Positive random fields for modeling material stiffness and compliance

Positive random fields with known marginal properties and known correlation function are not numerous in the literature. The most prominent example is the log\-normal field for which the complete distribution is known and for which the reciprocal field is also lognormal. It is of interest to supplement the catalogue of positive fields beyond the class of those obtained by simple marginal transformation of a Gaussian field, this class containing the lognormal field.As a minimum for a random field to be included in the catalogue itis required that an algorithm for simulation of realizations can be constructed and that the one-dimensional marginal moments up to at least the fourth order as well as the correlation function of the field are given explicitly.

This information makes it possible to use and check the so-called Winterstein approximations for finite element calculations of structures with material properties modeled in terms of the considered random fields.The paper addsthe gamma field, the Fisher field, the beta field, and their reciprocal fields to the catalogue.

These fields are all defined on the basis of sums of squares of independent standard Gaussian random variables.All the existing marginal moments and the correlation functions are obtained explicitly. Also an inverse Gaussian fieldis added to the catalogue. It is defined in terms of first passage times in correlated joint Brownian motions.

Finally an n-dimensional random vector of positive components is defined such that it can be used as an approximation to a discretization of a homogeneous Gaussian field with any specified correlation function and coefficient of variation less than 1/(square root of n).