SFEMaNS
version 4.1 (work in progress)
Reference documentation for SFEMaNS

This section starts with a presentation of the equations that are implemented in SFEMaNS. The restrictions on the domain of computation and an overview of the features available is given.
The following equations are implemented in SFEMaNS.
\begin{align*} \partial_t\bu+\left(\ROT\bu\right)\CROSS\bu  \frac{1}{\Re}\LAP \bu +\GRAD p &=\bef, \\ \DIV \bu &= 0, \end{align*}
with \(\bu\) the velocity field, \(p\) the pressure, \(\Re\) the kinetic Reynolds number and \(\bef\) a source term.\begin{align*} C \partial_t T+ \DIV(T \bu)  \DIV (\lambda \GRAD T) &= f_T, \end{align*}
with \(T\) the temperature, \(\bu\) the velocity field, \(C\) the volumetric heat capacity, \(\lambda\) the thermal conducitivty and \(f_T\) a source term.The Maxwell equations. In a conducting domain \(\Omega_c\), these equations can be written as follows:
\begin{align*} \partial_t (\mu^c \bH^c) + \nabla \times \left(\frac{1}{\Rm \sigma} \nabla \times \bH^c \right) = \nabla\times (\bu \times \mu^c \bH^c) + \nabla \times \left(\frac{1}{\Rm \sigma} \nabla\times \mathbf{j} \right), \\ \text{div} (\mu^c \bH^c) = 0 , \end{align*}
with \(\bH^c\) the magnetic field, \(\bu\) the velocity field, \(\textbf{j}\) a source term, \(\mu^c\) the magnetic permeability, \(\sigma\) the electrical conductivity and \(\Rm\) the magnetic Reynolds number. If the magnetic permeability presents jumps across a surface denoted \(\Sigma_\mu\), the following equations have to be satisfied on \(\Sigma_\mu\):
\begin{align*} \bH^c_1 \times \bn_1 + \bH^c_2 \times \bn_2 = 0,\\ \mu^c_1\bH^c_1 \cdot \bn_1 + \mu^c_2 \bH^c_2 \cdot \bn_2 = 0 ,\\ \end{align*}
where we define outward normals \(\bn_1, \bn_2\) to the surface \(\Sigma_\mu\).
In a simply connected insulating domain \(\Omega_v\), referred as vacuum, these equations can be written as follows:
\begin{align*} \mu^v \partial_t \LAP \phi = 0 , \end{align*}
with \(\phi\) a scalar potential such that \(\GRAD \phi\) is equal to the magnetic field \(\bH\) in the vacuum. The following continuity conditions across the interface \(\Sigma=\Omega_c \cap \Omega_v\) have to be satisfied:
\begin{align*} \bH^c \times \bn^c + \nabla \phi \times \bn^v = 0 , \\ \mu^c \bH^c \cdot \bn^c + \mu ^v \nabla \phi \cdot \bn^v = 0 , \end{align*}
with \(\bn^c\) and \(\bn^v\) the outward normals to the surface \(\Sigma\).
Remarks:
\begin{align*}  \nabla \times \textbf{E} = \partial_t (\mu \bH) \text{ in } \Omega,\\ \int_{\Gamma_{i,v}} \textbf{E} \cdot \textbf{n} = 0 \text{ for } 1 \leq i \leq J, \\ \end{align*}
where \((\Gamma_{i,v})_{1\leq i \leq J}\) are the connected components of \(\partial \Omega_v\) with outward normal denoted \(\textbf{n}\).The code SFEMaNS uses cylindrical coordinates \((r,\theta,z)\) so a spectral/finite element method can be applied. This method consists in using a Fourier decomposition in the azimuthal direction and approximates the problem in a meridian section with Lagrange finite elements. Due to the \(\theta\)periodicity, the domain of computation \(\Omega\) must be axisymmetric.
The domain of computation \(\Omega\) is splitted into the three following subdomains:
The insulating subdomain \(\Omega_v\) is assumed to be simply connected so the magnetic field can be written \(\bH=\GRAD\phi\). The scalar potential \(\phi\) can be proved to be the solution of the following equation in \(\Omega_v\):
\begin{align*} \mu^v \partial_t \LAP \phi = 0. \end{align*}
Remarks:
The following set ups can be considered by the code SFEMaNS:
The following extensions have been implemented in SFEMaNS:
The approximation methods of the above setting are described in the section Numerical approximation. We refer to this section
for more details on the quasistatic approximation of the MHD equations.