SFEMaNS
version 5.3
Reference documentation for SFEMaNS
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This section starts with a presentation of the equations that are implemented in SFEMaNS.
The following equations are implemented in SFEMaNS.
The incompressible Navier-Stokes equations. In a domain \(\Omega\), these equations are written as follows:
\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.
Remark: One can also consider the abovee equations with variable density and viscosity as described in Extension to multiphase flow problem
The heat equation. In a domain \(\Omega\), these equations are written as follows:
\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 are 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}\mathbf{j}^s \right), \\ \text{div} (\mu^c \bH^c) = 0 , \end{align*}
with \(\bH^c\) the magnetic field, \(\bu\) the velocity field, \(\textbf{j}^s\) a source term, \(\mu^c\) the magnetic permeability, \(\sigma\) the electrical conductivity and \(\Rm\) the magnetic Reynolds number. If the magnetic permeability is discontinuous 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 \(\bn_1, \bn_2\) are outward normal to the surface \(\Sigma_\mu\). \(\bn_1\) points from $ \(\Omega_1\) to $ \(\Omega_2\) and \(\bn_1=-\bn_2\).
\begin{align*} -\mu^v \partial_t \LAP \phi = 0 , \end{align*}
where \(\phi\) the scalar potential such that \(\bH=\GRAD \phi\) 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 \(\Sigma\).Remark: the above equations are supplemented by initial and boundaries conditions.
The code SFEMaNS uses cylindrical coordinates \((r,\theta,z)\) and assumes a priori that the computational domain is axi-symmetric. The approximation method consists of using a Fourier decomposition in the azimuthal direction and Lagrange finite elements in the meridian section.
The computational domain \(\Omega\) is divided into the following three sub-domains:
The insulating sub-domain \(\Omega_v\) is assumed to be simply connected so the magnetic field are 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 are also available in SFEMaNS but require a good knowledge of the code to be used properly:
The approximation methods of the above setting are described in the section Numerical approximation. The use is referred to this section
for more details on the quasi-static approximation of the MHD equations.