Evaluating Algorithm Efficiency in Large-Scale Dome Truss Optimization Under Frequency Constraints

Abstract

Incorporating frequency constraints into the optimum design of large-scale truss dome structures is crucial for maintaining seismic resilience, as the natural frequencies must remain within specified ranges. In this work, seven metaheuristic algorithms—including three variants of the Fitness–Distance–Balance-based Adaptive Guided Differential Evolution (FDB-AGDE), the Cheetah Optimizer (CO), the Bonobo Optimizer (BO), the Flood Algorithm (FLA), and the Lung Performance Optimization (LPO) are applied to solve high-dimensional truss sizing problems under strict frequency limitations. Their convergence characteristics and solution quality are systematically compared across multiple dome configurations. Besides traditional measures of computational efficiency and final weight minimization, a suite of statistical analyses is conducted: the Wilcoxon rank-sum test to assess pairwise performance significance, the Friedman test to establish overall rank ordering, and Cohen’s test to quantify effect sizes. The results reveal that LPO, BO, CO, and the first variant of FDB-AGDE consistently produce lighter feasible designs with lower variability, whereas FLA and other variants of FDB-AGDE exhibit heavier structures or higher dispersion. The findings underscore the value of robust, well-tuned metaheuristics and rigorous statistical evaluation in structural optimization, offering clear guidance for seismic-focused designers seeking both lightweight solutions and reliable performance across repeated runs.

1. Introduction

Dome structures are known for their aesthetic appeal, durability, and structural efficiency, making them a sustainable choice for modern construction. Their ability to cover large areas with minimal material usage offers economic advantages, and they can be adapted for various applications, including residential, commercial, cultural, and recreational spaces. Additionally, domes are versatile in size, with large-scale domes used for logistics and exhibition spaces, and smaller ones for shelters or kiosks.

The natural frequency of a truss is a critical design parameter, particularly when the structure is exposed to dynamic excitations. In practice, many engineering trusses are subjected to dynamic loading arising from operational conditions or unexpected events, which may result in undesirable vibrations and noise. Such effects can become hazardous if resonance occurs; therefore, natural frequency constraints must be imposed to ensure structural safety and performance [1,2,3,4]. Structural optimization aims to balance safety by meeting natural frequency constraints while minimizing material use and weight, enhancing both performance and efficiency. Metaheuristic algorithms [5,6,7,8,9,10,11,12,13,14,15] have become a valuable tool in optimizing such structures, offering efficient solutions to complex, nonlinear design problems. Numerous studies have been conducted by researchers on the optimal design of truss structures with frequency constraints using various metaheuristics [1,2,3,4,16]. A detailed chronological literature review focusing on the optimization of large-scale dome truss structures with frequency constraints in recent years is presented as follows:

Kaveh and Ilchi Ghazaan [17] employed the Enhanced Colliding Bodies Optimization (ECBO) method, using a cascading approach to break down the optimization process into autonomous stages. This approach improved design optimization across large-scale dome structures with multiple frequency constraints. Kaveh [18] used Democratic Particle Swarm Optimization (DPSO) for decomposing the eigenproblem of dome trusses, enabling more efficient optimization. This method significantly reduced computational time and demonstrated promising results in structural optimization. Dede et al. [19] introduced the Jaya algorithm for braced dome structures, focusing on weight minimization while adhering to frequency constraints. The integration of the Jaya algorithm with MATLAB and finite element analysis proved effective in solving constrained optimization problems. Liu et al. [20] enhanced the Fruit Fly Optimization Algorithm (FOA) by introducing a memory-based adaptive search radius and an elitist improved rule. These modifications reduced redundant analyses and improved computational efficiency. Applied to frequency-constrained truss optimization problems, the method outperformed traditional FOAs in both quality and efficiency. Degertekin et al. [21] proposed the parameter-free Jaya algorithm (PFJA), which adapts population size throughout the optimization process. PFJA demonstrated superior performance in terms of weight optimization, convergence speed, and statistical reliability. Kaveh et al. [22] developed the Enhanced Forensic-Based Investigation (EFBI) algorithm, improving communication between search components for dome-like trusses. EFBI showed enhanced performance compared to the original FBI and matched other metaheuristics in optimization tasks. Kaveh et al. [23] introduced a chaotic version of the Water Strider Algorithm (WSA) for large-scale frequency-constrained structural optimization. The chaotic WSA alternates between explorative and exploitative states, improving convergence speed and resulting in lighter designs compared to the basic WSA. An improved version of the Arithmetic Optimization Algorithm (IAOA) [24] addressed exploration and convergence issues in the original AOA. IAOA outperformed its predecessor and other state-of-the-art algorithms, optimizing benchmark dome trusses effectively. Dede et al. [25] applied Rao algorithms for optimizing dome trusses, using a MATLAB-GUI-based program integrated with finite element analysis. Their work highlighted the effectiveness of Rao-2, which produced optimal designs in fewer generations. Khodadadi and Mirjalili [26] proposed the Generalized Normal Distribution Optimization (GNDO) algorithm for truss design under frequency constraints. GNDO demonstrated superior reliability, stability, and efficiency compared to other metaheuristic algorithms in structural optimization tasks. Kaveh et al. [27] improved the Slime Mould Algorithm (ISMA) with an elitist strategy and enhanced exploration to overcome slow and premature convergence. Tested on large-scale dome trusses, ISMA outperformed the classical SMA and other methods. Öztürk and Kahraman [28] tackled challenges in truss optimization, such as inconsistent settings and a lack of stability analysis. They proposed a standardized simulation environment and a benchmark suite of nine problems. Their findings identified LSHADE-EpSin as the top-performing and most stable algorithm, achieving a 90% success rate across all truss problem types. Kaveh et al. [29] utilized the Success-History Based Adaptive Differential Evolution (SHADE) algorithm, which incorporates a historical memory of successful parameters, to optimize dome trusses under frequency constraints. This method demonstrated efficiency by achieving optimal designs with fewer structural analyses. Moosavian et al. [30] evaluated Differential Evolution (DE), its variants (IDE, LSHADE), and the covariance matrix adaptation evolution strategy (CMAES) for frequency-constrained truss optimization. CMAES showed superior performance and stability with minimal deviation, excelling in smaller populations but requiring more evaluations for larger-scale problems. Goodarzimehr et al. [31] introduced the Improved Marine Predators Algorithm (IMPA) for truss optimization with frequency constraints, featuring an adaptive factor for step size control. Applied to various benchmarks, IMPA outperformed state-of-the-art algorithms, proving highly effective and robust. Sheng-Xue [32] introduced the Medalist Learning Algorithm, a heuristic inspired by group learning behaviors, for truss optimization with frequency constraints. The algorithm identifies top-performing “medalists” for self-learning and allows common learners to imitate medalists and past experiences. Tested on six benchmark truss problems, it showed superior best and average weights with moderate computational cost, achieving minimal worst-case weights and fewer evaluations in certain cases. Truong and Chou [33] developed the Enterprise Development Optimizer (ED), a metaheuristic inspired by the enterprise development process. It updates solutions by switching activities at each step. Applied to steel structure optimization under frequency constraints, the algorithm efficiently optimized component weights for various dome structures, demonstrating optimal solutions with fewer function evaluations. Goodarzimehr and Topal [34] introduced the self-adaptive Bonobo Optimizer (SABO) to improve the original Bonobo Optimizer (BO) by incorporating adaptive mechanisms to enhance exploration and avoid premature convergence. SABO showed superior performance in minimizing truss weight under frequency constraints across five benchmark problems. Abbasi and Zakian explored [35] dome-shaped truss optimization with frequency constraints using seven metaheuristic algorithms: Social Network Search, Jellyfish Search Optimizer, Equilibrium Optimizer, Teaching-Learning-Based Optimization, Grey Wolf Optimizer, Colliding Bodies Optimization, and the Improved Grey Wolf Optimizer. They introduced a hyperparameter tuning procedure to improve algorithm performance, demonstrating its efficiency through statistical analysis on five benchmark problems.

This study systematically explores the capability of multiple metaheuristic algorithms—three variants of Fitness–Distance–Balance-based Adaptive Guided Differential Evolution (FDB-AGDE), Cheetah Optimizer (CO), Bonobo Optimizer (BO), Flood Algorithm (FLA), and Lung Performance Optimization (LPO)—to handle natural frequency constraints in large-scale dome truss designs, which are crucial for seismic resilience. In addition to evaluating performance based on best, mean, and worst weights, a comprehensive statistical assessment is performed using Wilcoxon rank-sum tests, Friedman ranking, and Cohen’s effect sizes. These analyses quantify both statistical significance and practical importance, offering a more rigorous evaluation than simple comparisons. Furthermore, by examining dome structures with up to 1410 members, this study demonstrates how selected metaheuristics effectively manage high-dimensional complexity. This provides valuable insights into the scalability and reliability of these algorithms for large-scale structural optimization problems.

Unlike earlier studies, this work delivers a more comprehensive and rigorous statistical comparison while systematically evaluating algorithms that originate from diverse conceptual inspirations: natural phenomena (CO and FO), human anatomy (LPO), animal social behavior (BO), and evolutionary computation (FDB-AGDE variants). This diversity enables an in-depth analysis of how performance differences emerge across fundamentally different algorithmic paradigms when applied to frequency-constrained dome truss optimization. This is the first study to jointly benchmark these seven metaheuristics on high-dimensional structural design problems, thereby offering unique insights into their comparative strengths, weaknesses, and scalability.

The remainder of this paper is as follows: Section 2 presents the formulation for the design optimization of truss domes under frequency constraints. Section 3 introduces the optimization algorithms employed in this study. In Section 4, the performance of the algorithms is evaluated through three well-established dome structure problems, and the results are compared with previous studies, including performance rankings based on statistical analysis. Finally, Section 5 summarizes the conclusions.

2. Optimization of Truss Structures

The main objective in truss optimization is to minimize the structural weight while adhering to all constraints imposed. In this study, natural frequencies are used as critical constraints to ensure structural performance. The objective function, defined in terms of the structure’s weight, is expressed as:

$$W(X)=\gamma \cdot {\displaystyle \sum _{i=1}^{nm}{A}_i}\cdot L_i i=1,2,\dots nm$$

(1)

where W denotes the total weight of structure, $A_i$ represents the cross-sectional area of the i-th member, $\gamma$ is the density of material, $L_i$ is the length of member, and “nm” is the total number of structural members.

Natural Frequency Constraints

Natural frequencies are critical for ensuring structural stability under dynamic loading conditions. These are determined by solving the eigenvalue problem derived from the mass and stiffness matrices of the structure. The natural frequencies $\left(\omega \right)$ are calculated by solving:

$$\left(K-{\omega }^{2}M\right)\varphi =0$$

(2)

This equation represents a standard eigenvalue problem:

$$\left[K-\lambda M\right]\varphi =0$$

(3)

where K is the global stiffness matrix, M denotes the global mass matrix, $\varphi$ represents mode shape vector (eigenvector) corresponding to the natural frequency, and $\lambda$ is an eigenvalue related to the square of the natural frequency $(\lambda ={\omega }^2)$.

The eigenvalues are computed as

$${\lambda }_k={\displaystyle \frac{{{\varphi }_k^T·K·\varphi }_k}{{{\varphi }_k^T·M·\varphi }_k}}$$

(4)

Here, k represents the k-th mode of vibration, ${\varphi }_k$ and ${\varphi }_k^T$ are k-th mode eigenvector and its transpose, respectively. The frequency of the k-th vibration mode is the square root of the eigenvalue ($w_k=\sqrt{{\lambda }_k}$).

The natural frequency constraints ensure that the calculated frequencies remain within the specified bounds:

$$\begin{gathered} w_j\le w_j^u \\ {w}_k\ge w_k^l \end{gathered}$$

(5)

where $w_j$ and $w_j^u$ denote the j-th natural frequency value and the maximum allowable frequency, respectively. In a similar way, $w_k$ and $w_k^l$ are k-th natural frequency and the minimum allowable frequency value. For size optimization, the cross-sectional areas must satisfy:

$${A_i^{min}\ge A}_i\ge A_i^{max}$$

(6)

where $A_i^{min}$ and $A_i^{max}$ are the bounds for cross-sectional areas. To differentiate between feasible and infeasible designs with respect to the natural frequency constraints, a penalized objective function $f_p$ is employed. The penalized objective function augments the structural weight with a multiplicative penalty term that grows as a function of the normalized constraint violation ($v)$. This mechanism assigns larger objective values to infeasible designs, thereby reducing their likelihood of survival in the search process, while at the same time permitting exploration in the vicinity of the constraint boundaries. The $f_p\left(A\right)$ is expressed as follows:

$$f_p\left(A\right)=W\left(A\right)\times {\left(1+v\right)}^e$$

(7)

where e is the exponential penalty coefficient, taken as 2 in this study. This selection is consistent with Degertekin et al. [10], who showed that a quadratic penalization scheme is effective for frequency-constrained structural optimization. It also falls within the practical range suggested by Kaveh and Yosufpour [21] and follows the general principle outlined by Kaveh et al. [3] that penalty parameters should balance exploration and exploitation.

The term ν represents the sum of the penalties for natural frequency constraints that are not satisfied, calculated as:

$$v=\sum _{j=1}^{nj}{v}_{wj}+\sum _{k=1}^{nk}{v}_{wk}$$

(8)

Here nj and nk are numbers of natural frequency constraints that are bounded from above and below, respectively. $v_{wj}$ and $v_{wk}$ are the penalty values for the j-th and k-th natural frequencies, determined as:

$$v_{wj}=\left\{\begin{array}{cc}0& {if w}_j\le w_j^u\\ \left|{\displaystyle \frac{{ w}_j-w_j^u}{w_j^u}}\right|& {if w}_j>w_j^u\end{array}\right.$$

(9)

$$v_{wk}=\left\{\begin{array}{cc}0& if w_j\ge w_j^l\\ \left|{\displaystyle \frac{{ w}_k-w_k^l}{w_k^l}}\right|& if w_k<w_k^l\end{array}\right.$$

(10)

3. Optimization Algorithms

This section provides an overview of seven metaheuristic algorithms used in this study, including the three variants of FDB-AGDE, CO, LPO, BO, and FLA. These algorithms were selected for their unique capabilities in solving complex optimization problems.

3.1. Full Distance-Based-Adaptive Guided Differential Evolution Algorithm (FDB-AGDE)

The FDB-AGDE algorithm was developed by Guvenc et al. [36], aiming to handle trapping local optima and premature convergence of the Adaptive Guided Differential Evolution (AGDE) [37] by efficiently selecting reference positions for guiding the search process and then integrating them into AGDE. By applying the fitness distance-based method (FDB) [38] to the mutation process, local solution traps can be avoided. AGDE eliminates the need for user-defined control parameters, and various strategies have been developed to integrate the FDB selection method systematically. To direct the AGDE search process efficiently, the top three strategies with the best search performance were implemented. The integration of a probabilistic FDB method—using a roulette-wheel approach based on FDB scores—transformed the greedy selection process. This method was employed to construct mutant vectors, as defined in three variations as given below:

$${\mathrm{v}}_{\mathrm{i}}^{G+1}={\mathrm{X}}_r^G+\mathrm{F}\cdot \left.\left({\mathrm{X}}_{pbest}^G-{\mathrm{X}}_{pworst}^G\right.\right)\times \left\{\begin{array}{ll}Case 1:& {\mathrm{X}}_r^G={\mathrm{X}}_{FDB\left(roulette wheel\right)}^G\\ Case 2:& {\mathrm{X}}_{pbest}^G={\mathrm{X}}_{FDB\left(roulette wheel\right)}^G\\ Case 3:& {\mathrm{X}}_{pworst}^G={\mathrm{X}}_{FDB\left(roulette wheel\right)}^G\end{array}\right.$$

(11)

${\mathrm{v}}_{\mathrm{i}}^{G+1}$ is the mutant vector at generation G + 1 for the i-th individual. The mutant vector is a candidate solution that is generated for the next iteration of the algorithm. ${\mathrm{X}}_r^G$ refers to the position of the randomly selected individual G. The position represents a solution in the search space. F is the scaling factor, typically a constant used to scale the difference between two positions in the mutation process. The scaling factor controls the step size and the impact of the mutation. ${\mathrm{X}}_{pbest}^G$ represents the position of the best solution of the i-th individual at generation G. It is used to guide the search towards better solutions. ${\mathrm{X}}_{pworst}^G$ refers to the position of the worst solution of the i-th individual at generation G. ${\mathrm{X}}_{FDB\left(roulette wheel\right)}^G$ represents a position selected using the fitness distance-based (FDB) method, which is guided by a roulette-wheel selection based on fitness distance scores. The roulette-wheel selection ensures that individuals with better fitness have a higher chance of being selected, thus guiding the mutation towards promising regions of the search space. n FDB-AGDE-1, the position of ${\mathrm{X}}_r^G$ (the randomly selected individual) is chosen using the roulette-wheel method based on FDB scores, prioritizing individuals with higher fitness. In FDB-AGDE-2, the position of ${\mathrm{X}}_{pbest}^G$ (the best solution) is selected using the same roulette-wheel selection method, focusing on exploiting the best-found solutions. In FDB-AGDE-3, the position of ${\mathrm{X}}_{pworst}^G$ (the worst solution) is chosen through the roulette-wheel method with FDB scores, promoting exploration by incorporating less-optimal solutions into the search process.

3.2. Cheetah Optimizer (CO)

The Cheetah Optimizer (CO) was proposed by Akbari et al. [12], inspired by hunting behaviors of cheetah using searching, sitting-and-waiting, attacking, and returning home strategies to solve complex optimization problems.

3.2.1. Searching

Cheetahs search their surroundings for prey. This searching strategy is modeled as a random exploration, adjusting their position according to the conditions of the prey and their environment. The position of a cheetah, which represents a solution, is updated by

$${\mathrm{X}}_{\mathrm{i},\mathrm{j}}^{t+1}={\mathrm{X}}_{\mathrm{i},\mathrm{j}}^t+{\widehat{r}}_{\mathrm{i},\mathrm{j}}^{-1}·{\mathrm{a}}_{i,j}^t$$

(12)

where ${\mathrm{X}}_{\mathrm{i},\mathrm{j}}^{t+1}$ and ${\mathrm{X}}_{\mathrm{i},\mathrm{j}}^t$ denote the next and current position of the cheetah i in arrangement of j, respectively. ${\widehat{r}}_{\mathrm{i},\mathrm{j}}^{-1}$ and ${\mathrm{a}}_{i,j}^t$ the random parameter and step length for i-th cheetah in arrangement j, respectively.

3.2.2. Sit-and-Wait Strategy

In this strategy, the cheetah remains in its position still to avoid alerting the prey, waiting for it to come closer. The update rule is

$${\mathrm{X}}_{\mathrm{i},\mathrm{j}}^{t+1} ={\mathrm{X}}_{\mathrm{i},\mathrm{j}}^t$$

(13)

in order not to change all cheetahs simultaneously in each group so that we can avoid premature convergence in the optimization.

3.2.3. Attack Strategy

Cheetahs use speed and flexibility to attack prey. When attacking, the cheetah follows the prey’s position and adjusts its path to intercept. Mathematically, the position update for the attacking cheetah is

$${\mathrm{X}}_{\mathrm{i},\mathrm{j}}^{t+1}={\mathrm{X}}_{\mathrm{B},\mathrm{j}}^t+{\widehat{r}}_{\mathrm{i},\mathrm{j}}·{\beta }_{i,j}^t$$

(14)

where ${\mathrm{X}}_{\mathrm{B},\mathrm{j}}^t$ is the prey’s current position, ${\widehat{r}}_{\mathrm{i},\mathrm{j}}$ is the random tuning factor, and ${\beta }_{i,j}^t$ is the interaction with the leader or neighboring cheetah.

3.2.4. Leave the Prey and Go Back Home

At this step, if the cheetah fails to hunt, it changes its position or returns to its territory. In a case with no hunting action for a while, it changes the position to the last known prey location to resume the search.

3.3. Bonobo Optimizer (BO)

The Bonobo Optimizer [13] is a metaheuristic algorithm inspired by bonobos’ social and reproductive behaviors. It uses a fission–fusion approach, dividing the population into smaller groups that reunite to balance exploration and exploitation. BO employs four mating strategies—restricting, promiscuous, extragroup, and consortship—to maintain diversity and avoid premature convergence. The top solution, the α-bonobo, guides the search, and the algorithm utilizes various parameters to optimize complex, non-convex problems effectively.

3.3.1. Bonobo Selection Using Fission–Fusion Social Strategy:

A bonobo is selected for mating using the fission–fusion social strategy, which mimics the dynamic group formations seen in bonobo communities. The population is divided into small temporary sub-groups of random sizes, based on a factor $\left({tsgs}_{factor}\right)$.

To select a mate, a bonobo is chosen by randomly selecting a temporary sub-group from the population (excluding the bonobo itself). The fittest bonobo from this sub-group is compared with the original bonobo (i-th-bonobo). If the selected bonobo has better fitness, it becomes the mating partner (p-th-bonobo). If not, a random bonobo from the temporary group is chosen as the p-th-bonobo. This process ensures diversity and effective mating for the offspring generation.

3.3.2. Promiscuous and Restrictive Mating Used During the Positive Phase

The phase probability parameter (pp) determines the mating strategy of the bonobo. If a randomly generated number is less than pp, a new bonobo is created using the following equation.

$${NewBonobo}_j={bonobo}_j^i+r_1\times scab\times \left(a_j^{bonobo}-{bonobo}_j^i\right)+\left(1-r_1\right)\times scsb\times flag\times \left({bonobo}_j^i-{bonobo}_j^p\right)$$

(15)

where $a_j^{bonobo}$, ${bonobo}_j^i$, and ${bonobo}_j^p$ are the j-th components of alpha bonobo, the i-th bonobo, and the p-th bonobo, respectively. $scab$ and $scsb$ are the sharing coefficients for the alpha bonobo. flag parameter is taken as 1 for promiscuous mating, and −1 for restrictive mating.

3.3.3. Consortship and Extra-Group Mating Is Used During the Negative Phase

The strategies for consortship and extra-group matings are determined randomly based on the phase probability and are expressed through the probability of extra-group mating (pxgm). The generation of a new bonobo is performed as follows:

$$\begin{gathered} if a_j^{bonobo}\ge {bonobo}_j^i and r_3\le p_d \\ {NewBonobo}_j={bonobo}_j^i+e^{\left(r_4^2+r_4-{\displaystyle \frac{2}{r_4}}\right)}\times \left(Va{r_{m}ax}_j-{bonobo}_j^i\right) \end{gathered}$$

(16)

$$\begin{gathered} if a_j^{bonobo}\ge {bonobo}_j^i and r_3>p_d \\ {NewBonobo}_j={bonobo}_j^i-e^{\left(-r_4^2+r_4-{\displaystyle \frac{2}{r_4}}\right)}\times \left({bonobo}_j^i-Va{r_{m}in}_j\right) \end{gathered}$$

(17)

$$\begin{gathered} if a_j^{bonobo}<{bonobo}_j^i and r_3\le p_d \\ {NewBonobo}_j={bonobo}_j^i-e^{\left(r_4^2+r_4-{\displaystyle \frac{2}{r_4}}\right)}\times \left({bonobo}_j^i-Va{r_{m}in}_j\right) \end{gathered}$$

(18)

$$\begin{gathered} if a_j^{bonobo}<{bonobo}_j^i and r_3>p_d \\ {NewBonobo}_j={bonobo}_j^i+e^{\left(-r_4^2+r_4-{\displaystyle \frac{2}{r_4}}\right)}\times \left(Va{r_{m}ax}_j-{bonobo}_j^i\right) \end{gathered}$$

(19)

where Var_max and Var_min are the upper and lower bounds.

In situations where $r_2$ is greater than pxgm, the consortship mating strategy is employed to generate the new bonobo, as described below:

$${NewBonobo}_j=\left\{\begin{array}{c}{bonobo}_j^i+flag\times e^{-r_5}\times \left({bonobo}_j^i-{bonobo}_j^p\right) if (flag=1 ||r_6\le p_d)\\ {bonobo}_j^p, otherwise\end{array}\right.$$

(20)

3.4. Flood Algorithm (FLA)

The Flood Algorithm (FLA) is a novel metaheuristic optimization algorithm presented by Ghasemi et al. [14] and inspired by the movement and flow patterns of water during flooding events in river basins. It mathematically models key natural phenomena such as water movement towards slopes, flow rates over time, soil permeability effects, and periodic changes in water levels due to precipitation and loss. By using these models, FLA guides the movement of a population of potential solutions to achieve optimal results.

FLA operates in two phases: the regular movement phase, where the population moves toward the best solution, and the flooding phase, which introduces random disturbances to enhance diversity. During the flooding phase, new solutions are periodically introduced while weaker solutions are discarded, mimicking natural water cycles.

As the water flow from the river increases, floods and turbulence may be induced due to variations in the movement of the water volume. The depletion coefficient or water flow is modeled as a function of the algorithm’s iterations, as shown in the following equation:

$$PK={\left(\sqrt{\left(MaxIt\cdot i{t}^2+1\right)}+{\displaystyle \frac{1}{\left(\frac{MaxIt}{4}\cdot it\right)}}\cdot \ln \left(\sqrt{\left(MaxIt\cdot i{t}^2+1\right)}+\left({\displaystyle \frac{MaxIt}{4}}\cdot it\right)\right)\right)}^{-{\displaystyle \frac{2}{3}}}\cdot {\displaystyle \frac{1.2}{it}}$$

(21)

where $PK$ is a scaling factor adjusting the magnitude of a movement, it and MaxIt represent the current and maximum iteration number, respectively.

3.4.1. Regular Movement Phase

The population (or water) moves naturally toward better solutions, similar to water flowing downhill toward a slope. The movement is modeled as follows:

$$P_{ei}={\left({\displaystyle \frac{f\left(S_i\right)-f_{\min }}{f_{\max }-f_{\min }}}\right)}^2$$

(22)

$$\begin{array}{c}if rand>rand+P_{ei}\hfill \\ {\mathrm{S}}_{\mathrm{i}}^{new}={\mathrm{S}}_{\mathrm{i}}+\left({\displaystyle \frac{{\left(Pk\right)}^{rand}}{Iter}}\right)\times rand\times \left({\mathrm{S}}_{max}-{\mathrm{S}}_{min}\right)+{\mathrm{S}}_{min}\\ \mathit{else}\hfill \\ {\mathrm{S}}_{\mathrm{i}}^{new}={\mathrm{S}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}+rand\times \left({\mathrm{S}}_{\mathrm{j}}-{\mathrm{S}}_i\right)\end{array}$$

(23)

$P_{ei}$ represents the fitness probability of the i-th solution, which is calculated by normalizing the cost function value $f\left(S_i\right)$ of the solution ${\mathrm{S}}_i$ with respect to the minimum and maximum values of the population represented by $f_{\min }$ and $f_{\max }$, respectively. rand denotes a random variable uniformly distributed between 0 and 1. ${\mathrm{S}}_{\mathrm{i}}^{new}$ is the new position of the i-th solution, ${\mathrm{S}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ and ${\mathrm{S}}_{\mathrm{i}}$ are the positions of the best solution and a random solution, respectively. ${\mathrm{S}}_{max}$ and ${\mathrm{S}}_{min}$ denote the upper and lower limits of the search space.

3.4.2. Flooding Phase: Increase and Decrease in Basin Water

In this phase, random disturbances are considered to maintain diversity within the population and prevent premature convergence. The algorithm selectively replaces weaker solutions with new, randomly generated solutions, mimicking the behavior of flooding that introduces new water into a system. The mathematical model of this phase is given below:

$$P_t=\left|\mathit{sin}\left({\displaystyle \frac{\mathit{rand}}{\mathit{Iter}}}\right)\right|$$

(24)

where P_t is a flooding probability, calculated based on the current iteration with a random parameter. If this probability is met, the flooding phase is triggered.

$$S_e^{\mathit{new}}=S_{\mathit{best}}+\mathit{rand}\times \left(\mathit{rand}\times \left(S_{\mathit{max}}-S_{\mathit{min}}\right)+S_{\mathit{min}}\right) e=1:Ne$$

(25)

Here, Ne is the number of evaporated water particles representing the weakest members.

3.5. Lung Performance Algorithm (LPO)

The Lung Performance Optimization (LPO), proposed by [15], is a novel meta-heuristic algorithm inspired by the natural processes of the lungs in the human body, specifically the efficiency of gas exchange in the respiratory system. The algorithm’s inspiration draws from the lungs’ ability to exchange oxygen and carbon dioxide, which is a highly efficient and adaptive process. This makes the lungs an ideal model for optimization problems that require the balancing of exploration and exploitation in a solution space. LPO operates through the mechanism of a “swarm”, analogous to the air and blood masses in the lungs, adjusting their positions based on feedback similar to the way oxygen is delivered to the bloodstream and carbon dioxide is removed. The optimization is based on Fick’s law of diffusion and the pressure changes in the respiratory system, which can be modeled as an electrical circuit with resistances and reactances.

3.5.1. The Entrance and Exit of Air Into and Out of the Lungs

The process of air entering and exiting the lungs is modeled by the position updates of the population particles. This is similar to the way air moves in and out of the lungs, and it is mathematically represented by:

$$M_i^{new,1}=M_i+M_i\times \left(\sqrt{{f\left(M_i\right)}^2+{\left({\displaystyle \frac{1}{2\pi \cdot Dim\cdot f\left(M_i\right)\cdot C_i}}\right)}^2}\times \sin \left(2\pi \cdot Dim\cdot iter\right)\times \sin \left(2\pi \cdot Dim\cdot iter+{\theta }_i\right)\right)$$

(26)

$${\theta }_i={\tan }^{-1}\left({\displaystyle \frac{1}{2\pi \times Dim\times f\left(M_i\right)\times C_i}}\right)$$

(27)

$$C_i={\displaystyle \frac{f\left(M_i\right)}{2}}\times \sin \left({\theta }_i\right)$$

(28)

In this equation, $M_i$ and $M_i^{new,1}$ are the current and the firstly updated position of the air and blood mass, respectively, $f\left(M_i\right)$ is the fitness value of the i-th solution, Dim denotes the number of design variables, and ${\theta }_i$ and $C_i$ represent the control parameters of the LPO algorithm.

3.5.2. Carbon Dioxide Separation from the Air and Blood Movement in the Veins

The oxygen extracted by the lungs enters the blood, similar to how particles in an optimization algorithm move through the problem space. These particles shift from areas of higher fitness (more pressure) to lower fitness (less pressure), seeking optimal solutions.

$$M_i^{\mathrm{new},2}=M_i^{\mathrm{new},1}+K_{i1}\times {\mathsf{\alpha }}_i\times \left(M_i^{\mathrm{new},1}-M_1\right)+K_{23}\times {\mathsf{\alpha }}_i\times \left(M_3-M_2\right)$$

(29)

where $K_{i1}$ and $K_{23}$ are direction vectors based on fitness comparisons, ${\mathsf{\alpha }}_i$ denotes a scalar factor controlling movement magnitude, $M_1{, M}_2, \mathrm{a}\mathrm{n}\mathrm{d} M_3$ are reference points in the solution space, and $M_i^{\mathrm{new},2}$ is the second updated position of the i-th solution.

3.5.3. Carbon Dioxide Separation from Blood

This is akin to the crossover and composition of the swarm, where particles (blood) are recombined to form new solutions. It is modeled as follows:

$${\mathrm{m}}_{\mathrm{i},\mathrm{j}}^{new,3}=\left\{\begin{array}{c} {\mathrm{m}}_{i,1}+{\mathrm{a}}_i\times \left({\mathrm{m}}_{3,j}-{\mathrm{m}}_{2,j}\right), if s_i>rand \\ {\mathrm{m}}_{\mathrm{i},\mathrm{j}}^{new,3}={\mathrm{m}}_{\mathrm{i},\mathrm{j}}^{new,2}, else \end{array}\right.$$

(30)

Here, $s_i$ is a probability that decreases with each cycle (exhalation), ${\mathrm{a}}_i$ is a scaling factor of the i-th iteration, and ${\mathrm{m}}_{i,j}$ denotes the j-th design variable of the i-th solution.

4. Numerical Examples

In this section, the efficiency of the proposed algorithms—three different versions of FDB-AGDE, BO, CO, FLA, and LPO—is demonstrated by applying them to three large-scale structural optimization problems involving frequency constraints. The problems include a 600-bar single-layer dome truss, a 1180-bar single-layer dome truss, and a 1410-bar double-layer dome truss. The material properties, bounds on cross-sectional areas, and frequency constraints for these problems are summarized in

Table 1. Material properties and frequency constraints of design examples.

Truss Example	Modulus of Elasticity, E (N/m2)	Material Density, q (kg/m3)	Allowable Cross-Sectional Areas, A (cm2)	Natural Frequency Constraints (Hz)
600-bar dome truss	2.1 × 1011	7800	1 ≤ A ≤ 100	${ \omega }_1\ge 5,{ \omega }_3\ge 7$
1180-bar dome truss	2.1 × 1011	7860	1 ≤ A ≤ 100	${ \omega }_1\ge 7,{ \omega }_2\ge 7,$ ${\omega }_3\ge 9$
1410-bar dome truss	2.1 × 1011	7860	1 ≤ A ≤ 100	${ \omega }_1\ge 7,{ \omega }_3\ge 9$

.

Given the stochastic nature of metaheuristic algorithms, 20 independent successful runs are conducted for each problem, using different randomly generated initial populations for each run. For each experiment, the best, worst, and mean weights, along with the standard deviation (SD) of the optimized weights across the runs, are recorded. Statistical tests (Wilcoxon rank-sum) are performed, and Cohen’s d effect sizes are calculated to assess both the significance and magnitude of differences among algorithms. The design variables corresponding to the best solutions are presented. Furthermore, the number of structural analyses (NSAs) representing the total objective function evaluations required to obtain the best design, and the percentage of constraint violations (CVs) are reported. Specific parameters employed for the seven distinct techniques are presented in

Table 2. Specific parameters of the utilized algorithms.

Algorithms	Specific Parameters
FDB-AGDE-1	-
FDB-AGDE-2	-
FDB-AGDE-3	-
BO	pxgm_init = 0.03; scab = 1.25; scsb = 1.3; rcpp = 0.0035; tsgsfactor = 0.05; pp = pd = 0.5
CO	m = 2
FLA	Ne = 5
LPO	Ne = 5

.

The Friedman rank test is also employed to rank the algorithms based on the best, mean, and worst weights, as well as the standard deviation of the optimized weights. The maximum number of structural analyses is set to 20,000, which serves as the termination criterion for the search process.

It is important to highlight that many algorithms in the literature fail to meet the frequency constraints of these problems. As a result, only proposed algorithms are considered for comparison.

4.1. The 600-Bar Spatial Dome Truss

The first case study, illustrated in Figure 1, examines a truss dome structure consisting of 600 members and 216 nodes. The design process considers the cross-sectional areas of the members as the primary variables. Exploiting the cyclic symmetry of the dome, the members within each substructure—defined by 9 nodes and 25 members—are assigned a uniform cross-sectional area, thereby reducing the number of independent design variables. The substructures are arranged with an angular spacing of 15 degrees between consecutive segments, and the nodal coordinates for each substructure are provided in

Table 3. Nodal coordinates of the substructure of the 600-bar dome truss [39].

Node Number	x, y, and z Coordinates (m)	Node Number	x, y, and z Coordinates (m)
1	1.0  0.0  7.0	6	9.0  0.0  5.0
2	1.0  0.0  7.5	7	11.0  0.0  3.5
3	3.0  0.0  7.25	8	13.0  0.0  1.5
4	5.0  0.0  6.75	9	14.0  0.0  7.0
5	7.0  0.0  6.0

. A nonstructural mass of 100 kgf is considered in all unconstrained nodes.

Table 4. Comparison of optimization results for the 600-bar truss.

Design Variables Ai (cm2)	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
A1	3.023791	1.129101	1.876414	1.629717	1.257963	6.728939	1.246882
A2	2.013871	1.163617	1.904015	1.241892	1.386239	1.231096	1.240454
A3	5.143588	4.281071	7.394453	4.395661	4.314845	2.382298	4.243337
A4	1.262915	1.338621	3.649760	1.490243	1.000000	1.002197	1.243932
A5	19.983319	17.854018	19.913364	18.396481	18.114738	21.181630	18.071250
A6	30.923535	36.261520	42.801216	35.300747	39.270116	31.480740	38.225070
A7	14.672528	14.122496	11.807665	15.161942	13.316925	13.863900	14.108590
A8	15.437327	16.829486	17.129579	16.811120	16.741512	18.784730	16.636750
A9	12.269978	13.446700	13.105120	12.208512	13.171590	21.821850	13.183480
A10	8.866545	9.582605	9.433643	10.005481	9.339057	8.423471	9.449940
A11	9.517119	9.400067	10.370055	9.183313	10.186414	8.751362	9.537473
A12	9.693147	10.015833	9.422258	9.687101	9.961688	14.226620	9.879310
A13	7.452563	7.191178	6.146992	7.224946	6.916795	8.373427	7.297250
A14	5.915914	5.583175	5.544019	5.544384	5.520614	4.685899	5.591398
A15	6.661043	6.997372	7.160914	6.526607	6.532547	21.243240	7.003780
A16	5.212128	5.039284	4.662089	5.471954	4.928692	4.140340	5.257701
A17	5.807982	3.820944	3.574381	3.683945	3.918022	2.571551	3.851501
A18	7.722593	7.667613	8.595977	7.728200	7.654024	7.157081	7.516086
A19	3.882874	4.194781	4.656853	3.843037	4.366477	5.477364	4.088303
A20	1.918084	2.176175	2.107502	2.058667	2.482042	1.377286	2.251530
A21	4.804600	4.982080	5.534956	4.577454	4.841515	3.915202	4.529711
A22	3.043238	3.324548	2.821205	3.311281	3.970928	2.733476	3.628636
A23	2.273651	1.752116	1.905604	1.999731	1.683172	5.349617	1.796958
A24	6.119064	4.977984	5.504228	5.050156	4.890802	18.268850	4.905784
A25	2.084542	1.673162	1.632660	1.572683	1.520068	3.372308	1.598683
Best weight (kg)	6598.6592	6339.9902	6507.9279	6352.8637	6345.9484	8260.9300	6335.1780
Mean weight (kg)	6706.9439	6400.5998	6625.1324	6719.0154	6563.4734	10,368.3000	6345.0550
Worst weight (kg)	6873.6474	7728.2814	6809.6114	7753.3576	7831.8871	12,274.3600	6361.9060
STD (kg)	72.5950	250.9175	73.0005	600.6546	451.1856	1221.3330	5.6611
CV (%)	0	0	0	0	0	0	0
NSA	20,000	20,000	20,000	20,000	20,000	20,000	20,000
Friedman Rank	6	2	5	4	3	7	1

compares proposed optimization algorithms—including three variants of the FDB-AGDE approach (FDB-AGDE-1, FDB-AGDE-2, and FDB-AGDE-3), CO, BO, FLA, and LPO—for the optimum design of a 600-bar dome truss. Among all methods, LPO achieves the lightest “best” design with 6335.178 kg, slightly outperforming the second-best method, FDB-AGDE-2 (6339.990 kg), and third-best, BO (6345.948 kg). CO follows closely with 6352.864 kg for its best solution, whereas FDB-AGDE-3 and FDB-AGDE-1 have best solutions of 6507.928 kg and 6598.659 kg, respectively. FLA yields a substantially higher best weight with 8260.930 kg, indicating that it was less effective in exploring the design space for this specific problem setup.

The last row in Table 4 provides a Friedman rank comparison, showing that LPO is ranked first overall (the best), followed in order by FDB-AGDE-2 (second), BO (third), CO (fourth), FDB-AGDE-3 (fifth), FDB-AGDE-1 (sixth), and FLA (seventh). This statistical ranking underscores the superior performance of LPO on the 600-bar truss example.

Table 5. Natural frequencies calculated for the optimized designs of the 600-bar truss structure.

	Mode	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
	1	5.223125	5.010574	5.024269	5.000221	5.000368	5.195123	5.010798
	2	5.223125	5.010574	5.024269	5.000221	5.000368	5.195123	5.010798
MATLAB	3	7.019655	7.000239	7.018063	7.000111	7.000001	7.024405	7.000080
	4	7.019655	7.000618	7.018063	7.000111	7.000010	7.024405	7.000133
	5	7.022560	7.000618	7.020028	7.000224	7.000010	7.082534	7.000133
	1	5.223125	5.010574	5.024269	5.000221	5.000368	5.195123	5.010798
	2	5.223125	5.010574	5.024269	5.000221	5.000368	5.195123	5.010798
SAP2000	3	7.019655	7.000239	7.018063	7.000111	7.000001	7.024405	7.000080
	4	7.019655	7.000618	7.018063	7.000111	7.000010	7.024405	7.000133
	5	7.022560	7.000618	7.020028	7.000224	7.000010	7.082534	7.000133

shows the natural frequencies for the first five modes calculated by two different finite-element solvers (MATLAB-based routines and SAP2000) for each optimized design. In all cases, the frequencies closely match the required constraints; CO and BO algorithms are pushing the first two modes near or at the 5.0 Hz boundary and the next three modes near 7.0 Hz. Overall, all listed designs remain feasible, satisfying the natural frequency constraints while achieving varied levels of weight reduction.

Figure 2 shows the weight reduction trajectories of seven metaheuristics over 200 iterations for the 600-bar dome truss problem. The proposed algorithms exhibit a rapid decrease in weight within the first 25–50 iterations, reflecting their ability to locate promising regions of the design space early on. For instance, FDB-AGDE-2, BO, and LPO all see significant weight drops in the first 25 iterations, suggesting strong global search components.

While all methods continue refining solutions over time, some plateau sooner than others. FLA levels off at a relatively higher weight (above ~10,000 kg) and does not appear to improve substantially after about 100 iterations. In contrast, LPO and FDB-AGDE-2 exhibit steady progress into the later stages, converging to lower final weights. By iteration 200, LPO and FDB-AGDE-2 appear to reach the best overall solutions, whereas FLA lags behind. CO, BO, and the FDB-AGDE-1 and 3 variants end up in intermediate positions. Overall, the convergence trends highlight a trade-off between how quickly different metaheuristics exploit “good” designs versus how effectively they continue refining solutions in the later stages of the run.

Figure 3 provides a statistical snapshot of final weights across multiple runs of each algorithm, FDB-AGDE-2 and LPO. Box plots are centered around the lowest weights. BO, FDB-AGDE-1, and FDB-AGDE-3 occupy the middle range, typically around 6500–7500 kg. CO is somewhat more scattered but mostly between 7000 and 8500 kg. By far the highest weights—and widest spread—are seen in FLA, reflecting both higher mean solutions and greater variability. A single run of FDB-AGDE-2 produced a slightly heavier design (>7000 kg), whereas BO shows a few runs dipping below its typical range. While FDB-AGDE-2 and LPO both feature lower medians, LPO’s box and whiskers are typically very narrow, suggesting it is consistently near its best solutions. FDB-AGDE-2, though capable of similar or better solutions, shows a slightly broader spread. The red marks in the figure denote outliers, individual runs that fall outside the expected whisker range In practical engineering contexts, lower spreads and smaller outliers can be advantageous, as they indicate greater reliability of the algorithm across repeated optimization runs.

Figure 4 provides the p-values from pairwise Wilcoxon rank-sum tests comparing the final weights of each pair of algorithms. Typically, a p-value < 0.05 is taken to indicate a statistically significant difference between two methods. The cells showing values on the order of 10⁻⁶ or 10⁻⁸ are highly significant. If a pairwise p-value is relatively large, such as 0.88 in the BO–CO comparison, it suggests there is insufficient evidence to conclude these two algorithms’ distributions differ. On the other hand, extremely small p-values < 10⁻⁶ indicate strong evidence that one algorithm outperforms the other on average. From the figure, LPO differs significantly from most other approaches, reflecting that its results are consistently among the best. FLA, by contrast, also exhibits highly significant differences from the rest, but in the opposite direction, and tends to yield consistently higher and more variable final weights.

Table 6. Cohen’s effect sizes for pairwise comparisons of metaheuristics in 600-bar dome truss optimization.

	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
FDB-AGDE-1		1.27973	0.81225	0.019112	0.560244	4.225151	6.156702
FDB-AGDE-2	1.27973		1.014166	0.621575	0.322801	4.43077	0.355874
FDB-AGDE-3	0.81225	1.014166		0.165109	0.35634	4.30046	6.188649
CO	0.019112	0.621575	0.165109		0.343111	3.791843	0.880433
BO	0.560244	0.322801	0.35634	0.343111		4.190787	0.660715
FLA	4.225151	4.43077	4.30046	3.791843	4.190787		4.658568
LPO	6.156702	0.355874	6.188649	0.880433	0.660715	4.658568

reports Cohen’s d effect sizes for each pairwise comparison of algorithms. Whereas the Wilcoxon test focuses on whether two distributions differ significantly, Cohen’s d gauges how large that difference is. Typical benchmarks for Cohen’s d are as follows:

d = 0.2: Small effect;

d = 0.5: Medium effect;

d = 0.8: Large effect.

LPO vs. FDB-AGDE-1 at d ≈ 6.16, FLA vs. FDB-AGDE-1 at d ≈ 4.23 far exceed 0.8, implying a very large practical difference. Such large d values confirm that the performance gap is not only statistically significant but also highly meaningful in engineering practice. In contrast, effect sizes near 0.02–0.34, such as CO vs. BO, BO vs. FDB-AGDE-3, suggest relatively small practical distinctions in their final weight distributions.

4.2. The 1180-Bar Spatial Dome Truss

In the second case study, illustrated in Figure 5, a dome-type truss with 1180 members and 400 nodes is investigated. The principal design variables are the cross-sectional areas of the truss members. To exploit the dome’s cyclic symmetry, the structure is divided into substructures, each composed of 20 nodes and 59 members. Within each substructure, all members share the same cross-sectional area, effectively reducing the total number of independent variables. The substructures are arranged with an 18-degree angular spacing between segments, and the nodal coordinates for one representative segment are listed in

Table 7. Nodal coordinates of the substructure of the 1180-bar dome truss [39].

Node Number	x, y, and z Coordinates (m)	Node Number	x, y, and z Coordinates (m)
1	3.1181    0.0  14.6723	11	4.5788    0.7252  14.2657
2	6.1013    0.0  13.7031	12	7.4077    1.1733  12.9904
3	8.8166    0.0  12.1354	13	9.9130    1.5701  11.1476
4	11.1476  0.0  10.0365	14	11.9860  1.8984  8.8165
5	12.9904  0.0  7.5000	15	13.5344  2.1436  6.1013
6	14.2657  0.0  4.6358	16	14.4917  2.2953  3.1180
7	14.9179  0.0  1.5676	17	14.8153  2.3465  0.0
8	14.9179  0.0  −1.5677	18	14.4917  2.2953  −3.1181
9	14.2656  0.0  −4.6359	19	13.5343  2.1436  −6.1014
10	12.9903  0.0  −7.5001	20	3.1181   0.0   13.7031

. Furthermore, an additional mass of 100 kgf is considered for all unconstrained nodes.

Table 8. Comparison of optimization results for the 1180-bar truss.

Design Variables Ai (cm2)	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
A1	7.874257	7.406661	7.137683	7.341092	8.033436	12.065870	7.586136
A2	10.063780	8.895251	11.788799	8.344326	10.403249	18.883870	8.903712
A3	4.910137	4.809313	13.614719	3.180261	2.711354	26.737830	11.403790
A4	14.695244	19.069249	23.679900	16.392227	15.006789	28.308650	15.534610
A5	2.909845	5.293858	6.986300	3.345957	3.896306	12.904980	3.764825
A6	7.115904	7.095440	6.228295	6.554237	5.851600	27.076660	6.674349
A7	6.846220	6.836420	6.387648	6.090645	7.289934	8.864943	7.834553
A8	8.800411	7.543361	9.590688	6.208024	6.949541	13.544940	6.571596
A9	1.673241	3.476135	2.475425	3.540716	1.879956	1.092358	2.190498
A10	14.991772	11.881658	7.964409	11.483886	11.311291	10.191710	11.688330
A11	16.363290	7.167235	11.791246	6.178595	7.471935	29.124340	7.865906
A12	8.481276	5.033267	11.763259	5.537235	6.065526	63.762680	6.436489
A13	6.170349	8.099839	7.245651	8.155172	7.101260	6.708742	7.969050
A14	9.015904	6.593022	11.534725	11.377883	6.898631	6.013308	7.292745
A15	9.383343	10.243248	10.153842	8.704600	9.752487	72.305020	10.477170
A16	5.084130	7.726060	8.501315	6.542469	6.545987	8.504965	5.577529
A17	9.901966	8.148965	6.434932	6.812164	7.210488	16.684040	9.067837
A18	7.212494	9.423108	8.026447	7.634667	8.751584	25.917830	7.722360
A19	14.900105	10.738194	9.968436	12.857575	14.054030	18.073100	13.969100
A20	4.471479	9.155714	11.479431	8.200010	6.429336	20.478890	9.843234
A21	9.834071	8.347413	8.419979	10.329897	10.047452	35.691520	11.025380
A22	12.464360	9.200902	5.763701	9.210465	9.070010	14.227740	9.142028
A23	13.279927	15.734809	20.932816	17.566533	17.172019	19.178010	21.280410
A24	12.909323	9.879389	8.879572	13.236654	10.765740	42.416180	10.116070
A25	19.849410	14.450982	16.254258	9.782828	13.643317	34.920030	12.228450
A26	17.116466	11.102753	6.027643	11.314389	10.427723	19.341960	12.398480
A27	25.230044	26.513425	21.529700	27.203381	22.742262	15.162700	25.406910
A28	11.384612	14.169903	19.223520	16.708121	11.949719	54.156250	13.552570
A29	13.256166	18.420711	19.726730	22.512111	15.483744	36.372350	21.023240
A30	18.706791	14.483310	16.383395	16.627003	16.013196	27.273130	17.371400
A31	42.960011	39.078312	48.417718	35.344354	37.503834	39.237400	33.663920
A32	16.490115	19.402189	25.786561	17.694735	19.678845	38.985940	19.147790
A33	30.200266	24.419041	28.723833	34.863078	29.031532	59.195030	26.353450
A34	26.428994	21.441246	23.842243	20.774481	21.351047	43.429560	20.196950
A35	54.489776	48.292333	59.666863	46.117090	53.202618	79.609570	43.312430
A36	23.641724	22.003935	38.880077	27.760575	27.033619	33.187020	28.010940
A37	29.134567	30.744031	26.408696	37.229955	32.893607	39.655420	34.872710
A38	22.696805	34.245250	48.070078	31.976388	30.070674	39.799000	27.073780
A39	41.181191	47.060922	36.881653	36.821762	35.539288	44.507250	33.745320
A40	5.913338	1.988946	2.655745	2.607668	1.000000	3.009441	1.408517
A41	7.547779	8.019023	8.536639	14.904613	10.478436	11.027630	8.988470
A42	16.820410	7.338677	10.579021	7.875747	5.840658	71.282890	6.986653
A43	7.469341	7.646755	8.213646	6.861987	5.949529	10.640270	7.601176
A44	9.131612	5.736607	7.635684	7.361532	6.340990	55.241720	6.781792
A45	5.796440	9.024426	3.739599	5.754250	6.893456	9.912174	7.375494
A46	6.831707	8.653040	7.966721	5.646005	5.600438	24.713670	6.003010
A47	10.743584	10.845934	7.118326	7.563096	8.920467	23.365030	9.659027
A48	13.541436	9.554500	13.673555	8.866247	5.665204	18.299500	7.527485
A49	11.731393	9.547094	10.036086	12.792507	10.709867	17.926910	11.049190
A50	12.890989	12.429990	14.567269	10.181782	9.280738	23.102630	11.502510
A51	25.563242	13.640546	11.930903	15.209169	13.411523	76.953520	14.859430
A52	10.118511	14.698245	13.804554	13.119193	12.723477	13.043160	14.238340
A53	19.802142	19.562892	16.259706	19.478679	19.150101	43.757370	17.925480
A54	27.251787	18.091759	16.667135	18.809209	21.334890	43.904440	16.480130
A55	39.500083	19.882811	19.703237	21.962420	26.309583	32.657630	28.848710
A56	20.852584	24.350124	30.319425	25.522552	24.625139	26.114530	26.045850
A57	47.364580	32.664712	41.999328	28.322514	34.803636	74.924180	35.628560
A58	40.438513	37.172432	47.688660	35.955349	36.755497	51.762490	37.183100
A59	11.901404	5.376399	5.221828	3.649674	5.166103	13.086730	6.384748
Best weight (kg)	42,941.683	38,767.648	43,042.693	38,963.136	38,030.272	76,891.100	38,610.050
Mean weight (kg)	46,075.984	40070.648	44,339.278	40,032.339	38,863.270	10,0367.200	39,999.660
Worst weight (kg)	48,104.622	41,353.299	46,513.243	40,761.896	39,511.108	137,741.500	41,774.340
STD (kg)	970.924	563.950	770.571	426.182	408.772	12,928.480	815.643
CV (%)	0	0	0	0	0	0	0
NSA	20,000	20,000	20,000	20,000	20,000	20,000	20,000
Friedman Rank	6	3	5	4	1	7	2

presents the optimization results of the proposed algorithms. The best-found design is achieved by BO with a value of 38,030 kg, followed closely by LPO at 38,610 kg, CO at 38,963 kg, and FDB-AGDE-2 at 38,768 kg. FLA ranks last with a design weight of 76,891 kg. BO also demonstrates stable performance, with a low mean value of 38,863 kg and a relatively small standard deviation of 409 kg. Both LPO and CO exhibit similar robustness in their results. In contrast, FLA shows a significantly higher average of 100,367 kg and a large standard deviation of 12,928 kg. Considering the overall distribution and performance, BO ranks first, followed by LPO in second place and FDB-AGDE-2 in third. FLA occupies the lowest rank, securing seventh place, as summarized in the final row of Table 8.

Table 9. Natural frequencies calculated for the optimized designs of the 1180-bar truss structure.

	Mode	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
MATLAB	1	7.009174	7.003915	7.003153	7.001028	7.000115	7.017611	7.003076
	2	7.009174	7.003915	7.003153	7.001028	7.000115	7.017611	7.003076
	3	9.043173	9.013461	9.105209	9.004496	9.000323	9.156227	9.047232
	4	9.043173	9.013461	9.105209	9.004496	9.000323	9.156227	9.047232
	5	9.462299	9.233525	9.200931	9.190641	9.000420	9.551542	9.172685
SAP2000	1	7.009174	7.003915	7.003153	7.001028	7.000115	7.017611	7.003076
	2	7.009174	7.003915	7.003153	7.001028	7.000115	7.017611	7.003076
	3	9.043173	9.013461	9.105209	9.004496	9.000323	9.156227	9.047232
	4	9.043173	9.013461	9.105209	9.004496	9.000323	9.156227	9.047232
	5	9.462299	9.233525	9.200931	9.190641	9.000420	9.551542	9.172685

demonstrates that all final designs maintain natural frequencies that are either near or slightly above the specified lower bound—approximately 7 Hz for the first two modes and around 9 Hz for the next three modes. Upon examining the results, it is evident that BO reaches these limit values with only a very small difference, indicating an effort to obtain the optimal solution. It is important to note that the MATLAB solver produces frequency values that are entirely consistent with those obtained from SAP2000, validating the accuracy of the finite-element modeling approach.

Figure 6 displays the convergence behaviors for seven metaheuristics over 200 iterations in optimizing the 1180-bar dome truss. All algorithms make substantial progress in the first 20–30 iterations, reflecting strong initial exploration of the design space. However, by iteration ~50, their performance diverges, with BO, CO, FDB-AGDE-2, and LPO continuing to trend downward more aggressively. FDB-AGDE-1 and FDB-AGDE-3 show a steady but slower improvement over the later iterations, leveling off at higher total weights than their top-performing counterparts. FLA converges prematurely to a considerably heavier design, illustrating that its parameter settings or exploration strategy may be less effective in this large-scale dome problem. BO, LPO, and CO continue to make modest refinements past iteration ~100. Although the improvement slows, these refinements cumulatively lead to lower final weights by iteration 200. FDB-AGDE-2 also shows a relatively sustained descent, though it levels off slightly sooner than BO or LPO.

Figure 7 provides a statistical comparison of the final solution quality across multiple runs. BO attains the smallest median weight (approximately 38,800 kg) with a very narrow spread, indicating both strong performance and high consistency. LPO and CO also show relatively tight distributions, though their means and medians are slightly higher. FDB-AGDE-2 yields competitive solutions, with some runs closely rivaling those of BO. However, the other variants, namely FDB-AGDE-1 and FDB-AGDE-3, display higher medians and wider spreads in their results. This performance gap may stem from differences in parameter tuning or the hybridization strategies used in each variant. FLA’s box reveals both high median weight (>70,000 kg) and large spread, confirming it struggles with this high-dimensional dome truss relative to the other algorithms. Notably, FLA exhibits a single outlier far above its median, indicating an exceptionally heavy design compared to the rest of its runs.These results underscore that not only is the final weight important, but so too is consistency across runs.

Figure 8 illustrates the p-values from Wilcoxon rank-sum tests, while

Table 10. Cohen’s effect sizes for pairwise comparisons of metaheuristics in 1180-bar dome truss optimization.

	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
FDB-AGDE-1		10.41499	2.225914	11.17907	13.43934	6.865008	8.20034
FDB-AGDE-2	10.41499		7.393442	0.082439	2.867457	7.586172	0.018019
FDB-AGDE-3	2.225914	7.393442		7.903801	10.01145	7.04714	5.924849
CO	11.17907	0.082439	7.903801		3.191828	7.592472	0.036963
BO	13.43934	2.867457	10.01145	3.191828		7.729345	1.907356
FLA	6.865008	7.586172	7.04714	7.592472	7.729345		7.576501
LPO	8.20034	0.018019	5.924849	0.036963	1.907356	7.576501

reports Cohen’s d-effect sizes for pairwise comparisons. Extremely low p-values between BO and other algorithms confirm that the performance differences in final weights are statistically significant. Where p-values approach 1.0 in some comparisons among CO, FDB-AGDE-2, and LPO, there is insufficient evidence to claim one method definitively outperforms the other. Several pairwise comparisons yield very large d-values, indicating not just statistical significance but also high practical importance. For instance, comparing BO and FLA yields d ≈ 7.73, emphasizing that FLA’s inferior results are both reliably and substantially worse. Conversely, when d < 0.1, the two methods’ distributions are nearly indistinguishable from a practical standpoint.

4.3. The 1410-Bar Spatial Dome Truss

In the final and the largest example, a dome structure comprising 390 nodes and 1410 members is utilized, as illustrated in Figure 9. The geometry is divided into repeating substructures, each containing 13 nodes and 47 members, with the nodal coordinates for one representative substructure provided in

Table 11. Nodal coordinates of the substructure of 1410-bar dome truss.

Node Number	x, y, and z Coordinates (m)	Node Number	x, y, and z Coordinates (m)
1	1.0    0.0    4.0	8	1.989    0.209    3
2	3.0    0.0    3.75	9	3.978    0.418    2.75
3	5.0    0.0    3.25	10	5.967    0.627    2.25
4	7.0    0.0    2.75	11	7.956    0.836    1.75
5	9.0    0.0    2	12	9.945    1.0453  1
6	11.0  0.0    1.25	13	11.934  1.2543  −0.5
7	13.0  0.00  0.0

. The primary objective is a sizing optimization, where the cross-sectional areas of the 47 member groups serve as the design variables. In addition, each free node is assigned a nonstructural mass of 100 kg.

Table 12. Comparison of optimization results for the 1410-bar truss.

Design Variables Ai (cm2)	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
A1	5.181748	3.594279	4.128842	5.098185	3.330343	2.679416	6.395886
A2	1.945098	5.373930	5.491109	4.511558	3.795250	4.360587	4.697734
A3	33.630189	23.011743	21.197120	24.711879	16.473644	3.405862	31.194970
A4	10.178946	8.717958	10.186617	6.766575	9.314022	11.670340	10.309150
A5	3.673742	5.951188	7.129077	5.348120	6.629964	3.058517	6.045048
A6	6.059775	2.842269	3.088278	2.392626	1.574074	4.369000	1.608069
A7	28.682730	18.055471	17.468484	12.740975	31.719067	69.743820	16.595200
A8	11.922395	11.440906	12.596099	7.903293	9.916940	17.442140	9.176643
A9	4.514635	3.251135	6.235259	2.959766	1.684694	8.080519	2.562129
A10	3.190210	3.878719	5.020697	2.821628	2.485740	2.276878	2.728512
A11	8.538754	10.711261	6.258136	10.721724	12.235064	18.486370	6.636622
A12	11.455126	10.641076	10.398511	11.869204	8.319166	10.645770	10.014390
A13	9.043816	2.509710	2.689918	2.486207	2.979454	1.028391	2.002495
A14	5.889539	5.267594	3.831996	4.990523	4.475278	9.597372	5.723452
A15	19.981151	15.628517	9.685841	14.253720	10.809117	19.895660	17.743890
A16	7.070823	8.065009	6.107678	9.215784	8.825649	15.175880	8.554423
A17	4.755079	5.327118	6.677618	3.906221	3.512708	4.770585	4.330146
A18	6.747342	4.719784	6.635954	6.187140	6.312047	11.168100	6.739172
A19	7.591766	5.327164	9.054030	10.391954	13.392728	12.466510	9.594120
A20	10.863922	15.010132	16.617751	15.560866	13.383526	15.074720	13.816910
A21	8.763071	4.511277	3.793691	5.317987	5.412711	8.739791	5.802691
A22	7.266203	7.157492	6.952273	6.431795	8.362446	8.053586	7.125749
A23	6.865031	5.736343	6.540901	2.101887	1.000000	1.614472	1.541427
A24	2.809976	3.728914	7.259128	6.379180	4.692243	6.361013	4.646534
A25	2.885440	3.774084	2.658645	2.439425	2.843081	1.703143	2.851894
A26	4.888448	3.582424	6.774899	5.023449	5.540875	1.378022	5.251167
A27	2.749199	6.736315	3.550336	4.951499	5.848941	4.942347	6.898313
A28	10.495416	11.462157	10.544361	9.336837	11.950240	10.126170	12.228630
A29	4.543833	4.397680	3.762613	4.254605	4.076259	4.422458	3.572397
A30	1.563968	1.554140	4.245685	2.773976	2.039153	7.784655	1.545229
A31	4.180324	2.373279	2.844768	3.699297	1.739721	9.756918	2.421220
A32	7.728548	3.410971	11.128707	5.240978	3.058070	8.822998	3.724947
A33	4.087358	5.998652	5.826317	5.734029	6.345732	2.524770	4.162181
A34	4.329644	3.830342	4.498434	3.876574	3.261677	1.836376	2.599018
A35	7.719226	4.005101	3.739867	2.500021	2.769724	4.488586	2.737338
A36	4.611545	4.390313	1.051559	1.565100	3.072166	2.160384	3.413748
A37	8.490724	6.892918	7.236847	9.557572	5.928834	9.411659	7.220829
A38	6.378797	4.884825	6.897034	4.543275	4.352731	3.515112	5.408577
A39	2.360077	4.696529	4.951945	4.204315	3.289214	7.944206	3.544814
A40	3.733957	1.032798	3.203468	1.187727	1.000000	1.211464	1.085991
A41	8.002200	7.661073	8.267425	7.971756	7.438468	4.951190	6.904313
A42	4.764519	5.650634	5.297260	5.537456	6.310769	3.008724	6.314960
A43	10.970546	5.004071	5.104024	5.390138	5.314027	3.478641	4.912087
A44	1.365050	1.381780	2.566834	1.155712	1.036771	1.036164	1.000000
A45	5.098131	7.429253	7.154799	7.956031	8.010678	11.350250	7.727939
A46	3.666173	4.737608	5.180218	3.926615	5.003753	5.912236	3.267847
A47	8.426747	1.666773	1.540386	1.274095	3.052483	1.907582	1.142857
Best weight (kg)	12,136.585	10,738.563	11,450.266	10,573.074	10,524.036	13,400.360	10,407.260
Mean weight (kg)	12,695.509	10,946.306	12,043.612	10,845.924	11,021.863	18,448.180	10,575.910
Worst weight (kg)	13,198.014	11,200.830	12,387.130	11,105.419	11,666.720	22,500.480	11,157.420
STD (kg)	287.632	139.758	230.137	135.411	313.799	2413.004	193.226
CV (%)	0	0	0	0	0	0	0
NSA	20,000	20,000	20,000	20,000	20,000	20,000	20,000
Friedman Rank	6	4	5	2	3	7	1

presents the best, mean, and worst structural weights found by each metaheuristic, along with their standard deviations, constraint violations (CVs), and Friedman ranks. LPO achieves the lightest best solution at 10,407 kg, closely followed by BO at 10,524 kg and CO at 10,573 kg. FDB-AGDE-2 also performs well, reaching a best solution of 10,738 kg, while FDB-AGDE-1 and FDB-AGDE-3 converge to slightly heavier best solutions, with values of 12,136 kg and 11,450 kg, respectively. FLA yields the heaviest best weight of 13,400 kg, indicating less effective exploration or local refinement in this high-dimensional problem. LPO again stands out with the lowest mean weight of 10,576 kg, reflecting consistently good performance across multiple runs. The methods ranked second and third in terms of mean weight are CO with 10,846 kg and FDB-AGDE-2 with 10,946 kg. FLA, on the other hand, not only exhibits a higher mean weight of 18,448 kg but also shows a large standard deviation of 2413 kg, indicating significant variability in its results across different runs. This high variability stands in stark contrast to the top-performing algorithms, which maintain standard deviations below approximately 300 kg. According to the Friedman ranking, LPO holds the top position (rank 1), followed by CO in second (rank 2), BO in third (rank 3), and FDB-AGDE-2 in fourth (rank 4). FLA ranks last (7), confirming its comparatively weaker performance under these conditions.

Table 13. Natural frequencies calculated for the optimized designs of the 1410-bar truss structure.

	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
MATLAB	7.004349	7.005965	7.014940	7.001217	7.000325	7.037261	7.002323
	7.004349	7.005965	7.014940	7.001217	7.000325	7.037261	7.002323
	9.062186	9.006729	9.008920	9.000524	9.000150	9.279117	9.004569
	9.140402	9.014389	9.046751	9.000524	9.000150	9.279117	9.004569
	9.140402	9.014389	9.046751	9.001757	9.000423	9.303549	9.017862
SAP2000	7.004349	7.005965	7.014940	7.001217	7.000325	7.037261	7.002323
	7.004349	7.005965	7.014940	7.001217	7.000325	7.037261	7.002323
	9.062186	9.006729	9.008920	9.000524	9.000150	9.279117	9.004569
	9.140402	9.014389	9.046751	9.000524	9.000150	9.279117	9.004569
	9.140402	9.014389	9.046751	9.001757	9.000423	9.303549	9.017862

confirms that all designs meet the frequency constraints, with only minor differences in the first five modes. Upon examining the table, it becomes clear that approaching the frequencies of the fourth and fifth modes is critical for attaining the optimum solution. While BO and CO achieve the limit in all five frequency modes, other algorithms fall short, particularly in approximating the limits of the fourth and fifth mode frequencies. The near-identical results from MATLAB and SAP2000 validate that each algorithm’s final design remains feasible in two different solvers.

Figure 10 shows how each of the seven metaheuristics reduces the dome’s weight over 200 iterations. Almost all algorithms lower the structural weight dramatically in the first 20–30 iterations, indicating strong early-stage exploration. In particular, BO, CO, and LPO drop below 40,000 kg after ~50 iterations, outpacing FLA and the FDB-AGDE variants at that stage. After ~75 iterations, differences among methods become clearer. LPO and CO continue steady improvement, with LPO approaching ~10,000 kg near iteration 150. FDB-AGDE-2 also exhibits consistent progress, stabilizing around ~11,000 kg by iteration 200. By the end, LPO, BO, and CO converge to lighter designs than FDB-AGDE-1/-3, FDB-AGDE-2, or FLA. FLA lags significantly behind, indicating possible difficulties in converging on the global optima.

Figure 11 provides box plots of final weights from repeated runs. LPO’s box is centered around the lowest overall weight (~10,500 kg), reflecting a strong and consistent performance. BO and CO follow closely, each with medians slightly above LPO’s. By contrast, FLA exhibits the highest median (>18,000 kg) and a large interquartile range, highlighting both heavier solutions and higher variability. FDB-AGDE-2 outperforms FDB-AGDE-1 and FDB-AGDE-3, with a somewhat lower median and narrower spread. Nevertheless, it still ranks below CO, BO, and LPO in terms of final weight. A few outliers appear for BO and LPO, suggesting occasional runs may deviate from their typical performance. However, these outliers remain relatively close to the main distribution, underscoring the general reliability of these methods.

Figure 12 presents pairwise p-values from the Wilcoxon rank-sum test for the final weights. The cells reveal extremely small p-values when comparing a top performer, such as LPO, with a weaker performer, like FLA or FDB-AGDE-1, confirming that the differences in final weights are statistically robust. In certain comparisons among CO, BO, and LPO, moderate to larger p-values are observed, such as 0.1017 or 2.302 × 10⁻⁵, indicating that while the methods are not always identical, their weight distributions are often quite similar, making it less conclusive in distinguishing the best among them. These test results suggest that LPO, BO, and CO generally form a statistically distinct top tier, while the FDB-AGDE variants and FLA typically rank lower.

Table 14. Cohen’s effect sizes for pairwise comparisons of metaheuristics in 1410-bar dome truss optimization.

	FDB-AGDE-1	FDB-AGDE-2	FDB-AGDE-3	CO	BO	FLA	LPO
FDB-AGDE-1		7.735565	2.502715	8.227756	5.560281	3.347824	8.650732
FDB-AGDE-2	7.735565		5.763521	0.729512	0.31106	4.389342	2.196544
FDB-AGDE-3	2.502715	5.763521		6.343328	3.713209	3.736632	6.907323
CO	8.227756	0.729512	6.343328		0.728022	4.448531	1.618359
BO	5.560281	0.31106	3.713209	0.728022		4.316072	1.711356
FLA	3.347824	4.389342	3.736632	4.448531	4.316072		4.599055
LPO	8.650732	2.196544	6.907323	1.618359	1.711356	4.599055

details Cohen’s d effect sizes for each pairwise comparison. Many pairs involving FLA exhibit d-values above 4.0 (e.g., FLA vs. LPO at 4.60), signifying a substantial practical difference in final weights. Similarly, comparing LPO to FDB-AGDE-1 yields d ≈ 8.65, reinforcing that these two methods lie on opposite ends of the performance spectrum. Comparisons between CO and BO, or BO and LPO, often produce d-values near or below 1.0, indicating that while still statistically different, in practical terms they may yield designs of similar quality.

Taken together, the effect sizes confirm that the performance gaps among the top algorithms like LPO, BO, and CO are smaller than those separating them from methods like FLA or FDB-AGDE-1.

5. Conclusions

This study assessed the performance of metaheuristic algorithms FDB-AGDE (FDB-AGDE-1, FDB-AGDE-2, FDB-AGDE-3), CO, BO, FLA, and LPO in designing large-scale truss and dome structures under frequency constraints. Three case studies were examined: a 600-bar truss, an 1180-bar dome, and a 1410-bar dome. Minimizing the overall structural weight subjected to prescribed natural frequency bounds. Across all three case studies, LPO and BO consistently produced the lightest designs and exhibited relatively small standard deviations, indicating both high solution quality and repeatability. FDB-AGDE-2 performed nearly as well in many trials, particularly for the 600-bar and 1180-bar domes. CO also achieved competitive performance, especially in the two larger dome cases. Meanwhile, FDB-AGDE-1 and FDB-AGDE-3 converged to feasible solutions but tended to plateau at heavier weights, suggesting that their balance between global exploration and local exploitation may require further refinement for high-dimensional truss problems. By contrast, FLA displayed noticeably heavier designs and higher variance, pointing to algorithmic limitations or parameter settings that were less suited to the complexity of these structures. The Friedman test rankings reveal that algorithmic performance is influenced by the problem size. Across all cases, LPO and BO consistently achieve the best rankings, while FLA remains the weakest performer. As the truss size increases from 600 to 1410 bars, CO improves in relative ranking, moving from mid-level performance to second place, which suggests that its adaptive search dynamics scale well with dimensionality. In contrast, the FDB-AGDE variants show a gradual decline in rank with problem size, indicating that their differential-based updating mechanisms are less effective in very large search spaces. These observations suggest a general trend: algorithms with strong adaptive exploration–exploitation control, such as CO, LPO, and BO, demonstrate better scalability, whereas methods with more rigid update rules, like FDB-AGDE variants, tend to lose efficiency as dimensionality increases. In addition to the best, mean, and worst weights, the Wilcoxon rank-sum and the Cohen’s d effect sizes are calculated to assess both the significance and magnitude of differences among algorithms. These tests confirmed that the top-tier methods (LPO, BO, CO, and occasionally FDB-AGDE-2) form a statistically distinct group in terms of solution quality and robustness. The frequency analyses showed that all optimized designs satisfied dynamic constraints without any discrepancies between MATLAB and SAP2000 solvers, thus validating the feasibility of each final design.

The findings highlight LPO, BO, and CO as especially promising methods, owing to their balanced exploitation–exploration behavior, robustness in converging to lightweight solutions, and comparatively low variance across runs. In practice, LPO and BO are more suitable when robustness and consistency are prioritized, whereas CO is advantageous in cases requiring faster convergence. For implementation, optimized continuous designs can be mapped to standard catalog sections, thereby yielding code-compliant and feasible solutions. Looking ahead, future research may focus on adaptive or hybrid strategies that combine the strengths of these algorithms, extend the framework to larger and more complex structural systems with discrete optimization, and incorporate multi-hazard considerations such as seismic and wind effects, as well as applications to different structural types such as steel frames.