aaaadaptive mhdsimulation of tokamak elms for...

ASTER meeting 12-14/11/2006

AAAAdaptive mhd Simulation

of Tokamak Elms for iteR

Projet ANR-CIS 2006

Meeting Bordeaux

12-14/11/2007


WP 2.2 Bezier Grid

• Development of the JOREK MHD simulation code

– Implementation of Bezier finite elements in new JOREK(2) code:

•Polar flux surface aligned grid

•X-point/separatrix aligned grid, open/closed field lines

•Report available on ASTER website


WP 2.2 Bezier Grid

• Bezier grid aligned on open/closed flux surfaces:

– Continuity of values enforced at axis and X-point

– Partial continuity enforced of gradients at X-point


• Verification on two resistive MHD instabilities:

– Linear growth rate n=1 resistive internal kink mode :

•Correct scaling (after correction of an error in grid generation)

WP 2.2 : Test Cases

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

h

error

h5

710η −= 8

10ν −=


WP 2.2 : Test Cases

• Linear growth rate n=1 tearing mode test case:

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

error5

h

h

710η −= 8

10ν −=


WP 2.2 : time evolution

• Implementation of new time evolution scheme in JOREK:

– Fully implicit but using iterative solution (GMRES)

•Matrix for each toroidal harmonic used as preconditioner

– Solved using PastiX or MUMPS

(using N MPI groups, new option in PastiX)

– Preconditioning only recalculated when GMRES requires too many

iterations

– Reduce time step when number of iterations still too large

•Much improved scaling parallelisation/memory use

– First tests on CCRT platine (up to 512 cores)

– Good scaling when (n_tor / n_cpu) constant

•To be compared with PastiX iterative/direct solution

– Partly explicit scheme:

•Each toroidal harmonic implicit (ignoring coupling between harmonics)

•Coupling terms explicit

•Scheme as in SFELES code


Parallelisation

Matrix: 49

Distr: 5.2

Fact: 46

GMRES:3.3/it

Matrix: 89

Distr: 6.9

Fact: 110

GMRES::-

51x64x16

Matrix: 34

Distr: 3.4

Fact: 17

GMRES: 1.8/it

Matrix: 43

Distr: 3.9

Fact: 24

GMRES: 2.6/it

Matrix: 88

Distr: 5.2

Fact: 46

GMRES:-

N(radial) 51

N(poloidal) 32

N(Fourier) 16

256128N_cpu=64

Largest matrix size : ndof = 2.4x106 (on 256 cpu’s)


GMRES

• GMRES number of iterations for two values of time step

– Larger time step not always the best choice

160

140

120

100

80

60

40

20

GM

RE

S i

tera

tio

ns

5004003002001000 no. timestep

t_step =10

t_step=20

residu 10-8

η = υ=10-5


GMRES

• Reduce time step when number of GMRES iterations too big.

250

200

150

100

50

0

ite

ratio

ns

8006004002000

no. steps

50

40

30

20

10

0

time

ste

p

η = υ = 10-6


WP 4

• Application of the methods developed in the project to

advance the simulation of ELMs in ITER plasmas

– Stanislas Pamela started thesis 1-10-2007

– First application using new jorek version:

• equilibrium flow in x-point plasmas

600

400

200

0

x1

0-6

4.03.53.02.52.0

R [m]

flow profile


Planning 2008

• JOREK2 developments (WP2):

– Include PastiX with distributed matrix, test direct/indirect version

– Change MHD model to full MHD, diamagnetic effects

– Include refinement in new JOREK version

– Post-doc Emiel van der Plas starts 1-2-2008

• JOREK applications (WP4)

– Equilibrium flow patterns due to X-point configuration

– Extend ELM simulations to higher toroidal resolution

•Saturation mechanism

•Crash amplitude


JOREK2

• New version of JOREK uses (isoparametric) Bezier finite

elements in the poloidal plane and Fourier harmonics in the toroidal (periodic direction)

– Leads to ‘sparse’ matrix with very large full blocks

•Blocksize (4 x N_tor(=16-64) x (n_var=6-8))

( )4

1 1 1

( , , ) Re( , )Ntor Nelm

in

ijn j

n i j

U s t u H s t eφφ

= = =

= ∑∑∑�

( )

( )

4

1

4

1

( , ) ,

( , ) ,

i ij j

j

i ij j

j

R s t R H s t

Z s t Z H s t

=

=

=

=

∑

∑


JOREK2

• Initialisation (serial):

Construct finite

element grid

Solve equilibrium equation

(~poisson equation, MUMPS)

Construct finite element

grid aligned on magnetic

field structure

Determine matrix structure

(CSR matrix setup)

Distribute ndof’s over all

available cpu’s

grid_bezier_square_polargrid_bezier_squaregrid_polar_bezier

equilibrium

grid_xpointgrid_flux_surface

distribute_nodes_elements (parallel)

global_matrix_structure (parallel)


JOREK2• Implicit time stepping (parallel):

Construct matrix

(distributed nodes/elements)

Redistribute reduced matrices

over N_tor masters

(distributed harmonics)

Analysis/Factorisation

Reduced matrices

GMRES

precond. (reduced matrix solve)

full matrix-vector product

Add solution

it_gmres<20it_gmres>20

it_gmres>200

tstep=tstep/2

Residue>10-8

construct_matrixelement_matrixelement_matrix_fft

distribute_harmonicssolve_matrix_n

gmres_drivergmres_preconditiongmres_matrix_vector

update_valuesupdate_deltas


JOREK2

• Direct with MUMPS or PastiX (not yet completed in JOREK2)

Construct matrix

(distributed nodes/elements)

Gather/scather full matrix

Add solution

PastiX:Analysis

PastiX:Factorisation

PastiX:Solve

It>1 It=1 or grid changed

construct_matrix

solve_mumps_allsolve_pastix_all

Update_valuesupdate_deltas


JOREK2type type_node ! type definition of a node

real*8 :: x(n_order+1,n_dim) ! x,y,z coordinates of points and

real*8 :: values(n_tor,n_order+1,n_var) ! variables

real*8 :: deltas(n_tor,n_order+1,n_var)

integer :: index(n_order+1) ! the index in the main matrix

integer :: boundary ! = 1, 2 or 3 for boundary nodes

endtype type_node ! x(:,1) : position, x(:,2) : vector u, ! x(:,3) : vector v, x(4) : vector w

type type_node_list ! type definition of a list of nodes

integer :: n_nodes ! the number of nodes in the list

integer :: n_dof ! the total number of degrees of freedom

type (type_node) :: node(n_nodes_max) ! an allocatable list of nodes

endtype type_node_list

type type_element ! type definition for one elements

integer :: vertex(n_vertex_max) ! the nodes of the corners

integer :: neighbours(n_vertex_max) ! the neighbouring elements

real*8 :: size(n_vertex_max,n_order+1) ! the size of the vectors at each vertex

endtype type_element

type type_element_list ! type definition for a list of elements

integer :: n_elements ! the number of elements in the list

type (type_element) :: element(n_elements_max) ! the list of elements

endtype type_element_list


JOREK2module parameters

parameter (n_var = 6) ! the number of variables

parameter (n_dim = 2) ! the number of dimensions

parameter (n_order = 3) ! order of the polynomial basis

parameter (n_tor = 3) ! the number of toroidal harmonics

parameter (n_period = 1) ! periodicity in toroidal direction

parameter (n_plane = 4) ! the number of toroidal angles

parameter (n_vertex_max = 4) ! the maximum number of corners of an element

parameter (n_nodes_max = 10001) ! the maximum number of nodes

parameter (n_elements_max = 10001) ! the maximum number of elements

parameter (n_pieces_max = 10001) ! max. no. of line pieces describing a flux surface

parameter (n_degrees = n_order+1) ! degrees of freedom per variable per node

endmodule


JOREK2 input&in1

restart = .f. ! start t=0 (.f.) or continue old run (.t.) using file .rst

tstep = 1000.d0 ! Time step (in Alfven times

nstep = 1 ! Number of timesteps

ellip = 1.d0 ! Shape of plasma boundary

tria_u = 0.d0

tria_l = 0.d0

quad_u = -0.d0

quad_l = -0.d0

xampl = +0.d0 ! Parameters to create x-point plasma (xampl=0 no x-point)

xsig = 0.5d0

xwidth = 1.d0

xtheta = 4.7d0

xshift = 0.0d0

xpoint = .f.

R_geo = 3.0d0 ! Major radius of plasma [m]

F0 = 3.0d0 ! Major radius times magnetic field (R_geo * B_vac)

amin = 1.d0 ! Minor radius of plasma

n_R = 0 ! Number of horizontal points in square grid

n_Z = 0 ! vertical

n_radial = 51 ! Number of radial points in polar grid

n_pol = 32 ! polar


JOREK2 inputn_flux = 0 ! Number radial points in flux surface grid

n_tht = 16 ! Number of angular points

n_open = 0 ! Number of radial points on open field lines

n_leg = 0 ! Number of points on the legs of the x-point

n_private = 0 ! Number of points in the private region (below x-point)

eta = 1.d-6 ! resistivity

visco = 1.d-6 ! viscosity

D_perp = 1.d-6 ! density diffusion

ZK_par = 1.d0 ! Temperature parallel diffusion

ZK_perp = 1.d-6 ! Perpendicular diffusion coefficient

FF_0 = 1.12d0 ! Parameters for current profile

FF_1 = 0.d0

aF = -1.d0

heatsource = 0.d-4 ! Heating power

particlesource = 0.d-6 ! Particle source

T_0 = 1.d-3 ! Temperature on axis

T_1 = 1.d-4 ! Temperature at the boundary

aT = -2.0d0 ! Profile coefficients

bT = +1.0d0

cT = 0.d0

dT = 9999.

eT = 9999.

r_0 = 1.d0 ! Density on axis

r_1 = 1.d0 ! Density at the boundary

&end


Equilibrium

• Static equilibrium:

• Force balance between pressure gradient and the Lorentz force:

• Poloidal flux ψ:

• Grad-Shafranov equation describes axisymmetric equilibrium:

0

1p

µ∇ = ×J B

0t

∂=

∂0=v

0∇ • =B

( )F 1= +

R Rϕ ϕ

ψψ∇ ×B e e

( ) ( ) ( )2 2

02

1' 'R RJ R p F F

Rϕψ µ ψ ψ ψ

∇ ∇ = − = − −

0ϕ

∂=

∂

JET equilibrium


Reduced MHD model

•Reduced MHD model in toroidal geometry

– fixed large toroidal magnetic field

– removes fast waves, easier on the numerics

– similar to reduction Navier-Stokes to potential flow

– Good for physics studies, not for detailed comparison with experiment

[ ]

[ ] [ ]

[ ] ( )

[ ] ( )

*

2

2

*

(1 ) ,

12 (1 ) , ,

(1 ) (1 )

(1 ) , 2

(1 ) , 2

;

T

ux u

t

w u Jw x w u J w

t y x x

ux u D S

t y

T ux T u T T T S

t y

J w u

ρ

ψε ψ η ψ ε

ϕ

εε ε ψ ν

ε ε ϕ

ρε ρ ερ ρ

ρ ε ρ ερ

ψ

⊥

⊥ ⊥

⊥ ⊥

⊥

∂ ∂= + + ∆ −

∂ ∂

∂ ∂ ∂= + + + − + ∇

∂ ∂ + + ∂

∂ ∂= + + + ∇ ⋅ ∇ +

∂ ∂

∂ ∂= + + + ∇ ⋅ Κ ∇ + Κ ∇ +

∂ ∂

= ∆ = ∇ ⋅∇

� �

0 0

0

R RB B e e

R Rϕ ϕψ= + ∇ ×

� ��

0 0

Rv u e

R Bϕ

−= ∇ ×

��

Magnetic flux:

Vorticity:

Density:

Temperature:

dimensionless form : ( )0 0( ) / ; / ; /x R R a y Z a a Rε= − = =

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