Shape and size optimization with frequency constraints is a highly nonlinear problem with mixed design variables, non-convex search space, and multiple local optima. Therefore, a hybrid sine cosine firefly algorithm (HSCFA) is proposed to acquire more accurate solutions with less finite element analysis. The full attraction model of firefly algorithm (FA) is analyzed, and the factors that affect its computational efficiency and accuracy are revealed. A modified FA with simplified attraction model and adaptive parameter of sine cosine algorithm (SCA) is proposed to reduce the computational complexity and enhance the convergence rate. Then, the population is classified, and different populations are updated by modified FA and SCA respectively. Besides, the random search strategy based on Lévy flight is adopted to update the stagnant or infeasible solutions to enhance the population diversity. Elitist selection technique is applied to save the promising solutions and further improve the convergence rate. Moreover, the adaptive penalty function is employed to deal with the constraints. Finally, the performance of HSCFA is demonstrated through the numerical examples with nonstructural masses and frequency constraints. The results show that HSCFA is an efficient and competitive tool for shape and size optimization problems with frequency constraints.

Since the natural frequency has an important influence on the vibration of the structural system, it is necessary to constrain the natural frequency in the structural design to avoid resonance and damage. Bellagamba et al. [

The structural optimization with frequency constraints aims to minimize the weight of the structure while ensuring the satisfaction of frequency constraints. Nevertheless, frequency constraints are highly nonlinear, nonconvex and implicit with respect to the design variables [

Firefly algorithm (FA) is one of the nature-inspired metaheuristic algorithms based on the flashing patterns and social behavior of fireflies, and it can be considered as a generalization to particle swarm optimization (PSO), differential evolution, and simulated annealing algorithms through parameter adjustment [

Sine cosine algorithm (SCA) [

In this paper, the computational complexity of FA is reduced and a hybrid sine cosine firefly algorithm (HSCFA) with adaptive penalty function is proposed to deal with shape and size optimization of truss structures with frequency constraints. HSCFA takes advantage of SCA, FA, and Lévy flight to achieve a better balance between exploration and exploitation.

The remainder of this article is organized as follows. The mathematical model of the discrete structural optimization problem is presented in

Generally, shape and size optimization for truss structures aims to minimize the weight while satisfying functional constraints. The design variables include the cross-sectional area of the members and the nodal positions of the critical members. Thus, the mathematical model can be expressed as the following

_{i} means the material density, _{i} indicates the cross-sectional area, _{i} is the length, _{q} is the nodal positions. _{j} and _{j,max} are the _{l} and _{l,min} are the _{i,min} and _{ i,max} are the lower and upper bounds of _{i}. _{q,min} and _{q,max} are the lower and upper bounds of _{q}.

FA was developed by Yang in 2008 [

_{0} is the attractiveness of _{ij} between fireflies

The movement of firefly

SCA is a novel population-based optimization algorithm proposed by Mirjalili [

^{t} means the best solution at _{1} is distributes in [0,2]. _{2} can be taken from [0,2π]. _{3} is a random number in [0,2]. It can determine the effect of destination ^{t} on the current movement. The parameter _{3} brings a random weight for the destination to stochastically emphasize (_{3} > 1) or deemphasize (_{3} < 1) the effect of destination in defining the distance. _{4} is a random number in [0,1] that decides the switch between sine and cosine components. The range of sine and cosine changes adaptively as the following

From _{1}sin(_{2}) are in [–1,1]. And it will be outside the space between X and P when the values of _{1}sin(_{2}) are in (1,2] and [–2, –1). Thus, both _{1} and _{2} determine the movement distance and search space.

Lévy flight was a non-Gaussian random process proposed by Chechkin et al. [

Then, the Lévy flight-based local search technique can be defined as the following

_{s} and the number of constraint violation _{g} and the allowable stagnation times _{a} are constants. The solutions will be updated by _{g} or _{s} = _{a}.

The elitist selection technique was proposed for the selection progress [

The penalty function method is one of the most popular constraint handling techniques [

_{i} and _{iall} are the actual value and allowable value of

Generally, the accuracy of the optimal solution and the computational cost are two core indices to evaluate the performance of metaheuristic algorithms. In this section, HSCFA is proposed to enhance the solution accuracy and computational cost of FA.

The full attraction model defines the movement of fireflies in FA during the search process [

As shown in

If the value of

The distance _{ij} between fireflies

Name | Function | Range | Minimum |
---|---|---|---|

Sphere | [–100,100] | 0 | |

Zakharov | [–5,10] | 0 | |

Griewank | [–600,600] | 0 | |

Michalewicz | [0, |
–4.687658 |

From

Moreover, it also can be concluded from the full attraction model that the total number of attractions at each generation for population ^{2} ·

Although the attraction can enhance the exploitation ability, too many attractions on a firefly lead to uncertain search direction and weaken the exploitation ability. Therefore, the attraction model is modified and only one solution in the top three is selected to update the worse solutions to reduce the time complexity. Furthermore, the search range is introduced into the calculation of _{1} of SCA is utilized to replace

_{k} and _{k} are the upper and lower bound of

From

A hybrid sine cosine firefly algorithm (HSCFA) integrating modified FA, SCA, Lévy flight and adaptive penalty function is proposed in this section. HSCFA takes advantage of modified FA’s exploration ability, SCA’s exploitation ability, and Lévy flight’s strong random search ability. In order to ensure varied population diversity, the population is divided into two equivalent parts that used modified FA and SCA for solution update respectively. The random search strategy based on Lévy flight is employed to update the solutions that stagnate for several iterations or severely violate constraints to improve the population diversity. Thus, the stagnation times of each solution _{s} is stored during the iteration, and the allowable stagnation times _{a} is set. The constraints number of each optimization problem is set as _{g}. In HSCFA, the elitist selection technique is adopted to replace the original selection way of FA and SCA to improve the convergence speed.

The detailed operation steps of HSCFA are presented in this section. The flowchart of HSCFA is shown in

As shown in

Step 1. Initialize the parameters _{0}, _{a}.

Step 2. Initial solutions and the fitness values.

Step 3. If _{g} or _{s} _{a}, the solutions update by

Step 4. The finite element analysis is applied.

Step 5. The results obtained by finite element analysis are treated by the adaptive penalty function method. Count the number of stagnations _{s} per solution.

Step 6. Use the elitist selection technique to select the solution.

Step 7. If the terminal condition is satisfied, end the iteration. Otherwise, it goes to Step 3.

In this section, the initial parameters of HSCFA are investigated and four well-known structural design problems including two size optimization and two shape and size optimization examples are tested. These structural design examples are all minimization problems with frequency constraints [

Since the performance of HSCFA is influenced by _{0}, _{a}, different initial parameters are tested by the benchmark functions in

From _{mean} decrease with the increase of _{0} for all the functions except for Michalewicz function. It is difficult to determine which value of _{0} is the most reasonable one for Michalewicz function. From _{mean} change irregularly with the increase of _{mean} and _{s} are also hard to be summarized from

Functions | _{0} = 0 |
_{0} = 0.1 |
_{0} = 0.2 |
_{0} = 0.3 |
_{0} = 0.4 |
_{0} = 0.5 |
_{0} = 0.6 |
_{0} = 0.7 |
_{0} = 0.8 |
_{0} = 0.9 |
_{0} = 1 |
---|---|---|---|---|---|---|---|---|---|---|---|

Sphere | 11.00 | 9.98 | 9.02 | 8.00 | 7.00 | 5.93 | 5.02 | 3.98 | 3.02 | 2.04 | 1.00 |

Griewank | 10.91 | 10.09 | 8.98 | 7.91 | 6.17 | 5.59 | 4.70 | 4.02 | 3.32 | 2.72 | 1.59 |

Zakharov | 10.87 | 9.93 | 8.69 | 7.64 | 6.80 | 5.91 | 5.18 | 4.24 | 3.16 | 2.36 | 1.22 |

Michalewicz | 4.11 | 2.63 | 4.37 | 5.64 | 6.51 | 7.70 | 7.77 | 6.39 | 7.09 | 7.10 | 6.69 |

Summation | 11 | 9.98 | 9.02 | 8 | 7 | 5.93 | 5.02 | 3.98 | 3.02 | 2.04 | 1 |

Functions | _{0} = 0 |
_{0} = 0.1 |
_{0} = 0.2 |
_{0} = 0.3 |
_{0} = 0.4 |
_{0} = 0.5 |
_{0} = 0.6 |
_{0} = 0.7 |
_{0} = 0.8 |
_{0} = 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

Sphere | 5.16E-09 | 5.17E-09 | 5.17E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 |

Griewank | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 7.25E-09 | 6.34E-09 | 8.30E-09 | 6.22E-08 | 5.25E-06 | 1.30E-03 |

Zakharov | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 9.48E-09 | 2.23E-08 | 2.42E-07 | 3.35E-05 |

Michalewicz | 4.57E-05 | 7.42E-07 | 1.29E-03 | 1.15E-01 | 9.50E-01 | 1.62E-01 | 8.22E-02 | 5.76E-01 | 5.13E-01 | 2.62E-01 |

Summation | 5.16E-09 | 5.17E-09 | 5.17E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 | 5.18E-09 |

Functions | |||||
---|---|---|---|---|---|

Sphere | 3.06 | 3.10 | 3.15 | 2.74 | 2.96 |

Griewank | 2.45 | 2.88 | 3.05 | 3.16 | 3.46 |

Zakharov | 2.44 | 3.09 | 3.18 | 3.02 | 3.26 |

Michalewicz | 2.44 | 2.81 | 3.30 | 3.15 | 3.29 |

Summation | 2.60 | 2.97 | 3.17 | 3.02 | 3.24 |

Functions | ||||
---|---|---|---|---|

Sphere | 9.93E-01 | 9.17E-01 | 3.94E-01 | 2.67E-01 |

Griewank | 4.55E-01 | 1.04E-01 | 2.23E-01 | 7.00E-03 |

Zakharov | 5.00E-03 | 3.80E-02 | 1.11E-01 | 1.82E-01 |

Michalewicz | 4.54E-02 | 7.32E-05 | 1.29E-03 | 7.23E-05 |

Summation | 3.96E-03 | 3.79E-04 | 7.73E-03 | 3.21E-03 |

Functions | ||||
---|---|---|---|---|

Sphere | 3.94E-01 | 3.14E-01 | 5.53E-01 | 9.06E-01 |

Griewank | 2.23E-01 | 4.92E-01 | 8.45E-01 | 4.30E-02 |

Zakharov | 1.11E-01 | 9.61E-01 | 2.14E-01 | 5.16E-01 |

Michalewicz | 1.00E-03 | 2.97E-01 | 2.91E-01 | 2.39E-01 |

Summation | 1.29E-03 | 2.97E-01 | 2.91E-01 | 2.39E-01 |

Functions | _{a} = 2 |
_{a} = 3 |
_{a} = 4 |
_{a} = 5 |
_{a} = 6 |
---|---|---|---|---|---|

Sphere | 2.85 | 3.11 | 2.72 | 3.00 | 3.32 |

Griewank | 3.09 | 3.08 | 2.87 | 2.53 | 3.44 |

Zakharov | 2.98 | 2.89 | 2.89 | 3.00 | 3.23 |

Michalewicz | 2.77 | 2.97 | 3.15 | 2.90 | 3.20 |

Summation | 2.92 | 3.01 | 2.91 | 2.86 | 3.30 |

Functions | _{a} = 2 |
_{a} = 3 |
_{a} = 5 |
_{a} = 6 |
---|---|---|---|---|

Sphere | 6.59E-01 | 3.04E-01 | 2.72E-01 | 1.20E-02 |

Griewank | 5.27E-01 | 3.91E-01 | 6.42E-01 | 3.00E-02 |

Zakharov | 9.71E-01 | 9.78E-01 | 6.53E-01 | 2.26E-01 |

Michalewicz | 2.06E-01 | 7.18E-01 | 7.75E-01 | 9.10E-01 |

Summation | 9.69E-01 | 4.71E-01 | 5.21E-01 | 4.04E-03 |

Functions | _{a} = 2 |
_{a} = 3 |
_{a} = 4 |
_{a} = 6 |
---|---|---|---|---|

Sphere | 6.66E-01 | 7.33E-01 | 2.72E-01 | 6.89E-01 |

Griewank | 1.31E-01 | 7.90E-02 | 6.42E-01 | 1.90E-02 |

Zakharov | 5.38E-01 | 6.25E-01 | 6.53E-01 | 5.00E-02 |

Michalewicz | 2.92E-01 | 8.67E-01 | 7.75E-01 | 4.06E-01 |

Summation | 1.92E-01 | 4.13E-01 | 5.21E-01 | 1.36E-02 |

From _{0} = 1. Furthermore, it can be concluded from _{0} = 1. From _{a} = 5 ranks first in Summation. From _{a} = 4 and _{a} = 5 are better than _{a} = 6 for two functions. Based on the Friedman and Wilcoxon tests, _{0} = 1, _{a} = 5 are the proper initial parameters corresponding to the optimal performance of HSCFA.

The 72-bar space truss including 16 size design variables is adopted as the first example. The geometry and support conditions are shown in

Parameters | Value |
---|---|

Modulus of elasticity ^{2}) |
6.895 × 10^{10} |

Material density ^{3}) |
2770 |

Added mass (kg) | 2268 |

Lower bound of cross sections (cm^{2}) |
0.645 |

Upper bound of cross sections (cm^{2}) |
25 |

Frequency constraints (Hz) | _{1} = 4, _{3} ≥ 6 |

^{2}) |
PSO |
HS |
PSRO |
HALC-PSO |
CPA |
SCA | FA | MFA | HSCFA-1 | HSCFA-2 | HSCFA |
---|---|---|---|---|---|---|---|---|---|---|---|

_{1-4} |
2.987 | 3.6803 | 3.840 | 3.3437 | 3.329 | 3.6389 | 3.6874 | 8.1938 | 5.2647 | 3.5810 | 3.4873 |

_{5-12} |
7.849 | 7.6808 | 8.360 | 7.8688 | 7.841 | 7.5260 | 7.2800 | 7.1640 | 9.9000 | 8.1125 | 8.0009 |

_{13-16} |
0.645 | 0.6450 | 0.645 | 0.6450 | 0.645 | 4.3052 | 1.0533 | 0.6450 | 0.6450 | 0.6450 | 0.6450 |

_{17-18} |
0.645 | 0.6450 | 0.699 | 0.6450 | 0.645 | 0.9348 | 2.3171 | 1.2907 | 0.6450 | 0.6450 | 0.6450 |

_{19-22} |
8.765 | 9.4955 | 8.817 | 8.1626 | 8.416 | 11.8417 | 12.3247 | 7.2060 | 8.7338 | 8.1744 | 8.2722 |

_{23-30} |
8.153 | 8.2870 | 7.697 | 7.9502 | 8.160 | 8.7219 | 7.3960 | 8.9467 | 7.3477 | 7.9444 | 7.9557 |

_{31-34} |
0.645 | 0.6450 | 0.645 | 0.6452 | 0.645 | 1.4480 | 0.6450 | 2.4139 | 0.6450 | 0.6455 | 0.6450 |

_{35-36} |
0.645 | 0.6461 | 0.651 | 0.6450 | 0.645 | 2.4655 | 1.8202 | 0.8822 | 0.6450 | 0.6478 | 0.6450 |

_{37-40} |
13.450 | 11.4510 | 12.136 | 12.2668 | 13.078 | 11.7052 | 11.9591 | 13.8707 | 13.2774 | 13.3592 | 13.0688 |

_{41-48} |
8.073 | 7.8990 | 8.839 | 8.1845 | 8.043 | 7.9695 | 8.5632 | 7.8524 | 7.0860 | 8.0817 | 8.0573 |

_{49-52} |
0.645 | 0.6473 | 0.645 | 0.6451 | 0.645 | 0.6450 | 0.6523 | 1.0628 | 0.8842 | 0.6450 | 0.6450 |

_{53-54} |
0.645 | 0.6450 | 0.645 | 0.6451 | 0.645 | 6.8792 | 4.9077 | 2.0244 | 0.6450 | 0.6450 | 0.6450 |

_{55-58} |
16.684 | 17.4060 | 17.059 | 17.9632 | 16.943 | 15.6249 | 15.1759 | 15.0925 | 15.2041 | 16.6489 | 16.9026 |

_{59-66} |
8.159 | 8.2736 | 7.427 | 8.1292 | 8.143 | 8.1066 | 9.5957 | 8.8736 | 8.6888 | 8.0145 | 8.1348 |

_{67-70} |
0.645 | 0.6450 | 0.646 | 0.6450 | 0.647 | 2.8991 | 1.2811 | 0.8432 | 0.6450 | 0.6455 | 0.6523 |

_{71-72} |
0.645 | 0.6450 | 0.645 | 0.6450 | 0.653 | 7.5037 | 6.6776 | 0.6556 | 1.2711 | 0.6581 | 0.6524 |

Best (kg) | 328.823 | 328.334 | 329.80 | 327.77 | 328.49 | 390.254 | 370.625 | 351.237 | 338.282 | 328.245 | 328.158 |

Mean (kg) | 332.24 | 332.64 | 334.95 | 327.99 | 330.91 | 424.13 | 405.29 | 360.59 | 341.24 | 331.93 | 330.37 |

SD (kg) | 4.23 | 2.39 | 2.86 | 0.19 | 1.84 | 22.35 | 16.10 | 8.15 | 3.46 | 3.74 | 1.71 |

NS | N/A | 50,000 | 6,000 | 8,000 | 12,800 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 |

Number
PSO
HS
PSRO
HALC-PSO
CPA
SCA
FA
MFA
HSCFA-1
HSCFA-2
HSCFA
1
4.000
4.0000
4.000
4.000
4.000
4.0000
4.0000
4.0000
4.0000
4.0000
4.0000
2
4.000
4.0000
4.000
4.000
4.000
4.0000
4.0000
4.0000
4.0000
4.0000
4.0000
3
6.000
6.0000
6.004
6.000
6.000
6.0041
6.0264
6.0374
6.0225
6.0002
6.0001
4
6.219
6.2723
6.249
6.418
6.238
7.5954
9.7073
7.9627
6.3633
6.2584
6.2496
5
8.976
9.0749
8.972
9.143
9.035
10.3349
10.0459
10.2556
9.4220
9.0940
9.0710

For this experiment, the population size is 10 and the iteration is 1000 for SCA, FA, MFA, HSCFA-1, HSCFA-2, and HSCFA.

The 120-bar dome truss including 7 size design variables is adopted as the second example. The geometry and support conditions are shown in

Parameters (unit) | Value |
---|---|

Modulus of elasticity ^{2}) |
2.1 × 10^{11} |

Material density ^{3}) |
7971.810 |

Added mass (kg) | _{1} = 3000, _{2} = 500, _{3} = 100 |

Lower bound of cross sections (cm^{2}) |
1.0 |

Upper bound of cross sections (cm^{2}) |
129.3 |

Constraints on frequencies (Hz) | _{1} ≥ 9, _{2} ≥ 11 |

^{2}) |
PSRO |
PSO |
HALC-PSO |
STMP-TLBO |
EFBI |
SCA | FA | MFA | HSCFA-1 | HSCFA-2 | HSCFA |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 19.972 | 18.4132 | 19.8905 | 19.5554 | 19.4744 | 20.891 | 15.916 | 19.8824 | 19.2110 | 19.6623 | 19.8064 |

2 | 39.701 | 47.8316 | 40.4045 | 40.2398 | 40.3940 | 36.038 | 52.489 | 37.7172 | 41.1982 | 40.0977 | 39.6874 |

3 | 11.323 | 15.6585 | 11.2057 | 10.5967 | 10.6238 | 10.147 | 10.304 | 11.6455 | 10.3171 | 10.6235 | 10.3824 |

4 | 21.808 | 28.7868 | 21.3768 | 21.1778 | 21.0395 | 22.604 | 17.138 | 21.1772 | 20.8726 | 21.3319 | 21.2291 |

5 | 10.179 | 9.1114 | 9.8669 | 9.8356 | 9.9007 | 10.443 | 28.038 | 10.4279 | 10.2465 | 9.7480 | 9.6936 |

6 | 12.739 | 15.1059 | 12.7200 | 11.8421 | 11.7354 | 20.197 | 33.376 | 11.2160 | 12.2075 | 11.8030 | 12.0102 |

7 | 14.731 | 14.4374 | 15.2236 | 14.7767 | 14.9079 | 14.358 | 28.980 | 15.2947 | 14.9217 | 14.6846 | 14.8588 |

Best (kg) | 8892.33 | 10163.99 | 8889.96 | 8708.894 | 8707.74 | 9179.50 | 12131.14 | 8769.327 | 8724.799 | 8713.531 | 8709.871 |

Mean (kg) | 8921.3 | 11134.77 | 8900.39 | 8710.040 | 8715.18 | 10120.17 | 16435.68 | 9038.162 | 8741.344 | 8734.964 | 8729.092 |

SD (kg) | 18.54 | 526.67 | 6.38 | 0.693 | 2.15 | 579.86 | 1930.97 | 206.79 | 10.48 | 16.07 | 11.75 |

NS | 4,000 | 18,800 | 17,000 | 20,000 | 5000 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 |

Number | PSRO |
PSO |
HALC-PSO |
STMP-TLBO |
EFBI |
SCA | FA | MFA | HSCFA-1 | HSCFA-2 | HSCFA |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 9.000 | 9.067 | 9.000 | 9.0004 | 9.0000 | 9.056 | 9.063 | 9.0004 | 9.0035 | 9.0052 | 9.0000 |

2 | 11.000 | 11.199 | 11.000 | 11.0001 | 11.0000 | 11.015 | 11.059 | 11.001 | 11.0026 | 11.0001 | 11.0000 |

3 | 11.005 | 11.214 | 11.000 | 11.0001 | 11.0000 | 11.015 | 11.059 | 11.001 | 11.0063 | 11.0001 | 11.0000 |

4 | 11.012 | 11.695 | 11.010 | 11.0001 | 11.0007 | 11.323 | 11.598 | 11.072 | 11.0063 | 11.0012 | 11.0003 |

5 | 11.045 | 11.726 | 11.050 | 11.0669 | 11.0679 | 11.387 | 11.720 | 11.140 | 11.0717 | 11.0675 | 11.0670 |

For this experiment, the population size is 10 and the iteration is 1000 for SCA, FA, MFA, HSCFA-1, HSCFA-2, and HSCFA.

The 37-bar planar truss including 5 shape and 14 size design variables is used as the third example. The geometry and support conditions are shown in ^{2}, and all the nodes in the lower chord attach a constant concentrated mass 10 kg. All nodes of the upper chord can vary from 1 m to 2.5 m in the y-axis. The results of NHPGA [

Design parameters (units) | Values |
---|---|

Young’s modulus (N/m^{2}) |
6.89 × 10^{10} |

Material density ^{3}) |
2770.0 |

Added mass (kg) | 10 |

Lower bound of cross sections (cm^{2}) |
1 |

Upper bound of cross sections (cm^{2}) |
10 |

Frequency constraints (Hz) | _{1} ≥ 20, _{2} ≥ 40, _{3} ≥ 60 |

^{2}) |
NHPGA |
HS |
DPSO |
PSRO |
HALC-PSO |
CPA |
STMP-TLBO |
SCA | FA | MFA | HSCFA-1 | HSCFA-2 | HSCFA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

_{1}-_{27} |
2.6246 | 3.2031 | 2.6208 | 2.6368 | 2.5000 | 2.9166 | 2.9972 | 3.794 | 3.820 | 4.8742 | 2.7357 | 3.2860 | 2.7040 |

_{2}-_{26} |
1.0000 | 1.1107 | 1.0397 | 1.3034 | 1.2319 | 1.0089 | 1.0490 | 2.637 | 6.873 | 1.1339 | 1.3855 | 1.2361 | 1.0015 |

_{3}-_{24} |
1.0018 | 1.1871 | 1.0464 | 1.0029 | 1.3669 | 1.0000 | 1.0000 | 2.533 | 2.464 | 1.0139 | 1.1186 | 1.0113 | 1.0006 |

_{4}-_{25} |
2.0759 | 3.3281 | 2.7163 | 2.3325 | 2.2801 | 2.3965 | 2.5917 | 1.715 | 9.977 | 2.8225 | 3.3667 | 2.3434 | 2.5844 |

_{5}-_{23} |
1.2207 | 1.4057 | 1.0252 | 1.2868 | 1.0011 | 1.3489 | 1.1576 | 1.067 | 2.206 | 1.0871 | 1.0043 | 1.0427 | 1.0373 |

_{6}-_{21} |
1.4892 | 1.0883 | 1.5081 | 1.0704 | 0.9750 | 1.2240 | 1.2046 | 2.123 | 3.893 | 1.0017 | 1.1144 | 1.2825 | 1.1459 |

_{7}-_{22} |
2.3085 | 2.1881 | 2.3750 | 2.4442 | 1.3577 | 2.5091 | 2.5445 | 1.229 | 1.341 | 2.5107 | 2.8253 | 2.0126 | 3.3189 |

_{8}-_{20} |
1.4324 | 1.2223 | 1.4498 | 1.3416 | 1.5520 | 1.2656 | 1.4090 | 3.844 | 1.000 | 1.4780 | 1.4898 | 1.4237 | 1.5469 |

_{9}-_{18} |
1.6468 | 1.7033 | 1.4499 | 1.5724 | 1.6920 | 1.4866 | 1.4821 | 1.446 | 6.325 | 1.3159 | 1.3095 | 1.2992 | 1.4883 |

_{10}-_{19} |
2.8707 | 3.1885 | 2.5327 | 3.1202 | 1.7688 | 2.5584 | 2.4796 | 5.008 | 1.291 | 1.3959 | 2.5477 | 2.6817 | 2.3607 |

_{11}-_{17} |
1.5041 | 1.0100 | 1.2358 | 1.2143 | 2.9652 | 1.1977 | 1.1702 | 2.059 | 2.867 | 1.4379 | 1.1787 | 1.1981 | 1.1884 |

_{12}-_{15} |
1.3133 | 1.4074 | 1.3528 | 1.2954 | 1.0114 | 1.4003 | 1.3042 | 2.748 | 1.412 | 1.4078 | 1.2065 | 1.2815 | 1.1774 |

_{13}-_{16} |
2.3228 | 2.8499 | 2.9144 | 2.7997 | 1.0090 | 2.5323 | 2.3958 | 4.545 | 2.272 | 2.4685 | 2.1471 | 2.4616 | 2.3834 |

_{14} |
1.0426 | 1.0269 | 1.0085 | 1.0063 | 2.4601 | 1.0000 | 1.0000 | 1.484 | 2.983 | 1.2587 | 1.1687 | 1.0178 | 1.0093 |

_{3}, _{19} |
1.0969 | 0.8415 | 0.9482 | 1.0087 | 1.2300 | 0.9592 | 0.9703 | 1.056 | 1.000 | 1.0526 | 1.0161 | 1.0110 | 1.0006 |

_{5}, _{17} |
1.4556 | 1.2409 | 1.3439 | 1.3985 | 1.2064 | 1.3480 | 1.3614 | 1.555 | 1.665 | 1.4063 | 1.3578 | 1.4082 | 1.3707 |

_{7}, _{15} |
1.5954 | 1.4464 | 1.5043 | 1.5344 | 2.4245 | 1.5236 | 1.5318 | 1.627 | 1.982 | 1.6133 | 1.5158 | 1.5421 | 1.5137 |

_{9}, _{13} |
1.7655 | 1.5334 | 1.6350 | 1.6684 | 1.4618 | 1.6617 | 1.6602 | 1.881 | 1.965 | 1.8633 | 1.6175 | 1.6581 | 1.6200 |

_{11} |
1.8741 | 1.5971 | 1.7182 | 1.7137 | 1.4328 | 1.7431 | 1.7404 | 2.024 | 2.129 | 1.8244 | 1.6948 | 1.7188 | 1.6997 |

Best (kg) | 363.032 | 368.84 | 360.4 | 360.97 | 359.93 | 359.93 | 359.854 | 391.12 | 424.20 | 365.280 | 360.520 | 359.870 | 359.650 |

Mean (kg) | 381.2 | N/A | 362.21 | 362.65 | 360.23 | 360.93 | 360.261 | 405.77 | 449.73 | 371.975 | 361.759 | 361.976 | 359.985 |

SD (kg) | 4.26 | N/A | 1.68 | 1.30 | 0.24 | 0.65 | 0.097 | 8.82 | 11.24 | 4.731 | 1.330 | 1.830 | 0.287 |

NS | 125,000 | N/A | 6,000 | 4,000 | 10,000 | 12,800 | 20,000 | 6,000 | 6,000 | 6,000 | 6,000 | 6,000 | 6,000 |

Number
NHPGA
HS
DPSO
PSRO
HALC-PSO
CPA
STMP-TLBO
SCA
FA
MFA
HSCFA-1
HSCFA-2
HSCFA
1
20.0819
20.193
20.019
20.1023
20.0216
20.0000
20.0055
20.965
20.951
20.1693
20.0074
20.0238
20.0077
2
40.0961
40.416
40.011
40.0804
40.0098
40.0002
40.0015
43.653
41.868
40.2835
40.1420
40.0283
40.0180
3
60.0321
61.849
60.008
60.0516
60.0017
60.0024
60.0306
66.424
65.042
60.3482
60.0453
60.0442
60.0652
4
73.4648
76.886
76.990
75.8918
76.7857
77.3492
76.0899
86.760
91.214
75.5901
75.6668
76.4078
74.0695
5
88.7942
98.073
97.222
97.2470
96.3543
96.4671
96.2735
116.132
123.555
95.8378
97.0759
96.5769
95.0637

For this experiment, the population size is 10 and the iteration is 600 for SCA, FA, MFA, HSCFA-1, HSCFA-2, and HSCFA. It can be concluded from

The 52-bar dome truss including 8 size and 5 shape design variables is adopted as the fourth example. The geometry and support conditions are shown in

Parameters (unit) | Value |
---|---|

Modulus of elasticity ^{2}) |
2.1 × 10^{11} |

Material density ^{3}) |
7800 |

Lower bound of cross sections (cm^{2}) |
1 |

Upper bound of cross sections (cm^{2}) |
10 |

Added mass (kg) | 50 |

Constraints on the first two frequencies (Hz) | _{1} ≤ 15.916, _{2} ≥ 28.648 |

^{2}), |
NHPGA |
HS |
DPSO |
PSRO |
HALC-PSO [ |
CPA |
STMP-TLBO [ |
EFBI |
SCA | FA | MFA | HSCFA-1 | HSCFA-2 | HSCFA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

_{1-4} |
1 | 1.0085 | 1.0001 | 1.0007 | 1.0001 | 1.000 | 1.0005 | 1.0002 | 1.000 | 1.603 | 1.0189 | 1.0000 | 1.0014 | 1.000 |

_{5-8} |
2.142 | 1.4999 | 1.1397 | 1.0312 | 1.1654 | 1.1077 | 1.1005 | 1.1620 | 1.293 | 1.389 | 1.0757 | 1.0000 | 1.0093 | 1.085 |

_{9-16} |
1.486 | 1.3948 | 1.2263 | 1.2403 | 1.2323 | 1.1988 | 1.1881 | 1.1992 | 1.437 | 1.511 | 1.7228 | 1.1400 | 1.1887 | 1.203 |

_{17-20} |
1.402 | 1.3462 | 1.3335 | 1.3355 | 1.4323 | 1.4899 | 1.4705 | 1.4108 | 1.135 | 1.508 | 1.7027 | 1.4717 | 1.4933 | 1.452 |

_{21-28} |
1.911 | 1.6776 | 1.4161 | 1.5713 | 1.3901 | 1.9337 | 1.4212 | 1.3945 | 1.449 | 1.568 | 1.3407 | 1.4046 | 1.5178 | 1.421 |

_{29-36} |
1.011 | 1.3704 | 1.0001 | 1.0021 | 1.0001 | 1.0001 | 1.0000 | 1.0000 | 1.000 | 1.000 | 1.0107 | 1.0000 | 1.0000 | 1.000 |

_{37-44} |
1.469 | 1.4137 | 1.575 | 1.3267 | 1.6024 | 1.5998 | 1.4751 | 1.4876 | 1.417 | 1.857 | 1.3126 | 1.4653 | 1.4755 | 1.556 |

_{45-52} |
2.141 | 1.9378 | 1.4357 | 1.5653 | 1.4131 | 1.4135 | 1.4714 | 1.4899 | 1.568 | 1.849 | 1.6947 | 1.5354 | 1.4083 | 1.391 |

_{A} |
5.885 | 4.7374 | 6.1123 | 6.252 | 5.9362 | 5.9227 | 6.0207 | 6.0445 | 5.819 | 4.126 | 4.0423 | 6.0152 | 6.0426 | 6.002 |

_{B} |
1.762 | 1.5643 | 2.244 | 2.456 | 2.2416 | 2.3048 | 2.3090 | 2.2002 | 2.146 | 1.548 | 2.7944 | 2.4943 | 2.4604 | 2.304 |

_{B} |
4.409 | 3.7413 | 3.8321 | 3.826 | 3.7309 | 3.7061 | 4.0113 | 4.0032 | 3.874 | 3.951 | 3.7616 | 3.7000 | 3.7182 | 3.733 |

_{F} |
3.441 | 3.4882 | 4.0316 | 4.179 | 3.9630 | 3.9768 | 3.7424 | 3.8088 | 4.049 | 2.347 | 4.3068 | 4.0706 | 4.0815 | 3.998 |

_{F} |
3.187 | 2.6274 | 2.5036 | 2.501 | 2.5000 | 2.5001 | 2.5000 | 2.5000 | 2.519 | 2.810 | 2.5000 | 2.5000 | 2.5007 | 2.500 |

Best (kg) | 236.046 | 214.940 | 195.351 | 197.186 | 194.85 | 194.826 | 193.432 | 193.60 | 198.23 | 222.25 | 211.111 | 194.6235 | 194.5406 | 193.20 |

Mean(kg) | 274.164 | 229.88 | 198.71 | 213.42 | 196.85 | 198.81 | 197.23 | 194.17 | 216.75 | 283.32 | 246.74 | 200.68 | 199.00 | 198.53 |

SD (kg) | 37.462 | 12.44 | 13.85 | 10.11 | 2.38 | 3.71 | 3.698 | 0.48 | 16.77 | 32.76 | 31.93 | 7.89 | 4.53 | 3.01 |

NS | 13,519 | 20,000 | 6,000 | 4,000 | 7,500 | 12,800 | 20,000 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 |

Number
NHPGA
HS
DPSO
PSRO
HALC-PSO
CPA
STMP-TLBO
EFBI
SCA
FA
MFA
HSCFA-1
HSCFA-2
HSCFA
1
13.114
12.2222
11.3115
12.311
11.434
11.736
11.7075
11.2011
11.812
14.836
10.3058
12.4623
12.9066
11.667
2
29.356
28.6577
28.648
28.648
28.648
28.648
28.6480
28.6479
28.673
28.846
28.6817
28.6517
28.6504
28.648
3
29.356
28.6577
28.648
28.649
28.648
28.648
28.6480
28.6479
28.673
28.856
28.6817
28.6517
28.6504
28.648
4
30.270
28.6618
28.650
28.715
28.648
28.654
28.6505
28.6503
28.759
28.856
31.3955
28.8062
28.7968
28.648
5
30.992
30.0997
28.688
28.744
28.685
28.690
28.6518
28.6578
29.856
29.518
31.4684
28.9906
28.9297
28.683

For this experiment, the population size is 10 and the iteration is 1000 for SCA, FA, MFA, HSCFA-1, HSCFA-2, and HSCFA.

A new hybrid metaheuristic method HSCFA is proposed to address the shape and size optimization of truss structures with nonstructural masses under multiple frequency constraints. The modified FA, SCA, Lévy flight, elitist selection technique and adaptive penalty function are integrated to construct the new method. The modified FA improves the attraction model of the standard FA by reducing the attraction number of each firefly, thus the solution accuracy is improved. The modified FA also uses the parameter of SCA instead of the original randomization parameter, which strengthens the exploitation ability of the algorithm. Lévy flight is utilized to improve the population diversity of the algorithm during the search process. Elitist selection technique is introduced into HSCFA for population selection to accelerate the convergence rate. An adaptive penalty function method considering the iteration stage, the degree and the number of constraint violations is adopted to deal with the frequency constraints. HSCFA takes advantage of the modified FA, SCA and Lévy flight to update different solutions. The modification enhances the exploration and exploitation abilities of FA and SCA with the reduction of computational complexity.

Four shape and size truss optimization problems with multiple frequency constraints are used to test the performance of HSCFA. The results demonstrate HSCFA performs better than other algorithms in the literature and achieves significant improvement compared to SCA and FA. HSCFA can obtain the lightest designs and cost the least computational time. Consequently, HSCFA provides an efficient and competitive tool for shape and size optimization problems with frequency constraints.