One of the fundamental model used for fusion plasma simulations is the magnetohydrodynamic (MHD) model. However, in practice, many theoretical and numerical works in this field use specific approximations of this model known as reduced MHD models. These models assume that in the presence of a strong magnetic field, the main dynamic reduces to incompressible motion in the plane perpendicular to the dominating magnetic field and to the propagation of Alfvén waves in the magnetic field direction. In the framework of the slab approximation for large aspect ratio tokamaks ($R/a>>1$ where $R$ and $a$ are respectively the major and minor radius of the machine) we have studied last year the validity of this assumption using techniques coming from the asymptotic theory of hyperbolic equations with a large parameter. In particular, we have proved that the solutions of the full MHD system converge in a weak sense to the solutions of an appropriate reduced model even in the presence of ill-prepared initial data. This work continues with a tentative to relax the large aspect ratio assumption that is not verified in modern machines.

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 . In particular, we have developped the geometric tools necessary for the construction of Powell-Sabin splines and have applied these methods for the approximation of some simple hyperbolic PDE systems (namely the Euler equation of fluid dynamics). The PhD thesis of Ali Elarif that has begun in october 2017 is devoted to the study of the applicability of these methods to more complex PDE models encountered in plasma physics and to an extension towards other triangular $C^1$ elements (Clough-Tocher elements).

We have studied numerically an extension designed for turbulent flows of the shallow water model. The model is able to describe the oscillatory nature of turbulent hydraulic jumps and as such correct the deficiency of the classical shallow water equations. The model equations, originally developed for horizontal flow or flows occurring over small constant slopes, are straightforwardly extended here for modeling flows over non-constant slopes and numerically solved by a second-order well-balanced finite volume scheme. Further, a new set of exact solutions to the extended model equations are derived and several numerical tests are performed to validate the numerical scheme and its ability to predict the oscillatory nature of hydraulic jumps under different conditions. The comparisons with experiments performed at Tainan University are very satisfactory given the simplicity of the model .

Due to the highly anisotropic character of strongly magnetized plasmas, a crucial point for numerical simulations is the construction of meshes that are aligned on the magnetic flux surfaces computed by Grad-Shafranov equilibrium solvers. In this work, we study an original method for the construction of flux aligned grids that respect the magnetic equilibrium topology and that can be applied to block-structured meshes using $C^1$ finite element methods (Hermite-Bézier/Cubic spline). This method relies on the analysis of the singularities of the magnetic flux function and the construction of the Reeb graph that allows the segmentation of the physical domain into sub-domains that can be mapped to a reference square domain. Once this domain decomposition has been done, the mapping of the sub-domain to reference patches can be done using integration along the streamlines of the flux function, . This work is performed in the framework of the EoCoE European project (see section ).

Incorporating boundary conditions at infinity into simulations on bounded computational domains is a repeatedly occurring problem in scientific computing. The combination of finite element methods (FEM) and boundary element methods (BEM) is the obvious instrument, and we adapt here for the first time the two standard FEM-BEM coupling approaches to the free-boundary equilibrium problem: the Johnson–Nédélec coupling and the Bielak–MacCamy coupling. We recall also the classical approach for fusion applications, dubbed according to its first appearance von-Hagenow–Lackner coupling and present the less used alternative introduced by Albanese, Blum and de Barbieri. We show that the von-Hagenow–Lackner coupling suffers from undesirable non-optimal convergence properties, that suggest that other coupling schemes, in particular Johnson–Nédélec or Albanese–Blum–de Barbieri are more appropriate for non-linear equilibrium problems. Moreover, we show that any of such coupling methods requires Newton-like iteration schemes for solving the corresponding non-linear discrete algebraic systems.

The modelization of polarimetry Faraday rotation measurements commonly used in tokamak plasma equilibrium reconstruction codes is an approximation to the Stokes model. This approximation is not valid for the foreseen ITER scenarios where high current and electron density plasma regimes are expected. In this work a method enabling the consistent resolution of the inverse equilibrium reconstruction problem in the framework of non-linear free-boundary equilibrium coupled to the Stokes model equation for polarimetry is provided. Using optimal control theory we derive the optimality system for this inverse problem. A sequential quadratic programming (SQP) method is proposed for its numerical resolution. Numerical experiments with noisy synthetic measurements in the ITER tokamak configuration for two test cases, the second of which is an H-mode plasma, show that the method is efficient and that the accuracy of the identification of the unknown profile functions is improved compared to the use of classical Faraday measurements.

In the framework of JET Task T17-15, the method has been implemented in the new code NICE and equilibrium reconstruction studies using real JET measurements are currently being performed.

Within the framework of JET Task T17-02 an update of the real-time equilibrium reconstruction code Equinox at JET in view of its coupling with the transport code Raptor has been performed. Mainly the computation of all the averaged geometric quantities which enter the transport equation have been added.

We have been involved in a benchmark study between the equilibrium reconstruction codes VACTH-EQUINOX, EQUAL and LIUQE on TCV equilibriums [EPS paper R. Coelho et al] The benchmark study lead us to include new functionalities to VACTH-EQUINOX such as the possibility to have an upper X-point. The adaptation of VACTH-EQUINOX to IMAS (Integrated Modelling & Analysis Suite), the ITER standard using IDS (Interface Data Structure) as data type, has been carried on. Equilibrium reconstructions using IMAS have been performed on real JET measurements and on the first recently available WEST measurements.

During the previous years, the free boundary equilibrium code CEDRES++ has been coupled to the transport solver ETS and a magnetic simulink controler using the Integrated Tokamak Modelling infrastructure (Project Eurofusion WPCD - Work Package Code Development). In 2017, we have started to port this tool on IMAS which is the integrated modelling infrastructure developped by ITER. CEDRES++ has been adapted to IMAS and has been coupled to a magnetic controller in the framework of the Eurofusion WPCD project. This activity has been a pilot project to test the C++ tools provided on IMAS.

Plasma breakdown in a tokamak requires a large toroidal electric field $E_{\phi }$ and a low poloidal magnetic field $B_p$, i.e. a so-called field null region. The latter should remain as extended as possible for a sufficient duration (typically a few tens of $ms$), all the more if one operates at low $E_{\phi }$ (e.g. in ITER where $E_{\phi }=0.3V/m$). Finding appropriate settings (i.e. premagnetization coils currents and voltage waveforms) to produce and maintain a good field null region is not a trivial task, in particular in the presence of highly conducting passive structures which make the problem dynamic. WEST is a good example of this situation, due to two toroidally continuous copper plates which have been added for vertical stabilization: indeed, the current in the plates ramps up fast when $E_{\phi }$ is applied, which tends to degrade the field null region.

Our automated approach to determining appropriate breakdown settings relies on a precise electromagnetic model of the machine (including the iron core) and solves a constrained optimization problem, where the objective function to be minimized quantifies the design goal: the averaged magnitude of $B_p$. After discretization we end up with finite dimensional convex constrained optimization problem, that can be solved efficiently with Sequential Quadratic Programming. The approach follows the lines of optimal control methods for plasma equilibria in and .

The automated approach was already beneficial for obtaining first breakdowns in WEST during the initial launch in December 2016. The data collected during these breakdowns allowed for improving the electromagnetic model and the simulations reproduce now very well magnetic measurements and the shapes observed on the fast camera during the experiments.

We introduced a novel method to compute approximations of integrals over implicitly defined hypersurfaces. The new method is based on a weak formulation in L2(0,1) that uses the coarea formula to circumvent an explicit integration over the hypersurfaces. As such it is possible to use standard quadrature rules in the spirit of hp/spectral finite element methods, and the expensive computation of explicit hypersurface parametrizations is avoided. We derived error estimates showing that high order convergence can be achieved provided the integrand and the hypersurface defining function are sufficiently smooth.

We rely on a combination of different finite element methods on composite meshes, for the simulation of axisymmetric plasma equilibria in tokamaks. One mesh with Cartesian quadrilaterals covers the burning chamber and one mesh with triangles discretizes the region outside the chamber. The two meshes overlap in a narrow region around the chamber. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the area that is covered by the plasma, while preserving accurate meshing of the geometric details in the exterior. The continuity of the numerical solution across the boundary of each subdomain is enforced by a mortar-like projection. Higher order regularity is very beneficial to improve computational tools for tokamak research. In , we showed that the numerical calculation of free boundary plasma equilibria highly benefits from approximating the poloidal flux through some higher regular FE functions in the interior of the limiter. In the present work we show how the composite meshes and higher regular finite element functions allow to single out snowflake configurations, that play an important role to mitigate heat load in divertors. Implementations and numerical test were carried out in FEEQS.M

The real-time control of plasma position, shape and current in a tokamak has to be ensured by the Poloidal Field (PF) system. A standard strategy is to feedback-control the currents in the PF coils in order to match reference currents, the latter being a combination of FeedForward (FF) and FeedBack (FB) terms. While the FB part allows a precise control, it can work only near the target and therefore the FF part is essential to “guide” the system, i.e. to approximately reach the target while remaining clear from hardware limits. An essential part of tokamak scenario design is therefore the construction of these FF waveforms. A tool for automatic FF waveforms optimisation (the inverse evolutive mode of the Free-Boundary Equilibrium [FBE] solver FEEQS.M) has been developed recently in the frame of a collaboration with IRFM, CEA. This tool reduces drastically the amount of human work needed to design optimized scenarios compatible with hardware limits. Xiao Song has performed first applications and validations of this tool on present and future machines such as WEST and ITER. This preliminary work focused on the choice of the cost-function and compared different choices for the same type of discharge. A second key aspect of this work was the treatment of inequality constraints via penalisation terms.

We extended FEEQS.M to work with higher order finite element functions (polynomials of degree 1, ... 11, or Powell-Sabin spline finite element functions). This feature is currently available only for the static modes. The free-boundary aspect is addressed by a subdivision approach, that needs to be improved in the future, to increase the accuracy.

The dynamics of plasma charged particles can be described by a two-fluid MHD model. This description considers a plasma as a mixture of ions fluid and electrons flow that are coupled by exchanged terms such as momentum transfer terms, ion and electron heating terms due to collisions, supplemented by the Maxwell's equations. This system is quite intricate so that it is usually reduced to more tractable models. We first derive the two-temperature model, the ideal and resistive MHD equations from the two-fluid MHD system, and show that they correspond to asymptotic regimes for weakly and strongly magnetized plasmas. We then propose a finite volume approximation to compute the solutions of these models in unstructured tessellations used to appropriate mesh the toroidal geometry of the tokamak, where flows the plasma. The formulation of the magnetic field as Euler potential ensures the divergence free constraint in cheap manner, while a relaxation scheme for the two temperature allows an accurate computation of the electron and ion temperatures.

S. Minjeaud and R. Pasquetti have addressed the Korteweg-de Vries equation as an interesting model of high order PDE, in order to show that it is possible to develop reliable and effective schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invariants, on the basis of a (only) $H^1$-conformal Galerkin approximation, namely the Spectral Element Method (SEM). The proposed approach relies on the introduction of additional variables that can be trivially eliminated, because the SEM mass matrix is diagonal, thus allowing to define discrete high order differentiation operators. Highly accurate RK IMEX schemes are used in time, with implicit treatment of the third order term and explicit treatment of the convective one. While the conservation of the mass invariant is natural, the conservation of the energy invariant is enforced by interpolation between embedded IMEX schemes, with preservation of the time discretization accuracy. Applications to several test problems have shown the robustness and accuracy of the proposed method, that is a priori easily extensible to other PDEs and to multidimensional problems (See ).

In a recent JCP paper (see ), a higher order triangular spectral element method ($T$SEM) is proposed to address seismic wave field modeling. The main interest of this $T$SEM is that the mass matrix is diagonal, so that an explicit time marching becomes very cheap. In , R. Pasquetti and F. Rapetti have compared this cubature points based method to the Fekete-Gauss one, that makes use of Fekete points for interpolation and of Gauss points for quadrature. Moreover, they have proposed an extension of this cubature $T$SEM to address elliptic PDEs with non homogeneous Neumann or Robin boundary conditions. More recently, the cubature $T$SEM has been experimented with isoparametric mappings to consider the case of non polygonal computational domains. In any cases it turns out that the cubature $T$SEM compares well with the Fekete-Gauss one.

We have implemented Bohm mach condition using penalty boundary integral over open field lines. We have added temperature dependent viscosity and parallel conductivity, Bohm condition on energy flux. The JET configuration has been considered with success to reach the equilibrium with Bohm conditions on the divertor. However, the solution needs to be improved at the separatrix close to the boundary by a proper numerical stabilization. These structures disappear with constant resistivity. First n=10 ballooning mode in JET has been considered and need to be compared to the reduced MHD results. In order to proceed to these comparisons, a model for the reduced model has been derived including modeling of the viscosity. Validations on circular geometries are on going.

In nonlinear MHD simulations of DIII-D QH-mode plasmas it has been found that low n kink/peeling modes (KPMs) are unstable and grow to a saturated kink-peeling mode. The features of the dominant saturated KPMs, which are localized toroidally by non-linear coupling of harmonics, such as mode frequencies, density fluctuations and their effect on pedestal particle and energy transport, are in good agreement with the observations of the Edge Harmonic Oscillation (EHO) typically present in DIII-D QH-mode experiments. The non-linear evolution of MHD modes including both kink-peeling modes and ballooning modes, is investigated through MHD simulations by varying the pedestal current and pressure relative to the initial conditions of DIII-D QH-mode plasma. The edge current and pressure at the pedestal are key parameters for the plasma either saturating to a QH-mode regime or a ballooning mode dominant regime. The influence of E×B flow and its shear on the QH-mode plasma has been investigated. E×B flow shear has a strong stabilization effect on the medium to high-n modes but is destabilizing for the n=2 mode. The QH-mode extrapolation results of an ITER Q=10 plasma show that the pedestal currents are large enough to destabilize n=1-5 kink/peeling modes, leading to a stationary saturated kink-peeling mode.

Diffuse interface methods with compressible fluids, considered through hyperbolic multiphase flow models, have demonstrated their capability to solve a wide range of complex flow situations in severe conditions (both high and low speeds). These formulations can deal with the presence of shock waves, chemical and physical transformations, such as cavitation and detonation. Compared to existing approaches able to consider compressible materials and interfaces, these methods are conservative with respect to mixture mass, momentum, energy and are entropy preserving. Thanks to these properties they are very robust. However, in many situations, typically in low transient conditions, numerical diffusion at material interfaces is excessive. Several approaches have been developed to lower this weakness. In the present contribution, a specific flux limiter is proposed and inserted into conventional MUSCL type schemes, in the frame of the diffuse interface formulation of Saurel et al. (2009). With this limiter, interfaces are captured with almost two mesh points at any time, showing significant improvement in interface representation. The method works on both structured and unstructured meshes and its implementation in existing codes is simple. Computational examples showing method capabilities and accuracy are presented.