Passive structural control has a long and successful history in mechanical and aerospace engineering with the support of applied mechanics. This rather new research field has been advanced together with innovative development of computer science and new material science in the latter half of the 20th century. However, in the field of civil engineering, it has a different background. Building and civil structures are often subjected to severe earthquake ground motions and wind disturbances with large uncertainties (Soong and Dargush, 1997; Soong and Constantinou, 2002; Christopoulos and Filiatrault, 2006; Takewaki, 2009; Lagaros et al., 2012). It is therefore required to take into account these uncertainties in a robust and reliable manner in the theory of structural control. Its application to actual structures is also important.

Professor Soong presented five important areas impacted by structural control in his seminal keynote lecture (Soong, 1998) in the second World Conference on Structural Control held in Kyoto, i.e., (a) systems approach, (b) deepening effect, (c) broadening effect, (d) experimental research and (e) creative engineering. Among these five areas, the broadening effect includes the effective use of passive dampers in building structures.

The early-stage development in the optimization of passive dampers was explained by Takewaki in his monograph (Takewaki, 2009). Then the general reviews including research works after his monograph were provided by Puthanpurayil et al. (2013), Whittle et al. (2013), De Domenico et al. (2019), Kookalani et al. (2021) and Takewaki and Akehashi. (2021), Zhu et al. (2022), El Ouni et al. (2022).

Figure 1 shows the role of dampers in the building structural design under load and model uncertainties. It is well known that energy dissipation by viscous, viscoelastic and hysteretic dampers can be performed stably compared to tuned-mass dampers under load and model uncertainties. Theoretical basis of effectiveness brought by passive control and energy dissipation via dampers can be provided from the classical viewpoint of earthquake input energy. If the classical earthquake input energy criterion, i.e., constant input energy (Housner, 1959; Takewaki, 2004; 2006; 2009; Takewaki and Fujita, 2009), holds regardless of the existence of supplemental dampers and the supplemental passive dampers can absorb the earthquake input energy as much as possible, the input energy to the frame can be reduced effectively.

The original point of this review is the classification of optimization methods based on the combination of several component algorithms or approaches. For this reason, it is not intended to cite as many papers as possible in this review. Mixed approaches considering input uncertainties are particularly highlighted.

There are several kinds of dampers, i.e., oil dampers (fluid viscous), viscoelastic dampers, hysteretic dampers, friction dampers, mass dampers, inertial dampers (see Figure 2). Because inertial dampers have been developed rather recently and possess a peculiar characteristic, the mechanism of those dampers is shown in Figure 2 in detail. In some cases, these dampers are used simultaneously in a building structure and system.

In discussing their reliability, the dependence on temperature, the manufacturing precision issue, the aging issue, the resistance for high-speed loading, etc., should be remarked. Since many advanced systems are developed recently in each damper system, it seems difficult to classify these damper systems in terms of their reliability degree.

In view of economic implications, oil dampers, viscoelastic dampers, hysteretic dampers, friction dampers have almost the same cost performance to the identical damping performance approximately when used in building structures. Therefore, the selection of dampers is usually made from the viewpoint of functionalities, i.e., the reduction of deformation or acceleration.

Since the original point of this review is to classify the methods of damper optimization from the viewpoint of mixed approaches of several component algorithms, the component algorithms are explained in the beginning. Figure 3 shows schematic diagrams of six component algorithms, i.e., (a) Evolutionary, (b) Heuristic, (c) Meta-heuristic, (d) Optimality criteria, (e) Sensitivity-based, (f) Input uncertainty.

There exist many review articles on passive dampers after the year of 2000. In contrast to the previous ones, the original point of the present review is the introduction of a new perspective that most passive damper algorithms can be classified based on the combination of several approaches with different objectives, i.e., evolutionary, heuristic, meta-heuristic, analytical, sensitivity-based. Figure 4 shows the classification of optimization methods based on combination of different approaches.

Fujita et al. (2021) presented a new robust optimal damper placement method for nonlinear oil dampers with uncertainty using the critical double impulse. They proposed a new index for the robust design and used a sequential quadratic programming for optimization. GA was used for demonstrating the validity of the proposed method.

Akehashi and Takewaki (2019) presented a new method of the optimal viscous damper placement for elastic-plastic MDOF structures under the critical double impulse using a gradient-based technique. Since elastic-plastic MDOF structures exhibit complicated responses, this kind of design problems may be complex. Furthermore, the uncertainty in the selection of random earthquake ground motions was tackled by introducing the concept of the worst-case analysis in which the resonant double impulse was regarded as the worst-case input for resonant response.

Kondo and Takewaki (2019) proposed a method for optimal damper placement considering the critical fault rupture slip distribution. They used the stochastic Green’s function method based on a plane-source model of the fault rupture to produce ground motions (Makita et al., 2018).

Akehashi and Takewaki (2020b) developed an algorithm of simultaneous optimization of elastic-plastic building structures and viscous dampers under the critical double impulse. Since the behaviors of elastic-plastic MDOF building structures are sensitive to the building frame member design and the great variation of damper distribution is allowed, the simultaneous optimization of elastic-plastic building structures and viscous dampers is quite tough. They overcame the difficulty by introducing an efficient sensitivity-based design algorithm. It was made clear that the order of changes of structural stiffness and damper damping coefficient is critical to the achievement of reasonable designs.

Cetin et al. (2019) developed a method of optimal design and optimum distribution of viscous dampers for a shear building under the critical excitation by using the random vibration theory in the frequency domain.

Akehashi and Takewaki (2022d) proposed a new optimal damper placement method for a group of recorded ground motions by using the critical double impulse input. They introduced the earthquake input energy through the lowest eigemode to structures with dampers as a criterion to normalize the amplitude of recorded ground motions. It was demonstrated that the response to the critical double impulse can bound the maximum response to the group of recorded ground motions after normalization.

Hosoda and Fujita (2024) presented a robust optimal placement method based on a robustness index by using a sensitivity-based evolutionary algorithm. They used the critical double impulse and the corresponding resonant one-cycle sine wave as the uncertain input motion. They considered variation of elastoplastic design criteria and input level.

In some research works, the comparison of optimal damper placement methods was made. The following papers are those research works. Such comparison enables the clarification of the characteristics of proposed methods.

Lopez Garcia (2001) presented a comparison of the optimal dampers among the Takewaki’s approach (1997a) based on minimum transfer function, SSSA (Lopez Garcia, 2001) and the uniform distribution as shown in Figure 8. He used four recorded ground motions. He concluded that the efficiency of the damper configurations given by the SSSA is similar to the efficiency of the damper configuration given by the optimal placement for minimum transfer function. It should be noted that the story stiffness distribution used by Lopez Garcia may be the one with the straight lowest eigenmode in which the effect of damper placement seems small (see Figure 5). If other stiffness distributions are employed, e.g., the uniform stiffness distribution, the difference among these damper placements will become remarkable.

Trombetti and Silvestri (2004) showed a comparison among the MP (mass-proportional)-damping scheme, the SP (stiffness-proportional)-damping scheme and the Takewaki’s method (1997a). The models of MP and SP are shown in Figure 9. Figure 10 shows the comparison of the maximum structural displacements for three base acceleration inputs, i.e., Imperial Valley, 1940, El Centro record, NS component, Kern County, 1952, Taft Lincoln School record, EW, and Kobe, 1995, Kobe University record, NS component. Although the MP-damping scheme exhibits a remarkably good performance, its applicability to actual building structures may need more effort. In other aspect, it can be seen that the Takewaki’s method exhibits a preferable result against the SP-damping scheme.

Figure 11 shows the average percentage deflection reductions for six different damper placements in six damping systems in a 18-story structure and Figure 12 indicates the average percentage deflection reductions for six damping systems in six different damper placements in a 18-story structure (Marko et al., 2006). It can be observed from Figure 11 that the damper locations do not affect so much in some damping systems.

Figure 13 presents the damping distributions by five approaches for a regular building and an irregular building (Whittle et al., 2012; 2013). Figure 14A illustrates the median of peak interstory drifts of the regular building under design basis earthquakes and the maximum considered earthquake (Whittle et al., 2012; 2013). On the other hand, Figure 14B indicates the median of peak interstory drifts of the irregular building under design basis earthquakes and the maximum considered earthquake (Whittle et al., 2012; 2013). It can be observed that the excessive interstory drifts in upper stories of the irregular building are controlled effectively by the passive dampers.

Figure 15 shows the comparative performance histograms of design methodologies of fluid viscous dampers (Uniform distribution, SSPD, SSSE, SSSEES, SPD, FEI/Filtered Energy Index) for three input motions, i.e., a) firm soil; b) soft soil; c) white-noise seismic input (De Domenico et al., 2019).

Akehashi and Takewaki (2020a) compared the optimization results between the transfer function method and the double impulse method.

Del Gobbo et al. (2020) compared six damper placement methods by paying attention to the condition that the total repair cost is limited. They demonstrated that the iterative methods can provide a greater total damping coefficient to the structures than the simple methods. This leads to a higher supplemental damping ratio and lower repair costs. They also showed that, if upfront funds are limited, or if architectural constraints prevent the placement of dampers in lower stories, then iterative methods provide the most favorable total-building seismic performance.

Recently, machine learning is used extensively in the damper optimization following the advancement of computer technologies and neural-network sophisticated development. The following is some examples of recent development.

Li and Zhao (2019) developed machine learning methods using Support Vector Machine and Multilayer Perceptron to analyze the optimal damper distribution. For different building structures, a genetic algorithm-based optimization method was applied to determine optimal damper distributions that are used as training samples. The structural features, the objective function, the number of dampers, etc., were input features, and the distribution of dampers was taken as an output result.

Fang et al. (2022) proposed a general optimization framework for the optimal design of multi-parameter hybrid braced structures. The framework consists of selecting optimization parameters, establishing sample database, formulating machine learning strategy, conducting automatic design and cost calculation, and optimizing the multi-objective problem through the genetic algorithm.

Bekdaş and Nigdeli (2022) edited a book which summarizes the latest developments in the optimization of tuned mass dampers covering all classical approaches and new trends including metaheuristic algorithms. Artificial intelligence and machine learning methods were also included to reach optimum results by skipping long optimization processes.

Yang et al. (2023) proposed a method including two machine learning-based algorithms to optimize the natural frequency and damping ratio of a tuned-mass-damper (TMD) device for seismically excited building. The first algorithm represents the single-objective optimization of TMD parameters in a linear building structure based on physics-enhanced generative and adversarial network architecture. The second physics-enhanced machine learning algorithm was used for the multi-objective optimization of a TMD parameters in a nonlinear building structure under various seismic excitations.

Fu et al. (2024) proposed a machine learning-based hybrid optimization method where tuned mass dampers were optimized considering soil-structure interaction. A total of 200,000 time history analyses of the tuned mass damper-structure-soil system were conducted and the results were used to construct the database of six machine learning models. Three meta-heuristic algorithms were used for optimization.

Although there are various advantages and shortcomings in the combined optimization methods, the principal ones are as follows.

The combined methods can broaden the degree of design spaces. For example, the combination of global-local search algorithms can reach a better optimal design with efficiency. On the other hand, too complicated algorithms may disturb the usage of the algorithms for practitioners who prefer simple ones. However, it is also true that the combination of some algorithms is inevitable to construct an optimization method which is robust for external disturbances.

As for other characteristics of various damper optimization algorithms, the readers can find something else in the review by Kookalani et al. (2021).

The optimization algorithms of passive dampers were classified in view of the types of mixture of algorithms. Six component algorithms were identified, i.e., (a) Evolutionary, (b) Heuristic, (c) Meta-heuristic, (d) Optimality criteria, (e) Sensitivity-based, (f) Input uncertainty. Then, the mixture of these component algorithms was discussed.

The first one is the sequential search algorithm (SSA) and its derivatives (evolutionary, heuristic). The second one is the optimality criteria method including the incremental inverse problem approach (OCIIP) or redesign approach (OCR) (analytical, sensitivity-based, evolutionary). The third one is the global-local hybrid search method (GLHS) (meta-heuristic, sensitivity-based, evolutionary). The fourth one is the steepest direction search method (SDS) and its derivatives such as mathematical programming (evolutionary, sensitivity-based). Except these four categories, the approach considering input uncertainty was discussed.

Finally, several researches focused on comparison among different optimization methods were discussed.

It is desired that the present review in terms of mixture types of algorithms will lead to a finding of new innovative approaches to more reliable and robust methods of damper optimization.