A-V Formulation in Axisymmetric coordinates for HTS

This section is based on the Axisymmetrical case of the A-V Formulation described for the Maxwell Quasi Static case for High critical Temperature Superconductor.

1. Transient Case - A-V Formulation

This section recalls the A-V Formulation :

Thus \(\Omega\) the domain contains the superconductor domain \(\Omega_c\) and non conducting materials \(\Omega_c^C\) (\(\mathbf{J} = 0\)) like the air for example. Also \(\Gamma = \partial \Omega\) is the bound of \(\Omega\), \(\Gamma_c = \partial \Omega_c\) the bound of \(\Omega_c\), \(\Gamma_D\) the bound with Dirichlet boundary condition and \(\Gamma_N\) the bound with Neumann boundary condition, such that \(\Gamma = \Gamma_D \cup \Gamma_N\).

We introduce :

  • Magnetic potential field \(\mathbf{A}\) : \(\textbf{B} = \nabla \times \textbf{A}\)

  • Electric potential scalar : \(\nabla \phi = - \textbf{E} - \frac{\partial \textbf{A}}{\partial t}\)

In this example we only consider a bulk cylinder without transport current so the electrical potential scalar can be ignored and we have :

  • Electrical field : \(\textbf{E}= - \frac{\partial \textbf{A}}{\partial t}\)

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

with \(\sigma\) expressed with the E-J power law for the HTS :

\[\sigma=\frac{J_c}{E_c} \left(\frac{||\textbf{E}||}{E_c}\right)^{(1-n)/n}=\frac{J_c}{E_c} \left(\frac{||-\partial_t \textbf{A}||}{E_c}\right)^{(1-n)/n}\]

with :

  • \(J_c\): the critical current density \((A/m^2)\)

  • \(E_c\): the threshold electric field \((V/m)\)

  • \(n\): material dependent exponent

1.1. Differential Formulation

The model use the A Formulation in axisymmetric coordinates.

We note \(\Omega^{axis}\), \(\Omega^{axis}_c\), \(\Gamma^{axis}\), \(\Gamma_D^{axis}\), \(\Gamma_N^{axis}\) and \(\Gamma_c^{axis}\) the representation of \(\Omega\), \(\Omega_c\), \(\Gamma\), \(\Gamma_D\), \(\Gamma_N\) and \(\Gamma_c\) in axisymmetric coordinates.

We note \(u = \begin{pmatrix} u_r \\ u_{\theta} \\ u_z \end{pmatrix}_{cyl}\) the coordinates of \(u \in \mathbb{R}^3\) in cylindrical base.

We note \(\mathbf{n}^{axis} = \begin{pmatrix} n^{axis}_r \\ n^{axis}_z \end{pmatrix}_{cyl}\) the exterior normal of \(\Gamma^{axis}\) on \(\Omega^{axis}\).

The magnetic flux density \(\textbf{B}\) as only two component, \(B_r\) and \(B_z\), and the magnetic potential field as only one component \(A_\theta\).

So, \(\mathbf{B} = \begin{pmatrix} B_r \\ 0 \\ B_z \end{pmatrix}_{cyl}\) and \(\mathbf{B} = \nabla \times \mathbf{A}(r,z)\), so \(\mathbf{A} = \begin{pmatrix} 0 \\ A_\theta \\ 0 \end{pmatrix}_{cyl}\)

As in the previous Axisymmetrical case, the A Formulation becomes :

A Formulation in axisymmetric coordinates
\[\text{(A Axi)} \left\{ \begin{matrix} \sigma \frac{\partial A_\theta}{\partial t} - \frac{1}{\mu} \Delta A_\theta +\frac{1}{\mu r^2} A_\theta = 0 \text{ on } \Omega^{axis} \hspace{2cm} &\text{(A Axi)} \\ A_\theta = 0 \text{ on } \Gamma_D^{axis} \hspace{2cm} &\text{(D Axi)} \\ \frac{\partial A_\theta}{\partial \mathbf{n}} = 0 \text{ on } \Gamma_N^{axis} \hspace{2cm} &\text{(N Axi)} \end{matrix} \right.\]

1.2. Weak Formulation

In this subsection, we write the weak formulation of equation (A Axi) the A-V Formulation in axisymmetric coordinates.

By multiplying by \(\phi \in H^1(\Omega)\) and integration of (A Axi) on \(\Omega\) :

\[ \int_{\Omega}{ \left( \sigma \frac{\partial A_\theta}{\partial t} - \frac{1}{\mu} \Delta A_\theta +\frac{1}{\mu r^2} A_\theta \right) \, \phi \ dxdydz } = 0\]

By Formula of Green (Second Form) :

\[ \scriptsize{\int_{\Omega}{ \sigma \frac{\partial A_\theta}{\partial t} \, \phi \ dxdydz } + \int_{\Omega}{ \frac{1}{\mu} \nabla A_\theta \cdot \nabla \phi \ dxdydz } +\int_{\Omega}{ \frac{1}{\mu r^2} A_\theta \cdot \phi \ dxdydz } = \int_{\Gamma}{ \frac{1}{\mu} \frac{\partial A_\theta}{\partial \mathbf{n}} d\Gamma }}\]
\[ \scriptsize{\int_{\Omega}{ \sigma \frac{\partial A_\theta}{\partial t} \, \phi \ dxdydz } + \int_{\Omega}{ \frac{1}{\mu} \nabla A_\theta \cdot \nabla \phi \ dxdydz }+\int_{\Omega}{ \frac{1}{\mu r^2} A_\theta \cdot \phi \ dxdydz } = \int_{\Gamma_D}{ \frac{1}{\mu} \frac{\partial A_\theta}{\partial \mathbf{n}} } + \int_{\Gamma_N}{ \frac{1}{\mu} \frac{\partial A_\theta}{\partial \mathbf{n}} }}\]

We switch to axisymmetrical coordinates :

\[\scriptsize{ \int_{\Omega^{axis}}{ \sigma \frac{\partial A_\theta}{\partial t} \, \phi \ rdrdz } + \int_{\Omega^{axis}}{ \frac{1}{\mu} \nabla A_\theta \cdot \nabla \phi \ rdrdz }+\int_{\Omega^{axis}}{ \frac{1}{\mu r} A_\theta \cdot \phi \ drdz } = \int_{\Gamma_D^{axis}}{ \frac{1}{\mu} \frac{\partial A_\theta}{\partial \mathbf{n}} } + \int_{\Gamma_N^{axis}}{ \frac{1}{\mu} \frac{\partial A_\theta}{\partial \mathbf{n}} } }\]

We impose the boundary conditions :

  • Dirichlet : \(A_\theta = 0\) on \(\Gamma_D^{axis}\) (D Axi)

  • Neumann : \(\frac{\partial A_\theta}{\partial \mathbf{n}} = 0\) on \(\Gamma_N^{axis}\) (N Axi)

Finally, we have :

Weak formulation of A Formulation in two dimensions approximation
\[ \int_{\Omega_c^{axis}}{\sigma \frac{\partial A_\theta}{\partial t} \, \phi \ rdrdz } + \int_{\Omega^{axis}}{ \frac{1}{\mu} \nabla A_\theta \cdot \nabla \phi \ rdrdz }+\int_{\Omega^{axis}}{ \frac{1}{\mu r} A_\theta \cdot \phi \ drdz } = 0 \hspace{1cm} \text{(Weak A Axi)}\]

1.3. Time Discretization

In this subsection, we use the time discretization by backward Euler method on the (Weak A Axi).

We discretize in time the problem with the 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 : \(\frac{\partial A_\theta}{\partial t} \approx \frac{A_\theta^{n+1}-A_\theta^n}{\Delta t}\).

The equations (Weak A Axi) becomes :

Time Discretization of A Formulation in axisymmetric coordinates
\[\small{ \int_{\Omega_c^{axis}}{\sigma \frac{A_\theta^{n+1}}{\Delta t} \, \phi \ rdrdz } + \int_{\Omega^{axis}}{ \frac{1}{\mu} \nabla A_\theta^{n+1} \cdot \nabla \phi \ rdrdz }+\int_{\Omega^{axis}}{ \frac{1}{\mu r} A_\theta^{n+1} \cdot \phi \ drdz } = \int_{\Omega_c^{axis}}{ \sigma \frac{A_\theta^n}{\Delta t} \, \phi \ rdrdz } \\ \hspace{14cm} \text{(Disc AV Axi)} }\]

2. 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