Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices

AYDIN, Selçuk Han; SEZGİN, Münevver Tezer

Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices

Selçuk Han AYDIN, (Orta Doğu Teknik Üniversitesi, , Ankara, Türkiye)

Münevver Tezer SEZGİN (Orta Doğu Teknik Üniversitesi, , Ankara, Türkiye)

Turkish Journal of Engineering and Environmental Sciences

4 0

Yıl: 2008 Cilt: 32 Sayı: 5 Sayfa Aralığı: 265 - 275 Metin Dili: İngilizce İndeks Tarihi: 29-07-2022

Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices

Öz:

The magnetohydrodynamic equilibrium in an axisymmetric plasma is described by the Grad-Shafranov (GS) equation in terms of the magnetic flux. Boundary element method (BEM) is suitable for plasma equilibrium since it requires the discretization of only the plasma boundary which changes shape during the operation of an actual fusion device. In this paper, numerical solutions of the GS equation are obtained by using the boundary element method, the finite element method (FEM) and the differential quadrature method (DQM) for a rectangular plasma when the source term (current density function) on the right hand side is assumed to be a monomial. Our aim is also to find the most applicable numerical procedure between those three methods for different plasma profiles. We transform the equation to the homogeneous one with a particular solution eliminating the domain integral in the BEM formulation. For the source term containing the magnetic flux (nonlinear right hand side) an iterative procedure is made use of in the BEM and FEM formulations. It is found that the FEM gives better accuracy for a D-shape tokamak plasma whereas the BEM is more suitable for a Solov’ev tokamak plasma. The solutions agree very well with the previously published numerical solutions for a rectangular plasma.

Anahtar Kelime:

Konular: Çevre Mühendisliği

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

Brebbia C.A. and Dominguez J., “Boundary Elements An Introductory Course”, Comp. Mech. Pub., Southampton:Boston, 1992.
Itagaki M. and Fukunaga T., “Boundary Element Modelling to Solve The Grad-Shafranov Equation as an Axisymmetric Problem”, Engineering Analysis with Boundary Elements, 30, 746–757, 2006.
Itagaki M., Kamisawada J., Oikawa S., “Boundaryonly Integral Equation Approach Based on Polynomial Expansion of Plasma Current Profile to Solve the Grad-Shafranov Equation”, Nuclear Fusion, 44, 427– 437, 2004.
Itagaki M., Yamaguchi M., Fukunaga T., “Boundary Integral Equation Approach Based on a Polynomial Expansion of the Current Distribution to Reconstruct the Current Density Profile in Tokamak Plasmas”, Nuclear Fusion, 45, 153–162, 2005.
Jardin S.C., “A triangular Finite Element with First- Derivative Continuity Applied to Fusion MHD Applications”, Jour. of Comput. Physics, 200, 133–152, 2004.
Leuer J.A., Schaffer M.J., Parks P.B., Brown M.R., “Calculation of Free Boundary SSX Doublet Equilibria Using the Finite Element Method”, 43rd APS/DPP Meeting, California, 2001.01).
J.N. Reddy J.N., “An Introduction to the Finite Element Method”, McGraw-Hill, 1993.
Sawai S., Tsuchimoto M., Igarashi H., Honma T., “Boundary Element Analysis of Free Boundary Field Reversed Configurations”, IEEE Transactions on Magnetics, 26(2), 571–574, 1990.
Shu C., “Differential Quadrature and Its Application in Engineering”, Springer Verlag, Londan Berlin Heidelberg, 2000.
Shu C. and Richards B.E., “Application of Generalized Differential Quadrature to Solve Two-Dimensional Incompressible Navier-Stokes Equations”, Int. J. Numer. Methods Fluids, 15, 791–798, 1992.
Tezer-Sezgin M., “Solution of MHD flow in a Rectangular Duct by Differential Quadrature Method”, Computers and Fluids, 33, 533–547, 2004.
Tezer-Sezgin M. and Dost S., “On the Fundamental Solutions of the Axisymmetric Helmholtz-type Equations”, Applied Mathematical Modeling, 17, 47–51, 1993.
Tsuchimoto M., Milya K., Honma T., Igarashi H., “Fundamental Solutions of the Axisymmetric Helmholtztype Equations”, Applied Mathematical Modeling, 14, 605–611, 1990.

APA	AYDIN S, SEZGİN M (2008). Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. , 265 - 275.
Chicago	AYDIN Selçuk Han,SEZGİN Münevver Tezer Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. (2008): 265 - 275.
MLA	AYDIN Selçuk Han,SEZGİN Münevver Tezer Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. , 2008, ss.265 - 275.
AMA	AYDIN S,SEZGİN M Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. . 2008; 265 - 275.
Vancouver	AYDIN S,SEZGİN M Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. . 2008; 265 - 275.
IEEE	AYDIN S,SEZGİN M "Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices." , ss.265 - 275, 2008.
ISNAD	AYDIN, Selçuk Han - SEZGİN, Münevver Tezer. "Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices". (2008), 265-275.

APA	AYDIN S, SEZGİN M (2008). Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. Turkish Journal of Engineering and Environmental Sciences, 32(5), 265 - 275.
Chicago	AYDIN Selçuk Han,SEZGİN Münevver Tezer Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. Turkish Journal of Engineering and Environmental Sciences 32, no.5 (2008): 265 - 275.
MLA	AYDIN Selçuk Han,SEZGİN Münevver Tezer Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. Turkish Journal of Engineering and Environmental Sciences, vol.32, no.5, 2008, ss.265 - 275.
AMA	AYDIN S,SEZGİN M Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. Turkish Journal of Engineering and Environmental Sciences. 2008; 32(5): 265 - 275.
Vancouver	AYDIN S,SEZGİN M Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. Turkish Journal of Engineering and Environmental Sciences. 2008; 32(5): 265 - 275.
IEEE	AYDIN S,SEZGİN M "Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices." Turkish Journal of Engineering and Environmental Sciences, 32, ss.265 - 275, 2008.
ISNAD	AYDIN, Selçuk Han - SEZGİN, Münevver Tezer. "Numerical solution of Grad-Shafranov equation for the distribution of magnetic flux in nuclear fusion devices". Turkish Journal of Engineering and Environmental Sciences 32/5 (2008), 265-275.