Computational Mechanics

, Volume 54, Issue 3, pp 789–801 | Cite as

New advances in the forced response computation of periodic structures using the wave finite element (WFE) method

  • J.-M. Mencik
Original Paper


The wave finite element (WFE) method is investigated to describe the harmonic forced response of one-dimensional periodic structures like those composed of complex substructures and encountered in engineering applications. The dynamic behavior of these periodic structures is analyzed over wide frequency bands where complex spatial dynamics, inside the substructures, are likely to occur. Within the WFE framework, the dynamic behavior of periodic structures is described in terms of numerical wave modes. Their computation follows from the consideration of the finite element model of a substructure that involves a large number of internal degrees of freedom. Some rules of thumb of the WFE method are highlighted and discussed to circumvent numerical issues like ill-conditioning and instabilities. It is shown for instance that an exact analytic relation needs to be considered to enforce the coherence between positive-going and negative-going wave modes. Besides, a strategy is proposed to interpolate the frequency response functions of periodic structures at a reduced number of discrete frequencies. This strategy is proposed to tackle the problem of large CPU times involved when the wave modes are to be computed many times. An error indicator is formulated which provides a good estimation of the level of accuracy of the interpolated solutions at intermediate points. Adaptive refinement is carried out to ensure that this error indicator remains below a certain tolerance threshold. Numerical experiments highlight the relevance of the proposed approaches.


Structural dynamics Wave finite element method Periodic structures Model reduction 


  1. 1.
    Allman DJ (1996) Implementation of a flat facet shell finite element for applications in structural dynamics. Comput Str 59(4):657–663CrossRefzbMATHGoogle Scholar
  2. 2.
    Argyris JH, Papadrakakis M, Apostolopoulou C, Koutsourelakis S (2000) The TRIC shell element: theoretical and numerical investigation. Comput Method Appl Mech Eng 182:217–245CrossRefzbMATHGoogle Scholar
  3. 3.
    Bai Z, Dewilde PM, Freund RW (2005) Reduced-order modeling. In: Schilders WHA, ter Maten EJW (eds) Numerical methods in electromagnetics. Handbook of numerical analysis, vol 13, Elseiver, Amsterdam, pp 825–895Google Scholar
  4. 4.
    Craig RR, Bampton MCC (1968) Coupling of substructures for dynamic analyses. AIAA J 6(7):1313–1319CrossRefzbMATHGoogle Scholar
  5. 5.
    Duhamel D, Mace B, Brennan MJ (2006) Finite element analysis of the vibrations of waveguides and periodic structures. J Sound Vib 294:205–220CrossRefGoogle Scholar
  6. 6.
    Gavric L (1995) Computation of propagative waves in free rail using a finite element technique. J Sound Vib 185(3):531–543CrossRefzbMATHGoogle Scholar
  7. 7.
    Gry L, Gontier C (1997) Dynamic modelling of railway track: a periodic model based on a generalized beam formulation. J Sound Vib 199(4):531–558CrossRefGoogle Scholar
  8. 8.
    Hetmaniuk U, Tezaur R, Farhat C (2013) An adaptive scheme for a class of interpolatory model reduction methods for frequency response problems. Int J Numer Method Eng 93:1109–1124CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hussein M (2009) Reduced Bloch mode expansion for periodic media band structure calculations. Proc Royal Soc A 465:2825–2848 Google Scholar
  10. 10.
    Mace B, Duhamel D, Brennan M, Hinke L (2005) Finite element prediction of wave motion in structural waveguides. J Acoust Soc Am 117:2835CrossRefGoogle Scholar
  11. 11.
    Mace B, Manconi E (2008) Modelling wave propagation in two-dimensional structures using finite element analysis. J Sound Vib 318(4–5):884–902CrossRefGoogle Scholar
  12. 12.
    Manconi E, Mace B, Gaziera R (2009) Wave finite element analysis of fluid-filled pipes. In: Proceedings of the NOVEM 2009 “noise and vibration: emerging methods”, Oxford, UKGoogle Scholar
  13. 13.
    Mead D (1973) A general theory of harmonic wave propagation in linear periodic systems with multiple coupling. J Sound Vib 27(2):235–260CrossRefzbMATHGoogle Scholar
  14. 14.
    Mencik JM (2010) On the low- and mid-frequency forced response of elastic systems using wave finite elements with one-dimensional propagation. Comput Str 88(11–12):674–689CrossRefGoogle Scholar
  15. 15.
    Mencik JM (2013) A wave finite element based formulation for computing the forced response of structures involving rectangular flat shells. Int J Numer Method Eng 95(2):91–120CrossRefMathSciNetGoogle Scholar
  16. 16.
    Mencik JM, Ichchou MN (2005) Multi-mode propagation and diffusion in structures through finite elements. Eur J Mech A/Solids 24(5):877–898CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Mencik JM, Ichchou MN (2007) Wave finite elements in guided elastodynamics with internal fluid. Int J Solids Str 44:2148–2167CrossRefzbMATHGoogle Scholar
  18. 18.
    Nilsson CM, Finnveden S (2008) Waves in thin-walled fluid-filled ducts with arbitrary cross-sections. J Sound Vib 310(1–2):58–76CrossRefGoogle Scholar
  19. 19.
    Renno J, Mace B (2010) On the forced response of waveguides using the wave and finite element method. J Sound Vib 329(26):5474–5488CrossRefGoogle Scholar
  20. 20.
    Salimbahrami B, Lohmann B (2006) Order reduction of large scale second-order systems using Krylov subspace methods. Linear Algebra Appl 415:385–405CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Signorelli J, von Flotow A (1988) Wave propagation, power flow, and resonance in a truss beam. J Sound Vib 126(1):127–144CrossRefGoogle Scholar
  22. 22.
    Waki Y, Mace B, Brennan M (2009) Numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides. J Sound Vib 327(1–2):92–108CrossRefGoogle Scholar
  23. 23.
    Zhong WX, Williams FW (1995) On the direct solution of wave propagation for repetitive structures. J Sound Vib 181(3):485–501CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.INSA Centre Val de LoireUniversité François Rabelais de Tours, (LMR EA 2640)Blois CedexFrance

Personalised recommendations