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Table 13 Design parameters of the 72-bar truss structure

Table 13 Design parameters of the 72-bar truss structure

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Fig. 1 Illustration of components of the cultural algorithm
Fig. 1 Illustration of components of the cultural algorithm
The belief space expresses the members’ behavioral pat- terns on population space, and it is created based on the The cultural algorithm is a stochastic optimization method originally proposed by Reynolds [30]. This algorithm con- tains a population space and a belief space which can evolve in parallel, by mutual influence on each other, and a com- munication protocol that expresses interaction between the two spaces. The basic framework of the cultural algorithm is presented in Fig. 1. The top members with less fitness value are acceptable to adjust the knowledge components of belief space [31]. The number of these members is determined by the acceptance function. There are two static and dynamic methods for this purpose. In the static method, during the optimization pro- cess, the number of accepted members is fixed, while in the dynamic method, this number reduces during the optimiza- tion process. Based on the accepted members, and using the following equations, the situational and normative compo- nents are updated:
The belief space expresses the members’ behavioral pat- terns on population space, and it is created based on the The cultural algorithm is a stochastic optimization method originally proposed by Reynolds [30]. This algorithm con- tains a population space and a belief space which can evolve in parallel, by mutual influence on each other, and a com- munication protocol that expresses interaction between the two spaces. The basic framework of the cultural algorithm is presented in Fig. 1. The top members with less fitness value are acceptable to adjust the knowledge components of belief space [31]. The number of these members is determined by the acceptance function. There are two static and dynamic methods for this purpose. In the static method, during the optimization pro- cess, the number of accepted members is fixed, while in the dynamic method, this number reduces during the optimiza- tion process. Based on the accepted members, and using the following equations, the situational and normative compo- nents are updated:
Fig. 2. Schematic movement of elephants of clan i in a two-dimensional space using EHOC algorithm The presented algorithm is a new hybrid method based on the elephant herding optimization (EHO) and cultural algo- rithm (CA). In EHO, separating operator eliminates some
Fig. 2. Schematic movement of elephants of clan i in a two-dimensional space using EHOC algorithm The presented algorithm is a new hybrid method based on the elephant herding optimization (EHO) and cultural algo- rithm (CA). In EHO, separating operator eliminates some
Table 1 Details of the mathematical optimization problems
Table 1 Details of the mathematical optimization problems
Fig. 3 Comparison of the best convergence curves of EHO and EHOC: a Sphere; b Schwefel 2.22; ¢ Ackley; d Griewank
Fig. 3 Comparison of the best convergence curves of EHO and EHOC: a Sphere; b Schwefel 2.22; ¢ Ackley; d Griewank
Fig. 5 Comparison of the best convergence curves of CA, EHO, and EHOC for the 10-bar truss structure (case1) Table 6 Design parameters of the 10-bar truss structure
Fig. 5 Comparison of the best convergence curves of CA, EHO, and EHOC for the 10-bar truss structure (case1) Table 6 Design parameters of the 10-bar truss structure
Table 4 Comparison of the detailed results of EHOC and some other algorithms in the mathematical optimization problems
Table 4 Comparison of the detailed results of EHOC and some other algorithms in the mathematical optimization problems
Bold values indicate that the best value for the parameter a = 2.5
Bold values indicate that the best value for the parameter a = 2.5
Table 7 Comparison of optimization results of EHOC and other algorithms for the 10-bar truss structure (case 1)
Table 7 Comparison of optimization results of EHOC and other algorithms for the 10-bar truss structure (case 1)
Table 8 Comparison of optimization results of EHOC and other algorithms for the 10-bar truss structure (case 2)
Table 8 Comparison of optimization results of EHOC and other algorithms for the 10-bar truss structure (case 2)
Fig.8 Comparison of the allowable values of constraints and the existing values of optimized 10-bar truss using EHOC (case 2
Fig.8 Comparison of the allowable values of constraints and the existing values of optimized 10-bar truss using EHOC (case 2
Fig. 7 Comparison of the best convergence curves of CA, EHO, and EHOC for the 10-bar truss structure (case 2)
Fig. 7 Comparison of the best convergence curves of CA, EHO, and EHOC for the 10-bar truss structure (case 2)
illustrated in Figs. 5 and 7, respectively. It is seen that EHOC is capable of accelerating the convergence rate effectively and it converges to the better solution compared to the CA and EHO, indicating the high exploration and exploitation ability of the proposed hybrid algorithm compared with the two other algorithms. As can be observed, the low explora- tion ability of the EHO algorithm leads to its weak results. In comparison, it appears that the CA algorithm has good reconnaissance ability. However, the best results obtained by CA compared to those gained by EHOC show that its exploi- tation ability is not very good. Tables 7 and 8 present the comparative information of optimal design results obtained for 50 independent runs for two cases, respectively. As it is observed, the number of required structural analyses by EHOC is only 4240 in load case 1 and 5000 in load case 2, which are less compared to those of the other algorithms. In addition, considering the best index, the proposed algorithm converges to a better solution compared to SASH, TLBO, PSO, MSPSO, HPSSO and EPSO algorithms. Figures 6 and 8 indicate the allowable values for stress and displacement constraints and the existing values related to the best solu- tion obtained by EHOC, for two load cases, respectively. In two cases, both the stress and displacement constraints are activated. The results related to the values of average weight Table 9 Design parameters of the 25-bar truss structure
illustrated in Figs. 5 and 7, respectively. It is seen that EHOC is capable of accelerating the convergence rate effectively and it converges to the better solution compared to the CA and EHO, indicating the high exploration and exploitation ability of the proposed hybrid algorithm compared with the two other algorithms. As can be observed, the low explora- tion ability of the EHO algorithm leads to its weak results. In comparison, it appears that the CA algorithm has good reconnaissance ability. However, the best results obtained by CA compared to those gained by EHOC show that its exploi- tation ability is not very good. Tables 7 and 8 present the comparative information of optimal design results obtained for 50 independent runs for two cases, respectively. As it is observed, the number of required structural analyses by EHOC is only 4240 in load case 1 and 5000 in load case 2, which are less compared to those of the other algorithms. In addition, considering the best index, the proposed algorithm converges to a better solution compared to SASH, TLBO, PSO, MSPSO, HPSSO and EPSO algorithms. Figures 6 and 8 indicate the allowable values for stress and displacement constraints and the existing values related to the best solu- tion obtained by EHOC, for two load cases, respectively. In two cases, both the stress and displacement constraints are activated. The results related to the values of average weight Table 9 Design parameters of the 25-bar truss structure
Table 10 Allowable stresses (ksi) for the 25-bar truss structure
Table 10 Allowable stresses (ksi) for the 25-bar truss structure
Fig. 10 Comparison of the best convergence curves of CA, EHO, and EHOC for the 25-bar truss structure Figure 10 shows the convergence curves related to the best solutions of CA, EHO, and EHOC algorithms. As it is observed, EHOC has high convergence rate especially in the initial steps of search and results in the better design compared to the two other algorithms, which shows the high exploration and exploitation ability of the EHOC compared with other two algorithms. Table 12 presents the comparative information regarding the optimal design results obtained for 50 independent runs. As it is indicated, considering the best index, the EHOC converges to a better solution compared to PSO, MSPSO, HPSSO, FPA, IGWO, CA and EHO algorithms. However, the best weight obtained from EHOC is a little bigger than SASH, TLBO, and HTS. The lightest design is obtained by TLBO which violated
Fig. 10 Comparison of the best convergence curves of CA, EHO, and EHOC for the 25-bar truss structure Figure 10 shows the convergence curves related to the best solutions of CA, EHO, and EHOC algorithms. As it is observed, EHOC has high convergence rate especially in the initial steps of search and results in the better design compared to the two other algorithms, which shows the high exploration and exploitation ability of the EHOC compared with other two algorithms. Table 12 presents the comparative information regarding the optimal design results obtained for 50 independent runs. As it is indicated, considering the best index, the EHOC converges to a better solution compared to PSO, MSPSO, HPSSO, FPA, IGWO, CA and EHO algorithms. However, the best weight obtained from EHOC is a little bigger than SASH, TLBO, and HTS. The lightest design is obtained by TLBO which violated
the design constraints, while the best design obtained by EHOC is feasible. The allowable values for stress and dis- placement constraints and the existing values related to the best solution obtained by EHOC are presented in Fig. 11. It can be seen that the stress constraints in load case | and the displacement constraints in direction z in both load cases are activated. Furthermore, the presented algorithm can achieve nearly the optimum designs with less number of analyses. EHOC needs 6640 structural analyses, while the number of analyses required by SASH, TLBO, PSO, MSPSO, HPSSO, FPA and HTS are 9051, 15,318, 18,400, 13,326, 10,800, 8149 and 7653, respectively. The weight of the IGWO with 5640 analyses was 545.4819, while EHOC was achieved in the lightest designs with the less number of analyses. For instance, the weight obtained by EHOC in iteration 105 (4200 analyses) was 545.272 lb, which is more optimum than the weight obtained by IGWO after 5640 analyses. It appears that EHOC is an efficient optimization algorithm in convergence speed. The results related to standard deviation show that the EHOC algorithm is reliable enough compared with the other metaheuristic algorithms. In terms of the aver- age index, TLBO, HPSSO, and HTS achieved the lightest weight than EHOC, but by a higher computational cost.
the design constraints, while the best design obtained by EHOC is feasible. The allowable values for stress and dis- placement constraints and the existing values related to the best solution obtained by EHOC are presented in Fig. 11. It can be seen that the stress constraints in load case | and the displacement constraints in direction z in both load cases are activated. Furthermore, the presented algorithm can achieve nearly the optimum designs with less number of analyses. EHOC needs 6640 structural analyses, while the number of analyses required by SASH, TLBO, PSO, MSPSO, HPSSO, FPA and HTS are 9051, 15,318, 18,400, 13,326, 10,800, 8149 and 7653, respectively. The weight of the IGWO with 5640 analyses was 545.4819, while EHOC was achieved in the lightest designs with the less number of analyses. For instance, the weight obtained by EHOC in iteration 105 (4200 analyses) was 545.272 lb, which is more optimum than the weight obtained by IGWO after 5640 analyses. It appears that EHOC is an efficient optimization algorithm in convergence speed. The results related to standard deviation show that the EHOC algorithm is reliable enough compared with the other metaheuristic algorithms. In terms of the aver- age index, TLBO, HPSSO, and HTS achieved the lightest weight than EHOC, but by a higher computational cost.
Table 12 Comparison of optimization results of EHOC and other algorithms for the 25-bar truss structur
Table 12 Comparison of optimization results of EHOC and other algorithms for the 25-bar truss structur
Fig. 11 Comparison of the allowable values of constraints and the existing values of optimized 25-bar truss using EHOC
Fig. 11 Comparison of the allowable values of constraints and the existing values of optimized 25-bar truss using EHOC
Table 14 Loading conditions for the 72-bar truss structure
Table 14 Loading conditions for the 72-bar truss structure
aand b are the constants related to the types of selected sec- tions for the truss members. In this case, similar to the
aand b are the constants related to the types of selected sec- tions for the truss members. In this case, similar to the
Fig. 13 Comparison of the best convergence curves of CA, EHO, and EHOC for the 72-bar truss structure
Fig. 13 Comparison of the best convergence curves of CA, EHO, and EHOC for the 72-bar truss structure
Table 15 Comparison of optimization results of EHOC and other algorithms for the 72-bar truss structure
Table 15 Comparison of optimization results of EHOC and other algorithms for the 72-bar truss structure
Fig. 14 Comparison of the allowable values of constraints and the existing values of optimized 72-bar truss using EHOC the optimum solution with a high convergence speed in the EHOC indicates the high exploration capability of the algo- rithm. Figure 17 presents the allowable values defined by constraints and the existing values related to the best solu- tion in this example. It can be observed that both stress and displacement constraints are activated. The displacements of the Ist to 13th nodes in z-direction and the stresses in the members of the 7th group are the effective factors in deter- mining the best cross-sections. presented. EHO is motivated by the herding behavior of elephant groups in which two updating and separating operators are implemented. Clan updating operator is used to determine the new position of elephants in each generation, moreover, the separating operator eliminates some elephants with the worst fitness value and replaces the members of population space randomly. Consider- ing the recommended separating operator by EHO, the eliminated elephants may be replaced with improper members, leading to worsening the search space. There- fore, in the recommended algorithm, the separating opera- tor is defined considering the belief space so that some elephants with worst fitness value are replaced by ran- dom members of belief space. In the CA, in the belief space, the cultural information regarding members on population space is modeled and it is formed with top
Fig. 14 Comparison of the allowable values of constraints and the existing values of optimized 72-bar truss using EHOC the optimum solution with a high convergence speed in the EHOC indicates the high exploration capability of the algo- rithm. Figure 17 presents the allowable values defined by constraints and the existing values related to the best solu- tion in this example. It can be observed that both stress and displacement constraints are activated. The displacements of the Ist to 13th nodes in z-direction and the stresses in the members of the 7th group are the effective factors in deter- mining the best cross-sections. presented. EHO is motivated by the herding behavior of elephant groups in which two updating and separating operators are implemented. Clan updating operator is used to determine the new position of elephants in each generation, moreover, the separating operator eliminates some elephants with the worst fitness value and replaces the members of population space randomly. Consider- ing the recommended separating operator by EHO, the eliminated elephants may be replaced with improper members, leading to worsening the search space. There- fore, in the recommended algorithm, the separating opera- tor is defined considering the belief space so that some elephants with worst fitness value are replaced by ran- dom members of belief space. In the CA, in the belief space, the cultural information regarding members on population space is modeled and it is formed with top
members with less fitness value. Then, the beliefs are used to update the position of members in the population space and to improve the population during the process. Using the belief space in separating operator of EHO can increase the variety of top members in search space and improve the search capability in the algorithm compared to the standard EHO. Using a set of eight mathematical space and to improve the population during the process.
members with less fitness value. Then, the beliefs are used to update the position of members in the population space and to improve the population during the process. Using the belief space in separating operator of EHO can increase the variety of top members in search space and improve the search capability in the algorithm compared to the standard EHO. Using a set of eight mathematical space and to improve the population during the process.
Table 16 Design parameters of the 120-bar dome truss structure
Table 16 Design parameters of the 120-bar dome truss structure
Fig. 16 Comparison of the best convergence curves of CA, EHO, and EHOC for the 120-bar dome truss structure
Fig. 16 Comparison of the best convergence curves of CA, EHO, and EHOC for the 120-bar dome truss structure
Fig. 17 Comparison of the allowable values of constraints and the existing values of optimized 120-bar truss using EHOC
Fig. 17 Comparison of the allowable values of constraints and the existing values of optimized 120-bar truss using EHOC
Table 17 Comparison of optimization results of EHOC and other algorithms for the 120-bar dome truss structure
Table 17 Comparison of optimization results of EHOC and other algorithms for the 120-bar dome truss structure
Related topics:
Computer ScienceMathematical OptimizationCultural Algorithm
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