Discrete mass-spring structure identification in nonlocal continuum space-fractional model

  • Krzysztof Szajek
  • Wojciech SumelkaEmail author
Open Access
Regular Article


This paper considers discrete mass-spring structure identification in a nonlocal continuum space-fractional model, defined as an optimization task. Dynamic (eigenvalues and eigenvectors) and static (displacement field) solutions to discrete and continuum theories are major constituents of the objective function. It is assumed that the masses in both descriptions are equal (and constant), whereas the spring stiffness distribution in a discrete system becomes a crucial unknown. The considerations include a variety of configurations of the nonlocal parameter and the order of the fractional model, which makes the study comprehensive, and for the first time provides insight into the possible properties (geometric and mechanical) of a discrete structure homogenized by a space-fractional formulation.


  1. 1.
    W. Sumelka, G.Z. Voyiadjis, Int. J. Solids Struct. 124, 151 (2017)CrossRefGoogle Scholar
  2. 2.
    R. Xiao, H. Sun, W. Chen, Int. J. Non-Linear Mech. 93, 7 (2017)ADSCrossRefGoogle Scholar
  3. 3.
    Y. Sun, Y. Shen, Int. J. Geomech. 17, 04017025 (2017)CrossRefGoogle Scholar
  4. 4.
    Yifei Sun, Yang Xiao, Int. J. Numer. Anal. Methods Geomech. 41, 555 (2017)CrossRefGoogle Scholar
  5. 5.
    Yifei Sun, Yang Xiao, Int. J. Solids Struct. 118-119, 224 (2017)CrossRefGoogle Scholar
  6. 6.
    Ruifan Meng, Deshun Yin, Chao Zhou, Hao Wu, Appl. Math. Model. 40, 398 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Xia Nan, Tiyip Tashpolat, Kelimu Ardak, Nurmemet Ilyas, Ding Jianli, Zhang Fei, Zhang Dong, J. Spectrosc. 2017, 1236329 (2017)Google Scholar
  8. 8.
    Mengke Liao, Yuanming Lai, Enlong Liu, Xusheng Wan, Acta Geotech. 12, 377 (2017)CrossRefGoogle Scholar
  9. 9.
    He Zhilei, Zhu Zhende, Wu Nan, Wang Zhen, Cheng Shi, Math. Probl. Eng. 2016, 8572040 (2016)Google Scholar
  10. 10.
    J.L. Suzuki, M. Zayernouri, M.L. Bittencourt, G.E. Karniadakis, Comput. Methods Appl. Mech. Eng. 308, 443 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    Caputo Michele, Ciarletta Michele, Fabrizio Mauro, Tibullo Vincenzo, Rend. Lincei - Mat. Appl. 28, 463 (2017)CrossRefGoogle Scholar
  12. 12.
    M. Faraji Oskouie, R. Ansari, Appl. Math. Model. 43, 337 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    K. Nishimoto, Fractional Calculus, Vol. I--IV (Descatres Press, Koriyama, Japan, 1984-1991)Google Scholar
  14. 14.
    I. Podlubny, Fractional Differential Equations, in Mathematics in Science and Engineering, Vol. 198 (Academic Press, 1999)Google Scholar
  15. 15.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)Google Scholar
  16. 16.
    A.B. Malinowska, T. Odzijewicz, D.F.M. Torres, Advanced Methods in the Fractional Calculus of Variations, in Springer Briefs in Applied Sciences and Technology (Springer, 2015)Google Scholar
  17. 17.
    J.A.T. Machado, V. Kiryakova, F. Mainardi, Commun. Nonlinear Sci. Numer. Simul. 16, 1140 (2011)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Fractional Calculus Appl. Anal. 16, 64 (2013)MathSciNetGoogle Scholar
  19. 19.
    C.S. Drapaca, S. Sivaloganathan, J. Elast. 107, 107 (2012)CrossRefGoogle Scholar
  20. 20.
    W. Sumelka, J. Therm. Stresses 37, 678 (2014)CrossRefGoogle Scholar
  21. 21.
    A.K. Lazopoulos, Arch. Appl. Mech. 86, 1987 (2016)ADSCrossRefGoogle Scholar
  22. 22.
    Beda Peter, Eng. Trans. 65, 209 (2017)Google Scholar
  23. 23.
    Blaszczyk Tomasz, J. Mech. Mater. Struct. 12, 23 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Sapora, P. Cornetti, B. Chiaia, E.K. Lenzi, L.R. Evangelista, J. Eng. Mech. 143, D4016007 (2017)CrossRefGoogle Scholar
  25. 25.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Meccanica 53, 1115 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    W. Sumelka, R. Zaera, J. Fernández-Sáez, Eur. Phys. J. Plus 131, 320 (2016)CrossRefGoogle Scholar
  27. 27.
    P. Rosenau, Phys. Rev. B 36, 5868 (1987)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    P. Rosenau, Phys. Lett. A 331, 39 (2003)ADSCrossRefGoogle Scholar
  29. 29.
    D.A. Fafalis, S.P. Filopoulos, G.J. Tsamasphyros, Eur. J. Mech. A/Solids 36, 25 (2012)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    W. Sumelka, J. Theor. Appl. Mech. 52, 671 (2014)Google Scholar
  31. 31.
    Wojciech Sumelka, Mech. Res. Commun. 86, 5 (2017)CrossRefGoogle Scholar
  32. 32.
    F.R. Hall, D.R. Hayhurst, Proc. Math. Phys. Sci. 433, 405 (1991)CrossRefGoogle Scholar
  33. 33.
    R.H.J. Peerlings, M.G.D. Geers, R. de Borst, W.A.M. Brekelmans, Int. J. Solids Struct. 38, 7723 (2001)CrossRefGoogle Scholar
  34. 34.
    Z. Odibat, Appl. Math. Comput. 178, 527 (2006)MathSciNetGoogle Scholar
  35. 35.
    J.S. Leszczyski, An Introduction to Fractional Mechanics, in Monographs, No 198 (The Publishing Office of Czestochowa University of Technology, 2011)Google Scholar
  36. 36.
    Harm Askes, Elias C. Aifantis, Int. J. Solids Struct. 48, 1962 (2011)CrossRefGoogle Scholar
  37. 37.
    W. Sumelka, T. Lodygowski, ASME J. Eng. Mater. Technol. 135, 021009 (2013)CrossRefGoogle Scholar
  38. 38.
    G. Strang, Linear Algebra and Its Applications, 2nd Ed. (Academic Press, Inc., Orlando, 1980)Google Scholar
  39. 39.
    R.H. Byrd, P. Lu, J. Nocedal, SIAM J. Sci. Stat. Comput. 16, 1190 (1995)CrossRefGoogle Scholar
  40. 40.
    C. Zhu, R.H. Byrd, J. Nocedal, ACM Trans. Math. Softw. 23, 550 (1997)CrossRefGoogle Scholar
  41. 41.
    SciPy Developers, SciPy, 2017Google Scholar
  42. 42.
    R.J. Allemang, Investigation of Some Multiple Input/Output Frequency Response Function Experimental Modal Analysis Techniques (University of Cincinnati, Department of Mechanical, and Industrial Engineering, 1980)Google Scholar
  43. 43.
    R.J. Allemang, D.L. Brown, A correlation coefficient for modal vector analysis, in 1st International Modal Analysis Conference (1982)Google Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Poznan University of TechnologyInstitute of Structural EngineeringPoznanPoland

Personalised recommendations