AFormulation

The $\vec A$ formulation is a $\vec B$ conformal formulation, which strongly enforces $\vec ∇ \cdot \vec B = 0$ in the solution. It is one of the time domain curl-curl formulations available from Hephaestus.

The governing equation for this formulations is given by Ampere’s law:

\[\vec ∇× \left(ν \vec ∇× \vec A\right) +σ \partial_t \vec A = \vec J^\mathrm{ext}\]

where $\vec A ∈ H(\mathrm{curl})$ and $\vec J^\mathrm{ext} ∈ H(\mathrm{div})$.

The material coefficients $σ$ and $ν$ vary spatially and represent the local electric conductivity and the magnetic reluctivity (the reciprocal of the magnetic permeability) respectively. In terms of these variables, the magnetic flux density $\vec B = \vec ∇ × \vec A$, the magnetic field $\vec H = ν \vec ∇× \vec A$, the (induced) electric field $\vec E = - \partial_t \vec A$, and the (total) electric current $\vec J = \vec J^\mathrm{ext} - σ\partial_t \vec A$.

Weak Form

The $\vec A$ formulation is solved using the weak form

\[\langle ν \vec ∇× \vec A, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle σ \partial_t \vec A, \vec v \rangle_{\vec L^2(\Omega)} - \langle \vec J^\mathrm{ext}, \vec v\rangle_{\vec L^2(\Omega)} - \langle \vec H × \vec n, \vec v\rangle_{\vec L^2(\partial \Omega)} = 0\]

where the test function $\vec v ∈ H(\mathrm{curl})$. Time discretisation is performed using a backwards Euler method,

\[\vec A_{n+1} = \vec A_{n} + \delta t \left(\partial_t \vec A\right)_{n+1}\]

Substituting into the weak form evaluated at timestep $n+1$;

\[\langle ν \delta t \vec ∇× \partial_t \vec A_{n+1}, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle σ \partial_t \vec A_{n+1}, \vec v \rangle_{\vec L^2(\Omega)} = \langle \vec J_{n+1}^\mathrm{ext}, \vec v\rangle_{\vec L^2(\Omega)} -\langle ν \vec ∇× \vec A_n, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle \vec H_{n+1} × \vec n, \vec v\rangle_{\vec L^2(\partial \Omega)}\]