Boundary-only integral equation approach based on polynomial expansion of plasma current profile to solve the Grad–Shafranov equation

Itagaki, Masafumi; Kamisawada, Jun-ichi; Oikawa, Shun-ichi

doi:10.1088/0029-5515/44/3/008

Cited by 14 publications

(23 citation statements)

References 15 publications

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“…with ( ) ( ) Itagaki et al [3] showed that the above Grad-Shafranov equation can be transformed into an equivalent boundary-only integral equation in terms of the plasma boundary Γ ,…”

Section: Boundary Integral Equation For the Grad-shafranov Equationmentioning

confidence: 99%

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

Itagaki¹,

Fukunaga²

2006

Engineering Analysis with Boundary Elements

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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

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“…with ( ) ( ) Itagaki et al [3] showed that the above Grad-Shafranov equation can be transformed into an equivalent boundary-only integral equation in terms of the plasma boundary Γ ,…”

Section: Boundary Integral Equation For the Grad-shafranov Equationmentioning

confidence: 99%

“…Itagaki et al [3,4] evaluated the total plasma current by directly integrating the polynomial-expanded inhomogeneous term. In the present work the authors propose an alternative technique based on Ampere's circuital law, i.e., a boundary integral of the poloidal field.…”

Section: Introductionmentioning

confidence: 99%

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

Itagaki¹,

Fukunaga²

2006

Engineering Analysis with Boundary Elements

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“…Itagaki et al [9] showed that the above Grad-Shafranov equation can be transformed into an equivalent boundary-only integral equation in terms of the plasma boundary Γ ,…”

Section: Boundary-only Integral Equation For the Grad-shafranov Equationmentioning

confidence: 99%

Boundary integral equation approach based on a polynomial expansion of the current distribution to reconstruct the current density profile in tokamak plasmas

2005

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A new approach has been proposed to reconstruct the current density profile in Tokamak plasma.The boundary-only integral equation derived from the Grad-Shafranov equation under the assumption of polynomial expansion of current density will have no unknowns except for the polynomial expansion coefficients, once the magnetic flux and its derivative have been given along the plasma boundary with the aid of Kurihara's Cauchy-condition surface method based on magnetic sensor data. In addition to the discretized form of the equation, some constraints are taken into account: ｔhe total plasma current, zero-current along the plasma boundary, a scalar relationship derived from the MHD equilibrium to connect the current density with the magnetic flux. It is also assumed that the poloidal field as a quantity closely related to the current density can be measured at a certain number of points inside the plasma. The whole set of the linear equations are solved using the singular value decomposition technique to determine the polynomial expansion coefficients. The validity of the present technique and the quality of the current density solution has been investigated through test calculations for some plasma configurations.

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“…This is in general a nonlinear elliptic partial differential equation derived from the ideal MHD equations. There are numerous extensive works in the literature for solving GS equation for the fixed and free boundary problems in the tokamak using the finite element, finite difference, spectral, boundary element and other mesh based methods [2][3][4][5][6][7][8][9]. However the work based on meshless methods applied to the GS equation is very limited.…”

Section: Introductionmentioning

confidence: 99%

Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions

Nath

Kalra

Munshi

2015

Computers & Mathematics with Applications

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a b s t r a c tIn this work a meshless method based on the approximate particular solutions is applied to the computation of fixed boundary tokamak equilibria using Grad-Shafranov (GS) equation. The GS equation is solved for different choices of the right hand side of the equation: (i) when it is not a function of magnetic flux (i.e., Solov'ev solutions), (ii) when it is a linear function of magnetic flux, and (iii) when it is a nonlinear function of magnetic flux. For all these cases the first order derivative term in the GS equation is transferred to the right hand side such that the left hand side consists only the Laplace operator. This enables us to use the Radial Basis Functions (RBFs) in the calculation of approximate particular solutions. A linear combination of these particular solutions is taken as the solution of the GS equation and the resulting system of algebraic equations is solved iteratively because of the presence of the magnetic flux on the right hand side in all three choices. Furthermore, we use least squares approach in solving the overdetermined system of algebraic equations which alleviates the problem of ill-conditioning to a certain extent. The numerical results obtained using this method are in good agreement with the analytical solutions (where available). We find that the method is convergent, accurate and easily applicable to the irregular geometries due to its meshless character.

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Boundary-only integral equation approach based on polynomial expansion of plasma current profile to solve the Grad–Shafranov equation

Abstract: A new type of boundary element method has been applied to solve the Grad-Shafranov equation and to give a distribution of magnetic flux function in a Tokamak nuclear fusion device. The quantity

Cited by 14 publications

References 15 publications

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

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

Boundary integral equation approach based on a polynomial expansion of the current distribution to reconstruct the current density profile in tokamak plasmas

Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions

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