Large-Scale Truss Topology and Sizing Optimization by an Improved Genetic Algorithm with Multipoint Approximation
Abstract
:1. Introduction
2. Problem Formulation
2.1. Optimization Model of a Large-Scale Truss
2.2. Further Explanations for the Model
3. Optimization Scheme
3.1. Flowchart for the Optimization Scheme
3.2. The First-Level Approximation Problems by BMA
3.3. The Label–Clip–Slice Strategy and GA to Address Mixed Variables
3.3.1. Labeling of the Topology Variables
3.3.2. Clipping of the Chromosome of the Last Best Individual
3.3.3. Generating of the Initial Population
- (a)
- The optimized individual that is obtained in previous iterations;
- (b)
- Individuals that randomly mutated from the optimized topology vector ;
- (c)
- Individuals that are calculated according to the optimized size vector .
3.3.4. Splicing of the Individuals in the Current Population
3.3.5. Fitness Calculation and Sizing Optimization
3.4. Analysis and Discussion
4. Numerical Examples
4.1. Example 1: 15-Bar Planar Truss
4.2. Example 2: A Cantilever Truss
4.3. Example 3: A Michell Truss
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Design Domain | Cross-Sectional Type | Topology Variables | Size Variables |
---|---|---|---|
Truss | BAR | αi (i = 1, 2, 3, 4, 5, 6, 7) | xi (i = 1, 2, 3, 4, 5, 6, 7) |
Design Variables | Data from [29] | Data from [30] | Present |
---|---|---|---|
A1 | 1.081 | 0.954 | 1.60 |
A2 | 0.539 | 0.539 | 1.59 |
A3 | 0 | 0.141 | 0 |
A4 | 1.081 | 0.954 | 2.39 |
A5 | 0.954 | 0.539 | 0.80 |
A6 | 0.440 | 0.287 | 0.79 |
A7 | 0 | 0.141 | 0 |
A8 | 0.141 | 0 | 0 |
A9 | 0 | 3.813 | 0 |
A10 | 0.270 | 0.440 | 1.13 |
A11 | 0.270 | 0.440 | 0.20 |
A12 | 0.539 | 0.220 | 0.20 |
A13 | 0.141 | 0.220 | 1.13 |
A14 | 0.440 | 0.347 | 1.13 |
A15 | 0 | 0.141 | 0 |
Iteration number | 7 | ||
Labeled variables | 5 | ||
Critical constraint | −0.00088 | −0.0649 | 0.000 |
Weight (lb.) | 77.84 | 74.33 | 75.376 |
Percentage difference (%) | 5.57 | 0.23 | — |
Structural analysis | 8000 | 8000 | 15 |
Percentage difference (%) | 0.19 | 0.19 | — |
With LCS | Without LCS (No Convergence) | |
---|---|---|
Iteration number | 56 | 100 |
Labeled variables | 51 | / |
Critical constraint | 0.000 | 5.492 × 10−2 |
Weight | 0.525 | 0.517 |
Structural analysis | 66 | 117 |
NO. | Area LCS/No LCS | No. | Area LCS/No LCS | No. | Area LCS/No LCS | |||
---|---|---|---|---|---|---|---|---|
1 | 0.1830 | 0.4293 | 15 | 0 | 0.1 | 43 | 0 | 1.0140 |
2 | 0 | 0.1 | 16–18 | 0 | 0 | 44–47 | 0 | 0 |
3 | 0.4113 | 0.2727 | 19 | 0.1 | 0 | 48 | 0.1 | 1.3853 |
4 | 0.5575 | 0.3942 | 20–26 | 0 | 0 | 49 | 1.4154 | 0.1386 |
5 | 0.4056 | 0.1349 | 27–28 | 0 | 0.1 | 50–51 | 0 | 0 |
6 | 0 | 0.1 | 29–35 | 0 | 0 | 52 | 0 | 0.1 |
7 | 0 | 0 | 36 | 0 | 0.1 | 53 | 0 | 0 |
8 | 0 | 0.1 | 37 | 0 | 0.4385 | 54 | 1.6756 | 0 |
9–11 | 0 | 0 | 38 | 0 | 0.1 | 55–56 | 0 | 0.1 |
12 | 0 | 0.1470 | 39 | 0 | 0 | 57–61 | 0 | 0 |
13 | 0 | 0 | 40 | 0.4103 | 0 | 62 | 0 | 0.1026 |
14 | 0.1826 | 0.1860 | 41–42 | 0 | 0 | 63 | 0 | 0.1414 |
Literature | Present | |
---|---|---|
Iteration number | 124 | |
Labeled variables | / | 314 |
Critical constraint | 0.000 | 0.000 |
Weight | / | 50.522 |
Structural analysis | 10,783 | 139 |
Percentage difference (%) | 1.29 | - |
No. | Area | No. | Area | No. | Area | No. | Area |
---|---|---|---|---|---|---|---|
1–12 | 0 | 56 | 0.9887 | 96 | 0.6406 | 171 | 0.1 |
13 | 0.2199 | 57–60 | 0 | 97–128 | 0 | 172–238 | 0 |
14–19 | 0 | 61 | 0.3213 | 129 | 1.4555 | 239 | 2.847 |
20 | 0.9186 | 62–63 | 0 | 130 | 0 | 240–317 | 0 |
21–35 | 0 | 64 | 0.3215 | 131 | 0.5037 | 318 | 0.1578 |
36 | 0.4054 | 65 | 0.7449 | 132 | 0 | 319–331 | 0 |
37–43 | 0 | 66–88 | 0 | 133 | 0.957 | ||
44 | 0.1221 | 89 | 0.6388 | 134–169 | 0 | ||
45–55 | 0 | 90–95 | 0 | 170 | 0.1438 |
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Dong, T.; Chen, S.; Huang, H.; Han, C.; Dai, Z.; Yang, Z. Large-Scale Truss Topology and Sizing Optimization by an Improved Genetic Algorithm with Multipoint Approximation. Appl. Sci. 2022, 12, 407. https://doi.org/10.3390/app12010407
Dong T, Chen S, Huang H, Han C, Dai Z, Yang Z. Large-Scale Truss Topology and Sizing Optimization by an Improved Genetic Algorithm with Multipoint Approximation. Applied Sciences. 2022; 12(1):407. https://doi.org/10.3390/app12010407
Chicago/Turabian StyleDong, Tianshan, Shenyan Chen, Hai Huang, Chao Han, Ziqi Dai, and Zihan Yang. 2022. "Large-Scale Truss Topology and Sizing Optimization by an Improved Genetic Algorithm with Multipoint Approximation" Applied Sciences 12, no. 1: 407. https://doi.org/10.3390/app12010407