Effect of Karhunen-Loève Eigensolution accuracy on the reliability of foundation engineering

Abstract

In the Karhunen–Loève (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 (N^1D). 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 N^1D 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 N^1D ≥ M + 2, N1D GP primarily influences eigenvalues (λ), while N^1D 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.