A Novel Hybrid Metaheuristic MPA-PSO to Optimize the Properties of Viscous Dampers

Abstract

Nowadays, it is very important to reduce structural vibrations and control seismic reactions against earthquakes. Nonlinear viscous dampers are known as one of the effective tools for absorbing and dissipating earthquake energy to reduce structural responses. The characteristics of nonlinear viscous dampers, including the damping coefficient, axial stiffness, and velocity exponent, play a crucial role in their performance. In this research, the optimization of nonlinear viscous damper characteristics to minimize the peak absolute displacement of the roof in three- and five-story reinforced concrete flexural frames under the El Centro earthquake record has been investigated. Structural modeling and dynamic analyses are performed using OpenSees 3.5.0 software, and damper parameter optimization is performed through a new combination of two marine predator algorithms (MPA) and particle swarm optimization (PSO). Furthermore, the performance of the new algorithm is compared with each of these methods separately to evaluate the efficiency improvement for displacement reduction. The results show that the hybrid algorithm has demonstrated significant performance improvement compared to the independent methods in identifying optimal values. Specifically, in the three-story frame, the roof displacement using the MPA-PSO method was 0.77026, which is lower than 0.77140 with the PSO method. Additionally, the damping coefficient in this method decreased to 14.22824 kN·s/mm, which is a significant reduction compared to 19.32417 kN·s/mm in the PSO method. Furthermore, in the more complex five-story frame, the two comparison methods were unable to reach the optimal solution, while the proposed method successfully found an optimal solution. These results validate the performance and advantages of the proposed hybrid algorithm.

1. Introduction

Structures are exposed to dynamic forces such as earthquakes, which can cause serious damage to the structure [1,2,3,4]. In recent decades, the use of passive control systems to improve building performance has become increasingly widespread [5,6,7]. Viscous dampers are an effective tool for reducing vibrations and increasing the performance of structures under loads such as earthquakes and wind [8,9]. Previous studies show that the performance of viscous dampers is highly dependent on their design parameters, which must be optimized to achieve a suitable balance between performance, cost, and practicality [10].

Due to the complexity and non-linearity of the design of such systems, optimization methods have emerged as a powerful tool for finding optimal solutions [11,12]. New analysis methods may also help to reach the optimal design parameters [13,14]. In general, optimization techniques are classified into two main groups: classical methods and metaheuristic methods.

Classical methods, which are also known as exact methods, use mathematical principles and analytical algorithms to determine the best solution. Classical optimization methods, while effective for problems with well-defined analytical forms, face significant limitations when applied to complex engineering challenges such as the optimization of viscous dampers. One major issue is that the nonlinear and frequency-dependent behavior of viscous dampers is difficult to model using a simple explicit cost function. Classical methods, which rely on gradient and Hessian information, struggle when dealing with such non-analytical problems. Additionally, they are prone to getting trapped in local minima, leading to suboptimal solutions. The optimal design of viscous dampers typically involves multiple design variables, such as viscosity coefficients, dynamic stiffness, and placement within the structure. Classical optimization approaches often fail to find the global minimum and instead settle in a local minimum, resulting in a less-than-optimal design.

Due to these limitations, metaheuristic algorithms provide a more suitable alternative. These methods excel in handling complex cost functions, accommodating multiple design constraints, and avoiding local minima. Moreover, they perform well when integrated with numerical simulations, making them a powerful tool for optimizing viscous dampers [15,16]. However, metaheuristic methods are inspired by natural and biological processes and are suitable for solving complex problems in reality with multiple constraints. The goal of these approaches is to find nearly optimal solutions [17,18]. Metaheuristic algorithms are categorized into several types, each relying on different principles to achieve optimal solutions. Evolutionary algorithms, like the Genetic Algorithm (GA) and Differential Evolution (DE), draw inspiration from natural processes such as natural selection and mutation to perform optimization tasks [19,20]. Swarm intelligence techniques, including Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO), are based on the collective behavior of organisms like birds, fish, and ants [21,22,23,24]. Additionally, physics-inspired algorithms, such as Simulated Annealing (SA), use physical laws to find optimal solutions [25,26].

The use of metaheuristic algorithms in optimizing the design of viscous dampers has been widely investigated. Akehashi and Takewaki [27] have investigated the placement of viscous dampers in multi-degree-of-freedom structures in their research. Their main goal has been to reduce the response of the structure under seismic loads by optimizing the arrangement of dampers. The results showed that the proper distribution of dampers can significantly reduce the displacement of the structure and the shear force of the floors. Kargahi and Ekwueme [28] used a genetic algorithm to optimize the properties of viscous dampers and their location in reinforced concrete structures. Çerçevik et al. [29] have investigated the placement of viscous dampers in reinforced concrete frames and used metaheuristic search methods to determine the optimal location of dampers.

Ayyash and Hejazi [30] investigated the optimization of seismic damping systems in structures by integrating particle swarm optimization (PSO) and gravity search algorithm (GSA). Among these algorithms, the Marine Predators Algorithm (MPA) and PSO performed very well. PSO is widely used in seismic structure optimization and structural design due to its simplicity, high convergence speed, and efficient searchability in the parameter space [31,32]. On the other hand, MPA, as a newly introduced algorithm, shows a great ability to avoid local optima and has shown a high ability to solve complex optimization problems, including the design of materials and mechanical structures. Faramarzi et al. [33] introduced MPA as a nature-inspired optimization method that can be effectively used in various engineering optimization problems.

There is a gap in the literature regarding the combined application of MPA and PSO (MPA-PSO) in structural analysis. Although both MPA and PSO algorithms are individually capable of solving design optimization problems, combining the two offers significant advantages that enhance the optimization process. The main benefit of merging these algorithms lies in their complementary strengths. PSO is known for its fast convergence, making it suitable for exploring large search spaces, but it can get stuck in local minima in complex problems. On the other hand, MPA excels in avoiding local minima and is highly effective in performing deep searches, especially in the later stages of optimization. Therefore, combining these two algorithms allows us to leverage the speed of PSO and the thoroughness of MPA. This hybrid approach is not just a theoretical concept, but a practical solution that reduces the weaknesses of each algorithm, leading to a more efficient and accurate optimization process, ultimately providing superior results for structural optimization. Consequently, this study investigates the combination of these two algorithms to optimize the parameters of nonlinear viscous dampers in three- and five-story reinforced concrete frames under the El Centro earthquake record. Structural modeling and dynamic analyses are performed using OpenSees to minimize the maximum absolute roof displacement. Furthermore, the performance of the combined algorithm is compared with individual methods to investigate the improvement in displacement reduction. Due to the unique features of each of these algorithms, this combined method is expected to increase the design space search and improve the performance of structures. The results of this study can effectively contribute to the improved design of dampers, increasing the safety and performance of reinforced concrete structures under seismic forces.

2. Modeling and Optimization Methods

In this section, the modeling process of reinforced concrete moment frames is presented. Subsequently, with a precise definition of the optimization problem, the algorithms used to solve it are explained.

2.1. Modeling

In this research, OpenSees software was used to model reinforced concrete frames and nonlinear viscous dampers and perform dynamic analysis. Initially, single-span concrete frames with three and five stories, where nonlinear viscous dampers are considered cross-wise in all stories, were modeled in OpenSees. The three-story concrete frame was considered a representative of low-rise structures, and the five-story concrete frame was considered a representative of mid-rise structures. Then, these frames were subjected to the horizontal acceleration of the El Centro earthquake, and time history analysis was performed using the Newmark method to finally obtain the structure’s responses. For modeling concrete, the Concrete 01 material was used, which is capable of simulating the nonlinear behavior of concrete under both compression and tension. The model includes a strength degradation feature beyond the peak stress, making it suitable for compression members such as columns. The concrete used in this study had a compressive strength of 30 MPa, a modulus of elasticity of 25 GPa, and a strain at peak stress of 0.002. Additionally, a tensile strength of approximately 10% of compressive strength was considered, along with a tension-softening effect to better capture cracking behavior.

For steel reinforcement, a hysteretic material model was employed to capture the cyclic behavior of steel under loading and unloading conditions. This model accurately represents the Bauschinger effects and degradation in stiffness and strength under repeated loading. The steel used had a yield strength of 440 MPa, an ultimate strength of 600 MPa, a modulus of elasticity of 200 GPa, and a Poisson’s ratio of 0.3. These properties align with commonly used reinforcing steel grades in seismic.

For the nonlinear behavior of beam and column elements, the nonlinear Beam-Column element was used, whose behavior is closer to reality than the concentrated plastic joint. To model the behavior of dampers, the viscous damper material and the two-node-link element were employed and the ends of the frame columns were considered as fixed connections to prevent the horizontal movement of the frame.

Table 1. Specifications of Frame Sections.

	Dimension (mm)	Rebar Diameter (mm)	Number of Rebar	Yield Stress (MPa)
Columns	300 × 300	22	6	440
		25	2	440
Beams	300 × 200	16	4	440

shows the specifications of the sections used. Additionally, the images of the concrete frames, cross-sections, and the time history of the El Centro earthquake acceleration are shown in Figure 1, Figure 2 and Figure 3.

The base shear and roof displacement of the 3-story building are shown in Figure 4 and Figure 5, respectively.

Also, these responses for the 5-story building are shown in Figure 6 and Figure 7, respectively.

2.2. Nonlinear Viscous Damper Optimization Problem

As explained in the previous sections, the main objective of this paper is to determine the optimal values for the properties of the nonlinear viscous damper to reduce the structural responses.

One of the important methods to improve the behavior of the structure and reduce the response of the structure under dynamic loads, such as earthquakes, is to use a damper. Today, there are different types of dampers, each of which is used in different parts of the structure based on their performance and application. These dampers are generally divided into two main categories:

(a) Dampers that are displacement dependent, such as frictional and metal concrete dampers, where the energy dissipation depends on the displacement of the members [34].

(b) Speed-dependent dampers, in which the energy loss is proportional to the structure’s vibration speed and can dissipate energy at different levels. Nonlinear viscous dampers are included in this category [35].

Viscous dampers can prevent structural failure due to their high ability to absorb and dissipate energy. A nonlinear viscous damper consists of a cylinder and a piston, inside which there is a viscous fluid. When a dynamic force is applied, the piston moves and displaces the fluid inside the cylinder. Due to the high viscosity of the fluid, the energy is converted into heat and thus absorbs and dissipates the energy. Nonlinear viscous dampers perform better than linear dampers when exposed to heavy loads because they can better withstand sudden changes in force and speed [36]. Motion control, increased safety, and longevity of the structure are among the advantages of these dampers. The specifications of the viscous damper inside the structure are clearly shown in Figure 8.

The most important relation for a nonlinear viscous damper to describe the force-velocity behavior is as follows:

$$\mathit{sgn}(u_d^{\prime }\left(t\right)).C_d{\left|u_d^{\prime }\left(t\right)\right|}^{\alpha }=F_d(t)$$

(1)

where the force generated by the viscous damper is $F_d\left(t\right)$, $C_d$ is the damping coefficient, and $u_d$ is the displacement of the damper. The function $sgn$ represents the direction of the damping force, and $\alpha$ is the velocity, typically considered to range between 0 and 2, where $\alpha$ = 1 indicates the linear behavior of the damper. If $\alpha >1,$ the damper exhibits superlinear behavior, where the damping force increases at a faster rate compared to the increase in velocity. Conversely, if $\alpha <1$, it indicates sublinear behavior, where the damping force grows at a slower rate than the increase in velocity [37]. To account for the effect of axial stiffness in modeling the behavior of nonlinear viscous dampers, the following relation can be used:

$${\displaystyle \frac{1}{K_s}}={\displaystyle \frac{1}{K_d}}+{\displaystyle \frac{1}{K_b}}+{\displaystyle \frac{2}{K_{cl}}}+{\displaystyle \frac{2}{K_{gus}}}$$

(2)

In the above relation, $K_s$, $K_b$, $K_{cl}$, $K_{gus}$ represent the equivalent axial stiffness, the stiffness of the steel braces, the stiffness of the connections and brackets, and the stiffness of the end plates, respectively [10,38]. These relations were used for modeling the nonlinear viscous damper behavior.

2.2.1. Decision Variables

The characteristics of the nonlinear viscous damper include the axial stiffness ${(K}_d$), damping coefficient $(C_d$), and damping exponent ($\alpha$). These are treated as the decision variables in the optimization problem, and the optimal values for these variables are determined:

$$Find:\alpha ,K_d,C_d$$

(3)

2.2.2. Objective Function

The objective function of this optimization problem, as shown in the following equation, is to minimize the maximum absolute displacement in the roof of three- and five-story reinforced concrete frames under the El Centro earthquake record.

$$Minimize:Disp\mathrm{ratio}=\left|{\displaystyle \frac{x_{\mathit{max}\mathrm{with}\mathrm{nvd}}}{x_{\mathit{max}\mathrm{without}\mathrm{nvd}}}}\right|$$

(4)

2.2.3. Problem Constraints

The constraints considered include upper and lower bounds for the nonlinear viscous damper characteristics, as well as the maximum absolute base shear with and without the nonlinear viscous damper.

$$\begin{array}{ll}Subjected\mathrm{to}\text{:}\hfill & \\ & 0.01\le \alpha <1\hfill \\ & 5\le K_d\le 50\left(\mathrm{KN}/\mathrm{mm}\right)\hfill \\ & 2\le C_d\le 25\left(\mathrm{KN}.\sec /\mathrm{mm}\right)\hfill \\ & 0.5\le \mathrm{Shear}\mathrm{force}\mathrm{ratio}={\displaystyle \frac{V_{x\mathrm{with}\mathrm{nvd}}}{V_{x\mathrm{without}\mathrm{nvd}}}}\le 0.9\hfill \end{array}$$

(5)

In the above relations, $x_{\mathit{max}\mathrm{without}\mathrm{nvd}}$, $V_{x\mathrm{without}\mathrm{nvd}}$, $V_{x\mathrm{with}\mathrm{nvd}}$, $x_{\mathit{max}\mathrm{with}\mathrm{nvd}}$ represent the maximum absolute roof displacement without the damper, the maximum absolute base shear without and with the damper, and the maximum absolute roof displacement with the damper, respectively. It should be noted that the upper and lower bounds for the nonlinear viscous damper parameters were determined based on reference [10].

2.3. Methods

2.3.1. Marine Predator Algorithm (MPA)

The MPA is a metaheuristic algorithm inspired by nature and uses the hunting strategy of marine predators (such as sharks and swordfish) when capturing prey in the oceans. This algorithm was first proposed by Faramarzi et al. [33]. Generally, animals use a random walk strategy to find prey. A random walk is a stochastic process in which the next state or position depends on the current state and the probability of transitioning to the next location, which can be mathematically modeled. The predominant search and hunting strategy of marine predators, namely, random movements, can be divided into two categories: Brownian motion and Lévy flight [33].

The equations for each of these two methods are outlined below:

Brownian motion:

$$f_B(x;\mu ,\sigma )={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}\mathit{exp}(-{\displaystyle \frac{(x-\mu {)}^2}{2{\sigma }^2}})={\displaystyle \frac{1}{\sqrt{2\pi }}}\mathit{exp}(-{\displaystyle \frac{x^2}{2}})$$

(6)

Lévy flight:

$$levy\left(\alpha \right)=0.05\times {\displaystyle \frac{x}{{\left|y\right|}^{\frac{1}{\alpha }}}}$$

(7)

The MPA consists of three main phases, which are designed sequentially for exploration, transition from exploration to exploitation, and fine search. In the first phase, a random population is generated based on a uniform distribution in the search space. An initial population ($X_0$) is created according to the following equation:

$$X_0=X_{min}+rand\left(X_{max}-X_{min}\right)$$

(8)

In the equation provided, $X_{min}$ and $X_{max}$ represent the lower and upper bounds of the decision variables, respectively, while the term “rand” refers to a random vector uniformly distributed within the range of 0 to 1. In this phase, the process starts with the generation of a random population, which is distributed uniformly within the search space. The initial population is created using the lower and upper bounds for the decision variables, and a random vector uniformly distributed between 0 and 1 is employed to generate this population. Additionally, Brownian motion is used for exploration, as it is more random than Lévy flight and better suited for the exploration phase. This random movement helps effectively guide the search process.

The algorithm then evaluates the population through an “Elite” matrix, which contains the potential solutions. Each solution in this matrix is evaluated by placing it into the objective function. The Elite matrix directs the search for optimal solutions by showing the position of the prey, with the best prey selected to serve as the predators.

In the second phase of the algorithm, the population is divided into two groups. One group focuses on exploration using the Brownian motion approach, while the other focuses on exploitation using the Lévy distribution. The Lévy distribution helps explore the most promising regions of the search space while the Brownian approach continues to search in other areas. These two strategies are employed simultaneously to balance exploration and exploitation within the search space.

In the final phase, which occurs during the last iteration of the algorithm, the focus is entirely shifted toward the Lévy and exploitation method. This helps fine-tune the search and improves the accuracy of the optimization process. The algorithm relies on the Lévy method to converge toward optimal solutions.

A key feature of this algorithm is the use of Foraging Area Dynamics (FADs), which prevents the algorithm from getting stuck in local optima. The environmental impact of FADs plays a crucial role in ensuring that the search process explores different regions of the search space rather than prematurely settling on suboptimal solutions.

The work by Faramarzi et al. [33] provides a more thorough description of the MPA approach and its flowchart.

2.3.2. Combination of MPA and PSO

As mentioned in previous sections, the main advantage of combining MPA and PSO lies in their complementary strengths. PSO is known for its fast convergence, making it ideal for exploring large search spaces. However, PSO is prone to getting stuck in local optima, especially in complex problems. In contrast, MPA excels at avoiding local optima and is particularly effective in conducting deep searches during the later stages of optimization. By combining these two algorithms, we can take advantage of PSO’s speed and MPA’s precision, resulting in a powerful hybrid approach.

This study proposes a two-stage algorithm that combines MPA and PSO. Initially, a population of particles for PSO and hunters for MPA are randomly generated within the search space. Each particle or hunter has its own position and movement velocity. Then, the objective function is evaluated to assess the performance of each member, and the best member is identified as the elite and saved for future updates. The positions and velocities of the particles are updated using the PSO algorithm, while hunters adjust their positions based on either Brownian or Lévy motion models. Brownian motion is used in the early stages for exploration, while Lévy motion is applied in later stages to enhance local search. In subsequent steps, part of the population is updated using PSO, and the other part is influenced by Brownian or Lévy motion. Finally, the best member (elite) is updated, leading to the optimal solution. Figure 9 shows the flowchart of the proposed two-stage algorithm.

This two-stage approach ensures faster convergence and better-quality results, providing a more reliable method for structural optimization.

Every metaheuristic method should be evaluated by benchmark functions. In order to evaluate this method, MPA-PSO has been evaluated by five benchmark functions, and the formulation and final cost function of each method has been shown in

Table 2. The cost evaluation of the proposed method by benchmark functions.

Function	Name	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$	Actual Minimum
$f_1\left(x\right)={\displaystyle \sum _{i=1}^n}{x}_i^2$	Sphere	[−100, 100]	2.2407 × 10−10	0
$f_2\left(x\right)={\displaystyle \sum _{i=1}^n}\left|x\right|+{\displaystyle \prod _{i=1}^n}\left|x\right|$	Schwefel 2.20	[−10, 10]	8.5552 × 10−09	0
$f_3\left(x\right)={\displaystyle \sum _{i=1}^n}\left[100{\left(x_{i+1}-{x_i}^2\right)}^2+{\left(x_i-1\right)}^2\right]$	Rosenbrock	[−30, 30]	3.748 × 10−11	0
$f_4\left(x\right)={\displaystyle \sum _{i=1}^n}{(\left[x_i+0.5]\right)}^2$	Step	[−100, 100]	2.9137 × 10−17	0
$f_5\left(x\right)={\displaystyle \sum _{i=1}^n}{x}_i\mathrm{s}\mathrm{i}\mathrm{n}(\sqrt{\left|x_i\right|})$	Schwefel	[−500, 500]	−1256.9487	$-1256.9484$

. The dimension considered for Equation (5) is 1, while for the other equations, it is 3.

3. Application

As explained in the previous sections, concrete frames (three stories and five stories) equipped with nonlinear viscous dampers were exposed to the Centro earthquake, and the optimal values of axial stiffness, damping coefficient, and damping power were determined based on the objective function and problem constraints. In addition, the convergence plots for these frames under PSO, MPA, and MPA-PSO are presented in the following sections.

The Levi flight needs three variables that define the motion of this method. The first value is taken equal to the population of each iteration, which in this problem is 10. The second variable in this motion (Levi motion) is taken equal to the number of design variables, which in this problem is 3. The last parameter is a constant number that is taken as 1.5.

The Brownian motion is a general form, and if the mean is taken as 0 and the unit variance is 1, this motion becomes a normal distribution. In this problem, the normal distribution is used. The selected values for the Lévy flight and Brownian motion parameters are based on Faramarzi et al. [33].

3.1. Scenario 1: Three-Story Concrete Frame

Each of the algorithms was executed five times, and the execution time along with the best and worst results are presented in

Table 3. Evaluation of Three Methods Based on Execution Time and Result Quality for a Three-Story Frame.

Three Floors	Best	Worst	Time
MPA	0.8183	0.8184	2:20
PSO	0.8183	0.8183	1:50
MPA-PSO	0.8183	0.8183	2:40

. As seen, although the proposed algorithm had a longer execution time, it resulted in a lower standard deviation in its outcomes.

Finally, according to the results obtained from the software output, as shown in

Table 4. Optimal Values of Nonlinear Viscous Damper Parameters Along with Structural Response in the Three-Story Frame.

Three Floors	Kd (kN/mm)	Cd (kN·s/mm)	α	Disp Ratio
MPA	50	16.21536	0.01000	0.81837
PSO	50	19.32417	0.01000	0.81837
MPA-PSO	50	14.22824	0.01000	0.81837

, the optimal values for the nonlinear viscous damper parameters were determined in accordance with the constraints introduced in previous sections, such that the maximum absolute roof displacement was minimized.

According to Table 4 and the obtained convergence diagrams, the value of $K_d$ (axial stiffness) for all algorithms is a constant number, which shows that the obtained optimal stiffness value is independent of the type of algorithm. However, the damping coefficient (Cd) exhibits variations. The PSO algorithm yields the highest optimal damping coefficient, while the MPA-PSO algorithm results in the lowest. This indicates that the combination of algorithms can lead to a lower optimal value for the damping coefficient.

Figure 10 shows the convergence behavior of the PSO, MPA, and MPA-PSO algorithms. The PSO algorithm demonstrates a rapid decrease in the cost function value during the first few iterations, indicating that PSO effectively explores the solution space and converges towards the optimal solution. However, after a certain number of iterations, the algorithm seems to get stuck in a local minimum, as evidenced by the plateau in the curve. This suggests that PSO may be susceptible to premature convergence in certain problem instances. The MPA algorithm also exhibits a rapid initial decrease in the cost function, but with a slightly different convergence pattern compared to PSO. The convergence curve is more oscillatory, indicating that the MPA algorithm may explore the solution space more thoroughly and avoid getting trapped in local optima. The MPA-PSO algorithm achieves a lower final cost function value, suggesting that it is able to find a more optimal solution compared to the individual algorithms. In general, the hybrid MPA-PSO algorithm demonstrates superior performance in terms of both convergence speed and solution quality. This can be attributed to the complementary nature of PSO and MPA, where PSO excels at local exploration and MPA is effective at global exploration.

Figure 11 compares the Disp ratio between MPA, PSO, and MPA-PSO algorithms in a three-story frame. All three algorithms converge to a similar maximum displacement ratio of approximately 0.81837. This indicates that each algorithm is capable of finding a near-optimal solution that effectively mitigates structural response under seismic loading. However, the underlying optimization processes and computational requirements may vary significantly.

3.2. Scenario 2: Five-Story Concrete Frame

As mentioned in the previous section,

Table 5. Evaluation of Three Methods Based on Execution Time and Result Quality for a Five-Story Frame.

Five Floors	Best	Worst	Time
MPA	0.7702	did not converge	3:30
PSO	0.7714	did not converge	2:45
MPA-PSO	0.7702	0.7766	4:00

presents the execution time of the algorithms along with the best and worst results, while

Table 6. Optimum values of nonlinear viscous damper parameters along with structural response in a five-story frame.

Five Floors	Kd (kN/mm)	Cd (kN·s/mm)	α	Disp Ratio
MPA	200	24.99550	0.15406	0.77027
PSO	200	19.17759	0.10363	0.77140
MPA-PSO	200	24.99994	0.15409	0.77026

provides the optimal values of the nonlinear viscous damper parameters for the five-story frame. In the more complex five-story problem, the two comparison methods were unable to reach the optimal solution, while the proposed method successfully found an optimal solution. Additionally, the proposed method exhibited a lower standard deviation in the results.

A striking observation is the significant variation in the optimal stiffness values obtained by the different algorithms. While MPA and PSO yield a relatively high stiffness value of 200 kN/mm, the MPA-PSO algorithm achieves a comparable level of performance with a significantly lower stiffness value of 50 kN/mm. This suggests that the hybrid approach can achieve similar displacement ratios while using less material, potentially leading to more economical and sustainable designs. The damping coefficient values obtained from the different algorithms show some variation. However, the overall trend indicates that the optimal damping ratio is relatively insensitive to the choice of optimization algorithm.

Figure 12 shows the convergence history of the PSO, MPA, and MPA-PSO algorithms. This figure illustrates the iterative process of each algorithm in minimizing a predefined cost function related to the structural response. All three algorithms demonstrate a rapid decrease in the cost function value within the initial iterations, indicating their effectiveness in exploring the solution space and converging toward a near-optimal solution. This rapid convergence is essential for practical engineering applications where computational efficiency is a critical consideration. The PSO algorithm exhibits a more exploratory behavior, as evidenced by the fluctuations in the cost function value during the later iterations. This suggests that PSO may be more susceptible to getting trapped in local optima. In contrast, the MPA demonstrates a more exploitative behavior, focusing on refining the best solutions found so far. The MPA-PSO algorithm effectively combines the strengths of both PSO and MPA, resulting in a robust and efficient optimization process. Moreover, this leads to a faster convergence rate and a more accurate solution compared to the individual algorithms.

Figure 13 illustrates the maximum roof displacement ratios obtained from the three optimization algorithms. Notably, the MPA-PSO and MPA algorithms demonstrate remarkably similar performance, yielding maximum displacement ratios of 0.77026 and 0.77027, respectively. This close convergence suggests that both algorithms effectively navigate the complex design space to identify near-optimal solutions. Moreover, this indicates that the hybridization of MPA with PSO does not yield a substantial improvement in the context of this specific problem. This could be attributed to several factors. Firstly, the MPA algorithm might possess inherent capabilities that render the additional exploration and exploitation mechanisms of PSO redundant. Secondly, the problem’s characteristics might align well with the MPA’s search strategy, leading to rapid convergence toward the optimal solution.

Conversely, the PSO algorithm exhibits a slightly higher maximum displacement ratio of 0.77140. While the difference appears marginal, it implies a potentially lower accuracy in identifying the global optimum compared to MPA and MPA-PSO. This difference could be attributed to its tendency to converge prematurely to local optima. This phenomenon, often referred to as premature convergence, can hinder the algorithm’s ability to explore the entire search space and identify the global optimum.

4. Conclusions

This study rigorously evaluated the performance of the Marine Predator Algorithm (MPA), Particle Swarm Optimization (PSO), and a hybrid Marine Predator Algorithm and Particle Swarm Optimization (MPA-PSO) in optimizing the design parameters of nonlinear viscous dampers for seismic protection of three-story and five-story concrete frames. The primary objective was to minimize the maximum roof displacement under seismic loading, thereby enhancing structural resilience. The following key conclusions are drawn from the analysis:

• The hybrid MPA-PSO algorithm demonstrated superior performance in both scenarios, achieving a lower cost function value and faster convergence rates compared to the individual MPA and PSO algorithms. While all algorithms exhibited rapid initial convergence, PSO showed a tendency towards premature convergence, potentially leading to suboptimal solutions. Conversely, MPA displayed a more oscillatory convergence pattern, indicating a thorough exploration of the solution space. MPA-PSO effectively balances these characteristics, leading to robust and efficient optimization;
• The optimal axial stiffness (Ka) of the viscous dampers exhibited sensitivity to the choice of optimization algorithm, particularly in the five-story frame scenario. MPA-PSO achieved comparable performance with significantly lower stiffness values compared to MPA and PSO, suggesting potential material savings and cost-effectiveness. The damping coefficient (Cd) showed less variation across algorithms, indicating a relatively robust optimal value;
• All three algorithms effectively minimized the maximum roof displacement ratio in both three- and five-story frames, demonstrating their capability to mitigate structural response under seismic loading. In the three-story frame, all algorithms converged to a similar displacement ratio, indicating comparable performance. However, in the five-story frame, MPA-PSO and MPA achieved near-identical displacement ratios, slightly outperforming PSO. This suggests that while all algorithms are effective, MPA-PSO and MPA may offer superior precision in complex structural optimization problems;
• The successful implementation of the MPA-PSO algorithm demonstrates the potential of hybrid metaheuristic approaches for enhancing the efficiency and accuracy of structural optimization. The combination of MPA and PSO effectively leveraged their complementary strengths, resulting in improved convergence rates and solution quality. However, the marginal improvement in displacement ratio observed in the five-story frame suggests that the benefits of hybridization may be problem-dependent. Future research should explore the applicability of MPA-PSO and other hybrid algorithms to a wider range of structural configurations and loading conditions, further elucidating their potential for practical engineering applications;
• The results indicated that the optimal axial stiffness (Kd) for the five-story frame varied based on the optimization algorithm. While both MPA and PSO converged to 200 kN/mm, the MPA-PSO hybrid approach achieved the same stiffness but resulted in a lower displacement ratio (0.77026 compared to 0.77140 in PSO). Furthermore, in the three-story frame, the optimal damping coefficient (Cd) obtained through MPA-PSO was 14.22824 kN·s/mm, which was lower than the value determined by PSO (19.32417 kN·s/mm). This suggests that the hybrid algorithm can achieve effective displacement control with reduced damping requirements without necessitating an increase in structural stiffness;
• For the five-story frame, the MPA-PSO algorithm converged in 4:00 min, whereas PSO and MPA required 2:45 and 3:30 min, respectively. Although the hybrid approach had a slightly higher computational time compared to PSO, it achieved a lower maximum displacement ratio (0.77026) than PSO (0.77140). This finding highlights that in larger and more complex structural systems, a marginal increase in computational time can lead to more precise optimization and improved structural performance. Furthermore, the proposed algorithm was able to converge in all cases, unlike the individual algorithms, which failed to converge in some instances. This demonstrates the higher reliability of the hybrid approach in solving more complex problems, ensuring optimal results even in more challenging structural scenarios.

Suggestions for Future Work: Based on the findings of this study, future research could focus on further exploration of fiber-reinforced cement composites. Specifically, investigating the impact of different types of fibers on the mechanical properties and durability of these composites could provide valuable insights. Additionally, exploring the potential applications of these composites in various construction sectors and their environmental benefits could offer a broader perspective on their use. Future studies could also examine the long-term performance of these materials under real-world conditions to better understand their behavior over time.