Non-Sympl Symplect ectic ic PIC IC for Long Term Spac ace-Charge - - PowerPoint PPT Presentation

▶
non sympl symplect ectic ic pic ic for long term spac ace
SMART_READER_LITE
LIVE PREVIEW

Non-Sympl Symplect ectic ic PIC IC for Long Term Spac ace-Charge - - PowerPoint PPT Presentation

Aug 15, 2023 430 likes •682 views

Co Comparis arison on of Symple lectic ctic PIC IC, Symplect lectic ic Gridle less ss Par Particle, le, an and Non-Sympl Symplect ectic ic PIC IC for Long Term Spac ace-Charge Charge Simula ulation tion Ji Qiang Accelerator

slide-1
SLIDE 1

Ji Qiang Accelerator Technology and Applied Physics Division Lawrence Berkeley National Laboratory

Co Comparis arison

  • n of Symple

lectic ctic PIC IC, Symplect lectic ic Gridle less ss Par Particle, le, an and Non-Sympl Symplect ectic ic PIC IC for Long Term Spac ace-Charge Charge Simula ulation tion

Space ce-Char Charge ge Work rksho hop p 2017 Oc

  • Oct. 4-6, 2017, TUD, Darmstadt,

tadt, Germany

slide-2
SLIDE 2

A Symplectic Multi-Particle Tracking Model (1)

2

H = H1+H2

A formal single step solution

space-charge Coulomb potential external focusing/acceleration

multi-particle Hamiltonian

  • J. Qiang, “A Symplectic Multi-Particle Tracking Model for Self-Consistent Space-Charge Simulation,” Phys. Rev. ST Accel.

Beams 20, 014203 (2017).

slide-3
SLIDE 3

3 3

A Symplectic Multi-Particle Tracking Model (2)

2nd order: 4th order: higher order: Symplectic condition:

Refs: E. Forest and R. D. Ruth, Physica D 43, p. 105, , 1990. . H. Yoshida, Phys. Lett. A 150, , p. 262, , 1990. .

M is the Jacobi Matrix of M

slide-4
SLIDE 4

4

A Symplectic Multi-Particle Tracking Model (3)

M1

  • symplectic map for H1 can be found from charged particle optics method

To satisfy the symplectic condition:

M2 will be sympl mplectic ctic if pi is updated from H2 an anal alyti ticall cally

M2

slide-5
SLIDE 5

5

Self-Consistent Space-Charge Transfer Map (1)

slide-6
SLIDE 6

Self-Consistent Space-Charge Transfer Map (2)

6

slide-7
SLIDE 7

7

Self-Consistent Space-Charge Transfer Map (3)

slide-8
SLIDE 8

8

Symplectic Gridless Particle Model

M2

w is the particle charge weight

slide-9
SLIDE 9

Symplectic PIC Model (1)

9

slide-10
SLIDE 10

10

Symplectic PIC Model (2)

slide-11
SLIDE 11

11

Symplectic PIC Model (3)

M2

slide-12
SLIDE 12

Non-Symplectic PIC Model

12

slide-13
SLIDE 13

Benchmark Case 1: FODO Lattice, Below 2nd Order Envelop Instability

13

  • 1 GeV proton beam
  • FODO lattice
  • 0 current phase advance: 85 degrees
  • Initial 4D Gaussian distribution
slide-14
SLIDE 14

Significant Difference in Final 4D Emittances Between the Symplectic and the Non-Symplectic Methods (Strong Space-Charge: Phase Advance Change 85 -> 42)

14

Two symplectic approaches show good agreement.

symplectic gridless symplectic PIC spectral PIC

slide-15
SLIDE 15

Final Beam X-Px Phase Spaces Have Similar Shapes Non-Symplectic Model Has Smaller Area

15

symplectic gridless symplectic PIC spectral PIC

slide-16
SLIDE 16

Final Y-Py Phase Space Show Similar Shapes

16

symplectic gridless symplectic PIC spectral PIC

slide-17
SLIDE 17

Finer Step Size Needed for Non-Symplectic PIC (Symplectic PIC vs. Non-Symplectic PIC)

17

nominal step size 1/2 step size 1/4 step size

slide-18
SLIDE 18

Final Transverse Phase Space: Symplectic PIC vs. Spectral PIC

18

Symplectic PIC

Spectral PIC

slide-19
SLIDE 19

Benchmark Case 2: 1 Turn = 10 FODOs + 1 Sextupole

19

  • 0 current tune 2.417
  • sextupole KL = 10 T/m/m
slide-20
SLIDE 20

Non-Symplectic PIC Shows Much Less Emittance Growth Compared with Two Symplectic Models (4D Emittance Evolution with Different Currents)

20

symplectic gridless symplectic PIC spectral PIC

10 A 20 A 30 A

slide-21
SLIDE 21

Final Beam X-Px Phase Spaces Have Similar Shapes

21

symplectic gridless symplectic PIC spectral PIC

slide-22
SLIDE 22

Final Beam Y-Py Phase Spaces Have Similar Shapes

22

symplectic gridless symplectic PIC spectral PIC

slide-23
SLIDE 23

Computational Complexity

  • Symplectic PIC/Spetral PIC: O(Np) + O(Ng log(Ng)),

parallelization can be a challenge

  • Symplectic gridless particle: O(Nm Np),

easy parallelization

23

  • Z. Liu and J. Qiang, “Symplectic multi-particle tracking on GPUs,”

submitted to Computer Physics Communications, 1997.

slide-24
SLIDE 24

Summary

24

  • Using the same step size, same number of modes, with

sufficient grid points, the symplectic PIC and the symplectic gridless particle model agree with each other very well.

  • Using same step size, the non-symplectic PIC yields significantly

different emittance growth.

  • All three models show similar final phase space shapes.
  • Using sufficient small step size, all three methods

converge to the similar emittance growth (Is this too optimistic?)

  • For small number of modes and particles used, the symplectic

gridless particle model can be computationally efficient;

  • therwise, the symplectic PIC model would be more efficient.

Thank You!