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Singularities Occuring in a Bimaterial with Transparent Boundary Conditions

  • Philippe Destuynder
  • Caroline Fabre
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 276)

Abstract

In many engineering problems one has to deal with two materials which can induce local singularities (infinite stresses) as soon as the interface reaches the boundary. This is the case of water pipes with an inner coating for avoiding rusting. Because the wave velocity is smaller in the coating, it appears Love waves which can be used in order to detect defects because they propagate further as in a wave guide. They can be cracks corresponding to a disconnection at the interface between the two materials. In order to detect them, one can use measurements performed at the extremities of the pipes even if the signal in the numerical model, is very much perturbed by the singularities appearing at the interface between the two media at the extremities of the pipe. The phenomenon is amplified when one considers an artificial truncation of the structure and adding transparent boundary conditions in order to avoid reflection for simulating what happens in the full structure. The goal of this paper is to focus on a numerical method which can be used for the analysis of the influence of the singularities on the signal processing analysis. First of all, we give a mathematical description of the singularity met in our problem. Then, we define the extension to our case of the method introduced for cracks by G. Fix. It consists in adding the singular function to finite element functional space used in a classical numerical simulation. The main point of the paper is then to analyze, in a mathematical framework, the error estimates on the coefficients of the singularities with respect to the mesh size. Few numerical tests illustrate the mathematical results obtained for the problem we are dealing with.

Keywords

Signal processing analysis Transparent boundary conditions in a bimaterial Solution singularities Error estimates of singularity coefficients 

Notes

Acknowledgements

This work has been presented at the international congress on Applications of Mathematics and Informatics in Natural Sciences and Engineering dedicated to Professor David Gordeziani for his 80th birthday. This congress organized at Vekua institute in Tbilissi, was a real fruitful opportunity to exchange new ideas on various open problems. Therefore the authors would like to thanks particularly Professor George Jaiani director of the Vekua institute and Doctor Natalia Chinchaladze who organised this fruitful congress.

References

  1. 1.
    Amara, M., Destuynder, P., Djaoua, M.: On a finite element scheme for plane crack problems. In: Numerical Methods in Fracture Mechanics, pp. 41–50. Pinridge Press, Swansea (1980)Google Scholar
  2. 2.
    Brezis, H.: Analyse Fonctionnelle. Masson, Paris (1983)zbMATHGoogle Scholar
  3. 3.
    Cawley, P., Cegla, F., Galvagni, A.: Guided waves for NDT and permanently installed monitoring. In: 18th World Conference on Nondestructive Testing, 16–20 Apr 2012, Durban, South Africa (2012)Google Scholar
  4. 4.
    Destuynder, Ph, Fabre, C.: Singularities occurring in multimaterials with transparent boundary conditions. Quart. Appl. Math. 74, 443–463 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Destuynder, Ph, Fabre, C.: Few remarks on the use of Love waves in non-destructive testing. Dis. Cont. Dyn. Syst. 9(2), 427–444 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Destuynder, P., Fabre, C.: Can we hear the echos of cracks? J. Elast. 1 (2017).  https://doi.org/10.1007/s10659-017-9632-7MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fung, Y.C.: Foundations of Solid Mechanics. Prentice-Hall (1965)Google Scholar
  8. 8.
    Hadamard, J.: Leçon sur le calcul des variations (1910 collège. de France). Gabay J. (ed.) Paris (2012)Google Scholar
  9. 9.
    Holmgren, E.: Über Systeme von linearen partiellen Differentialgleichungen, Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger 58, 91–103 (1901)Google Scholar
  10. 10.
    Jonquière, A.: Note sur la série: \( \sum _{n=1}^\infty \frac{x^n}{n^s}\). Bulletin de la Société Mathématique de France 17, 142–152 (1989, in French)Google Scholar
  11. 11.
    Leinov, E., Lowe, M., Cawley, P.: Investigation of guided wave propagation and attenuation in pipe buried in sand. J. Sound Vib. 347, 96–114 (2015)CrossRefGoogle Scholar
  12. 12.
    Lions, J.L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Elsevier/Masson, Paris (1988)zbMATHGoogle Scholar
  13. 13.
    Petcher, P.A., Burrows, S.E., Dixon, S.: Shear horizontal (SH) ultrasound wave propagation around smooth corners. Ultrasonics 54(4), 997–1004 (2014)CrossRefGoogle Scholar
  14. 14.
    Raviart, P.A., Thomas, J.M.: Introduction à la méthode des éléments finis. Dunod, Paris (1986)Google Scholar
  15. 15.
    Strang, G., Fix, G.: An Analysis of the Finite Element Method, 2nd edn. Wellesley, Cambridge Press (1988)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.M2N, CNAMParisFrance
  2. 2.LMO-UMR 8628, UnivParis-SudOrsayFrance
  3. 3.CNRS, Univ-SaclayParisFrance

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