Methods and Simulation Tools for Cavity Design

Transcription

1 Methods and Simulation Tools for Cavity Design Prof. Dr. Ursula van Rienen, Dr. Hans-Walter Glock SRF09 Berlin - Dresden Dresden

2 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools Practical Examples Some generalities Some selected examples 2

3 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools Practical Examples Some generalities Some selected examples 3

4 Superconducting accelerator cavities [Podl], IAP Frankfurt/M _dc_1.jpg... and some other types Jefferson Lab / => Often chains of repeated structures, combined with flanges / coupling devices. 4

5 What do you need to know? I) Accelerating Mode: ( ) v 2 f and L Do match? part res cell How much energy does the particle gain? ΔE kin = q part z= L cav z=0 E z (z) cos(2π f res z / v part + ϕ 0 ) dz Jefferson Lab How much energy is stored in the cavity? ε μ 0 Wstored = amplitude = amplitude 2 E dv 2 H dv V cav V How big is the power loss in the wall? And how big is the unloaded quality factor? R 1 = = ω μ = ω W P H da H da Q loss Where are the maxima of electric (field emission) and magnetic (quench) field strength at the surface? Which values do they reach? cav surface 2 2 tan tan ; 0 2 res 2 2σ res stored P cavity cavity loss 5

6 What do you need to know? II) Fundamental passband: All resonant frequencies! Which frequency spread does the fundamental passband have? How close is the next neighbouring mode to the accelerating mode? Cell-to-Cell coupling filling time f mode /MHz How strong is the beam interaction of all modes? Look at R over Q : R Q = ΔE kin q part 2 ( ω res W stored ) cell-to-cell phase advance ϕ III) Higher Order Modes (HOM): same questions as for fundamental passband wake potential kick factor special field profiles strongly confined far away from couplers: "trapped modes" 6

7 What do you need to know? IV) Input and HOM-coupler/absorber: Q loaded of all beam-relevant modes Field distortions due to coupler? Field distribution within the coupler CAD-plot: DESY V) Multipacting VI) Mechanical stability wrt. Lorentz Forces 7

8 What do you need to know? I) Accelerating mode II) Fundamental passband III) HOMs IV) Input and HOM-coupler/absorber V) Multipacting VI) Mechanical stability wrt. Lorentz Forces Jefferson Lab Eigenmodes provide most of the information needed: 100% of I) 100% of II) 80% of III) 25% of IV) 25% of V) 50% of VI) 8

9 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Mode Matching Technique Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools Practical Examples Some generalities Some selected examples 9

10 Need for Numerical Methods Most practical electrodynamics problems cannot be solved purely by means of analytical methods, see e.g.: radiation caused by a mobile phone near a human head shielding of an electronic circuit by a slotted metallic box mode computation in accelerator cavities, especially in chains of cavities In many of such cases, numerical methods can be applied in an efficient way to come to a satisfactory solution. 10

11 Numerical Methods Semi-analytical Methods Methods based on Integral Equations Method of Moments (MoM) Discretization Methods Finite Difference Method (FD) Boundary Element Method (BEM) Finite Element Method (FEM) Finite Volume Method (FVM) Finite Integration Technique (FIT) Possible Problems Need for geometrical simplifications MoM Violation of continuity conditions FD Unfavorable matrix structures BEM Unphysical solutions FEM if not mixed FEM Etc. U. van Rienen,

12 Discretization (I) of Solution itself Discretization error; 1D and 2D Picture source: 12

13 Discretization (II) of Boundary of Solution Domain In general: geometrical error Structured boundary-fitted grid Picture source: 13

14 Discretization (III) Spatial Grid Types Unstructured 2D grid: Picture source: FIDAP/erfahrung/node46.html 14

15 Discretization (V) Space and Time space time grid space grid time R 3 + R G T H = 4,96 A/m H =0,085 A/m Laptop with WLAN card f = 2, 4 GHz Computed with CST MWS U. van Rienen,

16 Grid Time Example Wave propagation in rectangular waveguide Energy density of wake fields in the TESLA cavity 16

17 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools Practical Examples Some generalities Some selected examples 17

18 FIT Grid 18

19 A FIT Discretization of Induction Law E ds = B da t A ^ = Ce = t b e l e k b n e i e j e + e e e = t b i j k l n ei.... e. j = bn t... e k. C. e l b e U. van Rienen,

20 FIT Discretization of Gauss Law B V d A = 0 ^ = Sb = 0 b m b i b k b n b l n. b + b + b b b b = 0 i j k l m n b j. bi bj.. bk = 0.. bl S bm b b U. van Rienen,

21 Dual FIT-Grid dual grid introduction of dual grid grid dual FIT cell h d h d : electric flux : magnetic voltage U. van Rienen,

22 FIT Discretization of Ampère s and Coulomb s Law FIT equations on the dual grid D H d s = + d t J A A A Ch = d+ j t D d A = ρ dv V V Sd = q h d U. van Rienen,

23 Maxwell s Grid Equations (MGE) B curle = t D curlh= + t div D = ρ div B = 0 D= ε E B= μ H J = κ E+ J e J FIT T. Weiland, 1977, 1985 Ce = b t Ch = d+ j t Sd = q Sb = 0 d= Mε e b= Mμ h j= M e+ j κ e U. van Rienen,

24 Shape Approximation on Cartesian Grids Modelling Curved Boundaries Staircase FDTD/FDFD (standard) : poor convergence FIT with diagonal filling: better convergence Weiland 1977 FIT on non-orthogonal grids 2 nd order accuracy R.Schuhmann, 1998 FIT with non-equidistant step size: 2 nd order convergence not always applicable Weiland 1977 Conformal FIT / PBA: 2 nd order convergence always applicable Krietenstein, Schuhmann, Thoma, Weiland

25 FIT on Triangular Grids Grid Dual grid URMEL-T U. van Rienen 1983 U. van Rienen,

26 FIT on Non-Orthogonal Grids Hilgner, Schuhmann, Weiland TU Darmstadt 1998 TESLA 9 cell struct. (Grid representation, N=13x69x13=11.661) Field in model of 1 cell 26

27 Maxwell s Grid Equations (MGE) Curl Curl Equation B curle = t D curlh= + t div D = ρ div B = 0 D= ε E B= μ H J = κ E+ J e J FIT T. Weiland, 1977, 1985 Ce = b t Ch = d+ j t Sd = q Sb = 0 d= Mε e b= Mμ h j= M e+ j κ e In full analogy to analytical derivation of wave equation U. van Rienen,

28 j 0 S σ = Curl-Curl-Eigenvalue Equation 0, ε, µ real loss-free ( no current excitation) ( ) M CM Ce= ω e 1 2 ε 1 µ eigenvalue problem Ax = λx A = M CM C 1 CC ε µ 1 ε curl µ curl E = ω E

29 Curl-Curl-Eigenvalue Equation 1/2 Use transformation e' = Mε e to derive an equation with symmetric system matrix M CM CM e e' 1/2 1/2 2 ε 1 ε ' = ω µ system matrix 1/2 1/2 A' = Mε ACC Mε = M CM CM = 1/2 1/2 ε µ 1 ε ( 1/2 1/2 )( 1/2 1/2 M ) ε CMµ Mε CMµ T is symmetric 29

30 Solution Space of Eigenvalue Problem system matrix 1/2 1/2 A' = Mε ACC Mε = M CM CM Eigenvalues of A (and thus of A CC as well) are 1. real 2. non-negative because 1. A is symmetric = 2. A = matrix (matrix) T 1/2 1/2 ε µ 1 ε ( 1/2 1/2 )( 1/2 1/2 M ) ε CMµ Mε CMµ Eigenfrequencies ω i are real and non-negative T is symmetric as it should be for a loss-less resonator 30

31 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools Practical Examples Some generalities Some selected examples 31

32 Coupled S-Parameter Calculation Coupled S-Parameter Calculation H.-W. Glock, K.Rothemund, UvR 98 32

33 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools not exhaustive Practical Examples Some generalities Some selected examples 33

34 CST STUDIO Suite CST MICROWAVE STUDIO -Transient Solver - Eigenmode Solver - Frequency Domain Solver - Resonant: S-Parameters and Fields - Integral Equations Solver - Predecessors for eigenmode calculation: MAFIA, URMEL, URMEL-T 34

35 HFSS 35

36 ACE3P Suite ACE3P - Advanced Computational Electromagnetic Simulation Suite developed at SLAC by the group around Kwok Ko, Cho NG et al. See examples in next part of the talk 36

37 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Mode Matching Technique Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools Practical Examples Some generalities Some selected examples 37

38 Workflow of eigenmode computation define geometry / import CAD-data take care of your grid define material properties define symmetries and boundaries check solver parameters run solver (and maybe: wait) frequency range ok??? precision ok??? (quasi) mode degeneration??? post process: visualization post process: derived quantities 38

39 Workflow of eigenmode computation define symmetries and boundaries 39

40 Passband Fields with E tan = 0 in middle of the cavity Make use of symmetry compute only ½ of the cavity Monopole-type E- and H- field, common to all modes and cells Computed with CST MWS TM π/5 (pseudo 0) TM 010-3π/5 TM 010 -π (accelerating) 40

41 Passband Fields with H tan = 0 in middle of the cavity Use of symmetry compute only ½ of the cavity Needs of course 2 (shorter) runs! Visualization of full cavity provided by CST MWS Even 1/8 part might be enough for computation TM 010-2π/5 TM 010-4π/5 Computed with CST MWS 41

42 Passband Fields altogether TM 010 -π/ MHz f mode /MHz TM 010-2π/ MHz TM 010-3π/ MHz TM 010-4π/ MHz TM 010 -π MHz cell-to-cell phase advance ϕ... which seems to obey some rule?! 42 Computed with CST MWS

43 Passband fields altogether TM 010 -π/ MHz f mode /MHz TM 010-2π/ MHz TM 010-3π/ MHz TM 010-4π/ MHz TM 010 -π MHz Computed with CST MWS In fact: κ cc : cell-to-cell coupling; compare K. Saito s talk 43 f mode f 0 + f π 2 cell-to-cell phase advance ϕ 1 κ cc 2 cos() ϕ

44 So, what are passbands? Cavities built by chains of identical cells show resonances in certain frequency intervalls, called passbands, determined only by the shape of the elementary cell. The distribution of resonances in the band depends on the number of cells in the chain: f mode /MHz 12-cell-chain resonances 5-cell-chain resonances Computed with CST MWS 44 cell-to-cell phase advance ϕ

45 Periodic Boundary Conditions In fact, it is possible, to calculate the spectrum of an infinite chain by discretizing a single cell (exploiting other symmetries as well)...: Δϕ... and to preset the cell-to-cell phase advance by application of an appropriate longitudinal boundary condition. Computed with CST MWS 45

46 Periodic Boundary Conditions This needs only one single run for each Δϕ, but gives eigenmode frequencies of several passbands with a very small grid (here 19,000 mp), e.g: f mode /MHz Computed with CST MWS 46 cell-to-cell phase advance ϕ

47 What are Monopole-, Dipole-, Quadrupole-Modes? Consider structures of axial circular symmetry. Then all fields belong to classes with invariance to certain azimuthal rotations: Quadrupole, ϕ =90 Oktupole, ϕ =45 Monopole, any ϕ Dipole, ϕ =180 Computed with CST MWS 47 Sextupole, ϕ =60 Dekapole, ϕ =36

48 Trapped Mode Analysis Search for strongly confined field distributions by simulating same structure with different waveguide terminations at beam pipe ends. Compare spectra! Small frequency shifts indicate weak coupling. H tan = 0 H_19: MHz H tan = 0 E tan = 0 E_18: MHz E tan = 0 H_18: MHz E_17: MHz Computed with CST MWS Remark: TE 11 -cut off of beam pipe at 1953 MHz 48

49 Overview Introduction Methods in Computational Electromagnetics (CEM) Examples of CEM Methods: Mode Matching Technique Finite Integration Technique (FIT) Coupled S-Parameter Simulation (CSC) Simulation Tools Practical Examples Some generalities Some selected examples 49

50 TESLA 9-Cell Structure Niobium; acceleration at 1.3 GHz Field on axis E z (normiert) Foto: DESY z/m Computed with MAFIA 2D-simulation of upper half; only azimuthal symmetry exploited 50

51 Filling of TESLA Superstructure - Semi-analytical calculation 9 λ / 2 3 λ / 2 7 λ / 2 λ / 2 Structure of CDR Superstructure, J. Sekutowicz 1997 E z /(V/m) 2500 t = T z/m E z /(kv/m) t = 1000 T E z /(MV/m) t = 10 6 T z/m -40 z/m rf period T = ns H.-W. Glock; D. Hecht; U. van Rienen; M. Dohlus. Filling and Beam Loading in TESLA Superstructures. Proc. of the 6th European Particle Accelerator Conference EPAC98, (1998): Computed with MAFIA 51

52 Higher Order Modes in TESLA Structure Dipole mode 28, f = GHz VAs/m 3 3 Dipole mode 29, f = GHz VAs/m VAs/m 3 3 Dipole mode 30, f = GHz VAs/m Computed with MAFIA VAs/m VAs/m 52

53 CSC - 9-Cell Resonator with Couplers HOM coupler Input coupler CAD-plot: DESY HOM coupler Resonator without couplers: N ~ 29,000 (2D) N ~ (3D) Resonator with couplers: N ~ (3D) N increases by ~ 500 CSC: Coupled S-Parameter Calculation allows for combination of 2D- and 3D-simulations K. Rothemund; H.-W. Glock; U. van Rienen. Eigenmode Calculation of Complex RF-Structures using S- Parameters. IEEE Transactions on Magnetics, Vol. 36, (2000):

54 CSC - Resonator Chain Variation of Tube Length Identical sections HOM1 HOM2 S Hom1Hom2 /db Weak dependence on position 54 Variation of coupler position 1,000 frequency points 31 lengths resulting S-matrix: 16 x 16 intern 84 x 84, 1h 12min, Pentium III, 1 GHz Computed with MAFIA, CST MWS and our own Mathematica Code for CSC

55 Effect of Changed Coupler Design* R=-1 R=-1 L2,u L1 L1 L2,d L1 = 45.0 mm L2,u = mm L2,d = 65.4 mm S-parameters of TESLA cavity: Modal analysis** S-parameters of HOM- & HOM-input-coupler: CST MicrowaveStudio CSC to determine S-parameters of various object combinations *New concept: M. Dohlus, DESY; ** Modal coeff. computed by M. Dohlus H.-W. Glock, K. Rothemund 55

56 Comparison: HOM(original) HOM(mirrowed) HOM (down) mirrowed Computed with MAFIA, CST MWS and our own Mathematica Code for CSC HOM (up) Q Q In cooperation with M. Dohlus, DESY ,46 GHz 2,5 GHz 2,54 GHz 2,58 GHz 2,46 GHz 2,5 GHz 2,54 GHz 2,58 GHz H.W. Glock; K. Rothemund; D. Hecht; U. van Rienen. S-Parameter-Based Computation in Complex Accelerator Structures: Q-Values and Field Orientation of Dipole Modes. Proc. ICAP

57 Some of his slides Courtesy to Martin Dohlus 57

58 Courtesy of M. Dohlus, DESY ICAP

59 Courtesy of M. Dohlus, DESY ICAP

60 Courtesy of M. Dohlus, DESY ICAP

61 Courtesy of M. Dohlus, DESY ICAP

62 Courtesy of M. Dohlus, DESY ICAP

63 Courtesy of M. Dohlus, DESY ICAP

64 Courtesy of M. Dohlus, DESY ICAP

65 Courtesy of Cho Ng, Lixin Ge 65

66 CLIC HDX Simulation HDX Surface E & B Fields Hs/Ea= A/m/(V/m) E H Courtesy of Cho Ng, Lixin Ge Es/Ea=2.80 Es/Ea=2.26 Hs/Ea= A/m/(V/m) Hs/Ea= A/m/(V/m) Fields enhanced around slot rounding 66

67 Electron Trajectory & Impact Energy Courtesy of Cho Ng, Lixin Ge emitted here Particles emitted from one of the irises At 85 MV/m gradient, energy of dark current electrons can reach ~0.4 MeV on impact 67

68 At the end some other "cavity" type. Hope, you feel better. Wilhelm Busch: Max und Moritz, sometimes in the 19th century. Widely published These are greetings of my co-author Hans-Walter Glock. 68