PSI - Issue 66

Umberto De Maio et al. / Procedia Structural Integrity 66 (2024) 502–510 Author name / Structural Integrity Procedia 00 (2025) 000–000

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plane strain conditions, using a hyperelastic Neo-Hookean material. Periodic boundary conditions were applied along the outer edges of the unit cell to represent the behavior of a continuous material. Two different COMSOL models were implemented using the same geometry. The first model was used to determine the buckling load due to uniaxial strain along the vertical direction, defined by a strain gradient equal to:

 

  

11 ( R F T

0) 0



11

( ) 

(3)

F

 

0

1





In this context, λ represents the dimensionless load parameter. The term 11 F corresponds to the horizontal component of the macroscopic deformation gradient that was imposed to obtain the macroscopic stress along the horizontal direction equal to zero. This approach simulates a uniaxial deformation test in the vertical direction, allowing the lateral faces of the material to expand freely in the horizontal direction. The second model was used to determine the nonlinear response in terms of normalized effective stress, obtained by imposing a geometric perturbation in the form of the first buckling mode in order to follow the primary buckling path. A key aspect of the optimization was the implementation of a convolutional neural network (CNN) to automate the buckling shape classification process. The CNN was trained on a dataset containing 836 images, obtained from a parametric analysis, divided into 80% intended for training and 20% for validation. The training showed 97.01% accuracy, with precision and recall metrics demonstrating excellent classification capabilities for all three categories. The confusion matrix reported in Fig. 3 provides a detailed overview of the classification performance across three categories that the model is trained and designed to predict. The matrix rows represent the actual (true) class labels, while the columns show the predicted class labels as determined by the model. For instance, with reference to the local class, 92.86% of the true instances were correctly classified, while 7.143% were misclassified as combined , but none of the local instances were incorrectly classified as global . Definitively, the confusion matrix indicates that the trained CNN is well-calibrated.

Fig. 3. Confusion matrix obtained as a report of the CNN training procedure.

To handle geometric optimization, a genetic algorithm (GA) was used to explore different combinations of the geometric parameters K 1 and K 2 to maximize the objective function, which is related to the critical buckling deformation along the vertical direction. The genetic algorithm explores the parameter space and ensures that the optimal configurations are based on local instabilities by considering an initial population of 50 candidate solutions, which pass through the selection, crossover, and mutation steps. The maximization of the objective function is performed by minimizing the cost function, which is evaluated by a MATLAB code. In Livelink with COMSOL, the latter can perform a linear buckling evaluation and classify the buckling mode shape by means of the trained CNN. If the buckling mode shape is classified as global or combined , the related geometrical configurations are penalized by setting the cost equal to infinite.