Application of Adaptive Mesh Refinement to PIC in inertial fusion

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Application of Adaptive Mesh Refinement to PIC
in inertial fusion
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
15th International Symposium on Heavy Ion
Inertial Fusion Princeton, New Jersey June
7-11, 2004
The Heavy Ion Fusion Virtual National Laboratory
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Abstract

• We have recently merged AMR with the
  Particle-In-Cell (PIC) method for simulation of
  plasmas and particle beams. The application of
  AMR to plasma modeling poses significant
  challenges, including the introduction of
  spurious forces on simulation particles. We have
  carried out a detailed analysis of the coupling
  of the two methods and, in collaboration with the
  developers of the popular Chombo package for AMR,
  have developed practical methods and demonstrated
  their effectiveness on electrostatic
  Particle-In-Cell simulations of intense ion
  beams. Initial successes include major savings of
  computational effort in simulations of
  time-dependent space-charge-limited flow (in a
  5-D phase space) and demonstrations of numerical
  convergence. Most recently, the merger of the PIC
  code WARP (developed for Heavy Ion Fusion
  studies) and Chombo has been accomplished. The
  application of AMR to electromagnetic plasma
  modeling is even more challenging we have
  introduced a new methodology using recently
  developed Absorbing Boundary Conditions, and are
  beginning to employ it on laser-plasma
  interaction in the context of fast-ignition.

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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)
4
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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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

As shown in the following slides, the choice of
algorithm matters.
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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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Illustration of the spurious self-force effect

• 2-grid set with metallic boundary

zoom

• particle trapped in fine gridded patch

• MR introduces spurious force,

as if there was a spurious image
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Self-force amplitude map and mitigation

• Magnitude of self force decreases rapidly with
  distance from edge
• self-force about one order of magnitude lower on
  collocated nodes
• with the 1-pass method, the self-force effect
  can be mitigated by defining a transition region
  surrounding the patch in which one
• deposit charge and gather force only at
  collocated nodes
• or deposit charge and solve but get field from
  underlying coarse patch

main grid
patch
transition region
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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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Electromagnetics we propose a method by
substitution
Rf
ABC
P2
Outside patch F F(G)
Rc
P1
Inside patch F F(G)-F(P1)F(P2)
Rc
G
Rc coarse resolution Rf fine resolution
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Illustration of instability in 1-D EM tests
o E, xB
Space only
SpaceTime
Most schemes relying on interpolations are
potentially unstable.
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Electrostatic PICAMR examples
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Study of steady-state regime of HCX triode
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3D WARP simulation of High-Current Experiment
(HCX)
Modeling of source is critical since it
determines initial shape of beam
WARP simulations show that a fairly high
resolution is needed to reach convergence
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Prototype MR implemented in WARPrz (f
axisymmetric )

• Three runs with single uniform grid
• One run at medium resolution MR patch

In this case speedup 4
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Prototype AMR implemented in WARPrz (2)

• Better results obtained with a dynamic AMR mockup
  refining emitter area beam edge

In this case speedup 4
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Prototype AMR implemented in WARPrz (3)

• Higher speedup obtained with a true dynamic AMR
  implementation

zoom
R (m)
R (m)
Z (m)
Z (m)
Z (m)
Z (m)
Refinement of gradients emitting area, beam edge
and front.
In this case speedup 10.5
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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)

• Simulations show head cleaner with shorter
  rise-time
• Question what is the optimal rise-time?

z (m)
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1D time-dependent modeling of ion diode
Emitter
Collector
d
V
V0
irregular patch in di
Ns 200
dx0/Dx10-5!
Insufficient resolution of beam front gt AMR
patch
Careful analysis shows that di too large by
gt104 gt irregular patch
Time (s)
MR patch suppresses long wavelength oscillation
Adaptive MR patch suppresses front peak
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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

No MR
With MR
Current history (Z0.62m)
Current history (Z0.62m)
Z (m)
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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Comparison single uniform high res. grid / low
res. patch
without patch
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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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.

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AMR in WARPrz efficiency considerations
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AMR in WARPrz efficiency considerations
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