DFLUX

A steady-state heat transfer analysis of a unit block is performed. The block is composed of six DC2D8 elements. Side A of the block (nodes 1–7) has its temperature, ${\theta }_A$, ramped up linearly over the course of a step. The opposite side of the block, side B (nodes 201–207), has a nonuniform distributed flux, $q_B$, applied to it via user subroutine DFLUX. The value of the distributed flux varies as a function of the current temperature of this side, ${\theta }_B$. This variation of applied flux is chosen to be $q_B\left({\theta }_B\right)=-k{\theta }_B$, where k is the conductivity of the block material. A thermal energy balance,

gives us a solution for ${\theta }_B$ such that ${\theta }_B={\theta }_A/2$.

The inclusion of $dq/d{\theta }_B$ in user subroutine DFLUX is essential for good convergence of the solution.

A steady-state mass diffusion analysis of a unit block is performed. The block is composed of six DC2D8 elements. Side A of the block (nodes 1–7) has its normalized concentration, ${\varphi }_A$, ramped up linearly over the course of a step. The opposite side of the block, side B (nodes 201–207), has a nonuniform distributed flux, $q_B$, applied to it via user subroutine DFLUX. The value of the distributed flux varies as a function of the current normalized concentration, ${\varphi }_B$; temperature, ${\theta }_B$; and equivalent pressure stress, $p_B$, of this side. This variation of applied flux is chosen to be $q_B\left({\varphi }_B,{\theta }_B,p_B\right)=-D\left(\theta ,p\right){\varphi }_B$, where $D\left(\theta ,p\right)$ is the diffusivity of the block material. The diffusivity is defined as

and diffusion is otherwise considered to be independent of temperature and equivalent pressure stress (i.e., ${\kappa }_p={\kappa }_s=$ 0). The temperature and pressure stress fields are specified at all nodes and are ramped up linearly over the course of the step. The mass balance,

gives a solution for ${\varphi }_B$ such that ${\varphi }_B={\varphi }_A/2$.

The inclusion of $dq/d{\varphi }_B$ in user subroutine DFLUX is essential for good convergence of the solution.