Energy-consistent stochastic DFE² modeling for heterogeneous piezoelectric structures with polynomial chaos surrogates

Heterogeneous piezoelectric structures exhibit coupled electromechanical responses that are sensitive to meso-scale material variability. Predicting how such uncertainty propagates to macro-scale quantities of interest remains computationally demanding, especially when direct numerical simulation (DNS) is used to explicitly resolve the underlying mesostructure. This study develops an energy-consistent stochastic DFE² DFE 2 –PCE framework for multiscale uncertainty analysis of heterogeneous piezoelectric structures. The proposed method extends the piezoelectric Direct FE² FE 2 ( DFE² DFE 2 ) formulation to stochastic settings by introducing randomness among the computational unit cells associated with macro-scale quadrature points. For each realization, the governing problem remains a deterministic two-scale electromechanical analysis, and the macro–meso coupling is preserved through an electric-enthalpy-based Hill–Mandel-consistent formulation, which ensures realization-wise energetic consistency across scales. On this basis, repeated deterministic DFE² DFE 2 simulations are used to train a non-intrusive polynomial chaos expansion (PCE) surrogate for efficient approximation of the stochastic response surface. The framework is assessed using porous and multiphase piezoelectric panels, with DNS adopted as the reference model for numerical validation. The results demonstrate that the stochastic DFE² DFE 2 model reproduces the DNS responses accurately while requiring substantially fewer degrees of freedom and significantly lower computational cost. The trained PCE surrogates further enable efficient univariate and bivariate parametric analyses, revealing distinct response sensitivities of mechanical and electrical quantities to uncertain elastic and piezoelectric parameters. Under matching sampling assumptions, the proposed stochastic DFE² DFE 2 formulation provides a statistically consistent approximation of the corresponding DNS ensemble response. The developed framework thus offers an efficient and physically consistent tool for stochastic multiscale analysis and uncertainty-aware design of heterogeneous piezoelectric structures.