Skip to main content
SpringerLink
Log in

A method based on small amplitude homogenization for detecting defects using elastic waves

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

This work is concerned with the resolution of inverse problems for the detection of defects inside a medium using the propagation of elastic waves, under the assumption of small contrast on the value of the stiffness between the matrix material and that of the defect. This is the so-called small amplitude, small contrast or small aspect ratio assumption. Following the framework developed for optimal design problems, we consider a formal second order asymptotic expansion with respect to the aspect ratio, which allows us to simplify the inverse problem considering it as an optimization problem. According to this and through solving the wave equation in the time domain, we can develop a gradient type algorithm that reduces, in the time interval being considered, the difference between the boundary values obtained from a problem with certain defect distribution that is numerically solved and those values obtained from an assumption on the distribution of the defect. An adaptive procedure is presented for locating the wave source in order to improve the results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Allaire G (2002) Shape optimization by the homogenization method. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  2. Allaire G, Gutiérrez S (2007) Optimal design in small amplitude homogenization. Math Model Numer Anal 41(3):543–574

    Article  MATH  MathSciNet  Google Scholar 

  3. Allaire G, Kelly A (2011) Optimal design of low-contrast two-phase structures for the wave equation. Math Model Methods Appl Sci 21(7):1499–1538

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonnet M, Constantinescu A (2005) Inverse problems in elasticity. Inverse Probl 21(2):R1–R50

    Article  MATH  MathSciNet  Google Scholar 

  5. Burger M, Osher SJ (2005) A survey on level set methods for inverse problems and optimal design. J Appl Math 16:263–301

    MATH  MathSciNet  Google Scholar 

  6. Gallego R, Rus G (2004) Identification of cracks and cavities using the topological sensitivity boundary integral equation. Comput Mech 33:154–163

    Article  MATH  Google Scholar 

  7. Gutiérrez S, Mura J (2008) Small amplitude homogenization applied to inverse problems. Comput Mech 41:699–706

    Article  MATH  MathSciNet  Google Scholar 

  8. Gutiérrez S, Mura J (2010) An adaptive procedure for defect identification problems in elasticity. Comptes Rendus Mecanique 338:402–411

    Google Scholar 

  9. Guzina BB, Bonnet M (2004) Topological derivative for the inverse scattering of elastic waves. Q J Mech Appl Math 57(2):161–179

    Article  MATH  MathSciNet  Google Scholar 

  10. Hecht F, Pironneau O, Le Hyaric A, Ohtsuka K (2010) FreeFem++. code and user manual. www.freefem.org. Accessed April 2011

  11. Mura J, Gutiérrez S (2011) Detection of weak defects in elastic bodies through small amplitude homogenization. Inverse Probl Sci Eng 19(2):233–250

    Article  MATH  MathSciNet  Google Scholar 

  12. Tartar L (1990) H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc R S Edinb 115A:193–230

    Article  MathSciNet  Google Scholar 

  13. Uhlmann G (1998) Inverse boundary value problems for partial differential equations. Documenta Mathematica III:77–86

    MathSciNet  Google Scholar 

  14. Vito A, Gutiérrez S, Santa María H (submitted) Experimental evaluation of a computational method for defect location in steel plates. Submitted

Download references

Acknowledgments

We thank the partial funding provided by the Fondecyt project 1090334, the valuable collaboration of Joaquín Mura, and the insightful comments and suggestions of the referees.

Author information

Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, Casilla 306 Correo 22, Santiago, Chile

    S. Gutiérrez & J. J. Uribe

Authors
  1. S. Gutiérrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. J. Uribe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Gutiérrez.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gutiérrez, S., Uribe, J.J. A method based on small amplitude homogenization for detecting defects using elastic waves. Comput Mech 53, 17–28 (2014). https://doi.org/10.1007/s00466-013-0889-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-013-0889-9

Keywords

Search

Navigation