, Volume 49, Issue 10, pp 2399–2418 | Cite as

Meshfree modeling of dynamic response of mechanical structures



Transient nature of the loading conditions applied to the structural components makes dynamic analysis one of the important components in the design-analysis cycle. Time-varying forces and accelerations can substantially change stress distributions and cause a premature failure of the mechanical structures. In addition, it is also important to determine dynamic response of the structural elements to the frequency of the applied loads. In this paper we describe the first application of the meshfree Solution Structure Method (SSM) to the structural dynamics problems. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the proposed approach we will use SSM to enforce initial and boundary conditions, while transient behavior is enforced by time marching schemes used to solve systems of ordinary differential equations. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward application to model vibrations of 3D structures.


Dynamic response Natural frequencies Finite element analysis Solution Structure Method Meshfree method Distance fields 



This research work was supported in part by the National Science Foundation Grant CMMI-0900219. The author Tomislav Kosta is also grateful for the following Grant 58940-RT-REP from the Army Research Office that enabled this work to be possible.


  1. 1.
    Bloomenthal J (1997) Introduction to implicit surfaces. Morgan Kaufmann Publishers, San FranciscoMATHGoogle Scholar
  2. 2.
    Causevic M, Mitrovic S (2011) Comparison between non-linear dynamic and static seismic analysis of structures according to European and US provisions. Bull Earthq Eng 9(2):467–489CrossRefGoogle Scholar
  3. 3.
    Deierlein G, Reinhorn A, Willford M (2010) Nonlinear structural analysis for seismic design, a guide for practicing engineers. NEHRP Seismic Design Technical Brief No. 4. NIST GCR 10-917-5Google Scholar
  4. 4.
    Duchon C (1979) Lanczos filtering in one and two dimensions. J Appl Meteorol Climatol 18(8):1016–1022CrossRefGoogle Scholar
  5. 5.
    Freytag M, Shapiro V, Tsukanov I (2006) Field modeling with sampled distances. Comput Aided Des 38(2):87–100CrossRefGoogle Scholar
  6. 6.
    Freytag M, Shapiro V, Tsukanov I (2007a) Differentiable distance fields from scanned data. Technical report, University of Wisconsin-Madison.
  7. 7.
    Freytag M, Shapiro V, Tsukanov I (2007b) Scan and solve: acquiring the physics of artifacts. In: Proceedings of the ASME 2007 international design engineering technical conferences and computers and information in engineering conference, Las Vegas, USAGoogle Scholar
  8. 8.
    Freytag M, Shapiro V, Tsukanov I (2011) Finite element analysis in situ. Finite Elem Anal Des 47:957–972CrossRefGoogle Scholar
  9. 9.
    Fryazinov O, Vilbrandt T, Pasko A (2013) Multi-scale space-variant FRep cellular structures. Comput Aided Des 45(1):26–34MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kantorovich LV, Krylov V (1958) Approximate methods of higher analysis. Interscience Publisher, New YorkMATHGoogle Scholar
  11. 11.
    Komzsik L (2003) The Lanczos method: evolution and application. Software, environments and tools. Society for Industrial and Applied MathematicsGoogle Scholar
  12. 12.
    Kosta T, Tsukanov I (2014a) Meshfree natural vibration analysis of 2D structures. Comput Mech 53(2):283–296. doi: 10.1007/s00466-013-0907-y CrossRefGoogle Scholar
  13. 13.
    Kosta T, Tsukanov I (2014b) Three-dimensional natural vibration analysis with meshfree solution structure method. J Vib Acoust (in print)Google Scholar
  14. 14.
    Krysl P (2006) A pragmatic introduction to the finite element method for thermal and stress analysis: with the Matlab toolkit Sofea. World Scientific, SingaporeCrossRefGoogle Scholar
  15. 15.
    Lanczos C (1988) Applied analysis. Dover books on mathematics. Dover Publications, New York.
  16. 16.
    Pasko A, Adzhiev V (2004) Function-based shape modeling: mathematical framework and specialized language. In: Winkler F (ed) Automated deduction in geometry, lecture notes in artificial intelligence, vol 2930. Springer, Berlin, pp 132–160CrossRefGoogle Scholar
  17. 17.
    Petyt M (2010) Introduction to finite element vibration analysis, 2nd edn. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  18. 18.
    Ricci A (1973) A constructive geometry for computer graphics. Comput J 16(2):157–160MATHCrossRefGoogle Scholar
  19. 19.
    Rvachev VL (1982) Theory of R-functions and some applications. Naukova Dumka, Kiev (in Russian)Google Scholar
  20. 20.
    Rvachev VL, Sheiko TI (1996) R-functions in boundary value problems in mechanics. Appl Mech 48:151–188Google Scholar
  21. 21.
    Rvachev VL, Sheiko TI, Shapiro V, Tsukanov I (2000) On completeness of RFM solution structures. Comput Mech 25:305–317MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Rvachev VL, Sheiko TI, Shapiro V, Tsukanov I (2001) Transfinite interpolation over implicitly defined sets. Comput Aided Geom Des 18:195–220MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Saito T, Toriwaki JI (1994) New algorithms for Euclidean distance transformation of an n-dimensional digitized picture with applications. Pattern Recognit 27(11):1551–1565CrossRefGoogle Scholar
  24. 24.
    Shapiro V (2007) Semi-analytic geometry with R-functions. Acta Numer 16:239–303MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Shapiro V, Tsukanov I (1999) Implicit functions with guaranteed differential properties. In: SMA ’99: proceedings of the fifth ACM symposium on solid modeling and applications. ACM, New York, pp 258–269Google Scholar
  26. 26.
    Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data. In: Proceedings of the 23rd ACM national conference, pp 517–524Google Scholar
  27. 27.
    Timoshenko S (1930) Strength of materials: elementary theory and problems. Strength of materials. D. Van Nostrand Company, Incorporated, New York.
  28. 28.
    Tsukanov I, Shapiro V (2002) The architecture of SAGE—a meshfree system based on RFM. Eng Comput 18(4):295–311CrossRefGoogle Scholar
  29. 29.
    Zienkiewicz O, Taylor R (2005) The finite element method set. Elsevier Science, Amsterdam.

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mechanical and Materials EngineeringFlorida International UniversityMiamiUSA

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