Boundary element modelling to solve the Grad–Shafranov equation as an axisymmetric problem

Abstract

The Grad-Shafranov equation describes the magnetic flux distribution of plasma in an axisymmetric system like a tokamak-type nuclear fusion device. The equation is transformed into an equivalent boundary integral equation by expanding the inhomogeneous term related to the plasma current into a polynomial. In the present research, the singularity of the fundamental solution, which consists of two elliptic integrals, and the properties of singular integrals have been minutely investigated. The discontinuous quadratic boundary elements have been introduced to give an accurate solution with a small number of boundary elements. Ampere's circuital law has been applied to estimate the total plasma current from the boundary integral of the poloidal field, and based on this idea, a new scheme to calculate the eigenvalue has also been proposed.3

FEM-BEM coupling methods for Tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains

“…Nevertheless, this task is well understood, and we refer to [11] for the technical details recalling also the asymptotic formulas for the fundamental solution G(x, y) when x − y → 0 derived in [24].…”

Speed-Up Technique of Extended Boundary Node Method for Large-Scale Simulation

“…The boundary element method (BEM) is a numerical method for solving boundary-value problems of partial differential equations and has been so far used in the field of the nuclear fusion science. For example, the BEM has been adopted to solve the Grad-Shafranov equation which describes the magnetohydrodynamics equilibrium of plasma in terms of the poloidal magnetic flux [1]. Although the BEM is well suited for solving the Grad-Shafranov equation, it has the inherent demerit: a boundary must be divided into a set of elements before executing the BEM code.…”

Reconstruction of the Eddy Current Distribution on the Vacuum Vessel in a Reversed Field Pinch Device based on the External Magnetic Sensor Signals

“…The above technique is exactly the same as that adopted for the CCS when the singular point i is not located on a boundary element [13,14]. For an element on the CCS which includes the singularity, the integration is performed sophisticatedly with the aid of the logarithmic Gaussian quadrature formula [12].…”