A constant recursive convolution technique for frequency dependent scalar wave equation based FDTD algorithm

Transcription

1 J Comput Electron (213) 12: DOI 1.17/s A constant recursve convoluton technque for frequency dependent scalar wave equaton bed FDTD algorthm M. Burak Özakın Serkan Aksoy Publshed onlne: 8 June 213 Sprnger ScenceBusness Meda New York 213 Abstract A scalar wave equaton bed recursve convoluton fnte-dfference tme-doman algorthm s developed for a frequency-dependent Debye medum n ths paper. Ths algorthm s bed on a recursve evaluaton of a convoluton ntegral n the tme doman. A numercal example s presented for a problem of wde-band reflecton from an ar-water nterface. The obtaned results are compared wth an analytcal soluton. The excellent agreement s observed between the numercal results. Memory and computatonal tme advantages of the proposed method over Maxwell s equatons bed soluton are also shown. Keywords Dspersve medum Fnte dfference tme doman method Recursve convoluton 1 Introducton The Fnte Dfference Tme Doman (FDTD) method s an effectve numercal tool for soluton of complex electromagnetc problems 1]. In partcular, the FDTD method s n rapd progress for modelng the dspersve materals such Debye, Lorentz types, etc. Fundamentally, three man Maxwell s equatons bed approaches n the dspersve applcaton of the FDTD method are Recursve Convoluton (RC), Auxlary Dfferental Equaton and Z-Transform technques. The RC technque h dfferent forms of Constant Recursve Convoluton (CRC) 2], Pecewse Lnear Recursve Convoluton 3], Pecewse Constant Recursve Convoluton 4], Trapezodal Recursve Convoluton 5], JE M.B. Özakın ( ) S. Aksoy Department of Electroncs Engneerng, Gebze Insttute of Technology, Kocael, Turkey e-mal: bozakn@gyte.edu.tr Convoluton 6] and Pecewse Lnear JE Recursve Convoluton 7]. Soluton of wave propagaton problems n a dspersve medum s a crucal topc because the wave velocty s a frequency dependent parameter n the dspersve medum. The materal dsperson strongly affects propagated wdeband sgnals. Ths leads to dstorton of ths sgnal n dspersve meda. In partcular, electromagnetc wave propagaton n the dspersve medum can be modeled ether wth the Maxwell s equatons or the wave equaton. Although the dfferent Maxwell s equatons bed approaches for the FDTD dspersve modelng are present mentoned before, the wave equaton bed approaches for the dspersve modelng are not wdespread n the lterature. However, the wave equaton s used n many dfferent scentfc dscplnes such acoustcs 8], flud dynamcs 9] and so on. It also smplfes an electromagnetc problem because of usng only one electromagnetc feld component n the soluton. Ths s especally mportant for applcaton of numercal methods because both calculaton tme and memory requrements become less. In ths paper, the electromagnetc wave propagaton n a lnear dspersve Debye medum s nvestgated by the scalar wave equaton bed recursve convoluton fnte-dfference tme-doman method. For ths am, frst, a recursve accumulator s extracted. Then, an FDTD update equaton bed on the CRC algorthm (generally known the RC technque) for the scalar wave equaton of the Debye medum s formulated and valdated wth an analytcal soluton of an example problem. Ths paper s organzed follows: In Sect. 2, a scalar wave equaton bed formulaton of the CRC-FDTD method s gven. In Sect. 3, a numercal example of an ar-water nterface for the Debye medum s shown by a comparson of an analytcal soluton. Memory and computatonal tme

2 J Comput Electron (213) 12: comparsons between the proposed and Maxwell s equatons bed solutons are also gven. Secton 4 gves summary and fnal remarks. 2 Formulaton of the problem A Fourer transformed Helmholtz equaton n a onedmensonal Cartesan coordnate s u(x, ω) k 2 (ω)u(x, ω) (1) where s the Laplace operator, u(x, ω) s a feld component and ω s an angular frequency. The key pont n ths equaton about treatment of the wave number k k(ω) for dspersve meda k 2 (ω) ω 2 ε(ω)μ. (2) Here, only, the frequency-dependent delectrc permttvty ε(ω) s used for dspersve medum modelng. The magnetc permeablty of the medum μ s sumed to be equal to a free space permeablty μ. In ths ce, ε(ω) can be wrtten n a more general form ε(ω) ε ε χ e (ω) ] (3) where ε and ε are the delectrc permttvty of the free space and the delectrc permttvty at a hgh-frequency lmt, respectvely. χ e (ω) shows the delectrc susceptblty of the dspersve medum. Substtutng Eq. (3) over Eq. (2) nto Eq. (1), the scalar wave equaton becomes u(x, ω) ε ε μ ω 2 u(x, ω) ε μ ω 2 χ e (ω)u(x, ω). (4) Here, one can note n Eq. (4) that the multplcaton of two functons n the frequency doman s present. It means that a convoluton ntegral wll appear n the tme doman. Applyng the nverse Fourer transform through Eq. (4) leads to the followng equaton u(x, t) ε 2 c 2 u(x, t) t2 1 2 { u(x, t) χe c 2 t 2 (t) } (5) where c 1/ ε μ s the speed of lght n the free space and the convoluton term can be wrtten u(x, t) ε 2 c 2 u(x, t) t2 1 2 { t } c 2 t 2 u(x, t τ)χ e (τ)dτ. (6) Now, let us defne the convoluton ntegral for smplcty ξ(x,t) t u(x, t τ)χ e (τ)dτ. (7) Therefore Eq. (5) can be wrtten u(x, t) ε c 2 2 t 2 u(x, t) 1 c 2 2 ξ(x,t). (8) t2 In order to extract the FDTD update equaton for the dspersve medum, the feld must be dscretzed over the tme and space varables u(x, t) u( x, n t). Here, and n are the ndces of dscrete space and tme varables, respectvely. x and t show the unt spatal and tme steps, respectvely. In ths ce, the scalar wave equaton becomes n ε 2 c 2 t 2 n 1 2 c 2 t 2 ξ n (9) where the man concern s to treat the convoluton ntegral n a dscrete form. In order to overcome ths problem, the feld component n the convoluton ntegral s sumed to be constant (the CRC technque) over the each tme nterval t. Therefore, the convoluton ntegral n the dscrete form wll be rearranged ξ n n t n1 n tτ χ e τ dτ (m1) t nm n1 χ e τ dτ nm χ e m. (1) Now, central fnte dfferences for the spatal and tme dervatves can be appled nto Eq. (9). Then, the FDTD update equaton for the scalar wave equaton becomes n 1 2n n 1 x 2 1 ξ n1 c 2 where ε n1 2 n n1 c 2 t 2 2ξ n ξ n1 t 2 (11) ξ n1 n1m χ e m, (12) n1 ξ n nm χ e m n χ e ξ n1 n n1m χ e m ( n1 χ e n χ e 1 χ e m1, (13) ) χ e m2. (14)

3 754 J Comput Electron (213) 12: Expresson of ξ n1 2ξ n ξ n1 n Eq. (11) can be wrtten by usng Eq. (12), Eq. (13) and Eq. (14) ξ n1 2ξ n ξ n1 ( n1 χ e n χ e 1 2n χ e ) n1m ( χe m2 2χ e m1 χ e m ). (15) Let us defne a parameter of χ e for smplcty χ e m χ e m2 2χ e m1 χ e m. (16) Then Eq. (15) wll be ξ n1 2ξ n ξ n1 n1 χ e n χ e 1 2n χ e χ e m. (17) Substtutng Eq. (17) nto Eq. (11), the dscrete scalar wave equaton becomes n 1 2n n 1 x 2 1 ( n1 ε n1 c 2 2 n n1 t 2 χ e n χ e 1 2n χ e ) 1 n1m χ e m. (18) After more smplfcatons n Eq. (18), the FDTD update equaton can be rearranged n1 x 2 (ε χ e )n 1 x 2 (ε χ e )n 1 x2 (2ε χ e 1 2χ e ) 2(c t)2 x 2 (ε χ e ) n ε ε χ e n1 1 ε χ e n1m χ e m (19) where, at a frst glance, t seems that the teraton progress n the summaton term wll requre storng a large number of pt feld tme values. However, n order to crcumvent ths problem, t s shown that a recursve relaton for χ e m can be obtanable for the Debye medum. Let us consder the Debye medum wth ts susceptblty functon χ e (ω) ε s ε (2) 1 jω where ε s s the statc delectrc permttvty at the low frequences and s the characterstc relaxaton tme of the medum (j 1). After applcaton of the nverse Fourer transform to Eq. (2), the tme doman representaton of χ e (t) wll be ( ) εs ε χ e (t) e t H(t) (21) where H(t) s the unt step functon, whch shows the satsfacton of the causalty prncple. χ e (t) can be used for the calculaton of χ e m n Eq. (16) χ e m (m1) t (m1) t χ e τ dτ ( ε s ε (ε s ε )e )e τ dτ Thus, by usng Eq. (22), Eq. (16) becomes χ e m (ε s ε )e t 1 3e (ε s ε )e 1 e t ]. (22) 3e 2 t e 3 t ] 1 e t ] 3. (23) Now, let us try to fnd the recursve relaton for the convoluton term n Eq. (1). For ths am, defne a new parameter ψ n ψ n χ e m. (24) One can examne a few tme steps for ψ n startng wth n 2 ψ 2 1m χ e m 1 χ e (25) and for n 3 1 ψ 3 2m χ e m 2 χ e 1 χ e 1 (26) where χ e m1 can be lnked to χ e m by usng Eq. (23) χ e m1 e t χ e m. (27) Then, the recursve relaton for ψ 3 wll be fnally obtaned ψ 3 2 χ e e t 1 χ e 2 χ e e t ψ 2 (28) where t s clear that a next tme value of ψ s found from ts prevous tme value. Therefore, a recursve update equaton can be reformulated n a more general form for n 3 ψ n n1 χ e e t ψ n1 (29)

4 J Comput Electron (213) 12: Fg. 2 Demonstraton of the ncdent Gaussan pulse whch propagates toward to the water regon at the 8th tme step Fg. 1 The flowchart of the scalar wave equaton bed CRC-FDTD method where ψ ψ 1 ψ 2. Here, t s worthy to note that the recursve update equaton accumulator for ψ n s extracted specally for an exponental form of χ e (t). As a result, substtutng ths recursve equaton nto Eq. (19), the fnal form of the FDTD update equaton for the scalar wave equaton becomes n1 x 2 (ε χ e )n 1 x 2 (ε χ e )n 1 x2 (2ε χ e 1 2χ e ) 2(c t)2 x 2 (ε χ e ) n ε ε χ e n1 1 ε χ e ψ n (3) where t s of nterest to note that the two-tme steps of χ e are present n the formulaton. However, χ e 1 can be calculated from χ e by usng Eq. (22). A flowchart of the scalar wave equaton bed CRC- FDTD algorthm can be drawn n Fg Numercal example wth valdaton In ths secton, the proposed method s valdated by an example problem of the wde-band reflecton from an ar-water nterface. The ar regon s modeled the free-space medum wth the parameters of ε 1, ε s 1, ε 1/36π 1 9 F/m, μ 4π 1 7 H/m and s. The water s modeled the Debye type dspersve medum wth the parameters of ε 1.8, ε s 81 and s. The well known CFL lmt s used n order to keep the stablty of the FDTD soluton. In addton, the problem space s termnated wth a frst order Mur type absorbng boundary condton. A one-dmensonal numercal example s consdered n 2]. It conssts of 1 cells separated by 499 cells and 51 cells for the free-space and water, respectvely. Each cell h equal length of x 37.5 µm and t.625 ps. A normalzed Gaussan pulse shown n Fg. 2 s used an exctaton source wth a spatal wdth of 4 cells and located at 1th cell ( ) πn t H(n t) n1 1 sn 2 ] H(n t T) (31) T where T s a spatal wdth of the pulse. After mpngng the electromagnetc wave on the arwater nterface, the pulse s separated nto the reflected and propagated parts. Ths phenomenon can be clearly seen n Fg. 3. In Fg. 4, a comparson between the proposed FDTD soluton and the analytcal soluton 2] of ths problem s shown for the reflecton coeffcent at 2th tme step. The reflecton coeffcent of the numercal example w obtaned by applyng a Ft Fourer Transform to the tme doman values of an observaton pont. An excellent agreement s observed between the results. In Table 1, the memory and the computatonal tme advantages of the proposed method over the Maxwell s equatons bed soluton are shown for the one-dmensonal example gven above. The comparsons are performed by an Intel Core2 Quad computer of 2.4 GHz CPU speed. The proposed formulaton needs % 33 less memory wth less computaton tme. Ths s due to the usng one feld component

5 756 J Comput Electron (213) 12: Summary and fnal remarks Fg. 3 Demonstraton of the pulses whch reflected from the nterface and propagates nto the water at the 2th tme step In ths study, the scalar wave equaton bed recursve convoluton formulaton of the frequency dependent fnte dfference tme doman method s gven for the lnear dspersve Debye medum. In the frst step, the scalar wave equaton n the tme doman for the dspersve medum s extracted from the Helmholtz equaton n the frequency doman. Relaton of the frequency-dependent parameters n that equaton s clearly revealed. Then, the nverse Fourer transform s employed n order to obtan the tme doman form. Ths leads to the convoluton ntegral n the formulaton. A tme doman recursve accumulator for ths convoluton ntegral s extracted and lnked wth the FDTD algorthm. The proposed formulaton s valdated wth an analytcal result over the numercal example of the wde-band reflecton coeffcent from the ar regon to water. The excellent agreement s observed between the results. The memory and the computatonal tme advantages of the proposed method over the Maxwell s equatons bed soluton are also shown. For the future work, we are plannng to extend the proposed scalar wave equaton bed FDTD formulaton for other types of the dspersve medums such Drude, Lorentz and so on wth ncludng the lossy term. References Fg. 4 Comparson of the frequency-dependent pulse reflecton coeffcent between the analytcal and the FDTD solutons Table 1 Comparson of the memory and the computatonal tme between the proposed and Maxwell s equatons bed solutons n onedmensonal ce Method (1 D) Memory (kb) Computatonal tme (s) Proposed Maxwell s eq. bed n the soluton, although the Maxwell s equatons bed soluton needs two feld components. These advantages wll be domnant n the ce of two and three dmensonal problems. 1. Taflove, A., Hagness, S.C.: Computatonal Electrodynamcs: The Fnte-Dfference Tme-Doman Method, 3rd edn. Artech House, London (25) 2. Luebbers, R., Hunsberger, F.P., Kunz, K.S., Standler, R.B., Schneder, M.: A frequency-dependent fnte-dfference tme-doman formulaton for dspersve materals. IEEE Trans. Electromagn. Compat. 32(3), (199) 3. Kelley, D.F., Luebbers, R.J.: Pecewse lnear recursve convoluton for dspersve meda usng FDTD. IEEE Trans. Antenn Propag. 44(6), (1996) 4. Schuster, J.W., Luebbers, R.J.: An accurate FDTD algorthm for dspersve meda usng a pecewse constant recursve convoluton technque. In: IEEE Antenn and Propagaton Socety Internatonal Symposum, Atlanta, GA, USA, June (1998) 5. Sushansan, R., LoVetr, J.: A comparson of numercal technques for modelng electromagnetc dspersve meda. IEEE Mcrow. Guded Wave Lett. 5(12), (1995) 6. Sushansan, R., LoVetr, J.: An FDTD formulaton for dspersve meda usng a current densty. IEEE Trans. Antenn Propag. 46(1), (1998) 7. Sushansan, R., LoVetr, J.: A Novel FDTD Formulaton for Dspersve Meda. IEEE Mcrow. Wrel. Compon. Lett. 13(5), (23) 8. Knsler, L.E., Frey, A.R., Coppens, A.B., Sanders, J.V.: Fundamentals of Acoustcs, 4th edn. Wley, New York (2) 9. Cebec, T., Shao, J.P., Kafyeke, F., Laurendeau, E.: Computatonal Flud Dynamcs for Engneers. Sprnger, Berln (25)