A-V Formulation using "Saddle-Point" Method with Gauge Condition

In the section General Case, Maxwell equation’s formulation is computed in a Maxwell Quasi Static approximation, the A-V Formulation with an imposed 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 the air for example. \(\Gamma = \partial \Omega\) is 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, 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 unknowns) :

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

In order to keep the curlcurl formulation and guarantee the unicity of the solution of our problem, we will use the Saddle-Point formulation as shown in Cecile Daversin - Catty. Reduced basis method applied to large non-linear multi-physics problems:application to high field magnets design. Electromagnetism. Université de Strasbourg, 2016.

As we impose the gauge condition \(\nabla \cdot \mathbf{A} = 0\) to the A-V Formulation, we need to add one more equation to the system, to represent this condition. This extra equation is handled by the constraint \(p\) which is an additional scalar unknown :

A-V Saddle-Point Formulation

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

1.3. Weak Formulation

Let express the weak formulation of equation (AV Saddle-Point Formulation).

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 :

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

We multiply the equation by \(\phi \in H^{curl}(\Omega)\) and integrate it on \(\Omega\) :

\[ \int_{\Omega}{ \left(\sigma \frac{\partial \textbf{A}}{\partial t} + \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \nabla p \right) \cdot \phi \ dxdydz} + \int_{\Omega_c}{ \sigma \nabla V \cdot \phi \ dxdydz} = 0 \hspace{1cm} \text{(Weak AV)}\]

By Formula of Green :

\[\scriptsize{ \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \phi \ dxdydz} + \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A}\cdot \nabla \times \phi \ dxdydz } - \int_{\Gamma}{ \frac{1}{\mu} \frac{\partial \mathbf{A}}{\partial \mathbf{n}} \cdot \phi }+ \int_{\Omega}{ \nabla p \cdot \phi \ dxdydz }+ \int_{\Omega_c}{ \sigma \nabla V \cdot\phi \ dxdydz } = 0 }\]

The imposed boundary conditions are :

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

So, the weak formulation is :

Weak formulation of A-V Saddle-Point Formulation

\[\scriptsize{ \text{(Weak AV)} \\ \left\{ \begin{eqnarray*} \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \phi \ dxdydz} + \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A} \cdot \nabla \times \phi \ dxdydz } &=& - \int_{\Omega}{ \nabla p \cdot \phi \ dxdydz } - \int_{\Omega_c}{ \sigma \nabla V \cdot\phi \ dxdydz } \quad\text{(Weak AV1)} \\ \int_{\Omega_c}{\sigma (\nabla V + \frac{\partial \mathbf{A}}{\partial t}) \cdot \nabla \mathbf{\psi}} &=& 0 \quad\text{(Weak AV-2)}\\ &&\forall \mathbf{\phi} \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in H^{1}(\Omega_c) \end{eqnarray*} \right. }\]

1.4. Time Discretization

In this subsection, we use the time discretization by backward Euler method on the (Weak AV 1).

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 : \(\frac{\partial A}{\partial t} \approx \frac{A^{n+1}-A^n}{\Delta t}\).

The equation (Weak AV 1) becomes :

Time Discretization of A-V Saddle-Point Formulation

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega}{ \sigma \frac{ \textbf{A}^{n+1}}{\Delta t} \cdot \phi \ dxdydz} + \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A}^{n+1} \cdot \nabla \times \phi \ dxdydz } &=& - \int_{\Omega}{ \nabla p^{n+1} \cdot \phi \ dxdydz } - \int_{\Omega_c}{ \sigma \nabla V^{n+1} \cdot\phi \ dxdydz } +\int_{\Omega}{ \sigma \frac{ \textbf{A}^{n}}{\Delta t} \cdot \phi \ dxdydz} \end{eqnarray*}}\]

2. Magnetostatic Case

In a stationary case, \(\frac{\partial f}{\partial t} = 0\).

2.1. Differential Equation

The differential formulation of A-V Saddle-Point Formulation becomes :

A-V Saddle-Point Formulation in Stationary Case

\[\text{(Magstat Saddle-Point)} \left\{ \begin{matrix} \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \nabla p + \sigma \nabla V &=& 0 &\text{ on } \Omega & \text{(AV-1)} \\ \nabla \cdot \textbf{A}&=&0 &\text{ on } \Omega & \text{(Gauge)} \\ \nabla \cdot \left( \sigma \nabla V \right) &=& 0 &\text{ on } \Omega_c & \text{(AV-2)} \\ \end{matrix} \right.\]

2.2. Weak Formulation

The weak formulation of A-V Saddle-Point Formulation becomes :

Weak formulation of A-V Saddle-Point Formulation in Stationary

\[\scriptsize{ \text{(Weak Magstat Saddle-Point)} \\ \left\{ \begin{eqnarray*} \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A} \cdot \nabla \times \phi \ dxdydz } &=& - \int_{\Omega}{ \nabla p \cdot \phi \ dxdydz } - \int_{\Omega_c}{ \sigma \nabla V \cdot\phi \ dxdydz } \quad\text{(Weak AV1)} \\ \int_{\Omega_c}{\sigma \nabla V \cdot \nabla \mathbf{\psi}} &=& 0 \quad\text{(Weak AV-2)}\\ &&\forall \phi \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in H^{1}(\Omega_c) \end{eqnarray*} \right. }\]

3. References

Cecile Daversin - Catty. Reduced basis method applied to large non-linear multi-physics problems: application to high field magnets design. Electromagnetism. Université de Strasbourg, 2016. p56-64 PDF