## 2 Motion-Based Design of Structures under Uncertainty Conditions  
### Motion-Based Design of Structures under Deterministic Conditions  
Structural optimization is a computational tool which can be used to assist engineering practitioners in the design of current structural systems [37]. Thus, this computational tool allows the optimum size, shape or topology of the structure to be found which meet the design requirements established by the designer/manufacturer/owner. Among the different structural optimization methods, the performance-based design method has been widely employed to design passive damping devices for civil engineering structures [23; 30]. When the design requirements are defined in terms of the vibration serviceability limit state of the structure, the performance-based design method is denominated the motion-based design method [29]. This general design method was adapted herein for the design of passive damping devices when they are used to control the dynamic response of civil engineering structures. As assumption, all the variables, involved in this problem, are deterministic. Thus, the motion-based design method under deterministic conditions transforms the design problem into a constrained multi-objective optimization problem. Therefore, the main objective of this problem is to find the optimum value of the characteristic parameters of the passive damping devices which guarantee an adequate serviceability structural behavior. For this purpose, a multi- objective function is minimized. The multi-objective function, \(\mathsf{f}(\mathsf{\theta})\), is defined in terms of the characteristic parameters, \(\mathsf{\theta}\), of the considered passive damping devices. Additionally, the space domain is constrained including two restrictions in the optimization problem: (i) an inequality constraint, \(\mathsf{g}_{\text{d}d}(\mathsf{\theta})\); and (ii) a search domain. \([\mathsf{\theta}_{\text{min}},\ \mathsf{\theta}_{\text{max}}]\). As the relation between the objective function and the design variables is nonlinear, global optimization algorithms are normally considered to solve this constrained multi-objective optimization problem. [38]. Accordingly, the motion-based design problem under deterministic conditions can be formulated as follows:  
\[ \begin{array}{c}\text{Find }\theta\text{ to Minimize }\text{f}(\theta)\\ \text{Subjected to }\left\{\begin{array}{c}\text{g}_{det}(\theta)\leq 0\\ \text{\theta}_{\min}<\theta<\theta_{\max}\end{array}\right.\end{array}  \tag{1}\]  
where \(\theta\) is the vector of the design variables; \(\text{f}(\theta)\) is the multi-objective function to be minimized; \(\theta_{\min}\) and \(\theta_{\max}\) are the lower and upper bounds of the search domain; and \(\text{g}_{det}(\theta)\) is a function which defines the inequality constraint. Therefore, the key aspect of this optimization problem is the definition of the inequality constraint. In the case of slender civil engineering structures, whose design is conditioned by their dynamic response [29], the compliance of the vibration serviceability limit state can be considered for this purpose. According to the most advanced design guidelines [6; 34], the vibration serviceability limit state of a structure is met if the movement of the structure, \(d_{s}(\theta)\), which can be characterized by its displacement, velocity or acceleration, is lower than an allowable value, \(d_{lim}\), defined in terms of the considered comfort requirements. Thus, the inequality constraint of the abovementioned optimization problem may be expressed as follows:  
\[\text{g}_{det}(\theta)=\frac{d_{s}(\theta)}{d_{lim}}-1\leq 0 \tag{2}\]  
Finally, as the result of this multi-objective optimization process, a set of possible solutions is obtained. This set of possible solutions is denominated the Pareto front. Accordingly, a subsequent decision-making problem must be solved, the selection of the best solution among the different elements of this Pareto front.