Magnetic equations with FreeFem++: The Grad-Shafranov equation &amp; the current hole

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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“…in the computational domain Ω surrounded by the boundary Γ (ψ = 0 =ψ) defined by the following [25]:…”

Section: The Gs Equation With Linear Right Hand Sidementioning

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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“…The method of construction of the sequence of regular solutions is based on a particular splitting in time of the equation, with the same structure has in the seminal Temam's work [39] for Navier-Stokes equations and related problems. It must be noticed that practical simulations in the context of the numerical simulation of the Current Hole [13,20,21] have been performed with Finite Elements Methods coupled with this spitting strategy [14]: they have indeed shown unconditional stability.…”

Section: Existence Of Weak Solutionsmentioning

confidence: 99%

“…Following [13] we consider that reduced MHD models can be helpful in this direction. The boundary conditions that we use have very little influence on the Current Hole simulations reported in [13,14]. In our work we use a mixed of Dirichlet and Neumann conditions (2.7), but it can be replaced with no harm with pure Dirichlet boundary conditions (2.8).…”

Section: Introductionmentioning

confidence: 99%

“…In our work we use a mixed of Dirichlet and Neumann conditions (2.7), but it can be replaced with no harm with pure Dirichlet boundary conditions (2.8). Even it is far to be the case in Tokamaks, we will consider that the exterior boundary is smooth because it fits with the Current Hole simulations of [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…The second aim is to study some mathematical properties of this general model: in this work we focus on the existence of weak solutions in order to establish a mathematical foundation for the simulations reported in [13,14]. The model is endowed with an important energy identity, see (4.1) and further generalization.…”

Section: Introductionmentioning

confidence: 99%

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Reduced resistive MHD in Tokamaks with general density

Després¹,

Sart²

2012

ESAIM: M2AN

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Abstract. The aim of this paper is to derive a general model for reduced viscous and resistive Magnetohydrodynamics (MHD) and to study its mathematical structure. The model is established for arbitrary density profiles in the poloidal section of the toroidal geometry of Tokamaks. The existence of global weak solutions, on the one hand, and the stability of the fundamental mode around initial data, on the other hand, are investigated.Mathematics Subject Classification. 93A30, 35Q35, 76E25, 82D10.

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Unconditionally stable numerical simulations of a new generalized reduced resistive magnetohydrodynamics model

Malapaka

Després

Sart

2013

Numerical Methods in Fluids

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SUMMARY Reduced‐resistive magnetohydrodynamics (MHD) models are used in understanding different phenomenon in various domains, for example, astrophysics to model magnetotail or for solar arcades [Finn Bozkaya JM, Guzdar PN. Loss of equilibrium and reconnection in tearing of two dimensional equilibrias. Physics of Fluids B 1993; 5:2870–2876], modeling plasma confinements in reverse field pinch [Strauss HR. The dynamo effect in fusion plasmas. Physics of Fluids 1985; 28:2786–2792] and tokamaks [Strauss HR. Reduced MHD in nearly potential magnetic fields. Journal of Plasma Physics 1997; 57(1):83–87; Freidberg J. Plasma Physics and Fusion Energy. Cambridge University Press: Cambridge, 2008]. In this context, recently, a new generalized reduced‐resistive MHD model, which can make use of an arbitrary density profile was proposed [Després B, Sart R. Reduced resistive MHD in Tokamaks with general density. ESAIM: Mathematical Modelling and Numerical Analysis 2012; 46(5):1081–1106. EDP Sciences, SMAI, 2012 online 2011]. We in this work show that this proposed theoretical model can be realized numerically as well, and that it is very robust if the equation set is written in a very particular form using the properties of FEM. To illustrate these points, we pick the current hole configuration [Fujita T, Oikawa T, Suzuki T, Ide S, Sakamoto Y, Koide Y, Hatae T, Naito O, Isayama A, Hayashi N, Shirai H. Plasma equilibrium and confinement in a tokamak with nearly zero central current density in JT‐60U. Physical Review Letters 2001; 87(11):245001; Hawkes NC, Stratton BC, Tala T, Challis CD, Conway G, DeAngelis R, Giroud C, Hobirk J, Joffrin E, Lomas P, Lotte P, Mailloux J, Mazon D, Rachlew E, Reyes‐Cortes S, Solano E, Zastrow K‐D. Observation of zero current density in the core of JET discharges with lower hybrid heating and current drive. Physical Review Letters 2001; 87(11):115001], which was modeled using reduced‐resistive MHD and remodel it using different combinations of current sources and density profiles. Our model can be implemented with reasonable computational resources at the price of solving a well‐posed global linear system and it is unconditionally stable. These features are also demonstrated as a part of our numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.

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Magnetic equations with FreeFem++: The Grad-Shafranov equation & the current hole

Cited by 12 publications

References 10 publications

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

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

Reduced resistive MHD in Tokamaks with general density

Unconditionally stable numerical simulations of a new generalized reduced resistive magnetohydrodynamics model

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