A spectral-fast Voronoi framework based on multivariate Karhunen–Loève expansion for simulation of two-dimensional random fields

This study proposes a high-precision, high-efficiency, and widely applicable algorithm for simulating two-dimensional random fields, capable of handling irregular, non-stationary, multivariate-correlated, and spatiotemporally-correlated random fields, as well as random fields with Lagrangian displacement. The primary innovations of this study are as follows: (1) By employing triangular spectral element methods (TSEM) and the Fast Multipole Method (FMM), the proposed algorithm significantly enhances the accuracy and efficiency of traditional Karhunen–Loève expansion (KLE) for spatiotemporally correlated random fields. It improves the computation of multivariate cross-covariance integrals while avoiding the costly eigenvalue decomposition of the full spatiotemporal covariance matrix; (2) To enable simulations in irregular domains, a global backtracking algorithm is proposed to address singular edge mismatches arising from quasi-interpolation in TSEM, and a hybrid binary–quadtree partitioning strategy is introduced to prevent malformed or sparse elements in FMM tree decomposition; (3) By leveraging the linear transformation property of KLE, a low-bias sampling strategy is employed to extract pointwise occurrence probabilities within each simulated sample. The algorithm is validated through three representative cases: a non-stationary multivariate slope field, a spatiotemporally correlated atmospheric field, and a spatially random material property field for a NACA airfoil. The results confirm the accuracy, efficiency, and broad applicability of the proposed method in modeling complex random fields.