3.2.4.10. Ampere Equation

Ampere’s equation in a conductive material is given by

\nabla \times \vector{H} = \vector{J} + \frac{\partial \vector{D}}{\partial t}

where \vector{H} is the magnetic field intensity, \vector{J} is the current density and \vector{D} is the electric displacement. Ignoring time dependent change in electric displacement a common approach of solving Ampere’s equation for a magnetic flux intensity {\bf H} in a conductive material is to recast the formulation in terms of magnetic field flux {\B} by employing the constitutive equation relating magnetic field intensity to magnetic field flux, permeability of free space \mu_{o} and magnetization vector \vector{M}, {\bf H}=\frac{1}{\mu_{o}} {\B} -\vector{M} subject to the constraint \nabla \cdot {\B}. An alternative formulation of this problem is obtained by introducing a magnetic vector potential {\A} where \nabla \times {\A} = {\B}. This approach circumvents the need to explicitly impose the divergence free constraint \nabla \cdot {\B} required in the formulation of the magnetic flux density. For static configurations the current density is a function of the electric field \vector{E} and the electrical conductivity \sigma, \vector{J} = \sigma \cdot \vector{E} whereas for conductors moving with velocity {\v} \vector{J} = \sigma \cdot \vector{E} + \sigma \cdot ( {\v} \times {\B}) . Hence the formulation for moving conductors will include contributions obtained from the solution of Ohm’s law for voltage V (see Voltage Equation).

(3.37)\pt{\A} - \v \times (\grad \times {\A}) + \frac{1}{\mu \sigma} \grad \times (\grad \times {\A}) - \grad ( \frac{1}{\mu \sigma} \grad \cdot {\A}) + \grad V - \grad \times \vector{M} = \vector{0}

and using the curl-curl identity and recalling that \grad \cdot {\A} = 0 then

(3.38)\pt{\A} - \v \times (\grad \times {\A}) + \grad^{2} \frac{1}{\mu \sigma} {\A} + \grad V - \grad \times \vector{M} = \vector{0} \;.

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

(3.39)\symRes_{m,k}^i & = \int\limits_\Vol \left[ \left(\density\pt{\A} - \v \times (\grad \times {\A}) + \grad V - \grad \times \vector{M} \right)\bcdot\e_k\phi^i + \frac{1}{\mu \sigma} \grad {\A}\bcdot \grad\left(\e_k\phi^i\right)\right]\dV \\ & - \int\limits_\Surf \frac{1}{\mu \sigma} (\n\bcdot\grad{\A}) \bcdot\e_k\phi^i\dS = 0 \;.

In Aria, each term in (3.39) is specified separately as identified in equation (3.40).

(3.40)\symRes_{m,k}^i & = \underbrace{\int\limits_\Vol \pt{\A}\bcdot\e_k\phi^i\dV}_\mathrm{MASS} - \underbrace{\int\limits_\Vol ({\v}\times\grad\times {\A})\bcdot\e_k\phi^i\dV}_\mathrm{ADV} + \underbrace{\int\limits_\Vol (\grad V - \grad \times \vector{M}) \bcdot\e_k\phi^i\dV}_\mathrm{SRC} \\ & + \underbrace{\int\limits_\Vol \frac{1}{\mu \sigma} \grad {\A} \bcdot \grad\left(\e_k\phi^i\right)\dV}_\mathrm{DIFF} - \int\limits_\Surf \frac{1}{\mu \sigma} (\n\bcdot\grad {\A}) \bcdot\e_k\phi^i\dS = 0 \;.