Optimum design of truss structures with frequency constraints by an enhanced particle swarm optimization method with gradient directions based on emigration philosophy

In this article, the combined method of particle swarm optimization (PSO) with gradient directions (PSOG) is further extended to increase the capabilities of exploration and exploitation. Four different enhancements are added to the PSOG, based on emigration ideas. To investigate the power of the presented methods, 29 complicated functions are optimized with several local optima. Then, the suggested approaches are used in structural design problems with multiple frequency constraints containing sizing and shape design variables for two- and three-dimensional truss structures. The results are compared with some of the methods in the literature and the superiority of the suggested techniques is demonstrated.


Introduction
The efficient optimum design of structures with frequency constraints is an important field for designers. The phenomenon of response to dynamic loads is an important factor in the design of structures. The design variables of a building should be chosen in such a way that the natural frequencies of the buildings are prevented from becoming close to the frequencies of the dynamic loads. Thus, the optimum design of structures under multiple-frequency constraints is an important issue in the design philosophy (Bellagamba and Yang 1981;Grandhi 1993;Wamsler 2009;Katariya and Panda 2019;Kalita et al. 2020). This subject has entered into the optimal design of structures since the early of computer advancement. The research has mainly focused on achieving a more robust and efficient optimal design process.
In the early stages of the optimum design of structures with frequency constraints by mathematical programming (MP) techniques, the computational efficiency of the approaches was achieved by approximation concepts. In these methods, in each search direction, the frequencies are approximated by some approximate functions. For many years, owing to the limitations in computing capabilities, researchers focused on the presentation of more efficient methods of function approximation (Salajegheh and Vanderplaats 1987;Vanderplaats and Salajegheh 1988;Canfield 1990;Salajegheh 1997, 2000a and. In the second generation of optimization techniques, metaheuristic methods evolved as a result of progress in the capabilities of computers. These methods are used extensively in the optimization of structures, in particular, for frequency constraints. The optimum design of structures under frequency limitations is a complicated problem owing to the highly nonlinear functions involved, with several local optima and disjoint design space. Thus, metaheuristic methods are suitable for such problems. A great number of approaches in this category has been developed, with different natures (Lin, Che, and Yu 1982;Konzelman 1986;Grandhi and Venkayya 1988;Sedaghati, Suleman, and Tabarrok 2002;Wang, Zhang, and Jiang 2004;Lingyun et al. 2005;Salajegheh, Gholizadeh, and Torkzadeh 2007;Gomes 2011;Wei et al. 2011;Kaveh and Zolghadr 2012;Miguel and Miguel 2012;Khatibinia and Naseralavi 2014;Zuo, Bai, and Li 2014;Ho-Huu et al. 2018;Lieu, Do, and Lee 2018). The methods are still under development (Seyedpoor et al. 2009;Dehghani, Mashayekhi, and Salajegheh 2016;Yang, Bai, and Chen 2017;Kaveh and Ilchi Ghazaan 2018) as a method suitable for all optimization problems cannot be found (Wolpert and Macready 1997). New developments for static and dynamic loads are under investigation. In this regard, efficient approaches related to the present study have been presented by Tejani, Savsani, and Patel (2016), Tejani et al. ("Truss," 2018, "Size," 2018, Savsani, Tejani, and Patel (2016), Gholizadeh, Salajegheh, and Torkzadeh (2008), Gholizadeh and Milany (2018), Gholizadeh, Danesh, and Gheyratmand (2020); Bigham and Gholizadeh (2020), Mortazavi, Toğan, and Nuhoğlu (2018), Mortazavi (2020Mortazavi ( , 2021, Mortazavi and Moloodpoor (2021) and Liu et al. (2020).
In the present work, the method of particle swarm optimization (PSO) combined with MP using gradient directions, referred to as PSOG (Salajegheh and Salajegheh 2019), is further enhanced to increase the efficiency of the approach. The method is used in the optimal design of two-and threedimensional truss structures with frequency constraints. The design variables are the cross-sectional areas of the members and the coordinates of the joints. In addition, the method is used to find the minimum of mathematical functions with several local optima. The numerical results are compared with other methods and it is observed that great efficiency is achieved with the enhanced approach.
The remainder of this article is organized as follows. In Section 2, the mathematical formulation of the optimum design of structures subject to frequency constraints is presented for constrained problems. Then, the problem is transferred into an unconstrained formulation suitable for metaheuristic approaches. The summary of the PSOG method presented by Salajegheh and Salajegheh (2019) is outlined in Section 3, and the enhancements are explained based on the emigration ideology. Section 4 presents the numerical results together with further details of the design problems for frequency constraints. Section 5 concludes the article.

Problem formulation
The optimum design of structures with frequency constraints can be mathematically expressed as: Minimize the weight of the structure, W(X) subject to where X is the vector of design variables containing the sizing and shape variables, f j is the jth frequencyf j is the limits on the frequencies, and m is the number of constraints. This is a constrained optimization problem and must be converted into an unconstrained problem. A penalty function F(X) is chosen as follows (Salajegheh and Salajegheh 2019): The parameters b and μ are constants. Therefore, minimization of W(X) with frequency constraints is equivalent to the minimization of the unconstrained function F(X), which is the subject of Section 3. The frequencies can be obtained by the solution of the following homogeneous equations: in which K is the stiffness matrix, M is the consistent mass matrix, λ j is the jth eigenvalue, φ j is the corresponding eigenvector, and f j is the frequency (Hz) of the structure under consideration.

Enhanced PSOG
PSO is an attractive metaheuristic optimization method that has been used and modified by many researchers (Kennedy and Eberhart 1995;Lynn and Suganthan 2017;Sengupta, Basak, and Peters 2019). The method of PSO has been combined with the gradient directions of MP techniques, and the numerical results, with mathematical functions and examples of structural engineering problems, have shown that great efficiency can be achieved. This method is referred to as PSOG (Salajegheh and Salajegheh 2019). In the present work, PSOG is further modified to enhance the approach. First, the basic steps in the PSOG are outlined and then the enhancements are introduced.
To minimize the unconstrained function F(X), some initial design points are chosen randomly in the specified design space. Each design point is referred to as a particle. The position of each particle is modified in such a way that the new particles move gradually towards the optimal location of the function F(X). The position of each particle is modified as where V is the search direction of each particle X for the kth iteration, which is also called the velocity of the particle. In PSOG, the velocity vector is specified as where V PSO is the velocity in the standard PSO, Pbest is the best X of the particle [if F(X k ) < F(X k − 1 ) then Pbest = X k ], Gbest is the best X of all particles in all iterations, g is the vector of gradient direction, c coefficients are constants, r multipliers are random numbers between 0 and 1, and ω is a factor that varies linearly between 0.9 and 0.4. In general, some functions possess several local optima, and an attractive feature of a successful metaheuristic method is that it does not become trapped in a local optimum. Thus, a suitable method of optimization should have both exploration (global search) and exploitation (local search) capabilities. Although the PSOG method is much better than the standard PSO, the numerical investigation indicates some failed results. To increase the exploration and exploitation aspects of the method, several enhancements are introduced to greatly increase the efficiency of the approach. In this regard, two methods are introduced to increase the capability of the PSOG and each method possesses two enhancements. These are outlined in Subsections 3.1-3.4.

Method 1a
In this method, first, the optimization process starts with the PSOG to search the domain space. Then, after a certain number of iterations (N), the particles are divided into two parts. The particles with better performance are named 'group 1' and the remainder 'group 2'. The particles in the second group should emigrate to some other places to gain some extra skills for better performance. Therefore, particles in group 2 can ultimately perform better than or equally to the first group of particles. After each N iteration, regrouping is repeated. With this idea, there are always two groups, each of which tries to find a better optimal solution than the other one.
For this purpose, after performing every N iterations, some of the particles with better performance (group 1) are retained and the remaining particles (group 2) are redistributed with random velocities in the design space. The retained particles (group 1) continue the exploitation capability and the redistributed particles (group 2) increase the exploration of the approach. The redistributed particles try to improve their performance further.
A compromise is introduced between exploration and exploitation capabilities. If the redistributed particles can find better results for Gbest, they attract other design points to achieve better exploration; otherwise, they move towards Gbest, which increases the exploitation. The formulation for each particle is: where n is the number of design variables, α is a factor between 0 and 1, rand (n) is a vector with n random numbers, and x u and x l are the upper and lower limits on the design variables, respectively. The value of N is chosen based on trial and error. In Equation (7), the best particles update by PSOG velocity (c 3 V PSO − c 4 r 4 |V PSO |g) while the other particles move to random positions by a random velocity (αrand(n)(x u − x l )) every N iterations to achieve better exploration.

Method 1b
The numerical results indicate that in many cases Pbest and Gbest move close to each other. Therefore, the convergence towards Gbest is increased and thus the exploration should be increased to prevent the power of premature exploitation. In such cases, in addition to method 1a, Pbest is modified as = a natural number, and group 2 (8) By this formulation, Pbest of the redistributed particles is recalculated. The resulting distance between the updated Pbest and Gbest causes the proper change in V PSO and better search performance. By Equation (8), the redistributed particles forget their previous Pbest by ((Pbest = X k ) and their velocity vectors have a direction towards the random locations (X k ) until these particles find better solutions (F(X k ) ≤ F(X k−1 )).
The basic idea of outlined in methods 1a and 1b is analogous to the emigration of residents in a community. The citizens of a community who can find a successful position will remain in the society and try to improve their position. However, those with an unsuccessful position attempt to emigrate and try to improve their position in another community. If those who have emigrated cannot improve their position in the new place, they will return to the neighbourhood of successful people. The emigration for a candidate can be repeated on several occasions. The flowchart of the combined methods is presented in Figure 1.

Method 2a
In the process of optimization, it is observed that some particles have slow velocity, indicating that these particles are becoming trapped in a local optimum or engaged in the process of exploitation. In this situation, these particles can be randomly retained or moved to other positions with a specified random number. Again, a compromise is achieved between a local search and a global search. The formulation is specified as follows: where ε 2 indicates the importance of the velocity, which has a preassigned value, and ε 3 is a decision number to retain or move the particle, the values of which are chosen based on trial and error. In Equation (9), the particles with small velocities (|V| < ε 2 ) are spread over the solution domain by random velocities (αrand(n)(x u − x l )) by a predefined chance (Random < ε 3 .). In this method, the particles that were undertaking exploitation, from now on, have their task is changed to exploration. As a result, in all iterations, most particles are exploring the design domain and only a few particles are exploiting for accurate optimal solutions. Since velocity vectors use gradient vectors, even a limited number of particles is sufficient for the exploitation task.

Method 2b
Similarly to method 1b, if Pbest and Gbest are approaching each other, in addition to method 2a, Pbest is modified as Methods 2a and 2b are based on the idea that the people in a community remain as long as they can improve their position. If their progress is stopped or slows down, there is a probability that these people will decide to take their chances in other regions. After the search for a better life in other places, if they are not successful, they will return to the original place. The flowchart for the cases of methods 2a and 2b is shown in Figure 2. With these revisions, the PSOG is greatly enhanced by four variants to achieve the following two main aims: (1) increasing the exploration power of the approach in the specified design space, in particular for multimodal problems; and (2) achieving a better exploitation capability around the local optima.

Numerical results
To investigate the efficiency and reliability of the proposed method, examples are chosen from the literature for comparison. The numerical results consist of two parts. In the first part, some complicated mathematical functions are selected from CEC 2017 (Awad et al. 2017). These functions are explicit without constraints. All the results are compared with PSOG. It was shown in Salajegheh and Salajegheh (2019) that the PSOG performs much better than many variants of PSO. Thus, comparison of the presented methods with PSOG is satisfactory for the present investigation. In the second part, five truss structures are optimized for frequency constraints and the results are compared with other methods.

Mathematical functions
There  Tables 1 and 2 for 10-and 30-dimensional variables, respectively. The results indicate the superiority of the present four variants of PSOG in terms of the best and average results. In these benchmarks, the results of method 2 are better than method 1. The last rank belongs to the original PSOG, indicating the great exploration power of the presented approaches.
The iteration histories of some of the 10-dimensional functions are shown in Figures 3-7. As expected, the suggested approaches have less exploitation power, in particular in the early stages of the optimization process, compared to PSOG. The methods can explore for better final results compared to PSOG. The convergence of the introduced methods can be further completed by using more iterations with continuing exploration and exploitation; however, the allowable 5000 iterations are imposed on the benchmark problems with the specified number of variables and the particles. From the statistical point of view, the standard deviation (Std.) indicates that the diversity of the final results of the independent runs is lower for the proposed approaches, showing the superiority of the methods.

Structural problems with frequency constraints
Five pin-jointed frame structures are chosen from other articles with similar data for comparison. The objective function is the weight and the constraints are the bounds on the frequencies. The element mass matrix is obtained by a consistent approach and the non-structural masses are added to the mass matrix in the appropriate degrees of freedom. The results are compared with the methods outlined in the literature, as follows: • Lin, Che, and Yu (1982) Note: PSOG = particle swarm optimization with gradient directions. The best results are marked in bold.        Note: PSOG = particle swarm optimization with gradient directions. The best results are marked in bold. In all of the problems the number of particles is chosen as 50 and the number of iterations is 300. The necessary parameters are selected as:

Ten-bar truss
The truss shown in Figure 8 is optimized with 10 cross-sectional areas of the members subjected to frequency constraints. The necessary data are given in Table 3. Four non-structural masses of 454 kg are applied at nodes 1-4. The results are presented in Table 4 together with the results of other articles.
The results indicate the superiority of the presented variants of PSOG. The numerical results show that the best rank belongs to method 2b. Approaches 2a and 1b are ranked second and third and PSOG lies in fourth position. However, the rank of the other methods is worse than the variants of PSOG.

Space truss with 72 members
The 72-member truss shown in Figure 9, with the information given in Table 5, is designed for two frequency constraints. Owing to the symmetry of the structure, 16 independent cross-sectional areas are considered, as shown in Table 6. Four non-structural masses of 2270 kg are imposed at nodes 1-4. The results are presented in Table 6 and compared with six other approaches from other articles. It can be observed that the results of the presented approaches are superior to the other methods. In this problem, the results of method 2 (2a and 2b) are better than method 1 (1a and 1b), and method 2 is better than PSOG.

Space truss with 52 members
The dome-type three-dimensional truss structure shown in Figure 10 with 52 elements is optimized under two frequency constraints with sizing and shape design variables. The properties of the dome are presented in Table 7. Owing to the symmetry of the structure, eight sizing and five shape variables are considered, as shown in Table 8. Non-structural weights of 50 kg are applied on all free joints. This problem is optimized with the variants of PSOG introduced in this article and the results are compared with those of other articles. The final results are presented in Table 8. Owing to the combination of sizing and shape variables with multi-frequency constraints, this is a highly nonlinear problem with several local optima.
From the numerical results, it can be observed that the method of Lin, Che, and Yu (1982), which is based on the gradients, could not find a suitable solution, and the results of PSOG are better than the standard PSO of Gomes (2011). The result of the newly introduced version of PSOG (method 1b) is better than most other approaches because of its exploration ability and its ability to escape from local optima.
The great differences between the results of the various approaches indicate the sensitivity of the final results to the design variables. The MP techniques resulted in 298 kg for the weight of the structure and the PSO produced 228.4 kg. The PSOG method gained 209.9 kg and PSOG (method 1b) converged to 195.7 kg as the second rank. The initial and final external shapes of the deflected front Note: PSOG = particle swarm optimization with gradient directions. The best results are marked in bold. views are shown in Figure 11. The numerical results and the deflected shapes indicate lower weights for the proposed methods.

Space truss with 120 members
The final structural problem is a three-dimensional dome with 120 elements, as shown in Figure 12. The structure is optimized for seven sizing variables and the necessary information is summarized   Table 9. The results are presented in Table 10. The best result corresponds to PSOG (method 1a).
Similarly to the other problems, the improved results indicate the good performance of the suggested methods.
In addition, a planar truss with 200 members is optimized for three frequency constraints as Problem 4.2.5. To limit the length of the article, the results are presented in the supplementary file.

Conclusions
There are two main categories of optimum design approaches. The traditional methods are based on gradient directions, in which the optimal solution is sought using the gradients of the function under consideration. These methods are known as mathematical programming (MP) techniques. The MP methods are effective and robust for problems with a single local optimum. For more complicated problems, the second category has been introduced, which comprises metaheuristic approaches. These methods are based on stochastic ideas and the search is performed over the whole of the design space to explore most of the local optima. The final results are the near-global optimum. A great number of metaheuristic approaches with different natures has been developed. Among these methods, PSO has received great attention because of its simplicity and efficiency.
The combination of MP and PSO was introduced by Salajegheh and Salajegheh (2019) as the PSOG method. PSOG possesses both local (exploitation) and global (exploration) power. However, further investigation indicates that the balance of exploitation and exploration capabilities is not properly adjusted. The uncontrolled power of the gradient directions leads the method towards a local optimum. To increase the ability of the exploration of the PSOG, two new ideas are incorporated into the approach based on emigration philosophy. In the first approach, some of the unsuccessful and trapped particles are randomly redistributed to other positions of the design space. This is analogous to the idea of the emigration of people to other places to facilitate better performance. In the second Note: PSOG = particle swarm optimization with gradient directions. The best results are marked in bold.  approach, the particles with a slow speed that are not effective in the optimization process emigrate to other places for better exploration and improvement. Each approach consists of two different versions, as explained in this article. All of the introduced approaches are examined by the mathematical benchmark problems of CEC 2017, with 29 functions. In addition, five structural design problems with multiple frequency constraints are chosen from the literature. The numerical results indicate that the enhancements in the second approach (methods 2a and 2b) led to great improvement and achieved powerful algorithms.
For problems with a single optimum, the difference between the results is not so obvious; however, for problems with many optima, the advantages of the enhanced PSOG method are clear.

Data availability statement
The data that support the findings of this study are available from the corresponding author.

Disclosure statement
No potential conflict of interest was reported by the authors.