A New Approach to Circulatory System-Based Optimization for the Shape and Sizing Design of Truss Structures

Abstract

This paper presents a new approach to the circulatory system-based optimization algorithm called NCSBO. The proposed method incorporates an adaptive parameter-tuning strategy and a refined mutation approach to improve the performance of optimizing the shape and size of truss structures. The adaptive parameter tuning dynamically adjusts the number of weaker solutions according to the diversity of the population, enabling the algorithm to intelligently balance exploration and exploitation. The mutation strategy selectively alters only one variable of the best solution, preserving its overall structure while conducting a detailed local search to exploit promising regions more efficiently. The validation of the proposed algorithm involves four different truss examples. The design constraints include stress, displacement, and buckling, and are considered in the optimum design. A comprehensive assessment of the NCSBO on various benchmark design examples is carried out and compared with state-of-the-art methods published in the literature. The results demonstrate the NCSBO algorithm’s superiority in achieving the optimum design, and statistical findings support its consistent ability to obtain competitive solutions.

1. Introduction

Various evolutionary optimization methods have been developed by drawing inspiration from natural phenomena such as physics, biology, human social behavior, astronomy, and evolutionary theory. The main aim of these methods is to improve the search process while minimizing computational costs to achieve global optimization. Metaheuristic optimization algorithms are powerful computational methods for solving engineering design problems. The number of metaheuristic optimization methods was relatively limited until the early 2010s. The most important ones in this period are the following: genetic algorithms [1], simulated annealing (SA) [2], tabu search [3], particle swarm optimization (PSO) [4], ant colony optimization (ACO) [5], harmony search (HS) [6], the big bang–big crunch algorithm (BB–BC) [7], the artificial bee colony (ABC) algorithm [8], cuckoo search (CS) [9], and teaching–learning-based optimization (TLBO) [10]. Since the early 2010s, there has been a significant increase in interest in metaheuristic optimization methods, leading to a tremendous rise in the number of novel methods proposed. Among these methods, some of the most notable ones are the following: the grey wolf optimizer (GWO) [11], moth–flame optimization (MFO) [12], the crow search algorithm [13], the Jaya algorithm (JA) [14], the butterfly optimization algorithm (BOA) [15], Henry gas solubility optimization (HGSO) [16], the equilibrium optimizer (EO) [17], the slime mold algorithm [18], the Aquila optimizer (AO) [19], the white shark optimizer [20], circulatory-system-based optimization (CSBO) [21], and the gazelle optimization algorithm (GOA) [22]. Due to the impossibility of citing hundreds of methods in this paper, a thorough comparison of various metaheuristic optimization techniques can be found in recent review papers [23,24,25,26,27].

Metaheuristic methods have been used to find optimal designs for various engineering problems. Structural optimization, involving many design variables and constraints, is a complicated task that researchers have studied for a long time. The optimum design of truss structures is a challenging problem, and efficient optimization algorithms have been implemented to solve this problem. Notable contributions to this topic include the following papers. Wu and Chow [28] utilized genetic algorithms to optimize the designs of trusses. The method included discrete variables for size optimization and continuous variables for shape refinement. Soh and Yang [29] subsequently improved upon this approach by incorporating genetic algorithms, the finite element method (FEM), and domain-specific knowledge to optimize bridge steel trusses. Kaveh and Talatahari [30] pursued layout optimization by utilizing an improved charged system search algorithm (CSS), further enriching the toolbox of optimization techniques. Miguel and Miguel [31] explored the potential of metaheuristic algorithms, specifically harmony search (HS) and the firefly algorithm (FA), to tackle the complicated optimization of steel trusses by considering size and geometry optimization while accommodating dynamic constraints. Azad et al. [32] engaged the modified big bang–big crunch algorithm to confront the size and shape optimization challenges for truss structures subjected to dynamic excitations. Grzywiński et al. [33] applied the teaching–learning-based optimization (TLBO) method with dynamic constraints for the shape and size optimization of truss structures. Varma et al. [34] proposed a measure-based evolutionary optimization method change by implicitly taking buckling constraints into the truss structures. Eid et al. [35] introduced a novel approach for optimizing truss structures with multi-objective criteria using the multi-objective water cycle algorithm (MOWCA). This algorithm is inspired by real-world water cycle dynamics and demonstrates promising performance in navigating complicated optimization landscapes. Azizi et al. [36] modified a recently developed metaheuristic algorithm called the material generation algorithm (MGA) for the optimum design of three benchmark steel trusses. The results demonstrated that the MGA could provide acceptable designs for the problems. Delyova et al. [37] demonstrate the effectiveness of genetic algorithms in optimizing truss structures, bypassing gradient calculations, and exploring vast design spaces efficiently. Their approach combines topological and size optimization, achieving improved weight and stress ratios. Pierezan et al. [38] presented a modified coyote optimization algorithm (MCOA) for solving structural optimization problems. Comparative analysis with the original COA using four truss examples demonstrated the robustness of MCOA. Kaveh and Zaerreza [39] investigated the efficacy of the shuffled shepherd optimization algorithm (SSOA) in the layout optimization of truss systems. SSOA, a multi-community algorithm inspired by the behavior of shepherds in nature, is examined across various layout optimization problems, including 15-bar, 18-bar, 25-bar, and 47-bar trusses, as well as a 272-bar transmission tower, demonstrating SSOA’s competitive performance. Pham and Tran [40] utilized Rao algorithms [41], a parameter-less novel metaheuristic method, for global optimization, focusing on the weight optimization of truss structures. The algorithm integrates Rao algorithms with feasible boundary search (FBS), ensuring the feasibility of solutions. Degertekin et al. [42] proposed a hybrid harmony search algorithm, LSSO-HHSJA, tailored for the weight minimization of large-scale truss structures. Combining harmony search with the Jaya method aimed to reduce structural analyses. LSSO-HHSJA utilizes gradient information and Jaya-based operators to refine trial designs, achieving competitive results in large-scale optimization problems. Kooshkbaghi and Kaveh [43] introduced the artificial coronary circulation system (ACCS), a novel optimization algorithm for the weight optimization of truss structures with continuous variables. The algorithm incorporates heart memory and boundary-handling strategies. ACCS’s effectiveness was assessed in solving the weight minimization problems of five truss structures. Panagant et al. [44] investigated the comparative performance of fourteen multi-objective metaheuristics in truss optimization, an area less explored in the literature than in single-objective cases. Eight classical trusses serve as test problems. Through comprehensive analysis, the study sheds light on the strengths and weaknesses of each algorithm, aiding in the development of tailored optimization approaches for truss design problems. Kaveh and Khosravian [45] examined the effectiveness of the improved vibrating particles system (IVPS) algorithm inspired by the free vibration of viscous-damped systems in optimizing the size and layout of truss structures. Results demonstrate competitive performance against other powerful algorithms, with better outcomes in some cases. Awad [46] explored the application of the political optimizer (PO) algorithm [47] in structural optimization. Results on various truss structures demonstrate the PO’s superiority regarding optimized weight, stability, and convergence speed for small- to medium-scale systems. Many papers on the optimization of truss structures are not mentioned here. A comprehensive literature survey on this topic can be found in review papers [48,49,50,51,52].

Circulatory-system-based optimization (CSBO) [21] is one of the most powerful recently proposed optimization methods. The algorithm achieves a highly effective optimization process by emulating biological circulatory systems. However, its efficiency in truss optimization remains suboptimal, indicating a need for further enhancements to improve overall performance. A new approach to circulatory-system-based optimization (NCSBO) that takes inspiration from how blood flows in the human circulatory system is introduced in this paper. The novelties of the NCSBO with regard to the existing literature are as follows:

• The NCSBO incorporates a mutation operator that preserves its structure while facilitating a detailed local search to better exploit promising candidates.
• NCSBO dynamically adjusts the number of weaker solutions within the population based on the evolving optimization landscape, allowing it to adapt to different stages of the search process. Early in the optimization, when exploration is crucial, weaker solutions may be modified to explore diverse regions of the design space. As the algorithm converges, fewer weaker solutions are modified, focusing on refining promising solutions and avoiding premature convergence.

These new features aim to create a more efficient optimization search process than standard CSBO. NCSBO seeks to improve the effectiveness of CSBO in addressing complicated optimization problems. The proposed method is tested for the size and shape optimization of steel truss structures. The effectiveness of the proposed method is assessed by minimizing the weight of four well-known classical truss examples.

2. Optimum Design of Steel Truss Structures

Structural optimization aims to identify the feasible design with the lowest weight while meeting all constraints. The optimization problem can be expressed with sizing and layout variables for steel trusses as follows:

$$W(A,X)=\sum _{i=1}^{N_m}{\gamma }_i·A_i·L_i$$

(1)

$$L_i=\sqrt{{(x_{i1}-x_{i2})}^2+{(y_{i1}-y_{i2})}^2+{(z_{i1}-z_{i2})}^2}$$

(2)

$${\sigma }_i^c\le {\sigma }_i\le {\sigma }_i^t$$

(3)

$${\delta }_{min}<{\delta }_j<{\delta }_{max}$$

(4)

$$A_{min}<A_k<A_{max}$$

(5)

$$g_k\le 0 k=1,2,3\dots ,N_c$$

(6)

where $A_i$ represents the cross-sectional area of each member; N_m is the total number of members; ${\gamma }_i$ is the unit weight of the material; and $L_i$ is the length of the i-th member. $x_{i1,2}$, $y_{i1,2}$, and $z_{i1,2}$ are the nodal coordinates of the i-th node and are considered shape variables. W is the total weight of the truss; ${\sigma }_i^c$ and ${\sigma }_i^t$ are the limit compression and tension stress values, respectively; ${\delta }_{min}$ and ${\delta }_{max}$ are the allowable minimum and maximum nodal displacements, respectively; $A_{min}$ and $A_{max}$ denote the upper/lower limits for cross-sectional areas; $N_c$ is the total number of design constraints; and $g_k$ is the normalized design constraint. As an example, the displacement constraint, gdisp, is calculated below.

$$g_{disp}={\displaystyle \frac{\left|{{\delta }_i-\delta }_{min}\right|}{{\delta }_{min}}} and {\displaystyle \frac{\left|{{\delta }_{max}-\delta }_i\right|}{{\delta }_{max}}}$$

(7)

A penalty function method handles design constraints during the optimization process. This method transforms the constrained problem into an unconstrained one by penalizing infeasible solutions, which helps to converge toward feasible regions. The penalized objective function, $W_p$, is then expressed with penalties as follows:

$$W_p(A,X)=W(A,X)·{(1+\psi )}^e$$

(8)

where $\psi$ is the sum of total constraint violations and $e$ is the penalty exponent set to 1.

3. Circulatory-System-Based Optimization (CSBO) Algorithm

The heart plays a crucial role in the orchestration of the cardiovascular system. It is the driving force behind the distribution of oxygen-rich blood throughout the body. Concurrently, blood vessels serve as conduits, facilitating this essential journey and ensuring the effective delivery of nutrients while reducing the removal of waste products. The circulatory system consists of two coordinated circuits: the pulmonary circuit, which revitalizes blood in the lungs, and the systemic circuit, which distributes oxygen-rich blood to bodily tissues. This interplay of circuits guarantees proficient oxygenation and revitalization [21].

Inspired by biological circulatory systems, the circulatory-system-based optimization (CSBO) algorithm has emerged as a promising method for refining solution optimization. It mimics the functions of the pulmonary and systemic circuits, executing discrete optimization cycles within distinct groups. This method reinforces the population by systematically removing less viable candidates, similar to how the circulatory system maintains vitality and physical wellbeing. Through this emulation, the CSBO algorithm elevates solution performance and steers the search process toward achieving optimal or closely optimal results [21]. The CSBO algorithm initiates by generating an initial population using a random function within the problem’s range, where each member represents a “blood droplet” with a specific mass ($X_i$). The blood droplet positions correspond to potential solutions within the search space of the optimization problem.

$$X_i=X_{min}+rand\left(1,Dim\right)\times \left(X_{max}-X_{min}\right) i=1:Npop$$

(9)

where $X_{min}$ and $X_{max}$ denote the lower and upper bounds of the search space, respectively, Dim is the number of design variables, $rand\left(1,Dim\right)$ is a randomly generated array with real numbers between 0 and 1, and $Npop$ is the population size, specifying the total number of candidate solutions generated.

The initial vector p, which determines the displacement magnitude and directs the blood mass toward an improved value in each circulation cycle, is generated using a random function within the range [0, 1], scaled according to the problem’s dimensionality.

$$p_i=rand\left(1,Dim\right) i=1,2,\dots Npop$$

(10)

3.1. Blood Mass Movement in Veins

Each blood mass ($X_i$) in the veins is driven by an external force or pressure to enhance optimization. The goal is to minimize an objective function representing these forces. The presence of clogged arteries, such as local optima, is unwanted. To counter this, the algorithm constantly optimizes the system. Particle positions and their objective function values dictate this movement and are expressed as follows:

$$X_i^{new}=X_i+K_{ia}\times p_i\times \left(X_i-X_a\right)+K_{bc}\times p_i\times \left(X_c-X_b\right) i=1,2\dots \mathit{Npop}$$

(11)

$$K_{ia}={\displaystyle \frac{F\left(X_a\right)-F\left(X_i\right)}{\left|F\left(X_a\right)-F\left(X_i\right)\right|+\epsilon }}=\left\{\begin{array}{c} 1 ; if F\left(X_i\right)<F\left(X_a\right) \\ -1 ; if F\left(X_i\right)>F\left(X_a\right)\\ 0 ; if F\left(X_i\right)=F\left(X_a\right)\end{array}\right.$$

(12)

where $X_i$ and $X_i^{new}$ are the current and modified blood masses indicating current and modified solutions, respectively. $K_{ia}$ indicates the movement direction of the i-th blood mass; $X_a, X_b, and X_c$ denote the first, second, and third blood masses, which are selected from the randomly generated R vector; $F\left(X_a\right)$ and $F\left(X_i\right)$ are the fitness values of the a-th and i-th solutions in the population, and $\epsilon$ provides the numerical stability and consistency, particularly in edge cases where the function values are equal or very close. $\epsilon$ is taken as 10⁻⁷. $randperm$ is a function that generates a random permutation of integers to shuffle or randomize a set of numbers without repetition.

$$\begin{array}{c}R=randperm\left(1,Dim\right);\\ \begin{array}{c}{X}_a=R\left(1\right); X_b=R\left(2\right); X_c=R\left(3\right)\end{array}\end{array}$$

(13)

3.2. Systemic Circulation

In CSBO, weaker individuals are defined as having deoxygenated blood in the circulation. The weak individuals (NR) are selected from the weakest portion of the sorted population and symbolically sent to the lungs for oxygenation. Meanwhile, the rest of the population, denoted as NL (where NL = N_pop − NR), consists of individuals with superior fitness levels, who are then routed into the systemic circulation. These fitter individuals are assigned new quantities and proceed through the emetic circulation, allowing them to circulate within the system. The following model illustrates this process:

$$X_{i,j}^{new}=\left\{\begin{array}{c} {X_{a,j}+p}_i\left(j\right)\times \left(X_{b,j}-X_{c,j}\right) ; if rand>0.9\\ \\ {\begin{array}{c}X\end{array}}_{i,j} else\\ \end{array} \right.\begin{array}{c}i=1,2\dots NL\\ j=1,2,\dots Dim\end{array}$$

(14)

In which $X_{i,j}^{new}$ and $X_{i,j}$ are the j-th design variables of the i-th iteration corresponding to the new and current solutions, respectively; $p_i\left(j\right)$ is the j-th variable of the p vector; $X_{a,j}$, $X_{b,j}$, and $X_{c,j}$ are the j-th variables of the a-th, b-th, and c-th solutions, which are described in Equation (13); and rand is a randomly generated real number between 0 and 1. Equation (14) incorporates a crossover probability condition. When a randomly generated value exceeds 0.9, the design variable is updated by Equation (14). Conversely, if the random value is less than or equal to 0.9, no modification is applied to the design variable. This probabilistic mechanism introduces a controlled degree of variability, enhancing the algorithm’s capacity for exploration and exploitation within the solution space. Within the systemic circulation, the value of $p_i$ for this subset of the population is updated according to the following expression:

$$p_i={\displaystyle \frac{F\left(X_i\right)-{F(X}_{worst})}{{F(X}_{best})-{F(X}_{worst})}} i=1,2,\dots NL$$

(15)

where $F\left(X_i\right)$, ${F(X}_{best}),$ and ${F(X}_{worst})$ denote the i-th, the best, and the worst fitness values of the population.

3.3. Pulmonary Circulation

Similar to how the pulmonary circulation manages deoxygenated blood, the CSBO algorithm handles the weaker segment of the population in an optimization process. The population is sorted at each iteration, and a subset of the least fit individuals (NR) is selected. These weaker individuals are then sent into the pulmonary circulation, metaphorically undergoing an “oxygenation” process. It mirrors how blood is oxygenated in the lungs, symbolizing the enhancement of the individuals’ fitness or quality within the algorithm. This mechanism aligns the algorithm’s sorting of weaker individuals with the concept of oxygenation in the pulmonary circulation, aiming to improve their overall performance.

$$X_i^{new}=X_i+\left({\displaystyle \frac{randn}{it}}\right)\times randc\left(1,Dim\right) i=\left(NL+1\right),\left(NL+2\right)\dots (Npop)$$

(16)

where $X_i$ and $X_i^{new}$ are the current and modified solutions of the i-th design, $randn$ is the random real number from a normal distribution; $randc$ is a random vector from a Cauchy probability distribution, Dim is the number of design variables, and it denotes the current iteration number.

In the pulmonary circulation phase, the value of $p_i$ for the weaker individuals is updated as follows:

$$p_i=rand\left(1,Dim\right) i=\left(NL+1\right), \left(NL+2\right),\dots \left(NL+NR\right)$$

(17)

These revised $p_i$ values are then transferred to the next iteration and employed in the systematic circulation. This adjustment ensures that the weaker solutions are equipped for continued optimization and refinement in the subsequent iteration of the algorithm. The CSBO algorithm iterates until the termination criterion is satisfied.

4. A New Approach to Circulatory-System-Based Optimization (NCSBO)

The original CSBO algorithm suffers from an inadequate balance between exploration and exploitation, making it difficult to find acceptable optimal solutions for certain complex problems. Additionally, CSBO has a relatively weak global search capability [53]. To overcome these issues, this paper introduces a refined version of the CSBO algorithm to enhance its performance and adaptability. The proposed method integrates a mutation operator from evolutionary algorithms and introduces a dynamic adaptive parameter to adjust the number of weak individuals within the population. These novelties collectively contribute to a more potent and versatile optimization tool that can navigate intricate optimization landscapes with greater efficacy.

4.1. Mutation

The mutation operation introduces controlled changes to individual solutions, similar to genetic mutations in nature. In this paper, the best solution is mutated to have a chance to become the new best solution. In every iteration, a single randomly chosen dimension of the target vector, denoted as $X_{best,r}$, is mutated via a parameter known as the mutation probability (MP). Changing only one variable randomly is due to the relatively lower likelihood of achieving a better result when multiple variables are altered simultaneously. The fitness value of $X_{mut\_best}$ is compared with the $\mathrm{i}\mathrm{f}$ it is smaller, then replaced with $X_{best}$ as given in Equation (19). By altering only one variable, the strategy ensures that the overall structure of the best solution remains mostly intact. This reduces the risk of disrupting well-performing features of the solution, maintaining the gains achieved in previous iterations without drastic shifts that could lead to poorer outcomes. Additionally, this allows the algorithm to exploit the best solution more efficiently and perform a more detailed search around a promising candidate, helping the algorithm better exploit local optima and converge to an even better solution. This operator is incorporated into the CSBO framework before each iteration to compute the following:

$$X_{mut\_best,r}=\left\{\begin{array}{c}randi\left(Dim\right) if rand\le MP \\ \\ X_{best,r} otherwise \end{array}\right.$$

(18)

$$X_{best}=\left\{\begin{array}{c}{X}_{mut\_best} if F\left(X_{mu{t}_{best}}\right)<F\left(X_{best}\right) \\ \\ X_{best} otherwise \end{array}\right.$$

(19)

where $X_{mut\_best,r}$ and $X_{best,r}$ are the r-th design variables of the mutated best solution and current best solution, respectively; $randi$ is a randomly generated integer between 1 and Dim; $rand$ is a randomly generated real number between 0 and 1; Dim is the dimension of the optimization problem; $X_{best}$ and $X_{mut\_best}$ are the current best and mutated best solutions, respectively; $F\left(X_{best}\right)$ and $F\left(X_{mu{t}_{best}}\right)$ are the fitness values of the current best design in the population and the mutated best design, respectively; and r is the randomly selected variable from the design variable vector.

4.2. Adaptive NR Parameter

In the conventional CSBO algorithm, the parameter NR denotes the count of individuals within a population characterized as ‘weak’. It is ascertained by setting it to one-third of the total population size (Npop), i.e., NR = Npop/3.

In the context of this paper, a novel adaptive approach for determining the NR parameter is introduced, deviating from the fixed value paradigm. This adaptive parameter varies per iteration, depending on the normalized differences between individual solutions and the best solution. The threshold is the mean value of the normalized difference vector. The number of solutions with a larger value than the threshold is considered the new NR parameter.

$${Normal\_dif}_i=\left({\displaystyle \frac{F\left(X_i\right)-F{(X}_{best})}{F{(X}_{best})}}\right)$$

(20)

$$threshold=mean\left(Normal\_dif\right)$$

(21)

$$N{R}_{adaptive}=Normal\_dif(Norma{l}_{dif}\ge threshold)$$

(22)

where $Normal\_dif$ is the normalized difference vector, $mean$ is a function calculating the average value of the array, and $N{R}_{adaptive}$ refers to the dynamically varying number of weak solutions maintained within the optimization process.

The threshold used to determine the number of weaker solutions allows the algorithm to intelligently decide how many solutions need modification at any given stage, depending on the diversity of the population. More solutions may be modified when there is high diversity, promoting exploration. When diversity decreases, fewer modifications are applied, reinforcing exploitation. The adaptive strategy ensures the algorithm does not stick to a fixed balance between exploration and exploitation. Instead, it dynamically shifts this balance based on the normalized differences between current and best solutions.

The pseudocodes of standard CSBO and NCSBO are proposed in Algorithm 1 and Algorithm 2, respectively. In addition, the corresponding concepts in the NCSBO algorithm to the steel truss optimization problem are provided in

Table 1. The equivalent concepts of NCSBO and steel truss optimization.

Parameter or Function in NCSBO	The Equivalent Concept in This Paper
Blood mass	Steel truss
The movement of blood in the body	Modifying the sizing and shape variables of truss designs
Circulation cycle	Iteration
Deoxygenated blood	Heavier truss designs
Oxygenated blood	Lighter truss designs
Purification of blood	Composition of high- and low-cost truss designs

.

Algorithm 1.  Pseudocode of CSBO

Initialize the population size (Np), maximum iteration number (itermax), and NR parameter Generate initial population While  it < itermax             Calculate objective function             Generate initial pi using Equation (10)             For  i = 1: Np       \\ movement of blood mass in the veins                    Calculate Kia and Kbc using Equations (12) and (13)                    Update the new position of Xi                    Calculate objective function, F(Xi,new)                    FE←FE + 1             End             For  i < 1:(Np-NR)      \\ Systemic circulation                    For  j < D                             if  rand > 0.9                             Perform the systemic circulation using Equation (14)                             else                             X_(i,j)^new = X_(i,j)^                             end                    end                    Calculate objective function, F(Xi,new)                    FE←FE + 1                    Calculate pi for weakest population using Equation (15)             end             For  i = (Np-NR) + 1:Np    \\ Pulmonary circulation                    Perform the pulmonary circulation using Equation (16)                    Update the new position of Xi                    Calculate objective function, F(Xi,new)                    FE←FE + 1                    Calculate pi using Equation (17)             end             Update the best solution end Return  the best solution

Algorithm 2.  Pseudocode of NCSBO

Initialize the population size (Np), maximum iteration number (itermax), NR Generate initial population While  it < itermax              Calculate objective function              Generate initial pi using Equation (10)              For  i = 1:Np \\ movement of blood mass in the veins                           Calculate Kia and Kbc using Equations (12) and (13)                           Update the new position of Xi                           Calculate objective function, F(Xi,new)                           FE←FE + 1              End              For  i < 1:(Np-NR) \\ Systemic circulation                           For  j < D                                        if  rand > 0.9                                        Perform the systemic circulation using Equation (14)                                        else                                        X_(i,j)^new = X_(i,j)^                                        end                           end                           Calculate objective function, F(Xi,new)                           FE←FE + 1                           Calculate pi for weakest population using Equation (15)              end              For  i = (Np-NR) + 1:Np \\ Pulmonary circulation                           Perform the pulmonary circulation using Equation (16)                           Update the new position of Xi                           Calculate objective function, F(Xi,new)                           FE←FE + 1                           Calculate pi using Equation (17)              end              Update the best solution              Perform crossover and mutation operators using Equations (18) and (19)              Perform adaptive NR approach using Equations (20) and (22) end Return  the best solution

in which FE denotes the functional evolution number.

5. Numerical Examples

This paper assesses the efficiency of the proposed algorithm for optimizing the sizing and layout of trusses utilizing well-known benchmark examples from the literature. The benchmark examples include a 15-bar planar truss, an 18-bar planar truss, a 47-bar planar truss, and a 25-bar spatial truss for which discrete size and continuous shape optimization are performed. The algorithm’s performance is assessed over 20 independent runs for each test example, with a population size of 45. The maximum iteration is set to 400 iterations for four examples. The distribution of stresses in the members of optimum trusses obtained by CSBO and NCSBO is illustrated using color maps. The colors in the color map vary between values of 0, corresponding to light blue for the minimum stress, and 1, corresponding to dark red for the maximum stress.

Sensitivity analyses were carried out for standard CSBO and the NCSBO. The goal was to identify the optimal values for the population size (Npop) and the initial number of weak individuals (NR) in standard CSBO. In NCSBO, since the initial number of weak individuals (NR) is adaptively determined, it was not necessary to identify this parameter. Instead, in addition to the population size, a sensitivity analysis for the mutation probability (MP) was performed to ascertain the most appropriate values.

The benchmark problem used for the sensitivity analysis was a 15-bar truss structure. This problem was optimized separately for nine different population sizes ranging from 15 to 300 for both CSBO and NCSBO. The results, presented in

Table 2. Sensitivity analyses to find the optimal N_pop value in CSBO and NCSBO for a 15-bar truss.

NCSBO
Npop	15	30	45	60	75	90	120	150	300
Best weight (lb *)	74.388	72.946	72.688	73.742	75.491	76.713	81.030	83.963	106.672
Mean (lb)	78.745	76.891	75.92	76.555	78.890	80.002	84.043	88.110	118.454
Std (lb)	2.197	2.061	1.966	2.131	1.516	1.854	1.686	1.899	5.744
Mean + Std (lb)	80.942	78.952	77.886	78.686	80.407	81.856	85.730	90.008	124.199
CSBO
Npop	15	30	45	60	75	90	120	150	300
Best weight (lb)	73.177	73.048	72.843	75.051	75.038	77.52	81.789	83.953	110.311
Mean (lb)	78.61	76.275	76.452	78.102	78.577	80.207	84.488	88.349	120.024
Std (lb)	2.845	2	2.116	1.917	2.188	1.882	1.894	2.187	5.863
Mean + Std (lb)	81.455	78.275	78.568	80.019	80.765	82.089	86.382	90.536	125.887

* lb = 4.44822 N.

and visualized in Figure 1, indicate that the most suitable population size is 45. After establishing N_pop = 45, a sensitivity analysis was performed for the NR parameter in CSBO. Since NR represents the weak solutions in the population, its possible values range from 0 to N_pop. To systematically investigate its impact, optimizations were conducted for NR values varying from 0 to 45 in increments of 5. The results are reported in

Table 3. Sensitivity analyses to find the optimal NR value in CSBO for a 15-bar truss.

NR	0	5	10	15	20	25	30	35	40	45
Best weight (lb)	74.434	73.019	72.963	72.348	72.641	73.275	73.842	74.881	77.250	75.932
Mean (lb)	77.031	77.735	76.835	76.024	76.088	77.136	77.410	77.654	78.890	80.458
Std (lb)	2.049	2.578	2.359	1.805	2.183	2.700	1.812	1.575	1.091	2.822
Mean + Std (lb)	79.080	80.313	79.194	77.829	78.271	79.836	79.222	79.229	79.982	83.280

and Figure 2. The optimal NR value was identified as 15. For the NCSBO method, the mutation probability (MP) was analyzed within the range of 0 to 1, with increments of 0.1. The optimization results demonstrated that an MP value of 0.3 yields the best performance in terms of solution quality, as reported in

Table 4. Sensitivity analyses to find the optimal MP value in NCSBO for a 15-bar truss.

MP	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Best weight (lb)	73.013	73.544	73.996	72.959	72.681	72.768	73.053	73.880	73.220	73.558	72.933
Mean (lb)	77.063	76.808	77.484	75.975	76.456	76.578	76.389	77.562	76.179	75.622	76.489
Std (lb)	2.331	2.281	1.731	2.294	2.850	2.055	2.073	1.956	2.031	2.065	2.403
Mean + Std (lb)	79.394	79.089	79.215	78.270	79.306	78.633	78.463	79.519	78.210	77.687	78.892

and Figure 3. The results of sensitivity analyses are considered applicable to the subsequent examples utilized in this paper as they provide insights into the parameter settings that are expected to be effective across different truss structures.

5.1. The 15-Bar Planar Truss

The first benchmark example is a 15-bar planar truss structure involving continuous variables (Figure 4). The modulus of elasticity is 10⁴ ksi (1 ksi = 6.8948 MPa) and the material density is 0.1 lb/in³. An applied concentrated load of P exerts a 10-kip downward force. This example has a total of twenty-three design variables, consisting of fifteen sizing and eight shape variables, as given below:

$$Sizing variables A=[A_1 A_{2 }{A}_3\cdots A_{15}]$$

$$Layout variables X=[x_2=x_6, x_3=x_7, y_2,y_3,y_4,y_6,y_7,y_8]$$

The allowable cross-sections for sizing variables are given in the discrete profile set outlined as {0.111, 0.141, 0.174, 0.220, 0.270, 0.287, 0.347, 0.440, 0.539, 0.954, 1.081, 1.174, 1.333, 1.488, 1.764, 2.142, 2.697, 2.800, 3.131, 3.565, 3.813, 4.805, 5.952, 6.572, 7.192, 8.525, 9.300,10.850, 13.330, 14.290, 17.170, 19.180} (in²) (1 in² = 6.4516 cm²). The maximum stress for all members must not exceed ±25 ksi. The layout constraints are considered as given below:

$$\begin{array}{c}100\le x_1\le 140 \left(in\right); 220\le x_3\le 260 \left(in\right); 100\le y_2\le 140 \left(in\right);\hfill \\ 100\le y_3\le 140 \left(in\right); 50\le y_4\le 90 \left(in\right); -20\le y_6\le 20 \left(in\right);\hfill \\ -20\le y_7\le 20 \left(in\right);20\le y_8\le 60 \left(in\right);\hfill \end{array}$$

Table 5. Comparison of optimized designs for the 15-bar truss.

No.	Design Variables	GA [54]	ARSAGA [55]	SCPSO [56]	IGA [57]	FA [58]	TLBO [59]	MHS [60]	D-ICDE [61]	ABC [62]	MLA [63]	CSBO	NCSBO
1	A1	1.081	0.954	0.954	1.081	0.954	1.081	0.954	1.081	0.954	0.954	0.954	0.954
2	A2	0.539	1.081	0.539	0.539	0.539	0.954	0.539	0.539	0.539	0.539	0.539	0.539
3	A3	0.287	0.44	0.27	0.287	0.22	0.141	0.22	0.141	0.347	0.347	0.141	0.174
4	A4	0.954	1.174	0.954	0.954	0.954	1.081	0.954	0.954	0.954	0.954	0.954	0.954
5	A5	0.539	1.488	0.539	0.954	0.539	0.539	0.539	0.539	0.539	0.539	0.539	0.539
6	A6	0.141	0.27	0.174	0.22	0.22	0.347	0.22	0.287	0.111	0.111	0.27	0.27
7	A7	0.111	0.27	0.111	0.111	0.111	0.111	0.111	0.111	0.111	0.111	0.111	0.111
8	A8	0.111	0.347	0.111	0.111	0.111	0.174	0.111	0.111	0.111	0.111	0.111	0.111
9	A9	0.539	0.22	0.287	0.287	0.287	0.141	0.44	0.141	0.539	0.347	0.539	0.347
10	A10	0.44	0.44	0.347	0.22	0.44	0.27	0.347	0.347	0.44	0.44	0.347	0.347
11	A11	0.539	0.22	0.347	0.44	0.44	0.22	0.347	0.440	0.44	0.44	0.347	0.347
12	A12	0.27	0.44	0.22	0.44	0.22	0.141	0.27	0.270	0.174	0.174	0.27	0.22
13	A13	0.22	0.347	0.22	0.111	0.22	0.44	0.27	0.270	0.174	0.174	0.22	0.22
14	A14	0.141	0.27	0.174	0.22	0.27	0.347	0.22	0.287	0.111	0.111	0.27	0.27
15	A15	0.287	0.22	0.27	0.347	0.22	0.141	0.22	0.174	0.347	0.347	0.141	0.174
16	X2	101.5775	118.346	137.2216	133.612	114.967	100.004	135.568	100.0309	110.209	108.1889	133.213	133.9027
17	X3	227.9112	225.209	259.9093	234.752	247.04	241.047	245.542	238.7010	249.819	246.2332	257.87	259.7541
18	Y2	134.7986	119.046	123.5006	100.449	125.919	118.823	123.13	132.8471	133.599	135.0565	118.672	120.6040
19	Y3	128.2206	105.086	110.002	104.738	111.067	100.083	120.696	125.3669	111.624	113.7962	111.769	106.7233
20	Y4	54.863	63.375	59.9356	73.762	58.298	50	57.9313	60.3072	55.1278	55.4635	53.0239	51.0915
21	Y6	−16.4484	−20	−5.1799	−10.067	−17.564	3.1411	−5.9742	−10.6651	−18.950	18.0985	−11.965	−9.57978
22	Y7	−16.4484	−20	4.2193	−1.339	−5.821	−9.6997	−2.9125	−12.2457	3.3411	1.9869	−5.4549	0.67778
23	Y8	54.8572	57.722	57.8829	50.402	31.465	46.8963	56.3256	59.9931	55.1423	55.4635	53.0262	51.2169
Best weight (lb)		76.6854	104.573	72.5143	79.82	75.55	76.6519	73.887	74.6818	72.715	72.478	72.66	72.224 [72.5143] b [73.887] c
Mean weight (lb)		N/A a	N/A	76.411	N/A	N/A	N/A	N/A	N/A	77.2558	75.2119	76.83	76.37
Worst weight (lb)		N/A	N/A	80.156	N/A	N/A	N/A	N/A	N/A	78.9446	78.6951	79.683	78.98
ODNSA		N/A	N/A	4500	8000	8000	30,640	5000	7980	5640	30,000	7620	5430
TNSA		N/A	N/A	4500	8000	8000	32,000	5000	7980	18,000	30,000	18,000	18,000
STD (lb)		N/A	N/A	1.922	N/A	N/A	N/A	N/A	N/A	2.4219	1.8314	2.482	1.73
CVP (%)		None	32.57	0	0.00089	None	None	None	None	None	None	None	None

^a N/A = Not available; ^b NCSBO achieved a weight of 72.5143 lb after 4260 structural analyses (the result of SCPSO [56]); ^c NCSBO achieved a weight of 73.887 lb after 3990 structural analyses (the result of MHS [60]).

presents the sizing and layout variables of NCSBO and provides a comprehensive comparison with results from previous methods, including GA, ARSAG, PSO, GCPSO, IGA, FA, TLBO, MHS, ABC, and MLA. An optimal weight of 72.224 lb was achieved, requiring 5430 structural analyses in NCSBO, whereas CSBO yielded an optimal weight of 72.66 lb, requiring 7620 structural analyses. NCSBO demonstrates the most effective weight minimization among all of the methods reported in Table 5. It is noteworthy that NCSBO is the most powerful method in terms of convergence speed. SCPSO and MHS required 4500 and 5000 structural analyses, respectively, to achieve their optimum designs. NCSBO, however, reached the optimized results for SCPSO and MHS after only 4260 and 3990 structural analyses, respectively. This comparison demonstrates the superior convergence capability of NCSBO. In the table, ODNSA represents the number of structural analyses that obtained the optimum design, while TNSA is the total number of structural analyses to terminate the optimization process. Moreover, STD refers to the standard deviation of independent runs, and CVP denotes the constraint violation percentage values for optimum designs.

Figure 5 displays the stress distributions within the members of the 15-bar planar truss, as determined by CSBO and NCSBO. The maximum stresses are recorded as 24.998597 ksi for CSBO and 24.999438 ksi for NCSBO, while the minimum stresses are 24.999924 ksi and −24.993868 ksi, respectively. The minimum and maximum stresses lie within the feasible range of [–25, 25] ksi. As seen in Figure 5, the stress values in almost all members closely approach the tensile or compressive strength limits.

Figure 6 presents the initial and optimized geometries, where the initial configuration is shown with dashed grey lines and the solid, colored lines represent the optimized layout. For enhanced clarity, the member between nodes 4 and 8 has been zoomed in on and included as an additional feature in the graph. After optimization using NCSBO, joint 4, which was initially positioned at (360.0, 120.0), shifts to (360.0, 51.0915), while joint 8, which was initially at (360.0, 0.0), relocates to (360.0, 51.2169). In the optimized geometry, member 9 experiences a substantial reduction in length, contributing directly to the overall weight reduction of the structure. Additionally, Figure 6 presents a color map illustrating the status of element stresses relative to their allowable stress limits.

Figure 7 illustrates the variation in weight values with respect to the number of analyses conducted using both the NCSBO and standard CSBO methods. A comparison of the convergence behavior of the two methods reveals that NCSBO exhibits a more rapid decrease in the initial exploration phase compared to CSBO.

5.2. The 18-Bar Planar Truss

The second benchmark example is an 18-bar planar truss structure involving continuous variables (Figure 8). The modulus of elasticity is 10⁴ ksi and the material density is 0.1 lb/in³. The structure is subjected to five concentrated loads (P), each applying a downward force of 20 kips to nodes 1, 2, 4, 6, and 8.

This truss example involves a total of twelve design variables, with four for sizing optimization and eight for shape optimization, as outlined below:

$$\begin{array}{c}Sizing variables \hfill \\ A={\left[\begin{array}{c}{A}_1=A_{4 }=A_{8 }=A_{12 }=A_{16 }\\ A_5=A_{9 }=A_{13}=A_{17 }\\ A_{2 }=A_{6 }=A_{10 }=A_{14}=A_{18 }\\ A_{3 }=A_7=A_{11 }=A_{15 }\end{array}\right]}_{nsize\times 1}\hfill \end{array}$$

$$\begin{array}{c}Layout variables\hfill \\ X=[x_3,x_5, x_{7,}{x}_9, y_3,y_5,y_7,y_9]\hfill \end{array}$$

The boundary conditions of layout variables are considered as given below:

$$\begin{array}{c}775\le x_3\le 1225\left(in\right); 525\le x_5\le 975\left(in\right);275\le x_7\le 725\left(in\right);\hfill \\ 25\le x_9\le 475\left(in\right);-225\le {y_3,y_5,y_7,y}_9\le 245\left(in\right);\hfill \end{array}$$

The allowable cross-sections for sizing variables are confined to the discrete profile set outlined as ={2, 2.25, 2.5, … 21.25, 21.5, 21.75} (in²). The maximum stress for all members must not exceed ±20 ksi. The allowable buckling stress is considered a design constraint and calculated as $4EA/L^2$ for each member subjected to compression, in which E is the modulus of elasticity, A denotes the cross-sectional area, and L is the member length.

Table 6. Comparison of optimized designs for the 18-member truss.

No.	Design Variables	SCPSO [56]	VGA [64]	GA [65]	GSO [66]	IGSO [66]	D-ICDE [61]	ABC [62]	MLA [63]	CSBO	NCSBO
1	G1	12.5	12.5	12.25	12.25	12.25	13	12.5	9.75	12.50	12.5
2	G2	17.5	16.25	18	18.25	18.25	17.5	17.75	18.75	17.50	17.5
3	G3	5.75	8	5.25	4.75	4.75	6.5	5.75	5.5	6.00	5.75
4	G4	3.75	4	4.25	4.25	4.25	3	3.75	3.5	3.75	3.75
5	X3	907.2491	891.9	913	916.9	920.812	914.06	912.997	931.4057	906.747	907.7788
6	Y3	179.8671	145.3	186.8	191.971	170.912	183.46	183.681	187.5341	179.523	180.4711759
7	X5	636.7873	610.6	650	654.224	640.506	640.53	642.714	662.7803	635.496	637.3638803
8	Y5	141.8271	118.2	150.5	156.1	139.87	133.74	143.892	162.0359	139.864	141.3239515
9	X7	407.9442	385.4	418.8	423.5	409.416	406.12	411.692	427.3530	406.841	408.0822344
10	Y7	94.0559	72.5	97.4	102.571	91.774	92.63	97.1476	99.4025	95.461	93.88007802
11	X9	198.7897	184.4	204.8	207.519	198.775	196.69	200.909	210.1569	198.719	198.7942297
12	Y9	29.5157	23.4	26.7	28.579	29.504	37.06	30.2191	28.3101	29.457	29.51760031
Best weight (lb)		4512.365	4616.8	4547.9	4538.7676	4553.12	4554.29	4537.06	4271.57	4530.13	4512.1244 [4537.06] a
Mean weight (lb)		4551.709	N/A	N/A	N/A	N/A	N/A	4585.110	4317.984	4539.63	4534.3
Worst weight (lb)		4661.268	N/A	N/A	N/A	N/A	N/A	4627.524	4348.598	4553.22	4548.19
ODNSA		4500	N/A	N/A	50,000	50,000	8025	2700	30,000	7585	5900
TNSA		4500	N/A	N/A	50,000	50,000	8025	18,000	30,000	18,000	18,000
STD (lb)		37.69	N/A	N/A	N/A	N/A	N/A	9.797	24.769	5.73	5.38
CVP (%)		0	0.0079	None	None	94.28	None	None	24.87	None	None

^a NCSBO achieved a weight of 4537.06 lb after 5150 structural analyses (the result of ABC [62]).

summarizes the optimization results obtained by CSBO, NCSBO, and the literature. NCSBO shows the best performance, achieving the lightest design at 4512.1244 lb after 5900 structural analyses. The optimal design found by CSBO, weighing 4530.13 lb after 7585 analyses, also shows strong performance relative to other methods. Furthermore, all solutions were carefully checked for constraint violations, and it was found that the allowable stress value for tension members in the MLA was 25 ksi instead of 20 ksi. As a result, a direct comparison between the optimized design obtained by the MLA and other methods found in the literature is not possible. Upon re-evaluating the design produced by the MLA against the 20 ksi stress limit, it was determined that it exceeded the design constraints by 24.87%. NCSBO shows the best convergence capability after ABC; however, NCSBO yielded the optimized results of ABC after 5150 vs. 2700 structural analyses. The STD for ten independent values is calculated at 5.39 lb, which is about 0.12% of the mean weight, as reported in Table 6. This result verifies the consistency of designs obtained by independent runs found by NCSBO.

Figure 9 illustrates the distribution of stresses in the members according to maximum, minimum, and buckling limit stresses. In the optimal design obtained with NCSBO, tensile stresses in members 16 and 17 are recorded as 19.999 ksi and 19.9113 ksi, respectively, approaching the maximum allowable tensile stress of 20 ksi. Additionally, members 2, 6, 7, 10, 11, and 14 are subjected to compressive forces closely approaching their buckling limit values. Similarly, in the CSBO design, members 16 and 17 exhibit tensile stresses of 19.997 ksi and 19.932 ksi, respectively, nearing the tensile stress limit.

Figure 10 depicts the optimized configurations derived from the initial layout of the 18-bar truss, accompanied by a stress color map indicating each member’s utilization relative to its stress limit. In the optimal design obtained with NCSBO, member 15 reaches near-maximum stress capacity, as represented by the orange coloration. Conversely, in the CSBO optimal design, member 15 exhibits a lower stress utilization, indicated by a yellow coloration, signifying that the stress capacity remains underutilized. The remaining members exhibit similar stress capacities in both optimal designs.

The reduction in weight relative to the number of analyses performed using both NCSBO and CSBO methods is shown in Figure 11. A comparison of the convergence behavior shows that NCSBO converges earlier; however, both methods exhibit similar reduction patterns with the number of analyses. For NCSBO, the final optimal solution is achieved after 5900 analyses, whereas for CSBO, the optimal result is reached after 7585 analyses. Both solutions remain stable until they reach the termination criterion of 18,000 analyses.

5.3. The 25-Bar Planar Truss

The third comparison problem revolves around optimizing the sizing and layout of a spatial 25-bar tower, as illustrated in Figure 12. The material density and the modulus of elasticity are considered to be 10,000 ksi and 0.1 lb/in³. Concentrated loads are applied to the tower truss, creating a unique load scenario. At node 1, forces of 1000 lbf in the positive X-direction, 10,000 lbf in the negative Y-direction, and 10,000 lbf in the negative Z-direction are exerted. Similarly, node 2 encounters a downward load of 10,000 lbf in the Y-direction and 10,000 lbf in the negative Z-direction. Moving to node 3, a load of 500 lbf in the positive X-direction is present, while node 6 bears a load of 600 lbf in the positive X-direction.

For optimization purposes, the cross-sectional areas of the truss’s bars are organized into eight discrete groups, resulting in eight sizing variables. In parallel, the layout variables are divided into five distinct groups, culminating in 13 design variables under consideration.

$$Sizing variables A={\left[\begin{array}{c}{A}_1\\ A_{2 }=A_3=A_4=A_{5 }\\ A_6=A_7=A_8=A_{9 }\\ A_{10 }=A_{11}\\ A_{12}=A_{13}\\ A_{14}=A_{15}=A_{16}=A_{17}\\ A_{18 }=A_{19}=A_{20}=A_{21}\\ A_{22 }=A_{23}=A_{24 }=A_{25 }\end{array}\right]}_{nsize\times 1}$$

$$Layout variables X={\left[\begin{array}{c}{x}_4=x_5={-x}_3=-x_6\\ y_3=y_4=-y_5=-y_6\\ z_3=z_4=z_5=z_6\\ x_8=x_9={-x}_7=-x_{10}\\ y_7=y_8=-y_9=-y_{10}\end{array}\right]}_{nshape\times 1}$$

The limit values of layout variables are defined as given below:

$$\begin{gathered} 20\le x_4=x_5={-x}_3=-x_6\le 60 \left(in\right) \\ 40\le y_3=y_4=-y_5=-y_6\le 80 \left(in\right) \\ 90\le z_3=z_4=z_5=z_6\le 130 \left(in\right) \\ 40\le x_8=x_9={-x}_7=-x_{10}\le 100 \left(in\right) \\ 100\le y_7=y_8=-y_9=-y_{10}\le 140 \left(in\right) \end{gathered}$$

The sizing variables are restricted to be selected in the following discrete profile set, S, containing 30 sections: S = {0.1, 0.2, 0.3…2.6, 2.8, 3, 3.2} (in²). The stress constraint for members has been set at ±40 ksi. In contrast to the previous two examples, nodal displacement constraints are also considered, and displacements of all nodes in the x, y, and z directions are limited to within ±0.35 in.

Optimum designs obtained by CSBO and NCSBO and various methods reported in the literature are compared in

Table 7. Comparison of optimized designs for the 25-bar truss.

No.	Design Variables	SCPSO [56]	GA [65]	GA [57]	MHS [60]	FA [58]	D-ICDE [61]	ABC [62]	MLA [63]	CSBO	NCSBO
1	A1 (in2)	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
2	A2	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
3	A3	1	1.1	1.1	1	1	0.9	1	1.0	1	1
4	A4	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
5	A5	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
6	A6	0.1	0.1	0.2	0.1	0.1	0.1	0.1	0.1	0.1	0.1
7	A7	0.1	0.1	0.2	0.1	0.1	0.1	0.1	0.1	0.1	0.1
8	A8	0.9	1	0.7	1	1	1.0	0.9	0.9	0.9	0.9
9	X4 (in)	36.952	36.230	35.470	37.820	37.320	36.830	36.211	37.759	37.498	37.67668
10	Y4	54.5786	58.560	60.370	55.485	55.740	58.530	54.637	54.625	54.672	54.50722
11	Z4	129.9758	115.590	129.070	128.730	126.620	122.670	129.965	129.625	130	130
12	X8	51.7317	46.460	45.060	52.068	50.140	49.210	52.074	51.927	51.880	51.87948
13	Y8	139.5316	127.950	137.040	139.590	136.400	136.740	139.980	139.678	139.494	139.5099
Best weight (lb)		117.227	124.0015	124.940	117.380	118.830	118.760	117.333	117.316	117.264	117.257
Mean weight (lb)		122.876	N/A	N/A	N/A	N/A	N/A	N/A	123.728	119.702	117.336
Worst weight (lb)		132.672	N/A	N/A	N/A	N/A	N/A	N/A	127.482	124.665	117.647
ODNSA		4500	N/A	6000	6000	6000	6000	5100	30,000	5310	4830
TNSA		4500	N/A	6000	6000	6000	6000	18,000	30,000	18,000	18,000
STD (lb)		3.671	N/A	N/A	N/A	N/A	N/A	2.22	2.384	0.352	0.142
CVP (%)		0.527	None	0.008	None	None	0.17	None	None	None	None

. The optimum design obtained by NCSBO, weighing 117.257 lb after 4830 structural analyses, is the lightest design reported in Table 7. NCSBO is also the most powerful method in terms of convergence speed since it requires only 4830 structural analyses to find the optimum design. It is worth mentioning that the standard deviation of 20 independent runs with NCSBO is 0.147 lb, as indicated in Table 7. This observation supports the robustness/consistency of the proposed algorithm. Meanwhile, CSBO produces a similarly competitive design with a weight of 117.264 lb after 5310 structural analyses and a comparatively low standard deviation of 0.352 lb. After reviewing the literature, it was found that designs generated by SCPSO, GA, and D-ICDE violate the displacement constraints by margins ranging from 0.008% to 0.527%.

For the optimal designs obtained via NCSBO and CSBO, all stress constraints for structural members and displacement limits for nodes have been effectively satisfied, as illustrated in Figure 13 and Figure 14, respectively. The color map in Figure 15 and Figure 16, representing the optimum configuration with member stress capacity, shows all members in light blue, indicating a considerable margin below maximum capacity. Figure 14 demonstrates that the displacement constraints in the x and y directions, particularly at nodes 1 and 2, reach their limits at ±0.349999 inches in both optimal designs. Examining these figures reveals that displacement constraints serve as the primary governing factor in the optimization process of the 25-bar truss.

In the optimization of the 25-bar truss, the convergence curve of the proposed NCSBO is compared with that of CSBO, as shown in Figure 17. The final optimal solutions are achieved after 4830 and 5310 analyses for NCSBO and CSBO, respectively. It is evident from the figure that NCSBO exhibits faster convergence, particularly in the early stages of the optimization process. In both cases, the optimization process continued until the maximum number of structural analyses (18,000) was reached, even without convergence.

5.4. The 47-Bar Planar Truss

The sizing and layout optimization of a 47-bar planar truss tower (Figure 18) aimed to simulate scenarios involving power lines and their potential failures. Mass density and Young’s modulus of elasticity are assigned as 0.3 lb/in³ and 3 × 10⁴ ksi. The tower is subjected to three independent loading conditions, which are as follows.

In the first case, a concentrated force of 6000 lbf in the positive X-direction and a concentrated force of 14,000 lbf in the negative Y-direction are applied at nodes 17 and 22. In the second loading condition, a concentrated force of 6000 lbf in the positive X-direction and a concentrated force of 14,000 lbf in the negative Y-direction is applied only at node 17. The third condition provides a concentrated force of 6000 lbf in the positive X-direction and a concentrated force of 14,000 lbf in the negative Y-direction at node 22 only.

The cross-sectional areas of the tower’s members were grouped into 27 sizing design variables, ranging from 0.1 to 5.0 in², which allowed for a comprehensive exploration of design possibilities. Additionally, 17 layout variables were considered, governing the positioning of nodes within the truss tower structure. These variables were intricately linked, ensuring the coherence and stability of the tower design.

Sizing variables

A3 = A1; A4 = A2; A5 = A6; A7; A8 = A9; A10; A12 = A11; A14 = A13; A15 = A16; A18 = A17; A20 = A19; A22 = A21; A24 = A23; A26 = A25; A27; A28; A30 = A29; A31 = A32; A33; A35 = A34; A36 = A37; A38; A40 = A39; A41 = A42; A43; A45 = A44; A46 = A47

Layout variables

x2 = −x1; x4 = −x3; y4 = y3; x6 = −x5; y6 = y5; x8 = −x7; y8 = y7; x10 = −x9; y10 = y9; x12 = −x11; y12 = y11; x14 = −x13; y14 = y13; x20 = −x19; y20 = y19; x21 = −x18; y21 = y18

Critical constraints were imposed to ensure the tower’s structural integrity under various loading conditions. Tension and compression stress limits were set at 20 ksi and 15 ksi, respectively. Moreover, the Euler buckling strength of each member was accounted for, with the buckling strength defined as 3.96 × EA/L².

The optimization of the 47-bar truss structure is particularly challenging due to multiple loading conditions and a high number of design variables, making it significantly more complicated than previous benchmark examples. A review of the existing literature, summarized in

Table 8. Comparison of optimized designs for the 47-bar truss.

No.	Design Variables	SCPSO [56]	BBM [67]	GA [68]	D-ICDE [61]	ABC [62]	MLA [63]	CSBO	NCSBO
1	A1 (in2)	2.5	2.7	2.5	2.7	2.4	2.7	2.9	2.8
2	A2	2.5	2.6	2.2	3	2.2	2.5	2.6	2.6
3	A3	0.8	0.7	0.7	0.5	1.1	0.7	0.6	0.7
4	A4	0.1	0.4	0.1	1.1	0.1	0.1	0.1	0.1
5	A5	0.7	0.8	1.3	0.7	1.2	1	1	0.9
6	A6	1.4	1.2	1.3	1.5	1.3	1.3	1.1	1.1
7	A7	1.7	1.7	1.8	2.1	1.7	2	2	2
8	A8	0.8	0.8	0.5	0.9	0.6	0.6	0.6	0.6
9	A9	0.9	1.1	0.8	0.8	0.8	0.9	0.9	0.9
10	A10	1.3	1.4	1.2	1.8	1.6	1.4	1.3	1.3
11	A11	0.3	0.4	0.4	0.4	0.3	0.3	0.5	0.5
12	A12	0.9	1	1.2	1	0.9	1.1	1.3	1.3
13	A13	1	1	0.9	1.3	1.2	1.1	1	1
14	A14	1.1	1.1	1	1.6	1	1.1	1	1
15	A15	5	0.8	3.6	1	1	0.9	0.7	0.7
16	A16	0.1	0.6	0.1	0.4	0.6	0.1	0.1	0.1
17	A17	2.5	2.7	2.4	3	2.8	2.6	2.8	2.8
18	A18	1	0.9	1.1	0.9	0.4	1	0.9	0.8
19	A19	0.1	0.1	0.1	0.2	0.1	0.1	0.1	0.1
20	A20	2.8	2.9	2.7	3.3	2.9	2.9	3	3
21	A21	0.9	1	0.8	0.4	1.5	0.8	0.9	0.8
22	A22	0.1	0.5	0.1	0.1	0.6	0.1	0.1	0.1
23	A23	3	3.1	2.8	3.3	3.1	3.1	3.1	3.1
24	A24	1	1.1	1.3	0.3	0.9	1	1	1
25	A25	0.1	0.1	0.2	0.1	0.1	0.1	0.1	0.1
26	A26	3.2	3.2	3	3.3	3.3	3.2	3.2	3.2
27	A27	1.2	1.1	1.2	0.5	0.8	1.3	1.2	1.2
28	X2 (in)	101.3393	106	114	106.2986	103.6063	104.929	102.916	103.16734
29	X4	85.9111	89	97	82.4936	81.5008	78.2568	83.58044	83.31772
30	Y4	135.9645	136	125	136.9634	143.0525	156.6791	150.4427	148.59871
31	X6	74.7969	66	76	62.7192	67.0169	67.7817	64.41907	65.00997
32	Y6	237.7447	255	261	244.4495	252.8466	254.8397	263.1986	261.78296
33	X8	64.3115	57	69	47.563	54.5203	59.7804	53.52898	54.22948
34	Y8	321.3416	342	316	332.7201	374.0126	323.7404	333.5202	329.84055
35	X10	53.3345	50	56	42.7377	39.8226	50.386	46.75877	47.56234
36	Y10	414.3025	415	414	401.7876	443.9461	414.8346	398.9399	398.27862
37	X12	46.0277	45	50	32.8229	30.9474	42.9782	42.57715	43.52248
38	Y12	489.9216	475	463	468.0985	491.9941	467.3015	445.722	450.88035
39	X14	41.8353	40	54	27.0026	36.7597	43.9541	44.47806	43.90031
40	Y14	522.4161	513	524	500.416	510	516.8386	497.4669	494.24315
41	X20	1.0005	17	1	11.9079	17.6763	12.2461	1.639016	2.05889
42	Y20	598.3905	598	587	581.5046	598.8911	586.3233	571.3341	571.56796
43	X21	97.8696	93	99	82.6543	77.6661	91.1605	86.42174	86.39172
44	Y21	624.0552	624	631	611.0089	619.8911	621.2209	629.3503	629.35088
Best weight (lb)		1864.10	1934.17	1925.79	1744.8	1871.843	1888.146	1881.965	1864.7753
Mean weight (lb)		1894.056	N/A	N/A	N/A	1887.838	1912.530	1909.574	1891.056
Worst weight (lb)		2007.563	N/A	N/A	N/A	1961.119	1929.989	1962.741	1918.475
ODNSA		25,000	N/A	100,000	17,745	2850	30,000	8910	7590
TNSA		25,000	N/A	100,000	17,745	18,000	30,000	18,000	18,000
STD (lb)		34.755	N/A	N/A	N/A	7.5649	11.534	25.38	15.47
CVP (%)		0	0.201	0	966.92	269.77	0	0	0

, reveals that NCSBO produced the second lightest design after SCPSO, weighing 1864.7753 lb compared to 1864.10 lb. However, NCSBO achieved an optimal design weighing 1864.7753 lb after 7580 analyses, while SCPSO found a slightly lighter design at 1864.10 lb (0.036% lighter) after 25,000 analyses. While D-ICDE results in a lower weight of 1744.80 lb, it fails to meet stress constraints. Similarly, the results yielded by the ABC and BBM violate stress constraints, as indicated in Table 8. Therefore, NCSBO demonstrates superiority among almost all optimization methods reported in Table 8 in achieving feasible and optimal outcomes for this test structure. The optimal design obtained by CSBO has a weight of 1881.795 lb after 8910 structural analyses, which is 0.9% heavier than the optimal design found by NCSBO. Moreover, CSBO required 1320 additional structural analyses to reach its result. These findings clearly demonstrate that NCSBO exhibits superior performance compared to standard CSBO.

Figure 19 illustrates the optimized configurations of the 47-bar truss, accompanied by a color map illustrating the stress capacity ratio of each element under the most unfavorable loading condition.

Figure 20 exhibits color map representations illustrating the stresses corresponding to each loading case. The maximum member stresses for the optimized design found by CSBO are 19.99994 ksi, 12.94114 ksi, and 19.99994 ksi for the first, second, and third loading conditions, respectively. The minimum stresses for these conditions are −14.99999 ksi, −14.99995 ksi, and −14.99998 ksi. For the NCSBO design, the maximum member stresses are 19.99956 ksi, 12.940895 ksi, and 19.99956 ksi across the same loading conditions, with minimum stresses of −14.99987 ksi, −14.99781 ksi, and −14.99997 ksi. These values indicate that the member stresses remain within the specified limits of [−15, 40 ksi], as illustrated in Figure 21. Furthermore, Figure 21 illustrates the stress states relative to the boundary buckling values, which vary according to the length and cross-sectional area of each element. None of the elements surpass the stress limits, indicating the structural integrity and robustness of the truss’s design under the considered loading conditions.

The convergence curve of NCSBO is compared with CSBO, as depicted in Figure 22. The optimum solutions are achieved after 7590 and 8910 analyses for NCSBO and CSBO, respectively. It was observed that NCSBO shows a faster convergence trend compared to CSBO in the early stages of the optimization process. However, in the later stages, the convergence behaviors of both methods become similar, with each eventually reaching the final optimum solution and remaining stable until the maximum number of analyses is reached.

6. Conclusions

In this paper, the performance of the proposed NCSBO algorithm is investigated by optimizing four well-known truss structures with sizing and shape variables. In almost all cases, the optimum results obtained by NCSBO are superior to other state-of-the-art metaheuristic optimization methods in the literature. The statistical performance of NCSBO demonstrates its high capability for convergence to the optimum. NCSBO consistently demonstrates superior performance in achieving lighter designs while satisfying structural constraints effectively. This development enables NCSBO to explore the design space efficiently, resulting in optimized truss configurations with reduced weights. The design examples presented in this paper confirm that NCSBO is an efficient and robust optimization tool that minimizes weight and satisfies design constraints. Furthermore, NCSBO shows promising potential for practical engineering applications and other structural design problems.