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!