Inria Project Lab FRATRES

Inria Project LabFRATRES

Fusion ReAcTors REsearchand Simulations

Herve Guillard September 21 2017

IPL FRATRES

I Creation 01/01/2015 (11/05/2015)I Partners

I Inria projectsI Castor : Control, Analysis and Simulations for TOkamak Research, SophiaI Tonus : TOkamaks and NUmerical Simulations, NancyI Ipso : Invariant Preserving SOlvers, Rennes

I External partnersI IRFM-Cea : Institut de Recherche sur la Fusion MagnetiqueI IPP : Garching - Max Planck InstituteI LJLL : Laboratoire Jacques Louis Lions, Paris 6I IMT : Institut de Mathematique de Toulouse

Herve Guillard - IPL-FRATRES September 21 2017 - 2

Energy and Fusion : the challenge of the XXI century

2015 Key worldEnergy statisticsreportInternational Energy Agency

Definite need foralternative energy sources


ITER : next step in Fusion research

Construction 2008 → 2019Cost : 19-20 B eWorld’s largest science experi-ment

I Fusion Power = 500MWI Power gain > 10I Temperature 100 M oKI International Consortium

I InternationalI AIEAI CERNI Principaute de Monaco


ITER Goals

I Demonstrate the possibility of controlled nuclear fusionI produce a plasma dominated by α-particle heatingI produce a significant fusion power amplification factor (Q ≥

10) in long-pulse operation (300 – 500 s)I aim to achieve steady-state operation of a tokamak (Q ≥ 5, ≤

3000 s)I retain the possibility of exploring ‘controlled ignition’ (Q ≥ 30)

I Technological purposesI demonstrate integrated operation of technologies for a fusion

power plantI test components required for a fusion power plantI test concepts for a tritium breeding module


Current status of the ITER program

I JET Joint European Torus, construction : 1979, 1rst fusionplasma 1991

I ITER Conceptual design : 1988, 1999,I construction begins : 2008,I Start of tokamak assembly : 2015I Tokamak assembly completion : 2019I First plasma : 2025I ITER Divertor : 2029I Start of D – T operation : 2035

I DEMO Conceptual design : 2017, Engineering design : 2024,

Long time perspective

Conception of ITER dates back to 1990’


ITER design

Numerical technologies of the 90’I based on Engineer’s techniquesI dimensionless scaling laws, with operational or global

parametersI Tokamaks Simulators (Integrated modeling codes)

I Suite of 0-D, 1-D (2-D) codesI Simplifying assumtionsI fitting of experimental data coming from existing tokamaks


IPL FRATRES Objectives

To help prepare a next generation of tokamak simulators based onfirst principle simulationsI Support of existing software : directly organized to support

the development of existing target codes used in the fusioncommunity

I Gysela : Gyrokinetic code for turbulent transportI Jorek : Fluid code for MHD stabilityI Cedres++ : Equilibrium – transport code for long time

evolution, control and optimisation of plasma dischargesI Development of new numerical algorithms, models and

mathematical results


Tokamak components


Numerical simulations of tokamak operationsI Core plasma : kinetic models

I Plasma and its close environment :MHD models

I The whole machine :Equilibrium-transport models


Core plasma simulations

Why numerical simulationsare needed ?

In a perfect world :Large magnetic fields causecharged particle to spiralaround field lines.

I Toroidal shape to ensurethat no losses at the end

I Only charged particles(D+,T +, α) are confinedNeutrons escape andrelease energy.


Core plasma : In the real world : Anomalous transportLoss of confinement

I Charged particles (D+,T +, α) also escape : plasma particlesare lost to vessel wall

I Transport exceeds theoretical expectations (Neo-classicaltheory) by an order of magnitude !

I micro-turbulence in the plasma is the main cause of this lossof confinement.


Simulation of plasma micro-turbulenceCore plasma are weakly collisional : ν∗ = qR/λmfp ∼ 10−3 thus :Kinetic description is required

I Particles description : Boltzmann-Vlasov equation fs(t, x, v) 6D pb !

∂fs

∂t +∇x (vfs) +∇v (es

ms[E + v× B]fs) = Cs =

∑Css′

I Simplified description of the fields, Constant magnetic field B = C ,Electrostatic Electric field E = −∇φ

−ε∆φ = e(

∫fid3v −

∫fed3v)


A gyro-kinetic code : Gysela

Target code for the IPL : Gysela :http://gyseladoc.gforge.inria.fr/ productioncode developped since 2006 and maintained by CEA for use on large scaleparallel machines

I Non-linear 5D simulationI Large range of time and space scales

I time ∆t ∼ 10−6 → tsim ∼ some τE ∼ 10sI space ρi → machine size a : ρ∗ = ρi/a << 1

I ITER ρ∗ = 1/512I Nb grid points ∼ ρ−3

∗

I typical ITER run : 300 B pointsI State of the art HPC techniques : IPL

C2S@Exa


IPL activities on Kinetic and Gyro-kinetic modelsI Improvement of Gysela : Gyro-average operator

I Pade approximation → InterpolationI Common work IPP-TonusI Implementation by CEA in Gysela

I New parallelisation scheme : interleaving of communicationand computations

I Improvement of Gysela : Realistic geometriesI Development of new numerical algorithms

I Oblique semi-lagrangian interpolation TONUS-CEA

I Development of multi-scale methodsI Common work IPSO-Tonus (IPL post-doc of Xiaofei Zhao)


Plasma interaction with its close environment

I More global description ofthe plasma

I∂B∂t mandatory

I (take into account the walland external coils )

I needs faster runs to performparametric studies withparameter scan

I ⇒ Magnetohydrodynamic(MHD) models


MHD model : Plasma instabilities

Very large number of possible instabilitiesI to help identify possible instabilitiesI to determine the stability domain constraining the operational range

of the design parameters

stability domain ofthe safety factorq = aBT

R0BPto avoid

the kink instability :


Global description of plasmas: MHD modelsuse the velocity moment of the distribution function.

∫ 1v

|v|2/2

× [∂fs∂t

+∇x (vfs ) +∇v (Ffs )− Cs ]dv = 0

∂ρ

∂t+∇ · (ρu) = Sρ ρ[

∂u∂t

+ (u · ∇)u] +∇p = J× B−∇ · π

∂Ti

∂t+ (u · ∇)Ti + Γi Ti∇ · u = −

Γi

ni(∇ · qi + πi : ∇ui −Qi )

∂Te

∂t+ (u · ∇)Te + ΓeTe∇ · u = −

Γe

ne(∇ · qe + πe : ∇ue −Qe )

∂B∂t

+∇× E = 0 µ0J = ∇× B

Less accurate description than kinetic modellingbut 6D → 3DStill a computationally intensive system !Physicists have introduced simplified models → reduced MHDmodels


Global description of plasmas : Reduced MHD models

I Physical basis : Bφ >> B⊥I 2D incompressible MHD in the transverse direction

I Alfven waves propagating in the direction of the dominantmagnetic field.

I Simplest example of reduced MHD model :

∂

∂τv⊥ + (v⊥ · ∇⊥)v⊥ − (B⊥ · ∇⊥)B⊥ +∇⊥π − ∂zB⊥

∂

∂τB⊥ + (v⊥ · ∇⊥)B⊥ − (B⊥ · ∇⊥)v⊥ − ∂zv⊥ = 0

∇⊥ · v⊥ = ∇⊥ · B⊥ = 0


A reduced MHD code : Jorek 303 model

Target code for the IPL : https://www.jorek.euI Use a reduced MHD model assuming

u⊥ = R2∇U ×∇φ,B = F∇φ +∇ψ ×∇φ

I Numerical scheme

I Flux-aligned 2D Bezier finite elements

I Toroidal Fourier expansion

I Fully implicit time stepping

I GMRES and Newton iterations

I Taylor-Galerkin stabilization

I MPI + OpenMP parallelization

Reference code for the European Fusion MHD community (Jorek isin use in 11 European institutions and labs (June 2017))


IPL activities on MHD models

I Improvements on the Jorek codeI Implicit physical pre-conditioningI Taylor-StabilizationI Discretization of the full MHD model

I Theoretical and numerical works on MHD modelsI Convergence full MHD → reduced MHD using singular

limit of hyperbolic PDEI High-order C1 triangular elements (Powell-Sabin,

Clough-Tocher)I Lattice Boltzmann type discretisation of the MHD equations


Taylor-Galerkin Stabilization

∂W∂t + L(∂W ,W ) = 0 ∈ Ω

Galerkin approximation with stabilization :Find Wh ∈ finite dimension Space, st, ∀W ∗

h∫Ω

(∂W∂t ,W ∗

h )dx +

∫Ω

(L(∂W ,W ),W ∗h ) == −

∫Ω

τ(A∂Wh).(A∂W ∗h )dx


Simulation of the whole machine : Equilibrium andtransport models

I Long time behavior of thesystem ⇒

I ε(ρ[∂u∂t

+ (u · ∇)u]−∇ · π) +∇p = J× B

I sequence of evolvingequilibria between Lorentzforce and plasma pressure

I Have to consider thevariation of the current inthe external coils

I Fast runs (Near real time)


Equilibrium and transport modelsShaping the plasma : H. Heumann, CASTOR

Transient optimal control problem :

Find evolution of voltages u(t) in coils that ensurea desired evolution of the shape of the boundary of

the plasma y(t).

Physical modeling : EQ(y, u)

Magnetic field Ansatz : B = F (ψ)∇φ+∇ψ ×∇φForce balance ∇p = J×B and toroidal symmetry yield

−∇(

1µr∇ψ

)=Jtoro =

Jplasma(x , ψ)) in plasmaJcoil(voltage, ∂tψ)) in coilsvanishing elsewhere


Equilibrium and transport modelsShaping the plasma : H. Heumann, CASTOR

Transient optimal control problem :

Find evolution of voltages u(t) in coils that ensurea desired evolution of the shape of the boundary of

the plasma y(t).

Physical modeling : EQ(y, u)

Magnetic field Ansatz : B = F (ψ)∇φ+∇ψ ×∇φForce balance ∇p = J×B and toroidal symmetry yield

−∇(

1µr∇ψ

)=Jtoro =

Jplasma(x , ψ)) in plasmaJcoil(voltage, ∂tψ)) in coilsvanishing elsewhere


IPL activity on Equilibrium-Transport and control

I Cedres++I Coupling with the European transport code ETSI Coupling with a position and shape controller

I FEEQS.M : Matlab implementation of Cedres++I Easy to use interface for engineering applicationsI FEEQS.M used in Cadarache for operating WEST


FRATRES & European programs on Fusion

FP7 & H2020 Projects : EUROfusion Grant agreementnumber 633053 Enabling Research program

I Enabling Research Project ER15-IPP05 (1/2015-12/2018)”Global non-linear MHD modeling in toroidal geometry ofdisruptions, edge localized modes, and techniques for theirmitigation and suppression” (PI : M. Hoelzl, IPP, Garching).

I CfP-WP14-ER-01 Synergetic experimental numericalapproach to fundamental aspects of turbulent transport in thetokamak edge (PI: P. Ricci, EPFL ).

I Enabling Research Project ER15-IPP01 (1/2015-12/2017)”Verification and development of new algorithms forgyrokinetic codes” (PI : E. Sonnendrucker, IPP, Garching).


FRATRES & European programs on Fusion

EUROfusion WPCD (Working Package Code Development)

I ACT1: Extended equilibrium and stability chain (participation)I ACT2: Free boundary equilibrium and control (participation

and coordination)

EINFRA-2014-2015 : Energy Oriented Center of Excellence(EoCoE)

I Responsable WP5.1 : Flux surface aligned grids


Animation activities

2015I June 12 - Minisymposium au congres SMAI Problemes numeriques de la

fusion controleeI July - August - Cemracs - CIRM Luminy 1 projet, CASTOR, TONUS,

IPP, IRFM-CEAI 12-14/10, Hot Plasmas II workshop BordeauxI Annual Meeting, october 15-16, Sophia Antipolis

2016I May 9 -13 - Minisymposium au CANUM Methodes numeriques pour les

equations de la MHDI April 12th to 15th, Jorek users meeting, Sophia AntipolisI July - August - Cemracs - CIRM Luminy, 3 projets, CASTOR, TONUS,

IPP, IRFM-CEAI Annual Meeting, November 16-18, Strasbourg


Animation activities

2017I Workshop ”Understanting disruptions”, Inria Sophia Antipolis, June 20

2017 Organizer : Holger Heumann,I Ecole CEA-EDF-INRIA, July, 3-7 2017, Paris Waves and Fusion Plasmas,

Organizers : Eric Sonnendrucker (IPP-Garching), Martin Campos-Pinto(CNRS-LJLL-UPMC), Lise-Marie Imbert-Gerard (Courant Institute-NewYork), Bruno Despres (LJLL-UPMC)

I Festival de Theorie 2017 : 2 projects (Castor–Tonus)I IPL Annual Meeting, November 27-28, Inria, Rennes-Bretagne Atlantique

Organizer : Nicolas Crouseilles



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