Transient Curl-Curl Formulations
The set of transient curl-curl formulations are intended to solve Maxwell’s equations in low frequency limits in the time domain. The governing equation for these formulations is given by
\[\vec ∇× \left(α \vec ∇× \vec u\right) +\partial_t \left(β \vec u \right) = \vec g\]where $\vec u ∈ H(\mathrm{curl})$ and $\vec g ∈ H(\mathrm{div})$. $α$ and $β$ are material-dependent coefficients. This is solved using the weak form
\[\langle\alpha \vec ∇× \vec u, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle\beta \partial_t \vec u, \vec v \rangle_{\vec L^2(\Omega)} - \langle\vec g, \vec v\rangle_{\vec L^2(\Omega)} - \langle(α \vec ∇× \vec u) × \vec n, \vec v\rangle_{\vec L^2(\partial \Omega)} = 0\]where the test function $\vec v ∈ H(\mathrm{curl})$.
Time Discretisation
Time discretisation is performed using a backwards Euler method,
\[\vec u_{n+1} = \vec u_{n} + \delta t \left(\partial_t \vec u\right)_{n+1}\]Substituting into the weak form evaluated at timestep $n+1$;
\[\langle\alpha \delta t \vec ∇× \partial_t \vec u_{n+1}, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle\beta \partial_t \vec u_{n+1}, \vec v \rangle_{\vec L^2(\Omega)} = \langle\vec g_{n+1}, \vec v\rangle_{\vec L^2(\Omega)} -\langle\alpha \vec ∇× \vec u_n, \vec ∇× \vec v \rangle_{\vec L^2(\Omega)} + \langle(α \vec ∇× \vec u_{n+1}) × \vec n, \vec v\rangle_{\vec L^2(\partial \Omega)}\]With suitable boundary conditions, one can therefore solve for $\partial_t \vec u_{n+1}$.
Formulations
Formulations that are solved using this weak form are:
- H Formulation - an $\vec H$ conformal formulation, taking the trial function $\vec u$ as as the magnetic field $\vec H$ to strongly enforce $\vec ∇ \cdot \vec J = 0$ in the solution.
- A Formulation - a $\vec B$ conformal formulation, taking the trial function $\vec u$ as the magnetic vector potential $\vec A$ to strongly enforce $\vec ∇ \cdot \vec B = 0$ in the solution.
- E Formulation - taking the trial function $\vec u$ as as the electric field $\vec E$ to strongly enforce $\vec ∇ \cdot \partial_t \vec B = 0$ in the solution.