In order to fulfill the increasing demand, alternative energy sources have to be developed. Indeed, the current rate of fossil fuel usage and its serious adverse environmental impacts (pollution, greenhouse gas emissions, ...) lead to an energy crisis accompanied by potentially disastrous global climate changes.

Controlled fusion power is one of the most promising
alternatives to the use of fossil resources, potentially
with a unlimited source of fuel. France with the ITER
(http://

The other route for fusion plasma is inertial confinement. In this latter case, large scale hydrodynamical instabilities prevent a sufficiently large energy deposit and lower the return of the target. Therefore, for both magnetic and inertial confinement technologies, the success of the projects is deeply linked to the theoretical understanding of plasma turbulence and flow instabilities as well as to mathematical and numerical improvements enabling the development of predictive simulation tools.

Castor gathers the activities in numerical simulation
of fusion plasmas with the activities in control and optimisation done in the laboratory
Jean-Alexandre Dieudonné of the University of Nice.
The main objective of the Castor team is to contribute
to the development of innovative numerical tools to improve
the computer simulations of complex turbulent or unstable flows in plasma physics and to develop
methods allowing the real-time control of these flows or the optimisation of scenarios of plasma discharges in tokamaks.
Castor is a common project between Inria (http://

The main reseach topics are:

Modelling and analysis

Fluid closure in plasma

Turbulence

Plasma anisotropy type instabilities

Free boundary equilibrium (FBE)

Coupling FBE – Transport

Numerical methods and simulations

High order methods

Curvilinear coordinate systems

Equilibrium simulation

Pressure correction scheme

Anisotropy

Solving methods and parallelism

Identification and control

Inverse problem: Equilibrium reconstruction

Open loop control

Applications

MHD instabilities : Edge-Localized Modes (ELMs)

Edge plasma turbulence

Optimization of scenarii

This paper reports numerical investigations on the identification of the electron temperature profile Te from interferometry and polarimetry Stokes vector measurements with the equilibrium code NICE (Newton direct and Inverse Computation for Equilibrium). This latter enables the consistent resolution of the inverse equilibrium reconstruction problem in the framework of nonlinear free-boundary equilibrium coupled to the Stokes model equation for polarimetry. We find that for ITER plasma with high Ip, Ne and Te the identification from noisy measurements is possible (Project EUROfusion / WP01Jet Campaigns (WPJET1) .

Twin experiments were performed with this tokamak geometry in the framework of the EUROfusion / WP10 JT-60SA (WPSA) project.

Plasma boundary reconstruction is one of the main tools to provide a reliable control and tokamak performance. We explore the feasibility for the ISTTOK tokamak (Portugal) of a reconstruction method based on calculated vacuum magnetic flux map and plasma intersection with the wall. We show that via square wave input response curves and pre-processing of the poloidal field coil currents, it is possible to build for ISTTOK a simple scaling model for the effective equilibrium magnetic fields, and perform plasma boundary reconstruction using the algorithm VacTH. This algorithm, included in the NICE numerical code suite, relies on the decomposition of the poloidal flux in toroidal harmonics. The reconstructed plasma boundary is shown for a given discharge and its shape and position are shown to evolve consistently with the typical timescale evolution of ISTTOK discharges. This provides an opportunity of using this plasma boundary reconstruction method as a diagnostic tool for ISTTOK.

Error bars on control variables are directly given by the inverse of the Hessian of the minimized cost function. This is not the case for other quantities such as the safety factor profile for example, and the computation of error bars for these important output quantities necessitate the non-trivial computation of their (discrete) derivatives with respect to the control variables as wall as the state variables.

CedricBoulbe

Developments have been done on the code NICE:

Implementation of a mode 'without plasma' for magnetostatic computations.

Implementation of pressure constraints in NICE for IMAS tested on JET data.

Regular updates of the relased NICE actor in IMAS (EUROfusion / WP13 Code Development for Integrated Modeling)

A Matlab interface has been developped to run the free boundary direct, evolutive, and inverse static modes of NICE.

In order to identify the plasma equilibrium operating space for future tokamaks, a new objective function is introduced in the inverse static free-boundary equilibrium code FEEQS.M. This function comprises terms which penalize the violation of the central solenoid and poloidal field coils limitations (currents and forces). The penalization terms do not require any weight tuning. Hence, this new approach automates to a large extent the identification of the operating space. As an illustration, the new method is applied on the ITER 15 and 17 MA inductive scenarios, and similar operating spaces compared to previous works are found. These operating spaces are obtained within a few (∼ 3) hours of computing time on a single standard CPU.

The real-time control of plasma position, shape and current in a tokamak has to been sured by a number of electrical circuits consisting of voltage suppliers and axisymmetric coils.Finding good target voltages/currents for the control systems is a very laborious, non-trivial task due to non-linear effects of plasma evolution. We introduce here an optimal control formulation to tackle this task and present in detail the main ingredients for finding numerical solutions:the finite element discretization, accurate linearizations and Sequential Quadratic Programming.Case studies for the tokamaks WEST and HL-2M highlight the flexibility and broad scope of the proposed optimal control formulation.

The free boundary equilibrium code NICE has been coupled to the fast transport solver METIS in a Matlab worflow. A first test case has been proposed on ITER. This work has been done for the project Eurofusion WPCD.

The implementation of high order curl- or div-conforming finite element spaces is quite delicate, especially in the three-dimensional case. I have worked on an implementation strategy, which has been applied in the open source finite element software FreeFem++. In particular, I have used the inverse of a generalized Vandermonde matrix to build a basis of generators in duality with the degrees of freedom, which then provides in FreeFem++ an easy-to-use but powerful interpolation operator. With Marcella Bonazzoli, now at the Inria Team DEFI in Saclay, I have carefully addressed the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain.

High order mixed finite element spaces generally lack natural choices of bases but they do have spanning families. I have worked on these FEs for simplicial meshes and proven theoretically their effectiveness. I have also commented on some aspects of a new set of degrees of freedom, the so-called weights on the small simplices, to represent discrete functions in these spaces .

I have worked to propose and analyze an efficient algorithm for the computation of a basis of
the space of divergence-free Raviart-Thomas finite elements. The algorithm is based
on graph techniques. The key point is to realize that, with very natural degrees of
freedom for fields in the space of Raviart-Thomas finite elements of degree r + 1 and
for elements of the space of discontinuous piecewise polynomial functions of degree

Merging ideas from compatible discretisations and polyhedral methods, I have worked with D. Di Pietro and J. Droniou to construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving exactness, we have shown that the usual sequence of Finite Element spaces forms, through appropriate interpolation operators, a commutative diagram with other proposed sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete L2-products completes the work .

In order to avoid some mesh singularities that arise when using quadrangular elements for complex
geometries and flux aligned meshes, the use of triangular elements is a possible option that
we have studied in the past years. However due to the appearence of fourth order terms in the PDE systems that we are
interested in, pure Galerkin methods require the use of finite element methods with

Acoustic streaming is a particularly interesting example of the interaction of phenomena occuring on two different time scales. From a practical point of view, it is mainly used to generate a slow motion in micro-fluidic devices by means of high frequency acoustic sources. Modelling of these interaction is a challenge : taking into account the high frequency phenomena is prohibitively expensive but on the other hand, there is no universal agreement on existing averaged models. In order to have reference simulations, we have constructed a numerical code solving the compressible Navier-Stokes equations with high-order accuracy using compact schemes. Comparison with asymptotic analytical results has been done and shows that the code is able to simulate acoustic waves propagation in a stable way on long time scale, a property that is essential for the study of this phenomenon.

Hermite-Bezier finite element modeling is the standard method used to discretize the MHD equations in codes such as JOREK. This finite element family allows for an accurate description of the magnetic topology using flux aligned grids where the iso-parametric curved elements match the magnetic flux level sets. However, the description of complex material geometries is difficult with this family of finite element. We have begun to study the use of discretization methods using overlapping meshes where one mesh is composed of quadrangular Hermite-Bezier finite element while the second one is made of triangular elements.

Charged particles in plasma in strong magnetic fields undergo a complicated motion, which is a combination of a fast cyclotron gyration around the magnetic field lines and a relatively slow dynamics along and across the magnetic field lines. Gyrokinetic equations, devised to describe plasma under such conditions, eliminate the fast cyclotron gyration from the equation of motion, thus reducing the space-velocity phase space dimension from six to five.

Originally, the gyrokinetic formulation was devised for a collisionless plasmas. The quest for retaining collisions in gyrokinetic equations is ongoing. Collisions are important if one wants to describe the transport properties of a magnetized plasma on a macroscopic level. A description of the transport of energy and momentum was proposed in Refs. , , , , . However, mathematical description of collisions in these works is too complicated for numerical implementation. We develop a simplified description of collision operators in the gyrokinetic formulation that preserve the pertinent conservation features and suitable for numerical modeling. A comparison of these operators with several test cases is under investigation.

We study a dispersive regularization of p-system. The governing equations are the Euler- Lagrange equations for a Lagrangian depending not only on the velocity and density, but also on the first material derivative of density. Such regularization arises, in particular, in the modeling of waves in solids, in bubbly fluids as well as in the theory of water waves. We show that such terms are not always regularizing. The solution can develop shocks even in the presence of dispersive terms. In particular, we construct such a shock solution that con- nects a constant state to a periodic wave train. The corresponding shock speed coincides with the velocity of the wave train. The generalized Rankine-Hugoniot relations (jump relations) are also obtained. The numerical evidence of the existence of such shocks is demonstrated in the case of the Serre-Green-Naghdi equations describing long surface water waves. In particular, it has been shown that such waves can dynamically be formed

The shear shallow water model provides a higher order approximation for shallow water flows by including the effect of vertical shear in the model. This model can be derived from the depth averaging process by including the second order velocity fluctuations, which are neglected in the classical shallow water approximation. The resulting model has a non-conservative structure, which resembles the 10-moment equations from gas dynamics. This structure facilitates the development of path conservative schemes and we construct HLL, 3-wave and 5-wave HLLC-type solvers. An explicit and semi-implicit MUSCL-Hancock type second order scheme is proposed for the time integration. Several test cases including roll waves show the performance of the proposed modeling and numerical strategy.

To avoid disruptions, the first thing to do is to operate as far as possible from disruptions operational limits. It means that plasma scenarios must be designed taking these limits into account. The challenge is to deal with peeling-ballooning instabilities called Edge Localized Modes (ELMs) which are characterized by the quasi-periodic relaxation of the pressure pedestal profile which results in the expelling of particles and energy from the bulk plasma to the edge. Injecting of impurities is one of the solutions to change the pedestal profile and mitigate MHD instabilities. The current design of the ITER DMS(Disruption Mitigation System) is a hybrid system using Massive Gas Injection (MGI) and Shattered Pellet Injection (SPI), methods which have demonstrated their efficiency on current tokamaks (JET, DIIID, …). Considering that the plasma is composed of impurities, main ion core and set of electrons, premixed “Full MHD” formulation has been proposed. This model assumes that, for any control volume, the plasma is locally neutral and at the thermal and coronal equilibrium. Properties of this model are under analysis, according to the tabulated equation of state. A numerical approximation in the Jorek Code is also under progress. This work has been done in the context of the JET program 2019.

Member of the ANR SISTEM, Oct. 2019 - Sept. 2023 coordinated by the M2P2 Institute of Aix-Marseille Univ. "SImulations with high-order schemes of tranSport and TurbulencE in tokaMak" programme Modeles numeriques 2019

Participants: Francesca Rapetti, Blaise Faugeras, Didier Auroux, Jacques Blum, Cédric Boulbe

Contact: F. Rapetti

EuroFusion Consortium

CASTOR participates to the following EuroFusion consortium projects :

EUROfusion WPCD (Working Package Code Development):

EWE-2: Enabling Workflow Exploitation Area - Enabling the exploitation of the equilibrium reconstruction and MHD stability workflow (participation)

WDEV-2: Workflow Development Area - Free boundary equilibrium and feedback control (participation and coordination)

EuroFusion Enabling Research CfP-AWP19-ENR-01, Strengthening the non-linear MHD code JOREK for application to key questions of the fusion roadmap.

EUROfusion WPSA(Work Package JT-60SA) 2018-2010

The team collaborates with TUC (Technical University of Crete, Prof. Argyris Delis) on the modelling of acoustic streaming phenomena. In this framework, Argyris Delis has visited the Castor team in November 2019.

J. Blum is member of the editorial board of the Journal Scientific Computing.

F. Rapetti is member of the editorial board of the Advances in Computational Mathematics (ACOM) journal by Springer

C. Boulbe is managing editor of the SMAI Journal of Computational Mathematics

*H. Guillard* has been reviewer for the Journal of Computational physics, Computers and Fluids and
International Journal for Numerical methods in Fluids.

F. Rapetti: Keynote speaker at the Enumath conference (Egmond aan Zee), oct 2019, "High-order Whitney forms on simplices"

H. Guillard is coordinator of the topic “Turbulence and transport of edge plasma” within the Fédération FR-FCM

Licence : F. Rapetti, Mathématiques 2,30h équivalent TD, L2, Université Cote d'Azur, France

Master : F. Rapetti, Méthodes numériques, 70h équivalent TD, M1, Université Cote d'Azur, France

Ecole d'ingénieur, C. Boulbe, Analyse Numérique 2, 45 équivalent TD, niveau L3, Université Cote d'Azur

Ecole d'ingénieur, C. Boulbe, Analyse numérique 1, 71h équivalent TD, niveau L3, Université Cote d'Azur

Ecole d'ingénieur, C. Boulbe, Algèbre linéaire et Scilab, 26h équivalent TD, Université Cote d'Azur

Ecole d'ingénieur, C. Boulbe, Projet 1, 41h équivalent TD, Université Cote d'Azur

Licence: A. Sangam, Mathématiques Fondements 1, 60h, Semestre 1 de la Licence, Université Nice Sophia Antipolis, France

Licence: A. Sangam, Méthodes Mathématiques - Approche Continue, 30h, Semestre 1 de la Licence, Université Nice Sophia Antipolis, France

Licence: A. Sangam, Mathématiques Compléments 1, 5h, Semestre 1 de la Licence, Université Nice Sophia Antipolis, France

Licence: A. Sangam, Mathématiques Fondements 2, 6h, Semestre 2 de la Licence, Université Nice Sophia Antipolis, France

Licence: A. Sangam, Analyse 2, 2h, L2, Université Nice Sophia Antipolis, France

Licence: A. Sangam, Analyse Numérique, 70h, L3, Université Nice Sophia Antipolis, France

Licence: A. Sangam, Calcul Différentiel, 20h, L3, Université Nice Sophia Antipolis, France

Ecole d'ingénieur/Master: B. Nkonga, Méthode des éléments finis, 24h, M2, Polytech Nice Sophia, France

Ecole d'ingénieur/Master: B. Nkonga, Eléments finis mixtes, 24h, M2, Polytech Nice Sophia, France

PhD : Xiao Song, Model-based Control-oriented Scenario Construction in Tokamaks, Université Cote d'Azur, 6 décembre 2019, Blaise Faugeras, Holger Heumann

PhD in progress : Ali Elarif, “Simulation numérique des instabilités magnétohydrodynamique dans les Tokamaks”, October 2017, Hervé Guillard

Hervé Guillard has been part of the jury for the Phd defence of Corentin Prigent, "Etude numérique et modélisation du modèle d'Euler bi-température : point de vue cinétique", Bordeaux University, 24/10/2019.

Hervé Guillard was referee for the Phd jury of Clément Colas, "Formulation intégrale implicite pour la modélisation d'écoulements fluides en milieux encombrés", AMU University, 14/11/2019.

Hervé Guillard has been part of the jury for the Phd defence of Quentin Carmouze, "Modélisation et simulation numérique des écoulements diphasiques denses et dilués", Nice University, 28/12/2019.

F. Rapetti: Juan Antonio Soler Vasco, Univ. Aix-Marseille, Sep. 2019.

F. Rapetti: Anouk Nicolopoulos-Salle, Univ. Sorbonne Paris, Déc. 2019.

B. Faugeras, H. Heumann, H. Guillard and J. Blum: Xiao Song, Université Cote d'Azur, Dec. 2019