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Tokamak Free-Boundary Plasma Equilibrium Computations in Presence of Non-Linear Materials

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Abstract

We consider the axisymmetric formulation of the equilibrium problem for a hot plasma in a tokamak. We adopt a non-overlapping mortar element approach, that couples \(\mathcal {C}^0\) piece-wise linear Lagrange finite elements in a region that does not contain the plasma and \(\mathcal {C}^1\) piece-wise cubic reduced Hsieh–Clough–Tocher finite elements elsewhere, to approximate the magnetic flux field on a triangular mesh of the poloidal tokamak section. The inclusion of ferromagnetic parts is simplified by assuming that they fit within the axisymmetric modeling and a new formulation of the Newton algorithm for the problem solution is stated, both in the static and quasi-static evolution cases.

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Notes

  1. All numerical simulations are here performed with the software NICE (see [11]).

  2. In the Standard International (SI) unit system, mass M (kg), length L (m), time T (s) and current intensity I (A) are base dimensions (resp., units).

  3. From \(\psi \, {\textbf{e}}_\varphi = r \,{\textbf{A}}_t\), we get \( {\textbf{A}}_t = (0, \, A_\varphi , \, 0)^\top \) where \(A_\varphi \) is equal to \( \frac{1}{r} \psi \). In cylindrical coordinates, for this vector \( {\textbf{A}}_t \) we have \( \textrm{curl}\, {\textbf{A}}_t = (-\partial _z A_\varphi , 0, \frac{1}{r}\partial _r (r \, A_\varphi ))^\top \).

  4. We wish to recall the fundamental contribution of Roland Glowinski to the analysis, the finite element approximation and numerical resolution by Newton-like methods of such non-linear problems.

  5. To approximate \(\texttt{J}_p(\cdot ,\cdot )\) in (12) by a quadrature formula we need to know the domain \(\varOmega _p(\psi )\) occupied by the plasma. This domain is an unknown of the equilibrium problem, as it depends on \(\psi \). An efficient technique to determine it is stated in [10].

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Acknowledgements

The authors are grateful to Jacques Blum for his precious enlightenments on plasma physics and to Holger Heumann for stimulating conversations on the derivation and approximation of the Grad–Shafranov–Schlüter’s equation. FR thanks INRIA for the delegation during which this work was accomplished.

Funding

GG is a PhD student from the Université Côte d’Azur. FR received funding from the French National Research Agency under Grant Agreement SISTEM (ANR-19-CE46-0005-03). CB and BF have worked within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200—EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

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  1. Dep. de Mathématiques, Univ. Côte d’Azur, CNRS, INRIA CASTOR Team, Nice, France

    Cédric Boulbe, Blaise Faugeras, Guillaume Gros & Francesca Rapetti

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  1. Cédric Boulbe
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  2. Blaise Faugeras
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  3. Guillaume Gros
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  4. Francesca Rapetti
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Correspondence to Francesca Rapetti.

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Boulbe, C., Faugeras, B., Gros, G. et al. Tokamak Free-Boundary Plasma Equilibrium Computations in Presence of Non-Linear Materials. J Sci Comput 96, 42 (2023). https://doi.org/10.1007/s10915-023-02265-8

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