1. www.consorzio-cometa.it FESR Consorzio COMETA - Progetto PI2S2 Finite Element Electromagnetic analysis of Travelling Wave Tubes in GRID environment G. Pollicino, A. Laudani, S. Coco Consorzio COMETA DIEES Università di Catania Grid Open Days all’Università di Palermo Palermo, 6-7.12.2007

2. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 2 Outline Introduction Finite Element Analysis of Vlasov-Poisson problems Electron gun simulation in GRID environment Finite Element Analysis of Slow-Wave Structures Slow-Wave structure simulation in GRID environment Conclusions

3. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 3 Use of TWTs: Satellite TLC systems.Satellite TLC systems. RADAR.RADAR. Military equipment for Electronic countermeasures (ECM)Military equipment for Electronic countermeasures (ECM) Travelling wave tubes (TWTs) are electronic vacuum devices used for high-power amplification of radio frequency (RF) signals Travelling Wave Tubes (1/2)

4. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 4 Characteristics: Frequency: 1 GHz  100 GHz Power:1 Watt  2 MWatt Travelling Wave Tubes (2/2)

5. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 5 The electron guns are widely used in TWTs and particle accelerators, in order to generate a electron beam having suitable characteristic (diameter, current, energy distribution, etc.) for the exchange of energy with the RF signal in the interaction region Electron Guns

6. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 6 The overall electromagnetic field (E, B), consists of a contribute coming out from the external applied field (E 0, B 0 ), and of a contribute due to the movement of the charged particles (E s, B s ), (self-consistent field or space charge field) governed by the Maxwell equations. Consequently the study of the charged particles dynamic for an assigned configuration of the electromagnetic field requires the solution of the system of Vlasov-Maxwell equations. f(x,p,t) is the function describing the space charge distribution in the phase space (x, p)  x  p in collisionless conditions. Electromagnetic Analysis of Electron Guns

7. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 7 In TWTs the flux of particles entering the interesting region can be schematized by means of a flux of particles independent from time. Since the boundary conditions of externally applied electromagnetic field are also indipendent from time, a steady state approach for the analysis is adopted. Consequently each macro-particle follows a defined trajectory independent from time, which can be viewed as a flux tube. Under these hypotheses the solution of Vlasov equations can be obtained by using an iterative scheme which alternates the solution of the electromagnetic problem with the mechanical one: The Steady-State Particle-In-Cell Method

8. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 8 Vlasov-Poisson solver Trajectories generator Trajectories tracer Charge distribution evaluation Convergence reached START END no yes Solver Algorithm The iterative solution of the non-linear coupled problem is performed by following these steps:

9. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 9 Vlasov-Poisson solver Trajectories generator Trajectories tracer Charge distribution evaluation Convergence reached START END no yes Solver Algorithm Solution of the Poisson problem for the electric potentialSolution of the Poisson problem for the electric potential The iterative solution of the non-linear coupled problem is performed following these steps:

10. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 10 Vlasov-Poisson solver Trajectories generator Trajectories tracer Charge distribution evaluation Convergence reached START END no yes Solver Algorithm Solution of the Poisson problem for the electric potentialSolution of the Poisson problem for the electric potential Generation of initial conditions for the macro-particles by using the Child formulasGeneration of initial conditions for the macro-particles by using the Child formulas The iterative solution of the non-linear coupled problem is performed following these steps:

11. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 11 Vlasov-Poisson solver Trajectories generator Trajectories tracer Charge distribution evaluation Convergence reached START END no yes Solver Algorithm Solution of the Poisson problem for the electric potentialSolution of the Poisson problem for the electric potential Generation of initial conditions for the macro-particles by using the Child formulasGeneration of initial conditions for the macro-particles by using the Child formulas Integration of dynamic equationsIntegration of dynamic equations The iterative solution of the non-linear coupled problem is performed following these steps:

12. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 12 Vlasov-Poisson solver Trajectories generator Trajectories tracer Charge distribution evaluation Convergence reached START END no yes Solver Algorithm Solution of the Poisson problem for the electric potentialSolution of the Poisson problem for the electric potential Generation of initial conditions for the macro-particles by using the Child formulasGeneration of initial conditions for the macro-particles by using the Child formulas Integration of dynamic equationsIntegration of dynamic equations Distribution of space charges on the mesh nodes.Distribution of space charges on the mesh nodes. The iterative solution of the non-linear coupled problem is performed following these steps:

13. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 13 Vlasov-Poisson solver Trajectories generator Trajectories tracer Charge distribution evaluation Convergence reached START END no yes Solution of the Poisson problem for the electric potentialSolution of the Poisson problem for the electric potential Generation of initial conditions for the macro-particles by using the Child formulasGeneration of initial conditions for the macro-particles by using the Child formulas Integration of dynamic equationsIntegration of dynamic equations Distribution of space charges on the mesh nodes.Distribution of space charges on the mesh nodes. These steps are followed until the difference between two successive solutions is under a target tolerance. The iterative solution of the non-linear coupled problem is performed following these steps: Solver Algorithm

14. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 14 Simulation in the GRID environment Mesh paremeters: Num. Tetrahedra = 91152 Num. nodes = 20857

15. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 15 Jobs submission Test.sh Template.prj Collgunm.jdl

16. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 16 Mesh tetrahedra: about 91.000 PC Computing time : 15 min. GRID Computing time: 9 min. 1.49 A1.47 AEmitted current 180 V Grid voltage 12200 V Cathode voltage Simulation results Experimental resuts Parameters Results

17. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 17 Results

18. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 18 Results

19. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 19 Current-Voltage relationship PC Computing time: 400 min. GRID computing time: 30 min. Simulation data: Cathode = 0 V Grid = 180 ÷ -300 V Anode = 12200 V Step = 20 V N° of simulations = 25

20. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 20 The TWT Slow-Wave Structure (SWS) is an helix structure used as waveguide for the RF signal. The aim of the SWS is to reduce the wave phase velocity in order to allow an energetic exchange between the electron beam and the RF signal. The Slow-Wave Structure The electromagnetic analysis can be performed by using the Cold Test, which consists of the eigen value analysis of the SWS without the electron beam in order to compute the characteristic parameters of the helix structure (cut-off frequency, phase velocity characteristic impedance, coupling impedance).

21. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 21 Inside the SWS the electromagnetic fields can be expressed by means of their space harmonic expansion: The electromagnetic problem is governed by the vector Helmholtz wave equation for the electric field Each propagation mode is constitued by an infinite number of space harmonics, characterized by the phase velocity: Cold Test Analysis

22. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 22 Brillouin Diagram for a sheath helix The  -  relationship is expressed by the Brillouin Diagram:

23. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 23 The FEM solution of the Helmholtz equation is obtained by means of the Galerkin method which leads to the following matrix equation: [A] and [B] are the global matrices obtained by connecting all the mesh elements. Finite Element Formulation

24. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 24 We consider the vector of the edge [e] elements subdivided into three parts, according to the class to which the edge belongs. If the mesh is built dicretizing in the same way the two periodic surfaces the eigenvalue problem assumes the following form: where the matrices depend on the coefficient β. In the construction of the mesh for the considered domain, the elements can be conveniently distinguished into three different classes: those belonging to the surface z=0 (class 1), those belonging to the surface z=L (class 3), and the remaining elements (class 2) FLOQUET Boundary Conditions

25. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 25 Radius of the cilinder containing the helix: 1.83 mm Helix parameter: Inner Radius = 0.9 mm Outer Radius = 1 mm Period = 0.9 mm 3-D View of the mesh used Simulation in the GRID environment

26. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 26 3D view of helix and rod discretization3D view of helix mesh Helix mesh

27. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 27 Jobs Submission Test.sh Template.jdl i.job.jdl

28. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 28 the first four modes of the helix computed by the developed tool are plotted for different values of β in the range [0, π/L]. In particular a step value of  /16L has been used in the simulations Results PC Computing time: 8 h. GRID Computing time: <1 h.

29. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 29 Computationally expensive FEM electromagnetic analyses of Travelling Wave Tubes have been advantageously carried out in the GRID environment. In particular two scripts have been implemented for the submission of jobs in order to parallelize the construction of the Current-Voltage relationship for an electron gun and of the Brillouin Diagram for the characterization of the SWS. The obtained results have shown the good performance of the GRID environment for the numerical analysis and optimization of TWTs. Conclusions

30. Palermo, Grid Open Days all’Università di Palermo, 6-7.12.2007 Any Questions ? Thank you very much for your kind attention!