G Giorgiani, H. Bufferand, F. Schwander, E. Serre, P. Tamain. A high-order non field-aligned approach for the discretization of strongly anistropic diffusion operators in magnetic fusion. Computer Physics Communications, Elsevier, 2020, 254, pp.107375. ⟨10.1016/j.cpc.2020.107375⟩. ⟨hal-02613709⟩ Plus de détails...
In this work we present a hybrid discontinuous Galerkin scheme for the solution of extremely anisotropic diffusion problems arising in magnetized plasmas for fusion applications. Unstructured meshes, non-aligned with respect to the dominant diffusion direction, allow an unequalled flexibility in discretizing geometries of any shape, but may lead to spurious numerical diffusion. Curved triangles or quadrangles are used to discretize the poloidal plane of the machine, while a structured discretization is used in the toroidal direction. The proper design of the numerical fluxes guarantees the correct convergence order at any anisotropy level. Computations performed on well-designed 2D and 3D numerical tests show that non-aligned discretizations are able to provide spurious diffusion free solutions as long as high-order interpolations are used. Introducing an explicit measure of the numerical diffusion, a careful investigation is carried out showing an exponential increase of this latest with respect to the non-alignment of the mesh with the diffusion direction, as well as an exponential decrease with the polynomial degree of interpolation. A brief assessment of the method with respect to two finite-difference schemes using non-aligned discretization, but classically used in fusion modeling, is also presented.
G Giorgiani, H. Bufferand, F. Schwander, E. Serre, P. Tamain. A high-order non field-aligned approach for the discretization of strongly anistropic diffusion operators in magnetic fusion. Computer Physics Communications, Elsevier, 2020, 254, pp.107375. ⟨10.1016/j.cpc.2020.107375⟩. ⟨hal-02613709⟩
Giorgio Giorgiani, H. Bufferand, G. Ciraolo, Eric Serre, P. Tamain. A magnetic-field independent approach for strongly anisotropic equations arising plasma-edge transport simulations. Nuclear Materials and Energy, Elsevier, 2019, 19, pp.340-345. ⟨10.1016/j.nme.2019.03.002⟩. ⟨hal-02177048⟩ Plus de détails...
A [Summary] The control of the power exhaust in tokamaks is still an open issue for the future fusion operations. The heat loads on divertor and limiter PFCs is largely determined by the physics of the Scrape-Off Layer (SOL), and therefore it depends mainly on the geometry of the magnetic surfaces and on the geometry of wall components. A better characterization of the heat exhaust mechanisms requires therefore to improve the capabilities of the transport codes in terms of geometrical description of the wall components and in terms of the description of the magnetic geometry. The possibility of dealing with evolving magnetic configurations becomes also critical: during start-up or control operations, for example, the evolution of particles and heat fluxes is little known, although being critical for the safety of the machine. Hence, among the new capabilities of future transport codes will be the possibility of accurately describe the reactor chamber, and the flexibility with respect the magnetic configuration. In particular, avoiding expensive re-meshing of the computational domain in case of evolving equilibrium is mandatory. In order to fulfill these requirements, in this work a fluid solver based on non-aligned discretization is used to solve the plasma-edge transport equations for density, momentum and energies. Preliminary tests on non-structured meshes and realistic geometries/physical parameters show the pertinency of this novel approach.
Giorgio Giorgiani, H. Bufferand, G. Ciraolo, Eric Serre, P. Tamain. A magnetic-field independent approach for strongly anisotropic equations arising plasma-edge transport simulations. Nuclear Materials and Energy, Elsevier, 2019, 19, pp.340-345. ⟨10.1016/j.nme.2019.03.002⟩. ⟨hal-02177048⟩
J. Soler, F. Schwander, G Giorgiani, J Liandrat, P Tamain, et al.. A new conservative finite-difference scheme for anisotropic elliptic problems in bounded domain A new conservative finite-difference scheme for anisotropic elliptic problems in bounded domain. Journal of Computational Physics, Elsevier, 2019, ⟨10.1016/j.jcp.2019.109093⟩. ⟨hal-02477007⟩ Plus de détails...
Highly anisotropic elliptic problems occur in many physical models that need to be solved numerically. A direction of dominant diffusion is thus introduced (called here parallel direction) along which the diffusion coefficient is several orders larger of magnitude than in the perpendicular one. In this case, finite-difference methods based on misaligned stencils are generally not designed to provide an optimal discretization, and may lead the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. This paper proposes an original scheme using non-aligned Cartesian grids and interpolations aligned along a parallel diffusion direction. Here, this direction is assumed to be supported by a divergence-free vector field which never vanishesand it is supposed to be stationary in time. Based on the Support Operator Method (SOM), the self-adjointness property of the parallel diffusion operator is maintained on the discrete level. Compared with existing methods, the present formulation further guarantees the conservativity of the fluxes in both parallel and perpendicular directions. In addition, when the flow intercepts a boundary in the parallel direction, an accurate discretization of the boundary condition is presented that avoids the uncertainties of extrapolated far ghost points classicaly used and ensures a better accuracy of the solution. Numerical tests based on manufactured solutions show the method is able to provide accurate and stable numerical approximations in both periodic and bounded domains with a drastically reduced number of degrees of freedom with respect to non-aligned approaches.
J. Soler, F. Schwander, G Giorgiani, J Liandrat, P Tamain, et al.. A new conservative finite-difference scheme for anisotropic elliptic problems in bounded domain A new conservative finite-difference scheme for anisotropic elliptic problems in bounded domain. Journal of Computational Physics, Elsevier, 2019, ⟨10.1016/j.jcp.2019.109093⟩. ⟨hal-02477007⟩
Giorgio Giorgiani, Hugo Bufferand, Guido Ciraolo, Philippe Ghendrih, Frédéric Schwander, et al.. A hybrid discontinuous Galerkin method for tokamak edge plasma simulations in global realistic geometry. Journal of Computational Physics, Elsevier, 2018, 374, pp.515-532. ⟨10.1016/j.jcp.2018.07.028⟩. ⟨hal-02114246⟩ Plus de détails...
Progressing toward more accurate and more efficient numerical codes forthe simulation of transport and turbulence in the edge plasma of tokamaks,we propose in this work a new hybrid discontinous Galerkin solver. Basedon 2D advection-diffusion conservation equations for the ion density and theparticle flux in the direction parallel to the magnetic field, the code simulatesplasma transport in the poloidal section of tokamaks, including the open fieldlines of the Scrape-off Layer (SOL) and the closed field lines of the core re-gion. The spatial discretization is based on a high-order hybrid DG schemeon unstructured meshes, which provides an arbitrary high-order accuracywhile reducing considerably the number of coupled degrees of freedom witha local condensation process. A discontinuity sensor is employed to identifycritical elements and regularize the solution with the introduction of artificialdiffusion. Based on a finite-element discretization, not constrained by a flux-aligned mesh, the code is able to describe plasma facing components of anycomplex shape using Bohm boundary conditions and to simulate the plasmain versatile magnetic equilibria, possibly extended up to the center. Nu-merical tests using a manufacturated solution show appropriate convergenceorders when varying independently the number of elements or the degree ofinterpolation. Validation is performed by benchmarking the code with thewell-referenced edge transport code SOLEDGE2D (Bufferandet al.2013,2015 [1, 2]) in the WEST geometry. Final numerical experiments show thecapacity of the code to deal with low-diffusion solutions.
Giorgio Giorgiani, Hugo Bufferand, Guido Ciraolo, Philippe Ghendrih, Frédéric Schwander, et al.. A hybrid discontinuous Galerkin method for tokamak edge plasma simulations in global realistic geometry. Journal of Computational Physics, Elsevier, 2018, 374, pp.515-532. ⟨10.1016/j.jcp.2018.07.028⟩. ⟨hal-02114246⟩
Matteo Valentinuzzi, Giorgio Giorgiani, Yannick Marandet, Hugo Bufferand, Guido Ciraolo, et al.. Fluid description of neutral particles in divertor regimes in WEST. Contributions to Plasma Physics, Wiley-VCH Verlag, 2018, 58 (6-8), pp.710-717. ⟨10.1002/ctpp.201700211⟩. ⟨hal-02116176⟩ Plus de détails...
A new neutral fluid code has been developed as a necessary step towards a hybrid fluid/kinetic neutral model, to be used in ITER or DEMO simulations, where part of the divertor will be very collisional for neutrals. The neutral fluid code, which is able to handle complex geometries in view of the coupling to Soledge2D, is tested on plasma backgrounds obtained by Soledge2D‐Eirene in WEST geometry, for different divertor regimes, and is found to be in fair agreement with the kinetic Monte Carlo solver Eirene. The differences are due to the simplifications introduced in the fluid model and to the fact that a fluid description is not fully valid in these cases.
Matteo Valentinuzzi, Giorgio Giorgiani, Yannick Marandet, Hugo Bufferand, Guido Ciraolo, et al.. Fluid description of neutral particles in divertor regimes in WEST. Contributions to Plasma Physics, Wiley-VCH Verlag, 2018, 58 (6-8), pp.710-717. ⟨10.1002/ctpp.201700211⟩. ⟨hal-02116176⟩
Giorgio Giorgiani, Hervé Guillard, Boniface Nkonga, Eric Serre. A stabilized Powell–Sabin finite-element method for the 2D Euler equations in supersonic regime. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2018, 340, pp.216-235. ⟨10.1016/j.cma.2018.05.032⟩. ⟨hal-01865708⟩ Plus de détails...
In this paper a Powell–Sabin finite-element (PS-FEM) scheme is presented for the solution of the 2D Euler equations in supersonic regime. The spatial discretization is based on PS splines, that are piecewise quadratic polynomials with a global continuity, defined on conforming triangulations. Some geometrical issues related to the practical construction of the PS elements are discussed, in particular, the generation of the control triangles and the imposition of the boundary conditions. A stabilized formulation is considered, and a novel shock-capturing technique in the context of continuous finite-elements is proposed to reduce oscillations around the discontinuity, and compared with the classical technique proposed by Tezduyar and Senga (2006). The code is verified using manufactured solutions and validated using two challenging numerical examples, which allows to evaluate the performance of the PS discretization in capturing the shocks.
Giorgio Giorgiani, Hervé Guillard, Boniface Nkonga, Eric Serre. A stabilized Powell–Sabin finite-element method for the 2D Euler equations in supersonic regime. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2018, 340, pp.216-235. ⟨10.1016/j.cma.2018.05.032⟩. ⟨hal-01865708⟩
Journal: Computer Methods in Applied Mechanics and Engineering
G. Ciraolo, H. Bufferand, J. Bucalossi, Ph. Ghendrih, P. Tamain, et al.. H-mode WEST tungsten divertor operation: deuterium and nitrogen seeded simulations with SOLEDGE2D-EIRENE. Nuclear Materials and Energy, Elsevier, 2017, 12, pp.187 - 192. ⟨10.1016/j.nme.2016.12.025⟩. ⟨hal-01702237⟩ Plus de détails...
Simulations of WEST H-mode divertor scenarios have been performed with SOLEDGE2D-EIRENE edge plasma transport code, both for pure deuterium and nitrogen seeded discharge. In the pure deuterium case, a target heat flux of 8 MW/m2 is reached, but misalignment between heat and the particle outflux yields 50 eV plasma temperature at the target plates. With nitrogen seeding, the heat and particle outflux are observed to be aligned so that lower plasma temperatures at the target plates are achieved together with the required high heat fluxes. This change in heat and particle outflux alignment is analysed with respect to the role of divertor geometry and the impact of vertical vs horizontal target plates on neutrals spreading.
G. Ciraolo, H. Bufferand, J. Bucalossi, Ph. Ghendrih, P. Tamain, et al.. H-mode WEST tungsten divertor operation: deuterium and nitrogen seeded simulations with SOLEDGE2D-EIRENE. Nuclear Materials and Energy, Elsevier, 2017, 12, pp.187 - 192. ⟨10.1016/j.nme.2016.12.025⟩. ⟨hal-01702237⟩
Giorgio Giorgiani, Hervé Guillard, Boniface Nkonga. A Powell-Sabin finite element scheme for partial differential equations . ESAIM: Proceedings, EDP Sciences, 2016, 53, pp.64-76. ⟨10.1051/proc/201653005⟩. ⟨hal-01377903⟩ Plus de détails...
In this paper are analyzed finite element methods based on Powell-Sabin splines, for the solution of partial differential equations in two dimensions. PS splines are piecewise quadratic polynomials defined on a triangulation of the domain, and exhibit a global C 1 continuity. Critical issues when dealing with PS splines, and described in this work, are the construction of the shape functions and the imposition of the boundary conditions. The PS finite element method is used at first to solve an elliptic problem describing plasma equilibrium in a tokamak. Finally, a transient convective problem is also considered, and a stabilized formulation is presented.
Giorgio Giorgiani, Hervé Guillard, Boniface Nkonga. A Powell-Sabin finite element scheme for partial differential equations . ESAIM: Proceedings, EDP Sciences, 2016, 53, pp.64-76. ⟨10.1051/proc/201653005⟩. ⟨hal-01377903⟩
H Guillard, M Bilanceri, C Colin, Philippe Ghendrih, G Giorgiani, et al.. Parallel Kelvin-Helmholtz instability in edge plasma. Journal of Physics: Conference Series, IOP Publishing, 2014, Joint Varenna-Lausanne International Workshop 2014, 561, pp.012009. ⟨10.1088/1742-6596/561/1/012009⟩. ⟨hal-01100365⟩ Plus de détails...
In the scrape-off layer (SOL) of tokamaks, the flow acceleration due to the presence of limiter or divertor plates rises the plasma velocity in a sonic regime. These high velocities imply the presence of a strong shear between the SOL and the core of the plasma that can possibly trigger some parallel shear flow instability. The existence of these instabilities, denoted as parallel Kelvin-Helmholtz instability in some works [1, 2] have been investigated theoretically in [3] using a minimal model of electrostatic turbulence composed of a mass density and parallel velocity equations. This work showed that the edge plasma around limiters might indeed be unstable to this type of parallel shear flow instabilities. In this work, we perform 3D simulations of the same simple mathematical model to validate an original finite volume numerical method aimed to the numerical study of edge plasma. This method combines the use of triangular unstructured meshes in the poloidal section and structured meshes in the toroidal direction and is particularly suited to the representation of the real complex geometry of the vacuum chamber of a tokamak. The numerical results confirm that in agreement with the theoretical expectations as well as with other numerical methods, the sheared flows in the SOL are subject to parallel Kelvin-Helmholtz instabilities. However, the growth rate of these instabilities is low and these computations require both a sufficient spatial resolution and a long simulation time. This makes the simulation of parallel Kelvin-Helmholtz instabilities a demanding benchmark.
H Guillard, M Bilanceri, C Colin, Philippe Ghendrih, G Giorgiani, et al.. Parallel Kelvin-Helmholtz instability in edge plasma. Journal of Physics: Conference Series, IOP Publishing, 2014, Joint Varenna-Lausanne International Workshop 2014, 561, pp.012009. ⟨10.1088/1742-6596/561/1/012009⟩. ⟨hal-01100365⟩
Giorgio Giorgiani, Sonia Fernández-Méndez, Antonio Huerta. Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier-Stokes equations ✩. Computers and Fluids, Elsevier, 2014. ⟨hal-01717504⟩ Plus de détails...
A degree adaptive Hybridizable Discontinuous Galerkin (HDG) method for the solution of the incompressible Navier-Stokes equations is presented. The key ingredient is an accurate and computationally inexpensive a posteriori error estimator based on the super-convergence properties of HDG. The error estimator drives the local modification of approximation degree in the elements and faces of the mesh, aimed at obtaining a uniform error distribution below a user-given tolerance in a given are of interest. Three 2D numerical examples are presented. High efficiency of the proposed error estimator is found, and an important reduction of the computational effort is shown with respect to non-adaptive computations, both for steady state and transient simulations.
Giorgio Giorgiani, Sonia Fernández-Méndez, Antonio Huerta. Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier-Stokes equations ✩. Computers and Fluids, Elsevier, 2014. ⟨hal-01717504⟩
Giorgio Giorgiani, David Modesto, Sonia Fernández-Méndez, Antonio Huerta. High-order continuous and discontinuous Galerkin methods for wave problems. International Journal for Numerical Methods in Fluids, Wiley, 2013, 73(10), pp.883-903. ⟨10.1002/fld.3828⟩. ⟨hal-01717513⟩ Plus de détails...
Three Galerkin methods —continuous Galerkin (CG), Compact Discontinuous Galerkin (CDG) and Hybridizable Discontinuous Galerkin (HDG)— are compared in terms of performance and computational efficiency in two-dimensional scattering problems for low and high-order approximations. The total number of degrees of freedom and the total runtime are used for this correlation as well as the corresponding precision. The comparison is carried out through various numerical examples. The superior performance of high-order elements is shown. At the same time, similar capabilities are shown for CG and HDG, when high-order elements are adopted, both of them clearly outperforming CDG.
Giorgio Giorgiani, David Modesto, Sonia Fernández-Méndez, Antonio Huerta. High-order continuous and discontinuous Galerkin methods for wave problems. International Journal for Numerical Methods in Fluids, Wiley, 2013, 73(10), pp.883-903. ⟨10.1002/fld.3828⟩. ⟨hal-01717513⟩
Journal: International Journal for Numerical Methods in Fluids
