Multi-Objective Optimization Based on Kriging Surrogate Model and Genetic Algorithm for Stiffened Panel Collapse Assessment
Abstract
:1. Introduction
2. Ultimate Strength of Stiffened Panel Under Combined Loading
2.1. Case Study: Bulk Carrier Stiffened Panels
2.2. Geometrical Initial Imperfections
- A1–A4 border: ;
- A2–A3 border: ;
- A1–A2 and A3–A4 border: ;
- C1–C4 and C2–C3 for plate nodes: ;
- C1–C4 and C2–C3 for stiffener web: ;
- B1–B2 and B3–B4: ;
2.3. The FE Description
3. Genetic Algorithm Multi-Objective Optimization Using Kriging Surrogate
3.1. Structural Collapse Assessment Using GA-MOEA Kriging Surrogate Model
3.2. Kriging Surrogate Model
3.3. Genetic Algorithm Optimization
3.4. Multi-Objective Optimization Problem Genetic Algorithm
4. Analysis of Results
4.1. FE Model Validation
4.2. FE Model Verification
4.3. Kriging Surrogate Model for Collapse Assessment
4.4. Kriging Hyperparameter Estimation
4.5. Multi-Objective Optimization in Ultimate Strength Behavior
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
a | Length of local rectangular plates |
A0 | Local plate initial deflection amplitude |
b | Width of local rectangular plates |
B | Width of the stiffened panels between girders |
B0 | Column-type initial deflection amplitude |
bf | Flange breadth |
C0 | Side-way initial deflection amplitude |
CD | Crowding-distance value |
cov | Covariance |
Dtest | Observed test dataset |
Dtrain | Observed dataset |
dw | Web height |
E | Young’s modulus |
f(x) | Function |
FEA | Finite element analysis |
G(x) | Regression basis function |
g(x) | Multi-objective constraint |
GA | Genetic Algorithm |
H’ | Strain hardening rate |
m | Number of semi-waves in the plate |
M | Mass |
MLE | Maximum Likelihood Estimation |
MOEA | Multi-objective Evolutionary Algorithm |
MOP | Multi-objective optimization problem |
MRE | Mean relative error |
NSGA | Non-Sorting Genetic Algorithm |
ntest | Observed test dataset size |
ntrain | Observed dataset size |
p | Kriging hyperparameter |
P | Stiffened panel specimen |
Q | Lateral pressure |
rx, ry, and rz | Rotational displacements around x, y, and z directions |
tf | Flange thickness |
tp | Plate thickness |
tw | Web thickness |
ux, uy and uz | Translational displacements in x, y, and z directions |
w0c | Column-type deflection of stiffeners |
w0p | Initial deflection of local plate panel |
w0s | Side-way deflection of stiffeners |
X0 | Observed data |
Y0 | Observed responses |
Z(x) | Regression Gaussian process error |
β | Plate slenderness |
η | Regression coefficient |
θ | Kriging hyperparameter |
μ | Mean |
σ2 | Variance |
σY | Yield stress |
σzu | Ultimate strength |
υ | Poisson’s ratio |
ψ | Correlation |
Ψ | Correlation matrix |
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Variable | Units | Description | Lower Bound | Upper Bound |
---|---|---|---|---|
tp | mm | Plate thickness | 8.0 | 30.0 |
dw | mm | Web height | 100.0 | 400.0 |
tf | mm | Flange thickness | 8.0 | 20.0 |
bf | mm | Flange breadth | 80.0 | 200.0 |
tw | mm | Web thickness | 8.0 | 16.0 |
Specimen | q (MPa) | |||
0.0 | 0.1 | 0.2 | 0.3 | |
MRE (%) | ||||
P(9.5)(383 × 12 + 100 × 17) | 1.96 | 1.22 | 0.83 | 0.37 |
P(22)(138 × 9 + 90 × 12) | 2.77 | 0.24 | 2.81 | 0.01 |
ntrain | Population Size | Elite Fraction (%) | Crossover Fraction | Objective Function (MLE) | Computational Time (seg) | Generations | Ncall |
---|---|---|---|---|---|---|---|
100 | 100 | 5 | 0.7 | 282.4 | 47 | 86 | 8275 |
200 | 100 | 5 | 0.6 | 583.3 | 162 | 82 | 7895 |
300 | 100 | 5 | 0.7 | 894.0 | 514 | 97 | 9320 |
400 | 200 | 5 | 0.8 | 1295 | 6981 | 81 | 15,600 |
500 | 200 | 5 | 0.8 | 1561.1 | 11,271 | 94 | 18,070 |
Scenario | Population Size | Crossover Fraction | Mutation Rate | Generations | Ncall | Time |
---|---|---|---|---|---|---|
1 | 200 (50) | 0.7 (0.8) | 0.3 (0.3) | 102 (112) | 20,400 (5600) | 30.55 (9.33) |
2 | 200 (50) | 0.7 (0.9) | 0.3 (0.4) | 102 (106) | 20,400 (5300) | 30.85 (9.84) |
3 | 200 (50) | 0.9 (0.8) | 0.4 (0.2) | 102 (102) | 20,400 (5100) | 30.65 (8.55) |
Scenario (MPa) | x* | ||||||
---|---|---|---|---|---|---|---|
tp (mm) | dw (mm) | tf (mm) | bf (mm) | tw (mm) | q (MPa) | M (kg) | |
0.0 ≤ q1 ≤ 0.1 | 28.83 | 222.48 | 12.74 | 138.83 | 8.00 | 0.09 | 6480.0 |
0.1 < q2 ≤ 0.2 | 28.68 | 183.46 | 13.24 | 117.84 | 8.00 | 0.13 | 6356.0 |
0.2 < q3 ≤ 0.3 | 30.00 | 184.00 | 13.38 | 144.40 | 11.41 | 0.20 | 6740.0 |
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Lima, J.P.S.; Vieira, R.L.; dos Santos, E.D.; Rocha, L.A.O.; Isoldi, L.A. Multi-Objective Optimization Based on Kriging Surrogate Model and Genetic Algorithm for Stiffened Panel Collapse Assessment. Appl. Mech. 2025, 6, 34. https://doi.org/10.3390/applmech6020034
Lima JPS, Vieira RL, dos Santos ED, Rocha LAO, Isoldi LA. Multi-Objective Optimization Based on Kriging Surrogate Model and Genetic Algorithm for Stiffened Panel Collapse Assessment. Applied Mechanics. 2025; 6(2):34. https://doi.org/10.3390/applmech6020034
Chicago/Turabian StyleLima, João Paulo Silva, Raí Lima Vieira, Elizaldo Domingues dos Santos, Luiz Alberto Oliveira Rocha, and Liércio André Isoldi. 2025. "Multi-Objective Optimization Based on Kriging Surrogate Model and Genetic Algorithm for Stiffened Panel Collapse Assessment" Applied Mechanics 6, no. 2: 34. https://doi.org/10.3390/applmech6020034
APA StyleLima, J. P. S., Vieira, R. L., dos Santos, E. D., Rocha, L. A. O., & Isoldi, L. A. (2025). Multi-Objective Optimization Based on Kriging Surrogate Model and Genetic Algorithm for Stiffened Panel Collapse Assessment. Applied Mechanics, 6(2), 34. https://doi.org/10.3390/applmech6020034