CSRCSR computation with computation with arbitrary cross-section of arbitrary cross-section of

the beam pipethe beam pipe

L.Wang, H.Ikeda, K.Ohmi, K. Oide D. Zhou

3.14.2012KEK Accelerator Seminar

Work in progress

OutlineOutlineParabolic equation and general formulation of

the problemIllustration of the method: numerical examplesExcise of micro-wave instability simulation

with Vlasov solverConclusions

2

MotivationMotivation Parabolic equation has been solved in FEL, CSR, and Impedance calculations, etc Most present codes are limited for simple boundary, for instance, rectangular cross-section for CSR and

zero boundary for FEL (need large domain) CSR is important for Super-KEKB damping ring. To estimate the threshold of micro-wave instability

Impedance calculation Impedance calculation Gennady Stupakov, New Journal of Physics 8 (2006) 280

Axis ymmetric geometry

2D parabolic solver for 2D parabolic solver for Impedance calculation Impedance calculation

L. Wang, L. Lee, G. Stupakov, fast 2D solver (IPAC10)

0 200 400 600 800 1000-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

f (GHz)

ReZ

, Im

Z (k

)

Real, ECHO2Imaginary, ECHO2Real, FEM codeImaginary, FEM code

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

z (mm)

r (m

m)

0 200 400 600 800 1000-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

f (GHz)

ReZ

, Im

Z (k

)

Real, ECHO2-Imaginary, ECHO2

dot-lines: FEM code

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

z (cm)

r (cm

)Echo2, sigl=0.1mm

FEL (for example, Genesis by sven reiche)FELFEL

Set the field ZERO out the domain of interest

CSRCSRFor example, CSR in bend magnet (Tomonori Agoh, Phys. Rev. ST Accel. Beams 7, 054403 (2004)

•Agoh, Yokoya, PRSTAB 054403•Gennady, PRSTAB 104401•Demin, PH.D thesis•K. Oide, PAC09

Nonlinear OpticsNonlinear Optics

8

BUNCH LENGTH MEASUREMENTS WITH LASER/SR CROSS-CORRELATION*

GENERALITYGENERALITY

IF We neglect the 1st term

Advantages of FEM Advantages of FEM Irregular grids

Arbitrary geometryEasy to handle boundarySmall beam in a large domain (FEL in undulator)CPU (strongly depends on the solver)Accuracy(higher order element, adaptive mesh, symmetry, etc)

Disadvantage & Challenge:Disadvantage & Challenge: Complexity in coding (irregular grid, arbitrary geometry, 3D…)

Why Finite Element Method (FEM)?

Impedance ofImpedance ofGrooved surfaceGrooved surface

Adaptive method

Speedup and improve accuracy

Mesh of chamber & beamMesh of chamber & beam

•Arbitrary geometry of beam pipe

•Any shape of beam

CSR computation

12

Straight beam pipeStraight beam pipe

Straight beam pipe+ Bend magnet + straight beam pipe…

Assumptions of our CSR Assumptions of our CSR problemproblem

We assume perfect conductivity of the walls and relativistic particles with the Lorentz factor =

The characteristic transverse size of the vacuum chamber a is much smaller than the bending radius R (a<<R)

Constant cross-section of beam chamber

13

Fourier transformFourier transformThe Fourier transformed components of the

field and the current is defined as

14

Where k=/c and js is the projection of the beam current ontos. The transverse component of the electric field is atwo-dimensional vector . , The longitudinalcomponent is denoted by

rE~

)~,~(~ ry

rx

r EEE

rsE~

CSR Parabolic EquationCSR Parabolic Equation

15

00

22 ~22 ne

ski

Rxk

E

br

Rxk

ski

Rxk

EE

222 2~22

br EEE

Initial field at the beginning of the bend magnet

0)0(~2 srEAfter the bend magnet

0~22

r

ski E

00

2 1 bE

Required for Self-consistentcomputation

(Agoh, Yokoya, PRSTAB 054403)

)(2

)(2

42

0r

r

r

ebr

rr

rreE

),( yx

222yx

FEM equationFEM equation

16

JEDEM rr ~~

0 nE

3D problem (Treat s as one space dimension)

JEM r~

2D problem (Treat s as time)

It’s general case, for instance, the cross-section can vary for the geometry impedance computation. Problem: converge slowly

It converges faster, the stability need to be treated carefully

0 nE

Current status: Bend only

0

),,(1)( dssyxEI

kZ ccs

The ways to improve the The ways to improve the accuracyaccuracy

17

0

y

Ex

Eki r

yrx

sEHigh order element

The (longitudinal) impedance is calculated from the transverse field, which is nonlinear near the beam as shown late,

Fine girds on the curved boundary

0 nE…..

Test of the codeTest of the code

18

Rectangular Cross-section with dimension 60mmX20mm Wang Zhou k=10e3m-1, (142.2, 119.1) (140.6, 119.5) (@end of the bend)

Bend Length =0.2meter, radius =1meter

Sample 1Sample 1

19

Bend Length[m] Bending angle # of elements

B1 .74248 .27679 32

B2 .28654 .09687 38

B3 .39208 .12460 6

B4 .47935 .15218 2

SuperKEKB damping ring bend magnet

Geometry: (1)rectangular with width 34mm and height 24mm(2)Ante-chamber(3)Round beam pipe with radius 25mm

Rectangular(34mmX24mm) Rectangular(34mmX24mm) k=1x10k=1x1044mm-1 -1 s=0.4ms=0.4m

20

21

k=1x10k=1x1044mm-1 -1 s=0.6ms=0.6m

k=1x10k=1x1044mm-1 -1 s=0.7ms=0.7m

22

k=k=2x102x1033mm-1 -1 s=0.55ms=0.55m

23

Ante-chamberAnte-chamber

24

Leakage of the field into ante-chamber, especially high frequency field

(courtesy K. Shibata)

k=1x10k=1x1044mm-1 -1 s=0.55ms=0.55m

25

k=k=2x102x1033mm-1 -1 s=0.55ms=0.55m

26

Round chamber, Round chamber, k=k=1x101x1044mm-1 -1 s=0.65ms=0.65m

27

Round chamber, radius=25mm

Coarse mesh used for the plot

Simulation of Microwave Simulation of Microwave Instability usingInstability using

28

Y. Cai & B. Warnock’s Code, Vlasov solver(Phys. Rev. ST Accel. Beams 13, 104402 (2010)

Simulation parameters:qmax=8, nn=300 (mesh), np=4000(num syn. Period), ndt=1024 (steps/syn.); z=6.53mm, del=1.654*10-4

CSR wake is calculated using Demin’s code with rectangular geometry (34mm width and 24mm height)

Geometry Wake is from Ikeda-san

Vlasov-Fokker-Planck code

Case I: Geometry wake onlyCase I: Geometry wake only

29

Case II: CSR wake only Case II: CSR wake only

30

Single bend wake computation

CSR only (single-bend model)CSR only (single-bend model)

Red: <z>, Blue: <delta_e>,Green: sigmaz; pink: sigmae In=0.03

In=0.04

In=0.05

Only CSR (1-cell model)Only CSR (1-cell model)

In=0.04

In=0.03 In=0.05

p4

Inter-bunch Communication through CSR inInter-bunch Communication through CSR inWhispering Gallery Modes Whispering Gallery Modes

Robert WarnockRobert Warnock

33

Next stepsNext steps

34

•Complete the field solver from end of the bend to infinite •Accurate computation with fine mesh & high order element, especially for ante-chamber geometry.

•Complete Micro-wave instability simulation with various CSR impedance options (multi-cell csr wake)

Integration of the field from Integration of the field from end of bend to infinityend of bend to infinity

35

•Agoh, Yokoya, Oide

Gennady, PRSTAB 104401 (mode expansion)

Other ways??

SummarySummary

36

A FEM code is developed to calculated the CSR with arbitrary cross-section of the beam pipe

Preliminary results show good agreement. But need more detail benchmark.

Fast solver to integrate from exit to infinite is the next step.

Simulation of the microwave instability with Vlasov code is under the way.

AcknowledgementsAcknowledgements

37

Thanks Prof. Oide for the support of this work and discussion; and Prof. Ohmi for his arrangement of this study and discussions

Thanks Ikeda-san and Shibata-san for the valuable information

Thanks Demin for detail introduction his works and great help during my stay.

Thanks Agoh -san for the discussions.

Backup slidesBackup slides

38

CSR + Geometry wakeCSR + Geometry wake

39

(Courtesy, H. Ikeda@ KEKB 17th review)

34mmX24mm rectangular

More complete model

401000 2000 3000 4000 5000 6000 7000 8000 9000 10000

-0.5

0

0.5

1

1.5

2

2.5

3x 10

4

k (m-1)

Zr/Z

i ()

round,real round,Imagrectangular, realrectangular, imag

Fast ion Instability in Fast ion Instability in SuperKEKB Electron ringSuperKEKB Electron ring

41

Ion Species

Mass Number

Cross-section (Mbarn)

Percentage in Vacuum

H2 2 0.35 25%H2O 18 1.64 10%CO 28 2.0 50%CO2 44 2.92 15%

Ion species in KEKB vacuum chamber (3.76nTorr=5E-7 Pa)Wang, Fukuma and Suetsugu

Beam current 2.6A2500 bunches, 2RF bucket spacingemiitacex, y 4.6nm/11.5pm

20 bunch trains20 bunch trains total P=1nTorr

0.5 1 1.5 2 2.5 3 3.5 4

10-3

10-2

10-1

100

101

Time (ms)

X,Y

()

x=1.0771 ms, y=0.1044ms

xy

2 4 6 8 1010

-3

10-2

10-1

100

101

Time (ms)

X,Y

()

x=0.264ms, y=0.0438 ms

xy

Total P=3.76nTorr

Growth time 1.0ms and 0.1ms in x and y

Growth time 0.26ms and 0.04ms in x and y

Agoh

43

HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS

Tetrahedron elements

1

9

8

7 10

2

5

6 3

4

10 nodes, quadratic:

1

13 12

7

15

2

9

6 3

4

5

8

10

11

14

16

17

18 195

20

20 nodes, cubic:

z

x

y

i

j

l

k 1 =

4 =

2 =

3 =

=0

=1

=1

=constant

P

Q

4 nodes, linear:

k=1x10k=1x1044mm-1 -1 s=0.46ms=0.46m

45