H Formulation

To describe the behavior of the magnetic field of superconductors, we solve the Maxwell equations with the Maxwell Quasi Static approximations. To solve numerically those equations, we will use the H Formulation.

1. Maxwell Equations

The Maxwell’s equations are approximated by the MQS (Maxwell Quasi Static) approximation, where the displacement current is neglected. In this context, the equations become :

Maxwell Equations in MQS
\[\begin{align*} \nabla \cdot \textbf{B} &= 0 & \text{ (Maxwell-Thomson)} \\ \nabla \times \textbf{H} &= \textbf{J} & \text{ (Maxwell-Ampere)}\\ \nabla \times \textbf{E} &= - \frac{\partial \textbf{B}}{\partial t} & \text{ (Maxwell-Faraday)} \\ \textbf{B} &= \mu \, \textbf{H} &\\ \textbf{J} &= \sigma \, \textbf{E}& \end{align*}\]

With the notations :

  • \(\textbf{H}\) magnetic field (\(A m^{-1}\))

  • \(\textbf{J}\) current density (\(A m^{-2}\))

  • \(\textbf{E}\) electric field (\(V m^{-1}\))

  • \(\sigma\) conductivity (\(S m^{-1}\))

  • \(\textbf{B}\) magnetic flux density (\(T\))

  • \(\mu\) permeability (\(H m^{-1}\))

2. H formulation

For this formulation, we will use the following notations :

  • \(\Omega\) the domain, consisting of the superconductor domain \(\Omega_c\) and the 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 \(\Omega_c\),

  • \(\Gamma_D\) the boundary with Dirichlet boundary condition

  • \(\Gamma_N\) the boundary with Neumann boundary condition

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

2.1. Differential Formulation

With the Maxwell-Faraday equation, \(\nabla\times \textbf{E}=-\partial_t \textbf{B}\), we replace \(\textbf{B}\) by \(\mu \textbf{H}\) and \(\textbf{E}\) by \(\rho \textbf{J}\) (\(\rho=1/\sigma\) the resistivity) :

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

Now we can modify the equation of Maxwell-Ampère :

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

We multiply the equation by the resistivity :

\[ \rho \nabla\times \textbf{H}=\rho \textbf{J}\]

Then the curlcurl expression appears :

\[ \nabla\times \left(\rho \nabla\times \textbf{H}\right)=\nabla\times \left(\rho \textbf{J}\right)\]

We put the equations together and we finally have the H Formulation :

H Formulation
\[\text{(H)} \left\{ \begin{matrix} -\nabla\times \left(\rho \nabla\times \textbf{H}\right)&=&\partial_t(\mu \textbf{H}) &\text{ on } \Omega & \text{(H)} \\ \mathbf{H} \times \mathbf{n} &=& 0 &\text{ on } \Gamma_D & \text{(D)} \\ \left( \nabla \times \mathbf{H} \right) \times \mathbf{n} &=& 0 &\text{ on } \Gamma_N & \text{(N)} \end{matrix} \right.\]

with \(\rho\) the resistivity following the E-J power law for the superconductor :

\[\rho=\frac{E_c}{J_c}\left(\frac{\mid\mid J \mid\mid}{J_c}\right)^{(n)}\]

In the air, we give the resistivity a high value like \(\rho_\text{air}=1\).

2.2. Weak Formulation

In this subsection, we express the weak formulation of H 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 \(\Phi \in H^{curl}(\Omega)\) and integrating it on \(\Omega\) :

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

By Formula of Green :

\[\int_{\Omega}{ \rho \nabla\times \textbf{H} \cdot \nabla\times\Phi} + \int_{\Gamma_D}{\rho \nabla\times \textbf{H} \cdot (\Phi \times \mathbf{n})} + \int_{\Gamma_N}{\rho (\nabla\times \textbf{H} \times \mathbf{n}) \cdot \Phi} = \int_{\Omega}{ \partial_t(\mu \textbf{H}) \cdot \Phi}\]

We impose the following boundary conditions :

  • Dirichlet : \(\textbf{H} \times \mathbf{n} = 0\) on \(\Gamma_D\)

  • Neumann : \(\left( \nabla \times \textbf{H} \right) \times \mathbf{n} = 0\) on \(\Gamma_N\)

We obtain :

Weak formulation of H Formulation
\[\text{(Weak H)} \begin{cases} \int_{\Omega}{ \rho \nabla\times \textbf{H} \cdot \nabla\times\Phi} = \int_{\Omega}{ \partial_t(\mu \textbf{H}) \cdot \Phi}\\ \forall \Phi \in H^{curl}(\Omega)\end{cases}\]

2.3. Time Discretization

In this section, we see the Weak formulation of H 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 Formulation becomes :

Time Discretization of H Formulation
\[\text{(Disc Weak H)} \begin{cases} \int_{\Omega}{ \rho \nabla\times \textbf{H}^{n+1} \cdot \nabla\times\Phi} - \int_{\Omega}{ \mu \frac{\textbf{H}^{n+1}}{\Delta t} \cdot \Phi} = -\int_{\Omega}{ \mu \frac{\textbf{H}^n}{\Delta t} \cdot \Phi}\\ \forall \Phi \in H^{curl}(\Omega)\end{cases}\]

3. References

  • Finite-Element Formulation for Systems with High-Temperature Superconductors, Julien Dular, Christophe Gauzaine, Benoît Vanderheyden, IEEE Transactions on Applied Superconductivity VOL. 30 NO. 3, April 2020, PDF