Enhanced Biogeography-based Optimization: A New Method for Size and Shape Optimization of Truss Structures with Natural Frequency Constraints

Seyed Heja Seyed Taheri Shahin Jalili About the authors

Abstract

The current study presents an enhanced biogeography-based optimization (EBBO) algorithm for size and shape optimization of truss structures with natural frequency constraints. The BBO algorithm is one of the recently developed meta-heuristic algorithms inspired by the mathematical models in biogeography science and is based on the migration behavior of species among the habitats in the nature. In this study, the overall performance of the standard BBO algorithm is enhanced by new migration and mutation operators. The efficiency of the proposed algorithm is demonstrated by utilizing four benchmark truss design examples with frequency constraints. Numerical results show that the proposed EBBO algorithm not only significantly improves the performance of the standard BBO algorithm, but also finds competitive results compared with recently developed optimization methods.

Keywords:
Biogeography-based optimization; meta-heuristic; truss structures; dynamic constraints; size and shape optimization

1 INTRODUCTION

The ability to control and modify the values of free vibrational frequencies and corresponding mode shapes in structures is a significant issue to keep their vibrational performance desirable. In most of the low frequency vibration problems, the response of the structure to dynamic excitation is primarily a function of its fundamental frequency and mode shapes (Grandhi, 1993Grandhi, R.V. 1993. Structural optimization with frequency constraints-a review. AIAA Journal 31:2296-2303.). Particularly, this is an important issue when a certain excitation frequency can cause resonance phenomena in the structure. Therefore, structural optimization with natural frequency constraints has important applications in manipulating the dynamic performance of the structures.

From optimization point-of-view, size optimization of truss structures with frequency constraints is described as highly non-linear and non-convex optimization problem with several local optimums in its search space. On the other hand, if the shape variables, beside the size variables, are considered to resolve this problem, the optimization problem will become more complex, and the associative performance will degrade seriously. The main reason for this complexity is related to the different physical representation of these variables, and sometimes their changes are of widely different orders of magnitude (Wang et al., 2004Wang, D., Zhang, W.H., Jiang, J.S., (2004). Truss optimization on shape and sizing with frequency constraints. AIAA Journal 42:1452-1456.). Therefore, the size and shape optimization of truss structures with frequency constraints is one of the active areas in the research of structural optimization at present.

Generally, two main approaches are available in the literature to address the structural optimization problem with frequency constraints. These approaches are the conventional optimization techniques based on the use of classical gradient-based optimization techniques, and the meta-heuristic search methods based on the use of nature-inspired stochastic optimization algorithms. Nevertheless, the conventional methods such as the optimality criteria (OC) and mathematical programming (MP), need complex and time-consuming dynamic sensitivity analysis and are easily trapped into local optimum (Lingyun et al., 2005Lingyun, W., Mei, Z., Guangming, W., Guang, M., 2005. Truss optimization on shape and sizing with frequency constraints based on genetic algorithm. Computational Mechanics 25:361-368.). On the other hand, the meta-heuristic search methods such as the genetic algorithms (GAs) (Goldberg, 1989Goldberg, D.E. 1989. Genetic Algorithms in Search Optimization and Machine Learning, Addison-Wesley, Boston.), particle swarm optimization (PSO) (Eberhart and Kennedy, 1995Eberhart, R.C., Kennedy, J., 1995. A new optimizer using particle swarm theory. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan.), big bang-big crunch (BB-BC) algorithm (Erol and Eksin, 2006Erol, O.K., Eksin, I., 2006. New optimization method: big bang-big crunch. Advances in Engineering Software 37:106-111.), gravitational search algorithm (GSA) (Rashedi et al., 2009Rashedi, E., Nezamabadi-pour, H., Saryazdi, S., 2009. GSA: a gravitational search algorithm. Information Science 179: 2232-2248.), biogeography-based optimization (BBO) (Simon, 2008Simon, D. 2008. Biogeography-based optimization. IEEE Transactions on Evolutionary Computation 12:702-713.; Jalili et al., 2015Jalili, S., Hosseinzadeh, Y., 2015. A cultural algorithm for optimal design of truss structures. Latin American Journal of Solids and Structures. 12:1721-1747.), Cultural Algorithm (CA) (Reynolds, 1999Reynolds, R.G., 1999. Cultural Algorithms: Theory and Application. In D. Corne, M. Dorigo, and F. Glover, edi-tors, New Ideas in Optimization, pages 367-378. McGraw-Hill.; Jalili and Hosseinzadeh, 2015Jalili, S., Hosseinzadeh, Y., Taghizadieh, N., 2015. A biogeography-based optimization for optimum discrete design of skeletal structures. Engineerng Optimization. DOI: 10.1080/0305215X.2015.1115028.
https://doi.org/10.1080/0305215X.2015.11...
), and charged system search (CSS) algorithm (Kaveh and Talatahari, 2010Kaveh, A., Talatahari, S., 2010. A novel heuristic optimization method: charged system search. Acta Mechanica 213:267-286.) require less computational effort and can also find near-optimum solutions in a relatively reasonable time. Therefore, meta-heuristic optimization techniques are usually preferred to conventional approaches, and in most cases, show much better performances. However, it is widely believed that the overall performance of heuristic search methods mainly depends on the type of the optimization problem and the features of its search space. Hence, extensive studies have been carried out to develop efficient heuristic optimization methods for size and shape optimization of truss structures with natural frequency constraints, ranging from hybrid techniques to enhanced versions of standard algorithms. The Democratic PSO (DPSO) algorithm (Kaveh and Zolghadr, 2014aKaveh, A., Zolghadr, A., 2014a. Democratic PSO for truss layout and size optimization with frequency constraints. Computers & Structures 130:102-103.), hybrid CSS-BBBC algorithm (Kaveh and Zolghadr, 2012Kaveh, A., Zolghadr, A., 2012. Truss optimization with natural frequency constraints using a hybridized CSS-BBBC algorithm with trap recognition capability, Computers and Structures 102-103:14-27.), enhanced CSS algorithm (Kaveh and Zolghadr, 2011Kaveh, A., Zolghadr, A., 2011. Shape and size optimization of truss structures with frequency constraints using enhanced charged system search algorithm. Asian Journal of Civil Engineering 12:487-509.), niche hybrid genetic algorithm (NHGA) (Lingyun et al., 2005Lingyun, W., Mei, Z., Guangming, W., Guang, M., 2005. Truss optimization on shape and sizing with frequency constraints based on genetic algorithm. Computational Mechanics 25:361-368.), and orthogonal multi-gravitational search algorithm (OMGSA) (Khatibinia and Naseralavi, 2014Khatibinia, M., Sadegh Naseralavi, S., 2014. Truss optimization on shape and sizing with frequency constraints based on orthogonal multi-gravitational search algorithm. Journal of Sound and Vibration. 333:6349-6369.) are some instances of these methods.

In this study, an enhanced biogeography-based optimization (EBBO) algorithm has been proposed to tackle the challenge of finding global optimum in size and shape optimization of truss structures with multiple frequency constraints. Biogeography-based optimization (BBO) is a recently developed population-based meta-heuristic algorithm based on the biogeography theory, which has been introduced by (Simon, 2008Simon, D. 2008. Biogeography-based optimization. IEEE Transactions on Evolutionary Computation 12:702-713.). The biogeography theory describes the geographical distribution of biological organisms in the nature. The BBO method is a successful meta-heuristic search technique that has been successfully applied to global optimization of numerical functions (Simon et al., 2011Simon, D., Rarick, R., Ergezer, M., Du, D. 2011. Analytical and numerical comparisons of biogeography-based optimization and genetic algorithms. Information Sciences 181(7):1224-1248.; Boussaid, 2012Boussaid, I., Chatterjee, A., Siarry, P., Ahmed-Nacer, M., 2012. Biogeography-based optimization for constrained optimization problems. Computers & Operations Research 39(12):3293-3304.) and has been used to solve numerous real-world optimization problems (Simon, 2008Simon, D. 2008. Biogeography-based optimization. IEEE Transactions on Evolutionary Computation 12:702-713.; Singh e al., 2010Singh, U., Kumar, H., Kamal, T., 2010. Design of Yagi-Uda Antenna Using Biogeography Based Optimization. IEEE Transactions on Antennas and Propagation. 58(10):3375-3379.; Bhattachary et al., 2010Bhattacharya, A., Chattopadhyay, P., 2010. Biogeography-Based Optimization for Different Economic Load Dispatch Problems, IEEE Transactions on Power Systems. 25(2):1064-1077.). However, despite having good exploitation ability, the standard BBO algorithm suffers from premature convergence; furthermore, its weak exploration ability is an issue in some cases. The main reason for this poor exploration ability arises from its simple migration operator. In addition, the simple and purely random mutation operator of the BBO may lead to revisiting non-productive regions of the search space. In this study, in order to enhance the performance of standard BBO algorithm, new migration and mutation operators are proposed. These new migration and mutation operators improve the convergence properties of the BBO algorithm and enhance the algorithm's ability to further escape stagnation and premature convergence. To evaluate the efficiency of the proposed algorithm, four benchmark truss design examples with frequency constraints are investigated and the results generated by the EBBO algorithm are compared with the results of several other state-of-the-art methods. Numerical results show that the proposed EBBO algorithm not only significantly improves the standard BBO algorithm, but also finds competitive results compared with recently developed optimization methods for truss optimum design problem with frequency constraints.

The remaining sections complete the presentation of this paper as followings. Section 2 formulates the optimum design problem of truss structures with frequency constraints. In Section 3, the BBO algorithm is first reviewed and then, the proposed EBBO algorithm is explained in detail. Four benchmark truss design examples are optimized by utilizing the proposed algorithm in Section 4. Finally, the concluding remarks are presented in Section 5.

2 TRUSS OPTIMUM DESIGN PROBLEM

The main purpose of this optimum design problem is to minimize the weight of the structure under some frequency constraints. In the layout and size optimization of truss structures, the cross sectional areas and the coordinates of nodes are considered as design variables. Thus, the optimal design of a truss structure with frequency constraints can be formulated as:

(1)

where {X} is the vector including design variables; n is the number of the design variables; W(.) is the weight of the structure; fp(.) is the penalty function; m is the number of structural members; γ is the material density; Ai and li are the cross-sectional area and length of member i, respectively; CU(.) and CL(.) are the upper and lower frequency constraints, respectively; ωj is the jth frequency of the structure; ωuj is the upper limit for jth frequency and ωlj denotes its lower limit; Lk and Uk are the lower and upper limits of kth design variable, respectively. As it can be seen, the simple penalty function approach is used for constraint handling in this study. For each candidate solution, the penalty function is calculated as follows:

(2)

where α is a constant value and nc and is the number of frequency constraints. The parameter α has a major effect on the algorithm's performance. At the initial stages of the optimization process, the value of this parameter should be small enough to explore the whole search space (exploration), while whatever the optimization process closes to the final stages, it should be large enough to provide more focus on the feasible solutions (exploitation). In this study, the value of α in (Eq. 2) starts from 2 and linearly increases to 7 by lapse of the iteration (Kaveh and Zolghadr, 2014bKaveh, A., Zolghadr, A., 2014b. Comparison of nine meta-heuristic algorithms for optimal design of truss structures with frequency constraints. Advances in Engineering Software 76:9-30.) as follows:

(3)

where It and Itmax are the current iteration number and maximum considered iterations, respectively.

3 OPTIMIZATION METHOD

3.1 Biogeography-Based Optimization

The Biogeography-based optimization (BBO) method inspired by biogeography science is a recently developed meta-heuristic algorithm which has been introduced by (Simon, 2008Simon, D. 2008. Biogeography-based optimization. IEEE Transactions on Evolutionary Computation 12:702-713.).The biogeography science describes the geographical distribution of biological organisms in nature. The overall framework of this algorithm is developed based on the probabilistic mathematical models of biogeography science. These mathematical models were developed by (MacArthur and Wilson, 1967MacArthur, R., Wilson, E., 1967. The Theory of Biogeography, Princeton University Press; USA.) and explain how species migrate between the habitats. In the BBO algorithm, each habitat (Hi) is a solution candidate for the optimization problem and the position of each habitat (Hi) in an n-dimensional search space represented by Suitability Index Variables (SIVs), which is an n-dimensional vector. The quality of each habitat is measured by the Habitat Suitability Index (HSI), which is directly proportional to the fitness function value. Thus, the habitats with high HSI values are better solutions than the ones with low HSI values. The BBO algorithm consists of two main operators: migration and mutation.

This algorithm utilizes migration operator as a powerful tool to share information between habitats in the solution space. So, it can be considered as an exploitation mechanism during the optimization process. The migration operator shares information between habitats based on immigration (λi ) and emigration (μi ) rates, probabilistically. Each habitat has its own immigration λi and emigration μi rates which are the functions of species in the habitat. For a given habitat, the immigration λi rate is inversely proportional to the HSI (fitness) value, while the emigration μi rate is directly proportional to HSI value. The habitats with high immigration rates (poor solutions) are more likely to accept information from the other habitats with high HSI values, while the habitats with low immigration rates (good solutions) share their information with other poor habitats with a high probability. The immigration and emigration rates are calculated for each habitat as follows (Simon, 2008Simon, D. 2008. Biogeography-based optimization. IEEE Transactions on Evolutionary Computation 12:702-713.):

(4)

(5)

where I is the maximum possible immigration rate; E is the maximum possible emigration rate; K is the number of species in the ith habitat; and Smax is the maximum number of species. These immigration and emigration rates are calculated based on the migration models. There are different migration models that can be utilized to calculate the immigration and emigration rates. Figure 1 shows the simple linear migration model for the case E=I. According to Figure 1, the habitats with a high HSI value tend to have a large number of species, while those with a low HSI have a small number of species (Simon, 2008Simon, D. 2008. Biogeography-based optimization. IEEE Transactions on Evolutionary Computation 12:702-713.). From Figure 1, it can be concluded that the habitat with few species (poor solution, low HSI) like S1, has a low emigration rate and a high immigration rate. This means that, the habitat with low HSI tends to take information about the good habitats with the high probability, while the probability of sharing its information for other habitats is relatively low. On the other hand, the habitat which has more species (good solution, high HSI) like S2, has a low immigration rate and a high emigration rate. Such habitats with high HSI values share their information with the other habitats with a high probability. By utilizing this mechanism, the migration operator of the BBO algorithm can achieve adequate exploitation ability between the habitats in the search space. For each variable of a given solution (Hi ), the immigration λi rate decides whether or not to immigrate. If the immigration condition is satisfied, the migration procedure occurs between the immigrating and emigrating habitats as follows:

(6)

(Eq. 6) explains that one of the variables of ith habitat is replaced by a variable of jth habitat. Here, Hi and Hj are the immigrating and emigrating habitats, respectively. It is worth mentioning that the emigrating habitat (Hj ) is selected based on the emigration rates (μi ). The probability of selecting jth habitat as emigrating habitat is calculated as follow:

(7)

where NP is the population size. Figure 2 demonstrates the details of the migration procedure in the BBO algorithm. Here, the roulette wheel selection technique is used to select emigrating habitat.

Figure 1
The simple linear migration model (E=I).

Figure 2
The migration procedure of the BBO algorithm.

In most cases, it is possible that a meta-heuristic algorithm is trapped to the local optimum by lapse of the iteration. In order to escape from the local traps in the search space, the BBO algorithm utilizes a mutation operator. Mutation operator is a probabilistic operator that modifies a habitat's SIV randomly based on mutation rate (pMutate), which is related to the habitat's probability. The mutation rate (pMutate) for each habitat is calculated as follows:

(8)

where mmax is a user-defined parameter and Pmax = max{Pi }. More details about the calculation of Pmax and Pi probabilities can be found in (Simon, 2008Simon, D. 2008. Biogeography-based optimization. IEEE Transactions on Evolutionary Computation 12:702-713.). Based on (Eq. 8), a variable of each habitat mutates randomly in search space with a given probability. For a better explanation, the mutation operator of the BBO algorithm can be described as in Figure 3. In this study, for simplicity, the probability of performing mutation operator for the all habitats is set to 0.1 (pMutate = 0.1).

Figure 3
The mutation procedure of the BBO algorithm.

Another feature of the BBO algorithm is that the elite habitats with high HSI values are selected to keep and transfer from previous generation to the current one. Therefore, the Keeprate parameter is defined for this purpose. In this study, 20% (Keeprate=0.2) of habitats with high HSI values are selected to keep in each generation. It means that the 20% of elite habitats from the previous population are transferred to the current generation and combined with new habitats. Finally, the habitats with high HSI values are selected from the combined population of habitats to form a new population.

For a better explanation, Figure 4 shows the flowchart of the standard BBO algorithm.

Figure 4
The flowchart of the standard BBO algorithm.

3.2 Enhanced Biogeography-Based Optimization

As mentioned before, the standard BBO algorithm may not be successful in finding better solutions for some non-linear complicated optimization problems. The main reason for this issue is that the basic BBO algorithm employs simple migration and mutation operators during the optimization process. Such simple operators may lead to some disadvantages such as a low exploration ability and premature convergence. In a migration operator, the immigrating habitat is updated by simply replacing one of the SIV of emigrating habitat randomly, which often implies a rapid loss of diversity in the population. With the aim of achieving a better exploitation capability and providing efficient information sharing between the habitats, the new migration operator is proposed as follows:

(9)

where Hi (SIV) and Hj (SIV) are the immigrating and emigrating habitats, respectively, Φ is a random number uniformly generated between the 0 and 1, and Hjbest (SIV) denotes the best position experienced by the emigrating habitat. As it can be seen from (Eq. 9), the new migration operator changes a variable of ith habitat by considering both current and best positions of the emigrating habitat. The proposed migration scheme has an important role in achieving an efficient exploitation ability.

On the other hand, the purely random mutation operator of the standard BBO algorithm may lead to revisiting non-productive regions of the search space, which leads to weak exploration ability, excessive computational efforts, and long computing time. Therefore, in order to enhance the exploration ability and eliminate the effect of the purely random mutation, following mutation operator is proposed:

(10)

where N(0,1) is a random number generated according to a standard normal distribution with mean zero and standard deviation equal to one; Hmax(.) and Hmin(.) are the upper and lower bounds of the search space, respectively; It and Itmax are the current iteration number and the maximum number of iterations, respectively. As it can be seen from (Eq. 10), the size of the search space considered for the mutation procedure decreases with respect to time. It is worth mentioning that, whenever the mutated position of a habitat goes beyond its lower or upper bound, the habitat will take the value of its corresponding lower or upper bound.

In order to better explain, the main steps of the proposed EBBO algorithm can be listed as below:

Step 1: Initialization

In the first step, the random habitats are generated in the search space as follows:

(11)

where Φ is the random number uniformly distributed between 0 and 1. Then, the value of HSI or cost function value is calculated for each habitat.

Step 2: Calculating immigration and emigration rates

In this step, the immigration λi and emigration μi rates are calculated for each habitat based on the migration model (Figure 1) and HSI values.

Step 3: Migration procedure

In the third step, the migration procedure is performed based on the immigration λi and emigration μi rates for each habitat by utilizing (Eq. 9).

Step 4: Mutation procedure

After migration procedure, the variables of each habitat mutate with constant probability (pMutation) by (Eq. 10).

Step 5: Evaluation of HSI values

In this step, the HSI values of the new generated habitats are computed.

Step 6: Formation of new population of habitats

A specific number of elite habitats from the previous population (KeepRate×NP ) are transferred to the current generation and combined with the new habitats. Finally, the habitats with high HSI values are selected from the combined population of habitats to form a new population.

Step 7: Finish or redoing

Repeat from Steps 2-6 until the stopping criteria is met and output the best solution.

4 NUMERICAL EXAMPLES

In this section, four commonly used benchmark truss design examples with frequency constraints, including a 10bar planar truss, a simply supported 37-bar planar truss, a 120-bar dome truss and a 200-bar planar truss are examined to verify the performance of the proposed algorithm. In all design examples, the parameters used for both standard BBO and EBBO algorithms are set as follows: the population size is 50, and the pMutation is 0.1. The percentage of the selected habitats to keep is 20% of the population. Moreover, due to stochastic nature of EBBO algorithm and to demonstrate real behavior of the algorithm, 100 independent runs are considered for each design example, with each run starting from a random population. In addition, for the purpose of comparing standard BBO algorithm with the EBBO algorithm, 10 independent runs are also performed for the standard BBO algorithm and the results are reported. Moreover, the maximum number of analyses is defined as termination criterion of each run. For each design example, the maximum number of analyses is set as follows: 20,000 for the first two examples and 30,000 for the last two examples. The algorithms and finite element analyses are coded in Matlab software and implemented on Dell Vostro 1520 with Intel CoreDuo2 2.66 GHz processor and 4 GB RAM memory.

4.1 A 10-Bar Planar Truss Structure

The 10-bar planar truss structure with fixed configuration shown in Figure 5 is the first example. Young's modulus is 6.89×1010 N/m2 and material density of truss members is 2770.0 kg/m3. As seen in Figure 5, a non-structural mass of 454.0 kg is attached to all free nodes of the structure. The lower and upper bounds for the cross-sectional areas are specified as 0.645 cm2 and 50 cm2, respectively. The three natural frequency constraints are considered as: ω1 ≥ 7 Hz, ω2 ≥ 15 Hz, ω3 ≥ 20 Hz. It is worth mentioning that, in some of the previous researches, the Young's modulus of the truss members is given as 6.98×1010 N/m2. So, for a fair comparison, we considered two cases as follows: E= 6.89×1010 N/m2 (Case 1) and E= 6.98×1010 N/m2 (Case 2).

Figure 5
Schematic of 10-bar planar truss structure.

For Case 1, the optimum designs obtained through various methods are tabulated in Table 1, wherein the best design is ensured by the EBBO algorithm. From Table 1, it can be seen that the EBBO algorithm not only yields a better design than the SGA method, but it also requires significantly less structural analyses. However, the values of standard deviation and average weight are larger compared to the SGA method. In addition, Table 2 presents the frequencies of the structure obtained by various methods at the optimum designs. According to Table 2, it can be concluded that the designs reported by (Sedaghati et al., 2002Sedaghati, R., Suleman, A., Tabarrok, B., 2002. Structural optimization with frequency constraints using finite element force method. AIAA Journal 40:382-388.) and HRPSO violate the design constraints.

Table 1
Comparison of the optimal designs obtained by different methods for the 10-bar planar truss structure (Case 1).

Table 2
Comparison of the frequencies (Hz) obtained by different methods for the 10-bar planar truss structure (Case 1).

For Case 2, the optimization results obtained by different methods are listed in Table 3. Based on Table 3, it can be easily observed that the best, average and standard deviation obtained by the EBBO algorithm are better than the PSO, CSS, enhanced CSS, CSS-BBBC, and HRPSO methods. Moreover, although the value of standard deviation yielded by the proposed algorithm is larger than the SGA method, the EBBO algorithm finds lighter structural weight than this method. It is also noteworthy to mention that, the proposed algorithm requires less structural analyses than the SGA method to reach optimum solution. Furthermore, the frequencies evaluated at the optimum designs are listed in Table 4. It can be seen that the design yielded by the EBBO algorithm is feasible.

Table 3
Comparison of the optimal designs obtained by different methods for the 10-bar planar truss structure (Case 2).

Table 4
Comparison of the frequencies (Hz) obtained by different methods for the 10-bar planar truss structure (Case 2).

Finally, the convergence characteristic of the BBO and EBBO algorithms for two cases are displayed in Figures 6 and 7. As it can be seen, the proposed algorithm reaches the near-optimum solution after 4000 analyses evaluated in 100 independent runs, while this value for standard BBO algorithm is about over 8000 analyses.

Figure 6
The Convergence diagrams of the EBBO and standard BBO algorithms for the 10-bar planar truss structure (Case 1).

Figure 7
The Convergence diagrams of the EBBO and standard BBO algorithms for the 10-bar planar truss structure (Case 2).

4.2 A Simply Supported 37-Bar Planar Truss Structure

The second design example is the size and shape optimization of a simply supported 37-bar planar truss structure shown in Fig. 8. Young's modulus and material density of truss members are 2.1×1011 N/m2 and 7800 kg/m3, respectively. A non-structural mass of 10 kg is attached to free nodes at the lower chord of the structure. The constant rectangular cross-sectional areas of 4×10-3 m2 are specified for all members of the lower chord and the cross-sectional areas of other members are considered as design variables. By considering geometrical symmetry, the y-coordinates of upper nodes are taken as layout variables and their vertical position can vary between ±1.5 m. Moreover, this structure is subject to the first three frequency constraints as follows: ω1 ≥ 20 Hz, ω2 ≥ 40 Hz, ω3 ≥ 60 Hz.

Figure 8
Schematic of the simply supported 37-bar planar truss structure.

The optimal designs obtained through various methods for this structure are tabulated in Table 5. From the results of comparisons given in Table 5, it can be concluded that the proposed algorithm is superior to the all other methods in solving this example by obtaining the lighter structural weight efficiently. The EBBO algorithm finds the minimum weight of 359.86 kg after 13,500 analyses, while the SGA method obtained the weight of 359.93 kg after 50,000 analyses. In addition, Table 5 also illustrates that: EBBO algorithm yields smaller standard deviation and mean weight than the HRPSO, OMGSA, DPSO, Enhanced CSS, CSS, and NHGA methods, and slightly larger than the values obtained by the SGA and FA methods. Moreover, the first five frequencies of the structure evaluated at the optimum design through various methods are listed in Table 6. As it can be seen, the design yielded by the EBBO algorithm is feasible and the design constraints are not violated.

Table 5
Comparison of the optimal designs obtained by different methods for the simply supported 37-bar planar truss structure; optimal nodal coordinates (m) and cross sectional areas (cm2).

Table 6
Comparison of the frequencies (Hz) obtained by different methods for the simply supported 37-bar planar truss structure.

Figure 9 demonstrates convergence diagrams of the standard BBO and EBBO algorithms for the simply supported 37-bar planar truss structures. With regard to Figure 9, it is clear that the convergence speed of the EBBO algorithm is faster than that of the standard BBO algorithm. The EBBO algorithm reaches to the vicinity of final optimum solution after about 6000 analyses, while this value for the standard BBO algorithm is about 14,000 analyses. In addition, Figure 10 indicates the comparison of the initial and optimized shapes at the best design for this design example.

Figure 9
The Convergence diagrams of the EBBO and standard BBO algorithms for the simply supported 37-bar truss structure.

Figure 10
Comparison of the optimized shape with the initial configuration of the simply-supported planar 37-bar truss structure.

4.3 A 120-Bar Dome Truss Structure

The third example is a 120-bar dome truss structure shown in Figure 11. The members of this structure are divided into 7 groups based on symmetry. The minimum and maximum cross-sectional areas for each group of members are 1 cm2 and 129.3 cm2, respectively. Young's modulus and material density of truss members are 2.1×1011 N/m2 and 7971.810 kg/m3, respectively. Non-structural masses are attached to all free nodes as follows: 3000 kg at node one, 500 kg at nodes 2 through 13 kg and 100 kg at the rest of the nodes. In addition, the structure is subject to frequency constraints as: ω1 ≥ 9 Hz, ω2 ≥ 11 Hz.

Figure 11
Schematic of the 120-bar dome truss structure: top and side views.

The obtained results through different optimization methods for this structure are summarized in Table 7. According to this table, one can observe that the design obtained by the EBBO algorithm is better than all the other methods. Although the value of standard deviation of the EBBO algorithm is larger than the values obtained by the OMGSA and DPSO methods, but the EBBO algorithm is more efficient than these methods in the term of the average of the results. In addition, it is important to note that the standard BBO algorithm obtained better results than the CSS, CSS-BBBC, PSO, and DPSO methods. However, it requires significantly more structural analyses than the EBBO algorithm. Moreover, the frequencies evaluated by different methods for the dome truss structure are listed in Table 8. As it can be seen, the values of the first two frequencies are very close to the allowable values.

Table 7
Comparison of the optimal designs obtained by different methods for the 120-bar dome truss structure.

Table 8
Comparison of the frequencies (Hz) obtained by different methods for the 120-bar dome truss structure.

Figure 12 illustrates the convergence diagrams of the standard BBO and EBBO algorithms for the 120-bar dome truss structure. As it can be observed, the convergence rate of the EBBO algorithm is considerably faster than the standard BBO algorithm. The proposed algorithm reaches to the vicinity of the final solution after about 5000 analyses evaluated in 100 independent runs without any abruption.

Figure 12
The Convergence diagrams of the EBBO and standard BBO algorithms for the 120-bar dome truss structure.

4.4 A 200-Bar Planar Truss Structure

The last design example is a 200-bar planar truss structure shown in Figure 13. The Young's modulus is 2.1×1011 N/m2 and the material density of truss members is 7860 kg/m3. The minimum permitted cross-sectional area for the truss members is considered as 0.1cm2. A non-structural mass of 100 kg is attached to the nodes at the top of the structure. In addition, the structure is subject to the first three frequency constraints as: ω1 ≥ 5 Hz, ω2 ≥ 10 Hz, ω3 ≥ 15 Hz. The members of the structure are divided into 29 groups as shown in Table 9. This example has 29 design variables and it is considered a high dimensional optimization problem.

Figure 13
Schematic of the 200-bar planar truss structure.

Table 9
Elements grouping for the 200-bar planar truss structure.

Table 10 compares the designs founded by the standard BBO and EBBO algorithms with other methods in the literature. With regard to the results given in Table 10, it is clear that the proposed EBBO algorithm obtained the lowest structural weight tan the CSS, CSS-BBBC, and OMGSA methods. In addition, the EBBO algorithm is more efficient than the OMGSA method in terms of average and standard deviation. Moreover, it can be seen from Table 10 that the standard BBO algorithm obtained better design than the CSS algorithm. Furthermore, the frequencies of the structure evaluated at the optimum designs are given in Table 11. Also, the design obtained by the EBBO algorithm is feasible, i.e., the constraints are not violated.

Table 10
Comparison of the optimal designs obtained by different methods for the 200-bar planar truss structure.

Table 11
Comparison of the frequencies (Hz) obtained by different methods for the 200-bar planar truss structure.

Figure 14 shows the convergence diagrams of the standard BBO and EBBO algorithms for this example. Again, it can be seen that the convergence rate of the EBBO algorithm is much faster than the standard BBO algorithm. Figure 14 reveals that the EBBO needs about 6000 analyses to reach near-optimum solutions evaluated in 100 independent runs.

Figure 14
The Convergence diagrams of the EBBO and standard BBO algorithms for the 200-bar planar truss structure.

5 CONCLUSIONS

The Biogeography-Based Optimization (BBO) is a simple and recently introduced meta-heuristic optimization algorithm. The basic concepts and ideas of the method are inspired by the mathematical models in biogeography science and are based on migration behavior of species among habitats in the nature. The algorithm consists of two main operators: migration and mutation. In most cases, the standard BBO algorithm may fail to find the best solution. The main reason for this issue is that the basic BBO algorithm employs simple migration and mutation operators during the optimization process. Herein, an effective algorithm, called the Enhanced BBO (EBBO), has been developed to mitigate premature convergence problem of the standard BBO algorithm. In the proposed algorithm, to enhance the overall performance of the standard BBO algorithm, the new migration and mutation operators are proposed. The new migration and mutation operators improve the convergence properties of the BBO algorithm and enhance the algorithm's ability to further escape stagnation and premature convergence. To evaluate the performance of the proposed algorithm, four benchmark truss design examples with frequency constraints are investigated and the results are compared with those of the standard BBO algorithm and other methods in literature. The computational experiments show that the presented EBBO algorithm can get better solutions, and it is more efficient than the standard BBO algorithm on the size and shape optimization of truss structures problems with frequency constraints. Moreover, it can be stated that the proposed algorithm is straightforward and free of computational complexity.

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Publication Dates

  • Publication in this collection
    July 2016

History

  • Received
    11 June 2015
  • Reviewed
    13 Mar 2016
  • Accepted
    16 Mar 2016
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