4 - Numerics for percolation and polydispersity via network models

Summary

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 R_i and R_j placed at a distance δ_ij from one another (see Figure 4.1).