Optimization and Analysis of Plates with a Variable Stiffness Distribution in Terms of Dynamic Properties
Abstract
:1. Introduction
2. Methods
2.1. Theoretical Background
- Straight lines normal to the mid-surface remain straight and normal after deformation.
- The plate thickness remains constant during deformation.
- Transverse shear deformations are negligible, εzz = 0.
2.2. Problem Solution Framework
2.2.1. Numerical Solution
2.2.2. Optimization Technique
2.2.3. Application Parameters
3. Results
3.1. Frequency Optimization
3.1.1. Optimization Process
3.1.2. Optimization Outcomes
3.2. Dynamic Response Analysis
3.3. Natural Forms
3.4. Gradient-Based Optimization Tool
4. Discussion
5. Conclusions
- The application of GAs proved to be effective in optimizing the plates for maximizing the gaps between natural frequencies. Optimized structural elements exhibited increased gaps between adjacent natural frequencies, according to the defined fitness function.
- Unlike prior studies focusing on infinite periodic plates or single-unit-cell models, this work introduces a GA tailored for finite plates with realistic boundary conditions. By enforcing symmetry constraints in chromosome encoding and integrating FEM-based dynamic analysis, the algorithm achieves up to 34.85% higher objective function values than gradient-based method (SLSQP) for complex cases.
- The GA’s mutation strategies and crossover mechanisms are explicitly defined, addressing a gap in the prior literature. This transparency enables reproducibility and adaptation for similar structural optimization problems.
- Faster convergence in the optimization process is observed when optimizing for lower natural frequencies compared to higher frequencies.
- For lower frequencies (Δω1), the thickness distribution is simpler, covering larger regions, whereas for higher frequencies (Δω2, Δω3), the patterns are more complex with varied thicknesses.
- The optimized plates show a structure related to vibration modes, which is especially visible at higher frequencies. This correlation could enable targeted stiffening of nodal regions.
- The optimization process had a complex impact also on the amplitude response.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Lx × Ly = 1 × 1 | |||||
---|---|---|---|---|---|
K | The Individual | Plate CFFF | Plate SSSS | ||
Δωk | Difference | Δωk | Difference | ||
1 | reference | 0.59 | 50.85% | 0.60 | 33.33% |
optimized | 0.89 | 0.80 | |||
2 | reference | 0.60 | 30.00% | 0.00 | - |
optimized | 0.78 | 0.58 | |||
3 | reference | 0.21 | 257.14% | 0.36 | 83.33% |
optimized | 0.75 | 0.66 |
Lx × Ly = 2 × 1 | |||||
---|---|---|---|---|---|
K | The Individual | Plate CFFF | Plate SSSS | ||
Δωk | Difference | Δωk | Difference | ||
1 | reference | 0.77 | 20.78% | 0.37 | 102.70% |
optimized | 0.93 | 0.75 | |||
2 | reference | 0.31 | 125.81% | 0.38 | 84.21% |
optimized | 0.70 | 0.70 | |||
3 | reference | 0.55 | 52.73% | 0.24 | 166.67% |
optimized | 0.84 | 0.64 |
Lx × Ly = 1 × 1 | ||||||
---|---|---|---|---|---|---|
k | Plate CFFF | Plate SSSS | ||||
Δωk | Difference | Δωk | Difference | |||
GA | SciPy | GA | SciPy | |||
1 | 0.89 | 0.88 | 1.12% | 0.80 | 0.78 | 2.50% |
2 | 0.78 | 0.75 | 3.85% | 0.58 | 0.38 | 34.48% |
3 | 0.75 | 0.77 | 2.67% | 0.66 | 0.43 | 34.85% |
Lx × Ly = 2 × 1 | ||||||
---|---|---|---|---|---|---|
k | Plate CFFF | Plate SSSS | ||||
Δωk | Difference | Δωk | Difference | |||
GA | SciPy | GA | SciPy | |||
1 | 0.93 | 0.89 | 4.30% | 0.75 | 0.74 | 1.33% |
2 | 0.70 | 0.69 | 1.43% | 0.70 | 0.70 | ≪1.00% |
3 | 0.84 | 0.77 | 8.33% | 0.64 | 0.62 | 3.13% |
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Domagalski, Ł.; Kowalczyk, I. Optimization and Analysis of Plates with a Variable Stiffness Distribution in Terms of Dynamic Properties. Materials 2025, 18, 2150. https://doi.org/10.3390/ma18092150
Domagalski Ł, Kowalczyk I. Optimization and Analysis of Plates with a Variable Stiffness Distribution in Terms of Dynamic Properties. Materials. 2025; 18(9):2150. https://doi.org/10.3390/ma18092150
Chicago/Turabian StyleDomagalski, Łukasz, and Izabela Kowalczyk. 2025. "Optimization and Analysis of Plates with a Variable Stiffness Distribution in Terms of Dynamic Properties" Materials 18, no. 9: 2150. https://doi.org/10.3390/ma18092150
APA StyleDomagalski, Ł., & Kowalczyk, I. (2025). Optimization and Analysis of Plates with a Variable Stiffness Distribution in Terms of Dynamic Properties. Materials, 18(9), 2150. https://doi.org/10.3390/ma18092150