We propose a theoretical framework for evaluation of electrostatic potentials in an unbounded isotropic matrix containing a number of arbitrarily dispersed elliptic cylinders subjected to a remotely prescribed potential field. The inclusions could be homogeneous or confocally multicoated, and may have different sizes, aspect ratios and different conductivities. The approach is based on a multipole expansion formalism, together with a construction of consistency conditions and translation operators. This procedure generalizes the approach of the classic work of Rayleigh  for a periodic array of circular disks or spheres to an arbitrarily dispersion of elliptic cylinders. We combine the methods of complex potentials with a re-expansion formulae and the generalized Rayleigh's formualtion to obtain a complete solution of the many-inclusion problem. We show that the coefficients of field expansions can be written in the form of an infinite set of linear algebraic equations. Numerical results are presented for several configuarions. We further apply the obtained field solutions to determine the effective conductivity of the composite.
- Complex potentials
- Multicoated elliptic cylinders
- Multipole expansions