MULTISCALE MODEL. SIMUL. c  2012 Society for Industrial and Applied Mathematics Vol. 10, No. 1, pp. 180–190 OPTIMAL DISTRIBUTION OF THE NONOVERLAPPING CONDUCTING DISKS ∗ VLADIMIR MITYUSHEV † AND NATALIA RYLKO † Abstract. Conducting nonoverlapping identical disks are embedded in a two-dimensional back- ground. The set of disks is infinite. The disks are distributed in such a way that the obtained composite is macroscopically isotropic. Let the conductivity of inclusions be higher than the conduc- tivity of the matrix. It is proved that the hexagonal (triangular) lattice of disks possess the minimal effective conductivity when the concentration is not high. Key words. effective conductivity, nonoverlapping disks, hexagonal lattice AMS subject classifications. 74Q15, 30E25 DOI. 10.1137/110823225 1. Introduction. Conducting nonoverlapping identical disks of conductivity λ 1 are embedded in a two-dimensional background of conductivity λ. The set of disks is infinite. The disks are distributed in such a way that the obtained composite is macroscopically isotropic. Let  λ denote the ratio of the scalar effective conductivity to λ. It depends on the contrast parameter ρ = λ1-λ λ1+λ , the concentration ν of the inclusions, and their location. Physical hypothesis. The effective conductivity  λ attains the minimum for the regular hexagonal (triangular) array for any fixed ν and ρ> 0. The hexagonal array (see Figure 2 or Figure 3(a)) is suggested as the optimal distribution due to the famous geometrical result by L. Fejes T´oth. Geometrical theorem (see [12]). The densest packing of the equal disks in the plane is the hexagonal array. The density of this arrangement (concentration of inclusions) is equal to π √ 12 . Kozlov [7] proposed an original method to study locations of inclusions for which the trace of the effective conductivity tensor attains the minimum. However, Kozlov’s statement and solution should be revised and corrected, since his brilliant idea is not properly realized in [7]. In order to simplify the presentation we consider a macroscopically isotropic medium. The effective conductivity can be written in the form of the series [9] (1)  λ =1+2ρν  1+ ∞  k=1 A k ν k  , where the coefficients A k depend on the location of the disks. (See formulae (15)– (16) in the next section.) Actually, Kozlov [7] used the series (1) up to O(ν 6 ), i.e., investigated the coefficients A 1 ,A 2 ,A 3 ,A 4 . These coefficients were calculated with the accuracy O(ρ 2 ). However, A 3 and A 4 have higher order terms in ρ. (See for- mulae (16), where the terms with ρ 3 and ρ 4 are absent in Kozlov’s paper.) Kozlov investigated “incomplete” coefficients A 3 and A 4 , arrived at an optimization problem ∗ Received by the editors February 2, 2011; accepted for publication (in revised form) December 5, 2011; published electronically March 8, 2012. http://www.siam.org/journals/mms/10-1/82322.html † Dept. Computer Sciences and Computer Methods and Dept. Technology, Pedagogical University, ul. Podchorazych 2, Krakow 30-084, Poland (vmityu@yahoo.com, nrylko@gmail.com). 180