Skip to main content
SpringerLink
Log in

Hybrid Simulation Methods: Combining Finite Element Methods and Analytical Solutions

  • Chapter
  • First Online:
Lamb-Wave Based Structural Health Monitoring in Polymer Composites

Part of the book series: Research Topics in Aerospace ((RTA))

  • 1449 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ahmad ZAB (2011) Numerical simulation of Lamb waves in plates using a semi-analytical finite element method. VDI Fortschritt-Berichte Reihe 20 Nr. 437

    Google Scholar 

  2. Ahmad ZAB, Gabbert U (2012) Simulation of Lamb wave reflections at plate edges using the semi-analytical finite element method. Ultrasonics 52:815–820

    Article  Google Scholar 

  3. Ahmad ZAB, Vivar Perez JM, Gabbert U (2013) Semi-analytical finite element method for modeling of Lamb wave propagation. CEAS Aeronaut J 4:21–33

    Article  Google Scholar 

  4. Bartoli I, Marzani A, di Scalea F, Viola E (2006) Modeling wave propagation in damped waveguides of arbitrary cross-section. J Sound Vib 295:685–707

    Article  Google Scholar 

  5. Boyd JP (2000) Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, Mineola

    Google Scholar 

  6. Chang Z, Mal A (1999) Scattering of Lamb waves from a rivet hole with edge cracks. Mech Mater 31(3):197–204

    Article  Google Scholar 

  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):198–203

    Article  Google Scholar 

  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 263:617–642

    Article  Google Scholar 

  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):1582–1591

    Article  Google Scholar 

  10. Finnveden S (2004) Evaluation of modal density and group velocity by a finite element method. J Sound Vib 273:51–75

    Article  MATH  Google Scholar 

  11. Fish J, Belytschko T (2007) A first course in finite elements. Wiley, Hoboken

    Book  MATH  Google Scholar 

  12. Fornberg B (1998) A practical guide to pseudospectral methods. Cambridge monograph on applied and computational mathematics, Cambridge University Press, Cambridge

    Google Scholar 

  13. Galan J, Abascal R (2002) Numerical simulation of Lamb wave scattering in semi-infinite plates. Int J Numer Methods Eng 53:1145–1173

    Article  Google Scholar 

  14. Gao H (2007) Ultrasonic guided wave mechanics for composite material structural health monitoring. PhD thesis, The Pennsylvania State University

    Google Scholar 

  15. Gavric L (1995) Computation of propagative waves in free rail using a finite element technique. J Sound Vib 185(3):531–543

    Article  MATH  Google Scholar 

  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 Diagnostics

    Google Scholar 

  17. Giurgiutiu V (2008) Structural health monitoring with piezoelectric active wafer sensors: fundamentals and applications. Elsevier, Amsterdam

    Google Scholar 

  18. Giurgiutiu V (2008) Structural health monitoring with piezoelectric wafer active sensors. Academic, Elsevier, Amsterdam

    Google Scholar 

  19. Glushkov EV, Glushkova NV, Seemann W, Kvasha OV (2006) Elastic wave excitation in a layer by piezoceramic patch actuators. Acoust Phys 52(4):398–407

    Article  Google Scholar 

  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):297–305

    Article  MathSciNet  MATH  Google Scholar 

  21. Han X, Liu GR, Xi ZC, Lam KY (2002) Characteristics of waves in a functionally graded cylinder. Int J Numer Methods Eng 53:653–676

    Article  MATH  Google Scholar 

  22. Hayashi T (2002) Guided wave animation using semi-analytical finite element method. NDT, pp 75–79

    Google Scholar 

  23. Hayashi T, Endoh S (2000) Calculation and visualization of Lamb wave motion. Ultrasonics 38:770–773

    Article  Google Scholar 

  24. Hayashi T, Inoue D (2014) Calculation of leaky Lamb waves with a semi-analytical finite element method. Ultrasonics 54:1460–1469

    Article  Google Scholar 

  25. Hayashi T, Kawashima K (2002) Multiple reflections of Lamb waves at a delamination. Ultrasonics 40:193–197

    Article  Google Scholar 

  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):1241–1248

    Article  Google Scholar 

  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 41:175–183

    Article  Google Scholar 

  28. Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  29. Inoue D, Hayashi T (2015) Transient analysis of leaky Lamb waves with a semi-analytical finite element method. Ultrasonics 62:80–88

    Article  Google Scholar 

  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):425–445

    MATH  Google Scholar 

  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):17–28

    Article  Google Scholar 

  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 181–187

    Google Scholar 

  33. Karunasena W (2008) Elastodynamic reciprocity relations for wave scattering by flaws in fiber-reinforced composite plates. J Mech Mater Struct 3(10):1831–1846

    Article  Google Scholar 

  34. Karunasena W, Shah A, Datta S (1991) Wave propagation in a multilayered laminated cross-ply composite plate. Trans. ASME 58:1028–1032

    Article  MATH  Google Scholar 

  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 125:221–233

    Article  Google Scholar 

  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):644–651

    Article  Google Scholar 

  37. Lagasse P (1973) Higher-order finite-element analysis of topographic guides supporting elastic surface waves. J Acoust Soc Am 53(4):1116–1122

    Article  Google Scholar 

  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 345:121–145

    Article  Google Scholar 

  39. Liu GR (2002) A combined finite element/strip element method for analyzing elastic wave scattering by cracks and inclusions in laminates. Comput Mech 28:76–81

    Article  MATH  Google Scholar 

  40. Liu G, Xi Z (2002) Elastic waves in anisotropic laminates. CRC Press, Boca Raton

    Google Scholar 

  41. Loveday P (2006) Numerical comparison of patch and sandwich piezoelectric transducers for transmitting ultrasonic waves. Proc SPIE 6166:616,612

    Article  Google Scholar 

  42. Loveday P (2007) Analysis of piezoelectric ultrasonic transducers attached to waveguides using waveguide finite elements. IEEE Trans Ultrason Ferroelectr Freq Control 54(10): 2045–2051

    Article  Google Scholar 

  43. Loveday PW (2009) Semi-analytical finite element analysis of elastic waveguides subjected to axial loads. Ultrasonics 49:298–300

    Article  Google Scholar 

  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–1

    Google Scholar 

  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 Diego

    Google Scholar 

  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 53:1227–1241

    Article  MATH  Google Scholar 

  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):1417–1425

    Article  Google Scholar 

  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):87

    Article  Google Scholar 

  49. Muller DE (1956) A method for solving algebraic equations using an automatic computer. Math Tables and Other Aids to Comput 10(56):208–215

    Article  MathSciNet  MATH  Google Scholar 

  50. Nelson R, Dong S (1973) High frequency vibrations and waves in laminated orthotropic plates. J Sound Vib 30(1):33–44

    Article  Google Scholar 

  51. Piersol A, Paez T (2009) Harris’s shock and vibration handbook, 6th edn. McGraw-Hill Professional, New York

    Google Scholar 

  52. Royer D, Dieulesaint E (2000) Elastic waves in solids I: free and guided propagation. Springer, Berlin

    Book  MATH  Google Scholar 

  53. Ryue J, Thompson D, White P, Thompson D (2009) Decay rates of propagating waves in railway tracks at high frequencies. J Sound Vib 320:955–976

    Article  Google Scholar 

  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 46:74–88

    Article  Google Scholar 

  55. Tian J, Gabbert U, Berger H, Su X (2004) Lamb wave interaction with delaminations in CFRP laminates. Comput Mater Continua 1(4):327–336

    MATH  Google Scholar 

  56. Trefethen LM (2000) Spectral Methods in MATLAB. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  57. Velichko A, Wilcox P (2007) Modeling the excitation of guided waves in generally anisotropic multilayered media. J Acoust Soc Am 121(1):60–69

    Article  Google Scholar 

  58. Vivar-Perez JM (2012) Analytical and Spectral Methods for the Simulation of Elastic Waves in Thin Plates. VDI Fortschritt-Berichte Reihe 20 Nr. 441

    Google Scholar 

  59. Vivar Perez JM, Ahmad ZAB, Gabbert U (2013) Membrane carrier wave function in the modelling of Lamb wave propagation. CEAS Aeronaut J 4:51–59

    Article  Google Scholar 

  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 13:587–614

    Article  Google Scholar 

  61. von Ende S, Schäfer I, Lammering R (2007) Lamb wave excitation with piezoelectric wafers – an analytical approach. Acta Mech 193(3–4):141–150

    Article  MATH  Google Scholar 

  62. Wilcox P (2004) Modeling the excitation of Lamb and SH waves by point and line sources. AIP Conf Proc 700:206–213

    Article  Google Scholar 

  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 67:1–20

    Article  Google Scholar 

  64. Yang J (2005) An introduction to the theory of piezoelectricity. Advances in mechanics and mathematics, vol 9. Springer, Berlin

    Google Scholar 

  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):1115–1123

    Article  Google Scholar 

  66. Zienkiewicz OC, Taylor RL (2000) The finite element method: volume 1 the basis. Butterworth Heinemann, Oxford

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    S. Duczek, Z. A. B. Ahmad, J. M. Vivar-Perez & U. Gabbert

Authors
  1. S. Duczek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. A. B. Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. M. Vivar-Perez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. U. Gabbert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. Duczek or U. Gabbert .

Editor information

Editors and Affiliations

  1. Institute of Mechanics, Helmut-Schmidt-University, University of the Federal Armed Forces Hamburg, Hamburg, Germany

    Rolf Lammering

  2. Institute of Mechanics, Otto von Guericke University Magdeburg, Magdeburg, Germany

    Ulrich Gabbert

  3. Institute of Adaptronics and Function Integration, Braunschweig University of Technology, Braunschweig, Germany

    Michael Sinapius

  4. Faculty of Mathematics, Saarland University, Saarbrücken, Germany

    Thomas Schuster

  5. DLR - German Aerospace Center, Braunschweig, Germany

    Peter Wierach

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Cite this chapter

Duczek, S., Ahmad, Z.A.B., Vivar-Perez, J.M., Gabbert, U. (2018). Hybrid Simulation Methods: Combining Finite Element Methods and Analytical Solutions. In: Lammering, R., Gabbert, U., Sinapius, M., Schuster, T., Wierach, P. (eds) Lamb-Wave Based Structural Health Monitoring in Polymer Composites. Research Topics in Aerospace. Springer, Cham. https://doi.org/10.1007/978-3-319-49715-0_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-49715-0_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-49714-3

  • Online ISBN: 978-3-319-49715-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation