Accurate plasma boundary calculation using linear triangular finite elements

Abstract

This paper presents a novel method for accurately tracing magnetic field lines. This procedure is of particular interest for axisymmetric nuclear fusion devices. The design of the plasma facing components in a tokamak strictly depends on the Scrape-Off Layer (SOL), a thin region of open field lines through which charged particles and energy flow out from the plasma core to the solid walls with a huge heat flux. The power exhaust issue is among the top priorities in the European Fusion Roadmap, hence the determination of the plasma boundary and SOL with a high accuracy is essential. Existing equilibrium codes using finite element formulations with linear triangles of mesh size $h$ provide a piecewise constant magnetic flux gradient, resulting in coarse accuracy of the field lines in the SOL, with an error of order $O\left(h\right)$, especially near the X-point where the flux gradient approaches zero. The proposed procedure is based on a method introduced in 2023, which achieves continuity and a convergence rate of order $O\left(h^2\right)$ for the magnetic flux gradient too. The magnetic poloidal flux is then approximated with a piecewise polynomial function of second or third degree in the triangles. A more accurate evaluation of the plasma boundary and SOL field lines is obtained, particularly near the X-point, which is no longer constrained to be a mesh node. The method has been successfully tested in two cases with available analytical solutions. It has also been used as a post-processor for the flat top configuration of the DTT tokamak obtained with the free boundary CREATE-NL+ equilibrium code. For a given accuracy, the computational cost of the procedure is significantly lower than alternative methods relying on finer first order discretization or techniques using triangular C¹ finite elements.