Weight Optimization of Truss Structures by a New ... - Springer Link

KSCE Journal of Civil Engineering (2014) 18(4):1105-1118 Copyright ⓒ2014 Korean Society of Civil Engineers DOI 10.1007/s12205-014-0438-x

Structural Engineering

pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205

TECHNICAL NOTE

Weight Optimization of Truss Structures by a New Feasible Boundary Search Technique Hybridized with Firefly Algorithm A. Baghlani* and M. H. Makiabadi** Received August 26, 2012/Accepted My 17, 2013

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Abstract In most metaheuristic optimization techniques, constraint handling is accomplished by traditional penalty function method which may generate infeasible solutions. The main objective of this paper is to introduce a new technique for constraint handling in size optimization of truss structures. The method restricts the searching space on the feasible boundary and hence it is referred to as Feasible Boundary Search (FBS) technique. By limiting the searching space on the feasible boundary, the problem constraints are automatically satisfied. The method generates absolutely feasible high-quality solutions. Its generality enables the scheme to be hybridized with most metaheuristics. By presenting some modifications on the standard Firefly Algorithm (FA), a powerful hybridized technique which is denoted by FBSFA is then developed for fast weight optimization of truss structures. Several design examples show the effectiveness, robustness and accuracy of FBSFA. Keywords: size optimization, truss structures, constraint handling, firefly algorithm, constrained problems ··································································································································································································································

1. Introduction Weight optimization of structures is important owing to many engineering and economical aspects. Firstly, it leads to a structure with optimum members having minimum weight and cost. Secondly, it makes it easier to transport the components and build up the whole structure. Thirdly, the force exerted on a lighter structure due to ground excitation is reduced. Weight or size optimization of truss structures concerns finding the minimum cross sectional areas of the members while ensuring that some design constraints are satisfied. In most cases, the constraints include the stress within the bars, displacements in any direction at the nodes, or dynamic response of the structure. There are other kinds of structural optimization problems including shape and topology optimization. This is a formidable task to optimize a structure with traditional calculus-based optimization methods. Recent developments in nature-inspired powerful optimization techniques have enabled the researches to optimize complicated problems in engineering. In the context of truss optimization, there exist various sophisticated approaches such as Genetic Algorithm (GA) (Rajeev and Krishnamoorthy, 1992, 1997; Adeli andCheng, 1993; Adeli and Kumar, 1995; Hajela and Lee, 1995; Oshaki, 1995; Sarma and Adeli, 2000; Wu and Chow, 1995; Kaveh and Kalatjari, 2002, 2003; Kaveh and Rahami, 2006; Togan and Daloglu, 2006), Ant Colony Optimization (ACO) (Camp and Bichon, 2004; Serra

and Venini, 2006; Kaveh et al., 2008; Luh and Lin, 2008; Kaveh and Talatahari, 2009a), Particle Swarm Optimization (PSO) (Schutte and Groenwold, 2003; He et al., 2004; Perez and Behdinan, 2007; Li and Huang, 2007, 2008; Gomes, 2011a; Luh and Lin, 2011), Harmony Search (HS) (Lee and Geem, 2004, 2005, 2005; Saka, 2007; Lamberti and Pappalettere, 2009; Degertekin, 2004, 2012; Kaveh and Talatahari, 2009b) and so on. Recently, Firefly Algorithm (FA) has been developed by Yang (2009) inspired by behavior of fireflies in nature. By altering one of the parameters of the algorithm, i.e. light absorption coefficient, the method lies between two extreme cases of random search and PSO. As indicated by Yang (2009), by adjusting this parameter, FA outperforms both random search and PSO methods. Because of the newness of FA compared to other techniques, few articles have been published concerning its application in structural optimization problems. Gandomi et al. (2011) utilized firefly algorithm to solve continuous and discrete structural optimization problems. Gomes (2011b) used FA for size and shape optimization of structures including dynamic constraints. Gandomi et al. (2012) introduced chaos in the algorithm to enhance its global search capability. There are two main drawbacks in using metaheuristic natureinspired algorithms such as PSO or FA to solve real-life problems. First, the rate of convergence of most of these evolutionary methods is low and the structure should be analyzed many times

*Assistant Professor, Faculty of Civil and Environmental Engineering, Dept. of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran (Corresponding Author, E-mail: [email protected]) **MS.c Student, Faculty of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran (E-mail: [email protected]) − 1105 −

A. Baghlani and M. H. Makiabadi

with no guarantee to achieve a solution. If the structure is large and complicated, the computational cost of the method is very high. In some cases, there are some tuning parameters in the algorithm which should be adjusted as well. Second, there is not a perfect general way to handle the constraints in these methods while preserving their fundamental searching strength at the same time. The simplest way is to maintain just feasible solutions during evolutions and discard the solutions that violate the constraints immediately. However, in some cases, finding feasible solutions for the problem is itself hard (Garey and Johnson, 1979). The other way is employing traditional penalty function method which undergoes some major disadvantages. The solutions found by this technique usually violate the problem-specified constraints to some extent. Moreover, the procedure requires some effort to tune the penalty function for a given problem. Most of the difficulty arises because the optimal solution is usually located on the boundary of the feasible region (Siedlecki and Sklansky, 1989). Many of the solutions most similar to the optimum solution in the evolution are hence infeasible. Therefore, limiting the search to only feasible solutions or imposing very strict penalties makes it difficult to find a scheme that leads the population toward the optimum. On the contrary, if the penalty is not severe enough, an extensive region is searched and much of the search time will be wasted to explore regions far from the feasible region. Then, the search will tend to halt outside the feasible region (Smith and Coit, 1995). The third way to handle the constraints is to use the recently-developed flyback mechanism (He et al., 2004). In this strategy, the solution candidates are first initialized in the feasible region and the constraints in the problem are then computed to find out whether they violate the design constraints. If one or a number of the candidates gives infeasible solutions, they will be forced to fly back to the feasible region. The candidate that flies back to the previous position may be closer to the boundary at the next iteration. This makes the candidates to fly to the global minimum in a great probability. These candidates having one or more constraints slightly infeasible are utilized in the searching process that might provide a new candidate that may be feasible. This is accomplished by using larger error values initially for the acceptability of the new design vectors and then reducing this value gradually during the evolution and using finally a permissible error value towards the end of iterations. Despite converging faster than other techniques (He et al., 2004; Le et al., 2007), because of accepting an error for acceptability of design vectors, the previous experiments show that some violations from the problem-specified constraints still exist (Degertekin, 2012). Due to difficulties which arise for constraint handling in aforementioned methods, the main objective of this study is to propose an effective novel strategy to handle problem-specified constraints and to generate absolutely feasible solutions. The method is referred to as Feasible Boundary Search (FBS). The formulation of FBS is general and it can be hybridized with many evolutionary methods. Combination of FBS technique

with firefly algorithm is presented in this paper. In order to overcome the intrinsic disadvantage of standard firefly algorithm and to improve the convergence rate of FA, some minor but effective modifications on FA are also proposed. Then, FBS is applied on improved FA to handle the constraints effectively. This hybridized algorithm is called FBSFA in this article. The design examples show that the proposed FBS in conjunction with enhanced firefly algorithm is a powerful tool for fast optimization of truss structures in which high-quality and absolutely feasible solutions are generated. Accordingly, the structure of the manuscript is arranged as follows: In section 2 the problem of size optimization of truss structures is defined. In section 3 the standard firefly algorithm is presented and some modifications to improve the convergence rate of the standard algorithm are presented. Section 4 illustrates the new feasible search technique. In section 5 some benchmark design problems are solved and the effectiveness of the proposed technique is explored. Finally, in section 6 conclusions are included.

2. Problem Formulation The aim of weight optimization of truss structures is to optimize cross sections Ai of the bars of the truss such that the overall weight of the structure W is minimized and some design constraints are satisfied. The problem can be mathematically expressed as follows: Minimize W ( A ) =

∑ k = 1 Ak ∑ i = 1 ρ iL i ng

mk

(1)

Subject to: σlow ≤ σi ≤ σup , i = 1, 2, …, nm

(2)

σ ≤ σi ≤ 0,

i = 1, 2, …, ncm

(3)

i = 1, 2, …, m

(4)

b i

δlow ≤ δi ≤ δup, Alow ≤ Ai ≤ Aup,

i = 1, 2, …, ng

(5)

in which A is the vector of design variables (i.e. cross sections A = { A1, A2, …, Ang } ), W ( A ) is the total weight of the truss structure, ρi is the material density of member i, Li is the length of member i, nm is the number of members in the structure , ncm is the number of compression members, nn is the number of nodes, ng is the total number of member groups (i.e. design variables), Ak is the cross sectional area of the members belonging to group k, mk is the total number of members in group k, σi is the stress of b the ith member, σi is the allowable buckling stress for the ith member, δ i is the displacement of the ith node, and low and up are the lower and upper bounds for stress, displacement and cross-sectional area.

3. Formulation of Firefly Algorithm One of the most recent metaheuristic algorithms is the Firefly Algorithm (FA). It is a nature-inspired algorithm, which was first developed by Yang (2009) inspired by the light attenuation over the distance and fireflies' mutual attraction. Algorithm considers

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KSCE Journal of Civil Engineering

Weight Optimization of Truss Structures by a New Feasible Boundary Search Technique Hybridized with Firefly Algorithm

what each firefly observes at the point of its position, when trying to move to a greater light-source, than is his own. This algorithm idealizes some of the characteristics of the firefly behavior in nature. They follow three rules: i) all the fireflies are unisex, ii) attractiveness is proportional to their flashing brightness which decreases as the distance from the other firefly increases due to the fact that the air absorbs light. The most attractive firefly is the brightest one which convinces neighbor fireflies to move toward him, and, iii) brightness of every firefly, which is proportional to the objective function, determines its quality of solution. Like other evolutionary optimization methods, the first steps of the FA are initializing a swarm of fireflies, evaluating the objective function for each firefly and hence finding out the flashing light intensity. The light intensities of fireflies are then compared mutually and the firefly with lower light intensity moves toward the higher one. The moving distance depends on the attractiveness. The less bright fireflies do not tend to move to brighter fireflies which are too far. Once a firefly is moved, the new firefly is evaluated and the light intensity is updated. In each loop, the best-so-far solution is iteratively updated. The pairwise comparison process is repeated until termination criteria are satisfied. Finally, the best-so-far solution is visualized and reported. As stated previously, the light intensity of each firefly is proportional to the quality of the solution it is currently located at. In order to improve its current location, the firefly needs to advance towards the fireflies that have brighter light emission than is his own. Each firefly observes decreased light intensity, than the one firefly actually emit, due to the air absorption over the distance. Attractiveness of a firefly is defined as (Yang, 2009): β = β0 exp( –γr )

(6)

in which β0 is the attractiveness in distance r = 0 and γ is light absorption coefficient in the range [0, ∞) . The distance r between firefly i and j at xi and xj is defined as Cartesian distance:

∑ k = 1( xi, k – xi, k ) d

r = r ij = xi – xj =

2

(7)

where xi, k is the kth component of the spatial coordinate xi of the ith firefly and d is the number of dimensions . Moreover, the movement of firefly i which is attracted by a more attractive or brighter firefly j is given by the following equation: 2

xi = xi + β0 exp( –γr ) ( xj – xi ) + α ( ε – 0.5 )

(8)

in which the second term is due to the attraction. The third term is randomization with α being the randomization parameter such that α ∈ [ 0, 1] , and ε is a vector of random numbers drawn from a Gaussian distribution or uniform distribution in the range [0, 1]. Furthermore, for most problems, one can adopt β0 = 1 . In the case of size optimization of trusses, the cross sectional areas of bars are considered as design variables to be optimized with d being the number of bars (or group of bars) in the truss. As Gandomi et al. (2011, 2012) have shown, firefly algorithm can be slightly modified if randomization parameter α in Eq. (9) Vol. 18, No. 4 / May 2014

is gradually reduced as the solution is approached: t α = α0θ

(9)

Using this modification, Eq. (8) changes to: 2

xi = xi + β0 exp ( –γr )( xj – xi ) + α 0 θ ( ε – 0.5 ) t

(10)

in which t ∈ [0, tmax ] is the iteration number where tmax is the maximum number of iterations, and θ ∈ ( 0, 1] is the randomization reduction constant. We also introduce a scaling factor λ and rewrite Eq. (10) as: 2

2

xi = xi + β0 exp ( –γr )( xj – xi ) + λα 0 θ ( ε – 0.5 )

(11)

where λ is defined for truss optimization problems as: λ = Aup – Alow

(12)

As the design examples reveal, the convergence rate of Eq. (11) is much faster than Eq. (8). Hybridization of Eq. (11) with our proposed Feasible Boundary Search (FBS) technique for accurate constraint handling is illustrated in the next section.

4. Feasible Boundary Search (FBS) Technique In Fig. 1, the solution domain of a typical constrained optimization problem has been schematically shown. As has been mentioned by Li et al. (2007) the whole space can be divided into 3 regions. In regions 1 and 2, firefly satisfies the problem-specified constraints, whereas in regions 1 and 3 firefly satisfies the variables boundary. In region 2, firefly satisfies the problem-specified boundary, but violates the variables boundary, whereas as in region 3 the opposite is true. Region 1 forms the feasible space in which both problem-specified constraints and variables constraint are satisfied. Most optimization algorithms search for the optimal solution within the feasible space. This space could be an enormous space in practical problems such as trusses with many bar elements. However, as mentioned earlier, optimal solution is frequently located on the boundary of the feasible space. Therefore, if by an appropriate process, the searching space is merely restricted to this boundary, not only the rate of convergence will be increased, but also the problem-specified constraints are automatically satisfied and they are conveniently handled. In most problems, the optimal solution is located on the common part of feasible boundary and problem-specified constraints boundary, since in this part the problem-specified constraints are located on the boundary and the variables boundary constraint is also satisfied. For convenience, this common part can be called solution boundary and it is shown on Fig. 1 as well. By employing some fundamental principles of structural analysis, a very appropriate approach is proposed to limit the search on the feasible boundary. Furthermore, No violations in satisfying problem-specified constraints are observed. This technique is referred to as Feasible Boundary Search (FBS) and the implementation of the technique in the enhanced firefly algorithm, which is called FBSFA, is presented. To illustrate the technique, the principle of superposition for linear elastic structures is brought to mind. Recall that under

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linear elastic behavior assumption, if with constant cross section of bar elements, the external loads are multiplied by a factor, design variables such as nodal displacements and internal stresses are multiplied in the same factor. However, in a truss optimization problem it is not the case; the external loads are constant and the cross sectional areas are variables to be determined. Nevertheless, the same principle can be employed by noting that in this case, for constant external loads, if the cross sectional areas are multiplied by a factor β, the aforementioned design constraints are multiplied by a factor 1/β. This means that for a truss optimization problem, if under specific load conditions, the maximum stress in a truss with known cross sectional areas of bars is for example σ max = 30 ksi , but the upper limit of the stress specified by the problem is α up = 40 ksi , the cross sectional areas of the truss can be reduced so that the maximum stress in the truss equals the upper limit of stress allowed by the design criteria, specified by the problem. Thus, this truss, which is represented by a firefly, can be easily mapped onto the problem-specified constraint boundary for the same loading by multiplying all of its cross sectional areas by a value equal to σmax 30 --------- = 0.75 (or similarly by dividing the cross sectional areas σup- = ----40 σup by a factor r = --------σ max- ). For this new truss, the maximum stress within all bars will be obviously less than σup except for specific bar(s) in which the internal stress is exactly σmax = σup . Hence, this mapped firefly not only satisfies the problem-specified

constraints, but also is exactly positioned on the problemspecified constraint boundary and could be a potential solution. Once the swarm of fireflies is generated, since the upper permissible limit of stress is known and the maximum stress in each firefly can be computed by analysis, the aforementioned procedure permits mapping each firefly onto the problem-specified constraints boundary. However, the mapped firefly may either satisfy or violate the variables boundary constraint, i.e. the new cross sections may fall beyond the acceptable range. In Fig. 1, two schematic fireflies have been depicted. When firefly x1 is mapped

Fig. 1. Definition of Various Regions in the Searching Space, Solution Boundary, Fireflies and Mapped Fireflies on the Feasible Boundary

Table 1. The Pseudo Cod for Feasible Boundary Search Firefly Algorithm (FBSFA) Objective function f ( x ), x = ( x1, x2, ……, xd) d=no. of design variables i Generate initial population of fireflies randomly X , i = 1, 2, …, n n=no. of fireflies Define light absorption coefficient γ Define randomness reduction constant θ Define initial randomization parameter α0 Define attractiveness at ( r = 0 ) , β0 Calculate scaling parameter λ = (upper variable boundary - lower variable boundary) while t < maximum number of generation or convergence criteria met Calculate α = α 0θ for i = 1 to n for j = 1 to nc nc=no. of specified conditions of the problem Calculate the ratio of allowable-to-maximum value ( rj = cjup ⁄ cjmax ) end for j Find rmin = min ( r1, r2, …, rj, …rnc ) Moving all fireflies on the problem-specified constraint boundary: divide each component (k) of the current vector xi by rmin to find mapped position vector y(i k) ( k) Check feasibility: check whether each component of the mapped position vector yi violates lower variables boundary or not. If it does, move these to the variable boundary (multiple each (k) component of the position vector(k)of these mapped particles to ratio s = Alow ⁄ min ( xi ) to find new mapped position vector yi ) T

(i)

Calculate light intensity Ij = f ( yi ) at of mapped fireflies on the feasible boundary end for i Rank the fireflies and find the current global best for i = 1 to n for j = 1 to n if ( Ii > Ij ) (k ) (k ) Calculate the distance rij =2 xi – xj Calculate β = β0 exp( –γrij ) Generate random number(kvector εi ) ( k) ( k) (k) Update design variable xi = xi + β( xj – xi ) + αλεi end if end for j end for i end while Postprocess results and visualization − 1108 −



on the boundary, the mapped firefly y1 satisfies the variable boundary constraints. In other words, it is located on the solution boundary. These new mapped fireflies are most likely near the optimal solution. When firefly x2 is mapped, the new firefly is located on the problem-specified boundary but violates the variable boundary constraint. In most truss optimization problems, the lower limit of cross sectional areas is defined. In these cases, the mapped firefly on the problem-specified boundary has at least a cross section which is less than lower limit of cross sectional area Alow . These mapped fireflies should be repositioned and remapped onto the feasible boundary. This can be readily accomplished by multiplying all cross sectional areas of the mapped firefly by a ratio s = Alow ⁄ Amin in which Amin is the minimum cross sectional area among all cross sectional areas of the truss to obtain the mapped firefly y2 which is located on the feasible boundary. If a standard searching algorithm is performed among candidates located on the feasible boundary, the quality of search is highly increased with high probability of finding an absolutely feasible solution in less iterations. Implementation of this searching technique for problems with several design constraints is straightforward by noting that in such problems, usually one of the design constraints is influential and controls the design; hence an extreme value for the mapping factor r should be used. Since the FBS technique described above is independent of the optimization algorithm used, it can be hybridized with any evolutionary optimization method. For more details, pseudo code shown in Table 1 clearly demonstrates implementation of this technique hybridized with the Enhanced Firefly Algorithm (FBSFA) for problems with several design constraints.

5. Design Examples To investigate the effectiveness of the proposed FBSFA in

Fig. 2. 10-bar Planar Truss Structure

weight optimization of truss structures, several benchmark planar and spatial trusses, which are widely tested in the literature, are optimized. A finite element code was developed to analyze the planar and spatial trusses. Since there exist no published articles concerning optimization of these problems by standard FA, the problems have been also optimized by this algorithm using standard penalty function method to handle the constraints; the results are reported and a comparison have been also made among FA, FBSFA and other approaches. In all tables of results, the optimal solution vectors, the value of objective function, problem specified constraints and amount of violation from the design constraints in each method have been reported in order to make a clear comparison among various techniques. In all examples solved in this study, after some calibrations, the following parameters were adopted: θ = 0.97 , γ = 0.05 , β0 = 1 , α 0 = 1 5.1 10-bar Planar Truss For the first example, the planar 10-bar truss shown in Fig. 2. which has been optimized by many researchers Lee and Geem

Table 2. Comparison of Optimal Design for the 10-bar Planar Truss Structure (Case 1) Optimal cross-sectional areas (in.2) Variables

1 A1 2 A2 3 A3 4 A4 5 A5 6 A6 7 A7 8 A8 9 A9 10 A10 Weight(lb) σmax (ksi) δmax (in.) Constraint violation (%)

Lee and Geem (2004) HS 30.15 0.102 22.71 15.27 0.102 0.544 7.541 21.56 21.45 0.100 5057.88 24.9974 2.00181 0.090

Vol. 18, No. 4 / May 2014

30.5218 0.1000 23.1999 15.2229 0.100 0.5514 7.4572 21.0364 21.5284 0.1000 5060.85 25.0000 2.00000

Kaveh and Talatahari (2009b) HPSACO 30.307 0.100 23.434 15.505 0.100 0.5241 7.4365 21.079 21.229 0.100 5056.56 24.9998 2.00198

None

0.099

Sedaghati (2005)

Farshi and Alinia-ziazi (2010)

Li et al. (2007)

Degertekin (2012)

This study

30.5208 0.1000 23.2040 15.2232 0.1000 0.5515 7.4669 21.0342 21.5294 0.1000 5061.4 24.9523 1.99987

PSO 33.469 0.110 23.177 15.475 3.649 0.116 8.328 23.340 23.014 0.190 5529.50 11.8503 1.99999

PSOPC 30.569 0.100 22.974 15.148 0.100 0.547 7.493 21.159 21.556 0.100 5061.00 24.9978 1.99999

HPSO 30.704 0.100 23.167 15.183 0.100 0.551 7.460 20.978 21.508 0.100 5060.92 24.9996 2.00000

EHS 30.208 0.100 22.698 15.275 0.100 0.529 7.558 21.559 21.491 0.100 5062.39 24.9639 1.99996

SAHS 30.394 0.100 23.098 15.491 0.100 0.529 7.488 21.189 21.342 0.100 5061.42 24.9602 2.00006

FBSFA 30.55155 0.10000 23.20861 15.23060 0.10000 0.55019 7.45737 21.03241 21.50054 0.10000 5060.86 25.0000 2.00000

None

None

None

None

None

0.003

None

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FA 30.968 0.100 23.215 15.043 0.100 0.591 7.453 20.866 21.461 0.100 5060.14 24.9998 2.00052 0.026


Table 3. Comparison of Optimal Design for the 10-bar Planar Truss Structure (Case 2) Optimal cross-sectional areas (in.2) Variables

1 A1 2 A2 3 A3 4 A4 5 A5 6 A6 7 A7 8 A8 9 A9 10 A10 Weight(lb) σmax (ksi) δmax (in.) Constraint violation (%)

Lee and Geem (2004) HS 23.25 0.102 25.73 14.51 0.100 1.977 12.21 12.61 20.36 0.100 4668.81 25.0406 2.00387 0.193

Schmit and Farshi (1974) 24.29 0.100 23.35 13.66 0.100 1.969 12.67 12.54 21.97 0.100 4691.84 25.0003 1.99994 0.001

Kaveh and Talatahari (2009b) HPSACO 23.194 0.100 24.585 14.221 0.100 1.969 12.489 12.925 20.952 0.101 4675.78 25.0019 2.00158 0.079

Farshi and Alinia-ziazi (2010) 23.5270 0.1000 25.2941 14.3760 0.1000 1.9698 12.4041 12.8245 20.3304 0.1000 4677.8 24.9996 1.99993 None

Li et al. (2007) PSO 22.935 0.113 25.355 14.373 0.100 1.990 12.346 12.923 20.678 0.100 4679.47 24.9996 2.00000 None

(2004), Sedaghati (2005), Li et al. (2007), Kaveh and Talatahari (2009b), Farshi and Alinia-ziazi (2010), Degertekin (2012) is considered. The material density of all members was 0.1 lb/in3 and the modulus of elasticity was 10,000 ksi. The maximum allowable stress in all bars was ± 25 ksi with nodal displacement limitations of ± 2.0 inches for both directions. The minimum cross-sectional area of each bar element was 0.1 in2. The weight optimization of truss have been studied for two cases: Case 1 with pi = 100 kips and p2 = 0 ; Case 2 with pi = 150 kips and p2 = 20 kips. In Tables 2 and 3 the results found by other researchers and those obtained by employing FA and FBSFA techniques have been presented for Case 1 and 2, respectively. For Case 1, the value of objective function is 5060.14 and 5060.86 for FA and FBSFA, respectively. For Case 2, the value of objective function is 4673.70 and 4677.08 for FA and FBSFA, respectively. As Table 2 shows for Case 1, the work done by Sedaghati (2005), PSO, PSOPC, HPSO, EHS and FBSFA give absolutely feasible solutions. The other methods give slightly infeasible solutions, including FA in conjunction with penalty function approach. The value of objective function found by FBSFA is almost the same as Sedaghati (2005) which is the best value among the methods that give absolute feasible solutions. It is worth pointing out that for this case, in the solution found by FBSA and Sedaghati (2005), the values of maximum stress and displacement coincide with the values of maximum allowable stress and displacement defined by the problem. This indicates that the solutions located exactly on the problem-specified constraint boundary are of better values for objective function. In some other problems, one of these constraints is influential. Table 3 reveals that the study in Farshi and Alinia-ziazi (2010), PSO, EHS, SAHS and FBSFA give absolute feasible solutions for the Case 2 and the solution

Degertekin (2012)

PSOPC 23.743 0.101 25.287 14.413 0.100 1.969 12.362 12.694 20.323 0.103 4677.70 25.0011 2.00002

HPSO 23.353 0.100 25.502 14.250 0.100 1.972 12.363 12.894 20.356 0.101 4677.29 25.0006 2.00000

0.004

0.002

This study

EHS 23.589 0.100 25.422 14.488 0.100 1.975 12.362 12.682 20.322 0.100 4679.02 24.9346 1.99986

SAHS 23.525 0.100 25.429 14.488 0.100 1.992 12.352 12.698 20.341 0.100 4678.84 24.9352 1.99997

FBSFA 23.48241 0.10056 25.30242 14.28922 0.10000 1.97025 12.41012 12.90688 20.31424 0.10015 4677.077 24.9991 2.00000

None

None

None

FA 23.326 0.100 25.375 14.616 0.100 1.975 12.358 12.814 20.217 0.100 4673.70 24.9854 2.00191 0.095

Fig. 3. Comparison of the Convergence Rates of the Two Algorithms for the 10-bar Planar Truss Structure (Case 1)

Fig. 4. Comparison of the Convergence Rates of the Two Algorithms for the 10-bar Planar truss Structure (Case 2)

obtained by FBSFA is the best among them. To see the effectiveness of the proposed FBSFA technique in

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reducing the rate of convergence compared to standard FA, Figs. 3 and 4 have been presented. As the figures show, approximately 2000 iterations are required for FA for both cases to lead to a slightly infeasible solution, whereas approximately 100 iterations are required for FBSFA to lead to an absolute feasible solution which indicates considerable improvement of FA. 5.2 17-bar Planar Truss Khot and Berke (1984), Adeli and Kumar (1995), Lee and Geem (2004), and Li et al. (2007) have been optimized the 17bar truss shown in Fig. 5. The material density of all members was 0.268 lb/in3 and the modulus of elasticity was 30,000 ksi. The maximum allowable stress in all bars was ± 50 ksi with nodal displacement limitations of ± 2.0 inches for both directions. Table 4 compares the results found by the current study and aforementioned ones. The weight of 2577.57 lb and a 2581.90 lb were found with optimal cross sectional areas shown in the table

Fig. 6. Comparison of the Convergence Rates of the Two Algorithms for the 10-bar Planar Truss Structure

by using FA and FBSFA, respectively. Khot’s approach, PSOPC and FBSFA give absolutely feasible solutions with no violations. As it is clear, nodal displacement constraint controls the design in this problem. The optimal solution found by FBSFA is better than the optimal solution found by PSOPC technique and it is very close to solution obtained by Khot. Fig. 6. shows that FBSFA technique is very effective in increasing the rate of convergence as well. 5.3. 25-bar Space Truss Structure In Fig. 7, the 25-bar spatial truss which has been extensively

Fig. 5. 17-bar Planar Truss Structure

Table 4. Comparison of Optimal Design for the 17-bar Planar Truss Structure

Variables 1 A1 2 A2 3 A3 4 A4 5 A5 6 A6 7 A7 8 A8 9 A9 10 A10 11 A11 12 A12 13 A13 14 A14 15 A15 16 A16 17 A17 Weight(lb) σmax (ksi) δmax (in.) Constraint violation (%)

Lee and Geem (2004) HS 15.821 0.108 11.996 0.100 8.150 5.507 11.829 0.100 7.934 0.100 4.093 0.100 5.660 4.061 5.656 0.100 5.582 2580.81 25.2376 2.00088

Vol. 18, No. 4 / May 2014

0.044

Adeli and Kumar (1995) 16.029 0.107 12.183 0.110 8.417 5.715 11.331 0.105 7.301 0.115 4.046 0.101 5.611 4.046 5.152 0.107 5.286 2594.42 27.1677 2.03386 1.693

Optimal cross-sectional areas (in.2) Li et al. Khot and (2007) berke (1984) PSO PSOPC HPSO 15.930 15.766 15.981 15.896 0.100 2.263 0.100 0.103 12.070 13.854 12.142 12.092 0.100 0.106 0.100 0.100 8.067 11.356 8.098 8.063 5.562 3.915 5.566 5.591 11.933 8.071 11.732 11.915 0.100 0.100 0.100 0.100 7.945 5.850 7.982 7.965 0.100 2.294 0.113 0.100 4.055 6.313 4.074 4.076 0.100 3.375 0.132 0.100 5.657 5.434 5.667 5.670 4.000 3.918 3.991 3.998 5.558 3.534 5.555 5.548 0.100 2.314 0.101 0.103 5.579 3.542 5.555 5.537 2581.89 2724.37 2582.85 2581.94 25.0022 32.2930 25.4221 25.1833 1.99997 2.00003 2.00000 2.00002 None

0.001

− 1111 −

None

0.001

This study FBSFA 15.89579 0.10001 12.09872 0.10001 8.06641 5.57929 11.93532 0.10001 7.93941 0.10001 4.04360 0.10001 5.66825 4.02092 5.53592 0.10001 5.57186 2581.90 25.0981 2.00000 None

FA 15.942 0.100 12.023 0.100 8.035 5.519 11.777 0.100 7.924 0.100 4.077 0.100 5.632 4.029 5.603 0.100 5.620 2577.57 25.3295 2.00355 0.178


Table 6. Member Stress Limits for the 25-bar Spatial Truss Structure Variables 1 2 3 4 5 6 7 8

A1 A2~A5 A6~A9 A10~A11 A12~A13 A14~A17 A18~A21 A22~A25

Compressive stress limitations (ksi) 35.092 11.590 17.307 35.092 35.092 6.759 6.959 11.802

Tensile stress limitations (ksi) 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

Fig. 7. 25-bar Spatial Truss Structure Table 5. Load Cases for the 25-bar Spatial Truss Structure Node 1 2 3 6

Px 0.0 0.0 0.0 0.0

Case 1 (Kips) Py Pz 20.0 -5.0 -20.0 -5.0 0.0 0.0 0.0 0.0

Px 1.0 0.0 0.5 0.5

Case 2 (Kips) Py 10.0 10.0 0.0 0.0

Pz -5.0 -5.0 0.0 0.0

Fig. 8. Comparison of the Convergence Rates of the Two Algorithms for the 25-bar Spatial truss Structure

used for testing the effectiveness of various techniques (e.g. Lee and Geem (2004), Li et al. (2007), Camp (2007), Kaveh and Talatahari (2009b), Farshi and Alinia-ziazi (2010), Degertekin (2012)) has been shown. For this truss structure, modulus of elasticity of the material was 10,000 ksi and its density was 0.1 lb/ in3. Table 5 reports the two load cases examined for this example. The structure should satisfy the problem-specified constraints for

both cases. The design variables of the structure are categorized in 8 groups, and the allowable stress values for all groups are listed in Table 6. All nodes in all directions are subjected to the displacement limits of ± 0.35 in. Moreover, the minimum crosssectional area for each group of elements was 0.01 in2. Table 7 gives the results. An extra row has been added in the table for this particular problem to report the allowable stress as

Table 7. Comparison of Optimal Design for the 25-bar Spatial Truss Structure Optimal cross-sectional areas (in.2) Variables

1 A1 2 A2~A5 3 A6~A9 4 A10~A11 5 A12~A13 6 A14~A17 7 A18~A21 8 A22~A25 Weight(lb) σallow σmax (ksi) δmax (in.) Constraint violation (%)

Lee and Kaveh and Camp Geem Farshi and Talatahari (2007) (2004) Alinia-ziazi (2009b) (2010) HS HPSACO BB-BC 0.047 0.0100 0.010 0.010 2.022 1.9981 2.054 2.092 2.950 2.9828 3.008 2.964 0.010 0.0100 0.010 0.010 0.014 0.0100 0.010 0.010 0.688 0.6837 0.679 0.689 1.657 1.6750 1.611 1.601 2.663 2.6668 2.678 2.686 544.38 545.37 544.99 545.38 -6.959 -6.959 -6.959 -6.959 -6.9734 -6.9606 -7.2042 -7.2273 0.35071 0.34988 0.35004 0.34994 0.206

0.024

3.523

3.855

Li et al. (2007)

Degertekin (2012)

This study

PSO 9.863 1.798 3.654 0.100 0.100 0.596 1.659 2.612 629.08 -6.959 -6.9215 0.34923

PSOPC 0.010 1.979 3.011 0.010 0.010 0.657 1.678 2.693 545.27 -6.959 -6.9553 0.35002

HPSO 0.010 1.970 3.016 0.010 0.010 0.694 1.681 2.643 545.19 -6.959 -6.9543 0.34997

EHS 0.010 1.995 2.980 0.010 0.010 0.696 1.679 2.652 545.49 -6.959 -6.9478 0.34984

SAHS 0.010 2.074 2.961 0.010 0.010 0.691 1.617 2.674 545.12 -6.959 -7.1664 0.34998

FBSFA 0.01000 1.99248 2.98329 0.01000 0.01000 0.68417 1.6771 2.66454 545.17 -6.959 -6.9547 0.35000

FA 0.0100 1.9722 3.0074 0.0100 0.0100 0.6852 1.6836 2.6564 545.25 -6.959 -6.9405 0.34999

None

0.005

None

None

2.980

None

None

− 1112 −



well. The optimal weight of 545.25 lb and 545.17 lb were found by FA and FBSFA, respectively. Most algorithms reported in Table 7, including PSO, HPSO, EHS, FA and FBSFA, give absolutely feasible solutions. The value of objective function

found by FBSFA is competitive to the value obtained by other methods. The rates of convergence of FA and FBSFA have been compared in Fig. 8. FBSFA gives better solutions in early iterations and it rapidly converges compared to FA. 5.4 72-bar Spatial Truss For the fourth example, the 72-bar spatial truss structure shown in Fig. 9. is considered. The modulus of elasticity of the material was 10,000 ksi and material density was 0.1 lb/in3. The cross-sectional areas of members as design variables are separated into 16 groups : (1) A1-A4, (2) A5-A12, (3) A13-A16, (4) A17-A18, (5) A19-A22, (6) A23-A30, (7) A31-A34, (8) A35-A36, (9) A37-A40, (10) A41-A48, (11) A49-A52, (12) A53-A54, (13) A55-A58, (14) A59-A66, (15) A67-A70, (16) A71-A72. The maximum allowable stress in all members was equal in tension and compression and it was ± 25 ksi. Maximum allowable displacement of uppermost nodes was ± 0.25 inches in both x and y directions. Table 8 gives the two load cases for this Table 8. Load Cases for the 72-bar Spatial Truss Structure Node 17 18 19 20

Fig. 9. 72-bar Spatial Truss Structure

Px 5.0 0.0 0.0 0.0

Case 1 (Kips) Py 5.0 0.0 0.0 0.0

Pz -5.0 0.0 0.0 0.0

Px 0.0 0.0 0.0 0.0

Case 2 (Kips) Py 0.0 0.0 0.0 0.0

Pz -5.0 -5.0 -5.0 -5.0

Table 9. Comparison of Optimal Design for the 72-bar Spatial Truss Structure (Case 1) Optimal cross-sectional areas (in.2) Variables

1 A1~A4 2 A5~A12 3 A13~A16 4 A17~A18 5 A19~A22 6 A23~A30 7 A31~A34 8 A35~A36 9 A37~A40 10 A41~A48 11 A49~A52 12 A53~A54 13 A55~A58 14 A59~A66 15 A67~A70 16 A71~A72 Weight(lb) σmax (ksi) δmax (in.) Constraint violation (%)

Lee and Perez and Kaveh and Geem Behdinan Talatahari (2004) (2009) (2009b) HS PSO HBB-BC 1.7901 1.7427 1.9042 0.521 0.5185 0.5162 0.100 0.1000 0.1000 0.100 0.1000 0.1000 1.229 1.3079 1.2582 0.522 0.5193 0.5035 0.100 0.1000 0.1000 0.100 0.1000 0.1000 0.517 0.5142 0.5178 0.504 0.5464 0.5214 0.100 0.1000 0.1000 0.101 0.1095 0.1007 0.156 0.1615 0.1566 0.547 0.5092 0.5421 0.442 0.4967 0.4132 0.590 0.5619 0.5756 379.27 381.91 379.66 25.0188 24.4857 24.9984 0.25054 0.24972 0.25000 0.217

Vol. 18, No. 4 / May 2014

None

None

Camp (2007)

Li et al. (2007)

Degertekin (2012)

BB-BC 1.8577 0.5059 0.1000 0.1000 1.2476 0.5269 0.1000 0.1012 0.5209 0.5172 0.1004 0.1005 0.1565 0.5507 0.3922 0.5922 379.85 24.9659 0.24999

PSO 41.794 0.195 10.797 6.861 0.438 0.286 18.309 1.220 5.933 19.545 0.159 0.151 10.127 7.320 3.812 18.196 6818.67 10.7732 0.24971

PSOPC 1.855 0.504 0.100 0.100 1.253 0.505 0.100 0.100 0.497 0.508 0.100 0.100 0.100 0.525 0.394 0.535 369.65 34.7627 0.24993

HPSO 1.857 0.505 0.100 0.100 1.255 0.503 0.100 0.100 0.496 0.506 0.100 0.100 0.100 0.524 0.400 0.534 369.65 34.7688 0.25000

EHS 1.967 0.510 0.100 0.100 1.293 0.511 0.100 0.100 0.499 0.501 0.100 0.100 0.160 0.522 0.478 0.591 381.03 24.6838 0.24968

None

None

39.051

39.075

None

− 1113 −

SAHS 1.860 0.521 0.100 0.100 1.271 0.509 0.100 0.100 0.485 0.501 0.100 0.100 0.168 0.584 0.433 0.520 380.62 23.5951 0.25065 0.259

This study FBSFA 1.90107 0.49688 0.10003 0.10003 1.26497 0.50802 0.10003 0.10003 0.57209 0.52720 0.10003 0.10237 0.15632 0.53325 0.43573 0.56554 379.93 24.9667 0.25000 None

FA 1.8898 0.5062 0.1023 0.1000 1.2781 0.5258 0.1000 0.1000 0.5506 0.5135 0.1000 0.1145 0.1573 0.5238 0.4576 0.5337 380.50 24.8539 0.25001 0.005


Table 10. Comparison of Optimal Design for the 72-bar Spatial Truss Structure (Case 2) Optimal cross-sectional areas (in.2) Variables

1 A1~A4 2 A5~A12 3 A13~A16 4 A17~A18 5 A19~A22 6 A23~A30 7 A31~A34 8 A35~A36 9 A37~A40 10 A41~A48 11 A49~A52 12 A53~A54 13 A55~A58 14 A59~A66 15 A67~A70 16 A71~A72 Weight(lb) σmax (ksi) δmax (in.) Constraint violation (%)

Lee and Lamberti Sarma Geem (2009) (2000) (2004) HS CMLPSA Simple GA Fuzzy GA 1.963 1.8866 2.141 1.732 0.481 0.5169 0.510 0.522 0.010 0.0100 0.054 0.010 0.011 0.0100 0.010 0.013 1.233 1.2903 1.489 1.345 0.506 0.5170 0.551 0.551 0.011 0.0100 0.057 0.010 0.012 0.0100 0.013 0.013 0.538 0.5207 0.565 0.492 0.533 0.5180 0.527 0.545 0.010 0.0100 0.010 0.066 0.167 0.1141 0.066 0.013 0.161 0.1665 0.174 0.178 0.542 0.5363 0.425 0.524 0.478 0.4460 0.437 0.396 0.551 0.5761 0.641 0.595 364.33 363.818 372.40 364.40 25.0128 24.9964 24.8372 24.0237 0.25023 0.25001 0.25000 0.25231 0.092

0.002

None

0.925

Li et al. (2007) PSO 40.053 0.237 21.692 0.657 22.144 0.266 1.654 10.284 0.559 12.883 0.138 0.188 29.048 0.632 3.045 1.711 5417.02 6.5302 0.22206

PSOPC 1.652 0.547 0.100 0.101 1.102 0.589 0.011 0.010 0.581 0.458 0.010 0.152 0.161 0.555 0.514 0.648 368.45 24.8744 0.24984

None

None

Degertekin (2012) HPSO EHS 1.907 1.889 0.524 0.502 0.010 0.010 0.010 0.010 1.288 1.284 0.523 0.526 0.010 0.010 0.010 0.010 0.544 0.528 0.528 0.525 0.019 0.010 0.020 0.063 0.176 0.173 0.535 0.550 0.426 0.444 0.612 0.592 364.86 364.36 25.0204 24.87265 0.24995 0.24997 0.082

None

This study

SAHS 1.889 0.520 0.010 0.010 1.289 0.524 0.010 0.010 0.539 0.519 0.015 0.105 0.167 0.532 0.425 0.579 364.05 24.9526 0.25000

FBSFA 1.93369 0.51918 0.01025 0.01001 1.36131 0.51022 0.01001 0.01001 0.52830 0.51135 0.01001 0.10299 0.16813 0.52129 0.42156 0.62427 364.17 24.9801 0.25000

None

None

FA 1.8539 0.5123 0.0100 0.0100 1.2889 0.5406 0.0100 0.0100 0.5151 0.5183 0.0100 0.1012 0.1686 0.5198 0.4131 0.6577 363.98 24.9506 0.25029 0.118

Fig. 10. Comparison of the Convergence Rates of the Two Algorithms for the 72-bar Spatial Truss Structure (Case 1)

Fig. 11.Comparison of the Convergence Rates of the Two Algorithms for the 72-bar spatial Truss Structure (Case 2)

example. This problem was considered for two different cases: Case 1 in which minimum cross-sectional area of each member was 0.1 in2, and Case 2 in which this value was 0.01 in2. Tables 9 and 10 compare the results found by FA and FBSFA and those of other studies for Case 1 and Case 2, respectively. As the tables indicate, the standard PSO was not capable of finding a reasonable solution. FBSFA gives absolute feasible solutions for both cases yet again. Among the methods that have generated absolute feasible solutions, FBSFA has produced competitive results in both cases. Taking the rate of convergence into account, the proposed FBSFA technique dramatically outperforms FA as it is obvious

in Figs. 10 and 11 for Case 1 and Case 2, respectively. 5.5 120-bar Spatial Truss For the final design example, the 120-bar dome truss depicted in Fig. 12. is optimized via the proposed technique. This truss structure has some specific design criteria and it is suitable for testing the robustness of optimization techniques in solving more complicated problems. Owing to complexity and nonlinearity of the design, the truss has been optimized by few researches such as Lee and Geem (2004), Kaveh and Talatahari (2009b). Design criteria of this truss structure is defined as follows: Allowable tensile and compressive stresses are considered

− 1114 −



where Fy is the yield stress of steel; E is Young’s modulus of elasticity of steel; λi is slenderness ratio ( λi = kLi ⁄ r i ) ; k is the effective length factor, Li is the length of each member i; ri is the radius of gyration of member i; and Cc is defined as: 2

2π E Cc = -----------Fy

Fig. 12. 120-bar Spatial Truss Structure

according to American Institute of Steel Construction (AISC) (1998) code as follows: ⎧ σup = 0.6Fy for σi > 0 ⎨ b for σi < 0 ⎩ σi 2 ⎧ λi -2 Fy ⎪ 1 – -------⎪ 2C c b σi = ⎨ ⎪ 12π 2 E - for ⎪ -------------2 ⎩ 23λi

(13) 3

λi ⎞ 5 3λ - for λi < Cc⎞ ⁄ ⎛ ⎛ --- + --------i + -------⎝ ⎝ 3 8Cc 8C 3⎠ ⎠ c λi ≥ C c

(14)

(15)

For this structure, the Young’s modulus of elasticity was 30,450 ksi; the material density was 0.288 lb/in3; and Fy was considered as 58.0 ksi. The radius of gyration was expressed in b terms of cross-sectional areas as ri = aAi (Saka, 1990). Here, a and b are the constants depending on the types of sections adopted for the members such as pipes, angles, and tees. For pipe sections considered in this study a = 0.4993 and b = 0.6777 were adopted. Design variables, i.e. cross sectional areas, were categorized in seven groups. The minimum cross-sectional area was 0.775 in2. The loading condition on the spatial truss was considered as vertically downward loads as -13.49 kips at node 1, -6.744 kips at node 2 through 14; and -2.248 kips at the rest of the nodes. Two cases are considered in this study: Case 1 in which no nodal displacement constraints were considered; and Case 2 in which a value of ± 0.1969 inches was considered for displacement limitations of all nodes in any direction. Tables 11 and 12 give the results obtained by Lee and Greem (2004), and Kaveh and Talatahari (2009b) for Case 1 and Case 2, respectively. The allowable stress and the critical cross sectional area in which the critical stress is occurred have been reported as well. The only method that has generated absolute feasible solutions in both cases is FBSFA. In most cases, the optimal solutions found by PSO, PSOPC and HPSACO highly violate the constraints. Figure 13 and 14 shows the usefulness of the proposed FBSFA in expediting the rate of convergence of FA. It is worthy of remark that in addition to FA, most of the other optimization approaches, including standard and modified ones, usually need

Table 11. Comparison of Optimal Design for the 120-bar Spatial Truss Structure (Case 1) Variables

1 2 3 4 5 6 7

A1 A2 A3 A4 A5 A6 A7 Weight(lb) Acritical σmax (ksi) σallow Constraint violation (%) Vol. 18, No. 4 / May 2014

Lee and Geem (2004) HS 3.295 2.396 3.874 2.571 1.150 3.331 2.784 19707.07 1.150 -2.9382 -2.9121 0.896

PSO 3.147 6.376 5.957 4.806 0.775 13.798 2.452 32432.9 3.147 -2.7233 -1.7056 16.303

Optimal cross-sectional areas (in.2) Kaveh and Talatahari (2009) PSOPC HPSACO 3.235 3.311 3.370 3.438 4.116 4.147 2.784 2.831 0.777 0.775 3.343 3.474 2.454 2.551 19618.7 19491.3 3.343 4.147 -7.1489 -4.8091 -6.3174 -4.4443 13.162 8.208 − 1115 −

This study FBSFA 3.297431 2.400493 3.873526 2.572376 1.155445 3.333398 2.78748 19721.08 3.29743 -2.5661 -2.5661 None

FA 3.298 2.399 3.873 2.572 1.157 3.338 2.789 19754.43 2.786 -2.7611 -2.7602 0.034


Table 12. Comparison of Optimal Design for the 120-bar Spatial Truss Structure (Case 2) Variables

1 2 3 4 5 6 7

A1 A2 A3 A4 A5 A6 A7 Weight(lb) Acritical σmax (ksi) σallow δmax (in.) Constraint violation (%)

Lee and Geem (2004) HS 3.296 2.789 3.872 2.570 1.149 3.331 2.781 19893.34 1.149 -2.9343 -2.9086 0.19686 0.881

PSO 15.987 9.599 7.467 2.790 4.324 3.294 2.479 41052.7 3.294 -7.0948 -6.1922 0.0752 0.1654

Optimal cross-sectional areas (in.2) Kaveh and Talatahari (2009) PSOPC HPSACO 3.083 3.779 3.639 3.377 4.095 4.125 2.765 2.734 1.776 1.609 3.779 3.533 2.438 2.539 20681.7 20078.0 3.0830 2.5390 -2.6979 -4.7018 -2.3430 -4.4124 0.1820 0.1922 15.150 6.558

This study FBSFA 3.298448 2.781744 3.873252 2.572269 1.158857 3.334045 2.787371 1.9911.26 2.572269 -2.25923 -2.25923 0.19684 None

FA 3.310 2.765 3.891 2.571 1.179 3.339 2.790 19952.01 2.571 -2.2575 -2.2577 0.19672 None

6. Conclusions

Fig. 13. Comparison of the Convergence Rates between the Two Algorithms for the 120-bar Spatial Truss Structure (Case 1)

The effectiveness of the recently-developed firefly algorithm in weight minimization of truss structures was first investigated in this paper. In all cases, FA in conjunction with penalty function approach for handling constraints was capable of finding a solution which was either absolutely feasible or slightly infeasible. The method may outperform the standard PSO algorithm, since in some problems PSO is incapable of finding an optimal solution. Some modifications on FA were then imposed to enhance the rate of convergence of the algorithm. In the next step, a new method to limit the search space on the feasible boundary and automatically handle the problem-specified constraints as well as generating absolutely feasible solutions was proposed. The technique was referred to as Feasible Boundary Search (FBS). Another advantage of the proposed method is its generality that enables the technique to be combined with any evolutionary method. Hybridization of FBS with the Enhanced firefly algorithm (FBSFA) was developed in this paper. The experiments revealed that for all design examples in the range of simple to severe design criteria, the proposed FBSFA generates high quality solutions that absolutely satisfy the constraints. Test examples also show that the hybridized FBSFA technique considerably outperforms the other techniques in terms of rate of convergence.

References Fig. 14. Comparison of the Convergence Rates between the Two Algorithms for the 120-bar Spatial Truss Structure (Case 2)

fairly large number of iterations to converge (e.g., Lee and Geem (2004), Li et al. (2007), Kaveh and Talatahari (2009b), and Degertekin (2012). This emphasizes the importance of the proposed FBSFA in fast optimization of truss structures as well.

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