1812 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 6, JUNE 2008 acoustic imaging. In this paper, we have illustrated such a case that the traditional MUSIC may fail to detect some small degenerate cylinders. We have proposed an analytic method to cast the problem of imaging the cylinders as a constrained optimization problem. We have illus- trated that the pseudospectrum thus generated can detect the degen- erate as well as the nondegenerate cylinders. The method works well in noise-free as well as noisy scenarios. Besides providing an algorithm to tackle the degenerate cases, another purpose of this paper is to call at- tention to the polarization effect on MUSIC imaging. We have also ob- served the effect of polarization on the resolution of MUSIC imaging, and a systematic research on it is in progress. One of the applications of the results presented in the paper is the nondestructive detection of a long crack with a small and sharp cross-section in a homogeneous material, where the free-space Green’s function presented in this paper will be replaced by the one corresponding to the experimental setup. Another application of the proposed method is to provide an initial guess for the optimization methods used in the inverse imaging of large sharp cylindrical scatterers. Although the presented MUSIC method cannot provide the exact contours of large cylindrical scatterers, it can provide some information about them, such as the number of cylinders and their approximate locations. This information can be used as an initial guess for the optimization methods, so that the computation cost is substantially reduced. REFERENCES [1] A. J. Devaney, E. A. Marengo, and F. K. Gruber, “Time-reversal-based imaging and inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am., vol. 118, pp. 3129–3138, Nov. 2005. [2] X. Chen and Y. Zhong, “A robust noniterative method for obtaining scattering strengths of multiply scattering point targets,” J. Acoust. Soc. Am., vol. 122, pp. 1325–1327, Sep. 2007. [3] E. A. Marengo and F. K. Gruber, “Noniterative analytical formula for inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am., vol. 120, pp. 3782–3788, Dec. 2006. [4] S. M. Hou, K. Solna, and H. K. Zhao, “A direct imaging algorithm for extended targets,” Inverse Problems, vol. 22, pp. 1151–1178, 2006. [5] H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, “MUSIC- type electromagnetic imaging of a collection of small three-dimen- sional inclusions,” SIAM J. Sci. Comput., vol. 29, pp. 674–709, 2007. [6] H. Ammari, E. Iakovleva, and D. Lesselier, “A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering am- plitude at a fixed frequency,” SIAM J. Sci. Comput., vol. 3, pp. 597–628, 2005. [7] Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres,” IEEE Trans. Antennas Propag., vol. 55, pp. 3542–3549, 2007. [8] L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Elec- tromagnetic Waves: Numerical Simulations. New York: Wiley-Inter- science, 2001, pp. 376–379, 476-483. [9] M. Lax, “Multiple scattering of waves,” Rev. Modern Phys., vol. 23, pp. 287–310, Oct. 1951. [10] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1986, pp. 414–417. [11] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory.. New York: Springer, 2006, pp. 3–4. A Comparison of Higher Order Nodal- and Edge-Basis Functions in the MFIE on Rational Bézier Geometries Andrew D. Hellicar, John S. Kot, Geoffrey James, and Gregory K. Cambrell Abstract—Higher order nodal basis functions for representing equiva- lent surface currents on antennas and scatterers are introduced. The per- formance of the nodal basis is evaluated by comparing two existing higher order edge bases, using the magnetic field integral equation (MFIE) for- mulation for scattering by a perfect electric conductor (PEC) sphere and icosahedron as test problems. Both nodal and edge bases are implemented on rational Bézier patches, giving an exact representation of the surfaces, free from geometrical error. The accuracy of the numerical solutions ob- tained with the three different bases for both the surface current and the radar cross section (RCS) are compared, and it is shown that in general the nodal bases give better accuracy than the edge bases for equal computa- tional cost. Index Terms—Boundary element methods (BEMs), electromagnetic scat- tering, magnetic field integral equation (MFIE). I. INTRODUCTION Boundary element methods (BEMs) result from the application of finite element methods to solve boundary integral equations (BIEs) arising in connection with certain elliptic boundary value problems in physics and engineering. In formulating electromagnetic scattering problems, the BIEs are expressed in terms of equivalent electric and magnetic currents flowing on the surface of the scatterer. Numerical solution via the BEM, also commonly referred to as the method of mo- ments, proceeds by expanding these surface currents onto a finite ele- ment basis, and then projecting the resulting integral equation onto a suitable testing space. Complexities in the method arise because of the need to accurately represent surface vector fields, and one successful approach is the use of special, edge-based finite element bases to rep- resent the surface currents. Rao, Wilton, and Glisson (RWG) [1] introduced a set of edge basis functions which had a constant edge-normal component along the shared edge of two adjacent triangular patches. These bases have seen wide use since their inception. The limitations of these elements are that they generate a zeroth order polynomial representation of the surface currents supported by flat facetted patches. Wandzura [2] introduced a set of basis functions on curvilinear patches, and since then, bases have been generated on a range of surface types including Bézier patches and NURBS [3]. Wang and Webb [4] introduced a higher order adaptive basis where a solution could be adaptively im- proved by targeted incrementation of the basis order. Graglia, Wilton, and Peterson (GWP) [5] introduced higher order elements by scaling the RWG basis with a set of higher order shifted interpolatory poly- nomials. Higher order bases on curvilinear patches gained popularity due to the improved convergence despite requiring more advanced methods of evaluating the integrals than those available for the first order flat facetted cases [6]. This paper has two goals. First, to introduce new interpolatory basis functions whose continuity matches that of the solution currents. That is, the representation of the current is continuous over smooth surfaces Manuscript received January 4, 2006; revised April 30, 2007. A. D. Hellicar, J. S. Kot, and G. James are with the CSIRO ICT Centre, Epping NSW 1710, Australia (e-mail: andrew.hellicar@csiro.au). G. K. Cambrell is with Monash University, Victoria 3800, Australia. Digital Object Identifier 10.1109/TAP.2008.922694 0018-926X/$25.00 © 2008 IEEE