3.2.4.9. Charge Density Equation

This equation solves for the volumetric free charge density, \rho_e, and is meant to be coupled with the voltage equation in Voltage Equation. This equation can also be used in place of the current equation (3.31), since this equation has the correct form (the current equation has the correct form with \rho_e when electric displacement is activated) and the charge density is treated as a variable. The charge density evolves as

(3.34)\frac{\partial \rho_e}{\partial t}+ \vector{v} \cdot \grad \rho_e = -\div \vector{J} = \div \left( \sigma \grad V \right)

where \vector{J}=-\sigma \grad V is assumed to obey Ohm’s law and \sigma represents the conductivity.

The G/FEM residual form of this equation is

(3.35)\symRes_V^i = \int\limits_\Vol \left( \frac{\partial \density_e}{\partial t} \phi^i +\grad\phi^i\bcdot\vector{v}+\grad\phi^i\bcdot \sigma \grad V \right)\dV + \int\limits_\Surf q_n\phi^i\dS = 0

In Aria, each term in (3.35) is specified separately as identified in equation (3.36).

(3.36)\symRes_V^i = \underbrace{\int\limits_\Vol \frac{\partial \density_e}{\partial t} \phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\vector{v}\dV}_\mathrm{ADV} + \underbrace{\int\limits_\Vol \grad\phi^i\bcdot \sigma \grad V \dV}_\mathrm{DIFF} + \int\limits_\Surf q_n\phi^i\dS = 0.