## 6 Single and Dual Optimization Formulations  
Generally, depending on its main aim, structural optimization can be classified into three main types [53]: sizing, shape and topology or layout optimization. There are numerous engineering applications based on single optimization formulations, while there are also dual approaches that combine the main types. For instance, a combined optimization approach was implemented by [PERSON] et al. [54] to simultaneously optimize the layout and members' size of several benchmark truss structures. As mentioned in the introduction, the literature review revealed that there is no relevant study that has focused on the combined optimization of both the key parameters (i.e., sizing optimization) and the layout (i.e., optimal placement) of the isolators installed at the base of buildings or tanks. For this reason, single and dual optimization formulations for base-isolated tanks are presented herein, which can be considered a significant novelty compared to the authors' recent study on sizing optimization of friction bearings [16]. In the proposed single and combined optimization formulations, the CS optimizer is used for the minimization of the eccentricity (i.e., the distance between the centers of mass and rigidity of the isolation system) of liquid storage tanks isolated either by SFPB or TFPB. In this manner, the dynamic response of the system is not influenced by the higher modes and the torsional distress is minimized. In order to assess the effectiveness of the optimized isolation schemes, a comparison is performed with a reference design with standard parameters for the isolators, referred to as SFPB1 and TFPB1, which are derived using the equivalent linear force (ELF) procedure shown in Figure 6.  
ELF is a process often applied for the preliminary design of base-isolation systems. More specifically, it is a repetitive procedure in which initial isolator parameters are selected. After the calculations, the isolator capacity should be equal to the target displacement. Otherwise, different design parameters are selected, and the process is repeated. This is schematically illustrated in Figure 6, where \(\mu_{i}\) denotes the friction coefficient, \(R_{i}\) is the radius of curvature, \(d_{cd}\) is the target displacement, \(K_{\mathit{eff}}\) and \(T_{\mathit{eff}}\) denote isolators effective stiffness and period, \(S_{a}\) is the spectral acceleration and \(d_{cd-new}\) is the current displacement capacity. For the examined tank, the following conditions are assumed, according to Eurocode 8: Soil type A, importance factor, \(\gamma_{i}=1.4\), and bedrock acceleration, \(a_{g}=0.36\) g.  
The first optimization approach, namely SFPB2 and TFPB2, focuses on the sizing optimization of the main parameters (i.e., radius of curvature and friction coefficients) of the two types of friction-based devices. In addition, combined optimization formulations, denoted as SFPB3 and TFPB3, have been developed. These approaches achieve the optimal placement of the minimum number of required bearings in conjunction with the sizing optimization of the friction coefficient and radius of curvature of sliding surface(s). More specifically, in each optimization cycle of the dual optimization formulations, CS randomly assigns isolators with zero parameters to assess if their placement is fully necessary to best serve the problem needs, according to the objective and constraints functions which are presented in the sequence. This process is repeated in every iteration and results in the minimum possible number of isolators for the specific prearranged layout, as shown in Figure 7.  
It should be noted that both single and dual optimization approaches result in isolators with different parameters (i.e., \(\mu_{i}\), \(R_{i}\)). When friction coefficients are not the same for all friction devices, the center of superstructure mass may not coincide with the isolator center of rigidity [55]. In general, variations of isolator properties can cause accidental torsion of the superstructure.