This paper presents in detail the originally developed Quadratic Point Estimate Method (QPEM), aimed at efficiently and accurately computing the first four output moments of probabilistic distributions, using 2n^2+1 sample (or sigma) points, with n, the number of input random variables. The proposed QPEM particularly offers an effective, superior, and practical alternative to existing sampling and quadrature methods for low- and moderately-high-dimensional problems. Detailed theoretical derivations are provided proving that the proposed method can achieve a fifth or higher-order accuracy for symmetric input distributions. Various numerical examples, from simple polynomial functions to nonlinear finite element analyses with random field representations, support the theoretical findings and further showcase the validity, efficiency, and applicability of the QPEM, from low- to high-dimensional problems.

翻译：本文详细阐述了原创开发的二次点估计法（QPEM），该方法利用2n²+1个样本（或sigma）点（其中n为输入随机变量个数），旨在高效且精确地计算概率分布的前四阶输出矩。所提出的QPEM特别为低维和中高维问题提供了一种有效、优越且实用的替代方案，以替代现有的采样法和求积法。本文提供了详细的理论推导，证明所提方法对于对称输入分布可实现五阶或更高阶的精度。从简单多项式函数到含随机场表示的非线性有限元分析等多种数值算例，均支持了理论发现，并进一步展示了QPEM在低维到高维问题中的有效性、效率及适用性。