EFormulation

The $\vec E$ formulation is a $\partial_t \vec B$ conformal formulation, which strongly enforces $\vec ∇ \cdot \partial_t \vec B = 0$ 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:

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

where $\vec E ∈ H(\mathrm{curl})$ 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 rate of change of the magnetic flux density $\partial_t \vec B = -\vec ∇ × \vec E$, the rate of change of the magnetic field $\partial_t \vec H = -ν \vec ∇× \vec E$, and the (total) electric current $\vec J = \vec J^\mathrm{ext} + σ \vec E$.

Weak Form

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

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

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

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

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

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