Equation 9–3, along with the boundary conditions and loads, governs the diffusion process in a homogeneous domain. In an inhomogeneous domain, different materials have different saturated concentrations. This difference in the saturation levels is responsible for the discontinuity in the concentration across the material interface. To be able to use Equation 9–3 in a finite element analysis, a continuous variable, normalized concentration , is introduced [405].

(9–5)

where is the saturated concentration of the material (input as CSAT on MP command).

Substituting the concentration from Equation 9–5 into Equation 9–3 produces a governing equation for the diffusion analysis in terms of normalized concentration :

(9–6)

When is not specified, it defaults to 1.0 and Equation 9–6 becomes Equation 9–3.

can be temperature-dependent for the coupled-field elements PLANE223, SOLID226, and SOLID227 only with thermal and diffusion DOFs. In this case, applying the chain rule of differentiation with respect to temperature T to Equation 9–6 without the transport term produces:

(9–7)

For more information, see Thermal-Diffusion Coupling.