Conductivity of a medium containing a dense array of perfectly conducting square cylinders

In his landmark paper, Keller (J Appl Phys 34:991–993, 1963) obtained an approximation for the effective conductivity of a composite medium made out of a densely packed square array of perfectly conducting circular cylinders embedded in a conducting medium. We here examine Keller’s problem for the case of square cylinders, considering both a full-pattern and a checkerboard configurations. The dense limit is handled using matched asymptotic expansions, where the conduction problem is separately analyzed in the the “local” narrow gap between adjacent cylinders and the “global” region outside the gap. The conduction problem in both regions is solved using conformal-mapping techniques.