3.2.4.8. Current Equation

An alternate formulation in solving for the electrical potential (see Voltage Equation) is to solve the “current” equation which is a conservation equation for electrical charge. The electrical current $\vector{J}$ is frequently related to the electric field $\vector{E}$ using Ohm’s law as $\vector{J} = \eleccond\vector{E}$ where $\eleccond$ is the electrical conductivity. The electric potential or voltage $V$ is used in determining the electric field, $\vector{E} = -\grad V$ . However, we choose to leave the electrical current as a more general constitutive model to be provided as a material model input (see Current Density). More complex version for charged ion fluxes (Nerst Planck, etc. can be utilized with the species equation).

(3.31) $-\div\vector{J} = S$

Here $S$ is a current source term, for example from electrochemical reactions in a battery simulation.

Using equation (3.3), the G/FEM residual form is

(3.32) $\symRes_V^i = \int\limits_\Vol \left(-S \phi^i -\grad\phi^i\bcdot\vector{J}\right)\dV + \int\limits_\Surf q_n\phi^i\dS = 0$

In Aria, each term in (3.32) is specified separately as identified in equation (3.33).

(3.33) $\symRes_V^i = - \underbrace{\int\limits_\Vol S \phi^i\dV}_\mathrm{SRC} - \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\vector{J}\dV}_\mathrm{DIFF} + \int\limits_\Surf q_n\phi^i\dS = 0$