- CEDRES++: In Tokamaks, at the slow resistive diffusion time scale, the magnetic configuration in the plasma can be described by the MHD equilibirum equations inside the plasma and the Maxwell equations outside. Moreover, the magnetic field is often supposed not to depend on the azimutal angle. Under this assumption of axisymmetric configuration, the equilibrium in the whole space reduces to solving a 2D problem in which the magnetic field in the plasma is described by the well known Grad Shafranov equation. The unknown of this problem is the poloidal magnetic flux. The P1 finite element code CEDRES++ solves this free boundary equilibrium problem in direct and inverse mode. The direct problem consists in the computation of the magnetic configuration and of the plasma boundary, given a plasma current density profile and the total current in each poloidal field coils (PF coils). The aim of the inverse problem is to find currents in the PF coils in order to best fit a given plasma
- FEEQS.M: FEEQS.M, MATLAB-Code for plasma equilibrium problems, started in 2014 is a MATLAB/OCTAVE implementation of the methods for toroidal free-boundary plasma equilibria that are described in . The code is fully vectorized and therefore the running time is comparable to C/C++ -implementations. It contains more than 10 different main modes that address different applications. Novel numerical schemes can be tested and validated within the same framework. Moreover, It contains currently a couple of additional features that are not yet published, such as a fixed boundary mode, mesh refinement or static equilibrium calculations with high order FEM or Powel Sabin. A lite version of FEEQS.M is publicly available. A snapshot of the current development version can be download by selected collaborators. FEEQS.M was intended to be a fast prototyping test bed but started to be a major research tool for physicists and engineers. FEEQS.M is used for pre- and post shot simulations. Optimal control formulations are used to set up experiments.
- EQUINOX: This code is dedicated to the numerical reconstruction of the equilibrium of the plasma in a Tokamak. The problem solved consists in the identification of the plasma current density, a non-linear source in the 2D Grad-Shafranov equation which governs the axisymmetric equilibrium of a plasma in a Tokamak. The experimental measurements that enable this identification are the magnetics on the vacuum vessel, but also polarimetric and interferometric measurements on several chords, as well as motional Stark effect measurements.
The reconstruction can be obtained in real-time and the numerical method implemented involves a finite element method, a fixed-point algorithm and a least-square optimization procedure.
- VACTH implements a method based on the use of toroidal harmonics and on a modelization of the poloidal field coils and divertor coils for the 2D interpolation and extrapolation of discrete magnetic measurements in a tokamak. The method is generic and can be used to provide the Cauchy boundary conditions needed as input by a bounded domain equilibrium reconstruction code like EQUINOX. It can also be used to extrapolate the magnetic measurements in order to compute the plasma boundary itself.
- NICE This code is under development. Its goal is to gather in a single modern, modular and evolutionary C++ code, the different numerical methods and algorithms from VacTH, EQUINOX and CEDRES++ which share many common features. It also integrates new methods as for example the possibility to use the Stokes model for equilibrium reconstruction using polarimetry measurements.
- FBGKI The Full Braginskii solver (FBGKI) considers the equations proposed by Braginskii (1965), in order to describe the plasma turbulent transport in the edge part of tokamaks.
These equations rely on a two fluid (ion – electron) description and on the electroneutrality of the plasma. If assuming the electric field electrostatic, one has then a set of 10 coupled non-linear and strongly anisotropic PDEs. FBGKI makes use in space of high order methods:
Fourier in the toroidal periodic direction and the spectral element method (SEM) in the poloidal plane, so that one can easily address the divertor configuration. The integration in time is based on a Strang splitting and Runge-Kutta schemes, with implicit treatment of the Lorentz terms (DIRK scheme) and a subcycling technique for the explicit part. The spectral vanishing viscosity (SVV) technique is implemented for stabilization. Static condensation is used to reduce the computational cost. In its sequential version, a matrix free solver is used to compute the potential. A parallel version of the code, whose linear algebra makes use of the PETSC library routines, is also availab
- JOREK-INRIA is a new version of the JOREK software, for MHD modeling of plasma dynamic in tokamak geometries. The numerical approximation is derived in the context of finite elements where 3D basic functions are tensor products of 2D basis functions in the poloidal plane by 1D basis functions in the toroidal direction. More specifically, Jorek uses curved bicubic isoparametric elements in 2D and a spectral decomposition (sine, cosine) in the toroidal axis. Continuity of derivatives and mesh alignment to equilibrium surface fluxes are enforced. Resulting linear systems are solved by the PASTIX software developed at Inria-Bordeaux.