Shape and size optimization of trusses with multiple ...

Shape and size optimization of trusses with multiple frequency constraints using harmony search and ray optimizer for enhancing the particle swarm optimization algorithm A. Kaveh & S. M. Javadi

Acta Mechanica ISSN 0001-5970 Volume 225 Number 6 Acta Mech (2014) 225:1595-1605 DOI 10.1007/s00707-013-1006-z

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Author's personal copy Acta Mech 225, 1595–1605 (2014) DOI 10.1007/s00707-013-1006-z

A. Kaveh · S. M. Javadi

Shape and size optimization of trusses with multiple frequency constraints using harmony search and ray optimizer for enhancing the particle swarm optimization algorithm Received: 25 February 2013 / Revised: 7 August 2013 / Published online: 5 November 2013 © Springer-Verlag Wien 2013

Abstract In this paper, size and shape optimization of truss structures is performed using an efficient hybrid method. This algorithm uses a particle swarm strategy and ray optimizer, and utilizes additional harmony search for a better exploitation. Here, multiple frequency constraints are considered making the optimization a highly nonlinear problem. Some basic benchmark problems are solved by this hybrid method, and the numerical results demonstrate the efficiency and robustness of this method compared to other mathematical and heuristic algorithms. 1 Introduction The behavior of a structure is dependent on the shape and size, which have an important effect on its static and dynamic characteristics. Multiple frequency constraints force the structure to avoid potential resonant frequencies due to machinery or actuators on the structure [1]; however, to attain a safe condition, there is a great obstacle: optimization of structures with multiple natural frequency constraints is a highly nonlinear optimization problem [1,2]. The coupling of two different types of design variables, nodal coordinates and cross-sectional areas, often leads to divergence while multiple frequency constraints often cause difficult dynamic sensitivity analysis [3]. Therefore, mathematical programming approaches can be hard and time-consuming to apply to these optimization problems. Furthermore, a good starting point is vital for these methods to be executed successfully [4], and for their gradient-based nature, they may converge only to the local optima. In the last decades, researchers have employed different mathematical and evolutionary algorithms for truss optimization on shape and sizing with multiple frequency constraints. Khot [5] proposed an algorithm that used the scaling procedure on some design variables and expressed the weight of the structure as a function of the Lagrange multipliers and the active value of the constraints; Wang et al. [6] used an optimality criterion based on the differentiation of the Lagrangian function; Sesdaghati [7] utilized mathematical programming based on the sequential quadratic programming (SQP) technique and finite elements based on the integrated force method; Lingyun et al. [3] introduced a hybridization of the genetic algorithms and simplex search for better exploitation called niche genetic hybrid algorithm (NGHA); Gomes [8] used a particle swarm algorithm; and Kaveh and Zolghadr [9] utilized a hybridization of the charged system search and the Big Bang-Big Crunch algorithms with trap recognition capability. In this paper, harmony search and a ray optimizer are utilized for enhancing the particle swarm optimization algorithm (HRPSO). This hybrid algorithm is applied to the shape and size optimization of trusses with multiple frequency constraints. In fact, PSO acts as the main engine of the algorithm, RO boosts the movement vector of the particles, and HS is used to enhance the local search for better exploitation. A. Kaveh (B) · S. M. Javadi Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran E-mail: [email protected]

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2 A brief introduction to the PSO, HS, and RO 2.1 Particle swarm optimization Particle swarm optimization (PSO) is a simple and effective algorithm for optimizing a wide range of functions. Conceptually, it seems to lie somewhere between genetic algorithms and evolutionary programming [10]. The PSO uses real-number randomness and the global communication among the swarm particles. In this sense, it is also easier to implement as there is no encoding or decoding of the parameters into binary strings as in genetic algorithms, which can also use real-number strings [11]. On each iteration, the swarm is updated by the following equations [12,13]: Vik+1 = ωVik + c1r1 (Pik − X ik ) + c2 r2 (Pgk − X ik ),

(1)

X ik+1

(2)

=

X ik

+

Vik+1 ,

where Pi is the best previous position of the ith particle and Pg is the best position of the particles that was ever found. Here, ω is an inertia weight to control the influence of the previous velocity, c1 and c2 are two acceleration constants, and r1 and r2 are two random numbers uniformly distributed in the range (0, 1). 2.2 Harmony search The harmony search algorithm was conceptualized using the musical process of searching for a perfect state of harmony. Musical performances seek to find pleasing harmony as determined by an aesthetic standard, just as the optimization process seeks to find a global solution as determined by an objective function. The pitch of each musical instrument determines the aesthetic quality [14]. The optimization procedure of the HS algorithm consists of the following steps [15]: Step 1: Initialize the optimization problem and algorithm parameters such as the specification of each decision variable, possible value range for each decision variable, harmony memory size (HMS), harmony memory considering rate (HMCR), pitch adjusting rate (PAR), harmony memory (HM), and termination criterion. Step 2: Improvise a new harmony from the HM. A new harmony vector is generated from the HM based on the memory considerations rate (HMCR), pitch adjustments, and randomization (PAR). The HMCR sets the rate of choosing one value from the historic values stored in the HM, and (1-HMCR) sets the rate of randomly choosing one value from the possible range of values. The HMCR varies between 0 and 1, and the pitch adjusting process is performed only after a value is chosen from the HM. The value (1-PAR) sets the rate of doing nothing. If the pitch adjustment decision for xi is yes, then xi ← xi + bw · u (−1, 1) where bw is an arbitrary distance bandwidth for the continuous design variable and u(−1, 1) is a uniform distribution between −1 and 1. The HMCR and PAR parameters introduced in the harmony search help the algorithm find globally and locally improved solutions, respectively [14]. Step 3: Update the HM. In Step 4, if the new harmony is better than the worst harmony in the HM, the new harmony is included in the HM and the existing worst harmony is excluded from the HM. The HM is then sorted by the objective function value. Step 4: Repeat Steps 2 and 3 until the termination criterion is satisfied. The computations are terminated when the termination criterion is satisfied. If not, Steps 2 and 3 are repeated.

2.3 Ray optimization Ray optimization (RO) has recently been developed by Kaveh and Khayatazad [16]. This method is inspired by the transition of a ray from one medium to another from physics and uses Snell’s refraction law of light. The transition of the ray is utilized for finding the global or near-global solution.

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The pseudocode of RO is presented in the following [17]: Level 1: Scattering and evaluation Step 1. Initialization. Initialize the parameter of the RO. Initialize an array of agents with random positions. According to the number and type of groups that belong to the agent positions, make an arbitrary array of the velocity vector. Each of these two or three variable velocity vectors should be a normalized vector. Step 2. Evaluation. For each agent, evaluate the value of the goal function in the current position. Save the position of the best agent as the global best. Save the position of each agent as its local best. Level 2: Movement vector and motion refinement Step 1. Movement vector. Add the solution vectors with the corresponding movement vector. Step 2. Motion refinement. If any agent violates a variable boundary, refine its movement vector. After motion refinement and evaluation of the goal function, again the so far best agent at this stage is selected as the global best, and for each agent, the so far best position by this stage (belonging to itself) is selected as its local best. Level 3: Origin making and converging Step 1. Origin making. Find the origin of the each agent. Step 2. Converging. Calculate the new movement vector for each agent. Level 4: Finish or redo. Repeat the optimization process until a terminating criterion is satisfied. 3 Mixed particle swarm, ray optimization, and harmony search algorithm Compared to other algorithms, the PSO has a versatility to hybridize with other metaheuristics and is simple to implement. However, standard PSO has some weaknesses, and Shi and Eberhart [13] introduced a parameter known as the inertia weight into the original particle swarm optimizer to decrease the computational time and to improve the ability to find the global optimum. However, there is no information sharing among individuals except that the global best broadcasts the information to the other individuals. Therefore, the population may lose diversity and is more likely to confine the search around local minima if committed too early in the search to the global best found so far; thus, He et al. [18] introduced a new PSO with passive congregation (PSOPC), and by introducing the passive congregation, information can be transferred among individuals that will help individuals to avoid misjudging information and becoming trapped by poor local minima. Therefore, in the PSOPC, there are parameters such as C1 , C2 , and C3 , each of them having an important role on the performance of the algorithm. On the other hand, the ray optimization algorithm has an origin making part playing an important role in this algorithm. In the RO, first, the point to which each particle moves must be determined. This point is named origin and it is specified by Oik =

(ite + k).GB + (ite − k) . LBi , 2.ite

(3)

where Oik is the origin of the ith agent or particle for the kth iteration, ite is the total number of iterations of the optimization process, GB and LBi are the global best and local best of the ith agent, respectively [16]. In HRPSO, ray origin making is used to update the positions of the particles by the following equations: Vik+1 = ωVik + rand.Oik

(4)

Thus, in this algorithm, parameters such as C1 , C2 and C3 in standard PSO and PSO with the passive congregation (PSOPC) are substituted with the origin making relation, which is independent from parameter tuning.

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Fig. 1 The flowchart of the HRPSO

In this equation, the inertia weight is considered as a decreasing function of time which gradually decreases from 1 by each iteration, and rand is a random number between 0 and 1. On the other hand, for enhancing the exploitation, the HS introduces a parameter named pitch adjustment, which helps the algorithm find locally improved solutions [14]; thus, the PAR is used to reinforce the HRPSO for better local search. By these techniques, there is no dependency on the parameters like as C1 , C2 , and C3 in the PSO and PSOPC. The flowchart of the HRPSO is shown in Fig. 1. 4 Formulation of the optimization problem Size and shape optimization of truss structures involves arriving at optimum values for member cross-sectional areas and desirable node coordinates that minimize the structural weight. This minimum design also has to satisfy inequality constraints that limit design variable sizes and structural responses [15]. Thus, the optimal design problem may be expressed as: minimize W ({X }) =

n 

γi · Ai · L i (x)

i=1

subject to : ω j ≤ ω∗j ωk∗

ωk ≥ Amin ≤ Ai ≤ Amax , Amin ≤ Ai ≤ Amax ,

for some natural frequencies j for some natural frequencies k i = 1, 2, . . . , ng i = 1, 2, . . . , snc

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Fig. 2 A 10-bar planar truss with added masses Table 1 Optimal design comparison for the 10-bar planar truss (cm2 ) Element number

Optimal cross-sectional areas (cm2 ) Grandhi and Venkayya [19]

Sedaghati [7]

Wang et al. [6]

Lingyun et al. [3]

Gomes [8] PSO

E = 68.9 GPa

E = 68.9 GPa

E = 68.9 GPa

E = 69.8 GPa

32.456 16.577 32.456 16.577 2.115 4.467 22.810 22.810 17.490 17.490 553.8

42.234 18.555 38.851 11.222 4.783 4.451 21.049 20.949 10.257 14.342 542.75 4.864

1 36.584 38.245 2 24.658 9.916 3 36.584 38.619 4 24.658 18.232 5 4.167 4.419 6 2.070 4.194 7 27.032 20.097 8 27.032 24.097 9 10.346 13.890 10 10.346 11.4516 Weight (kg) 594 537.01 Standard deviation for 10 independent runs

Present paper

E = 69.8 GPa

Kaveh and Zolghadr CSS-BBBC [9] E = 69.8 GPa

E = 68.9 GPa

E = 69.8 GPa

37.712 9.959 40.265 16.788 11.576 3.955 25.308 21.613 11.576 11.186 537.98

35.274 15.463 32.11 14.065 0.645 4.880 24.046 24.340 13.343 13.543 529.09

35.54022 15.29310 35.78427 14.60570 0.64554 4.62572 24.77893 23.31005 12.48229 12.67468 532.11 2.374

34.79250 15.24510 35.56230 13.83640 0.64640 4.58270 25.5346 22.3002 11.6142 13.0716 524.88 2.253

where W ({X }) is the weight of the structure; n is the number of members making up the structure; ng is the number of groups (number of design variables); γi is the material density of member i; L i is the length of member i; Ai is the cross-sectional area of member i chosen between Amin and Amax ; min is the lower bound and max is the upper bound; X min and X max are used for node coordinates in the shape optimization problem; and snc is the number of some node coordinates that are allowed to change. 5 Numerical examples In this section, four truss structures are optimized utilizing the present method. Then, the final results are compared to the solutions of other mathematical and recent advanced heuristic methods to examine the viability of the proposed algorithm. In the proposed algorithm, the maximum velocity is set as the difference between the upper and lower bounds, which guarantees that the particles rationally survey the search space, and the pitch adjusting rate (PAR), is considered as 0.2. The maximum number of iterations is set to 300, a population of 70 particles is used for the first example, a population of 80 particles is used for the second example, and a population of 90 particles is employed for the last two examples. 5.1 A ten-bar truss The 10-bar truss problem has become a common problem in the field of structural design with multiple frequency constraints for verifying the efficiency of many different optimization methods and is shown in Fig. 2. This example was first solved for Grandhi and Venkayya [19], and in this structure, the material density is ρ = 2,770 kg/m3 (0.1 lbm/in3 ). The lower bound of the section area is set to 0.645 cm2 (0.1 in2 ) for all

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Fig. 3 The comparative convergence history diagrams for the 10-bar truss

Table 2 Natural frequencies (Hz) of the 10-bar planar truss Frequency number

1 2 3 4 5 6 7 8

Grandhi and Venkayya [19]

Sedaghati [7]

Wang et al. [6]

Lingyun et al. [3]

Gomes [8] PSO

Kaveh and Zolghadr CSS-BBBC [9]

Present paper

E = 68.9 GPa

E = 68.9 GPa

E = 68.9 GPa

E = 69.8 GPa

E = 69.8 GPa

E = 69.8 GPa

E = 68.9 GPa

E = 69.8 GPa

7.059 15.895 20.425 21.528 28.976 30.189 54.286 56.546

6.992 17.599 19.973 19.977 28.173 31.029 47.628 52.292

7.011 17.302 20.001 20.100 30.869 32.666 48.282 52.306

7.008 18.148 20.000 20.508 27.797 31.281 48.304 53.306

7.000 17.786 20.000 20.063 27.776 30.939 47.297 52.286

7.0028 16.7429 20.0548 20.3351 28.5232 29.2911 49.0342 51.7451

6.9999 16.1752 19.9999 20.0060 28.5156 28.9837 48.5734 51.0823

7.0000 16.1686 20.0015 20.0050 28.1466 29.2724 48.5235 50.9950

Fig. 4 Initial configuration design for the 37-bar truss structure with added masses

elements. A non-structural mass of 454 kg (1,000 lbm) is attached to each of the four free nodes, and the frequency constraints are as follows: ω1 ≥ 7 Hz; ω2 ≥ 15 Hz; ω3 ≥ 20 Hz. This problem is solved for two different values for the elastic modulus as 6.89 × 1010 and 6.98 × 1010 N/m2 (107 psi). For better judgment between different solutions, we decided to solve this problem with two different values for the elastic modulus. A comparison with the results of other references considering the cross-sectional area of each group and the final weight reached for the 10-bar planar truss is provided in Table 1. The comparative convergence history diagrams for this example are shown in Fig. 3, and Table 2 represents the natural frequencies of the optimized structures obtained by different algorithms. 5.2 A simply supported bridge The initial configuration of the structure is shown in Fig. 4. This example was first investigated by Wang et al. [6]. In this example, the elastic modulus is 210 GPa and the material density is ρ = 7,800 kg/m3 for all elements. Members of the lower chord are represented by bar elements with fixed rectangular cross section

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Table 3 Optimal design comparison for the simply supported bridge Variable group

Initial design

Wang et al. [6]

Y3 , Y19 (m) 1.0 1.2086 Y5 , Y17 (m) 1.0 1.5788 1.0 1.6719 Y7 , Y15 (m) Y9 , Y13 (m) 1.0 1.7703 1.0 1.8502 Y11 (m) 1.0 3.2508 A1 , A27 (cm2 ) 1.0 1.2364 A2 , A26 (cm2 ) A3 , A24 (cm2 ) 1.0 1.0000 1.0 2.5386 A4 , A25 (cm2 ) 1.0 1.3714 A5 , A23 (cm2 ) 1.0 1.3681 A6 , A21 (cm2 ) 1.0 2.4290 A7 , A22 (cm2 ) A8 , A20 (cm2 ) 1.0 1.6522 1.0 1.8257 A9 , A18 (cm2 ) 1.0 2.3022 A10 , A19 (cm2 ) 1.0 1.3103 A11 , A17 (cm2 ) 1.0 1.4067 A12 , A15 (cm2 ) A13 , A16 (cm2 ) 1.0 2.1896 1.0 1.0000 A14 (cm2 ) Weight (kg) 336.30 366.50 Standard deviation for 10 independent runs

Lingyun et al. [3]

Gomes [8] PSO

Present work

1.1998 1.6553 1.9652 2.0737 2.3050 2.8932 1.1201 1.0000 1.8655 1.5962 1.2642 1.8254 2.0009 1.9526 1.9705 1.8294 1.2358 1.4049 1.0000 368.84 9.0325

0.9637 1.3978 1.5929 1.8812 2.0856 2.6797 1.1568 2.3476 1.7182 1.2751 1.4819 4.6850 1.1246 2.1214 3.8600 2.9817 1.2021 1.2563 3.3276 377.20

1.07444 1.49568 1.73243 1.89449 1.96970 2.85176 1.00000 1.83410 1.88766 1.06267 1.80266 1.93387 1.24946 1.87404 1.95716 1.24410 1.77792 1.80643 1.00000 364.72 5.776

Table 4 Natural frequencies (Hz) of the simply supported bridge Frequency number

Initial design

Wang et al. [6]

Lingyun et al. [3]

Gomes [8] PSO

Present work

1 2 3 4 5

8.8778 29.2135 48.5539 67.7487 84.2484

20.0850 42.0743 62.9383 74.4539 90.0576

20.0013 40.0305 60.0000 73.0444 89.8244

20.0001 40.0003 60.0001 73.0440 89.8240

20.0000 40.0160 60.0101 79.3488 100.2331

area of 4 × 10−3 m2 . A non-structural mass m = 10 kg is attached at each of the nodes on the lower chord. Nodes on the upper chords can be shifted vertically, while nodes on the lower chords remain fixed. In addition, shape and size are linked to maintain structural symmetry about the y–z plane and the allowable minimum area of the cross section is 1 × 10−4 m2 . There are three constraints in the first three natural frequencies so that ω1 ≥ 20 Hz, ω2 ≥ 40 Hz, ω3 ≥ 60 Hz. Table 3 shows the results for this case and compares these results with those previously reported in the literature. In addition, Table 4 shows the corresponding natural frequencies.

5.3 A seventy-two-bar spatial truss A 72-bar spatial truss is shown in Fig. 5. This problem was investigated by Konzelman [20]. The material density is considered as 2,770 kg/m3 (0.1 lb/in3 ). The 72 structural members of this spatial truss are categorized into 16 groups using symmetry. Four non-structural masses of 2,270 kg (5,000 lbm) are attached to the nodes 1–4. There are two constraints in the natural frequencies, which are ω1 = 4 Hz and ω3 ≥ 6 Hz. The allowable minimum cross-sectional area of all members is set to 6.45 × 10−5 m2 (0.1 in2 ). Similar to the ten-bar planar truss, this problem is solved by different elastic modulus as 6.89 × 1010 and 6.98 × 1010 N/m2 (107 psi) for better comparison. The results are compared to those of the other references considering the cross-sectional area of each group, and the natural frequencies of the optimized structures for the seventy-two-bar space truss are shown in Tables 5 and 6, respectively.

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Fig. 5 A 72-bar space truss with added masses Table 5 Optimal design comparison for the 72-bar space truss Element group

Optimal cross-sectional areas (cm2 ) Konzelman [20]

Sedaghati [7]

E = 68.9 GPa

E = 68.9 GPa

1 A1–A4 3.499 3.499 2 A5–A12 7.932 7.932 3 A13–A16 0.645 0.645 4 A17–A18 0.645 0.645 5 A19–A22 8.056 8.056 6 A23–A30 8.011 8.011 7 A31–A34 0.645 0.645 8 A35–A36 0.645 0.645 9 A37–A40 12.812 12.812 10 A41–A48 8.061 8.061 11 A49–A52 0.645 0.645 12 A53–A54 0.645 0.645 13 A55–A58 17.279 17.279 14 A59–A66 8.088 8.088 15 A67–A70 0.645 0.645 16 A71–A72 0.645 0.645 Weight (lb) 327.605 327.605 Standard deviation for 10 independent runs

Gomes [8]

Present work

E = 68.9 GPa

Kaveh and Zolghadr [9] E = 69.8 GPa

E = 68.9 GPa

E = 69.8 GPa

2.987 7.749 0.645 0.645 8.765 8.153 0.645 0.645 13.450 8.073 0.645 0.645 16.684 8.159 0.645 0.645 328.823

2.854 8.301 0.645 0.645 8.202 7.043 0.645 0.645 16.328 8.299 0.645 0.645 15.048 8.268 0.645 0.645 327.507

3.9494 7.9680 0.6452 0.6479 7.5252 7.8638 0.6451 0.6520 12.9665 8.3473 0.645 0.6451 17.3896 8.0068 0.645 0.6451 328.589 3.959

3.63529 7.83480 0.64507 0.64558 8.41172 7.96728 0.64503 0.64510 13.29653 7.87893 0.645 0.645 15.9834 8.07824 0.64501 0.64609 324.497 3.948

Table 6 Natural frequencies (Hz) of the 72-bar space truss Frequency number

Konzelman [20]

Sedaghati [7]

E = 68.9 GPa

E = 68.9 GPa

1 2 3 4 5

4.000 4.000 6.000 6.247 9.074

4.000 4.000 6.000 6.247 9.074

Gomes [8]

Present work

E = 68.9 GPa

Kaveh and Zolghadr [9] E = 69.8 GPa

E = 68.9 GPa

E = 69.8 GPa

4.000 4.000 6.000 6.219 8.976

4.000 4.000 6.004 6.2491 8.9726

4.0000 4.0000 6.0002 6.2639 9.1166

4.0000 4.0000 6.0003 6.2958 9.1215

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Fig. 6 Initial configuration design for the 52-bar truss structure with added masses Table 7 Element grouping Group number

Members

1 2 3 4 5 6 7 8

1–4 5–8 9–16 17–20 21–28 29–36 37–44 45–52

5.4 A fifty-two-bar dome truss The initial topology of a 52-bar dome truss is shown in Fig. 6. This is a simultaneous shape and size optimization problem, where both the cross-sectional area of the members and the nodal coordinates are considered as variables. The elastic modulus is equal to 210 GPa, and the material density is considered as 7,800 kg/m3 . A non-structural mass of 50 kg is attached at each of the free nodes. These 52 bars are linked into eight groups, as shown in Table 7. All free nodes could be shifted in the way that the symmetry kept in the design process and each movable node is allowed to vary by ±2 cm. The frequency constraints in this benchmark case study are considered as ω1 ≤ 15.916 Hz and ω2 ≥ 28.648 Hz. In addition, the cross-sectional areas are permitted to vary in the range of 0.0001–0.001 m2 .

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Table 8 Optimal design comparison for the 52-bar dome truss Variable

Initial

Lin et al. [21]

ZA (m) 6.000 4.3201 XB (m) 2.000 1.3153 ZB (m) 5.700 4.1740 XF (m) 4.000 2.9169 ZF (m) 4.500 3.2676 A1 (cm2 ) 2.0 1.00 A2 (cm2 ) 2.0 1.33 A3 (cm2 ) 2.0 1.58 A4 (cm2 ) 2.0 1.00 2.0 1.71 A5 (cm2 ) A6 (cm2 ) 2.0 1.54 A7 (cm2 ) 2.0 2.65 A8 (cm2 ) 2.0 2.87 Weight (kg) 338.69 298.0 Standard deviation for 10 independent runs

Lingyun et al. [3]

Gomes [8] PSO

Kaveh and Zolghadr CSS-BBBC [9]

Present paper

5.8851 1.7623 4.4091 3.4406 3.1874 1.0000 2.1417 1.4858 1.4018 1.911 1.0109 1.4693 2.1411 236.046 37.462

5.5344 2.0885 3.9283 4.0255 2.4575 0.3696 4.1912 1.5123 1.5620 1.9154 1.1315 1.8233 1.0904 228.381

5.331 2.134 3.719 3.935 2.500 1.0000 1.3056 1.4230 1.3851 1.4226 1.0000 1.5562 1.4485 197.309

5.82857 2.24360 3.72064 3.95665 2.50008 1.00000 1.13655 1.22183 1.48666 1.39548 1.00000 1.55152 1.41820 193.361 17.637

Table 9 Natural frequencies (Hz) of the 52-bar space truss Frequency number

Initial

Lin et al. [21]

Lingyun et al. [3]

Gomes [8] PSO

Kaveh and Zolghadr CSS-BBBC [9]

Present paper

1 2 3 4 5

22.69 25.17 25.17 31.52 33.80

15.22 29.28 29.28 31.68 33.15

12.81 28.65 28.65 29.54 30.24

12.751 28.649 28.649 28.803 29.230

12.987 28.648 28.679 28.713 30.262

11.6853 28.6486 28.6486 28.6509 29.1298

Fig. 7 The comparative convergence history diagrams for the 52-bar truss

Table 8 shows the final optimized coordinates and cross-sectional areas. In addition, Table 9 shows the natural frequencies obtained by various methods for the 52-bar dome-like truss, and the comparative convergence history diagrams for this example are shown in Fig. 7. 6 Concluding remarks In this paper, the HRPSO algorithm is employed to solve shape and sizing truss optimization problem with multiple frequency constraints. This is a highly nonlinear optimization problem in the field of the structure. HRPSO is an efficient hybrid algorithm in which PSO acts as the main engine of the algorithm, RO enhances

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the movement vector, and HS improves the local search ability for a better exploitation. Results show that the present algorithm outperforms the other previously developed mathematical and heuristic algorithms. Acknowledgments The first author is grateful to the Iran National Science Foundation for the support.

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