ABSTRACT
This chapter details a slow-scale reference boundary value problem (BVP) with the classical finite element method and briefly reviews the formulation of the classical finite element method (FEM). The fast-scale fluctuation BVP is tackled by formulating the novel Green-function-based FEM. Two numerical examples are presented to demonstrate unique capacity and algorithmic features of the multiscale stochastic finite element method (MSFEM). The examples demonstrate that the multiscale stochastic FEM provides certain important insights into stress and strain analysis of a finite body random heterogeneous material. According to the MSFEM results, an optimized composite is to emphasize high strength of the reinforcing phase near all the boundaries and of the matrix phase at the center. With respect to an optimal particle size, there is no general rule, and MSFEM computation should be run on each individual case, since the scale-coupling resonance depends on a combination of several factors including geometry, boundary conditions, and interphase contrast of the elastic moduli.
