Scalable Poisson solver for gyrokinetic simulations

Transcription

1 Scalable Poion olver for gyrokinetic imulation V. Grandgirard1, G. Latu1, N. Croueille2, A. Ratnani3,E. Sonnendru cker3 1 3 Virginie G CEA, IRFM, Cadarache, France IPP Garching, Germany N Groupe Calcul N 2 4 IRMAR Renne, France Nancy univ., France 26th January

2 Magnetic Confinement Turbulence govern Fuion plama performance In a plama, particle follow magnetic field line: Particle may be confined in a toroidal magnetic field In the un, plama i confined by gravity In a tokamak, plama i confined by a magnetic field magnetic toroidal geometry (r, θ, ϕ Scaling law in tokamak: plama volume τe cte with τe = energy confinement time meaure of thermal inulation. à Two main poibilitie to increae tokamak performance: increae the ize of the machine or/and increae τe Turbulence govern τe à Generate lo of heat and particle à & Confinement propertie of the magnetic configuration Undertanding, predicting and controlling turbulence for optimizing experiment like ITER and future reactor i a ubject of utmot importance. Virginie G N Groupe Calcul N 26th January

3 Gyrokinetic theory: large phae pace reduction 6D to 5D Virginie GRANDGIRARD Groupe Calcul 26th January Kinetic theory: 6D ditribution function of particle (3D in pace and 3D in velocity F (r, θ, ϕ, v, v, α Fuion plama turbulence i low frequency: ω turb ω ci Phae pace reduction: fat gyro-motion i averaged out Adiabatic invariant: magnetic moment µ = m v 2 /(2B Velocity drift of guiding center Tranvere drift Large reduction memory/cpu time Complexity of the ytem Tranvere & parallel dynamic: (with B v G// v G Gyrokinetic Projection theory: on the 5D tranvere ditribution plane ( function of guiding-center : F (r, θ, ϕ, v G, µ where µ parameter

4 Virginie GRANDGIRARD Groupe Calcul 26th January Gyrokinetic code require tate-of-the-art HPC Gyrokinetic code require tate-of-the-art HPC technique and mut run efficiently on everal thouand proceor. non-linear 5D imulation multi-cale problem in pace and time time: t γ t imul few τ E 10 pace: ρi machine ize a ρ ρ i a 1 GK code are extremely CPU time conuming (ρiter 10 3 GK code development i an highly international competitive activity US: 8 code - EU: 5 code - Japan: 2 code European collaboration Eurofuion project ( ? Validation and verification of european GK code - GYSELA (France - GENE (Germany - ORB5 (Switzerland

5 Virginie GRANDGIRARD Groupe Calcul 26th January Particle Guiding-center Complex interplay between field and particle Gyrokinetic complexity due to the fact the Poion i olved with the charge denity of particle and the Vlaov equation decribe the guiding-center evolution. Particle Guiding-center Integral in phae pace _ J F (r,,,v //, gyroaverage _ F (r,,,v //, r.h. for Poion equation _ (r,, J F r v d 3 v gyroaverage Quai-neutrality, j E, B J (r,, Kinetic plama repone derivative gyrokinetic Boltzmann 5D equation

6 WEAK caling: (on JUQUEEN - Juelich Relative efficiency of 91% on core Virginie GRANDGIRARD Groupe Calcul 26th January Hybrid MPI/OpenMP parallelim Relative efficiency of 91% on core (performed on the totality of JUQUEEN/Blue Gene machine (Juelich Execution time, one Gyela (Weak Scaling - Juqueen Vlaov olver Field olver Derivative computation Diagnotic Total for one run Nb. of Kcore (x 1000 Relative efficiency, one run (Weak caling - Juqueen Vlaov olver Field olver Derivative computation Diagnotic Total for one run Nb. of Kcore (x 1000 Poion olver 15% of the total time Efficiency of Poion olver 41% Work till under progre.

7 Numerical method for 3D quai-neutrality equation olving Virginie GRANDGIRARD Groupe Calcul 26th January Solving the 3D quai-neutrality equation i equivalent to: Find φ(r, θ, ϕ uch that: 1 ( n,eq Z φ + e ( φ φ = n e0 BΩ T e,eq FS 1 Z n e0 } {{ }} {{ } polarization term adiabatic electron due to between guiding-center and particle J 0 ( F F,eq d 3 v Numerical method: Fourier projection in periodic direction θ and ϕ Finite difference in radial direction

8 Numerical method for 3D quai-neutrality equation olving Virginie GRANDGIRARD Groupe Calcul 26th January Solving the 3D quai-neutrality equation i equivalent to: Find φ(r, θ, ϕ uch that: 1 ( n,eq Z φ + e ( φ φ = n e0 BΩ T e,eq FS 1 Z n e0 } {{ }} {{ } polarization term adiabatic electron due to between guiding-center and particle J 0 ( F F,eq d 3 v Difficultie: R.H.S = integral over the velocity pace Parallel communication ++

9 Numerical method for 3D quai-neutrality equation olving Virginie GRANDGIRARD Groupe Calcul 26th January Solving the 3D quai-neutrality equation i equivalent to: Find φ(r, θ, ϕ uch that: 1 ( n,eq Z φ + e ( φ φ = n e0 BΩ T e,eq FS 1 Z n e0 } {{ }} {{ } polarization term adiabatic electron due to between guiding-center and particle J 0 ( F F,eq d 3 v Difficultie: R.H.S = integral over the velocity pace Parallel communication ++ φ FS = φj x dθdϕ/ J x dθdϕ Pb in Fourier due to coupling between θ and ϕ flux urface average of φ

10 Virginie GRANDGIRARD Groupe Calcul 26th January Quai-neutrality: Treatment of φ FS Quai-neutrality Poion equation form: Lφ(r, θ, ϕ + α(r ( φ(r, θ, ϕ φ FS (r = ρ(r, θ, ϕ (1 with L = 1 ( n,eq Z φ and FS = J x dθdϕ/ n e0 BΩ J x dθdϕ Compute ρ(r, θ, ϕ and ρ θ,ϕ (r with θ, ϕ (r = dθdϕ/l θ L ϕ Solve for all ϕ, olve with Fourier Projection in θ and Finite difference in r Compute Φ FS (L + α(r Φ = ρ ρ θ, ϕ with Φ = φ φ θ, ϕ (2 Solve the 1D radial ytem with Finite difference of econd order Finally, φ i recontructed a : φ = Φ }{{} L φ θ, ϕ + α(r ( φ θ, ϕ φ FS = ρ θ, ϕ (3 olution of (2 Φ FS } {{ } compute from Φ + φ FS }{{} olution of (3

11 More realitic magnetic configuration for GYSELA(1/2 Virginie GRANDGIRARD Groupe Calcul 26th January

12 More realitic magnetic configuration for GYSELA(2/2 Virginie GRANDGIRARD Groupe Calcul 26th January

13 Collaboration: ANR GYPSI ( , Strabourg, Nancy, Mareille ANR Nufue ( , France, Germany, Japan, US, UK ADT INRIA Selalib ( , Strabourg, Bordeaux Action - IPL INRIA (march , Nice, Bordeaux New project following AEN INRIA Fuion (evaluation in progre, Strabourg, Lyon, Nice Collaboration with IPP Garching (Germany ince 2012 Collaboration with Maion de la Simulation - Saclay (Pari ince 2012 Virginie G N Groupe Calcul N 26th January