Recent Developments on Beam Dynamics and Wake Field Simulations at TEMF

Transcription

1 Recent Developments on Beam Dynamics and Wake Field Simulations at TEMF Institut für Theorie Elektromagnetischer Felder (TEMF) Technische Universität Darmstadt Joint DESY and University of Hamburg Accelerator Physics Seminar December 18, 27, DESY, Hamburg Technische Universität Darmstadt, Fachbereich Elektrotechnik und Informationstechnik Schlossgartenstraße 8, Darmstadt, Germany - URL:

2 Outline Wakefield Computations PBCI Wakefield Simulations at TEMF Self-Consistent PIC Simulations Higher Order Finite Elements Current Work V-Code Basic principle based on the moment approach Implementation using symbolic algebra transformation 1

3 Wakefield Computations Motivation for improved wakefield codes 1. A new generation of LINACs with ultra-short electron bunches - bunch size for ILC: 3 µm - bunch size for LCLS: 2 µm 2. Geometry of tapers, collimators far from rotational - 8 rectangular collimators at ILC-ESA in the design process - 3 rectangular-to to-round transitions in the undulator of LCLS 3. Many (semi-) analytical approximations become invalid - based on rotationally symmetric geometry - low frequency assumptions (Yokoya, Stupakov) 2

4 Wakefield Computations beam view side view mm 38.1mm 2.75mm 38.5mm 17.65mm mm Courtesy: N. Watson, Birmingham 1 m 38.5mm ILC-ESA collimator #3 ILC-ESA collimator #8 bunch length 3µm collimator length ~1.2m catch-up distance ~2.4m

5 Wakefield Computations vacuum vessel gap outgoing pipe bunch length taper length 1cm 5cm tube shielding step PITZ diagnostics double cross bunch length bunch width structure length 2.5mm Tapered III elliptic pipe vacuum slots 2.5mm 325mm 4 Courtesy: R. Wanzenberg, DESY

6 There is an actual demand for: Wakefield Computations 1. Wake field simulations in arbitrary 3D-geometry 3D-codes 2. Utilizing large computational resources parallelized codes 3. Specialized algorithms for long accelerator structures moving window dispersion free codes 5

7 Wakefield Computations An (incomplete) survey of available codes years 22 5 years 27 Dimensions Nondispersive Parallelized Moving window BCI / TBCI 2.5D No No Yes NOVO 2.5D Yes No No ABCI 2.5D No No Yes MAFIA 2.5/3D No No Yes GdfidL 3D No Yes No Tau3P 3D No Yes No ECHO 2.5/3D Yes No Yes CST 3D No No No PBCI 3D Yes Yes Yes NEKCEM 3D No Yes No

8 Based on the FIT discretization Numerical Method 7 A A V V r r uur r E ds = µ H da t A r r ur r r H ds = ε E + J da t A uur r µ H da= ur r εe da= V ρ dv FIT ) d Ce = M µ h ) dt ) d ) ) Ch = M ε e + j dt SM % ) e A moving window algorithm is not directly applicable: - Stability limit t < x x / c - Longitudinal dynamics has numerical dispersion ε = SMµ h ) = Important works of I. Zagorodnov on TE/TM splitting for the ECHO code q

9 Numerical Method 8 Using the conventional leapfrog time integration ) n+ 1/2 1 T ) n 1/2 n e t e ) 1 M 1 ε C t ε n+ 1 = T n M j ) t µ t ) h M C 1 Mµ CMε C h Behavior of numerical phase velocity vs. propagation angle c t 1 σ : = z Stable but large dispersion z c t σ : = = 1 z 3 33 No dispersion but unstable z

10 Using a split-operator method (LT) Numerical Method Behavior of numerical phase velocity vs. propagation angle c t σ : = = 1 z No longitudinal dispersion z z Neutrally stable Note: The elimination of longitudinal dispersion does not necessarily improve accuracy. Splitting error may become important. Non-split algorithms are currently under investigation.

11 Example: Tapered transition for PETRA III Parallelization 1 moving grid window Grid points Total Min Max Dev. < 1.% Number of Grid Points Distribution of grid points Domain partitioning pattern for 7 processors Processor Number

12 Parallelization Parallel performance tests Number of grid cells 1E+6 Ideal speedup Ideal 1E+6 cells Number of Processors Speedup

13 Parallelization Parallel performance tests Number of grid cells 1E+6 Ideal speedup Ideal 1E+6 cells Number of Processors Speedup

14 Parallelization Parallel performance tests Number of grid cells 5E+6 Ideal speedup Ideal 5E+6 cells Number of Processors Speedup

15 Parallelization Parallel performance tests Number of grid cells 1E+6 Ideal speedup Ideal 1E+6 cells Number of Processors Speedup

16 Parallelization Parallel performance tests Number of grid cells 2E+6 Ideal speedup Ideal 2E+6 cells Number of Processors Speedup

17 Parallelization 16 Speedup Parallel performance tests 1e5 Grid points 1e6 Grid points 5e6 Grid points 1e7 Grid points 2e7 Grid points Ideal Number of Processors TEMF Cluster: on 2 INTEL 3.4GHz, 8GB RAM, 1Gbit/s Ethernet Network

18 y Modal Termination of Pipes z = Modal approach 17 C C1 direct 1 z+ s Wz() s = dzez(, z t ) Q = c = n modal n 1 i( ω / c) s z(, ) ω n( ω) i ω c k ω e x y d C e ( / zn, ( )) 1 z+ s 1 n = dz Ez(, z t ) ez(, x y) Wn() s Q = c Q = + = = z s i n ikn ( ω) z c dz Ez( x, y, z, t ( z s)/ c) dz dω Cn( ω) ez( x, y) e e ω + n C n-th (TM) mode contribution z n spectral coefficient of n-th (TM) mode Indirect methods for wake potential integration, I. Zagorodnov et al, PRSTAB 9, 6 Eigenmode expansion method in the indirect calculation of wake potential in 3D in structures, X. Dong et al, ICAP 6

19 Modal Termination of Pipes Using modal reconstruction in long intermediate pipes 18 diagnostics cross injector section 1mm step full Time Domain E z / [kv /m] FD analysis reconstruction lowest eigenmodes bunch bunch In a typical simulation ~3 modes

20 ILC-ESA Collimators 19 Wz / [V / pc] ILC-ESA collimator # Convergence vs. grid step σ / z = 2.5 σ / z = 5 σ / z = 1 σ / z = 15 bunch size 3µm no. of grid points ~45M no. of processors 24 simulation time s / σ 85hrs Moving window: 3 mm length

21 38.5 mm W z / [V / pc] 2 ILC-ESA Collimators ILC-ESA collimator #8 Direct vs. transition wakes bunch 15.5 mm mm 38.1 mm 2.75 mm mm 3 Direct potential (z = 15mm) Transition potential Steady state potential s / σ

22 Tapered Transitions for PETRA III PETRA III Tapers 21 The two variants of the vacuum-to-undulator tapered transition Wake Computations for Undulator Vacuum Chambers of PETRA III, R. Wanzenberg et al, PAC 7

23 Tapered Transition for PETRA III PETRA III Tapers 22 Variant 1 Variant 2 Variant / Code Variant1 / MAFIA Variant1 / PBCI Variant2 / PBCI k / (V/nC) k (1) / (V/pC m) k / (V/pC m) factor ~1.8 higher than PBCI due to insufficient resolution

24 23 PITZ Photoinjector PITZ diagnostics double cross

25 PITZ diagnostics double cross PITZ Photoinjector E z / [kv /m] bunch size 2.5mm σ / z 3 no. of grid points ~5e6 no. of processors 24 simulation time 32hrs Vacuum Vessel Gap Beam Tube bunch Tube Shielding Step

26 25 PITZ Photoinjector PITZ diagnostics double cross Wake field impact of the single components 1 gap (1 mm) -1 cross with shielding tube step (no taper) -2 Wz / [V / pc] full model -3 no shielding tube s [mm]

27 26 PITZ Photoinjector PITZ diagnostics double cross Laser mirror

28 PITZ Photoinjector PITZ diagnostics double cross WxHsL mmd Transversal wake potentials for different beam offsets 2mm 1. mm. mm 1. mm 2. mm 3. mm 4. mm 5. mm 6. mm 7. mm 8. mm 9. mm 1. mm 11. mm

29 PITZ Photoinjector xhmml Optimal offset at 8mm from tube axes κxhvêpc mml

30 TESLA / HOM coupler TESLA 9-cell cavity E z / [kv /m] bunch length 1mm bunch charge 1nC cavity length ~1m no. of grid points ~76M no. of processor cores 48 simulation time ~4hrs bunch

31 3 TESLA / HOM coupler TESLA 9-cell cavity Longitudinal wake potential 5 PBCI ECHO2D -5 Wz / (V / pc) s / σ

32 31 TESLA / HOM coupler HOM / HOM-RF coupler (present DESY design) Upstream coupler HOM Downstream coupler RF + HOM Beam view (from Igor)

33 TESLA / HOM coupler Wx / [kv / nc] Wy / [kv / nc],5 -,5 -,1 -,15 -,2 -,25 -,3 Grid points / sigma ,35 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm],5 -,5 -,1 -,15 -,2 Grid points / sigma Wz / [kv / nc],5 -,5 -,1 -,15 -,2 -,25 Upstream coupler Grid points / sigma ,25 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm] -,3 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm]

34 33 TESLA / HOM coupler Upstream coupler Wt / (kv / nc ) ECHO3D Wx Wy bunch length 1mm coupler length 6mm bunch charge 1nC PBCI grid points (max) ~65M processor cores 48 simulation time ~24hrs s / σ

35 34 TESLA / HOM coupler Downstream coupler Wt / ( kv / nc ) -.15 Wx Wy PBCI Wz / ( kv / nc ) PBCI ECHO3D s / σ s / σ

36 35 TESLA / HOM coupler Present DESY Design Longitudinal wake potential Grid points / sigma 12 Bunch Beam view ,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm] Wz / [kv / nc]

37 36 TESLA / HOM coupler Present DESY Design,1 Transversal wake potentials,5 Bunch Wx Wy -,5 -,1 -,15 Is this result correct? Beam view -,2 -,6 -,4 -,2,2,4,6 s / [cm] Wx / [kv / nc]

38 TESLA / HOM coupler Simplified Design 37 Wx / [kv / nc] -,5 -,1 -,15 -,2 -,25 -,3 -,35 -,4 Couplers connected by homogeneous beam tube Transversal wake potentials W x W x 1 +W x 2 -,45 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm] Beam view

39 TESLA / HOM coupler Simplified Design 38 Wy / [kv / nc] -,5 -,1 -,15 -,2 -,25 -,3 -,35 -,4 Couplers connected by homogeneous beam tube Transversal wake potentials Expected additive potentials W y W y 1 +W y 2 -,45 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm] Beam view

40 TESLA / HOM coupler,1,5 Bunch W x W y 39 Wx / [kv / nc] -,5 -,1 -,15 -,2 -,25 -,3 -,35 -,4 Wx / [kv / nc] -,5 -,1 -,15 -,2 -,6 -,4 -,2,2,4,6 s / [cm] -,5 W x W 1 2 x +W x -,1 + -,3 Preliminary: -,35 Transversal wakes -,4 are affected -,45 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm] Wy / [kv / nc] -,15 -,2 -,25 by the crymodule? W y W y 1 +W y 2 -,45 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm]

41 4 TESLA / HOM coupler FNAL Design couplers are interchanged,3,2,1 Wx Wy Transversal Longitudinal -2 Wz -,1 -,2 -, ,4 -,5 -,6 -,5 -,4 -,3 -,2 -,1,1,2,3,4,5 s / [cm] s / [cm] Wt / [kv / nc] Wz / [kv / nc] ,5 -,4 -,3 -,2 -,1,1,2,3,4,5

42 Wakefield Computations Current / future work on wakefield codes at TEMF 1. Non-split algorithms for moving windows: A new finite volume based algorithm is available. 2. Implementation of impedance boundary conditions for resistivity wakes. 3. Handling of long (super-periodic) structures with a modal approach 41

43 Self-Consistent Simulations The European X-Ray Laser Project XFEL at DESY, Hamburg 42 x RMS / mm electron gun Beam size z-position / m z < 1.6 m soft beam diagnostics x Emit / mm mrad booster Beam emittance frozen emittance injection phase z-position / m

44 Self-Consistent Simulations HF-Noise in PIC Simulations (PITZ Gun) 43 Ez / V/m Longitudinal electric field srength / (MV / m) -9,3,6,9,12,15, z / mm 9 distance / cm

45 Self-Consistent Simulations Numerical approaches for time domain simulations Time-adaptive mesh refinement On the fly -discretization of bunch and structure Higher order FEM on unstructured grids Accuracy Local mesh refinement 44

46 45 Self-Consistent Simulations Examples of local mesh refinement e-bunch Discretization of the RF PITZ gun 32-fold refinement 1.5 m

47 The DG-FEM Method 46 The DG-FEM formulation for Maxwell Equations εe H = t µ H + E = t Approximate field components with high order, piecewise polynomial basis functions. E (,) r t E (,) r t E () t ϕ () r = x x x j; p j; p j, p E (,) r t E (,) r t E () t ϕ () r = y y y j; p j; p j, p M H (,) r t H z (,) r t z H () t ϕ () r = z j; p j; p j, p Discretization of computational domain Er (,) t = Ej; p() t ϕ j; p() r j, p Hr (,) t = H j; p() t ϕ j; p() r j, p I j Ω

48 The DG-FEM Method Test approximate equations (e.g., Ampere s law) 47 I I I I I i i i i i ( ) ( ) 3 εe 3 3 εe 3 3 d r ϕ d r Hϕ = d r ϕ d r Hϕ + d r ϕ H= t t ε E d r ϕ d r n F ϕ+ d r ϕ H = numerical flux ( ) ( ) H t I I I i i i Ansatz: centered flux for tangential fields at element boundaries: 1 + n F E = n (, t) + (, t), 2 E r E r I i 1 + n F H = n (, t) + (, t) 2 H r H r I i n F H I n + Hr (, t) Hr (, t) j

49 The DG-FEM Method Matrix representation of semidiscrete equations 48 d dt M ε Me ε C e + = T Mh µ C h ex e= ey, ez h h x = hy h M ε x = Mε, y Mε z z skew-symmetric symmetric, positive, (block-) diagonal M semidiscrete DG formulation Recovers FDTD for piecewise constant basis functions, on Cartesian grids M µ x = M, y M µ z µ µ symmetric: C T = Pz Py C= Pz P Py Px C x

50 49 The DG-FEM Method Periodic solution in resonator box E z 1,5,9,75,6,45,3 Dispersion error for EM 111 mode First Order Second Order Third Order Fourth Order H y Normalized RMS-Error,15 1,5 3 4,5 6 7,5 9 1, ,5 15 Time / [A.u.]

51 The DG-FEM Method Dispersion analysis. The P x P x P -case 5 ~ grid points / wave length Exact k D=pê 8 k D=pê 4 k D=pê 2 k D=p wsinhjl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ck y-axis φ x-axis w coshjl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ck

52 The DG-FEM Method Dispersion analysis. The P 1 x P 1 x P 1 -case k D=pê2 Mode #1 Mode #2 Mode #3 Mode #4 wsinhjl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ck Dispersion curves in the xy-plane physical mode w coshjl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ck spurious mode Exact k D=pê 8 k D=pê 4 k D=pê 2 k D=p wsinhjl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ck w coshjl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ck 51

53 The DG-FEM Method 52 The Problem of Charge Conservation Define a (consistent) divergence operator as: d d d = ε = = dt dt dt ( Sd) ( SMe) ( SCh)? d d d = µ = = dt dt dt ( Sb) ( SM h) ( SCe)? Sufficient and necessary conditions for charge conservation: SC= [ ] ( x y z ) S= P P P Continuity equations for electric and magnetic charges: (source free case) Px, P y = Px, P z = Py, P z = pairwise commuting derivatives

54 53 Conservation of Charge The tensor product case Domain of influence of product derivative operators y y x P P x

55 54 Conservation of Charge The tensor product case Domain of influence of product derivative operators y x y P P x

56 Conservation of Charge 55 The tensor product case y x P P y x Domain of influence of product derivative operators Conservation of discrete charges on regular grids (PIC). Gjonaj, Lau, Weiland, ICEAA 27 P P x y

57 Self-Consistent Simulations Example of a long-distance PIC simulation with DG-FEM (later ) Current work on PIC codes at TEMF 1. Application for the PITZ injector 2. Development of moving grid algorithms (S. Schnepp) 3. Combined (hp-) grid / order refinement 4. Dynamic parallel partitioning for higher order 56

58 V-Code: Motivation 57 Basic principle Description of the particle distribution by moments Distribution: First order moments: r r x = x f dd p x y z z position momentum Second order moments: 2 r r σ x = x x f dd p ( ) 2 σ z σ σ x, y z

59 Motivation Evolution of the moments in time 58 Bunch parameter at different time instants x y t= t t= t σ zt, = t t = t : t = t : 1 z t = t σ σ Simplified description t = t x,... σ,... x t x, t= t yt, = t t = t + t x,... σ,... x t t t x y t= t1 t= t1 t t σ zt, = t1 z t = t 1 t = t 1 x,... σ,... x σ σ x, t= t1 yt, = t1 z z

60 Possible ensemble parameter (1) Motivation 59 Raw moments i j k r r µ ijk ( xyz,, ) = x y z µ = µ f dd p Order : Property µ = 1 µ = x, µ = y, µ = z Order 1: Order 2: Order 3: µ = x, µ = y, µ = z µ 11 = xy, µ 11 = xz, µ 11 = yz K µ ( x a, y b, z c) µ ( x, y, z) ijk ijk (not translatory invariant)

61 Possible ensemble parameter (2) Motivation 6 Centralized moments i j k r r σ ijk ( xyz,, ) = x x y y z z µ = µ f dd p Order : Property ( ) ( ) ( ) σ = 1 Order 1: σ 1 =, σ1 =, σ1 = σ Order 2: Order 3: ijk σ = σ, σ = σ, σ = σ x 2 y 2 z σ = σ, σ = σ, σ = σ K 11 xy 11 xz 11 yz ( x a, y b, z c) = σ ( x, y, z) ijk (variance) (covariance) (translatory invariant)

62 Motivation 61 Appropriate moments in 6-D phase space Order 1: M µ = µ µ { xyzp,,, x, py, pz} ( M,,,,, ) x M y Mz M p M p M p x y z Order 2: M µν = ( µ µ ) ( ν ν ) Order 3: Order 4: M M M M M M M yx M M M M M Mzx Mzy Mzz M M M M pxxm pxym pxzm M M M pyxm pyym pyzm pyp M M x M pz xm M M M M K K xx xy xz xpx xpy xpz yy yz ypx ypy ypz zpx zpy zpz px px pxpy pxpz pypy pypz pzy pzz pzpx pzpy pz pz { xyzpx py pz} µν,,,,,, ( 6) w 1 C 6 = = 6 1 ( 7) w 2 C 6 = = 2 ( 8) 21 w 3 C 6 = = 56 3 ( 9) w 4 C 6 = = Σ6 Σ27 Σ83 Σ29

63 62 Motivation Number of moments in 6-D phase space Order Moments per order Total number of moments

64 Ensemble Model Method to solve the VLASOV equation Basic formulation Moment approach Integration over phase space reduces PDE to ODE Approximation: limited amount of moments used to describe the particle distribution function time evolution 63

65 Ensemble Model 64 Evolution of raw and centralized moments in time µ r r = µ f dd p µ τ τ µ f r r µ f r r = dd p = f + µ dd p τ τ τ r r p r r = f grad r ( µ ) + grad r ( µ ) d dp p τ τ r r r p F F r r µ grad r ( f ) + grad r p( f ) + f div r dd 2 p p 2 γ mc mc r µ p = grad r ( µ ) + grad r ( µ ) p τ γ r r p F + grad r ( µ ) + grad r p( µ ) 2 γ mc { ( ) 2 } 2 xyzp,,, x, py, pz, x, K, x x, K r F mc 2 τ = ct r r, p ( f ) lim =

66 Ensemble Model Evolution of raw and centralized moments in time 65 General form: r µ p = grad r ( µ ) + grad r ( µ ) p τ γ r r p F + grad r ( µ ) + grad r p( µ ) 2 γ mc x 2 r F mc First component, order 1: Aim: x px Expand all arguments of the right = τ γ hand side such that the resulting p expression can be interpreted as x F = a moment description form τ mc 2 τ = ct

67 Ensemble Model 66 Incorporation of the influencing forces TAYLOR series expansion F µ F constant ν ν F ν µ = = a linear F = ν a F ν Fν = bµ F ν F µ F ν ν µµ µ quadratic = F = cm = bm µ F = c M ν F ν µµ 2 ν = cµ Series expansion of the forces for all beam line elements Dipole magnets, quadrupoles, solenoids, resonators,... µµµ µ

68 Linear space charge model Ensemble Model 67 r r F Configuration r r r r F r P r ( ) G r r 2γσ z eq 2 σ x σ + y γ V homogeneous space charge density Q G r 1 = ( 1 exp ) 1 γ / u ( ) ( u) G u r r Calculation r r Q P Q dq

69 Implementation Implementation using a symbolic algebra program 68 Mathematica (sequence of algebraic transformations) Visual.cpp Tracking.cpp CDipole.cpp CQuadrupole.cpp C++ Compiler (user interface, time integration, administration of BLE) V-Code.exe

70 Implementation Decomposition in individual beam line elements V-Code Gun Globals Macros CEnsemble CField CVisual BeamLineElements DriftSpace Solenoid BeamLineElements CRFGun CDriftSpace CSolenoid CRFCavity CSteerer CBendMagnet DriftSpace Cavity l l l l l K z 69

71 7 Implementation Solving the assembled moment equations Time dependent ensemble parameter Initial condition and time stepping technique

72 71 V-Code user interface Implementation

73 72 Layout Application Courtesy: Georg Hofstaetter, Cornell University dc gun solenoid Bunch: T rms = ps XY rms = mm Q bunch = nc z/m buncher cavity

74 73 Comparison Application transversal bunch dimension ASTRA V-Code M2 DG-FEM Order 5 Xrms / mm

75 74 Comparison Application transversal bunch dimension ASTRA V-Code M4 DG-FEM Order 5 Xrms / mm

76 75 Comparison ASTRA Application longitudinal bunch dimension V-Code M2 DG-FEM Order 5 Zrms / mm

77 76 Comparison Application longitudinal bunch dimension ASTRA V-Code M4 DG-FEM Order 5 Zrms / mm

78 77 Comparison V-Code M2 ASTRA Application transversal emittance DG-FEM Order 5 Xemittance / mm mrad

79 78 Comparison ASTRA Application transversal emittance V-Code M4 on 1 CPU ~seconds DG-FEM Order 5 on 2 CPU ~1 week Xemittance / mm mrad

80 Current Work and Outlook Current Work related to V-Code Implementation of multi-ensemble formulation Incorporation of VCode into the control system at the S-DALINAC to assist operation Extension of the BLE library to enable alpha-magnet simulation for the new S-DALINAC injector Outlook Improved and consistent space charge model for single and multi-ensemble formulation Automatic ensemble decomposition 79

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