Application of Adaptive Mesh Refinement to ParticleInCell simulations of plasmas and beams

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Application of Adaptive Mesh Refinement to
Particle-In-Cell simulations of plasmas and beams
J.-L. Vay, P. Colella, J.W. Kwan, P.
McCorquodale, D. Serafini Lawrence Berkeley
National Laboratory A. Friedman, D.P. Grote, G.
Westenskow Lawrence Livermore National Laboratory
J.-C. Adam, A. Héron CPHT, Ecole Polytechnique,
France I. Haber University of Maryland
45th Annual Meeting of the Division of Plasma
Physics Albuquerque, New Mexico October 27-31,
2003
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Outline

• Motivations for coupling PIC with AMR
• Issues
• Examples electrostatic and electromagnetic
  PIC-AMR
• Joint project at LBNL to develop AMR library for
  PIC
• Conclusion

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Goal end-to-end modeling of a Heavy Ion Fusion
driver
challenging because length scales span a wide
range mm to km(s)
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The Adaptive-Mesh-Refinement (AMR) method

• addresses the issue of wide range of space
  scales
• well established method in fluid calculations

3D AMR simulation of an explosion (microseconds
after ignition)
AMR concentrates the resolution around the edge
which contains the most interesting scientific
features.
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Mesh Refinement in Particle-In-Cell Issues

• Asymmetry of grid may imply asymmetry
• of field solution for one particle
• spurious self-force strongest at interface
• Some implementations may violate Gauss Law
• Total charge may not be conserved exactly
• EM shortest wavelength resolved on fine grid not
  resolved on coarse grid reflect at interface with
  factorgt1
• May cause instability by multiple reflections

However, with a careful implementation, PIC-AMR
can be used effectively.
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Electrostatic PICAMR examples
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3D WARP simulation of High-Current Experiment
(HCX)
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Study of steady-state regime of HCX triode
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Prototype MR implemented in WARPrz (f
axisymmetric )

• Three runs with single uniform grid
• Low res. (56x640) Np
• Medium res. (112x1280) Np x 4
• High res. (224x2560) Np x 16

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Prototype AMR implemented in WARPrz (f
axisymmetric )

• Three runs with single uniform grid
• Low res. (56x640) Np
• Medium res. (112x1280) Np x 4
• High res. (224x2560) Np x 16
• Medium MR
• MR factor 2 Np x 4

Medium res.MR High res. result when refining
regions of high gradients emitter, beam edge
4x saving in computational cost (gt 16x in 3-D)
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Time-dependent modeling of ion source risetime
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3D WARP simulation of HCX shows beam head
scrapping
Rise-time t 800 ns beam head particle loss lt
0.1
x (m)
z (m)
Rise-time t 400 ns zero beam head particle loss
x (m)

• Can we get even cleaner head with faster
  rise-time?
• Optimum?

z (m)
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1D time-dependent modeling of ion diode
Emitter
Collector
d
V
V0
MR patch suppresses long wavelength oscillation
Adaptive MR patch suppresses front peak
Careful analysis shows that di too large by
gt104 gt irregular patch
Insufficient resolution of beam front gt AMR patch
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Application to three dimensions

• Specialized 1-D patch implemented in 3-D
  injection routine (2-D array)
• Extension Lampel-Tiefenback technique to 3-D
  implemented in WARP
• predicts a voltage waveform which extracts a
  nearly flat current at emitter
• Run with MR predicts very sharp risetime (not
  square due to erosion)
• Without MR, WARP predicts overshoot

Optimized Voltage
Current at Z0.62m
STS500 experiment
X (m)
V (kV)
Z (m)
T (ms)
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Comparison with experiment

• Experimental voltage lowered so that particle
  transit time risetime
• Overshoot predicted without MR is not present in
  experimental current history which is well
  recovered when using MR
• Discrepancy of steady-state current within
  experimental errors of high voltage probe
  calibration (will be addressed soon)

Mesh Refinement essential to recover experimental
results Ratio of smaller mesh to main grid mesh
1/1000
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Electromagnetic PICMR example
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Laser-plasma interaction in the context of fast
ignition

• A laser impinges on a cylindrical target which
  density is far greater than the critical density.
  
• The center of the plasma is artificially cooled
  to simulate a cold high-density core.
• Patch boundary surrounds plasma. Laser launched
  outside the patch.

• Implemented new MR technique in EM PIC code
  Emi2d (E. Polytech.)

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We propose a new method by substitution
R2
ABC
P2
Outside patch F F(G)
R1
P1
Inside patch F F(G)-F(P1)F(P2)
R1
G
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Comparison single uniform high res. grid / low
res. patch
without patch

• no instability nor spurious wave reflection
  observed at patch border

• while still working on improvements, test case
  satisfactory and method will soon be used for
  production

with patch
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AMR library for PIC
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Effort to develop AMR library for PIC at LBNL

• Researchers from AFRD (PIC) and ANAG (AMR-Phil
  Colellas group) collaborate to provide a library
  of tools that will give AMR capability to
  existing PIC codes (on serial and parallel
  computers)
• The base is the existing ANAGs AMR library
  Chombo
• The way it works
• WARP is test PIC code but library will be usable
  by any PIC code

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Example of WARP-Chombo injector field calculation

• Chombo can handle very complex grid hierarchy

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Conclusion

• PIC and AMR are numerical techniques that have
  proven to be very valuable in various fields and
  their combination may lead to more powerful tools
  for plasma modeling.
• The implementation must be done with care
  (beware of potential spurious self-forces,
  violation of Gauss Law, reflection of smallest
  wavelengths).
• Prototypes of AMR methods were implemented in
  existing PIC codes and test runs demonstrated the
  effectiveness of the method in ES-PIC and a
  proof-of-principle of a new method was performed
  in EM-PIC.
• There is an ongoing effort at LBNL to build an
  AMR library which will ultimately provide AMR
  capabilities to existing PIC codes.

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Backup slides
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Time and length scales in driver and chamber span
a wide range
Time scales
depressed
betatron
betatron
electron drift
t
pb
out of magnet

transit
lattice
thru
electron
period
fringe
beam
cyclotron
pulse
fields
residence
in magnet
log of timescale
pulse
beam
t
pe
in seconds
residence
t
pi
t
pb
Length scales

• electron gyroradius in magnet 10 mm
• lD,beam 1 mm
• beam radius cm
• machine length km's

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Electrostatic possible implementations

• Given a hierarchy of grids, there exists several
  ways to solve Poisson
• Two considered
• 1-pass
• solve on coarse grid
• interpolate solution on fine grid boundary
• solve on fine grid
• different values on collocated nodes
• back-and-forth
• interleave coarse and fine grid relaxations
• collocated nodes values reconciliation
• same values on collocated nodes

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Self-force test

• 2-grid set with metallic boundary

• particle trapped in fine gridded patch

• MR introduces spurious force,

as if
Can we reduce its magnitude?
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Self-force logE

• 1 pass self-force about one order of magnitude
  lower on collocated nodes
• can reduce self-force by depositing charge and
  gathering force only at collocated nodes in
  transition zone

• 1 pass also offers possibility to use coarse
  grid solution in transition zone

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Global error

• global error larger with BF than 1P
• BF Gauss law not satisfied error transmitted
  to coarse grid solution

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Electrostatic issues summary

• Mesh Refinement introduces spurious self-force
  that has a repulsive effect on a macroparticle
  close to coarse-fine interface in fine grid, but
• real simulations involve many macroparticles
  dilution of the spurious force
• for some coarse-fine grid coupling, the magnitude
  of the spurious effect can be reduced by an order
  of magnitude by interpolating to and from
  collocated nodes in band in fine grid along
  coarse-fine interface
• we may also simply discard the fine grid solution
  in band and use coarse grid solution instead for
  force gathering (or ramp)
• some scheme may violate Gauss law and may
  introduce unphysical non-linearities into
  mother grid solution hopefully there is also
  dilution of the effect in real simulations
• we note that our tests were performed for a
  node-centered implementation and our conclusion
  applies to this case only. For example, a
  cell-centered implementation does strictly
  enforce Gauss Law and results may differ.