Multi-population particle swarm optimization algorithm for automatic design of steel frames

Abstract

Steel structures are widely used; however, their traditional design method is a trial-and-error procedure which is neither efficient nor cost effective. Therefore, a multi-population particle swarm optimization (MPPSO) algorithm is developed to optimize the weight of steel frames according to standard design codes. Modifications are made to improve the algorithm performances including the constraint-based strategy, piecewise mean learning strategy and multi-population cooperative strategy. The proposed method is tested against the representative frame taken from American standards and against other steel frames matching Chinese design codes. The related parameter influences on optimization results are discussed. For the representative frame, MPPSO can achieve greater efficiency through reduction of the number of analyses by more than 65% and can obtain frame with the weight for at least 2.4% lighter. A similar trend can also be observed in cases subjected to Chinese design codes. In addition, a migration interval of 1 and the number of populations as 5 are recommended to obtain better MPPSO results. The purpose of the study is to propose a method with high efficiency and robustness that is not confined to structural scales and design codes. It aims to provide a reference for automatic structural optimization design problems even with dimensional complexity. The proposed method can be easily generalized to the optimization problem of other structural systems.

Graphical abstract

Keywords

steel frame / multi-population particle swarm optimization / automatic structural optimization design

1 Introduction

Steel structures have advantages of good earthquake-resistant performance, high ductility and construction convenience. The amount of steel used for construction in developed countries such as the United States and Japan accounts for about 50% of total steel output. By comparison in China, the matching proportion is only about 20%. Furthermore, while the output value of steel structures accounts for less than 3% of that of total construction in 2020, the application and development of this type of structure have been accelerated in recent years.

Steel structures are traditionally designed by a trial-and-error procedure which is based on the principles of mechanics and numerical analysis [1]. In practice, the design process is often performed by using finite element (FE) software. With specified seismic precautionary objectives and design parameters of the ground motion (e.g., seismic fortification intensity), this performance-based analysis simulates output variables such as the inter-story drift ratio (IDR) and displacement [2–5]. Then, to satisfy safety demands, the structural form and the configuration of structural components are manually adjusted based on the expertise of workers according to the standard design codes and architectural functions [6–9]. Although designers may have abundant practical experience, such a method still requires days or even months of work due to the tediousness of repeated parameter tuning of the calculation model in order to obtain a desirable design. In addition, the method cannot guarantee the economy of the designed building because manual design is always conservative. Thus, traditional structural design is neither efficient nor cost effective.

To improve the above issue, structural optimization using intelligent tools has become increasingly popular [10]. The aim is to achieve prescribed objectives, such as minimization of total cost, while the optimal design satisfies the code-stipulated constraints including the stiffness requirements at the structural level and the strength requirements at the member level. The use of optimization algorithms for structures can effectively save a lot of repetitive labor work, satisfying the safety demands and achieving economic benefit of the designed structure. In general, there are two common techniques used in optimization problems: the classical search method based on a mathematical formula and a metaheuristic algorithm. The former approaches the solution using gradient information (such as gradient descent) and nonlinear programming [11]. However, the optimization problem of actual steel frames is nonconvex and highly nonlinear, involving high-dimensional design variables. The direct relationship between design variables and objectives is often missing or its derivation requires substantial computing resources. Moreover, the final solution relies on the initially selected variable so that the global optimum cannot be necessarily obtained. Therefore, the mathematical programming method is difficult to handle for a complex engineering problem such as the structural optimization of steel frames. Thus, with the development of computer technology, metaheuristic algorithms which do not require gradient data have emerged for adoption in optimization problems. The efficiency of such method derives from its good global search ability in the exploration stage and the local search ability in exploitation stage. Those algorithms are mainly inspired by different natural resources, such as a natural evolution based genetic algorithm [12,13] or animal behavior based particle swarm algorithm [14]. By employing the population of agents in a feasible region based on specified rules, the steel structure can be optimized and its total mass can be reduced, with outputs, such as displacement, IDR and strength, being less than the limiting values [15–20]. The engineering demand parameters of the structure (e.g., base shear) have also been optimized by such method [12,21]. In addition, many optimization heuristics and machine learning algorithms have been used to evaluate and predict the mechanical properties of reinforced concrete structures, and these studies have provided new approaches to structural optimization [22–26]. Despite the advantages of the population-based method, there are several limitations to structural optimization. For example, the random search feature cannot ensure the robustness of the algorithm, making search agents sometimes converge to an invalid solution (such that the design is unable to satisfy all structural constraints). In addition, the final result highly relies on the initial parameter settings (such as population size), and establishment of such settings requires a large number of FE analyses and a trial-and-error process. These barriers lead to the inefficiency of optimizations and large computational cost. The application of metaheuristic algorithms in complex problems such as the optimization of steel frames is still a challenging task.

To overcome the aforementioned limitations, multi-population algorithms are used in many fields [27–31]. Instead of one group of individuals, multiple sets of parallel populations are used to search for the result, with each population acting as an independent optimizer with specific performances due to specific parameter settings. To accelerate the convergence rate, multi-population collaboration is performed periodically to exchange the information of optimizers. This mechanism extends the explorative abilities of the algorithm and is capable of decreasing the number of analyses without reducing the probability of finding the optimum solution [32,33]. Consequently, the multi-population algorithm can significantly reduce the impact of the disadvantages, such as the result reliance on initial parameter setting and the random search feature, of the metaheuristic algorithm. For example, a multi-population cooperative particle swarm optimization was introduced and tested by benchmark functions, showing faster convergence speed and better convergence compared to the more basic particle swarm optimization [29]. However, studies on the structural optimization using the more recent method based on coevolution are still limited. Various difficulties involved in structural designs remain, including.

1) Member size is one design variable in optimizing a steel frame, which is usually selected from a designated member list. However, the member shape itself has several parameters (e.g., flange width, flange thickness, web depth, and web thickness for an I-shape), which are related to the optimization. With these numerous design variables, the optimization task becomes cumbersome; the effectiveness of available algorithms needs to be examined.

2) Most previous studies have investigated the optimization of two-dimensional steel frames based on AISC LRFD Specifications. The available algorithms need to be checked for structural designs with various design parameters and different design codes.

3) The multi-population algorithm has multiple parallel optimizers, with each of them having different optimization performance. The influence of the settings in each optimizer and the parameters of migration strategies (such as the migration interval and the population number) needs to be investigated to provide a basis for practical applications.

In view of the above, the optimization of steel structures subjected to various structural constraints is investigated in this work, based on a multi-population particle swarm optimization (MPPSO) algorithm with several improved strategies. The main contributions of such method include.

1) A large-scale optimization problem with more dimensional complexity is studied and all involving parameters of the member size are separately considered. This gives the structural design more available results and makes it more likely to obtain economic structure. It can be easily extended to many design practices.

2) Different examples of steel frames, satisfying different design codes, as well as different spatial complexities of the structure (planar or spatial steel frames), are applied to investigate the performance of the optimization algorithm, to exhibit the generality of the proposed method.

3) The influences of the migration interval and the population number on optimization performances are evaluated, which helps to provide recommendations for algorithm parameter settings of the proposed method in practical application of steel frame optimization design problems.

The paper is constructed as follows. Section 2 illustrates the structural optimization problem of steel frames. Section 3 presents the details of different optimization algorithms and modification strategies. Section 4 provides the optimization results of a representative frame taken from the AISC Manual and of another two steel frames matching the Chinese code. The performances of different modified algorithms are compared with each other. Section 5 discusses the influences of parameter settings on the performances of the MPPSO.

2 Structural optimization problem

The benchmark problem is described in Subsection 4.1, while the optimization of two studied structures is introduced in this section.

2.1 Design variables

The structure is designed according to the standard design codes including GB 50011-2010 [34], JGJ 99-2015 [35], and GB 50017-2017 [36]. The members in a steel frame are columns and beams made of I-type sections. The section size is represented by four parameters: section height, flange width, web thickness, and flange thickness, whose ranges and increments are listed in Tab.1.

2.2 Objective function

The objective function of steel structures is the minimum material weight defined as:

(1)W=\sum _{i=1}^{N_m}{\rho }_{i}A{{}_{i}l}_i,

where A_i and l_i are the area and length of member i; ρ_i is the density of the steel, respectively; N_m is the total number of structural members. The external penalty method is used to consider all structural constraints and transform the optimization problem into an unconstrained one. The penalized objective function P is defined as:

(2)P=W+100 \sum _{i=1}^n{G}_i,

where n is the total number of structural constraints; G_i is an auxiliary function used to measure the violation of the ith constraint and is formulated as:

(3)G_i= \{\begin{array}{cc}0,& g_i\le 1,\\ 1,& g_i>1,\end{array}

where g_i is the constraint value defined in Subsection 2.3. When the structure meets all corresponding design requirements, the sum of G_i is equal to 0 and P is equal to W. Thus, the penalty objective function is a comprehensive evaluation index considering the extent of the constraint violation. A smaller value represents a better solution with less weight of the structure.

2.3 Design constraints

The structural constraint and the component constraints are considered in the optimization of steel frames.

The IDR constraint g_i^{\mathrm{d}} is defined as:

(4)g_i^{\mathrm{d}}=\frac{\left|d_i\right|}{d_i^a},

where d_i is the maximum IDR of story i; and d_i^{\mathrm{a}} is the limit IDR value (= 1/250).

The strength constraint g_i^{\sigma } is defined as:

(5)g_i^{\sigma }=\frac{\left|{\sigma }_i\right|}{{\sigma }_i^a} ,

where σ_i is the maximum stress of members in group i; and {\sigma }_i^{\mathrm{a}} is the yielding strength (= 235 MPa).

For stability constraint g_i^{x/y} of members in group i is

(6)g_i^x= \frac{N}{{\phi }_{x}Af}+\frac{{\beta }_{\mathrm{m}x}{M}_x}{{\gamma }_x{W}_x\left(1-0.8N/N_{\mathrm{E}x}^{\prime }\right)f}+\eta \frac{{\beta }_{\mathrm{t}y}{M}_y}{{\phi }_{\mathrm{b}y}{W}_{y}f},

(7)g_i^y= \frac{N}{{\phi }_{y}Af}+\eta \frac{{\beta }_{\mathrm{t}x}{M}_x}{{\phi }_{\mathrm{b}x}{W}_{x}f}+\frac{{\beta }_{\mathrm{m}y}{M}_y}{{\gamma }_y{W}_y\left(1-0.8N/N_{\mathrm{E}y}^{\prime }\right)f},

where N, A, f, and η are the axial force, cross-sectional area, design strength, and section influence coefficient, respectively; and M, γ, φ, φ_b, β_m, β_t, W, and N_{\mathrm{E}}^{\prime } are the moment, coefficient of section plastic development, global stability coefficient under axial compression, global stability coefficient under bending moment, in-plane equivalent moment coefficient, out-of-plane equivalent moment coefficient, section modulus, and Euler force, in either x or y direction, respectively.

The geometry constraint g_i^{\mathrm{w}} is defined as:

(8)g_i^{\mathrm{w}}=\frac{{b{h}_{\mathrm{w}i}/bt}_{\mathrm{w}i}}{\mathrm{m}\mathrm{i}\mathrm{n}\left(85-120{\rho }_i,75\right)},

(9)g_i^{\text{λ}} =\frac{{\lambda }_i}{{\text{λ}}_i^{\mathrm{a}}},

where bh_wi, bt_wi, and ρ_i are the web height, web thickness, and axial compression ratio in beam group i, respectively; and λ_i and {\lambda }_i^{\mathrm{a}} are the slenderness ratio and its limiting value (= 100) in column group i, respectively.

Some geometrical constraints tested before the structural analysis are directly considered by the constraint-based strategy in algorithms described in Subsection 3.1. For columns, the constraint is described by:

(10)c{b}_{\mathrm{f}i}/c{t}_{\mathrm{f}i}\le 13\times \epsilon ,

(11)c{h}_{\mathrm{w}i}/c{t}_{\mathrm{w}i}\le 52\times \epsilon ,

where cb_fi, ch_wi, ct_fi, and ct_wi are the width of flange overhang, web height, flange thickness, and web thickness in column group i, respectively. For beams, the constraint is expressed by:

(12)b{b}_{\mathrm{f}i}/b{t}_{\mathrm{f}i}\le 11\times \epsilon ,

where bb_fi and bt_fi are the width and thickness of flange overhang in beam group i, respectively.

2.4 Automatic optimization design process

The automatic optimization design of structures is composed of mechanical analysis and optimization algorithm, as shown in Fig.1. The first part is completed by the FE software MSC.Marc. Repeated modeling and massive calculation are required for structures whose parameters are generated by an optimization algorithm. The second part is performed by Python-based codes to collect analysis results to realize the updated structural parameters defined by specified rules. These two parts proceed alternatively until the termination criterion is met and the optimized structure is obtained. The multi-core parallel computing technique is adopted to improve the design efficiency. The automatic optimization process is performed on a computer with 32 GB RAM and Intel i7 2.90 GHz processor.

3 Optimization algorithm

Five modified algorithms with the constraint-based strategy are introduced into the optimization design of steel frames and each one is repeated 20 times. The optimization is terminated when the reduction rate of the structural weight is less than 0.002 for 20 consecutive adjacent successive optimizations.

3.1 Constraint-based strategy

During the initialization and variable updating of algorithms, the newly generated individuals cannot always satisfy all the constraints, which increases the number of invalid individuals and results in meaningless calculations. Thus, the constraint-based strategy (Fig.2) is used in all discussed algorithms to be applied in the optimization of steel frames. Based on Eqs. (10)–(12) and integer constraint, the design parameters of a newly produced member are pretested before the structural analysis in the next iteration and regenerated until satisfying those constraints. When such an operation repeats a certain number of times, the parameters are randomly selected within a specified range. The regeneration of variables is provided by numerical calculation based on mathematical formulas. It is conducted by python code and such process only takes, approximately, less than a millisecond. To set the max times of variable regeneration, the purpose is to avoid the program getting stuck in repeating dead-end circle in which the variable cannot always meet the pre-test constraints.

3.2 Simulated annealing

Simulated annealing (SA) is a stochastic method based on the Metropolis sampling criterion characterized by a probabilistic sudden jump. As the high initial annealing temperature drops, the optimal solution is constantly searched within a feasible region. The algorithm has a certain probability of accepting a worse solution, which can avoid the trapping in local optima. The main process is shown in Algorithm 1.

3.3 Direction-based adaptive genetic algorithm

To improve the performances of the GA, the direction-based adaptive genetic algorithm (DBAGA) (Algorithm 2) is proposed with several strategies.

Fitness scaling. To control the probability of the individuals with large fitness to be selected in the initial stage and amplify the difference of individuals with similar fitness in a later stage, the linear scaling method is applied for the fitness f. The scaled fitness f^{\prime } is defined as:

(13)f^{\prime }=af+b,\text{where}{f}_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\prime } =f_{\mathrm{a}\mathrm{v}\mathrm{e}},\text{and}{f^{\prime }}_{max}=c{f}_{\mathrm{a}\mathrm{v}\mathrm{e}},

where a and b are the scaled coefficients; and c is the expected replication coefficient ( = 2); in this equation, f^{\prime } can be taken as f_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\prime } and f_{max}^{\prime }, and f can be taken as f_{\mathrm{a}\mathrm{v}\mathrm{e}}.

Direction-based crossover. The moving directions of two selected individuals are guided by the better one to make them search a region with larger probability to find the optimal solution. The direction-based crossover is defined as:

(14)X_i^{t+1}=X_i^t+ \lambda r_1\left(X_j^t-X_i^t\right), X_j^{t+1}=X_j^t+ \lambda r_2\left(X_j^t-X_i^t\right),

where X_i^t and X_j^t are respectively the selected individuals with better fitness and worse fitness in iteration t; λ is the step size (= 1); and r₁ and r₂ are the random vectors with values in [0, 1].

Nonuniform mutation. Instead of replacing the gene value x_i by a random number, a small random perturbation of the gene is made to enhance the local search ability of the algorithm, and the altered gene value is:

(15)x_i^{\prime }=\{\begin{array}{cc}{\mathrm{x}}_i+\left(x_i^{\mathrm{u}}-x_i\right)\left(1-r^{\left(1-t/T_{\mathrm{m}}\right)b}\right),& \mathrm{i}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\left(0,1\right)=0,\\ x_i-\left(x_i-x_i^{\mathrm{d}}\right)\left(1-r^{\left(1-t/T_{\mathrm{m}}\right)b}\right),& \mathrm{i}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\left(0,1\right)=1,\end{array}

where x_i^{\mathrm{u}} and x_i^{\mathrm{d}} are the upper and lower limits of x_i, respectively; T_m is the maximum iterations; b is the scale coefficient.

Adaptive probability. The two-phase crossover probability and three-phase mutation probability are adopted for individuals. A larger probability is assigned to the individual with lower fitness to facilitate the search ability, while a smaller one is applied to the individual with greater fitness, to improve the convergence rate. The relevant equation is:

(16)

where f_i and f_a are the fitness of individual i and the average fitness of individuals, respectively; pc₁ and pc₂ are the large and small crossover probabilities, respectively; pm₁, pm₂, and pc₃ are the large, medium, and small mutation probabilities, respectively.

Elite strategy and substitution of duplicate item. The historical optimal solution is saved and used to replace the worst solution in the current iteration to enhance the convergence rate of the algorithm. However, such strategy may lead to premature convergence, which can be improved by substituting the duplicate item with the elite after mutations.

3.4 Adaptive particle swarm optimization

The adaptive particle swarm optimization (APSO) (Algorithm 3) searches for the optimal solution by updating the positions of particles based on their own historical best solution Pbes{t}_i^t and the global historical optimal solution Gbest^t, which is defined at iteration t as:

(17)\begin{array}{rl}& V_i^{t+1}=\omega V_i^t+c_1{r}_1\left(Pbes{t}_i^t-X_i^t\right)+c_2{r}_2\left(Gbes{t}^t-X_i^t\right),\\ & X_i^{t+1}=X_i^t+V_i^{t+1},\end{array}

where V_i^t and X_i^t are the velocity and the position of particle i, respectively; c₁ is the linearly decreasing learning factor of individual behavior and c₂ is the linearly increasing factor of social behavior; r₁ and r₂ are the random numbers within the range [0,1]; and ω is the linearly decreasing inertia weight defined as:

(18)\omega =\{\begin{array}{ll}{\omega }_{min}+\left({\omega }_{max}-{\omega }_{min}\right)\left(f_i-f_{min}\right)/\left(f_{\mathrm{a}}-f_{min}\right),& f_i\le f_{\mathrm{a}},\\ {\omega }_{max},& f_i>f_{\mathrm{a}},\end{array}

where ω_max and ω_min are the maximum and minimum weight, respectively; and f_min is the minimum fitness of individuals.

3.5 Multi-population genetic algorithm

The multi-population genetic algorithm (MPGA) is obtained by introducing the multi-population cooperative strategy into the DBAGA. The main steps of the MPGA are shown in Algorithm 4.

Multi-population cooperative strategy. To improve the problem of the dependence of the result on the parameter setting, a multi-population cooperative strategy is used with the migration mechanism. Multiple groups (mp) of individuals are adopted for parallel and independent optimizations, which have different parameter settings randomly selected within a specified range. For every defined number of iterations t_m, i.e., migration interval, the information is exchanged by alternatively replacing the worst solution of the neighbor group with the historical optimal solution of the current group, as shown in Fig.3.

3.6 Multi-population particle swarm optimization

Compared to the APSO, there are two additional rules supplemented to the MPPSO (Algorithm 5).

Piecewise mean learning strategy. In each iteration, the population is divided into two groups: the elite group (EP) and the general group (GP) with a specified proportion. The dimension of particles is divided into m partitions (m is equal to total number of member groups of the studied steel frame). In each partition m, two steps are conducted for particle updating: 1) a 4-dimensional model particle is generated by k particles randomly selected from the EP, as expressed by Eqs. (19), to improve the learning ability of the algorithm; 2) the mth partition of each particle, i.e., a 4-dimensional variable, in the GP is updated using Eqs. (20) which is a modified version of Eqs. (17).

(19)p{l}_m^t= \sum _{i=1}^k\frac{f_i{X}_{mi}^t}{{\displaystyle \sum _{j=1}^k{f}_j}},

(20)X_{mi}^{t+1}=X_{mi}^t+ \omega V_{mi}^t+c_1{r}_1\left(p{l}_m^t-X_{mi}^t\right)+c_2{r}_2\left(Gbes{t}_m^t-X_{mi}^t\right),

where m is the partition number; X_{mi}^t and V_{mi}^t are the mth partitioned position and velocity of the particle X_i in iteration t, respectively; Gbes{t}_m^t is the mth partitioned position of Gbest^t.

Multi-population cooperative strategy. It is the same as MPGA described above in Subsection 3.5.

This study discusses the performance improvement of the algorithm by a multi-population cooperative strategy. Therefore, the SA, DBAGA, and APSO algorithms adopt the most basic and commonly used parameter settings (Tab.2). For MPGA and MPPSO, the mechanism of multiple populations is allowed to set a variable range of parameters and randomly generate multiple optimizers with different parameters. These optimizers can use information sharing to complement their performance differences. Therefore, two types of parameters are involved, which are: 1) parameters related to the basic optimizer which are randomly selected within the given range to ensure that multiple parallel optimizers have different parameter settings; 2) fixed parameters related to multi-population cooperative strategy which are determined based on sensitivity analysis described in Section 5. As is well known, increase of population size leads to improved performance of the algorithm, but also requires more computational resources. To better exhibit the excellent performance improvement by multiple population mechanisms, the population size for MPGA and MPPSO is set to a small value of 50, while the one for others is set to 100.

4 Results and discussion of different algorithms

A FE model for steel frames is established using MSC.Marc, where the structural members are represented by beam elements. The optimization is performed with different design codes.

4.1 Three-bay 24-story frame according to the AISC LRFD [37]

Fig.4 shows the geometry and applied loads for the three-bay 24-story frame [38–40] with 4 beam groups chosen from 267 W-shapes and 16 column groups selected from W14 sections. The size of the search space is 6.27 × 10³⁴. The loads are W = 25.628 kN, w₁ = 4.378 kN/m, w₂ = 6.362 kN/m, w₃ = 6.917 kN/m, and w₄ = 5.954 kN/m. The material has elastic modulus of 205 GPa, and the yielding strength of 230.3 MPa is the stress limit. The displacement constraints are: maximum IDR = 1/300 and limiting roof displacement = 292.6 mm. The strength constraint is described by:

(21)

where P_u and P_n are the required and nominal axial strengths, respectively; M_ux and M_uy are the required flexural strengths in x and y directions, respectively; M_nx and M_ny are the nominal flexural strengths in x and y directions, respectively; ϕ is the resistance factor ( = 0.90 for tensile yielding and 0.85 for compression); ϕ_b is the flexural resistance factor (= −0.90). Other detailed information can be found in Refs. [38–40].

Tab.3 lists the optimization results from different algorithms. Compared to other relevant algorithms, the standard deviation (std.) of the MPPSO is the largest. When compared with the ACO [38], the difference of std. reaches 2422 lb. However, such difference is only 1.1% of the total frame weight and considered acceptable for practical designs. In addition, the MPPSO has the minimum weight of 209814 lb which is 2.4%–4.8% smaller and the mean weight of 219823 lb which is 1.3%–4.2% less, compared to other results. The total analysis numbers of the MPPSO are respectively 66.8%, 68.3%, and 66.5% smaller than those of the SBO [32], ACO [38,39] and HS [40]. With the multi-population cooperative strategy, the MPPSO requires lower computational cost and can obtain an optimal structure with smaller weight, demonstrating the efficiency of the proposed algorithm.

4.2 Planar frame according to the design codes [26–28]

The material of steel frames has density of 7850 kg/m³ and elastic modulus of 206 GPa. The representative gravity load (RG) and equivalent horizontal earthquake (EX) are considered. The RG consists of the dead load of 4.5 kN/m² and the live load of 2 kN/m², which act on the beams as vertical line loads. The EX is determined by the equivalent base shear method, which is applied to the columns as the horizontal point load. The seismic precautionary intensity, design characteristic period of ground motions, peak ground acceleration, and damping ratio are respectively 6 degrees, 0.35s, 0.018g, and 0.04. The load combination is 1.3RG + 1.3EX. The structural members are divided into groups with each group having four design variables (Tab.1).

Fig.5(a) shows a 10-story planar frame with three standard stories (SS). Each SS has 3 structural member groups including corner columns (JZ), side columns (BZ) and beams (L), resulting in a total of 9 element groups for the structure. The size of the search space is 1.8 × 10⁶².

Fig.6 and Tab.4 show the optimization results of the planar frame using different algorithms. The penalty costs by the SA and DBAGA show large discreteness with large std. values of 78.0 and 89.1 t, respectively. Their penalty cost curves reach a plateau in the early stage and drop afterwards, illustrating the weak local search ability of algorithms for complicated problems. The optimized structures cannot meet all design requirements in most runs. By contrast, the convergence of the other 3 algorithms is better and the largest std. is only 1.2 t. Compared to the MPGA, the APSO and MPPSO have smaller std.-to-optimal ratios (less than 3.4%) and converge to a value close to the final result with less than 1500 analyses. The PSO-based algorithms have better astringency in the optimization of steel frames due to learning behaviors. Moreover, in comparison with the APSO and MPGA, the MPPSO has the optimal cost of 4.2%–15.0% less (11.3 t) and the average cost of 5.7%–25.6% less (11.6 t), and requires significantly fewer analyses in a single run (average = 3710).

The obtained optimal frames by all algorithms satisfy all constraints (Fig.7). The results are different in each case due to the complicated force allocation of structural members. However, compared to others, the structural performance the optimal structure achieved by the MPPSO is generally superior and approaches to the limiting values. In particular, the stability values are almost within 0.9–1.0 and the maximum IDR reaches 1/250, which shows a better utilization of structural resources, so that the frame meet the safety requirements with less material.

In addition, in comparison with the APSO, MPGA, and DBAGA, the MPPSO has better economical factors (optimal, average, worst, and std. cost) and material utilization of the structure due to its multi-population cooperative strategy. Such mechanism improves the explorative ability and the overall efficiency of the algorithm by reducing the dependence of the result on the parameter setting.

4.3 Spatial frame according to the design codes [26–28]

Fig.5(b) shows a 6-story spatial frame with three SS, and each SS has 5 element groups including JZ, BZ, middle columns (ZZ), side beams (BL), and middle beams (ZL), resulting in a total of 15 element groups for the structure. The size of the search space is 5.7 × 10¹⁰³.

Fig.8 and Tab.4 show the optimization results of the spatial frame using different algorithms. As the problem becomes much more complicated than in the case of the planar frame, 80% of optimal results for the SA and DBAGA are non-feasible as the obtained structures cannot satisfy all constraints. Their average costs reach 192.8 t and 162.5 t respectively and their penalty costs show a premature convergence of solutions. Compared to the DBAGA, with the multi-population cooperative strategy, the MPGA obtains feasible results in all runs and has good astringency with a much lower average cost of 43.1 t. However, the PSO-based algorithms present the superior search capability and better cost than other algorithms because of their learning behaviors among particles. In contrast with the MPGA, the average cost and std. of the APSO are respectively 27.6% and 74.6% lower. Furthermore, the optimal cost of the MPPSO is the smallest of all algorithms at 24.7 t, which is 10.2% less than that of the APSO. The piecewise mean learning strategy makes the GP learn from multiple particles of the EP, based on the structural member level. The multiple populations provide parallel optimization and information exchange. These two supplemental rules improve the global and local search abilities of the algorithm.

The structural performances of optimal results are different in each algorithm (Fig.9). In general, the MPPSO has all the values of the optimal frame closer to ultimate values, compared to other algorithms, implying that limited materials can maximize the mechanical properties of structure and show the economical factor. More than half of groups have stress exceeding 74.5% of a the yielding strength and more than 86.7% of groups have stability value reaching 0.8. The maximum IDR of the structure is 0.0037, closely approaching 1/250. Higher performance values for the MPGA can also be seen compared to the DBAGA due to its multi-population mechanism. In addition, the MPPSO presents a small computational cost in optimizing the spatial frame and requires 5597 analyses on average, which are respectively 10.3% and 59.6% lower than those of the MPGA and APSO.

5 Results and discussion

In this section, the influences of parameters in the MPPSO on optimization results are discussed. In the multi-population method, the independent optimizer and migration are the two important concepts which strongly affect the performance of the algorithm. The first part is a PSO-based optimizer and its efficiency has been shown in Section 4. Due to the learning behavior among particles and the piecewise mean learning strategy, it presents a good search ability including the convergence and obtained objective function value. However, the optimized result highly depends on the initial parameter setting of the algorithm. Therefore, the parameters in the independent optimizer are specified with a range in order to improve such problems. In comparison, the second part is more significant and the information of multiple populations is exchanged during the evolution process. The better solution can propagate among them, and is related to the convergence rate and the quality of the result in optimizations. Various aspects have a great impact on optimization performances (e.g., the migration interval and the number of populations). It is still a challenge to determine those parameters and such work is always based on a trial-and-error procedure. Thus, the spatial frame optimization problem described in Subsection 4.3 is used to investigate the influences of the parameters and the results are discussed. The penalty cost curves and the results are shown in Fig.10 and Tab.5, respectively.

5.1 Influence of migration interval

The migration interval t_m is the iteration step between two migration operations. Previous studies took the migration interval as 1, but two different values, 5 and 10, are tested in this study. The other parameters listed in Tab.2 remain unchanged. As seen, 3618 analyses are required on average for Case t_m = 10, which is less than in the other two cases. The optimization using the larger t_m shows a premature convergence of solutions. When the iteration does not reach the migration interval, the optimization of independent optimizers is conducted based on its specified pattern without exchanging information. The optimization with larger t_m may not bypass the local optima because the larger t_m leads to greater randomization of search. As a result, the migration mechanism is not effective and the performance of the MPPSO is similar to that of an independent optimizer. By contrast, when using a smaller t_m, the optimization results are better and all four cost indexes are reduced. The optimal cost and std. for Case t_m = 1 are 3.5% and 61.7% lighter than those for Case t_m = 10, respectively. This is because a smaller migration interval facilitates the information exchange and the timely propagation of superior particles among subpopulations, guiding the evolution direction of other particles toward a better solution. This helps to improve the quality of results and the convergence of the algorithm.

5.2 Influence of multiple populations

Previous studies adopted the number of populations mp as 5, while two other values, 8 and 3, are investigated in this study. In the three cases, the number of particles in each population is the same as 10 and the other parameters listed in Tab.2 remain unchanged. As seen, the cost index for Case mp = 8 outperforms the others in general. For example, both the penalty cost curves and the std. show that the discreteness of optimization results in 20 runs is smaller. The optimal and average costs are reduced by 7.8% and 12.3%, respectively, compared to those for Case mp = 3. This is because more particles increase the diversity of populations and make the search in feasible regions more adequate. The optimization may converge to a global optimal solution and avoid the premature problem. As a result, the performance of the proposed method is better when using a larger mp. In addition, when compared to Case mp = 5, the cost indexes for Case mp = 8 are very similar but the average value for 20 analyses is 8.7% larger. Therefore, considering both the economic factors of the optimized structure and the computing cost of the studied optimization problem, the MPPSO with 5 populations is preferred.

6 Concluding remarks

A modified multi-population cooperative particle swarm optimization algorithm is adopted for the automatic design of steel frames subjected to code-specified requirements. The external penalty function method is applied to solve a non-constrained optimization problem. The design process is established to minimize the penalty cost of the frame, with the analysis completed by the FE method. To reduce the randomness of the results in a single run, the multiple populations with different parameter settings are defined to provide the parallel optimization and information sharing mechanism to extend the explorative capacity of the algorithm. Each population is divided into the GP and the EP. A piecewise mean learning strategy is employed to make the general particles learn from the multiple elites to enhance the intra-population collaboration. The optimization design of three steel frames is made to test the performance of the MPPSO and the results are compared to other metaheuristic algorithms.

Compared to other algorithms, the proposed method requires fewer analyses to converge to an optimal solution, thus reducing the computational cost. The achieved optimal frame is more efficient and effective. The proposed method is capable of handling large-scale optimization problems and can achieve the good structural performance, meeting all design code requirements. The MPPSO is robust and effective for optimizing the design of a steel frame with large search space. However, there are still some limitations. This paper only focuses on the seismic design and single objective optimization of steel frames. More factors involved in practical structural design projects, such as the wind resistance of the structure and multi-objective optimization, are not considered. Although the proposed method can be easily generalized to optimization design of other structural systems, its applicability needs to be examined. These will be investigated in further studies.

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