Optimal Discretization of Random Fields

TLDR

The method is based on principles of optimal linear estimation theory. A new method for efficient discretization of random fields is introduced. The method represents the field as a linear function of nodal random variables and shape functions chosen to minimize error variance, measures efficiency by the number of variables needed for a given accuracy, and further improves efficiency via spectral decomposition of the nodal covariance matrix. The method outperforms existing discretization techniques and is more practical than Karhunen–Loève expansions, making it especially useful for stochastic finite element analyses of random media by reducing the number of required random variables and computational effort.

Abstract

A new method for efficient discretization of random fields (i.e., their representation in terms of random variables) is introduced. The efficiency of the discretization is measured by the number of random variables required to represent the field with a specified level of accuracy. The method is based on principles of optimal linear estimation theory. It represents the field as a linear function of nodal random variables and a set of shape functions, which are determined by minimizing an error variance. Further efficiency is achieved by spectral decomposition of the nodal covariance matrix. The new method is found to be more efficient than other existing discretization methods, and more practical than a series expansion method employing the Karhunen‐Loève theorem. The method is particularly useful for stochastic finite element studies involving random media, where there is a need to reduce the number of random variables so that the amount of required computations can be reduced.