Hybrid FDTD/FETD Technique Using Parametric Quadratic Programming for Nonlinear Maxwell s Equations

Transcription

1 Progress In Electromagnetcs Research M, Vol. 54, , 217 Hybrd FDTD/FETD Technque Usng Parametrc Quadratc Programmng for Nonlnear Maxwell s Equatons Hongxa L 1,BaoZhu 2 and Jefu Chen 3, * Abstract A nonlnear hybrd FDTD/FETD technque based on the parametrc quadratc programmng method s developed for Maxwell s equatons wth nonlnear meda. The proposed technque allows nonconformng meshes between nonlnear FETD and lnear FDTD subdomans. The coarser structured cells of FDTD are used n regular, large and lnear meda, whereas smaller unstructured elements of FETD based on the parametrc quadratc programmng method are used to smulate complcated structures wth nonlnear meda. Ths hybrd technque s partcularly sutable for structures wth small nonlnear regons n an otherwse lnear medum. Numercal results demonstrate the valdty of the proposed method. 1. INTRODUCTION Wth growng complexty of nonlnear devce structures, approxmate analytcal technques become nadequate. More accurate and effcent numercal technques are sought. A wdely used modelng tool n ths area s the fnte-dfference tme-doman (FDTD) soluton of Maxwell s equatons [1 3]. A smple applcaton of the FDTD method based on the nonteratve procedure for nonlnear meda employs an explct tme-steppng scheme [3]. Ths scheme places a severe stablty constrant on the tme step and mesh szes, and t ntroduces an artfcal tme lag n the medum response. To elmnate ths artfcal tme lag, Joseph and Taflove [4] proposed an teratve FDTD of the nonlnear Maxwell equatons. Van and Chaudhur [5] proposed a hybrd mplct-explct FDTD method to elmnate the restrctve stablty condton. These methods allow for full vectoral wave solutons, but starcasng errors are ntroduced when they are appled to problems wth curved geometry [6, 7]. On the other hand, the fnte-element tme-doman (FETD) method allows an unstructured mesh. However, the nonlnear FETD scheme requres updatng system matrces or sparse matrx solver durng every tme steppng [8], whch can be qute expensve for computatonal cost of mult-scale problems. In order to adopt the advantages of both methods and avod ther weaknesses, a hybrd FDTD/FETD technque [1] has been developed for lnear Maxwell s equatons, However, the hybrd method has yet to be extended to transent electromagnetc problems wth nstantaneous nonlnear meda. In ths paper, a nonlnear hybrd FDTD/FETD technque s proposed for solvng the nonlnear Maxwell s equatons, whch allows nonconformng meshes [9, 1] between dfferent subdomans wth lnear and nonlnear meda based on the doman decomposton technque [11]. The cell or element sze of the proposed method can be chosen by balancng the computatonal cost. Larger structured cells are used n regular, large and lnear meda wth explct tme ntegraton scheme, whereas smaller unstructured elements based on the parametrc quadratc programmng method [12] are used to smulate the nonlnear meda wth mplct tme ntegraton scheme. Ths method s partcularly sutable for Receved 22 November 216, Accepted 21 January 217, Scheduled 13 February 217 * Correspondng author: Jefu Chen (jchen84@uh.edu). 1 School of Mechancal Engneerng, Dalan Unversty of Technology, Dalan, Laonng 11623, Chna. 2 Surface Engneerng Laboratory, School of Materals Scence and Engneerng, Dalan Unversty of Technology, Dalan, Laonng 11623, Chna. 3 Department of Electrcal and Computer Engneerng, Unversty of Houston, Houston, Texas 774, USA.

2 114 L, Zhu, and Chen structures wth small nonlnear regons n an otherwse lnear medum. The nonlnear FETD scheme s not based on teraton, but on the base exchanges n the soluton of a standard quadratc programmng problem, as the nonlnear constructve law s transformed nto a set of lnear complementary problems (LCP) [13] wth parametrc varables. These LCPs can be solved by a number of mature mathematcal tools, effcently [14 16]. Ths proposed scheme need not requre updatng system matrces and tedous teratve procedure at every tme step, so the couplng between FDTD and nonlnear FETD s easy to be mplemented based on the framework of lnear case [1]. Numercal results demonstrate the valdty of the proposed method. 2. NONLINEAR HYBRID METHOD FORMULATION 2.1. Nonlnear DG-FETD Formulaton We start from the frst-order Maxwell s equatons of two-dmensonal TM z case ( z = ) for the nonlnear DG-FETD scheme by defnng t = yŷ (1) H t H xˆx + H y ŷ (2) E z = E z ẑ, D z = D z ẑ (3) where superscrpt t denotes the transverse components. Nonlnear Maxwell s equatons for the two-dmensonal TM z case become (ε non E z ) t = t H t J s (4) μ H t t = t E z M s (5) wth an nstantaneous nonlnear permttvty ε non ε, E E 1 ε non = ε 1, E 1 < E <E 2... ε N, E >E N (6) Fgure 1. Lnearzaton of the nonlnear consttutve relaton. The nonlnear consttutve relaton can be lnearzed nto several lne segments by ntroducng the parametrc varatonals λ. Take a fve-segment approxmaton as shown n Fg. 1 as an nstance, where E 1 = E 1, E 2 = E 2,, E N = E N, D 1 = D 1, D 2 = D 2,, D N = D N. Assumng that <ε <ε 1 <ε 2 < <ε N,let D z = ε non E z = ε (E z + λ) (7)

3 Progress In Electromagnetcs Research M, Vol. 54, where ( 1 λ 1 =(D z D 1 ) 1 ), D z D 1 ε ε 1 ( 1 λ N =(D z D N ) ε N 1 1 ), D z D N ε N ( 1 λ 1 =(D z D 1 ) 1 ) (8), D z D 1 ε 1 ε ( 1 λ N =(D z D N ) 1 ), D z D N ε N ε N 1 λ = λ 1 + λ λ N λ 1 λ 2 λ N Then we wll have λ, λ 1, λ 2,,λ N (9) λ 1, λ 2,,λ N We wll have f <, f λ =; f =, f λ >, = N,..., 1, 1,...,N (1) Defne f 1 = D z D 1 λ 1 C 1 f N = D z D N λ N C N f 1 = D 1 D z λ 1 C 1 f N = D N D z λ N C N (11) C 1 = ε 1ε,,C N = ε Nε N 1 ε ε 1 ε N 1 ε N C 1 = ε ε 1,,C N = ε N 1ε N ε 1 ε ε N ε N 1 The consttutve relatonshps can be rewrtten as f 1 <, λ 1 =; f 1 =,λ 1 f N <, λ N =; f N =,λ N (12) f 1 <, λ 1 =; f 1 =,λ 1 f N <, λ N =; f N =,λ N Introducng slack varables v to make above nequalty become equatons, the nonlnear consttutve relatons becomes f + v = (13) λ, v, λ v = We can see that treatng nonlnearty s essentally solvng a standard lnear complementary problem (LCP) [13]. Several hghly effcent computatonal methods [14 16] from computatonal mathematcs can be used to solve the fnal quadratc programmng model. Therefore, the teratve process n each tme step s avoded n the proposed method. For a general nonlnear materal, the nonlnear consttutve relaton can be lnearzed nto arbtrary number of lne segments wth a correspondng number of parameter varables, and the correspondng dervaton of the equaton can be referred to [12]. The correspondng parametrc varatonals weak forms of nonlnear Maxwell s equatons are ε N e (E z + λ) ds = N e t H t ds N e J s ds (14) S t S S S μn h H t t ds = N h t E z ds N h M s ds (15) S S

4 116 L, Zhu, and Chen where N e and N h are test-and-basc functons for E z and H t, respectvely, and S denotes the area of the -th subdoman. Usng ntegraton by parts and Gauss s theorem n Equatons (14) and (15), by applyng some smple denttes, we obtan ε N e (E z + λ) ds = t N e H t ds N e J s + N e (ˆn H t ) dl (16) S t S S L μn h H t S t ds = t N h E z ds N h M s N h (ˆn E z ) dl (17) S S L where ˆn s the outward unt normal of the boundary, and L denotes the subdoman surface. The relatonshp of the nterface felds between subdoman and ts neghbor subdomans can be evaluated as the two numercal fluxes ˆn E z and ˆn H t. As the bass functons N e and N h are not requred to be contnuous across the nterface between two adjacent subdomans, a nonconformng mesh can be used between subdomans. The Remann solver [17 19] s used here to calculate the above numercal fluxes. Take the -th subdoman as the local subdoman, and assume that t s adjacent to the j-th subdoman. The corrected felds on the nterface wll be ˆn E z = ˆn Y E z + Y E j z Y + Y j ˆn ˆn H t H j t Y + Y j (18) ˆn H t = ˆn Z H t + Z H j t Z + Z j + ˆn ˆn E z E j z Z + Z j (19) where E z and H t are felds from the local (-th) subdoman; E j z and H j t are from the neghbor (j-th) subdoman; and Z and Y are wave mpedance and admttance of the medum Z = 1 μ Y = ε, Zj = 1 μ Y j = j ε j (2) for the -th and j-th subdomans, respectvely. The dscretzed system by the nonlnear DG-FETD method for the -th subdoman s gven n M de ee dt = K eee + K eh h + j L j eee j + j L j eh hj J M dλ ee dt (21) M dh hh dt = K he e + K hh h + j L j he ej + j L j hh hj M (22) M ee mn S = ε N e m N e n ds, M hh mn = μn S h m N h n ds (23) K ee mn L = N e m {ˆn ˆn N e n }/ ( Z + Z j) dl (24) K hh mn L = N h m {ˆn ˆn N h n }/ ( Y + Y j) dl (25) K eh = N e m N h n ds + N e m {ˆn N h n }Z / ( Z + Z j) dl (26) S L K he mn S = N h m N e n ds + N h m {ˆn N e n }Y / ( Y + Y j) dl (27) L L j ee mn L = N e m {ˆn ˆn N j e n }/ ( Z + Z j) dl (28) L j hh = N h mn m {ˆn ˆn N j h n }/ ( Y + Y j) dl (29) L

5 Progress In Electromagnetcs Research M, Vol. 54, L j eh = N e mn m {ˆn N j h n }Z / ( Z + Z j) dl (3) L L j he = N h mn m {ˆn N j e n }Y / ( Y + Y j) dl (31) L [ J ] m = N [ e m J s ds, M ] m = N h m M s ds (32) S S where e and h are vectors of the dscretzed electrc and magnetc felds for the -th subdoman; the matrces and vectors n Equatons (21) and (22) have the same defntons as n [2]. As shown n Equatons (21) and (22), a large system wth small nonlnear regons can be dvded nto several smaller lnear and nonlnear systems by ths doman decomposton method. By dong ths, we can solve the mult-scale system wth small nonlnear regons effcently usng nonconformng meshes between the electrcally fne and electrcally coarse structures. The nonconformng meshes are used for the transton from nonlnear FETD meshes to lnear FDTD grds n the hybrd method. Theoretcally, the electrcally fne structures can be dvded nto any number of subdomans wth DGM n the proposed method. In ths nonlnear hybrd FETD/FDTD method, we adopt the conventonal lnear FDTD scheme wth the leap-frog scheme on a staggered Cartesan grd. The couplng between FDTD and nonlnear FETD s n the next secton Couplng between the Nonlnear FETD and FDTD Methods The dscretzed Equatons (21) and (22) for nonlnear DG-FETD can be rewrtten as a frst order lnear ordnary dfferental equaton: M dφ Kφ = f Γdλ (33) dt dt where M ee K ee K eh L j ee L j eh M = M hh M j ee, K = K he K hh L j he L j hh M j L j ee L j eh K j ee K j (34) eh hh L j he L j hh K j he K j hh λ = [ λ λ j ] T f = [ f f j ] T Γ Γ = Γ j φ = [ e h e j h j ] T If the -th subdoman does not contan nonlnear medum, Γ =. The Crank-Ncolson method can be used to solve ths dscretzed frst-order equatons, and the smple central fnte dfference s used for the parameter vector λ. We wll get a tme steppng scheme as (M KΔt/2) φ n+1 (M + KΔt/2) φ n = f n+1 + f n 2 subject to the LCPs equatons f (e,λ)+v = v T λ =, v,λ + Γ λ n+1 λ n dt where φ n and φ n+1 are the values of felds φ at tme steps t n and t n+1, respectvely. We can see that ths proposed method fnally leads to a seres of standard LCP problems. Several mature algorthms (35) (36) (37) (38) (39) (4)

6 118 L, Zhu, and Chen Fgure 2. Couplng between coarser grd and denser grd wth nonconformng mesh. The rectangular grd marked by dotted lne s the buffer zone. The lnear FDTD subdoman s on the left of the buffer, and the nonlnear FETD subdoman s on the rght. Unknowns assocated wth edges marked by sold arrows, H FD, are exported from the lnear FDTD grd to the nonlnear FETD mesh, and those marked by empty arrows, H FE, are exported from the nonlnear FETD mesh back to the lnear FDTD grd. The E FD feld s located at the cell center n the lnear FDTD part marked by sold crcle. such as the aggregate-functon smoothng algorthm can be used to solve Eq. (4) wth hgh effcency. Detals of ths algorthm can be referred to [15] and wll not be elaborated here. The edges of the explct FDTD grd are updated usng the FDTD scheme. Ths provdes Equaton (39) wth a Drchlet boundary condton [2]. The soluton of Equaton (39) s exported to the explct grd to complete the full cycle of one tme step (see Fg. 2). Ths process s repeated untl the desred tme wndow s completed. For the example of 2-D TMz case, let us assume that E n 1/2 and H n are known n FDTD grd, and E n and H n are known on nonlnear DG-FETD mesh, where the superscrpts denote the tme step ndex. To advance the felds to the next tme step, we use the followng procedures: 1. Use the FDTD method to advance the electrc feld to E n+1/2 wth the known E n 1/2 and H n n the FDTD grd n Fg Use the FDTD method to advance the magnetc feld to H n+1 wth the known H n and E n+1/2 n the FDTD grd n Fg Pass H n+1 FD components, marked by sold arrows n Fg. 2, from the FDTD grd to the nonlnear FETD buffer zone as the boundary values. 4. Use the Crank-Ncolson method to advance the felds n the nonlnear FETD subdoman from n-th tme step to (n + 1)-th tme step wth the boundary values updated from FDTD n Step 3 above. After the soluton of FETD subdoman for one full tme step, the magnetc feld values at the crcles n Fg. 2, denoted by H n+1 FE, are known together wth the other feld values n the nonlnear FETD sub-domans. 5. Pass H n+1 FE, marked by empty arrows n Fg. 2, to the FDTD subdoman so that the felds n the whole FDTD subdoman are now known for the (n + 1)-th tme step.

7 Progress In Electromagnetcs Research M, Vol. 54, Go to Step 1 and repeat the process untl the specfed tme wndow has been reached. The nterfaces between adjacent subdomans can be nonconformng because the DG scheme permts dscontnuous bass functons between adjacent subdomans. The elemental mass matrces for the rectangular lnear edge elements n the buffer zone are obtaned by the trapezodal ntegraton rule [2]. 3. NUMERICAL RESULTS 3.1. Wave Propagaton n 2D Structure wth Nonlnearty As shown n Fg. 3(a), the frst example s about plane wave propagaton n a rectangular zone nserted wth one array of delectrc cylnders. The wdth and heght of zone are 7.5 mm and 3.75 mm, respectvely. The perodc boundary condtons [21] are appled to the upper and lower edges, and the perfectly matched layers [22] are used to truncate the computatonal doman along the x-drecton. The radus of each delectrc cylnder s.164 mm, and the nonlnear relatve permttvty s ε r = E 2. The lattce constants are both.492 mm along the x and y drectons. Ths nonlnear problem s solved by both nonlnear FDTD method [6] and the proposed scheme. Fg. 3 shows the hybrdzaton of FDTD grds and fnte elements for the proposed scheme. The staggered Cartesan grd of Conventonal FDTD (2 4 cells) s used for the dscretzaton of the homogeneous around the array of delectrc cylnders, and the unstructured element s appled to the zone flled wth delectrc cylnders. Dscontnuous Galerkn technque s used to hybrdze FDTD grd and unstructured element meshes. Nonconformng dscretzaton s allowed on the nterfaces between adjacent subdomans. Ths feature greatly mproves the flexblty and effcency of the proposed scheme n modelng the structures comprsng of small nonlnear regons n an otherwse lnear medum. An ncdent plane wave wth transverse magnetc (TM) polarzaton s mposed at the left sde of the computatonal doman. A recever s located mm away from the left boundary of the computatonal doman, and the tme steps for the two methods are both 55 fs. Fgure 3. A rectangular zone flled wth a array of delectrc cylnders. Fgure 4. Hybrd mesh for the nonlnear structure. Ez(V/m) Benchmark FDTD wth coarse grd FDTD wth fne grd Ez(V/m) Benchmark FDTD wth coarse grd FDTD wth fne grd Fgure 5. Tme varyng receved sgnals by nonlnear FDTD and the proposed method.

8 12 L, Zhu, and Chen Relatve error FDTD wth coarse grd FDTD wth fne grd Fgure 6. Tme varyng relatve error by nonlnear FDTD and the proposed method. (a) (b) Fgure 7. Electrc feld (V/m) near the delectrc crcular rods at.33 ns. (a), (b) nonlnear FDTD. (a) (b) Fgure 8. Electrc feld (V/m) near the delectrc crcular rods at.495 ns. (a), (b) nonlnear FDTD. Fgure 5 shows the smulated sgnals by the recever for the two numercal technques. The results obtaned by nonlnear FDTD wth a fne mesh (16 32) are used here as reference. Good agreements can be observed between the results by the proposed method and nonlnear FDTD method. Fg. 6 shows the comparsons of relatve errors by these two methods. From the fgure we clearly see that the proposed scheme can acheve hgher level of accuracy only by refnng the mesh of nonlnear regons. Fgs. 7 9 show the electrc feld near the delectrc crcular rods at dfferent tmes, and the results agree well wth nonlnear FDTD and the proposed method. Table 1 shows the computatonal costs of the nonlnear FDTD and the proposed method, whch demonstrate the effcency of the proposed method.

9 Progress In Electromagnetcs Research M, Vol. 54, (a) (b) Fgure 9. Electrc feld (V/m) near the delectrc crcular rods at.66 ns. (a), (b) nonlnear FDTD. Table 1. Computatonal cost of the nonlnear FDTD and the proposed. Coarse FDTD Fne FDTD Benchmark Grd 2*4 8*16 see Fg. 3 16*32 dt (fs) Tme length (ps) Computatonal tme (s) Nonlnear Structure wth Mult-Subdomans As shown n Fg. 1, the second example s about plane propagaton n a rectangular zone nserted wth two arrays of delectrc cylnders. The wdth and heght of zone are mm and 3.75 mm, respectvely. The radus of each delectrc cylnder s.164 mm, and the nonlnear relatve permttvty s ε r = E 2. The lattce constants are both.492 mm along the x and y drectons. The separaton of the two arrays s 3.75 mm. The computatonal doman s meshed by an FDTD regon outsde (2 65 cells), whle the two arrays of delectrc cylnders are dscretzed by two subdomans wth unstructured fnte element meshes. Fg. 1 shows the structure of ths case, whch s decomposed nto three subdomans dscretzed separately. The Fgure 1. A rectangular zone flled wth two arrays of delectrc cylnders. Ez(V/m) Benchmark FDTD wth coarse grd FDTD wth fne grd Ez(V/m) Benchmark -.4 FDTD wth coarse grd FDTD wth fne grd Fgure 11. Tme varyng receved sgnals by nonlnear FDTD and the proposed method.

10 122 L, Zhu, and Chen Relatve error.1 FDTD wth coarse grd FDTD wth fne grd Fgure 12. Tme varyng relatve error by nonlnear FDTD and the proposed method. Table 2. Computatonal cost of the nonlnear FDTD and the proposed. Coarse FDTD Fne FDTD Benchmark Grd 2*65 8*26 see Fg. 1 16*52 dt (fs) Tme length (ps) Computatonal tme (s) meshes could be nonconformng on the nterfaces between adjacent subdomans. The nonlnear FDTD method and proposed method are used to solve ths problem. A recever s located mm away from the left boundary of the computatonal doman, and the tme steps for the two methods are both 55 fs. The results obtaned by nonlnear FDTD wth a fne mesh (16 52) are used here as a benchmark. Fg. 11 shows the calculated sgnals by the recever for nonlnear FDTD and the proposed method. Good agreements can be observed between the results by the proposed method and the benchmark, whch demonstrate the valdty of the proposed method wth mult-subdomans. Fg. 12 shows the comparsons of relatve errors by the nonlnear FDTD method and proposed method, and Table 2 shows the computatonal costs of the nonlnear FDTD and proposed method. We clearly see that the relatve error of the proposed method s approxmate to the FDTD wth fne grd, but the computatonal tme of proposed method s less. 4. CONCLUSIONS We have demonstrated the valdty of the nonlnear hybrd FDTD/FETD scheme by presentng numercal results for nstantaneous nonlnear meda. Usng the parametrc quadratc programmng technque, ths proposed scheme does not requre updatng system matrces and tedous teratve procedure at every tme step. The proposed technque allows nonconformng meshes between dfferent subdomans wth lnear and nonlnear meda, makng t sutable for structures comprsed of small nonlnear regons n an otherwse lnear medum. ACKNOWLEDGMENT Ths work s supported by the Natonal Natural Scence Foundaton of Chna (Grant Nos , and ) and Postdoctoral Scence Foundaton of Chna (Grant No. 214M561222). REFERENCES 1. Yee, K. S., Numercal soluton of ntal boundary value problems nvolvng Maxwell s equatons n sotropc meda, IEEE Transactons on Antennas and Propagaton, Vol. 14, 32 37, 1966.

11 Progress In Electromagnetcs Research M, Vol. 54, Taflove, A., Computatonal Electrodynamcs: The Fnte-Dfference Tme-Doman Method, Artech House, Norwood, MA, Kunz, K. S. and R. J. Luebbers, The Fnte Dfference Tme Doman Method for Electromagnetcs, CRC, Boca Raton, FL, Joseph, R. M. and A. Taflove, Spatal solton deflecton mechansm ndcated by FDTD Maxwell s equatons modelng, IEEE Photoncs Technology Letters, Vol. 6, , Van, V. and S. K. Chaudhur, A hybrd mplct-explct FDTD scheme for nonlnear optcal wavegude modelng, IEEE Transactons on Mcrowave Theory and Technques, Vol. 47, , Joseph, R. M. and A. Taflove, FDTD Maxwell s equatons models for nonlnear electrodynamcs and optcs, IEEE Transactons on Antennas and Propagaton, Vol. 45, , Zolkowsk, R., Full-wave vector Maxwell equaton modelng of the self-focusng of ultrashort optcal pulses n a nonlnear kerr medum exhbtng a fnte response tme, Journal of the Optcal Socety of Amerca B, Vol. 1, , Fsher, A., D. Whte, and G. Rodrgue, An effcent vector fnte element method for nonlnear electromagnetc modelng, Journal of Computatonal Physcs, Vol. 225, , Chen, J., Q. H. Lu, M. Cha, and J. A. Mx, A nonspurous 3-D vector dscontnuous Galerkn fnte-element tme-doman method, IEEE Mcrowave and Wreless Components Letters, Vol. 2, 1 3, Zhu, B., J. Chen, W. Zhong, and Q. H. Lu, A hybrd FETD-FDTD method wth nonconformng meshes, Communcatons n Computatonal Physcs, Vol. 9, , Tobon, L. E., Q. Ren, Q. T. Sun, J. Chen, and Q. H. Lu, New effcent mplct tme ntegraton method for DGTD appled to sequental multdoman and multscale problems, Progress In Electromagnetcs Research, Vol. 151, 1 8, Zhu, B., H. Yang, and J. Chen, A novel fnte element tme doman method for nonlnear Maxwell s equatons based on the parametrc quadratc programmng method, Mcrowave and Optcal Technology Letters, Vol. 57, , Cottle, R. W. and G. B. Dantzg, Complementary pvot theory of mathematcal programmng, Lnear Algebra Applcatons, Vol. 1, , Ferrs, M. C. and J. S. Pang, Engneerng and economc applcatons of complementarty problems, SIAM Revew, Vol. 39, , Zhang, H. W., S. Y. He, and X. S. L, Two aggregate-functon-based algorthms for analyss of 3D frctonal contact by lnear complementarty problem formulaton, Computer Methods n Appled Mechancs and Engneerng, Vol. 194, , Fscher, A., A specal Newton-type optmzaton method, Optmzaton, Vol. 24, , Lu, T., P. Zhang, and W. Ca, Dscontnuous Galerkn methods for dspersve and lossy Maxwell equatons and PML boundary condtons, Journal of Computatonal Physcs, Vol. 2, , Lu, T., W. Ca, and P. Zhang, Dscontnuous Galerkn tme-doman method for GPR smulaton n dspersve meda, IEEE Transactons on Geoscence and Remote Sensng, Vol. 43, 72 8, Mohammadan, A. H., V. Shankar, and W. F. Hall, Computaton of electromagnetc scatterng and radaton usng a tme-doman fnte-volume dscretzaton procedure, Computer Physcs Communcatons, Vol. 68, , Zhu, B., J. Chen, W. Zhong, and Q. H. Lu, A hybrd fnte-element/fnte-dfference method wth an mplct-explct tme-steppng scheme for Maxwell s equatons, Internatonal Journal of numercal Modellng: Electronc Networks, Devces and Felds, Vol. 25, , Luo, M. and Q. H. Lu, Spectral element method for band structures of three-dmensonal ansotropc photonc crystals, Physcal Revew E, Vol. 8, 1 7, Fan, G. and Q. H. Lu, A strongly well-posed PML n lossy meda, IEEE Antennas and Wreless Propagaton Letters, Vol. 2, 97 1, 23.