# Truss Sizing Optimization with a Diversity-Enhanced Cyclic Neighborhood Network Topology Particle Swarm Optimizer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Optimization Framework

**Step 0. Transformation of the minimum weight problem into a pseudo-function-based problem.**

**Remark**

**1.**

**Step 1. Initialization of PSO parameters and start of the optimization procedure.**

**Remark**

**2.**

**Step 2. Apply the CNNT-PSO algorithm to update the positions ${\mathit{x}}_{i}^{\ell}$ and velocities ${\mathit{v}}_{i}^{\ell}$ of all particles.**

**Remark**

**3.**

**Remark**

**4.**

**Step 3. Termination criterion.**

## 3. Test Problems and Optimization Results

#### 3.1. Planar 10-Bar Truss Structure

#### 3.2. Spatial 25-Bar Truss Structure

- (a)
- Condition 1: 20 kips acting in the positive Y-direction and 5 kips acting in the negative Z-direction at node 1, 20 kips acting in the negative Y-direction and 5 kips acting in the negative Z-direction at node 2;
- (b)
- Condition 2: 1 kip acting in the positive X-direction at node 1, 0.5 kip acting in the positive X-direction at nodes 3 and 6, 10 kips acting in the positive Y-direction at nodes 1 and 2, 5 kips acting in the negative Z-direction at nodes 1 and 2.

**Remark**

**5.**

#### 3.3. Spatial 72-Bar Truss Structure

- (a)
- Condition 1: 5 kips acting in the positive X- and Y-directions, and in the negative Z-direction at node 17;
- (b)
- Condition 2: 5 kips acting in the negative Z-direction at nodes 17 through 20.

#### 3.4. Planar 200-Bar Truss Structure

- (a)
- Condition 1: 1 kip acting in the positive X-direction at nodes 1, 6, 15, 20, 29, 34, 43, 48, 57, 62, and 71;
- (b)
- Condition 2: 10 kips acting in the negative Y-direction at nodes 1–6, 8, 10, 12, 14–20, 22, 24, 26, 28–34, 36, 38, 40, 42–48, 50, 52, 54, 56–62, 64, 66, 68, and 70–75;
- (c)
- Condition 3: loading conditions 1 and 2 acting together.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**Convergence curve of the best trial run carried out for 200-bar truss problem (${n}_{s}=60$).

**Table 1.**Survey of literature studies that have analyzed the test cases considered in this research.

Problem | Investigated Optimization Techniques |
---|---|

10-bar planar truss | Harmony search (HS) algorithm [33] |

(Cases 1 and 2) | Particle swarm optimization (PSO) [7,8] |

Particle swarm optimization with passive congregation (PSOPC) [8] | |

Heuristic particle swarm optimization (HPSO) [8] | |

Heuristic particle swarm ant colony optimization (HPSACO) [34] | |

Improved harmony search (IHS) algorithm [35] | |

Artificial bee colony algorithm with an adaptive penalty function approach (ABC-AP) [36] | |

Efficient harmony search (EHS) algorithm [3] | |

Self-adaptive harmony search (SAHS) algorithm [3] | |

Teaching-learning-based optimization (TLBO) [1] | |

25-bar spatial bar | Harmony search (HS) algorithm [33] |

Particle swarm optimization (PSO) [8] | |

Particle swarm optimization with passive congregation (PSOPC) [8] | |

Heuristic particle swarm optimization (HPSO) [8] | |

Big bang-big crunch (BB-BC) optimization [37] | |

Hybrid big bang-big crunch (HBB-BC) optimization [38] | |

Improved harmony search (IHS) algorithm [35] | |

Corrected multi-level & multi-point simulated annealing (CMLPSA) algorithm [6] | |

Efficient harmony search (EHS) algorithm [3] | |

Self-adaptive harmony search (SAHS) algorithm [3] | |

Teaching-learning-based optimization (TLBO) [1] | |

72-bar spatial truss | Harmony search (HS) algorithm [33] |

Particle swarm optimization (PSO) [7,8] | |

Particle swarm optimization with passive congregation (PSOPC) [8] | |

Heuristic particle swarm optimization (HPSO) [8] | |

Big bang-big crunch (BB-BC) optimization [37] | |

Hybrid big bang-big crunch (HBB-BC) optimization [38] | |

Efficient harmony search (EHS) algorithm [3] | |

Self-adaptive harmony search (SAHS) algorithm [3] | |

Teaching-learning-based optimization (TLBO) [1] | |

200-bar planar truss | Harmony search (HS) algorithm [33] |

Particle swarm optimization with passive congregation (PSOPC) [34] | |

Heuristic particle swarm ant colony optimization (HPSACO) [34] | |

Corrected multi-level & multi-point simulated annealing (CMLPSA) algorithm [6] | |

Efficient harmony search (EHS) algorithm [3] | |

Self-adaptive harmony search (SAHS) algorithm [3] | |

Teaching-learning-based optimization (TLBO) [1] |

Optimal Design | Lee & | Perez & | Kaveh & | Lamberti & | Sonmez | Degertekin & | This Study | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

Variables | Geem: | Li et al. [8] | Behdinan: | Talatahari: | Pappalettere: | Degertekin [3] | [36] | Hayalioglu: | (${\mathit{n}}_{\mathit{s}}=20$) | ||

${\mathit{A}}_{\mathit{i}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbf{in}}^{2}\right)$ | HS [33] | PSOPC | HPSO | PSO [7] | HPSACO [34] | IHS [35] | EHS | SAHS | ABC-AP | TLBO [1] | CNNT-PSO |

${A}_{1}$ | 30.15 | 30.569 | 30.704 | 33.500 | 30.307 | 30.5222 | 30.208 | 30.394 | 30.548 | 30.4286 | 30.5171 |

${A}_{2}$ | 0.102 | 0.100 | 0.100 | 0.100 | 0.100 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 | 0.1000 |

${A}_{3}$ | 22.71 | 22.974 | 23.167 | 22.766 | 23.434 | 23.2005 | 22.698 | 23.098 | 23.180 | 23.2436 | 23.1999 |

${A}_{4}$ | 15.27 | 15.148 | 15.183 | 14.417 | 15.505 | 15.2232 | 15.275 | 15.491 | 15.218 | 15.3677 | 15.2198 |

${A}_{5}$ | 0.102 | 0.100 | 0.100 | 0.100 | 0.100 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 | 0.1000 |

${A}_{6}$ | 0.544 | 0.547 | 0.551 | 0.100 | 0.5241 | 0.5513 | 0.529 | 0.529 | 0.551 | 0.5751 | 0.5512 |

${A}_{7}$ | 7.541 | 7.493 | 7.460 | 7.534 | 7.4365 | 7.4572 | 7.558 | 7.488 | 7.463 | 7.4404 | 7.4574 |

${A}_{8}$ | 21.56 | 21.159 | 20.978 | 20.467 | 21.079 | 21.0367 | 21.559 | 21.189 | 21.058 | 20.9665 | 21.03931 |

${A}_{9}$ | 21.45 | 21.156 | 21.508 | 20.392 | 21.229 | 21.5288 | 21.491 | 21.342 | 21.501 | 21.5330 | 21.5310 |

${A}_{10}$ | 0.100 | 0.100 | 0.100 | 0.100 | 0.100 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 | 0.1000 |

Best weight (lb) | 5057.88 | 5061.00 | 5060.92 | 5024.21 | 5056.56 | 5060.82 | 5062.39 | 5061.42 | 5060.880 | 5060.96 | 5060.8540 |

Worst weight | N/A | N/A | N/A | N/A | N/A | N/A | 5065.83 | 5063.39 | N/A | 5063.23 | 5060.8743 |

Average weight | N/A | N/A | N/A | N/A | N/A | N/A | 5063.73 | 5061.95 | N/A | 5062.08 | 5060.8581 |

Standard deviation | N/A | N/A | N/A | N/A | N/A | N/A | 1.98 | 0.71 | N/A | 0.79 | 0.0054 |

Analysis of the optimization results using ANSYS/MATLAB programs | |||||||||||

Weight (lb) | 5058.336 | 5040.669 | 5060.906 | 5024.248 | 5056.59 | 5060.93 | 5062.39 | 5061.275 | 5060.888 | 5060.956 | 5060.8540 |

Feasibility | Infeasible | Infeasible | Feasible | Infeasible | Infeasible | Feasible | Feasible | Feasible | Feasible | Feasible | Feasible |

Infeasible node | $-2.0018$ | $-2.0085$ | None | $-2.0389$ | $-2.002$ | None | None | None | None | None | None |

displacement (in) ${}^{1}$ | (1st node) | (1st node) | (1st node) | (1st node) | |||||||

Infeasible bar | None | 25.00118 | None | 25.01712 | None | None | None | None | None | None | None |

stress (ksi) ${}^{2}$ | (5th bar) | (5th bar) |

Optimal Design | Lee & | Kaveh & | Sonmez | Degertekin & | This study | ||||
---|---|---|---|---|---|---|---|---|---|

Variables | Geem: | Li et al. [8] | Talatahari: | Degertekin [3] | [36] | Hayalioglu: | (${\mathit{n}}_{\mathit{s}}=20$) | ||

${\mathit{A}}_{\mathit{i}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbf{in}}^{2}\right)$ | HS [33] | PSOPC | HPSO | HPSACO [34] | EHS | SAHS | ABC-AP | TLBO [1] | CNNT-PSO |

${A}_{1}$ | 23.250 | 23.473 | 23.353 | 23.194 | 23.589 | 23.525 | 23.4692 | 23.524 | 23.5186 |

${A}_{2}$ | 0.1020 | 0.101 | 0.100 | 0.100 | 0.101 | 0.100 | 0.1005 | 0.1000 | 0.1000 |

${A}_{3}$ | 25.730 | 25.287 | 25.502 | 24.585 | 25.422 | 25.429 | 25.2393 | 25.441 | 25.2897 |

${A}_{4}$ | 14.510 | 14.413 | 14.250 | 14.221 | 14.488 | 14.488 | 14.354 | 14.479 | 14.3685 |

${A}_{5}$ | 0.1000 | 0.100 | 0.100 | 0.100 | 0.100 | 0.100 | 0.1001 | 0.1000 | 0.1000 |

${A}_{6}$ | 1.9770 | 1.969 | 1.972 | 1.969 | 1.975 | 1.992 | 1.9701 | 1.995 | 1.9697 |

${A}_{7}$ | 12.210 | 12.362 | 12.363 | 12.489 | 12.362 | 12.352 | 12.4128 | 12.334 | 12.3894 |

${A}_{8}$ | 12.610 | 12.694 | 12.984 | 12.925 | 12.682 | 12.698 | 12.8925 | 12.689 | 12.8299 |

${A}_{9}$ | 20.360 | 20.323 | 20.356 | 20.952 | 20.322 | 20.341 | 20.3343 | 20.354 | 20.3371 |

${A}_{10}$ | 0.100 | 0.103 | 0.1010 | 0.101 | 0.100 | 0.100 | 0.1000 | 0.1000 | 0.1000 |

Best weight (lb) | 4668.81 | 4677.70 | 4677.29 | 4675.78 | 4679.02 | 4678.84 | 4677.077 | 4678.31 | 4676.9239 |

Worst weight | N/A | N/A | N/A | N/A | 4684.28 | 4682.26 | N/A | 4681.23 | 4679.2153 |

Average weight | N/A | N/A | N/A | N/A | 4681.61 | 4680.08 | N/A | 4680.12 | 4677.0500 |

Standard deviation | N/A | N/A | N/A | N/A | 2.51 | 1.89 | N/A | 1.016 | 0.4491 |

Analysis of the optimization results using ANSYS/MATLAB | |||||||||

Weight (lb) | 4669.365 | 4667.76 | 4681.93 | 4675.797 | 4679.0148 | 4678.8476 | 4677.0754 | 4678.3149 | 4676.9239 |

Feasibility | Infeasible | Infeasible | Infeasible | Infeasible | Feasible | Feasible | Feasible | Feasible | Feasible |

Infeasible node | $-2.00399$ | $-2.005$ | None | $-2.00158$ | None | None | None | None | None |

displacement (in) | (2nd node) | (2nd node) | (2nd node) | ||||||

Infeasible bar | 25.04062 | 25.00876 | 25.07655 | 25.00189 | None | None | None | None | None |

stress (ksi) | (5th bar) | (6th bar) | (5th bar) | (6th bar) |

Neighbor Size: ${\mathit{n}}_{\mathit{s}}$ | Objective Function Value | ||||
---|---|---|---|---|---|

(Ratio of ${\mathit{n}}_{\mathit{s}}$ to ${\mathit{n}}_{\mathit{p}}$) | Best | Average | Worst | Standard Deviation | |

10 | (4.0%) | 5060.9167908101 | 5061.2950060723 | 5062.0421888195 | 0.2907511199 |

20 | (8.0%) | 5060.8540025711 | 5060.8580488412 | 5060.8742859840 | 0.0054030957 |

40 | (16.0%) | 5060.8536755940 | 5061.4278657355 | 5076.6697042583 | 2.8820669209 |

80 | (32.0%) | 5061.0075312501 | 5093.2458964858 | 5173.6500952290 | 30.7951400369 |

140 | (56.0%) | 5061.8921893369 | 5168.3643472361 | 5409.6582554952 | 85.3588142997 |

200 | (80.0%) | 5097.2427471776 | 5282.2050683960 | 5778.8410590943 | 189.9139754508 |

250 | (100.0%) | 5066.4836471986 | 5336.3103411734 | 5895.1836530033 | 180.8318776546 |

Neighbor Size: ${\mathit{n}}_{\mathit{s}}$ | Objective Function Value | ||||
---|---|---|---|---|---|

(Ratio of ${\mathit{n}}_{\mathit{s}}$ to ${\mathit{n}}_{\mathit{p}}$) | Best | Average | Worst | Standard Deviation | |

10 | (4.0%) | 4677.1658402493 | 4678.2375183454 | 4680.5386535189 | 0.9051300720 |

20 | (8.0%) | 4676.9238853154 | 4677.0500355970 | 4679.2153171228 | 0.4490860051 |

40 | (16.0%) | 4676.9228600423 | 4677.2863340068 | 4680.5057282055 | 0.8084333711 |

80 | (32.0%) | 4679.1222079698 | 4717.6958257250 | 4846.8397665392 | 39.7234590669 |

140 | (56.0%) | 4690.6747765927 | 4876.4393422376 | 5155.7746654222 | 144.7862819775 |

200 | (80.0%) | 4721.7716442903 | 5011.4643856148 | 5512.8166815800 | 210.2604407706 |

250 | (100.0%) | 4825.5787830425 | 5176.9813415952 | 5635.2466296128 | 232.7394293553 |

Member Group | Compressive Stress Limit, Ksi (MPa) | Tensile Stress Limit, Ksi (MPa) | |
---|---|---|---|

1 | ${A}_{1}$ | 35.092 (241.96) | 40.0 (275.80) |

2 | ${A}_{2}$∼${A}_{5}$ | 11.590 (79.913) | 40.0 (275.80) |

3 | ${A}_{6}$∼${A}_{9}$ | 17.305 (119.31) | 40.0 (275.80) |

4 | ${A}_{10}$∼${A}_{11}$ | 35.092 (241.96) | 40.0 (275.80) |

5 | ${A}_{12}$∼${A}_{13}$ | 35.092 (241.96) | 40.0 (275.80) |

6 | ${A}_{14}$∼${A}_{17}$ | 6.7590 (46.603) | 40.0 (275.80) |

7 | ${A}_{18}$∼${A}_{21}$ | 6.9590 (47.982) | 40.0 (275.80) |

8 | ${A}_{22}$∼${A}_{25}$ | 11.082 (76.410) | 40.0 (275.80) |

Optimal Design | Lee & | Camp | Kaveh & | Kaveh & | Lamberti & | Lamberti | Degertekin & | This study | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Variables | Geem: | Li et al. [8] | [37] | Talatahari: | Talatahari: | Pappalettere: | [6] | Degertekin [3] | Hayalioglu: | (${\mathit{n}}_{\mathit{s}}=20$) | ||

${\mathit{A}}_{\mathit{i}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbf{in}}^{2}\right)$ | HS [33] | PSOPC | HPSO | BB-BC | HPSACO [34] | HBB-BC [38] | IHS [35] | CMLPSA | EHS | SAHS | TLBO [1] | CNNT-PSO |

${A}_{1}$ | 0.047 | 0.010 | 0.010 | 0.010 | 0.010 | 2.6622 | 0.0100 | 0.0100 | 0.010 | 0.010 | 0.0100 | 0.0100 |

${A}_{2}$–${A}_{5}$ | 2.022 | 1.979 | 1.970 | 2.092 | 2.054 | 1.993 | 1.9871 | 1.9870 | 1.995 | 2.074 | 2.0712 | 1.9870 |

${A}_{6}$–${A}_{9}$ | 2.950 | 3.011 | 3.016 | 2.964 | 3.008 | 3.056 | 2.9935 | 2.9935 | 2.980 | 2.961 | 2.9570 | 2.9935 |

${A}_{10}$–${A}_{11}$ | 0.010 | 0.100 | 0.010 | 0.010 | 0.010 | 0.010 | 0.0100 | 0.0100 | 0.010 | 0.010 | 0.0100 | 0.0100 |

${A}_{12}$–${A}_{13}$ | 0.014 | 0.100 | 0.010 | 0.010 | 0.010 | 0.010 | 0.0100 | 0.0100 | 0.010 | 0.010 | 0.0100 | 0.0100 |

${A}_{14}$–${A}_{17}$ | 0.688 | 0.657 | 0.694 | 0.689 | 0.679 | 0.665 | 0.6839 | 0.6894 | 0.696 | 0.691 | 0.6891 | 0.6839 |

${A}_{18}$–${A}_{21}$ | 1.657 | 1.678 | 1.681 | 1.601 | 1.611 | 1.642 | 1.6769 | 1.6769 | 1.679 | 1.617 | 1.6209 | 1.6769 |

${A}_{22}$–${A}_{25}$ | 2.663 | 2.693 | 2.643 | 2.686 | 2.678 | 2.679 | 2.6622 | 2.6621 | 2.652 | 2.674 | 2.6768 | 2.6622 |

Best weight (lb) | 544.38 | 545.27 | 545.19 | 545.38 | 544.99 | 545.16 | 545.15 | 545.15 | 545.49 | 545.12 | 545.09 | 545.1627 |

Worst weight | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | 548.04 | 546.60 | 546.33 | 545.1627 |

Average weight | N/A | N/A | N/A | 545.78 | 545.52 | 545.66 | N/A | N/A | 546.52 | 545.94 | 545.41 | 545.1627 |

Standard deviation | N/A | N/A | N/A | 0.491 | 0.315 | 0.367 | N/A | N/A | 1.05 | 0.91 | 0.42 | $6.94\times {10}^{-6}$ |

Analysis of the optimization results using ANSYS/MATLAB | ||||||||||||

Weight (lb) | 544.36 | 547.96 | 545.238 | 545.52 | 544.991 | 565.03 | 545.1658 | 545.554 | 545.486 | 545.118 | 545.08 | 545.1627 |

Feasibility | Infeasible | Feasible | Feasible | Infeasible | Infeasible | Infeasible | Feasible | Infeasible | Feasible | Infeasible | Infeasible | Feasible |

Infeasible node | 0.350709 | None | None | None | 0.35004015 | 0.35001849 | None | None | None | None | None | None |

displacement (in) | (1st bar) | (1st node) | (1st node) | |||||||||

Infeasible bar | 6.973365 | None | None | 7.227284 | 7.2042322 | None | None | 6.9597203 | None | 7.166396 | 7.1501796 | None |

stress (ksi) ${}^{1}$ | (19th bar) | (19th bar) | (19th bar) | (19th bar) | (19th bar) | (19th bar) |

Neighbor Size: ${\mathit{n}}_{\mathit{s}}$ | Object. Funct. Value | ||||
---|---|---|---|---|---|

(Ratio of ${\mathit{n}}_{\mathit{s}}$ to ${\mathit{n}}_{\mathit{p}}$) | Best | Average | Worst | Standard Deviation | |

6 | (3.0%) | 545.2022914740 | 545.3321179916 | 545.5017575323 | 0.0826258622 |

10 | (5.0%) | 545.1629167122 | 545.1648971493 | 545.1687823043 | 0.0014564516 |

20 | (10.0%) | 545.1627105408 | 545.1627143990 | 545.1627412660 | 0.0000069387 |

50 | (25.0%) | 545.1627102443 | 545.1628795084 | 545.1655031932 | 0.0005456476 |

100 | (50.0%) | 545.1627427736 | 545.8373723082 | 549.1823298747 | 1.1695878525 |

150 | (75.0%) | 545.2296934557 | 552.9448115665 | 583.3440908564 | 10.1769607107 |

200 | (100.0%) | 545.8290039641 | 569.5171789211 | 607.8854689086 | 16.7379238301 |

Optimal Design | Lee & | Prez & | Camp | Kaveh & | Degertekin & | This Study | ||||
---|---|---|---|---|---|---|---|---|---|---|

Variables | Geem: | Behdinan: | Li et al. [8] | [37] | Talatahari: | Degertekin [3] | Hayalioglu: | (${\mathit{n}}_{\mathit{s}}=30$) | ||

${\mathit{A}}_{\mathit{i}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbf{in}}^{2}\right)$ | HS [33] | PSO [7] | PSOPC | HPSO | BB-BC | HBB-BC [38] | EHS | SAHS | TLBO [1] | CNNT-PSO |

${A}_{1}$–${A}_{4}$ | 1.790 | 1.7427 | 1.855 | 1.857 | 1.8577 | 1.9042 | 1.967 | 1.860 | 1.9064 | 1.8861 |

${A}_{5}$–${A}_{12}$ | 0.521 | 0.5185 | 0.504 | 0.505 | 0.5059 | 0.5162 | 0.510 | 0.521 | 0.50612 | 0.5123 |

${A}_{13}$–${A}_{16}$ | 0.100 | 0.1000 | 0.100 | 0.100 | 0.1000 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

${A}_{17}$–${A}_{18}$ | 0.100 | 0.1000 | 0.100 | 0.100 | 0.1000 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

${A}_{19}$–${A}_{22}$ | 1.229 | 1.3079 | 1.253 | 1.255 | 1.2476 | 1.2582 | 1.293 | 1.271 | 1.2617 | 1.2685 |

${A}_{23}$–${A}_{30}$ | 0.522 | 0.5193 | 0.505 | 0.503 | 0.5269 | 0.5035 | 0.511 | 0.509 | 0.5111 | 0.5117 |

${A}_{31}$–${A}_{34}$ | 0.100 | 0.1000 | 0.100 | 0.100 | 0.1000 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

${A}_{35}$–${A}_{36}$ | 0.100 | 0.1000 | 0.100 | 0.100 | 0.1012 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

${A}_{37}$–${A}_{40}$ | 0.517 | 0.5142 | 0.497 | 0.496 | 0.5209 | 0.5178 | 0.499 | 0.485 | 0.5317 | 0.5236 |

${A}_{41}$–${A}_{48}$ | 0.504 | 0.5464 | 0.508 | 0.506 | 0.5172 | 0.5214 | 0.501 | 0.501 | 0.51591 | 0.5171 |

${A}_{49}$–${A}_{52}$ | 0.100 | 0.1000 | 0.100 | 0.100 | 0.1004 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

${A}_{53}$–${A}_{54}$ | 0.101 | 0.1095 | 0.100 | 0.100 | 0.1005 | 0.1007 | 0.100 | 0.100 | 0.100 | 0.1000 |

${A}_{55}$–${A}_{58}$ | 0.156 | 0.1615 | 0.100 | 0.100 | 0.1565 | 0.1566 | 0.160 | 0.168 | 0.1562 | 0.1565 |

${A}_{59}$–${A}_{66}$ | 0.547 | 0.5092 | 0.525 | 0.524 | 0.5507 | 0.5421 | 0.522 | 0.584 | 0.54927 | 0.5456 |

${A}_{67}$–${A}_{70}$ | 0.442 | 0.4967 | 0.394 | 0.400 | 0.3922 | 0.4132 | 0.478 | 0.433 | 0.40966 | 0.4103 |

${A}_{71}$–${A}_{72}$ | 0.590 | 0.5619 | 0.535 | 0.534 | 0.5922 | 0.5756 | 0.591 | 0.520 | 0.56976 | 0.5697 |

- Best weight (lb) | 379.27 | 381.91 | 369.65 | 369.65 | 379.85 | 379.66 | 381.03 | 380.62 | 379.63 | 379.6148 |

Worst weight | N/A | N/A | N/A | N/A | N/A | N/A | 385.50 | 383.89 | 380.83 | 379.6155 |

Average weight | N/A | N/A | N/A | N/A | 382.08 | 381.85 | 383.51 | 382.42 | 380.20 | 379.6149 |

Standard deviation | N/A | N/A | N/A | N/A | 1.912 | 1.201 | 1.92 | 1.38 | 0.41 | 0.0002 |

Analysis of the optimization results using ANSYS/MATLAB | ||||||||||

Weight (lb) | 379.217 | 381.936 | 369.65 | 369.64 | 379.84 | 379.65 | 381.026 | 380.837 | 379.63494 | 379.6148 |

Feasibility | Infeasible | Feasible | Infeasible | Infeasible | Feasible | Feasible | Feasible | Feasible | Feasible | Feasible |

Infeasible node | 0.2505456 | None | 0.3114043 | 0.3115011 | None | None | None | None | None | None |

displacement (in) | (17th node) | (17th node) | (17th node) | |||||||

Infeasible bar | 25.01883 | None | 34.762759 | 34.768775 | None | None | None | None | None | None |

stress (ksi) | (55th bar) | (55th bar) | (55th bar) |

Neighbor Size: ${\mathit{n}}_{\mathit{s}}$ | Objective Function Value | ||||
---|---|---|---|---|---|

(Ratio of ${\mathit{n}}_{\mathit{s}}$ to ${\mathit{n}}_{\mathit{p}}$) | Best | Average | Worst | Standard Deviation | |

10 | (1.250%) | 379.6318052789 | 379.6653403456 | 379.7761760177 | 0.0330033631 |

30 | (3.750%) | 379.6148044971 | 379.6149401277 | 379.6155036251 | 0.0001752131 |

60 | (7.500%) | 379.6148633151 | 379.6495513185 | 379.8339175611 | 0.0568846942 |

100 | (12.50%) | 379.7268230162 | 387.4123385006 | 407.4333925023 | 7.6311956102 |

300 | (37.50%) | 390.9034206475 | 471.0343138592 | 570.4458234840 | 43.2075800681 |

400 | (50.00%) | 426.3544544514 | 513.5732153123 | 606.2598561574 | 46.7613059395 |

600 | (75.00%) | 459.8827786937 | 532.3907876752 | 637.1533325741 | 49.0785590185 |

800 | (100.0%) | 461.0964821247 | 569.2890375109 | 688.3641284045 | 54.5771993620 |

Design Variables | Member Number | Design Variables | Member Number |
---|---|---|---|

1 | 1, 2, 3, 4 | 16 | 82, 83, 85, 86, 88, 89, 91, 92, |

103, 104, 106, 107, 109, 110, | |||

2 | 5, 8, 11, 14, 17 | 17 | 112, 113, 115, 116, 117, 118 |

3 | 19, 20, 21, 22, 23, 24 | 18 | 119, 122, 125, 128, 131 |

4 | 18, 25, 56, 63, 94, 101, | 19 | 133, 134, 135, 136, 137, 138 |

132, 139, 170, 177 | |||

5 | 26, 29, 32, 35, 38 | 20 | 140, 143, 146, 149, 152 |

6 | 6, 7, 9, 10, 12, 13, 15, 16, | 21 | 120, 121, 123, 124, 126, 127, |

27, 28, 30, 31, 33, 34, 36, | 129, 130, 141, 142, 144, 145, | ||

37 | 147, 148, 150, 151 | ||

7 | 39, 40, 41, 42 | 22 | 153, 154, 155, 156 |

8 | 43, 46, 49, 52, 55 | 23 | 157, 160, 163, 166, 169 |

9 | 57, 58, 59, 60, 61, 62 | 24 | 171, 172, 173, 174, 175, 176 |

10 | 64, 67, 70, 73, 76 | 25 | 178, 181, 184, 187, 190 |

11 | 44, 45, 47, 48, 50, 51, 53, | 26 | 158, 159, 161, 162, 164, 165, |

54, 65, 66, 68, 69, 71, 72, | 167, 168, 179, 180, 182, 183, | ||

74, 75 | 185, 186, 188, 189 | ||

12 | 77, 78, 79, 80 | 27 | 191, 192, 193, 194 |

13 | 81, 84, 87, 90, 93 | 28 | 195, 197, 198, 200 |

14 | 95, 96, 97, 98, 99, 100 | 29 | 196, 199 |

15 | 102, 105, 108, 111, 114 |

Optimal Design Variables | Lee & Geem [33] | Kaveh & Talatahari [34] | Lamberti [6] | Degertekin [3] | Degertekin & Hayalioglu [1] | This Study (${\mathit{n}}_{\mathit{s}}=60$) | ||
---|---|---|---|---|---|---|---|---|

${\mathit{A}}_{\mathit{i}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbf{in}}^{2}\right)$ | HS | PSOPC | HPSACO | CMLPSA | EHS | SAHS | TLBO | CNNT-PSO |

1 | 0.1253 | 0.7590 | 0.1033 | 0.1468 | 0.150 | 0.154 | 0.146 | 0.1482 |

2 | 1.0157 | 0.9032 | 0.9184 | 0.9400 | 0.946 | 0.941 | 0.941 | 0.9405 |

3 | 0.1069 | 1.1000 | 0.1202 | 0.1000 | 0.101 | 0.100 | 0.100 | 0.1000 |

4 | 0.1096 | 0.9952 | 0.1009 | 0.1000 | 0.100 | 0.100 | 0.101 | 0.1000 |

5 | 1.9369 | 2.1350 | 1.8664 | 1.9400 | 1.945 | 1.942 | 1.941 | 1.9408 |

6 | 0.2686 | 0.4193 | 0.2826 | 0.2962 | 0.296 | 0.301 | 0.296 | 0.2975 |

7 | 0.1042 | 1.0041 | 0.1000 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

8 | 2.9731 | 2.8052 | 2.9683 | 3.1042 | 3.161 | 3.108 | 3.121 | 3.1067 |

9 | 0.1309 | 1.0344 | 0.1000 | 0.1000 | 0.102 | 0.100 | 0.100 | 0.1000 |

10 | 4.1831 | 3.7842 | 3.9456 | 4.1042 | 4.199 | 4.106 | 4.173 | 4.1067 |

11 | 0.3967 | 0.5269 | 0.3742 | 0.4034 | 0.401 | 0.409 | 0.401 | 0.4057 |

12 | 0.4416 | 0.4302 | 0.4501 | 0.1912 | 0.181 | 0.191 | 0.181 | 0.1897 |

13 | 5.1873 | 5.2683 | 4.96029 | 5.4284 | 5.431 | 5.428 | 5.423 | 5.4343 |

14 | 0.1912 | 0.9685 | 1.0738 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

15 | 6.2410 | 6.0473 | 5.9785 | 6.4284 | 6.428 | 6.427 | 6.422 | 6.4340 |

16 | 0.6994 | 0.7825 | 0.78629 | 0.5734 | 0.571 | 0.581 | 0.571 | 0.5745 |

17 | 0.1158 | 0.5920 | 0.73743 | 0.1327 | 0.156 | 0.151 | 0.156 | 0.1366 |

18 | 7.7643 | 8.1858 | 7.3809 | 7.9717 | 7.961 | 7.973 | 7.958 | 7.9803 |

19 | 0.1000 | 1.0362 | 0.66740 | 0.1000 | 0.100 | 0.100 | 0.100 | 0.1000 |

20 | 8.8279 | 9.2062 | 8.3000 | 8.9717 | 8.959 | 8.974 | 8.958 | 8.9802 |

21 | 0.6986 | 1.4774 | 1.19672 | 0.7049 | 0.722 | 0.719 | 0.720 | 0.71089 |

22 | 1.5563 | 1.8336 | 1.0000 | 0.4196 | 0.491 | 0.422 | 0.478 | 0.4659 |

23 | 10.9806 | 10.6110 | 10.8262 | 10.8636 | 10.909 | 10.892 | 10.897 | 10.9110 |

24 | 0.1317 | 0.9851 | 0.1000 | 0.1000 | 0.101 | 0.100 | 0.100 | 0.1000 |

25 | 12.1492 | 12.5090 | 11.6976 | 11.8606 | 11.985 | 11.887 | 11.897 | 11.9112 |

26 | 1.6373 | 1.9755 | 1.3880 | 1.0339 | 1.084 | 1.040 | 1.080 | 1.0712 |

27 | 5.0032 | 4.5149 | 4.9523 | 6.6818 | 6.464 | 6.646 | 6.462 | 6.5030 |

28 | 9.3545 | 9.8000 | 8.8000 | 10.8113 | 10.802 | 10.804 | 10.799 | 10.7210 |

29 | 15.0919 | 14.5310 | 14.6645 | 13.8404 | 13.936 | 13.870 | 13.922 | 13.9310 |

Best weight (lb) | 25,447.1 | 28,537.8 | 25,156.5 | 25,445.63 | 25,542.5 | 25,491.9 | 25,488.15 | 25,453.0957 |

Worst weight | N/A | N/A | N/A | N/A | 25,838.2 | 25,799.3 | 25,563.05 | 25,466.0958 |

Average weight | N/A | N/A | N/A | N/A | 25,659.71 | 25,610.2 | 25,533.14 | 25,459.1089 |

Standard deviation | N/A | N/A | N/A | N/A | 164.17 | 141.85 | 27.44 | 3.1544 |

Analysis of the optimization results using ANSYS/MATLAB programs | ||||||||

Weight (lb) | 25,447.5276 | 28,571.4343 | 25,155.6741 | 25,445.9597 | 25,537.0548 | 25,491.9226 | 25,488.1788 | 25,453.0957 |

Feasibility | Infeasible | Infeasible | Infeasible | Infeasible | Feasible | Feasible | Feasible | Feasible |

Infeasible bar | 10,369.318165 | −10,744.977441 | −10,996.947443 | 1007.084660 | None | None | None | None |

stress (ksi) | (122th bar) | (76th bar) | (184th bar) | (120th bar) |

Neighbor Size: ${\mathit{n}}_{\mathit{s}}$ | Objective Function Value | ||||
---|---|---|---|---|---|

(Ratio of ${\mathit{n}}_{\mathit{s}}$ to ${\mathit{n}}_{\mathit{p}}$) | Best | Average | Worst | Standard Deviation | |

30 | (3.0%) | 25457.3759759502 | 25476.9767631906 | 25514.0491597601 | 17.0656154436 |

60 | (6.0%) | 25453.0957113131 | 25459.1089220335 | 25466.0958098744 | 3.1543853566 |

100 | (10.0%) | 25453.8020671454 | 25463.0508675004 | 25474.0057600994 | 5.2856406250 |

250 | (25.0%) | 25465.3248022958 | 25495.9601387794 | 25519.0074637713 | 14.5041309227 |

500 | (50.0%) | 25489.1260391797 | 25534.2209378792 | 25574.0068587098 | 23.1527113502 |

750 | (75.0%) | 25513.2968909123 | 25574.0489259354 | 25607.9003950622 | 24.8296616471 |

1000 | (100.0%) | 25551.3127913481 | 25613.7785004692 | 25651.3237014500 | 28.1973072143 |

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## Share and Cite

**MDPI and ACS Style**

Kim, T.-H.; Byun, J.-I.
Truss Sizing Optimization with a Diversity-Enhanced Cyclic Neighborhood Network Topology Particle Swarm Optimizer. *Mathematics* **2020**, *8*, 1087.
https://doi.org/10.3390/math8071087

**AMA Style**

Kim T-H, Byun J-I.
Truss Sizing Optimization with a Diversity-Enhanced Cyclic Neighborhood Network Topology Particle Swarm Optimizer. *Mathematics*. 2020; 8(7):1087.
https://doi.org/10.3390/math8071087

**Chicago/Turabian Style**

Kim, Tae-Hyoung, and Jung-In Byun.
2020. "Truss Sizing Optimization with a Diversity-Enhanced Cyclic Neighborhood Network Topology Particle Swarm Optimizer" *Mathematics* 8, no. 7: 1087.
https://doi.org/10.3390/math8071087