Research Article Random 2D Composites and the Generalized Method of Schwarz Vladimir Mityushev Pedagogical University, ul. Podchorazych 2, 30-084 Krakow, Poland Correspondence should be addressed to Vladimir Mityushev; mityu@up.krakow.pl Received 3 September 2015; Accepted 26 November 2015 Academic Editor: Xavier Leoncini Copyright © 2015 Vladimir Mityushev. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Two-phase composites with nonoverlapping inclusions randomly embedded in matrix are investigated. A straightforward approach is applied to estimate the efective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, that is, for all events of the considered probabilistic space C by the generalized method of Schwarz. Second, the efective properties are calculated in analytical form and averaged over C. Tis method is related to the traditional method based on the average probabilistic values involving the -point correlation functions. However, we avoid computation of the correlation functions and compute their weighted moments of high orders by an indirect method which does not address the correlation functions. Te efective properties are exactly expressed through these moments. It is proved that the generalized method of Schwarz converges for an arbitrary multiply connected doubly periodic domain and for an arbitrary contrast parameter. Te proposed method yields an algorithm which can be applied with symbolic computations. Te Torquato-Milton parameter  1 is exactly written for circular inclusions. 1. Introduction Te efective properties of random heterogeneous materials and methods of their computation are of considerable interest [1–9]. Randomness in such problems is revealed through ran- dom tensor-functions locally describing the physical proper- ties of medium. Despite the considerable progress made in the theory of disordered media, the main tool for studying such systems remains numerical simulations. Frequently, it is just asserted that it is impossible to get general analytical formulae for the efective properties except dilute composites when interactions among inclusions are neglected and except regular composites having simple geometric structures. Tis opinion sustained by unlimited belief in numerics has to be questioned by the recent pure mathematical investigations devoted to explicit solution to the Riemann-Hilbert problem for multiply connected domains [10] and by signifcant progress in symbolic computations [11–13]. In the present paper, we demonstrate that the theoretical results [10] can be efectively implemented in symbolic form that yields analytical formulae for random composites. First applications of the theory were performed in [14, 15]. For conductivity problems governed by Laplace’s equa- tion, the local conductivity tensor (x) can be considered as a random function of the spatial variable x = ( 1 , 2 , 3 ). In the present paper, we restrict ourselves to two-phase composites with nonoverlapping inclusions when a collection of particles with fxed shapes and sizes is embedded in matrix (see Figure 1). More precisely, let a set of hard particles {D  ,= 1, 2, . . .} be given, where each D  has a fxed geometry. Let all the particles be randomly located in the space and each particle D  occupies a domain   without deforma- tions. Tus, the deterministic elements D  are introduced independently but the joint set is introduced { 1 , 2 , . . .} randomly. Te diversity of random locations is expressed by joint probabilistic distributions of the nonoverlapping domains   . Various methods were applied to such random com- posites. Te most known theoretical approach is based on the -point correlation functions presented by Torquato [9] and summarized below in Section 3. Tis theory yields analytic formulae and bounds for the efective properties. Te main shortage of this theory is computational difculties to calculate the -point correlation functions for large . Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 535128, 15 pages http://dx.doi.org/10.1155/2015/535128