The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Application of Adaptive Mesh Refinement to PIC simulations 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 15 th International Symposium on Heavy Ion Inertial Fusion Princeton, New Jersey June 7-11, 2004

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Outline Motivations for coupling PIC with AMR Issues Examples of electrostatic and electromagnetic PIC-AMR Joint project at LBNL to develop AMR library for PIC Conclusion

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 challenging because length scales span a wide range:  m to km(s) Goal: end-to-end modeling of a Heavy Ion Fusion driver

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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)

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Electrostatic PIC+AMR: issues

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Mesh Refinement in Particle-In-Cell: Issues Asymmetry of grid implies asymmetry of field solution for one particle  spurious self-force 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 As shown in the following slides, the choice of algorithm matters.

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Illustration of the spurious self-force effect 2-grid set with metallic boundary; Patch grid “Mother” grid Metallic boundary Test particle one particle attracted by its image Spurious “image” as if there was a spurious image zoom  MR introduces spurious force, particle trapped in patch

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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)

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Illustration of instability in 1-D EM tests Space onlySpace+Time o: E, x:B

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Rc ABC Rc P1 Rf P2 G Outside patch: F = F(G) Inside patch: F = F(G)+F(P2)-F(P1) Electromagnetics: we proposed a method by “substitution” Rc: coarse resolution Rf: fine resolution

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Electrostatic PIC+AMR examples

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Study of steady-state regime of HCX triode

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 3D WARP simulation of High-Current Experiment (HCX) Modeling of source is critical since it determines initial shape of beam 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

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Prototype MR implemented in WARPrz (  axisymmetric ) Low res. Medium res. High res. Medium res. +MR Three runs with single uniform grid One run at medium resolution + MR patch RunGrid sizeNb particles Low res.56x640~1M Medium res.112x1280~4M High res.224x2560~16M Medium res. + MR112x1280~4M In this case: speedup ~ 4 Low res. Medium res. High res. Medium res. +MR

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 In this case: speedup ~ 4 Prototype AMR implemented in WARPrz (  axisymmetric ) Low res. Medium res. High res. Medium res. +AMR Better results obtained with a dynamic AMR mockup refining emitter area + beam edge RunGrid sizeNb particles Low res.56x640~1M Medium res.112x1280~4M High res.224x2560~16M Medium res. + AMR112x1280~4M In this case: speedup ~ 4 Low res. Medium res. High res. Medium res. +AMR

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Prototype AMR implemented in WARPrz (  Higher speedup obtained with a “true” dynamic AMR implementation Z (m) R (m) In this case: speedup ~ 11.3 RunGrid sizeNb particles Low res.56x640~1M Medium res.112x1280~4M High res.224x2560~16M Low res. + AMR56x640~1M Low res. +AMR (N transit =2) Low res. High res. Low res. +AMR (N transit =0) Medium res. Z (m) Refinement of gradients: emitting area, beam edge and front. zoom Z (m) R (m)

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Time-dependent modeling of ion source rise-time

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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?

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 1D time-dependent modeling of ion diode: fast rise-time 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 Irregular MR patch covering emission area suppresses long wavelength oscillation Adaptive MR patch following the beam head suppresses front peak Irregular MR patch covering emission area suppresses long wavelength oscillation Adaptive MR patch following the beam head suppresses front peak

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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 Without MR, WARP predicts overshoot Run with MR predicts very sharp risetime (not square due to erosion) Application to three dimensions T (  s) V (kV) “Optimized” VoltageCurrent at Z=0.62m X (m) Z (m) STS500 experiment

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Experimental voltage lowered so that risetime = particle transit time Mesh Refinement essential to recover experimental results Ratio of smaller mesh to main grid mesh ~ 1/1000 Z (m) No MRWith MR Current history (Z=0.62m) Comparison with experiment

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Electromagnetic PIC+MR example

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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, 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.)

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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 (well understood, possible cures) no instability nor spurious wave reflection observed at patch border can be used as is for various applications and we are also exploring improvements and variants

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 AMR library for PIC

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Example of WARP-Chombo injector field calculation Interactions with particles is being implemented

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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 beams and plasmas modeling in inertial fusion (and beyond). 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.

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Backup slides

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 Time and length scales in driver and chamber span a wide range Length scales: 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:

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04  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

The Heavy Ion Fusion Virtual National Laboratory Vay 06/08/04 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.