Axisymmetrical Case

In the previous section :

General Case : we approximate the Maxwell Equations in Maxwell Quasi Static and exprim the result in A-V Formulation.
Two Dimensions Case : we add an approximation in two dimensions of A-V Formulation and we have see that this greatly reduces the problem.

In this section, we add an approximation on A-V Formulation in axisymmetric, like in section Two Dimensions Case. 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. Axisymmetrical Geometry

1.3. Differential Formulation

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

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

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

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

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 \(\nabla \times \mathbf{A} = \begin{pmatrix} -\partial_z A_{\theta} \\ 0 \\ \frac{1}{r} \partial_r (r A_{\theta}) \end{pmatrix}_{cyl}\) and \(\nabla \times \left( \nabla \times A \right) = \begin{pmatrix} 0 \\ -\frac{\partial^2 A_{\theta}}{\partial z^2} - \frac{1}{r} \frac{\partial^2 (r A_{\theta})}{\partial r^2} + \frac{1}{r^2} \, \frac{\partial (r A_{\theta})}{\partial r} \\ 0 \end{pmatrix}_{cyl}\)

We pose \(\Phi = r A_{\theta}\), we have :

\[ \sigma \frac{\partial \Phi}{\partial t} - \frac{1}{\mu} \frac{\partial^2 \Phi}{\partial z^2} - \frac{1}{\mu} \frac{\partial^2 \Phi}{\partial r^2} + \frac{1}{\mu \, r} \frac{\partial \Phi}{\partial r} + \sigma \frac{\partial V}{\partial \theta} = 0\]

The equation (AV-1) becomes :

With \(\Delta \Phi = \frac{1}{r} \frac{\partial \left( r \frac{\partial \Phi}{\partial r} \right)}{\partial r} + \frac{\partial^2 \Phi}{\partial z^2}\), we have :

\[ \sigma \frac{\partial \Phi}{\partial t} - \frac{1}{\mu} \Delta \Phi + \frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} + \sigma \frac{\partial V}{\partial \theta} = 0 \hspace{2cm} \text{(AV-1 Axis)}\]

In the other hand, \(\nabla V = \begin{pmatrix} 0 \\ \frac{1}{r} \frac{\partial V}{\partial \theta} \\ 0 \end{pmatrix}\), thus, the equation (AV-2) becomes :

\[ \nabla \cdot \left( \sigma \begin{pmatrix} 0 \\ \frac{1}{r} \frac{\partial V}{\partial \theta} \\ 0 \end{pmatrix} + \sigma \begin{pmatrix} 0 \\ \frac{\partial A_{\theta}}{\partial t} \\ 0 \end{pmatrix} \right) = 0 \hspace{2cm} \text{(AV-2)}\]

So :

\[\begin{eqnarray*} \frac{\sigma}{r^2} \frac{\partial^2 V}{\partial \theta^2} &=& 0 \\ \frac{\partial^2 V}{\partial \theta^2} &=& 0 \hspace{2cm} \text{(AV-2 Axis)} \end{eqnarray*}\]

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

Now, let’s compute the new doundary conditions.

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

\[\begin{align} \mathbf{A} \times \mathbf{n} = 0 \Longleftrightarrow & \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} \times \begin{pmatrix} n_x \\ n_y \\ n_z \end{pmatrix} = 0 \\ \Longrightarrow & \begin{pmatrix} A_y \, n_z - A_z \, n_y \\ - A_x \, n_z + A_z \, n_x \\ A_x \, n_y - A_y \, n_x \end{pmatrix} = 0 \\ \Longrightarrow & \begin{pmatrix} A_y \, n_z \\ - A_x \, n_z \\ A_x \, n_y - A_y \, n_x \end{pmatrix} = 0 & A_z = 0 \\ \Longrightarrow & \begin{pmatrix} cos \theta \, A_{\theta} \, n_z \\ sin \theta \, A_{\theta} \, n_z \\ \small{ \begin{matrix} \left( cos\theta \, A_r - sin \theta \, A_{\theta} \right) \left( sin \theta \, n_r + cos \theta \, n_{\theta} \right) \\ - \left( sin\theta \, A_r + cos \theta \, A_{\theta} \right) \left( cos \theta \, n_r - sin \theta \, n_{\theta} \right) \end{matrix} } \end{pmatrix}_{cyl} = 0 & \text{passage in cylindric} \\ \Longrightarrow & \begin{pmatrix} cos \theta \, A_{\theta} \, n_z \\ sin \theta \, A_{\theta} \, n_z \\ -sin^2 \theta \, A_{\theta} \, n_r - cos^2 \theta \, A_{\theta} \, n_r \end{pmatrix}_{cyl} = 0 & A_r = 0 \text{ and } n_{\theta} = 0 \\ \Longrightarrow & \begin{pmatrix} cos \theta \, A_{\theta} \, n_z \\ sin \theta \, A_{\theta} \, n_z \\ - A_{\theta} \, n_r \end{pmatrix}_{cyl} = 0 \end{align}\]

However, \(\mathbf{n}^{axis}=\begin{pmatrix} n_r \\ n_z \end{pmatrix}\) isn’t zeros, thus :

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

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

To conclude, the A-V Formulation becomes :

A-V Formulation in axisymmetric coordinates

\[\text{(AV Axis)} \left\{ \begin{matrix} \sigma \frac{\partial \Phi}{\partial t} - \frac{1}{\mu} \Delta \Phi + \frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} + \sigma \frac{\partial V}{\partial \theta} = 0 & \text{ on } \Omega^{axis} & \text{(AV-1 Axis)} \\ \frac{\partial^2 V}{\partial \theta^2} = 0 & \text{ on } \Omega_c^{axis} & \text{(AV-2 Axis)} \\ A_{\theta} = 0 & \text{ on } \Gamma_D^{axis} & \text{(D Axis)} \\ \frac{\partial A_{\theta}}{\partial \mathbf{n}^{axis}} = 0 & \text{ on } \Gamma_N^{axis} & \text{(N Axis)} \end{matrix} \right.\]

With \(\Delta \Phi = \frac{1}{r} \frac{\partial \left( r \frac{\partial \Phi}{\partial r} \right)}{\partial r} + \frac{\partial^2 \Phi}{\partial z^2}\)

1.4. Weak Formulation

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

By making product \(\phi \in H^1(\Omega)\) and integration on equation (AV-1 Axis) on \(\Omega\) :

\[ \int_{\Omega}{ \left( \sigma \frac{\partial \Phi}{\partial t} - \frac{1}{\mu} \Delta \Phi + \frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} \right) \, \phi \ dxdydz} + \int_{\Omega_c}{ \sigma \frac{\partial V}{\partial \theta} \, \phi \ dxdydz} = 0 \hspace{1cm} \text{(Weak AV Axis)}\]

By Formula of Green :

\[\scriptsize{ \int_{\Omega}{ \sigma \frac{\partial \Phi}{\partial t} \phi \ dxdydz} + \int_{\Omega}{ \frac{1}{\mu} \nabla \Phi \cdot \nabla \phi \ dxdydz } - \int_{\Gamma}{ \frac{1}{\mu} \frac{\partial \Phi}{\partial \mathbf{n}} \, \phi \ d\Gamma} + \int_{\Omega}{\frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} \, \phi \ dxdydz} + \int_{\Omega_c}{ \sigma \frac{\partial V}{\partial \theta} \phi \ dxdydz } = 0 \hspace{1cm} \text{(Disc AV Axis)} }\]

We pass on axisymmetric coordinates (see Formula of changement in axisymmetric coordinates) :

\[\scriptsize{ \int_{\Omega^{axis}}{ \sigma \frac{\partial \Phi}{\partial t} \phi \ r\ drdz} + \int_{\Omega^{axis}}{ \frac{1}{\mu} \tilde{\nabla} \Phi \cdot \tilde{\nabla} \phi \ r \ drdz } - \int_{\Gamma^{axis}}{ \frac{1}{\mu} ( \tilde{\nabla} \Phi \cdot \mathbf{n}^{axis} ) \, \phi \ r \ d\Gamma} + \int_{\Omega^{axis}}{\frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} \, \phi \ r \ drdz} + \int_{\Omega_c^{axis}}{ \sigma \frac{\partial V}{\partial \theta} \phi \ r \ drdz } = 0 }\]

With \(\tilde{\nabla} = \begin{pmatrix} \partial_r \\ \partial_{\theta} \\ \partial_z \end{pmatrix}_{cyl}\)

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega^{axis}}{ \sigma \frac{\partial \Phi}{\partial t} \phi \ r\ drdz} + \int_{\Omega^{axis}}{ \frac{1}{\mu} \tilde{\nabla} \Phi \cdot \tilde{\nabla} \phi \ r \ drdz } - \int_{\Gamma_D^{axis}}{ \frac{1}{\mu} ( \tilde{\nabla} \Phi \cdot \mathbf{n}^{axis} ) \, \phi \ r \ d\Gamma} & & \\ - \int_{\Gamma_N^{axis}}{ \frac{1}{\mu} ( \tilde{\nabla} \Phi \cdot \mathbf{n}^{axis} ) \, \phi \ r \ d\Gamma} + \int_{\Omega^{axis}}{\frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} \, \phi \ r \ drdz} + \int_{\Omega_c^{axis}}{ \sigma \frac{\partial V}{\partial \theta} \phi \ r \ drdz } &=& 0 \end{eqnarray*} }\]

We impose the boundary conditions :

Dirichlet : \(A_{\theta} = 0\) (so \(\Phi = 0\)) on \(\Gamma_D^{axis}\) (D Axis)
Neumann : \(\frac{\partial A_{\theta}}{\partial n} = 0\) (so \(\Phi = A_{\theta} \, n_r^{axis}\)) on \(\Gamma_N^{axis}\) (N Axis)

\(\Phi\) is not equal to \(0\).

We have :

Weak formulation of A-V Formulation in axisymmetric coordinates

\[\scriptsize{ \text{(Weak AV Axis)} \\ \left\{ \begin{eqnarray*} \int_{\Omega^{axis}}{ \sigma \frac{\partial \Phi}{\partial t} \phi \ r\ drdz} + \int_{\Omega^{axis}}{ \frac{1}{\mu} \tilde{\nabla} \Phi \cdot \tilde{\nabla} \phi \ r \ drdz } - \int_{\Gamma_N}{ \left( \Phi n_r^{axis} \right) \, \phi \ drdz} + \int_{\Omega^{axis}}{\frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} \, \phi \ r \ drdz} + \int_{\Omega_c^{axis}}{ \sigma \frac{\partial V}{\partial \theta} \phi \ r \ drdz } = 0 \\ \text{for } \phi \in H^1(\Omega) \end{eqnarray*} \right. }\]

1.5. Time Discretization

Time Discretization of A-V Formulation in axisymmetric

\[\begin{eqnarray*} \int_{\Omega^{axis}}{ \sigma \frac{\Phi^{n+1}}{\Delta t} \phi \ r \ drdz} + \int_{\Omega^{axis}}{ \frac{1}{\mu} \tilde{\nabla} \Phi^{n+1} \cdot \tilde{\nabla} \phi \ r \ drdz } - \int_{\Gamma_N}{ \left( \Phi^{n+1} n_r^{axis} \right) \, \phi \ drdz} \\ + \int_{\Omega^{axis}}{\frac{2}{\mu \, r} \frac{\partial \Phi^{n+1}}{\partial r} \, \phi \ r \ drdz} + \int_{\Omega_c^{axis}}{ \sigma \frac{\partial V^{n+1}}{\partial \theta} \phi \ r \ drdz } = \int_{\Omega^{axis}}{ \sigma \frac{\Phi^n}{\Delta t} \phi \ r \ drdz} \end{eqnarray*}\]

2. Magnetostatic Case

This section presents the AV-Formulation in axisymmetric 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 axisymmetric coordinates in Stationary Case

\[\text{(Magstat Axis)} \left\{ \begin{matrix} \frac{1}{\mu} \Delta \Phi + \frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} + \sigma \frac{\partial V}{\partial \theta} = 0 & \text{on } \Omega^{axis} & \text{MagStat-1 Axis} \\ \frac{\partial^2 V}{\partial \theta^2} = 0 & \text{ on } \Omega_c^{axis} & text{MagStat-2 Axis} \\ A_{\theta} = 0 & \text{ on } \Gamma_D^{axis} & \text{D Axis} \\ \frac{\partial A_{\theta}}{\partial \mathbf{n}^{axis}} = 0 & \text{ on } \Gamma_N^{axis} & \text{N Axis} \end{matrix} \right.\]

With \(\Delta \Phi = \frac{1}{r} \frac{\partial \left( r \frac{\partial \Phi}{\partial r} \right)}{\partial r} + \frac{\partial^2 \Phi}{\partial z^2}\)

2.2. Weak Formulation

The weak formulation of A-V Formulation in Axisymmetric becomes :

Weak formulation of A-V Formulation in axisymmetric coordinates in Stationary

\[\scriptsize{ \int_{\Omega^{axis}}{ \frac{1}{\mu} \tilde{\nabla} \Phi \cdot \tilde{\nabla} \phi \ r \ drdz } - \int_{\Gamma_N}{ \left( \Phi n_r^{axis} \right) \, \phi \ drdz} + \int_{\Omega^{axis}}{\frac{2}{\mu \, r} \frac{\partial \Phi}{\partial r} \, \phi \ r \ drdz} + \int_{\Omega_c^{axis}}{ \sigma \frac{\partial V}{\partial \theta} \phi \ r \ drdz } = 0 \\ \text{for } \phi \in H^1(\Omega) }\]