Numerical Modeling of Heat Transfer and Material Flow During Wire-Based Electron-Beam Additive Manufacturing

  • A. V. Shcherbakov
  • D. A. Gaponova
  • R. V. RodyakinaEmail author
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


The development of a mathematical model provides an analysis of heat transfer and metal flow during wire-based electron-beam additive manufacturing described. The procedure for solving the heat equation for the metal in the solid phase and the Navier–Stokes equations in the liquid phase, based on the use of the finite-difference method and the predictor–corrector procedure, is described. An algorithm for numerical approximation of the motion of the free surface of the melt, using the concept of the volume of fluid (VOF) is described. The original method for calculating the effect of surface tension forces, based on the numerical calculation of the surface curvature index, is proposed. The results of simulating the melting of a wire element localized above a substrate made of 316L steel are described. Experiments showed the predominant role of surface tension force in the formation of deposited layer and also that metal’s flow has a laminar structure. These results were obtained by simulating a short-time beam exposure (t = 0.1 s) with a power of 6 kW. Thus, even when the wire and the substrate are exposed with a more intense beam than often used in practice, the metal transfer is not characterized by the formation of intense vortex flows. This can simplify the solution of the problem of additive manufacturing modeling in the future.


Additive manufacturing Electron beam Melting Mathematical model Navier–Stokes equations Heat transfer Fluid flow Surface tension 



This work was carried out in National Research University “Moscow Power Engineering Institute”; it was supported by grant from the Russian Science Foundation (Project 17-79-20015).


  1. 1.
    Zlenko MA, Nagaytsev MV, Dovbysh VM (2015) Additive technologies in mechanical engineering. Manual for Engineers. Moscow: State Science Center of Russia Federal State Unitary Enterprise “NAMI”Google Scholar
  2. 2.
    Gibson I, Rosen D, Stucker B (2015) Additive manufacturing technologies. 3D printing, rapid prototyping, and direct digital manufacturing. Springer Science + Business Media, New YorkGoogle Scholar
  3. 3.
    Ding D, Pan Z, Cuiuri D, Li H (2015) Wire-feed additive manufacturing of metal components: technologies, developments and future interests. Int J Adv Manuf Technol 81(1–4):465–481. Scholar
  4. 4.
    Ji L, Lu J, Tang S, Wu Q et al (2018) Research on mechanisms and controlling methods of macro defects in TC4 alloy fabricated by wire additive manufacturing. Materials 11:1104. Scholar
  5. 5.
    Williams SW, Martina F, Addison AC, Ding J et al (2016) Wire + arc additive manufacturing. Mater Sci Technol 32(7):641–647CrossRefGoogle Scholar
  6. 6.
    Feng Y, Bin Z, He J, Wang K (2018) The double-wire feed and plasma arc additive manufacturing process for deposition in Cr-Ni stainless steel. J Mater Process Technol 259:206–215. Scholar
  7. 7.
    Jhavar S, Jain N (2014) Development of Micro-plasma wire deposition process for layered manufacturing. In: Katalinic B (ed) DAAAM International scientific book, pp 239–256Google Scholar
  8. 8.
    Ding Y, Akbari M, Kovacevic R (2018) Process planning for laser wire-feed metal additive manufacturing system. Int J Adv Manuf Technol 95(1–4):355–365CrossRefGoogle Scholar
  9. 9.
    Fuchs J, Schneider C, Enzinger N (2018) Wire-based additive manufacturing using an electron beam as heat source. Weld World 62(2):267–275CrossRefGoogle Scholar
  10. 10.
    Taminger KMB, Hafley RA (2003) Electron beam freeform fabrication: a rapid metal deposition process. In: Proceedings of third annual automotive composites conference. Society of Plastic Engineers, Troy, Michigan, USA, pp 9–10Google Scholar
  11. 11.
    Zalameda JN, Burke ER, Hafley RA, Taminger KM et al (2013) Thermal imaging for assessment of electron-beam freeform fabrication (EBF3) additive manufacturing deposits. Proc SPIE 8705(87050M):8Google Scholar
  12. 12.
    Taminger KM, Domack CS, Zalameda JN, Taminger BL et al (2016) Thermal imaging of the electron beam freeform fabrication process. In: Proceedings of SPIE Commercial + Scientific Sensing and Imaging. Baltimore, MD, United States, Report No. NF1676L-22574Google Scholar
  13. 13.
    Noh WF; Woodward P (1976) SLIC (Simple line interface calculation). In: van de Vooren AI, & Zandbergen PJ (ed) Proceedings of 5th international conference of fluid dynamics. Lecture Notes in Physics 59, pp 330–340Google Scholar
  14. 14.
    Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225CrossRefGoogle Scholar
  15. 15.
    Patankar SV (1980) Numerical Heat transfer and fluid flow. Series in computational methods in mechanics and thermal sciencesGoogle Scholar
  16. 16.
    Kampanis Nikolaos A, Ekaterinaris John A (2006) A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations. J Comput Phys 215(2):589–613MathSciNetCrossRefGoogle Scholar
  17. 17.
    Versteeg HK, Malalasekera W (1995) An introduction to computational fluid dynamics: the finite volume method. Longman Scientific & TechnicalGoogle Scholar
  18. 18.
    Gopala Vinay R, van Wachem Berend GM (2008) Volume of fluid methods for immiscible-fluid and free-surface flows. Chem Eng J 141(1–3):204–221CrossRefGoogle Scholar
  19. 19.
    Rhee SH, Makarov BP, Krishinan H, Vladimir I (2005) Assessment of the volume of fluid method for free-surface wave flow. J Mar Sci Technol 10(4):173–180CrossRefGoogle Scholar
  20. 20.
    Hargreaves DM, Morvan HP, Wright NG (2007) Validation of the volume of fluid method for free surface calculation: the broad-crested weir. Eng Appl Comput Fluid Mech 1(2):136–146. Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • A. V. Shcherbakov
    • 1
  • D. A. Gaponova
    • 1
  • R. V. Rodyakina
    • 1
    Email author
  1. 1.National Research University MPEIMoscowRussia

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