Fast Explicit and Unconditionally Stable FDTD Method for Electromagnetic Analysis Jin Yan, Graduate Student Member, IEEE, and Dan Jiao, Fellow, IEEE

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1 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 Fast Explicit and Unconditionally Stable FDTD Metod for Electromagnetic Analysis Jin Yan, Graduate Student Member, IEEE, and Dan Jiao, Fellow, IEEE Abstract In tis paper, a fast explicit and unconditionally stable finite-difference time-domain (FDTD) metod is developed, wic does not require a partial solution of a global eigenvalue problem. In tis metod, a patc-based single-grid representation of te FDTD algoritm is developed to facilitate bot teoretical analysis and efficient computation. Tis representation results in a natural decomposition of te curl curl operator into a series of rank-1 matrices, eac of wic corresponds to one patc in a single grid. Te relationsip is ten teoretically analyzed between te fine patces and unstable modes, based on wic an accurate and fast algoritm is developed to find unstable modes from fine patces wit a bounded error. Tese unstable modes are ten upfront eradicated from te numerical system before performing an explicit time marcing. Te resultant simulation is absolutely stable for te given time step irrespective of ow large it is, te accuracy of wic is also ensured. In addition, bot lossless and general lossy problems are addressed in te proposed metod. Te advantages of te proposed metod are demonstrated over te conventional FDTD and te state-ofte-art explicit and unconditionally stable FDTD metods by numerical experiments. Index Terms Explicit metods, fast metods, finite-difference time-domain (FDTD) metod, stability, unconditionally stable metods. I. INTRODUCTION THE finite-difference time-domain (FDTD) metod is one of te most popular time-domain metods for electromagnetic analysis. Tis is mainly because of its simplicity and optimal computational complexity at eac time step. However, te time step of a conventional FDTD [1], [2] is restricted by space step for stability, wic is known as te Courant Friedric Levy condition. Suc a coice of time step is also te time step required by accuracy, if te space step can be determined solely from te accuracy perspective for capturing te working wavelengt. However, wen tere are fine features relative to te working wavelengt in te problem being simulated, te smallest space step can be muc smaller tan tat determined by accuracy. Hence, te resultant time step for ensuring stability can be muc smaller tan tat required by accuracy. As a consequence, a large number of time steps must be simulated to finis one simulation, wic is time consuming. Manuscript received October 12, 2016; revised December 23, 2016 and February 18, 2017; accepted February 25, Tis work was supported in part by te National Science Foundation (NSF) under Award and in part by te Defense Advanced Researc Projects Agency (DARPA) under Award HR Te autors are wit te Scool of Electrical and Computer Engineering, Purdue University, West Lafayette, IN USA ( djiao@purdue.edu). Color versions of one or more of te figures in tis paper are available online at ttp://ieeexplore.ieee.org. Digital Object Identifier /TMTT To overcome te aforementioned barrier, researcers ave developed implicit unconditionally stable FDTD metods, suc as te alternating-direction implicit metod [3], [4], te Crank Nicolson (CN) metod [5], te CN-based split step sceme [6], te locally 1-D FDTD [7], [8], te pseudospectral time-domain metod [9], te Laguerre FDTD [10], [11], te associated Hermite type FDTD [12], a series of fundamental scemes [13], and many oters, but te advantage of te conventional FDTD is sacrificed in avoiding a matrix solution. Researc as also been pursued [14] [18] to address te time step problem in te original explicit time-domain metods. In [17] and [18], te source of instability in FDTD is identified to be te unstable eigenmodes wose eigenvalues are larger tan wat can be accurately simulated by a given time step. Te unstable modes are subsequently deducted from te underlying numerical system in [17] and [18] before performing an explicit time marcing. As a result, an explicit FDTD can also be made unconditionally stable. However, to find te unstable modes, [17] and [18] require a partial solution of a global eigenvalue problem. Te resulting computational cost may still be ig wen te matrix size and/or te number of unstable modes are large. Since te largest eigenvalue of a discretized curl curl operator is inversely proportional to te smallest space step, te unstable modes exist indeed because of fine space discretizations relative to working wavelengts. It may not be necessary to perform a global eigenvalue solution to find te unstable modes. Along tis line of tougt, in tis paper, we first represent te FDTD metod into a patc-based single-grid format, in contrast to a conventional matrix representation tat can be viewed as an edge-based dual-grid one. Tis new representation allows us to use patces in a single grid to formulate te FDTD, regardless of weter te grid is 2-D or 3-D. For eac patc, one only needs to generate a column vector of four nonzero entries, and a row vector tat is te transpose of te column vector in a uniform grid. Te resultant rank-1 matrix is positive semidefinite, wose eigenvalue can also be found analytically, wic we sow to be inversely proportional to te patc area. Tis new representation elps us readily identify a teoretical and quantitative relationsip between te fine patces and te largest eigenmodes of te discretized cur curl operator. We tus prove tat once tere exists a difference between te time step required by stability and te time step determined by accuracy, i.e., a difference between te fine-patc size and te regular-patc size, te largest eigenmodes of te original system matrix can be extracted from fine patces wit a bounded error. Te larger te contrast ratio between te two time steps (or space steps), IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See ttp:// for more information.

2 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. 2 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES te more accurate te eigenmodes extracted in tis way. Tis finding can also be utilized by oter researc requiring finding te largest eigenmodes. Based on tis teoretical finding, we propose an efficient algoritm to find te unstable modes directly from te fine patces. We ten deduct tese unstable modes from te numerical system of te FDTD. As a result, an explicit time marcing can be performed wit unconditional stability and fast. Te preliminary result of tis paper as been reported in [19]. In tis paper, we complete te teory beind te proposed work, wic is not made clear in [19]. We also provide a detailed description of ow to reformulate te FDTD metod into a patc-based single-grid format, wic is not given in [19]. In addition, a significant amount of work is devoted to extend te proposed metod to analyze general lossy problems were bot lossy dielectrics and conductors can coexist. In suc a scenario, te unstable modes are muc more difficult to be obtained efficiently and accurately, since te underlying eigenvalue problem is quadratic. Extensive numerical experiments are also carried out to simulate bot lossless and lossy problems. Tis paper is organized as follows. In Section II, we present a new way of representing te FDTD metod, i.e., a patcbased single-grid representation. In Section III, we describe te proposed metod for analyzing general lossless problems, were we provide a detailed teoretical analysis, elaborate step-by-step te proposed metod, explain ow it works, and analyze its computational efficiency. In Section IV, we develop an algoritm for analyzing general lossy problems. In Section V, a number of numerical examples are presented to demonstrate te accuracy and efficiency of te proposed new metod in comparison wit te original FDTD and te stateof-te-art explicit FDTD metods tat are unconditionally stable. In Section VI, we summarize our findings. II. PATCH-BASED SINGLE-GRID REPRESENTATION OF THE FDTD AND ITS IMPLICATION To facilitate te development of te proposed metod, we first present a new way of representing te FDTD. In te original FDTD formulation, eac difference equation is written for obtaining one field unknown, eiter primary field unknown or dual field one, and eac of tese field unknowns is defined along te edge of a primary or a dual grid. Suc a matrix-less formulation as also been compactly written into a matrixbased form. We can view te original FDTD formulation, eiter its matrix-less formulation or its matrix-based one, an edge-based dual-grid formulation, since eac edge in te primary and dual grid is associated wit one field unknown. Te new representation presented in tis section is a patcbased single-grid one. We use only one grid. In tis grid no matter it is a 2-D or 3-D grid, we loop over all te rectangular patces present in te grid. For eac patc, we formulate a column vector and a row vector, wose product is a rank-1 matrix. Te row vector describes ow te E (H) unknowns along te contour of te patc produce te normal H (E) field at te patc center. Te column vector describes ow te normal H (E) field at te patc center is used to obtain te E (H) unknowns. Te two are transpose of eac oter in a uniform grid, but can be very different in a nonuniform grid or a grid wit subgrids. Wit te two vectors generated for eac patc, we can marc on in time to find te electric and magnetic field solutions. In te following presentation of te proposed formulation, we place te normal H at eac patc center and E along te edges of te grid. But te two can also be reversed. Consider a single 2-D or 3-D grid. Denote te total number of E and H unknowns by N e and N respectively. For eac patc in te grid, we obtain te normal magnetic field at te patc center, i, as follows: ] [ 1Li, 1Li, 1Wi, 1Wi i [e] = μ i (1) were subscript i denotes te patc index, wic is also te H unknown index. Te L i and W i are, respectively, te two side lengts of patc i, andμ i is te permeability at te patc center. Defining a global unknown vector {e} containing all electric field unknowns, (1) can be rewritten as S (i) i e 1 N e {e} = μ i (2) were S (i) e is a row vector of lengt N e, wit te superscript denoting te patc index. Tis row vector as only four nonzero entries sown in (1), wose column indexes are te indexes of te four local electric field unknowns in global vector {e}. In te original FDTD formulation, Ampere s law is discretized on a grid dual to te grid used for discretizing Faraday s law, resulting in {e} (S ) Ne N {} =D ɛ +{j} (3) were {} contains all of te unknowns, D ɛ is a diagonal matrix of permittivity, and { j} denotes a current source vector. Te above can be rewritten as follows: S (1) 1 + S (2) 2 + +S (N ) N = D ɛ {e} +{j} (4) were S (i) denotes te it column of S,andteS {} in (3) is realized as te sum of weigted columns, instead of a rowbased computation tat one is more used to. Eac row of (3) represents a curl of H operation producing an electric field unknown, but eac column does not. However, by doing so, (4) allows us to discretize Ampere s law in te original grid and on te same patc producing i. Eac column vector S (i) as only nonzero entries at te rows corresponding to te E unknowns generated from i. In a uniform grid, S (i) as only four nonzero entries, and it is simply ( (i) S )N e 1 = S(i) T e. (5) In a nonuniform grid, te S (i) e stays te same since i is still centered by E unknowns along te patc contour; te S (i) can be altered for a better accuracy. In our implementation, we observe tat a lengt or widt averaged between adjacent patces yields a muc better accuracy tan its nonaveraged counterpart. Hence, te L i and W i in S (i) are canged to te

3 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. YAN AND JIAO: FAST EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD FOR ELECTROMAGNETIC ANALYSIS 3 average ones between patc i and its adjacent patc tat sares te same E unknown. Note tat eac row of S (i) corresponds to one electric field unknown. So for an arbitrary jt row of S (i), te oter patc, wic sares te corresponding e j -unknown wit patc i, is known. Now, if we take a time derivative of (4), and substitute (2) into it, we obtain were 2 {e} { j} D ɛ 2 + C{e} = N C = i=1 1 μ ( i S (i) ) N e 1 (6) ( ) S (i) e 1 N e (7) wic is clearly te sum of a rank-1 matrix obtained from eac patc. After column vector S (i) and row vector S (i) e are obtained for eac patc, we can use tem to perform a leap-frog time marcing based on (2) and (4). We can also directly solve (6) as a second-order differential equation in time using a centraldifference sceme. Te two can be proved to be equivalent to eac oter. Since te rank-1 matrix as a form of [a][b] T,were[a] is a column vector and [b] T is a row vector, its eigenvalue can be found analytically as te following. Basically, by observing te following eigenvalue problem of [a][b] T,wereλ denotes an eigenvalue, and w is te eigenvector: [a][b] T w = λw (8) we can immediately identify its eigenvalue solution as λ =[b] T [a] (9) w =[a]. (10) Since a rank-1 matrix as only one nonzero eigenvalue, if λ =[b] T [a] is greater tan 0, ten te matrix must be positive semidefinite as te rest eigenvalues are zero. Now consider te rank-1 matrix of an arbitrary patc i in an FDTD grid S i = ( S (i) ) N e 1 ( S (i) e ) 1 N e. (11) From (1), (5), and (9), it can be readily seen tat S i s eigenvalue is λ patc(i) = 2 2 L i W. (12) i For a square patc, it is simply 4/ i, were i is te area of patc i. Tis eigenvalue is also te 2-norm of te rank-1 matrix, wic is also te 2-norm of column vector S (i) multiplied by tat of row vector S (i) e. Obviously, te contribution of every patc in an FDTD system matrix is positive semidefinite. Tus, te sum of tem is also positive semidefinite. Furtermore, te smaller te patc, te larger te eigenvalue of its rank-1 matrix. From (7) and te analysis of eac rank-1 matrix, it becomes possible to find te largest k eigenvectors of te FDTD system matrix from its k columns and k rows aving te largest norm. Tese columns and rows correspond to exactly tose contributed by fine patces, as analyzed in te above. Let te sequence of S (1), S(2),... be in a descending order of vector norm, wit S (1) s norm being te largest and S(k+1) s norm ɛ a times smaller tan S (1) s norm. Since S(i) s 2-norm is (i) S 2 2 = L (13) i W 2 i tis also means tat te area of patc k is about ɛa 2 times larger tan tat of patc 1. C can ten be well approximated as C = k i=1 μ i 1 S (i) S(i) e (14) wit te error of C C / C bounded by O(ɛa 2 ). Hence, C can be sufficient for finding m k largest eigenvalues and teir corresponding eigenvectors wit good accuracy, altoug it cannot be used to find all eigenpairs. Tis also indicates tat te field distribution of largest eigenmodes is actually localized in fine patces. Te above analysis can still be conceptual. In te following section, we will provide a detailed study. III. PROPOSED METHOD FOR LOSSLESS PROBLEMS A. Teoretical Analysis As sown in [17], [18], and [20], te time step for a stable FDTD simulation, t s, is required to satisfy te following criterion: 2 t s (15) ρ(s) were ρ(s) denotes te spectral radius of S = D (1/ɛ) C,wic is te largest eigenvalue of S. Using te eigenvalue sown in (12), dividing it by ɛ and μ, we can obtain te eigenvalue of S for a single patc i. Tis is clearly 4c 2 / i for a square patc, were c is te speed of ligt. Based on (15), we obtain t s i /c. Te largest eigenvalue of S is bounded by its norm. Because of a factorized form sown in (14), te S s norm can be readily found as no greater tan te largest c 2 S (i) S(i) e = c 2 S (i) 2, and ence te largest 4c 2 / i as can be seen from (13). Tus, t s min /c, were min is te smallest patc area. Hence, using te proposed new representation of te FDTD, we can also readily see tat (15) dictates tat te maximum time step permitted by stability is proportional to te smallest space step. In [17] and [18], te eigenvectors of S corresponding to te largest eigenvalues, wic are beyond te stability criterion, are identified as te root cause of instability. Te Arnoldi algoritm is ten employed to find tese unstable eigenvectors. For a sparse matrix of size N e, finding its largest k eigenpairs may take many more tan k Arnoldi steps, wit te computational complexity being O(k 2 N), werek > k. Wen N is large, te computational overead for obtaining a complete set of unstable modes in [18] could still be too ig to tolerate. In tis section, we identify a relationsip between te unstable modes and te fine patces present in te space discretization, from wic a global eigenvalue problem can be avoided.

4 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. 4 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Given a time step t, S can be partitioned into two components S = S f + S c (16) were S f consists of all fine patces, te rank-1 matrix of wic as an eigenvalue sown in (12) larger tan tat permitted by te given time step, and S c is composed of te rest. Consider an eigenvector, F i,ofs f. It satisfies S f F i = λ i F i. (17) Next, we prove tat it also satisfies te following wit a bounded error ɛ acc : ɛ SF acc i = λi F i (18) and tus te eigenvectors obtained from te fine patces are also te eigenvectors of te entire problem domain wit ɛ acc accuracy. Proof: To prove (18), we evaluate te following: ɛ acc = SF i λ i F i. (19) SF i Since SF i = (S f + S c )F i,andf i satisfies (17), (19) can be rewritten as S c F i ɛ acc = S c F i + λ i F i. (20) Since S c is positive semidefinite, te above satisfies ɛ acc S c F i λ i F i = S c λ i. (21) Since S c is Hermitian, its norm is also its spectral radius, i.e., te largest eigenvalue of S c. Tis number determines te maximum time step tat can be used in te regular patces for a stable simulation, denoted by t c. Similarly, te maximum λ i of S f determines te time step t f tat can be used in te fine patces for a stable simulation, wic is also equal to te t s in (15) for te entire computational domain. As a result, from (21), we obtain ( ) t 2 ( ) 2 f ts ɛ acc =. (22) t c t Te last equality in te above olds true because te ratio of t f to t c is also te ratio of time step required by stability t s to tat determined by solution accuracy ( t), assuming te regular-cell region is discretized based on accuracy. From (22), it is evident tat once t s is smaller tan t, wic is exactly te scenario wen te time step issue sould be solved, te unstable eigenmodes can be obtained from fine patces wit a bounded error. Meanwile, te larger te contrast ratio of regular-patc size to te fine-patc size, te better te accuracy of te unstable eigenmodes extracted from fine patces. In addition, from (21), it can be seen among te eigenvalues λ i obtained from te fine patces, te larger te eigenvalue, te better te accuracy. Based on te above finding, we develop a fast metod to acieve unconditional stability in an explicit FDTD time marcing, as sown in te next section. B. Proposed Algoritm Te proposed metod includes tree steps. First, we find unstable modes accurately from fine patces wit controlled accuracy. Second, we upfront deduct te unstable modes from te system matrix, and perform an explicit marcing on te updated system matrix wit absolute stability. Finally, we add back te contribution of unstable modes if necessary. 1) Step I: Finding Unstable Modes Accurately From Fine Patces: Given any desired time step t, we categorize te patces in te grid into two groups. One group, denoted by C c, consists of te regular patces, wic allow for te use of te desired time step witout becoming unstable. To be specific, te single nonzero eigenvalue of te rank-1 matrix of a regular patc divided by its permittivity and permeability is no greater tan 4/ t 2. Te oter group C f includes all te fine patces and te patces immediately adjacent to te fine patces. Te rank-1 matrix of eac fine patc residing in a fine grid as an eigenvalue larger tan tat of te regular patc. Te same is true for te patces immediately adjacent to te fine patces. Tis is because in tese patces, te lengt parameters L i and W i are averaged from te fine patc and te regular patc saring te E-unknown, wen calculating te four nonzero elements of column vector S (i). As a result, te eigenvalue of te rank-1 matrix of a patc adjacent to te fine patc is also larger tan tat of te regular one. Clearly, C f patces require a smaller t to ensure te stability. Tese patces do not ave to be connected. Tey can be arbitrarily located in te grid. Accordingly, S can be split as sown in (16), were S f is S assembled from C f and S c is from C c. To identify S f, te new FDTD representation presented in Section II provides a convenient and efficient approac. Based on (7), we obtain S f by looping over all te fine patces. For eac patc, we obtain a rank-1 matrix S (i) S(i) e. We ten sum tem up to obtain k S f = D 1 1 S (i) ɛ N e 1 S(i) T e 1 N e (23) i=1,i C f μ i in wic k is te patc number in C f. Denote te E unknown number in C f by n. Tisisalso te edge number in C f.letteh unknown number in C f be k. Tis is also te patc number in C f. It is evident tat n < N e and k < N. Te column vector S (i) in (23) as only four nonzero entries, wose row indexes correspond to te fine-patc electric field unknown indexes. Similarly, te row vector S (i) e as four nonzero elements at te columns of te fine-patc unknowns. Te matrix in (23), ence, can be rewritten as a small n by n matrix S ( f ) f n n = A n kb T k n (24) were A is composed of te k columns of S (i),andbt as te k rows of S (i) T e, were te zeros corresponding to te unknowns in regular patces are removed. Te permittivity is included in A, wile te permeability is included in B. Since k is less tan n, tes ( f ) f is intrinsically low rank. We ten extract l unstable eigenmodes, F i, from S ( f ) f, te complexity

5 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. YAN AND JIAO: FAST EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD FOR ELECTROMAGNETIC ANALYSIS 5 of doing so is only O(l 2 n). Here, l < k, i=1, 2,...,l. Since l < k and n N e, tis cost is muc smaller tan O(k 2 N e ) in [18], wic is te complexity of an Arnoldi-based global eigenvalue solution. Given an accuracy tresold ɛ, if te following requirement is satisfied, te F ( f ) i is accurate enoug to be cosen as an unstable mode: ɛ acc = SF i λ i F i <ɛ (25) SF i were F i is F ( f ) i extended to lengt N e based on te global unknown ordering. Among l eigenvectors, assume k r of tem are accurate. Tey are also te k r largest eigenvalues. We ten ortogonalize tem as V for te use of te next step. Wen calculating ɛ acc, 2-norm is used in tis paper. Te coice of te accuracy tresold ɛ is a user-defined parameter. Since te larger te eigenvalue, te better te accuracy of te eigenmode extracted from S f, we compute te eigenvalues of S f starting from te largest to smaller ones. For eac eigenpair computed, we calculate ɛ acc defined in (19) until it is greater tan prescribed ɛ. Teɛ acc calculated for te largest eigenpair represents te best accuracy one can acieve in te given grid, wic also dictates te smallest ɛ one can coose. 2) Step II: Explicit and Unconditionally Stable Time Marcing: After te unstable modes are found, before performing te explicit time marcing, we directly deduct V modes from S as follows: S l = S V V H S (26) wic permits te use of te desired time step, regardless of ow large it is. Te explicit marcing can ten be carried out using te updated system matrix as {e} n+1 = 2{e} n {e} n 1 t 2 S l {e} n + t 2 { f } n. (27) At eac time step after finding {e} n+1, te following treatment is added to ensure tat {e} is free of V modes: {e} n+1 ={e} n+1 V V H {e}n+1. (28) In te special case were all patces are fine patces, te proposed metod is equally valid. In tis case, only te nullspace of S, V 0, is left in (S V V H S), i.e., (S V V H S)=V 0V0 H S. Since te product of S and V 0 is zero, te S l term vanises in (27). Te {e} s solution, wic contains only nullspace modes as ensured by (28), becomes te time integration of te rigt-and side performed twice. Te resultant solution is correct, as sown in [18]. Note tat one cannot just take (6) and vanis te S-related term terein for obtaining a field solution of nullspace modes (also known as DC modes). If one does so, te solution would be wrong. Only te nullspace component of te field solution makes te S-related term vanis. 3) Step III: Adding Back te Contribution of Unstable Modes if Necessary: Tis step is not needed wen te time step is cosen based on accuracy, since te unstable modes removed are not required by accuracy as analyzed in [18]. In te case wen time step cosen is larger tan tat required by accuracy, some eigenvectors tat are important to te field solution are also removed from te numerical system, and terefore te solution computed from (27) and (28) is no longer an accurate solution of te original problem in (6) any more. In tis case, te proposed algoritm allows users to add te V contribution back to guarantee accuracy. Basically, we can express te field solution {e} of (6) as {e} =V{y} =V l {y l }+V {y }={e l }+{e } (29) were V=[V l, V ] is ortogonal of full rank N e. Since from (27) and (28), {e l } as been obtained, we need to find only {e }.SinceV as been computed, front multiplying V H on bot sides of (6), te {y } can be readily found from 2 {y } 2 + S r {y }=V T ( { f } S{el } ) (30) were S r =V H SV, wose size k r is te number of unstable modes. Since (30) is of small size k r, it can be efficiently solved by te algoritm in [17], or by implicit metods. C. How Does It Work? Apparently, since te proposed algoritm also allows one to add te V contribution back, it seems tat any ortogonal space V can be used. Tis is not true. To obtain a correct solution from (27) and (28), V sould satisfy Vl T SV = 0. Tis can be found as follows. Since {e} can be expanded as (29), (6) can be rewritten as 2 (V l {y l }+V {y }) 2 + S (V l {y l }+V {y }) ={f }. (31) By multiplying te above by Vl H, we can obtain te V l -component of {e} by solving 2 {y l } 2 + Vl H S ( V l {y l }+V {y } ) = Vl H { f }. (32) If Vl H SV is not equal to zero, (32) cannot be reduced to an equation of {y l } only. Only wen Vl H SV = 0, (32) can be reduced to (27), were {e} ={e l } due to (28), and I V V H = V l Vl H. Since (25) is satisfied, F is an accurate eigenvector of S. Wit V ortogonalized from F, te property of Vl H SV = 0 is satisfied. Tis is because SV = SF Z = F Z = V Z 1 Z,andVl H V = 0. Here, we use te relationsip of V = F Z were Z is a full-rank transformation matrix, as V is ortogonalized from F. D. Computational Efficiency In te proposed metod, we avoid finding te eigensolutions of te original global system matrix S. Instead, we work on a muc smaller matrix S f. Terefore, compared wit te approac developed in [18], te proposed metod can acieve unconditional stability more efficiently witout sacrificing accuracy. Te complexity of finding unstable modes is reduced significantly from te original O(k 2 N e ) to O(l 2 n) wit n N e and l < k. Tis small cost is also a one-time cost, wic is performed before time marcing. Since te unstable modes found in tis paper are frequency and time independent, once found, tey can be reused for different simulations of te same pysical structure. In te second step of explicit time marcing, te matrix-free property of te FDTD is preserved. Te time marcing as a strict linear (optimal) complexity at eac time step.

6 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. 6 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES IV. PROPOSED METHOD FOR LOSSY PROBLEMS In te previous section, we focus on lossless problems. Wen tere exist lossy dielectrics and conductors, we need to add one more term to (6) as follows: 2 {e} 2 + D {e} + S{e} ={f } (33) were D is diagonal wit its it entry being σ i /ɛ i at te point of te it E unknown. Different from a lossless problem, (33) is governed by te following quadratic eigenvalue problem: (λ 2 + λd + S)v = 0. (34) Te treatment of suc a problem is different from tat of a generalized eigenvalue problem. We ence use a separate section to describe our solution to general lossy problems. A. Teoretical Analysis Te second-order differential equation (33) can be transformed to te following first-order equation in time witout any approximation: {ẽ} M{ẽ} ={ f } (35) were { f }=[0 f ] T, {ẽ} =[e ė] T,inwicė denotes te first-order time derivative of e, and matrix M is [ ] 0 I M = (36) S D were I is an identity matrix. Obviously, {ẽ} s upper part is te original field solution of (33). Te solution of (35) is governed by te following generalized eigenvalue problem: Mx = λx. (37) Tis problem is also equivalent to (34) using te relationsip of x=[v λv] T. Since I is positive definite, D is positive semidefinite, and S is positive semidefinite, te eigenvalues of (37) eiter are nonpositive real or come as complex conjugate pairs wose real part is less tan zero. Similar to lossless problems, to acieve unconditional stability, we also need to remove te unstable modes from te system matrix, now M. Tese modes are analyzed in [21]. Tey ave eigenvalues wose magnitude satisfies λ > 2 t. (38) Again, given a desired time step, te unstable modes ave te largest eigenvalues in magnitude. Compared wit lossless problems, now it is even more computationally expensive to find tese unstable modes since M is double sized and can be igly ill-conditioned wen conductor loss is involved. Terefore, similar to wat we do for lossless problems, we propose to find te unstable modes efficiently from te fine patces only. B. Proposed Metod Wen dealing wit lossless problems, all te patces in te computational domain are divided into two groups, C f and C c, based on te time step permitted by teir grid size. For lossy problems, we incorporate into C f not only te fine patces and teir immediately adjacent patces, but also all te patces filled wit conductive metals. Tis is because te conductive materials contribute eigenvalues as large as conductivity divided by permittivity. To explain, te lowest eigenmode of (34) satisfies Sv=0, wic is a gradient field. For tis field, in addition to zero eigenvalues, tere is a set of eigenvalues wose magnitude is approximately D, wicis σ over permittivity. Hence, te conductive region is included since unstable modes correspond to te largest eigenvalues. After C f is identified, we can form a matrix M f as follows: [ ] 0 I M f = (39) S f D f were S f can be found in te same way as (24) and D f is obtained by selecting te diagonal entries of D corresponding to te field unknowns in C f. As a result, M f is a 2n 2n matrix, wic is muc smaller tan te original size of M. We ten extract te largest eigenpairs of M f using te Arnorldi metod. Similarly, an accuracy ceck similar to (25) (wit S replaced by M) is performed to select accurate unstable modes obtained from M f.letk r be te unstable eigenmodes obtained from M f, te complexity of finding tem is simply O ( kr 2n). We ten ortogonalize te unstable modes obtained, and augment tem wit zeros based on te global unknown indexes to build V. Using V, we upfront deduct teir contributions from te system matrix before time marcing as follows: M l = M V V H M. (40) We ten perform a time marcing of (35) using te updated system matrix M l as follows: {ẽ} M l {ẽ} ={ f }. (41) If we perform a forward-difference-based time marcing on (41), te resultant update equation is definitely explicit. However, te stability requirement on te time step is t 2Re(λ)/ λ 2 were λ is te eigenvalue of M l. Tis results in a time step smaller tan t 2/ λ, wic is te time step required by a central-difference discretization of te original second-order (33), for stably simulating te same set of λ. To solve tis problem, we propose to perform a backward difference as sown in te following: (I tm l ){ẽ} n+1 ={ẽ} n + t{ f } n+1. (42) A z-transform of te above results in z=1/(1 λ t).sinceλ of M l as a nonpositive real part, te stability of (42) is ensured for any large time step. Using te accuracy determined time step t, and wit te corresponding unstable modes removed, all te eigenvalues of M l satisfy λ 1 t. (43)

7 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. YAN AND JIAO: FAST EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD FOR ELECTROMAGNETIC ANALYSIS 7 Hence, te inversion of te left-and matrix of (42) can be replaced by a series expansion wit a small number of terms. Tus, (42) can be explicitly marced on in time as follows: {ẽ} n+1 (I+ tm l +( tm l ) 2 + +( tm l ) p ){ỹ} (44) were {ỹ} represents te rigt-and-side term in (42). In te above, tere is no need to compute te matrix matrix product. Instead, (44) is a summation of p vectors, and every vector can be obtained by multiplying te previous vector by M l. Hence, te computational cost of (44) is simply p matrix vector multiplications, and p < 10. To make sure tat te solution is free of unstable modes, we need to add te following treatment after (44) at eac time instant: {ẽ} n+1 ={ẽ} n+1 V V H {ẽ}n+1. (45) C. Matrix Scaling Wen conductor loss and/or multiscale structures are involved, I, D, ands can be orders of magnitude different in teir matrix norm. Te solution of te generalized eigenvalue problem (37) may ave a poor accuracy. To improve te accuracy of finding unstable modes from M f, we adopt an optimal scaling tecnique introduced in [22]. Based on tis tecnique, te I and S in (36) are scaled to were Ĩ = αi S = S/α (46) α = S 2. (47) Consequently, te first-order double-sized system (35) is updated as follows: were {ẽ }=[e {ẽ } M{ẽ }={ f } (48) ė/α] T, { f }=[0 f/α] T,and M is [ ] 0 Ĩ M =. (49) S D Te M f formulated for fine patces is also scaled accordingly. As can be seen in (48), te upper alf of te solution vector {ẽ } is te same as tat of (35). V. NUMERICAL RESULTS In tis section, we simulate a number of 2-D and 3-D examples involving inomogeneous materials and lossy conductors to demonstrate te validity and efficiency of te proposed fast unconditionally stable FDTD metod. Bot small and largest contrast ratios are considered between fine and regular patces. A nonuniform grid is used to discretize fine features. As sown in [23] and [24], a naive nonuniform grid can produce errors and numerical artifacts in te field solution of te FDTD simulation. A good nonuniform grid sould minimize te solution error. In te proposed implementation, te lengt parameters used in te column vector of eac patc are averaged from adjacent patces. Tey ave sown to produce a better accuracy, compared wit nonaveraged ones. In addition, te proposed metod can also be applied to subgridding. Its validity is independent of ow te rank-1 matrix for eac fine patc is generated. As sown in Section II, using te proposed new way of representing FDTD, te difference between a uniform grid, a nonuniform grid, and a grid wit subgrids is simply te difference in te column vector and te row vector generated from eac patc. Te proposed metod remains te same regardless of te content of te column and te row vector generated from eac patc. A. Validation of te Proposed Metod First, we did a detailed case study to validate te proposed metod from a variety of aspects. A wave propagation as well as a cavity problem in a 2-D rectangular region is considered. Te grid is sown in Fig. 1(a), were fine patces are introduced to examine te unconditional stability of te proposed metod. Along te y-axis, te cell size is uniform of 0.1 m widt. Along te x-axis, we define Contrast Ratio = x c / x f were x c = 0.1 m,and x f is controlled by Contrast Ratio. Tere are tree fine patces along te x-axis wose cell size is x f. Te total number of E unknowns is 258. Te incident electric field is E inc =ŷ2(t t 0 x/c)e (t t 0 x/c) 2 /τ 2 wit c = m/s, τ = s, and t 0 = 4τ. Te regular grid size, x c = 0.1 m, satisfies accuracy for capturing frequencies present in te input spectrum, wic is about 1/20 of te smallest wavelengt. Te computational domain is terminated by an exact absorbing boundary condition, wic is te known total field. Tis is because for any problem, te total fields on te boundary serve as an exact absorbing boundary condition to truncate a computational domain. For most of te problems, suc fields are unknown. However, in a free-space problem studied in tis example, te total field is known since it is equal to te incident field. Wen coosing Contrast Ratio = 100, x f =0.001 m, wic is two orders of magnitude smaller tan tat required by accuracy. Hence, tere is a two orders of magnitude difference between te time step required by accuracy and tat by stability. Te conventional FDTD metod must use a time step no greater tan s to perform a stable simulation. In contrast, te proposed metod is able to use a time step of s solely determined by accuracy to carry out te simulation. Te fine patces and teir adjacent patces are identified, wic are marked in red in Fig. 1(a). Tey involve 50 internal E unknowns. Terefore, te size of S f is 50 by 50, from wic 15 unstable eigenmodes are found accurately for a prescribed accuracy of ɛ = Te ɛ acc for te 16t eigenmode in (25) is Hence, te 16t eigenmode and tereafter are not selected since teir accuracy does not meet te required accuracy. Te 15 unstable modes are ten deducted from te system matrix, permitting a two-orders-of-magnitude larger time step. In Fig. 1(b), te electric fields at two observation points marked by blue cross in Fig. 1(a) are plotted as a function of time. Obviously, tey sow excellent agreement wit reference analytical solutions.

8 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. 8 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES TABLE I LARGEST 16 EIGENVALUES OBTAINED FROM S f WHEN Contrast Ratio = 100 Fig. 1. Simulation of a wave propagation in a 2-D grid. (a) Space discretization. (b) Simulated electric fields at two observation points wen Contrast Ratio = 100. (c) Entire solution error versus time wit different Contrast Ratio from 2, 5, and 10 to 100. In tis example, we ave also numerically examined weter te eigenmodes extracted from te fine patces are accurate approximations of te eigenmodes of te entire problem. In Table I, we list te eigenvalues of te 15 unstable modes and also te 16t one we extract from S f wit Contrast Ratio = 100. It is clear to see tat te largest 15 eigenvalues are at least two orders of magnitude larger tan te 16t one. Once tey are removed, a muc larger time step can be used for a stable simulation. In Table II, we list te accuracy of eac unstable eigenmode wit respect to different Contrast Ratios from 2, 5, and 10 to 100, by calculating te relative error sown in (19). Obviously, for all tese contrast ratios, te eigenmodes extracted from fine patces are sown to be accurate eigenmodes of te entire S. Furtermore, te larger te contrast ratio of fine patces to te coarse ones, te better te accuracy of te eigenmodes found from fine patces. Moreover, te eigenmodes wose eigenvalues are larger are more accurate. All of tese ave verified our teoretical analysis given in Section III. Note tat wen Contrast Ratio = 2, te number of unstable eigenmodes tat can be accurately extracted is smaller. However, we still can obtain a set of eigenmodes accurately for suc a small contrast ratio. To examine te solution accuracy at all points in te grid, we define te entire solution error at eac time instant as Entire Solution Error = {e} N e 1 {e} analne 1 (50) {e} anal Ne 1 were {e} Ne 1 consists of all electric field unknowns generated by te proposed metod and {e} anal is te analytical solution to all te unknowns. For example, considering an E unknown located at r i wit direction ˆt i, its analytical solution for tis wave propagation problem is simply E inc (r i ) ˆt i. Two-norm is used to calculate (50) in tis paper. Meanwile, we examine te solution accuracy as a function of Contrast Ratio.Te entire solution error is plotted in Fig. 1(c) for four different Contrast Ratios 2, 5, 10, and 100, respectively. It is evident tat te solution accuracy of te proposed metod is satisfactory for all tese contrast ratios. Furtermore, te larger te contrast ratio, te better te accuracy. In tis example, we also use te conventional FDTD metod wit t = s to simulate te case wit Contrast Ratio = 100, and plot te entire solution error versus time in Fig. 2. Comparing Fig. 1(c) wit Fig. 2 for Contrast Ratio = 100, it is obvious tat te proposed metod can acieve te same level of accuracy as te conventional FDTD metod. As for efficiency, te CPU time speedup is 1.58, 3.08, and 28.16, respectively, for contrast ratios of 5, 10, and 100. However, no speedup is observed wen contrast ratio is two, because of te small time step difference and te additional overead of te proposed metod. Te proposed metod takes s including te CPU time

9 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. YAN AND JIAO: FAST EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD FOR ELECTROMAGNETIC ANALYSIS 9 TABLE II ACCURACY OF EACH UNSTABLE EIGENMODE OBTAINED FROM S f WITH DIFFERENT Contrast Ratio Fig. 2. Conventional FDTD for wave propagation problem: entire solution error versus time wit Contrast Ratio = 100. of every step from finding te unstable eigenmodes to explicit time marcing, wile te conventional FDTD metod only requires s to finis te simulation. We ave also studied a cavity problem using te same mes [25], wic again reveals an excellent agreement between te proposed metod and te conventional FDTD. In Fig. 3, we plot tree eigenvectors of S wose eigenvalues are, respectively, te largest, te fift largest, and te 15t largest eigenvalues of global S, for a contrast ratio of 100. As can be seen from Fig. 3, te field distributions of tese eigenvectors are localized in te fine patces, wit te fields in te regular patces many orders of magnitude smaller. For example, for te 15t largest eigenmode wose field distribution is more spread over tan te first two, its eigenmode (eigenvector) still as a field value in te immediately adjacent coarse patces being tree orders of magnitude smaller tan tat in te fine patces. Fig. 3 furter confirms tat te igest eigenmodes can be accurately extracted from fine patces. Altoug it is plotted for contrast ratio 100, similar localizations ave been observed for oter smaller contrast ratio, wic can also be seen from te small error of eigenvectors extracted from S f listed in Table II. Numerically, suc a localization is because te rapid field variation of te large-eigenvalue modes cannot be captured by a coarse discretization. Tis is similar to te fact tat if one uses a coarse grid to extract te cavity resonance frequencies, te frequencies (eigenvalues) one can numerically identify are muc smaller tan te ones e can find using a fine grid. Analytically, all tese eigenvalues sould exist in te solution domain. However, numerically, only finer patces can capture larger eigenvalues. It sould also be mentioned tat removing unstable modes is not te same as removing fine features or patces, since te stable eigenmodes kept in te numerical system also capture te fine features, i.e., tese modes are Fig. 3. Field distribution of te eigenvectors of S for a contrast ratio of 100 plotted in log scale. (a) Eigenvector aving te largest eigenvalue. (b) Eigenvector aving te fift largest eigenvalue. (c) Eigenvector aving te 15t largest eigenvalue. different wen tere are fine features and wen tere are not. Note tat eac eigenmode is a source-free solution in te given problem satisfying all boundary conditions at te material

10 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. 10 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 4. Simulation of a cavity wit fins. (a) Structure illustration. (b) Electric fields at two observation points. (c) Entire solution error as a function of time in comparison wit reference solution. interface. Te eigenvalue of an eigenmode does not reflect te spatial frequency. For example, a zero-eigenvalue mode (a static field distribution) also as rapid space variations. B. Demonstration of Accuracy and Efficiency After an extensive validation of te proposed metod, we simulate a variety of lossless and lossy examples to demonstrate te accuracy and efficiency of te proposed new metod compared wit te conventional FDTD and te state-of-te-art unconditionally stable explicit FDTD metod like [18]. 1) Conductive Fins Separated by a Narrow Gap: In tis example, we simulate a 2-D PEC cavity wit two conductive fins separated by a narrow gap. Te details of te structure are sown in Fig. 4(a). Te conductive fins are treated as perfect conductors. Between te two fins, tere is a small gap discretized into tree fine grids of widt 0.01 mm eac along te x-direction. Te regular grid size is 0.1 mm along bot te x- and y-directions. Te discretization results in 1261 edges and 602 patces. An x-orientated current source is launced between te two fins and its waveform is I = τ 2 exp (t t 0 /τ) 2,wereτ= sandt 0 = 4τ. For tis example, 28 unstable modes are extracted from S f wit ɛ = S f is a square matrix of size 123 assembled from fine patces only. After unstable modes are removed from te system matrix, we can enlarge te time step from to s. Te total CPU time required is s including te time for finding unstable modes. As a comparison, te explicit unconditionally stable FDTD metod [18] can also increase te time step to te same value, but it takes s to finis te simulation. Te traditional FDTD metod costs s to finis te simulation. Terefore, te proposed metod is more efficient. To assess accuracy, te electric fields at two points located at (1.0, 0.4) and (2.0, 1.1) mm are plotted in comparison wit te reference result obtained from te traditional FDTD in Fig. 4(b). It is evident tat te results from te proposed metod agree wit te reference results very well. Meanwile, te entire solution error (50), wic includes te error at all points in te grid, is plotted versus time in Fig. 4(c). It is sown to be small across te wole time window, validating te accuracy of te proposed metod. 2) 3-D Wave Propagation: A wave propagation problem is simulated in a 3-D free space. Te incident field is te same as tat of te first example. We also supply an exact absorbing boundary condition to all te unknowns on te boundary. Unlike te first example tat as an abruptly canged grid size, a progressively canged grid size is adopted for space discretization. Te space step is 0.1 m along bot te y- andz-directions. Tere are five cells along eac of te two directions. Te grid along te x-direction as 13 cells, eac of wic is of widt 0.1 m except for te tree cells in te middle wose space step is 0.01, 0.001, and 0.01 m, respectively. Te time step of a conventional FDTD is less tan s, wereas te time step of te proposed metod is s cosen based on sampling accuracy. Te electric fields obtained from te proposed metod are plotted in Fig. 5(a) at two points, (0.51, 0.45, 0.2) and (0.57, 0.4, 0.2) m, respectively. Excellent agreement wit analytical solutions can be seen. In Fig. 5(b), we plot te entire solution error, {e} {e} anal / {e} anal, compared wit te analytical solution, wic reveals tat te proposed metod is accurate at all points and across te wole time window simulated. It is wort mentioning tat te large errors at early and late time are expected, since te teoretical error is infinity at tese times due to a zero denominator, since te field solution is zero. In tis simulation, te number of fine patces is 350. Te A (B) sown in (24) is of size 320 by 350. Given ɛ=10 2, we obtain 120 unstable eigenmodes accurately from S f. It takes te proposed metod s to finis te simulation. To simulate te same example, a conventional FDTD costs s. Te state-of-te-art unconditionally stable explicit FDTD metod in [18] spends s in finding te unstable modes and s for explicit marcing. Hence, te proposed

11 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. YAN AND JIAO: FAST EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD FOR ELECTROMAGNETIC ANALYSIS 11 Fig. 5. Simulation of a 3-D wave propagation problem. (a) Electric fields at two observation points. (b) Entire solution error wit respect to time. metod is faster tan bot metods. Tis is because te metod in [18] finds unstable modes from a global S matrix. In addition, te V in [18] is dense. In contrast, te V in tis new metod is zero in regular patces, tus speeding up te explicit time marcing step as well. 3) Inomogeneous 3-D Pantom Head Beside a Wire Antenna: Previous examples are in free space. Te tird example is a large-scale pantom ead [26] beside a wire antenna, wic involves many inomogeneous materials. Te permittivity distribution of te ead at z=2.8 cmissownin Fig. 6(a). Te wire antenna is located at (24.64, 12.32, 13.44) cm as marked by te wite dot in Fig.6(a), te current on wic as a pulse waveform of J=2(t t 0 )e (t t 0) 2 /τ 2 wit τ= sandt 0 =4τ. Te size of te pantom ead is cm cm cm. Te coarse step size along te x-, y-, and z-directions is 17.6, 17.6, and 1.4 mm, respectively, wic results in unknowns. To capture te fine tissues located at te center of tis ead, tree layers of fine grid wose lengts are 1.4 μm are added in te middle along te z-direction. As a result, te conventional FDTD metod can use only a time step less tan s to ensure stability. In te proposed metod, 768 fine patces are identified, wic involve 4709 electric field unknowns and 4256 magnetic field unknowns. Given ɛ=10 7, 1088 unstable eigenmodes are obtained accurately from S f. Wit te contribution of unstable eigenmodes removed, te time Fig. 6. Simulation of a pantom ead beside a wire antenna. (a) Relative permittivity distribution in a cross section of te pantom ead at z =2.8cm. (b) Simulated electric field at two observation points in comparison wit reference FDTD solutions. step is increased to s. In Fig. 6(b), te electric fields at two points (12.32, 3.52, 13.44) and (12.32, 24.64, 13.44) cm are plotted in comparison wit reference FDTD results. Again, very good agreement is observed. As for CPU time, te proposed metod takes s to extract unstable eigenmodes and s for explicit time marcing. However, te conventional FDTD needs s to finis te same simulation. Meanwile, altoug te metod developed in [18] can also boost te time step up to te same value as te proposed metod, it requires s instead in CPU time. Terefore, te proposed metod is not only muc faster tan te conventional FDTD metod, it is also more efficient tan [18] since te proposed metod requires te fine region only instead of te entire computational domain to extract unstable eigenmodes. 4) Inomogeneous and Lossy 3-D Microstrip Line Structure: A microstrip line illustrated in Fig. 7(a) is simulated. It as lossy conductors and inomogeneous dielectrics. A 3-D view of te structure can be seen in Fig. 7(a), and te structure is 4 mm long along te y-direction. A current source is injected between te bottom plate and te strip. It as a pulse of I = ẑ2(t t 0 )e (t t 0) 2 /τ 2 A, were τ =10 10 sand t 0 =4τ s. Te space step is 0.4 mm in all directions, but to capture skin effects in lossy conductors, te microstrip is discretized into 0.15, 0.15, and 34.7 μm intez-direction,

12 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. 12 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES accurate not only at te two sampled points, but also in te entire computational domain across te entire simulated time window. In tis simulation, 1925 E unknowns are involved to assemble M f. Te total time of te proposed metod is s, out of wic s is used to extract 1317 unstable modes for a prescribed accuracy ɛ=10 6, and s is spent on explicit time marcing. In contrast, a conventional FDTD based on (6) requires s to finis te same simulation. Te explicit unconditionally stable FDTD metod in [18] requires s to extract unstable modes and s to perform time marcing. Tus, te proposed metod is more efficient, despite te additional computational overead for solving lossy problems suc as te requirement of a series expansion sown in (44). Fig. 7. Microstrip line wit fine features. (a) Structure. (b) Voltages at two ports. (c) Entire solution error in comparison wit reference FDTD solutions. respectively. Te total number of E unknowns in tis structure is Due to te small space step to capture te skin effects, a time step of s is required in te conventional FDTD metod. In contrast, te proposed metod is able to use a time step of s. Te number of terms kept in (44) is six. In Fig. 7(b), te voltage drops extracted at bot near and far ends of te strip line are plotted in comparison wit te results obtained from a conventional FDTD metod. It is clear tat te results of te proposed metod agree very well wit te reference solutions. Te entire solution error at eac time instant is evaluated as {e} {e} FDTD / {e} FDTD, and plotted in Fig. 7(c). Obviously, te proposed metod is VI. CONCLUSION In tis paper, a fast explicit and unconditionally stable FDTD metod is developed. It does not require a global eigenvalue solution. In tis metod, first, we represent te FDTD metod into a patc-based single-grid matrix representation, in contrast to a conventional matrix representation tat can be viewed as an edge-based dual-grid one. Tis new representation elps us identify te relationsip between unstable eigenmodes and te fine patces. We find tat te largest eigenmodes of te system matrix obtained from te entire computational domain can be extracted from te system matrix assembled from te fine patces wit controlled accuracy. Te larger te contrast ratio of te fine-patc size to te regular one, te more accurate te extracted eigenmodes. Tis finding can also be used in oter researc. Based on tis teoretical finding, we develop an accurate and fast algoritm for finding unstable modes from fine patces. We ten upfront eradicate tese unstable modes from te numerical system before performing an explicit time marcing. Te resultant simulation eliminates te sortcoming of te original FDTD in time step s dependence on space step, witout sacrificing te merit of te FDTD in an explicit time marcing. Te proposed metod is also extended to andle general lossy problems were dielectrics and conductors are inomogeneous and lossy. Numerical experiments including bot lossless and lossy problems ave demonstrated te accuracy, efficiency, and unconditional stability of te proposed metod, by comparing wit conventional FDTD as well as te state-of-te-art explicit and unconditionally stable metod. Te essential idea of tis paper is also applicable to oter time-domain metods. It is also wort mentioning tat altoug te unstable modes are extracted from fine patces and subsequently removed for a stable simulation, tis does not mean tat te resultant field solution in te fine patces is zero or as a large error. Tis is because te stable eigenmodes preserved in te numerical system ave teir field distributions all over te grid, including bot regular and fine patces. Tese modes are different wen tere are fine features and wen tere are not. REFERENCES [1] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media, IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp , May 1966.

13 Tis article as been accepted for inclusion in a future issue of tis journal. Content is final as presented, wit te exception of pagination. YAN AND JIAO: FAST EXPLICIT AND UNCONDITIONALLY STABLE FDTD METHOD FOR ELECTROMAGNETIC ANALYSIS 13 [2] A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd ed. Norwood, MA, USA: Artec House, [3] T. Namiki, A new FDTD algoritm based on alternating-direction implicit metod, IEEE Trans. Microw. Teory Tecn., vol. 47, no. 10, pp , Oct [4] F. Zeng, Z. Cen, and J. Zang, A finite-difference time-domain metod witout te Courant stability conditions, IEEE Microw. Guided Wave Lett., vol. 9, no. 11, pp , Nov [5] C. Sun and C. W. Trueman, Unconditionally stable Crank Nicolson sceme for solving two-dimensional Maxwell s equations, Electron. Lett., vol. 39, no. 7, pp , Apr [6] J. Lee and B. Fornberg, A split step approac for te 3-D Maxwell s equations, J. Comput. Appl. Mat., vol. 158, no. 2, pp , [7] J. Sibayama, M. Muraki, J. Yamauci, and H. 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Circuits Syst., vol. 31, no. 3, pp , Mar [23] W. Heinric, K. Beilenoff, P. Mezzanotte, and L. Roselli, Optimum mes grading for finite-difference metod, IEEE Trans. Microw. Teory Tecn., vol. 44, no. 9, pp , Sep [24] L. N. Trefeten, Group velocity in finite difference scemes, SIAM Rev., vol. 24, no. 2, pp , [25] J. Yan and D. Jiao, Fast explicit and unconditionally stable FDTD metod for electromagnetic analysis, P.D. dissertation, Scool Elect. Comput. Eng., Purdue Univ., West Lafayette, IN, USA, [26] I. G. Zubal, C. R. Harrell, E. O. Smit, Z. Rattner, G. Gindi, and P. B. Hoffer, Computerized tree-dimensional segmented uman anatomy, Med. Pys., vol. 21, no. 2, pp , Jin Yan (GS 13) received te B.S. degree in electronic engineering and information science from te University of Science and Tecnology of Cina, Hefei, Cina, in Se is currently pursuing te P.D. degree in electrical engineering at Purdue University, West Lafayette, IN, USA. Se is currently wit te On-Cip Electromagnetics Group, Purdue University. Her current researc interests include computational electromagnetics, ig-performance VLSI CAD, and fast and igcapacity numerical metods. Ms. Yan was a recipient of an Honorable Mention Award of te IEEE International Symposium on Antennas and Propagation in 2015 and te Best Student Paper Award Finalist of te IEEE MTT-S International Microwave Symposium in Dan Jiao (M 02 SM 06 F 16) received te P.D. degree in electrical engineering from te University of Illinois at Urbana Campaign, Campaign, IL, USA, in Se was wit te Tecnology Computer-Aided Design (CAD) Division, Intel Corporation, until 2005, as a Senior CAD Engineer, Staff Engineer, and Senior Staff Engineer. In 2005, se joined te Scool of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA, as an Assistant Professor, were se is currently a Professor. Se as autored 3 book capters and over 260 papers in refereed journals and international conferences. Her current researc interests include computational electromagnetics, ig-frequency digital, analog, mixed-signal, and RF integrated circuit design and analysis, ig-performance VLSI CAD, modeling of microscale and nanoscale circuits, applied electromagnetics, fast and ig-capacity numerical metods, fast time-domain analysis, scattering and antenna analysis, RF, microwave, and millimeter-wave circuits, wireless communication, and bioelectromagnetics. Dr. Jiao as served as te reviewer for many IEEE journals and conferences. Se is an Associate Editor of te IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY. Se received te 2013 S. A. Scelkunoff Prize Paper Award of te IEEE Antennas and Propagation Society, wic recognizes te Best Paper publised in te IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION during te previous year. Se was among te 21 women faculty selected across te country as te Fellow of Executive Leadersip in Academic Tecnology and Engineering at Drexel, a national leadersip program for women in te academic STEM fields. Se as been named a University Faculty Scolar by Purdue University since Se was among te 85 engineers selected trougout te nation for te National Academy of Engineering s 2011 U.S. Frontiers of Engineering Symposium. Se was a recipient of te 2010 Rut and Joel Spira Outstanding Teacing Award, te 2008 National Science Foundation CAREER Award, te 2006 Jack and Catie Kozik Faculty Start-up Award (wic recognizes an outstanding new faculty member of te Scool of Electrical and Computer Engineering, Purdue University), a 2006 Office of Naval Researc Award under te Young Investigator Program, te 2004 Best Paper Award presented at te Intel Corporation s annual corporatewide tecnology conference (Design and Test Tecnology Conference) for er work on generic broadband model of ig-speed circuits, te 2003 Intel Corporation s Logic Tecnology Development (LTD) Divisional Acievement Award, te Intel Corporation s Tecnology CAD Divisional Acievement Award, te 2002 Intel Corporation s Components Researc te Intel Hero Award (Intel-wide se was te tent recipient), te Intel Corporation s LTD Team Quality Award, and te 2000 Raj Mittra Outstanding Researc Award presented by te University of Illinois at Urbana Campaign.