A Spectral Formulation of Stochastic Finite Elements

Abstract

Until recently, stochastic structural mechanics has addressed the issue of deterministic structures subjected to random loading. With the availability of more accurate analysis and design tools, however, quantifying the sensitivity of model predictions to uncertainty in the mechanical properties of structures has become possible. It has been observed, with the help of these tools, that this type of uncertainty can be more significant to the overall predictions of a particular structural model than the more traditional fluctuations attributed to external loads. In view of that, recent procedures have been developed for representing uncertainties in the parameters of a structural model, as well as, for propagating this uncertainty to obtain the associated uncertainty in the predicted response. The stochastic finite element method is a procedure for performing such an analysis, whereby the spatial extent of the structure has been represented within the context of the finite element method. This chapter describes a recent implementation of the stochastic finite element method that combines theoretical rigor with generality and efficiency of implementation. Specifically, the Spectral Stochastic Finite Element Method (SSFEM) presented in this chapter addresses the situation where the uncertain material properties are realizations of a spatially fluctuating random field. The formulation relies on discretizing the random processes using spectral expansions, thus eliminating the correlation between the requisite mesh size to meet energy-based convergence criteria, and the scales of fluctuation of the random material properties involved.