Two Dimensions Case

In the previous section (General Case), we approximate the Maxwell Equations in Maxwell Quasi Static and exprim the result in A-V Formulation.

In this section, we add an approximation on A-V Formulation in two dimensions. After, we compute the result equation, in Differential Formulation, in Weak Formulation, we do a Time Discretization and we deal the magnetostatic problem.

1. Transient Case

1.1. A-V Formulation

This section recalls the A-V Formulation :

Thus \(\Omega\) the domain, comprising the conductor (or superconductor) domain \(\Omega_c\) and non conducting materials \(\Omega_n\) (\(\mathbf{J} = 0\)) like air. Let \(\Gamma = \partial \Omega\) 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 : \(\nabla V = - \textbf{E} - \frac{\partial \textbf{A}}{\partial t}\)

We want to resolve the electromagnetism problem ( with \(\mathbf{A}\) and \(V\) the unknows) :

A-V Formulation
\[\left\{ \begin{matrix} \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \sigma \frac{\partial \textbf{A}}{\partial t} + \sigma \nabla V &=& 0 \text{ on } \Omega & \text{(AV-1)} \\ \nabla \cdot \left( \sigma \nabla V + \sigma \frac{\partial \textbf{A}}{\partial t} \right) &=& 0 \text{ on } \Omega_c & \text{(AV-2)} \\ \mathbf{A} \times \mathbf{n} &=& 0 \text{ on } \Gamma_D & \text{(D)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 \text{ on } \Gamma_N & \text{(N)} \end{matrix} \right.\]

1.2. Differential Formulation

In this subsection, we express the A-V Formulation on geometry in two dimensions.

On equations A-V Formulation, we suppose the geometry, the parameters, \(B\) are independent by \(z\) and we suppose \(V\) uniquely dependant by \(z\).

We note \(\Omega^{2d}\) (respectively \(\Omega^{2d}_c\), \(\Gamma^{2d}\), \(\Gamma_D^{2d}\), \(\Gamma_N^{2d}\) and \(\Gamma_c^{2d}\)) the projection of \(\Omega\) (respectively \(\Omega_c\), \(\Gamma\), \(\Gamma_D\) and \(\Gamma_N\)) in xy plane.

We note \(\mathbf{n}^{2d} = \begin{pmatrix} n_x^{2d} \\ n_y^{2d} \end{pmatrix}\) the output normal of \(\Gamma^{2d}\) on \(\Omega^{2d}\).

So, \(\mathbf{B} = \begin{pmatrix} B_x(x,y) \\ B_y(x,y) \\ 0 \end{pmatrix}\) and \(\mathbf{B} = \nabla \times \mathbf{A}(x,y)\), so \(\mathbf{A} = \begin{pmatrix} 0 \\ 0 \\ A_z \end{pmatrix}\)

The problem goes from 4 (3 for \(\mathbf{A}\) and 1 for \(V\)) components to 2 (1 for \(\mathbf{A}\) and 1 for \(V\)).

We have \(\nabla \times \mathbf{A} = \begin{pmatrix} \frac{\partial A_z}{\partial y} \\ -\frac{\partial A_z}{\partial x} \\ 0 \end{pmatrix}\) and \(\nabla \times \left( \nabla \times \mathbf{A} \right) = \begin{pmatrix} 0 \\ 0 \\ -\Delta A_z \end{pmatrix}\)

The first equation (AV-1) in A-V Formulation becomes :

\[ \sigma \frac{\partial A_z}{\partial t} - \frac{1}{\mu} \Delta A_z + \sigma \frac{\partial V}{\partial z} = 0 \hspace{2cm} \text{(AV-1 2D)}\]

In the other hand \(\nabla V = \begin{pmatrix} 0 \\ 0 \\ \frac{\partial V}{\partial z} \end{pmatrix}\), thus, the second equation (AV-1) becomes :

\[ \nabla \cdot \left( \sigma \begin{pmatrix} 0 \\ 0 \\ \frac{\partial V}{\partial z} \end{pmatrix} + \sigma \begin{pmatrix} 0 \\ 0 \\ \frac{\partial A_z}{\partial t} \end{pmatrix} \right) = 0\]

So :

\[ \sigma \frac{\partial^2 V}{\partial z^2} = 0\]

\(V\) becomes a polynomial of the second degree on \(z\).

Now, let’s compute the new boundary conditions.

With Dirichlet conditions (D), we have on \(\Gamma_D\) :

\[\begin{eqnarray*} \mathbf{A} \times \mathbf{n} = 0 \Longleftrightarrow \begin{pmatrix} - A_z \, n_y \\ A_z \, n_x \\ 0 \end{pmatrix} = 0 \end{eqnarray*}\]

However, \(\mathbf{n}^{2d} = \begin{pmatrix} n_x \\ n_y \end{pmatrix}\) isn’t zero :

\[ A_z = 0 \text{ on } \Gamma_D^{2d} \hspace{2cm} \text{(D 2D)}\]

With Neumann conditions (N), we have on \(\Gamma_N\) :

\[\begin{align} \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} = 0 \Longleftrightarrow & \begin{pmatrix} \partial_y A_z \\ \partial_x A_z \\ 0 \end{pmatrix} \times \begin{pmatrix} n_x \\ n_y \\ 0 \end{pmatrix} = 0 \\ \Longleftrightarrow & \begin{pmatrix} 0 \\ 0 \\ \partial_y A_z \, n_y + \partial_x A_z \, n_x \end{pmatrix} = 0 \\ \Longleftrightarrow & \hspace{1.5cm} \frac{\partial A_z}{\partial \mathbf{n}^{2d}} = 0 & \text{(N 2D)} \end{align}\]

The A-V Formulation becomes :

A-V Formulation in two dimensions approximation
\[\text{(AV 2D)} \left\{ \begin{matrix} \sigma \frac{\partial A_z}{\partial t} - \frac{1}{\mu} \Delta A_z + \sigma \frac{\partial V}{\partial z} = 0 \hspace{2cm} \text{(AV-1 2D)} \\ A_z = 0 \text{ on } \Gamma_D^{2d} \hspace{2cm} \text{(D 2D)} \\ \frac{\partial A_z}{\partial \mathbf{n}^{2d}} = 0 \text{ on } \Gamma_N^{2d} \hspace{2cm} \text{(N 2D)} \end{matrix} \right.\]

And \(V\) is a polynomial of the second degree on \(z\).

1.3. Weak Formulation

In this subsection, we write the weak formulation of equation (AV 2D) the A-V Formulation in two dimensions.

By making product \(\phi \in H^1(\Omega^{2d})\) and integration of (AV-1 2D) on \(\Omega^{2d}\) :

\[ \int_{\Omega^{2d}}{ \left( \sigma \frac{\partial A_z}{\partial t} - \frac{1}{\mu} \Delta A_z \right) \, \phi \ dxdy } + \int_{\Omega_c^{2d}}{ \sigma \frac{\partial V}{\partial z} \, \phi \ dxdy } = 0\]

By Formula of Green (Second Form) :

\[ \int_{\Omega^{2d}}{ \sigma \frac{\partial A_z}{\partial t} \, \phi \ dxdy } + \int_{\Omega^{2d}}{ \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdydz } + \int_{\Omega_c^{2d}}{ \sigma \frac{\partial V}{\partial z}\, \phi \ dxdy } = \int_{\Gamma^{2d}}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}^{2d}} d\Gamma^{2d} }\]
\[ \int_{\Omega}{ \sigma \frac{\partial A_z}{\partial t} \, \phi \ dxdydz } + \int_{\Omega}{ \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdydz } + \int_{\Omega_c}{ \sigma \frac{\partial V}{\partial z}\, \phi \ dxdydz } = \int_{\Gamma_D^{2d}}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}^{2d}} } + \int_{\Gamma_N^{2d}}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}^{2d}} }\]

We impose the boundary conditions :

  • Dirichlet : \(A_z = 0\) on \(\Gamma_D^{2d}\) (D 2D)

  • Neumann : \(\frac{\partial A_z}{\partial \mathbf{n}^{2d}} = 0\) on \(\Gamma_N^{2d}\) (N 2D)

Finally, we have :

Weak formulation of A-V Formulation in two dimensions approximation
\[ \int_{\Omega^{2d}}{ \sigma \frac{\partial A_z}{\partial t} \, \phi \ dxdy } + \int_{\Omega^{2d}}{ \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdy } + \int_{\Omega_c^{2d}}{ \sigma \frac{\partial V}{\partial z}\, \phi \ dxdy } = 0 \hspace{1cm} \text{(Weak AV 2D)}\]

1.4. Time Discretization

In this section, we see the (Weak AV 2D) (the weak 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 : \(\frac{\partial A_z}{\partial t} \approx \frac{A_z^{n+1}-A_z^n}{\Delta t}\).

The equations (Weak AV 2D) becomes :

Time Discretization of A-V Formulation in two dimensions approximation
\[\small{ \int_{\Omega^{2d}}{ \sigma \frac{A_z^{n+1}}{\Delta t} \, \phi \ dxdy } + \int_{\Omega^{2d}}{ \frac{1}{\mu} \nabla A_z^{n+1} \cdot \nabla \phi \ dxdy } + \int_{\Omega_c^{2d}}{ \sigma \frac{\partial \left( V^{n+1} \right)}{\partial z}\, \phi \ dxdy } = \int_{\Omega^{2d}}{ \sigma \frac{A_z^n}{\Delta t} \, \phi \ dxdy } \\ \hspace{14cm} \text{(Disc AV 2D)} }\]

2. Magnetostatic Case

This section presents the AV-Formulation in two dimensions in stationary case. In this section we consider all parameters be dependant by time. Thus the derivates become : \(\frac{\partial f}{\partial t} = 0\).

The equations become :

2.1. Differential Equation

The differential formulation of A-V Formulation becomes :

A-V Formulation in two dimensions approximation in Stationary Case
\[\text{(MagSta 2D)} \left\{ \begin{matrix} - \frac{1}{\mu} \Delta A_z + \sigma \frac{\partial V}{\partial z} = 0 \hspace{2cm} \text{(MagSta-1 2D)} \\ A_z = 0 \text{ on } \Gamma_D^{2d} \hspace{2cm} \text{(D 2D)} \\ \frac{\partial A_z}{\partial \mathbf{n}^{2d}} = 0 \text{ on } \Gamma_N^{2d} \hspace{2cm} \text{(N 2D)} \end{matrix} \right.\]

And \(V\) is a polynomial of the second degree on \(z\).

2.2. Weak Formulation

The weak formulation of A-V Formulation in two dimensions becomes :

Weak formulation of A-V Formulation in two dimensions approximation in Stationary Case
\[ \int_{\Omega^{2d}}{ \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdy } + \int_{\Omega_c^{2d}}{ \sigma \frac{\partial V}{\partial z}\, \phi \ dxdy } = 0 \hspace{1cm} \text{(Weak MagSta 2D)}\]