On the numerical evaluation of electrostatic fields in dense random dispersions of cylinders

We consider the interface boundary value problem which arises in the evaluation of electrostatic fields in composite materials consisting of dense random dispersions of cylinders in a uniform background. This is a well-studied problem from the viewpoint of homogenization and effective medium theory, but one for which accurate numerical simulations have been difficult to obtain. Size effects, in particular, have been neglected due to the expense of solving the field equation in the presence of large numbers of close-to-touching inclusions. Such features require very fine discretizations, even with the use of adaptive gridding, and cause the linear systems which arise to be highly ill-conditioned. In this paper, we present a new integral equation method for the solution of the interface problem which uses a recently developed method of images to resolve the close-to-touching interactions and the fast multipole method to compute far field interactions. Only minutes of workstation time are needed to solve the field equation with thousands of inclusions, allowing us to carry out large-scale statistical studies of the effective conductivity of random two-phase materials at a variety of volume fractions and contrast ratios.