DURING the past decade, several fast algorithms for efficiently. Time Domain Adaptive Integral Method for Surface Integral Equations

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1 2692 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 Time Domain Adaptive Integral Method for Surface Integral Equations Ali E. Yılmaz, Student Member, IEEE, Jian-Ming Jin, Fellow, IEEE, and Eric Michielssen, Fellow, IEEE Abstract An efficient marching-on-in-time (MOT) scheme is presented for solving electric, magnetic, and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium. The proposed scheme is the extension of the frequency-domain adaptive integral/pre-corrected fast-fourier transform (FFT) method to the time domain. Fields on the scatterer that are produced by space-time sources residing on its surface are computed: 1) by locally projecting, for each time step, all sources onto a uniform auxiliary grid that encases the scatterer; 2) by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global, multilevel/blocked, space-time FFTs; 3) by locally interpolating these fields back onto the scatterer surface. As this procedure is inaccurate when source and observer points reside close to each other; and 4) near fields are computed classically, albeit (pre-)corrected, for errors introduced through the use of global FFTs. The proposed scheme has a computational complexity and memory requirement of 2 ( log ) and 3 2 ( ) when applied to quasiplanar structures, and of ( log ) and 2 ( ) when used to analyze scattering from general surfaces. Here, and denote the number of spatial and temporal degrees of freedom of the surface current density. These computational cost and memory requirements are contrasted to those of classical MOT solvers, which scale as 2 ( ) and 2 ( ), respectively. A parallel implementation of the scheme on a distributed-memory computer cluster that uses the message-passing interface is described. Simulation results demonstrate the accuracy, efficiency, and the parallel performance of the implementation. Index Terms Electromagnetic scattering, fast solvers, integral equations, parallel processing, time domain analysis. I. INTRODUCTION DURING the past decade, several fast algorithms for efficiently solving integral equations [1], [2] pertinent to the analysis of electromagnetic scattering from perfect electrically conducting (PEC) surfaces have been proposed [3] [8]. For example, a host of fast multipole [3] [5] and fast-fourier transform (FFT) based accelerators [6] [12] have been shown to increase by orders of magnitude the number of unknowns that classical frequency-domain integral equation solvers can cope with. Likewise, the recently developed plane wave time domain (PWTD) scheme [13], viz. time-domain fast multipole method, Manuscript received May 9, This work was supported in part by ARO program DAAD , by the DARPA VET Program under AFOSR Contract F , and in part by the MURI Grant F Analysis and design of ultrawide-band and high power microwave pulse interactions with electronic circuits and systems. The authors are with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana Champaign, Urbana IL USA ( ayilmaz@uiuc.edu). Digital Object Identifier /TAP permits the analysis of transient scattering problems that cannot be tackled using classical time-domain integral equation solvers [14], [15]. To date, however, FFT-based schemes for analyzing transient scattering from arbitrarily shaped surfaces have not been presented in the archival literature. (Various conference papers are available on the subject [16] [19].) Hence, this paper completes the matrix of fast integral equation methods for analyzing surface scattering phenomena by describing an FFT-based time-domain integral equation solver. Specifically, this study extends the frequency domain adaptive integral [7]/pre-corrected FFT [8] method to the time domain. FFT-based schemes for solving frequency-domain integral equations, such as the conjugate gradient [6] and adaptive integral/pre-corrected FFT methods (CG-FFT, AIM), provide an appealing avenue for tackling a broad class of surface scattering problems. The classical, method of moments (MOM) based, iterative solution of such problems requires CPU cycles per matrix-vector multiplication and bytes of memory; here, and in what follows, denotes the number of spatial basis functions that discretize the scatterer current density. Both the CG-FFT and AIM algorithms reduce the computational complexity of a matrix-vector multiplication by employing uniform spatial grids that facilitate the use of (spatial) FFTs for rapidly convolving source distributions with Green s functions. To this end, the CG-FFT method assumes a uniform scatterer discretization, whereas the AIM introduces a uniform auxiliary grid that does not constrain the primary surface discretization. The AIM reduces the above matrix-vector multiplication cost to operations and the associated memory requirement to bytes, where is the number of nodes of the auxiliary grid. For general three-dimensional (3-D) surfaces devoid of geometric details, ; for quasiplanar surfaces whose transverse dimensions are much larger than their (fixed) height,. The computational complexity and memory requirement of AIM-based solvers generally scale less favorably than those of multilevel fast multipole driven ones. Nonetheless, the AIM scheme often remains a viable alternative, especially when the structure under study is relatively small, quasiplanar, or densely packed. The AIM also has the advantage of being immediately applicable to all problems involving Green s functions that exhibit translational, rotational, or reflection invariance [3]. Recently proposed higher order methods [10], [11], which employ auxiliary uniform grids on facets to reduce, and effective parallelization strategies based on either domain decomposition [7] or parallel spatial FFTs [9] further improve the viability of AIM-based solvers for large-scale time-harmonic analysis X/04$ IEEE

2 YILMAZ et al.: TD-AIM FOR SURFACE INTEGRAL EQUATIONS 2693 FFT-based schemes for solving time-domain integral equations, in contrast, are in their infancy. The classical, marching-on-in-time (MOT) based solution of time-domain surface integral equations requires operations and memory, where is the number of time steps in the analysis. Recently, various extensions of the CG-FFT scheme to the time domain have been proposed [20] [23]. Similar to the CG-FFT method, these algorithms assume a uniform discretization of the scatterer surface and therefore are of limited use. This paper presents the time-domain counterpart of the AIM; predecessors of the present scheme and related methods were outlined in [16] [19], [24], [25]. The proposed time-domain AIM (TD-AIM) scheme has CPU and memory requirements of and, respectively. Here, denotes the maximum number of time steps for a pulsed field to travel across the scatterer; for surface scatterers, is generally proportional to [23]. Just as the frequency-domain AIM is asymptotically less efficient than the multilevel fast multipole method, the proposed TD-AIM is asymptotically inferior to multilevel PWTD schemes. Yet, it is relatively easy to implement, and efficient enough to remain competitive with the PWTD scheme for many practical problems. Furthermore, as the scheme relies solely on the translational invariance of the Green s function to achieve computational savings, it is applicable with little modification to problems involving lossy [25], layered [26], or dispersive media [16]. Such problems are, however, not treated herein; rather, this paper provides a detailed description of the TD-AIM scheme for analyzing free-space scattering problems using time-domain combined field integral equations. In addition, the paper presents a derivation of the scheme s computational complexity and proposes avenues for minimizing its CPU and memory costs. It also elucidates a parallel implementation of TD-AIM scheme and its application to the analysis of large-scale transient scattering phenomena. This paper is organized as follows. Section II formulates several time-domain surface integral equations, classical MOT-based schemes for solving them, and the proposed TD-AIM scheme. Section III demonstrates the efficiency, accuracy, and parallel performance of the algorithm, which is followed by our conclusions. II. FORMULATION This section expounds the proposed TD-AIM scheme. First, time-domain electric, magnetic, and combined field integral equations for analyzing electromagnetic scattering from PEC structures residing in an unbounded homogenous medium are presented and their classical solution by MOT is reviewed. Next, the TD-AIM scheme is introduced and its theoretical computational complexity and memory requirements are explored. Finally, a parallel implementation of the scheme is described. A. Time-Domain Integral Equations and MOT Let denote the surface of a closed PEC body that resides in an unbounded, lossless, homogenous dielectric medium with permittivity and permeability. A transient electromagnetic field is incident on ; it is assumed that this field is (approximately) temporally bandlimited to and that it vanishes for. A surface current is induced on that in turn generates the scattered field where represents the time derivative and and are the vector and scalar potentials Here, is the surface current density, is the distance between source point and observation point, and is the speed of light. Enforcing the temporal derivatives of the fundamental boundary conditions on the electric and magnetic fields tangential to yields time-domain electric and magnetic field integral equations (EFIE and MFIE). The combined-field integral equation (CFIE) is a linear combination of the EFIE and the MFIE [27], [28]. The time-domain EFIE, MFIE, and CFIE read Here, is an outward directed unit vector normal to and is the intrinsic impedance of the medium. The discussions hereafter will focus on the CFIE; equations pertaining to the EFIE and the MFIE can be derived from those for the CFIE by setting and, respectively. The EFIE is valid for open surfaces as well, whereas the MFIE and the CFIE are pertinent to closed surfaces. To numerically solve the CFIE, space-time basis functions as (1) (2) (3) is discretized using Here, is an unknown expansion coefficient and is the time-step size with typically.in what follows, the spatial and temporal basis functions and are assumed local. For example, the and can be Rao Wilton Glisson (RWG) functions [29] or their higher order extensions [30] and shifted Lagrange interpolants [31], respectively. Furthermore, it is assumed that (4)

3 2694 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 for and, i.e., that the temporal basis functions are (discretely) causal and of duration. Finally, spatial discretization lengths on are assumed to be of ; this assumption is valid whenever is electrically large and devoid of small geometric details. Upon substituting (4) into the CFIE and testing the resulting equation with the spatial functions at times, equations for the expansion coefficients result The entries of the vectors and and those of the matrices are The integer in (5) is defined such that approximates the longest possible transit time of the field produced by a temporal basis function across : where is the maximum distance between any two points on and rounds to the smaller integer. It follows that only the matrices through are nonzero because the basis functions are local and the Green s function implicit in (8) is impulsive in time [25]. That is, the fields radiated by currents on various parts of the scatterer at a given time step interact with the scatterer for at most time steps. Once the spatial and temporal basis functions and the time-step size that support the frequency content of the incident field are determined, the unknown coefficients in (5) can be found recursively. First, at time is found; this, in turn, permits computation of. This vector is then added to the tested incident field and (5) is solved for the current at time. At the next time step, and are used to compute, which together with permits the computation of, and so forth. This recursive solution algorithm is known as the MOT scheme. (5) (6) (7) (8) The dominant cost of the MOT method is the computation of the vectors, which involves a space-time convolution of the currents with the Green s function for each time-step. Indeed, under the above assumptions, for, that is, as soon as the retarded fields from every part of the scatterer reach all the other parts, the computation of a single requires operations. Compared to the cost of computing, the cost of iteratively solving (5) typically is insignificant because contains only nonzero elements, i.e., it is highly sparse, and the iterative solution typically converges in very few ( 15) iterations. Thus, the goal of the proposed TD-AIM scheme is to accelerate the computation of the fields on the right-hand side of (5), the classical evaluation of which requires operations for all time steps. B. TD-AIM The space-time translational invariance of the Green s function inherent in (8) can be exploited to rapidly evaluate for. The scheme proposed here is a marriage between the fast time- and frequency-domain integral equation solvers in [25] and [7]. In [25], a fast scheme for evaluating the temporal convolution of the impedance matrices with space-time currents was introduced assuming that was a uniformly meshed plate. Temporal convolutions were evaluated via blocked (multilevel) FFTs while spatial convolutions could be evaluated by a CG-FFT like scheme because of the uniform surface mesh used. In [7], a fast scheme for evaluating fields generated by time-harmonic sources residing on an arbitrarily shaped and meshed surface was described. The use of FFTs to evaluate these fields was facilitated by introducing an auxiliary grid. Here, the extension of the scheme in [25] to nonuniformly meshed surfaces using ideas similar to [7] is elucidated. The proposed TD-AIM scheme embeds within an auxiliary 3-D Cartesian grid with nodes. Node separation along the three Cartesian directions are assumed identical and equal to for ease of presentation; the effects of node separation on accuracy are investigated in Section III but is always of. The proposed TD-AIM scheme computes in four stages: Stage 1: At each time step, all primary sources on are locally projected onto the auxiliary grid. The auxiliary sources that are associated with a given primary source are chosen to reproduce the transient fields of the primary source with high accuracy, at least for observation points that are separated by more than a prespecified distance from the primary source. Stage II: Future transient fields produced by these auxiliary sources are computed at all the nodes on the auxiliary grid; this is achieved via global, blocked, space-time FFTs. Stage III: The fields at the next time step are locally interpolated from the auxiliary grid onto the primary surface mesh. Stage IV: Near fields due to all primary sources on are computed wherever needed and (pre-) corrected for errors induced through the use of global FFTs in Stage II. Specifically, the proposed scheme approximates each of the impedance matrices through as, where and account for Stages I III and Stage IV

4 YILMAZ et al.: TD-AIM FOR SURFACE INTEGRAL EQUATIONS 2695 can be ex- of the algorithm, respectively. The matrices pressed as where and are sparse projection matrices, represents transpose, and the matrices propagate onto all of the auxiliary-grid nodes (temporal derivatives of) magnetic fields, magnetic vector potentials, and electric scalar potentials due to sources on the auxiliary grid. In (9), the curl operator of (8) is transferred onto neither the spatial basis functions [32] nor the spatial testing functions [7]; rather, is left on the Green functions to facilitate its numerical computation as described in Section II-D and [12]. The various entities in (9) are discussed in more detail below. The projection matrices are computed by first choosing points on the auxiliary grid that each function is projected onto, denoted by the set. These points can be chosen to reside inside or on the surface of the smallest cube that snaps to the auxiliary grid and that encloses. Then, for each, the corresponding entries of the seven projection matrices and are found by matching, in the least square sense, the multipole moments of the point sources to those of the functions,,,,,, and, respectively. The time-domain accuracy of this moment-matching scheme, which is the same as that used for the frequency-domain AIM scheme [7], is examined in Section III. Other matching criteria can be used to fill and [17]. An advantage of moment matching is that the resulting projection operation is instantaneous/dispersion-free and accurate enough for most applications. Other, more accurate procedures exhibit dispersion, which renders the projection step itself a convolution [17]. The are sparse propagation matrices of size whose entries are (9) (10) In (10), denotes node on the auxiliary grid and is the time signature of the auxiliary sources (typically, although not necessarily, ). Note that, to avoid singularities,. The efficient multiplication of these matrices by past current coefficients via blocked FFTs is discussed in Section II-C. The point sources on the auxiliary grid can accurately reproduce the transient fields of only for observers separated from the source by several grid lengths. Their erroneous contributions to the fields at nearby observers are corrected with the help of matrices. The matrices are sparse approximations to and are defined as if if (11) where, is the minimum grid-distance between the auxiliary points representing the functions and [7], i.e. (12) and is a threshold distance beyond which represents the scattered fields with the desired accuracy; in practice,. In general, the larger is the more costly and the more accurate the TD-AIM scheme becomes. C. Multilevel/Blocked Space-Time FFTs Stage II of the above TD-AIM scheme requires the multiplication of the matrices by vectors containing past coefficients of auxiliary sources. The scheme exploits the spatial Toeplitz structure of these matrices as well as that of their temporal arrangement implicit in (5) to accelerate these multiplications via blocked 4-D FFTs. In what follows, the mechanics of this operation are demonstrated; for the sake of brevity, only scalar potential contributions are considered. To our knowledge, the use of space-time FFTs for solving time-domain integral equations was first proposed in the context of a global solution scheme, i.e., one that does not march in time but instead solves for all space-time unknowns simultaneously [20]. Nevertheless, space-time FFTs can also be used while marching in time by employing the multilevel/blocking framework of [23] or [25]; the latter was originally proposed in [33]. In this paper, the scheme of [25] is adopted because of its simplicity. Fig. 1 illustrates the mechanics of computing the scalar potential on the auxiliary grid for the first 16 time steps. Denote the matrix in Fig. 1 by ; for future reference, denote the corresponding magnetic field and vector potential matrices by and, respectively. To compute, for example at time step eight, the scalar potentials generated by on, the block matrix comprising the leftmost seven matrices on row eight of, i.e.,, has to be multiplied by the vector. Note that (and its extension to all time steps) is block-toeplitz on four levels; the temporal Toeplitz structure is immediately visible in Fig. 1 and the 3-D spatial Toeplitz structure of each of the matrices follows trivially from their construction. Therefore, can be rapidly multiplied by a vector using 4-D FFTs. When operating within an MOT framework, however, the current vectors are unknown at time step

5 2696 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 It is important to note that the largest nonzero block aggregate in contains only block rows and columns. That is, Fig. 1 should not be interpreted to imply that the size of the largest block aggregate in scales as ; indeed, the impulsive nature of the Green s function renders the very large block aggregates zero. This implies that the number of different square boxes is only, with the largest one of size. Thus, the largest temporal FFT size is proportional not to the duration of the analysis but to the maximum transit time across the scatterer. D. Computational Cost and Complexity/Comparison to Other Time Domain Methods Fig. 1. Computation of the scalar potential contribution in (9). At each time step l, rowl of the matrix-vector multiplication has to be computed. Rather than computing the multiplication row-by-row as in the MOT method, one of the blocked aggregates indicated by squares is multiplied at each time step. Causality is not violated: only past currents are used to find the present and future fields. seven; moreover, multiplication of the entire by the vector (with filled with zeros) to compute the scattered fields at time step eight is clearly inefficient (as a similar procedure would have to be repeated each and every time step). Nevertheless, there exist various space-time submatrices of that are themselves block-toeplitz on four levels and that can be multiplied by current coefficients without violating causality. That is, only currents that are already known are required to multiply these submatrices when marching in time. These submatrices are the aggregates of the individual matrices indicated by the square boxes superimposed on in Fig. 1 and are further referred to as block aggregates. Stage II of the TD-AIM scheme computes future scattered fields by multiplying these block aggregates with current vectors. The cost of multiplying the block aggregates varies from time step to time step. For example, at time step nine, the scheme calls for the multiplication of a block aggregate of size, whereas at time step ten only a matrix of size is multiplied. Furthermore, larger block aggregates are multiplied less frequently. For example, the block aggregate in is multiplied only once for every eight time steps, whereas the block is multiplied once for every two time steps (Fig. 1). Using this scheme, future scattered fields are constructed partially at each time step, and the complete scattered field at a given time step is available only at that time step. That is, at each time step, temporal chunks of past auxiliary current coefficients are transformed to the spectral-frequency domain, multiplied with (the 4-D FFTs of) the corresponding block aggregate in Fig. 1, and then inverse transformed to produce temporal chunks of future scattered fields. Stages I, III, and IV of the algorithm require operations because each of the seven projection matrices and as well as each near-field matrix contains nonzero elements, and because matrices are zero for, with. The sparsity of the projection matrices follows directly from the fact that each basis function is represented by a fixed number of nodes on the auxiliary grid where is independent of. That each near-field correction matrix contains no more than nonzero entries and that only a fixed and small number of the matrices are nonzero follows from the following facts: 1) is independent of ; 2) the temporal functions are localized, with on the order of the discretization length on ; and 3) the free-space Green s function is impulsive in nature. Therefore, each basis function communicates with at most of its neighbors through the near-field correction matrices. Because the projection and near-field matrices are so sparse, the computational cost of the algorithm is dominated by that of Stage II, i.e., the multiplication of the matrices by vectors of past coefficients of auxiliary sources. To estimate the computational cost of Stage II, recall that there are different nonzero block aggregates in. The th block aggregate,, is of size and can be multiplied in operations using space-time FFTs by storing the currents and fields in the spectral domain ( space). Notice that the above complexity estimate is significantly lower than that resulting from a straightforward application of a 4-D FFT, which would require operations. The complexity is reduced by observing that the spatial FFTs need not be repeatedly computed (This observation is also important for parallel efficiency as discussed in Section II-E). For example, at time step nine, spatial FFTs of auxiliary current coefficients at time steps one through eight are needed, but those of steps one through seven are already computed at previous time steps. Hence, the number of spatial FFTs can be minimized by precomputing the 3-D forward spatial FFTs of each matrix, by computing 3-D forward spatial FFTs of the currents only at the previous time step, and by computing 3-D inverse spatial FFTs of the fields only at the next time step. Because the th block aggregate is multiplied once every time steps, the per-time step cost of multiplication is

6 YILMAZ et al.: TD-AIM FOR SURFACE INTEGRAL EQUATIONS 2697 is. Thus, the total computational complexity of Stage II (13) For quasiplanar surfaces,, while for general 3-D surface scatterers. Therefore, because, the total number of operations of TD-AIM scales as for quasiplanar surfaces and as for general surfaces. The memory requirement of the TD-AIM scheme is dictated by the largest block to be FFTed, which requires bytes of storage space, i.e., bytes for quasiplanar surfaces and bytes for general 3-D surfaces. Next, to quantify the performance of TD-AIM for the three surface integral equations in (3), the number of spatial FFTs required for solving each one is counted. By computing and storing the spatial FFTs of the propagation matrices prior to time marching, a total of forward and inverse spatial FFTs are computed for the EFIE. Similarly, forward and inverse spatial FFTs are required for the MFIE. A straightforward combination for the CFIE would thus require forward and inverse spatial FFTs. In our implementation, the number of spatial FFTs required for the CFIE is reduced to forward and inverse spatial FFTs using an approach similar to that in [12]. The CFIE cost is reduced by observing that both the EFIE and the MFIE kernels are formulated in terms of the vector potential, and that each Cartesian component of the temporal derivatives of are available on the uniform grid at the end of Stage II. Consequently, the MFIE contributions can be obtained from the EFIE computations without any additional FFTs, i.e., space-time samples of are computed from the space-time samples of using a higher order finite difference approach and numerical time integration. (The time integration translates to a trivial complex scalar multiplication in the time-harmonic formulations). The time integral is evaluated efficiently in the time-marching context by storing the value of the integral at each auxiliary point at the previous time step and adding to it at each time step. Hence, the CFIE computation requires the same number of spatial (and temporal) FFTs as that of the EFIE and a total of extra operations and extra storage. A comparison with other time-domain schemes is in order. Irrespective of the nature of the scatterer, the CPU cost and memory requirements of classical MOT solvers scale as and, respectively. Therefore, the CPU and memory costs of the TD-AIM scheme always scale better than those of classical solvers, with the exception of the TD-AIM memory cost for 3-D surfaces, which scales only on par with that of classical MOT solvers. In [21], [22] algorithms that ignore the temporal Toeplitz structure of (5) and employ only spatial FFTs were considered. The computational complexity of those schemes when applied to Stage II scales as, i.e., as for quasiplanar surfaces and as for general surfaces (under the above delineated conditions). Therefore, the CPU cost of the present algorithm always scales better than time-domain schemes that use only spatial FFTs. The memory requirements of those schemes, however, are of the same order as those of the present algorithm. Finally, the CPU and memory requirements of general purpose, multilevel PWTD schemes scale as and, respectively [13]. Therefore, when compared to multilevel PWTD acceleration, the TD-AIM scheme asymptotically scales worse for general 3-D surfaces and has the same computational cost for quasiplanar structures. Nonetheless, in our experience, TD-AIM remains competitive with multilevel PWTD for 3-D surfaces up to tens of thousands of spatial unknowns, and generally outperforms it for quasiplanar structures. E. Parallelization The practical bottleneck of the above described TD-AIM algorithm is its memory requirement. Compared to its frequency-domain counterpart, the TD-AIM scheme requires times more memory because it needs to store the temporal history of sources to compute retarded fields throughout the auxiliary mesh. To moderate the impact of this memory bottleneck, this section presents a TD-AIM parallelization strategy targeting distributed memory computers that communicate via message passing. While the TD-AIM scheme s memory apetite serves as the main motivation behind the parallelization effort, parallel efficiency is not ignored and load balancing and scalability issues are also addressed. The following approach to parallelization is based on the balanced distribution of the uniform grid and the FFT operations related to it among the available processors. Because zeropadding (doubling) is required to multiply a (block-)toeplitz matrix via FFTs, all 4-D FFTs are done on arrays of size at most (the temporal array size varies from time step to time step), which are either derived from the blocked aggregates in or are the zero-padded currents and fields on the auxiliary grid. To reduce the memory demand per processor, the arrays are distributed among the processors by 1-D slab-decomposition: each processor stores spatial entries over time as depicted in Fig. 2. Because now only part of the auxiliary grid is available to each processor, all four stages of the algorithm should be modified accordingly as illustrated in Figs. 3 and 4 and described below. Stage I: (Fig. 3): At time step, each processor projects the current density onto its section of the auxiliary grid. Because approximately half of the processors do the projection and because the primary mesh is not uniform, the distribution of this effort among processors is unbalanced. Nonetheless, this does not significantly affect the parallel efficiency because of the minute amount of CPU resources required to compute local projections as compared to FFTs. Stage II: [Fig. 4(a) (c)]: The space-time FFTs are computed in parallel using the FFTW library [34]. Specifically, each processor computes 2-D FFTs in the and dimensions of its section of the zero-padded current-arrays. Next, all arrays are transposed, which requires an all-to-all processor communication. Finally, each processor computes FFTs in the dimension

7 2698 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 Fig. 2. One dimensional slab-decomposition for P =6. The auxiliary mesh is of size N 2 N 2 N (z dimension is not shown) and is distributed by dividing the x dimension into slabs. Due to doubling, each processor stores N =3 2 2N 2 2N spatial entries over time. Fig. 3. Stage I of the algorithm: parallel projection onto the uniform mesh. Here, P =6, N =18, and N =12(z dimension is not shown). Only the part of the uniform mesh that is projected to by the active processors is shown. Three RWG functions and the related nodes are highlighted. Each processor projects all the RWG functions that affect its section of the mesh. to complete the spatial FFTs [9], [24], [34]. Hence, the auxiliary current coefficients at time is transformed to the spectral domain ( space). These spectral-domain arrays are stored to minimize repeated computations and communication. Then, the processors concurrently compute temporal FFTs at each of their spatial nodes according to the blocking scheme of Section II-C to transform the current to the spectral-frequency domain. Next, each processor multiplies its part of the transformed currents and block aggregates to obtain a part of fields in the spectral-frequency domain. Once these fields are found, the above steps are reversed: Each processor computes inverse temporal FFTs to find temporal chunks of future fields in the spectral domain ( space), and these partial future fields are added to those obtained from previous time steps at each one of the spatial points. Finally, the inverse spatial FFTs of the fields only at the next time step are computed to obtain the complete field in the space-time domain at time Fig. 4. Stage II of the algorithm: parallel space-time FFTs. Here, P =6, N =18, and N =12(z dimension is not shown). (a) Each processor computes six FFTs of size 24 in the y dimension. This is followed by an all-to-all communication step that transposes the arrays. (b) Each processor then computes four FFTs of size 36 in the x dimension for the transposed arrays. (c) Temporal FFTs, the size of which depends on the time step, are computed. Here, each processor computes 144 FFTs of size 4.. As described here, the computational cost of Stage II is and its per-processor memory

8 YILMAZ et al.: TD-AIM FOR SURFACE INTEGRAL EQUATIONS 2699 Fig. 5. RWG functions and some of the auxiliary points used. requirement is. Because there are only two global communications for transposing the arrays at each time step, the total communication cost is. The work in this stage is equally distributed if and are integers. Stage III: Similar to Stage I, each processor interpolates the fields on its section of the auxiliary grid to the surface mesh, that is, the processors consider all the unknowns on the primary mesh that are affected by their sections of the arrays. Stage IV: The near-field interactions are corrected in parallel. Let denote the number of nonzero elements of the matrices, i.e., the number of near-field corrections. To multiply the matrices in parallel, the nonzero elements are distributed among the processors using a column-based decomposition, i.e., each processor stores and multiplies approximately matrix elements plus or minus one column of the matrices. Hence, each processor stores a short history ( time steps) of parts of the primary currents that correspond to its columns of. Once all four stages of the TD-AIM algorithm are completed at a given time step and is formed, (5) is solved in parallel for the current density at the next time step. Because matrix in general has nonzero elements, the communication cost dominates the parallel solution time, which does not exhibit good scalability, and hampers the overall parallel efficiency [35]. The solution time becomes subdominant, however, as the problem becomes electrically large, i.e., as,, and increase. III. NUMERICAL RESULTS This section presents several numerical examples that demonstrate the operation of the proposed TD-AIM scheme. First, the accuracy of the moment-matching approach is investigated in detail. Then, the computational complexity, memory requirement, and parallel efficiency of the TD-AIM scheme are examined by simulating two types of scatterers that represent the best- and worst-case scenarios for the algorithm. Finally, various broadband scattering problems are solved by TD-AIM and the results are validated by comparing the radar cross section (RCS) patterns to analytical solutions and to those generated by frequency-domain integral equation solvers over a range of frequencies. In all simulations, the temporal basis functions are fourth-order Lagrange interpolants with, and the MFIE contributions, when needed, are computed using the numerical scheme presented in Section II-D, with fifth-order accurate time integration and fourth-order accurate central differencing scheme. All the results in this section are obtained using a cluster of 1-GHz Pentium III processors with 1 GB of memory. A. Moment-Matching Accuracy To investigate the accuracy of the moment-matching criteria for time-domain surface integral equations, the interaction between two triangular RWG functions is computed and the error incurred by the TD-AIM scheme is examined. Specifically, a bent RWG basis function that is anchored at the origin is treated as an impressed current source and its radiated fields are measured by a flat RWG function. Both RWG functions have an area of 25 and consist of two isosceles right triangles that intersect on the plane as shown in Fig. 5. The auxiliary grid is centered at the origin and. For the MFIE computations, the spatial functions are assumed to reside on a closed body which lies above the plane. The relative energy error in the tested fields is defined in (14) shown at the bottom of the page. To analyze the performance of the moment-matching scheme for a broad range of frequencies, the bent RWG function is assigned a Gaussian time-history: The current source is, where is the unit-step function and more than % of the power of the Gaussian pulse is at frequencies below. In what follows, quantifies the maximum frequency of interest and is the wavelength at this frequency. Figs. 6 and 7 present as a function of the center-to-center distance in between and and investigate the parameters involved in the TD-AIM computation. (14)

9 2700 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 Fig. 6. Relative energy error 1 for the RWG functions in Fig. 5 as a function of their center-to-center distance. The bent RWG function has a baseband Gaussian pulse history whose maximum frequency is such that (a) =50cmand (b) = 100 cm. The multipole moments are matched from zero up to order one for filled circles (using M = M =8auxiliary points) and up to order three (using M = M = 64auxiliary points) for empty circles. In Fig. 6(a),, the time-step size is, and in order to capture the significant part of the pulse for even the farthest interaction,. In Fig. 6(b),,, and. In both figures, the auxiliary-grid spacing is, i.e., in Fig. 6(a) and in Fig. 6(b). The figures indicate that when up to third-order multipole moments are matched, the error made by the moment-matching scheme falls below 1% after approximately 30 cm, which corresponds to. This is to be expected, i.e., the approximation (9) cannot be accurate for for the CFIE, for three reasons: 1) the point projection and interpolation operations use one cell to the right and one cell to the left of the RWG functions in Fig. 5; 2) the self-term in (10) is set to zero to avoid the singularity of the propagators implying that the approximation can be accurate only after one cell; and 3) the fourth-order curl operation is accurate only if Fig. 7. (a) Effect of the auxiliary-mesh spacing on 1 and (b) the effect of the time-step size on 1. The multipole moments are matched up to order three, and = 100 cm. the fields at the nodes of the two neighboring cells are accurate. These conditions imply that the minimum separation between the auxiliary points corresponding to the above source and testing functions should be and to accurately compute EFIE and MFIE terms in (9), respectively, which is reflected by the CFIE error in Fig. 6. To further quantify the effects of the auxiliary-grid spacing and the time-step size on accuracy, the excitation pulse is fixed to in Fig. 7. In Fig. 7(a), only the auxiliary-grid spacing is varied from to while the time-step size is kept constant at (, ). In contrast, in Fig. 7(b), only the time-step size is varied from (, )to (, ) while the auxiliary-grid spacing is kept constant at. Taken as a whole, Figs. 6 and 7 show that the error made by (9) decreases as either the auxiliary-grid spacing or the time-step size becomes smaller for a given pulse, or as the incident pulse has lower frequency content for a given space-time discretization.

10 YILMAZ et al.: TD-AIM FOR SURFACE INTEGRAL EQUATIONS 2701 TABLE I PARAMETERS FOR SCATTERING ANALYSIS OF PLATES TABLE II PARAMETERS FOR SCATTERING ANALYSIS OF SPHERES B. Computational Complexity Verification The theoretical computational complexity, memory requirement, and parallel efficiency of the proposed TD-AIM scheme were derived in Sections II-D and II-E. Here, the practical efficiency and accuracy of the scheme are systematically evaluated by analyzing scattering from two types of canonical surfaces, viz. plates and spheres, which are representative of general quasiplanar and 3-D surfaces. In what follows, all scatterers are centered at the origin of the coordinate system, and the plate side-lengths and sphere diameters are denoted by and, respectively. In all the simulations, the incident electromagnetic field is a polarized modulated Gaussian plane-wave propagating toward direction that is parameterized as (15) The bandwidth of the Gaussian pulse is measured by ; less than 0.003% of the power of the incident field is outside the frequency band, and the surface-discretization density, auxiliary-grid spacing, and timestep size are chosen according to. The accuracy of the TD-AIM scheme is measured with respect to a reference scheme by computing the root-mean-square RCS error shown in (16) at the bottom of the page. Here,, the VV-polarized bistatic RCS pattern of the scatterer, is computed at three characteristic frequencies:,, and, and is defined as (17) where and denote the Fourier transforms of the incident and scattered electric fields, respectively. For all the plate and sphere simulations, the incident field is the same, i.e.,,,,, and. Hence, the error measure in (16) does not include the forward scattering and emphasizes the error in back scattered (16)

11 2702 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 Fig. 8. Average wall-clock time to compute V at each time step for (a) plates and (b) spheres. Dashed lines are ideal speed-up tangents. RCS (,, which is in general smaller and more prone to numerical errors. Scattering from square plates is analyzed using the EFIE, whereas scattering from spheres is analyzed using the CFIE. The surface mesh density is the same for all scatterers (average triangular patch areas are approximately ), while the remaining TD-AIM parameters are chosen to minimize the CPU time and memory product without violating an error constraint, (Tables I and II). For the plate simulations, the reference solution is a frequency-domain MOM solver or, when MOM is not feasible, an accurate frequency-domain AIM-based solver (for which,,, and up to third-order moments are matched). For the sphere simulations, the errors are computed with respect to both frequency-domain numerical methods and analytical results. All numerical schemes use the same primary surface mesh as TD-AIM. Fig. 8(a) and (b) plot the average TD-AIM wall-clock time per time-step (i.e., the total wall-clock time spent to construct divided by ). Fig. 9(a) and (b) plot the corresponding measured peak memory requirement of the algorithm which also includes operating system overhead. Fig. 9. Observed peak storage requirement among all processors for (a) plates and (b) spheres. Dashed lines are ideal speed-up tangents. The figures show the parallel efficiency of the algorithm as the surface areas of the plates and spheres are increased 625 and 144 times, respectively, i.e., varies from 1 m to 25 m and varies from 1 m to 12 m. The scalability of the algorithm is restricted for small problems because of the limited number of auxiliary-grid points and the system memory overheads. Nonetheless, both the CPU time and the memory requirement of the algorithm show good parallel efficiency as the size of the scatterers increase. The computational complexity of TD-AIM with respect to the number of surface unknowns is verified in Fig. 10(a) and (b) using the data points of Figs. 8 and 9: The total CPU time per time-step and the memory requirement of TD-AIM are computed by multiplying the data points at Figs. 8 and 9 with the corresponding number of processors. Thus, if the implementation had ideal parallel efficiency all the data points for a given would coincide in Fig. 10. The computational complexity of the implementation is in good agreement with the trends predicted in Section II. Fig. 10(a) and (b) also compares the computational demands of TD-AIM with those of a classical (parallel) MOT scheme: The TD-AIM scheme outperforms the MOT solver for as little as for plate and for sphere simulations.

12 YILMAZ et al.: TD-AIM FOR SURFACE INTEGRAL EQUATIONS 2703 Fig. 10. TD-AIM versus MOT for plates and spheres. (a) Computational complexity. The lines are parallel to N, N, and N log N. (b) Memory requirement. The lines are parallel to N, N, and N. C. Applications Lastly, the practical application of the TD-AIM scheme to large-scale broadband scattering analysis is demonstrated. For this purpose, the transient responses of a conesphere, a NASA almond [36], and a generic aircraft [37] are characterized. The main axes of all three scatterers lie in the plane and their tips point in the direction (, ). For all scatterers, the current found by the TD-AIM solver is Fourier transformed and the RCS patterns are computed at,, and, which are then compared to those obtained by frequencydomain solvers that use the same surface mesh as the time-domain scheme. The auxiliary-grid parameters for frequency-domain AIM solvers are set to be the same as the TD-AIM ones in all cases. Note that, the frequency domain iterative solvers are terminated when the relative residual error is less than. The conesphere is illuminated by a polarized Gaussian plane-wave propagating toward direction, with and. The cone part is cm Fig. 11. Bistatic RCS (VV) of the benchmark conesphere in the x 0 z plane: (a) f =2:5 GHz, (b) f = 6 GHz, and (c) f =9:5GHz. long, the sphere part has a radius of cm, and the half angle is 7 degrees [36]. The surface of the conesphere, which has an area of approximately 0.18, is discretized using triangular patches with average area of 4.37 resulting in a total of spatial RWG basis functions. The auxiliary-grid spacing

13 2704 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 is,, up to third-order moments are matched, and. The time-step size is,, and. The scattering is analyzed using the CFIE. Fig. 11(a) (c) compare the RCS patterns in the plane. Note the strong physical-optics reflection from the side walls of the cone at 14 degrees and the good agreement between the schemes. For this analysis, the TD-AIM scheme required 1,122 s to fill the matrices and a total of 1,547 s (on average 1.5 s per time step) to find the solution using processors. Of the 1.5 s, 0.6 s were spent in Stages I III, 0.6 s in Stage IV, and 0.3 s in the iterative solution of (5). The peak memory requirement was 706 MB with 468 MB used by the far-field operations. On the other hand, the frequency-domain AIM scheme required 445, 268, and 225 s to fill the matrices and a total of 95 s (61 iterations), 73 s (46 iterations), and 61 s (39 iterations) to solve for the current at 2.5, 6, and 9.5 GHz, respectively (also using processors). The peak memory requirement of the frequency-domain scheme was 80 MB for all frequencies with only 3 MB used by the far-field operations. In short, when the matrix-fill times are not included, TD-AIM required about 16 to 25 times more time and 9 times more memory than AIM. The NASA almond is also illuminated by a polarized Gaussian plane-wave propagating toward direction, but with and. The almond is about cm long and its largest cross-section is an ellipse which is approximately cm wide and cm thick. The surface of the almond, which has an area of about 397, is discretized using triangular patches with average area of 0.47 resulting in a total of spatial RWG basis functions. The auxiliary-grid spacing is,, up to second-order moments are matched, and. The time-step size is,, and. The scattering is analyzed using the CFIE. Fig. 12(a) (c) compare the RCS patterns in the plane, which show good agreement. The TD-AIM scheme required 740 s to fill the matrices and a total of 1,538 s (on average 1.9 s per time step) to find the solution using processors. Of the 1.9 s, 0.9 s were spent in Stages I III, 0.7 s in Stage IV, and 0.4 s in the iterative solution of (5). The peak memory requirement was 770 MB with 555 MB used by the far-field operations. In contrast, the frequency-domain AIM scheme required 483, 393, and 275 s to fill the matrices and a total of 112 s (53 iterations), 68 s (27 iterations), and 70 s (26 iterations) to solve for the current at 4, 14, and 24 GHz, respectively (also using processors). The peak memory requirement of the frequency-domain scheme was 159 MB with only 5 MB used by the far-field operations for all frequencies. To sum up, when the matrix-fill times are not included, TD-AIM solution took about 14 to 23 times longer and required about 5 times more memory than AIM. To further verify the accuracy and efficiency of the TD-AIM scheme, broadband scattering from a generic aircraft is analyzed next. The geometrical, electromagnetic, and algorithmic description of the aircraft simulation follows. The wing span of the aircraft is 0.81 m, its tip-to-tail length is 0.82 m, its height is 0.24 m, and its surface area is approximately The aircraft is illuminated head-on by a polarized Gaussian Fig. 12. Bistatic RCS (VV) of the NASA almond in the x0z plane: (a) f = 4GHz, (b) f = 14 GHz, and (c) f = 24 GHz. plane-wave traveling toward direction with and. Thus, the aircraft fits into a rectangular prism of size approximately and has a surface area of at. The surface of the aircraft is discretized using triangular patches with average area of 48 resulting in a total of

14 YILMAZ et al.: TD-AIM FOR SURFACE INTEGRAL EQUATIONS 2705 Fig. 13. Amplitude of the transient current density close to the tip and tail of the generic aircraft. spatial RWG basis functions. The auxiliary grid spacing is,, up to third-order moments are matched, and. The time-step size is,, and. The scattering analysis is carried out using the CFIE. The current densities at two sample basis functions located close to the tip and tail of the aircraft are plotted in Fig. 13 to demonstrate the stability of the scheme. The RCS patterns are computed at the plane and are compared with those obtained from the MOM solution in Fig. 14(a) (c). The schemes exhibit good agreement. Finally, to demonstrate the advantage of the TD-AIM scheme, the range (radar) profile of the above aircraft is computed. The range profile quantifies the time-domain radar return (back scattered RCS) of a target [38] (18) where the low-pass functions and are the in-phase and quadrature portions and is the envelope of the back scattered far-field. The range profile can be computed for various bandlimited incident pulses; here, it is computed for a modulated sinc pulse with center frequency and bandwidth, i.e. (19) Note that, the 3 db width of the sinc pulse corresponds to a distance of 0.15 m, and that in order not to violate the temporal bandlimitedness assumption in Section II, the back scattered RCS is extracted from TD-AIM simulations that use the Gaussian pulse of (15). Fig. 15 compares HH and VV polarized range profiles obtained from frequency and time domain solvers. Each backscattered RCS is computed at 64 Fig. 14. Bistatic RCS (VV) of the aircraft in the x 0 y plane: (a)f = 1GHz, (b) f = 2 GHz, and (c) f = 3 GHz. frequencies equally spaced in the band [1 GHz, 3 GHz] for this analysis. Using processors with a peak memory of 370 MB, TD-AIM computed each range profile in 18 min (6 min to fill the matrices and 12 min to solve for the transient current). Each frequency-domain AIM simulation, on the other hand, required on average 91 min (66 min to fill the matrices and 25 min

15 2706 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 its time-harmonic counterpart in reducing the computational burden of the solution. The practical bottleneck of TD-AIM is its high far-field memory demand that is typically times larger than that of frequency-domain AIM. The relative ease of parallelization of multidimensional FFTs, however, alleviates the impact of the scheme s memory requirement; moreover, distributed-memory parallelization of TD-AIM exhibits good scalability and permits large-scale transient electromagnetic analysis in practical amounts of time as demonstrated in Section III. The TD-AIM scheme is especially efficient for analyzing broadband phenomena that involve quasiplanar structures where the ratio of the volume enclosed by the auxiliary grid to the surface area of the scatterer is minimal. The scheme also exhibited remarkable speedup and memory-savings even for general 3-D structures, as demonstrated through a series of examples including spheres, a conesphere, an almond, and a generic aircraft, due to its leverage on high efficiency of FFT computation in modern computers. The TD-AIM will be even more powerful for analyzing inhomogeneous objects that require volume discretization in which case the computational complexity and memory requirements will be reduced from and for the MOT to and, respectively, where denotes the number of spatial degrees of freedom resulting from volume discretization. We expect the flexibility, generality, and ease of implementation of the algorithm to lead to increased research activity in FFT-based time-domain integral equation solvers and their applications to electromagnetic analysis. ACKNOWLEDGMENT Fig. 15. Range profile of the aircraft for (a) VV polarization and (b) HH polarization. to solve for the time-harmonic current) on processor with a peak memory of 453 MB. Hence, even if the embarrassingly parallel nature of frequency-domain computations is utilized and the solutions at all 64 frequencies are found concurrently on 64 processors, TD-AIM would still require one fifth the time to compute the range profile. If the fill times are ignored (for example if the range profile is simulated at multiple angles), TD-AIM would still be twice faster. Note that, while these observations are suggestive, they do not prove that TD-AIM is necessarily more efficient; for example, frequency-domain schemes can use coarser meshes to compute the RCS at lower frequencies or can use frequency interpolation to reduce the number of simulation frequencies. Nonetheless, the efficiency of the TD-AIM scheme for broadband computations is evident. IV. CONCLUSION This paper presented the time-domain counterpart of the AIM scheme for efficiently solving time-domain surface integral equations. The performance of the time-domain scheme mirrors The authors acknowledge the Computational Science and Engineering Department, University of Illinois at Urbana Champaign, for the access to their parallel cluster. A. E. Yılmaz would like to thank Drs. K. Aygün and M. Lu for countless discussions on time-domain integral equations and Dr. T. Rylander for sharing the aircraft mesh which was originally constructed by AerotechTelub according to a design provided by the Swedish Defense Research Agency. REFERENCES [1] D. R. Wilton, Review of current status and trends in the use of integral equations in computational electromagnetics, Electromagn., vol. 12, pp , [2] E. K. Miller, A selective survey of computational electromagnetics, IEEE Trans. Antennas Propagat., vol. 36, pp , Sept [3] W. C. Chew, J. M. Jin, C. C. Lu, E. Michielssen, and J. M. Song, Fast solution methods in electromagnetics, IEEE Trans. Antennas Propagat., vol. 45, pp , Mar [4] V. Rokhlin, Diagonal forms of translational operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal, vol. 1, no. 1, pp , [5] J. M. Song and W. C. Chew, Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering, Microwave Opt. Tech. Lett., vol. 10, no. 1, pp , [6] M. F. Catedra, R. F. Torres, J. Basterrechea, and E. Gago, The CG-FFT Method: Application of Signal Processing Techniques to Electromagnetics. Norwood, MA: Artech House, [7] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, AIM: adaptive integral method for solving large-scale electromagnetic scattering and radiation problems, Radio Sci., vol. 31, no. 5, pp , 1996.

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Yılmaz (S 96) was born in Ankara, Turkey, in He received the B.S. degree in electrical engineering from Bilkent University, Turkey, in 1999 and the M.S. degree in electrical and computer engineering from the University of Illinois at Urbana- Champaign (UIUC), in 2001, where he is currently pursuing the Ph.D. degree. His research interests include computational electromagnetics, with emphasis on time-domain integral equations and their fast solutions, high performance computing, and coupled electromagnetic/circuit simulations. Mr. Yilmaz is the recipient of the 2003 Raj Mittra Outstanding Research Award from the electrical and computer engineering department and the Interdisciplinary Graduate Fellowship from the Computational Science and Engineering Department at UIUC. Jian-Ming Jin (S 87 M 89 SM 94 F 01) received the B.S. and M.S. degrees in applied physics from Nanjing University, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in He is a Professor of electrical and computer engineering and Associate Director of the Center for Computational Electromagnetics at the University of Illinois at Urbana-Champaign (UIUC). He has authored and co-authored over 140 papers in refereed journals and 15 book chapters. He has also authored The Finite Element Method in Electromagnetics (New York: Wiley, 1st edition 1993, 2nd edition 2002) and Electromagnetic Analysis and Design in Magnetic Resonance Imaging (Boca Raton, FL: CRC, 1998) and Computation of Special Functions (New York: Wiley, 1996), and co-edited Fast and Efficient Algorithms in Computational Electromagnetics (Norwood, MA: Artech, 2001). His current research interests include computational electromagnetics, scattering and antenna analysis, electromagnetic compatibility, bioelectromagnetics, and magnetic resonance imaging. His name is often listed in the UIUC s List of Excellent Instructors. He was elected by ISI as one of the world s most cited authors in Dr. Jin is a Member of Commision B of USNC/URSI and Tau Beta Pi. He was a recipient of the 1994 National Science Foundation Young Investigator Award and the 1995 Office of Naval Research Young Investigator Award. He also received the 1997 Xerox Junior Research Award and the 2000 Xerox Senior Research Award presented by the College of Engineering, University of Illinois at Urbana-Champaign, and was appointed as the first Henry Magnuski Outstanding Young Scholar in the Department of Electrical and Computer Engineering in He was a Distinguished Visiting Professor in the Air Force Research Laboratory in He served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and Radio Science and is also on the Editorial Board for Electromagnetics and Microwave and Optical Technology Letters. He was the Symposium Co-chairman and Technical Program Chairman of the Annual Review of Progress in Applied Computational Electromagnetics in 1997 and 1998, respectively.

17 2708 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 10, OCTOBER 2004 Eric Michielssen (M 95 SM 99 F 02) received the M.S. degree in electrical engineering (summa cum laude) from the Katholieke Universiteit Leuven (KUL), Belgium in 1987 and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign (UIUC) in He served as a Research and Teaching Assistant in the Microwaves and Lasers Laboratory at KUL and the Electromagnetic Communication Laboratory at UIUC from 1987 to 1988 and from 1988 to 1992, respectively. He joined the Faculty of the Department of Electrical and Computer Engineering, UIUC in 1993, where he serves as Professor of electrical and computer engineering and as Associate Director of the Center for Computational Electromagnetics. He has authored or coauthored over 100 journal papers and book chapters and over 160 papers in conference proceedings. From 1997 to 1999, he was as an Associate Editor for Radio Science. His research interests include all aspects of theoretical and applied computational electromagnetics. His principal research focus has been on the development of fast frequency and time-domain integral-equation-based techniques for analyzing electromagnetic phenomena, and the development of robust, genetic algorithm driven optimizers for the synthesis of electromagnetic/optical devices. Prof. Michielssen is a Member of International Union of Radio Scientists (URSI) Commission B. He received a Belgian American Educational Foundation Fellowship in 1988 and a Schlumberger Fellowship in He received a 1994 URSI Young Scientist Fellowship, a 1995 National Science Foundation CAREER Award, and the 1998 Applied Computational Electromagnetics Society (ACES) Valued Service Award. In addition, he was named 1999 URSI- United States National Committee Henry G. Booker Fellow and was selected as the recipient of the 1999 URSI Koga Gold Medal. He received UIUC s 2001 Xerox Award for Faculty Research, was appointed 2002 Beckman Fellow in the UIUC Center for Advanced Studies, named 2003 Scholar in the Tel Aviv University Sackler Center for Advanced Studies, and selected as the UIUC 2003 University Scholar. He was Technical Chairman of the 1997 ACES Symposium (Review of Progress in Applied Computational Electromagnetics, March 1997, Monterey, CA), and from 1998 to 2001 and 2002 to 2003, served on the ACES Board of Directors and as ACES Vice-President from 1998 to He currently is an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.