Application of a multiscale finite-element approach to calculate the effective conductivity of particulate media

Particulate media, such as composite materials and fluid-solid dispersions, are found in several important engineering applications. In this paper, we extend and apply a multiscale finite-element approach to the three-dimensional heat conduction problem in particulate media composed of monodisperse solid spherical particles distributed in a continuous phase. The approach is based on homogenization theory, variational calculus and the finite element method; the geometric stiffness, which arises in configurations where particles are very close together, is dealt with by incorporating isotropic microscale models to the mesoscale problem. The continuous and numerical formulations are applicable to both ordered and random media. The technique is here employed to calculate the effective thermal conductivity of the simple cubic array of spheres for the entire range of concentrations, including maximum packing; the accurate set of results that we present is new in the literature.