1. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Application of Adaptive Mesh Refinement to Particle-In-Cell simulations Workshop on Multiscale Processes in Fusion Plasmas UCLA, California - January 10-15, 2005 J.-L. Vay Heavy Ion Fusion-Virtual National Lab./Lawrence Berkeley National Lab. In collaboration with: A. Friedman, D.P. Grote - HIF-VNL/Lawrence Livermore National Laboratory P. Colella, McCorquodale, D. Serafini - Lawrence Berkeley National Laboratory J.-C. Adam, A. Héron - CPHT, Ecole Polytechnique, France 1.Motivations for coupling PIC with AMR 2.Issues 3.Examples of electrostatic and electromagnetic PIC-AMR 4.Joint project at LBNL to develop AMR library for PIC

2. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 1 - Motivations for coupling PIC with AMR

3. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory challenging because length scales span a wide range:  m to km(s) Goal: end-to-end modeling of a Heavy Ion Fusion driver

4. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 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:

5. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory The Adaptive-Mesh-Refinement (AMR) method addresses the issue of wide range of space scales well established method in fluid calculations goal: develop successful PIC-AMR AMR concentrates the resolution around the edge which contains the most interesting scientific features. 3D AMR simulation of an explosion (microseconds after ignition)

6. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 2 - Issues

7. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Mesh Refinement in Particle-In-Cell: Issues 1.Asymmetry of grid implies asymmetry of field solution for one particle  spurious self-force 2.Some implementations may violate Gauss’ Law  total charge may not be conserved exactly 3.EM: shortest wavelength resolved on fine grid not resolved on coarse grid may reflect at interface  if reflection factor <=1, spurious waves  if reflection factor >1, may cause instability by multiple reflections Remark: BTW, these are general and apply also to PIC on irregular grids!

8. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Electrostatic: possible implementations Given a hierarchy of grids, there exists several ways to solve Poisson. Two considered here: 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

9. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Patch grid “Mother” grid Metallic boundary Illustration of the spurious self-force effect 1 grid with metallic boundary + 1 refinement patch Test particle one particle attracted by its image Spurious “image” as if there was a spurious image  MR introduces spurious force, zoom particle trapped in patch

10. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Self-force amplitude map and mitigation y Linear Quadratic 1-pass x multipass Log(E) Magnitude of self force decreases rapidly with distance from edge with the 1-pass method, the self-force effect can be mitigated by defining a transition region surrounding the patch in which deposit charge and solve but get field from underlying coarse patch patch main grid transition region

11. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Electromagnetics: usual scheme Rc Rf G Rc: coarse resolution Rf: fine resolution P the solution is computed as usual in the main grid and in the patch interpolation is performed at the interface unfortunately, most schemes relying on interpolations have instability issues at short wavelengths (the ones that may be generated in the patch but cannot propagate in the main grid)

12. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Illustration of instability in 1-D EM tests Space onlySpace+Time o: E, x:B Most schemes relying on interpolations are potentially unstable. * J.-L. Vay, JCP (2001) *

13. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 2 - Examples of ES and EM PIC-AMR

14. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 3D WARP simulation of High-Current Experiment (HCX) Modeling of source is critical since it determines initial shape of beam 0.00.10.20.30.4 0.2 0.4 0.6 0.8 1.0 Low res. Medium res. High res. Very High res. 4  NRMS (  mm.mrad) Z (m) RunGrid sizeNb particles Low res.56x640~1M Medium res.112x1280~4M High res.224x2560~16M Very High res.448x5120~64M WARP simulations show that a fairly high resolution is needed to reach convergence

15. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Example of AMR calculation with WARPrz: speedup ~10.5 RunGrid sizeNb particles Low res.56x640~1M Medium res.112x1280~4M High res.224x2560~16M Low res. + AMR56x640~1M R (m) Z (m) Refinement of gradients: emitting area, beam edge and front. Z (m) R (m) zoom 0.00.10.20.30.4 0.2 0.4 0.6 0.8 1.0 Low res. Medium res. High res. Low res. + AMR 4  NRMS (  mm.mrad) Z(m)

16. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 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 Simulations show head cleaner with shorter rise-time Question: what is the optimal rise-time?

17. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory I (A) Time (s) N = 160  t = 1ns d = 0.4m “L-T” waveform N s = 200 irregular patch in d i  x 0 /  x~10 -5 ! Time (s) 1D time-dependent modeling of ion diode EmitterCollector VV=0 d virtual surface didi ViVi time current 0.01.0 t/  0.0 1.0 V/Vmax Lampel-Tiefenback 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 Insufficient resolution of beam front => AMR patch

18. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory MR patch key in simulation of STS500 Experiment MR off Current history (Z=0.62m) MR on Current history (Z=0.62m) Mesh Refinement essential to recover experimental results Ratio of smaller mesh to main grid mesh ~ 1/1000 MR patch * J.-L. Vay et al, PoP (2003)

19. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory New MR method implemented in EM PIC code Emi2d coarse Extended PML M coarse fine Outside patch: F = F M Inside patch: F = F M -F C +F F F C Mesh refinement by substitution* 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 Applied to Laser-plasma interaction in the context of fast ignition * J.-L. Vay, J.-C. Adam, A. Heron, CPC (2004)

20. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory same results except for: small residual incident laser at exit of patch when patch englobes target dip in density on patch border when patch inside target Comparison patch on/off MR off MR on

21. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Partial cancellation due to numerical dispersion

22. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Use less dispersive Maxwell solver Inject inject residual of waves on main grid at patch interface Do not use coarse patch and solve on fine patch with source term  J as a correction to J Go back to usual scheme with a hole in the main grid –put PML inside hole and on fine patch border –couple using clean cross-injections Possible paths for better scheme coarse PML M fine Outside patch: J Inside patch:  J F

23. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 2 - Joint project at LBNL to develop AMR for PIC

24. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory 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

25. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory Example of WARP-Chombo injector field calculation * P. McCorquodale, P. Colella, D.P. Grote, J.-L. Vay, JCP (2004)

26. Vay 01/10/05 The Heavy Ion Fusion Virtual National Laboratory AMR can be of great help for PIC multiscale plasma simulations but scheme must be derived with care –spurious self-force –conservation of charge –reflection of waves –non-cancellations due to numerical errors (dispersion) –… in electrostatic, ‘problem solved’ in electromagnetic, existing schemes can be successfully applied to some problems but more research is needed to get better scheme(s) AMR on regular cartesian grids is not the solution to everything: sometimes need to apply irregularly gridded patch which maps to some conductor, field line, … Conclusion