A Chebyshev polynomial-based Galerkin method for the discretization of spatially varying random properties

Abstract

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.