A parallel FFT accelerated transient field-circuit simulator

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 9, SEPTEMBER 2005 2851

A Parallel FFT Accelerated TransientField-Circuit Simulator

Ali E. Yılmaz, Member, IEEE, Jian-Ming Jin, Fellow, IEEE, and Eric Michielssen, Fellow, IEEE

Abstract—A novel fast electromagnetic field-circuit simulatorthat permits the full-wave modeling of transients in nonlinearmicrowave circuits is proposed. This time-domain simulator iscomposed of two components: 1) a full-wave solver that modelsinteractions of electromagnetic fields with conducting surfacesand finite dielectric volumes by solving time-domain surface andvolume electric field integral equations, respectively, and 2) acircuit solver that models field interactions with lumped circuits,which are potentially active and nonlinear, by solving Kirchoff’sequations through modified nodal analysis. These field and circuitanalysis components are consistently interfaced and the resultingcoupled set of nonlinear equations is evolved in time by a multi-dimensional Newton–Raphson scheme. The solution procedure isaccelerated by allocating field- and circuit-related computationsacross the processors of a distributed-memory cluster, whichcommunicate using the message-passing interface standard.Furthermore, the electromagnetic field solver, whose demand forcomputational resources far outpaces that of the circuit solver, isaccelerated by a fast Fourier transform (FFT)-based algorithm,viz. the time-domain adaptive integral method. The resulting par-allel FFT accelerated transient field-circuit simulator is applied tothe analysis of various active and nonlinear microwave circuits,including power-combining arrays.

Index Terms—Parallel processing, microwave circuits, nonlinearcircuits, time-domain integral equations, transient analysis.

I. INTRODUCTION

AS OPERATING frequencies increase, accurate and ef-ficient hybrid full-wave field-circuit simulation tools

are becoming increasingly indispensable in the design ofmicrowave circuits as well as in the assessment of their vulner-ability to unintentional coupling, crosstalk, packaging effects,and intentional electromagnetic interference. Microwave cir-cuits can be analyzed using either frequency- or time-domainsimulators; however, when the circuit under study containsnonlinear components, time-domain methods enjoy the ad-vantage of allowing for the direct analysis of field-circuitinteractions without resorting to harmonic balance or portextraction methods [1]. Although early efforts at formulating

Manuscript received September 27, 2004; revised January 3, 2005. Thiswork was supported in part by the Army Research Office under ARO ProgramDAAD19-00-1-0464, by the Defense Advanced Research Projects Agency(DARPA) VET Program under AFOSR Contract F49620-01-1-0228, by theMURI Grant F49620-01-1-04, “Analysis and design of ultra-wide band andhigh power microwave pulse interactions with electronic circuits and systems”,and by the Computational Science and Engineering (CSE) Fellowship at theUniversity of Illinois at Urbana-Champaign.

The authors are with the Center for Computational Electromagnetics, De-partment of Electrical and Computer Engineering, University of Illinois at Ur-bana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMTT.2005.854260

hybrid time-domain field-circuit simulators relied on inte-gral-equation schemes—the goal often consisted in the analysisof electromagnetic interactions with nonlinearly loaded wires[2]–[4]—most of the ensuing simulators invoked time-domaindifferential-equation methods. By now, various extensions toboth the finite-difference time-domain (FDTD) [5]–[11] andfinite-element time-domain (FETD) [12], [13] methods aimedat incorporating device physics/behavior into electromagneticanalysis environments have been proposed. The most rig-orous of these schemes permit the simultaneous solution ofthe Maxwell and semiconductor carrier transport equationsby casting them as a strongly coupled nonlinear system ofdifferential equations on the same grid [14]–[16]. To minimizecomputational cost, whenever possible, these differential equa-tion solvers account for device and circuit behavior (as opposedto physics) through their description in terms of equivalentlumped elements and macromodels [17]. However, lumpedloads and circuits, be they passive or active, linear or nonlinear,static or time-varying, can be quite easily accounted for intime-domain integral-equation solvers as well [1]–[4].

Indeed, the recent development of stable [18]–[20], ac-curate [21], and fast [22]–[30] marching on in time-basedintegral-equation solvers for analyzing large-scale transientscattering and radiation phenomena calls for a study into theapplicability of these solvers to the analysis of microwave cir-cuits. Modern-day fast time-domain integral-equation solversare either accelerated by plane wave time domain (PWTD)[23] or by fast Fourier transform (FFT)-based [24]–[30] algo-rithms. Use of either algorithm permits the analysis of transientelectromagnetic phenomena with far more degrees of freedomthan possible by classical time-domain integral-equation ap-proaches. Recently, a PWTD accelerated electromagnetic fieldsolver was coupled to a SPICE-like circuit simulator and ap-plied to the analysis of transients on microwave circuits withnonlinear loads [31]. Here, we report, instead, on the hybridiza-tion (formulation and implementation) of an FFT acceleratedsolver with a modified nodal analysis-based circuit simulator;an initial implementation of this scheme was described in [32].The main reason for pursuing FFT accelerated field-circuitanalysis tools is as follows. Just like their frequency-domaincounterparts, PWTD accelerated time-domain integral-equa-tion solvers are asymptotically superior to FFT acceleratedones when analyzing electromagnetic transients on arbitrarilyshaped three-dimensional (3-D) surfaces. FFT acceleratedsolvers, however, generally outperform PWTD acceleratedones when analyzing transients on quasiplanar, volumetric,or densely packed structures, which are often encountered in

0018-9480/$20.00 © 2005 IEEE

2852 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 9, SEPTEMBER 2005

microwave circuits. This paper supports this trend. Furthermore,the availability of parallel FFT algorithms [33] and the relativeease of load balancing parallel FFT accelerated time-marchingsolvers render the FFT route even more appealing. The specificFFT-based algorithm used in this paper is the time-domainadaptive integral method (TD-AIM) [30], which allows theefficient analysis of electromagnetic field interactions withnonuniformly discretized microwave structures/circuits.

The hybridization of this FFT accelerated field solver witha modified nodal analysis-based circuit solver results in acoupled nonlinear system of equations, which is solved by aNewton–Raphson algorithm to compute the time evolution ofthe fields, currents, and voltages on the microwave circuit. Inthis paper, the hybrid field-circuit simulator is implementedon a distributed-memory computer cluster that communicatesthrough the message-passing interface. The computationalwork is divided among multiple processors using a simple buteffective parallelization paradigm: field and circuit unknownsand associated operations are assigned to separate groups ofprocessors. It is shown that this strategy allows for the separatedevelopment and optimization of field and circuit solvers andresults in near-optimal parallel scalability for the hybrid solver.The proposed scheme is described in Section II and applied tothe analysis of microwave circuits in Section III. Section IVoutlines the conclusions of this study.

II. FORMULATION

This section details the proposed parallel FFT acceleratedtransient field-circuit simulator. Subsections II-A and II-B for-mulate the field and circuit equations, respectively. These twosubsections introduce notation that enables the description ofthe field and circuit solvers’ hybridization (including dc anal-ysis), the method for solving the coupled system of equations(including complexity analysis), and the acceleration of the hy-brid simulator (including parallelization); these topics are cov-ered in Subsections II-C–E.

A. Electromagnetic Field Equations

Let denote the conducting surfaces and the potentially in-homogeneous dielectric volumes that comprise the microwavestructure under study. The microwave structure resides in freespace with permittivity . The permeability of the structure andthe surrounding free space is denoted by . In the following, allconductors are assumed perfect and all dielectrics are assumedlinear, isotropic, nonmagnetic, nondispersive, lossless, and ofpermittivity . Extensions to lossy conductors and dielectricsare possible, see [27], [34]–[36].1 A known transient electro-magnetic field excites ; it is assumed that this field’s spec-trum essentially vanishes for frequencies and that thefield is nearly zero for . The incident field in-duces surface currents on and volume (polarization)

1The hybrid simulator formulation in this paper is general enough to supportsuch extensions; moreover, the TD-AIM acceleration in Subsection II-E wouldremain valid and efficient with little modification [27], [30]. Indeed, this versa-tility is one of the advantages of FFT-accelerated simulators over PWTD-accel-erated ones.

currents in . These currents, in turn, generate the scat-tered electric field

(1)

where represents the time derivative and and are thevector and scalar potentials

(2)

Here, is the distance between source point andobservation point and is the free-space speed oflight. The volume current density relates to the electric flux den-sity as ,where is the contrast ratio [37], [38]. In-tegral equations for the surface and volume currents (or better,the flux—see below) are arrived at by: 1) forcing the temporalderivative of the sum of the incident and scattered electric fieldstangential to to vanish, and 2) expressing the temporal deriva-tive of the total field as the sum of the temporal derivatives ofthe incident and scattered fields throughout

(3)

Upon expressing in (3) using(1)–(2), as well as the above-stated relationships linking

, and , a coupled set of sur-face-volume time-domain integral equations in and

is obtained. These integral equations are solvednumerically by discretizing and usingand space-time basis functions, respectively, as

(4)

Here, and are unknown expansion coefficients(electromagnetic unknowns) and is the time-stepsize.2 In this paper, the surface basis functions areRao–Wilton–Glisson (RWG) functions [39] defined on pairsof triangular patches that approximate . The volume basis

2Unlike explicit FDTD/FETD/integral-equation-based hybrid simulators,whose time-step size is constrained by the minimum spatial discretization size,the choice of �t for the proposed implicit integral-equation-based simulatoris dictated by the maximum frequency of interest and the desired accuracy;typically 0:04=f � �t � 0:1=f .

YILMAZ et al.: PARALLEL FFT ACCELERATED TRANSIENT FIELD-CIRCUIT SIMULATOR 2853

functions are zeroth-order divergence conformingfunctions [37] defined over tetrahedral elements, which ap-proximate the dielectric volumes , and over each of which

(hence ) is assumed constant; this implies that.

This choice of basis functions enforces the continuityof the normal component of the surface current densityacross patches and the electric flux density across tetrahe-drons, respectively. The temporal basis functions areshifted Lagrange interpolants [19]. It is important to notethat the composite basis functions in (4) are localized inspace-time. Upon substituting (4) into (3) and testing theresulting equation at times with the spatial functions

and atotal of equations for expan-sion coefficients result

for (5)

Here, and denotes the longest transittime of a free-space propagating electromagnetic field across

, expressed in terms of time steps [30]. Expressions forthe entries of the vectors and and matricesare provided in the Appendix. The system of (5) is recast intothe following form and solved by forward substitution (i.e., bymarching on in time)

for (6)

The matrix , which represents immediate electromagneticinteractions, is a sparse but nondiagonal impedance matrix ofsize , with typically nonzero elements.The vectors and hold the unknown current/fluxcoefficients and the known tested incident field values at time

, respectively. The dominant computational cost of the fieldsolver involves the evaluation of the space-time convolutionappearing on the right-hand side of (6), viz. the computation ofthe scattered electromagnetic fields, which requiresoperations per time step [30].

B. Circuit Equations

The proposed solver allows the microwave structure de-scribed above to contain an arbitrary number of lumped circuitswith independent reference/ground nodes. Equations governingcircuit behaviors are formulated, starting from the circuittopologies and Kirchoff’s laws, via modified nodal analysisusing the SPICE2 approach [40] as detailed next; specificdetails of how the circuits are coupled to the electromagneticsystem are further discussed in Subsection II-C. The circuitunknowns are node voltages and voltage-source currents;hence, the total number of circuit unknowns, denoted as ,is equal to the total number of nonground nodes and voltagesources in the circuits. A total of equations in terms ofthese unknowns are obtained by imposing Kirchoff’s voltagelaw at the voltage-defined elements and current law at allnodes except the grounds. Branch equations relating currents

to voltages are obtained from element stamps and companionmodels that are formulated using the trapezoidal integrationrule. The circuit unknowns are evolved in time using the sametime step size as the field solver , which is assumed constantthroughout the simulation. While contemporary circuit solversemploy variable time stepping schemes, the fixed but small

dictated by the field solver was found to be sufficientlyaccurate for the applications considered in this paper. Whenanalyzing circuits composed of linear and nonlinear resistors,capacitors, inductors, as well as dependent and independentvoltage and current sources for time steps, this procedureyields equations for unknowns

for (7)

The matrix , which represents branch equations of linear andtime-invariant circuit elements, is a sparse admittance matrixof size with typically nonzero ele-ments. The vectors and hold the circuit unknownsand source currents and voltages, respectively, and the vector

represents the branch equations of nonlinearand time-varying elements at time .

C. Coupled System of Equations

The lumped circuits are connected to the conducting surfacesthrough electromagnetic surface unknowns and are modeled

as local voltage sources. To illustrate this procedure, assumethat electromagnetic surface unknown is loaded by a one-portcircuit whose terminals are at nodes and 0 [Fig. 1(a)]. Thefield solver requires as one of its inputs the time-derivative ofthe port voltage (the voltage difference between the two termi-nals) and the circuit solver requires the port current

, where is the length of the edge.3 Thus, the fol-lowing coupled system of equations results when the two solversare interfaced:

(8)

Here, is the vector of unknowns attime . The matrix , which represents circuit to field cou-pling, is formed according to Fig. 1(b) and (c) and is used toselect voltages observed at the circuit ports. The transpose ma-trix , which represents field to circuit coupling, is used toselect the currents observed at the circuit ports. The nonzeroentries of are . In (8), the derivatives of the port voltagesare computed by a third-order backward differentiation formulaand are . For circuitswith three or more terminals, a straightforward extension of theprocedure in Fig. 1 [31] is employed by defining the number of

3Multiple surface unknowns k ; . . . ; k can also be loaded by the same cir-cuit, in which case n identical voltage sources excite k ; . . . ; k and the portcurrent is defined as I (k )d .


Fig. 1. Coupling of electromagnetic and circuit systems. (a) The surfaceunknown k of edge-length d is connected to a one-port circuit at terminalsa and 0. (b) Coupling from circuit system point of view. (c) Coupling fromelectromagnetic system point of view.

ports of a circuit to be one less than the number of its termi-nals, i.e., each circuit has a unique ground terminal connectedto the electromagnetic system. In general, if there areports, will have nonzero elements since one terminalof each port is grounded (assuming each port loads only onesurface unknown). The above coupling scheme inherently as-sumes that the physical separation between circuit terminals issmall with respect to the minimum wavelength of interest andthe ground nodes of each circuit have well defined locations onthe microwave structure. Other coupling schemes that definea global ground (at infinity) exist [41]. Although the couplingscheme of [41] introduces additional unknowns at the circuit ter-minals (contacts), the off-diagonal matrix blocks representingcoupling between the circuit and field formulations/solvers re-main sparse.

Notice that, while the circuit solver can incorporate initialconditions (typically computed from a dc analysis) for the circuitunknowns (i.e., andfor ), the field solver assumes zero initial conditions (i.e.,

and for ). In this paper, microwavecircuits with dc sources are analyzed in two ways: 1) Zeroinitial conditions are assumed everywhere; the dc sources areturned on gradually (in order not to violate the finite bandwidthassumption), and the transient analysis is performed only afterthe system reaches steady state. Depending on the specifics ofthe simulation being conducted, this procedure may require(too) many time steps and lead to low-frequency stabilityproblems, which are avoided by the second scheme. 2) The dcand transient responses are separated, similar to [10] and [11],using the linearity of the electromagnetic system as follows.On the one hand, the total electromagnetic response at time

is represented as and the field solver modelsonly the transient response . On the other hand, the totalcircuit response at time is represented as and thecircuit solver models both the dc response and the transientresponse (by enforcing the dc solution as initial conditionsthrough and ). To maintain consistency, dc voltagesand currents are introduced at the interface [Fig. 1(b)]. Hence,the hybrid field-circuit solver uses the (precomputed) vectors

, and the entries of at the circuit ports toaccount for the dc conditions in the second scheme. Thesevalues can be computed through a dc analysis of the circuits

with the microwave structure characterized either explicitly,e.g., through fast electrostatic solvers [42], or approximately,e.g., as a small resistor modeling the dc resistance of theconductors between circuits [11].

D. Solution Algorithm/Computational Complexity Analysis

At each time step , the following five-stageNewton–Raphson algorithm is used to solve the coupled non-linear system of (8):

Stage (i): Compute the right-hand-side vector . Then foreach Newton iteration

Stage (ii): Evaluate the residual vector, where is the so-

lution vector from the previous Newton step andthe initial guess is , i.e., the solutionat the previous time step.

Stage (iii): If then stop; the solutionat time is .

Stage (iv): Compute the Jacobian submatrix.

Stage (v): Iteratively solve

(9)

for the Newton step and find the next solutionvector .

While the above Newton–Raphson algorithm exhibitsquadratic convergence for sufficiently close to the solution(assuming the discretization of (3) and the nonlinear devicemodels result in nonsingular ), it can fail if the initial guessis far from the solution. The algorithm’s robustness can beimproved by combining it with line-search methods that takeless than the full Newton step ([43] for details). Note that, ingeneral all four submatrices of the Jacobian matrix in (9)are sparse and the algorithm needs only one Newton iterationper time step if there are no nonlinear circuit elements. It shouldbe emphasized that the above solution algorithm is differentfrom that of [31]. While the algorithm in [31] solves a smallersystem of equations and therefore might potentially requirefewer Newton iterations, here the Jacobian matrix is computedsignificantly faster. This is because, unlike [31], the above algo-rithm does not require a matrix solution to compute the entriesof ; indeed they can be computed analytically. Furthermore,the algorithm here is more amenable to the parallelizationframework discussed in Subsection II-E. The computationalcomplexity of the above algorithm is analyzed next.

The algorithm evaluates and stores , which is inde-pendent of the Newton iteration, in op-erations. Then, at each Newton iteration , the algorithmrequires operations to evaluate the vectors

and operations to update , andoperations to solve for the Newton

step. Here, is the number of nonzero entries of andis the average number of iterations needed to solve (9).

In general, and . Thus, the twodominant computational operations are the evaluation of the


scattered electromagnetic fields in stage (i) and the iterativesolution of the Newton step in stage (v). For each timestep, the computations involved in stages (i) and (v) require

and operations,respectively, where denotes the average number of Newtoniterations. In our experience, the dominant computationaloperation for typical microwave circuits is incurred in stage (i),the calculation of .

E. Parallelization and TD-AIM Acceleration

The solution of (8) is accelerated by parallel processing. Oneparallelization approach might be to simultaneously distributeall unknowns among the available processorswithout separating electromagnetic and circuit unknowns. Thisapproach, however, forces all processors handling both types ofunknowns to run both electromagnetic and circuit solvers; this,in turn, leads to load balancing problems. Furthermore, it is notclear how to retrofit existing parallel field and circuit solversto operate in unison inside such a framework. In this work, theparallelization strategy is to distribute electromagnetic- and cir-cuit-related unknowns and operations to separate sets of proces-sors. Of the available processors, processorsare dedicated to computations governing the updates of thefield unknowns and processors are used for operationsrelated to updating the circuit unknowns. The two setsof processors communicate only when evaluating and itsnorm and while iteratively solving (9) in the Newton–Raphsonalgorithm of Subsection II-D. Moreover, the two systems in-teract only at the loading ports and hence the two sets of pro-cessors exchange only numbers when they communi-cate. Thus, the total amount of communication between the twosets of processors is bytes at each time stepand is negligible compared to other communication and compu-tation costs. This strategy allows for the separate developmentof field and circuit solvers and enables hybridization of alreadydeveloped and optimized parallel field and circuit solvers inthe Newton–Raphson framework without loss of their load-bal-ancing features. Indeed, in this paper the circuit solver is hy-bridized with a highly scalable parallel FFT-based field solver[30] as described next. It should be noted that our implementa-tion employs the message-passing interface standard; this is inpart due to the availability of parallel FFT libraries based on thisstandard [33].

The FFT-acceleration is employed to reduce thecost of computing the first entries of at each time step,which quickly overwhelms the processors. FFT-basedalgorithms for accelerating electromagnetic analysis origi-nated with the -space method [44], [45] and were initiallyused for frequency-domain analysis involving uniformly dis-cretized structures. They were later extended to incorporatenonuniformly discretized structures through the introduction ofauxiliary uniform grids [42], [46] and parallelized [46]–[48] toallow large-scale static and time-harmonic analysis. Recently,FFT-based algorithms have been adopted to permit the par-allel analysis of electromagnetic transients on large arbitrarilyshaped structures [24]–[30]. Here, the TD-AIM algorithm [30]is used.

The TD-AIM scheme embeds the microwave structure in anauxiliary 3-D Cartesian grid with nodesthat are separated by and in the three orthogonaldirections. Each of the impedance matrices is approximatedas by using these auxiliary grid points.The matrices help preserve accuracy by reproducing theoriginal entries of : their entries are nonzero for onlynear-field interactions, for which they are equal to .The matrices , on the other hand, are approximations of theoriginal matrices that are efficiently multiplied with the vectors

using multidimensional FFTs. The TD-AIM algorithmcomputes the first entries of in four steps: 1) At each timestep , all current-coefficients are locally projected ontothe auxiliary grid, such that sources that reside on the auxiliarygrid accurately approximate the transient fields radiated by theoriginal sources outside a near-field region. In order to useonly one auxiliary grid and the same propagation operators forboth, the projection step for surface and volume sources is notidentical. Because the fields radiated by volume sources requirean additional temporal derivative [see (14) in the Appendix],a finite difference scheme is used to compute the numericalderivatives of the volume coefficients, which are then projectedon to the auxiliary grid. 2) Present and future transient fieldsproduced by the sources on the auxiliary grid are computed onthe same grid using vector-and scalar-potential propagators asdescribed in [30] in a multilevel approach via global space-timeFFTs. 3) The fields at time step are locally interpolated fromthe vector- and scalar-potential values on the auxiliary gridonto the primary mesh. 4) The errors in the near-field regionare corrected by computing and addingit to the fields computed via steps 1)–3). In this paper, theprojection and interpolation operators of steps 1) and 3) arefound by matching the multipole moments of point sourceson the auxiliary grid to those of , and componentsand the gradients of the functions and [30],[46]. Hence, four projection matrices are used for surface andvolume basis functions. Because the projection operations andthe correction matrices are localized in space and time,the dominant computational burden of the TD-AIM schemeis the computation of 4-D space-time FFTs in step 2). Usinga multilevel algorithm and employing parallel FFTs (e.g., viathe FFTW library [33]), the space-time FFTs are computedin operations per processorper time step. A detailed description of the parallel FFTs inthis context is given in [30]. For volumetric or quasiplanarstructures , whereas (for volumetric)or (for quasiplanar) [30].

To sum up, the two computationally dominant stagesof the proposed algorithm, stages (i) and (v), are bothaccelerated by parallelization while stage (i) is furtheraccelerated by the TD-AIM algorithm. For typical mi-crowave circuits, neglecting the communication costs,the time spent in stages (i) and (v) at each time stepscales as and

per processor, re-spectively. Ignoring the circuit processors and operations, theprocessors exchange, at each time step, a total of


bytes while computing FFTs in stage (i) and depending onthe numbering and distribution of the unknowns amongstthem to bytes in stage (v). Hence,communication costs, which are subdominant to computationcosts for stage (i), may become the bottleneck for stage (v) asthe number of processors is increased.

III. APPLICATIONS

The accuracy and efficiency of the proposed scheme aredemonstrated by analyzing various microwave systems andcircuits. First, an active antenna and a microwave amplifier areanalyzed and the results obtained are compared to measure-ment and simulation data available in the literature. Next, a gridamplifier is simulated and the results are compared with bothfrequency-domain simulations and measurements available inthe literature. In all simulations, the TD-AIM kernel matchesup to third-order moments when projecting sources onto theauxiliary grid, and defines the near-field region of a basisfunction as the space extending four (auxiliary grid) cells ineach direction away from it (that is, as defined in [30] is 4).Bandwidths of Gaussian pulses are specified as two-sided andquantify the frequency range over which spectral power densi-ties are no less than 45 dB below their peak values. The resultsin this section are obtained using a cluster of 1-GHz PentiumIII processors with . In each and every case, theparallel efficiency of the scheme is examined by observing itsrun time and memory requirements. While a direct performancecomparison with other simulators is not performed, the belowapplications clearly demonstrate the viability of the solver foranalyzing large and detailed microwave systems and circuits.

A. Active Patch Antennas

The first microwave system analyzed consists of an array oftwo (nonlinear) Gunn diode loaded patch antennas. The an-tennas reside on a 0.789-mm-thick dielectric substrate of per-mittivity that measures 41.4 37.21 mm; thesubstrate is backed by an equally sized ground plane [Fig. 2(a)].The construction and experimental characterization of this an-tenna were reported in [5] and [49]; prior efforts at analyzingit relied on FDTD [5] and FETD [12] methods. In our model,the Gunn diodes connect across the centers of strips bonding thepatches to the ground plane [Fig. 2(b)]. Each diode is modeledby an equivalent circuit identical to that used in [5] and [12] anddetailed in Fig. 3, leading to circuit unknowns. Thefield solver processes a mesh of the array comprised of trian-gles and tetrahedrons with average edge length of approximately1 mm, leading to surface and volumeunknowns. The analysis is carried out for time steps,with ps and . The TD-AIM accelerator usesan auxiliary grid with spacings mm, mm,and mm, resulting in auxiliarygrid points.

The array is excited by a normally incident -polarizedGaussian plane wave pulse, with 0.1-mV/m peak amplitude,8-GHz center frequency, and 8-GHz bandwidth. The oscilla-tions due to this low-power broadband plane wave are allowed

Fig. 2. Active patch antenna. Dimensions of the structure are in millimeters.The geometry dimensions and diode locations are shown in (a) top view and(b) side view.

Fig. 3. The i-v characteristics of the Gunn diodes and their circuit model,which is identical to those in [5] and [12].

to build up [Fig. 4(a)] and the resulting steady-state voltagesacross the diodes are plotted in Fig. 4(b). The oscillations acrossthe two diodes are out of phase, as was observed in [5] and [12].The fundamental frequency of the oscillations computed bythe simulator is 12 GHz [Fig. 4(c)], which agrees well with thevalues of 12.2 and 12.08 GHz computed via differential-equa-tion-based schemes of [5] and [12] and the measured valueof 11.8 GHz. Fig. 4(d) compares the normalized polarizedelectric-field pattern of the antenna in the planecomputed by the proposed method at 12 GHz to measurementsat 11.8 GHz [49]. Good agreement is observed.


Fig. 4. (a) Transient voltage on the first diode. (b) Steady stage voltages across the diodes. (c) Frequency spectrum of the voltage across diode 1. (d) Radiationpattern of the active patch antenna at 12.0 GHz compared to measured values at 11.8 GHz.

B. Microwave Amplifier

To further verify the accuracy of the simulator, a non-linear microwave amplifier circuit is analyzed next. Microstripmatching networks are connected to a packaged transistor,which resides on a 0.7874 mm thick dielectric substrate of per-mittivity that measures 17.526 16.256 mm; thesubstrate is backed by an equally sized ground plane [Fig. 5(a)].This circuit was previously analyzed by FDTD-based [8],FETD-based [13], and PWTD accelerated integral-equa-tion-based [31] methods. Here, the circuit solver models theMESFET using the nonlinear large signal circuit model inFig. 5(b), which is similar to those in [8], [13], and [31]; atotal of circuit unknowns result. The field solverprocesses a mesh of the microwave amplifier comprised oftriangles and tetrahedrons with average edge length of approx-imately 1 mm, leading to surface andvolume unknowns. The TD-AIM auxiliary grid spacings are

mm, mm, and mm, resulting inauxiliary grid points.

First, the nonlinear behavior of the amplifier is studied. Theinput (1) and output (2) ports are driven by dc sources with 50internal impedance and amplitudes and , respectively.The dc sources are turned on from zero, according to the firstscheme described in Subsection II-C, using a three-derivativesmooth window function [50]

ns V ns V

(10)

After 3 ns, when a steady state is reached, the transient sourceis activated at the input

ns GHz V

(11)


Fig. 5. Microwave amplifier. (a) The dimensions of the structure inmillimeters. (b) The circuit models for the three-terminal MESFET, the sourceport termination, and the loading port termination.

The analysis is carried out for time steps, withps and . Fig. 6(a) shows the transient input

and output voltage waveforms at the transistor terminals whenthe amplifier is driven by the 6-GHz single-tone excitation of(11) at an input power level of 5.95 dBm (i.e., V).The power dissipated in the loading resistor is calculated fromthe Fourier transform of the steady-state current minus the dccurrent at the output port. The output power is observed at theharmonic frequencies of 6 GHz as shown in Fig. 6(b) for variousdifferent input power levels. Fig. 7(a)–(c) compares the powerlevels of the first three harmonics with those simulated by theFDTD scheme of [8] and FETD scheme of [13]. Good agree-ment is observed.

Next, the broadband behavior of the amplifier is analyzedusing the large signal circuit model of Fig. 5(b) under smallsignal operation. The input port is driven by a unit amplitudeGaussian pulse modulated at 7 GHz with 12-GHz bandwidth.For this analysis, the bias conditions are modeled with dcsources at the ports and initial conditions at the circuits, ac-cording to the second scheme described in Subsection II-C.In this case, only time steps are needed, with

ps and . The input and output voltage wave-forms at the ports shown in Fig. 5(a) are plotted in Fig. 8(a).

Fig. 6. (a) Voltage waveforms across the transistor terminals for an inputpower level of at 6 GHz. (b) Output power spectrum due to the same input.

The -parameters of the amplifier are extracted at the portsover the frequency range of 2–10 GHz and compared with thoseobtained using the differential-equation-based schemes of [8]and [13] in Fig. 8(b). The results obtained using the TD-AIMscheme are in good agreement with those obtained using thedifferential equation algorithms.

C. Reflection-Grid Amplifier

Finally, a quasi-optical grid amplifier is analyzed [51], [52].During the past decade, various grid amplifiers that employup to hundreds of transistors have been demonstrated [53].While most grid amplifiers use a transmission architecture,here, a grid-amplifier with an “active mirror” architecture thataids in heat-sinking [54], [55] is analyzed. The grid amplifier,which is modeled after those reported in [54] and [55], iscomposed of 16 six-terminal differential amplifiers that areconnected by a microstrip network mounted on a 0.8-mm-thickdielectric substrate of permittivity that measures39.47 27.46 mm; the substrate is backed by an equally sized


Fig. 7. Output power at the harmonic frequencies of (a) 6 GHz, (b) 12 GHz,and (c) 18 GHz.

ground plane, whose distance from the bottom of the dielec-tric varies between 1.6 and 31.6 mm in the cases analyzed

Fig. 8. Small signal analysis using the large-signal circuit model and nonzeroinitial conditions. (a) The broadband input signal and the transient waveformsat the input and output ports. (b) S and S of the microwave amplifier.

[Fig. 9(a)–(c)]. The amplifier chips measure 0.4 0.4 mm andare modeled using the nonlinear large signal circuit model ofFig. 10(a), which is identical to that specified in [55], resultingin circuit unknowns. Note that, because the de-tailed circuit models for the transistors used in [55], whichwere custom built, were not available, here the transistors aremodeled using the simple Ebers–Moll model of Fig. 10(b)[40]. Ebers–Moll model parameters are given in Table I. Thefield solver processes a mesh of the reflection-grid ampli-fier using triangles and tetrahedrons with average edge lengthof approximately 0.8 mm, leading to surface and

volume unknowns. The simulations are carried outfor to time steps depending on the incidentfield angle and the ground plane position, the time step size is

ps, and varies between 43 and 51 depending on theground plane position. The TD-AIM auxiliary grid spacingsare mm, mm, and mm,resulting in to auxiliarygrid points depending on the ground plane position.


Fig. 9. Reflection-grid amplifier. The dimensions in millimeters of (a) theentire structure, (b) a single element, and (c) the connections to the chipterminals.

As shown in Fig. 10(a), the amplifier bias conditions aremodeled by dc sources at the ports. After the dc conditions atthe circuits are established via the second scheme describedin Subsection II-C, the grid is illuminated by apolarized Gaussian plane wave, with 1 V/m peak-amplitude,10 GHz center frequency, and 10 GHz bandwidth. In the

(a)

(b)

Fig. 10. Circuit models. (a) The six-terminal differential-amplifier model andthe dc bias values at the interface. (b) The equivalent circuit model for thetransistors (the transport version of the Ebers–Moll model).

TABLE IEBERS-MOLL MODEL PARAMETERS

following simulations, the plane wave is incident from thedirection and the gain of the grid is defined

as the (cross) polarized radar cross section of the structureat the specular angle ( direction) divided bythe top-surface area of the grid (37.47 25.46 mm ); thisreplicates the gain definitions of the measurement setup in[53]–[55]. First, for verification purposes, the gain of thegrid under normal illumination computed by the proposedscheme is compared to that computed by a frequency-domainhybrid simulator. The frequency-domain simulator’s circuit-solver replaces the transistors with their small-signal models atthe operating point, whereas its field-solver uses a (frequency-domain) AIM accelerated integral-equation scheme [46], whichuses the same surface mesh, volume mesh, auxiliary grid, andmoment-matching order as the time-domain solver. Fig. 11(a)


Fig. 11. Reflection-grid amplifier gain. (a) Gain with no bias or 3.5-V bias when the ground plane is 13.6 mm below the dielectric and the grid is illuminatedat normal incidence. (b) Gain with respect to the ground plane position with normal incidence and 3.5-V bias. (c) Gain with respect to the incidence angle withground plane 13.6 mm below the dielectric and with 3.5-V bias. (d) Gain with respect to the frequency with � = 20 incidence angle, with the ground plane13.6 mm below the dielectric, and with 3.5-V bias.

shows the grid gain computed by the frequency- and time-domain simulators, with the transistors either unbiased or biasedat 3.5 V. In both cases, good agreement between the resultsobtained using both schemes is observed. When the transistorsare biased, approximately 15-dB gain is observed around 8.4GHz, which is in good agreement with the measurement of 15dB at 8.4 GHz reported in [55]. Next, the effects of changesin various design parameters on the reflection-grid amplifier’sperformance are studied. Fig. 11(b) plots the grid gain withrespect to the ground-plane position for various frequenciesunder normal illumination. Fig. 11(c) shows the performance ofthe amplifier for off-normal illuminations at various frequenciesand compares the gain with the obliquity factor. The datapresented in both figures support the predicted and measuredtrends reported in [54]. Finally, the measurement setup of [55] isreplicated by illuminating the array from direction andthe computed and measured gains are compared as a functionof frequency in Fig. 11(d). The disagreement between the

measured and simulated results at higher frequencies may beattributed to a variety of differences between the measured andmodeled structures, such as the thickness of the substrate, theground-plane position and size, the details of chip connectionsto the grid, and the transistor characteristics. Nevertheless, thecorrelation between the simulation results obtained using theproposed solver and the measured data of [54] and [55] isevident.

D. Parallel Performance

Finally, Fig. 12 illustrates the parallel performance of thefield-circuit solver in the above simulations. Fig. 12(a) showsthat in all of the above applications most of the computationtime is spent in stages (i) and (v) of the Newton–Raphson algo-rithm of Subsection II-D, as expected. The times in Fig. 12(a)include communication as well as computation times; it is clearthat communication costs do not hamper the scalability of thescheme even when tens of processors are used. Fig. 12(b) shows


Fig. 12. Parallel performance of the simulator for the active patch antenna, themicrowave amplifier, and the reflection-grid amplifier. (a) Average time spent invarious stages of the algorithm at each time step. (b) The memory used by thecircuit processor and the maximum and minimum memory used by the Pprocessors. The dashed lines are ideal speed-up tangents.

the memory requirement of the algorithm and its distributionamong the processors. The memory scalability of the algorithmis limited for the microwave amplifier, which has the smallestnumber of unknowns among the three cases. This is mainly dueto operating system overheads and data replication among pro-cessors, such as geometry description. These factors, however,scale at most linearly with number of unknowns and hence be-come subdominant for problems with more unknowns. In short,Fig. 12 verifies that both the computation time and the memoryrequirement of the proposed scheme show good parallel scala-bility for all three simulations presented.

IV. CONCLUSION

This paper has outlined a parallel FFT accelerated hybridfield-circuit simulator and highlighted its application to theanalysis of various nonlinear microwave systems. The proposed

simulator models fields on distributed passives by time-do-main integral equations and currents and voltages in devicesapproximated in terms of lumped elements by modified nodalanalysis equations. The resulting nonlinear hybrid systemof equations in field and circuit unknowns is solved usinga Newton–Raphson algorithm, which is accelerated by dis-tributing the computational work across multiple processorsand by employing the TD-AIM algorithm to expedite fieldcomputations. The simulator can characterize a broad varietyof microwave circuits accurately, as was demonstrated throughcomparison of data derived from it with independent measure-ments and simulations of a nonlinearly loaded patch antenna,a microwave amplifier, and a reflection-grid amplifier. Further-more, as demonstrated in Section III, the distributed-memoryparallelization of the algorithm shows near-ideal scalability,allowing the simulator to efficiently characterize nonlinearcomponents on electrically large platforms.

The versatility of the proposed simulator can be increasedthrough various extensions that are currently being developed.These include the incorporation of variable time-stepping andreduced-order macromodeling algorithms, as well as accuratelossy- and layered-media formulations for the time-domain in-tegral equations.

APPENDIX

The entries of the vectors and , which are of length, are

for

else

forelse (12)

The entries of the impedance matrices , for, each of which are of size , are

for

for

for

for .(13)


Notice that the first and second conditions in (3) are enforcedon conductor surfaces and dielectric volumes by testing themwith surface and volume functions, respectively. In (13), the en-tries of are defined in terms of the first and second timederivatives of scalar and vector potentials due to each spatialbasis function, respectively

(14)

ACKNOWLEDGMENT

The authors would like to thank the CSE Department andthe National Center for Supercomputing Applications (NCSA)for access to their parallel clusters. The first author thanks B.Fischer and Dr. K. Aygün for a reference circuit-solver programand Dr. M. Lu for discussions on the coupling scheme.

REFERENCES

[1] E. K. Miller, “Time-domain modeling in electromagnetics,” J. Electro-magn. Waves Appl., vol. 8, no. 9/10, pp. 1125–1172, 1994.

[2] H. Schuman, “Time-domain scattering from a nonlinearly loaded wire,”IEEE Trans. Antennas Propagat., vol. 22, no. 4, pp. 611–613, Jul. 1974.

[3] T. K. Liu and F. M. Tesche, “Analysis of antennas and scatterers withnonlinear loads,” IEEE Trans. Antennas Propagat., vol. 24, no. 2, pp.131–139, Mar. 1976.

[4] J. A. Landt, E. K. Miller, and F. J. Deadrick, “Time domain modelingof nonlinear loads,” IEEE Trans. Antennas Propagat., vol. 31, no. 1, pp.121–126, Jan. 1983.

[5] B. Toland, J. Lin, B. Houshmand, and T. Itoh, “FDTD analysis of anactive antenna,” IEEE Microw. Guided Wave Lett., vol. 3, no. 11, pp.423–425, Nov. 1993.

[6] V. A. Thomas, M. E. Jones, M. Piket-May, A. Taflove, and E. Harrigan,“The use of SPICE lumped circuits as sub-grid models for FDTD anal-ysis,” IEEE Microw. Guided Wave Lett., vol. 4, no. 5, pp. 141–143, May1994.

[7] P. Ciampolini, P. Mezzanotte, L. Roselli, and R. Sorrentino, “Accurateand efficient circuit simulation with lumped-element FDTD technique,”IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2207–2215, Dec.1996.

[8] C. Kuo, B. Houshmand, and T. Itoh, “Full-wave analysis of packagedmicrowave circuits with active and nonlinear devices: An FDTD ap-proach,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 819–826,May 1997.

[9] K.-P. Ma, B. Houshmand, Y. Qian, and T. Itoh, “Global time-domainfull-wave analysis of microwave circuits involving highly nonlinear phe-nomena and EMC effects,” IEEE Trans. Microw. Theory Tech., vol. 47,no. 6, pp. 859–866, Jun. 1999.

[10] G. Kobidze, A. Nishizawa, and S. Tanabe, “Ground bouncing in PCBwith integrated circuits,” in Proc. IEEE Int. Symp. EMC, vol. 1, Aug.2000, pp. 349–352.

[11] N. Orhanovic, R. Raghuram, and N. Matsui, “Full wave analysis ofplanar interconnect structures using FDTD-SPICE,” in Proc. IEEE Elec-tronic Components and Technology Conf., 2001, pp. 489–494.

[12] K. Guillouard, M. F. Wong, V. F. Hanna, and J. Citerne, “A new globaltime-domain electromagnetic simulator of microwave circuits includinglumped elements based on finite-element method,” IEEE Trans. Microw.Theory Tech., vol. 47, no. 10, pp. 2045–2048, Oct. 1999.

[13] S.-H. Chang, R. Coccioli, Y. Qian, and T. Itoh, “A global finite-elementtime-domain analysis of active nonlinear microwave circuits,” IEEETrans. Microw. Theory Tech., vol. 47, no. 10, pp. 2410–2416, Dec.1999.

[14] M. A. Alsunaidi, S. M. S. Imtiaz, and S. M. El-Ghazaly, “Electromag-netic wave effects on microwave transistors using a full-wave time-do-main model,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 6, pp.799–808, Jun. 1996.

[15] R. O. Grondin, S. M. El-Ghazaly, and S. Goodnick, “A review ofglobal modeling of charge transport in semiconductors and full-waveelectromagnetics,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6,pp. 817–829, Jun. 1999.

[16] P. Ciampolini, L. Roselli, G. Stopponi, and R. Sorrentino, “Global mod-eling strategies for the analysis of high-frequency integrated circuits,”IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 950–955, Jun.1999.

[17] S. Grivet-Talocia, I. S. Stievano, and F. G. Canavero, “Hybridization ofFDTD and device behavioral-modeling techniques,” IEEE Trans. Elec-tromagn. Compat., vol. 45, no. 1, pp. 31–42, Feb. 2003.

[18] A. Sadigh and E. Arvas, “Treating the instabilities in marching-on-in-time method from a different perspective,” IEEE Trans. Antennas Prop-agat., vol. 41, no. 12, pp. 1695–1702, Dec. 1993.

[19] G. Manara, A. Monorchio, and R. Reggiannini, “A space-time dis-cretization criterion for a stable time-marching solution of the electricfield integral equation,” IEEE Trans. Antennas Propagat., vol. 45, no.4, pp. 527–532, Apr. 1997.

[20] J. Garret, A. E. Ruehli, and C. R. Paul, “Accuracy and stability improve-ments of integral equation models using the partial element equivalentcircuit (PEEC) approach,” IEEE Trans. Antennas Propagat., vol. 46, no.12, pp. 1824–1832, Dec. 1998.

[21] D. S. Weile, G. Pisharody, N.-W. Chen, B. Shanker, and E. Michielssen,“A novel scheme for the solution of time-domain integral equations ofelectromagnetics,” IEEE Trans. Antennas Propagat., vol. 52, no. 1, pp.283–295, Jan. 2004.

[22] S. P. Walker and C. Y. Leung, “Parallel computation of time-domainintegral equation analyses of electromagnetic scattering and RCS,” IEEETrans. Antennas Propagat., vol. 45, no. 4, pp. 614–619, Apr. 1997.

[23] B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysisof transient electromagnetic scattering phenomena using the multilevelplane wave time domain algorithm,” IEEE Trans. Antennas Propagat.,vol. 51, no. 3, pp. 628–641, Mar. 2003.

[24] J. L. Hu, C. H. Chan, and Y. Xu, “A fast solution of time domain integralequation using fast Fourier transformation,” Microw. Opt. Tech. Lett.,vol. 25, no. 3, pp. 172–175, 2000.

[25] A. E. Yılmaz, D. S. Weile, J. M. Jin, and E. Michielssen, “A fast Fouriertransform accelerated marching-on-in-time algorithm for electromag-netic analysis,” Electromagn., vol. 21, pp. 181–197, 2001.

[26] , “A hierarchical FFT algorithm for fast analysis of transient elec-tromagnetic scattering phenomena,” IEEE Trans. Antennas Propagat.,vol. 5, no. 7, pp. 971–982, Jul. 2002.

[27] A. E. Yılmaz, D. S. Weile, B. Shanker, J. M. Jin, and E. Michielssen,“Fast analysis of transient scattering in lossy media,” IEEE AntennasWireless Propagat. Lett., vol. 1, no. 1, pp. 14–17, 2002.

[28] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “A new fast time do-main integral equation solution algorithm,” in IEEE APS Symp. Dig.,vol. 4, 2001, pp. 176–179.

[29] A. E. Yılmaz, K. Aygün, J. M. Jin, and E. Michielssen, “Matching cri-teria and the accuracy of time domain adaptive integral method,” in IEEEAPS Symp. Dig., vol. 2, 2002, pp. 166–169.

[30] A. E. Yılmaz, J. M. Jin, and E. Michielssen, “Time domain adaptiveintegral method for surface integral eqautions,” IEEE Trans. AntennasPropagat., vol. 52, no. 10, pp. 2692–2708, Oct. 2004.


[31] K. Aygün, B. C. Fisher, J. Meng, and E. Michielssen, “A fast hybridfield-circuit simulator for transient analysis of microwave circuits,”IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 573–583, Feb.2004.

[32] A. E. Yılmaz, J. M. Jin, and E. Michielssen, “A parallel time-domainadapative integral method based hybrid field-circuit simulator,” in IEEEAPS Symp. Dig., vol. 3, 2004, pp. 3309–3312.

[33] M. Frigo and S. G. Johnson, “FFTW: An adaptive software architecturefor the FFT,” in Proc. IEEE Int. Conf. Acoustics, Speech, and SignalProcessing (ICASSP), vol. 3, 1998, pp. 1381–1384.

[34] M. J. Bluck, S. P. Walker, and M. D. Pocock, “The extension of time-domain integral equation analysis to scattering from imperfectly con-ducting bodies,” IEEE Trans. Antennas Propagat., vol. 49, no. 6, pp.875–879, Jun. 2001.

[35] Q. Chen, M. Lu, and E. Michielssen, “Integral-equation-based analysisof transient scattering from surfaces with an impedance boundarycondition,” Microw. Opt. Tech. Lett., vol. 42, no. 3, pp. 213–220,2004.

[36] B. Shanker, K. Aygün, and E. Michielssen, “Fast analysis of transientscattering from lossy inhomogeneous dielectric bodies,” Radio Sci., vol.39, no. 2, pp. 1–14, 2004.

[37] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral mod-eling method for electromagnetic scattering by arbitrarily shaped inho-mogenous dielectric bodies,” IEEE Trans. Antennas Propagat., vol. 32,no. 1, pp. 77–85, Jan. 1984.

[38] N. T. Gres, A. A. Ergin, and E. Michielssen, “Volume-integral-equa-tion-based analysis of transient electromagnetic scattering from three-di-mensional inhomogeneous dielectric objects,” Radio Sci., vol. 36, no. 3,pp. 379–386, May/Jun. 2001.

[39] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scatteringby surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol.30, no. 3, pp. 409–418, May 1982.

[40] A. Vladimirescu, The SPICE Book. New York: Wiley, 1994.[41] Y. Wang, D. Gope, V. Jandhyala, and C.-J. R. Shi, “Generalized

Kirchoff’s current and voltage law formulation for coupled circuit-electromagnetic simulation with surface integral equations,” IEEETrans. Microw. Theory Tech., vol. 52, no. 7, pp. 1673–1682, Jul.2004.

[42] J. R. Phillips and J. K. White, “A precorrected-FFT method for elec-trostatic analysis of complicated 3-D structures,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 16, no. 10, pp. 1059–1072, Oct.1997.

[43] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu-merical Recipes in FORTRAN. New York: Cambridge Univ. Press,1992.

[44] N. N. Bojarski, “K-space formulation of the electromagnetic scatteringproblem,” in URSI Dig., 1971, p. 117.

[45] , “The k-space formulation of the scattering problem in the timedomain,” J. Acoust. Soc. Am., vol. 72, no. 2, pp. 570–584, Apr. 1982.

[46] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptiveintegral method for solving large-scale electromagnetic scattering andradiation problems,” Radio Sci., vol. 31, no. 5, pp. 1125–1251, 1996.

[47] H. T. Anastassiu, M. Smelyanskiy, S. Bindiganavale, and J. L. Volakis,“Scattering from relatively flat surfaces using the adaptive integralmethod,” Radio Sci., vol. 33, no. 1, pp. 7–16, 1998.

[48] N. R. Aluru, V. B. Nadkarni, and J. White, “A parallel precorrected FFTbased capacitance extraction program for signal integrity analysis,” inProc. Design Automation Conf., 1996, pp. 363–366.

[49] B. Toland, J. Lin, B. Houshmand, and T. Itoh, “Electromagnetic simula-tion of mode control of a two element active antenna,” in IEEE MTT-SSymp. Dig., 1994, pp. 883–886.

[50] R. W. Ziolkowski and E. Heyman, “Wave propagation in media havingnegative permittivity and permeability,” Phys. Rev. E., vol. 64, p. 056625,2001.

[51] J. W. Mink, “Quasioptical power combining of solid-state millimeter-wave sources,” IEEE Trans. Microw. Theory Tech., vol. 34, no. 2, pp.273–279, Feb. 1986.

[52] R. M. Weikle II, M. Kim, J. B. Hacker, M. P. Delisio, Z. B. Popovic, andD. Rutledge, “Transistor oscillator and amplifier grids,” Proc. IEEE, vol.80, no. 11, pp. 1800–1809, Nov. 1992.

[53] M. P. Delisio, S. W. Duncan, D.-W. Tu, C.-M. Liu, A. Moussessian, J. J.Rosenberg, and D. B. Rutledge, “Modeling and performance of a 100-element pHEMT grid amplifier,” IEEE Trans. Microw. Theory Tech., vol.44, no. 12, pp. 2136–2144, Dec. 1996.

[54] F. Lecuyer, R. Swisher, I.-F. F. Chio, A. Guyette, A. Al-Zayed, W. Ding,M. Delisio, K. Sato, A. Oki, A. Gutierrez, R. Kagiwada, and J. Cowles,“A 16-element reflection grid amplifier,” in IEEE MTT-S Symp. Dig.,2000, pp. 809–812.

[55] A. Guyette, R. Swisher, F. Lecuyer, A. Al-Zayed, A. Kom, S.-T. Lei,M. Oliviera, P. Li, M. Delisio, K. Sato, A. Oki, A. Gutierrez-Aitken,R. Kagiwada, and J. Cowles, “A 16-element reflection grid amplifierwith improved heat sinking,” in IEEE MTT-S Symp. Dig., 2001, pp.1839–1842.

Ali E. Yılmaz (S’96–M’05) was born in Ankara,Turkey, in 1978. He received the B.S. degree inelectrical engineering from Bilkent University,Turkey, in 1999, and the M.S. degree in electricaland computer engineering from the University ofIllinois at Urbana-Champaign (UIUC), in 2001,where he is currently pursuing the Ph.D. degree.

His research interests include computationalelectromagnetics, with emphasis on time-domainintegral equations and their fast solutions, highperformance computing, and coupled electromag-

netic/circuit analysis.Mr. Yılmaz is the recipient of the 2003 Raj Mittra Outstanding Research

Award from the Department of Electrical and Computer engineering and the2003–2004 Interdisciplinary Graduate Fellowship from the Computational Sci-ence and Engineering Program at UIUC. He was the third place winner of theStudent Paper Competition Contest at the IEEE APS Symposium, June 2004,Monterey, CA.

Jian-Ming Jin (S’87–M’89–SM’94–F’01) receivedthe B.S. and M.S. degrees in applied physics fromNanjing University, Nanjing, China, in 1982 and1984, respectively, and the Ph.D. degree in electricalengineering from the University of Michigan, AnnArbor, in 1989.

He is a Professor of electrical and computerengineering and Associate Director of the Center forComputational Electromagnetics at the Universityof Illinois at Urbana-Champaign (UIUC). He hasauthored and co-authored over 140 papers in ref-

ereed journals and 15 book chapters. He has also authored The Finite ElementMethod in Electromagnetics (New York: Wiley, 1st ed. 1993, 2nd ed. 2002)and Electromagnetic Analysis and Design in Magnetic Resonance Imaging(Boca Raton, FL: CRC, 1998), co-authored Computation of Special Functions(New York: Wiley, 1996), and co-edited Fast and Efficient Algorithms inComputational Electromagnetics (Norwood, MA: Artech, 2001). His currentresearch interests include computational electromagnetics, scattering andantenna analysis, electromagnetic compatibility, bioelectromagnetics, andmagnetic resonance imaging. His name is often listed in the UIUC’s List ofExcellent Instructors. He was elected by ISI as one of the world’s most citedauthors in 2002.

Dr. Jin is a member of Commision B of USNC/URSI and Tau Beta Pi. Hewas a recipient of the 1994 National Science Foundation Young InvestigatorAward and the 1995 Office of Naval Research Young Investigator Award. Healso received the 1997 Xerox Junior Research Award and the 2000 XeroxSenior Research Award presented by the College of Engineering, UIUC, andwas appointed as the first Henry Magnuski Outstanding Young Scholar inthe Department of Electrical and Computer Engineering in 1998. He was aDistinguished Visiting Professor in the Air Force Research Laboratory in 1999.He served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS

AND PROPAGATION and Radio Science. He is also on the Editorial Board forElectromagnetics and Microwave and Optical Technology Letters. He was theSymposium Co-chairman and Technical Program Chairman of the AnnualReview of Progress in Applied Computational Electromagnetics in 1997 and1998, respectively.


Eric Michielssen (M’95–SM’99–F’02) receivedthe M.S. degree in electrical engineering (summacum laude) from the Katholieke Universiteit Leuven(KUL), Belgium, in 1987, and the Ph.D. degree inelectrical engineering from the University of Illinoisat Urbana-Champaign (UIUC) in 1992.

He served as a Research and Teaching assistant inthe Microwaves and Lasers Laboratory at KUL andthe Electromagnetic Communication Laboratory atUIUC from 1987 to 1988 and from 1988 to 1992, re-spectively. He joined the Faculty of the Department

of Electrical and Computer Engineering at UIUC in 1993, where he serves asProfessor of electrical and computer engineering and as Associate Director ofthe Center for Computational Electromagnetics. He has authored or co-authoredover 100 journal papers and book chapters and over 160 papers in conferenceproceedings. His research interests include all aspects of theoretical and appliedcomputational electromagnetics. His principal research focus has been on thedevelopment of fast frequency- and time-domain integral-equation-based tech-niques for analyzing electromagnetic phenomena, and the development of ro-bust optimizers for the synthesis of electromagnetic/optical devices.

Prof. Michielssen is a a member of International Union of Radio Scientists(URSI) Commission B. He received a Belgian American Educational Founda-tion Fellowship in 1988 and a Schlumberger Fellowship in 1990. He receivedthe 1994 URSI Young Scientist Fellowship, a 1995 National Science Founda-tion CAREER Award, and the 1998 Applied Computational ElectromagneticsSociety (ACES) Valued Service Award. In addition, he was named 1999 URSIUnited States National Committee Henry G. Booker Fellow and selected as therecipient of the 1999 URSI Koga Gold Medal. Recently, he was awarded theUIUC’s 2001 Xerox Award for Faculty Research, appointed 2002 BeckmanFellow in the UIUC Center for Advanced Studies, named 2003 Scholar in theTel Aviv University Sackler Center for Advanced Studies, and selected as UIUC2003 University Scholar. He was Technical Chairman of the 1997 Applied Com-putational Electromagnetics Society (ACES) Symposium (Review of Progressin Applied Computational Electromagnetics, March 1997, Monterey, CA), andfrom 1998 to 2001 and 2002 to 2003, served on the ACES Board of Directorsand as ACES Vice-President from 1998 to 2001. He was an Associate Editorfor Radio Science from 1997 to 1999, and he currently is an Associate Editorfor the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Documents

A parallel FFT accelerated transient field-circuit simulator