B. H. Jung Department of Information and Communication Engineering Hoseo University Asan, Chungnam , Korea

Transcription

1 Progress In Electromagnetcs Research, PIER 79, , 2008 SOLVING TIME DOMAIN HELMHOLTZ WAVE EQUATION WITH MOD-FDM B. H. Jung Department of Informaton and Communcaton Engneerng Hoseo Unversty Asan, Chungnam , Korea T. K. Sarkar Department of Electrcal Engneerng and Computer Scence Syracuse Unversty Syracuse, NY , USA Abstract In ths work, we present a marchng-on n degree fnte dfference method (MOD-FDM) to solve the tme doman Helmholtz wave equaton. Ths formulaton ncludes electrc and magnetc current denstes that are expressed n terms of the ncdent feld for scatterng problems for an open regon to mplement a plane wave exctaton. The unknown tme doman functonal varatons for the electrc feld are approxmated by an orthogonal bass functon set that s derved usng the Laguerre polynomals. These temporal bass functons are also used to expand current denstes. Wth the representaton of the dervatves of the tme doman varable n an analytc form, all the tme dervatves of the feld and current densty can be handled analytcally. By applyng a temporal testng procedure, we get a matrx equaton that s solved n a marchng-on n degree technque as the degree of the temporal bass functons s ncreased. Numercal results computed usng the proposed formulaton are presented and compared wth the solutons of the conventonal tme doman fnte dfference method (TD-FDM) and analytc solutons. 1. INTRODUCTION The fnte dfference method n the tme doman proposed by Yee has been extensvely employed to analyze transent scattered felds from conductng and delectrc materals [1 7]. Ths technque s often called the fnte-dfference tme-doman (FDTD) method [2, 3]. Recently, a

2 340 Jung and Sarkar marchng-on n degree fnte dfference method (MOD-FDM) wth the entre doman temporal orthogonal bass usng the assocated Laguerre functons was proposed to obtan an uncondtonally stable soluton [8]. Ths s n contrast to the conventonal FDTD analyss. The MOD- FDM methodology has been successfully mplemented n tme doman ntegral equatons [9 11], and tme doman fnte element method [12]. Based on ths scheme [8], varous works have been publshed [13 20]. However all the prevous works were developed to solve the frst-order Maxwell s curl equatons. In ths paper, we present a new method to solve the second-order tme doman Helmholtz wave equaton n a lossy meda wth the MOD- FDM to obtan transent electromagnetc scatterng responses [21, 22]. The formulaton uses the volume electrc and magnetc current denstes for scatterng problems n an open regon to mplement a plane wave exctaton [23]. These current denstes are expressed n term of the ncdent feld. The tme doman unknown coeffcent of the electrc feld s approxmated by a set of orthogonal bass functons that are derved from the Laguerre polynomals [24]. The Laguerre polynomals are defned only over the temporal nterval from zero to nfnty, and hence, are consdered to be more suted for the transent problems, as they naturally enforce causalty. The temporal bass functons used n ths work are completely convergent to zero as tme ncreases. Therefore ths bass may never cause any late tme nstabltes n the soluton as s often the case n solvng tme doman problems. So, the transent response spanned by these bass functons s also convergent to zero as tme progresses. These temporal bass functons are used to expand the varous current denstes. Snce all these temporal bass functons have analytcal dervatves, n the tme doman formulaton, all the tme dervatves of the felds and current denstes can be handled analytcally. Use of ths temporal expanson functon characterzng the tme varable also decouples the space-tme contnuum n an analytc fashon. By applyng a temporal testng procedure, we get a matrx equaton that s solved usng a marchngon n degree as the degree of temporal bass functons s ncreased. Ths paper s organzed as follows. In the next secton we present the conventonal fnte dfference scheme usng the Yee algorthm for the Helmholtz wave equaton brefly, and then derve the MOD-FDM formulaton wth a plane wave ncdence for scatterng problems n an open regon. In Secton 3, numercal results computed usng the proposed formulaton are presented and compared wth the solutons of the conventonal tme doman fnte dfference methods (TD-FDM) and compared wth analytc solutons. Fnally, n Secton 4, we present some conclusons drawn from ths work.

3 Progress In Electromagnetcs Research, PIER 79, FORMULATION In the tme doman, the general Maxwell s equatons n a lossy delectrc meda can be wrtten as H(r,t)=ε (r) E(r,t) + σ (r) E(r,t)+J(r,t) (1) t E(r,t)= µ (r) H(r,t) ρ (r) H(r,t) M(r,t) (2) t where ε, µ, σ and ρ are the permttvty, the permeablty, the electrc conductvty, and magnetc conductvty, respectvely [2, 3]. These parameters n the frst place are consdered ndependent of tme. E s the electrc feld and H s the magnetc feld. J and M are the volume electrc and magnetc current denstes, respectvely, that are ncluded to mplement a plane wave exctaton. From the two curl Equatons (1) and (2), we obtan a vector Helmholtz wave equaton for the electrc feld n a dvergence free regon wthout any net charge as [21, 22] 2 E(r,t) µ(r)ε(r) 2 E(r,t) t 2 [µ(r)σ(r)+ε(r)ρ(r)] E(r,t) t σ(r)ρ(r)e(r,t)=µ(r) J(r,t) + ρ(r)j(r,t)+ M(r,t). (3) t For smplcty, we consder the one-dmensonal problem wth the feld component E y propagatng along the x-drecton. Then the wave Equaton (3) becomes 2 E y (x, t) x 2 µ(x)ε(x) 2 E y (x, t) t 2 [µ(x)σ(x)+ε(x)ρ(x)] E y(x, t) t σ(x)ρ(x)e y (x, t) =µ(x) J y(x, t) + ρ(x)j y (x, t) M z(x, t). (4) t x Followng Yee s notaton and dscretzng the space and the tme wth the cell sze x and tme step t, respectvely [1], we can express (4) at x = x = x and t = t n = n t as 2 x 2 E y n µ 2 ε t 2 E y n (µ σ + ε ρ ) t E y n σ ρ E y n = µ t J y n + ρ J y n x M z n (5) where E y n = E y (x,t n ). Applyng the second-order central fnte dfference approxmaton to (5) yelds E y n +1 2 E y n + E y n 1 ( x) 2 µ ε E y n+1 2 E y n + E y n 1 ( t) 2

4 342 Jung and Sarkar (µ σ + ε ρ ) E y n+1 J y n+1/2 = µ J y n 1/2 t E y n 1 2 t σ ρ E y n + ρ J y n M z n +1/2 M z n 1/2. (6) x Solvng for E y n+1 and rearrangng terms, we get an alternate equaton wth changng n +1ton as E y n = 2( t) 2 [ 1 2µ ε + t (µ σ + ε ρ ) ( x) 2 E y n 1 1 ( µ ε + ( t) 2 2 ) ( x) 2 σ ρ E y n ( x) 2 E y n µ ε t (µ σ +ε ρ ) 2( t) 2 ρ J y n x E y n 2 µ t ( J y n 1/2 J y n 3/2 ) ( M z n 1 +1/2 M z n 1 1/2) ]. (7) Ths s generally the equaton used n a TD-FDM to obtan the electrc feld. Now we derve a MOD-FDM formulaton. To carry out the tme dervatves analytcally, we expand all the temporal quanttes n terms of the assocate Laguerre polynomals gven by φ p (st) =e st/2 L p (st) (8) where s s a tme scale parameter whch takes care of the unts along the tme axs [25], and L p s the Laguerre polynomal wth degree p [24]. Ths temporal bass functons are orthogonal as φ p (st)φ q (st)d(st) =δ pq (9) 0 where δ pq s Kronecker delta wth value 1 when p = q and 0 otherwse. A contnuous functon F (x, t) defned for any value of tme t 0 can be expanded by the assocate Laguerre bass functons as F (x, t) = F p (x)φ p (st) (10) where F p s the coeffcent whch can be obtaned from F p (x) = F (x, t)φ p (st)d(st). (11) 0

5 Progress In Electromagnetcs Research, PIER 79, The frst and second dervatves of the functon F (x, t) can be wrtten as [9] d dt F (x, t) =s 1 p 1 2 F p(x)+ F m (x) φ p (st) (12) d 2 F (x, t) =s2 1 p 1 dt2 4 F p(x)+ (p m)f m (x) φ p (st). (13) By expandng E y, J y, and M z wth (10), (12), and (13), and puttng them n (4) we have 2 x 2 Ey(x)φ p p (st) µ(x)ε(x)s 2 1 p 1 4 Ep y(x)+ (p m) Ey m (x) φ p (st) [µ(x)σ(x) +ε(x)ρ(x)] s 1 p 1 2 Ep y(x)+ Ey m (x) φ p (st) σ(x)ρ(x) Ey(x)φ p p (st) =µ(x)s 1 p 1 2 J y p (x)+ Jy m (x) φ p (st) +ρ(x) Jy p (x)φ p (st) Mz p (x)φ p (st) (14) x where Ey, p Jy p and, Mz p are the coeffcents of the assocate Laguerre polynomals bass for E y, J y, and M z, respectvely. To elmnate the varable t and the nfnte summaton n (14), we test ths equaton n a Galerkn s methodology wth φ q (st). Due to the orthogonal property n (9) we have d 2 dx 2 Eq y(x) µ(x)ε(x)s 2 1 q 1 4 Eq y(x)+ (q m) Ey m (x) [µ(x)σ(x)+ε(x)ρ(x)] s 1 q 1 2 Eq y(x)+ Ey m (x) σ(x)ρ(x)ey(x) q = µ(x)s 1 q 1 2 J y q (x)+ Jy m (x) + ρ(x)jy q (x) d dx M z q (x). (15)

6 344 Jung and Sarkar Usng the fnte dfference n space to approxmate the spatal dervatves s smlar to the tradtonal FDTD method. By usng second-order spatal dfference n (15) at x = x, wehave E y q +1 2 E y q + E y q 1 ( x) 2 µ ε s 2 1 q 1 4 E y q + (q m) E y m (µ σ + ε ρ ) s 1 q 1 2 E y q + E y m σ ρ E y q = µ s 1 q 1 2 J y q + J y m where E y q = Eq y (x ), + ρ J y q M z q +1/2 M z q 1/2 x (16) J y q = J y (x,t) φ q (st)d(st) (17) 0 ) M z q ±1/2 = M z (x ±1/2,t φ q (st)d(st). (18) Rewrtng (16) n a smple form, we have 0 E y q 1 + α E y q + E y q +1 = βq (19) where [ ( ) s x 2 α = 2+µ ε + (µ σ + ε ρ ) s( x) σ ρ ( x) 2 ] (20) q 1 β q = µ ε (s x) 2 ( +( x) 2 µ s 2 + ρ ) q 1 (q m) E y m +(µ σ + ε ρ ) s( x) 2 E y m q 1 J y q + µ ( ) s J y m x M z q +1/2 M z q 1/2.(21) We can get a matrx equaton form from (19) (21) wth a proper boundary condton as [ ] [α j ] E y q j =[β q ], q =0, 1, 2... (22) Here we use the dsperson boundary condton derved wth the assocate Laguerre bass functons n [8]. By solvng ths matrx

7 Progress In Electromagnetcs Research, PIER 79, Equaton (22) recursvely n a MOD manner and usng (10), the electrc feld s expressed as E y (x,t n )= M 1 E y p φ p (st n ) (23) where Ms a fnte number of temporal bass functons. Usng the plane wave njector scheme, the correspondng electrc and magnetc current denstes are expressed n term of the ncdent electrc feld [23]. When a plane wave wth y-polarzaton s ncdent to the x-drecton, the current denstes n (17) and (18) are gven by [20] J y (x, t) = Enc (x, t) (24) η x M z (x, t) = Enc (x, t) (25) x where η s the wave mpedance of free space. We can also construct a Helmholtz wave equaton for the magnetc feld usng the two famlar curl Equatons (1) and (2), and derve the matrx equaton correspondng to (22) n a smlar way. In ths case, the matrx [α j ] s same as n (22), but [β q ] s slghtly dfferent due to the dualty. Therefore we can obtan the coeffcents from the magnetc feld wthout an addtonal matrx nverson n ths procedure when computng the electrc feld coeffcents. 3. NUMERICAL EXAMPLES The geometry to be analyzed here s a one-dmensonal delectrc slab backed by a perfectly electrc conductor (PEC). Ths slab has a relatve permttvty 4, σ = 0.05 S/m, and s 9cm thck. The permeablty s that of free space and the magnetc loss s assumed to be zero. The problem space conssts of 310 cells wth x =1.5mm and cells for the slab. The PEC boundary condton s appled at the boundary x =46.5cm. The ncdent feld n (24) and (25) used n ths work s the Gaussan pulse plane wave as defned n [26] E nc 4 (x, t) =E 0 T π e γ2, γ = 4 T (ct ct 0 x + x s ) (26) where T s the pulse wdth, c s the velocty of propagaton n free space, t 0 s a tme delay whch represents the tme at whch the pulse peak at the orgn, and x s s the source poston of the plane wave

8 346 Jung and Sarkar ncdence. In the computaton of the TD-FDM usng (7), we set the tme step sze t = x/2c. In the numercal computaton, the pulse wdth s T/c = 400 ps, and x s / x = 50. We set the number of Laguerre bass functons as M = 500 and the tme scale parameter (a) (b) Fgure 1. Electrc feld along x-poston wth the ncdence of Gaussan pulse plane wave. (a) 400 tme steps, (b) 800 tme steps.

9 Progress In Electromagnetcs Research, PIER 79, s set to s = We compare the electrc feld computed by the proposed MOD-FDM wth the soluton obtaned usng the TD-FDM and from the analytc soluton. The analytc soluton s obtaned by the nverse dscrete Fourer transform of the frequency doman soluton as descrbed n [27]. For the frst example, Fg. 1 shows the electrc feld along the x- drecton for an ncdent Gaussan pulse at n = 400 and 800 tme steps of the TD-FDM computaton. We set E 0 = T π/4. The agreement between the conventonal TD-FDM and the proposed MOD-FDM s very good. Fg. 2 shows the transent electrc feld at x = 30cm computed by the MOD-FDM and TD-FDM, and nverse Fourer transform of the analytc soluton. All the three solutons agree well, as s evdent from the fgure. Fgure 2. Transent electrc feld at x = 30 cm versus tme wth the ncdence of Gaussan pulse plane wave. In the next example, Fg. 3 shows the electrc feld as a functon of the x-coordnates at temporal locatons of n = 400 and 800 tme steps when the dervatve of Gaussan pulse wth E 0 = T 2 π/(32c) s ncdent on the slab. Agreement between the solutons obtaned usng the proposed MOD-FDM and the conventonal TD-FDM s excellent. Fg. 4 shows the transent electrc feld at x = 30 cm for an ncdent feld proportonal to the dervatve form of the Gaussan pulse. We can see that the agreement between the solutons obtaned by the TD-FDM and MOD-FDM, and usng the analytc soluton s very good.

10 348 Jung and Sarkar (a) Fgure 3. Electrc feld along x-poston wth the ncdence of the dervatve of Gaussan pulse plane wave. (a) 400 tme steps, (b) 800 tme steps. (b)

11 Progress In Electromagnetcs Research, PIER 79, Fgure 4. Transent electrc feld at x = 30 cm versus tme wth the ncdence of the dervatve of Gaussan pulse plane wave. 4. CONCLUSION We have proposed a marchng-on n degree fnte dfference method to solve the tme doman Helmholtz wave equaton. Ths formulaton uses the electrc and the magnetc current denstes to excte a plane wave source n an open regon. The tme doman unknown coeffcents for the electrc feld are approxmated by a set of orthogonal bass functons that are derved from the Laguerre polynomals. Wth the representaton of the dervatves of the tme doman coeffcents n an analytc form, all the tme dervatves of the feld and current densty can be handled analytcally. By applyng a temporal testng procedure, we get a matrx equaton that s solved usng a marchngon n degree technque as the degree of the temporal bass functons s ncreased. The agreement between the solutons obtaned usng the proposed method and the tradtonal tme doman fnte dfference method, and the nverse Fourer transform of the frequency doman analytc soluton s excellent as a functon of both the spatal and the temporal varables. The proposed formulaton can be extended to twoand three-dmensonal scatterng problems drectly. ACKNOWLEDGMENT Ths work was supported by the Korea Research Foundaton Grant (KRF D00341) funded by the Korean Government (MOEHRD).

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