3D Templates = Elliptic Discs or Rings:

Space Charge Templates for 3D beam simulationContribution to HB 2008, ORNL/SNS, Tennessee

L.G.Vorobiev* APC, Fermilab, R.C.York FRIB, NSCL/MSU* Corresponding author; lgv@fnal.gov

Disk-Template (Basis/Library)Ring-Template=Superposition of positive (BLUE) & negative (RED) derived from Disc-Templates

Motivation:

• Fast: orders of magnitude faster, compared to PICs

• Accurate: exact for a large family of 3D distributions

• Modest Memory Demands

• Increased the accelerator/beam design space

Templates: Concept

Template code deals with macro-particles, as standard 3D PICsTemplate substitute Grid (density & Poisson) Solvers in particle tracking codes:

3D beam (non-elliptical!)Within conductingBoundary (not shown)

)()(

, )(4)(

xx

xxx

beambeam

beambeam

Uu

u

3D Templates withinthe same boundary

)()(

, )(4)(

xx

xxx

tmptmp

tmptmp

Uu

u

-0.10 -0.05 0.00 0.05 0.10 0.15

-0.02

-0.01

0.00

0.01

0.02

Z [m]

Sx,

y(z)

[m

] Instead of 3D Poisson Equation for Beam, Convolve Template Potentials

zdzzSzzyxuzyxu yxtmpbeam~))~(),~(,~,,(),,( ,

Shell) (3DFunction Shape )(, zS yx

newold }Particles{Integrator

esTrajectori

Method

Template}Particles{

r,p

+rm(0)0

r,0KV - uniform)

r,0.5

0

0,0

+rm(0)

r,1(W-B)

EXPr

-rm(0)

Templates: Disc/Ring 2D Densities

p

mmm py

y

px

xppyx

)()(1)(),,(

2

2

2

2

-0.02 -0.01 0.00 0.01 0.02-0.02

-0.01

0.00

0.01

0.02

X [m]

Y [m

]

Beam Templates

Conducting Chamber

Library of Templates Disc Template Assigned densities

Ring template More General densities

EXP

ARB

-r_m 0 +r_m

Template potentials/fileds are scaled proportionally to the number of comprised particles

Templates: Library, Reproduction of total SC

• Free Space & R-Z symmetry: Analytics

• Boundaries & Arbitrary Geometries: Pre-calculation + Parameterization

-0.10 -0.05 0.00 0.05 0.10 0.15Z [m]

0

20

40

60

80

u tmp(x

0 ,y

0 ,

z) [V

]

x0=0.0025 m, y0=0.004 m

-0.10 -0.05 0.00 0.05 0.10 0.15Z [m]

-5000

-1000

3000

7000

Ez(

x 0 ,y

0 ,

z) [V

/m]

x0=0.0025 m, y0=0.004 m

-0.10 -0.05 0.00 0.05 0.10 0.15Z [m]

-2*106

0

2*106

4*106

6*106

E' z,

tmp(

x,y,

z) [V

/m2 ]

x0=0.0025 m, y0=0.004 m

-0.10 -0.05 0.00 0.05 0.10 0.15Z [m]

0.0

0.4

0.8

1.2

u(x 0

,y0 ,

z) [V

]

x0=0.0025 m, y0=0.004 m

-0.10 -0.05 0.00 0.05 0.10 0.15Z [m]

-50

-30

-10

10

30

50

Ez(

x 0 ,y

0 ,

z) [V

/m]

x0=0.0025 m, y0=0.004 m

-0.10 -0.05 0.00 0.05 0.10 0.15Z [m]

-6*103

-4*103

-103

2*103

4*103

E' z(

x,y,

z) [V

/m2 ]

x0=0.0025 m, y0=0.004 m

Template Potentials Template Fields Ez Template Ezz

Beam Potential Beam Field Ez Beam Ezz

0

0.02

0.04

0.06

0.08

-0.05 0 0.05

Potential [V]

Z [m]

x tmp

-6

-4

-2

0

2

4

6

-0.05 0 0.05

E_z [V/m]

Z [m]

tmp

0

2000

4000

6000

-0.05 0 0.05

E_{zz} [V/m^{2}]

Z [m]

x tmp

Template Potentials (Library) Template Fields Ez (Library) Template Ezz (Library)

Templates: Example Application

Example: FFS for Heavy Ion Fusion

Mission critical: Losses→ min

Templates increase performances by 2-3 orders of magnitude in comparison withstandard PICs: 30 min vs 2 days

Optimization in automatic mode

Templates: Summary

Template’s Foundation: Principle of Superposition in Electrodynamics

Templates - Special functions Pre-calculated Macro Green’s functions

Range from 2D to 3D beams including boundaries

Scalable & Exact for a large family of distributions

Modest Memory Demands due to parameterization

Performance 102-103 times faster than conventional PICs

Optional SC Solvers in existing PICs

Limitations Clusters, filaments, complex boundaries

Design & Optimization in Automatic Mode

Increased design space for one-passage structures & Rings

Templates: Heirarchy/Bibliography

• Template foundations: MSUCL-1117 (1998); PAC’99 p.2781 Link: http://accelconf.web.cern.ch/AccelConf/p99/PAPERS/THA79.PDF

• 2D envelopes within boundaries PAC’01, p.3075Link: http://accelconf.web.cern.ch/AccelConf/p01/PAPERS/RPAH098.PDF

• Sub 3D (one or more thick slices) Phys. Rev. ST, AB, 114201 Link: http://prst-ab.aps.org/abstract/PRSTAB/v3/i11/e114201

• 3D fully parametrized PAC’03, p.3533 Link: http://accelconf.web.cern.ch/accelconf/p03/PAPERS/FPAG035.PDF

• 3D Ring & Disk Templates FERMILAB-PUB-08-236-APCLink: http://lss.fnal.gov/archive/2008/pub/fermilab-pub-08-236-apc.pdf