Abstract
In this study, a novel hybrid metaheuristic method combining thermal exchange optimization and particle swarm optimization has been introduced. The suggested method has been applied to a base isolation optimization problem. The optimization of base-isolated systems is to find the optimum design of lead–rubber which minimizes the acceleration of the roof. Unlike previous studies, which aimed to reduce the structural acceleration response at a specific hazard level (e.g., under earthquakes with return periods of 475 years and 2,475 years), this study considers a continuous hazard level. The endurance time method is employed to represent this continuous hazard level. In this method, the structure is subjected to a series of progressively intensifying acceleration functions and the structural response at each time instant is interpreted as the response corresponding to a specific intensity level. The cost function of the optimization problem is to find the minimum area under the curve of roof acceleration versus time generated by the endurance time function. The constraints come from both structural factors, like inter-story drift, and geometric limitations. The proposed method, comprising the optimization algorithm and objective function, was applied to two four-, and eight-story structures. The optimization results show that compared with the non-isolated case, the base isolation system reduces roof acceleration by 66% and 38% (for the four- and eight-story frames, respectively), whereas inter-story drift displacement is reduced by approximately 38% and 33% (compared with the fixed-base and manually designed cases), demonstrating effective seismic shock absorption across a continuous hazard range.
1 Introduction
Base isolation is one of the most effective passive control techniques for protecting structures against earthquake-induced ground motions (Sadeghi Movahhed et al., 2023; ). By decoupling the superstructure from the ground through flexible bearings or sliding interfaces, base-isolated buildings experience significantly reduced accelerations and inter-story drifts, thereby limiting both structural and non-structural damage (Sadeghi-Movahhed et al., 2025). In recent years, this technique has been widely implemented in critical facilities such as hospitals, bridges, and heritage buildings, where maintaining functionality after an earthquake is essential.
However, designing an effective base isolation system is a complex multi-parameter problem. The seismic behavior of isolated systems can be viewed as a combined response of the isolation bearing and the superstructure; failure one of either component can lead to failure of the entire system (Sadeghi-Movahhed et al., 2024b; ; ; ). Some failure modes of isolation bearings include strength deterioration, axial buckling, cavitation, and rupture of the lead core. Moreover, interaction effects such as coupling between shear deformation and end rotation of elastomeric bearings () and potential internal pounding between structural components in isolated buildings () further complicate the seismic response of base-isolated systems. Numerous studies have been conducted on base isolation systems. Several studies on lead–rubber bearings (LRBs) have been conducted such as those of , , Vitiello et al. (2017), and Zhang and Shu (2018). All of these studies have used Bouc–Wen-based models, which cannot model degradation. Another type of bearing used in base isolation systems is the friction pendulum bearings (FPBs). The FPBs can be modeled using friction-based models. Some studies that used friction-based models are those of Zhang and Shu (2018), Xu et al. (2021), , and Sadeghi-Movahhed et al. (2024a). These models do not account for failure modes of isolation bearings, such as strength deterioration, axial buckling, cavitation, and rupture of the lead.
Common design procedures usually rely on prescriptive rules or trial and error and may not necessarily achieve the most efficient or economical solution for a given project. To address this challenge, engineers and researchers have increasingly turned to formal optimization methods to make their works more reliable and more efficient (). In general, optimization problems are problems in which one value should be minimized or maximized under some conditions. The value that should be minimized or maximized is called the cost function or objective function. Factors that influence the cost function are called design variables. The conditions that must be satisfied are called constraints in optimization problems. To handle the constraints, the penalty function method can be used.
To solve optimization problems, there are two procedures. The first procedure is classical optimization methods, and the second one is metaheuristics. Although the classical methods need information on the formula of the problem, such as the Hessian matrix or other information that cannot be obtained in real problems, the metaheuristics rely on a population in each iteration. Additionally, classical methods can easily get stuck in local minima, whereas metaheuristics can escape from local minima. For example, in the study of Tang et al. (2025), we can see that the metaheuristic algorithm outperforms classical methods.
Metaheuristic algorithms are inspired by natural processes. Most of the natural processes would reach the optimal design without solving complex mathematical problems. The natural processes are too hard to solve with mathematical formulas, but these algorithms suggest a way to solve the natural processes to achieve the best scores. The emergence of these algorithms is due to the need to overcome the weaknesses of classical methods, such as getting stuck in local minima and requiring some information about the cost function. The first known metaheuristic algorithm is the genetic algorithm (GA) developed by Rechenberg (1965). This algorithm mimics natural selection and other natural processes like mutation and crossover.
Metaheuristic algorithms are generally divided into three groups: 1. evolutionary algorithms (EAs), 2. swarm intelligence (SI), and 3. physics-based algorithms.
EAs: These algorithms are inspired by natural evolution and use techniques like mutation and crossover. Numerous EAs have been developed and used in solving several real problems. GA, which is the considered as the first metaheuristic algorithm, is known as an EA. Differential evolution (DE) (Qin et al., 2008) is another EA that combines multiple solution vectors, which, therefore, can investigate the search space in a more efficient manner. Tabu search (), using memory to guide the search process represents another EA. Simulated annealing (SA), forest optimization algorithm (FOA) (), and bird mating optimization (BMO) () are other famous EAs. One recent algorithm is a work that is mimics how coronavirus spreads and infects other healthy people ().
SI: This group of algorithms is based on the collective behavior of flocks of birds or insects. The flocks will share their information with other populations, and the populations will communicate with each other. The most famous algorithm from this group is the particle swarm optimization (PSO) (); by modeling how particles move and interact within the search space, it seeks optimal solutions. The marine predator algorithm (MPA) () is another SI algorithm that mimics the foraging and hunting behaviors of marine predators to solve optimization problems. Some other algorithms that could be mentioned here are Cat Swarm Optimization proposed by , the dragonfly algorithm by , and Harris hawks optimization (HHO) by .
Physics-based algorithms form an important category of metaheuristic methods, as they are inspired by physical laws and natural phenomena. Among them, the thermal exchange optimization (TEO) algorithm (), described in Section 5, is based on the thermal exchange processes between objects. Another example is galactic swarm optimization (GSO) (), which models the gravitational interactions governing the motion of stars and galaxies. Ray optimization () is also included in this group, drawing its mechanism from Snell’s law of light refraction when transitioning between media with different densities. Additional physics-motivated examples include yin–yang pair optimization (YYPO) (Punnathanam and Kotecha, 2016) and lightning attachment procedure optimization (LAPO) ().
Although the majority of metaheuristic algorithms can be placed within well-defined categories such as physics-based, swarm-based, or evolutionary methods, some algorithms fall outside these conventional groups. For instance, battle royale optimization (BRO) (Rahkar Farshi, 2021) inspired by the shrinking zone mechanics of battle royale video games, does not align with any of the aforementioned categories. Another example is the work of , which is based on the improvisation process of music players rather than natural, biological, or physical phenomena.
Metaheuristic algorithms benefit from two fundamental mechanisms: exploration and exploitation. Exploration refers to searching new regions of the solution space with less dependence on the solutions obtained in previous iterations. In contrast, exploitation focuses on intensifying the search around promising solutions found in earlier iterations. At the first iterations, the exploration is high and exploitation is low (because, at the start of the algorithm, there are not many solutions to benefit from), but as the algorithm moves on, the exploration becomes less and exploitation becomes greater. The main reason behind the development of many metaheuristic algorithms is that each metaheuristic has good potential to solve certain problems and one algorithm cannot solve all problems.
Metaheuristics have been used to solve real problems. For example, the PSO algorithm has been used for producing endurance time (ET) functions (). The imperialist competitive algorithm (ICA) has been utilized for simulating ET functions. Zamani et al. (2025) implemented several metaheuristic algorithms to optimize the design of double tuned mass dampers. The optimum parameters of passive tuned mass dampers have been calculated using three different algorithms in .
By combining the strengths of different metaheuristic algorithms, hybrid metaheuristics can achieve improved optimization performance. For example, a new HHO and PSO has been integrated together to find optimum solution for castellated beams () or has utilized the HHO–PSO algorithm to identify damage in truss structures. The optimum properties of viscous dampers have been calculated using the MPA–PSO algorithm (Shemshaki et al., 2025).
Additionally, these algorithms can be beneficial for obtaining optimum parameters of isolators in base isolation systems. Pal et al. (2019) calculated optimum parameters of base isolation systems using GA. The criteria for base isolation systems have been noted in . Reliability-based optimization has been used for active base isolation systems (). Moreover, value-based design has been compared with cost-based design for three different isolated models (Rahgozar et al., 2023a), and optimum design of a triple friction pendulum for VBD has been calculated (Rahgozar et al., 2023b). The optimum solution of base isolation has been calculated for random system parameters in a study by Roy and Chakraborty (2015).
The previous studies mainly focus on the optimal design of base isolators for a desired intensity, and the algorithms that have been used to achieve the best solutions are mainly metaheuristics. However, in this study, all of the intensities have been considered because of the use of ET excitation functions (ETEFs) and a novel metaheuristic algorithm has been introduced and used to achieve the best solutions. Moreover, the present study utilizes the time history of roof acceleration as the cost function. In contrast, previous studies typically employed the maximum roof acceleration for this purpose. To this end, two models of four- and eight-story buildings have been modeled, and a base isolation system has been added to each one of them. The parameters of the isolators have been optimized using a novel hybrid metaheuristic algorithm for each structure.
2 Methodology
In this study, ET has been used to evaluate the performance of the building. In the ET method, dynamic analysis is carried out using artificial acceleration functions whose intensity gradually increases, serving as the excitation inputs (). The ET method is a dynamic analysis that can estimate engineering demand parameters (EDPs) adequately with lower computational effort compared to conventional time history analysis.
The efficiency of the ET method stems from the unique characteristics of its predesigned excitation functions, known as ETEFs. These functions are developed to intentionally increase their intensity over time, such that each time instant represents the structural response to ground motion scaled to a specific seismic hazard level. Therefore, a single ET analysis can reproduce the outcomes of multiple time history analyses performed under a suite of scaled ground motions across different seismic intensity levels.
The primary advantage of ET lies in its computational effort while maintaining an acceptable level of accuracy. This feature makes it especially suitable for use in optimization-based structural design, where numerous analyses are typically required, thereby significantly reducing the computational burden of seismic response evaluations ().
Several studies have confirmed the effectiveness of the ET method in estimating the seismic response of both fixed-base buildings and isolated building structures (; ; ). Furthermore, some studies (; ) have demonstrated its reliability within the FEMA P-58 performance assessment framework for estimating seismic consequences.
This study employs the fifth-generation ETEFs, referred to as the “kd” series (). These functions are hysteretic-based intensifying excitations (), developed to ensure consistency in the estimation of cumulative damage in non-linear structural components. Unlike previous generations (), which did not incorporate an energy-based measure, the “kd” series was formulated using the modified Park–Ang damage index (). Hence, the accumulated damage grows progressively and consistently as the intensity of the ETEFs increases over time.
Figure 1 illustrates the general procedure for predicting EDPs using the ET method, with its main steps briefly outlined. More comprehensive explanations of the ET concept, the latest generation of ETEFs, and detailed implementation guidelines are available on the official ET method website ().
FIGURE 1
3 Case studies
The case study structures consist of two steel moment-resisting frames (MRFs), with four and eight stories, originally developed and verified by the National Institute of Standards and Technology (). Each MRF is isolated using LRBs. In the non-linear dynamic simulation of multi-story, three-dimensional frame structures, and plan- and elevation-symmetric buildings can be efficiently modeled using a leaning column system. This modeling approach allows the primary MRFs to carry lateral loads and inelastic deformations, whereas the leaning column represents the gravity system and captures P-Δ effects, mass participation, and vertical load distribution without requiring a full 3D gravity frame model. Because the considered buildings are symmetric in both plan and height, the leaning column idealization provides an accurate and computationally efficient representation of the global dynamic behavior. These structural models are subjected to ETEFs, which are intensifying ground motion records. In an ETEF, the earthquake amplitude increases progressively with time, and each second of the record corresponds to a specific intensity level associated with a distinct return period. This enables the evaluation of structural performance at gradually increasing seismic intensity within a single dynamic analysis, allowing the identification of collapse capacity and performance limits under escalating hazard levels. The isolation bearing parameters for all three MRFs have been optimally designed with a hybrid metaheuristic algorithm. The isolated MRFs are modeled in OpenSees version 3.2.1 (OpenSees, 2013) and are represented by two main components: the superstructure and the isolation system.
3.1 Superstructure
The superstructures used in this study are benchmark response spectrum analysis models (RSA-Dmax). The plan of the models is shown in Figure 2. As shown, the plans are symmetrical and regular in plan. The dead load for the roof and floors is 4.31 kN/m2. The live loads are 0.957 kN/m2 for the roof and 2.39 kN/m2 for the floors, respectively. Moreover, the steel has a yield stress of 380 MPa and an elastic modulus of 200 GPa. The considered direction is also shown in Figure 2. The two MRFs have been modeled, so half of the weight of the building has been assigned to one MRF.
FIGURE 2
The four-story model of the building and sections is shown in Figure 3. The non-linear behavior of the columns and beams is simulated using the modified Ibarra–Medina–Krawinkler (IMK) deterioration model, which represents members as elastic beam–column elements equipped with bilinear hysteretic hinges at their ends (). Furthermore, the hinge properties are enhanced by incorporating a cyclic stiffness and strength deterioration material proposed by . The leaning column has been used to simulate the 3-D model effect. Moreover, the leaning column configuration is shown in Figure 3. The sections of the four-story building are listed in Table 1. The building is using reduced beam sections (RBS) for both ends. The behavior of IMK springs is shown in Figure 4.
FIGURE 3
TABLE 1
| Story | Elevation (m) | Beam size | Column size | |
|---|---|---|---|---|
| Interior | Exterior | |||
| 4 | 16.5 | W21 × 57 | W24 × 62 | W24 × 62 |
| 3 | 12.5 | W21 × 57 | W24 × 62 | W24 × 62 |
| 2 | 8.5 | W21 × 73 | W24 × 103 | W24 × 103 |
| 1 | 4.6 | W21 × 73 | W24 × 103 | W24 × 103 |
Sections of four-story building.
FIGURE 4
Each of the parameters in the above picture can be computed using equations that are predicted by regression (). For RBS the equations are expressed as Equations 1-3.
In the above equations, , , and are section dimensions; is column face to the nearest lateral brace, and is the radius of gyration about the y-axis of the beam. Also, is the length of the beam, and is the depth of the section. and are related to the units that are selected. If metric units are used, both are equal to 1; if inches and kilopounds per square inch (ksi) are selected, they are equal to 25.4 and 6.895, respectively. The equations for columns are different from the above equations and can be found in the work of as expressed in Equations 4–6. Moreover, Pye is expressed in Equations 7.where is the tributary area load, which, for exterior and interior columns, can be observed in Figure 5, and is the axial yield strength and is calculated based on expected steel material properties.
FIGURE 5
Moreover, the eight-story building can be modeled using these formulas, and sections of this building can be found in Table 2.
TABLE 2
| Story | Elevation (m) | Beam size | Column size | |
|---|---|---|---|---|
| Interior | Exterior | |||
| 8 | 32.5 | W21 × 68 | W24 × 94 | W24 × 94 |
| 7 | 28.5 | W24 × 84 | W24 × 94 | W24 × 94 |
| 6 | 24.5 | W24 × 84 | W24 × 131 | W24 × 131 |
| 5 | 20.5 | W27 × 94 | W24 × 131 | W24 × 131 |
| 4 | 16.5 | W27 × 94 | W24 × 131 | W24 × 162 |
| 3 | 12.5 | W30 × 116 | W24 × 131 | W24 × 162 |
| 2 | 8.5 | W30 × 116 | W24 × 131 | W24 × 162 |
| 1 | 4.6 | W30 × 108 | W24 × 131 | W24 × 162 |
Sections of eight-story building.
3.2 Base isolation system
For modeling the base isolation system, an advanced model named LeadRubberX is employed. As shown in
Figure 2, there is one isolator under each column, which is shown with one spring. In contrast to materials such as Bouc–Wen (
) and bilinear elastomeric materials (
Ye et al., 2019), which assume a stable hysteretic response, experimental investigations (
;
;
) suggest various forms of degradation for LRBs. These degradations are as follows (
):
coupled bi-directional horizontal motion effects, which influence the interaction of shear forces in orthogonal directions;
coupled vertical–horizontal motion effects, where simultaneous vertical and horizontal excitations alter the bearing’s stiffness and damping characteristics;
cavitation and post-cavitation behavior in tension (), representing the separation and subsequent re-contact between bearing layers under tensile loading;
strength deterioration under cyclic tensile loading due to cavitation, leading to reduced load-bearing capacity over repeated cycles;
variation in the critical buckling load capacity caused by lateral (sideways) displacements, affecting overall bearing stability; and
strength deterioration resulting from heating of the lead core, which reduces the material’s yield strength and energy dissipation capability during prolonged cyclic loading.
All six degradation mechanisms are incorporated within the LeadRubberX bearing element implemented in OpenSees, as formulated by Kumar and Whittaker (). This advanced constitutive model can simultaneously reproduce the axial and shear hysteretic responses of lead–rubber bearings, as illustrated in Figure 6. The shear response is consistently correlated with the axial behavior, thereby ensuring a realistic representation of the coupled interaction between vertical and horizontal load effects. In Figure 6a, is the axial load in the bearing, and is the force corresponding to the onset of cavitation, which is a critical mode of failure in this model. Furthermore, is the total vertical displacement, and is the deformation at which cavitation initiates. is a parameter controlling the slope of the post-cavitation plateau, and it is heavily dependent on (total thickness of rubber layers). and are the tangent stiffness, and and are related to the reloading and unloading characteristics of the element, respectively. In Figure 6b, is the yield strength of the lead core. During the dynamic analysis, the ratio of the instantaneous axial force to the variable rupture or buckling capacity of each LRB element is continuously evaluated, allowing for an accurate simulation of potential instability and strength degradation under combined axial–lateral loading conditions.
FIGURE 6
4 Analysis
Defining the design variables is the first step in modeling the optimization problem. The number of design variables in this problem is six, which are represented by
–
. The goal of the optimization problem is to determine the value of these variables. The design variables in this problem are as follows (
Figure 7):
: shear modulus of LRB ()
: inner diameter ()
: outer diameter ()
: maximum displacement ()
: shear strain ratio ()
: shape factor (S)
FIGURE 7
The upper and lower bounds of the design variables are shown in Table 3. The units are in megapascals and millimeters. The bounds of the design variables have been derived from factory limitations and fabrication constraints from specifications provided by FIP in Italy () and DIS in the United States ().
TABLE 3
| Design variable | ||||||
|---|---|---|---|---|---|---|
| Lower bound | ||||||
| Upper bound |
Lower and upper bounds of design variables.
The number of rubber layers in the design is obtained by dividing the total rubber thickness by the thickness of one rubber layer (). As shown in Equation 8.
In addition, and can be computed using Equations 9, 10.
Equations 9, 10 can be rewritten in terms of the design variables. Equations 11, 12 show the formulas in terms of the design variables.
4.1 Cost function
The value of a desired EDP that is to be minimized or maximized is the cost function. In this problem, the cost function is formulated based on the acceleration response of the roof, consistent with the approach adopted in previous research, such as that adopted Pal et al. (2019), , and Ocak et al. (2023). The cost function is the area of the acceleration time history, which means that this value should be minimized.
4.2 Constraints
In any optimization problem, there are some limitations and restrictions about the design variables. These limitations mainly arise from geometry limitations or some code requirements. For example, some codes do not accept a drift more than 2%, for structure or size of components cannot be negative (e.g., the height of a building cannot be negative).
For this study, four constraints are considered. Two of them are related to codes and the limitations of codes, and the other two are related to geometry limitations.
The first constraint is that the base isolation system must make the structure more flexible than the bare superstructure. Therefore, as expressed in Equation 13, the period of the isolated structure must be larger than that of the fixed-base structure ().
1.
In the above equation, corresponds to the effective first-mode vibration period of the superstructure, and refers to the effective vibration period of the isolation bearing evaluated at the maximum considered earthquake (MCE) displacement. This limitation has been mentioned in ASCE 7-16.
2. Most of the codes will not allow the drift ratio to be bigger than 2%. The constraint for this limitation has been shown in Equation 14.
In Equation 14, is the maximum inter-story drift ratio.
3, 4. Considering the minimum practical geometric manufacturing limits, the design parameters were constrained to ensure feasibility and producibility.The formula of these constraints are shown in Equations 15, 16.
4.3 Penalty function
There are several ways to make sure that the constraints are not violated. One simple and effective approach is to include a penalty function within the cost function.
In this problem, whenever a constraint is violated, a large penalty is added to the cost function, for instance, a big number like . This makes that solution very unattractive to the optimizer, because the cost becomes extremely high. Consequently, any candidate design that violates a constraint is automatically ruled out from being the best solution.
The mathematical format of the penalty function is shown in Equation 17.
5 Thermal exchange optimization
There are many metaheuristic algorithms developed in recent years. These algorithms are being utilized to solve real problems. Each of them is efficient for solving certain types of problems. One of the recent metaheuristic algorithms is TEO, which was developed by . This algorithm is inspired by Newton’s law of cooling. This law, in Newton’s own words, is mentioned in and .
5.1 Initialization
The population of this algorithm is called thermal objects (TO). The position of each will be calculated using Equation 18.
In Equation 18, and are lower and upper bound of the design variables, respectively, and random is a random number between 0 and 1.
After initialization, the entire population must be evaluated to determine the cost function for each solution.
Incorporating a memory mechanism that stores a set of historically best solution vectors and their corresponding objective function values can enhance the performance of the algorithm without increasing its computational cost. To this end, a thermal memory () is employed to retain a limited number of the best-so-far solutions, thereby improving the balance between exploration and exploitation during the optimization process.
5.2 Creating groups
There are two equal groups; each agent is assigned to one group, and every agent has another agent in a different group, for example, the first agent ( is incorporating with the agent with one number above half of the population (. The process of this step is shown in Figure 8.
FIGURE 8
5.3 Parameters of TEO
The two most important parameters of TEO are and . is the time for each iteration, and is a natural characteristic of elements. The elements with lower will exchange thermal heat more slowly. Formulations of these parameters are given in Equations 19, 20.
5.4 Escaping from local minima
5.4.1 Escaping from local minima I
The advantage of metaheuristic algorithms is that they will not get stuck in local minima and can escape from the local minima. In this algorithm, one way of escaping the trap (local minima) is to use Equation 21 and change the initial temperature to a new one using this formula.
In the above equation, and are controlling variables that can be selected as 0 or 1. is the previous temperature of the object, and is the new temperature.
5.4.2 Escaping from local minima II
Another mechanism is introduced in the algorithm to help escape from local minima. This mechanism has an advantage over the previous one. In the previous approach, only and determine whether the position of an agent changes, and these parameters can take only binary values (0 or 1). In the proposed mechanism, one of the agent’s variables is randomly selected and modified. For example, if the problem has variables, the th variable may be selected for modification. To implement this process, a parameter called , with a value between 0 and 1, is defined. If a randomly generated number is less than Pro, the ith variable is changed. The modification of the variable is expressed in Equation 22.
The above formula will check if the random number is less that for each dimension of the agent and will change the variable if is the condition true. Moreover, is the th variable of the th agent. and are respectively the lower and upper bounds of the dimension in the search space.
5.5 Iteration
For each iteration, the temperature of each object should be updated, and to update the temperature, Equation 23 is used. This equation is based on Newton’s law of cooling.
The flowchart of this algorithm can be observed in Figure 9.
FIGURE 9
6 Proposed hybrid TEO–PSO algorithm
Generally, metaheuristic algorithms are defined to solve particular problems, and each metaheuristic algorithm cannot solve all problems. To overcome this limitation, a hybrid metaheuristic framework combining two distinct optimization algorithms is adopted. The integration aims to leverage the complementary strengths of each method, enhancing exploration of the global search space while maintaining effective exploitation of promising regions. This synergy improves convergence stability and solution diversity, thereby enabling the hybrid algorithm to address a wider spectrum of complex optimization problems more efficiently than either metaheuristic used independently. In this study, a combination of PSO and TEO is suggested. In this algorithm, each TO is a particle in PSO. First, each TO finds its location using the PSO algorithm, where the location is calculated using the velocity of each particle and the previous location of the particle. The velocity of a particle can be obtained using Equation 24.
In the above equation, d is the dimension of the problem, and are the parameters of PSO and are different from the parameters in the TEO algorithm, and is the weight of inertia. Furthermore, and are the best solution that the particle has experienced and the best solution that has been found so far during that iteration. The location of each particle can be calculated using Equation 25.
In the proposed algorithm, each particle is represented by a TO in TEO. After the PSO algorithm has found the location of each particle, these particles will change their location and find a better solution using TEO, which is explained in Section 5. The flowchart of the proposed algorithm has been presented in Figure 10.
FIGURE 10
7 Application of proposed algorithm on different problems
The performance of the proposed algorithm has been evaluated by solving different problems. In this section, two different problems have been solved by this algorithm: benchmark tests and base isolation optimization.
7.1 Benchmark tests
Some famous benchmark functions are used to evaluate the performance of each optimization algorithm (classical and metaheuristics). Four of these functions have been selected and optimized using the proposed algorithm. All information about the benchmark tests and their solutions has been presented in Table 4.
TABLE 4
| Function | Name | Range | Dimension | Actual minimum | |
|---|---|---|---|---|---|
| Sphere | [−30, 30] | 4 | 0 | ||
| Schwefel | [−500, 500] | 4 | −418.4829 × dim | ||
| Rosenbrock | [−30, 30] | 6 | 0 |
Benchmark functions and their information.
The preliminary benchmarks (problems 1 and 3) presented fewer computational challenges, which led to the use of the Schwefel function as a more robust metric for differentiation. By introducing this higher level of complexity, we were able to conduct a more nuanced comparison between the standard PSO and the TEO–PSO algorithms. As shown in the boxplots in Figure 11, the comparative data highlight significant variances in performance, offering a clearer picture of how each algorithm handles multimodal optimization landscapes.
FIGURE 11
7.2 Base isolation problem
In this section, the best solution for the problem described in Section 4 is presented. As mentioned previously, two structures with four- and eight-story have been selected, and the optimization algorithm has been implemented on them. This section is divided into three subsections, with each one of them corresponding to one structure. Furthermore, the parameters of the proposed algorithm have been provided in Table 5.
TABLE 5
| Parameter | Performance | Value |
|---|---|---|
| Number of thermal objects (TO) in each iteration | ||
| The random number used as PRO in Section 5.4.2 | ||
| Maximum number of iterations | 100 | |
| The number of saved memories in each iteration | 5 | |
| Inertia weight | 1 | |
| Damping ratio of inertia weight | 0.89 | |
| Personal and social acceleration coefficient | 1.14 |
Parameters of the TEO–PSO algorithm.
7.2.1 Four-story building
Table 6 presents a comparison between the non-optimized (manual design) and optimized design parameters for the base isolation system in the four-story building. The manual design shows a large outer diameter and a relatively high maximum displacement, indicating that it is suitable for absorbing earthquake energy at medium to high levels. This design focuses on a balance between stiffness and flexibility. In the optimized design, compared with the manual design, a higher shear modulus (increased rubber stiffness), smaller inner and outer diameters, a lower maximum displacement, and a lower shear strain ratio are observed. The shape factor has also reached its minimum bound (10). These changes indicate the algorithm’s focus on reducing the physical dimensions of the isolator to decrease cost and weight while increasing stiffness to better control the dynamic response of the building.
TABLE 6
| Design type | ||||||
|---|---|---|---|---|---|---|
| Manual design | 0.75 | 112 | 539 | 493 | 125 | 20 |
| Optimized design | 0.9598 | 56.61 | 300 | 100 | 113 | 10 |
Parameters of manual design and optimized design of base isolation (four-story building).
Figure 12 shows the convergence trend of the TEO–PSO algorithm for the four-story structure, where the horizontal axis represents the iteration number from 0 to 100, and the vertical axis shows the cost function value on a logarithmic scale. The algorithm exhibits a sharp decrease in cost during the initial iterations (0–10), indicating efficient exploration of the design space and the discovery of good initial solutions. From iteration 10 onward, the changes diminish, and the curve becomes nearly horizontal, signifying convergence to the optimal point. This rapid convergence (in fewer than 50 iterations) demonstrates the high efficiency of the hybrid algorithm compared with traditional methods, as it leverages the combination of metaheuristics to avoid getting trapped in local minima. The low final cost indicates that the roof acceleration response (on which the cost function is based on its time history) has been minimized, which means that the optimized isolator absorbs vibrations more effectively.
FIGURE 12
Figure 13 shows the ET curve of absolute roof acceleration. The fixed-base model shows a sharp increase in acceleration from 5 s onward, reaching over 25 m/s2, indicating the building’s high vulnerability to increasing ET intensities in the absence of an isolator. The manual design controls acceleration to less than 25 m/s2 but still exhibits a gradual increase. The optimized design shows the lowest acceleration (approximately 15 m/s2). This comparison emphasizes that optimization not only reduces the peak acceleration but also smooths the time history, which is beneficial for reducing cumulative damage.
FIGURE 13
Figure 14 illustrates the story drift ratio for structures subjected to an earthquake with a 2475-year return period. In the fixed-base building, the first-story drift exceeds 0.03, signifying severe deformation and a considerable risk of collapse. The manually designed isolation system reduces the drift to approximately 0.022, whereas the optimized design achieves a further reduction to 0.02. These results indicate that the acceleration-based optimization not only reduces peak responses but also yields a lower maximum drift, even with a notable reduction in isolator dimensions compared with the manual design.
FIGURE 14
7.2.2 Eight-story building
Table 7 shows that the TEO–PSO algorithm identifies different parameters for the eight-story structure. The increase in shear modulus compared with the manual design (and relative to the four-story structure) indicates that a stiffer isolator is required for taller buildings to prevent overall instability. The outer diameter is significantly increased (to 1,336 mm), providing a larger contact area for load-bearing. However, this reduces the isolator’s displacement compared with the manual design. The convergence process is similar to that of four-story structure but may have more fluctuations in the early stages, indicating the greater complexity of the optimization problem for the taller structure. The algorithm eventually reaches a stable value (Figure 15).
TABLE 7
| Design type | ||||||
|---|---|---|---|---|---|---|
| Manual design | 0.75 | 151 | 769 | 837 | 125 | 20 |
| Optimized design | 1.171 | 119.67 | 1,336.23 | 240.467 | 120.5 | 32.13 |
Parameters of manual design and optimized design of base isolation (eight-story building).
FIGURE 15
Although the manual design performs similarly to the optimized design for the initial 15 s, the acceleration differs at higher intensities. The manual design has higher roof acceleration than the optimized design, but it is less than that of the fixed-base building. This indicates that parameter optimization improves energy dissipation, especially in taller buildings, where P-Delta effects are more significant (Figure 16). It should be noted that the relatively high acceleration levels observed in Figure 16 are primarily related to the intensity of the input excitation and the dynamic amplification inherent to multi-story structural behavior. Under such elevated excitation levels, the system enters a more non-linear response regime, which can amplify the dynamic behavior and lead to larger peak accelerations in the numerical results. In taller configurations, the P-Delta effects and the interaction between stories further intensify the transient accelerations.
FIGURE 16
As shown in Figure 17, the optimized design limits the maximum drift of the structure to approximately 0.013, whereas the manual design results in a maximum drift of 0.019. Furthermore, the optimized design achieves a more uniform drift distribution across the floors, in contrast to the manual design, which exhibits a sharp increase in drift in the upper floors relative to the lower ones.
FIGURE 17
7.2.3 Effect of height
With increasing height (from four to eight stories), the isolator dimensions in the manual design become larger (e.g., from 539 to 769 mm) to withstand higher loads. In the optimized design, Gr increases for both the four- and eight-story structures, indicating that the isolator stiffness should be adjusted according to the period and dynamic characteristics of the building.
consistently decreases during optimization, but as the structure’s height increases, it is observed that for taller buildings (with greater weight), the TEO–PSO algorithm tends to enlarge the outer diameter of the isolator. This adjustment increases the contact area and, consequently, enhances its load-bearing capacity.
Although taller buildings typically require larger displacements, the optimization process successfully reduces the isolator’s displacement requirements significantly, even as height increases.
Increasing height also increases the complexity of the design space (due to more vibrational modes and intricate dynamic loads), requiring the algorithm to take more time to converge. Nevertheless, it remains efficient overall.
In both structures, incorporating the isolator, even in the manual design, leads to a substantial reduction in roof acceleration and floor drifts compared with the fixed-base case. However, the TEO–PSO algorithm’s effectiveness in reducing acceleration is more pronounced in the four-story structure. Here, the difference between the manual design and optimized design curves is far greater than in the eight-story structure. In other words, the TEO–PSO method proves more beneficial for shorter structures in ensuring the safe performance of acceleration-sensitive non-structural components.
8 Conclusion
In this study, a novel hybrid metaheuristic algorithm that combines TEO and PSO has been developed. The proposed algorithm was applied to two different base-isolated structures. The constraints of this problem were structural limitations and geometric limitations. In order to deal with the constraints, a penalty function was used. Furthermore, benchmark tests were solved by the algorithm. The results obtained are summarized as follows:
The TEO–PSO hybrid algorithm demonstrated better performance in solving complex optimization problems such as seismic isolator design. In standard benchmarks such as the Schwefel function, it exhibited lower variance compared with standard PSO and achieved more optimal solutions with higher stability, making it a powerful tool for structural engineering applications.
The optimization of base isolator parameters using the TEO–PSO algorithm led to a significant reduction in the physical dimensions of the isolator (such as outer diameter and maximum displacement) while increasing its stiffness (shear modulus). This not only reduced the cost and weight of the system but also improved the seismic response of the buildings, particularly by reducing the absolute roof acceleration of the four-story building by up to 60% compared with its manual design under a 2,475-year return period earthquake.
For buildings of different heights, the TEO–PSO algorithm tended to increase isolator dimensions as the height increased (from four to eight stories) to handle larger dynamic loads while still significantly reducing maximum displacement.
The comparison of results showed that optimization not only limited the roof acceleration during high-risk earthquakes (e.g., 2,475 years) but also made the distribution of story drifts more uniform and reduced the maximum drift by up to 38% compared with the fixed-base structure and up to 32% compared with the manual design. These improvements are particularly effective at protecting acceleration-sensitive non-structural components.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding authors.
Author contributions
MH: Conceptualization, Formal Analysis, Investigation, Methodology, Software, Visualization, Writing – original draft. MM: Conceptualization, Investigation, Methodology, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing. AM: Conceptualization, Methodology, Validation, Writing – review and editing. AS-M: Conceptualization, Methodology, Validation, Writing – review and editing. MMR: Conceptualization, Methodology, Validation, Writing – review and editing.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was used in the creation of this manuscript. During the preparation of this work, the first author used ChatGPT in order to improve readability and language. After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.
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Summary
Keywords
base isolation, endurance time, particle swarm optimization, performance-based seismic design, thermal exchange optimization
Citation
Haddad MH, Mashayekhi M, Majdi A, Sadeghi-Movahhed A and Movahedi Rad M (2026) A hybrid TEO–PSO framework for improving the performance of base-isolated structures under continuous seismic hazard. Front. Built Environ. 12:1835647. doi: 10.3389/fbuil.2026.1835647
Received
21 March 2026
Revised
08 May 2026
Accepted
14 May 2026
Published
15 July 2026
Volume
12 - 2026
Edited by
Paolo Castaldo, Polytechnic University of Turin, Italy
Reviewed by
Rodolfo Labernarda, University of Calabria, Italy
Junhong Xu, Nanjing Forestry University, China
Updates
Copyright
© 2026 Haddad, Mashayekhi, Majdi, Sadeghi-Movahhed and Movahedi Rad.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ali Majdi, ali.majdi1978@gmail.com; Majid Movahedi Rad, majidmr@sze.hu
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.