1. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Mesh Refinement for Particle-In-Cell Plasma Simulations: application to Heavy-Ion-Fusion 18 th International Conference on Numerical Simulation of Plasmas Cape Cod, Massachusetts September 10, 2003 J.-L. Vay, P. Colella, P. McCorquodale, D. Serafini, B. Van Straalen Lawrence Berkeley National Laboratory A. Friedman, D.P. Grote Lawrence Livermore National Laboratory

2. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Outline Issues in coupling Electrostatic PIC with AMR Examples Joint project to couple electrostatic PIC and AMR at LBNL Conclusion

3. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Electrostatic PIC+AMR: issues

4. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 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

5. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 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

6. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 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

7. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03  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

8. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 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.

9. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Electrostatic PIC+AMR: examples

10. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Time-dependent modeling of ion diode risetime

11. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 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? How good is our ability to model it?

12. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 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 MR patch suppresses long wavelength oscillation; Adaptive MR patch suppresses front peak N s = 200 irregular patch in d i Time (s)  x 0 /  x~10 -6 ! time current AMR ratio = 16 irregular patch in d i + AMR following front Time (s)

13. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Current rise-time in STS500 experiment vs WARP run Applied voltage measured from experiment input in WARP run using aggressive (x1000) non-uniform mesh refinement in emitter area allows high-fidelity modeling of fast current rise-time Exp. WARP Exp. WARP Z (m) X (m) T (  s) I (mA) T (  s) V (kV) Applied voltage T (  s) V (kV) Applied voltage No MRWith MR Current history (Z=0.62m)

14. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Shorter rise-time using optimized voltage waveform T (  s) V (kV) T (  s) I (mA) Novel technique based on decomposition of field solution in WARP predicts a voltage waveform which extracts a flat current at emitter Despite slight beam head erosion, rise-time very sharp at exit of diode We were able to answer our questions by using mesh refinement Existing Optimized Existing Optimized Voltage Current at Z=0.62m

15. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Example of PIC-AMR calculation using WARP-RZ Steady-state study of a diode

16. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Prototype axisymmetric AMR implemented in WARPrz Base grid 56x640 Multipliers: –cells along each axis, ngf –number of particles, npf Mesh refinement: factor-of-2 finer grid in emitter patch

17. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Prototype axisymmetric AMR implemented in WARPrz Z(m) R(m) Base grid 56x640 Multipliers: –cells along each axis, ngf –number of particles, npf Mesh refinement: factor-of-2 finer grid in emitter patch ~ 4x saving in computational cost for quasi same result

18. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Effort to couple PIC and AMR at LBNL: Chombo library

19. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Researchers from AFRD (PIC) and NERSC (AMR-Phil Colella’s group) collaborate to provide a library of tools that give AMR capability to PIC codes (on serial and parallel computers) The way it works First beta version released a few months ago: being tested with WARP (Heavy Ion Fusion main Particle-In-Cell code) There is a LDRD effort at LBNL to couple PIC and AMR PIC Advance particles Do other things Receive forces Send particles Setup grid hierarchy Deposit charge Solve fields Gather forces AMR

20. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Example of WARP-Chombo injector field calculation Chombo grid hierarchy can handle very complex geometry

21. The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Conclusion PIC and AMR are numerical techniques that have proven to be very valuable in various fields and combination may lead to more powerful tools for plasma modeling The implementation must be done with care; at the least, when interpreting simulation results, we must have in mind that: – refinement introduces spurious self-forces – Gauss’ law violations, spuriously anharmonic forces may be associated with some schemes Using 1-D and 2-D axisymmetric prototypes, we have shown that AMR can be used in PIC simulations with great efficiency There is an ongoing LDRD effort (AFRD+NERSC) to introduce AMR in PIC in a form of a library (Chombo) which can be linked with existing codes Electromagnetic PIC poses additional challenges due to EM waves (see poster)