HFormulation

The $\vec H$ formulation is an $\vec H$ conformal formulation, which strongly enforces $\vec ∇ \cdot \vec J = 0$ 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 Faraday’s law:

\[\vec ∇× \left(ρ \vec ∇× \vec H\right) + \mu\partial_t \vec H = -\partial_t \vec B^\mathrm{ext}\]

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

The material coefficients $\mu$ and $ρ$ vary spatially and represent the local magnetic permeability and the electric resistivity respectively. In terms of these variables, the electric current $\vec J = \vec ∇ × \vec H$, the electric field $\vec E = ρ \vec ∇ × \vec H$, and the (total) magnetic flux density $\vec B = \vec B^\mathrm{ext} + \mu \vec H$.

Weak Form

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

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

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

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

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

\[\langle ρ \delta t \vec ∇× \partial_t \vec H_{n+1}, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle μ \partial_t \vec H_{n+1}, \vec v \rangle_{\vec L^2(\Omega)} = -\langle\vec \partial_t \vec B_{n+1}^\mathrm{ext}, \vec v\rangle_{\vec L^2(\Omega)} -\langleρ \vec ∇× \vec H_n, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle\vec E_{n+1} × \vec n, \vec v\rangle_{\vec L^2(\partial \Omega)}\]