On the Karhunen-Loeve expansion and spectral representation methods for the simulation of Gaussian stochastic fields

Abstract

The parameters describing a structure are uncertain quantities and the analysis and safe design of most engineering systems must take into account these uncertainties. Uncertain structural parameters are usually modeled as random fields. Despite the fact that most of the uncertain quantities appearing in practical engineering problems are non-Gaussian in nature (e.g. material and geometric properties, wind loads), the Gaussian assumption is often used due to the lack of relevant experimental data. From the wide variety of methods developed for the simulation of Gaussian stochastic processes and fields, two are most often used in practice: the spectral representation method and the Karhunen- Loeve (K-L) expansion. The K-L expansion can be seen as a special case of the orthogonal series expansion where the orthogonal functions are chosen as the eigenfunctions of a Fredholm integral equation of the second kind with the autocovariance as kernel. In this paper, a wavelet-Galerkin scheme is adopted for the efficient solution of the Fredholm equation. A one-dimensional homogeneous Gaussian random field with two types of autocovariance function (power spectrum), exponential and square exponential, is used as test example. The numerical instabilities arising in some cases during the calculation of eigenvalues of both kernels at high wavelet levels ( 6 m = ) are reported. The influence of the scale of correlation on the simulation quality is quantified by using several values of correlation length parameter b. In this work, a comparison of the accuracy achieved and the computational effort required by the K-L expansion and the spectral representation for the simulation of the stochastic field is pursued. The accuracy obtained by the two methods is examined by comparing their ability to produce sample functions that match the target correlation structure and the Gaussian probability distribution or, alternatively, its low order statistical moments (mean, variance and skewness).