Keywords: random fields, Karhunen-Loeve, discretization, one and two-dimensional structures.

The reliability analysis of a structure aims to estimate the probability of failure, which depends on the limit state function and the joint probability density function of the uncertain model parameters. Often each uncertain quantity is described by means of a single random variable.

However, the limit state function may show that the random spatial variability of the structural properties may significantly influence the reliability of the structure. In this case it is recommended to characterize the spatial fluctuations by a continuous random field.

The probabilistic characterization of a continuous homogeneous random field is given usually in terms of the marginal distribution, the mean value, the standard deviation and the covariance function. This function models the random spatial fluctuations, while the remaining items of the probabilistic model are common to the random variable model. The exponential function is one of the most used covariance models to represent the decrement of the correlation with the distance.

Even if the random field model can be fully characterized, it is necessary to reduce a continuous random field to a finite set of random variables. This reduction is commonly called random field discretization. The number of random variables is responsible for the accurate representation of the properties of the random field.

Several methods can be applied to the discretization of random fields for engineering applications (point discretization, average discretization and series expansion methods).

The paper describes a numerical approach, that adopts a truncated Karhunen-Loeve series expansion [1,2,3] to represent the random field and the finite element method to discretize the domain. The accuracy and efficiency of the discretization are the main objectives of the proposed procedure.

The accuracy is assessed by means of an error estimator related to the variance of the random field and the discretization is performed in order to achieve an error lower than a prescribed value.

The efficiency is gained by means of an adaptive refinement of the finite element mesh used to discretize the structural domain. A significant reduction of the computational effort is obtained.

1

B. Sudret, A. Der Kiureghian, "Stochastic finite elements and reliability: a state-of-the-art report", Technical Report UCB/SEMM-2000/08, University of California, Berkeley, 2000.

2

R.G. Ghanem, P.D. Spanos, "Stochastic finite elements: a spectral approach", Springer-Verlag, New York, 1991.

3

D.L. Allaix, V.I. Carbone, "Discretization of 2D random fields: A genetic algorithm approach", Eng. Struct., 31, 1111-1119, 2009. doi:10.1016/j.engstruct.2009.01.008

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