Interval field model and interval finite element analysis

Abstract

Spatially uncertain parameters are traditionally represented by random field models. However, the large amount of information required for construction of precise probability distribution is often difficult to obtain for many practical engineering problems. In this paper, an interval field model is proposed to represent spatial uncertainties with insufficient information, in which the variation of the parameter at any location is quantified by an interval with upper and lower bounds. The spatial dependency is measured by a covariance function or a correlation coefficient function, defined for the interval variables at different locations. An interval Karhunen–Loève expansion is formulated for the proposed interval field model, in which the continuous spatial uncertainty is expressed through a series of deterministic functions with uncorrelated interval coefficients. Furthermore, by incorporating the interval field model into the finite element method, interval finite element analysis with spatially uncertain parameters is carried out. Perturbation-based interval finite element methods are developed to evaluate the upper and lower bounds of structural responses such as displacement and stress. A Monte Carlo simulation method is also presented to provide a reference solution for the structural analysis with interval fields. Finally, two numerical examples are investigated to demonstrate the effectiveness of the interval field model and the interval finite element methods.