Journal of Low Temperature Physics

, Volume 186, Issue 1–2, pp 63–73 | Cite as

An Enhanced Facet Determination Scheme in 3D

  • M. Reivinen
  • E.-M. Salonen
  • I. Todoshchenko
  • V. P. Vaskelainen


An enhanced scheme for the determination of equilibrium crystal shapes is introduced. The emphasis of the present study is put on the proper detection of the facets appearing on the crystal surface. The original formulation presented in [1] employs the principle of the virtual work, and the enhanced scheme additionally makes use of a penalty formulation approach. The derivation of the penalty terms is discussed in detail. Finally, the corresponding discrete method is applied to a certain problematic example case presented in [1] resulting now in the correct crystal shape.


Crystal shape Facets Virtual work Penalty method 



We would like to thank Dr. H. Junes for helpful discussions.


  1. 1.
    M. Reivinen, E.-M. Salonen, I. Todoschchenko, V.P. Vaskelainen, Equilibrium crystal shapes by virtual work in 3D. J. Low Temp. Phys. 180, 394–415 (2015). doi: 10.1007/s10909-015-1313-y
  2. 2.
    M. Reivinen, E.-M. Salonen, I. Todoschchenko, V.P. Vaskelainen, Equilibrium crystal shapes by virtual work. J. Low Temp. Phys. 170, 75–90 (2013). doi: 10.1007/s10909-012-0666-8
  3. 3.
    E.-M. Salonen, M. Reivinen, Notes on surface stiffness. J. Low Temp. Phys. 174, 76–86 (2014). doi: 10.1007/s10909-013-0914-6
  4. 4.
    O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1, 4th edn. (McGraw-Hill, London, 1989)Google Scholar
  5. 5.
    M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming: theory and Algorithms (Wiley, New York, 1993)zbMATHGoogle Scholar
  6. 6.
    J.B. Hudson, Surface Science: an Introduction (Wiley, New York, 1998)Google Scholar
  7. 7.
    P. Nozieres, in Solids Far from Equilibrium, ed. by C. Godreche. Shape and growth of crystals (Cambridge University Press, Cambridge, 1991), pp. 1–154Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of EngineeringAalto UniversityAaltoFinland
  2. 2.Low Temperature LaboratoryAalto UniversityAaltoFinland
  3. 3.Department of Communications and NetworkingAalto UniversityAaltoFinland

Personalised recommendations