PyECLOUD development: accurate space charge module+
Preliminary results on buildup in SPS quadrupoles
G. Iadarola, G. Rumolo
Electron cloud meeting – 27/06/2014
Many thanks to:H. Bartosik, K.Li, G. Miano, A. Romano
Introduction
128 mm
52 mme-cloud
Before launching extensive convergence scans (especially for quadrupole simulations), we addressed possible accuracy issues coming from boundary conditions in the electrons space evaluation
Example: two different models of the SPS MBB dipole
E- distribution significantly different even if geometry is very similar in the multipacting region
Can it be an artifact coming from the grid of the space charge solver?
Nominal 25 ns - 26 GeV - SEY = 1.6
Electron space charge evaluation in PyECLOUD
Standard Particle In Cell (PIC) 4 stages:
1. Charge scatter from macroparticles (MPs) to grid
2. Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method
3. Calculation of the electric field at the nodes (gradient evaluation)
4. Field gather from grid to MPs
Internal nodes
External nodes
Uniform square grid
Electron space charge evaluation in PyECLOUD
Standard Particle In Cell (PIC) 4 stages:
1. Charge scatter from macroparticles (MPs) to grid
2. Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method
3. Calculation of the electric field at the nodes (gradient evaluation)
4. Field gather from grid to MPs
Electron space charge evaluation in PyECLOUD
Standard Particle In Cell (PIC) 4 stages:
1. Charge scatter from macroparticles (MPs) to grid
2. Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method
3. Calculation of the electric field at the nodes (gradient evaluation)
4. Field gather from grid to MPs
Internal nodes:
External nodes:
Electron space charge evaluation in PyECLOUD
Standard Particle In Cell (PIC) 4 stages:
1. Charge scatter from macroparticles (MPs) to grid
2. Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method
3. Calculation of the electric field at the nodes (gradient evaluation)
4. Field gather from grid to MPs
Internal nodes:
External nodes:
Can be written in matrix form:
A is sparse and depends only on chamber geometry and grid size It can be computed and LU factorized in the initialization stage to speed up calculation
Electron space charge evaluation in PyECLOUD
Standard Particle In Cell (PIC) 4 stages:
1. Charge scatter from macroparticles (MPs) to grid
2. Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method
3. Calculation of the electric field at the nodes (gradient evaluation)
4. Field gather from grid to MPs
Electron space charge evaluation in PyECLOUD
Standard Particle In Cell (PIC) 4 stages:
1. Charge scatter from macroparticles (MPs) to grid
2. Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method
3. Calculation of the electric field at the nodes (gradient evaluation)
4. Field gather from grid to MPs
Electron space charge evaluation in PyECLOUD
With this approach a curved boundary is approximated with a staircase
Can we do better?
The Shortley - Weller method
Sorry for the change of notation…
Usual 5-points formula at internal nodes:
Refined approximation of Laplace operator at boundary nodes:
O(h2) truncation error is preserved (see: N. Matsunaga and T. Yamamoto, Journal of Computational and Applied Mathematics 116 – 2000, pp. 263–273)
The Shortley - Weller method
Sorry for the change of notation…
Usual central difference for gradient evaluation at internal nodes:
Refined gradient evaluation at boundary nodes:
Tricky implementation:
• Boundary nodes need to be identified, distances from the curved boundary need to be evaluatedo PyECLOUD impact routines have been employed (some
refinement was required since they are optimized for robustness while here we need accuracy)
• Nodes too close to the boundary can lead to ill conditioned A matrix we identify them and impose U=0
o Special treatment for gradient evaluation is needed at these nodes
• Since chamber geometry and grid size stay constant along the simulation most of the boundary treatment can be handled in the initialization stage
The Shortley - Weller method
Tricky implementation:
• Boundary nodes need to be identified, distances from the curved boundary need to be evaluatedo PyECLOUD impact routines have been employed (some
refinement was required since they are optimized for robustness while here we need accuracy)
• Nodes too close to the boundary can lead to ill conditioned A matrix we identify them and impose U=0
o Special treatment for gradient evaluation is needed at these nodes
• Since chamber geometry and grid size stay constant along the simulation most of the boundary treatment can be handled in the initialization stage
• Field map extrapolated outside the chamber to simplify field gather for particle close to the chamber’s wall
The Shortley - Weller method
Test: uniform charge distribution in a circular chamber
Old space charge module New space charge module
Electrostatic potential [a.u] Electrostatic potential [a.u]
Test: uniform charge distribution in a circular chamber
Old space charge module New space charge module
Ex
[a.u
]
Ex
[a.u
]
Tests: uniform charge distribution in a circular chamber
Old space charge module New space charge module
Ey
[a.u
]
Ey
[a.u
]
Test: uniform charge distribution in a circular chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 1 mm
Test: uniform charge distribution in a circular chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 1 mm
Test: uniform charge distribution in a circular chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 1 mm
Test: uniform charge distribution in a circular chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 1 mm
Test: uniform charge distribution in a circular chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 0.5 mm
Tests: uniform charge distribution in a circular chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 0.2 mm
Test: Gaussian beam in an elliptic chamber
Ey [a.u] Ey [a.u]
Old space charge module New space charge module
Test: Gaussian beam in an elliptic chamber
Ey [a.u] Ey [a.u]
Old space charge module New space charge module
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 1 mm
Test: Gaussian beam in an elliptic chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 0.5 mm
Test: Gaussian beam in an elliptic chamber
Field close to the boundary significantly more accurate
Analytic Numerical
Old space charge m
oduleN
ew space charge m
odule
Dh = 0.2 mm
Test: Gaussian beam in an elliptic chamber
Test: LHC beam screen – uniform e- distribution
Old space charge module New space charge module
Electrostatic potential [a.u] Electrostatic potential [a.u]
Test: LHC beam screen – uniform e- distribution
Old space charge module New space charge module
Ex
[a.u
]
Ex
[a.u
]
Test: LHC beam screen – uniform e- distribution
Old space charge module New space charge module
Ey
[a.u
]
Ey
[a.u
]
128 mm
52 mme-cloud
First test within buildup simulations
Two different models of the SPS MBB dipole
Nominal 25 ns - 26 GeV - SEY = 1.6Nominal 25 ns - 26 GeV - SEY = 1.6
Old space charge module New space charge module
SPS quadrupoles - simulated scenarios
8 72 8 7272 8 72
25 ns beam
Intensity 1.25 x 1011ppb
26GeV:
σz=0.22 m
0.82 T/m
450GeV:
σz=0.12 m
14 T/m
Two energy values
Beam transverse size is calculated
assuming εn=2.5μm
SPS quadrupoles - QF
Quite low thresholds
Distribution shrinks at higher energy
SPS quadrupoles - QD
Even lower thresholds than QF
Distribution shrinks at higher energy
Thanks for your attention!