T-A Formulation

To describe the behavior of magnetic field of High Temperature Superconductor magnet with induced current, we will use the T-A formulation, which is a combination of two formulation based on the Maxwell equations with Maxwell Quasi Static approximations.

The following notations will be used in the following :

\(\Omega\) the domain, consisting of the superconductor domain \(\Omega_c\) and non conducting materials domain \(\Omega_c^C\) 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 and
\(\Gamma_N\) the boundary with Neumann boundary condition

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

1. Differential Formulation

For the T-A Formulation, we introduce a magnetic potential field \(\textbf{A}\) defined as :

\[ \textbf{B} = \nabla \times \textbf{A}\]

and an electrical potential field \(\textbf{T}\) defined as :

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

To ensure \(A\) uniqueness, we need to add a gauge condition :

\[ \nabla \cdot \textbf{A} = 0\]

From the A-Formulation part, the Maxwell-Ampere equation (\(\nabla \times \textbf{H} = \textbf{J}\)) is rewritten as:

\[ \nabla\times\left(\frac{1}{\mu}\nabla\times\textbf{A}\right) = \nabla\times\textbf{T}\]

As for the T-Formulation, the Maxwell-Faraday equation (\(\nabla \times \textbf{E} = - \frac{\partial \textbf{B}}{\partial t}\)) is rewritten as :

\[ \nabla \times \left(\rho\nabla\times\textbf{T} \right) = -\frac{\partial (\nabla\times\textbf{A})}{\partial t}\]

where \(\rho=1/\sigma\) is the resistivity. As the resistivity is zero in the non-conducting material, the T-formulation is only defined on \(\Omega_c\). In the superconductors, the resistivity follows the E-J power law :

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

In conclusion, the T-A Formulation consists in the following two equations :

T-A Formulation

\[\text{(T-A)}\left\{ \begin{matrix} \nabla\times\left(\frac{1}{\mu}\nabla\times\textbf{A}\right) &=& \nabla\times\textbf{T} \text{ on } \Omega & \text{(A-form)} \\ \nabla \times \left(\rho\nabla\times\textbf{T} \right) &=& -\frac{\partial (\nabla\times\textbf{A})}{\partial t} \text{ on } \Omega_c & \text{(T-form)} \\ \mathbf{A} \times \mathbf{n} &=& 0 \text{ on } \Gamma_D & \text{(D)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 \text{ on } \Gamma_N & \text{(N)} \end{matrix} \right.\]

with :

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

1.1. Weak Formulation

In this subsection, we express the weak formulation of T-A 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 :

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

1.1.1. First equation

We express the weak formulation of first equation of (A-form) (for \(\mathbf{A}\) unknown).

By multiplying the (A-form) by \(\mathbf{ϕ} \in H^{curl}(\Omega)\) and integrating over \(\Omega\), we obtain :

\[\int_{\Omega}{ \nabla \times (\frac{1}{\mu} \nabla\times \textbf{A}) \cdot \mathbf{ϕ}} = \int_{\Omega_c}{ \nabla \textbf{T} \cdot \mathbf{ϕ}}\]

By Formula of Green :

\[\int_{\Omega}{ \nabla \times (\frac{1}{\mu} \nabla\times \textbf{A}) \cdot \mathbf{ϕ}d\Omega} + \int_{\Gamma_D}{\frac{1}{\mu} (\mathbf{ϕ} \times (\nabla \times \textbf{A})) \cdot \mathbf{n}d\Gamma} + \int_{\Gamma_N}{\frac{1}{\mu} (\mathbf{ϕ} \times (\nabla \times \textbf{A})) \cdot \mathbf{n}d\Gamma} = \int_{\Omega_c}{ \nabla \textbf{T} \cdot \mathbf{ϕ}d\Omega}\]

Noting that \(\textbf{u} \cdot (\textbf{v} \times \textbf{w}) = \textbf{w} \cdot (\textbf{u} \times \textbf{v}) = \textbf{v} \cdot (\textbf{w} \times \textbf{u})\) on \(\Gamma_D\) or \(\Gamma_N\) :

\[\small{ \int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\nabla \times \mathbf{ϕ})}d\Omega + \int_{\Gamma_D}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\mathbf{ϕ} \times \mathbf{n})}d\Gamma + \int_{\Gamma_N}{\frac{1}{\mu} ((\nabla \times \textbf{A}) \times \mathbf{n}) \cdot ϕ}d\Gamma = \int_{\Omega_c}{ \nabla \textbf{T} \cdot \mathbf{ϕ}}d\Omega }\]

We impose the following boundary conditions :

Dirichlet : \(\mathbf{A} \times \mathbf{n} = 0\) on \(\Gamma_D\)
Neumann : \(\left( \nabla \times \mathbf{A} \right) \times \mathbf{n} = 0\) on \(\Gamma_N\)

We obtain :

Weak formulation of A Formulation

\[\text{(Weak A-form)}\begin{cases}\int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\nabla \times \mathbf{ϕ})}d\Omega = \int_{\Omega_c}{ \nabla \textbf{T} \cdot \mathbf{ϕ}}d\Omega \\ \forall \mathbf{ϕ} \in H^{curl}(\Omega)\end{cases}\]

1.1.2. Second equation

We express the weak formulation of (T-form) (for \(T\) unknown).

By mutliplying by (T-form) \(\mathbf{\psi} \in H^{curl}(\Omega_c)\) and integrating on \(\Omega_c\), we have :

\[-\int_{\Omega_c}{ \nabla\times\left(\rho\nabla\times\textbf{T} \right)\cdot \mathbf{\psi}} = \int_{\Omega_c}{ \frac{\partial (\nabla\times\textbf{A})}{\partial t}\cdot \mathbf{\psi}}\]

By Formula of Green :

\[\int_{\Omega_c}{\rho\nabla\times\textbf{T} \cdot \nabla\times\mathbf{\psi}d\Omega_c} + \int_{\Gamma_D}{\frac{1}{\mu} (\mathbf{\psi} \times (\nabla \times \textbf{T})) \cdot \mathbf{n}d\Gamma} + \int_{\Gamma_N}{\frac{1}{\mu} (\mathbf{\psi} \times (\nabla \times \textbf{T})) \cdot \mathbf{n}d\Gamma} = \int_{\Omega_c}{ \frac{\partial (\nabla\times\textbf{A})}{\partial t}\cdot \mathbf{\psi}d\Omega_c}\]

Noting that \(\textbf{u} \cdot (\textbf{v} \times \textbf{w}) = \textbf{w} \cdot (\textbf{u} \times \textbf{v}) = \textbf{v} \cdot (\textbf{w} \times \textbf{u})\) on \(\Gamma_D\) or \(\Gamma_N\) :

\[\small{ int_{\Omega_c}{\rho\nabla\times\textbf{T} \cdot \nabla\times\mathbf{\psi}d\Omega_c} + \int_{\Gamma_D}{\frac{1}{\mu} (\nabla \times \textbf{T}) \cdot (\mathbf{\psi} \times \mathbf{n})}d\Gamma + \int_{\Gamma_N}{\frac{1}{\mu} ((\nabla \times \textbf{T}) \times \mathbf{n}) \cdot \mathbf{\psi}}d\Gamma = \int_{\Omega_c}{ \frac{\partial (\nabla\times\textbf{A})}{\partial t}\cdot \mathbf{\psi}d\Omega_c} }\]

We impose the following boundary conditions :

Dirichlet : \(\mathbf{T} \times \mathbf{n} = 0\) on \(\Gamma_D\)
Neumann : \(\left( \nabla \times \mathbf{T} \right) \times \mathbf{n} = 0\) on \(\Gamma_N\)

We obtain :

Weak formulation of T Formulation

\[ \text{(Weak T-form)}\begin{cases}\int_{\Omega_c}{\rho\nabla\times\textbf{T} \cdot \nabla\times\mathbf{\psi}d\Omega_c} = \int_{\Omega_c}{ \frac{\partial (\nabla\times\textbf{A})}{\partial t}\cdot \mathbf{\psi}d\Omega_c}\\ \forall \mathbf{\psi} \in \in H^{curl}(\Omega_c)\end{cases}\]

1.1.3. The T-A weak Formulation

From (Weak A-form) and (Weak T-form), we obtain the weak formulation of the T-A Formulation :

Weak formulation of T-A Formulation

\[\text{(Weak TA)} \left\{ \begin{align*} \int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\nabla \times \mathbf{ϕ})d\Omega} = \int_{\Omega_c}{ \nabla \textbf{T} \cdot \mathbf{ϕ}d\Omega} \hspace{2cm} \text{(Weak A-form)}\\ \int_{\Omega_c}{\rho\nabla\times\textbf{T} \cdot \nabla\times\mathbf{\psi}d\Omega_c} = \int_{\Omega_c}{ \frac{\partial (\nabla\times\textbf{A})}{\partial t}\cdot \mathbf{\psi}d\Omega_c} \hspace{2cm} \text{(Weak T-form)} \\ \forall \mathbf{ϕ} \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in \in H^{curl}(\Omega_c) \end{align*} \right.\]

1.2. Time Discretization

In this section, we see the Weak formulation of T-A 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 : \(\frac{d \mathbf{A}}{d t} \approx \frac{\mathbf{A}^{n+1}-\mathbf{A}^n}{\Delta t}\).

The equations Weak formulation of T-A Formulation becomes :

Time Discretization of T-A Formulation

\[\small{ \left\{ \begin{align*} \int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}^{n+1}) \cdot (\nabla \times \mathbf{ϕ})d\Omega} = \int_{\Omega_c}{ \nabla \textbf{T}^{n+1} \cdot \mathbf{ϕ}d\Omega} \hspace{2cm} \text{(Weak A-form)}\\ \int_{\Omega_c}{\rho\nabla\times\textbf{T}^{n+1} \cdot \nabla\times\mathbf{\psi}d\Omega_c} -\int_{\Omega_c}{ \frac{\nabla\times\textbf{A}^{n+1}}{\Delta t}\cdot \mathbf{\psi}d\Omega_c} = -\int_{\Omega_c}{ \frac{\nabla\times\textbf{A}^n}{\Delta t}\cdot \mathbf{\psi}d\Omega_c} \hspace{2cm} \text{(Weak T-form)} \\ \forall \mathbf{ϕ} \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in \in H^{curl}(\Omega_c) \end{align*} \right. }\]

2. References

Real-time simulation of large-scale HTS systems: multi-scale and homogeneous models using the T–A formulation, Edgar Berrospe-Juarez et al 2019 Supercond. Sci. Technol. 32 065003, PDF