Skip to main content
SpringerLink
Log in

A Fast Preconditioned Iterative Algorithm for the Electromagnetic Scattering from a Large Cavity

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

Abstract

In this paper, a fast preconditioned Krylov subspace iterative algorithm is proposed for the electromagnetic scattering from a rectangular large open cavity embedded in an infinite ground plane. The scattering problem is described by the Helmholtz equation with a nonlocal artificial boundary condition on the aperture of the cavity and Dirichlet boundary conditions on the walls of the cavity. Compact fourth order finite difference schemes are employed to discretize the bounded domain problem. A much smaller interface discrete system is reduced by introducing the discrete Fourier transformation in the horizontal and a Gaussian elimination in the vertical direction, presented in Bao and Sun (SIAM J. Sci. Comput. 27:553, 2005). An effective preconditioner is developed for the Krylov subspace iterative solver to solve this interface system. Numerical results demonstrate the remarkable efficiency and accuracy of the proposed method.

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

Similar content being viewed by others

References

  1. Ammari, H., Bao, G., Wood, A.W.: Analysis of the electormagnetic scattering from a cavity. Jpn. J. Ind. Appl. Math. 19, 301–310 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anastassiu, H.T.: A review of electromagnetic scattering analysis for inlets, cavities and open ducts. IEEE Antennas Propag. Mag. 45, 27–40 (2003)

    Article  Google Scholar 

  3. Bao, G., Sun, W.: A fast algorithm for the electromagnetic scattering from a large cavity. SIAM J. Sci. Comput. 27, 553–574 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baruch, G., Fibich, G., Tsynkov, S.: High-order numerical method for the nonlinear Helmholtz equation with material discontinuities. J. Comput. Phys. 227, 820–850 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baruch, G., Fibich, G., Tsynkov, S., Turkel, E.: Fourth order schemes for time-harmonic wave equations with discontinuous coefficients. Commun. Comput. Phys. 5, 442–455 (2009)

    MathSciNet  Google Scholar 

  6. Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for elliptic boundary value problems: a survey. Numer. Algorithms 56, 253–295 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erlangga, Y.A.: Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Methods Eng. 15, 37–66 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Erlangga, Y.A., Vuik, C., Oosterlee, C.W.: On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math. 50, 409–425 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fang, Q., Nicholls, D.P., Shen, J.: A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering. J. Comput. Phys. 224, 1145–1169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feng, X., Li, Z., Qiao, Z.: High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients. J. Comput. Math. 29, 324–340 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu, Y.: Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers. J. Comput. Math. 26, 98–111 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Gustafsson, B., Mossberg, E.: Time compact high order difference methods for wave propagation. SIAM J. Sci. Comput. 26, 259–271 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, J., Wood, A.W., Havrilla, M.J.: A hybrid finite element-Laplace transform method for the analysis of transient electromagnetic scattering by an over-filled cavity in the ground plane. Commun. Comput. Phys. 5, 126–141 (2009)

    MathSciNet  Google Scholar 

  14. Ihlenburg, F.: Finite Element Analysis of Acoustic Scattering. In: Applied Mathematical Sciences, vol. 132. Springer, New York (1998)

    Google Scholar 

  15. Ito, K., Toivanen, J.: Preconditioned iterative methods on sparse subspaces. Appl. Math. Lett. 19, 1191–1197 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ito, K., Toivanen, J.: A fast iterative solver for scattering by elastic objects in layered media. Appl. Numer. Math. 57, 811–820 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jin, J.: The Finite Element Method in Electromagnetics. Willey, New York (1993)

    MATH  Google Scholar 

  18. Kuffuor, D.O., Saad, Y.: Preconditioning Helmholtz linear systems. Appl. Numer. Math. 60, 420–431 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Larid, A.L., Giles, M.B.: Preconditioned iterative solution of the 2D Helmholtz equation. Report NA-02/12, Oxford University Computing Laboratory, 2002

  20. Lee, J., Zhang, J., Lu, C.C.: Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems. J. Comput. Phys. 185, 158–175 (2003)

    Article  MATH  Google Scholar 

  21. Liu, J., Jin, J.: A special higher order finite-element method for scattering by deep cavities. IEEE Trans. Antennas Propag. 48, 694–703 (2000)

    Article  MATH  Google Scholar 

  22. Plessix, R.E., Mulder, W.A.: Separation of variables as a preconditioner for an iterative Helmholtz solver. Appl. Numer. Math. 44, 385–400 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Saad, Y.: ILUT: a dual threshold incomplete LU factorization. Numer. Linear Algebra Appl. 4, 387–402 (1994)

    Article  MathSciNet  Google Scholar 

  24. Wang, Y., Du, K., Sun, W.: A second-order method for the electromagnetic scattering from a large cavity. Numer. Math. Theor. Meth. Appl. 1, 357–382 (2008)

    MathSciNet  Google Scholar 

  25. Wood, A.: Analysis of electromagnetic scattering from an overfilled cavity in the ground plane. J. Comput. Phys. 215, 630–641 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xiang, Z., Chia, T.: A hybrid BEM-WTM approach for analysis of the EM scattering from large open-ended cavities. IEEE Trans. Antennas Propag. 49, 165–173 (2001)

    Article  MATH  Google Scholar 

  27. Zhang, D., Ma, F., Dong, H.: A finite element method with rectangular perfectly method matched layers for the scattering from cavities. J. Comput. Math. 26, 98–111 (2008)

    MathSciNet  Google Scholar 

  28. Zhao, M., Qiao, Z., Tang, T.: A fast high order method for electromagnetic scattering by large open cavities J. Comput. Math. 29, 287–304 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their valuable suggestions. The authors would also like to thank Professor Tao Tang at Hong Kong Baptist University and Dr. Jari Toivanen at Stanford University for their beneficial discussions. The research of the first author was partially supported by NSFC (Nos. 11161014 and 11001062) and Guangxi Science and Technology Development Foundation (No. 0731018). The research of the second author was partially supported by the Hong Kong RGC grant (No. 201710).

Author information

Authors and Affiliations

  1. Guilin University of Electronics Technology, Guilin, Guangxi, 541004, China

    Chenliang Li

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Zhonghua Qiao

Authors
  1. Chenliang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhonghua Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhonghua Qiao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, C., Qiao, Z. A Fast Preconditioned Iterative Algorithm for the Electromagnetic Scattering from a Large Cavity. J Sci Comput 53, 435–450 (2012). https://doi.org/10.1007/s10915-012-9580-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9580-0

Keywords

Search

Navigation