Micromechanically based stochastic finite elements

A correct stochastic finite element formulation must account for the actual random fields of material properties of heterogeneous media. In this study, a methodology for solution of a stochastic finite element problem is introduced and illustrated with the help of conductivity (out-of-plane elasticity) problem of a matrix-inclusion composite under spatially inhomogeneous boundary and source (body force) conditions. Two versions of the method, both implemented in a Monte Carlo sense, are based on determination of the stiffness matrices at the meso-scale (finite element) length. In one version, an exact calculation of all the element's stiffness matrices over the finite element mesh is carried out, while in the second one, the statistics of a meso-scale continuum random field is used to generate these matrices. In this paper we review various results obtained recently in this area [1, 2, 3].