A deep energy method with normalized hard constraints for mechanical buckling analysis of FGM sandwich plates using higher-order shear deformation theory

This study proposes a novel Two-Stage Deep Energy Framework for the buckling analysis of Functionally Graded Material (FGM) sandwich plates, governed by Reddy's Higher-order Shear Deformation Theory (HSDT). To address the limitations of conventional mesh-dependent methods and the convergence issues of standard Physics-Informed Neural Networks (PINNs), the proposed framework approximates the kinematic field using Deep Neural Networks (DNNs) by minimizing the Rayleigh quotient. A distinguishing innovation of this work is the development of a Normalized Hard Constraint architecture. By utilizing polynomial distance functions on a normalized computational domain, this technique guarantees the strict, a priori satisfaction of geometric boundary conditions, effectively eliminating the spectral bias and hyperparameter sensitivity associated with penalty-based soft constraints. Furthermore, to ensure high-precision eigenvalue convergence, a strategic two-stage optimization protocol is implemented. This hybrid approach couples the global exploration capabilities of the Adam optimizer with the local refinement power of the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm. The energy functionals are evaluated via Quasi-Monte Carlo integration, with spatial derivatives computed exactly using Automatic Differentiation (AD). Comprehensive numerical benchmarks confirm that the proposed model achieves excellent agreement with analytical Navier solutions and established finite element methods, offering a robust, mesh-free paradigm with acceptable computational cost for the stability analysis of complex composite structures.