Igor Zagorodnov Second Topical Workshop on Instabilities ...€¦ · * Dohlus M., Zagorodnov I.,...

Algorithms and Applications

Wakefield Code ECHO

Igor Zagorodnov

Second Topical Workshop on

Instabilities, Impedance and

Collective Effects

Abingdon, Oxfordshire

8. February 2016

Igor Zagorodnov| Workshop on Instabilities, Impedance and Collective Effects , England | 8. February 2016 | Seite 2

Overview

Motivation

Numerical Methods

low-dispersive schemes

boundary approximation

indirect integration algorithm

modelling of conductive walls

Code Status and Applications

geometries of revolution (ECHOz1, ECHOz2 )

rectangular structures (ECHO2D)

fully 3D geometries (ECHO3D)

Particle-In-Cell code


Motivation

First codes in time domain ~ 1980

A. Novokhatski (BINP),

T. Weiland (CERN)

j c

Wake field calculation – estimation of the effect of the

geometry variations on the bunch

Wake potential

|| 0 0

1( , , ) , , ,z

z sW s E z dz

Q c

r r r r

0 || 0( , , ) ( , , )s W ss

W r r r r


Motivation

MAFIA*

rotationally symmetric and 3D

longitudinal and transverse wakes

triangular geometry

approximation;

arbitrary materials

moving window

dispersion error

NOVO**

rotationally symmetric

only longitudinal wake

“staircase” geometry

approximation;

only PEC

moving mesh

low dispersion error

** A. Novokhatski, M. Timm,

T.Weiland, Transition Dynamics

of the Wake Fields of Ultra

Short Bunches. Proc. of the

ICAP 1998, Monterey,

California, USA.

* MAFIA Collaboration, MAFIA Manual,

CST GmbH, Darmstadt, 1997.


Motivation

New projects with

short bunches;

long structures;

tapered collimators

Solutions

zero dispersion in longitudinal

direction;

“conformal” meshing;

moving mesh and “explicit” or

“split” methods

(Picture from W.Ackermann, TU Darmstadt, 2012)


Low-dispersive schemes

1 1

, ,

, ,

,

d d

dt dt

Ce b Ch d j

Sb 0 Sd q

e M d h M b

z y

z x

y x

curl

0 P P

C P 0 P

P P 0

* * *

x y zdiv S P P P

Dual grid

mm jd

,

ih

ie

mb

Primary grid

Maxwell Grid Equations*

* T. Weiland, A discretization method for

the solution of Maxwell’s equations for

six-component fields, Electronics and

Communication (AEÜ) 31, p. 116 (1977)



r

z

transversal plane longitudinal direction

r

r

T

0 0 P

0 0 P

P P 0

z

z

0 P 0

P 0 0

0 0 0

L

z

z r

r

0 P P

C P 0 P

P P 0



n

x

n n

y

n

z

h

h h

h

0.5

0.5 0.5

0.5

n

x

n n

y

n

z

e

e e

e

FDTD (1966)

1

1

1 2 1 2

1

*

1 2

*( )n n

n n n

n nn n

t

t

e eM h h j

h hh M Ce

T L

Implicit Scheme* (2002)

1 1(1 2 )n n n n h h h h

1

1

1 2 1 2*

11 2

( )n n

n n

n nn n

t

t

e eM C h j

h hh M Ce

* I. Zagorodnov, R. Schuhmann, T. Weiland, Long-Time Numerical

Computation of Electromagnetic Fields in the Vicinity of a

Relativistic Source, Journal of Computational Physics 191, No.2 ,

pp. 525-541 (2003)



For fully rotationally symmetric problems (monopole mode m=0) our

scheme in staircase approximation with =0.5 coincides with scheme

realized in code NOVO**.

1 1 1

1 1

1 2 1 12 ,2r z

n n

n n n T n T n

z z r rt

a a

M a a a P M P a P M P F

1 1 1 1

1 1 1

1 1 1

2 2

1

* *

1

2 (1 2 )

,

nt

n

s

n n n n n n n

d

t t

a h

I T a a a T a a L a F

T M CM L M CMT L

Implicit Scheme (2002)

** A. Novokhatski, M. Timm, T. Weiland, Transition Dynamics of the

Wake Fields of Ultra Short Bunches. Proc. of the ICAP 1998,

Monterey, California, USA.


n

x

n n

y

n

z

h

h h

h

xe

ye

zh ze

xh

0 0 / 2 0

yh

0.5

0.5 0.5

0.5

n

x

n n

y

n

z

e

e e

e

E/M splitting (Yee’s Scheme)

0.5

0.5 0.5

0.5

n

x

n n

y

n

z

h

u h

e

n

x

n n

y

n

z

e

v e

h

TE/TM splitting

Subdue the updating procedure

to the bunch motion

E/M and TE/TM* splitting (2004)


* Zagorodnov I.A, Weiland T., TE/TM

Field Solver for Particle Beam

Simulations without Numerical

Cherenkov Radiation, Phys. Rev. ST

Accel. Beams 8, 042001 (2005)



1 1

0 0

1 1/ 2 1/ 2 1 0

0

1 1

z y

z x

y x

c

0 P P

C M CM P 0 P

P P 0

i

* *

, 0,1

i

y

i

x

i i

y x

i

0 0 P

T 0 0 P

P P 0

0

1

z

z

0 P 0

L P 0 0

0 0 0

transversal plane longitudinal

direction



0.5 0.5 0.5 0.5

0 ,2

n n n nn n

u

u u u uT Lv j

1 1* 0.5 0.5

12

n n n nn n

v

v v v vT L u j

0.5 0.5*

0

10.5

0

,n n

n n

n nn

e eC h j

h hC e

n

x

n n

y

n

z

h

h h

h

0.5

0.5 0.5

0.5

n

x

n n

y

n

z

e

e e

e

0.5

0.5 0.5

0.5

n

x

n n

y

n

z

h

u h

e

n

x

n n

y

n

z

e

v e

h

E/M splitting (1966) TE/TM splitting (2004)

dispersion error dispersion error suppressed for z



# #

1

0.5 0.5* *

z

n ne nz z

y x x y z

e eW M P h P h j

# #

1

1

z

n nh z z

y x x y

h hW M P e P e

1 1 1 1

2 2* *

4 4z x z y

e e

CN y y x x

W W I M P M P M P M P

Implicit TE/TM formulation



TE/TM-ADI scheme

4

2 ( )e e

CN ADI O W W

- splitting error

Explicit TE/TM scheme*

e h W W I

2( )e

CN O W I

ADI and Explicit TE/TM formulations

- splitting error

* Dohlus M., Zagorodnov I., Explicit TE/TM Scheme for Particle Beam

Simulations// Journal of Computational Physics 225, No. 8, pp. 2822-

2833 (2009)



n+1 nn n-

+ =

y yB Ay f

The stability condition

2 2 2

1

x y z

z

E/M splitting TE/TM implicit

(FDTD scheme)

Stability, energy and charge conservation

2 2

1min , z

x y

TE/TM explicit

0.5 0 Q B A

dispersion error suppressed for z



22222

2 2 2 2

sinsinsinsincos

yxzKKK

z x y

2222 22

2 2 2 2 2

sinsinsinsin1 sin

yxzz

KKKK

z x y z

No dispersion in z-direction + no dispersion along XY diagonals

Dispersion relation in the transverse plane

Implicit TE/TM scheme

Explicit TE/TM scheme

-1 0 10.88

0.9

0.92

0.94

0.96

0.98

1

-1 0 10.88

0.9

0.92

0.94

0.96

0.98

1

1tan x yk k 1tan x yk k

kc

kc

5N 10N

explicit

implicit

explicit

implicit



0 5 10 15 20 25 30 35 40 45 50

-5

0

5

-4 -2 0 2 40

20

40

60

80

100

120

140

160

180

-4 -2 0 2 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Transverse Deflecting Structure

Gaussian bunch with sigma=300m

explicit

implicit

VW

pC m

/s /s

[%]

exp

exp exp*100%

max min

impW W

W W

5h


Boundary approximation

Standard Conformal Scheme

zijsjyil 1

xijl

PEC

reduced cell area time step must

be reduced

Dey S, Mittra R. A locally conformal finite-difference time-domain (FDTD)

algorithm for modeling threedimensional perfectly conducting objects.

IEEE Microwave and Guided Wave Letters 7(9):273–275 (1997)

Thoma P. Zur numerischen Lösung der Maxwellschen Gleichungen im

Zeitbereich. Dissertation Dl7: TH Darmstadt,1997.



New Conformal Scheme* (2002)

PEC

PEC l

L

1i jb

,i jijb

1

iju

virtual cell

* Zagorodnov I., Schuhmann R.,Weiland T., A Uniformly Stable Conformal

FDTD-Method on Cartesian Grids, International Journal on Numerical

Modeling, vol. 16, No.2, pp. 127-141 (2003)

Time step is not reduced!



Square rotate by angle 𝜋/8.

2

2

h

h

z zL

z L

H H

H

PFC (Partially Filled Cells) ~ Dey-Mittra

USC (Uniformly Stable Conformal)

101

10210

-4

10-3

10-2

10-1

100

( )O h

3( )O h

2( )O h

USC

/d h

PFC

staircase

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

PFC

USC

/d h

Ctt /



1 10 100

0.01

0.1

1

h

( )O h

2( )O h

staircase

conformal

38 mm

z

r

1.9 mm

335 mrad

1e-3

1e-4

1e-5

1 10 100

0.01

0.1

1

h

( )O h

2( )O h

staircase

conformal

38 mm

z

r

1.9 mm

335 mrad

1e-3

1e-4

1e-5

Error in loss factor for a taper

The error δ relative to the extrapolated loss factor L=-7.63777

Vp/C for bunch with σ=1 mm is shown.

0 5 10 15 20 25 30 35 405.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

7.8V

pC

staircase

conformal

h

1

calcL L L


Indirect Integration Algorithm

0C1C

0r 2C

z0z z

1 0

s

zC

s

zC

m e dQ z e dzW

0 1 20 0

1

2

m m

D S S SmC Cz m C

s ra

e z rda

-1C

I. Zagorodnov, R. Schuhmann, T. Weiland, Journal of

Computational Physics 191, No.2 , pp. 525-541 (2003)

O. Napoly, Y. Chin, and B. Zotter, Nucl. Instrum. Methods Phys. Res., Sect.

A 334, 255 (1993)



Zagorodnov I., Indirect Methods for Wake Potential

Integration, Phys. Rev. STAB 9, 102002 (2006)

H. Henke and W. Bruns, in Proceedings of EPAC 2006,

Edinburgh, Scotland (WEPCH110, 2006)



Gaussian bunch with rms length 25 µm


Modelling of Conductive Walls

Tsakanian A., Dohlus M., Zagorodnov I., Hybrid TE-TM

scheme for time domain numerical calculations of wakefields

in structures with walls of finite conductivity, Phys.

Rev. STAB 15, 054401 (2012)

Zagorodnov I., Bane K., Stupakov G. Calculations of

wakefields in 2D rectangular structures, Phys. Rev. STAB

18, 104401 (2015)


Modelling of Conductive Walls

2w= 10 cm, T=5 cm, 2b=2cm, L=12 cm.

Gaussian bunch with rms length 25mm

Longitudinal wake potential of tapered collimator


Code Status

ECHOz1, ECHOz2 (rotationally symetric)

ECHO2D (rectangular and rotationally

symetric)

ECHO3D (fully 3D )

ECHO-PIC (Particle-In-Cell for rotationally

symmetric and rectangular)

https://www.desy.de/~zagor/WakefieldCode_ECHOz/

https://www.desy.de/~zagor/WakefieldCode_ECHOz/


ECHOz1

,

,

0

,

cos( )

r mr

m

m

z z m

EE

H H m

E E

,

,

0

,

sin( )

r mr

m

m

z z m

HH

E E m

H H

|| 0 0 0 0

0

( , , , , ) ( ) cos ( )m m

m

m

W r r s W s r r m

Rotationally Symmetric Geometries


ECHOz1

ECHO

Only fully rotationally symmetric problems (m=0

mode)

vector potential wave equation (slide 9)

only PEC

stand-alone Windows GUI application


ECHOz2

Rotationally symmetric problems (all modes)

TE/TM implicit (slide 10)

surface conductivity

stand-alone Windows GUI application


ECHOz2

Short-Range Wake Functions for TESLA Cryomodule

Longitudinal wake functions was found earlier with code NOVO

Novokhatski A., Timm M., Weiland T., Single Bunch Enetgy Spread in the

TESLA Cryomodule, DESY TESLA-99-16, 1999Transverse wake functions is found with code ECHO

T. Weiland, I. Zagorodnov, The short-range transverse wake function for

TESLA accelerating structure, TESLA Report 2003-19


ECHOz2

0

||( )

s

sW s Ae

1

1

1

( ) 1 1

s

ssW s A e

s

0

|| 0 0( ) 1

s

sW s A e

Novokhatski A., Timm M.,

Weiland T. (1999, NOVO)

Weiland T., Zagorodnov I.

(2003, ECHO)*

Bane K.L.F., Short-Range

Dipole Wakefields in

Accelerating Structures for the

NLC, SLAC-PUB-9663, LCC-

0116, 2003


ECHO2D

Rectangular Geometries with Constant Width

2a

2wT L

2b

Zagorodnov I., Bane K., Stupakov G.

Calculations of wakefields in 2D rectangular

structures, Phys. Rev. STAB 18, 104401

(2015)

The idea can be found, for example, in

A. Tremaine, J. Rosenzweig, and P.

Schoessow, Phys. Rev. E 56, 7204 (1997)

,

,

1

,

1cos

2

x mx

y y m

m

z z m

EE

H H mxw w

H E

,

,

1

,

1sin

2

x mx

y y m

m

z z m

HH

E E mxw w

E E


ECHO2D

|| 0 0 0 , 0 ,

0

1( , , , , ) ( , , )sin sinm x m x m

m

W x y x y s W y y s k x k xw

,

0 , 0 , 0

,

cosh( )( ) ( )( , , ) cosh( ) sinh( )

sinh( )( ) ( )

cc csx mm m

m x m x m sc ssx mm m

k yW s W sW y y s k y k y

k yW s W s

General case

With symmetry in y

0 , 0 ,

, 0 ,

( , , ) ( )cosh( )cosh( )

( )sinh( )sinh( )

cc

m m x m x m

ss

m x m x m

W y y s W s k y k y

W s k y k y

Wake field expansion is new (to our knowledge)


ECHO2D

0zH

2

Q

0zE

2

Q

0 , 0 , , 0 ,( , , ) ( )cosh( )cosh( ) ( )sinh( )sinh( )cc ss

m m x m x m m x m x mW y y s W s k y k y W s k y k y

0 0 0( , , ) ( , , ) ( , , )H E

m m mW y y s W y y s W y y s

0 0

2

, 0

( , , )( )

cosh( )

Hcc m

m

x m

W y y sW s

k y 0 0

2

, 0

( , , )( )

sinh( )

Ess m

m

x m

W y y sW s

k y


ECHO2D

Rectangular and rotationally symmetric problems (all modes)

TE/TM implicit (slide 10)

surface conductivity

stand-alone console application (Windows, Mac OS, Linux)

Parallelized (MPI and threads)

Zagorodnov I., Bane K.,

Stupakov G. Calculations of

wakefields in 2D rectangular

structures, Phys. Rev. STAB

18, 104401 (2015)


ECHO2D

Pohang Dechirper Experiment

2w= 5 cm, p=0.5 mm,

h=0.6mm,

t=p/2, L=1 m, 2a= 6mm.

Gaussian bunch with rms

length 0.5mm


ECHO2D

Pohang Dechirper Experiment*

measured0.75

analyticalr

ECHO

(1m structure)

ECHO

(periodic)

||( , , )

V/pC

W s0 0

*P. Emma et al., Phys. Rev. Lett. 112, 034801 (2014)


ECHO2D

SLAC Dechirper 2w= 1.2 cm, p=0.5 mm, h=0.5mm,

t=p/2, L=2 m, 2a= 1.4 mm.

0

||( , , )

s

sW s Ae

0 0

Analytical* Periodic

(ECHO)

Length 2m

(ECHO)

Length 2m

(NOVO**)

𝐴 [V/pC] 9.05 9.08 9.29 9.80

𝑠0 [mm] 0.25 0.21 0.22

00 2

*16

Z cA L

a

**Novokhatski A., Wakefield potentials of corrugated

structures, PR-STAB 18, 104402 (2015)

*Bane K., Stupakov G., Dechirper Wakefields for

Short Bunches, Report LCLS-II TN-16-01(2016)


0 0.01 0.02 0.03 0.04 0.0528

30

32

34

36

38

40

42

44

46

ECHO2D

SLAC Dechirper

00.5A||( , , )

kV/pC

W s0 0ECHO (2m structure)

ECHO

(periodic)

[mm]


ECHO2D

SLAC Dechirper

0

0

0

( , , )

~ 1 12

( , , )

sy

s

y

W sy A s

es

W sy

0 0

0 0

Quadrupole Dipole

𝐴 [V/pC] 8.08 6.94

𝑠0 [mm] 0.086 0.066 -2 -1 0 1 2 3 4 5

-2

-1.5

-1

-0.5

0x 10

4

( , , )

kV/pC/mm

yW sy

0 0

s

50 m

2 m 0 0

0

( , , ) ( , , ) ( , , )y y yW y y s y W s y W sy y

0 0 0 0


ECHO3D


20 TESLA cells structureMoving mesh

3 geometry elements

The geometric elements are loaded at the instant when the moving

mesh reach them. During the calculation only 2 geometric elements

are in memory.

3D simulation. Cavity

ECHO3D


Comparison of the wake potentials obtained by different methods

for structure consisting of 20 TESLA cells excited by Gaussian

bunch

-0.5 0 0.5 1 1.5-40

-30

-20

-10

0

-0.5 0 0.5 1 1.5

-40

-30

-20

-10

0

/s cm

||

/

W

V pC

/ 2.5

( / 5)

( /10)

E M D

z

z

2.5POT D

/s cm

||

/

W

V pC

/ 3

( / 3 / 3 / 2.5)

TE TM D

z x y

2.5POT D

1mm

3

~zL

~z E/M splitting TE/TM splitting

E/M ECHO-3D (TE/TM)

ECHO3D


ECHO 3D

Coupler Kick


ECHO-PIC

Stupakov G., Using pipe with corrugated walls for a

subterahertz free electron laser, PR-STAB 18,

030709 (2015)


ECHO-PIC

Wakefield code ECHO

(with resistivity)

Particle-in-Cell code ECHO-PIC

(only longitudinal dynamics)


ECHO-PIC

Pipe length 𝐿, m 1.5-2

Bunch enegry 𝐸0, MeV 5-20

Gausian bunch rms 𝜎, mm 0.3*8

Charge 𝑄, nC 0.96

-5 0 50

10

20

30

40

50

𝑠 [mm]

𝐼 [A]


ECHO-PIC

𝐸0 = 20 MeV

𝑠 [mm]

𝐼 [A]

∆𝐸 [eV]

𝐸0 = 5 MeV

𝑠 [mm]

𝐼[A]

∆𝐸 [eV]


ECHO-PIC

0 0.5 1 1.5 20

0.05

0.1

0.15

𝐸0 = 20MeV

𝑓 ≈ 0.3THz

𝐸0 = 5 MeV

𝑓 ≈ 0.35 THz

𝑧 [m]

∆𝐸 [mJ]140 µJ


ECHO-PIC

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

410 µJ

𝐸0 = 20 MeV

𝑓 ≈ 0.3 THz

𝐸0 = 8 MeV

𝑓 ≈ 0.33 THz

𝑧 [m]

∆𝐸 [mJ]

125 µJ

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Igor Zagorodnov Second Topical Workshop on Instabilities ...€¦ · * Dohlus M., Zagorodnov I.,...

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