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

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 2 Overview Motivation Numerical Methods low-dispersive schemes boundary approximation indirect integration algorithm modelling of conductive walls Examples Code Status

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 3 Motivation First codes in time domain ~ 1980 A. Novokhatski (BINP), T. Weiland (CERN) Wake field calculation – estimation of the effect of the geometry variations on the bunch Maxwell‘s Equations

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 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 *** A. Novokhatski, M. Timm, T. Weiland, Transition Dynamics of the Wake Fields of Ultra Short Bunches. Proc. of the ICAP 1998, Monterey, California, USA. * 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.

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 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

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 6 Low-dispersive schemes Dual grid 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) (1977) 31, p. 116.

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 7 Low-dispersive schemes transversal plane longitudinal direction

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 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.

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 9 Low-dispersive schemes Our scheme in staircase approximation with =0.5 is reduced to the scheme realized in code NOVO. Implicit Scheme (2002)

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 10 E/M splitting TE/TM splitting Subdue the updating procedure to the bunch motion MAFIA ECHO E/M and TE/TM* splitting (2004) Low-dispersive schemes * 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)

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 11 Low-dispersive schemes transversal plane longitudinal direction

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 12 Low-dispersive schemes E/M splitting TE/TM splitting dispersion error dispersion error suppressed for

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 13 Low-dispersive schemes Implicit TE/TM scheme Implicit and explicit TE/TM formulations

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 14 Low-dispersive schemes TE/TM-ADI scheme - splitting error Explicit TE/TM scheme Implicit and explicit TE/TM formulations - splitting error

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 15 Low-dispersive schemes Explicit TE/TM scheme E/M scheme (FDTD)

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 16 Low-dispersive schemes The stability condition E/M splitting TE/TM implicit (FDTD scheme) Stability, energy and charge conservation TE/TM explicit* * 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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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 17 Low-dispersive schemes 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 Explicit TE/TM vs FDTD Explicit TE/TM scheme E/M scheme (FDTD)

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 18 Low-dispersive schemes No dispersion in z-direction + no dispersion along XY diagonals Dispersion relation in the transverse plane Implicit TE/TM scheme Explicit TE/TM scheme explicit implicit explicit implicit

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 19 Low-dispersive schemes Transverse Deflecting Structure Gaussian bunch with sigma=300 m explicit implicit

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 20 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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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 21 Boundary approximation New Conformal Scheme* (2002) PEC 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 , Time step is not reduced!

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 22 Boundary approximation Square rotate by angle p/8. PFC (Partially Filled Cells) ~ Dey-Mittra USC (Uniformly Stable Conformal)

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 23 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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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 24 Boundary approximation e h ()Oh 2 ()Oh staircase conformal r z e r z r z 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. staircase conformal

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 25 Boundary approximation 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 staircase conformal

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 26 Indirect Integration Algorithm 0 C 1 C 0 r z C O. Napoly, Y. Chin, and B. Zotter, Nucl. Instrum. Methods Phys. Res., Sect. A 334, 255 (1993)

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 27 Indirect Integration Algorithm 0 C 1 C 0 r 2 C z C I. Zagorodnov, R. Schuhmann, T. Weiland, Journal of Computational Physics, vol. 191, No.2, pp , 2003.

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 28 Indirect Integration Algorithm

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 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)

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 30 Indirect Integration Algorithm

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 31 Modelling of Conductive Walls

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 32 Modelling of Conductive Walls

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 33 Modelling of Conductive Walls

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite TESLA cells structure Moving 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 Examples

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 35 Comparison of the wake potentials obtained by different methods for structure consisting of 20 TESLA cells excited by Gaussian bunch E/M splittingTE/TM splitting E/M ECHO-3D (TE/TM) Examples

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 36 3D simulation. Collimator Examples

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 37 Examples 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 % 82.5% 30% 12.2% 5.4% 12.4% 2.8% 0.8%0.36%ref. E/M TE/TM

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 38 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 E/M ECHO-3D (TE/TM) Examples

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 39 Examples Coupler Kick

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 40 Code Status ECHO 2D https://www.desy.de/~zagor/WakefieldCode_ECHOz/

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Igor Zagorodnov| ICFA mini-Workshop, Erice | April 2014 | Seite 41 Code Status ECHO 3D

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