In order to compare simulation results performed with different codes a simulation test bench is defined and four different codes are compared with each other to validate their compatibility

Biarrotte, JL

Kaltchev, D

Schuh, M

Tückmantel, J

CERN Document Server

sLHC-Project-Note-001025/01/2010sLHC-Project-Note-00102010-01-25marcel.schuh@cern.chCode Benchmarking of Higher Order Modes simulation codesM. Schuh1, J. Tu¨ckmantel1, J.-L. Biarrotte2, D. Kaltchev31CERN, Geneva, Switzerland, 2 IPNO, Orsay, France, 3 TRIUMF, Vancouver, B.C., CanadaKeywords: SPL, superconducting cavities, HOM, beam dynamics, benchmarkSummaryIn order to compare simulation results performed with different codes a simulation test bench isdefined and four different codes are compared with each other to validate their compatibility.1 IntroductionA mini-workshop on the subject of Higher Order Modes (HOMs) in a superconducting protonlinac was held at CERN on the 25th and 26th June 2009 [1, 2]. The purpose of this workshopwas to discuss more specifically the effects of HOMs on beam dynamics in the SPL [3–5] andcomparable accelerators and to provide guidelines for the specification of HOM dampers, ifnecessary.Several beam dynamic studies [6–9] done with different codes (see Tab. 1) were presented. Mostof the codes are based on [10], where a drift kick model is used to simulate the longitudinalbeam dynamics. In this model the point-like bunches are accelerated and get an addition kick bya HOM. This causes an energy error which leads to an arrival time error in the next cavity sincethe particles are not highly relativistic. This dynamic behaviour from one cavity (m) to the next(m+1) can be expressed by the following equationsdE(m+1) = dE(m) + dURF + dUH (1)dt(m+1) = dt(m) + (dt/dE)(m)E · dE (2)where dURF is the RF-error caused by the arrival time (phase) error in the cavity and dUH theeffect of the HOM. The arrival time error depends on the energy error and causes a time of flighterror which is given by(dt/dE) = − Lc ·m0c2 · (γ2 − 1)3/2 (3)This is an internal CERN publication and does not necessarily reflect the views of the CERNmanagement.using the relativistic γ, the drift space L between the cavity centre to the next cavity centre, andthe particle rest mass m0. The effect of the HOM isdUH = q(<(VH) cos(ωHdt)−=(VH) sin(ωHdt)− 12Vb)(4)where VH is the HOM voltage, ωH its angular frequency and Vb the voltage induced by thebunch itself. The HOM is excited by the beam itself and the evolution of VH from bunch n tobunch n+ 1 can be expressed asV n+1H = (VnH + Vb) · exp(iωHTb) · exp (−ωHTb/(2Qex)) (5)with an nominal time between two bunches Tb and the damping of the HOM in terms of Qex.Table 1: Benchmarked codesCode Author Implementation Code based onJT J. Tu¨ckmantel (CERN) C, Cocoa -MS M. Schuh (CERN) C++, ROOT JTJB J.-L. Biarrotte (IPN Orsay) Matlab JTDK D. Kaltchev (TRIUMF) Mathematica TALOBBU[11]In order to compare and classify the results of the different codes a benchmark was performed.All codes were used to simulate a generic linac described in [10] with same set of input param-eters and the same bunch pattern. Afterwards the output was compared bunch by bunch.2 Benchmark setupFor the sake of simplicity a test setup with the generic linac model of [10] is used in all simu-lations. It consists of 150 equal cavities with a constant drift space L in between. Each cavityprovides an accelerating voltage VRF = 20.7 MV which correspond to Vacc = 20.0 MV in op-eration1). The focus is set to the beam HOM interaction and effects like changing E0T with βor RF errors are neglected. One HOM with constantR/Q andQex is present in each cavity. TheHOM frequency has a certain cavity to cavity spread which is generated once using a randomnumber generator with a Gaussian distribution (fHOM ,σfHOM ) and then used in all simulations.The same is done with the input charge error pattern2) of the bunches. In total one pulse with50,000 bunches is tracked through the linac with no deviation in energy and phase at the en-trance. The output energy and phase error of each bunch in this pulse is then compared for allcodes using separate scatter plots for energy and phase. All simulation parameters of this setupare listed in Tab. 2. In this setup only the longitudinal behaviour is tested and protons are usedinstead of H−.1)The linac operates with a synchronous phase of φ = −15◦: Vacc = VRF cos(φ)2)10% white noise bunch to bunch charge scatter.2Table 2: Benchmark setup parametersParameter Value VarianceCavity cavities 150 -L [m] 1.75 -fRF [MHz] 704.4 -Vacc [MV] 20.7 -φ [deg] -15 -HOM fHOM [MHz] 2283.8 0.1(R/Q) [Ω]† 100 -Qex 108 -Beam bunches 50,000 -Tb [ns] 1000/352.2 ≈ 2.84 0Ib [mA] 400 40Win [MeV] 150 0Wout [MeV] 3150 0dtin [ps] 0 0†linac definition3 Benchmark resultsIn total 4 codes listed in Tab. 1 were compared with each other. The code JT is chosen asreference for this comparison. In case of perfect agreement of the codes there should be nodeviation in the scatter plots (see. Fig. 1-3) which correspond to a straight line with slope equalone.The output of code MS (Fig. 1) shows very small deviation in energy and phase which are mostlikely due to numerical imperfections in the implementation.In the code JB (Fig. 2) the drift space after the last cavity is not modelled. The output datais generated at the last cavity. The drift space afterwards leads to a change in the arrival time(phase) error, but no change in the energy error. This explains the perfect match in the energybut mis-match in the phase.All codes up to now are based on J.T.’s code. The code of DK is a modified version of TALOBBU[11]. TALOBBU was used at SNS3) for such studies. Looking at the output of code DK (Fig. 3)shows, that this code behaves like the others and models the same physics4). In this code also nodrift space is modeled after the last cavity, which leads to a small deviation in the phase error.3)Superconducting proton linac for the Spallation Neutron Source at the Oak Ridge National Laboratory4)DK performed also a benchmark of TALOBBU against R.L. Gluckstern et. all, ”Cumulative Beam Breakup inRF Linacs”. There was only a good agreement found, if the first bunch passes the linac unaffected. Further detailssee [12]3Comparing DK with JB shows a perfect match (Fig. 4). The relative error is in the order of 10−4,which corresponds to the input value accuracy. E [MeV]∆JT: -1.5 -1 -0.5 0 0.5 1 1.5 E [MeV]∆MS: -1.5-1-0.500.511.5Scatter Plot Energy: MS vs JT [ps]φJT: -3 -2 -1 0 1 2 3 [ps]φMS: -3-2-10123Scatter Plot Phase: MS vs JTFigure 1: Comparison of the output phase space of MS against JT. Minor deviation in energy and phasevisible. Relative error: 10−3 E [MeV]∆JT: -1.5 -1 -0.5 0 0.5 1 1.5 E [MeV]∆JB: -1.5-1-0.500.511.5Scatter Plot Energy: JB vs JT [ps]φJT: -3 -2 -1 0 1 2 3 [ps]φJB: -3-2-10123Scatter Plot Phase: JB vs JTFigure 2: Comparison of the output phase space of JB against JT. No deviation in energy and only slightdeviation in phase visible.4 E [MeV]∆JT: -1.5 -1 -0.5 0 0.5 1 1.5 E [MeV]∆DK: -1.5-1-0.500.511.5Scatter Plot Energy: DK vs JT [ps]φJT: -3 -2 -1 0 1 2 3 [ps]φDK: -3-2-10123Scatter Plot Phase: DK vs JTFigure 3: Comparison of the output phase space of DK against JT. No deviation in energy and only slightdeviation in phase visible. E [MeV]∆JB: -1.5 -1 -0.5 0 0.5 1 1.5 E [MeV]∆DK: -1.5-1-0.500.511.5Scatter Plot Energy: DK vs JB [ps]φJB: -3 -2 -1 0 1 2 3 [ps]φDK: -3-2-10123Scatter Plot Phase: DK vs JBFigure 4: Comparison of the output phase space of DK against JB. No deviation visible. Relative error:10−44 ConclusionAll codes show the same behaviour and model the same physics in a very well defined test setup.The chosen input data for HOM spread and bunch pattern are irregular random choices, hence5it is reasonable to conclude that the codes would also agree for any other random choice. Allresults of these codes can be treated equally as long as the used input parameters and used linacgeometry modelling is the same. The different results presented at the SPL HOM workshop aredue to different input parameters and not due to differences in the codes.This enables also the possibility of sharing future work between different groups with eachgroup using their own well understood code.The defined test bench can also be used for furthercomparisons with other codes and can be easily extended to a test bench for transversal codes,which is not discussed here.References[1] SPL HOM Workshop (CERN), http://indico.cern.ch/conferenceDisplay.py?confId=57247[2] A. Lombardi et al., ”Summary of the workshop on HOM in SPL, held at CERN June25-26.”, CERN-sLHC-Project-Note-0003.[3] R. Garoby et al., ”SPL at CERN”, in Proceedings SRF 2009, Berlin, 2009.[4] F. Gerigk et al., ”Higher Order Modes in the Superconducting Cavities of the SPL”, inProceeding of the Particle Accelerator Conference (PAC09), Vancouver, 2009.[5] M. Schuh et al., ”Higher Order Mode Beam Breakup Limits in the Superconducting Cavi-ties of the SPL”, in Proceedings SRF 2009, Berlin, 2009.[6] J. Tu¨ckmantel, ”HOMs”, at SPL HOM Workshop[7] M. Schuh, ”Studies for the SPL at CERN”, at SPL HOM Workshop[8] R. Baartman, ”Studies for the SPL at TRIUMF”, at SPL HOM Workshop[9] J.-L. Biarrotte, ”Studies for the SPL at IPN Orsay”, at SPL HOM Workshop[10] J. Tu¨ckmantel, ”Beam simulations with initial bunch noise in superconducting rf protonlinacs”, Phys. Rev. ST Accel. Beams 13, 011001[11] D. Jeon et al., ”Cumulative Beam Break-Up Study of the Spallation Neutron Source Su-perconducting Linac”, Nuc. Instr. A (2002) 495.[12] D. Kaltchev, ”BBU Simulation Notes”, http://trshare.triumf.ca/~kaltchev/CERN-HOM/notes/6

Code Benchmarking of Higher Order Modes Simulation Codes

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