A-V Formulation with gauge condition

In section General Case, we compute a formulation of Maxwell equation in Maxwell Quasi Static approximation, the A-V Formulation with gauge condition on potential magnetic field :

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

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
\[\text{(AV)} \left\{ \begin{matrix} \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \sigma \frac{\partial \textbf{A}}{\partial t} + \sigma \nabla V &=& 0 \text{ on } \Omega & \text{(AV-1)} \\ \nabla \cdot \left( \sigma \nabla V + \sigma \frac{\partial \textbf{A}}{\partial t} \right) &=& 0 \text{ on } \Omega_c & \text{(AV-2)} \\ \mathbf{A} \times \mathbf{n} &=& 0 \text{ on } \Gamma_D & \text{(DA)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 \text{ on } \Gamma_N & \text{(NA)} \\ V &=& V_0 & \text{ on } \Gamma_{DV} & \text{(DV)} \\ \frac{\partial V}{\partial \mathbf{n}} &=& 0 & \text{ on } \Gamma_{NV} & \text{(NV)} \end{matrix} \right.\]

1.2. Differential Formulation

We simplificate the A-V Formulation with the gauge condition :

  • \(\nabla \cdot \mathbf{A} = 0\)

With this conditions, we have :

\[\begin{eqnarray*} \nabla \times \left( \frac{1}{\mu} \nabla \times \mathbf{A} \right) &=& \frac{1}{\mu} \, \nabla \times \left( \nabla \times \mathbf{A} \right) \\ &=& \frac{1}{\mu} \begin{pmatrix} \partial_x \left( \partial_x A_x + \partial_y A_y + \partial_z A_z \right) - \partial_{xx} A_x - \partial_{yy} A_x - \partial_{zz} A_x \\ \partial_y \left( \partial_x A_x + \partial_y A_y + \partial_z A_z \right) - \partial_{xx} A_y - \partial_{yy} A_y - \partial_{zz} A_y \\ \partial_z \left( \partial_x A_x + \partial_y A_y + \partial_z A_z \right) - \partial_{xx} A_z - \partial_{yy} A_z - \partial_{zz} A_z \end{pmatrix} \\ &=& \frac{1}{\mu} \begin{pmatrix} \partial_x \left( \nabla \cdot \mathbf{A} \right) - \Delta A_x \\ \partial_y \left( \nabla \cdot \mathbf{A} \right) - \Delta A_y \\ \partial_z \left( \nabla \cdot \mathbf{A} \right) - \Delta A_z \\ \end{pmatrix} \\ &=& - \frac{1}{\mu} \begin{pmatrix} \Delta A_x \\ \Delta A_y \\ \Delta A_z \end{pmatrix} \\ &=& - \frac{1}{\mu} \Delta \mathbf{A} \end{eqnarray*}\]

With \(\Delta \mathbf{A}\) the vectorial Laplacian of \(\mathbf{A}\).

The A-V Formulation becomes :

A-V Formulation with gauge condition
\[\text{(AV+gauge)} \left\{ \begin{matrix} \frac{1}{\mu} \Delta \textbf{A} + \sigma \frac{\partial \textbf{A}}{\partial t} &=& - \sigma \nabla V & \text{ on } \Omega & \text{(AV+gauge-1)} \\ \nabla \cdot \left( \sigma \nabla V \right) &=& - \nabla \cdot \left( \sigma \frac{\partial \mathbf{A}}{\partial t} \right) & \text{ on } \Omega_c & \text{(AV-2)} \\ \mathbf{A} \times \mathbf{n} &=& 0 & \text{ on } \Gamma_D & \text{(DA)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 & \text{ on } \Gamma_N & \text{(NA)} \\ V &=& V_0 & \text{ on } \Gamma_{DV} & \text{(DV)} \\ \frac{\partial V}{\partial \mathbf{n}} &=& 0 & \text{ on } \Gamma_{NV} & \text{(NV)} \end{matrix} \right.\]

1.3. Weak Formulation

By making scalar product with \(\mathbf{ϕ} \in H^{1}(\Omega)\) and integration on first equation of (AV+gauge-1) :

\[ \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \mathbf{ϕ} } + \int_{\Omega}{\frac{1}{\mu} \Delta \textbf{A} \cdot \mathbf{ϕ} } = - \int_{\Omega_c}{ \sigma \nabla V \cdot \mathbf{ϕ} }\]

With Formula of Green (Second Form) :

\[ \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \mathbf{ϕ} } + \int_{\Omega}{\frac{1}{\mu} \nabla \textbf{A} \cdot \nabla \mathbf{ϕ} } - \int_{\Gamma}{ \frac{1}{\mu} \frac{\partial \mathbf{A}}{\partial \mathbf{n}} \cdot \mathbf{ϕ} } = - \int_{\Omega_c}{ \sigma \nabla V \cdot \mathbf{ϕ} }\]

We impose the boundary conditions :

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

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

\[ \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \mathbf{ϕ} } + \int_{\Omega}{\frac{1}{\mu} \nabla \textbf{A} \cdot \nabla \mathbf{ϕ} } = - \int_{\Omega_c}{ \sigma \nabla V \cdot \mathbf{ϕ} } \hspace{1cm} \text{(Weak+gauge-1)}\]

The weak formulation of second equation of (AV+gauge) is the same like second equation of (AV).

Thus, we obtain :

Weak formulation of A-V Formulation with gauge condition
\[\begin{align*} \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \mathbf{ϕ} } + \int_{\Omega}{\frac{1}{\mu} \nabla \textbf{A} \cdot \nabla \mathbf{ϕ} } &= - \int_{\Omega_c}{ \sigma \nabla V \cdot \mathbf{ϕ} } & \text{(Weak AV+gauge-1)} \\ \int_{\Omega_c}{\sigma (\nabla V + \frac{\partial \mathbf{A}}{\partial t}) \cdot \nabla \mathbf{\psi}} &= 0 & \text{(Weak AV-2)} \\ \forall \mathbf{\mathbf{ϕ} } \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in H^{1}(\Omega_c) \end{align*}\]

2. Magnetostatic Case

This section presents the AV-Formulation 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 with gauge condition in Stationary Case
\[\left\{ \begin{matrix} \frac{1}{\mu} \Delta \textbf{A} + \sigma \nabla V &=& 0 \text{ on } \Omega & \text{(Magneto+gauge-1)} \\ \nabla \cdot \left( \sigma \nabla V \right) &=& 0 \text{ on } \Omega_c & \text{(Magneto-2)} \\ \mathbf{A} \times \mathbf{n} &=& 0 & \text{ on } \Gamma_D & \text{(DA)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 & \text{ on } \Gamma_N & \text{(NA)} \\ V &=& V_0 & \text{ on } \Gamma_{DV} & \text{(DV)} \\ \frac{\partial V}{\partial \mathbf{n}} &=& 0 & \text{ on } \Gamma_{NV} & \text{(NV)} \end{matrix} \right.\]

2.2. Weak Formulation

The weak formulation of A-V Formulation becomes :

Weak formulation of A-V Formulation with gauge condition in Stationary Case
\[\begin{align*} \int_{\Omega}{\frac{1}{\mu} \nabla \mathbf{A} \cdot \nabla \mathbf{ϕ} } + \int_{\Omega_c}{ \sigma \nabla V \cdot \mathbf{\mathbf{ϕ} } } &= 0 \hspace{2cm} \text{(Weak Magneto-1)} \\ \int_{\Omega_c}{\sigma (\nabla V) \cdot \nabla \mathbf{\psi}} &= 0 \hspace{2cm} \text{(Weak Magneto-2)} \\ \forall \mathbf{\mathbf{ϕ} } \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in H^{1}(\Omega_c) \end{align*}\]