An improved magnetic charged system search for optimization of truss structures with continuous and discrete variables

Applied Soft Computing 28 (2015) 400–410

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

An improved magnetic charged system search for optimization of truss structures with continuous and discrete variables A. Kaveh a,∗ , B. Mirzaei b , A. Jafarvand b a b

Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran 16, Iran Department of Civil Engineering, University of Zanjan, Zanjan, Iran

a r t i c l e

i n f o

Article history: Received 2 June 2013 Received in revised form 2 November 2014 Accepted 2 November 2014 Available online 9 December 2014 Keywords: Magnetic charged system search Harmony search Truss structures Size optimization Discrete variables

a b s t r a c t In this study, an improved magnetic charged system search (IMCSS) is presented for optimization of truss structures. The algorithm is based on magnetic charged system search (MCSS) and improved scheme of harmony search algorithm (IHS). In IMCSS some of the most effective parameters in the convergence rate of the HS scheme have been improved to achieve a better convergence, especially in the final iterations and explore better results than previous studies. The IMCSS algorithm is applied for optimal design problem with both continuous and discrete variables. In comparison to the results of the previous studies, the efficiency and robustness of the proposed algorithm in fast convergence and achieving the optimal values for weight of structures, is demonstrated. © 2014 Elsevier B.V. All rights reserved.

1. Introduction In the last three decades, many meta-heuristic algorithms have been proposed and applied for optimization of structures. Every meta-heuristic method consists of a group of search agents that explore the feasible region based on both randomization and some specified rules. The rules are usually inspired by natural phenomena laws. Genetic algorithms (GA) proposed by Holland [1] and Goldberg [2] are inspired by Darwin’s theory about biological evolutions. Niche Hybrid Parallel Genetic Algorithm (NHPGA) proposed by Wei et al. [3], Generative Algorithms (GA) by Allison et al. [4], Particle swarm optimization (PSO) proposed by Eberhart and Kennedy [5] simulates the social behaviour, and it is inspired by the movement of organisms in a bird flock or fish school. Ant colony optimization (ACO) formulated by Dorigo et al. [6] imitates foraging behaviour of ant colonies. Many other physical-inspired algorithms such as simulated annealing (SA) proposed by Kirkpatrick et al. [7], Harmony Search (HS) presented by Geem et al. [8], Gravitational Search Algorithm (GSA) proposed by Rashedi et al. [9], Big Bang–Big Crunch algorithm (BB–BC) proposed by Erol and Eksin [10] which improved by Kaveh and Talatahari [11], Bat-Inspired Algorithm proposed by Yang [12], Ray Optimization (RO) developed by Kaveh and Khayatazad [13], Krill Herd (KH) presented by

∗ Corresponding author. Tel.: +98 21 44202710. E-mail address: [email protected] (A. Kaveh). http://dx.doi.org/10.1016/j.asoc.2014.11.056 1568-4946/© 2014 Elsevier B.V. All rights reserved.

Gandomi and Alavi [14], Dolphin Echolocation method by Kaveh and Farhoudi [15], Colliding Bodies Optimization (CBO) by Kaveh and Mahdavi [16], and Interior search algorithm (ISA) by Gandomi [17], in recent years. A new meta-heuristic algorithm has been proposed recently, by Kaveh and Talatahari which is called Charged System Search (CSS) [18]. The CSS algorithm is based on the Coulomb and Gauss laws from physics and the governing laws of motion from the Newtonian mechanics. This algorithm can be considered as a multi-agent approach, where each agent is a Charged Particle (CP). Each CP is considered as a charged sphere with a specified radius, having a uniform volume charge density which can insert an electric force to the other CPs. After a while the CSS algorithm was modified to Magnetic Charged System Search (MCSS) by Kaveh et al. [19]. This algorithm utilizes the governing laws for magnetic forces and includes magnetic forces in addition to electrical forces, so the movements of CPs due to the total force (Lorentz force) are determined using Newtonian mechanical laws. In this paper, an improved magnetic charged system search (IMCSS) is proposed for optimization of some truss structures as the well-known benchmark problems, in order to comparison the efficiency of the IMCSS algorithm with recently presented meta-heuristic algorithms. In this algorithm, an improved harmony search scheme (HIS) is utilized and some of the most effective parameters in the convergence rate of the algorithm are improved.

A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410

In a recently presented research by the authors, optimal design of double layer barrel vaults has been proposed, in which a 384-bar barrel vault and a 693-bar braced barrel vault have been optimized via the IMCSS algorithm [20]. The authors have also introduced an optimization approach (IMCSS-OAPI) for the problem of simultaneous shape-size optimization of large-scale barrel vault frames, which deals with the interface between the IMCSS as the optimization algorithm and SAP2000 as the structural analysis software through the open application programming interface (OAPI) [21]. In this work a 173-bar and a 292-bar single-layer barrel vault frame have been optimized. The present paper is organized as follows: in Section 2, the statement of the optimization problem is expressed and formulated. CSS and MCSS algorithm are reviewed in Section 3. In Section 4, the improved form of MCSS algorithm is introduced and also its discrete version is described. Section 5 contains several illustrative examples with continuous and discrete variables to determine whether the efficiency of the new algorithm could be enough, and finally in Section 6, some concluding remarks are derived. 2. Statement of the optimization problem The principal objective of size optimization process is aimed at achieving the optimum values for member cross-sectional areas of structure Ai in order to minimize the structural weight W and simultaneously satisfying the constraints that the optimization problem imposes. Hence, the problem of size optimization of truss structures can be expressed as:

to minimize Mer(X) = fpenalty (X) × W (X) subject to

min < i < max

ımin < ıi < ımax

i = 1, 2, . . ., nm

(1)

i = 1, 2, . . ., nn

where X is the vector containing the design variables; for a discrete optimum design problem, the variables xi are selected from an allowable set of discrete values; n is the number of member groups; Mer(X) is the merit function; W(X) is the cost function, which is taken as the weight of the structure; fpenalty (X) is the penalty function which results from the violations of the constraints corresponding to the response of the structure; nm is the number of members forming the structure; nn is the number of nodes;  i and ıi are the stress and nodal displacements, respectively; min and max mean the lower and upper bounds of constraints, respectively. The cost function can be expressed as: W (X) =

nm 

i · Ai · Li

(2)

i=1

where i , Li , and Ai are the material density, length and the crosssectional area of member i, respectively. The penalty function can be defined as:



fpenalty (X) =

1 + ε1 ·

np 

ε2

k k ((i) + ı(i) )

,

(3)

i=1

where np is the number of multiple loadings. In this paper ε1 is taken as unity and ε2 is set to 1.5 in the first iterations of the search process, but gradually it is increased to 3 [22]. k is the summation of stress penalties and ık is the summation of nodal displacement penalties for kth charged particle which mathematically expressed as:  =

nm  i=1

max

     i  ¯  − 1, 0 , i

i=1

   ıi    max   − 1, 0 , ı¯

(5)

i

where  i , ¯ i are the stress and allowable stress in member i, respectively, and ıi , ı¯ i are the displacement of the joints and the allowable displacement, respectively. 3. Introduction to CSS and MCSS The CSS algorithm has been proposed by Kaveh and Talathari [18] for optimization of structures. This meta-heuristic optimization algorithm takes its inspiration from the physic laws governing a group of charge particles (CP). These CPs are sources of the electric fields, and each CP can exert electric force on other CPs. The movement of each CP due to the electric force can be determined using the Newtonian mechanic laws. In physics, it has been shown that when a charged particle moves, produces a magnetic field. This magnetic field can exert a magnetic force on other CPs. Thus, for considering this force in addition to electric force, the CSS algorithm is modified to MCSS algorithm by Kaveh et al. The MCSS algorithm can be summarized as follows [19]: Level 1. Initialization Step 1: Initialization. Initialize CSS algorithm parameters; the initial positions of CPs are determined randomly in the search space (0)

xi,j = xi,min + rand · (xi. max − xi,min ),

i = 1, 2, . . ., n.

(6)

(0)

X = [x1 , x2 , x3 , . . ., xn ]

Find

ı =

nn 

401

where xi,j determines the initial value of the ith variable for the jth CP; xi,min and xi,max are the minimum and the maximum allowable values for the ith variable; rand is a random number in the interval [0,1]; and n is the number of variables. The initial velocities of charged particles are set to zero

v(0) = 0, i,j

i = 1, 2, ..., n.

(7)

The magnitude of the charge is defined as follows: qi =

fit(i) − fitworst , fitbest − fitworst

i = 1, 2, . . ., N.

(8)

where fitbest and fitworst are the best and the worst fitness of all particles, respectively; fit(i) represents the fitness of the agent i; and N is the total number of CPs. The separation distance rij between two charged particles is calculated as: rij =

||Xi − Xj || ||(Xi + Xj )/2 − Xbest || + ε

,

(9)

where Xi and Xj are the positions of the ith and jth CPs, Xbest is the position of the best current CP, and ε is a small positive number to avoid singularities. Step 2. CP ranking. Evaluate the values of the fitness function for the CPs, compare with each other and sort them in an increasing order. Step 3. CM creation. Store CMS number of the first CPs and their related values of the objective function in the CM (based on CMS size). Level 2: Search Step 1: Force determination. The probability of the attraction of the ith CP by the jth CP is expressed as:

⎧ ⎨ 1 fit(i) − fitbest > rand or fit(j) > fit(i), fit(j) − fit(i) pij = ⎩

(10)

0 else.

(4)

where rand is a random number uniformly distributed in the range of (0,1). The resultant electrical force FE,j acting on the jth CP can be

402

A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410

calculated as follow: FE,j = qj ·







q qi rij · i1 + 2i i2 a3 rij

i,i = / j

· pij (Xi − Xj ),

⎧ i = 1, i2 = 0 ⇔ rij < a, ⎪ ⎨1

×

⎪ ⎩

i1 = 0, i2 = 1 ⇔ rij ≥a,

(11)

pmij =

j = 1, 2, . . ., N.

1 fit(j) > fit(i),

(12)

0 else.

where fit(i) and fit(j) are the fitness values of the ith and jth CP, respectively. This probability determines that only a good CP can affect a bad CP by the magnetic force. The magnetic force FB,ji exerted on the jth CP due to the magnetic field produced by the ith virtual wire (ith CP) can be expressed as: FB,j = qj ·

 I i R

i,i = / j

×

r · z1 + 2 ij

Ii · z2 rij



· pmij (Xi − Xj ),

⎧ z = 1, z2 = 0 ⇔ rij < R, ⎪ ⎨ 1 ⎪ ⎩

z1 = 0, z2 = 1 ⇔ rij ≥R,

(13)

where qi is the charge of the ith CP, R is the radius of the virtual wires, Ii is the average electric current in each wire, and pmij is the probability of the magnetic influence (attracting or repelling) of the ith wire (CP) on the jth CP. The average electric current in each wire Ii can be expressed as:

  dfi,k  − dfmin,k

dfmax,k − dfmin,k

,

dfi,k = fitk (i) − fitk−1 (i),

(14) (15)

where dfi,k is the variation of the objective function of the ith CP in the kth movement (iteration). Here, fitk (i) and fitk−1 (i) are the values of the objective function of the ith CP at the start of the kth and k − 1th iterations, respectively. dfmax,k and dfmin,k will be the maximum and minimum values among these absolute values of df, considering that absolute values of dfi,k for all of the current CPs. A modification can be considered to avoid trapping in part of search space (Local optima) because of attractive electrical force in CSS algorithm F = pr × FE + FB ,

(16)

where pr is the probability that an electrical force is a repelling force which is defined as pr =

⎧   ⎨ 1 rand > 0.1 · 1 − iter , ⎩

itermax

(17)

where rand is a random number uniformly distributed in the range of (0,1), iter is the current number of iterations, and itermax is the maximum number of iterations. Step 2: Finding new solution. Move each CP to the new position and calculate the new velocity as follows:

Vj,new =

Fj

· t 2 + randj2 · kv · Vj,old · t + Xj,old ,

mj

Xj,new − Xj,old t





,

kv = c2 · 1 −

,

iter itermax



,

(20)

where c1 and c2 are two constants to control the exploitation and exploration of the algorithm, respectively. Step 3. Modification of CP position. If each CP violates from its allowable boundary, its position is corrected using harmony searchbased approach. In this paper the process of position correction has been improved via IMCSS algorithm which expressed in Section 4. Step 4: CP ranking. Evaluate and compare the values of the fitness function for the new CPs, and sort them in an increasing order. Step 5: CM updating. If some new CP vectors are better than the worst ones in the CM (means CPs with better merit function), include the better vectors in the CM and exclude the worst ones from the CM. Level 3: Controlling the terminating criterion. Repeat the search level steps until a terminating criterion is satisfied. The terminating criterion is considered to be the number of iterations.

In the process of position correction of CPs using harmony search-based approach (Step 3), The CMCR and PAR parameters help the algorithm to find globally and locally improved solutions, respectively. PAR and bw in HS scheme are very important parameters in fine-tuning of optimized solution vectors, and can be potentially useful in adjusting convergence rate of the algorithm to optimal solution [23]. The traditional HS scheme which is utilized in MCSS algorithm, uses fixed value for both PAR and bw. Small PAR values with large bw values can led to poor performance of the algorithm and considerable increase in iterations needed to find optimum solution. Although small bw values in final iterations increase the fine-tuning of solution vectors, but in the first iterations bw must take a bigger value to enforce the algorithm to increase the diversity of solution vectors. Furthermore large PAR values with small bw values usually led to the improvement of best solutions in final iterations which algorithm converged to optimal solution vector. To improve the performance of the HS scheme and eliminate the drawbacks lies with fixed values of PAR and bw, IMCSS algorithm uses IHS scheme with varied PAR and bw in correction step (Step 3, Level 2). PAR and bw change dynamically with iteration number as shown in Fig. 1 and expressed as follow [23]: PAR(iter) = PARmin +

(PARmax − PARmin ) · iter itermax

(21)

and bw(iter) = bwmax exp(c · iter),

−1 else.

Xj,new = randj1 · ka ·

iter itermax

4. Improved magnetic charged system search

j = 1, 2, . . ., N.

(Iavg )ik = sign(dfi,k ) ×



ka = c1 · 1 +

The probability of the magnetic influence (attracting or repelling) of the ith wire (CP) on the jth CP is expressed as:



where randj1 and randj2 are two random numbers uniformly distributed in the range of (0,1). Here, mj is the mass of the jth CP which is equal to qj . t is the time step and is set to unity (It could be utilize a continuous concept of time in every iteration as another modification for MCSS algorithm). ka is the acceleration coefficient; kv is the velocity coefficient to control the influence of the previous velocity. In this paper ka and kv are considered as:

c=

ln(bwmin /bwmax ) , itermax

(22) (23)

where bw(iter) is the bandwidth for each iteration, bwmin and bwmax are the minimum and maximum bandwidth, respectively. In this paper PARmin and PARmax are set to 0.3 and 0.99, respectively.

(18)

4.1. A discrete IMCSS

(19)

IMCSS algorithm is also applied to optimal design problem with discrete variables. One way to solve discrete problems using a

A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410

403

Table 1 The allowable steel pipe sections taken from AISC code [25].

bw

bw max

bw min First Iter

Iterati on

Iter max

(a)

PAR

PAR max

PAR min

First Iter

Iter max

No.

Area (in2 )

Area (mm2 )

No.

Area (in2 )

Area (mm2 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0.111 0.141 0.196 0.25 0.307 0.391 0.442 0.563 0.602 0.766 0.785 0.994 1 1.228 1.266 1.457 1.563 1.62 1.8 1.99 2.13 2.38 2.62 2.63 2.88 2.93 3.09 1.13 3.38 3.47 3.55 3.63

71.613 90.968 126.451 161.29 198.064 252.258 285.161 363.225 388.386 494.193 506.451 641.289 645.16 792.256 816.773 939.998 1008.385 1045.159 1161.288 1283.868 1374.191 1535.481 1690.319 1696.771 1858.061 1890.319 1993.544 729.031 2180.641 2238.705 2290.318 2341.931

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

3.84 3.87 3.88 4.18 4.22 4.49 4.59 4.8 4.97 5.12 5.74 7.22 7.97 8.53 9.3 10.85 11.5 13.5 13.9 14.2 15.5 16 16.9 18.8 19.9 22 22.9 24.5 26.5 28 30 33.5

2477.414 2496.769 2503.221 2696.769 2722.575 2896.768 2961.284 3096.768 3206.445 3303.219 3703.218 4658.055 5141.925 5503.215 5999.988 6999.986 7419.43 8709.66 8967.724 9161.272 9999.98 10,322.56 10,903.2 12,129.01 12,838.68 14,193.52 14,774.16 15,806.42 17,096.74 18,064.48 19,354.8 21,612.86

There are 10 design variables in this example and a set of pseudo variables ranging from 0.1 to 35.0 in2 (0.6452 cm2 to 225.806 cm2 ). In this problem two cases are considered:

Iterati on

(b) Fig. 1. Variation of (a) bw and (b) PAR versus iteration number [20].

continuous algorithm is to utilize a rounding function which changes the magnitude of a result to the nearest discrete value [24], as follow



Xj,new = Fix randj1 · ka ·

Fj mj



· t 2 + randj2 · kv · Vj,old · t + Xj,old , (24)

where Fix(X) is a function which rounds each elements of vector X to the nearest permissible discrete value. Using this position updating formula, the agents will be permitted to select discrete values.

Case 1: P > P1 = 100 kips (444.8 kN) and P2 = 0, Case 2: P1 = 150 kips (667.2 kN) and P2 = 50 kips (222.4 kN). Fig. 2 shows the geometry and support conditions for this two dimensional, cantilevered truss with loading condition. The material density is 0.1 lb/in3 (2767.990 kg/m3 ) and the modulus of elasticity is 10,000 ksi (68,950 MPa). The members are subjected to the stress limits of ±25 ksi (172.375 MPa) and all nodes in both vertical and horizontal directions are subjected to the displacement limits of ±2.0 in (5.08 cm).

5. Numerical examples In this section, common truss optimization examples as benchmark problems are used for optimization using the proposed algorithm. This algorithm is applied to problem with both continuous and discrete variables. The final results are compared to the solutions of other methods to demonstrate the efficiency of the present approach. The discrete variables are selected from AISC code shown in Table 1 [25]. In the proposed algorithm, for all of examples a population of 25 CPs is used and the value of CMCR is set to 0.95. 5.1. A 10-bar planar truss structure The 10-bar truss structure is a common problem in the field of structural optimization which is used a lot, to verify the efficiency of a novel proposed optimization algorithm.

Fig. 2. A 10-bar planar truss structure.

404

A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410 6500

6500

MCSS IMCSS

MCSS IMCSS 6000

Weight(lb)

Weight(lb)

6000

5500

5500 5000

5000

50

100

150

200

250

300

4500

350

50

100

150

200

250

300

Iteration

Iteration

(b)

(a)

Fig. 3. Convergence history for the 10-bar planar truss structure using MCSS, IMCSS (a) Case 1 and (b) Case 2.

Table 2 Optimal design comparison for the 10-bar planner truss (Case 1). Element group

1 2 3 4 5 6 7 8 9 10

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

Weight (lb) Displacement constraint No. of analyses

Camp et al. [26]

Lee and Geem [27]

Li et al. [29]

Kaveh and Talatahari [30]

Present work

GA

HS

PSO

PSOPC

HPSO

HPSACO

MCSS

IMCSS

28.92 0.1 24.07 13.96 0.1 0.56 7.69 21.95 22.09 0.1

30.15 0.102 22.71 15.27 0.102 0.544 7.541 21.56 21.45 0.1

33.469 0.11 23.177 15.475 3.649 0.116 8.328 23.34 23.014 0.19

30.569 0.1 22.974 15.148 0.1 0.547 7.493 21.159 21.556 0.1

30.704 0.1 23.167 15.183 0.1 0.551 7.46 20.978 21.508 0.1

30.307 0.1 23.434 15.505 0.1 0.5241 7.4365 21.079 21.229 0.1

29.5766 0.1142 23.8061 15.8875 0.1137 0.1003 8.6049 21.6823 20.3033 0.1117

30.0258 0.1 23.6277 15.9734 0.1 0.5167 7.4567 21.4374 20.7443 0.1

5076.31 – N/A

5057.88 – 20,000

5529.5 – 150,000

5061 – 150,000

5060.92 5.53E−07 N/A

5056.56 9.92E−04 10,650

5086.9 1.49E−05 8875

5064.6 5.85E−08 8475

For MCSS and IMCSS algorithms a comparison of the convergence history of both cases is given in Fig. 3. Tables 2 and 3 are also provided for comparison with the results of previous studies. As seen in the results, for both cases the PSO and PSOPC algorithms obtain the best solutions after 3000 iterations (150,000 analyses), the HS algorithm after 20,000 analyses. The HPSACO algorithm finds its best solution after 10,650 and 9925 analyses, for Case 1 and Case 2, respectively. The MCSS and IMCSS algorithms achieve its best solutions after 355 iterations (8875 analyses) and 339 iterations (8475 analyses), respectively. For Case 1, the best weights of GA, HS, PSO, PSOPC, HPSO, HPSACO and MCSS algorithms are 5076.31, 5057.88, 5529.88, 5061, 5060.92, 5056.56 and 5086.9 lb, respectively, while for IMCSS

is 5064.6 lb. For Case 2, the best weight of IMCSS is 4679.15 lb, but for MCSS, HPSACO, HPSO, PSOPC, PSO and HS is 4686.47, 4675.78, 4677.7, 4679.47 and 4668.81 lb. As seen in Tables 2 and 3, however the best weights obtained from IMCSS in both cases are a little bigger than the HPSACO, but IMCSS has no penalty values rather than HPSACO, and therefore has a lower merit function than HPSACO. 5.2. A 52-bar planar truss The 52-bar planar truss structure shown in Fig. 4 has been analyzed by Wu and Chow [28], Lee and Geem [27] and Li et al. [29]. The members of this structure are divided into 12 groups:

Table 3 Optimal design comparison for the 10-bar planner truss (Case 2). Element group

1 2 3 4 5 6 7 8 9 10 Weight (lb) Displacement constraint Stress constraint No. of analyses

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

Lee and Geem [27]

Li et al. [29]

Kaveh and Talatahari [30]

Present work

HS

PSO

PSOPC

HPSO

HPSACO

MCSS

IMCSS

23.25 0.102 25.73 14.51 0.1 1.977 12.21 12.61 20.36 0.1

22.935 0.113 25.355 14.373 0.1 1.99 12.346 12.923 20.678 0.1

23.473 0.101 25.287 14.413 0.1 1.969 12.362 12.694 20.323 0.103

23.353 0.1 25.502 14.25 0.1 1.972 12.363 12.894 20.356 0.101

23.194 0.1 24.585 14.221 0.1 1.969 12.489 12.925 20.952 0.101

22.863 0.120 25.719 15.312 0.101 1.968 12.310 12.934 19.906 0.100

23.299 0.1 25.682 14.510 0.1 1.969 12.149 12.360 20.869 0.1

4668.81 – – N/A

4679.47 – – 150,000

4677.7 – – 150,000

4677.29 0 2.49E−05 N/A

4675.78 7.92E−04 7.97E−05 9625

4686.47 0 0 7350

4679.15 0 0 6625

A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410

405

7000

MCSS IMCSS

6500 6000

Weight(lb)

5500 5000 4500 4000 3500 3000 2500 2000 1500

20

40

60

80

100

120

140

160

180

Iteration Fig. 5. Convergence history for the 52-bar planar truss structure using MCSS, IMCSS.

concluded that the IMCSS algorithms achieve good optimal results than previous methods like MCSS, PSO, PSOPC, HPSO and DHPSACO algorithms. 5.3. A 72-bar spatial truss In the 72-bar spatial truss structure which is shown in Fig. 6, the material density is 0.1 lb/in3 (2767.990 kg/m3 ) and the modulus of elasticity is 10,000 ksi (68,950 MPa). The nodes are subjected to the displacement limits of ±0.25 in (±0.635 cm) and the members are subjected to the stress limits of ±25 ksi (±172.375 MPa). The 72 structural members of this spatial truss are categorized into 16 groups using symmetry: 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, and (16) A71–A72. Two optimization cases are considered as follows:

A1–A4, (2) A5–A10, (3) A11–A13, (4) A14–A17, (5) A18–A23, (6) A24–A26, (7) A27–A30, (8) A31–A36, (9) A37–A39, (10) A40–A43, (11) A44–A49, (12) A50–A52.

Case 1: The discrete variables are selected from the set D = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2} (in2 ) or {0.65, 1.29, 1.94, 2.58, 3.23, 3.87, 4.52, 5.16, 5.81, 6.45, 7.10, 7.74, 8.39, 9.03, 9.68, 10.32, 10.97, 12.26, 12.90, 13.55, 14.19, 14.84, 15.48, 16.13, 16.77, 17.42, 18.06, 18.71, 19.36, 20.00, 20.65} (cm2 ). Case 2: The discrete variables are selected from AISC code in Table 1. The values and directions of the two load cases applied to the 72-bar spatial truss are listed in Table 5.

The material density is 7860.0 kg/m3 and the modulus of elasticity is 2.07 × 105 MPa. The members are subjected to stress limitations of ±180 MPa. As seen in Fig. 4, loads of Px = 100 kN and Py = 200 kN, are applied to the structure. Table 4 and Fig. 5 are provided for the comparison of the optimal design results with the previous studies and the convergence rates of 52-bar planar truss structure for MCSS and IMCSS algorithms As it can be seen in the Table 4 and Fig. 5, the best weight of MCSS and IMCSS algorithms are 1904.05 and 1902.61 lb, respectively, while for DHPSACO which has the best results in previous studies, is 1904.83 lb. The MCSS and IMCSS algorithms find the best solutions after 4225 and 4075 analyses respectively however the DHPSACO reach its best solution in 5300 analyses. From Table 4, it can be

Tables 6 and 7 are provided for comparison the results of IMCSS with the results of the previous studies for both cases. The Convergence history for MCSS, IMCSS algorithms is shown in Fig. 7. In Case 1, the best weight of the IMCSS and DHPSACO algorithm is equal to 385.54 lb (174.88 kg), while it is 389.49 lb 388.94 lb, 387.94 lb, 400.66 lb for the MCSS, HPSO, HS, and GA, respectively. The PSO and PSOPC algorithms, do not get optimal results when the maximum number of iterations is reached. The IMCSS algorithm gets its best solution after 145 iterations (3625 analyses) while it takes 216 iterations (5400 analyses) and 213 iterations (5330 analyses) for MCSS and DHPSACO, respectively. In Case 2, the best weight obtained from IMCSS is 389.6 lb, but it is 393.13, 389.87, 392.84, 393.06 and 393.38 lb from MCSS, CS, ICA, CSS and HPSACO algorithms, respectively. IMCSS algorithm finds

Fig. 4. A 52-bar planar truss.

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Table 4 Optimal design comparison for the 52-bar planar truss. Element group

1 2 3 4 5 6 7 8 9 10 11 12

A1–A4 A4–A10 A11–A13 A14–A17 A18–A23 A24–A26 A27–A30 A31–A36 A37–A39 A40–A43 A44–A49 A50–A52

Weight (kg) No. of analyses

Wu and Chow [28]

Lee and Geem [27]

Li et al. [29]

Kaveh and Talatahari [31]

Present work

GA

HS

PSO

PSOPC

HPSO

DHPSACO

MCSS

IMCSS

4658.055 1161.288 645.16 3303.219 1045.159 494.193 2477.414 1045.159 285.161 1696.771 1045.159 641.289

4658.055 1161.288 506.451 3303.219 940 494.193 2290.318 1008.385 2290.318 1535.481 1045.159 506.451

4658.055 1374.19 1858.06 3206.44 1283.87 252.26 3303.22 1045.16 126.45 2341.93 1008.38 1045.16

5999.988 1008.38 2696.38 3206.44 1161.29 729.03 2238.71 1008.38 494.19 1283.87 1161.29 494.19

4658.055 1161.288 363.225 3303.219 940 494.193 2238.705 1008.385 388.386 1283.868 1161.288 792.256

4658.055 1161.288 494.193 3303.219 1008.385 285.161 2290.318 1008.385 388.386 1283.868 1161.288 506.451

4658.055 1161.288 363.225 3303.219 939.998 506.451 2238.705 1008.385 388.386 1283.868 1161.288 729.031

4658.055 1161.288 494.193 3303.219 939.998 494.193 2238.705 1008.385 494.193 1283.868 1161.288 494.193

1970.142 N/A

1906.76 N/A

2230.16 N/A

2146.63 N/A

1905.49 50,000

1904.83 5300

1904.05 4225

1902.61 4075

Fig. 6. A 72-bar spatial truss. 1100

2,000

MCSS IMCSS

1000

MCSS IMCSS

900

Weight(lb)

Weight(lb)

1,500 800 700

1,000

600 500

500 400 300

20

40

60

80

100

120

Iteration

(a)

140

160

180

200

220

20

40

60

80

100

120

Iteration

(b)

Fig. 7. Convergence history for the 72-bar truss structure using MCSS, IMCSS (a) Case 1 and (b) Case 2.

140

160

180

200

A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410

407

Table 5 Loading conditions for the 72-bar spatial truss. Node

17 18 19 20

Case 1

Case 2

PX kips (kN)

Py kips (kN)

Pz kips (kN)

PX kips (kN)

Py kips (kN)

Pz kips (kN)

5.0 (22.25) 0.0 0.0 0.0

5.0 (22.25) 0.0 0.0 0.0

−5.0 (−22.25) 0.0 0.0 0.0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0

−5.0 (−22.25) −5.0 (−22.25) −5.0 (−22.25) −5.0 (−22.25)

Table 6 Optimal design comparison for the 72-bar truss (Case 1). Element group

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

A1–A4 A5–A12 A13–A16 A17–A18 A19–A22 A23–A30 A31–A34 A35–A36 A37–A40 A41–A48 A49–A52 A53–A54 A55–A58 A59–A66 A67–A70 A71–A72

Weight (kg) No. of analyses

Wu and Chow [28]

Lee and Geem [27]

Li et al. [29]

GA

HS

PSO

PSOPC

1.5 0.7 0.1 0.1 1.3 0.5 0.2 0.1 0.5 0.5 0.1 0.2 0.2 0.5 0.5 0.7

1.9 0.5 0.1 0.1 1.4 0.6 0.1 0.1 0.6 0.5 0.1 0.1 0.2 0.5 0.4 0.6

2.6 1.5 0.3 0.1 2.1 1.5 0.6 0.3 2.2 1.9 0.2 0.9 0.4 1.9 0.7 1.6

3 1.4 0.2 0.1 2.7 1.9 0.7 0.8 1.4 1.2 0.8 0.1 0.4 1.9 0.9 1.3

400.66 N/A

387.94 N/A

1089.88 N/A

1069.79 150,000

the best solutions after 173 iterations (4325 analyses), while MCSS, CS, ICA, CSS and HPSACO algorithms need respectively 4775, 4840, 4500, 7000 and 5330 analyses to find their best solutions. 5.4. A 120-bar dome shaped truss The 120-bar dome truss was first analyzed by Soh and Yang [34] to obtain the optimal sizing and configuration variables in the process of structural configuration optimization, but for this study only sizing variables are considered to minimize the structural weight, similar to Lee and Geem [27] and Keles¸o˘glu and Ülker [35].

Kaveh and Talatahari [31]

Present work

HPSO

DHPSACO

MCSS

IMCSS

2.1 0.6 0.1 0.1 1.4 0.5 0.1 0.1 0.5 0.5 0.1 0.1 0.2 0.5 0.3 0.7

1.9 0.5 0.1 0.1 1.3 0.5 0.1 0.1 0.6 0.5 0.1 0.1 0.2 0.6 0.4 0.6

1.8 0.5 0.1 0.1 1.3 0.5 0.1 0.1 0.7 0.6 0.1 0.1 0.2 0.6 0.4 0.4

2 0.5 0.1 0.1 1.3 0.5 0.1 0.1 0.5 0.5 0.1 0.1 0.2 0.6 0.4 0.6

388.94 50,000

385.54 5330

389.49 5400

385.54 3625

The geometry of this structure is shown in Fig. 8. The modulus of elasticity is 30,450 ksi (210,000 MPa) and the material density is 0.288 lb/in3 (7971.81 kg/m3 ). The yield stress of steel is taken as 58.0 ksi (400 MPa). The allowable tensile and compressive stresses are used according to the AISC-ASD (1989) [25] code, as follows:



i+ = 0.6Fy i−

i ≥0

for for

(25)

i < 0

Table 7 Optimal design comparison for the 72-bar truss (Case 2). Element group

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Weight (lb) No. of analyses

A1–A4 A5–A12 A13–A16 A17–A18 A19–A22 A23–A30 A31–A34 A35–A36 A37–A40 A41–A48 A49–A52 A53–A54 A55–A58 A59–A66 A67–A70 A71–A72

Wu and Chow [28]

Li et al. [29]

Kaveh and Talatahari

Kaveh and Bakhshpoori [33]

Present work

GA

PSO

PSOPC

HPSO

HPSACO [30]

CSS [24]

ICA [32]

CS

MCSS

IMCSS

0.196 0.602 0.307 0.766 0.391 0.391 0.141 0.111 1.8 0.602 0.141 0.307 1.563 0.766 0.141 0.111

7.22 1.8 1.13 0.2 3.09 0.79 0.56 0.79 3.09 1.23 0.11 0.56 0.99 1.62 1.56 1.27

4.49 1.457 0.111 0.111 2.62 1.13 0.196 0.111 1.266 1.457 0.111 0.111 0.442 1.457 1.228 1.457

4.97 1.228 0.111 0.111 2.88 1.457 0.141 0.111 1.563 1.228 0.111 0.196 0.391 1.457 0.766 1.563

1.8 0.442 0.141 0.111 1.228 0.563 0.111 0.111 0.563 0.563 0.111 0.25 0.196 0.563 0.442 0.563

1.99 0.442 0.111 0.111 0.994 0.563 0.111 0.111 0.563 0.563 0.111 0.111 0.196 0.563 0.442 0.766

1.99 0.442 0.111 0.141 1.228 0.602 0.111 0.141 0.563 0.563 0.111 0.111 0.196 0.563 0.307 0.602

1.8 0.563 0.111 0.111 1.266 0.563 0.111 0.111 0.563 0.442 0.111 0.111 0.196 0.602 0.391 0.563

1.8 0.563 0.111 0.111 1.457 0.563 0.111 0.111 0.563 0.442 0.111 0.111 0.196 0.563 0.307 0.766

1.8 0.563 0.111 0.111 1.228 0.563 0.111 0.111 0.391 0.563 0.111 0.111 0.196 0.563 0.307 0.563

427.20 N/A

1209 N/A

941.82 150,000

933.09 50,000

393.38 5330

393.06 7000

392.84 4500

389.87 4840

393.128 4775

389.60 4325

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A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410

radius of gyration. The radius of gyration (ri ) can be expressed in terms of cross-sectional areas, i.e., ri = aAbi [36]. Here, a and b are the constants depending on the types of sections adopted for the members such as pipes, angles, and tees. In this example, pipe sections (a = 0.4993 and b = 0.6777) were adopted for bars. All members of the dome are categorized into 7 groups, as shown in Fig. 8. The dome is considered to be subjected to vertical loading at all the unsupported joints. These loads were taken as −13.49 kips (60 kN) at node 1, −6.744 kips (30 kN) at nodes 2–14, and −2.248 kips (10 kN) at the rest of the nodes. The minimum cross-sectional area of all members is 0.775 in2 (2 cm2 ). In this example, two cases of constraints are considered: Case 1: With stress constraints and no displacement constraints Case 2: With stress constraints and displacement limitations of ±0.1969 in (5 mm) imposed on all of the nodes in x- and ydirections. For two cases, the maximum cross-sectional area is set to 5.0 in2 (32.26 cm2 ). Fig. 9 shows the convergence history for both cases and Table 8 gives the best solution vectors and weights for all cases. In Case 1, the best weight of MCSS and IMCSS is 19607.39 lb and 19476.92 lb, respectively, while for the Ray, HPSACO and PSOPC are 19476.19 lb, 19491.30 lb and 19618.7 lb. The MCSS and IMCSS find their best solutions respectively in 314 iterations (7850 analyses) and 299 iterations (7475 analyses), but Ray and HPSACO algorithms take respectively 19,950 and 10,025 analyses to reach their best solutions. In Case 2, the MCSS and IMCSS algorithms need 386 iterations (9650 analyses) and 324 iterations (8100 analyses) to find their best solutions, respectively, while for Ray and HPSACO algorithms, 19,950 and 10,075 analyses is required. The best weights obtained from MCSS and IMCSS algorithms are 19,928 and 19,796.71 lb, respectively, but from the Ray and HPSACO are 20,071.9 and 20,078 lb, respectively. As seen in the results of both cases, the IMCSS algorithm finds a better result in a lower number of analyses than previous studies.

Fig. 8. A 120-bar dome shaped truss.

where i− is calculated according to the slenderness ratio

i− =

⎧ ⎪ ⎪ ⎨

(1 − (2i /2Cc2 ))Fy (5/3) + (3i /8Cc ) − (3i /8Cc3 )

122 E ⎪ ⎪ ⎩ 2 23i

for i < Cc (26) for i ≥Cc

where E is the modulus of elasticity, Fy is the yield stress of steel, Cc is the slenderness ratio (i ) dividing the elastic and inelastic buckling regions (Cc = 22 E/Fy ), ␭i is the slenderness ratio (i = kLi /ri ), k is the effective length factor, Li is the member length and ri is the

6. Discussion In this research, an improved magnetic charged system search is presented for optimization of truss structures. This algorithm is based on a novel meta-heuristic algorithm called magnetic charged system search (MCSS). In this algorithm, in order to get a better convergence and better results especially in final iteration of the algorithm, an improved scheme of harmony search (IHS) is utilized

Fig. 9. Comparison of the convergence rates between the MCSS and IMCSS for the 120-bar dome truss structure, (a) Case 1 and (b) Case 2.

A. Kaveh et al. / Applied Soft Computing 28 (2015) 400–410

409

Table 8 Optimal design comparison for the 120-bar dome truss (two cases) optimal cross-sectional areas (in2 ). Element group

Case 1 Lee and Geem [27]

Kaveh and Talatahari [30]

Kaveh and Khayatazad [37]

Present work

HS

PSO

PSOPC

HPSACO

Ray

MCSS

IMCSS

1 2 3 4 5 6 7

3.295 3.396 3.874 2.571 1.15 3.331 2.784

3.147 6.376 5.957 4.806 0.775 13.798 2.452

3.235 3.37 4.116 2.784 0.777 3.343 2.454

3.311 3.438 4.147 2.831 0.775 3.474 2.551

3.128 3.357 4.114 2.783 0.775 3.302 2.453

3.1108 3.3903 4.106 2.7757 0.9674 3.2981 2.4417

3.1208 3.3566 4.111 2.7811 0.8055 3.3001 2.4451

Weight (lb) No. of analyses

19,707.77 35,000

32,432.9 N/A

19,618.7 125,000

19,491.3 10,025

19,476.19 19,950

19,607.39 7850

19,476.92 7475

Element group

Case 2 Lee and Geem [27]

Kaveh and Talatahari [30]

Kaveh and Khayatazad [37]

Present work

HS

PSO

PSOPC

HPSACO

Ray

MCSS

IMCSS

1 2 3 4 5 6 7

3.296 2.789 3.872 2.57 1.149 3.331 2.781

15.978 9.599 7.467 2.79 4.324 3.294 2.479

3.083 3.639 4.095 2.765 1.776 3.779 2.438

3.779 3.377 4.125 2.734 1.609 3.533 2.539

3.084 3.360 4.093 2.762 1.593 3.294 2.434

3.309 2.6316 4.2768 2.7918 0.9108 3.5168 2.3769

3.3187 2.4746 4.2882 2.8103 0.7753 3.523 2.3826

Weight(lb) No. of analyses

19,893.34 35,000

41,052.7 N/A

20,681.7 125,000

20,078 10,075

20,071.9 19,950

19,928.0 9650

19,796.71 8100

and two of the most effective parameters (PAR and bw) in the convergence rate of algorithm are improved. The PAR and bw in HS scheme are very important parameters in fine-tuning of optimized solution vectors and important in adjusting the convergence rate of the algorithm to the optimal solution. The IMCSS algorithm is applied to some illustrated numerical example, as well-known benchmark problems for truss structures with both continuous and discrete design variables, to compare the efficiency of the presented algorithm in the process of structural optimization with a wide variety of novel meta-heuristic algorithms such as MCSS, CSS, CS, ICA, HPSACO, DHPSACO, PSOPC, PSO, HS, GA and Ray (RO). As seen in all of the results, the IMCSS algorithm has found better solutions for the optimum weight of the structures within a lower number of structural analyses and consequently within a lower time for the whole optimization process. Although in some cases of the benchmark problems the IMCSS has given a worse value for the best weight of the structures, but in all problems the IMCSS has no penalty for ignoring the problem constraints which means it has the best merit function than the previous studies. The present algorithm has not limitation on the size of the structures and can deal with practical problems. However for very large structures special techniques should be developed such as those of Jarre et al. [38] and [39]. 7. Concluding remarks In this research, the robustness and efficiency of the IMCSS algorithm in the process of structural optimization, in fast convergence and achieving more optimal solutions for the weight of the structure within a lower computation time, than recently presented algorithms such as MCSS, RO, CSS, CS, ICA, HPSACO, DHPSACO, PSOPC, PSO, HS, GA algorithms, has been demonstrated. Since the structural analysis is a time-consuming process in the process of structural optimization, especially in the large-scale optimization problems, and on the other hand as the number and

type of variables or optimization schemes increases, computation time significantly increases, therefore presentation of a high performance meta-heuristic algorithm or approach would reduce the number of analyses and consequently reduce the computation time needed for the optimization process. The following can be considered as the future research prospects: 1. Studying the effectiveness of the IMCSS algorithm in topology optimization of truss structures. 2. Solving the same problem using the novel heuristic optimization methods such as Colliding Bodies Optimization (CBO), Interior search algorithm (ISA) and comparing the results. 3. Solving the same problem considering natural frequency constraints.and 4. Utilizing the IMCSS algorithm for optimization of large-scale frame structures. Acknowledgement The first author is grateful to Iran National Science Foundation for the support. References [1] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975. [2] D.E. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning, Addison-Wesley, Boston, 1989. [3] L. Wei, T. Tang, X. Xie, W. Shen, Truss optimization on shape and sizing with frequency constraints based on parallel genetic algorithm, Struct. Multidisc. Optim. 43 (2011) 665–682. [4] T.J. Allison, A. Khetan, D. Lohan, Managing variable-dimension structural optimization problems using generative algorithms, in: Proceedings of the 10th World Congress on Structural and Multidisciplinary Optimization, 19–24 May, 2013. [5] R.C. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 1995. [6] M. Dorigo, V. Maniezzo, A. Colorni, The ant system: optimization by a colony of cooperating agents, IEEE Trans. Syst. Man Cybern. 26 (1996) 29–41.

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