1. Wakefield Code ECHO 2(3)D
How it works, limitations, pros/cons and new/current developments
Igor Zagorodnov
ICFA mini-Workshop on “Electromagnetic wake fields and impedances in particle accelerators“
Erice, Sicily
April 2014

2. Overview Motivation Numerical Methods Examples Code Status
low-dispersive schemes
boundary approximation
indirect integration algorithm
modelling of conductive walls
Examples
Code Status

3. Motivation
Wake field calculation – estimation of the effect of the geometry variations on the bunch
Maxwell‘s Equations
First codes in time domain ~ 1980
A. Novokhatski (BINP),
T. Weiland (CERN)

4. Motivation TBCI*, MAFIA*** rotationally symmetric and 3D
longitudinal and transverse wakes
triangular geometry approximation;
arbitrary materials
moving window
dispersion error
NOVO***
only rotationally symmetric
only longitudinal wake
“staircase” geometry approximation;
only PEC
moving mesh
low dispersion error
* T. Weiland, TBCI and URMEL - New Computer Codes for Wake field and cavity mode calculations, IEEE Trans. Nuclear Science, 30 (1983) 2489
** MAFIA Collaboration, MAFIA Manual, CST GmbH, Darmstadt, 1997.
*** A. Novokhatski, M. Timm, T. Weiland, Transition Dynamics of the Wake Fields of Ultra Short Bunches. Proc. of the ICAP 1998, Monterey, California, USA.

5. Low-dispersive schemes
New projects – new needs
short bunches;
long structures;
tapered collimators
New methods are required
without dispersion error accumulation;
“staircase” free;
fast 3D calculations on PC
Solutions
zero dispersion in longitudinal direction;
“conformal” meshing;
moving mesh and “explicit” or “split” methods

6. Low-dispersive schemes
Dual grid
Maxwell Grid Equations*
Primary grid
* T. Weiland, A discretization method for the solution of Maxwell’s equations for six-component fields, Electronics and Communication (AEÜ) (1977) 31, p. 116.

7. Low-dispersive schemes
transversal plane
longitudinal
direction

8. Low-dispersive schemes
FDTD
Implicit Scheme* (2002)
* I. Zagorodnov, R. Schuhmann, T. Weiland, Long-Time Numerical Computation of Electromagnetic Fields in the Vicinity of a Relativistic Source, Journal of Computational Physics, vol. 191, No.2 , pp , 2003.

9. Low-dispersive schemes
Implicit Scheme (2002)
Our scheme in staircase approximation with =0.5 is reduced to the scheme realized in code NOVO.

10. E/M and TE/TM* splitting (2004)
Low-dispersive schemes
E/M and TE/TM* splitting (2004)
Subdue the updating procedure
to the bunch motion
* Zagorodnov I.A, Weiland T., TE/TM Field Solver for Particle Beam Simulations without Numerical Cherenkov Radiation, Phys. Rev.  ST Accel. Beams, vol. 8, (2005)
E/M splitting
TE/TM splitting
MAFIA
ECHO

11. Low-dispersive schemes
transversal plane
longitudinal
direction

12. Low-dispersive schemes
E/M splitting
TE/TM splitting
dispersion error
dispersion error suppressed for

13. Low-dispersive schemes
Implicit and explicit TE/TM formulations
Implicit TE/TM scheme

14. Low-dispersive schemes
Implicit and explicit TE/TM formulations
TE/TM-ADI scheme
- splitting error
Explicit TE/TM scheme
- splitting error

15. Low-dispersive schemes
Explicit TE/TM scheme
E/M scheme (FDTD)

16. Low-dispersive schemes
Stability, energy and charge conservation
The stability condition
E/M splitting
TE/TM implicit
TE/TM explicit*
(FDTD scheme)
* Dohlus M., Zagorodnov I., Explicit TE/TM Scheme for Particle Beam Simulations// Journal of Computational Physics, vol. 225, No. 8, pp , 2009.

17. Low-dispersive schemes
Explicit TE/TM vs FDTD
Explicit TE/TM scheme
E/M scheme (FDTD)
Explicit TE/TM requires the same memory 6N as FDTD
Explicit TE/TM requires only 18% more computational time than FDTD (the same transverse mesh)
Explicit TE/TM allows “magical” time step to avoid the dispersion error

18. Low-dispersive schemes
Dispersion relation in the transverse plane
Implicit TE/TM scheme
Explicit TE/TM scheme
explicit
implicit
No dispersion in z-direction + no dispersion along XY diagonals

19. Low-dispersive schemes
Transverse Deflecting Structure
Gaussian bunch with sigma=300mm
explicit
implicit

20. Standard Conformal Scheme*
Boundary approximation
Standard Conformal Scheme*
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 1997; 7(9):273–275..
Thoma P. Zur numerischen Lösung der Maxwellschen Gleichungen im Zeitbereich. Dissertation Dl7: TH Darmstadt,1997.

21. Time step is not reduced!
Boundary approximation
New Conformal Scheme* (2002)
PEC
PEC
virtual cell
Time step is not reduced!
* 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 , 2003.

22. Square rotate by angle p/8.
Boundary approximation 20 40 60 80
100
0.2
0.4
0.6
0.8 1
PFC
USC 10 1 2 -4 -3 -2 -1
USC
PFC
staircase
Square rotate by angle p/8.
PFC (Partially Filled Cells) ~ Dey-Mittra
USC (Uniformly Stable Conformal)

23. Simple Conformal (SC) Scheme*
Boundary approximation
Simple Conformal (SC) Scheme*
* Zagorodnov I., Schuhmann R., Weiland T., Conformal FDTD-methods to avoid Time Step Reduction with and without Cell Enlargement , Journal of Computational Physics, vol. 225,  No. 2, pp , 2007.

24. Error in loss factor for sphere
Boundary approximation ? 1 1
staircase
staircase () Oh
conformal
0.01
0.01
conformal 2 () Oh r r r z 1e 1e - - 4 4 z z h ? 10 10
100
100
Error in loss factor for sphere
Gaussian bunch with σ=0.5 cm is passing through a spherical resonator. The sphere has the diameter 1.8 cm. The analytical loss factor L is equal to Vp/C
The error for stationary mesh is demonstrated by lines. The results for moving mesh are shown by triangles and circles.

25. Error in loss factor for a taper
Boundary approximation 5 10 15 20 25 30 35 40
5.8 6
6.2
6.4
6.6
6.8 7
7.2
7.4
7.6
7.8
staircase
conformal
Error in loss factor for a taper
The error δ relative to the extrapolated loss factor L= Vp/C for bunch with σ=1 mm is shown.

26. Indirect Integration Algorithm1 C -1 C C r z
O. Napoly, Y. Chin, and B. Zotter, Nucl. Instrum. Methods Phys. Res., Sect. A 334, 255 (1993)

27. Indirect Integration Algorithm1 C -1 C C r 2 C z
I. Zagorodnov, R. Schuhmann, T. Weiland, Journal of Computational Physics, vol. 191, No.2 , pp , 2003.

28. Indirect Integration Algorithm

29. Indirect Integration Algorithm
Zagorodnov I., Indirect Methods for Wake Potential Integration, Phys. Rev.  STAB vol. 9, (2006)
H. Henke and W. Bruns, in Proceedings of EPAC 2006, Edinburgh, Scotland (WEPCH110, 2006)

30. Indirect Integration Algorithm

31. Modelling of Conductive Walls

32. Modelling of Conductive Walls

33. Modelling of Conductive Walls

34. 3D simulation. Cavity Examples 3 geometry elements
Moving mesh
20 TESLA cells structure
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.

35. Examples E/M ECHO-3D (TE/TM)
Comparison of the wake potentials obtained by different methods for structure consisting of 20 TESLA cells excited by Gaussian bunch
E/M splitting
TE/TM splitting

36. 3D simulation. Collimator
Examples
3D simulation. Collimator

37. Examples 10 20 30 40
500
1000
1500
2000
2500
3000
250.6%
82.5%
30%
12.2%
5.4%
12.4%
2.8%
0.8%
0.36%
ref.
E/M
TE/TM -5 5
-500
The transverse dipole wake function (left) and loss factor for the collimators with and L=20cm (right). The solid lines show the results for the E/M scheme (Yee’s scheme) and the dashed lines display the results for the TE/TM scheme. The relative errors are given regarding the reference value (marked as ref. on the graphs) calculated by TE/TM method with the finest mesh.

38. Examples E/M ECHO-3D (TE/TM)
Comparison of the wake potentials obtained by different methods for round collimator excited by Gaussian bunch
TE/TM method – fast, stable and accurate with coarse mesh

39. Examples
Coupler Kick

40. Code Status
ECHO 2D

41. Code Status
ECHO 3D