Skip to main content
SpringerLink
Log in

Detecting Near Resonances in Acoustic Scattering

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose and study a method for finding quasi-resonances for a linear acoustic transmission problem in frequency domain. Starting from an equivalent boundary-integral equation we perform Galerkin boundary element discretization and look for the minima of the smallest singular value of the resulting matrix as a function of the wave number k. We develop error estimates for the impact of Galerkin discretization on singular values and devise a heuristic adaptive algorithm for finding the minima in prescribed k-intervals. Our method exclusively relies on the solution of eigenvalue problems for real k, in contrast to alternative approaches that rely on extension to the complex plane.

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

Similar content being viewed by others

Data Availability

The codes with which the numerical tests reported in the manuscript have been conducted is available from https://github.com/DiegoRenner/HelmholtzTransmissionProblemBEM. Specific computations can be launched through make targets as described in the top-level REAME.md file.

Notes

  1. For boundary integral operators we suppress their \(\kappa \)-/k-dependence in the notation.

  2. Some details of the algorithm have been omitted, in particular the treatment of special cases.

  3. The C++ codes used for all numerical experiments are available from https://github.com/DiegoRenner/HelmholtzTransmissionProblemBEM. The accompanying documentation outlines how to re-run the computations.

  4. https://github.com/m-reuter/arpackpp, last accessed March 2021.

References

  1. Amini, S., Maines, N. D.: Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation. Int. J. Numer. Methods Eng. 41(5), 875–898 (1998). ISSN: 0029-5981. url: https://doi.org/10.1002/(SICI)1097-0207(19980315)41:5%3C875::AID-NME313%3E3.0.CO;2-9

  2. Andrew, A. L., Tan, R. C. E.: Computation of derivatives of repeated Eigenvalues and the corresponding eigenvectors of symmetric matrix pencils. SIAM J. Matrix Anal. Appl. 20(1): 78–100 (1998). eprint: https://doi.org/10.1137/ S0895479896304332.

  3. Asakura, J. et al.: A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 1, 52–55 (2009). ISSN: 1883-0609. url: https://doi.org/ 10.14495/jsiaml.1.52

  4. Babich, V., Buldyrev, V.: Short-Wavelength Diffraction Theory. Asymptotic Methods. Vol. 4. Springer Series on Wave Phenomena. Berlin: Springer (1991)

  5. Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, Vol. II. North-Holland, Amsterdam, pp. 641–787 (1991)

  6. Balac, S., Dauge, M., Moitier, Z.: Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying index. IMA J. Appl. Math. 86(6), 1212–1265 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balac, S., et al.: Mathematical analysis of whispering gallery modes in graded index optical micro-disk resonators. Eur. Phys. J. D 74(11), 221 (2020)

    Article  Google Scholar 

  8. Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. 436(10), 3839–3863 (2012). https://doi.org/10.1016/j.laa.2011.03.030

    Article  MathSciNet  MATH  Google Scholar 

  9. Bunse-Gerstner, A., et al.: Numerical computation of an analytic singular value decomposition of a matrix valued function. Numer. Math. 60(1), 1–39 (1991). https://doi.org/10.1007/BF01385712

    Article  MathSciNet  MATH  Google Scholar 

  10. Chaumont Frelet, T.: Finite element approximation of Helmholtz problems with application to seismic wave propagation. Theses. INSA de Rouen, Dec. 2015. url: https://tel.archives-ouvertes.fr/tel-01246244

  11. Claeys, X., Hiptmair, R., and Jerez-Hanckes, C.: Multi-trace boundary integral equations. In: I. Graham et al. (eds) Direct and Inverse Problems in Wave Propagation and Applications, Vol. 14. Radon Series on Computational and Applied Mathematics. Berlin/Boston: De Gruyter, pp. 51–100 (2013)

  12. Claeys, X., Hiptmair, R., Spindler, E.: A second-kind Galerkin boundary element method for scattering at composite objects English. BIT Numer. Math. 55(1), 33–57 (2015). https://doi.org/10.1007/s10543-014-0496-y

    Article  MATH  Google Scholar 

  13. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. 2nd. Vol. 93. Applied Mathematical Sciences. Heidelberg: Springer (2013)

  14. Gavin, B., Miedlar, A., Polizzi, E.: FEAST eigensolver for nonlinear eigenvalue problems. J. Comput. Sci. 27, 107–117 (2018). https://doi.org/10.1016/j.jocs.2018.05.006

    Article  MathSciNet  Google Scholar 

  15. Gomes, F. M., Sorensen, D. C.: ARPACK++ - An object-oriented version of ARPACK eigenvalue package. (1998). url: https://github.com/m-reuter/arpackpp/blob/master/ doc/arpackpp.pdf

  16. Greenbaum, A., Li, R.-C., Overton, M.L.: First-order perturbation theory for eigenvalues and eigenvectors. SIAM Rev. 62(2), 463–482 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Heider, P.: Computation of scattering resonances for dielectric resonators. Comput. Math. Appl. 60(6), 1620–1632 (2010). https://doi.org/10.1016/j.camwa.2010.06.044

    Article  MathSciNet  MATH  Google Scholar 

  18. Heuser, H.: Funtional Analysis, 2nd edn. Teubner-Verlag, Stuttgart (1986)

    Google Scholar 

  19. Hiptmair, R., Jerez-Hanckes, C.: Multiple traces boundary integral formulation for Helmholtz transmission problems. Adv. Comput. Math. 37(1), 39–91 (2012). https://doi.org/10.1007/s10444-011-9194-3

    Article  MathSciNet  MATH  Google Scholar 

  20. Hiptmair, R., Moiola, A., Spence, E.A.: Spurious Quasi-resonances in boundary integral equations for the Helmholtz transmission problem. SIAM J. Appl. Math. 82(4), 1446–1469 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang, R., et al.: Recursive integral method for transmission eigenvalues. J. Comput. Phys. 327, 830–840 (2016). https://doi.org/10.1016/j.jcp.2016.10.001

    Article  MathSciNet  MATH  Google Scholar 

  22. Kress, R.: Linear Integral Equations. Applied Mathematical Sciences, vol. 82. Springer, Berlin (1989)

    MATH  Google Scholar 

  23. Lietaert, P., et al.: Automatic rational approximation and linearization of nonlinear eigenvalue problems. IMA J. Numer. Anal. 42(2), 1087–1115 (2022). https://doi.org/10.1093/imanum/draa098

    Article  MathSciNet  MATH  Google Scholar 

  24. Mäkitalo, J., Kauranen, M., Suuriniemi, S.: Modes and resonances of plasmonic scatterers. Phys. Rev. B 89, 165429 (2014). https://doi.org/10.1103/PhysRevB.89.165429

    Article  Google Scholar 

  25. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, UK (2000)

    MATH  Google Scholar 

  26. Misawa, R., Niino, K., Nishimura, N.: Boundary integral equations for calculating complex eigenvalues of transmission problems. SIAM J. Appl. Math. 77(2), 770–788 (2017). https://doi.org/10.1137/16M1087436

    Article  MathSciNet  MATH  Google Scholar 

  27. Moiola, A., Spence, E.A.: Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions. Math. Models Methods Appl. Sci. 29(2), 317–354 (2019). https://doi.org/10.1142/S0218202519500106

    Article  MathSciNet  MATH  Google Scholar 

  28. Osborn, J.E.: Spectral approximation for compact operators. Math. Comput. 29, 712–725 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  29. Parlett, B. N.: The symmetric eigenvalue problem. Vol. 20. Classics in Applied Mathematics. Corrected reprint of the 1980 original. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1998) pp. xxiv+398. ISBN: 0-89871-402-8. url: https://doi.org/10.1137/1.9781611971163

  30. Popov, G., Vodev, G.: Resonances near the real axis for transparent obstacles. Comm. Math. Phys. 207(2), 411–438 (1999). https://doi.org/10.1007/s002200050731

    Article  MathSciNet  MATH  Google Scholar 

  31. Pradovera, D.: Interpolatory rational model order reduction of parametric problems lacking uniform inf-sup stability. SIAM J. Numer. Anal. 58(4), 2265–2293 (2020). https://doi.org/10.1137/19M1269695

    Article  MathSciNet  MATH  Google Scholar 

  32. Sauter, S., Schwab, C.: Boundary Element Methods. Vol. 39. Springer Series in Computational Mathematics. Heidelberg: Springer (2010)

  33. Stefanov, P.: Quasimodes and resonances: sharp lower bounds. Duke Math. J. 99(1), 75–92 (1999). https://doi.org/10.1215/S0012-7094-99-09903-9

    Article  MathSciNet  MATH  Google Scholar 

  34. Sun, J.-G.: Stability and accuracy: perturbation analysis of algebraic eigenproblems. Technical Report UMINF 98.07. Umeå University, Department of Computer Science (1998). url: https://people.cs.umu.se/jisun/Jiguang-Sun-UMINF98-07-rev2002-02-20.pdf

  35. Tang, S.-H., Zworski, M.: From quasimodes to resonances. Math. Res. Lett. 5, 261–272 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Torres, R.H., Welland, G.V.: The Helmholtz equation and transmission problems with Lipschitz interfaces. Indiana Univ. Math. J. 42(4), 1457–1485 (1993). https://doi.org/10.1512/iumj.1993.42.42067

    Article  MathSciNet  MATH  Google Scholar 

  37. Werner, D.: Functional Analysis. Springer, Berlin (1995)

    MATH  Google Scholar 

  38. Xiao, J., et al.: Solving large-scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh-Ritz method. Int. J. Numer. Methods Eng. 110(8), 776–800 (2017). https://doi.org/10.1002/nme.5441

    Article  MathSciNet  MATH  Google Scholar 

  39. Yokota, S., Sakurai, T.: A projection method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 5, 41–44 (2013). https://doi.org/10.14495/jsiaml.5.41

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

LG was supported by the Hrvatska Zaklada za Znanost (Croatian Science Foundation) under the Grant IP-2019-04-6268 -Randomized low-rank algorithms and applications to parameter dependent problems.

Funding

Author LG was supported by the Hrvatska Zaklada za Znanost (Croatian Science Foundation) under the Grant IP-2019-04-6268 -Randomized low-rank algorithms and applications to parameter dependent problems. The other authors did not receive any fund, grants, or other support for their work.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia

    Luka Grubišić

  2. Seminar für Angewandte Mathematik, Department of Mathematics, ETH Zürich, Zurich, Switzerland

    Ralf Hiptmair

  3. Rautistrasse 345, 8048, Zurich, Switzerland

    Diego Renner

Authors
  1. Luka Grubišić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ralf Hiptmair
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Diego Renner
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

This study work was initiated by RH. All authors contributed to the study conception and design. The code for numerical experiments was written by DR, and the code review has been performed by RH. The first draft of the manuscript was written by RH and LG and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Luka Grubišić.

Ethics declarations

Competing interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grubišić, L., Hiptmair, R. & Renner, D. Detecting Near Resonances in Acoustic Scattering. J Sci Comput 96, 81 (2023). https://doi.org/10.1007/s10915-023-02284-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02284-5

Keywords

Mathematics Subject Classification

Search

Navigation