Multi-scale mathematical and mathematical-physical experiment in material science – foundry

  • 1 Bulgarian Academy of Science, Sofia, Bulgaria Institute of Metal Science, Equipment and Technologies with Hydro- and Aerodynamics Centre „Acad. Angel Balevski “,Sofia, Bulgaria


Multiscale mathematical modelling requires cross-cutting work with all areas of mathematics. The simplest proof is the mathematical experiment. An example is the finite element method, which is already used in quantum mechanics. The presented results are interesting for us, such as the 3D printer application and epitaxy.Numerical experiments for micron-sized hardening are shown.



  1. A. Balevski, Metal science, Technics, Sofia, 1962. (In Bulgarian)
  2. I. M. Parshorov, Dissertation for scientific degree Dr. Tech. Sc., Processes and states related to dynamic relaxation in alloyed martensite, Bulgarian academy of sciences, Institute of metal science, Sofia, 2001.
  3. A. Velikov, S. Stanev, A. Maneva, R. Dikov, Experimental mold for studying the pressure on heat transfer between a casting and a mold during casting by the method of gas pressing (GP-PROCESS). VIII MEZINARODNI VEDECKO-PRACTICKA KONFERENCE PRAHA, 2012, 28-32. (In Russian)
  4. G. Georgiev, A. Velikov, S. Stanev, A. Maneva, Thermal processes in the formation of aluminum casting as per the „GP “method. XXII INTRENATIONAL SCIENTIFIC TECHNICAL CONFERENCE „FOUNDRY 2015“, 16- 17.04.2015, PLEVEN, BULGARIA.
  5. A. Velikov, S. Stanev, A. Maneva, K. Petrov, Forecasting of temperature changes in obtaining castings with complex configuration by the method „Gas pressing “. Proceedings - International Conference „70 Years of MTF “, 2015, ISBN: 978-619-167-178-6, 141-144 International Non-Academic Publishing House (In Bulgarian)
  6. K. Kostov, M. Manilova, A. Velikov, At. Baikushev, Influence of interrupted recrystallization on the mechanical properties of aluminum alloys. XXIX International Conference “Defectoscopy’14”, “Days of Non-Destructive Testing 2014”, Year XXII, issue 1 (150), June 2014, 1, 150, 2014, 186-188. ISBN: 131-3946 (In Bulgarian)
  7. M. Manilova, A. Velikov, K. Kostov, At. Baikushev, Change of the mechanical properties of aluminum alloys with heat treatment while preserving the coarse-grained cast structure. NDT days 2014, "Days of non-destructive testing 2014", Year XXII, issue 1 (150), June 2014, 1, 150, 2014. ISBN: 131-3946
  8. Ch. Kittel, Introduction to Solid State Physics, Fourth Edition, John Wiley and Sons, Inc., Science, Moscow, 1978. (In Rusian)
  9. Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip, P. J.; Probert, M. J.; Refson, K.; Payne, M. C. "First principles methods using CASTEP", Zeitschrift fuer Kristallographie, 220 (5-6), 567-570 (2005). › wiki › CASTEP and CASTEP
  10. Enrico Fermi, Thermodynamics, Prentice-Hall, New York, 1937, second stereotype edition, Publishing House of Kharkov University, 1973. (In Russian)
  11. M. Flemings, Solidification processing, Peace, Moscow, 1977. (In Russian)
  12. M. Borisov, K. Marinova, Introduction to Solid State Physics Part One, Science and Art, Sofia, 1977. (In Bulgarian)
  13. A. Polikarov, Methodology of scientific knowledge, Science and Art, Sofia, 1972.
  14. A. Voigt, Multiscale Modeling in Epitaxial Growth, Birkhäuser Verlag, 2005.
  15. N. V. Vitanov,
  16. L. Litov, Quantum physics,
  17. R. E. Smallman, R. J. Bishop, Modern Physical Metallurgy and Materials Engineering, Butterworth-Heinemann, Oxford, 1999. ISBN 0 7506 4564 4
  18. B. Engquist, O. Runborg, Yen-Hsi R. Tsai., Numerical Analysis of Multiscale Computations, Springer-Verlag Berlin Heidelberg, 2012.
  19. A. Cohn, M. Rabinowitz, Classical Tunneling, arXiv:physics/0306009v1 [physics.gen-ph]
  20. W. Ubachs – Lectures MNW-Quant-Tunneling,
  21. Weinan E, Jianfeng L Multiscale modeling (invited mini-review article), Scholarpedia, 6(10):11527 (2011)..
  22. Ch. Wert, R. Thomson, Physics of Solids, Peace, Moscow (1969). (In Russian)
  23. W. Moy, M. A. Carignano, S. Kais, Finite Element Method for Finite-Size Scaling in Quantum Mechanics, J. Phys. Chem. A 2008, 112, 5448–5452
  24. H. J. W. Müller–Kirsten, D. K. Park, J. M. S. Rana, A condition for first order phase transitions in quantum mechanical tunneling models, arXiv:cond-mat/9902184v1[cond-mat.mes-hall],12 Feb 1999.
  25. S. A. Owerre, M. B. Paranjape, Macroscopic quantum tunneling and quantum-classical phase transitions of the escape rate in large spin systems, arXiv:1403.4208v3 [cond-mat.other], 2 Oct 2014
  26. P.M.W. Gill, Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-Consistent Field, University Cambridge, UK, 1994, 678-689 http://rsc/
  27. List of quantum chemistry and solid state physics software … Tunnel effect
  28. J. Stefan. ¨Uber die verdampfung und die aufl¨osung als vorgūange der diffusion. Annalen der Physik und Chemie, 277(12):725–747, 1890.
  29. G. Schulze, Metal physics, Peace, Moscow, 1971. (In Russian)
  30. P. G. Shewmon, Diffusion in solids, McGraw-Hill, New York, 1963.
  31. D. Elwell and H.J. Scheel, Crystal Growth from High- Temperature Solutions, Academic press, London, 1975 with additional Chapter II and Appendices A and B, 2011.
  32. M. Volmer and A. Weber, Z. Phys. Chem. 119 (1926) 277
  33. Kossel, W. (1927) Nachr. Gesell. Wiss. Göttingen, Math-Phys. Kl., 135.
  34. Stranski, I. N. Ann. Univ. Sofia 24, 297 (1927). I. N. Stranski, (1928) Z. Phys. Chem. 136, 259. I. Stranski, Z. Phys. Chem. A142, 453 (1929). I. Stranski, and K. Kuleliev, Z. Phys. Chem. A142, 467 (1929)
  35. Stranski, I. N. (1932) Z. physik. Chem. B 17,127.
  36. Stranski, I. N. and Kaischev, R. (1931) Z. Krist. 78,373.
  37. Stranski, I. N. and Kaischev, R. (1934) Z. physik. Chem. B 26,100,312. I. Stranski, Z. Phys. Chem. A142, 453 (1929).
  38. Volmer, M. (1939) "Kinetik der Phasenbildung" Steinkopff, Dresden, Leipzig.
  39. Stranski, I. N., et a/. (1949) Disc. Faraday Soc. 5, 13-32.
  40. Stranski, I. N. (1956) Bul/. Soc.franr. Mineral. Crist. 79,359.
  41. Burton, W. K. and Cabrera, N. (1949) Disc. Faraday Soc. 5, 33.
  42. F. C. Frank and J. H. van der Merwe, Proc. R. Soc. London, Ser. A 198, 205 (1949).
  43. 4F. C. Frank and J. H. van der Merwe, Proc. R. Soc. London, Ser. A 198, 216 (1949).
  44. Burton, W. K., Cabrera, N. and Frank, F. C. (1951) Phil. Trans. Roy. Soc. A243, 299.
  45. Cabrera, N. and Vermilyea, D. A. (1958) In "Growth and Perfeetion of Crystals" (R. H. Doremus, B. W. Roberts and D. Turnbull, eds.) p. 393. Wiley, New York.
  46. Cabrera, N. (1959) In "Metal and Semiconductor Surfaces" (H. C. Gates, ed.) p. 71, Wiley, New York.
  47. Frank, F. C. (1958) In "Growth and Perfeetion of Crystals" (R. H. Doremus, 13. W. Roberts and D. Turnbull, ecls.) pp. 3, 304. Wiley, New York.
  48. Chernov, A. A. (1961) Sov. Phys. Usp. 4, 129.
  49. Chernov, A. A. (1963) Sm•. Phys. Cryst. 8, 63.
  50. D. Kashchiev, Nucleation, Butterworth/Heinemann, Oxford, 2000. ISBN 0 7506 4682 9
  51. I. V. Markov, Crystal Growth for Beginners: Fundamentals crystal growth and epitaxy, 2-nd edition, World Scientific Publishing Co. Pte. Ltd, 2003. ISBN 981-238-245-3
  52. S. Bushev, Theoretical model of structure-formation in die casting, XXII International Scientific Technical Conference, 16-17.04.2015, Pleven, Bulgaria, Scientific notices, publisher: scientific technical union of mechanical engineering, XXIII y., ISSUE 3 (166) April 2015. ISSN: 1310 – 3946 (In Bulgarian)
  53. S. Bushev, Multi-scale approach for the complexity description – sustainability, Scientific proceedings „NDT days 2014“, XXII y., Number 1 (150), June 2014. ISSN 1310-3946 (In Bulgarian)
  54. A. Popov, Semiconductors materials and structures for nanoeleetronics, University Publishing House „Kliment Ohridski “, Sofia, 2007.

Article full text

Download PDF