1. DELPHI and Vlasov solvers used at CERN
D.Amorim, S.Antipov, N.Biancacci, E.Métral, N.Mounet, B.Salvant
ABP-Computing Working Group
16 March 2017
ABP-CWG

2. DELPHI Discrete Expansion over Laguerre Polynomials and HeadtaIl modes
Code written by N.Mounet
Semi-analytic Vlasov solver: computes the complex coherent frequency shifts caused by a beam coupling impedance and/or a damper
Vlasov equation > Perturbation formation > Sacherer Integral
Sacherer Integral + Laguerre Polynomials > Eigensystem
Eigenvalues give the modes frequency shifts and the associated growth rates
Eigenvectors allow to reconstruct the signal which could be observed at the pick-ups
Convergence of the eigenvalues is obtained by automatically increasing the number of azimuthal and radial modes computed, thus increasing the impedance/damper matrix size
Can treat impedance, damper, chromaticity, single or multi-bunch
Functions to account for Landau damping are present and currently under review
High impact on CERN studies: assess the stability thresholds for different machine configuration, input for the calculation of the octupole current threshold
Impact on CERN studies: next step after impedance model,
Assess stability in the machines and impact of different scenarios
Compute octupole current needed to stabilize
Complementary to tracking simulations (PyHEADTAIL)
ABP-CWG

3. DELPHI Code initially implemented in C/C++
Compilation provides a stand-alone executable and a functions library
The stand-alone executable is now depreciated
Python functions are routinely used. They use the C++ functions library compiled beforehand
The code runs on Linux platforms. The C++ core requires three libraries (already installed on LXPLUS):
LAPACK (
BLAS (
GSL (
No parallelization strategy implemented
Simulations can be run on LSF
Possibility to perform scans in chromaticity, bunch intensity, number of bunches, damper gain… Each simulation can be run in parallel on LSF
Used with SPS/LHC/HL-LHC/FCC-hh/FCC-ee impedance models
An individual simulation can last from a few seconds to several minutes, depending on the convergence criterion and the number of points in the impedance file
Number of users: in the order of 10 people
ABP-CWG

4. DELPHI
No performance limit reached so far: current hardware infrastructure matches our needs
Open-source code, maintained on CERN IRIS repository, hosted on CERN’s GitLab. A mirrored repository is also available for external users without a CERN account
Documentation on the code usage is available in the repository
For the theoretical development, see N.Mounet presentation
No further developments on the C++ core
Foreseen future evolutions for the Python code:
Check the functions associated to Landau damping
Include new parameters such as Q’’, linear coupling, detuning, space-charge (will need some work on the theory side)
Implement job submission to HTCondor to address LSF degrading performance
Rewrite some features to be more Object Oriented
Code available in GitLab + Mirror
Add some plots (benchmarks) to visualize the output ?
Future: migration to HTCondor ?
Documentation available inside repository
License
ABP-CWG

5. DELPHI N.Biancacci L.Carver et al.
Octupole current threshold with the LHC impedance model:
Single bunch, fixed intensity
Scan in chromaticity and damper gain
The eigenvalues given by DELPHI have been postprocessed to give the current threshold
TMCI threshold with the LHC impedance model:
Single bunch, zero chromaticity, no damper
Scan in bunch intensity
Real part of the eigenvalues on the upper plot
Imaginary part on the lower plot
TMCI threshold
Octupole current vs chroma
Effect of damper
ABP-CWG

6. MOSES
MOde-coupling Single bunch instability in an Electron Storage ring
Code written by Y.H.Chin
Semi-analytic Vlasov solver: computes the complex coherent frequency shifts caused by a beam coupling impedance
Output the eigenvalues, giving the modes frequency shifts and the associated growth rates
The user inputs the number of azimuthal and radial modes computed: there is no convergence check on the eigenvalues
Can only treat a resonator impedance, with a Gaussian longitudinal distribution
Include chromaticity, single bunch only
Used for studies with simple impedance models
Possible benchmark
ABP-CWG

7. MOSES Code written in Fortran
A Windows executable is provided, as well as a software to plot the results
Source code is also provided
The code runs on Windows (tested with Windows 7)
No parallelization strategy implemented
Possible to perform a scan in bunch intensity.
If only a few modes are computed, an individual simulation is almost instantaneous
A scan in bunch intensity can last up to several minutes depending on the number of steps
No performance limitation
Code freely available on Y.H. Chin page, no information on licensing
Documentation on the code usage is provided with the sources
The documentation includes some theoretical development
ABP-CWG

8. MOSES
B.Salvant
Scan in bunch intensity performed with MOSES (red) and HEADTAIL (white), for the SPS impedance model (broad-band resonator, 𝑓 𝑟 =1𝐺𝐻𝑧, 𝑅 𝑠 =10𝑀Ω/𝑚, 𝑄=1)
ABP-CWG

9. Nested Head-Tail Vlasov Solver
Author: A.Burov
What’s included:
Single-bunch or single bunch + coupled bunch
Damper: flat or arbitrary frequency response
Beam-beam: several IP’s
Landau damping: analytic estimate, small betatron tune spread, no synchrotron tune spread
Out of scope:
Intra-beam scattering
Synchrotron radiation damping
Space charge
Mathematica notebook
Runs on any laptop or desktop, supporting Mathematica v. 10
Reference:
ABP-CWG

10. NHT: Physics
Solving for eigenvalues and eigenfunctions of transverse modes α and R(r).
Air-bag approximation:
Unperturbed solution
Nested air-bags: equal population
A. Chao, Physics of collective beam instabilities, 6.6 Transverse modes
ABP-CWG

11. NHT: Simulation procedure
Generate nested air-bags
Compute the impedance matrix of Z = Z(no modes, chromaticities)
Most time-consuming, overnight on a laptop
Can be saved to be reused in future studies
Solve for the eigenvalues
One study scenario takes ~ 1 sec on a laptop
Can make a scan through gains and chromaticities
Growth rates of individual modes
Most unstable mode vs gain and chromaticity
ABP-CWG

12. Backup slides
ABP-CWG

13. Vlasov equation Phase space coordinates Phase space distribution
Transverse
Longitudinal
Phase space distribution
Vlasov equation: Phase space conservation
From A.W. Chao
ABP-CWG

14. Perturbation formalism
Perturbed phase space distribution
Unperturbed distribution
Perturbation term
Transverse distribution
Longitudinal distribution
Assume that a mode is developing
At complex frequency
While staying close to the unperturbed distribution
Re: mode frequency shift
Im: mode growth rate
ABP-CWG

15. N.Mounet
ABP-CWG

16. Decomposition in Laguerre polynomials
Decompose the longitudinal functions
Unperturbed longitudinal distribution
Perturbed longitudinal distribution
Orthogonal polynomials
ABP-CWG

17. Eigenvalues problem Eigenvalues problem
Vlasov equation: how the particle distribution evolves + Perturbation formalism: how the disturbance is treated + Laguerre decomposition : how to treat the problem
Eigenvalues problem
Eigenvalues
Impedance and damper matrix
Eigenvectors
l,n: azimuthal and radial mode numbers
time
time
From A.W. Chao
ABP-CWG