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DELPHI and Vlasov solvers used at CERN

DELPHI solver computes complex coherent frequency shifts due to beam impedance and damper. MOSES solver analyzes mode-coupling instability in an electron storage ring. Both solvers offer detailed insights for CERN studies.

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DELPHI and Vlasov solvers used at CERN

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  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 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 functionsare 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 (http://www.netlib.org/lapack) • BLAS (http://www.netlib.org/blas) • GSL (http://www.gnu.org/software/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-eeimpedance models • An individual simulationcan 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 CERNIRIS repository, hosted on CERN’s GitLab. A mirrored repository is also available for external users without a CERN account • https://gitlab.cern.ch/IRIS/DELPHI • https://gitlab.com/IRIS_mirror/DELPHI_mirror • Documentation on the code usage is available in the repository • For the theoretical development, see N.Mounet presentation • https://espace.cern.ch/be-dep/ABP/HSC/Meetings/DELPHI-expanded_Part2.pdf • No further developments on the C++ core • Foreseen future evolutions for the Python code: • Check the functions associated to Landaudamping • Include new parameters such as Q’’,linearcoupling, detuning, space-charge (will need some work on the theory side) • Implement job submission to HTCondorto address LSFdegradingperformance • Rewrite some features to be more ObjectOriented 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 beenpostprocessedto 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 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 shiftsand the associated growthrates • 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 ABP-CWG

  7. MOSES • Code written in Fortran • A Windows executable is provided, as well as a software to plot the results • Source codeis 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 individualsimulation 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 • http://abci.kek.jp/moses.htm • 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, ,,) ABP-CWG

  9. Nested Head-Tail Vlasov Solver • Author: A.Burov • What’s included: • Single-bunchor 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 • Mathematicanotebook • Runs on any laptop or desktop, supporting Mathematica v. 10 • Reference: • https://arxiv.org/ftp/arxiv/papers/1309/1309.0044.pdf ABP-CWG

  10. NHT: Physics • Solving for eigenvalues and eigenfunctions oftransverse 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(nomodes, 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 • Vlasov equation: Phase space conservation Longitudinal Transverse • Phase space distribution From A.W. Chao ABP-CWG

  14. Perturbation formalism • Perturbed phase space distribution • Assume that a mode is developing • At complex frequency • While staying close to the unperturbed distribution Perturbation term Unperturbed distribution Longitudinal distribution Transverse 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 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 anddamper matrix Eigenvectors l,n: azimuthal and radial mode numbers time time From A.W. Chao ABP-CWG

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