Finite element methods on composite meshes for tuning plasma equilibria in tokamaks

Abstract

Abstract The fundamental concept of magnetically confined nuclear fusion devices is the magnetohydrodynamic (MHD) equilibrium: the pressure gradient due to highly energetic charged particles is balanced by the Lorenz force due to strong magnetic fields. Hence, numerical methods for MHD equilibria are also fundamental for fusion engineering applications. We rely here on a finite element method on composite meshes for the simulation of axisymmetric equilibria in tokamaks, torus-shaped nuclear fusion devices. One mesh with Cartesian quadrilaterals covers the domain accessible by the plasma and one mesh with triangles discretizes the region outside the chamber. The two meshes overlap in a narrow region. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the principal unknown, the poloidal magnetic flux, while preserving accurate meshing of the geometric details in the exterior. We show that higher order regularity allows to formulate appropriate optimal control problems that help to find a special type of equilibria, called snowflake equilibria, that are a very promising concept to mitigate high heat loads due to plasma escaping particles.

Keywords

• Mortar element method
• Thermonuclear fusion
• Optimal control
• Non-linear elliptic partial differential equations