Direct probability integral assisted machine learning framework for uncertain vibration response of porous bi-directional functionally graded sandwich structures

This study presents a computationally efficient stochastic framework for analyzing the dynamic behavior of bi-directionally functionally graded (FG) porous sandwich structures with sigmoid material gradation. The proposed methodology combines the Direct Probability Integral Method (DPIM) with the XGBoost machine-learning algorithm to achieve fast and accurate prediction of stochastic vibration characteristics. The analysis considers two distinct sandwich configurations: FG face-sheets with a ceramic core, and homogeneous face-sheets with an FG core. A higher-order layerwise finite element model is employed to perform free vibration analysis, while uncertainties in material properties are systematically incorporated using the DPIM approach. For the stochastic analysis of bi-directional FG sandwich plates, DPIM alone requires only 6–10% of full Monte Carlo samples, while the integrated DPIM-XGBoost approach achieves similar accuracy with just 1–2% of the data, leading to a substantial reduction in computational effort. The accuracy of the developed stochastic model is validated through comparison with Monte Carlo simulation results, and the model is subsequently employed to conduct comprehensive stochastic analyses. The numerical results indicate that transverse sigmoid material gradation has a dominant influence in bi-directional FG face-sheets and ceramic core configurations, whereas longitudinal gradation plays a more significant role in sandwich plates with an FG core and homogeneous face-sheets. Furthermore, the proposed stochastic framework is used to identify the most sensitive parameters under various boundary conditions, volume fraction indices, and porosity levels, offering useful insights to reduce variability in vibration responses to support safe structural design. • A new meta-model integrating the direct probability integral method with the extreme gradient boosting machine-learning algorithm is developed for efficient stochastic vibration analysis of porous bi-directional functionally graded sandwich structures with sigmoid material variation. • The proposed uncertainty framework is highly efficient, achieving accuracy comparable to a full Monte Carlo simulation while using only 6–10% of the samples of the direct probability integral method and 1–2% of the samples of the direct probability integral method combined with extreme gradient boosting, resulting in significantly reduced computational cost and time. • Material gradation influences the two configurations differently: transverse gradation has a stronger effect on bi-directional sigmoid functionally graded face-sheet plates, while longitudinal gradation mainly dominates the dynamic behavior of bi-directional sigmoid functionally graded core sandwich structures. • Sensitivity analysis shows that metal mass density is the most influential parameter affecting natural frequencies, while even porosity distribution exhibits the highest coefficient of variation, thus highlighting the need for careful porosity modeling to ensure safe and reliable structural design.