608 zyxwvutsrqponmlk IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 39, NO. 3, MAKCH 1991 IV. CONCLUSIONS In this work, the surface integral equation has been used as a boundary condition for the finite element solution of the multi- port waveguide discontinuity problem. The major advantage offered by the use of the surface integral equation approach is that it allows for placing the mesh-terminating outer boundaries of the finite element region as close to the junction discontinuity as possible, thus minimizing the size of the finite element matrix. This advantage is achieved despite the fact that the evanescent modes have significant amplitudes in the region close to the discontinuity. The accuracy of the surface integral equation formulation and its simplicity make it an efficient and versatile tool in the analysis of waveguide discontinuity prob- lems. REFERENCES 111 S. W. Lee, W. R. Jones, and J. J. Campbell, “Convergence of numerical solutions of iris-type discontinuity problems,” IEEE zyxwvutsr Trans. Microwaue Theory Tech., vol. MTT-19, pp. 528-536, June 1971. [2] R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waues. (31 P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, 2nd ed. Cambridge, England: Cambridge University Press, 1989. I41 V. Kanellopoulos and J. P. Webb, “A complete E-plane analysis of waveguide junctions using the finite element method,” IEEE Trans. Microwaue Theory Tech., vol. 38, pp. 290-295, Mar. 1990. [SI zyxwvutsrqpo B. H. McDonald and A. Wexler, “Finite element solution of unbounded field problems,” IEEE Trans. Microwaue Theory Tech., vol. MTT-20, pp. 841-847, Dec. 1972. [61 J. Jin and V. Liepa, “Application of hybrid finite element method to electromagnetic scattering from coated cylinders,” IEEE Trans. Antennas Propagat., vol. 36, pp. 50-54, Jan. 1988. (71 0. M. Ramahi, “Boundary conditions for the solution of open-re- gion electromagnetic scattering problems,” Ph.D. dissertation, Uni- versity of Illinois, Urbana, 1990. 181 S. Kagami and I. Fukai, “Application of the boundary-element method to electromagnetic field problems.” IEEE Trans. Mi- crowaue Theory Tech., vol. MTT-32, pp. 455-461, Apr. 1984. [91 M. Koshiba and M. Suzuki, “Application of the boundary-element method to waveguide discontinuities,” IEEE Trans. Microwaue Theory Tech., vol. MTT-34, pp. 301-307: Feb. 1984. [lo] G. Strang and G. Fix, An Analysis of the Finite Element Method. Englewood Cliffs, NJ: Prentice-Hall, 1973. ill] J. N. Reddy, An Introduction to the Finite-Element Method. New York, NY: McGraw-Hill, 1984. [121 J. Stratton, Electromagnetic Theory. New York, NY: McGraw-Hill, 1941. New York, NY: MacMillan, 1971. On the Use of Shanks’s Transform to Accelerate the Summation of Slowly Converging Series Surendra Singh and Ritu Singh Abstruct --It is shown that the application of Shanks’ transform results in accelerating the convergence of slowly converging series. The transform is applied to a periodic Green’s function involving a single summation. The convergence properties of this series are reported for Manuscript received July 5, 1990; revised October 8, 1990. The authors are with the Department of Electrical Engineering, IEEE Log Number 9041945. University of Tulsa, Tulsa, OK 74104. zyxwvutsrqp ~5x10-4 ~2x10-4 zyxwvuts 0 Direct Sum A Shanks’ Tx. 1x10-4, 5x10-5. 2x10-5, 1x10-5 zyxwv 10-5 100 101 102 103 104 105 Number of Terms Fig. 1. Relative error magnitude versus number of terms for the series in (1) for x = ~/2. the “on-plane” case, in which the series converges extremely slowly. Numerical results indicate that by employing Shanks’s transform the computation time can be reduced by as much as a factor of 200. I. INTRODUCTION In the analysis of periodic structures, one usually encounters a Green’s function which converges very slowly. As repeated eval- uations of the Green’s function series are needed in determining the radiation or scattering from a periodic array using the method of moments with subsectionally defined basis functions, the slow convergence of the series would result in a considerable amount of computation time. In order to reduce this time, we look for ways to accelerate the convergence of the Green’s function series. A method for improving the convergence of a doubly infinite periodic Green’s function series has previously been suggested [1]-[3]. It has been successfully applied by a number of investigators to singly and doubly periodic Green’s function series [4]-[8]. This paper reports the use of Shanks’s transform in accelerating the convergence of a periodic Green’s function involving a single infinite summation. Although the use of Shanks’s transform in conjunction with Kummer’s and Poisson’s transformations has been shown in [3] to improve the convergence of a doubly periodic Green’s function, it is reported here that a simple application of this transform alone to very slowly converging series enhances their convergence rremen- dously. Another advantage of using the transform is that no analytical work need be done to the series. This is an attractive feature, as the transform can be applied to a wide variety of series. 11. ILLUSTRATIVE EXAMPLE OF SHANKS’S TRANSFORM If a sequence of partial sums of a series behaves as a “mathematical transient” as defined by Shanks in [9], then it is possible to extract the base of this “transient” by an application of Shanks’s transform [9]. The transform is applied successively to the partial sums of the series until a predefined convergence criterion is satisfied. An algorithm to compute different orders of Shanks’s transform is given in [lo]. It is interesting to note that although the partial sums show no indication of converging to the sum of the series, the application of Shanks’s transform is able to extract the sum from these partial sums. This is illus- trated by taking the following series: sin(2n - 1)x s= ’E zyx n=, 4a(2n-1) . 0018-9480/91/0300-0608$01 .OO 01991 IEEE