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

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.


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, 1993). 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., 2004). 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., 2005). On the other hand, the meta-heuristic search methods such as the genetic algorithms (GAs) (Goldberg, 1989), particle swarm optimization (PSO) (Eberhart and Kennedy, 1995), big bang-big crunch (BB-BC) algorithm (Erol and Eksin, 2006), gravitational search algorithm (GSA) (Rashedi et al., 2009), biogeography-based optimization (BBO) (Simon, 2008;, Cultural Algorithm (CA) (Reynolds, 1999;, and charged system search (CSS) algorithm (Kaveh and Talatahari, 2010) 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, 2014a), hybrid CSS-BBBC algorithm (Kaveh and Zolghadr, 2012), enhanced CSS algorithm (Kaveh and Zolghadr, 2011), niche hybrid genetic algorithm (NHGA) (Lingyun et al., 2005), and orthogonal multi-gravitational search algorithm (OMGSA) (Khatibinia and Naseralavi, 2014) 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 (2008). 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., 2011;Boussaid, 2012) and has been used to solve numerous real-world optimization problems (Simon, 2008;Singh e al., 2010;Bhattachary et al., 2010). 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 Latin American Journal of Solids and Structures 13 (2016) 1406-1430 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.

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: where is the vector including design variables; n is the number of the design variables; W(.) is the weight of the structure; . is the penalty function; m is the number of structural members; is the material density; and are the cross-sectional area and length of member i, respectively; . and . are the upper and lower frequency constraints, respectively; is the jth frequency of the structure; is the upper limit for jth frequency and denotes its lower limit; and 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: 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, 2014b) as follows:

=2+5
(3) where It and Itmax are the current iteration number and maximum considered iterations, respectively.

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 (2008).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 (1967) 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 and emigration 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, 2008): 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, 2008). 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: Eq. (6) explains that one of the variables of ith habitat is replaced by a variable of jth habitat. Here, and are the immigrating and emigrating habitats, respectively. It is worth mentioning that the emigrating habitat ( ) is selected based on the emigration rates μ . The probability of selecting jth habitat as emigrating habitat is calculated as follow: ∑ , for j=1, 2, 3, …, NP 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. 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 ( ), which is related to the habitat's probability. The mutation rate for each habitat is calculated as follows: where is a user-defined parameter and = max . More details about the calculation of and probabilities can be found in (Simon, 2008). 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 0.1 . 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.

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: where and are the immigrating and emigrating habitats, respectively, Φ is a random number uniformly generated between the 0 and 1, and 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: where 0,1 is a random number generated according to a standard normal distribution with mean zero and standard deviation equal to one; . and . are the upper and lower bounds of the search space, respectively; It and 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: 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 λ and emigration μ 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 λ and emigration μ 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.

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.

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 6.89×10 10 N/m 2 and material density of truss members is 2770.0 kg/m 3 . 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 cm 2 and 50 cm 2 , respectively. The three natural frequency constraints are considered as: 7 Hz, 15 Hz, 20 Hz. It is worth mentioning that, in some of the previous researches, the Young's modulus of the truss members is given as 6.98×10 N/m . So, for a fair comparison, we considered two cases as follows: E= 6.89×10 10 N/m 2 (Case 1) and E= 6.98×10 10 N/m 2 (Case 2).
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. (2002) and HRPSO violate the design constraints.
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.     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.

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 2.1×10 11 N/m and 7800 kg/m 3 , 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 m 2 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 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: ω 20 Hz, ω 40 Hz, ω 60 Hz. 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 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.

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 2.1×10 11 N/m 2 and the material density of truss members is 7860 kg/m 3 . The minimum permitted cross-sectional area for the truss members is considered as 0.1cm 2 . 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: ω 5 Hz, ω 10 Hz, ω 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. 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.   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

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.