3.1. Domain Definition

To fix the notation, consider Fig. 3.1, which is a schematic representation of a typical problem. The entire domain is represented by $\symDomain$ , which, for example, lies in three-dimensional coordinate space, with spatial coordinates $\symCoordVect$ . In this particular case, $\symDomain$ consists of two separate subdomains, $\symDomain = \symDomain_1 \cup \symDomain_2$ . These subdomains may consist of different materials. The entire boundary of $\symDomain$ is indicated by $\symDomainBoundary$ , subject to one or more boundary conditions on subsections we denote with a subscript on $\Gamma$ . For example, let $\Gamma_q$ be that portion of $\symDomainBoundary$ along which a specified heat flux normal to the boundary is applied; similarly, let $\Gamma_T$ be subject to an applied temperature; let the surface $\Gamma_a$ be adiabatic (no heat flux); let $\Gamma_r$ , be subject to an applied radiation heat flux; and let $\Gamma_h$ be subject to a convective heat flux, which is modeled by Newton’s law of cooling. Note that the boundary conditions are of two types: either the flux or the temperature is specified. Finally, the interface between $\Omega_1$ and $\Omega_2$ is denoted $\partial\Omega_{1-2}$ . The interface conditions applied along a boundary such as $\partial\Omega_{1-2}$ are that both the temperature and the normal component of the heat flux are continuous. Given appropriate initial conditions, the problem is to determine the time-evolution of the temperature field.

Schematic of the domain and boundary conditions — Fig. 3.1 A schematic diagram of the mathematical thermal model, showing the domain $\Omega$ ; the subdomains $\Omega_i$ and their interfaces $\partial\Omega_{i-j}$ ; and the boundary conditions on the surface $\Gamma$ .

Fig. 3.1 A schematic diagram of the mathematical thermal model, showing the domain $\Omega$ ; the subdomains $\Omega_i$ and their interfaces $\partial\Omega_{i-j}$ ; and the boundary conditions on the surface $\Gamma$ .

Note that when referencing a generic volume or surface, $\Vol$ and $\Surf$ will be used respectively.