Optimized ANN Model for Predicting Buckling Strength of Metallic Aerospace Panels Under Compressive Loading
Abstract
1. Introduction
2. Materials and Methods
2.1. Finite Element Setup
2.2. Artificial Neural Network
2.3. Decision Tree
2.4. KNN
2.5. Random Forest Regression
2.6. XGBoost
2.7. CatBoost
2.8. LightGBM
3. Data Acquisition and Preparation
3.1. Latin Hypercube Sampling
3.2. Data Processing
3.2.1. One Hot Encoding
3.2.2. MinMax Scalar
3.2.3. Performance Evaluation Metrics
3.2.4. Hyperparameter Optimization for ML and ANN
3.3. Overfitting Control and Generalization
4. Performance Optimization of ANN Model
4.1. Activation Function
4.2. Neuron Architecture
4.3. Batch Size, Optimizers, and Learning Rates
4.3.1. Batch Size 4
4.3.2. Batch Size 8
4.3.3. Batch Size 12
4.3.4. Batch Size 16
5. Comparing ANN and ML Models
Feature Importance Analysis
6. Limitations and Future Scope
6.1. Direction 1: Advanced Models
6.2. Direction 2: Expanding Data Size
6.3. Direction 3: Complex Neural Network Architecture
6.4. Direction 4: Generalization and Transfer Learning
7. Conclusions
- ANN models performed better with a 200/100/50/25/12/6/3/1 neural configuration. For this study, the best activation configuration is selu/relu/relu/relu/linear/linear/selu/1.
- ANN models with the Nadam optimizer performed significantly better than others in this study.
- Normalization or feature scaling significantly improves the model’s prediction, which this study endorses. Normalization improves MSE values by up to 53% and MAE values by up to 32.5%.
- The Nadam optimizer, with a learning rate of 0.0025 and batch size of 8, outperformed all other ANN configurations and ML algorithms. The best model configuration achieves an MSE value of 2.9584, an MAE value of 0.9875, an RMSE value of 1.72, and an R2 score of 0.9998. It significantly outperforms other models.
- This proposed ANN model significantly outperforms other classical algorithms. Compared to the best-performing classical model (CatBoost), our proposed ANN model has an almost 6-fold better MSE value, a 2.68-fold better MAE value, a 2.5-fold better RMSE value, and a nearly identical R2 score.
- Taylor’s diagram highlights that all ANN models (three different optimizers with the best results) had better training and testing performance, indicating these ANN models have generalized well to the unseen data and have not been overfitted.
- SHAP analysis reveals that stiffener height and mass are the most influential features affecting critical buckling load, highlighting the importance of geometric optimization for enhancing panel stability.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Density (kg/m3) | Elastic Modulus (GPa) | Poisson’s Ratio |
---|---|---|
2780 | 73.7 | 0.33 |
AF | Description | Advantages | Formula |
---|---|---|---|
Sigmoid | S-shaped function, compresses input values to a range of 0 to 1. | Suitable for binary classification. | |
tanh [45] | S-shaped function; output range is −1 to 1. | Zero–centered output, helps in gradient descent optimization. | |
ReLU (Rectified Linear Unit) [46,47,48] | Activates the positive inputs in linear fashion, and outputs 0 for negative inputs. | Can learn the nonlinearity of the dataset; addresses the vanishing gradient. | |
ELU (Exponential Linear Unit) [46] | Like ReLU, utilizes the positive input but also uses the negative inputs to push the average unit activation value close to zero. | Speeds up learning process; has the advantages of ReLU but can handle negative values. | |
SELU (Scaled ELU) | Self-normalizes by scaling ELU outputs to maintain a mean close to zero and variance close to one during training | Simple activation function; prevents dying neurons; faster convergence during training. | |
Linear | Outputs values directly without transforming the input values | Typically, suitable for regression tasks. | |
Softmax [49] | converts calculated outputs in interpretable probabilities that sum to one | Suitable for multi-class classification. |
Input Variables | Upper Bound (mm) | Lower Bound (mm) | Sample Size for Each Grid |
---|---|---|---|
Pt | 3.5 | 1.75 | 1000 |
St | 4 | 2 | |
Sh | 40 | 20 | |
Mplate | Calculated by Ansys based on Pt, St, Sh |
Anisogrid | Bi-Grid | Isogrid Type I | Isogrid Type II | Ortho Grid | X-Grid Panel | Other Feature |
---|---|---|---|---|---|---|
1 | 0 | 0 | 0 | 0 | 0 | Anisogrid Feature |
0 | 1 | 0 | 0 | 0 | 0 | Bi-grid Features |
0 | 0 | 1 | 0 | 0 | 0 | Isogrid I Features |
0 | 0 | 0 | 1 | 0 | 0 | Isogrid II Features |
0 | 0 | 0 | 0 | 1 | 0 | Orthogrid Features |
0 | 0 | 0 | 0 | 0 | 1 | X-grid Features |
Anisogrid | Bi-Grid | Isogrid Type I | Isogrid Type II | Orthogrid | X-Grid Panel | Panel Thickness | Stiffener Thickness | Stiffener Height | Mass |
---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 1 | 0 | 0 | 0.742743 | 0.301301 | 0.676677 | 0.361479 |
0 | 0 | 0 | 0 | 0 | 1 | 0.293293 | 0.083083 | 0.125125 | 0.102885 |
0 | 0 | 1 | 0 | 0 | 0 | 0.125125 | 0.293293 | 0.083083 | 0.149352 |
0 | 1 | 0 | 0 | 0 | 0 | 0.083083 | 0.293293 | 0.125125 | 0.271401 |
0 | 0 | 0 | 0 | 1 | 0 | 0.083083 | 0.293293 | 0.125125 | 0.024996 |
1 | 0 | 0 | 0 | 0 | 0 | 0.083083 | 0.293293 | 0.125125 | 0.182503 |
Name of the Algorithm | Hyperparameter | Range |
---|---|---|
Decision Tree | max_depth | Default (none), 5 to 20 |
min_samples_split | 2 to 20 | |
min_samples_leaf | 1 to 20 | |
KNN | n_neighbors | 1 to 50 |
weights | uniform, distance | |
metrics | euclidean, manhattan, minkowski | |
algorithm | auto, ball_tree, kd_tree, brute | |
Random Forest | n_estimators | 2 to 200 |
max_depth | None, 1 to 100 | |
max_features | None, Sqrt, Log2 | |
CatBoost | iterations | 100 to 1000 |
learning_rate | 0.01 to 0.2 | |
depth | 4 to 12 | |
l2_leaf_reg | 1 to 10 | |
XG Boost | n_estimators | 100 to 1000 |
learning_rate | 0.01 to 0.2 | |
max_depth | 4 to 12 | |
reg_lambda | 0.01 to 4 | |
reg_alpha | 0 to 4 | |
LightGBM | n_estimators | 100 to 1000 |
learning_rate | 0.01 to 0.2 | |
max_depth | 4 to 12 | |
reg_lambda | 0.01 to 4 | |
reg_alpha | 0 to 4 |
Model | Parameters | Values |
---|---|---|
ANN | Dense Layers | 200/100/50/25/12/6/3/1 128/64/32/16/8/4/2/1 100/50/25/12/6/3/1 64/32/16/8/4/2/1 50/25/12/6/3/1 32/16/8/4/2/1 |
Activation Functions | tanh; sigmoid; selu; relu; linear; elu; softmax | |
Dense Layer–Activation Function Combination | relu/relu/relu/relu/relu/relu/relu/1 linear/linear/linear/linear/linear/linear/linear/1 selu/selu/selu/selu/selu/selu/selu/1 sigmoid/sigmoid/sigmoid/sigmoid/sigmoid/sigmoid/sigmoid/1 elu/elu/elu/elu/elu/elu/elu/1 softmax/softmax/softmax/softmax/softmax/softmax/softmax/1 tanh/tanh/tanh/tanh/tanh/tanh/tanh/1 selu/relu/relu/relu/linear/linear/selu/1 selu/relu/relu/elu/linear/linear/selu/1 sigmoid/relu/relu/sigmoid/linear/linear/sigmoid/1 tanh/relu/relu/elu/linear/linear/tanh/1 softmax/relu/relu/relu/linear/linear/softmax/1 elu/relu/relu/relu/linear/linear/elu/1 selu/elu/elu/elu/linear/linear/selu/1 | |
Optimizers | Adagrad; Adadelta; Adam; Nadam; RMSprop | |
Initial Learning Rate | 0.01, 0.001, 0.0025, 0.005 | |
Batch Size | 4, 8, 12, 16 | |
Callbacks | Early Stopping | |
Reduce LR on Plateau |
AF Configuration | MSE | MAE | R2 | RMSE |
---|---|---|---|---|
selu/relu/relu/relu/linear/linear/selu/1 | 7.36985 | 1.7741 | 0.9996 | 2.7147 |
elu/elu/elu/elu/elu/elu/elu/1 | 12.0609 | 1.9229 | 0.9994 | 3.4728 |
elu/relu/relu/relu/linear/linear/elu/1 | 18.8675 | 2.6274 | 0.9990 | 4.3436 |
linear/linear/linear/linear/linear/linear/linear/1 | 1760.8873 | 31.4167 | 0.9126 | 41.9629 |
relu/relu/relu/relu/relu/relu/relu/1 | 652.4689 | 16.3624 | 0.9676 | 25.5434 |
selu/elu/elu/elu/linear/linear/selu/1 | 46.1173 | 3.61097 | 0.9977 | 6.7909 |
sigmoid/sigmoid/sigmoid/sigmoid/sigmoid/sigmoid/sigmoid/1 | 20,209.3946 | 113.31977 | −0.0025 | 142.1597 |
sigmoid/relu/relu/sigmoid/linear/linear/sigmoid/1 | 20,194.4674 | 113.4345 | −0.0018 | 142.1072 |
softmax/softmax/softmax/softmax/softmax/softmax/softmax/1 | 20,194.9277 | 113.4306 | −0.0018 | 142.1088 |
Model No. | Architecture | Optimizer | Batch Size | Initial Learning Rate | MSE | MAE | R2 | RMSE |
---|---|---|---|---|---|---|---|---|
1 | 200/100/50/25/12/6/3/1 | Adam | 8 | 0.01 | 7.3698 | 1.7741 | 0.9996 | 2.7147 |
2 | 128/64/32/16/8/4/2/1 | 11.5979 | 2.1179 | 0.9994 | 3.4055 | |||
3 | 100/50/25/12/6/3/1 | 26.8587 | 3.3608 | 0.9986 | 5.1825 | |||
4 | 64/32/16/8/4/2/1 | 28.9180 | 3.0904 | 0.9985 | 5.3775 | |||
5 | 50/25/12/6/3/1 | 42.760 | 3.6686 | 0.9978 | 6.5391 | |||
6 | 32/16/8/4/2/1 | 74.0971 | 4.7582 | 0.9963 | 8.6079 |
Parameters | With Scaling | Without Scaling | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Optimizer | Initial LR | Batch Size | MSE | MAE | R2 | RMSE | MSE | MAE | R2 | RMSE |
Adam | 0.01 | 4 | 5.8561 | 1.4106 | 0.9997 | 2.4199 | 30.0566 | 2.8713 | 0.9985 | 5.4823 |
0.001 | 4.4609 | 1.3466 | 0.9997 | 2.1120 | 11.9957 | 2.0400 | 0.9994 | 3.4634 | ||
0.0025 | 3.6402 | 1.2371 | 0.9998 | 1.9079 | 19.753 | 2.5970 | 0.9990 | 4.4444 | ||
0.005 | 4.6033 | 1.1867 | 0.9997 | 2.1455 | 9.2475 | 1.8212 | 0.9995 | 3.0409 | ||
Nadam | 0.01 | 5.7836 | 1.6023 | 0.9997 | 2.4049 | 14.7046 | 2.3814 | 0.9992 | 3.8347 | |
0.001 | 3.1768 | 0.9563 | 0.9998 | 1.7823 | 6.6568 | 1.5852 | 0.9996 | 2.5800 | ||
0.0025 | 4.9861 | 1.2169 | 0.9997 | 2.2329 | 7.8822 | 1.8031 | 0.9996 | 2.8075 | ||
0.005 | 3.5690 | 1.0203 | 0.9998 | 1.8892 | 17.2828 | 2.6176 | 0.9991 | 4.1572 | ||
RMSprop | 0.01 | 3.7165 | 1.1636 | 0.9998 | 1.9278 | 32.0571 | 2.6332 | 0.9984 | 5.6619 | |
0.001 | 9.8788 | 1.7071 | 0.9995 | 3.1431 | 23.8586 | 2.1916 | 0.9988 | 4.8845 | ||
0.0025 | 3.3712 | 1.1537 | 0.9998 | 1.8361 | 12.0302 | 1.8103 | 0.9994 | 3.4684 | ||
0.005 | 5.9408 | 1.5111 | 0.9997 | 2.4374 | 18.3429 | 2.2477 | 0.9990 | 4.2828 |
Parameters | With Scaling | Without Scaling | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Optimizer | Initial LR | Batch Size | MSE | MAE | R2 | RMSE | MSE | MAE | R2 | RMSE |
Adam | 0.01 | 8 | 3.5577 | 1.0204 | 0.9998 | 1.8862 | 7.36985 | 1.7741 | 0.9996 | 2.7147 |
0.001 | 4.8397 | 1.2346 | 0.9997 | 2.19995 | 45.3998 | 3.5643 | 0.9977 | 6.7379 | ||
0.0025 | 4.9690 | 1.3566 | 0.9997 | 2.22913 | 31.7479 | 3.0205 | 0.9984 | 5.6345 | ||
0.005 | 8.0687 | 1.7114 | 0.9996 | 2.84055 | 14.3205 | 2.3642 | 0.9992 | 3.7842 | ||
Nadam | 0.01 | 3.8987 | 1.1248 | 0.9998 | 1.97452 | 7.3496 | 1.7316 | 0.9996 | 2.7110 | |
0.001 | 3.3759 | 1.1163 | 0.9998 | 1.83737 | 13.0716 | 2.0399 | 0.9993 | 3.6154 | ||
0.0025 | 2.9584 | 0.9875 | 0.9998 | 1.72 | 4.4485 | 1.2591 | 0.9997 | 2.1091 | ||
0.005 | 3.5690 | 1.0203 | 0.9998 | 1.88918 | 9.3814 | 1.7891 | 0.9995 | 3.0629 | ||
RMSprop | 0.01 | 3.6200 | 1.1531 | 0.9998 | 1.90265 | 11.5274 | 2.1125 | 0.9994 | 3.3952 | |
0.001 | 3.4163 | 1.1475 | 0.9998 | 1.84834 | 20.4865 | 2.2389 | 0.9989 | 4.5262 | ||
0.0025 | 3.7629 | 1.1067 | 0.9998 | 1.93984 | 9.21970 | 1.7027 | 0.9995 | 3.0363 | ||
0.005 | 6.1268 | 1.4562 | 0.9997 | 2.47525 | 16.0527 | 2.1852 | 0.9992 | 4.0065 |
Parameters | With Scaling | Without Scaling | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Optimizer | Initial LR | Batch Size | MSE | MAE | R2 | RMSE | MSE | MAE | R2 | RMSE |
Adam | 0.01 | 12 | 5.3417 | 1.4411 | 0.9997 | 2.3112 | 74.0218 | 4.3468 | 0.9963 | 8.6035 |
0.001 | 4.5793 | 1.3601 | 0.9997 | 2.1399 | 39.4716 | 3.4821 | 0.9980 | 6.2826 | ||
0.0025 | 6.4505 | 1.6290 | 0.9996 | 2.5398 | 8.3510 | 1.8200 | 0.9995 | 2.8898 | ||
0.005 | 5.8938 | 1.4961 | 0.9997 | 2.4277 | 21.8676 | 2.6690 | 0.9989 | 4.6762 | ||
Nadam | 0.01 | 4.1654 | 1.2360 | 0.9997 | 2.0409 | 13.6589 | 2.1114 | 0.9993 | 3.6958 | |
0.001 | 4.0695 | 1.2570 | 0.9998 | 2.0173 | 5.3638 | 1.4808 | 0.9997 | 2.3159 | ||
0.0025 | 4.3573 | 1.3708 | 0.9997 | 2.0874 | 5.0690 | 1.3691 | 0.9997 | 2.2514 | ||
0.005 | 3.7474 | 1.1351 | 0.9998 | 1.9358 | 6.0024 | 1.4751 | 0.9997 | 2.4499 | ||
RMSprop | 0.01 | 6.2747 | 1.6636 | 0.9996 | 2.5049 | 18.1875 | 2.6995 | 0.9990 | 4.2646 | |
0.001 | 3.6928 | 1.1818 | 0.9998 | 1.9217 | 18.5165 | 2.1939 | 0.9990 | 4.3030 | ||
0.0025 | 4.6259 | 1.2895 | 0.9997 | 2.1508 | 6.7574 | 1.5942 | 0.9996 | 2.5995 | ||
0.005 | 8.0380 | 1.5777 | 0.9996 | 2.8351 | 16.6946 | 2.2337 | 0.9991 | 4.0859 |
Parameters | With Scaling | Without Scaling | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Optimizer | Initial LR | Batch Size | MSE | MAE | R2 | RMSE | MSE | MAE | R2 | RMSE |
Adam | 0.01 | 16 | 4.0550 | 1.1191 | 0.9998 | 2.0137 | 7.4855 | 1.7720 | 0.9996 | 2.7359 |
0.001 | 6.9332 | 1.6727 | 0.9996 | 2.6331 | 72.6474 | 4.3359 | 0.9963 | 8.5233 | ||
0.0025 | 4.9058 | 1.2197 | 0.9997 | 2.2149 | 7.81778 | 1.7047 | 0.9996 | 2.7960 | ||
0.005 | 5.5738 | 1.3099 | 0.9997 | 2.3608 | 10.9850 | 1.9726 | 0.9994 | 3.3143 | ||
Nadam | 0.01 | 5.5604 | 1.3844 | 0.9997 | 2.3580 | 59.7409 | 3.8024 | 0.9970 | 7.7292 | |
0.001 | 4.9870 | 1.4503 | 0.9997 | 2.2331 | 8.7131 | 1.8291 | 0.9995 | 2.9518 | ||
0.0025 | 3.0523 | 1.0181 | 0.9998 | 1.7471 | 6.5314 | 1.4655 | 0.9996 | 2.5556 | ||
0.005 | 5.2322 | 1.3344 | 0.9997 | 2.2874 | 5.2145 | 1.3334 | 0.9997 | 2.2835 | ||
RMSprop | 0.01 | 8.6232 | 1.7105 | 0.9996 | 2.9365 | 12.3067 | 2.2218 | 0.9993 | 3.5080 | |
0.001 | 3.4655 | 1.1386 | 0.9998 | 1.8615 | 26.2176 | 2.7088 | 0.9986 | 5.1203 | ||
0.0025 | 8.7005 | 1.6513 | 0.9996 | 2.9496 | 15.2302 | 1.9491 | 0.9992 | 3.9025 | ||
0.005 | 4.6465 | 1.1963 | 0.9997 | 2.1555 | 12.1344 | 2.1356 | 0.9993 | 3.4834 |
Model Name | MSE | MAE | RMSE | R2 |
---|---|---|---|---|
ANN 1 (Adam) | 3.5577 | 1.0204 | 1.8862 | 0.9998 |
ANN 2 (Nadam) | 2.9584 | 0.9875 | 1.72 | 0.9998 |
ANN 3 (RMSProp) | 3.3712 | 1.1537 | 1.8361 | 0.9998 |
RF | 93.2298 | 6.3288 | 9.6556 | 0.9953 |
DT | 270.1035 | 10.6451 | 16.4348 | 0.9866 |
KNN | 127.8302 | 7.8980 | 11.3062 | 0.9936 |
CatBoost | 17.6968 | 2.5899 | 4.2067 | 0.9991 |
XGBoost | 53.9406 | 4.8025 | 7.3444 | 0.9973 |
LightGBM | 42.1116 | 4.4702 | 6.4893 | 0.9979 |
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Khan, S.; Rayhan, S.B.; Rahman, M.M.; Sultana, J.; Varga, G. Optimized ANN Model for Predicting Buckling Strength of Metallic Aerospace Panels Under Compressive Loading. Metals 2025, 15, 666. https://doi.org/10.3390/met15060666
Khan S, Rayhan SB, Rahman MM, Sultana J, Varga G. Optimized ANN Model for Predicting Buckling Strength of Metallic Aerospace Panels Under Compressive Loading. Metals. 2025; 15(6):666. https://doi.org/10.3390/met15060666
Chicago/Turabian StyleKhan, Shahrukh, Saiaf Bin Rayhan, Md Mazedur Rahman, Jakiya Sultana, and Gyula Varga. 2025. "Optimized ANN Model for Predicting Buckling Strength of Metallic Aerospace Panels Under Compressive Loading" Metals 15, no. 6: 666. https://doi.org/10.3390/met15060666
APA StyleKhan, S., Rayhan, S. B., Rahman, M. M., Sultana, J., & Varga, G. (2025). Optimized ANN Model for Predicting Buckling Strength of Metallic Aerospace Panels Under Compressive Loading. Metals, 15(6), 666. https://doi.org/10.3390/met15060666