In this chapter we explain how to construct a discrete network for nonlinear high-contrast densely-packed composites. We use this presentation to demonstrate the so-called perforated medium technique in the analysis of high-contrast composites. We also use this investigation to demonstrate how the discrete network approximation arises from the interplay between geometry and asymptotic analysis. More specifically, the key mathematical feature of partial differential equations that describe high-contrast densely-packed composites is that their solutions exhibit asymptotically singular behavior, when particles are close to touching (high concentration). The singularities of the solutions occur exactly in the necks between almost touching particles. The location of these singularities can be characterized naturally by the geometric patterns of the distribution of the particles in the materials. Thus a geometric construction of a network is completely natural. As it is illustrated in this book, a rigorous mathematical justification is based on geometric and asymptotic arguments. These two arguments are coupled together. As a result, most of the constructions of asymptotic discrete network approximations for high-contrast composites are complicated, thus they are not attractive for practitioners. It is possible, however, to separate the geometric and the asymptotic arguments. It makes the construction of the network more transparent, and allows us to strengthen some of the previous results. In particular, it turns out that the validity of discrete network approximations could be verified for a class of composites, which is larger than the one that satisfies the δ-N close-packing condition.

In this chapter we review several ways of applying network models to inhomogeneous continuum media and systems of inclusions.

Discrete networks have been used as analogs of continuum problems in various areas of physics and engineering for a long time (see, e.g., Acrivos and Chang (1986); Ambegaokar et al. (1971); Bergman et al. (1990); Curtin and Scher (1990b); Koplik (1982); Newman (2003); Schwartz et al. (1984)). However, as demonstrated in Kolpakov (2006a), such analogs may or may not provide a correct approximation. In recent decades, the problem of the development of network models as rigorous approximations of continuum models was posed and solved for certain physical problems.

The objectives of our book are two-fold. First, we will develop an approach that allows us to derive network models by structural discretization (structural approximation). The key feature of this approach is that it is based on a rigorous asymptotic analysis with controlled error estimates, and thus we obtain the limits of validity for the network approximation. Secondly, we show that our network models are efficient tools in the study and prediction of properties of disordered particle-filled composites of various kinds.

Examples of real-world problems leading to discrete network models

Our interest is motivated by real-word problems and we next present three examples of highly packed composites which can be modeled by networks.

This chapter follows closely the work of Berlyand and Kolpakov (2001). The approach presented here was applied to the modeling of particle-filled composite materials. It is based on dual variational bounds and has been applied to both two-and three-dimensional problems (Berlyand et al., 2005). Further development of this approach allowed us to obtain error estimates for the network approximation (Berlyand and Novikov, 2002). It also provides answers to several unsettled physical questions, such as polydispersity at high concentration (Berlyand and Kolpakov, 2001; Berlyand and Mityushev, 2005), weak and strong blow up of the effective viscosity of disordered suspensions (Berlyand and Panchenko, 2007), and it establishes a connection between the notion of capacitance and the network approximation (Kolpakov, 2005, 2006a). Subsequently this approach was generalized for fluids. Next a new “fictitious fluid” approach was introduced in Berlyand et al. (2005). This approach led to a complete description of all singular terms in the asymptotics of the viscous dissipation rate of such suspensions and provided a comprehensive picture of microflows in highly packed suspensions. Note that previous works addressed only certain singularities and therefore provided a partial analysis of such microflows. It also allowed us to predict an anomalous singularity in two-dimensional problems (thin films) which has no analog in three-dimensions (Berlyand and Panchenko, 2007).

In this section, we present an application of the network model developed in Chapter 3 to the numerical analysis of high-contrast composite materials.

In Chapter 3, we expressed the leading term A of the conductivity of high-contrast composite materials through the solution of the network problem (3.4.11). The dimension of the network problem (3.4.11) is significantly smaller than the dimension of a non-structural (for example, finite elements or finite differences) approximation of the original problem (3.2.3)–(3.2.7). We demonstrate that the network approximation also provides us with an effective tool for the numerical analysis of high-contrast composite materials.

We consider models of a composite material filled with mono- and polydispersed particles (once again, we will model particles by disks). A composite material is called monodispersed if all disks have the same radii. If the radii of the disks vary, then the composite material is called polydispersed.

Computation of flux between two closely spaced disks of different radii using the Keller method

In order to analyze polydispersed composite materials, we need to know the flux between two disks (from one disk to another) of different radii if the potential on each disk is constant. A simple approximate formula for this flux was obtained in Keller (1987) for identical disks. We employ Keller's method to derive an approximate formula for the flux between two disks (the i-th and the j-th) of arbitrary radii Ri and Rj placed at a distance δij from one another (see Figure 4.1).

In this chapter, we present a method that allows one to obtain an a priori error estimate for the discrete network approximation independent of the total number of filling particles. Such estimates are referred to as homogenization estimates. These estimates can be derived under the natural δ-N close-packing condition (Berlyand and Novikov, 2002), which, loosely speaking, allows for “holes” (regions containing no particles) to be present in the medium of order NR (see Figure 5.1). Here, R is the radius of the particles and N is the number of particles in the perimeter of the largest hole in the conducting cluster (see Figure 5.2). We demonstrate that the error of the network approximation is determined not by the total number of particles in the composite material but by the perimeter of these “holes”. The explicit dependence of the network approximation and its error on the irregular geometry of the particle array is explicitly evaluated.

Formulation of the mathematical model

We consider here the composite material described in Section 3.1.1. It is a two-dimensional rectangular specimen of a two-phase composite material that consists of a matrix filled by a large number of ideally conducting disks that do not intersect. In this chapter, we do not assume any restriction on the number of particles and prove the network approximation theorem independent of the total number of particles.