1398 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 6, JUNE 2008 IE-FFT Algorithm for a Nonconformal Volume Integral Equation for Electromagnetic Scattering From Dielectric Objects Nilufer A. Ozdemir and Jin-Fa Lee , Fellow, IEEE UCL-TELE, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA This study extends the integral equation fast Fourier transform (IE-FFT) algorithm to the method of moments solution of a noncon- formal volume integral equation. The algorithm relies on the interpolation of Green’s function by Lagrangian polynomials on a uniform Cartesian tensor grid. Hence, the matrix-vector product in the iterative solver can be computed via the fast Fourier transform. The memory requirement and the computational complexity of the algorithm tend to stay close to O(N) and O(NlogN), respectively, where N is the number of unknowns. Index Terms—Electromagnetic scattering, fast Fourier transform (FFT), fast methods, Green’s function, interpolation, method of mo- ments (MoM), nonconformal volume integral equation (VIE). I. INTRODUCTION E LECTROMAGNETIC scattering and radiation problems that involve dielectric structures find a wide range of applications, such as radome design [1] and interaction of elec- tromagnetic fields with biological media [2]. These problems can be formulated in the form of either differential or volume integral equations (VIEs). VIEs are largely superseded by differential equation methods due to their high computational demands. When VIEs are solved by the method of moments (MoM), the solution of the matrix equation by a Krylov-space iterative solver requires memory and compu- tation time, where N is the number of unknowns and p is the number of iterations to achieve the desired accuracy. In an attempt to expedite the iterative solution of VIEs, several methods have been proposed and successfully applied. The most widely used approach to solve VIEs is the conjugate gradient (CG) fast Fourier transform (FFT) method [3]. This method requires the volume of the object to be discretized into uniform hexahedral cells in order to use the Toeplitz property of the coefficient matrix and hence to compute the matrix-vector product via the FFT. Therefore, the method requires O(N) memory and solution time. How- ever, the staircasing error introduced by the uniform grid to achieve the Toeplitz system matrix can be a major limitation for objects with curved surfaces and thin coatings. In order to avoid this limitation, irregular mesh-based algorithms—such as the multilevel fast multipole algorithm (MLFMA) [1], the precorrected FFT (p-FFT) method [4], and the adaptive integral method (AIM) [5]—have been extended to speed up the solu- tion of VIEs. All of these methods require O(N) memory and solution time to solve VIEs. The IE-FFT algorithm [6] is yet another irregular mesh-based grid algorithm. In this algorithm, the integral kernel, which is the free space Green’s function, and its gradient-gradient are Digital Object Identifier 10.1109/TMAG.2008.915842 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. sampled on a uniform Cartesian tensor grid and interpolated by Lagrangian polynomials. The action of this kernel representa- tion on a vector can be computed efficiently by means of the FFT. Unlike the p-FFT and AIM, the uniform grid does not rep- resent equivalent sources, but instead corresponds to an interpo- lation tool for Green’s function. For near interactions, a correc- tion term is required, since the accuracy of the sampled repre- sentation is insufficient. Hence, the IE-FFT algorithm performs as efficiently as other grid-based methods; only minor modifi- cations are needed to extend the interpolation to other standard polynomials such as Newtonian or trigonometric polynomials. The next section explains the basic steps of the algorithm for a nonconformal VIE. II. IE-FFT ALGORITHM FOR A NONCONFORMAL VOLUME INTEGRAL EQUATION The MoM solution of the nonconformal VIE leads to a coef- ficient matrix with entries (1) where is the free space Green’s function for the scalar Helmholtz equation with and as the observation and the source locations, respectively; are piecewise constant basis functions in the source (obser- vation) tetrahedron is the free space wave number; and denotes the relative permittivity tensor [7]. The IE-FFT algorithm speeds up the matrix-vector product in the iterative solution of , where denotes the unknown vector; and represents the incident electric field intensity at location . The starting point of the IE-FFT algorithm is to interpolate Green’s function and its gradient-gradient by Lagrangian poly- nomials on a uniform Cartesian tensor grid. Interpolation of 0018-9464/$25.00 © 2008 IEEE