Karhunen-Loève expansion based on an analytical solution over a bounding box domain

Abstract

This paper explores the accuracy and the efficiency of analytical solution of Fredholm integral equation to represent a random field on complex geometry. Because no analytical solution is available for arbitrary domains, it is proposed to use the analytical solution on simple bounding domains that enclose complex two- or three-dimensional geometries. It is a simple, accurate and robust approach for discretising a random field. The effect of the size of the bounding box on the resulting random field variance is investigated carefully and compared with the numerical solution given by the finite element method. The error variance is particularly localised near to the support domain boundary. Therefore, it is suggested to expand the bounding domain. This paper proposes a calibration of the correlation length to the ratio of the domain to maintain the convergence rate and the variance accuracy of the KLE without enlarging the stochastic dimension.