1. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Application of Adaptive Mesh Refinement to Particle-In-Cell simulations of plasmas and beams 45 th Annual Meeting of the Division of Plasma Physics Albuquerque, New Mexico October 27-31, 2003 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

2. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 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

3. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 challenging because length scales span a wide range:  m to km(s) Goal: end-to-end modeling of a Heavy Ion Fusion driver

4. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 The Adaptive-Mesh-Refinement (AMR) method addresses the issue of wide range of space scales well established method in fluid calculations AMR concentrates the resolution around the edge which contains the most interesting scientific features. 3D AMR simulation of an explosion (microseconds after ignition)

5. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 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 factor>1  May cause instability by multiple reflections However, with a careful implementation, PIC-AMR can be used effectively. However, with a careful implementation, PIC-AMR can be used effectively.

6. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Electrostatic PIC+AMR examples

7. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 3D WARP simulation of High-Current Experiment (HCX) Modeling of source is critical since it determines initial shape of beam

8. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Study of steady-state regime of HCX triode

9. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Low res. Medium res. High res. Medium res. + MR Prototype MR implemented in WARPrz (  axisymmetric ) Three runs with single uniform grid –Low res. (56x640); Np –Medium res. (112x1280); Np x 4 –High res. (224x2560); Np x 16 Low res. Medium res. High res. Medium res. +MR Low res. Medium res. High res. Medium res. +MR MR patch Medium + MR – MR factor = 2; Np x 4

10. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Low res. Medium res. High res. Medium res. + AMR Prototype AMR implemented in WARPrz (  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 Low res. Medium res. High res. Medium res. +AMR Low res. Medium res. High res. Medium res. +AMR Medium res.+MR = High res. result when refining regions of high gradients: emitter, beam edge ~ 4x saving in computational cost (=> 16x in 3-D) Medium res.+MR = High res. result when refining regions of high gradients: emitter, beam edge ~ 4x saving in computational cost (=> 16x in 3-D)

11. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Time-dependent modeling of ion source risetime

12. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 3D WARP simulation of HCX shows beam head scrapping Rise-time  = 800 ns beam head particle loss < 0.1% z (m) x (m) Rise-time  = 400 ns zero beam head particle loss Can we get even cleaner head with faster rise-time? Optimum?

13. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 1D time-dependent modeling of ion diode EmitterCollector VV=0 d virtual surface didi ViVi I (A) Time (s) N = 160  t = 1ns d = 0.4m “L-T” waveform N s = 200 irregular patch in d i Time (s)  x 0 /  x~10 -5 ! time current AMR ratio = 16 irregular patch in d i + AMR following front Time (s) Careful analysis shows that d i too large by >10 4 => irregular patch Careful analysis shows that d i too large by >10 4 => irregular patch Insufficient resolution of beam front => AMR patch Insufficient resolution of beam front => AMR patch MR patch suppresses long wavelength oscillation Adaptive MR patch suppresses front peak MR patch suppresses long wavelength oscillation Adaptive MR patch suppresses front peak

14. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 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 Application to three dimensions T (  s) V (kV) “Optimized” VoltageCurrent at Z=0.62m X (m) Z (m) STS500 experiment

15. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Comparison with experiment Exp. WARP Exp. WARP Z (m) T (  s) I (mA) No MRWith MR Current history (Z=0.62m) 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 Mesh Refinement essential to recover experimental results Ratio of smaller mesh to main grid mesh ~ 1/1000

16. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Electromagnetic PIC+MR example

17. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 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. core Laser beam =1  m, 10 20 W.cm -2 (P osc /m e c~8,83) 2  =28/k 0 10n c, 10keV Patch Implemented new MR technique in EM PIC code Emi2d (E. Polytech.)

18. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 R1 ABC R1 P1 R2 P2 G Outside patch: F = F(G) Inside patch: F = F(G)-F(P1)+F(P2) We propose a new method by “substitution”

19. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Comparison single uniform high res. grid / low res. + patch without patch with patch same results except for small residual incident laser outside region of interest 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

20. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 AMR library for PIC

21. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Researchers from AFRD (PIC) and ANAG (AMR-Phil Colella’s 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 ANAG’s AMR library Chombo The way it works WARP is test PIC code but library will be usable by any PIC code Effort to develop AMR library for PIC at LBNL

22. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Example of WARP-Chombo injector field calculation Chombo can handle very complex grid hierarchy

23. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 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.

24. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Backup slides

25. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Time and length scales in driver and chamber span a wide range Length scales: -11-12-10-9-8-7-6-5-4-3-20 electron cyclotron in magnet pulse electron drift out of magnet beam residence  pb  lattice period betatron depressed betatron  pe transit thru fringe fields beam residence pulse log of timescale in seconds In driverIn chamber  pi  pb electron gyroradius in magnet ~10  m D,beam ~ 1 mm beam radius ~ cm machine length ~ km's Time scales:

26. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Electrostatic: possible implementations Given a hierarchy of grids, there exists several ways to solve Poisson Two considered: 1.‘1-pass’ solve on coarse grid interpolate solution on fine grid boundary solve on fine grid  different values on collocated nodes 2.‘back-and-forth’ interleave coarse and fine grid relaxations collocated nodes values reconciliation  same values on collocated nodes Patch grid “Mother” grid

27. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 Self-force test particle trapped in fine gridded patch Can we reduce its magnitude? 2-grid set with metallic boundary; Patch grid “Mother” grid Metallic boundary  MR introduces spurious force, Test particle one particle attracted by its image Spurious “image” as if

28. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 y x y Linear Quadratic 1 pass x Back and forth 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 Self-force log|E| 1 pass also offers possibility to use coarse grid solution in transition zone

29. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03  global error larger with BF than 1P  BF: Gauss’ law not satisfied; error transmitted to coarse grid solution y Linear Quadratic 1 pass x y x Back and forth x Global error

30. The Heavy Ion Fusion Virtual National Laboratory Vay 10/31/03 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.