A Fast Volume-Surface Integral Equation Solver for ... - Semantic Scholar

818

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 2, FEBRUARY 2005

A Fast Volume-Surface Integral Equation Solver for Scattering From Composite Conducting-Dielectric Objects Xiao-Chun Nie, Ning Yuan, Le-Wei Li, Fellow, IEEE, Yeow-Beng Gan, Senior Member, IEEE, and Tat Soon Yeo, Fellow, IEEE

Abstract—This paper presents a fast hybrid volume-surface integral equation approach for the computation of electromagnetic scattering from objects comprising both conductors and dielectric materials. The volume electric field integral equation is applied to the material region and the surface electric field integral equation is applied on the conducting surface. The method of moments (MoM) is used to convert the integral equation into a matrix equation and the precorrected-FFT (P-FFT) method is employed to reduce the memory requirement and CPU time for the matrix solution. The present approach is sufficiently versatile in handling problems with either open or closed conductors, and dielectric materials of arbitrary inhomogeneity, due to the combination of the surface and volume electric field integral equations. The application of the precorrected-FFT method facilitates the solving of much larger problems than can be handled by the conventional MoM. Index Terms—Electromagnetic scattering, method of moments, precorrected-FFT method, volume-surface integral equation.

I. INTRODUCTION

N

UMERICAL studies of electromagnetic scattering from conducting-dielectric objects, such as conducting targets coated with lossy dielectric materials and microstrip structures on finite substrates, etc., are of considerable interest due to their various useful applications. A typical computational method for this problem is based on the surface integral equation (SIE) formulation [1] or the hybrid volume-surface integral equation (VSIE) formulation [2], [3], in conjunction with the method of moments (MoM). In the SIE approach, both the conducting and dielectric surfaces are formulated using the SIE, while in the VSIE approach, the volume integral equation (VIE) is applied to the material region, with the SIE enforced on the conducting surface. In comparison to the SIE approach, the VSIE approach has several unique advantages. First, the VSIE approach can conveniently handle composite objects with arbitrarily inhomogeneous dielectric materials, due to the use of the VIE, while the SIE can only consider piecewise homogeneous dielectric mateManuscript received October 16, 2003; revised May 21, 2004. X.-C. Nie, N. Yuan, and Y.-B. Gan are with Temasek Laboratories, National University of Singapore, 119260 Singapore, Singapore (e-mail: tslniexc@nus.edu.sg; tslyuann@nus.edu.sg; tslganyb@nus.edu.sg). L.-W. Li is with the Department of Electrical and Computer Engineering, National University of Singapore, 119620 Singapore and with the High Performance Computation for Engineered Systems Programme, Singapore-MIT Alliance, 119620 Singapore, Singapore (e-mail: lwli@nus.edu.sg). T. S. Yeo is with the Department of Electrical and Computer Engineering, National University of Singapore, 119620 Singapore, Singapore (e-mail: eleyeots@nus.edu.sg). Digital Object Identifier 10.1109/TAP.2004.841323

rials. Even for piecewise homogeneous materials, the efficiency of the SIE decreases as the number of the subdomains increases, since boundary condition must be enforced on the surface of each homogeneous subdomain, and the Green’s function in different homogeneous medium must be used in the formulation. Besides, for special objects such as a conducting sphere of two halves coated with different materials, despite the simple geometry, the SIE approach requires special treatments on the conducting–dielectric and dielectric–dielectric junctions to obtain accurate results [4], [5]. On the other hand, the VSIE approach retains the same simple form regardless of the complexity of the object and materials. Hence, the implementation is relatively convenient and simpler, as compared to the SIE. Also, no special treatments are required for problems with junctions. In this paper, we choose the VSIE instead of the SIE to model the composite conducting–dielectric objects. However, for either SIE or VSIE, the traditional MoM incurs very high computational cost and memory requirement as the electrical size of the scatterer increases. This problem is extremely serious for the VSIE, since the entire dielectric volume needs to be meshed, resulting in more unknowns as compared to the SIE. Fortunately, the fast algorithms such as the fast multiple method (FMM) [6] and its extension, the multilevel fast multipole algorithm (MLFMA) [7], the conjugate gradient fast Fourier transform method (CG-FFT) [8], the adaptive integral method (AIM) [9], and the precorrected-FFT (P-FFT) method [10] and others are capable of overcoming this difficulty to some reasonable extent. The initial focus of these fast methods (except the CG-FFT) was to solve the SIE for scattering problems. So far, the MLFMA has been extended to the solution of the VSIE [11], which reduces both the computational complexity . In [12] and [13], the and memory requirement to finite element/boundary integral (FE/BI) method is used to analyze the problems involving both metallic and dielectric materials, with the solution accelerated using the array decomposition method (ADM) combined with the CG-FFT and AIM method, respectively. In this paper, the P-FFT method is applied to solve the hybrid VSIE for composite conducting and dielectric objects. In comparison to the P-FFT accelerated SIE approach [14], [15], the P-FFT algorithm for VSIE requires considerably reduced number of FFTs in each matrix-vector multiplication, the only repeated operator throughout the method. Therefore, although more unknowns are needed in the VSIE approach, it is more efficient than the SIE approach, especially for scatterers with complex dielectric materials.

0018-926X/$20.00 © 2005 IEEE

NIE et al.: FAST VOLUME-SURFACE INTEGRAL EQUATION SOLVER

819

II. FORMULATION A. Formulation and Discretization of the Volume-Surface Integral Equation Consider a mixed conducting and dielectric scattering target illuminated by an incident field . It is assumed that the dielecfor all regions. tric materials are nonmagnetic, namely, However, the approach can also be extended to magnetic material, but for simplicity, we will focus on electric materials first. Using the equivalence principle, the conducting bodies are reand the dielectric maplaced by equivalent surface currents . All the terials are replaced by equivalent volume current currents radiate in free space, and hence the free-space Green’s is the function is used in the formulation. The scattered field total contribution of the surface current and volume current , which can be calculated by (1) , and are the vector and where scalar potentials produced by the surface and volume current, respectively, given by (2)

(3) The volume current by

is related to the total electric flux density (4)

where

is the contrast ratio defined as (5)

being the permittivity of the dielectric material. with On all conductor surfaces ( can be either closed or open), the boundary condition requires that the total tangential electric field is zero, i.e.,

the discretized model of a target, the following criteria should be satisfied. First, the surface patches should coincide with the external faces of the volume cells on the conductor–dielectric interface, namely partial overlap is not permitted. Secondly, since the dielectric properties in each volume cell are assumed to be constant, all the boundaries between different dielectric materials should be identified in the volume mesh. and electric flux density The unknown surface current can be represented by the vector basis functions introduced by Rao, Wilton, Glisson, and Schaubert, respectively [16], [17] (7a) (7b) is the number of edges that make up the triangulated where is the number of faces that make up the model of , while and are the unknown expansion tetrahedron model of . represents the th surface basis function, coefficients. which is defined on two triangle patches associated with edge in the surface mesh while denotes the th volume basis function defined on two adjoining tetrahedrons associated with face in the volume mesh. The details of the basis functions and can be found in [16] and [17]. The factor is included in (7b) to ensure that the magnitudes of the expanand are of the same order. When the sion coefficients th face is located on the boundary of region , an auxiliary cell sharing the th face with the interior cell should be introduced in the exterior region. For convenience, we usually make the free vertex of the auxiliary cell coincide with the center of the th face [17]. Substituting (7) into (6) and testing (6a) with the surface , and (6b) with the volume basis function basis function (this is the extended Galerkin’s method), a linear independent equations is system consisting of obtained

(6a) This is the surface electric field integral equation. In the dielectric region, the total electric field is equal to the sum of the incident field and the scattered field. Hence, the volume integral equation is given by

(8a)

(6b) Equations (6a) and (6b), together with (1)–(5), constitute a hybrid volume-surface integral equation in terms of the surface on the conducting surface and the electric flux current density in the dielectric region. In the above equations, time convention is used and suppressed. the is discretized into To solve (6), the conducting surface small triangular patches, while the dielectric region is divided into tetrahedral elements. However, the triangle-tetrahedron mesh is not the only choice for the present method. Other types of meshes, such as quadrangle for surface and hexahedron for volume, can also be used. Curvilinear-faced meshes can also be considered, since the basic idea of the present method retains the same regardless of the mesh type. When setting up

(8b) Equation (8) can be written as a submatrix form in the following:

(9)

820


The formulas of the submatrices and the excitation vector can be readily derived , and are the contributions to from (8). and from a single surface and volume basis function, given by

(10a)

by these grid sources do not match well with those radiated by the original sources. Therefore, to obtain more accurate results, near-field interactions are evaluated directly and the errors introduced are corrected. This step is the “precorrection.” The first step of the algorithm is the construction of the grid projection operators. Assume the th surface basis function or volume basis function is contained in a given cell . Select test points on the surface of a sphere of radius whose center coincides with the center of the th cell. For the electric or charges (corresponding to the divergence operators ), by matching the scalar potential produced by the grid charges and that produced by the original charge distributions at test points, we obtain the following projection operator: all (11)

(10b)

where is the th column of and represents . is the mapping between the the generalized inverse of grid charges and the test-point potentials, and is the mapping between the actual charge distributions and the test-point potentials. Their expressions are given by (12)

(10c)

for

(13a)

(10d) is the length of the th edge, and are the where supports of the two triangles associated with the th edge, and is the area of the triangular . represents the position vector of an arbitrary point in with respect to the free vertex . All the above symbols are defined in the surface mesh; of can be defined in the same other symbols such as is the constant value of way but in the volume mesh [17]. in . B. The Precorrected-FFT Solution of the VSIE To implement the precorrected-FFT algorithm, the entire object, including the space occupied by both conductor and dielectric materials, is enclosed in a uniform rectangular grid that is grid further grouped into small cells. Each cell consists of points and contains only a few triangular patches and/or tetrahedral elements. Next, the original current and charge distributions are replaced by an approximate equivalent set of point-like currents and charges located at the nodes of the grid, referred to as “projection.” This projection allows the vector and scalar potentials at the grid points produced by the grid sources to be computed by the efficient FFT. Knowledge of these potentials is then used to obtain the potentials on the triangular or tetrahedral elements via interpolation. So far, the far-field interactions have been well approximated, but the near fields radiated

for

(13b)

where and are the position vectors at the th test point and the th grid point, respectively. A common constant 1/4 is omitted in (12) and (13) for convenience of description and implementation. Similarly, by matching the vector potential due to the grid currents and that due to the surface current distributions in the triangular patches or the volume current distributions in and ) at the tetrahedral elements (corresponding to the test points, we can obtain the projection operators for the electric currents (14) where

for

(15a)


for

821

(15b)

As shown in (14) and (15), the electric currents are first decomposed into three components, followed by separate projection of each component. Similarly, a common constant 4 is omitted in (14) and (15). The accuracy of the above projection approach is determined by three parameters: the grid order , the number of test points , and the radius of the sphere . The grid order is one of the most important parameters and affects not only the accuracy of the projection procedure but also the computation time and memory requirement. On one hand, when the size of the rectangular cell is fixed, should be sufficiently large so that the grid spacing is small enough and the potentials produced by the source distributions on the scatterer are well approximated by the point sources on the grids. On the other hand, larger results in more grid points in each cell, consequently larger dimension of FFT, and thus requires more memory and computation time in the projection, FFT, or interpolation procedure. For an overall consideration, is usually set to be three or four, which is a good compromise between the accuracy and the costs. However, the and is relatively flexible. It is found that reachoice of is in the rage of 4–70, sonable results can be obtained when and can be arbitrarily selected, as long as all cell vertices are entirely enclosed in the sphere. In our analysis, 25 or 36 test points are selected according to the high-order quadrature rule [10]. Detailed error analysis can be found in [18] and [19]. Once the volume and surface source distributions have been projected onto uniform grids, the vector and scalar potentials at other grid locations produced by these grid-projected sources can be efficiently computed using the FFT. Upon computing the grid potentials, the potentials in the triangular patches and the tetrahedral elements can be obtained , the through interpolation. For the surface testing function interpolation operators are essentially the same as the projection operators [10], [14]. However, for the volume testing function , the interpolation operators are different from the projection operator, although they can be defined in the same way. The difference lies in the fact that in the construction of the inneed not terpolation operators, the dielectric contrast ratio be included, but in the construction of the projection operators, this factor must be included. Thus, the interpolation operators for the volume testing function can be expressed in the same and form as the projection operator of the basis function its divergence , i.e. (16) where

(17a)

Fig. 1. Bistatic RCS ( polarization) of a conducting sphere (ka = 2:6858) with a discontinuous coating (ka = 3:0; " = 3 and 4).

(17b) A special case is that when the th testing function is defined on a boundary face of region , the volume integral over the auxiliary cell will decrease to a surface integral on the th face in (17a) and vanish in (17b), since we define the free vertex of the auxiliary cell coincide with the center of the th face. Finally, a procedure referred to as “precorrection” is necessary to get more accurate results. In this procedure, the near-zone interactionsare directly computedas in the conventional MoM, andthe inaccurate contribution from the use of the grid is removed. For the VSIE, the memory requirement and computational complexity of the P-FFT method depend on the structure of the object considered. For some objects, such as a conducting sphere coated with a thin dielectric layer, the number of unknowns in the dielectric volume and that on the conducting surfaces are of the same order, and the memory requireand ment and computational complexity scales as , respectively (the same as those of the SIE). But for most objects, since the number of unknowns in the dielectric volume and on the conducting surfaces is proportional to and , respectively, and the former usually dominates in the total number of unknowns, the memory requirement and comand , respectively putational complexity is (the same as those of the VIE). III. NUMERICAL RESULTS In this section, some numerical results are presented to validate the algorithm and demonstrate the efficiency of the method. For the first example, we consider a conducting sphere covered by two hemispherical shells of different permittivity. The geometry is shown in the inset of Fig. 1. The radii of the conducting sphere and the outer surface of the dielectric shell are

822


TABLE I CPU TIME AND MEMORY REQUIREMENTS OF THE P-FFT METHOD AND THE MoM

and , respectively, where is the wave number in free space. The permittivity of the coating materials is and , respectively. The surface of the conducting sphere is discretized into 1568 triangular patches and the volume of the dielectric shell into 10 816 tetrahedrons, yielding a total number of 25 792 unknowns. Fig. 1 shows the bistatic radar cross-section (RCS) for a normally incident plane wave on the sphere. It is observed that the results obtained by the present method are in excellent agreement with those of the MoM for bodies of revolution in [4]. In this example, a junction between two different dielectric materials exists. When using the surface integral equation approach to solve this problem, some triangular patches are shared by more than three basis functions, and extra efforts must be made to enforce the continuity of the current at the junction. This makes the implementation of the SIE more complicated. However, the present approach circumvents this difficulty simply by using the volume integral equation in the dielectric region. The CPU time and memory requirement of the P-FFT method are presented in Table I and compared with the estimated MoM values. The computations are carried out on a Pentium 2.4-G PC. All the computations are in a single precision, and the normalized residual is set to be 1 10 . In the second example, the monostatic RCS of a metal-dielectric cylindrical scatterer as shown in Fig. 2(a) is considered. The top half of the cylinder is perfectly conducting, while the bottom . The object is 7.62 cm in diameter and half is dielectric 15.24 cm in length, with the conducting segment of length 10.16 cm. The working frequency is 3 GHz and the total number of unknowns is 13 248, with the dielectric volume modeled by 5184 tetrahedrons and the metal surface modeled by 1584 triangles. In this example, junctions existed around the periphery where the dielectric, perfectly electric conductor, and free space meet. The results calculated using the precorrected-FFT accelerated VSIE are shown in Fig. 2(b) and compared with the results of the SIE approach and measured data given in [4]. It is observed that the results of this method are in better agreement with measured data than those of the SIE. In this example, the step of the incident angle is set to be 2 , resulting in totally 91 incident angles. To improve the convergence, the solution of the current distribution from the previous angle is used, with the necessary phase correction, as the initial guess for the next incident angle. The CPU time and memory requirements of the P-FFT method and the MoM are also presented in Table I. It should be noted that the iteration number shown in the table is the average value of all the incident angles, but the corresponding CPU time is the total time to get the solution for all 91 incident angles. Finally, we consider a dielectric rod covered by a doubly periodic 8 8 cylindrical-square patch array, as shown in Fig. 3(a). , a height of The dielectric cylinder has a radius of , and a permittivity of . The dimension of

(a)

(b) Fig. 2. (a) An inhomogeneous conducting–dielectric cylinder " = 2:6; a = 10:16 cm, b = 15:24 cm, d = 7:62 cm. (b) Monostatic RCS ( polarization) for an inhomogeneous conducting-dielectric cylinder (f = 3 GHz).

the conducting patch is , with the inter-el4. Although the surfaces of the patches ement spacing at are open, this method remains applicable since the surface electric field integral equation is applied on the conducting surface. The incident wave is along the direction of , and the bistatic RCS versus in the plane are investigated. The results obtained from the present method and the precorrected-FFT accelerated SIE approach are shown in Fig. 3(b). Again, good agreements are observed. In the computation, the volume of the dielectric cylinder and the surface of the conducting patches are discretized into 46 656 tetrahedrons and 3200 triangles, respectively, yielding a total of 99 524 unknowns. The memory requirement and CPU time for this example are also given in Table I. It is estimated that the conventional MoM will require over 73.8 Gbyte memory and cost 311 h


(a)

(b)

2

Fig. 3. (a) Dielectric cylinder covered by a 8 8 conducting patch array. (b) Bistatic RCS ( polarization) versus in the xoy plane.

to obtain the final solution, provided sufficient memory is available. However, the present method requires only 0.489 Gbyte memory, which is about 0.66% that of the conventional MoM and also yields a reduction of over 95% in CPU time. IV. CONCLUSION The precorrected-FFT method has been applied to solve the hybrid volume-surface integral equation for scattering from composite conducting and dielectric objects with arbitrary shape and inhomogeneity. The application of the P-FFT method significantly reduces the memory requirement and computational complexity of the conventional MoM. Through the use of the irregular volume and surface meshes, this technique offers good flexibility to model arbitrarily shaped structures while maintaining the efficiency of the FFTs. REFERENCES [1] S. M. Rao, C. C. Cha, R. L. Cravey, and D. L. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antennas Propag., vol. 39, no. 5, pp. 627–631, May 1991. [2] T. K. Sarkar and E. Arvas, “An integral equation approach to the analysis of finite microstrip antennas: Volume/surface formulation,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 305–313, Mar. 1990.

823

[3] C. C. Lu and W. C. Chew, “A coupled surface-volume integral equation approach for the calculation of electromagnetic scattering from composite metallic and material targets,” IEEE Trans. Antennas Propag., vol. 48, no. 12, pp. 1866–1868, Dec. 2000. [4] L. N. Medgyesi-Mitschang and J. M. Putnam, “Electromagnetic scattering from axially inhomogeneous bodies of revolution,” IEEE Trans. Antennas Propag., vol. AP-32, no. 8, pp. 797–806, 1984. [5] M. A. Carr, E. Topsakal, J. L. Volakis, and D. C. Ross, “Adaptive integral method applied to multilayer penetrable scatterers with junctions,” in IEEE Antennas Propag. Soc. Int. Symp., Jul. 2001, pp. 858–861. [6] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag., vol. 35, pp. 7–12, Jun. 1993. [7] J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1488–1493, Oct. 1997. [8] T. K. Sarkar, E. Arvas, and S. M. Rao, “Application of FFT and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies,” IEEE Trans. Antennas Propag., vol. AP-34, no. 5, pp. 635–640, May 1986. [9] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, pp. 1225–1251, 1996. [10] J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. ComputerAided Design Integr. Circuits Syst., vol. 16, pp. 1059–1072, 1997. [11] C. C. Lu and C. Yu, “Simulation of radiation and scattering by large microstrip patch arrays on curved substrate by a fast algorithm,” in Proc. 3rd Int. Conf. Microwave Millimeter Wave Technology, 2002, pp. 401–404. [12] R. W. Kindt, K. Sertel, E. Topsakal, and J. L. Volakis, “Array decomposition method for the accurate analysis of finite arrays,” IEEE Trans. Antennas Propag., vol. 51, no. 6, pp. 1364–1372, Jun. 2003. [13] J. L. Volakis, T. F. Eibert, D. S. Filipovic, Y. E. Erdemli, and E. Topsakal, “Hybrid finite element methods for array and FSS analysis using multiresolution elements and fast integral techniques,” Electromagn., vol. 22, no. 4, pp. 297–313, 2002. [14] X. C. Nie, L. W. Li, N. Yuan, T. S. Yeo, and Y. B. Gan, “A fast analysis of electromagnetic scattering by arbitrarily shaped homogeneous dielectric objects,” Microwave Opt. Technol. Lett., vol. 38, no. 1, pp. 30–35, 2002. [15] S. Gedney, A. M. Zhu, W. H. Tang, G. Wang, and P. Peter, “A fast, high-order quadrature sampled precorrected fast Fourier transform for electromagnetic scattering,” Microwave Opt. Technol. Lett., vol. 36, no. 5, pp. 343–349, Mar. 2003. [16] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, no. 3, pp. 409–418, May 1982. [17] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. AP-32, no. 1, pp. 77–85, Jan. 1984. [18] J. R. Phillips, “Rapid solution of potential integral equations in complicated 3-dimensional geometries,” Ph.D. dissertation, Massachusetts Inst. of Technology, 1997. , “Error and complexity analysis for a collocation-grid-projection [19] plus precorrected-FFT algorithm for solving potential integral equations with Laplace or Helmholtz kernels,” in Proc. Copper Mountain Conf. Multigrid Methods, Copper Mountain, CO, Apr. 2–7, 1995.

Xiao-Chun Nie received the B.Eng. degree from Xi’an Jiaotong University, China, in 1988, the M.Eng. degree from the University of Electronic Science and Technology of China, Chengdu, in 1993, and the Ph.D. degree from Xidian University in 2000, all in electrical engineering. From 1993 to 1997, he was a Lecturer with the University of Electronic Science and Technology of China, Chengdu, China. Since September 2000, he has been a Research Fellow with the Singapore-MIT Alliance, National University of Singapore. In September 2002, he joined Temasek Laboratories, National University of Singapore, where he is currently a Research Scientist. His main interests include numerical analysis of scattering, radiation problems, microwave circuits, and antennas.

824


Ning Yuan received the B.Eng. and M.Eng. degrees from University of Electronic Science and Technology of China, Chengdu, in 1993 and 1996, respectively, and the Ph.D. degree from Xidian University, China, in 1999, all in electrical engineering. From September 1999 to July 2000, she was a Postdoctoral Fellow in telecommunications and industrial physics, CSIRO, Sydney, Australia. From August 2000 to June 2002, she was with the Department of Electrical and Computer Engineering, National University of Singapore, as a Research Fellow. Currently, she is a Research Scientist with Temasek Laboratories, National University of Singapore. Her main interests are in computational electromagnetics.

Le-Wei Li (S’91–M’92–SM’96–F’04) received the of B.Sc. degree in physics from Xuzhou Normal University, Xuzhou, China, in 1984, the M.Eng.Sc. degree from China Research Institute of Radiowave Propagation (CRIRP), Xinxiang, in 1987, and the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. In 1992, he was with La Trobe University (jointly with Monash University), Melbourne, as a Research Fellow. Since 1992, he has been with the Department of Electrical and Computer Engineering, National University of Singapore (NUS), where he is currently a Professor and Director of the Centre for Microwave and RF. Since 1999, he has also been with the High Performance Computations on Engineered Systems Programme of the Singapore-MIT Alliance (SMA), where he is an SMA Fellow. His current research interests include electromagnetic theory, radiowave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. In these areas, he has coauthored Spheroidal Wave Functions in Electromagnetic Theory (New York: Wiley, 2001), 42 book chapters, more than 200 international refereed journal papers, 25 regional refereed journal papers, and more than 200 international conference papers. He is an Editor of the Journal of Electromagnetic Waves and Applications, an Associate Editor of Radio Science, and an Overseas Editorial Board Member of Chinese Journal of Radio Science. Dr. Li is a member of The Electromagnetics Academy based at MIT. He received the Best Paper Award from the Chinese Institute of Communications for his paper published in Journal of China Institute of Communications in 1990 and the Prize Paper Award from the Chinese Institute of Electronics for his paper published in Chinese Journal of Radio Science in 1991. He was selected to receive a Ministerial Science and Technology Advancement Award by the Ministry of Electronic Industries, China, in 1995 and a National Science and Technology Advancement Award with medal by the National Science and Technology Committee, China, in 1996.

Yeow-Beng Gan (M’90–SM’01) received the B.Eng. (Hons.) and M.Eng. degrees from the National University of Singapore in 1994 and 1989, respectively, both in electrical engineering. He has been with DSO National Laboratories (formerly the Defence Science Organization) since 1989, where he started and built up the area of antenna analysis and design. He became a Principal Member of Technical Staff in 1998. In 2001, he was seconded to Temasek Laboratories, National University of Singapore, where he is currently a Principal Research Scientist. His research interests include periodic arrays for antennas and radomes, wave propagation and scattering, computational electromagnetics, and modeling of composite materials. Dr. Gan is a member of the Materials Research Society, Singapore.

Tat Soon Yeo (M’79–SM’93–F’03) received the B.Eng. degree (Hons. I) from the University of Singapore in 1979, the M.Eng. degree from the National University of Singapore in 1981, and the Ph.D. degree from the University of Canterbury, New Zealand, in 1985. He is currently a Professor in the Electrical and Computer Engineering Department and a Vice Dean of the Fcaulty of Engineering, National University of Singapore. He is also concurrently the Director of Radar and Signal Processing Laboratory, and of the Antennas and Propagation Laboratory, in the Department of Electrical Engineering, National University of Singapore. He is also Director of Temasek Defence Systems Institute, a teaching institute established jointly by NUS and the U.S. Naval Postgraduate School. His current research interests are scattering analysis, synthetic aperture radar, antenna and propagation study, numerical methods in electromagnetics, and electromagnetic compatibility. Dr. Yeo received a Colombo Plan Scholarship. He received a Singapore Ministry of Defence–National University of Singapore 1997 Joint R&D Award, the IEEE Millennium Medal in 2000, and the Singapore Standard Council’s Distinguished Award in 2002. He is the past Chairman and Executive Committee Member of the MTT/AP and EMC Chapters, IEEE Singapore Section, and Chairman of the Singapore EMC Technical Committee.