A Nyström method with edge condition (EC) is developed for electromagnetic scattering by two-dimensional (2D) open structures. Since EC correctly describes the edge behavior of currents on the scatterers, the use of it in Nystr Ìˆom method can dramatically coarsen the discretization near the edges. In the implementation of the scheme, we derive the closed-form expressions for the singular or near- singular integrations of Hankel functions multiplied by the polynomials with or without EC. This allows us to control the numerical errors efficiently by approximating the Hankel functions with more series terms and selecting higher-order polynomials to represent the currents in the local correction. The numerical results illustrate that the solutions with the use of EC converge much faster than without the use of EC. Also, EC is more essential in TM polarization than in TE polarization due to the singular behavior of current near edges.
1. Kot, J. S., "Solution of thin-wire integral equations by Nyström methods," Microw. Opt. Tech. Lett., Vol. 3, No. 11, 393-396, 1990.
2. Canino, L. F., J. J. Ottusch, M. A. Stalzer, J. L. Visher, and S. Wandzura, "Numerical solution of the Helmholtz equation in 2D and 3D using a high-order Nyström discretization," J. Comput. Phys., Vol. 146, 627-663, 1998. doi:10.1006/jcph.1998.6077
3. Gedney, S. D., "On deriving a locally corrected Nyström scheme from a quadrature sampled moment method," IEEE Trans. Antennas Propagat., Vol. 51, No. 9, 2402-2412, 2003. doi:10.1109/TAP.2003.816305
5. Meixner, J., "The behavior of electromagnetic fields at edges," IEEE Trans. Antennas Propagat., Vol. AP-20, No. 4, 442-446, 1972. doi:10.1109/TAP.1972.1140243
6. Fara ji-Dana, R. and Y. Chow, "Edge condition of the field and a.c. resistance of a rectangular strip conductor," IEE Proceedings, Vol. 137, No. 2, 1990.
7. Lavretsky, E. L., "Taking into account the edge condition in the problem of scattering from the circular aperture in circular-to- rectangular and rectangular-to-rectangular waveguide junctions," IEE Proc.-Microw. Antennas Propag., Vol. 141, No. 1, 1994.
8. Balanis, C. A., Advanced Engineering Electromagnetics, 2nd edition, Wiley, New York, 1989.
9. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1964.
10. Liu, K. and C. A. Balanis, "Simplifed formulations for two- dimensional TE-polarization field computations," IEEE Trans. Antennas Propagat., Vol. 39, No. 2, 259-262, 1991. doi:10.1109/8.68193
11. Tong, M. S. and W. C. Chew, "A higher-order Nystr Ìˆom scheme for electromagnetic scattering by arbitrarily shaped surfaces," IEEE Antennas and Wireless Propagation Letters, Vol. 4, 277-280, 2005. doi:10.1109/LAWP.2005.853000
12. Stroud, A. H. and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, 1966.
13. Ma, J.-H., V. Rokhlin, and S. M. Wandzura, "Generalized Gaussian quadrature rules for systems of arbitrary functions," SIAM J. Numerical Anal., Vol. 33, No. 3, 971-996, 1996. doi:10.1137/0733048
14. Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetter- ling, Numerical Recipes, The Art of Scientific Computing, Cam- bridge University Press, Cambridge, 1987.
15. Dwight, H. B., Tables of Integrals and Other Mathematical Data, 4th edition, Macmillan, New York, 1961.