{cr}\right)^{2}-\left(y_{cw}-y_{cr}\right)^{2}} \tag{1}\]  
in which \(e_{min}\) refers to the minimum eccentricity of the base-isolated tank, since \(x_{cw}\) and \(y_{cw}\) denote the coordinates of the center of mass, while \(x_{cr}\) and \(y_{cr}\) are the coordinates of the center of rigidity, which are calculated as follows:  
\[x_{cw}=\frac{\sum x_{i}\cdot P_{vertical,i}}{\sum P_{vertical,~{}i}} \tag{2}\]  
\[y_{cw}=\frac{\sum y_{i}\cdot P_{vertical,~{}i}}{\sum P_{vertical,~{}i}} \tag{3}\]  
\[x_{cr}=\frac{\sum K_{eff,~{}i}\cdot x_{i}}{\sum K_{eff,~{}i}} \tag{4}\]  
\[y_{cr}=\frac{\sum K_{eff,~{}i}\cdot y_{i}}{\sum K_{eff,~{}i}} \tag{5}\]  
where \(x_{i},y_{i}\) are the coordinates of the selected nodes at the tank's base, \(P_{vertical,i}\) refers to the applied vertical load and \(K_{eff,i}\) denotes the effective stiffness of each isolator. Figure 7: Optimal placement for: (**a**) SFPB3 and (**b**) TFPB3 isolators, in which black dots denote active bearings, while “zero-isolators” are represented with red dots. The constraints in both optimization schemes are related to the isolated system response, according to the recommendations of [PERSON] [60], CEN [31] and [PERSON] et al. [61]:  
Figure 7.0  
Figure 7.1  
Figure 7.2  
in which \(T\) is the fundamental period of the fixed-base system, \(T_{eff}\) refers to the effective period and \(\beta_{eff}\) denotes the effective damping of the isolation system. Lastly, an additional constraint is imposed on the load capacity of each isolator, as follows:  
Figure 7.3  
where \(P_{critical}\) is the critical load defined by [PERSON] et al. [62]. This constraint is vital to the dual optimization process; as the number of \"zero\" isolators is reduced, the load on the remaining active bearings is increased up to the critical limit.