Analysis of MHD equilibrium solution by finite element method

In analyzing the experimental results or the design for large Tokamak facility in nuclear fusion reactor development, handling of MHD equations is considered to be extremely important concerning reactor core equilibrium and stability analysis. In this paper, the Grad-Shafranov equation which the stationary equilibrium solution for ideal MHD fluid satisfies in a torus-form region is assumed, and its axisymmetric solution is determined, using the finite element method. First, the equation which is to be satisfied by an axisymmetric stationary equilibrium solution and to be analyzed in this paper is derived from the ideal MHD equation system. This is a non-linear eigenvalue problem. In analyzing this, it is proposed to employ a numerical calculation which traces the equilibrium solution using both the increment method in which the eigenvalues or the values at specific nodal points are used as the control parameters and the iteration method in accordance with Newton-Raphson method. As the finite element models, triangular first order elements and second order isoparametric elements are used, and numerical calculation accuracy is examined for convergence and free boundary processing of lower and higher order elements. The above analysis method is applied to a torus, which is considered to be the shape of shells for Tokamak nuclear fusion reactors, and the effectiveness of this technique is shown by performing plasma analysis for circular and square cross sections. (Wakatsuki, Y.)