## TÜBİTAK-Research Project: The Boundary Element Solutions of The Nuclear Fusion Reactor Problems

The magnetohydrodynamic equilibrium in an axisymmetric plasma is described by the Grad-Shafranov (GS) equation in terms of the magnetic flux. Due to the non-linear nature of the Grad-Shafranov equation a general analytical solution is not possible. However, for a given current density (the non-homogeneouity), the Grad-Shafranov equation can be solved numerically. In this project two different recent and powerful numerical methods, namely the finite element method with residual free bubble functions and the boundary element method are going to be used without the need of iteration. The residual free bubble functions are used for stabilizing the numerical results obtained from the finite element method. The boundary element method is well-suited for plasma equilibrium analysis that requires efficient data preparation and computation following the change in plasma shape during the operation of an actual fusion device. The nonhomogenouity in the Grad-Shafranov equation is going to be approximated by using radial basis functions.

The GS equation is the the axisymmetric case of the MHD equations, in terms of flux function with some simplifications. Therefore, the general case of MHD equations will be solved numerically. The two-dimensional MHD equations are defined in terms of momentum equations(Navier-Stokes equations), Maxwell equation and continuity equation. Firstly, the Navier-Stokes equations are solved in primitive variables form with finite element method. Oscillations coming from the solution of the coupled form are been solved by using stabilized methods. Finally MHD terms are added and full form of the equations are solved with two level finite element method using residual free bubble functions.

##### Collaborators

- M.Tezer-Sezgin, Department of Mathematics & Institute of Applied Mathematics, METU (Coordinator)
- A.I. Neslitürk, Department of Mathematics, Izmir Institute of Technology & Institute of Applied Mathematics, METU
- S. Han Aydin, Institute of Applied Mathematics, METU
- S. Gümgüm, Institute of Applied Mathematics, METU

Funded by TUBITAK-105T091, Scientific and Technical Research Council of Turkey