Data-driven framework for vibration analysis of cracked bidirectional gradient plates: An extended scaled boundary finite element method

Vibration analysis of cracked bidirectional functionally graded material (BDFGM) plates remains a complex task due to the interaction between material heterogeneity, geometric coupling, and crack-tip singularities. Although traditional numerical methods ensure accuracy, they incur high computational costs when applied in extensive parametric sweeps, thereby restricting their utility in rapid structural design and uncertainty quantification. To address this, we propose a hybrid framework combining the extended scaled boundary finite element method (XSBFEM) with machine learning (ML). This approach utilizes XSBFEM to resolve crack-tip fields with high precision while minimizing mesh sensitivity and computational load. We constructed a dataset of 3000 vibration scenarios by systematically varying crack geometry, bidirectional grading indices, structural dimensions, and boundary conditions. Seven machine learning regression models were trained on this data, utilizing both Bayesian optimization and the parameter-free balancing composite motion optimization (BCMO) algorithm for hyperparameter tuning. Results indicate that the BCMO-XGBR (extreme gradient boosting regression) model achieves exceptional predictive accuracy (R² = 0.999, RMSE = 0.015 for the dimensionless normalized frequency), allowing for real-time natural frequency prediction without the need for expensive finite element simulations. Furthermore, SHapley Additive exPlanations (SHAP) based interpretability analysis elucidates the model's underlying mechanics, quantifying how boundary conditions, geometry, grading indices, and crack length collectively influence the dynamic response. This study presents an integrated workflow where numerical rigor generates data, machine learning generalizes the physics, and explainability provides necessary engineering insights for the analysis of cracked functionally graded structures.