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The Heavy Ion Fusion Science Virtual National Laboratory 1Vay - APS-DPP 2011
Novel Simulation Methods in the Particle-In-Cell Framework Warp
J.-L. Vay*, C.G.R. GeddesLawrence Berkeley National Laboratory, CA, USA
D.P. Grote, A. FriedmanLawrence Livermore National Laboratory, CA, USA
53rd Annual Meeting of the APS Division of Plasma PhysicsSalt Lake City, Utah, USA – November 14-18, 2011
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The Heavy Ion Fusion Science Virtual National Laboratory*[email protected]
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Markers
2Vay - APS-DPP 2011
Warp is a versatile parallel 3D Particle-In-Cell framework developed by the Heavy Ion Fusion Science Virtual National Laboratory
Standard PIC
Non-standard PIC
Laboratory frame Moving window Lorentz Boosted frameExamples: Beam generation
Neutralization in plasma
Example: Beam transport
Example: Laser plasma acceleration
Steady flow
Example: Fast injector design
Quasi-static
Example: electron cloud studies2-D slab of electrons
3-D beam
s
SPS - CERN
p+ bunches
e- clouds
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But short wavelength instability observed at front of plasma for large (≥g 100)
3Vay - APS-DPP 2011
Conjectured that instability related to numerical dispersion, i.e. a kind of numerical Cerenkov.
Warp 2D simulation 10 GeV LPA (ne=1017cc, =130)
Longitudinal electric fieldlaser
plasma
Modeling of 10 GeV laser plasma accelerator stage is challenging
1 Vay, PRL 2007
llaser ≈ lwake ≈ Lacceleratellaser << lwake << Laccelerate
Predicted speedup: >10,000 for 10 GeV stage; > 1,000,000 for 1 TeV stage.
Boosted frame g = gwakeLab frameCalculation in boosted frame at ≈wake minimizes scale differences1
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An electromagnetic solver based on Non-Standard Finite-Difference (NSFD) was implemented in Warp
NSFD: weighted average of quantities transverse to FD ().
NSFD=FD if =0
Yee Cole-Karkkainen (CK)
Yee/CK allows for perfect dispersion along 3D/principal axes.
x=y=z)
Cole1 and Karkkainen2 have applied NSFD to source free Maxwell equations
Warp3: switched FD/NSFD to B/E.
=> FD on source terms, i.e. standard exact current deposition schemes still valid.
*
FD
NSFD
a
b
bb
b
g
g
g
g-a
-b
-b -
b
-b
-g
-g
-g
-g
Dx
NSFD
FD
FD
NSFD
3J.-L. Vay, et al., J. Comput. Phys. 230 (2011) 5908.
x=y=z)
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1J. B. Cole, IEEE Trans. Microw. Theory Tech. 45 (1997), J. B. Cole, IEEE Trans. Antennas Prop. 50 (2002).2M. Karkkainen et al., Proc. ICAP, Chamonix, France (2006).
NSFD offers tunability of numerical dispersion
perfect dispersion 2D diagonal isotropic
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Example: Reflection of circular pulse
using 5 cells PML
with quadratic progression
and standard coefficients
or improved coefficients2
5
Perfectly Matched Layer1,2 (PML) implemented with NSFD solver - for absorption of outgoing waves -
Same high efficiency as with Yee.1JP Berenger, J. Comput. Phys. 127 (1996) 3632J.-L. Vay, J. Comput. Phys. 183 (2002) 367
NSFDNSFD
FDFD
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After testing: instability mostly insensitive to numerical dispersion… …but very sensitive to time step!
6Vay - APS-DPP 2011
Sharp decrease of instability level at ct=z/√2
Tunable NSFD solver allows ct=z/√2 time step for (near) cubic cells• ct=z/√2 time step restricted to “pancake” cells in 3D using Yee FDTD solver
Use of special time step was helpful but not sufficient for large g boost
Pow
er s
pect
rum
(a.
u.)
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Digital filtering of current density and/or fields -- commonly used for improving stability and accuracy
Multiple pass of bilinear filter + compensation routinely used
100% absorption at Nyquist freq.Bilinear (B)Bilinear (B) + compensation (C)
1/2 1/41/4
Bilinearfilter
Wideband filtering difficult in parallel (footprint limited by size of local domains) or expensive
Example: wideband filters using N repetitions of bilinear filter
1×B + C
4×B + C
20×B + C
50×B + C
80×B + C
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“Strided” bilinear filters enable efficient and versatile filtering1
4×BC stride 1 (G1)4×BC stride 2 (G2)4×BC stride 3 (G3)4×BC stride 4 (G4)
Bilinear filterwith stride 2
1/21/4 1/4
Using a stride N shifts the 100% absorption frequency to Fnyquist/N
Combination of filters with strides allows for more efficient filtering:• G1G2 20*B+C; speedup ×2• G1G2G3 50*B+C; speedup ×3.5• G1G2G4 80*B+C; speedup ×5.5
G1 × G2G1 × G2 × G3G1 × G2 × G420×B+C50×B+C80×B+C
Vay - APS-DPP 2011
1J.-L. Vay, et al., J. Comput. Phys. 230 (2011) 5908.
Nice, but is wideband filtering possible without altering the physics?
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Tim
eT
ime
Laser field
Laser field
Lab frame
Wake frame
Hyperbolic rotation from Lorentz Transformation converts laser…
…spatial oscillations into
time beating
Vay - APS-DPP 2011
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Lab frame Frame of wake (=130)
spectrum spectrum
Spectrum very different in boosted and lab frames
Time history of laser spectrum (relative to laser l0 in vacuum)
Dephasing time
Content concentrated around l0
0 0
Content concentrated at much larger l
More filtering possible without altering physics*.*J.-L. Vay, et al., PoP Lett. 18 (2011).
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Controlling the numerical instability with tunable EM solver & filtering
Laser injection
Particle injection
Diagnostics
led to over 1 million x speedup2
+ new laser/particle injection and diagnostics through planes1:
2J.-L. Vay, et al., PoP Lett. 18 (2011) & PoP (in press).1J.-L. Vay, et al., J. Comput. Phys. 230 (2011).
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A. Friedman, D. P. Grote, and I. Haber, Phys. Fluids B 4, 2203 (1992) – code description, warped coordinates
J.-L. Vay et al., Phys. Plasmas 11 (2004) – mesh refinement
D.P. Grote et al., AIP Conf. Proc. 749, 55 (2005) – updated Warp description
R. Cohen et. al., Phys. Plasmas 12, 056708 (2005) – “Drift-Lorentz” particle pusher
J.-L. Vay, Phys. Plasmas 15, 056701 (2008) – ultra-relativistic pusher
R. Cohen et al. Nucl. Instr. & Methods 608, 53 (2009) – direct implicit Drift-Lorentz
For questions on Warp, email to
Warp contains a lot more not described here, see
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BACKUP
13Vay - APS-DPP 2011
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as well as Friedman damping algorithm- for noise control -
B push modified to
with where is damping parameter.
Yee-Friedman (YF) Cole-Karkkainen-Friedman (CKF)
Dispersion degrades with higher values of
Damping more potent on axis and more isotropic for CKF than YF.
Vay - APS-DPP 2011
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BELLA Project: state-of-the-art PWfacility for laser accelerator science
BELLA Laser
Control RoomGowning Room
Final focus< 100 cm
e- beam
~10 GeV Laser
Plasma
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