3228 ieee transactions on microwave theory and … · structure-preserving reduction of...

3228 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 62, NO. 12, DECEMBER 2014

Structure-Preserving Reduction of Finite-DifferenceTime-Domain Equations With Controllable

Stability Beyond the CFL LimitXihao Li, Student Member, IEEE, Costas D. Sarris, Senior Member, IEEE, and Piero Triverio, Member, IEEE

Abstract—The timestep of the finite-difference time-domainmethod (FDTD) is constrained by the stability limit known as theCourant–Friedrichs–Lewy (CFL) condition. This limit can makeFDTD simulations quite time consuming for structures containingsmall geometrical details. Several methods have been proposed inthe literature to extend the CFL limit, including implicit FDTDmethods and filtering techniques. In this paper, we propose anovel approach, which combines model-order reduction and aperturbation algorithm to accelerate FDTD simulations beyondthe CFL barrier. We compare the proposed algorithm againstexisting implicit and explicit CFL extension techniques, demon-strating increased accuracy and performance on a large numberof test cases, including resonant cavities, a waveguide structure, afocusing metascreen, and a microstrip filter.

Index Terms—Finite difference time domain (FDTD),model-order reduction (MOR), numerical stability.

I. INTRODUCTION

T HE finite-difference time-domain (FDTD) method (Yee’sFDTD) is one of the most popular algorithms for solving

Maxwell’s equations [1]. Standard FDTD updates electric andmagnetic field values with a leapfrog scheme, which ensuressecond-order accurate approximations of time and spatialderivatives. A remarkable feature of FDTD is that it avoidsexpensive matrix inversions due to an explicit discretizationof time derivatives. As a consequence, FDTD timestep isconstrained by the Courant–Friedrichs–Lewy (CFL) stabilitycondition [1]

(1)

Manuscript received June 25, 2014; revised September 09, 2014 and October23, 2014; accepted October 23, 2014. Date of publication November 14, 2014;date of current version December 02, 2014. This work was supported in part bythe Natural Sciences and Engineering Research Council (NSERC) of Canadaunder the Discovery Grant Program. This work was supported in part by theCanada Research Chairs Program. This paper is an expanded version from theIEEEMTT-S InternationalMicrowave Symposium, Tampa Bay, FL, USA, June1–6 2014.The authors are with the Edward S. Rogers Sr. Department of Electrical

and Computer Engineering, University of Toronto, Toronto, ON, CanadaM5S 3G4 (e-mail: [email protected]; [email protected];[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMTT.2014.2366140

where is the wave velocity in the medium and , ,denote the cell size in the three dimensions. Limit (1) dictatesa very small timestep in problems containing small geometricfeatures relative to the wavelength. In such cases, both the largenumber of unknowns and the small timestep can make FDTDsimulations very time consuming.The CFL limit can be overcome in several ways. Implicit

methods are unconditionally stable for any timestep, but requirematrix inversions [2]–[4]. The direct use of implicit techniquesis prohibitive even for medium-size problems. Model-order re-duction (MOR) has been used to reduce the computational com-plexity of an implicit approach in combination with subgridding[5], [6]. The alternating-direction-implicit finite difference timedomain (ADI-FDTD) [7], [8] has been proposed in an attemptto maintain some of the efficiency of explicit FDTD while en-suring unconditional stability. The ADI-FDTD splits the timestepping process into implicit and explicit half steps, thus guar-anteeing unconditional stability [7]. This step incurs additionalcomputation costs, although partially mitigated through the useof a larger timestep.Recently, alternative methods to overcome the CFL limit

have been proposed. In spatial filtering [9], unstable harmonicsthat arise above the CFL limit are removed at runtime usinga fast Fourier transform and low-pass filtering. This approachis simple to implement, but requires special care at materialboundaries to avoid aliasing in the Fourier transform process. In[10] and [11], stable FDTD simulations beyond the CFL limithave been obtained by first running a short FDTD simulationat a stable timestep in order to identify the dominant and stableeigenmodes of the structure. This information is then used toremove the unstable modes from the FDTD equations and runabove the CFL limit.In this paper, we propose a new way to accelerate FDTD sim-

ulations beyond the CFL limit, combiningMOR and eigenvalueperturbation. MOR is applied to FDTD equations to reduce theirorder. For a timestep below the CFL limit, we prove that thereduced model is stable by construction, differently from ourprevious approach [12]. For timesteps above the CFL limit, wepropose a perturbation algorithm to enforce late time stability.The reduction process and the extension of the stability limitmake the proposed technique faster than standard FDTD, withvery small loss of accuracy. Moreover, the proposed reductionprocess preserves the structure of the original FDTD equations,making the reducedmodel easy to integrate in an existing FDTDcode.

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LI et al.: STRUCTURE-PRESERVING REDUCTION OF FDTD EQUATIONS WITH CONTROLLABLE STABILITY BEYOND CFL LIMIT 3229

This paper makes the following two contributions.1) We show how the CFL limit can be extended withouthaving to perform time-consuming filtering operationsat runtime, as in spatial filtering [9], or having to iden-tify the dominant eigenmodes of the structure with apre-processing step that requires careful monitoring ofconvergence [10], [11].

2) We propose a MOR method for FDTD equations thatworks directly in the discrete time domain, as opposedto previous methods that worked in the continuous timedomain [5], [6], [2]. This approach makes the enforce-ment of late time stability straightforward, and preservesthe structure of FDTD equations, differently from ourprevious method [12].

Some preliminary results from the proposed approach were pre-sented without proofs in [13]. In this paper, a complete deriva-tion with proofs is provided, together with new ideas for an op-timal implementation that make the method scalable to 3-D sim-ulations.This paper is organized as follows. In Section II, we cast

FDTD equations in matrix form and develop the proposedreduction algorithm. We also prove that the reduced modelis stable by construction for timesteps below the CFL limit.Section III shows how stability can be enforced for timestepsabove the CFL limit. In Section IV, we discuss implementationdetails. Finally, in Section V we apply the proposed methodto five test structures, demonstrating its excellent accuracy,computational efficiency, and scalability to 3-D problems ofpractical interest.

II. STABILITY-PRESERVING MOR OF FDTD EQUATIONS

A. Matrix Formulation of FDTD EquationsWe start from Yee’s FDTD equations [1] that, in one dimen-

sion, read

(2a)

(2b)

While, for the sake of readability, we stated FDTD equations in1-D, all results presented in the paper are valid in the general3-D case. In (2a) and (2b), denotes the electric field at timeand position , while denotes the magnetic field.

With , , , and , we denote permittivity, permeability,electric conductivity, and magnetic conductivity, respectively.Terms and denote electric and magneticsources. FDTD equations can be arranged into matrix form [14]

(3)

where• and are diagonal matrices containing the electricpermittivity and magnetic permeability value for each cell;

• and are diagonal matrices containing the electricand magnetic conductivity values for each cell;

• matrix arises from the discretization of the curl operatorsin Maxwell’s equations, and contains terms in the form( , , );

• vector includes all sources;• matrix contains 1’s corresponding to electric and mag-netic source locations.

Representation (3) holds for general 3-D problems, with nonuni-form material properties and a nonuniform Cartesian grid [14].To compact the notation, we rewrite (3) as [14]

(4)

with

(5)

and where

(6)

is a vector of size , where is the number ofelectric field unknowns, and is the number of magnetic fieldunknowns.

B. Stability ConditionsWritten in form (4), a system of FDTD equations can be in-

terpreted as a discrete time system with input and state. The stability of the FDTD system (4) is controlled by the

eigenvalues of the update matrix [14]

(7)

that must be all inside or on the boundary of the unit circle inthe complex plane [14], [15]

(8)

The eigenvalues of the update matrix are equal to the polesof the discrete time system (4). In [14], it is shown that the fol-lowing two inequalities are equivalent to (8)

(9)

(10)

where denotes a positive semidefinite matrix, and de-notes a positive definite matrix. A symmetric matrix is pos-itive definite if for any vector we have . Itis positive semidefinite if . Conditions (9) and (10)were proposed in [14] and have an intuitive physical explana-tion. Inequality (9) simply requires all conductivities to be pos-itive. Inequality (10) can be shown [14] to be equivalent to theCFL limit (1) and limits the maximum timestep that can beused in a stable FDTD simulation.


C. Model-Order ReductionWe now reduce FDTD equations (4) using the structure-pre-

serving reduced-order interconnect macromodeling (SPRIM)MOR technique [16]. Firstly, from the matrices in (4), wegenerate a projection matrix

(11)

using the robust Arnoldi process [16], [17].Matrices andare orthonormal and of size and , respectively,with much smaller than and . We then approximatethe full vector of unknowns with a reduced vector as

(12)

Substituting (12) into (4), and multiplying on the left by ,we obtain

(13)

and, after carrying out matrix multiplications,

(14)

where , , and are “com-pressed” versions of , , and , respectively. The order of(14) is , which is much lower than order of the originalsystem (4). The reduced model order, and consequently, its ac-curacy, can be controlled by choosing the number of columnsof the projection matrices and generated by the Arnoldialgorithm. Due to the small size, (14) can be solved very quicklyto find the reduced unknowns . Once is available, thefields at any point in the system are computed through (12).Using (5) and (12), the matrices in (14) can be written as

(15)

where

(16)(17)(18)

Owing to the block-diagonal nature of the projection ma-trix (11), the reduced matrices (15) have exactly the sameblock structure of the original FDTD matrices (5). Equa-tions (16)–(18) are compressed counterparts of the originalpermittivity, permeability, conductivity, and “curl” matrices.Although, after the reduction process these blocks are full, theycan be easily diagonalized since their size is small. Therefore,the reduction process preserved the structure of the originalFDTD equations (4), which is a novel result. Being in the sameform as FDTD equations, the solution of (14) can be computedin a leapfrog manner for increased efficiency.

D. Stability Preservation Below the CFL LimitWe now discuss the stability of the reduced model (4). First,

we consider the case when is below the CFL limit of the

original FDTD equations, showing that the obtained reducedmodel is stable by construction. Since we have preserved thestructure of the FDTD equations, stability conditions (9) and(10) can be also applied to the reduced model

(19)(20)

The first condition can be rewritten as

(21)

Since the original model (4) satisfies (9), the last expression in(21) is positive semidefinite by construction, as it is the con-gruence of a positive semidefinite matrix [18]. Similarly, since

because of (10), and is full rank, we have

(22)

Therefore, the proposed approach preserves stability by con-struction, avoiding the need for an additional post-processingstep to enforce its stability as in [12].

III. STABILITY ENFORCEMENT ABOVE THE CFL LIMIT

A. Theoretical DerivationIf the chosen is beyond the CFL limit of the original

FDTD equations, conditions (9) and (10) will be violated andreduced model (14) may contain unstable eigenvalues. How-ever, due to its small size, its stability can be easily enforced,effectively breaking the CFL barrier.From stability criteria (9) and (10), we see that changing

will only affect the second condition since the first one does notdepend on . In order to make the reduced model stable, weneed to enforce

(23)

which can be achieved by perturbing as follows. Using theSchur complement [19], we can state two conditions equivalentto (32)

(24)

(25)

It can be seen that (24) always holds, while (25) is the onlysource of potential instability at refined CFL numbers. Rear-ranging terms in (25), we arrive at the following inequality:

(26)

If we denote the singular values [18] of asfor , we have that (26) holds if and only if [14]

for (27)


Above the CFL limit, some singular values may violate (27),and make the reduced model unstable. In order to enforce itsstability, we propose the following procedure.1) Compute the singular value decomposition [18]

(28)

where is a diagonal matrix containing the singular values. This operation is cheap since it is performed on amatrix

of size . Among the five test cases of Section V, thelargest is used for the waveguide filter, where we have

. The SVD decomposition of a 100 100 matrixtakes less than 20 ms on a 2.7-GHz CPU.

2) Perturb the singular values , which exceed (27)

ifotherwise (29)

where is slightly less than 1. In the examples ofSection V, we used . Form a new diagonalmatrix with the perturbed singular values .

3) Obtain the perturbed matrix

and replace with in (15).This procedure leads to a reduced model, which satisfies (19)and (20) by construction, and is thus stable for a timestep abovethe CFL limit. Numerical tests, presented in Section V, willshow that the proposed reduction and perturbation process re-sults in a very small error compared to a standard FDTD, of theorder of 1%. This finding is consistent with previous works thatdemonstrated that the CFL limit can be extended without im-pairing accuracy [9]–[12]. However, in the proposed method,the CFL limit is extended without resorting to an implicit formu-lation or filtering operations at runtime [9], which reduce com-putational efficiency.

B. Demonstration of Stability EnforcementWe illustrate the proposed stability enforcement method on

a simple example. We consider a 1 m 1 m 1 m perfectelectric conductor (PEC) cavity discretized with a 3-D FDTDgrid with cell size m along each dimension. We letthe excitation be a Gaussian pulse with a maximum frequencyof 0.3 GHz, which leads to an effective of 10 at 0.3 GHz.There are two resonant frequencies within the excitation band-width, one at 0.21199 GHz and one at 0.25963 GHz. Fig. 1 plotsthe eigenvalues of (4) for and . With , wedenote the CFL extension factor

(30)

where is the maximum timestep compatible with theCFL limit (1). In the first case, the timestep is below the CFLlimit and all eigenvalues fall inside the unit circle, as shown inthe left panel of Fig. 1. In the second case, since the timestepviolates the CFL constraint, some eigenvalues move into theunstable region, as shown in Fig. 1 (right panel). We thereforeapply the stability enforcement procedure of Section III-A, per-turbing the matrix in (4). Fig. 2 shows the eigenvalues of the

Fig. 1. Example of Section III-B: eigenvalues of FDTD equations (4) belowthe CFL limit (left panel, ) and above the CFL limit (right panel,

). The stability region is given by the red circle (in the online version).

Fig. 2. Example of Section III-B: eigenvalues of FDTD equations (14) forafter stability has been enforced using the method of Section III-A.

perturbed system, which are all stable since they fall on the unitcircle.In this example, due to the small size of the problem, we

have enforced stability directly on the original FDTD equa-tions. For larger problems, such as those that will be presentedin Section V, enforcement will be performed after the size ofthe problem has been reduced through MOR.

IV. PRACTICAL IMPLEMENTATIONIn this section, we discuss how the proposed method has been

implemented for maximum computational efficiency.

A. Complex Frequency HoppingSPRIM [16], like all order reduction methods based on mo-

ment matching, generates a reduced model accurate near a givenexpansion point in the complex frequency plane. As the orderof the reduced model is increased through , the bandwidthof validity of the reduced model around the expansion pointgrows. In terms of eigenvalues, as increases, more and moreeigenvalues of the original system will be matched by the re-duced model eigenvalues, starting from those closer to the ex-pansion point. By choosing the position of the expansion point,one can thus optimize the accuracy of the reduced model, andguide its convergence towards the most relevant eigenvalues ofthe system.In order to ensure accuracy over a large bandwidth, it is

common practice to take multiple expansion points usingso-called “complex frequency hopping” [20]. In this work, weuse the following distribution of expansion points:

(31)


for . This formula places expan-sion points along a circular arc of radius centered at theorigin of the complex plane. The magnitude of the pointsis chosen to be slightly higher than 1, and will be usedin all numerical examples of Section V. With this distribution,one expansion point is always placed near . Through thisexpansion point, we capture the low-frequency response of thesystem since corresponds to static conditions. The otherexpansion points are placed near the unit circle up to the max-imum frequency of interest . With them, we capture theeigenvalues along the unit circle, starting from those that are inmagnitude close to one. These eigenvalues indeed have a sig-nificant impact on the system response since they correspond toweakly damped modes. Eigenvalues well inside the unit circleare instead highly damped, and their contribution to the systemresponse quickly fades away. It has been experimentally deter-mined that distribution (31) significantly improves accuracy fora fixed reduced model size. An additional benefit is the reduc-tion of the Gram–Schmidt orthogonalizations required to gen-erate the Krylov subspace [16], [17].

B. Linear System Solution

A linear system must be solved for each new moment gener-ated with the Arnoldi process used to generate (11). The systemis in the form

(32)

where is the current expansion point. When , sinceis upper triangular, the system can be solved very ef-

ficiently. For , the system can be solved with one LUdecomposition [21] in 2-D and small 3-D cases, similarly towhat done in [5] and [6]. For large 3-D cases, iterative methodsmust be used, and we adopted the conjugate gradient squaredmethod [22] available in MATLAB. We determined, throughthe test cases of Section V, that with a normalized residue limitof 10 , the iterative solver leads to results comparable to thoseobtained with a direct solver.In solving (32), we also exploit the 2 2 block structure of

and . Let us denote the four blocks of the system matrix as

(33)

where and are diagonal matrices. Using the Schurcomplement [19], we first solve for in (32)

(34)

and then solve for ,

(35)

which can be done very quickly since is diagonal. With theimplementation discussed in this section, we were able to applythe proposed method to 3-D simulations of practical relevancewith more than 1 million unknowns.

TABLE ICAVITIES OF Section V-A: SIMULATION PARAMETERS

C. Choice of Reduced Order

In the test cases of Section V, the reduced model order forall methods based on MOR was selected manually to achievethe same accuracy from all techniques, and meaningfully com-pare computational costs. Finding the optimal reduced orderin MOR is an active research topic, and several approacheshave been proposed based on empirical error measures [23],eigenvalue monitoring [24], and false nearest neighbor [25].Rigorous methods exist, however, only for the truncated bal-anced realization approach to MOR [26]. Due to its high com-putational cost, truncated balanced realization is not suitablefor large electromagnetic systems. Future work will investigatewhich method, among those cited, is most suitable to estimatethe reduced model order in the proposed algorithm.

V. NUMERICAL RESULTS

A. 2-D and 3-D Cavities

The proposed method to reduce FDTD equations and extendthe CFL limit was implemented in MATLAB, and applied toseveral test structures. First, we consider two empty cavitieswith PEC walls, one in a 2-D setting, and the other one in a3-D setting. The sidelength of the cavity is 1 m in both cases. Asingle source and probe are placed to capture all resonant fre-quencies within the interested range. The input is a Gaussianpulse with a bandwidth of 0.5 GHz. The proposed method hasbeen compared against four other approaches in the literature:an implicit integration of Maxwell’s equations combined withMOR [5], ADI-FDTD [7], spatial filtering [10], and the reduc-tion algorithm of [11].We investigated their overall run time, ac-curacy, and approximate numerical dispersion below and abovethe CFL limit. Analytical resonant frequencies were calculatedand used as accuracymetric. All simulations were run for 10 000timesteps in order to achieve sufficient resolution in the fre-quency domain. They were also run for 10 timesteps to de-termine their late time stability. The most relevant simulationsettings are summarized in Table I.For the proposed method, we set the order of the reduced

model to 80. Five expansion points were used, distributedin the complex plane according to (31) with and

GHz. A direct linear system solver (LU decompo-sition) was used in the 2-D case, while the iterative conjugategradient squared method was used for the 3-D case. The set-tings of the other methods were selected in order to obtain acomparable accuracy. A reduced order of 40 was used for [5],while the method of Gaffar and Jiao required 500 timesteps


Fig. 3. 2-D cavity of Section V-A: frequency response of the cavity obtainedfrom the results of Yee’s FDTD (—), the implicit method of [5] [(- - -), red inonline version], and ADI-FDTD [(- – -), pink in online version]. The proposedmethod, spatial filtering [9], and the method of Gaffar and Jiao [10] gave thesame results and are depicted with a single curve [(- - -), blue in online version].Yee’s FDTD was run below the CFL limit . All other methods wererun above the CFL limit .

Fig. 4. As in Fig. 3, but with focus on the resonance. Color coding isthe same as in Fig. 3. The different methods were run at (left panel)and at (right panel). Yee’s FDTD was run in both cases at .

to accurately identify the important system eigenvalues. Theweighting coefficient, of Gaffar and Jiao was set at 10 .We investigate the accuracy of the different methods by

looking at resonant frequencies. The frequency response ofthe 2-D cavity is depicted in Fig. 3, where we can observethat all methods accurately capture the resonant frequencies ofthe structure, even when run above the CFL limit. Only smalldeviations can be observed in the highest resonances. A zoomon the resonance at 0.45 GHz is provided in Fig. 4. The smallincrease in dispersion due to the CFL extension can be observedby comparing the two panels of Fig. 4. Fig. 5 shows the relativeerror on the first resonances for the different methods, whichmay be attributed to numerical dispersion. When run above theCFL limit, all methods introduce some additional dispersionwith respect to Yee’s FDTD run below the CFL limit. It can beobserved that explicit methods (proposed method, [9] and [10])introduce less dispersion than implicit alternatives [7], [5].Fig. 6 refers to the 3-D cavity and shows the relative error onthe first six resonant frequencies obtained with Yee’s FDTD andthe proposed method run at different CFL extension factors,which remains well below 1% in all cases.

Fig. 5. 2-D cavity of Section V-A: relative error on the first six resonant fre-quencies obtained with Yee’s FDTD run at — , the implicit methodof [5] [ — , red in online version], and ADI-FDTD [ —– , blue in online ver-sion]. The proposed method, spatial filtering [9], and the method of Gaffar andJiao [10] gave the same results and are depicted with a single curve [ — , pinkin online version]. Yee’s FDTD was run below the CFL limit . Allother methods were run above the CFL limit . Gold standard: ana-lytical formulas.

Fig. 6. 3-D cavity of Section V-A: relative error on the first six resonant fre-quencies. Yee’s FDTD — at and proposed method at[ — , red in online version], [ —– , blue in online version], and

[ — , pink in online version]]. Gold standard: analytical formulas.

Table II shows the simulation time breakdown for the 2-Dcase. Below the CFL limit , the proposed method en-sures a speed up of 4.32 with respect to the FDTD, thanksto the reduced size of the generated model, which makes itssolution very cheap. The proposed method is also faster thanthe other tested methods. CFL extension techniques such asthe ADI-FDTD and spatial filtering are not necessary for

, but have been included to illustrate runtime scaling. Whenrun above the CFL limit, at , all methods deliveredstable results and achieved a higher speed-up with respect tothe FDTD. The proposed method offers a speed up of about 5with respect to the FDTD, which is comparable to the speed upobtained with the method of Gaffar and Jiao [10].Table III shows the simulation time break down for the pro-

posed method and FDTD for the 3-D case. It can be observedthat the proposed method demonstrates a significant speed upover standard FDTD. In this case, due to the large number ofunknowns, MOR time dominates over run time. As a result, thetotal time required for a single simulation improves only slightlyas the CFL limit is extended. In such cases, overcoming the CFLlimit can still be beneficial if multiple system simulations are re-quired since, once generated, the proposedmodels can be solvedmultiple times with different inputs. Extending the CFL limit


TABLE II2-D CAVITY OF Section V-A: EXECUTION TIME BREAKDOWNFOR THE DIFFERENT METHODS. ALL TIMES ARE IN SECONDS

TABLE IIICAVITY OF Section V-A: EXECUTION TIME BREAKDOWN FOR FDTD ANDTHE PROPOSED METHOD FOR DIFFERENT CFL EXTENSION FACTORS.

ALL TIMES ARE IN SECONDS

decreases the cost of each simulation, as can be observed in thecolumn “Run” of Table II.

B. 2-D Waveguide with IrisesA2-Dwaveguide filter operating in the TMmode of size 5 cm50 cm is discretized into a 41 401 mesh withmm. The layout of the waveguide is shown in Fig. 7. A

Gaussian current line source with bandwidth of 3 GHz is placedat one end of the waveguide, while a line probe is placed onthe other end. The waveguide is filled with a dielectric materialwith . Five irises (length: 1.25 cm, aperture size: 1 cm,separation: 5 cm) are evenly placed within the waveguide. Thewaveguide is terminated at both ends on a fourth-order matchedabsorber with a thickness of five cells. A matched absorber isused for simplicity, although we have shown in [12] that a splitPML with auxiliary equations can also be used. The minimum

is 80 at 3 GHz. The original system size is 48 440, andthe size of the reduced model generated with the proposed algo-rithm is 200. Due to the wide band of the excitation, five expan-sion points were placed on the complex plane according to (31)with and GHz. A direct solver (LU decom-position) was used to solve (32). The FDTD and the proposedtechnique were run for 20 000 timesteps, in order to allow forthe input power to dissipate in the structure.

Fig. 7. Layout of the waveguide considered in Section V-B. For readability,axes are not in scale.

Fig. 8. Waveguide of Section V-B: Magnetic field at the probes (top) andtransmission coefficient (bottom) for Yee’s FDTD at and for theproposed method at and .

Fig. 8 shows the simulation results for both standard FDTDand the proposed method run below and above the CFL limit.Top panel depicts the magnetic field received by the probes,and shows the stability of the proposed method, even whenrun five times above the CFL limit. The bottom panel givesthe transmission coefficient over frequency. Excellent ac-curacy in both the time and frequency domain may be observed.Fig. 9 plots the relative error between the proposed method andFDTD, calculated on the transmitted waveform for a CFL ex-tension factor of and . The peak error re-mained under 1.65% for all cases, while the average errors re-mained under 0.21%. Timing results for FDTD and the proposedmethod are summarized in Table IV. The proposed method sub-stantially accelerates the analysis of the waveguide, giving aspeed up of 11 at the CFL limit , and of almost16 when run with a timestep, which is nine times larger thanthe CFL limit . These results confirm the advantagesof the proposed method, which combines MOR and eigenvalueperturbation to accelerate FDTD simulations beyond the CFLlimit.

C. 3-D Focusing MetascreenThe proposed method is applied to the focusing metascreen

structure first proposed in [27] and subsequently investigated


Fig. 9. Waveguide of Section V-B: relative error on the transmitted waveformbetween the proposed method run at , , and andYee’s FDTD run at . Only the first 3000 time steps are shown forclarity (error has its maximum in the time interval shown).

TABLE IVWAVEGUIDE OF Section V-B: EXECUTION TIME BREAKDOWN FOR YEE’SFDTD AND THE PROPOSED METHOD AT REFINED CFL NUMBERS.

ALL TIMES ARE IN SECONDS

using the FDTD in [28]. This test case involves the transmis-sion of a plane wave through a metallic screen with a central slotfor focusing onto an image plane. The 3-D simulation domainis of size 61 61 61 with mm, andis terminated on all sides with fourth-order five-cell matchedabsorbers. A plane of uniform sinusoidal sources at 10 GHz isplaced on one side of the PEC screen, which is one-cell thick.The screen has a single focusing slot of size 13.2 mm 1.2 mm.Probes are placed on the other side of the screen along the centeraxis of the image plane, at a distance of 0.15 from the screen.A very high ratio of 100 is required due to the resonatingnature of the slot. The fine mesh makes the size of the orig-inal FDTD equations quite high . Due tothe single-frequency excitation, only a pair of expansion pointswere used, given by

(36)

with and GHz. The proposed method wasused to generate a reduced model of order 40 and simulationswere run for 10 000 timesteps until a steady state was reachedon the image plane.Fig. 10 compares the electric field on the image plane

calculated with the proposed method and with standardFDTD. Fig. 11 plots the time-domain error between the pro-posed method and FDTD calculated on the average electricfield amplitude on the image plane. The maximum error at

and is 0.28%, 0.13%, and 0.28%, respec-tively. An excellent accuracy can be observed, even when theproposed method is run at five times the maximum timestepallowed in the conventional FDTD. Table V shows the simula-tion time breakdown for the standard FDTD and the proposed

Fig. 10. Focusing metascreen of Section V-C: average amplitude of the electricfield on the image plane. Comparison of Yee’s FDTD — atagainst the proposed method at [ — , red in online version],[ —– , blue in online version], and [ — , pink in online version].

Fig. 11. Focusing metascreen of Section V-C: relative error on the average am-plitude of the electric field on the image plane. The proposed method (runat , , and ) is compared against Yee’s FDTD (runat ).

TABLE VFOCUSING METASCREEN OF Section V-C: EXECUTION TIME BREAKDOWN

FOR YEE’S FDTD AND PROPOSED METHOD AT REFINEDCFL NUMBERS. ALL TIMES ARE IN SECONDS

method, which leads to a speed up of 4.28 . In this case, theextension of the CFL limit has a small influence on the totalsolution time for the proposed method since the reduction stepdominates the solution of the reduced model due to the fairlylarge size of the problem. Future investigations will focus onimproving the efficiency of the reduction process for large-scaleproblems.

D. 3-D Microstrip FilterThe final test case is an application of the proposed method

to a 3-D multi-port microstrip filter from [29]. The structure isshown in Fig. 12. The simulation domain is of size 81 91 14cells with mm and mm. A one-cellthick PEC microstrip rests on a dielectric substrate withand thickness of three cells. The PEC microstrip is seven cellwide (2.8 mm) and two ports are placed at its ends. The domain


Fig. 12. Microstrip filter example of Section V-D: layout.

Fig. 13. Microstrip filter of Section V-D: time-domain reflected (top) and trans-mitted (bottom) waveforms computed with Yee’s FDTD — at andwith the proposed method at [ — , red in online version],[ —– , blue in online version], and [ — , pink in online version].

is terminated on five sides with five-cell fourth-order matchedabsorbers. A PEC wall is used for the sixth side to model theground plane. The size of the original FDTD (4) is 619 164. Thesimulation utilizes uniform line sources and probes. A Gaussianpulse of 20-GHz bandwidth is used to extract the andparameters. A reduced model (14) of order 100 was generatedusing five expansion points (31) with andGHz. Simulations were run for 8000 timesteps, when most

of the input power was dissipated.Fig. 13 depicts the time-domain reflected and transmitted

waveforms, while Fig. 15 plots the and parametersextracted from the time-domain analysis. Fig. 14 plots therelative error between the proposed method and Yee’s FDTD.The proposed method was run for a CFL extension factor of

and , and resulted in a maximum erroralways lower than 1.85%, on both reflected and transmittedwaveforms. These results further confirm the excellent accu-racy of the proposed technique and its stable behavior abovethe CFL limit. Finally, Table VI shows the simulation timebreakdown for the proposed method compared to Yee’s FDTDat various CFL extension factors. Also, in this case we observe

Fig. 14. Microstrip filter of Section V-D: relative error between the proposedmethod at , , and and Yee’s FDTD at .The reflected (top) and transmitted (bottom) waveforms are considered.

Fig. 15. Microstrip filter of Section V-D: (top) and (bottom) parame-ters obtained with Yee’s FDTD — at and with the proposed methodat [ — , red in online version], [ —– , blue in online ver-sion], and [ — , pink in online version].

TABLE VIMICROSTRIP FILTER OF Section V-D: EXECUTION TIME BREAKDOWN FORYEE’S FDTD AND THE PROPOSED METHOD. ALL TIMES ARE IN SECONDS

a speed up with respect to a standard FDTD, with very smallloss of accuracy.

VI. CONCLUSIONWe have proposed a new method to perform stable FDTD

simulations beyond the CFL limit. Themethod is based onMOR


and eigenvalue perturbation. In comparison to the state-of-the-art, the proposed technique preserves the structure of the FDTDequations in the reduction process and can handle 3-D problemswith more than 1 million unknowns. Stability below the CFLlimit is guaranteed by construction by the reduction process.Stability above the CFL limit is instead achieved through anovel approach based on eigenvalue perturbation. The proposedmethod was applied to microstrip-based circuits, waveguidesand resonant cavities. Speed ups of up to 16 were shown withrespect to FDTD. Examples also demonstrated the accuracy ofthe proposedmethod, which always resulted in errors lower than2% with respect to standard FDTD. Future work will focus ondevising an algorithm to determine the optimal order of the re-duced model, and on further improving the computational costof the reduction step, which currently limits the efficiency of themethod for very large structures.

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Xihao Li (S’12) received the B.A.Sc. degree in elec-trical engineering from the University of Toronto,Toronto, ON, Canada, in 2012, and is currentlyworking toward the M.A.Sc. degree in electricalengineering at the University of Toronto.His research interests are finite-difference electro-

magnetic simulation and model-order reduction tech-niques.

Costas D. Sarris (SM’08) received the Diploma de-gree (with distinction) in electrical and computer en-gineering from the National Technical University ofAthens, Athens, Greece, in 1997, and M.Sc. degreein applied mathematics and Ph.D. degree in electricalengineering from the University of Michigan, AnnArbor, MI, USA, both in 2002.He is currently a Full Professor and the Eugene V.

Polistuk Chair in Electromagnetic Design with theDepartment of Electrical and Computer Engineering,University of Toronto, Toronto, ON, Canada. His re-

search interests are in the area of numerical electromagnetics with an emphasison high-order multi-scale computational methods, modeling under stochasticuncertainty, as well as applications of numerical methods to wireless channelmodeling, wave propagation in complex media and metamaterials, wirelesspower transfer, RF therapeutics and electromagnetic compatibility/interference(EMI/EMC) problems.Prof. Sarris was an associate editor for the IEEE TRANSACTIONS ON

MICROWAVE THEORY AND TECHNIQUES (2009–2013) and IEEE MICROWAVEAND WIRELESS COMPONENTS LETTERS (2007–2009). He is the TechnicalProgram Committee (TPC) chair for the 2015 IEEE Antennas and Propa-gation (AP-S) International Symposium on Antennas and Propagation andCNC/USNC Joint Meeting, Vancouver, BC, Canada. He was the recipient of the


IEEE Microwave Theory and Techniques Society (IEEE MTT-S) OutstandingYoung Engineer Award in 2013 and an Early Researcher Award from theOntario Ministry for Research and Innovation in 2007. His students were therecipients of paper awards of the 2009 IEEE MTT-S International MicrowaveSymposium (IMS), the 2008 Applied Computational Electromagnetics SocietyConference, and the 2008 and 2009 IEEE International Symposia on Antennasand Propagation.

Piero Triverio (S’06–M’09) received the M.Sc. andPh.D. degrees in electronic engineering from the Po-litecnico di Torino, Turin, Italy in 2005 and 2009, re-spectively.He is currently an Assistant Professor with the

Department of Electrical and Computer Engineering,University of Toronto, Toronto, ON, Canada, wherehe holds the Canada Research Chair in Modelingof Electrical Interconnects. From 2009 to 2011, hewas a Research Assistant with the ElectromagneticCompatibility Group, Politecnico di Torino. He has

been a Visiting Researcher with Carleton University, Ottawa, ON, Canada, andwith the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA.His research interests include signal integrity, electromagnetic compatibility,and model-order reduction.Dr. Triverio was the recipient of several international awards, including

the 2007 Best Paper Award of the IEEE TRANSACTIONS ON ADVANCEDPACKAGING, the EuMIC Young Engineer Prize of the 13th European Mi-crowave Week, and the Best Paper Award of the IEEE 17th Topical Meetingon Electrical Performance of Electronic Packaging (EPEP 2008).

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