Simulation of simply cross correlated random fields by series expansion methods

A practical framework for generating cross correlated random fields with a specified marginal distribution function, an autocorrelation function and cross correlation coefficients is presented in the paper. The approach relies on well-known series expansion methods for simulation of a Gaussian random field. The proposed method requires all cross correlated fields over the domain to share an identical autocorrelation function and the cross correlation structure between each pair of simulated fields to be simply defined by a cross correlation coefficient. Such relations result in specific properties of eigenvectors of covariance matrices of discretized field over the domain. These properties are used to decompose the eigenproblem, which must normally be solved in computing the series expansion, into two smaller eigenproblems. Such a decomposition represents a significant reduction of computational effort.Non-Gaussian components of a multivariate random field are proposed to be simulated via memoryless transformation of underlying Gaussian random fields for which the Nataf model is employed to modify the correlation structure. In this method, the autocorrelation structure of each field is fulfilled exactly while the cross correlation is only approximated. The associated errors can be computed before performing simulations and it is shown that the errors happen only in the cross correlation between distant points and that they are negligibly small in practical situations.Some comments on available techniques for simulation of underlying random variables in connection with the accuracy of basic fields’ statistics at a given sample size are made. For this purpose a simple error assessment procedure is presented.Simulated random fields can be used both for representation of spatially correlated properties of structure or random load in the stochastic finite element method (SFEM). An example of this application is related to size effect studies in the nonlinear fracture mechanics of concrete, and is used to illustrate the method.