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Title: Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code

Abstract

Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity. The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here. Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field. Several well-known instabilities are simulated to demonstratemore » the capabilities of the new method. The linear growth rate of an internal kink mode and a tearing mode are benchmarked against the results of a linear MHD code. The evolution of a tearing mode and the resulting magnetic islands are simulated well into the nonlinear regime. The results are compared with predictions from the reduced MHD model. Finally, a simulation of a ballooning mode illustrates the possibility to use our method as an ideal MHD method without the need to add any physical dissipation.« less

Authors:
 [1];  [2];  [3];  [4];  [3];  [5]
  1. Centrum Wiskunde & Informatica, P.O. Box 94079, 1090 GB Amsterdam (Netherlands)
  2. (Netherlands)
  3. Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands)
  4. ITER Organization, Route de Vinon sur Verdon, 13115 Saint Paul Lez Durance (France)
  5. Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven (Netherlands)
Publication Date:
OSTI Identifier:
22572322
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 316; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BALLOONING INSTABILITY; COMPUTERIZED SIMULATION; DIFFUSION; EQUATIONS; FINITE ELEMENT METHOD; HOT PLASMA; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; NONLINEAR PROBLEMS; TEARING INSTABILITY; THERMONUCLEAR REACTORS; TOKAMAK DEVICES

Citation Formats

Haverkort, J.W., Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven, Blank, H.J. de, Huysmans, G.T.A., Pratt, J., and Koren, B., E-mail: b.koren@tue.nl. Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code. United States: N. p., 2016. Web. doi:10.1016/J.JCP.2016.04.007.
Haverkort, J.W., Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven, Blank, H.J. de, Huysmans, G.T.A., Pratt, J., & Koren, B., E-mail: b.koren@tue.nl. Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code. United States. doi:10.1016/J.JCP.2016.04.007.
Haverkort, J.W., Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven, Blank, H.J. de, Huysmans, G.T.A., Pratt, J., and Koren, B., E-mail: b.koren@tue.nl. 2016. "Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code". United States. doi:10.1016/J.JCP.2016.04.007.
@article{osti_22572322,
title = {Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code},
author = {Haverkort, J.W. and Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven and Blank, H.J. de and Huysmans, G.T.A. and Pratt, J. and Koren, B., E-mail: b.koren@tue.nl},
abstractNote = {Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity. The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here. Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field. Several well-known instabilities are simulated to demonstrate the capabilities of the new method. The linear growth rate of an internal kink mode and a tearing mode are benchmarked against the results of a linear MHD code. The evolution of a tearing mode and the resulting magnetic islands are simulated well into the nonlinear regime. The results are compared with predictions from the reduced MHD model. Finally, a simulation of a ballooning mode illustrates the possibility to use our method as an ideal MHD method without the need to add any physical dissipation.},
doi = {10.1016/J.JCP.2016.04.007},
journal = {Journal of Computational Physics},
number = ,
volume = 316,
place = {United States},
year = 2016,
month = 7
}
  • The linear stability and nonlinear evolution of the resistive m = 1 mode in tokamaks is studied using a full set of resistive magnetohydrodynamic (MHD) equations in toroidal geometry. The modification of the linear and nonlinear properties of the mode by a combination of strong toroidal effects and low resistivity is the focus of this work. Linearly there is a transition from resistive kink to resistive tearing behavior as the aspect ratio and resistivity are reduced, and there is a corresponding modification of the nonlinear behavior, including a slowing of the island growth and development of a Rutherford regime, asmore » the tearing regime is approached. In order to study the sensitivity of the stability and evolution to assumptions concerning the equation of state, two sets of full nonlinear resistive MHD equations (a pressure convection set and an incompressible set) are used. Both sets give more stable nonlinear behavior as the aspect ratio is reduced. The pressure convection set shows a transition from a Kadomtsev reconnection at large aspect ratio to a saturation at small aspect ratio. The incompressible set yields Kadomtsev reconnection for all aspect ratios, but with a significant lengthening of the reconnection time and development of a Rutherford regime at an aspect ratio approaching the transition from a resistive kink mode to a tearing mode. The pressure convection set gives an incomplete reconnection similar to that sometimes seen experimentally. The pressure convection set is, however, strictly justified only at high beta.« less
  • We measure the loss of power incurred by the bending of a single mode step-indexed optical fiber using vector finite element modeling of the full-wave Maxwell equations in the optical regime. We demonstrate fewer grid elements can be used to model light transmission in longer fiber lengths by using high-order basis functions in conjunction with a high order energy conserving time integration method. The power in the core is measured at several points to determine the percentage loss. We also demonstrate the effect of bending on the light polarization.
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  • No abstract prepared.
  • This paper presents an application of a magnetostatic 3D non-linear FE code to the computation of the axial forces on the windings of a large-power autotransformer due to short circuit currents. The analyzed five-limb core autotransformer has an internal tertiary delta-connected winding. A comparison between the axial forces due to the peak currents of both a time-varying configuration fault and two conventional faults - two-phase earth fault and three-phase earth faults - has been made. First, the authors compute the transient fault currents and the corresponding MMFs of each winding, by simulation with ATP (Alternative Transients Program) of a specificmore » circuital non-linear model of the autotransformer. Then, by imposing the peak current density J to each winding, they compute the magnetic field H with the code Maxwell 3DFS. Finally, they evaluate the axial force on the autotransformer windings. The FE model takes only 1/4 of the actual geometry of the autotransformer, due to the symmetry. The obtained results show that the time-varying fault yields greater axial forces than the conventional faults, particularly on the internal tertiary delta-connected winding.« less