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**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

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**Overview Motivation Numerical Methods Examples Code Status**

low-dispersive schemes boundary approximation indirect integration algorithm modelling of conductive walls Examples Code Status

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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)

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**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.

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**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

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**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.

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**Low-dispersive schemes**

transversal plane longitudinal direction

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**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.

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**Low-dispersive schemes**

Implicit Scheme (2002) Our scheme in staircase approximation with =0.5 is reduced to the scheme realized in code NOVO.

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**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

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**Low-dispersive schemes**

transversal plane longitudinal direction

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**Low-dispersive schemes**

E/M splitting TE/TM splitting dispersion error dispersion error suppressed for

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**Low-dispersive schemes**

Implicit and explicit TE/TM formulations Implicit TE/TM scheme

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**Low-dispersive schemes**

Implicit and explicit TE/TM formulations TE/TM-ADI scheme - splitting error Explicit TE/TM scheme - splitting error

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**Low-dispersive schemes**

Explicit TE/TM scheme E/M scheme (FDTD)

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**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.

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**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

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**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

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**Low-dispersive schemes**

Transverse Deflecting Structure Gaussian bunch with sigma=300mm explicit implicit

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**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.

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**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.

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**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)

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**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.

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**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.

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**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.

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**Indirect Integration Algorithm**

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

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**Indirect Integration Algorithm**

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

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**Indirect Integration Algorithm**

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**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)

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**Indirect Integration Algorithm**

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**Modelling of Conductive Walls**

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**Modelling of Conductive Walls**

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**Modelling of Conductive Walls**

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**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.

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**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

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**3D simulation. Collimator**

Examples 3D simulation. Collimator

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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.

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**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

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Examples Coupler Kick

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Code Status ECHO 2D https://www.desy.de/~zagor/WakefieldCode_ECHOz/

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Code Status ECHO 3D

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