Hybrid Simulation Methods: Combining Finite Element Methods and Analytical Solutions

  • S. DuczekEmail author
  • Z. A. B. Ahmad
  • J. M. Vivar-Perez
  • U. GabbertEmail author
Part of the Research Topics in Aerospace book series (RTA)


In the context of wave propagation analysis the computational efficiency of numerical and semi-analytical methods is essential, as very fine spatial and temporal resolutions are required in order to describe all phenomena of interest, including scattering, reflection, mode conversion, and many more. These strict demands originate from the fact that high-frequency ultrasonic guided waves are investigated. In this chapter, our focus is on developing semi-analytical methods based on higher order basis functions and demonstrating their range of applicability. Thereby, we discuss the semi-analytical finite element method (SAFE) and a hybrid approach coupling spectral elements with analytical solutions in the frequency domain. The results illustrate that higher order methods are essential in order to decrease the numerical costs. Moreover, it is demonstrated that the proposed approaches are the methods of choice when we want to compute dispersion diagrams or if large parts of the structure are undisturbed and, therefore, can be described by analytical solutions. If, however, complex geometries are considered or the whole structure has to be investigated, only purely FE-based approaches seem to be a viable option.


  1. 1.
    Ahmad ZAB (2011) Numerical simulation of Lamb waves in plates using a semi-analytical finite element method. VDI Fortschritt-Berichte Reihe 20 Nr. 437Google Scholar
  2. 2.
    Ahmad ZAB, Gabbert U (2012) Simulation of Lamb wave reflections at plate edges using the semi-analytical finite element method. Ultrasonics 52:815–820CrossRefGoogle Scholar
  3. 3.
    Ahmad ZAB, Vivar Perez JM, Gabbert U (2013) Semi-analytical finite element method for modeling of Lamb wave propagation. CEAS Aeronaut J 4:21–33CrossRefGoogle Scholar
  4. 4.
    Bartoli I, Marzani A, di Scalea F, Viola E (2006) Modeling wave propagation in damped waveguides of arbitrary cross-section. J Sound Vib 295:685–707CrossRefGoogle Scholar
  5. 5.
    Boyd JP (2000) Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, MineolaGoogle Scholar
  6. 6.
    Chang Z, Mal A (1999) Scattering of Lamb waves from a rivet hole with edge cracks. Mech Mater 31(3):197–204CrossRefGoogle Scholar
  7. 7.
    Chapuis B, Terrien N, Royer D (2010) Excitation and focusing of Lamb waves in a multilayered anisotropic plate. J Acoust Soc Am 127(1):198–203CrossRefGoogle Scholar
  8. 8.
    Chitnis M, Desai Y, Shah A, Kant T (2003) Comparisons of displacement-based theories for waves and vibrations in laminated and sandwich composite plates. J Sound Vib 263:617–642CrossRefGoogle Scholar
  9. 9.
    Damljanovic V, Weaver R (2004) Forced response of a cylindrical waveguide with simulation of the wavenumber extraction problem. J Acoust Soc Am 115(4):1582–1591CrossRefGoogle Scholar
  10. 10.
    Finnveden S (2004) Evaluation of modal density and group velocity by a finite element method. J Sound Vib 273:51–75CrossRefzbMATHGoogle Scholar
  11. 11.
    Fish J, Belytschko T (2007) A first course in finite elements. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  12. 12.
    Fornberg B (1998) A practical guide to pseudospectral methods. Cambridge monograph on applied and computational mathematics, Cambridge University Press, CambridgeGoogle Scholar
  13. 13.
    Galan J, Abascal R (2002) Numerical simulation of Lamb wave scattering in semi-infinite plates. Int J Numer Methods Eng 53:1145–1173CrossRefGoogle Scholar
  14. 14.
    Gao H (2007) Ultrasonic guided wave mechanics for composite material structural health monitoring. PhD thesis, The Pennsylvania State UniversityGoogle Scholar
  15. 15.
    Gavric L (1995) Computation of propagative waves in free rail using a finite element technique. J Sound Vib 185(3):531–543CrossRefzbMATHGoogle Scholar
  16. 16.
    Giurgiutiu V (2002) Lamb wave generation with piezoelectric wafer active sensors for structural health monitoring. In: SPIE’s 10th Annual International Symposium on Smart Structures and Materials and 8th Annual International Symposium on NDE for Health Monitoring and DiagnosticsGoogle Scholar
  17. 17.
    Giurgiutiu V (2008) Structural health monitoring with piezoelectric active wafer sensors: fundamentals and applications. Elsevier, AmsterdamGoogle Scholar
  18. 18.
    Giurgiutiu V (2008) Structural health monitoring with piezoelectric wafer active sensors. Academic, Elsevier, AmsterdamGoogle Scholar
  19. 19.
    Glushkov EV, Glushkova NV, Seemann W, Kvasha OV (2006) Elastic wave excitation in a layer by piezoceramic patch actuators. Acoust Phys 52(4):398–407CrossRefGoogle Scholar
  20. 20.
    Glushkov Y, Glushkova N, Krivonos A (2010) The excitation and propagation of elastic waves in multilayered anisotropic composites. J Appl Math Mech 74(3):297–305MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Han X, Liu GR, Xi ZC, Lam KY (2002) Characteristics of waves in a functionally graded cylinder. Int J Numer Methods Eng 53:653–676CrossRefzbMATHGoogle Scholar
  22. 22.
    Hayashi T (2002) Guided wave animation using semi-analytical finite element method. NDT, pp 75–79Google Scholar
  23. 23.
    Hayashi T, Endoh S (2000) Calculation and visualization of Lamb wave motion. Ultrasonics 38:770–773CrossRefGoogle Scholar
  24. 24.
    Hayashi T, Inoue D (2014) Calculation of leaky Lamb waves with a semi-analytical finite element method. Ultrasonics 54:1460–1469CrossRefGoogle Scholar
  25. 25.
    Hayashi T, Kawashima K (2002) Multiple reflections of Lamb waves at a delamination. Ultrasonics 40:193–197CrossRefGoogle Scholar
  26. 26.
    Hayashi T, Kawashima K, Sun Z, Rose JL (2003) Analysis of flexural mode focusing by a semianalytical finite element method. J Acoust Soc Am 113(3):1241–1248CrossRefGoogle Scholar
  27. 27.
    Hayashi T, Song W, Rose J (2003) Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example. Ultrasonics 41:175–183CrossRefGoogle Scholar
  28. 28.
    Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Upper Saddle RiverzbMATHGoogle Scholar
  29. 29.
    Inoue D, Hayashi T (2015) Transient analysis of leaky Lamb waves with a semi-analytical finite element method. Ultrasonics 62:80–88CrossRefGoogle Scholar
  30. 30.
    Karmazin A, Kirillova E, Seemann W, Syromyatnikov P (2010) Modelling of 3d steady-state oscillations of anisotropic multilayered structures applying the Green’s functions. Adv Theor Appl Mech 3(9):425–445zbMATHGoogle Scholar
  31. 31.
    Karmazin A, Kirillova E, Seemann W, Syromyatnikov P (2011) Investigation of Lamb elastic waves in anisotropic multilayered composites applying the Green’s matrix. Ultrasonics 51(1):17–28CrossRefGoogle Scholar
  32. 32.
    Karunasena W (2004) Numerical modeling of obliquely incident guided wave scattering by a crack in a laminated composite plate. In: Atrens A, Boland J, Clegg R, Griffiths J (eds) Structural integrity and fracture international conference (SIF04), pp 181–187Google Scholar
  33. 33.
    Karunasena W (2008) Elastodynamic reciprocity relations for wave scattering by flaws in fiber-reinforced composite plates. J Mech Mater Struct 3(10):1831–1846CrossRefGoogle Scholar
  34. 34.
    Karunasena W, Shah A, Datta S (1991) Wave propagation in a multilayered laminated cross-ply composite plate. Trans. ASME 58:1028–1032CrossRefzbMATHGoogle Scholar
  35. 35.
    Karunasena W, Liew K, Kitipornchai S (1995) Hybrid analysis of Lamb wave reflection by a crack at the fixed edge of a composite plate. Comput Methods Appl Mech Eng 125:221–233CrossRefGoogle Scholar
  36. 36.
    Karunasena W, Liew KM, Kitipornchai S (1995) Reflection of plate waves at the fixed edge of a composite plate. J Acoust Soc Am 98(1):644–651CrossRefGoogle Scholar
  37. 37.
    Lagasse P (1973) Higher-order finite-element analysis of topographic guides supporting elastic surface waves. J Acoust Soc Am 53(4):1116–1122CrossRefGoogle Scholar
  38. 38.
    Li W, Dwight RA, Zhang T (2015) On the study of vibration of a supported railway rail using the semi-analytical finite element method. J Sound Vib 345:121–145CrossRefGoogle Scholar
  39. 39.
    Liu GR (2002) A combined finite element/strip element method for analyzing elastic wave scattering by cracks and inclusions in laminates. Comput Mech 28:76–81CrossRefzbMATHGoogle Scholar
  40. 40.
    Liu G, Xi Z (2002) Elastic waves in anisotropic laminates. CRC Press, Boca RatonGoogle Scholar
  41. 41.
    Loveday P (2006) Numerical comparison of patch and sandwich piezoelectric transducers for transmitting ultrasonic waves. Proc SPIE 6166:616,612CrossRefGoogle Scholar
  42. 42.
    Loveday P (2007) Analysis of piezoelectric ultrasonic transducers attached to waveguides using waveguide finite elements. IEEE Trans Ultrason Ferroelectr Freq Control 54(10): 2045–2051CrossRefGoogle Scholar
  43. 43.
    Loveday PW (2009) Semi-analytical finite element analysis of elastic waveguides subjected to axial loads. Ultrasonics 49:298–300CrossRefGoogle Scholar
  44. 44.
    Loveday P, Long C (2007) Time domain simulation of piezoelectric excitation of guided waves in rails using waveguide finite elements. Proc SPIE 6529:65,290V–1Google Scholar
  45. 45.
    Matt HM (2006) Structural diagnostics of CFRP composite aircraft components by ultrasonic guided waves and built-in piezoelectric transducers. PhD thesis, University of California San DiegoGoogle Scholar
  46. 46.
    Mazzotti M, Bartoli I, Marzani A, Viola E (2013) A coupled SAFE-2.5D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides for arbitrary cross-section. Ultrasonics 53:1227–1241CrossRefzbMATHGoogle Scholar
  47. 47.
    Morvan B, Wilkie-Chancellier N, Duflo H, Trinel A, Duclos J (2003) Lamb wave reflection at the free edge of a plate. J Acoust Soc Am 113(3):1417–1425CrossRefGoogle Scholar
  48. 48.
    Moulin E, Assaad J, Delebarre C (2000) Modeling of Lamb waves generated by integrated transducers in composite plates using a coupled finite element-normal modes expansion method. J Acoust Soc Am 107(1):87CrossRefGoogle Scholar
  49. 49.
    Muller DE (1956) A method for solving algebraic equations using an automatic computer. Math Tables and Other Aids to Comput 10(56):208–215MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Nelson R, Dong S (1973) High frequency vibrations and waves in laminated orthotropic plates. J Sound Vib 30(1):33–44CrossRefGoogle Scholar
  51. 51.
    Piersol A, Paez T (2009) Harris’s shock and vibration handbook, 6th edn. McGraw-Hill Professional, New YorkGoogle Scholar
  52. 52.
    Royer D, Dieulesaint E (2000) Elastic waves in solids I: free and guided propagation. Springer, BerlinCrossRefzbMATHGoogle Scholar
  53. 53.
    Ryue J, Thompson D, White P, Thompson D (2009) Decay rates of propagating waves in railway tracks at high frequencies. J Sound Vib 320:955–976CrossRefGoogle Scholar
  54. 54.
    Terrien N, Osmont D, Royer D, Lepoutre F, Déom A (2007) A combined finite element and modal decomposition method to study the interaction of Lamb modes with micro-defects. Ultrasonics 46:74–88CrossRefGoogle Scholar
  55. 55.
    Tian J, Gabbert U, Berger H, Su X (2004) Lamb wave interaction with delaminations in CFRP laminates. Comput Mater Continua 1(4):327–336zbMATHGoogle Scholar
  56. 56.
    Trefethen LM (2000) Spectral Methods in MATLAB. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  57. 57.
    Velichko A, Wilcox P (2007) Modeling the excitation of guided waves in generally anisotropic multilayered media. J Acoust Soc Am 121(1):60–69CrossRefGoogle Scholar
  58. 58.
    Vivar-Perez JM (2012) Analytical and Spectral Methods for the Simulation of Elastic Waves in Thin Plates. VDI Fortschritt-Berichte Reihe 20 Nr. 441Google Scholar
  59. 59.
    Vivar Perez JM, Ahmad ZAB, Gabbert U (2013) Membrane carrier wave function in the modelling of Lamb wave propagation. CEAS Aeronaut J 4:51–59CrossRefGoogle Scholar
  60. 60.
    Vivar Perez JM, Duczek S, Gabbert U (2014) Analytical and higher order finite element hybrid approaches for an efficient simulation of ultrasonic guided waves I: 2D-analysis. Smart Struct Syst 13:587–614CrossRefGoogle Scholar
  61. 61.
    von Ende S, Schäfer I, Lammering R (2007) Lamb wave excitation with piezoelectric wafers – an analytical approach. Acta Mech 193(3–4):141–150CrossRefzbMATHGoogle Scholar
  62. 62.
    Wilcox P (2004) Modeling the excitation of Lamb and SH waves by point and line sources. AIP Conf Proc 700:206–213CrossRefGoogle Scholar
  63. 63.
    Willberg C, Vivar Perez JM, Duczek S, Ahmad ZAB (2015) Simulation methods for guided-wave based structural health monitoring: A review. Appl Mech Rev 67:1–20CrossRefGoogle Scholar
  64. 64.
    Yang J (2005) An introduction to the theory of piezoelectricity. Advances in mechanics and mathematics, vol 9. Springer, BerlinGoogle Scholar
  65. 65.
    Zheng-Sheng Y, Yao-Zhi C, Min-Jae O, Tae-Wan K, Qun-Sheng P (2006) An efficient method for tracing planar implicit curves. J Zheijang Univ Sci A 7(7):1115–1123CrossRefGoogle Scholar
  66. 66.
    Zienkiewicz OC, Taylor RL (2000) The finite element method: volume 1 the basis. Butterworth Heinemann, OxfordzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of MechanicsOtto von Guericke University MagdeburgMagdeburgGermany

Personalised recommendations