劉麗娜,朱 峰,徐常偉

(西南交通大學(xué)電氣工程學(xué)院,成都610031)

三維LOD-FDTD方法在PMC邊界處的精確格式*

劉麗娜,朱 峰**,徐常偉

(西南交通大學(xué)電氣工程學(xué)院,成都610031)

證明了局部一維時(shí)域有限差分(LOD-FDTD)方法實(shí)現(xiàn)理想磁導(dǎo)體(PMC)邊界時(shí)的待求場(chǎng)分量系數(shù)與傳統(tǒng)的LOD-FDTD方法系數(shù)不同。通過(guò)在獲得該系數(shù)前應(yīng)用理想導(dǎo)體邊界條件,得到對(duì)應(yīng)的修正系數(shù)。計(jì)算了單個(gè)PMC立方體和對(duì)稱的兩個(gè)PMC立方體的雙站RCS。計(jì)算結(jié)果表明, PMC邊界作為理想導(dǎo)體表面時(shí),傳統(tǒng)LOD-FDTD方法計(jì)算誤差較大,采用修正系數(shù)的計(jì)算結(jié)果與傳統(tǒng)FDTD方法計(jì)算結(jié)果更為吻合;PMC邊界作為截?cái)嘤?jì)算空間的對(duì)稱面,采用修正系數(shù)的計(jì)算結(jié)果與傳統(tǒng)LOD-FDTD方法計(jì)算結(jié)果相同。采用修正系數(shù)處理PMC邊界無(wú)需區(qū)分PMC邊界是理想磁導(dǎo)體表面還是截?cái)嘤?jì)算空間的對(duì)稱面,具有統(tǒng)一的表達(dá)式,計(jì)算理想磁導(dǎo)體表面較傳統(tǒng)LOD-FDTD方法誤差更小。

理想磁導(dǎo)體邊界;時(shí)域有限差分方法;局部一維時(shí)域有限差分方法

1 引 言

時(shí)域有限差分(FDTD)方法及其改進(jìn)算法在很多領(lǐng)域得到了廣泛應(yīng)用[1-2],但是Courant-Friedrich -Levy(CFL)穩(wěn)定條件限制了時(shí)間步長(zhǎng),導(dǎo)致傳統(tǒng)FDTD方法在數(shù)值模擬復(fù)雜模型時(shí),所需計(jì)算時(shí)間過(guò)長(zhǎng)。基于劃分子時(shí)間步隱式計(jì)算思想的FDTD改進(jìn)算法,包括局部一維時(shí)域有限差分(LOD-FDTD)方法[3]和交替時(shí)間隱式時(shí)域有限差分(ADI-FDTD)方法[4]等方法,它們能夠?qū)崿F(xiàn)無(wú)條件穩(wěn)定,時(shí)間步長(zhǎng)的選取不受CFL條件的限制。LOD-FDTD方法將n→n+1的過(guò)程分成n→n+1/2和n+1/2→n+1的兩個(gè)子時(shí)間過(guò)程,被證明與ADI-FDTD方法有相同的計(jì)算精度,且計(jì)算時(shí)間比ADI-FDTD方法減少55%[5]。LOD-FDTD方法的無(wú)條件穩(wěn)定[6],計(jì)算效率[6]、計(jì)算誤差[7]、相關(guān)吸收邊界以及與減縮時(shí)域有限差分(R-FDTD)方法[8]等相結(jié)合的探討都是研究的熱點(diǎn)。人工磁導(dǎo)體在天線領(lǐng)域的應(yīng)用備受關(guān)注[9],PMC邊界還可以作為對(duì)稱面截?cái)嘤?jì)算空間,有效減少內(nèi)存使用量和計(jì)算時(shí)間[10]。PMC邊界上的切向磁場(chǎng)均為零,當(dāng)LOD-FDTD計(jì)算區(qū)域中出現(xiàn)PMC邊界時(shí),由于部分場(chǎng)分量被置零,導(dǎo)致待求場(chǎng)分量系數(shù)發(fā)生變化,如果不對(duì)系數(shù)進(jìn)行修正,場(chǎng)分量的計(jì)算結(jié)果會(huì)出現(xiàn)誤差。

本文考慮三維LOD-FDTD情況,在得到待求場(chǎng)分量系數(shù)之前應(yīng)用PMC邊界條件,推導(dǎo)出LODFDTD方法在理想導(dǎo)體邊界處待求場(chǎng)分量的修正系數(shù)。對(duì)于PMC邊界作為理想導(dǎo)體表面和截?cái)嘤?jì)算空間的對(duì)稱面的不同情況加以區(qū)分,討論了修正系數(shù)與傳統(tǒng)LOD-FDTD系數(shù)的區(qū)別。采用修正系數(shù)LOD-FDTD方法計(jì)算了單個(gè)PMC立方體和具有對(duì)稱結(jié)構(gòu)的兩個(gè)PMC立方體的雙站RCS,計(jì)算結(jié)果與傳統(tǒng)LOD-FDTD方法和FDTD方法的計(jì)算結(jié)果進(jìn)行比較,結(jié)果表明對(duì)于理想磁導(dǎo)體表面,采用修正系數(shù)LOD-FDTD方法的計(jì)算結(jié)果與傳統(tǒng)FDTD方法計(jì)算結(jié)果更為吻合;PMC邊界條件作為對(duì)稱面截?cái)嘤?jì)算空間的情況,采用修正系數(shù)LOD-FDTD方法的計(jì)算結(jié)果與傳統(tǒng)LOD-FDTD方法計(jì)算結(jié)果相同。修正系數(shù)LOD-FDTD方法處理PMC邊界的優(yōu)點(diǎn)在于具有統(tǒng)一的表達(dá)式,降低了編程復(fù)雜度,無(wú)需區(qū)分PMC邊界是理想磁導(dǎo)體表面還是截?cái)嘤?jì)算空間的對(duì)稱面,計(jì)算理想磁導(dǎo)體表面較傳統(tǒng)LOD-FDTD方法有較小的計(jì)算誤差。

2 理論分析

2.1 LOD-FDTD方法

LOD-FDTD方法將FDTD方法中n→n+1的過(guò)程分成n→n+1/2和n+1/2→n+1的兩個(gè)子時(shí)間過(guò)程:

其中,U=[Ex,Ey,Ez,Hx,Hy,Hz]T,I為單位矩陣,矩陣A和B如文獻(xiàn)[11]中式(33)和式(34)所示。

將各場(chǎng)分量代入矩陣關(guān)系式,可以得到場(chǎng)分量隱式表達(dá)式。在n→n+1/2子時(shí)間步,Ex和Hz相關(guān)聯(lián),沿y方向推進(jìn);Ey和Hx相關(guān)聯(lián),沿z方向推進(jìn); Ez和Hy相關(guān)聯(lián),沿x方向推進(jìn)。由式(1)可以得到n→n+1/2子時(shí)間步,僅與Ex和Hz相關(guān)的方程:

離散以上兩式,得到Ex和Hz的隱式差分方程:

將式(5)中兩個(gè)n+1/2時(shí)刻Hz分量由式(6)表達(dá),整理得到傳統(tǒng)LOD-FDTD計(jì)算Ex分量的表達(dá)式

其中:

2.2 PMC邊界的LOD-FDTD實(shí)現(xiàn)

LOD-FDTD方法在n→n+1/2和n+1/2→n+1的兩個(gè)子時(shí)間過(guò)程均需要隱式計(jì)算電場(chǎng),再由計(jì)算所得電場(chǎng)值,顯式計(jì)算得到磁場(chǎng)值。LOD-FDTD方法隱式更新電場(chǎng)的特點(diǎn),決定了PEC邊界實(shí)現(xiàn)較容易,但PMC邊界實(shí)現(xiàn)較復(fù)雜。下面展開(kāi)論述三維LOD-FDTD方法實(shí)現(xiàn)PMC邊界的具體問(wèn)題。

PMC邊界滿足切向磁場(chǎng)和法向電場(chǎng)為零的條件,在y=(j0-1/2)Δy處,垂直于y軸設(shè)置PMC邊界面,分別采用傳統(tǒng)LOD-FDTD方法和修正系數(shù)LODFDTD方法兩種方法處理PMC邊界條件。

(1)傳統(tǒng)LOD-FDTD方法

得到式(7)之后將PMC邊界條件代入:

①如果PMC邊界作為理想磁導(dǎo)體表面,則有

②如果PMC邊界作為截?cái)嘤?jì)算空間的對(duì)稱面,應(yīng)用鏡像原理,PMC截?cái)嗝鎯蓚?cè)有可以得到

(2)修正系數(shù)LOD-FDTD方法

得到式(7)之前,即在式(6)代入式(5)之前考慮PMC邊界條件僅將用式(6)展開(kāi)代入式(5)可以得到

修正系數(shù)LOD-FDTD方法不需要區(qū)分PMC邊界是理想磁導(dǎo)體表面還是截?cái)嘤?jì)算空間的對(duì)稱面,得到了實(shí)現(xiàn)PMC邊界的統(tǒng)一格式。對(duì)比修正系數(shù)LOD-FDTD方法與傳統(tǒng)LOD-FDTD方法比較的結(jié)果,在處理理想磁導(dǎo)體表面時(shí),等式左端的系數(shù)與不同;等式右端的系數(shù),在d?中為,在d′中為。對(duì)于PMC邊界作為對(duì)稱面截?cái)嘤?jì)算空間的情況,截?cái)嗝娴谋砻骐m然同樣滿足PMC邊界條件,但是由于截?cái)嗝嬉酝獾膱?chǎng)分量不再為零,修正系數(shù)LOD-FDTD方法與傳統(tǒng)LOD-FDTD方法表達(dá)式相同。n+1/2→n+1的子時(shí)間步有類似的情況。

3 數(shù)值計(jì)算結(jié)果

采用修正系數(shù)LOD-FDTD方法計(jì)算單個(gè)PMC立方體和對(duì)稱的兩個(gè)PMC立方體的雙站RCS,與傳統(tǒng)LOD-FDTD方法和FDTD方法的計(jì)算結(jié)果進(jìn)行比較。平面波頻率3 GHz沿z方向入射,x方向極化。計(jì)算空間由PML吸收邊界截?cái)?。FDTD算法的時(shí)間步長(zhǎng)Δt=8.333×10-12s,空間步長(zhǎng)Δx=Δy=Δz =0.005 m,修正系數(shù)LOD-FDTD方法和傳統(tǒng)LODFDTD方法的時(shí)間步長(zhǎng)ΔtLOD=3Δt。

算例1:理想磁導(dǎo)體表面的情況

計(jì)算一個(gè)PMC立方體的雙站RCS,PMC立方體每個(gè)面都采用PMC邊界條件,立方體邊長(zhǎng)為L(zhǎng)= 0.1 m。計(jì)算模型如圖1所示,3種方法對(duì)比計(jì)算PMC立方體的雙站RCS如圖2所示。分析對(duì)比計(jì)算結(jié)果,對(duì)于設(shè)置在理想磁導(dǎo)體表面的PMC邊界,3種方法的計(jì)算結(jié)果差別不大,修正系數(shù)LOD-FDTD方法的計(jì)算結(jié)果更接近FDTD方法的計(jì)算結(jié)果,說(shuō)明了理想磁導(dǎo)體表面采用修正系數(shù)計(jì)算的正確性和必要性。

圖1 算例1的計(jì)算模型Fig.1 The calculation model of Example 1

圖2 不同方法計(jì)算一個(gè)PMC立方體的雙站RCSFig.2 Different methods comparison of bistatic RCS of a PMC box

算例2:PMC邊界截?cái)鄬?duì)稱空間的情況

計(jì)算對(duì)稱的兩個(gè)PMC立方體的雙站RCS,每個(gè)立方體邊長(zhǎng)為L(zhǎng)=0.1 m,兩個(gè)PMC立方體中心間距0.2 m。PMC立方體表面均采用修正系數(shù)計(jì)算。在與y軸垂直的整個(gè)計(jì)算區(qū)域的對(duì)稱面處設(shè)置PMC邊界截?cái)?只計(jì)算FDTD方法的一半空間。計(jì)算模型如圖3所示,分別用3種方法處理PMC截?cái)嗝?對(duì)比計(jì)算得到的兩個(gè)PMC立方體的雙站RCS如圖4所示。

圖3 算例2的計(jì)算模型Fig.3 The calculation model of Example 2

圖4 不同方法計(jì)算對(duì)稱的兩個(gè)PMC立方體的雙站RCSFig.4 Different methods comparison of bistatic RCS of two symmetric PMC boxes

分析對(duì)比計(jì)算結(jié)果,對(duì)于PMC邊界作為對(duì)稱面截?cái)嘤?jì)算空間的情況,修正系數(shù)LOD-FDTD方法和傳統(tǒng)LOD-FDTD方法的計(jì)算結(jié)果吻合。截?cái)嗝娴谋砻嫱瑯訚M足PMC邊界條件,但是由于截?cái)嗝嬉酝獾膱?chǎng)分量不再為零,所以,修正系數(shù)LOD-FDTD方法有與傳統(tǒng)LOD-FDTD方法相同的表達(dá)式。數(shù)值計(jì)算結(jié)果也表明,PMC邊界作為對(duì)稱面截?cái)嘤?jì)算空間,修正系數(shù)LOD-FDTD方法和傳統(tǒng)LOD-FDTD方法等同。

4 結(jié) 論

本文考慮三維LOD-FDTD方法,在得到待求場(chǎng)分量系數(shù)之前應(yīng)用理想導(dǎo)體邊界條件,推導(dǎo)出在理想磁導(dǎo)體邊界處與傳統(tǒng)LOD-FDTD方法不同的待求場(chǎng)分量的修正系數(shù)。計(jì)算了單個(gè)PMC立方體和對(duì)稱的兩個(gè)PMC立方體的雙站RCS,采用修正系數(shù)LOD-FDTD方法與傳統(tǒng)LOD-FDTD方法和FDTD方法的計(jì)算結(jié)果進(jìn)行了對(duì)比。數(shù)值計(jì)算結(jié)果表明, PMC邊界作為理想磁導(dǎo)體表面時(shí),修正系數(shù)LODFDTD方法與傳統(tǒng)LOD-FDTD方法相比誤差更小; PMC邊界作為對(duì)稱面截?cái)嘤?jì)算空間時(shí),應(yīng)用鏡像原理,修正系數(shù)LOD-FDTD方法與傳統(tǒng)LOD-FDTD方法等同。修正系數(shù)方法處理PMC邊界的優(yōu)點(diǎn)在于具有統(tǒng)一的表達(dá)式,降低了編程復(fù)雜度,無(wú)需區(qū)分PMC邊界是理想磁導(dǎo)體表面還是截?cái)嘤?jì)算空間的對(duì)稱面,計(jì)算理想磁導(dǎo)體表面較傳統(tǒng)LOD-FDTD方法有較小的計(jì)算誤差。

[1] 馮延彬,李國(guó)林,路翠華,等.基于混合信號(hào)仿真技術(shù)的高功率微波與無(wú)線電引信耦合效應(yīng)分析[J].電訊技術(shù),2013,53(6):807-811.

FENG Yan-bin,LI Guo-lin,LU Cui-hua,et al.Analysis of high power microwaves effect on radio fuse by mixed signal simulation technology[J].Telecommunication Engineering,2013,53(6):807-811.(in Chinese)

[2] 馮延彬,李國(guó)林,李春榮,等.基于PSO/FDTD的波導(dǎo)縫隙天線優(yōu)化設(shè)計(jì)[J].電訊技術(shù),2013,53(5): 645-649.

FENG Yan-bin,LI Guo-lin,LI Chun-rong,et al.Optimal Design of Waveguide Slot Antenna Based on PSO/ FDTD[J].Telecommunication Engineering,2013,53 (5):645-649.(in Chinese)

[3] Shibayama J,Muraki M,Yamauchi J,et al.Efficient implicit FDTD algorithm based on locally one-dimensional scheme[J].Electronics Letters,2005,41(19):1046-1047.(in Chinese)

[4] Namiki T.3-D ADI-FDTD method-unconditionally stable time-domainalgorithmforsolvingfullvector Maxwell′s equations[J].IEEE Transactions on Microwave Theory and Techniques,2000,48(10):1743-1748.

[5] Liu Q F,Chen Z Z,Yin W Y.An efficient unconditionally stable three-dimensional LOD-FDTD method[C]//Proceedings of 2008 IEEE MTT-S International Microwave Symposium Digest.Atlanta,GA:IEEE,2008:45-48.

[6] Ahmed I,Chua E K,Li E P,et al.development of the three-dimensional unconditionally stable LOD-FDTD method[J].IEEE Transactions on Antennas and Propagation,2008,56(11):3596-3600.

[7] Ahmed I,Chua E K,Li E P.Numerical dispersion analysis of the unconditionally stable three-dimensional LOD -FDTD method[J].IEEE Transactions on Antennas and Propagation,2010,58(12):3983-3989.

[8] 張品,陳亦望,傅強(qiáng).一種提高內(nèi)存使用效率的時(shí)域有限差分算法[J].電波科學(xué)學(xué)報(bào),2011,26(4):814 -819.

ZHANG Pin,CHEN Yi-wang,FU Qiang.A memory-efficient FDTD algorithm[J].Chinese Journal of Radio Science,2011,26(4):814-819.(in Chinese)

[9] 魯磊,屈紹波,馬華,等.寬帶雷達(dá)散射截面減縮人工磁導(dǎo)體復(fù)合結(jié)構(gòu)[J].物理學(xué)報(bào),2013,62(3):170-175.

LU Lei,QU Shao-bo,MA Hua,et al.A broadband artificial magnetic conductor composite structure for radar cross section reduction[J].Acta Physica Sinica,2013, 62(3):170-175.(in Chinese)

[10] 張巖,呂善偉,苗俊剛,等.FDTD-PWS法用于分析毫米波透鏡天線焦面場(chǎng)[J].北京航空航天大學(xué)學(xué)報(bào),2007,33(6):682-685. ZHANG Yan,LV Shan-wei,MIAO Jun-gang,et al.

Hybrid FDTD-PWS method for focal field analysis of lens antenna at millimeter-wave band[J].Journal of Beijing University of Aeronautics and Astronautics, 2007,33(6):682-685.(in Chinese)

[11] Tan E L.Fundamental schemes for efficient unconditional-ly stable implicit finite-difference time-domain methods[J].IEEE Transactions on Antennas and Propagation,2008,56(1):170-177.

LIU Li-na was born in Tangshan,Hebei Province,in 1981.She received the B.S.degree in 2000.She is currently working toward the Ph.D.degree.Her research concerns electromagnetic scattering and electromagnetic computation.

Email:linaapple329@163.com

朱 峰(1963—),男,安徽人,教授、博士生導(dǎo)師,主要研究方向?yàn)殡姶艌?chǎng)與電磁波、電磁場(chǎng)數(shù)值計(jì)算;

ZHU Feng was born in Anhui Province,in 1963.He is now a professor with the Ph.D.degree and also the Ph.D.supervisor.His research concerns electromagnetic computation,electromagnetic field and wave.

Email:zhufeng@swjtu.cn

徐常偉(1984—),男,河南人,2004年獲學(xué)士學(xué)位,現(xiàn)為西南交通大學(xué)博士研究生,主要研究方向?yàn)殡姶派⑸?、電磁?chǎng)數(shù)值計(jì)算。

XU Chang-wei was born in Henan Province,in 1984.He received the B.S.degree in 2004.He is currently working toward the Ph.D.degree.His research concerns electromagnetic scattering and electromagnetic computation.

Accurate Algorithm on PMC Boundary for 3D LOD-FDTD Method

LIU Li-na,ZHU Feng,XU Chang-wei

(College of Electrical Engineering,Southwest Jiaotong University,Chengdu 610031,China)

The field coefficient on perfect magnetic conductor boundary is proved to be different from that in the conventional locally one-dimensional finite-difference time-domain(LOD-FDTD)calculation.The correction coefficient is derived by setting PMC boundary condition before the conventional field coefficient is obtained from the implicit equations.Bistatic RCS calculations of a PMC cube and two symmetrical PMC cubes are provided by using correction coefficient method,conventional LOD-FDTD method and FDTD method,respectively.For the surface of perfect conductor,numerical results of correction coefficient method agree better with those of conventional FDTD.For the symmetry plane truncated computing space,numerical results of correction coefficient method agree well with those of conventional LOD-FDTD.The theory proposed in this paper is validated.Correction coefficient method has unified expressions and it is found that less calculation errors occur than conventional LOD-FDTD method is used.

PMC boundary;finite-difference time-domain(FDTD)method;locally one-dimensional finite-difference time-domain(LOD-FDTD)method

The National Natural Science Foundation of China(No.60971041)

date:2013-09-10;Revised date:2013-11-19

國(guó)家自然科學(xué)基金資助項(xiàng)目(60971041)

**通訊作者:zhufeng@swjtu.cn Corresponding author:zhufeng@swjtu.cn

TN04;TM153

:A

:1001-893X(2013)12-1638-05

劉麗娜(1981—),女,河北唐山人,2000年獲學(xué)士學(xué)位,現(xiàn)為西南交通大學(xué)博士生研究生,主要研究方向?yàn)殡姶派⑸?、電磁?chǎng)數(shù)值計(jì)算;

10.3969/j.issn.1001-893x.2013.12.019

2013-09-10;

2013-11-19