Mathematical Models and Computer Simulations

, Volume 10, Issue 2, pp 198–208 | Cite as

Numerical Modeling of Plasma Devices by the Particle-In-Cell Method on Unstructured Grids

  • A. S. Dikalyuk
  • S. E. Kuratov


The paper considers methods and algorithms providing the basis for a computer program implementing an axial-symmetric electrostatic version of the particle-in-cell method on unstructured triangular grids. In the presented implementation, the Poisson equation is approximated using the finite volume method. A discrete analog of the Poisson equation is solved by the multigrid method. Charged particle trajectories are calculated using the Boris method. Methods for interpolating electrostatic fields on unstructured grids and obtaining the charge density in the computational domain are considered. Special attention is paid to the specifics of implementing these methods in axisymmetric geometry. The developed computer code is tested on the problem of a flat diode operating in the space charge mode.


particle-in-cell method unstructured triangular grids two-dimensional Child–Langmuir problem 


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  1. 1.
    F. M. Penning, “Ein neues Manometer fur niedrige Gasdrucke, insbesondere zwischen 10-3 and 10-5 mm,” Physica 4, 71–75 (1937).CrossRefGoogle Scholar
  2. 2.
    F. M. Penning and J. H. A. Moubis, “Eine Neutronenrohre ohne Pumpvorrichtung,” Physica 4, 1190–1199 (1937).CrossRefGoogle Scholar
  3. 3.
    J. L. Rovey, B. P. Ruzic, and T. J. Houlahan, “Simple penning ion source for laboratory research and development applications,” Rev. Sci. Instrum. 78, 106101-1–106101-3 (2007).Google Scholar
  4. 4.
    J. L. Rovey, “Design parameter investigation of a cold-cathode penning ion source for general laboratory applications,” Plasma Sources Sci. Technol. 17, 035009-1–035009-7 (2008).Google Scholar
  5. 5.
    Yu. A. Berezin and V. A. Vshivkov, Particle-in-cell Method in Rarefied Plasma Dynamics (Nauka, Novosibirsk, 1980) [in Russian].Google Scholar
  6. 6.
    R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (CRC, New York, 1987).zbMATHGoogle Scholar
  7. 7.
    C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (McGraw-Hill, New York, 1985).Google Scholar
  8. 8.
    Yu. N. Grigoriev, V. A. Vshivkov, and M. P. Fedoruk, Numerical 'Particle-In-Cell’ Methods: Theory and Applications (VSP, Utrecht, Boston, 2002; Sib. Otdel. RAN, Novosibirsk, 2004).CrossRefGoogle Scholar
  9. 9.
    Yu. P. Raizer and S. T. Surzhikov, “Magnetohydrodynamic description of collisionless plasma expansion in upper atmosphere,” AIAA J. 33, 486–490 (1995).CrossRefGoogle Scholar
  10. 10.
    S. T. Surzhikov, “Expansion of multi-charged plasma clouds into ionospheric plasma with magnetic field,” AIAA Paper No. 97–2361 (AIAA, 1997).Google Scholar
  11. 11.
    S. T. Surzhikov, “Collisionless expansion of a plasma with doubly charged ions in a rarefied magnetized plasma,” Plasma Phys. Rep. 26, 759–771 (2000).CrossRefGoogle Scholar
  12. 12.
    C. K. Birdsall, “Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with neutral atoms, PIC-MCC,” IEEE Trans. Plasma Sci. 19, 65–85 (1991).CrossRefGoogle Scholar
  13. 13.
    Z. Donko, “Particle simulation methods for studies of low-pressure plasma sources,” Plasma Sources Sci. Technol. 20, 024001(2011).CrossRefGoogle Scholar
  14. 14.
    V. Vahedi and M. Surendra, “A Monte Carlo collision model for the particle in cell method: applications to argon and oxygen discharges,” Comput. Phys. Commun. 87, 179–198 (1995).CrossRefGoogle Scholar
  15. 15.
    V. Vahedi, G. DiPeso, C. K. Birdsall, M. A. Lieberman, and T. D. Rognlien, “Capacitive RF discharge modelled by particle-in-cell Monte Carlo simulation. I: Analysis of numerical techniques,” Plasma Sources Sci. Technol. 2, 261–272 (1993).CrossRefGoogle Scholar
  16. 16.
    V. Vahedi, C. K. Birdsall, M. A. Lieberman, G. DiPeso, and T. D. Rognlien, “Capacitive RF discharge modelled by particle-in-cell Monte Carlo simulation. II: Comparisons with laboratory measurements of electron energy distribution function,” Plasma Sources Sci. Technol. 2, 273–278 (1993).CrossRefGoogle Scholar
  17. 17.
    H. Burau, R. Widera, W. Honig, G. Juckeland, A. Debus, T. Kluge, U. Schramm, T. E. Cowan, R. Sauerbrey, and M. Bussmann, “PIConGPU: a fully relativistic particle-in-cell code for a GPU cluster,” IEEE Trans. Plasma Sci. 38, 2831–2839 (2010).CrossRefGoogle Scholar
  18. 18.
    I. A. Surmin, S. I. Bastrakov, E. S. Efimenko, A. A. Gonoskov, A. V. Korzhimanov, and I. B. Meyerov, “Particle-in-cell laser-plasma simulation on xeon phi coprocessors,” Comput. Phys. Commun. 202, 204–210 (2016).CrossRefGoogle Scholar
  19. 19.
    M. Pfeiffer, A. Mirza, C.-D. Munz, and S. Fasoulas, “Two statistical particle split and merge methods for particle-in-cell codes,” Comput. Phys. Commun. 191, 9–24 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    G. B. Jacobs and J. S. Hesthaven, “High-order nodal discontinuous Galerkin particle-in-cell method on unstructured grids,” J. Comput. Phys. 214, 96–121 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics, 2nd ed. (Pearson Education, Harlow, 2007).Google Scholar
  22. 22.
    R. P. Fedorenko, “A relaxation method for solving elliptic difference equations,” Zh. Vychisl. Mat. Mat. Fiz. 1, 922–927 (1961).zbMATHGoogle Scholar
  23. 23.
    R. P. Fedorenko, “The speed of convergence of one iterative process,” Zh. Vychisl. Mat. Mat. Fiz. 4, 559–564 (1964).Google Scholar
  24. 24.
    D. J. Mavriplis, “Multigrid techniques for unstructured meshes,” Report NASA-CR-195070 (NASA, Hampton, 1995).Google Scholar
  25. 25.
    I. E. Sutherland and G. W. Hodgman, “Reentrant polygon clipping,” Commun. ACM 17, 32–42 (1974).CrossRefzbMATHGoogle Scholar
  26. 26.
    J. D. Ramshaw, “Conservative rezoning algorithm for generalized two-dimensional meshes,” J. Comput. Phys. 59, 193–199 (1985).CrossRefzbMATHGoogle Scholar
  27. 27.
    D. Elbery, Intersection of Convex Objects: The Method of Separating Axes (Geometric Tools, Redmond WA, 2008).Google Scholar
  28. 28.
    G. L. Delzanno and E. Camporeale, “On particle movers in cylindrical geometry for particle-in-cell simulations,” J. Comput. Phys. 253, 259–277 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    D. G. Holmes and S. D. Connell, “Solution of the 2-D Navier-Stokes equations on unstructured adaptive grids,” AIAA Paper No. 89-1392 (AIAA, 1989).Google Scholar
  30. 30.
    R. D. Rausch, J. T. Batina, and H. T. Y. Yang, “Spatial adaption procedures on unstructured meshes for accurate unsteady aerodynamic flow computation,” AIAA Paper No. 1991-1106 (AIAA, 1991).Google Scholar
  31. 31.
    A. M. Spirkin, “A three-dimensional particle-in-cell methodology on unstructured Voronoi grids with applications to plasma microdevices,” PhD Thesis (Worcester Polytech. Inst., Worcester, 2006).Google Scholar
  32. 32.
    K. Nanby, “Probability theory of electron-molecules, ion-molecule, molecule-molecule, and Coulomb collisions for particle modeling of materials processing plasmas and gases,” IEEE Trans. Plasma Sci. 28, 971–990 (2000).CrossRefGoogle Scholar
  33. 33.
    B. P. Bromley, “Computational modeling of the axial-cylindrical inertial electrostatic confinement fusion neutron generator,” PhD Thesis (Univ. Illinois at Urbana-Champaign, Urbana-Champaign, 2001).Google Scholar
  34. 34.
    J. W. Luginsland, Y. Y. Lau, and R. M. Gilgenbach, “Two-dimensional Child-Langmuir law,” Phys. Rev. Lett. 77, 4668–4670 (1996).CrossRefGoogle Scholar
  35. 35.
    Y. Li, H. Wang, C. Liu, and J. Sun, “Two-dimensional Child-Langmuir law of planar diode with finite-radius emitter,” Appl. Surf. Sci. 251, 19–23 (2005).CrossRefGoogle Scholar
  36. 36.
    Y. Y. Lau, “Simple theory of the two-dimensional Child-Langmuir law,” Phys. Rev. Lett. 87, 278301-1–278301-3 (2001).Google Scholar
  37. 37.
    G. Jaffe, “On the currents carried by electrons of uniform initial velocity,” Phys. Rev. 65, 91–98 (1944).CrossRefGoogle Scholar
  38. 38.
    K. G. Kostov and J. J. Barroso, “Space-charge-limited current in cylindrical diodes with finite-length emitter,” Phys. Plasmas 9, 1039–1042 (2002).CrossRefGoogle Scholar
  39. 39.
    J. J. Watrous, J. W. Lugisland, and G. E. Sasser III, “An improved space-charge-limited emission algorithm for use in particle-in-cell codes,” Phys. Plasmas 8, 289–296 (2001).CrossRefGoogle Scholar
  40. 40.
    B. Ragan-Kelley, J. Verboncoueur, and Y. Feng, “Two-dimensional axisymmetric Child-Langmuir scaling law,” Phys. Plasmas 16, 103102-1–103102-6 (2009).Google Scholar
  41. 41.
    S. V. Irishkov, “Fully kinetic model of plasma dynamics in discharge of plasma thruster with closed electron drift,” Mat. Model. 18 (6), 70–84 (2006).zbMATHGoogle Scholar
  42. 42.
    A. Dikalyuk and S. T. Surzhikov, “The modeling of dust particles in a normal glow discharge: the comparison of two charged models,” AIAA Paper No. 2010-4310 (AIAA, 2010).Google Scholar
  43. 43.
    L. V. In’kov, “Calculation of selfconsistent electrostatic field in kinetic simulation of dusty plasma,” Mat. Model. 15 (7), 46–54 (2003).MathSciNetzbMATHGoogle Scholar
  44. 44.
    Iu. E. Kreindel and A. S. Ionov, “Characteristic features of low-pressure pressure discharges in penning tubes,” Sov. Tech. Phys. 9, 930 (1964).Google Scholar
  45. 45.
    N. V. Mamedov, N. N. Shchitov, and I. A. Kanshin, “Investigation of the dependency of penning ion source operational characteristics on its geometric parameters,” Fiz.-Khim. Kinet. Gaz. Din. 16 (4), 1 (2015).Google Scholar
  46. 46.
    V. G. Markov, D. E. Prokhorovich, A. G. Sadilkin, and N. N. Shchitov, “Determination of the corpuscular emission energy characteristics for the ion sources of gas-filled neutron tubes,” Usp. Prikl. Fiz. 1, 23–29 (2013).Google Scholar
  47. 47.
    A. N. Dolgov, V. G. Markov, A. A. Okulov, D. E. Prokhorovich, A. G. Sadilkin, D. I. Yurkov, I. V. Vizgalov, V. I. Rashchikov, N. V. Mamedov, and D. V. Kolodko, “Integrated approach in the investigation of corpuscular beam dynamics in ion-optical system of the neutron tube,” Usp. Prikl. Fiz. 2, 267–272 (2014).Google Scholar
  48. 48.
    D. A. Storozhev and S. T. Surzhikov, “Numerical simulation of the two-dimensional structure of glow discharge in molecular nitrogen in light of vibrational kinetics,” High Temp. 53, 307–318 (2015).CrossRefGoogle Scholar
  49. 49.
    D. A. Storozhev, “Numerical simulation of the kinetics of ionization and dissociations of hydrogen in penning discharge plasma in the LTE approach,” Fiz.-Khim. Kinet. Gaz. Din. 15 (3), 3 (2014).Google Scholar
  50. 50.
    S. T. Surzhikov, “Application of the modified drift-diffusion theory to study of the two-dimensional structure of the penning discharge,” AIAA Paper No. 2015-1832 (AIAA, 2015).Google Scholar
  51. 51.
    S. T. Surzhikov, “The two-dimensional structure of penning discharge in a cylindrical chamber with axial magnetic field at pressure of about 1 torr,” Tech. Phys. Lett. 43, 169(2017).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.All-Russia Research Institute of Automatics (VNIIA)MoscowRussia
  2. 2.Institute for Problems of Mechanics (IPMech)Russian Academy of Sciences (RAS)MoscowRussia
  3. 3.Moscow Institute of Physics and Technology (MIPT)MoscowRussia

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