An Introduction to the NIMROD Fusion Magnetohydrodynamics Simulation

Project

Prof. Carl Sovinec

Department of Engineering PhysicsUniversity of Wisconsin-Madison

presented at

Argonne National Laboratory

November 21, 2002

CEMMCEMM

Goals for NIMROD(Non-Ideal Magnetohydrodynamics with Rotation, an Open Discussion Project)

Develop a simulation code package for studying three-dimensional, nonlinear electromagnetic activity in laboratory fusion experiments.

Allow flexibility in the geometry and physics models used in simulations.

Allow efficient computation on a wide range of platforms from PCs to massively parallel supercomputers.

Provide user-friendly features, such as a graphical interface and documentation, and make the code publicly available. [http://nimrodteam.org]

Apply techniques such as integrated product development and quality function deployment in design and development.

The project has been a multi-institutional effort since 1996.

Curt Bolton, OFESDan Barnes, LANLDylan Brennan, GAJames Callen, Univ. of WIMing Chu, GATom Gianakon, LANLAlan Glasser, LANLChris Hegna, Univ. of WIEric Held+students, Utah StateCharlson Kim, CU-BoulderMichael Kissick, Univ. of WI

Scott Kruger, SAIC-San DiegoJean-Noel Leboeuf, UCLARick Nebel, LANLScott Parker, CU-BoulderSteve Plimpton, SNLNina Popova, MSUDalton Schnack, SAIC-San DiegoCarl Sovinec+students, Univ. of WIAlfonso Tarditi, NASA-JSC

Presently, there is non-team-member use of the NIMROD code at LLNL, IFS, Univ. of WA, UCLA, and AIST-Japan.

Characteristics of Magnetically Confined

Fusion Plasmas • Reactor-grade conditions require a mix of deuterium and tritium at nE1020 m-3s and T10 KeV. Both species are fully ionized at these conditions.

• The Lorentz force, qVB, confines the perpendicular motion of charged particles in a magnetic field.

• A toroidal container can have lines of B completely enclosed, but the field must be twisted in order to avoid rapid perpendicular particle drifts.

Ideally, Magnetic Configurations Consist of Nested Toroidal Flux Surfaces

• Each charged particle will tend to remain confined on a magnetic surface.

• Surfaces provide insulation for hot and dense conditions in the center.

Equilibrium Flux Surfaces and Pressure from the General Atomics DIII-D Tokamak

Pressure and current density gradients can drive asymmetric collective modes unstable, changing the magnetic topology.

Puncture-plots show simulated magnetic topology changing gradually over ~300 wave transit times around the DIII-D tokamak. Initial conditions are taken from experimental measurements.

Integrated modeling is the new horizon. Simulations of the Pegasus tokamak at the Univ. of WI are suggestive of what is possible.

• Plasma current and separatrix evolve self-consistently with applied loop voltage and vertical-field ramp.

• Transport has a strong influence on dynamics.

• High-order spatial accuracy is essential for distinguishing closed flux and open flux through modeled transport effects.

This study integrates MHD and transport effects with realistic geometry and experimental parameters.

plasma current

loop volts

Cur

rent

(kA

)

EM

F (V

)

Time (ms)

• Study has focused on 2D evolution, but 3D tearing-mode simulation is a straightforward extension for NIMROD.• Results emphasize interaction between MHD, transport effects, and overall performance.

Pegasus data courtesy of A. Sontag. Axisymmetric simulation results.Time (s)

Cur

rent

(A)

EM

F(V

)

0 0.01 0.020.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

1.2E+05

1.4E+05

1.6E+05

0

1

2

3

4

5

6

plasma current

loop volts

BEB

divbt

JBVE

BJ 0

nDnt

n

V

VBJVVV

p

t

QTnpTt

Tn

IbbVV

ˆˆ1 ||

• Density and magnetic-divergence diffusion are for numerical purposes.

Physical models for these macroscopic dynamics are based on fluid-like magnetohydrodynamic (MHD) descriptions.

• Conditions of interest possess two properties that pose great challenges to numerical approaches—anisotropy and stiffness.

• Anisotropy produces subtle balances of large forces, nearly singular behavior at rational surfaces, and vastly different parallel and perpendicular transport properties.

• Stiffness reflects the vast range of time-scales in the system, and targeted physics is slow (~transport scale).

The NIMROD code has a unique combination of advanced numerical methods for solving systems of PDEs that describe high-temperature plasmas:

• High-order finite element representation of the poloidal plane:

• accuracy for MHD and transport anisotropy at realistic parameters: S>106, ||/perp>109

• flexible spatial representation

• Temporal advance with semi-implicit and implicit methods:

• multiple time-scale physics from ideal MHD (s) to transport (10-100 ms)

• Coding modularity for physics model development

• Large-scale parallel computing capability

The finite element method provides an approach to spatial discretization that has the needed flexibility and accuracy.

NIMROD uses 2D finite elements (that are general with respect to the degree of polynomials used for basis functions) for the poloidal plane and finite Fourier series for the periodic direction, which may be toroidal, azimuthal, or a periodic linear coordinate.

ptp

ttt

t

BJBBB

JBBV

000

000

11

00 BVBVB

tt

t

VVVV

0000 ppt

ppt

tt

p

Using the alternative differential approximation to the resulting wave equation leads to

VLBJBBVLV

tpttt

2

100

0

22

0BVB

t

VV

00 ppt

p

where L is the ideal MHD force operator. We may drop the t-term on the rhs to avoid numerical dissipation and arrive at a semi-implicit advance stable for all t where V is leap-frogged with B and p.

The semi-implicit advance is derived through the differential approximation for an implicit time advance for ideal linear MHD with arbitrary time centering,

A linear resistive tearing study in a periodic cylinder shows that nearly singular behavior can be reproduced with packed finite elements and a large time-step.

• This S=106 computation has a 32x32 mesh of bicubic elements and t=100A (1.8x105 times explicit). is within 2%.

sqrt( )

Vz(arbitrary)

0 0.25 0.5 0.75 1

sqrt( )

Vr(arbitrary)

0 0.25 0.5 0.75 1

Accuracy while varying the mesh and degree of polynomial basis functions meets expectations for biquadratic and bicubic elements.

cells

gamma

3296160224

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1.0E-03

1.1E-03

bilinearbiquadraticbicubic

ln(h)

ln(

||di

vb(b

)||

/||b

||)

-5 -4 -3 -2-8.0E+00

-7.0E+00

-6.0E+00

-5.0E+00

-4.0E+00

-3.0E+00

-2.0E+00

bilinearbiquadraticbicubic

• Divergence errors are too large with bilinear elements for these S=106 conditions and the numerical parameters.

R

Z

1 1.5 2 2.5

-1

-0.5

0

0.5

1

A nonlinear simulation of a classical tearing-mode demonstrates full application inlow-dissipation conditions.

NIMROD Simulation• S=R=106

• Pm=R=0.1• A=1s• 1% to avoid GGJ stabilizationDIII-D L-mode Startup Plasma[R. LaHaye, Snowmass Report]• S=1.6x106

• Pm=4.5• A=0.34s• E=0.03s

1/2

q

0 0.25 0.5 0.75 1

2

3

4

5

6

7

8

Small ’ (linear A=5x10-4) leads to nonlinear evolution over the energy confinement time-scale.

time (s)

En

(J)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

10-6

10-4

10-2

100

102

104

106

n=2

n=1

n=0 + ss

R

Z

1.5 2

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Magnetic Energy vs. TimeSaturation of Coupled

Island Chains

• 5th-order accurate biquartic finite elements resolve anisotropies.

• 20,000 semi-implicit time-steps evolve solution for times > E.

• Explicit computation is impossible2x108 time-steps.

Thermal conduction also exercises spatial accuracy for realistic ratios of thermal conductivity coefficients (~109).

• Adaptive meshing alone cannot provide the needed accuracy in nonlinear 3D simulations; magnetic topology changes across islands and stochastic regions.

• High-order finite elements provide a solution.

A simple but revealing quantitative test is a box, 1m on a side, with source functions to drive the lowest eigenmode, cos(x) cos(y), in T(x,y) and Jz (x,y). Mass density is large to keep V negligible.• Analytic solution is independent of

)cos()cos(),( 1 yxyxT

• Computed T-1(0,0) measure effective cross-field conductivity.

• Any simple rectangular mesh has poor alignment.

Convergence of the steady state solution shows that even bicubic elements are sufficiently accurate for realistic parameters.

Effective perp-1 for || = 106

x

|ef

f-1|

0.1 0.2 0.310-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103bilinearbiquadraticbicubicbiquartic

Effective perp-1 for || = 109

x

|ef

f-1|

0.1 0.2 0.310-5

10-4

10-3

10-2

10-1

100

101bicubicbiquarticbiquintic

• Bilinear elements have severe difficulties with the test by conductivity-ratio values of 106.

Simulations of realistic configurations bring together the MHD influence on magnetic topology and rapid transport along field lines to show the net effect on confinement.

SWINDLE: these plots were handy but the computation ran the MHD first, then thermal conduction.

Tests of anisotropic thermal conduction at various times during the nonlinear classical tearing evolution reproduce an analytic wd

-4 scaling. [Fitzpatrick, PoP 2, 825

(1995)]

• Conductivity ratio is scaled until an inflection in T within (2,1) island is achieved.

• Power-law fit is ||

perp=3.0x103 (wd /a)-4.2.

• Result is for toroidal geometry.

• High-order spatial convergence is required for realistic anisotropy.• Implicit thermal conduction is required for stiffness.

wd (cm)

||/

per

p

2 3 4 5 6 7

108

109

1010

Solving ill-conditioned matrices is often the most performance-limiting aspect of the algorithm.

22max

2 tkvA

• The condition number of the velocity-advance matrix can be estimated as

which can be > 1011 in some computations.

• We have been using a home-grown conjugate gradient method solver with a parallel line-Jacobi preconditioner.

• It has been running out of wind on some of the more recent applications, forcing a reduction of time-step.

• We are presently implementing calls to Sandia’s AZTEC library, but we are interested in other possibilities (PETSc), too.

Conclusions

• Test results and past and present physics applications show the effectiveness of combining the semi-implicit method with a variational approach to spatial representation.

• Improved performance is expected from algorithm refinements.

• Iterative solution methods

• Adaptive meshing

• Advection (not discussed here)

Directions for the Project

• Hall and other two-fluid terms are in the NIMROD code, but the implementation requires small time-steps for accuracy.

• We are working on improved formulations.

• The ability to solve nonsymmetric matrices is important for this.

• Kinetic physics:

• Parallel electron streaming effects [E. Held, USU]

• Gyrokinetic hot ion effects [C. Kim and S. Parker, CU]

• Resistive wall and external vacuum fields [T. Gianakon, S. Kruger, and D. Schnack]