In the Karhunen-Loe`ve (KL) series expansion framework for random field discretization, computational errors in common KL methods (e.g., the orthogonal series expansion (OSE) and Jacobi-Legendre-Galerkin (JLG) methods) arise not only from the truncation error due to the number of retained terms (M), but also from eigensolution accuracy related to the number of Gauss integration points (N1D GP) and basis function terms (N1D). The OSE method with a full tensor-product construction and the JLG method with total-order truncation were employed for separable and nonseparable autocorrelation functions, respectively, thereby revealing the decoupled effects of the number of N1D GP and the number of N1D on the eigenvalues and eigenfunctions. The results indicate that the global covariance error may not fully capture local reconstruction errors within the physical domain. For N1D >= M + 2, N1D GP primarily influences eigenvalues (lambda), while N1D dictates fluctuations in eigenfunctions (f (x)). Accordingly, parameter selection criteria are established for one-dimensional and multi-dimensional random fields, respectively. Engineering case studies indicate that eigensolution accuracy affects the range of variation in load-settlement curves for shallow foundations and that the tilting failure mode of a pile group is particularly sensitive to it. Insufficient accuracy of the eigensolution causes drifts at the most unfavorable failure points and shifts in parameter sensitivity rankings within pile group systems. Furthermore, the convergence pattern of eigensolution errors directly maps onto the convergence behavior of the reliability assessment for pile group systems. This research provides a vital theoretical basis and accuracy parameter selection benchmarks for geotechnical reliability analysis within the KL framework.
This study examines the practical consequence of distance model selection in Gaussian random field generation over manifolds. An exponential correlation model is considered on three representative surfaces: a half-cylinder, a hemisphere, and a saddle surface. For each manifold, three distance models are compared: the reference geodesic distance computed on the triangulated surface using the MMP algorithm, the approximate geodesic distance obtained by the heat method, and the Euclidean distance computed from Cartesian coordinates. Errors are evaluated successively at the distance matrix, covariance matrix, and random field levels. The results show that higher distance-matrix accuracy does not necessarily lead to lower field-level discrepancy. Although the heat-method distance is consistently closer to the reference geodesic distance than the Euclidean distance, this geometric advantage is not preserved after covariance construction and field generation. In all investigated cases, the Euclidean distance model produces smaller field-level discrepancies and requires the lowest computational cost. These results suggest that distance model selection for manifold based random field simulation should be based on the propagated effect on the covariance structure and generated random fields, rather than on geometric distance accuracy alone.
