Higher order weighted integral stochastic finite element method and simplified first-order application

It is well known that expansion-based stochastic methods are approximate schemes, as they are based on a first or, at most, second order series expansion on the basic variable, e.g. displacement. Therefore, expansion-based stochastic analysis schemes are bound to show small response variability when compared with Monte Carlo simulation (MCS) results, and application of these schemes is limited to stochastic problems with relatively small variability. In order to overcome these general drawbacks of the expansion methods, we suggest a higher-order stochastic field function that can be employed in the expansion-based stochastic analysis scheme of the weighted integral method. We then propose a new weighted integral formulation using the higher-order stochastic field function. The new formulation is not only applicable to stochastic problems with a high degree of uncertainty but also can reproduce the phenomenon of accelerated increase in the response variability when the coefficient of variation of the stochastic field increases, as observed in the MCS. In order to show the validity of the proposed formulation, we provide two numerical examples and the results are discussed in detail.