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Derivation of Analytic Formulas and Numerical Verification of Weakly Singular Integrals for Near-Field Correction in Surface Integral Equations

  • Rim, Jae-Won (Department of Electronic Engineering, Inha University) ;
  • Koh, Il-Suek (Department of Electronic Engineering, Inha University)
  • Received : 2017.01.05
  • Accepted : 2017.04.04
  • Published : 2017.04.30

Abstract

An accurate and efficient evaluation for hypersingular integrals (HIs), strongly singular integrals (SSIs), and weakly singular integrals (WSIs) plays an essential role in the numerical solutions of 3D electromagnetic scattering problems. We derive analytic formulas for WSIs based on Stokes' theorem, which can be expressed in elementary functions. Several numerical examples are presented to validate these analytic formulas. Then, to show the feasibility of the proposed formulations for numerical methods, these formulations are used with the existing analytical expressions of HIs and SSIs to correct the near-field interaction in an iterative physical optics (IPO) scheme. Using IPO, the scattering caused by a dihedral reflector is analyzed and compared with the results of the method of moments and measurement data.

Keywords

References

  1. W. C. Gibson, The Method of Moments in Electromagnetics. Boca Raton, FL: Chapman and Hall/CRC, 2007.
  2. A. J. Poggio and E. K. Miller, "Integral equation solutions of three-dimensional scattering problems," in Computer Techniques for Electromagnetics, R. Mittra, Ed. Oxford: Pergamon Press, 1973, pp. 159-260.
  3. C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin: Springer-Verlag, 1969.
  4. D. Dunavant, "High degree efficient symmetrical Gaussian quadrature rules for the triangle," International Journal for Numerical Methods in Engineering, vol. 21, no. 6, pp. 1129-1148, 1985. https://doi.org/10.1002/nme.1620210612
  5. S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Transactions on Antenna and Propagation, vol. 30, no. 3, pp. 409-418, 1982. https://doi.org/10.1109/TAP.1982.1142818
  6. L. F. Canino, J. J. Ottusch, M. A. Stalzer, J. L. Visher, and S. M. Wandzura, "Numerical solution of the Helmholtz equation in 2D and 3D using a high-order Nystrom discretization," Journal of Computational Physics, vol. 146, no. 2, pp. 627-663, 1998. https://doi.org/10.1006/jcph.1998.6077
  7. M. S. Tong and W. C. Chew, "Super-hyper singularity treatment for solving 3D electric field integral equations," Microwave and Optical Technology Letters, vol. 49, no. 6, pp. 1383-1388, 2007. https://doi.org/10.1002/mop.22443
  8. M. S. Tong and W. C. Chew, "A novel approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics," IEEE Transactions on Antennas and Propagation, vol. 58, no. 11, pp. 3593-3601, 2010. https://doi.org/10.1109/TAP.2010.2071370
  9. D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, "Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains," IEEE Transactions on Antennas and Propagation, vol. 32, no. 3, pp. 276-281, 1984. https://doi.org/10.1109/TAP.1984.1143304
  10. C. Schwab and W. L. Wendland, "On numerical cubatures of singular surface integrals in boundary element methods," Numerische Mathematik, vol. 62, no. 1, pp. 343-369, 1992. https://doi.org/10.1007/BF01396234
  11. R. D. Graglia, "On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle," IEEE Transactions on Antennas and Propagation, vol. 41, no. 10, pp. 1448-1455, 1993. https://doi.org/10.1109/8.247786
  12. A. Herschlein, J. V. Hagen, and W. Wiesbeck, "Methods for the evaluation of regular, weakly singular and strongly singular surface reaction integrals arising in method of moments," Applied Computational Electromagnetics Society Journal, vol. 17, no. 1, pp. 63-73, 2002.
  13. B. M. Johnston and P. R. Johnston, "A comparison of transformation methods for evaluation two-dimensional weakly singular integrals," International Journal for Numerical Methods in Engineering, vol. 56, no. 4, pp. 589-607, 2003. https://doi.org/10.1002/nme.589
  14. M. G. Duffy, "Quadrature over a pyramid or cube of integrands with a singularity at a vertex," SIAM Journal on Numerical Analysis, vol. 19, no. 6, pp. 1260-1262, 1982. https://doi.org/10.1137/0719090
  15. F. Obelleiro-Basteiro, J. L. Rodriguez, and R. J. Burkholder, "An iterative physical optics approach for analyzing the electromagnetic scattering by large open-ended cavities," IEEE Transactions on Antennas and Propagataion., vol. 43, no.4, pp. 356-361, 1995. https://doi.org/10.1109/8.376032
  16. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. New York, NY: Macmillan, 1961.
  17. P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction. Hoboken, NJ: John Wiley & Sons, 2007.