Convolutional PML (CPML): An Efficient FDTD Implementation of the CFS-PML for Arbitrary Media

Transcription

1 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Convolutonal PML (CPML): An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda J. Alan Roden Personal Systems Group IBM Corporaton Research Trangle Park, NC Emal: FAX: and Stephen D. Gedney Department of Electrcal and Computer Engneerng Unversty of Kentucky Lexngton, KY Emal: FAX: Key Terms: fnte-dfference tme-doman (FDTD), perfectly matched layer (PML), stretched coordnates Abstract A novel mplementaton of perfectly matched layer (PML) meda s presented for the termnaton of FDTD lattces. The mplementaton s based on the stretched coordnate form of the PML, a recursve convoluton and the use of complex frequency shfted (CFS) PML parameters. The method, referred to here as the convolutonal PML (CPML) offers a number of advantages over the tradtonal mplementatons of the PML. Specfcally, the applcaton of the CPML s completely ndependent of the host medum. Thus, no modfcatons are necessary when applyng t to nhomogeneous, lossy, ansotropc, dspersve or non-lnear meda. Secondly, t s shown that the CFS-PML s hghly absorptve of evanescent modes and can provde sgnfcant memory savngs when computng the wave nteracton of elongated structures, sharp corners, or low frequency exctatons. Submtted to Mcrowave and Optcal Technology Letters, June

2 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton I. Introducton The perfectly matched layer (PML) absorbng meda [1] has proven to be the most robust and effcent technque for the termnaton of FDTD lattces [2]. Nevertheless, a lmtaton of the PML formulatons commonly used s that they are neffectve at absorbng evanescent waves [3-5]. As a result, the PML must be placed suffcently far from an obstacle such that the evanescent waves have suffcently decayed. A further symptom s that the PML wll suffer from late tme reflectons when termnatng hghly elongated lattces or when smulatng felds wth very long tme sgnatures [3, 6]. Ths s partly due to the weakly causal nature of the PML as orgnally proposed [7]. In [8], Kuzuoglu and Mttra ntroduced a strctly causal form of the PML by smply shftng the frequency dependent pole off the real axs and nto the negatve-magnary half of the complex plane. Ths s referred to here as the complex frequency shfted (CFS) PML. The applcaton of ths technque wthn the FDTD has been presented n [2, 9]. Unfortunately, these technques are lmted n threedmensons by the fact that 3 auxlary varables need to be ntroduced to represent the feld updates n a well-posed manner. Subsequently, ths partcular formulaton has not receved much attenton. However, further studes have found that the CFS-PML suggested n [8] s hghly effectve at absorbng evanescent waves and sgnals of long tme-sgnature [10]. Thus, gven a more effcent mplementaton, ths technque could lead to a sgnfcant savngs n memory over the tradtonal PML by greatly reducng the overall lattce dmenson. In ths paper, a novel mplementaton of the CFS-PML s ntroduced based on a stretched coordnate formulaton and a recursve convoluton. Ths method s referred to here as the convolutonal PML (CPML). It s shown that ths technque requres only two auxlary varables per dscrete feld pont. The mplementaton s also hghly robust n the sense that t wll absorb waves n sotropc and Submtted to Mcrowave and Optcal Technology Letters, June

3 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton homogeneous meda, as well as n nhomogeneous, lossy, dspersve, ansotropc or non-lnear meda wthout any further generalzaton as s needed by prevous formulatons (c.f. [11-13]). Further, for general meda, the number of addtonal unknowns that are requred for the CPML formulaton s dentcal to the most effcent technques proposed for the orgnal splt-feld and unaxal PML formulatons. Thus, gven the fact that the method s hghly absorptve of evanescent waves, a sgnfcant memory savngs can be realzed. In the followng secton, the CPML formulaton s presented n a general form. The specfc applcaton to the CSF-PML as proposed by Kuzouglu and Mttra s then presented n Secton III. In Secton V, the technque s appled to the study of the wave nteracton wth a hghly elongated plate n a lossy meda. The effectveness of the method s studed as a functon of the consttutve parameters of the PML. It s shown that the technque s ndeed hghly absorptve of evanescent waves and has lead to a four-fold reducton n memory as compared to termnaton wth a tradtonal formulaton of the PML. II. General Formulaton A PML medum s assumed to termnate a space occuped by a generalzed host medum. Here, for sake of example, a lossy medum s assumed. The PML formulaton s posed n the stretched coordnate space [14]. The x-projecton of Ampere's law s thus specfed as: 1 1 jωεe + σe = H H s y s z x x z y y z (1) where s are the stretched coordnate metrcs, whch was orgnally proposed by Berenger to be: σ s = 1 +, ( = xy,, or z) jωε o (2) Submtted to Mcrowave and Optcal Technology Letters, June

4 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Equaton (1) s next transformed to the tme doman. Due to the frequency dependence of the stretched coordnate metrcs, ths results n a convoluton on the rght-hand-sde. Namely, ε E + σe = s () t H s () t H t y z x x y z z y (3) 1 where st () s the nverse Laplace transform of s [15]. The convoluton on the rght-hand-sde of (3) can be accelerated by the use of the recursve convoluton (RC) method orgnally proposed by Luebbers, et al. [16, 17]. Specfcally, snce the nverse of the stretched coordnate parameters s smply a ratonal functon, ts mpulse response s analytcally known. As a result, the convoluton n (3) can be both effcently and accurately performed. Ths s llustrated n the next secton. III. The Dscrete Form of the CPML The CPML method s based on the tme-dependent form of the stretched coordnate formulaton of Maxwell's equatons as proposed n (3). In ths secton, the dscrete form of these equatons wll be derved. The choce of the complex stretchng varable wll be that proposed by Kuzouglu and Mttra. Specfcally, t s assumed that: σ s = κ +,( = x, y, or z) α + jωε o (4) where α and σ are assumed to be postve real and κ s real and 1. Defne 1 s = s. Then, usng Laplace transform theory, t can be shown that s has the mpulse response: σ α + ( t) σ t εκ ε δ ( t) δ o o s( t) = e ut () = + ζ () 2 t (5) κ εκ κ o Submtted to Mcrowave and Optcal Technology Letters, June

5 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton where δ ( t ) s the unt mpulse functon and ut ( ) s the step functon. Insertng (5) nto (3) leads to: E εε + σe = H H + ζ () t H ζ () t H x r 0 x z x y z z x t κy y κz z y κz z (6) To enable the effcent representaton of the convoluton n (6), the dscrete mpulse response for ζ () t s defned as ( ) + + t 0 = ( ) = m 2 εκ t m t o σ α + τ εκ o εo ( m 1) t σ ( m 1) Z m ζ τ dτ e dτ = ae σ m + α t κ εo (7) where σ σ + α t κ εo a = 1.0 e. (8) 2 ( σκ + κ α ) Utlzng (7) and (8), (6) s then dscretzed n both space and tme accordng to a staggered Yeescheme leadng to: εε r 0 n+ 1 n n+ 1 n n+ 1 n+ 1 n+ 1 n x 1 x,, 1 x +,, 1 x,, 1 z,, 1, 1 z , + 1, 1 y, + 1,, + 1 y jk jk jk jk j k j k jk + 1, jk, E E E E H H H H + σ = 2 κ κ t y y z z n m+ 1 n m+ 1 n m+ 1 n m N 1 Hz H z 1 1 N 1 H y H 2 ( ), j+ 2, k + 2, j 2, k 1,, 1 y + jk+ + 1, jk, o ( ) y oz m= 0 y m= 0 z + Z m Z m (9) The dscrete convoluton n (9) s qute costly n ts present form. Fortunately, due to the smple exponental form of Z ( m ), these sums can be performed recursvely usng the recursve convoluton o method [16, 17]. To ths end, the set of auxlary expressons as: ψ s ntroduced, and (9) s mplemented Submtted to Mcrowave and Optcal Technology Letters, June

6 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton εε r n+ 1 n n+ 1 n n+ n+ n+ n x 1 x,, 1 x +,, 1 x,, 1 z,, 1, 1 z , + 1, 1 y, + 1,, + 1 y jk jk jk jk j k j k jk + 1, jk, E E E E H H H H + σ = 2 κ κ t y y z z 1 1 n+ n ψ ψ exy 1 exz +, jk, + 1, jk, 2 2 (10) where ψ 1 1 n+ n n+ n+ 2 2 e = ψ + xy y e 1 1,, 1, 1, 1, 1 +, 2,, xy + jk y z + j + k z jk + j k y b a H H, (11) ψ 1 1 n+ n n+ n+ 2 2 e = ψ + xz z e 1 xz z y 1 1,, 1 y 1,, 1 z +, jk, +, jk, + jk+ + jk b a H H, (12) σ + α t κ εo b = e, ( = xy,, or z) (13) and a s gven by (8). Smlar expressons are derved for the remanng felds. Gven the dscrete feld equatons as posed by (10), an explct tme-marchng scheme s ntroduced. Ths scheme s second-order accurate and s stable wthn the Courant lmt for all postve real values of σ and α and for all real values of κ 1. IV. Dscusson The convolutonal terms are recursvely updated at each tme step usng (11) and (12). To reduce reflecton error n the dscrete space, the consttutve parameters σ, α, and κ are spatally scaled along ther respectve axes. Snce these coeffcents smply have one-dmensonal spatal varaton, the coeffcents a and b are also one-dmensonal functons. Thus, these coeffcents are convenently computed pror to the feld computaton and effcently stored n one-dmensonal vector arrays. It s further noted that the ψ e are zero outsde of ther assocated PML. Thus, they only need to be stored wthn the PML regons. From a practcal standpont, the electrc feld s thus updated n all space usng Submtted to Mcrowave and Optcal Technology Letters, June

7 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton standard FDTD updates. Then, the separate loops. To reduce the dscretzaton error, the parameters ψ e are smply added n wthn the perspectve PML regons n σ, κ are scaled such that they are 0 and 1 at the PML/workng volume nterface, respectvely, and are maxmum at the exteror boundary [1, 2]. However, α s not scaled n the same manner. Specfcally, to reduce reflecton error of evanescent modes α must be non-zero at the front boundary nterface. However, for the CFS-PML to absorb purely propagatng modes at low frequency, α should actually decrease to zero away from the boundary nterface [10]. Fnally, t s noted that the CPML mplementaton s ndependent of the host materal. It s well known that the PML perfectly matches general meda [2, 12, 13]. However, most mplementatons of PML requre a modfcaton of the mplementaton to accommodate general meda. It s obvous from (10) that the CPML mplementaton s ndependent of the materal medum. Subsequently, the same formulaton presented n (10) s also vald for dspersve meda, ansotropc meda, or non-lnear meda. In each of these cases, the left-hand-sde must be modfed to treat the specfc host medum as approprately presented n the lterature. Yet, the applcaton of the CPML remans unchanged. V. Results To demonstrate the proposed method, the problem of the electromagnetc scatterng by a hghly elongated object s studed. Specfcally, a thn 100 mm 25 mm plate s mmersed n a background meda wth consttutve parameters ε and σ as llustrated n Fgure 1. For the purposes of ths study, consttutve parameters for sol were assumed, gvng σ = and ε r = The plate s llumnated by a vertcally polarzed electrc current element placed just above one corner of the plate. Submtted to Mcrowave and Optcal Technology Letters, June

8 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton The current source was gven a dfferentated Gaussan tme sgnature wth a 6 GHz bandwdth. The space s dscretzed wth a FDTD lattce wth x = y = z = 1 mm. 10 cell thck PML layers termnate the grd and are placed only 3 cells from the scatterer on all sdes. Ths results n a cell lattce. Wthn the PML, σ and κ are scaled usng an m-th order polynomal scalng [2]. It s noted that α s not scaled, and s constant through the PML (due to the applcaton of the dfferentated Gaussan exctaton, t s not necessary to reduce α deeper nto the PML). To study the reflecton error due to the CPML, a reference problem was also smulated. To ths end, the same mesh s extended 75 cells out n all dmensons, leadng to a cell lattce. CPML was used to termnate ths lattce wth optmal PML parameters to mnmze any spurous reflecton. The felds wthn the lattce were then excted by an dentcal source and the tme-dependent felds were recorded wthn the regon representng the orgnal lattce. The error relatve to the reference soluton (n db) was computed as a functon of tme usng: Error = 20log db 10 χ()- t χ () t χ ref refmax (14) where χ () t represents the tme-dependent dscrete feld (e.g., E () x t ) computed wthn the workng volume of the lattce, χ () t represents the same dscrete feld computed by the ref reference problem, and χ represents the maxmum value of the reference feld over the full tme- refmax smulaton. The reflecton error was frst studed for the tradtonal PML method (whch s equvalent to the proposed method wth α = 0). To ths end, the maxmum relatve error n the electrc feld near the Submtted to Mcrowave and Optcal Technology Letters, June

9 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton opposng corner of the plate from the source was recorded over 2000 tme teratons for κ max = 11.0, and σ max = 0.7σ opt, where σ opt s predcted n [2] to be: σ opt m = 150π + 1 ε r x (15) and m s the order of polynomal scalng. For ths example, m = 4 and ε r = The relatve error computed va (14) are recorded n Fg. 2. Observng the α = 0 case n Fg. 2, t s seen that the error perssts nto the late tme. Ths error s predomnately due to low-frequency evanescent felds near the edges of the plate that are nteractng wth the PML nterface. As predcted by Berenger [3-5] these felds suffer from large reflectons n the late tme. Ths same example s repeated wth α = Agan, the reflecton error versus tme was recorded over 2000 tme steps and s llustrated n Fg. 2. Observng these results, t s seen that when α s ncreased, the reflecton error peaks n early tme due to dscretzaton error, and then t contnually reduces. Ths exemplfes the ablty of the CFS-PML to absorb evanescent waves. Next, t s nstructve to observe the maxmum reflecton error as a functon of the PML consttutve parameters κ max, σ max and α. To ths end, Fgures 3 and 4 llustrate contour plots of the reflecton error versus κ max and σ max wth α = 0.0 and α = It s demonstrated here that the maxmum error wth α = 0.0 s on the order of db. Wth α = 0.05 the error s greatly mproved and s on the order of -67 db. Ths s almost a 20 db mprovement over the tradtonal PML. Furthermore, the optmal error s realzed over a much broader range of κ max and σ max, makng these values easer to predct. In order to demonstrate the savngs possble usng the CFS-PML, a set of parameter sweeps was conducted wth α set to zero and as PML nterface was contnually moved away from the plate n all Submtted to Mcrowave and Optcal Technology Letters, June

10 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton drectons. Ths was contnued untl the error profle was smlar to the best case acheved usng the present method. Ths was found to occur when the PML boundares were pushed out at least an addtonal 14 cells from the edge of the plate. Fgure 5 demonstrates the error profle for ths case. Amazngly, just pushng out the boundary 14 cells n each dmenson ncreases the total memory requrements by a factor of four. Subsequently, the concluson s that the CFS-PML wth the CPML mplementaton can provde less than -66 db of reflecton error wth one-fourth of the memory as compared to prevous PML formulatons for ths problem. IV. Concluson A novel approach referred to as the convolutonal PML (CPML) has been ntroduced. The method s based on the stretched coordnate form of the PML and a recursve convoluton. The method s proposed n a manner that allows for a general representaton for the complex stretchng varables. Of specfc nterest was the mplementaton of the CFS-PML formulaton proposed by Kuzouglu and Mttra [8]. It was shown that the CPML mplementaton based on these complex stretchng varables requres only two auxlary varables per feld component. Ths s less than that reported by prevous mplementatons of ths method [2, 9]. Of more sgnfcance s that the formulaton s ndependent of the materal medum. Subsequently, when treatng more generalzed meda such as lossy, nhomogeneous, dspersve, ansotropc, or non-lnear meda, the CPML formulaton s unchanged. Furthermore, for such generalzed meda, the CPML formulaton can requre the same memory requrements, f not less, than requred by prevous formulatons, yet wth the added capablty of effectvely absorbng evanescent waves. Submtted to Mcrowave and Optcal Technology Letters, June

11 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton The method was valdated by observng the wave nteracton of a large flat plate embedded n a lossy meda. It was demonstrated that the CFS form of the stretched coordnate varables greatly enhances the ablty of the PML to absorb evanescent waves. A 10 cell thck PML meda was placed only 3 cells away from the surface of the plate. It was found that maxmum errors on the order of -66 db were recorded for ths problem, compared to db for the tradtonal PML formulaton. In order to acheve nearly comparable performance usng the tradtonal PML formulaton, the PML boundary had to be pushed at least 17 cells from the surface of the plate. Snce the PML formulatons for lossy meda also requre 2 addtonal varables per feld component [2, 12], ths results n a four-fold reducton n memory by usng the CPML for ths problem. It s concluded that the CPML formulaton wth the CFS stretched coordnate varables provdes for an effcent absorbng boundary that effectvely absorbs propagatng and evanescent waves for generalzed meda. References [1] J.-P. Berenger, "A perfectly matched layer for the absorpton of electromagnetc waves," Journal of Computatonal Physcs, vol. 114, pp , [2] S. D. Gedney, "The Perfectly Matched Layer Absorbng Medum," n Advances n Computatonal Electrodynamcs: The Fnte Dfference Tme Doman, A. Taflove, Ed. Boston: Artech House, 1998, pp [3] J. P. Berenger, "Improved PML for the FDTD soluton of wave-structure nteracton problems," IEEE Transactons on Antennas and Propagaton, vol. 45, pp , [4] J. P. Berenger, "An effectve PML for the absorpton of evanescent waves n wavegudes," IEEE Mcrowave and Guded Wave Letters, vol. 8, pp , [5] J. P. Berenger, "Evanescent waves n PML's: Orgn of the numercal reflecton n wavestructure nteracton problems," IEEE Transactons on Antennas and Propagaton, vol. 47, pp , [6] J. P. Berenger, "Perfectly matched layer for the FDTD soluton of wave-structure nteracton problems," IEEE Transactons on Antennas and Propagaton, vol. 44, pp , Submtted to Mcrowave and Optcal Technology Letters, June

12 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton [7] F. L. Texera and W. C. Chew, "On causalty and dynamc stablty of perfectly matched layers for FDTD smulatons," IEEE Transactons on Mcrowave Theory and Technques, vol. 47, pp , [8] M. Kuzuoglu and R. Mttra, "Frequency dependence of the consttutve parameters of causal perfectly matched ansotropc absorbers," IEEE Mcrowave and Guded Wave Letters, vol. 6, pp , [9] M. S. Tong, Y. C. Chen, M. Kuzuoglu, and R. Mttra, "A new ansotropc perfectly matched layer medum for mesh truncaton n fnte dfference tme doman analyss," Internatonal Journal of Electroncs, vol. 86, pp , [10] J. A. Roden and S. D. Gedney, "A convolutonal PML for the effectve absorpton of evanescent waves n arbtrary meda," IEEE Transactons on Antennas and Propagaton, submtted for revew, June [11] S. D. Gedney, "An ansotropc perfectly matched layer-absorbng medum for the truncaton of FDTD lattces," IEEE Transactons on Antennas and Propagaton, vol. 44, pp , [12] F. L. Texera and W. C. Chew, "A general approach to extend Berenger's absorbng boundary condton to ansotropc and dspersve meda," IEEE Transactons on Antennas and Propagaton, vol. 46, pp , [13] F. L. Texera and W. C. Chew, "General closed-form PML consttutve tensors to match arbtrary bansotropc and dspersve lnear meda," IEEE Mcrowave and Guded Wave Letters, vol. 8, pp , [14] W. C. Chew and W. H. Weedon, "A 3d Perfectly Matched Medum from Modfed Maxwells Equatons wth Stretched Coordnates," Mcrowave and Optcal Technology Letters, vol. 7, pp. 5994, [15] J. A. Roden and S. D. Gedney, "An Effcent FDTD Implementaton of the PML wth CFS n General Meda," presented at 2000 IEEE Internatonal Symposum on Antennas and Propagaton, Salt Lake Cty, UT, July, [16] R. J. Luebbers and F. Hunsberger, "FDTD for Nth-Order Dspersve Meda," IEEE Transactons on Antennas and Propagaton, vol. 40, pp , [17] J. H. Beggs, R. J. Luebbers, K. S. Yee, and K. S. Kunz, "Fnte-Dfference Tme-Doman Implementaton of Surface Impedance Boundary-Condtons," IEEE Transactons on Antennas and Propagaton, vol. 40, pp , Submtted to Mcrowave and Optcal Technology Letters, June

13 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Lst of Fgures Fgure 1 Elongated geometry consstng of 25 mm by 100 mm pec plate mmersed n a general meda. Fgure 2 Error n the electrc feld ntensty relatve to the feld's maxmum ampltude versus tme for α = 0 (wth σ = σ, κ = 11, = xyz,, ) and α = 0.05 (wth σ = σ, max max 0.7 opt max max 1.1 opt κ = 7, = xyz,, ). The PML s 10 cells thck and s placed 3 cells from the plate on all sdes. Fgure 3 Maxmum relatve error for the frst 2000 tme steps as a functon of k max and s max /s opt, wth α=0. The PML s 10 cells thck and s placed 3 cells from the plate on all sdes. Fgure 4 Maxmum relatve error for the frst 2000 tme steps as a functon of k max and s max /s opt, wth α=0.05. The PML s 10 cells thck and s placed 3 cells from the plate on all sdes. Fgure 5 Maxmum relatve error for the frst 2000 tme steps as a functon of k max and s max /s opt, wth α=0. The PML s 10 cells thck and s placed 17 cells from plate on all sdes. Submtted to Mcrowave and Optcal Technology Letters, June

14 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Fgure 1 a J s b c 2 c εσ, CPML Submtted to Mcrowave and Optcal Technology Letters, June

15 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Fgure Reflecton Error (db) e-9 4.0e-9 6.0e-9 8.0e-9 1.0e-8 1.2e-8 Tme (seconds) PML, α=0.0 CPML, α=0.05 Submtted to Mcrowave and Optcal Technology Letters, June

16 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Fgure κ max σ max /σ opt Submtted to Mcrowave and Optcal Technology Letters, June

17 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Fgure κ max σ max /σ opt Submtted to Mcrowave and Optcal Technology Letters, June

18 CPML: An Effcent FDTD Implementaton of the CFS-PML for Arbtrary Meda Not for Dstrbuton Fgure κ max σ max /σ opt Submtted to Mcrowave and Optcal Technology Letters, June