A-V Formulation

To describe the behavior of magnetic field of magnet, we solve the Maxwell equations with Maxwell Quasi Static approximations. To solve numerically those equations, we will use the classical A-V Formulation.

1. Maxwell Equations

1.1. In general case

The Maxwell equations govern the electromagnetism.

Differential form of Maxwell equations

For \(\mathbf{x} \in \Omega\) and time \(t \geq 0\) :

\[\begin{align*} \nabla \times \textbf{H}(\textbf{x}, t) &= \textbf{J}(\textbf{x}, t) + \frac{\partial \textbf{D}}{\partial t}(\textbf{x}, t) & \text{ (Maxwell-Ampere)} \\ \nabla \times \textbf{E}(\textbf{x}, t) &= - \frac{\partial \textbf{B}}{\partial t}(\textbf{x}, t) & \text{ (Maxwell-Faraday)} \\ \nabla \cdot \textbf{B}(\textbf{x}, t) &= 0 & \text{ (Maxwell-Thomson)} \\ \nabla \cdot \textbf{D}(\textbf{x}, t) &= 0 & \text{ (Maxwell-Gauss)} \\ \textbf{B}(\textbf{x}, t) &= \mu \left[ \textbf{H}(\textbf{x},t) + \textbf{M}(\textbf{x},t) \right] \\ \textbf{J}(\textbf{x}, t) &= \sigma \left[ \textbf{E}(\textbf{x},t) + \mathbf{E_i}(\textbf{x},t) \right] \\ \textbf{D}(\textbf{x},t) &= \epsilon \textbf{E}(\textbf{x},t) \end{align*}\]

with the notations :

  • \(\textbf{H}(\textbf{x},t)\) magnetic field intensity (\(A m^{-1}\))

  • \(\mathbf{E_i}(\textbf{x},t)\) impressed electric field (\(V m^{-1}\))

  • \(\textbf{E}(\textbf{x},t)\) electric field intensity (\(V m^{-1}\))

  • \(\mathbf{J_i}(\textbf{x},t)\) impressed current density (\(A m^{-2}\))

  • \(\textbf{B}(\textbf{x},t)\) magnetic induction density or magnetic field (\(T\))

  • \(\mu(\textbf{x},t)\) permeability (\(H m^{-1}\))

  • \(\textbf{D}(\textbf{x},t)\) electric induction density (\(C m^{-1}\))

  • \(\sigma(\textbf{x},t)\) conductivity (\(S m^{-1}\))

  • \(\textbf{J}(\textbf{x},t)\) current density (\(A m^{-2}\))

  • \(\epsilon(\textbf{x},t)\) electric permittivity (\(F m^{-1}\))

\(\mu\), \(\sigma\) and \(\epsilon\) depend on the material.

The current density \(\mathbf{J}\) is a function of electric field intensity \(\mathbf{E}\). This function varies according to the material :

  • Non-conducting materials : \(\textbf{J} = 0\)

  • Ressitive materials : \(\textbf{J} = \sigma \textbf{E}\) with \(\sigma\) constant (Ohm law)

  • Superconductors : \(\textbf{E}-\textbf{J}\) power law

1.2. Approximation in Maxwell Quasi Static

The MQS (Maxwell Quasi Static) approximation consists in neglecting the so-called displacement current, (\(\frac{\partial \textbf{D}}{\partial t} \approx 0\)). In this context, the equations become :

Maxwell Equations in MQS
\[\begin{align*} \nabla \times \textbf{H}(\textbf{x}, t) &= \textbf{J}(\textbf{x}, t) & \text{ (Maxwell-Ampere)}\\ \nabla \times \textbf{E}(\textbf{x}, t) &= - \frac{\partial \textbf{B}}{\partial t}(\textbf{x}, t) & \text{ (Maxwell-Faraday)} \\ \nabla \cdot \textbf{B}(\textbf{x}, t) &= 0 & \text{ (Maxwell-Thomson)} \\ \textbf{B}(\textbf{x}, t) &= \mu \, \textbf{H}(\textbf{x}, t) \\ \textbf{J} = \sigma \, \textbf{E} \end{align*}\]

With the notations :

  • \(\textbf{H}(\textbf{x},t)\) magnetic field intensity (\(A m^{-1}\))

  • \(\textbf{J}(\textbf{x},t)\) current density (\(A m^{-2}\))

  • \(\textbf{E}(\textbf{x},t)\) electric field intensity (\(V m^{-1}\))

  • \(\sigma(\textbf{x},t)\) conductivity (\(S m^{-1}\))

  • \(\textbf{B}(\textbf{x},t)\) magnetic induction density or magnetic field (\(T\))

  • \(\mu(\textbf{x},t)\) permeability (\(H m^{-1}\))

2. A-V formulation

In this paragraphe, we introduce magnetic potential field \(\mathbf{A}\) and electric potential scalar \(V\).

In the sequel, we will use the following notations :

  • \(\Omega\) the domain, consisting of conductor (or superconductor) domains \(\Omega_c\) and non conducting materials \(\Omega_n\) (\(\mathbf{J} = 0\)) 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 \(\Omega\).

2.1. Differential Formulation

From Maxwell-Thomson, we can define the magnetic potential field \(\textbf{A}\) on \(\Omega\) such as:

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

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

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

The Maxwell-Faraday equation may then be rewritten as :

\[ \nabla \times \left(\textbf{E} + \frac{\partial \textbf{A}}{\partial t} \right) = 0\]

Similarly, we can define the electric scalar potential \(V\) as :

\[ \textbf{E} + \frac{\partial \textbf{A}}{\partial t} = - \nabla V\]

It follows :

\[ \textbf{E} = - \frac{\partial \textbf{A}}{\partial t} - \nabla V\]

with Maxwell-Ampere :

\[\begin{eqnarray*} \nabla \times \left( \frac{1}{\mu} \textbf{B} \right) &=& \sigma ( -\frac{\partial \textbf{A}}{\partial t} - \nabla V) \\ \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) &=& \sigma \left( -\frac{\partial \textbf{A}}{\partial t} - \nabla V \right) \\ \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \sigma \frac{\partial \textbf{A}}{\partial t} + \sigma \nabla V &=& 0 \\ \end{eqnarray*}\]

Finally, Maxwell-Ampere and conservation of current density (\(\nabla \cdot J = 0\)), on \(\Omega_c\), give :

\[\begin{eqnarray*} \nabla \cdot \textbf{J} &=& 0 \\ \nabla \cdot \left( \sigma \left( -\nabla V - \frac{\partial \textbf{A}}{\partial t} \right) \right) &=& 0 \end{eqnarray*}\]

In conclusion, the A-V Formulation (with boundary conditions) consists in the following set of equations with associated boundary conditions:

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

2.2. Weak Formulation

In this subsection, we express the weak formulation of A-V Formulation with Dirichet 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 :

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

2.2.1. First equation

We express the weak formulation of first equation of (AV-1) (for \(\mathbf{A}\) unknown).

By multiplying (AV-1) by \(\mathbf{ϕ} \in H^{curl}(\Omega)\) and integrating over \(\Omega\), we obtain :

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

Since \(\nabla \cdot (\textbf{u} \times \textbf{v}) = \textbf{v} \cdot (\nabla \times \textbf{u}) - \textbf{u} \cdot (\nabla \times \textbf{v})\) :

\[\int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\nabla \times \mathbf{ϕ})} + \int_{\Omega}{\frac{1}{\mu} \nabla \cdot (\mathbf{ϕ} \times (\nabla \times \textbf{A}))} = - \int_{\Omega_c}{ \sigma \nabla V \cdot \mathbf{ϕ} } - \int_{\Omega}{ \sigma \frac{d \mathbf{A}}{d t} \cdot \mathbf{ϕ} }\]

Using the Divergence Theorem, it follows :

\[\small{ \int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\nabla \times \mathbf{ϕ})} + \int_{\Gamma_D}{\frac{1}{\mu} (\mathbf{ϕ} \times (\nabla \times \textbf{A})) \cdot \mathbf{n}} + \int_{\Gamma_N}{\frac{1}{\mu} (\mathbf{ϕ} \times (\nabla \times \textbf{A})) \cdot \mathbf{n}} = - \int_{\Omega_c}{ \sigma \, \nabla V \cdot \mathbf{ϕ} } - \int_{\Omega}{ \sigma \, \frac{d \mathbf{A}}{d t} \cdot \mathbf{ϕ} } }\]

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 \(\Gamma_N\) :

\[\small{ \int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\nabla \times \mathbf{ϕ})} + \int_{\Gamma_D}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\mathbf{ϕ} \times \mathbf{n})} + \int_{\Gamma_N}{\frac{1}{\mu} ((\nabla \times \textbf{A}) \times \mathbf{n}) \cdot ϕ} + \int_{\Omega_c}{ \sigma \, \nabla V \cdot \mathbf{ϕ} } + \int_{\Omega}{ \sigma \frac{d \mathbf{A}}{d t} \cdot \mathbf{ϕ} } = 0 }\]

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 :

\[\small{ \int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}) \cdot (\nabla \times \mathbf{ϕ})} + \int_{\Omega_c}{ \sigma \nabla V \cdot \mathbf{ϕ} } + \int_{\Omega}{ \sigma \frac{d \mathbf{A}}{d t} \cdot \mathbf{ϕ} } = 0 \hspace{2cm} \text{(Weak AV-1)} }\]

2.2.2. Second equation

We express the weak formulation of (AV-2) (for \(V\) unknown).

By mutliplying by (AV-2) \(\mathbf{\psi} \in H^1(\Omega_c)\) and integrating on \(\Omega_c\), we have :

\[\int_{\Omega_c}{ \nabla \cdot \left(\sigma \left(\nabla V + \frac{d \mathbf{A}}{d t}\right) \right) \, \mathbf{\psi}} = 0\]

Since \(\nabla \cdot (u \cdot v) = v \cdot \nabla u + u \nabla \cdot v\) :

\[- \int_{\Omega_c}{\nabla \cdot \left( \sigma \ \left( \nabla V + \frac{d \mathbf{A}}{d t}\right) \, \mathbf{\psi} \right) } + \int_{\Omega_c}{\sigma ( \nabla V + \frac{d \mathbf{A}}{d t}) \cdot \nabla \mathbf{\psi}} = 0\]

Appling the Divergence Theorem :

\[- \int_{\Gamma_c}{ \left( \sigma ( \nabla V + \frac{d \mathbf{A}}{d t}) \cdot \mathbf{n} \right) \, \mathbf{\psi}} + \int_{\Omega_c}{\sigma (\nabla V + \frac{d \mathbf{A}}{d t}) \cdot \nabla \mathbf{\psi}} = 0\]

As \(\mathbf{J} \cdot \mathbf{n} = 0\) on \(\Gamma_c\), we finally get using \(\mathbf{J} = \sigma \mathbf{E} = - \sigma \, \left( \nabla V + \frac{d \mathbf{A}}{d t} \right)\):

\[ \int_{\Omega_c}{\sigma (\nabla V + \frac{d \mathbf{A}}{d t}) \cdot \nabla \mathbf{\psi}} = 0 \hspace{2cm} \text{(Weak AV-2)}\]

2.2.3. The MQS system

From (Weak AV-1) and (Weak AV-2), we obtain the weak formulation of A-V Formulation :

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

2.3. Time Discretization

In this section, we see the Weak formulation of A-V 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 A-V Formulation becomes :

Time Discretization of A-V Formulation
\[\small{ \left\{ \begin{align*} \int_{\Omega}{\frac{1}{\mu} (\nabla \times \textbf{A}^{n+1}) \cdot (\nabla \times \mathbf{ϕ})} + \int_{\Omega_c}{ \sigma \nabla V^{n+1} \cdot \mathbf{ϕ} } + \int_{\Omega}{ \sigma \frac{\mathbf{A}^{n+1}}{\Delta t} \cdot \mathbf{ϕ} } &= \int_{\Omega}{ \sigma \frac{\mathbf{A}^n}{\Delta t} \cdot ϕ } & \text{(Disc AV-1)} \\ \int_{\Omega_c}{\sigma \left( \nabla V^{n+1} + \frac{\mathbf{A}^{n+1}}{\Delta t} \right) \cdot \nabla \mathbf{\psi}} &= \int_{\Omega_c}{\sigma \, \frac{\mathbf{A}^n}{\Delta t}} \cdot \nabla \psi & \text{(Disc AV-2)} \\ \forall \mathbf{ϕ} \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in H^{1}(\Omega_c) \end{align*} \right. }\]

3. Magnetostatic Case

This section presents the AV-Formulation for the stationary regime as a special case of MQS. We assume that all parameters are time independent.

3.1. Differential Equation

As all time derivates are null by hypothesis, the differential formulation of A-V Formulation becomes :

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

3.2. Weak Formulation

The weak formulation of A-V Formulation becomes :

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

4. Documentation

  • Potential Formulations in Magnetics applying the Finite Element Mehtod, Miklòs Kuczmann, Laboratory of Electromagneic Fields "Széchenyi István" University Gyorn, 2009, Download the PDF

  • 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, Download the PDF