The Heavy Ion Fusion Virtual National Laboratory Vay 9/10/03 Mesh Refinement for Particle-In-Cell...

The Heavy Ion Fusion Virtual National Laboratory

Vay 9/10/03

Mesh Refinement for Particle-In-Cell Plasma Simulations: application to Heavy-Ion-Fusion

18th International Conference on Numerical Simulation of Plasmas

Cape Cod, Massachusetts

September 10, 2003

J.-L. Vay, P. Colella, P. McCorquodale, D. Serafini, B. Van StraalenLawrence Berkeley National Laboratory

A. Friedman, D.P. GroteLawrence Livermore National Laboratory


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Outline

• Issues in coupling Electrostatic PIC with AMR

• Examples

• Joint project to couple electrostatic PIC and AMR at LBNL

• Conclusion


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Electrostatic PIC+AMR: issues


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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


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Self-force test

particle trapped in fine gridded patch

Can we reduce its magnitude?

0 100 200 300 400 500

25

26

X

reference case linear - 1p linear - bf quad. - 1p quad. - bf

X

T

0

10

20

30

• 2-grid set with metallic boundary;

Patch grid

“Mother” grid

Metallic boundary

MR introduces spurious force,

Test particle

v

one particle attracted by its image

Spurious “image”

as if


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y

x

y

Line

arQ

uadr

atic

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


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global error larger with BF than 1P BF: Gauss’ law not satisfied; error transmitted to coarse grid solution

y

Line

arQ

uadr

atic

1 pass

x

y

x

Back and forth

x

Back and forth

VV

S

d/dSdD

N// refrefGlobal error


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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.


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Electrostatic PIC+AMR: examples


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Time-dependent modeling of ion diode risetime


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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

Can we get even cleaner head with faster rise-time? Optimum?

How good is our ability to model it?


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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

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

Ns = 200

irregular patch in di

Time (s)

x0/x~10-6!

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)


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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) T (s)

I (m

A)

I (m

A)

T (s)

V (

kV)

Applied voltage

T (s)

V (

kV)

Applied voltage

No MR With MRCurrent history (Z=0.62m) Current history (Z=0.62m)


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Shorter rise-time using optimized voltage waveform

T (s)

V (

kV)

T (s)

I (m

A)

• 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

ExistingOptimized

Existing Optimized

Voltage Current at Z=0.62m


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Example of PIC-AMR calculation using WARP-RZSteady-state study of a diode


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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

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0 ngf=1, npf=1 (base grid) ngf=2, npf=4 ngf=4, npf=16 ngf=2, npf=4 (MR bloc)

4* n

orm

aliz

ed R

MS

em

ittan

ce ( m

m.m

rad)

Z(m)


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Prototype axisymmetric AMR implemented in WARPrz

Z(m)

R(m

)

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0 ngf=1, npf=1 (base grid) ngf=2, npf=4 ngf=4, npf=16 ngf=1, npf=4 (AMR)

4* n

orm

aliz

ed R

MS

em

ittan

ce ( m

m.m

rad)

Z(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


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Effort to couple PIC and AMR at LBNL: Chombo library


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• 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 fieldsGather forces

AMR


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Example of WARP-Chombo injector field calculation

Chombo grid hierarchy can handle very complex geometry


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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)

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