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

the beam pipethe beam pipe

Work in progress

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

3.14.2012

KEK Accelerator Seminar

OutlineOutline Parabolic equation and general formulation of

the problem

Illustration of the method: numerical examples

Excise of micro-wave instability simulation Excise of micro-wave instability simulation

with Vlasov solver

Conclusions

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)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 10 20 30 40 50 600

0.5

1

1.5

2

2.5

z (mm)

r (m

m)

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

z (cm)

r (c

m)Echo2, sigl=0.1mm

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 ΩΩ ΩΩ)

Real, ECHO2Imaginary, ECHO2Real, FEM codeImaginary, FEM code

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 ΩΩ ΩΩ)

Real, ECHO2-Imaginary, ECHO2

dot-lines: FEM code

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 geometry

Easy to handle boundary

Small beam in a large domain (FEL in undulator)

CPU (strongly depends on the solver)

Why Finite Element Method (FEM)?

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…)

Mesh of chamber & beamMesh of chamber & beam

•Arbitrary geometry of beam pipe

•Any shape of beam

Impedance ofImpedance ofGrooved surfaceGrooved surface

Adaptive method

Speedup and improve accuracy

Mesh of chamber & beamMesh of chamber & beam

CSR computation

Straight beam pipe+ Bend magnet + straight beam pipe…

12

Straight beam pip

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 γ=∞factor γ=∞

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 transform The Fourier transformed components of the

field and the current is defined as

14

Where k=ω/c and js is the projection of the beam current onto

s. The transverse component of the electric field is a

two-dimensional vector . , The longitudinal

component is denoted by

r

⊥E~

)~

,~

(~ r

y

r

x

r EEE =⊥

r

sE~

CSR Parabolic EquationCSR Parabolic Equation

0

0

22 ~

22

ne

ski

R

xk⊥⊥⊥ ∇−=

∂

∂+

+∇

εE

br xkki

xk−=

∂

+

+∇ EE22

2 2~2

2

br

⊥⊥⊥ += EEE

0

0

2 1ρ

ε⊥⊥⊥ ∇=∇ bE

Required for Self-

consistent

computation

(Agoh, Yokoya, PRSTAB 054403)

15

br

Rski

R⊥⊥⊥ −=

∂

+

+∇ EE2 2

Initial field at the beginning of the bend magnet

0)0(~2 ==∇ ⊥⊥ s

rE

After the bend magnet

0~

22 =

∂

∂+∇ ⊥⊥

r

ski E

<

>

=

)(2

)(2

42

0r

r

r

e

b

r

rr

rre

E

σσ

σ

λπε

),( yx ∂∂=∇⊥

222

yx ∂+∂=∇⊥

FEM equationFEM equation

0=×⊥ nr

E

3D problem (Treat s as one space dimension)

JEM =⊥r~

It’s general case, for instance, the cross-section can vary

for the geometry impedance computation.

Problem: converge slowly

16

JEDEM =+ ⊥⊥rr &~~

2D problem (Treat s as time)

Problem: converge slowly

It converges faster, the stability need to be treated carefully

0=×⊥ nr

E

Current status: Bend only ∫∞

−=0

),,(1

)( dssyxEI

kZ ccs

The ways to improve the The ways to improve the

accuracyaccuracy

0=

∂

∂+

∂

∂=

y

E

x

E

k

ir

yr

xsE

High order element

The (longitudinal) impedance is

calculated from the transverse field,

which is nonlinear near the beam as

shown late,

17

shown late,

Fine girds on the curved boundary

0=×⊥ nr

E…..

Test of the codeTest of the code

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

18

Sample 1Sample 1

Bend Length[m] Bending angle # of elements

B1 .74248 .27679 32

B2 .28654 .09687 38

B3 .39208 .12460 6

SuperKEKB damping ring bend magnet

19

B4 .47935 .15218 2

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

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

21

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

22

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

23

AnteAnte--chamberchamber

(courtesy K. Shibata)

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

Round

chamber,

radius=25mm

Coarse mesh

used for the

plot

27

Simulation of Microwave Simulation of Microwave

Instability usingInstability using

Y. Cai & B. Warnock’s Code, Vlasov solver

(Phys. Rev. ST Accel. Beams 13, 104402 (2010)

Simulation parameters:

Vlasov-Fokker-Planck code

28

Simulation parameters:

qmax=8, nn=300 (mesh), np=4000(num syn. Period),

ndt=1024 (steps/syn.); σz=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

Case I: Geometry wake onlyCase I: Geometry wake only

29

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

Single bend wake computation

30

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

Red: <z>, Blue: <delta_e>,

Green: sigmaz; pink: sigmae In=0.03

In=0.04

In=0.05

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

In=0.04

p4

In=0.03 In=0.05

InterInter--bunch Communication through CSR inbunch Communication through CSR in

Whispering Gallery Modes Whispering Gallery Modes

Robert WarnockRobert Warnock

33

Next stepsNext steps•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

34

•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•Agoh, Yokoya, Oide

35

Gennady, PRSTAB 104401 (mode expansion)

Other ways??

SummarySummary

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.

36

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

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

37

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 wake34mmX24mm rectangular

More complete model

39

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

2

2.5

3x 10

4

round,real

round,Imag

rectangular, real

40

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.5

0

0.5

1

1.5

k (m-1)

Zr/

Zi ( ΩΩ ΩΩ

)

rectangular, real

rectangular, imag

Fast ion Instability in Fast ion Instability in

SuperKEKBSuperKEKB Electron ringElectron ring

Ion

Species

Mass

Number

Cross-section

(Mbarn)

Percentage in

Vacuum

Ion species in KEKB vacuum chamber (3.76nTorr=5E-7 Pa)

Wang, Fukuma and Suetsugu

41

Species Number (Mbarn) Vacuum

H2 2 0.35 25%

H2O 18 1.64 10%

CO 28 2.0 50%

CO2 44 2.92 15%

Beam current 2.6A

2500 bunches, 2RF bucket spacing

emiitacex, y 4.6nm/11.5pm

20 bunch trains20 bunch trainstotal P=1nTorr

100

101

τx=1.0771 ms, τ

y=0.1044ms

101

τx=0.264ms, τ

y=0.0438 ms

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

0.5 1 1.5 2 2.5 3 3.5 4

10-3

10-2

10-1

Time (ms)

X,Y

(σσ σσ)

x

y

2 4 6 8 10

10-3

10-2

10-1

100

Time (ms)

X,Y

(σσ σσ)

x

y

Agoh

43

HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS

Tetrahedron elements

10 nodes, quadratic: 20 nodes, cubic:4 nodes, linear:

1

9

8

710

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

19

5

20

z

x

y

i

j

l

k

1 =

4 =

2 =

3 =

ξ=0

ξ=1

ξ=1

ξ=constant

P

Q

4 nodes, linear:

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

45