some numerical techniques developed in the heavy-ion fusion program

Vay 08/25/04

The Heavy Ion Fusion Virtual National Laboratory

Some numerical techniques developed in the Heavy-Ion Fusion program

Short-Pulse Laser Matter Computational Workshop Pleasanton, California - August 25-27, 2004

J.-L. Vay - Lawrence Berkeley National Laboratory

Collaborators: A. Friedman, D.P. Grote - Lawrence Livermore National Laboratory J.-C. Adam, A. Héron - CPHT, Ecole Polytechnique, France P. Colella, P. McCorquodale, D. Serafini - Lawrence Berkeley National Laboratory

The Heavy-Ion Fusion program has developed, and continues to work on, numerical techniques that have broad applicability:

Absorbing Boundary Conditions (ABC)Adaptive Mesh Refinement (AMR) for Particle-In-Cell (PIC)Advanced Vlasov methods (moving grid,AMR)Cut-cell boundaries

Vay 08/25/04


y

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MaxwellMaxwell => Split Maxwell Split Maxwell => Berenger PMLBerenger PML => Extended PML

Extended Perfectly Matched Layer

Principles of PML• field vanishes in layer surrounding domain,• layer medium impedance Z matches vacuum’s Z0.

If with u=(x,y), Z=Z0 => no reflection.0

*u

0

u

If and with u=(x,y), Z=Z0 => no reflection.0

*

00

*

0

,

uuuu *

uu cc

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Extended PML implemented in EM PIC code Emi2d

Extended PML

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The Adaptive-Mesh-Refinement (AMR) method

• addresses the issue of wide range of space scales• well established method in fluid calculations

• potential issues with PIC at interface– spurious self-force on macro-particles– violation of Gauss’ Law– spurious reflection of short wavelengths with amplification

AMR concentrates the resolution around the edge which contains the most interesting scientific features.

3D AMR simulation of an explosion (microseconds after ignition)

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3D WARP simulation of High-Current Experiment (HCX)

Modeling of source is critical since it determines initial shape of beam

0.0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1.0 Low res. Medium res.

High res. Very High res.

4N

RM

S ( m

m.m

rad)

Z(m)

Run Grid size Nb 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

Vay 08/25/04


Example of AMR calculation with WARPrz: speedup ~10.5

Run Grid size Nb particles

Low res. 56x640 ~1M

Medium res. 112x1280

~4M

High res. 224x2560

~16M

Low res. + AMR 56x640 ~1M

R (

m)

Z (m) Z (m)Refinement of gradients: emitting area,

beam edge and front.

Z (m)

R (

m)

zoom

0.0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1.0 Low res. Medium res.

High res. Low res. + AMR

4N

RM

S ( m

m.m

rad)

Z(m)

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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

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New MR method implemented in EM PIC code Emi2d

coarse

Extended PML

M

coarse

fine

Outside patch:F = FM

Inside patch:F = FM-FC+FF

F

C

Mesh refinement by substitution coreLaser beam

=1m,1020W.cm-2

(Posc/mec~8,83)2=

28/

k 0

10nc, 10keV

Patch

Applied to Laser-plasma interaction in the context

of fast ignition

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Comparison patch on/off very encouraging

M

R o

ff

MR

on

• same results except for small residual incident laser outside region of interest• no instability nor spurious wave reflection observed at patch border

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Backup slides

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challenging because length scales span a wide range: m to km(s)

Goal: end-to-end modeling of a Heavy Ion Fusion driver

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Time and length scales in driver and chamber span a wide range

Length scales:

-11-12 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

electroncyclotronin magnet pulse

electron driftout of magnet

beamresidence

pb

latticeperiod

betatrondepressedbetatron

pe

transitthru

fringefields

beamresidence

pulse log of timescalein seconds

In driver

In chamberpi

pb

• electron gyroradius in magnet ~10 m• D,beam ~ 1 mm• beam radius ~ cm• machine length ~ km's

Time scales:

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Illustration of instability in 1-D EM tests

10 1001E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10 'jump' (n=3) finite volume (n=3) 'directional' (n=3)

R

2c/xfine grid (xcoarse grid=nxfine grid)10 100

1E-5

1E-4

1E-3

0.01

0.1

1

10 'jump' space-time (n=3) energy conserving (n=2) 'directional' (n=2)

R

2c/xfine grid (xcoarse grid=nxfine grid)

Space only Space+Time

x o x o x o x o x oj-5/2 j-2 j-3/2 j-1 j-1/2 j j+1/2 j+1 j+3/2 j+5/2

x1 x2n.x1

Boundary

grid 1 grid 2

o: E[+],E[-]

x: B[+],B[-]

x o x o x o x o x oj-5/2 j-2 j-3/2 j-1 j-1/2 j j+1/2 j+1 j+3/2 j+5/2

x1 x2n.x1

Boundary

grid 1 grid 2

o: E[+],E[-]

x: B[+],B[-]

o: E, x:B

Most schemes relying on interpolations are potentially unstable.

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3D WARP simulation of HCX shows beam head scrapping Rise-time = 800 ns

beam head particle loss < 0.1%

z (m)

z (m)

x (

m)

x (

m)

Rise-time = 400 nszero beam head particle loss

• Simulations show: head cleaner with shorter rise-time

• Question: what is the optimal rise-time?

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1D time-dependent modeling of ion diodeEmitter Collector

V V=0d

virtual surface

di

Vi

tIQ

d

VχI

2i

2/3i

I (A

)

Time (s)

N = 160t = 1nsd = 0.4m

“L-T” waveform

Ns = 200

irregular patch in di

Time (s)

x0/x~10-5!

3

max

t4

3

t

V

V(t)

time

curr

ent

0.0 1.00.0

1.0

t/

V/V

max

Lampel-Tiefenback

AMR ratio = 16

irregular patch in di + AMR following front

Time (s)

Careful analysis shows that di too large by >104

=> irregular patch

Insufficient resolution of beam front => AMR patch

MR patch suppresses long wavelength oscillation Adaptive MR patch suppresses front peak

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• 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” Voltage Current at Z=0.62m

X (

m)

Z (m)

STS500 experiment

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• 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

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Example of WARP-Chombo injector field calculation

• Chombo can handle very complex grid hierarchy

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Split Maxwell

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**

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Extended PML

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Berenger PML

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Extended Perfectly Matched Layer

y

E

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xyz

zy

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Maxwell

If and

=> Z=Z0: no reflection.

0

*

00

*

0

,

uuuu *

uu cc

Principle of PML:Field vanishes in layer surrounding domain. Layer medium impedance Z matches vacuum’s Z0

If with u=(x,y)

=> Z=Z0: no reflection.

0

*u

0

u

some numerical techniques developed in the heavy-ion fusion program

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