## 1 Introduction  
The devastating Mw7.9-magnitude earthquake in Wenchuan county, Sichuan Province in China on 12 May 2008 [1, 2], which killed more than 87,000 people, was one of the most loss-making earthquakes in the 21 st century. This event led to modifications of the design earthquake intensities in about 60 regions in the Sichuan Province in the Chinese seismic design code. Such revisions of seismic design codes often change the design earthquake intensity in a region from low to high seismic hazards. On the other hand, some types of public buildings (e.g., hospitals, schools) are required to use a higher design earthquake intensity than the code-specific intensity [3]. The reason for such changes may be due to several aspects; e.g., the new observations from recent earthquakes, technical improvements of hazard analysis, data accumulation, and societal and economic increases. One of the issues is that a large number of existing buildings have insufficient resistant capacities to satisfy the increased design earthquake intensity. In terms of economic benefit and environmental sustainability, the best choice is to retrofit existing structures rather than demolish them and rebuild new ones. Among many retrofitting methods [4; 5; 6; 7; 8; 9; 10; 11], the placement of viscous dampers may be one of the easy ways to improve the seismic performance of structures, because it only induces limited downtime. To date, many studies have focused on the retrofitting of structures with viscous dampers. [PERSON] et al. [12] showed that the setting of the viscous damper coefficient of the story should be according to the story shear, and placements of dampers at upper stories may be not effective. [PERSON] and [PERSON] [13] studied the effectiveness of using viscous dampers for the retrofit of a frame building before the Northridge earthquake. [PERSON] and [PERSON] [14] presented a parametric analysis to compare the performances offered by various systems of added viscous dampers in shear-type structures. [PERSON] and [PERSON] [15] studied the seismic performance of buildings retrofitted with adjacent reaction towers and viscous dampers. [PERSON] [16] studied frame structures with viscous dampers and highlighted that plastic hinges may occur in columns under earthquakes. Some studies focused on the design methods of retrofitting. [PERSON] et al. [17] proposed a design procedure for viscous dampers based on the capacity spectrum method and verified by a 10-story and a 20-story frame structure. [PERSON] et al. [18] presented a displacement-based seismic design method for regular buildings with viscous dampers. [PERSON] [19] proposed a multi-mode design method based on energy for the seismic retrofitting of structures with passive energy dissipation systems. [PERSON] et al. [20] proposed a \"direct five-step procedure\" for the seismic design of structures with viscous dampers. [PERSON] et al. [21] proposed a retrofitting method for frame structures with viscous dampers in order to make the structure appropriate for high-intensity earthquake environments. Although the addition of viscous dampers may induce larger residual displacements compared to other retrofitting methods (e.g., isolation, brace, etc.) in a few cases [22], the above studies demonstrate the effectiveness of structural retrofitting using dampers. The optimum design of viscous dampers is a key issue in the retrofitting procedure based on viscous dampers. [PERSON] and [PERSON] [23] presented a sequential searching procedure for the optimal placement of viscous dampers in structures. [PERSON] and [PERSON] [24] used a gradient-based method to optimally place viscous dampers in structures. In the study of [PERSON] [25], the optimal damping placement utilizing a generalized objective function is presented by accounting for displacements, absolute accelerations and base shear. [PERSON] and [PERSON] [26] presented a computational framework for the optimal placement of dampers in frame structures.