PyECLOUD development: accurate space charge module +

PyECLOUD development: accurate space charge module + Preliminary results on buildup in SPS quadrupoles G. Iadarola, G. Rumolo Many thanks to: H. Bartosik, K.Li, G. Miano, A. Romano Electron cloud meeting – 27/06/2014

Introduction • 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 Nominal 25 ns - 26 GeV - SEY = 1.6 e-cloud 52 mm 128 mm • 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?

Electron space charge evaluation in PyECLOUD • Standard Particle In Cell (PIC)  4 stages: • Charge scatter from macroparticles (MPs) to grid • Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method • Calculation of the electric field at the nodes (gradient evaluation) • 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: • Charge scatter from macroparticles (MPs) to grid • Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method • Calculation of the electric field at the nodes (gradient evaluation) • Field gather from grid to MPs

Electron space charge evaluation in PyECLOUD • Standard Particle In Cell (PIC)  4 stages: • Charge scatter from macroparticles (MPs) to grid • Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method • Calculation of the electric field at the nodes (gradient evaluation) • Field gather from grid to MPs Internal nodes: External nodes:

Electron space charge evaluation in PyECLOUD • Standard Particle In Cell (PIC)  4 stages: • Charge scatter from macroparticles (MPs) to grid • Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method • Calculation of the electric field at the nodes (gradient evaluation) • 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: • Charge scatter from macroparticles (MPs) to grid • Calculation of the electrostatic potential at the nodes with Finite Difference (FD) method • Calculation of the electric field at the nodes (gradient evaluation) • 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… • Refined approximation of Laplace operator at boundary nodes: • Usual 5-points formula at internal 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… • Refined gradient evaluation at boundary nodes: • Usual central difference for gradient evaluation at internal nodes:

The Shortley - Weller method • Tricky implementation: • Boundary nodes need to be identified, distances from the curved boundary need to be evaluated • 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 • 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 evaluated • 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 • 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

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 • Old space charge module Dh = 1 mm • Analytic • Numerical • New space charge module

Test: uniform charge distribution in a circular chamber • Field close to the boundary significantly more accurate • Old space charge module Dh = 0.5 mm • Analytic • Numerical • New space charge module

Tests: uniform charge distribution in a circular chamber • Field close to the boundary significantly more accurate • Old space charge module Dh = 0.2 mm • Analytic • Numerical • New space charge module

Test: Gaussian beam in an elliptic chamber • Old space charge module • New space charge module • Ey [a.u] • Ey [a.u]

Test: Gaussian beam in an elliptic chamber • Field close to the boundary significantly more accurate • Old space charge module Dh = 1 mm • Analytic • Numerical • New space charge module

Test: Gaussian beam in an elliptic chamber • Field close to the boundary significantly more accurate • Old space charge module Dh = 0.5 mm • Analytic • Numerical • New space charge module

Test: Gaussian beam in an elliptic chamber • Field close to the boundary significantly more accurate • Old space charge module Dh = 0.2 mm • Analytic • Numerical • New space charge module

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]

First test within buildup simulations e-cloud • Two different models of the SPS MBB dipole 52 mm 128 mm • Old space charge module • New space charge module Nominal 25 ns - 26 GeV - SEY = 1.6 Nominal 25 ns - 26 GeV - SEY = 1.6 • 

SPS quadrupoles - simulated scenarios 72 8 72 8 72 8 72 • 25 ns beam Intensity 1.25 x 1011ppb • Two energy values • 26GeV: • σz=0.22 m • 0.82 T/m • 450GeV: • σz=0.12 m • 14 T/m • 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!

PyECLOUD development: accurate space charge module +