IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015 4003
An Explicit and Unconditionally Stable FDTDMethod for the Analysis of General
3-D Lossy ProblemsMd Gaffar and Dan Jiao, Senior Member, IEEE
Abstract—The root cause of the instability of an explicit finitedifference time-domain (FDTD) method is quantitatively iden-tified for the analysis of general lossy problems where bothdielectrics and conductors can be lossy and inhomogeneous. Basedon the root cause analysis, an efficient algorithm is developed toeradicate the root cause of instability, and subsequently achieveunconditional stability in an explicit FDTD-based simulation ofgeneral lossy problems. Numerical experiments have demon-strated the unconditional stability, accuracy, and efficiency of theproposed method.
T HE FINITE difference time-domain (FDTD) method hasbeen one of the most popular methods for time-domain
analyses [1], [2]. It has gained a wide-spread popularity notonly in electromagnetic simulations but also for photonic, ther-mal, biological, aerodynamic, and many other applications. Anexplicit FDTD method requires no matrix solution. However,its time step is traditionally restricted by the smallest spacestep to ensure the stability of a time-domain simulation, as dic-tated by the Courant–Friedrich–Levy (CFL) condition. Whenthe space step of a given problem can be determined solely froman accuracy point of view, the time step required by the CFLcondition has a good correlation with the time step determinedby accuracy. However, when the problem involves fine spacefeatures relative to working wavelength, which is common inmany engineering problems, the time step dictated by the stabil-ity condition can be orders of magnitude smaller than the timestep required by accuracy. As a result, a tremendous number oftime steps need to be simulated to complete one simulation, ren-dering the overall FDTD simulation computationally expensive,although the computational cost at each time step is trivial.
In contrast to traditional explicit methods that are condi-tionally stable, in an unconditionally stable method, the choiceof time step does not depend on space step, and one can use
Manuscript received October 28, 2014; revised April 05, 2015; accepted June08, 2015. Date of publication June 23, 2015; date of current version September01, 2015. This work was supported in part by the NSF under Grant 0747578 andGrant 1065318 and in part by the DARPA under Grant HR0011-14-1-0057.
The authors are with the School of Electrical and Computer Engineering,Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAP.2015.2448751
an infinitely large time step without making a time-domainsimulation unstable. In the past decade, a number of implicitunconditionally stable methods have been developed such asthe alternating-direction implicit (ADI) method [3], [4], theCrank–Nicolson (CN) method [5], the CN-based split step (SS)scheme [6], the pseudospectral time-domain (PSTD) method[7], the locally one-dimensional (LOD) FDTD [8], [9], theLaguerre FDTD method [10], [11], the associated Hermite(AH) type FDTD [13], a series of fundamental schemes [14],a recent one-step unconditionally stable method [24], and oth-ers. In these methods, the time discretization scheme is differentfrom that of an explicit FDTD. It yields an error amplificationfactor bounded by one irrespective of time step, thus ensuringstability. However, unlike an explicit FDTD method that is freeof matrix solution, an implicit FDTD requires solving a matrix.Therefore, it suffers from the issue of computational efficiencywhen the problem size, and hence matrix size, is large. In [16],a spatial filtering technique has been developed to extend theCFL limit for electromagnetic analysis. The filtering techniquesin [15] and [16] have not produced an unconditionally stablemethod.
Recently, based on the success of an explicit and uncon-ditionally stable time-domain finite-element method [19], anexplicit and unconditionally stable FDTD method has beensuccessfully developed in [20] and [21]. This method is sta-ble for an arbitrarily large time step irrespective of space step,and accurate for a time step solely determined by samplingaccuracy. Furthermore, it retains the matrix-free property ofthe original FDTD method, and hence no matrix solution isrequired. The method does not belong to the class of commonlyunderstood unconditionally stable methods. The essential ideaof this method to achieve unconditional stability is to iden-tify the root cause of the instability associated with an explicitmarching, and subsequently adapt the underlying numericalsystem to eradicate the root cause of the instability. As a result,an explicit method can also be made unconditionally stable. Onthe contrary, the root cause of instability has not been elim-inated in existing implicit methods, which is the set of theeigenmodes of the underlying numerical system, whose eigen-values (characterizing the rate of the space variations of theeigenmodes) are so high that they cannot be accurately simu-lated by the given time step. This set of unstable eigenmodesexists because of fine discretizations. The fine discretizationscannot be avoided in a structure having fine features relative toworking wavelengths. As a consequence, an implicit method
4004 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015
has to rely on various time integration techniques that havea bounded error amplification factor to control the stability.However, even though the stability is controlled, the presenceof these unstable modes, since they cannot be accurately sim-ulated, can still negatively impact the overall solution accuracyand stability in time domain. This will become clear in thesequel.
Despite the success in removing the dependence of the timestep on space step, neither [19] nor [20], [21] has addressed theanalysis of general lossy problems where dielectrics and con-ductors are not only inhomogeneous but also lossy. Differentfrom lossless problems where the field solution is governedby a symmetric positive definite generalized eigenvalue prob-lem, whose eigenvalue solutions are real, the field solution ofa lossy problem is governed by a quadratic eigenvalue prob-lem whose eigenvalues and eigenvectors are complex-valued.The over-damped eigenmodes, critically damped eigenmodes,and damped oscillations could coexist in the numerical system.Traditionally, the stability analysis of either a purely losslesssystem or a lossy problem having a uniform conductivity isused to estimate the stability of a lossy problem. However,this approach is not suitable for analyzing the root cause ofthe instability of an explicit scheme in a general lossy setting,where the lossy materials are inhomogeneous. The root causeof the instability, thus, remains to be thoroughly understoodfor the analysis of general lossy problems. Furthermore, basedon the theory given in [19]–[21], we need to remove certaineigenmodes for the given time step. However, the governingquadratic eigenvalue problem of a lossy problem yields manycomplex-valued eigenvalues and eigenvectors. Which eigen-values, and thereby eigenvectors, to remove is unknown forensuring stability without sacrificing accuracy. In addition, newexplicit algorithms need to be devised to achieve unconditionalstability for general lossy problems. This paper is written toaddress these unsolved problems. Numerical examples involv-ing lossy and inhomogeneous dielectrics and conductors, inboth closed- and open-region settings, are presented to demon-strate the accuracy, efficiency, and unconditional stability of theproposed explicit method.
The preliminary work of this paper has been reported in ourconference papers [22] and [23]. In this paper, we provide acomprehensive and thorough description of the proposed workfrom theory to methods to numerical experiments. The disper-sion error of the proposed method is also theoretically analyzedfor not only free-space scenario but also inhomogeneous set-tings. In addition, we have compared the proposed methodwith two representative implicit unconditionally stable meth-ods in accuracy, dispersion error, stability, and computationalefficiency.
II. PROPOSED THEORY ON MAKING AN EXPLICIT FDTDUNCONDITIONALLY STABLE FOR ANALYZING GENERAL
LOSSY PROBLEMS
A. Root Cause of Instability
Consider Maxwell’s equations governing a generallossy problem having space-dependent conductivity σ,
permittivity μ, and permittivity ε in a source-free region
∇×E = −μ∂H
∂t
∇×H = σE+ ε∂E
∂t. (1)
In the FDTD method, the above continuous equations areessentially discretized into
Hn+ 12 = Hn− 1
2 −ΔtDEEn (2)(
I+Δt
2Dσ
)En+1 =
(I− Δt
2Dσ
)En +ΔtDHHn+ 1
2
(3)
where H denotes the vector of unknown magnetic field compo-nents, E the vector of unknown electric field components, Δtis the time step, Dσ is a diagonal matrix of element σ/ε, DE isthe sparse matrix representing the discretized 1/μ∇× operator,DH represents the discretized 1/ε∇× operator, and I standsfor an identity matrix. The superscripts n, n± 1/2, and n+ 1denote the time instants. Following (3), the E’s value at the nthtime step can be written as(I+
1
2ΔtDσ
)En =
(I− 1
2ΔtDσ
)En−1 +ΔtDHHn−1/2.
(4)
Subtracting the above from (3), and substituting (2) into theresultant, we obtain(
I+1
2ΔtDσ
)En+1 = 2En −
(I− 1
2ΔtDσ
)En−1
−Δt2DHDEEn (5)
which is nothing but a central-difference-based discretization ofthe following second-order vector wave equation for E
∂2E
∂t2+Dσ
∂E
∂t+ME = 0 (6)
where
M = DHDE . (7)
Therefore, from the aforementioned derivation, it can also beseen that the leap-frog based FDTD solution of the first-orderMaxwell’s equations is essentially a central-difference-baseddiscretization of the second-order wave equation.
The solution of (6), and thereby (5), is governed by thefollowing quadratic eigenvalue problem(
λ2 +Dσλ+M)V = 0 (8)
in which λ denotes the eigenvalue and V is the eigenvector.Since Dσ is positive semidefinite, so is M, the λ of (8) is eitherreal or comes with complex conjugate pairs, and the real partof λ is no greater than zero [17]. The eigenvectors V of (8) arealso either real or come with complex conjugate pairs. We canrepresent E at any time instant rigorously by
E(t) = Vy(t) (9)
GAFFAR AND JIAO: EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD 4005
with V being the eigenvector matrix which represents the col-umn space of the space variation of the field, and y(t) theunknown coefficient vector that is time dependent. In otherwords, the field solution at any time is a linear superpositionof the eigenvectors of the quadratic eigenvalue problem (8).
Now, let us consider an arbitrary eigenvector (eigenmode) of(8) Vi and examine which condition is necessary and sufficientto make this mode be stably simulated in the FDTD-based timemarching. For the Vi mode, (5) becomes(
I+1
2ΔtDσ
)Viy
n+1i = 2Viy
ni −
(I− 1
2ΔtDσ
)Viy
n−1i
−Δt2DHDEViyni . (10)
Front multiplying the above by V Hi , we have(
1 +1
2Δtbi
)yn+1i = 2yni −
(1− 1
2Δtbi
)yn−1i −Δt2ciy
ni
(11)
in which bi and ci are scalars given by
bi =V Hi DσVi
V Hi Vi
ci =V Hi MVi
V Hi Vi
. (12)
Since both Dσ and M are positive semidefinites, bi and ci arereal and no less than zero.
Performing a z-transform of (11), we obtain
(z − 1)2 + 0.5Δtbi(z2 − 1) + Δt2ciz = 0. (13)
The roots of the above quadratic equation of z can be readilyfound as follows:
z =(2−Δt2ci)±
√Δt4c2i +Δt2(b2i − 4ci)
2(1 + 0.5biΔt). (14)
To make (11) stable, |z| < 1 needs to be satisfied. To obtainthe modulus of z, we need to consider all possible scenarios ofb2i − 4ci as follows.
Since eigenpair (λi, Vi) satisfies (8), after front multiplying(8) by V H
i , we obtain
λ2i + biλi + ci = 0. (15)
Thus, the eigenvalue λi can be written as
λi =−bi ±
√b2i − 4ci2
. (16)
There are three kinds of eigenvalues corresponding to thecase of b2i − 4ci > 0, b2i − 4ci = 0, and b2i − 4ci < 0, respec-tively. The corresponding time-domain solution represents anover-damped, a critically damped, and an under-damped solu-tion, respectively. All of these three cases can exist in theeigenvalue solution of (8). Hence, we must consider all of thethree cases when analyzing the roots of z in (14). For eachcase, we have derived the necessary and sufficient conditionthat ensures the modulus of (14) bounded by 1, from which we
have found that for any given time step Δt, no matter how largeit is, the eigenmodes whose eigenvalues satisfy the followingcondition can be stably simulated by the given time step:
√ci ≤ 2
Δt. (17)
To understand the meaning of (17) more clearly, it is neces-sary to reveal the relationship between
√ci and the magnitude
of eigenvalues. Consider the under-damped case where b2i −4ci < 0, from (16), it is evident that
√ci = |λi| (18)
i.e.,√ci is nothing but the magnitude of λi. For the critically
damped case where b2i − 4ci = 0, we have
|λi| = bi/2 =√ci. (19)
Hence,√ci is also the magnitude of λi. As for the over-
damped case of b2i − 4ci > 0, (16) has two distinct val-
ues λi1 =−bi+
√b2i−4ci2 and λi2 =
−bi−√
b2i−4ci2 , respectively,
both of which are negative. It is clear that the following equationholds true: √
|λi1||λi2| = √ci. (20)
The above relationship also holds true for the cases shown in(18) and (19). As a result, (17) can be rewritten as√
|λi1||λi2| ≤ 2
Δt. (21)
To understand what happens when (21) is violated andthereby the root cause of instability, it is important to realizethat (11) is nothing but a central-difference-based discretizationof the following second-order differential equation in time
d2y
dt2+ bi
dy
dt+ ciy = 0. (22)
Based on the value of b2i − 4ci, there are three types ofsolutions of the above equation as follows:
Now it is ready to analyze the root cause of the instability ofan explicit FDTD-based simulation of general lossy problems.To make it clear, we analyze the root cause for each of the threecases one by one.
1) Critically Damped Case: For the critically damped case,√|λi1||λi2| = |λi|. It can be seen from (23) that |λi| denotesthe decay rate of the time-domain solution. As a result, theeigenmodes whose eigenvalues violate (21) have a decay ratefaster than that can be accurately sampled by the given timestep, and hence they cannot be accurately simulated, thuscausing instability. These eigenmodes are the root cause ofinstability.
4006 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015
2) Under-Damped Case: For the under-damped case, wehave
√|λi1||λi2| = |λi|. The |λi| represents the upper boundof the oscillation frequency ωi. The eigenmodes that violate(21) are also those modes whose oscillation frequencies are toohigh to be accurately simulated by the given time step Δt, thuscausing instability.
3) Over-Damped Case: For the over-damped case, if bothλi1 and λi2 satisfy |λi| < 2/Δt, (21) would hold true.Therefore, when (21) is violated, at least one of them, specif-ically λi2, has a magnitude beyond 2/Δt. Since λi for theover-damped case represents the decay rate of the time-domainsolution, when |λi| > 2/Δt, the corresponding mode decays sofast in time domain that it cannot be accurately captured by thegiven time step Δt, causing instability.
As a result, we have found that the eigenmodes whoseeigenvalues’ magnitude exceed 2/Δt are the root cause of theinstability in an explicit FDTD-based simulation of generallossy problems. These eigenmodes exist because of fine spacediscretization relative to working wavelength. The smaller thespace step, the higher the modulus of the eigenvalue of (8).The fine discretization cannot be avoided in structures havingsmall features relative to working wavelength such as integratedcircuits operating from zero to microwave frequencies.
B. Making an Explicit FDTD Solution of General LossyProblems Unconditionally Stable
When the time step is chosen based on accuracy, we have
Δt ≤ 1
2fmax(24)
where fmax denotes the maximum frequency that exists in thesystem response based on a prescribed accuracy. The eigen-modes violating (21) have
|λi| > 2/Δt. (25)
Substituting (24) into (25), we obtain
|λi| > 4fmax. (26)
By performing a Fourier transform of (23), it is evidentthat the unstable eigenmodes satisfying (26) have a frequencybeyond the maximum frequency required to be captured byaccuracy, and hence they can be removed without sacrificingaccuracy. As a result, to make an explicit FDTD-based solutionof a general lossy problem unconditionally stable, what we onlyneed to do is to discard the unstable eigenmodes for the giventime step. By doing so, we obtain stability without sacrificingaccuracy. It is important to note that this statement does not holdtrue for the spatial Fourier-mode-based expansion like the oneused in the Von–Neumann stability analysis, as analyzed in thefollowing section. In addition, if an infinitely large time step isthe time step required by accuracy such as simulating a dc prob-lem, we only need to keep the eigenmodes whose eigenvaluesare zero, and discard others.
C. Comparison With Von–Neumann Analysis and AccuracyAnalysis
In the Von–Neumann stability analysis, the field solution atany time is essentially expanded into the following form:
E(r, t) =
N∑i=1
ejki·rF (ki, t) (27)
where the space dependence of the field solution is expandedinto Fourier modes. Since the time-dependence of each Fouriermode F (ki, t) is not analytically known in a general nonfree-space problem, theoretically, it is not feasible to truncate spatialFourier modes without affecting the accuracy. Take the dc modewhose λi = 0 as an example, this single eigenmode would alsohave to be represented by many spatial Fourier modes since ithas a complicated space dependence in a general inhomoge-neous problem. With (27), it is not feasible to use an infinitelylarge time step to simulate λi = 0 mode stably. In contrast, inthe proposed method, the dc mode, whose ci is zero because adc mode V has a zero curl and hence satisfying MV = 0, canbe simulated by an infinitely large time step without becomingunstable. As can be seen from (17), the time step required forstably simulating ci = 0 modes is infinity.
In the proposed method, by expanding the space depen-dence of the field solution using the eigenmodes of a governingeigenvalue problem, the time dependence of each mode hasan analytical expression as shown in (23). As a result, themodes whose λi exceed the maximum frequency required tobe captured by accuracy can be removed without affectingthe desired accuracy. The aforementioned accuracy analysisis for a source-free problem. For a problem with sources,following the same analysis given in [19], it can be shownthat removing eigenmodes that satisfy (26) does not affectthe accuracy desired for simulating an fmax-based numericalsystem.
III. PROPOSED METHODS
Based on the aforementioned theoretical analysis, the pro-posed explicit and unconditionally stable FDTD method hastwo straightforward steps for the analysis of general lossyproblems. The first step is a preprocessing step to find the eigen-modes that can be stably simulated by the given time step nomatter how large the time step is. In this step, we develop anefficient algorithm to find the stable eigenmodes instead of solv-ing (8) as it is. This step retains the advantage of the originalFDTD in being matrix free. In the second step, we expand thefield solution in the solution domain strictly in the space ofthe stable eigenmodes, and also project the FDTD numericalsystem onto the space of the stable eigenmodes. As a result,the FDTD-based explicit time marching is absolutely stable forthe given time step regardless of the time step size. Next, wefirst explain the second step, then proceed to elaborate the algo-rithm of the preprocessing step. In the third part of this section,we also present a diagonal-preserving method to achieve thedesired features.
GAFFAR AND JIAO: EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD 4007
A. Explicit FDTD Marching With Unconditional Stability
We divide the E and H unknowns into two groups. Onegroup is inside the solution domain denoted by subscript S, andthe other is outside the solution domain denoted by subscriptO, such as unknowns on the boundary or inside an artificialabsorber like perfectly matched layer (PML). Subsequently, thesparse matrices DE and DH can be cast into the followingform:
DE =
[DE,SS DE,SO
DE,OS DE,OO
], DH =
[DH,SS DH,SO
DH,OS DH,OO
].
(28)
With the above, (2) can be rewritten for Hs and HO asfollows:
Hn+1/2S = H
n−1/2S −ΔtDE,SSE
nS −ΔtDE,SOE
nO (29)
Hn+1/2O = H
n−1/2O −ΔtDE,OOE
nO −ΔtDE,OSE
nS . (30)
Similarly, (3) can be rewritten as(I+
1
2ΔtDσ
)En+1S =
(I− 1
2ΔtDσ
)EnS +ΔtDH,SSH
n+ 12
S
+ ΔtDH,SOHn+1/2O −ΔtDε
−1jn+1/2
(31)(I+
1
2ΔtDσ
)En+1O =
(I− 1
2ΔtDσ
)EnO +ΔtDH,OOH
n+ 12
O
+ ΔtDH,OSHn+ 1
2
S (32)
where Dε is a diagonal matrix of element ε. The arrangementshown in (29)–(32) is made in view of the fact that the artificial-absorber region such as PML is filled with a single materialwhose space discretization can be performed solely based onaccuracy, while it is the fine feature present in the solutiondomain that makes the time step smaller than that requiredby accuracy. Hence, we leave the FDTD solution in the PMLregion as it is, while performing the time marching in the solu-tion domain strictly in the space of the stable modes for thegiven time step.
For an unconditionally stable simulation of (29) and (31), weexpand the unknown fields HS and ES strictly in the space ofstable eigenmodes for the given time step as follows:
ES(t) = VE,stye(t)
HS(t) = VH,styh(t) (33)
where ye and yh are unknown coefficient vectors which are timedependent, VE,st is the matrix whose columns are the stableeigenmodes for E, and VH,st is the same for H . Both VE,st
and VH,st are independent of time, representing the space vari-ation of the fields only. From (1), it can be seen that the H’sspace dependence is related to E’s dependence by DE operator.Therefore, VH,st can be written as
VH,st = DEVE,st. (34)
We also orthogonalize VE,st to obtain VE,st so thatVH
E,stVE,st = I. The same is performed on VH,st to obtain
orthogonalized VH,st. Substituting (33) with orthogonalizedstable eigenmodes into (29) and (31), we obtain
yn+1/2h = y
n−1/2h −ΔtVH
H,stDE,SSVE,styne
−ΔtVHH,stDE,SOE
nO (35)
yn+1e = VH
E,st (I+ 0.5ΔtDσ)−1
(I− 0.5ΔtDσ) VE,styne
+ΔtVHE,st (I+ 0.5ΔtDσ)
−1DH,SSVH,sty
n+1/2h
+ΔtVHE,st (I+ 0.5ΔtDσ)
−1DH,SOH
n+1/2O
+ΔtVHE,st (I− 0.5ΔtDσ)
−1Dε
−1jn+1/2. (36)
After the unknown coefficient vectors ye and yh are foundfrom (35) and (36), the entire field solution can be recovered atany point of interest from (33).
B. Preprocessing for Finding Stable Eigenmodes for the GivenTime Step
In the proposed algorithm for finding stable eigenmodes, westart the conventional FDTD simulation of (29)–(32), which isonly done for a small time window [when to stop is adaptivelycontrolled by the following (42) and (43)]. At selected timeinstants such as every pth step (p ≥ 1 and it is usually chosen asthe ratio of the time step determined by sampling accuracy andthe time step dictated by stability), we add E field solution ES
in matrix FE (initialized to be zero) as one column vector, andalso orthogonalize FE . The column dimension of the orthogo-nalized FE is denoted by k′ and its row dimension is denoted byNe. With FE , we transform the original large-scale eigenvalueproblem (8) to a reduced eigenvalue problem as follows:
(λ2 + λDσ,r +Mr)Vr = 0 (37)
in which both Dσ,r and Mr are small matrices of size k′, whichare given by
Dσ,r = FTEDσFE
Mr = FTEMFE . (38)
The small quadratic eigenvalue problem (37) can be rigor-ously transformed to a standard eigenvalue problem⎡
⎣0k′×k′ Ik′×k′
− Mr︸︷︷︸k′×k′
−Dσ,r︸︷︷︸k′×k′
⎤⎦{
Vr
λVr
}= λ
{Vr
λVr
}(39)
which can be solved with negligible cost due to its small size.When progressively adding a solution vector into FE in
the above process, repeating eigenvalues will appear from theeigenvalue solution of (39). These repeating eigenvalues, whenthe weights of their eigenvectors become dominant in the fieldsolution, can be identified as the physically important eigenval-ues of the original system as analyzed in [19]. To determine theweights, denote the upper half (k′ rows) of the eigenmodes of(39) corresponding to the repeating eigenvalues by Vr,l, andthose of the rest eigenmodes by Vr,h. As the matrix in (39)
4008 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015
is not symmetric positive semidefinite, Vr,l and Vr,h are notorthogonal by themselves. To find their weights in the fieldsolution, we cannot use the same procedure as that in [21]developed for the lossless problems. Therefore, we first orthog-onalize Vr,l to obtain Vr,l, and then exclude the projectionof Vr,h onto Vr,l by computing Vr,h = Vr,h − Vr,lV
Hr,lVr,h,
which is then orthogonalized to a new Vr,h. The weights of therepeating eigenmodes and the nonrepeating ones, yel and yeh,can then be determined from
[yel yeh]T= VH
r FHEES (40)
where
Vr =[Vr,l Vr,h
]. (41)
The dominance of the repeating eigenmodes Vr,l can beassessed by the ratio of their weights to the weights of thenonrepeating ones as the following:∣∣yHehyeh∣∣ / ∣∣yHel yel∣∣ ≤ ε1. (42)
The preprocessing step is terminated when the above, as wellas the following, is satisfied:∣∣∣λj+1
i,l − λji,l
∣∣∣ / ∣∣∣λji,l
∣∣∣ ≤ ε2 (43)
in which λji,l is the ith nonzero repeating eigenvalue observed
at the jth step.When the preprocessing step is terminated, a complete and
accurate set of the stable eigenmodes has been identified.Among the repeating eigenmodes observed from (39) step afterstep, we simply select those whose eigenvalues satisfy (17) toform VE,st in (33). As a result, we obtain
VE,st = FEVr,st (44)
where Vr,st are the repeating eigenmodes identified from thesmall generalized eigenvalue problem (39) whose eigenvaluessatisfy (17).
C. Diagonal-Preserving Formulation
The updating equations shown in (35) and (36) involve theorthogonalization of VE,st, VH,st, and a few matrix–matrixmultiplications in the right-hand side. These computations donot depend on time, and hence can be prepared in advanceand used for all time steps. The computational complexity ofthe orthogonalization as well as one matrix–matrix productinvolved in (35) and (36) is O(k2N), where k is the numberof stable eigenmodes, which is much smaller than the numberof E or H unknowns N as analyzed in [21]. In addition, thematrices in (35) and (36) associated with inverses are diagonalmatrices, thus the inversion cost is negligible. Despite the afore-mentioned facts, the computational efficiency of (35) and (36)is still not desirable when k is large. In this section, we presenta diagonal-preserving formulation to facilitate the analysis ofproblems with a large k.
Basically, in lossless problems, the eigenmodes are orthogo-nal by themselves. They also diagonalize the underlying systemmatrix. As a result, the resultant time-marching equation is afully decoupled diagonal system of equations when the FDTDnumerical system is projected onto the space of stable eigen-modes. Unlike lossless problems, the eigenmodes in a generallossy problem are not orthogonal by themselves; they do notdiagonalize the underlying system matrix either. These are thesources of the additional computational cost incurred in (35)and (36) as compared to the updating equations of losslessproblems given in [20] and [21].
To overcome the aforementioned problem, we first rewrite(6) as a first-order system as follows:
∂
∂t
{E∂E∂t
}−[
0 I−M −Dσ
]{E∂E∂t
}=
{0
f ′
}(45)
which can further be compactly written as
∂U
∂t−MAU = f (46)
where
MA =
[0 I
−M −Dσ
](47)
and
U =
{E∂E∂t
}. (48)
Meanwhile, we can cast (8) into the following generalizedeigenvalue problem:[
0 I−M −Dσ
]{V
λV
}= λ
{V
λV
}(49)
which can be rewritten in short as
MAVA = λVA. (50)
From (49), it can be seen that the upper half of the eigen-vector of MA, V , is the eigenvector of the original (8). If weexpand U in (46) by VA,st, the stable subset of VA satisfying(17), and front multiply (46) by VH
A,st. Utilizing the propertyof MAVA,st = VA,stΛst, where Λst is the diagonal matrix ofstable eigenvalues, we can obtain
dy
dt−Λsty = (VH
A,stVA,st)−1VH
A,stf (51)
which is a diagonal system of equation. Therefore, the diagonalproperty is preserved in the projected space of stable modes.Furthermore, the computation involved in the right hand sidecan be done once for all time steps. More important, the over-all cost of (51) is less than that of (35) and (36). If there aremany stable eigenmodes, as given in [21], we can divide thefrequency band of interest into multiple smaller bands. We thenfind the stable eigenmodes in each small band, the number ofwhich is not large so that the product VH
A,stVA,st of O(k2 N)
GAFFAR AND JIAO: EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD 4009
computational cost can be efficiently computed, and the resul-tant k × k matrix can be readily inverted. The union of thesable eigenmodes found in each small band forms the stableeigenmodes in the entire frequency band of interest.
To find VA,st efficiently without solving (50), similar to thepreprocessing algorithm described in the previous section, wefirst obtain a set of time-domain solutions. From these solutionsand the system matrix, we determine the stable eigenmodes. Tobe specific, we perform the time marching of (45) in a leap-frogway, obtaining(
I+Δt
2Dσ
)Wn+1 =
(I− Δt
2Dσ
)Wn
−ΔtMEn+ 12 −Δtf ′ (52)
En+ 12+1 = En+ 1
2 +ΔtWn+1 (53)
where W denotes ∂E∂t . The resultant solution [E W ]T is
then stored as a column vector in FE , and FE is orthogonal-ized, which is then used to obtain a reduced matrix MAr =FT
EMAFE . When progressively adding a solution vector intoFE , repeating eigenvalues will be observed from the eigenvaluesolution of MAr. These repeating eigenmodes that satisfy (17),when their weights become dominant in the field solution basedon (42), and the difference between repeating eigenvalues iden-tified at adjacent steps that is within an error tolerance ε2 shownin (43), can be multiplied in front by FE to obtain a completeset of stable modes VA,st.
D. Dispersion Analysis
In a vacuum or free space, the following dispersion relationholds true:
ω2 = c2(k2x + k2y + k2z) (54)
where ω is angular frequency; c is the speed of light; kx, ky , andkz are wavenumber along x-, y-, and z-directions, respectively.
However, the above relationship does not hold true in aninhomogeneous problem because the solution of Maxwell’sequations subject to all the boundary conditions at the materialinterfaces is not a plane wave. In fact, for a lossless inhomo-geneous problem, we can derive the following relationship fordispersion analysis:
ω2 = ξ (55)
where ξ is the eigenvalue of the 1/ε∇× (1/μ∇×) operatorobtained from the inhomogeneous problem. This relationshipcan be readily obtained by starting from the second-order vec-tor wave equation in an inhomogeneous problem, and findingits source-free solution in frequency domain. Numerically, thisleads to a frequency-domain counterpart of (6) except that Dσ
term is not present in lossless cases. Hence, the eigenvalue ξis the eigenvalue of M only shown in (7). In a vacuum or freespace, the ξ would revert to the right-hand side of the commonlyused dispersion relation (54). But for general inhomogeneousproblems, it does not. For lossy inhomogeneous problems, from
the frequency-domain counterpart of (6) which is the same as(46), we can see the following relationship holds true:
jω = λ (56)
where λ is the eigenvalue of the quadratic eigenvalue problemshown in (8).
Now, we can analyze the dispersion error of the proposedmethod for both single-material and inhomogeneous cases.For lossless problems, we solve the following equation in theproposed explicit time marching:
d2yidt2
+ ξiyi = 0 (57)
as shown in [21], where ξi is the ith eigenvalue. To obtain theabove, we expand the field solution E(t) by stable eigenmodesfound in the preprocessing step as VE,sty(t), and multiplythe numerical system (6) by VT
E,st on both sides. The leap-frog-based first-order solution given in [21] also naturally leadsto (57) by eliminating the magnetic field unknown. Since thecentral-difference scheme is used in the proposed method,for a time-harmonic input of ejωt, we obtain the followingdiscretization of (57)
−4sin2(ωΔt/2) + Δt2ξi = 0. (58)
When ωΔt approaches zero, i.e., the time step is chosenbased on good sampling accuracy, it is evident that we willobtain (55). In addition, it can be seen that if ξi is large, a verysmall time step needs to be used to control dispersion error.However, if the modes having large ξi are completely removedas done in the proposed method, a large time step can be usedwithout degrading the dispersion accuracy. For inhomogeneouslossy problems, similarly, by discretizing (51) in time, we cansee when ωΔt approaches zero, we obtain (56). As for theaccuracy of ξ in (55) and λ in (56), they are obtained by central-difference-based space discretization since the FDTD scheme isused. Hence, the accuracy obeys the accuracy of the central dif-ference. For example, in free space, ξ in the proposed method
is equal to c2( 4sin2(kxΔx/2)Δx2 +
4sin2(kyΔy/2)Δy2 + 4sin2(kzΔz/2)
Δz2 ).As a result, it becomes c2(k2x + k2y + k2z) when the space stepis small. Similarly, it also approaches to the exact value inhomogeneous problems when space discretization is refined.
IV. NUMERICAL RESULTS
A. Demonstration of Unconditional Stability
First, we demonstrate the unconditional stability of the pro-posed method by simulating an example that has an analyticalsolution. The example is a three-dimensional (3-D) parallel-plate structure filled with a lossy dielectric of conductivity0.3 S/m. The height, width, and length of the structure are 1,6, and 900 µm, along each of which the space step is 0.2, 1.2,and 100 µm, respectively. A current source is launched from thebottom plate to the top plate at the near end of the parallel platestructure, while voltages are sampled at both the near end andthe far end. The current waveform is a Gaussian derivative pulseof I(t) = 2(t− t0)e
−(t−t0)2/τ2
with τ = 0.2 s and t0 = 4τ s.
4010 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015
Fig. 1. Simulation of a lossy parallel-plate structure.
Due to the small space step of the structure, the traditionalFDTD must use a time step as small as 6.58e− 16 s to ensurethe stability of the time-domain simulation. In contrast, for thesame space step, the proposed method can use an arbitrarilylarge time step without becoming unstable. For example, touse an infinitely large time step, we only need to keep zero-eigenvalue modes while discarding others. In Fig. 1, we plot theresults generated with the time step of 0.1, 0.01, and 0.001 s,respectively, in comparison with analytical data. It is evidentthat the proposed method is stable, while with the same timesteps, the conventional FDTD simply diverges. Moreover, whenthe time step satisfies accuracy requirements for the given inputspectrum, such as Δt = 0.01 s and Δt = 0.001 s, the resultsgenerated by the proposed method are not only stable but alsoaccurate, as can be seen from Fig. 1. Notice that the time step of0.01 s is 13 orders of magnitude larger than that of the CFL timestep, which also verifies the capability of the proposed methodin controlling dispersion error. Both the algorithm described inSections III-A and III-B and the diagonal-preserving algorithmin Section III-C are used to simulate this example. The resultsare on top of each other.
B. Demonstration of the Efficiency and Accuracy of theProposed Method
We have simulated a suite of examples to examine the per-formance of the proposed methods. The algorithm describedin Sections III-A and III-B is used to simulate all examples.Some examples are also simulated by the diagonal-preservingalgorithm described in Section III-C.
1) Inhomogenous On-Chip Lossy Interconnect: The sec-ond example is a 3-D inhomogeneous on-chip interconnectstructure with a large metal conductivity of 5e+ 7 S/m andmultiple layers of dielectrics. The dimension of the structure is2000 µm×300 µm×100 µm, along which the mesh size is 200,50, and 16.67 µm, respectively. The input current source is thesame as that in the first example, but with τ = 3e− 11 s. In thepreprocessing step, the accuracy-control parameters ε1 and ε2are chosen as 10−4 and 10−5, respectively. The FDTD solutionsare sampled every 20 steps, i.e., p = 20. Six stable modes areidentified from the preprocessing step, whose eigenvalues are−3.044× 1013,−1.979× 1012,−3.098× 105 ± 6.317× 108j,
Fig. 2. Simulation of a lossy on-chip interconnect of metal conductivity σ =5e+ 7 S/m.
and −2.92× 106 ± 4.8× 1011j, respectively. In conventionalFDTD, the time step, constrained by the smallest space step,must be chosen as small as 5.2577× 10−14 s for a stablesimulation. In contrast, the time step in the proposed methodis 2.9412× 10−12 s solely determined by accuracy. Thetotal CPU time of the proposed method including both thepreprocessing step and the explicit marching step is 28.422 s,whereas the total time of the conventional FDTD is 362.588 s.In Fig. 2, we compare the solution obtained from the proposedmethod with that from the FDTD at the near and far end ofthe interconnect. Excellent agreement is observed at bothends. Notice that in this example, the near-end and far-endwaveforms are different due to a relatively high frequency, andhence a nonstatic effect. We also compare the entire solutionwith the FDTD solution by assessing ||u− uref ||/||uref ||across the whole time window, where u denotes the vector ofthe electric and magnetic field solutions at all points obtainedfrom the proposed method, and uref is the same but fromthe conventional FDTD. The average ||u− uref ||/||uref || isshown to be 0.226%, revealing an excellent agreement betweenthe proposed method and the conventional FDTD at all pointsin space and across the whole time window.
2) Lossy Parallel Plate Structure Excited by a High-Frequency Pulse: In this example, we consider the same 3-Dparallel plate structure simulated in Section IV-A but with σ =0.1 S/m, and a fast Gaussian derivative pulse having a max-imum input frequency of 34 GHz. To simulate this example,a conventional explicit FDTD method requires a time step assmall as 6.5805× 10−16 s to maintain the stability of the time-domain simulation because the smallest space step is 0.1 µm.In contrast, the proposed explicit method is able to use a largetime step of 2.9412× 10−12 s solely determined by accuracyto generate accurate and stable results. In the preprocessingstep, the FDTD solutions are sampled every 40 steps, i.e., p =40. The accuracy-control parameters ε1 and ε2 are chosen as10−4 and 10−5, respectively. Two stable eigenmodes are iden-tified, whose eigenvalues are −1.0993× 108 and −1.1184×1010, respectively. In Fig. 3, the voltage waveforms simulatedby the proposed method are shown to agree very well withthose generated by the conventional explicit FDTD. The totalCPU time required by the proposed method is 113.50898 sincluding both the preprocessing step and the explicit marching
GAFFAR AND JIAO: EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD 4011
Fig. 3. Simulation of a lossy parallel plate waveguide of σ = 0.1 S/m.
Fig. 4. Simulation of an antenna with a lossy dielectric cylinder.
step, whereas the CPU time of the conventional FDTD is19 079.793260 s, yielding a speedup of approximately 170. Thediagonal-preserving formulation described in Section III-C wasalso used to simulate this example, the speedup of which isshown to be about 186.
3) Radiation of a Dipole Antenna in the Presence of LossyDielectrics: Next example is the radiation of a dipole antennain the presence of a lossy dielectric cylinder of conductivity0.1 S/m, the computational domain of which is truncated bya PML. The dipole of length 75 µm is placed at the centerof a solution domain of dimension 900 µm×300 µm×100µm. The PML region has 20 grid cells all around the solutiondomain with a uniform cell size of 81.8, 33.33, and 25 µm inx-, y-, and z-directions, respectively. The lossy rectangularcylinder has a length of 81.8 µm, width 33.33 µm, and height75 µm. The smallest mesh size is approximately 1.6364 µm.The source is the same as that in the second example. Thesolutions are sampled every nine steps in the preprocessing.The ε1 and ε2 are chosen as 10−3 and 10−5, respectively. Theconventional FDTD uses a time step of 5.44× 10−15 s andtakes 12 592.07 s to complete the whole simulation, whereasthe proposed method uses a time step of 6.4805× 10−14 s andtakes 1053.05 s to complete the simulation. There are 32 stableeigenmodes identified from the proposed method, whose eigen-values are −7.23× 105 ± 2.31990× 1012,−2.8007× 107 ±4.1638× 1012j,−3.0126× 105 ± 5.2944× 1012j,−6.8298×107 ± 6.782805× 1012j,−2.16576× 106 ± 7.522289× 1012j,and others. The speedup of the proposed method is
Fig. 5. Simulation of a phantom head beside a wire antenna. (a) Relative per-mittivity distribution in a cross section of the phantom head at a height of 8.4 cm(XY-cut). (b) Electric conductivity distribution in a cross section of the phantomhead at a height of 8.4 cm (XY-cut). (c) Simulated electric field at two points ofthe phantom head in comparison with reference FDTD solutions.
approximately 12, without sacrificing accuracy as can beseen from Fig. 4.
4) Phantom Head Beside a Wire Antenna: In previousexamples, the structures simulated involve fine features rela-tive to working wavelength. This is understood because only in
4012 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015
TABLE ICPU TIME AND MEMORY COMPARISON
these structures, there is a need to enlarge the time step, sincethe time step required by the CFL stability condition is smallerthan that required by accuracy. In the last example, to examinethe performance of the proposed method in a comprehensivefashion, we simulate an example in two settings: without finefeatures relative to working wavelength and with fine features.This example is also much larger than previous examples inboth unknown number and electrical size.
The example is a large-scale phantom head example[18] beside a wire antenna, having more than 48 mil-lion unknowns. The dimension of the phantom head is28.16 cm×28.16 cm×17.92 cm. The relative permittivity andconductivity distributions of the phantom head are shownin Fig. 5(a) and (b), respectively, at the height of 8.4 cm.The number of discretization cells in the solution domain is255×255×127, while the number of cells used in PML is30 along each direction. The total number of cells is thus315×315×187. The smallest space step along x-, y-, and z-directions is, respectively, 1.1, 1.1, and 1.4 mm. The inputcurrent source has a waveform of Gaussian derivative, locatedat x = 14.52 cm and y = 26.18 cm. In the first simulation,we do not consider fine tissues in this example, and let thespace step solely determined by accuracy. As a result, thetime step required by stability and that by accuracy are atthe same level. Thus, both traditional FDTD and the pro-posed method employ the same time step of 2.2680e− 12 s.In the preprocessing step, 74 stable eigenmodes are identi-fied with a choice of ε1 = 10−5 and ε2 = 10−7. The small-est eight nonzero eigenvalues are, respectively, −8.5423e6±15474e10j, −1.489e7± 5.414e10j, −3.612e7± 1.2275e11j,and −1.859e7± 1.708e11j. The total CPU time cost by thetraditional FDTD is 68 881.738268 s, whereas the total CPUtime of the proposed method including both preprocessingand explicit time-marching is only 38 343.45917 s. Thus, thespeedup of the proposed method is 1.7937, although the struc-ture does not involve fine features and the same time step isused in the proposed method. The theoretical reason for thisspeedup is the same as that analyzed in [21] for lossless cases.The electric field waveforms simulated from the proposedmethod at point (14.52, 27.72, 10.92) cm and point (14.52, 0.22,10.92) cm are compared with that of the conventional FDTD inFig. 5(c). Excellent agreement is observed.
Next, we consider the small tissues involved in the humanhead, for which the smallest space step is reduced to 0.044 mm.As a result, the conventional FDTD has to reduce the time stepaccordingly to 1.4672e− 13 s, whereas the proposed methodis able to use the same time step as before. It takes the pro-posed method 90 106 s to finish the entire simulation, whereasthe conventional FDTD cannot finish the simulation in 10
days even though we enlarge the cell size in the regions with-out fine tissues by two times. Based on the CPU time costof the FDTD at each time step, the projected run time ofFDTD is 2.0330e+ 06 s; thus, the speedup of the proposedmethod over the conventional FDTD is greater than 23 in thisexample.
In Table 1, we summarize the CPU time and memory usedby the proposed method in comparison with the conventionalFDTD for each example, where N denotes the total numberof electric field and magnetic field unknowns. The additionalmemory used in the proposed method is mainly the storage ofFe matrix, whose size is Ne by k′. Notice that the VE,st in(44) is stored in Fe and Vr,st (a small matrix of size k′ × k),instead of being separately stored as a new matrix. The expres-sion of (44) is used for computation instead of forming VE,st
explicitly. For example, the VE,st in (35) and (36) is nothingbut Fe multiplied by orthogonalized Vr,st. Furthermore, it isworth mentioning that the reduced eigenvalue problem formu-lated in this work is of a small size O(k′), and hence its storageis negligible as compared to the storage of Fe.
5) Comparisons With Implicit Unconditionally Stable FDTDMethods: In this section, we compare the accuracy, stability,and efficiency of the proposed explicit method with those of twoimplicit FDTD methods. One is the ADI-FDTD method [3],[4], the other is a recently developed one-step implicit uncondi-tionally stable FDTD method [24]. This new implicit method isvery convenient for implementation since only one time instantneeds to be changed in the conventional FDTD method to makethe FDTD unconditionally stable. The unconditional stabilityof this new method is also theoretically proved in [24].
The example considered is a free-space wave propagationproblem, the analytical solution of which is known. Hence, wecan use this example to accurately assess the performance ofthe three unconditionally stable methods. The computationaldomain has a length (L) of 9 mm and a width (W ) of 9 mm. Theincident field is a y-polarized electric field propagating alongx-direction, whose expression is E = yf(t− x/c), where thepulse f(t) = exp(−(t− t0)
2/τ2), with τ = (1/3)× 10−10 s,and t0 = 4τ s. Since the computational domain is filled by air,the boundary condition on the four outermost boundaries isanalytically known, which is the tangential incident field sincethe scattered field is zero. Hence, we impose such an analyt-ical absorbing boundary condition on x = 0, L, and y = 0,W , respectively. The computational domain is discretized into50 cells along both x- and y-directions, which is determinedbased on the input spectrum. The 26th cell size along x and ydirections is 20 times smaller than the rest of the cell size thatis L/50, and hence being L/1000. As a result, the time steprequired by CFL stability condition is 20 times smaller than the
GAFFAR AND JIAO: EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD 4013
time step required by accuracy. We define the following timestep:
Δt =CFLN
c√
1/min(dx)2 + 1/min(dy)2(59)
where CFLN is clearly the ratio between the time step usedin simulation to the time step required by the CFL condition.Based on the time step required by accuracy, the CFLN canbe as large as 20. The observation point is chosen at the centerof the computational domain, specifically, point (x = 4.32, y =4.41) mm, to examine the dispersion error, as well as entiresolution accuracy of the three unconditionally stable methods.
In Fig. 6(a), we plot the Ey fields obtained from the ADImethod at the observation point, with CFLN = 2 and 4, respec-tively. If we enlarge CFLN to be 5 or larger, the ADI scheme isfound to be unstable in this example, which may be attributedto the relatively rapid change in the space step in the com-putational domain, as the scheme is found to be stable if thediscretization is made uniform. When the ADI is stable, as canbe seen from Fig. 6(a), the ADI results agree well with the ref-erence solution which is the analytical solution in this example.In Fig. 6(b), we plot the Ey field obtained from the proposedmethod at the same observation point. Obviously, the proposedmethod is stable, and also accurate not only for CFLN = 4 butalso for large CFLN such as 10, 15, and 20. In Fig. 6(c), weplot the fields obtained from the one-step implicit method [24]with CFLN ranging from 4 to 20. The results are also shownto be stable and accurate. However, compared to the proposedmethod, the accuracy is lower.
In addition to comparing the time-domain waveforms, wehave also quantitatively examined the dispersion error of thethree methods. The time-domain field at the observation pointand the incident field at the left boundary are Fourier trans-formed, and the phase difference is extracted between the twoFourier transforms, from which we find the average phasevelocity in a range of frequencies from 3 to 15 GHz as a func-tion of CFLN. In Fig. 7, we plot the ratio of the numerical phasevelocity (Vp) to the ideal phase velocity for proposed methodin comparison with the ADI and the one-step implicit uncon-ditionally stable method. The dispersion error of the proposedmethod is shown to be much less than that of the other twomethods, with phase velocity ratio in the range of 0.9924 and0.9969. As far as the theoretical reason is concerned, in the pro-posed explicit unconditionally stable time marching, only stableeigenmodes are kept in the numerical system, and these stableeigenmodes have an eigenvalue ξ that can be accurately sam-pled by the given time step, and hence ensuring accuracy. Incontrast, in implicit unconditionally stable methods, the unsta-ble eigenmodes for the given time step that have large ξ are notremoved. Instead, they are kept in the numerical system, andpresent at each time instant. When a large time step is used,although the unstable modes are suppressed to be stable by animplicit time integration scheme, they cannot be accurately sim-ulated by the given time step based on sampling accuracy, andhence deteriorating the overall accuracy of the field solution. Ascan be observed from Fig. 7, the dispersion error of the implicit
Fig. 6. Simulated fields as a function of CFLN. (a) ADI-FDTD. (b) Proposedmethod. (c) One-step implicit unconditionally stable FDTD [24].
4014 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015
Fig. 7. Comparison of dispersion error characterized by phase velocity ratioamong three methods.
Fig. 8. Comparison of solution error among three methods.
methods increases when CFLN increases, whereas the proposedmethod has a well-controlled dispersion error.
Dispersion error is only one aspect of the time-domain solu-tion error. To assess the entire error, in Fig. 8, we plot thesolution error measured by ||E(t)− Eref (t)||/||Eref (t)|| as afunction of CFLN for three different methods, where Eref (t) isthe analytical solution at the observation point at all time. Lessthan 0.35% error is observed in the proposed method, whereasthe two implicit methods are shown to have a larger error, whichalso increases with CFLN.
In addition to stability and accuracy, we have also com-pared the computational efficiency of the proposed method withthe implicit unconditionally stable methods. For the phantomhead example having over 48 million unknowns, the one-stepimplicit method [24] with a GMRES-based iterative matrixsolver takes 86 656.4 s to finish the entire simulation, whereasthe proposed method only costs 38 343.4 s. Therefore, preserv-ing the matrix-free property of the original FDTD is anotherimportant advantage one can benefit from the proposed explicitand unconditionally stable method. For the phantom headexample having fine tissues simulated in Section IV-B4, theimplicit method [24] is found to be unstable at the late time.
V. CONCLUSION
In this paper, we have theoretically analyzed the root causeof the instability of an explicit FDTD method for the analysis ofgeneral 3-D lossy problems, where the materials are inhomoge-neous, and both dielectrics and conductors can be lossy. Basedon this root cause analysis, we develop an explicit FDTD that isunconditionally stable for analyzing general 3-D lossy electro-magnetic problems. In this method, we fix the instability fromthe root by completely eliminating the source of instability; andwe retain the strength of an explicit FDTD in avoiding matrixsolutions. The dispersion error of the proposed method is alsotheoretically analyzed for general inhomogeneous settings andnumerically examined. Numerical experiments and compar-isons with implicit unconditionally stable methods, as well asthe conventional FDTD, have demonstrated the unconditionalstability, accuracy, and efficiency of the proposed method.The essential idea of the proposed method for handling gen-eral lossy problems is also applicable to other time-domainmethods.
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Md Gaffar received the B.Sc. degree in electri-cal engineering from the Bangladesh Universityof Engineering and Technology (BUET), Dhaka,Bangladesh, in 2009. Since 2011, he has been pur-suing the Ph.D. degree in electrical engineering atthe School of Electrical and Computer Engineering,Purdue University, West Lafayette, IN USA.
His research interests include computational elec-tromagnetics and semiconductor physics.
Mr. Gaffar was the recipient of academic awardsin recognition of his research achievements, includ-
ing the Best Poster Award (among all groups) and Best Project Award inCommunication and Electromagnetic in EEE Undergraduate Project Workshop(EUProW) 2009. At Purdue, his research has been recognized by the IEEEInternational Microwave Symposium Best Student Paper Finalist Award in2013 and 2015, and the 2014 IEEE International Symposium on Antennas andPropagation Honorable Mention Paper Award.
Dan Jiao (S’00–M’02–SM’06) received the Ph.D.degree in electrical engineering from the Universityof Illinois at Urbana-Champaign, Champaign, IL,USA, in 2001.
She then worked with the Technology Computer-Aided Design (CAD) Division, Intel Corporation,Santa Clara, CA, USA, until September 2005, as aSenior CAD Engineer, Staff Engineer, and SeniorStaff Engineer. In September 2005, she joined PurdueUniversity, West Lafayette, IN, USA, as an AssistantProfessor with the School of Electrical and Computer
Engineering, where she is currently a Professor. She has authored 3 book chap-ters and over 200 papers in refereed journals and international conferences.Her research interests include computational electromagnetics, high-frequencydigital, analog, mixed-signal, and RF integrated circuit (IC) design and anal-ysis, high-performance VLSI CAD, modeling of microscale and nanoscalecircuits, applied electromagnetics, fast and high-capacity numerical methods,fast time-domain analysis, scattering and antenna analysis, RF, microwave, andmillimeter-wave circuits, wireless communication, and bio-electromagnetics.
Dr. Jiao has served as a Reviewer for many IEEE journals and conferences.She is an Associate Editor of the IEEE TRANSACTIONS ON COMPONENTS,PACKAGING, AND MANUFACTURING TECHNOLOGY. She was among the85 engineers selected throughout the nation for the National Academy ofEngineering’s 2011 U.S. Frontiers of Engineering Symposium. She has beennamed a University Faculty Scholar by Purdue University since 2013. Shewas the recipient of the 2013 S. A. Schelkunoff Prize Paper Award of theIEEE Antennas and Propagation Society, which recognizes the Best Paperpublished in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
during the previous year, the 2010 Ruth and Joel Spira Outstanding TeachingAward, the 2008 National Science Foundation (NSF) CAREER Award, the2006 Jack and Cathie Kozik Faculty Start up Award (which recognizes anoutstanding new faculty member of the School of Electrical and ComputerEngineering, Purdue University), a 2006 Office of Naval Research (ONR)Award under the Young Investigator Program, the 2004 Best Paper Award pre-sented at the Intel Corporations annual corporate-wide technology conference(Design and Test Technology Conference) for her work on generic broadbandmodel of high-speed circuits, the 2003 Intel Corporations Logic TechnologyDevelopment (LTD) Divisional Achievement Award, the Intel CorporationsTechnology CAD Divisional Achievement Award, the 2002 Intel CorporationsComponents Research the Intel Hero Award (Intel-wide she was the tenthrecipient), the Intel Corporations LTD Team Quality Award, and the 2000 RajMittra Outstanding Research Award presented by the University of Illinois atUrbana-Champaign.
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