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Optoelectron., Vol. 7, No. 6, 1992, pp. 705721. 8. B. Culshaw, F. A. Muhammad, G. Stewart, S. Murray, D. Pinch- beck, J. Norris, S. Cassidy, D. Williams, I. Crisp, M. Wilkinson, R. V. Ewyk, and A. McGhee, ‘‘Evanescent Wave Methane De- tection Using Optical Fibre,’’ Electron. Lett., Vol. 28, No. 24, 1992, pp. 22322234.  1998 John Wiley & Sons, Inc. CCC 0895-247798 ADAPTIVE INTEGRAL SOLUTION OF COMBINED FIELD INTEGRAL EQUATION Chao-Fu Wang, 1 Feng Ling, 1 Jiming Song, 1 and Jian-Ming Jin 1 1 Electromagnetics Laboratory and Center for Computational Electromagnetics Department of Electrical and Computer Engineering University of Illinois at Urbana  Champaign Urbana, Illinois 61801-2991 Recei  ed 4 May 1998 ( ) ABSTRACT: The adapti  e integral method AIM is applied to the ( ) solution of the combined field integral equation CFIE of scattering by ( ) ( ) a three-dimensional 3-D perfect electric conductor PEC . The employ- ment of CFIE eliminates the interior resonance problem suffered by both ( ) the electric field integral equation EFIE and the magnetic field integral ( ) equation MFIE . Furthermore, it significantly impro  es the efficiency of AIM by reducing the number of iterations for con ergence. It is shown that the memory requirement and computational complexity per iteration ( 1.5 ) ( 1.5 ) of the AIM solution of CFIE are O N and O N log N , respecti  ely, for 3-D PEC surface scattering problems.  1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 19: 321328, 1998. Key words: integral equation; electromagnetic scattering; electromagnetics I. INTRODUCTION The computational of electromagnetic scattering from an arbitrarily shaped electric conductor is important in a num- ber of applications. This computation can be carried out by solving a pertinent integral equation using the method of Ž . moments MoM . In MoM, the integral equation is first discretized into a matrix equation, which is then solved either by a direct or iterative method. A direct method, such as Ž . Gaussian elimination of LU decomposition LUD , requires Ž 3 . ON floating-point operations for the matrix equation of order N, and an iterative method, such as the conjugate Ž . Ž 2 . gradient CG method, requires ON operations per itera- Ž 2 . tion. The memory requirement for both methods is ON . Such a computational complexity and memory requirement is too high for large scattering problems, and must be reduced to make the solution more efficient. One of the most powerful methods for the efficient MoM Ž .   solution is the fast multipole method FMM 1, 2 , which reduces the computational complexity and memory require- Ž 2 . Ž 1.5 . ment from ON to ON . The computational complexity and memory requirement can be further reduced to Ž . ON log N by using the multilevel fast multipole algorithm Ž . MLFMA with the help of translation, interpolation, anter-   polation, and a grid-tree data structure 3, 4 . Another power- ful method for the efficient MoM solution is the adaptive Ž . integral method AIM that has been developed by Bleszyn-   ski, Bleszynski, and Jaroszewicz 5, 6 . Compared to the conventional MoM, AIM reduces the computational com- plexity and memory requirement with the aid of auxiliary Ž . basis functions and the fast Fourier transform FFT . The Ž 1.5 . computational complexities of AIM are ON log N and Ž . ON log N for surface and volumetric scatterers, respec- Ž 1.5 . tively. The corresponding memory requirements are ON Ž . and ON , respectively. The AIM has been successfully ap- plied to the analysis of scattering and radiation from arbitrar- Ž . ily shaped three-dimensional 3-D and planar structures   5  7. The problem of scattering by a closed conducting body can be solved using either the electric field integral equation Ž . Ž . EFIE , the magnetic field integral equation MFIE , or the Ž . combined field integral equation CFIE . It has been found that, in addition to the elimination of the interior resonance problem suffered by both EFIE and MFIE, CFIE usually yields a better conditioned matrix equation, and its iterative   solution converges much faster than EFIE and MFIE 8  11 . In this paper, we apply AIM to the solution of CFIE, and study the convergence behavior of AIM as applied to EFIE, MFIE, and CFIE. The memory requirement and computa- Ž 1.5 . tional complexity of the AIM solution of CFIE are ON Ž 1.5 . and ON log N for 3-D surface scattering problems. The numerical results demonstrate that the AIM solution of CFIE is efficient for the analysis of scattering from arbitrarily shaped 3-D conducting structures. II. FORMULATION Consider an arbitrarily shaped 3-D conducting object illumi- inc Ž. nated by an incident field E r . The EFIE is given by 1 inc Ž . Ž . Ž. Ž. n  Gr , r' Jr' dr'  n  E r on S 1 ˆ ˆ HH jk  S 0 0 where S denotes the conducting surface of the object, n is an ˆ Ž . outwardly directed normal, and Gr, r' is the well-known free-space dyadic Green’s function given by  e jk 0  rr '  Ž . Ž . Ž . Ž. Gr , r'  I  g r , r' , g r , r'  2 2 ž /   4 r  r' k 0 Ž. with I being the unit dyad. Also, Jr denotes the unknown surface current, k is the free-space wavenumber, and  is 0 0 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 19, No. 5, December 5 1998 321