A Chebyshev polynomial-based Galerkin method for the discretization of spatially varying random properties
A Chebyshev polynomial-based Galerkin method for the discretization of spatially varying random properties
复制标题
用于空间变化随机属性离散化的基于切比雪夫多项式的伽辽金方法
DOI：
10.1007/s00707-017-1819-2
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发表时间：
2017
期刊：
影响因子：
2.7
通讯作者：
Zhang Xufang
中科院分区：
文献类型：
--
作者：
Liu Qian;Zhang Xufang
与空间变化的随机属性相关的不确定性需要被描述为随机场，其在定义上由附加到随机场的连续域的每个点的无限数量的随机变量组成。Karhunen-Loève展开的效用允许将随机场表示为一系列确定性函数和随机变量的总和。通过这种方式，用有限个随机变量来表示随机场。然而，K-L展开式主要依赖于第二类Fredholm型积分方程本征解。因此，实际的随机场模拟的兴趣主要是包含尽可能少的随机变量，但同时需要保证再现结果的小的逼近误差。为此，本文引入了切比雪夫多项式来求解积分特征值问题。作为Galerkin投影的一类有效基函数，基于切比雪夫多项式的Galerkin方法通过全局方差和协方差误差来估计收敛速度。并通过再现几个平稳和非平稳、高斯和非高斯随机场来检验其精度。结合混凝土抗压强度场的实际模拟，数值结果表明基于切比雪夫多项式的Galerkin格式具有工程应用价值。
The uncertainty related to the spatially varying random property requires to be described as a random field, which in definition consists of an infinite number of random variables that are attached to each point of a continuous domain of the random field. The utility of Karhunen–Loève expansion allows to represent the random field as the summation of a series of deterministic functions and random variables. By this way, the random field is represented by using a finite number of random variables. However, the K–L expansion depends crucially upon the eigen-solutions of the Fredholm integral equation of the second kind. Therefore, the interest of practical random field simulations is mainly concerned with containing as few random variables as possible, but needs to guarantee small approximation errors for the reproduced results in the meantime. To this end, the Chebyshev polynomial is introduced in the paper for the solution of the integral eigenvalue problem. As an effective class of basis functions for the Galerkin projection, the convergence rate of the Chebyshev polynomial-based Galerkin method is assessed by means of the global variance and covariance errors. And its accuracy is examined by reproducing several stationary and non-stationary, Gaussian, and non-Gaussian random fields. Together with the practical simulations of the concrete compressive strength field, numerical results show that the Chebyshev polynomial-based Galerkin scheme has engineering applications.