Bézier surfaces and finite elements for MHD simulations

Czarny, Olivier; Huysmans, G.

doi:10.1016/j.jcp.2008.04.001

Cited by 134 publications

(208 citation statements)

References 24 publications

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“…The advantage of this is that the finite elements are aligned with the equilibrium magnetic flux surfaces. 21 In the toroidal direction, a spectral method is used. For further details of the numerical models in the current JOREK code, we refer to Franck et al, 32 H€ olzl et al…”

Section: B a Reduced Mhd Fluid Model In The Jorek Codementioning

confidence: 99%

“…These codes include NIMROD, 12,13 BOUTþþ, 14,15 M3D(-C1), 16,17 XTOR(-2F), 18,19 and JOREK. 20,21 The JOREK code has been used to simulate MHD instabilities in tokamaks successfully using several nonlinear reduced MHD models 22,23 as well as full MHD models. 24,25 In this work we use JOREK and a reduced MHD fluid model to simulate the evolution of classical 2/1 tearing modes from their first development, through their saturation and eventual partial suppression using electron cyclotron current drive (ECCD).…”

Section: Introductionmentioning

confidence: 99%

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Early evolution of electron cyclotron driven current during suppression of tearing modes in a circular tokamak

2016

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When electron cyclotron (EC) driven current is first applied to the inside of a magnetic island, the current spreads throughout the island and after a short period achieves a steady level. Using a two equation fluid model for the EC current that allows us to examine this early evolution in detail, we analyze high-resolution simulations of a 2/1 classical tearing mode in a low-beta large aspectratio circular tokamak. These simulations use a nonlinear 3D reduced-MHD fluid model and the JOREK code. During the initial period where the EC driven current grows and spreads throughout the magnetic island, it is not a function of the magnetic flux. However, once it has reached a steady-state, it should be a flux function. We demonstrate numerically that if sufficiently resolved toroidally, the steady-state EC driven current becomes approximately a flux function. We discuss the physics of this early period of EC evolution and its impact on the size of the magnetic island.

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Section: B a Reduced Mhd Fluid Model In The Jorek Codementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Early evolution of electron cyclotron driven current during suppression of tearing modes in a circular tokamak

2016

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“…In this part we study the Grad-Shafranov equation (2) [1,13] that models a plasma equilibrium where the pressure gradient balances the Lorentz force, Ampere's law and the fact that the magnetic eld B is divergencefree (see for example [3,6,8]). The function ψ is the magnetic ux and the plasma boundary corresponds to a surface of constant ψ.…”

Section: The Grad-shafranov Operatormentioning

confidence: 99%

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

et al. 2011

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Abstract. FreeFem ++ [11] is a software for the numerical solution of partial dierential equations. It is based on nite element method. The FreeFem ++ platform aims at facilitating teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. Our goal in this work is to evaluate the FreeFem ++ tool on basic magnetic equations arising in Fusion Plasma in the context of the ITER project.First we consider the Grad-Shafranov equation, which is derived from the static ideal MHD equations assuming axisymetry. Some of the properties of the equation and its analytical solutions are discussed. Second we discretize a reduced resistive MHD model which admits solutions of the Grad-Shafranov equation as stationary solutions. Then the physical stability of these stationary solutions is investigated through numerical experiments and the numerical stability of the algorithm is discussed.

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“…In the world some codes have been developed to simulate this problem and solve the MHD equations. One of those codes is the JOREK code which implements and solves both some reduced and full MHD models [10] based on assumptions on the velocity and magnetic fields in a toroidal geometry. The equations in the poloidal plane (circular, d-shape or x-point meshes) [6] [5] [7] are discretized using cubic bezier splines and the toroidal discretization use Fourier expansion.…”

Section: Introductionmentioning

confidence: 99%

Anisotropic Diffusion in Toroidal geometries

Ratnani

Nkonga²,

Franck

et al. 2016

ESAIM: Proc.

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Abstract. In this work, we present a new finite element framework for toroidal geometries based on a tensor product description of the 3D basis functions. In the poloidal plan, different discretizations, including B-splines and cubic Hermite-Bézier surfaces are defined, while for the toroidal direction both Fourier discretization and cubic Hermite-Bézier elements can be used. In this work, we study the MHD equilibrium by solving the Grad-Shafranov equation, which is the basis and the starting point of any MHD simulation. Then we study the anisotropic diffusion problem in both steady and unsteady states.

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Bézier surfaces and finite elements for MHD simulations

Cited by 134 publications

References 24 publications

Early evolution of electron cyclotron driven current during suppression of tearing modes in a circular tokamak

Early evolution of electron cyclotron driven current during suppression of tearing modes in a circular tokamak

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

Anisotropic Diffusion in Toroidal geometries

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