2.17: Methodology

As descriptions of subregions composing a heterogeneous system, transfer relations (illustrated for quasistatic fields in Sec. 2.16) are building blocks for describing complicated interactions. By appropriate identification of variables, the same relations can be used to describe different regions.

As an example, three planar regions are shown in Figure 2.20.1. The symbols in parentheses denote positions adjacent to the surfaces demarking subregions. At the surfaces, variables can be discontinuous. Hence it is necessary to distinguish variables evaluated on adjacent sides of a boundary. The transfer relations describe the fields within the subregions and not across the boundaries.

Figure 2.20.1. Convention used to denote surface variables.

The transfer relations of Table 2.16.1 can be applied to the upper region by identifying \((\alpha) \rightarrow (d), \quad (\beta) \rightarrow (e), \quad \Delta \rightarrow a\) and \(\varepsilon\) or \(\mu \rightarrow \varepsilon_a\) or \(\mu_a\). Similarly, for the middle region,\((\alpha) \rightarrow (f), \quad (\beta) \rightarrow (g), \quad \Delta \rightarrow b\) and \(\varepsilon\) or \(\mu \rightarrow \varepsilon_b\) or \(\mu_b\). Boundary conditions and jump relations across the surfaces then provide coupling condition on the surface variables. Once the surface variables have been self-consistently determined, the field distributions within the region can be evaluated using the bulk distributions evaluated in terms of the surface coefficients. With appropriate surface amplitudes and \(x \rightarrow x^{'}\), where the latter is defined for each region in Figure 2.20.1, Equation 2.16.15 describes the potential distribution.

This approach will be used not only in other geometries but in representing mechanical and electromechanical processes.