Mathematical model of crack diagnosis: inverse acoustic scattering problem and its high-precision numerical solution

Victor A. Kovtunenko1

1Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrich str. 36, 8010 Graz, Austria

1Lavrent’ev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090, Novosibirsk, Russia

1Corresponding author

Vibroengineering PROCEDIA, Vol. 22, 2019, p. 31-35. https://doi.org/10.21595/vp.2019.20513
Received 10 January 2019; accepted 20 January 2019; published 15 March 2019

Copyright © 2019 Victor A. Kovtunenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

The inverse acoustic scattering model for crack diagnosis is described by Helmholtz problem within mathematic framework and investigated for the sake of scientific computing. Minimizing the misfit from given measurements leads to an optimality condition-based imaging function which is used for non-iterative identification of the center of an unknown crack put in a test domain. The numerical tests are presented for the cracks of T-junction shape and are carried out based on the Petrov-Galerkin generalized FEM using wavelets basis and level-sets. This shows high-precision identification result and stability to noisy data of the diagnosis, which is illustrated for sound-soft as well as moderately sound-hard cracks when varying the coefficient of surface impedance.

Highlights
  • The inverse acoustic scattering problem is investigated for scientific computing of crack diagnosis
  • Minimizing the misfit from given measurements leads to an optimality condition-based imaging
  • Imaging function is used for non-iterative identification of the center of a crack put in test domain
  • Numerical tests are based on Petrov-Galerkin GFEM and wavelet basis
  • T-junction shaped cracks are sound-soft as well as sound-hard with moderate surface impedance
  • High-precision identification result and stability to noisy data of the diagnosis are reported

Keywords: crack, acoustic scattering, inverse Helmholtz problem, optimization, imaging, variational method, level set, Petrov-Galerkin generalized FEM, wavelet basis, noisy data.

Acknowledgements

The author is supported by the Austrian Science Fund (FWF) Project P26147-N26: ‘Object Identification Problems: Numerical Analysis’ (PION) and the Austrian Academy of Sciences (OeAW), the RFBR and JSPS research Project 19-51-50004.

References

  1. Kovtunenko V. A., Kunisch K. High precision identification of an object: optimality conditions based concept of imaging. SIAM Journal of Control and Optimization, Vol. 52, Issue 1, 2014, p. 773-796. [Publisher]
  2. Cakoni F., Kovtunenko V. A. Topological optimality condition for the identification of the center of an inhomogeneity. Inverse Problems, Vol. 34, Issue 3, 2018, p. 035009. [Publisher]
  3. Kovtunenko V. A. Two-parameter topological expansion of Helmholtz problems with inhomogeneity. Mathematical Analysis of Continuum Mechanics and Industrial Applications. Mathematics for Industry, Vol. 26, 2017, p. 51-81. [CrossRef]
  4. Kovtunenko V. A. High-order topological expansions for Helmholtz problems in 2d. Topological Optimization and Optimal Transport, Radon Series on Computational and Applied Mathematics, Vol. 17, 2017, p. 64-122. [CrossRef]
  5. Khludnev A. M., Kovtunenko V. A. Analysis of Cracks in Solids. WIT-Press, Southampton, Boston, 2000. [CrossRef]
  6. Kovtunenko V. A., Kunisch K. Revisiting generalized FEM: a Petrov-Galerkin enrichment based FEM interpolation for Helmholtz problem. Calcolo, Vol. 55, 2018, p. 38. [Publisher]