Precorrected-FFT solution of the volume Integral ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 1, JANUARY 2005

313

Precorrected-FFT Solution of the Volume Integral Equation for 3-D Inhomogeneous Dielectric Objects Xiao-Chun Nie, Le-Wei Li, Senior Member, IEEE, Ning Yuan, Tat Soon Yeo, Fellow, IEEE, and Yeow-Beng Gan, Senior Member, IEEE

Abstract—This paper presents a fast solution to the volume integral equation for electromagnetic scattering from three-dimensional inhomogeneous dielectric bodies by using the precorrectedfast Fourier transform (FFT) method. The object is modeled using tetrahedral volume elements and the basis functions proposed by Schaubert et al. are employed to expand the unknown electric flux density. The basis functions are then projected onto a fictitious uniform grid surrounding the nouniform mesh, enabling the FFT to be used to speed up the matrix-vector multiplies in the iterative solution of the matrix equation. The resultant method greatly reduces the memory requirement to ( ) and the computational complexity to ( log ), where is the number of unknowns. As a result, this method is capable of computing electromagnetic scattering from large complex dielectric objects. Index Terms—Electromagnetic scattering, method of moments (MoM), precorrected-fast Fourier transform (FFT) method, volume integral equation.

I. INTRODUCTION

E

LECTROMAGNETIC scattering from inhomogeneous dielectric objects is of great interest due to its wide applications in communications, target identification, geophysical prospecting, microwave imaging, etc. The volume integral equation (VIE) in conjunction with the method of moments (MoM) is an accurate and flexible method to calculate the electromagnetic scattering from dielectric bodies of arbitrary shape and arbitrary inhomogeneity. However, the conventional MoM suffers from tremendously high computational cost and memory requirement as the electrical size of the scatterers increases. Recent developments in fast algorithms have alleviated this problem to some extent. The most widely used approach to solve the VIE is the conjugate gradient fast Fourier transform (CG-FFT) method [1]–[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.

Manuscript received May 6, 2002; revised March 5, 2004. X.-C. Nie, N. Yuan, and Y.-B. Gan are with the Temasek Laboratories, National University of Singapore, Kent Ridge, Singapore 119260, Singapore (e-mail: [email protected]). L.-W. Li is with the Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge, Singapore 119260, and also with the High Performance Computation for Engineered Systems Programme, Singapore-MIT Alliance (SMA), Kent Ridge, Singapore 117576, Singapore. T. S. Yeo is with the Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge, Singapore 119260, Singapore. Digital Object Identifier 10.1109/TAP.2004.838803

Thus, when modeling an arbitrary geometry, very dense cells are required which result in a large number of unknowns, and the unavoidable staircase geometry error will degrade the accuracy of the final solution. To overcome this drawback, some irregular-mesh-based fast algorithms such as the fast multipole method (FMM), the precorrected-FFT method and the adaptive integral method (AIM) can be used. These fast methods were initially implemented for surface integral equations. Recently, Lu et al. [4], [5] extended the multilevel fast multiple algorithm (MLFMA), a multilevel variant of the FMM, to solve the hybrid volume surface integral equation (VSIE) for composite conducting and dielectric objects. Zhang et al. [6] applied the AIM to solve the VIE for dielectric objects. In their work, a kind of linear normal constant tangential (LNCT) basis function was used to expand the unknown electric flux density. Similar to FMM and AIM, the precorrected-FFT (P-FFT) method also works on approximating the far-zone interactions, and its key idea is similar to that of the AIM. This technique was originally proposed by Philips and White in 1994 [7], [8] to solve the electrostatic integral equation for capacitance extraction. Later, the present authors made appropriate modifications and adapted it to the solution of surface integral equations for various electromagnetic scattering problems [9]–[11]. In this paper, it is further extended to solve the VIEs efficiently. Unlike the CG-FFT, the present approach uses the flexible tetrahedral volume elements to model the scattering body. The basis functions developed by Schaubert et al. [12] are employed to expand the unknown electric flux density. These special basis functions are constant normal and linear tangential (CNLT) and ensure that the normal electric field satisfies the correct jump condition at the interfaces between different media. Next, the entire object is enclosed in a uniform rectangular grid, and all basis functions are projected onto the surrounding grid points. By such a procedure, the interactions between the point sources on the grid are described by three-dimensional convolutions, which can be performed rapidly in the Fourier space by a volumetric Fourier transform. Thus, the matrix-vector product can operations. The memory requirebe performed in ment of the method is proportional to , since only an order- subset of the square coefficient matrix which corresponds to near-zone interactions are computed and stored. Such complexity and requirement make the volume precorrected-FFT method highly competitive with the MLFMA. In contrast to

0018-926X/$20.00 © 2005 IEEE

314


the CG-FFT, this technique offers good flexibility to model arbitrarily shaped structures while keeping the efficiency of the FFTs. In addition, the density of the uniform grid depends on the requirement of the solution accuracy and can retain a desired coarse level even if a complex structure is analyzed. II. FORMULATION A. The Formulation and Discretization of the Volume Integral Equation Consider a lossy, inhomogeneous dielectric body illuminated by an incident field . Assume that the material is di) and has complex dielectric constant of electric ( , where and are the relative permittivity and conductivity at position . By invoking the equivalence principle, the dielectric body is removed and replaced by a volume polarization current . According to the fact that the total electric field is the sum of the incident field and the scattered field due to , we can obtain the following VIE (1)

is the number of faces that make up the tetrahedral where denotes the unknown expansion coeffimodel of , and cients. represents the th basis function, which is defined on two adjoining tetrahedrons associated with face . It should be noted that when the 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 so that the two constitute a pair of cells required by the volume basis function. However, in practical implementation, we make the free vertex of the auxiliary cell coincide with the center of the th face, thus the volume of the auxiliary cell equals zero [4], [12]. This is possible because there is no limit to the size of the auxiliary cell as long as it shares the th face with the cell in the dielectric region. With such auxiliary cells, no extra grid need to be introduced in the volume P-FFT method and the complexity of the mesh generation is also reduced to a minimal level. Substituting (6) into (1) and applying the Galerkin’s testing independent procedure yield a linear system consisting of equations shown in (7) at the bottom of the page. Equation (7) can be written in a matrix form as (8)

and are the vector and scalar potentials prowhere time convention duced by the volume current given by ( is used)

(3)

The formulas for the computation of the matrix and the excitation vector can be derived from (7) readily. In the derivation, the gradient operation on the scalar potential in (7) can be transferred to the testing function using the vector cal. Considering culus identity the properties of on the faces of , the gradient terms in (7) can be rewritten as

is related to the total electric flux density

(9)

(2)

The volume current by

(4) where

is the contrast ratio defined by (5)

Thus, (1), taken with (2) and (3), constitutes an integral equation , which has continuous normal for the electric flux density component at media interfaces. To solve (1), the volume is discretized into a number of tetrahedral elements, in each of which the dielectric properties are approximated as constant. The unknown electric flux density can be represented by the volume Schaubert–Wilton–Glisson basis functions [12]: (6)

and are omitted here for The resultant formulas for simplicity. When computing the elements of , the contributions to and from a single basis function are needed. For easier description of the following precorrected-FFT approach, we give their expressions as follows:

(10)

(11)

(7)

NIE et al.: PRECORRECTED-FFT SOLUTION OF THE VIE

where is the area of the th face, and are the two tetrahedrons associated with the th face, is the volume of . represents the position vector of an arbitrary point in with respect to the free vertex of . is the contrast value of in . Note that the third term in (11) represents surface charges which exist only on the interfaces of different media. is a dense matrix, requiring comThe matrix puter memory to store it. The CPU requirement is if the matrix equation is solved by a direct solver, or if an iterative solver is used. For volumetric inhomogeneities, even a moderate size volume will make this computation intractable. To alleviate this difficulty, we use a generalized minimal residual (GMRES) algorithm to solve the matrix equation iteratively, and apply the precorrected-FFT method to speed up the matrix-vector products in iterations and reduce the memory requirement. In next section, we will describe the implementation of the volume precorrected-FFT method with emphasis on the differences from that of the surface P-FFT.

315

grid charges to match that produced by the original charge distributions on the two tetrahedral elements and the th triangular face (only applicable when the th face is an interface of different media), we obtain (12) where and denote the scalar potentials at the th test point due to the original charge distributions and the grid charges respectively. Equation (12) is enforced for all test points. Substituting (11) into (12), we obtain (13) where denote the vectors consisting of the charges grid points, represent the mappings between the at the represent the grid charges and the test-point potentials, and mappings between the actual charge distributions and the testpoint potentials, respectively, given by (14)

B. The Precorrected-FFT Solution of the VIE The precorrected-FFT method considers separately the nearand far- field interactions when evaluating a matrix-vector multiplication. To compute far-field interactions, sources supported by the scatterer are projected onto a regular grid by matching their vector and scalar potentials at some given test points to guarantee the approximate equality of their far fields. Next, the potentials at other grid locations produced by these grid-projected sources are evaluated by a 3-D convolution. Knowledge of these fields permits the computation of fields on the scatterer through interpolation. The projection and interpolation operators are represented by sparse matrices, while the convolution can be performed using FFT. However, the near fields radiated by these grid currents do not match those radiated by the original sources. Therefore, although the far-field interactions are approximated by the above three steps very well, the near-field interactions are poorly approximated. To get more accurate results, the near-field interactions are to be evaluated directly and corrected for errors introduced by the far-field operator. The implementation of the four-step ( ) precorrected-FFT algorithm for the VIE will be described in the following paragraph. To implement the method, the entire object is enclosed in a uniform rectangular grid which is further subdivided into small cells, with each cell consisting of grid points and containing only a few tetrahedral elements. Assume the th volumetric SWG basis function is contained in a given cell . Select test points according to high-order quadrature rule on the surface of a sphere of radius ( can be arbitrarily selected as long as all the cell vertices are entirely enclosed in the sphere) whose center coincides with the center of the cell . For the projection of the electric charges, i.e., the divergence operator , enforcing the scalar potential produced by the

(15) where and are the position vectors at the th test point and the th grid point, respectively. With the above two matrices, we can obtain the projection operator for the divergence operator of the th basis function as (16) where denotes the th column of and indicates the generalized inverse of . For any basis function in cell , this projection operator generates a subset of the grid charges . The contribution to from the charges in cell can be computed by summing over all the actual charges in this cell, i.e. (17) Following the above procedure, we can project the charges onto the grid points surrounding cell . It should be noted that in contrast to the SIE where only surface charges exist on the triangular elements, in the VIE there exist not only volume charges in the tetrahedron elements but also surface charges on the interfaces of different media. This is because that the induced currents and charges are represented by and (except for a constant) respectively in the VIE, in contrast to and in the SIE, so the electric currents are discontinuous across the th face if and have different values, resulting in the surface charges on the interface.

316


In the implementation of the projection operator, in stead of projecting the volume and surface charges separately, we consider their total contributions and perform their projections simultaneously in one step. This is a convenient and efficient scheme, specifically developed for the VIE. Since the relative positions of the grid points and the test points can be constructed only need to be computed to be identical for each cell, once throughout the method, making the computation of the projection operators very efficient. Similarly, by matching the vector potential due to the grid currents and that due to the volume current distributions in the tetrahedron elements at the test points, we can obtain the projection operators for the electric currents . Once the volume and surface source distributions have been projected onto uniform grids, the relationship between the vector and scalar potentials at the grid points and the grid sources is in fact a 3-D convolution. The convolutions can be efficiently computed using the FFT due to the Toeplitz property of the Green’s function matrix. In practice, each convolution requires one forward and one inverse three-dimensional FFT, and the FFT of the kernel Green’s function matrix need to be computed only once. After the grid potentials are computed, the potentials in the tetrahedral elements can be obtained through interpolation, and the matrix-vector product can be computed from the information of these potentials. In this paper, we use a new interpolation operator which is specifically suitable and efficient for the VIE. In the SIE, the transpose of the projection matrix can be used directly as the interpolation operator [8], [13], [14]. However, due to the existence of the constant in the VIE, we can not use the transpose of the projection matrix directly but have to construct the interpolation operator for the vector and scalar potentials by reand its divergence spectively projecting the testing function onto the uniform grids. This requires extra efforts to calculate some numerical integrations and extra memory to store the interpolation matrices. To overcome this disadvantage, we propose a simpler method but still with comparable precision to construct the interpolation operator for the VIE. First, we choose one sampling point in each of the two supporting tetrahedrons , usually the centroid of the tetrahedrons. Then, instead of the testing function and its divergence, we project a unit point charge located at the sampling points onto the uniform grids. Following a similar procedure of deriving (12) to (16), we obtain the following interpolation operator (18) where

potentials at the sampling points once the potentials at the vertexes of cell (the cell in which is enclosed) are calculated by FFTs. Then the inner products in (7) can be calculated according to [12, eqs. (24) and (26) in the Appendix]. The above three steps, projection, convolution, and interpolation, in that order, resulted in the grid approximation to the inner products in (7) (20) (21) signifies that the values obtained are where the subscript , , based on grid approximation of the accurate values. and are the generalized projection, convolution and interpolation operators. So far, the far-zone interactions have been well approximated, while the near-zone interactions are poorly approximated since the near fields radiated by the grid sources do not match those radiated by the original sources. Therefore, to obtain more accurate results, it is necessary to compute the nearby interactions directly as in the conventional MoM and remove the errors introduced by the far-field operator. In the implementation, if the and the testing function distance between the basis function is less than a predefined threshold, the precorrection opercan be expressed by the following formula ator (22) Otherwise, and between pressed as

will be set to zero. Next, the interaction obtained by the P-FFT method can be ex-

(23) By treating the near-zone and far-zone interactions separately as shown above, the precorrected-FFT algorithm achieves its memory saving and CPU reduction. For the memory, the nearzone interactions are represented by a very sparse matrix, restorage; the far-zone interactions are represented quiring . For the CPU by a Toeplitz matrix, with the storage also of time, the computation of the near-zone interactions cost CPU time while the far-zone interactions cost CPU time because the cost of the FFTs dominates the total cost in the computation of the far-field interactions. Therefore, the precorrected-FFT is a and algorithm when implemented on the VIE, in contrast to and when implemented on the SIE. III. NUMERICAL RESULTS

(19) where is the position vector of the centroid of the tetrahedron , is the same as in (16). This interpolation operator depends only on the relative position between the sampling points and the testing points, and can be used for either the scalar or vector potential. With it, we can obtain the scalar and vector

In this section, some numerical results are presented to validate the implementation and demonstrate the efficiency of the method. All results are obtained on a Pentium 2.4 G PC with 1 G memory. In the first example, we consider a rectangular dielectric box with a relative dielectric constant of . The volume is discretized into 18 000 tetrahedrons, leading to 37 720 unknowns. A plane wave is incident along the


Fig. 1.

Bistatic RCS of a 5

2 1 2 0:6

317

Fig. 3.

Bistatic RCS of a layered sphere. a =

Fig. 4.

Geometry of a five-period dielectric slab.

=3, a = 2 =3, a =

.

dielectric rectangular box.

Fig. 5. Bistatic RCS of a five-period dielectric slab with k Fig. 2. Geometry of a layered sphere.

axis from the top. The bistatic RCS calculated by the precorrected-FFT method in conjunction with the VIE and SIE respectively are shown in Fig. 1, and compared with the AIM solution of the SIE [13]. Good agreements are observed for both polarizations, validating the present method. The second example is a layered spherical scatterer as shown in Fig. 2. The radius of the three layers from inside to outside are respectively , , and . The corresponding permittivity are , , and , respectively. The bistatic RCS obtained from the present method is compared with the Mie series solution [2] in Fig. 3. Again, good agreement is observed. In the computation,

h = 9.

the object is divided into 46 656 tetrahedrons, yielding a total of 94 608 unknowns. In the third example, a five-period dielectric slab shown in Fig. 4 is considered. The parameters of the slab are , , , , , , where is the wavenumber in free space. The calculated bistatic RCS under a -polarized incident plane wave from the direction of ( , ) is shown in Fig. 5. For comparison, the CG-FFT result in [3] is given in the figure. There are 100 800 tetrahedrons used to model the slab and the number of unknowns is up to 206 200. In the precorrected-FFT approach, we set the grid order and the near field threshold . Thus, the number of nonezero entries in the pre-correction matrix (corresponding

318

Fig. 6. Monostatic RCS of a 3:5 box.


2 2:0 2 0:25

dielectric rectangular

Fig. 7. Comparison of iteration numbers required by the GMRES solution of the VIE and SIE.

to the near-zone interactions need to be computed directly) is , about 0.237% that of . The total memory required by the present method is only about 0.311% that of the conventional MoM, enables considerably large problems to be computed on a normal PC. To reach a normalized residual of , the number of iterations required by the current GMRES is 98. The present method takes about 38 hours to obtain the final solution, whereas conventional MoM is estimated to require 990 hours, even if a computer with enough large memory is available. Finally, we investigate the convergence property of the VIE. Consider a rectangular dielectric box with the size of and the dielectric constant of . The volume is divided into 18 144 tetrahedrons, yielding 38,700 unknowns. We compute the monostatic RCS using the precorrected-FFT accelerated VIE and SIE approach respectively. (It should be noted that the application of the precorrected-FFT approach does not change the convergence rate of the original MoM system.) Since the induced current distributions change a little for adjacent incident angles, the current solution from the previous angle with phase correction is used as the initial guess for the next angle. The calculated results for different po-

larizations are shown in Fig. 6. The iteration numbers to achieve residual for every incident angle is shown in Fig. 7. It is observed that the number of iterations required by the VIE is only a quarter that of the SIE when the same GMRES solver is used for both VIE and SIE. IV. CONCLUSION The precorrected-FFT method has been applied to solve the VIE for the scattering from dielectric objects with arbitrary shape and inhomogeneity. The application of the P-FFT significantly reduces the memory requirement and computational complexity of the conventional MoM. The CPU time of this method is and the memory requirement is . Owing to the use of irregular meshes, this technique offers good flexibility to model arbitrarily shaped structures while maintaining the efficiency of the FFTs. In addition, more flexible discretization elements, such as curvilinear tetrahedrons, can also be used without complex modifications. This method is an attractive choice for large-scale scattering problems, particularly superior when handling inhomogeneous dielectric scatterers.


REFERENCES [1] 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. 34, no. 5, pp. 635–640, May 1986. [2] H. Gan and W. C. Chew, “A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems,” J. Electromagn. Waves Appl., vol. 9, no. 10, pp. 1339–1357, 1995. [3] H. Y. Yao, X. Q. Sheng, K. N. Yuang, and Z. P. Nie, “Analysis of scattering by finite periodic dielectric structures using TFQMR-FFT,” Microwave Opt. Technol. Lett., vol. 34, pp. 438–442, 2002. [4] C. C. Lu, “A fast algorithm based on volume integral equation for analysis of arbitrarily shaped dielectric radomes,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 606–612, Sep. 2003. [5] 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 and Millimeter Wave Technology, 2002, pp. 401–404. [6] Z. Q. Zhang and Q. H. Liu, “A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 102–105, Jul. 2002. [7] J. K. White, J. R. Phillips, and T. Korsmeyer, “Comparing precorrected-FFT method and fast multipole algorithms for solving three-dimensional potential integral equations,” in Proc. Colorado Conf. Iterative Methods, Apr. 5–9, 1994. [8] J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. ComputerAided Design of Integr. Circuits and Syst., vol. 16, pp. 1059–1072, Oct. 1997. [9] X. C. Nie, L. W. Li, N. Yuan, and T. S. Yeo, “Precorrected-FFT algorithm for solving combined field integral equations in electromagnetic scattering,” J. Electromagn. Waves Appl., vol. 16, no. 8, pp. 1171–1187, 2002. [10] 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, Jul. 2003. [11] N. Yuan, T. S. Yeo, X. C. Nie, and L. W. Li, “A fast analysis of scattering and radiation of large microstrip antenna arrays,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2218–2226, Sep. 2003. [12] 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. 32, no. 1, pp. 77–85, Jan. 1984. [13] E. Topsakal, M. Carr, J. Volakis, and M. Bleszynski, “Galerkin operators in adaptive integral method implementations,” Proc. Inst. Elect. Microw. Antennas Propag., vol. 148, no. 2, pp. 79–84, Apr. 2001. [14] 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.

Xiao-Chun Nie received the B.Eng. degree from Xi’an Jiaotong University, Xi’an, China, the M.Eng. degree from the University of Electronic Science and Technology of China, Chengdu, and the Ph.D. degree from the Xidian University, Xidian, China, in 1988, 1993, and 2000, respectively, all in electrical engineering. From 1993 to 1997, he was a Lecturer in the University of Electronic Science and Technology of China. In September 2000, he worked as a Research Fellow at Singapore-MIT Alliance, National University of Singapore. In September 2002, he joined Temasek Laboratories of 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.

319

Le-Wei Li (S’91–M’92–SM’96) received the B.Sc. degree in physics from Xuzhou Normal University, Xuzhou, China, in 1984, the M.Eng.Sc.degree in electrical engineering from the China Research Institute of Radiowave Propagation (CRIRP), Xinxiang, China, in 1987, and the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. In 1992, he worked at La Trobe University (jointly with Monash University), Melbourne, Australia, 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 Director of the NUS Centre for Microwave and RF (CMRF) and a CMRF Chair Professor of Electromagnetics. Since 1999, he has also been with High Performance Computations on Engineered Systems (HPCES) Programme of the Singapore-MIT Alliance (SMA) where he is an SMA Fellow. He is a Member of The Electromagnetics Academy based at MIT. He has coauthored Spheroidal Wave Functions in Electromagnetic Theory by (New York: Wiley, 2001), 42 book chapters, over 210 international refereed journal papers, 25 regional refereed journal papers, and over 230 international conference papers. He is an Editor of the Journal of Electromagnetic Waves and Applications, an Associate Editor of Radio Science, and an Editorial Board Member of the Chinese Journal of Radio Science. His current research interests include electromagnetic theory, radio wave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. Dr. Li was a recipient of 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 & 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. He is an Editorial Board Member of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He is currently Chairman of IEEE Singapore Section MTT/AP Joint Chapter.

Ning Yuan received the B.Eng. and M.Eng. degrees in electrical engineering from the University of Electronic Science and Technology of China, Chengdu, in 1993 and 1996, respectively, and the Ph.D. degree in electrical engineering from Xidian University, Xidian, China, in 1999. From September 1999 to July 2000, she worked as 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 at Temasek Laboratories, National University of Singapore. Her main research interests are in computational electromagnetics.

320


Tat Soon Yeo (M’80–SM’93–F’02) received the B.Eng. (Hons I) and M.Eng. degrees from the National University of Singapore, Singapore, in 1979 and 1981, respectively, and the Ph.D. degree from the University of Canterbury, New Zealand, in 1985, under a Colombo Plan Scholarship. He is currently a Professor of the Electrical and Computer Engineering Department and a Vice Dean of the Faculty of Engineering at the National University of Singapore. He is also concurrently the Director of the Radar and Signal Processing Laboratory and the Antennas and Propagation Laboratory, Department of Electrical Engineering, National University of Singapore (NUS). He is also the Director of Temasek Defence Systems Institute, a teaching institute established jointly by NUS and the U.S. Naval Postgraduate School (NPS). His current research interests are scattering analysis, synthetic aperture radar, antenna and propagation study, numerical methods in electromagnetics, and electromagnetic compatibility. Dr. Yeo was the recipient of the 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 the Chairman of Singapore EMC Technical Committee.

Yeow-Beng Gan (M’90–SM’01) received the M.Eng. and B.Eng. (Hons) degrees in electrical engineering from the National University of Singapore, in 1994 and 1989, respectively. In 1989, he joined the DSO National Laboratories (formerly the Defence Science Organization), Singapore, where he was primarily responsible for the buildup of technical capabilities in the analysis and design of antennas, and in 1998, he became a Principal Member of the Technical Staff. In May 2001, he joined the 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 (MRS), Singapore.