Spectral Finite Element Method

  • Srinivasan Gopalakrishnan
  • Massimo Ruzzene
  • Sathyanarayana Hanagud
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


This chapter presents the procedures for the development of various types of spectral elements. The chapter begins with basic outline of spectral finite element formulation and illustrates its utility for wave propagation studies is complex structural components. Two variants of spectral formulations, namely the Fourier transform-based, and Wavelet transform-based spectral FEM are presented for both 1D and 2D waveguides. A number of examples are solved using the formulated elements to show the effectiveness of the spectral FEM approach to solve problems involving high frequency dynamic response.


Nodal Displacement Lamb Wave Spectral Element Nodal Force Group Speed 
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  1. 1.
    Amaratunga K, Williams JR (1995) Time integration using wavelet. In: Proceedings of SPIE, wavelet application for dual use, 2491, Orlando, FL, pp 894ᾢ902Google Scholar
  2. 2.
    Amaratunga K, Williams JR (1997) Wavelet-Galerkin solution of boundary value problems. Arch Comput Methods Eng 4(3):243ᾢ285MathSciNetCrossRefGoogle Scholar
  3. 3.
    Belykin G (1992) On wavelet based algorithms for solving differential equations. Department of Mathematics, University of Colorado, BoulderGoogle Scholar
  4. 4.
    Chakroborty A (2004) Wave propagation in anisotropic and inhomogeneous medium, October 2004. Ph.D. thesis, Indian Institute of Science, BangaloreGoogle Scholar
  5. 5.
    Chakraborty A, Gopalakrishnan S (2006) An approximate spectral element for the analysis of wave propagation in inhomogeneous layered media. AIAA J 44(7):1676ᾢ1685CrossRefGoogle Scholar
  6. 6.
    Chen MQ, Hwang C, Shih YP (1996) The computation of wavelet-Galerkin approximation on a bounded interval. Int J Numer Methods Eng 39:2921ᾢ2944MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19:297ᾢ301MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Daubechies I (1988) Orthonormal bases of compactly supported wavelets. Commun Pure Appl Math 41:906ᾢ966Google Scholar
  9. 9.
    Doyle JF (1997) Wave propagation in structures. Springer, New YorkMATHCrossRefGoogle Scholar
  10. 10.
    Gopalakrishnan S, Doyle JF (1994) Wave propagation in connected waveguides of varying cross-section. J Sound Vib 175(3):347ᾢ363MATHCrossRefGoogle Scholar
  11. 11.
    Gopalakrishnan S, Doyle JF (1995) Spectral super-elements for wave propagation in structures with local non-uniformities. Comput Methods Appl Mech Eng 121:77ᾢ90MATHCrossRefGoogle Scholar
  12. 12.
    Gopalakrishnan S, Chakraborty A, Roy Mahapatra D (2008) Spectral finite element method. Springer, LondonMATHGoogle Scholar
  13. 13.
    Graff KF (1975) Wave motion in elastic solids. Dover Publications Inc., New YorkMATHGoogle Scholar
  14. 14.
    Heideman MT, Johnson DH, Burrus CS (1984) Gauss and the history of the fast Fourier transform. IEEE ASSP Mag 1(4):14ᾢ21CrossRefGoogle Scholar
  15. 15.
    Moulin E, Assaad J, Delebarre C, Grondel S, Balageas D (2000) Modeling of integrated Lamb waves generation systems using a coupled finite element normal mode expansion method. Ultrasonics 38:522ᾢ526CrossRefGoogle Scholar
  16. 16.
    Nayfeh AH (1995) Wave propagation in layered anisotropic media. North Holland, AmsterdamMATHGoogle Scholar
  17. 17.
    Powel MJD (1970) A Fortran subroutine for solving systems of nonlinear. Numerical methods for nonlinear algebraic equations. Rabinowitz P (ed) Ch. 7, pp 115ᾢ161, Algebraic EquationsGoogle Scholar
  18. 18.
    Reddy JN (1985) Finite element method. McGraw Hill, New YorkGoogle Scholar
  19. 19.
    Rizzi SA (1989) A spectral analysis approach to wave propagation in layered solids potential in a wave guide. PhD thesis, Purdue UniversityGoogle Scholar
  20. 20.
    Rose JL (1999) Ultrasonic waves in solid media. Cambridge University Press, CambridgeGoogle Scholar
  21. 21.
    Sneddon IN (1951) Fourier transforms. McGraw-Hill, New YorkGoogle Scholar
  22. 22.
    Sneddon IN (1964) Partial differential equations. McGraw-Hill, New YorkGoogle Scholar
  23. 23.
    Solie LP, Auld BA (1973) Elastic waves in free anisotropic plates. J Acoust Soc Am 54(1):50ᾢ65CrossRefGoogle Scholar
  24. 24.
    Varadan VK, Vinoy KJ, Gopalakrishnan S (2006) Smart material systems and MEMS. Wiley, ChichesterCrossRefGoogle Scholar
  25. 25.
    Veidt M, Liub T, Kitipornchai S (2002) Modelling of Lamb waves in composite laminated plates excited by interdigital transducers. NDT E Int 35(7):437ᾢ447CrossRefGoogle Scholar
  26. 26.
    Verdict GS, Gien PH, Burge CP (1996) Finite element study of Lamb wave interactions with holes and through thickness defects in thin metal plates. NDT E Int 29(4):248Google Scholar
  27. 27.
    Viktorov IA (1967) Rayleigh and Lamb wave. Plenum Press, New YorkGoogle Scholar
  28. 28.
    Williams JR, Amaratunga K (1997) A discrete wavelet transform without edge effects using wavelet extrapolation. J Fourier Anal Appl 3(4):435ᾢ449MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Zhao G, Rose JL (2003) Boundary element modeling for defect characterization. Int J Solids Struct 40(11):2645ᾢ2658MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Srinivasan Gopalakrishnan
    • 1
  • Massimo Ruzzene
    • 2
  • Sathyanarayana Hanagud
    • 3
  1. 1.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia
  2. 2.School of Aerospace Engineering, School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

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