H-\(\phi\) Formulation

The H-\(\phi\) Formulation combines the magnetic field and a magnetic scalar potential \(\phi\). This formulation avoid the use of \(\rho_\text{air}\) in the H Formulation.

For this formulation, we will use the following notations :

\(\Omega\) the domain, consisting of the superconductor domain \(\Omega_c\) and non conducting materials \(\Omega_n\) (\(\mathbf{J} = 0\)) like the air.
\(\Gamma = \partial \Omega\) the boundary of \(\Omega\),
\(\Gamma_c = \partial \Omega_c\) the boundary of \(\Omega_c\),
\(\Gamma^c_D\) the boundary between the conductor and the non-conductor with Dirichlet boundary condition
\(\Gamma^c_N\) the boundary between the conductor and the non-conductor with Neumann boundary condition

\(\mathbf{n}\) denotes the exterior normal on \(\Gamma\) to \(\Omega\).

1. Differential Formulation

In current free domain, for simply connected materials, we have :

\[\nabla\times \textbf{H}=0\]

So the magnetic field H can be written as :

\[\textbf{H}=-\nabla\phi\]

This definition forces the electric current density J to be zero in non conducting domains since the curl of a gradient is equal to zero :

\[\textbf{J}=\nabla\times \textbf{H}=\nabla\times(-\nabla\phi)=0\]

Using the Maxwell-Thompson equation \(\nabla\times\textbf{B}=0\) and the fact that \(\textbf{B}=\mu_0\textbf{H}\), we have :

\[-\nabla\cdot\nabla\phi=0 \quad \text{ in }\Omega_n\]

In the conducting domain, the governing equation stays the H formulation :

\[-\nabla\times \left(\rho \nabla\times \textbf{H}\right)=\partial_t(\mu \textbf{H})\]

where the resistivity is non-linear for superconductors and follows the E-J power law :

\[\rho=\frac{E_c}{J_c}\left(\frac{\mid\mid \nabla\times\textbf{H} \mid\mid}{J_c}\right)^{(n)}\]

In order to couple the equations, we use the boundary condition of the interface between the conducting domain and the non-conducting domain. On the conductor part, we use Dirichlet boundary conditions :

\[n\times \textbf{H}=n\times - \nabla\phi\]

On the non-conductor part, we use Neumann boundary condition :

\[\frac{\partial\phi}{\partial n} =\nabla\phi\cdot n=\textbf{H}\cdot n\]

We put the equations together and we finally have the H-\(\phi\) Formulation :

H-\(\phi\) Formulation

\[\text{(H-phi)} \left\{ \begin{matrix} -\nabla\times \left(\rho \nabla\times \textbf{H}\right)=\partial_t(\mu \textbf{H}) & \text{ on } \Omega_c & \text{(H-form)} \\ -\nabla\cdot\nabla\phi=0 & \text{ on } \Omega_n & (\phi\text{-form)} \\ \textbf{H} = - \nabla\phi & \text{ on } \Gamma^c_D & \text{(D)} \\ \frac{\partial \phi}{\partial \mathbf{n}} = \textbf{H}\cdot n & \text{ on } \Gamma^c_N & \text{(N)} \end{matrix} \right.\]

1.1. Weak Formulation

In this subsection, we express the weak formulation of H-\(\phi\) Formulation with Dirichlet and Neumann boundary conditions.

We introduce the set :

\[H^{curl}(\Omega) = \{ \textbf{v} \in L^2(\Omega), \nabla \times \textbf{v} \in L^2(\Omega) \}\]

The set \(H^{curl}(\Omega)\) is a Hilbert space with the scalar product :

\[<\mathbf{v_1},\mathbf{v_2}>_{H^{curl}(\Omega)} = \int_{\Omega}{ \mathbf{v_1} \cdot \mathbf{v_2} + \nabla \times \mathbf{v_1} \cdot \nabla \times \mathbf{v_2} } \text{ for all } \mathbf{v_1}, \mathbf{v_2} \in H^{curl}(\Omega)\]

We multiply H formulation equation by \(\xi \in H^{curl}(\Omega_c)\) and integrating it on \(\Omega_c\) :

\[-\int_{\Omega_c}{\nabla\times \left(\rho \nabla\times \textbf{H}\right) \cdot \xi} = \int_{\Omega_c}{ \partial_t(\mu \textbf{H}) \cdot \xi}\]

By Formula of Green :

\[\int_{\Omega_c}{ \rho \nabla\times \textbf{H} \cdot \nabla\times\xi} + \int_{\Gamma^c_D}{\rho \nabla\times \textbf{H} \cdot (\xi \times \mathbf{n})} + \int_{\Gamma^c_N}{\rho (\nabla\times \textbf{H} \times \mathbf{n}) \cdot \xi} = \int_{\Omega_c}{ \partial_t(\mu \textbf{H}) \cdot \xi}\]

We impose the following boundary conditions :

Dirichlet : \(\textbf{H} \times \mathbf{n} = \nabla\phi \times \mathbf{n}\) on \(\Gamma^c_D\)
Neumann : \(\left( \nabla \times \textbf{H} \right) \times \mathbf{n} = 0\) on \(\Gamma^c_N\)

We obtain :

Weak formulation of H Formulation

\[\text{(Weak H)} \begin{cases} \int_{\Omega_c}{ \rho \nabla\times \textbf{H} \cdot \nabla\times\xi}+ \int_{\Gamma^c_D}{ \nabla\phi \times \mathbf{n} \cdot (\xi \times \mathbf{n})} = \int_{\Omega_c}{ \partial_t(\mu \textbf{H}) \cdot \xi}\\ \forall \xi \in H^{curl}(\Omega_c)\end{cases}\]

Then, we multiply the \(\phi\) formulation equation by \(\gamma \in H^{1}(\Omega_n)\) and integrating it on \(\Omega_n\) :

\[-\int_{\Omega_n}{\nabla\cdot\nabla\phi \cdot \gamma} = 0\]

By Formula of Green :

\[\int_{\Omega_n}{ \nabla\phi \cdot \nabla\gamma} + \int_{\Gamma^c_D}{\nabla\phi \times \mathbf{n} \cdot \gamma} + \int_{\Gamma^c_N}{\nabla\phi \times \mathbf{n} \cdot \gamma} = 0\]

We impose the following boundary conditions :

Dirichlet : \(\nabla\phi \times \mathbf{n} = 0\) on \(\Gamma^c_D\)
Neumann : \(\nabla\phi \times \mathbf{n} = \textbf{H}\times \mathbf{n}\) on \(\Gamma^c_N\)

We obtain :

Weak formulation of phi Formulation

\[\text{(Weak phi)} \begin{cases} \int_{\Omega_n}{ \nabla\phi \cdot \nabla\gamma} + \int_{\Gamma^c_N}{\textbf{H} \times \mathbf{n} \cdot \gamma} = 0\\ \forall \gamma \in H^{1}(\Omega_n)\end{cases}\]

We combine both formulation and we have :

Weak formulation of H-phi Formulation

\[\text{(Weak H-phi)} \left\{ \begin{matrix} \int_{\Omega_c}{ \rho \nabla\times \textbf{H} \cdot \nabla\times\xi}+ \int_{\Gamma^c_D}{ \nabla\phi \times \mathbf{n} \cdot (\xi \times \mathbf{n})} &= \int_{\Omega_c}{ \partial_t(\mu \textbf{H}) \cdot \xi}\\ \int_{\Omega_n}{ \nabla\phi \cdot \nabla\gamma} + \int_{\Gamma^c_N}{\textbf{H} \times \mathbf{n} \cdot \gamma} &= 0\\ \forall \xi \in H^{curl}(\Omega_c), \forall \gamma \in H^{1}(\Omega_n) \end{matrix} \right.\]

1.2. Time Discretization

In this section, we see the Weak formulation of H-\(\phi\) Formulation with time discretization by backward Euler method.

We discretize in time the problem with time step \(\Delta t\).

We note \(f^n(\mathbf{x}) = f(n\Delta t, \mathbf{x})\), for \(n \in \mathbb{N}\).

We have the approximation with backward Euler method : \(\frac{d \textbf{H}}{d t} \approx \frac{\textbf{H}^{n+1}-\textbf{H}^n}{\Delta t}\).

The equations Weak formulation of H-\(\phi\) Formulation becomes :

Time Discretization of H Formulation

\[\left\{ \begin{matrix} \int_{\Omega_c}{ \rho \nabla\times \textbf{H}^{n+1} \cdot \nabla\times\xi}+ \int_{\Gamma^c_D}{ \nabla\phi^{n+1} \times \mathbf{n} \cdot (\xi \times \mathbf{n})}-\int_{\Omega_c}{ \frac{\mu \textbf{H}^{n+1}}{\Delta t} \cdot \xi} &= \int_{\Omega_c}{ \frac{\mu \textbf{H}^{n}}{\Delta t} \cdot \xi}\\ \int_{\Omega_n}{ \nabla\phi^{n+1} \cdot \nabla\gamma} + \int_{\Gamma^c_N}{\textbf{H}^{n+1} \times \mathbf{n} \cdot \gamma} &= 0\\ \forall \xi \in H^{curl}(\Omega_c), \forall \gamma \in H^{1}(\Omega_n) \end{matrix} \right.\]

2. References

COMSOL Implementation of the H-ϕ-Formulation With Thin Cuts for Modeling Superconductors With Transport Currents, A. Arsenault, B. d. S. Alves and F. Sirois, in IEEE Transactions on Applied Superconductivity, vol. 31, no. 6, pp. 1-9, Sept. 2021, Art no. 6900109, doi: 10.1109/TASC.2021.3097245. PDF