趙 博,龔書喜,王 興,董海林

(西安電子科技大學(xué)天線與微波技術(shù)重點(diǎn)實(shí)驗(yàn)室,陜西西安 710071)

快速分析非均勻介質(zhì)體寬帶散射特性的算法

趙 博,龔書喜,王 興,董海林

(西安電子科技大學(xué)天線與微波技術(shù)重點(diǎn)實(shí)驗(yàn)室,陜西西安 710071)

應(yīng)用漸近波形估計(jì)技術(shù),快速分析了基于體積分方程的任意非均勻介質(zhì)體的寬帶電磁散射特性.對于任意非均勻介質(zhì)體的散射分析,需要考慮介質(zhì)分界面兩側(cè)體電流的法向不連續(xù)性,采用體積分方程并對電通量密度進(jìn)行SWG基函數(shù)展開,很好地解決了這個(gè)問題.而對于非均勻介質(zhì)體的寬帶散射分析,逐個(gè)頻點(diǎn)計(jì)算雷達(dá)散射截面是非常耗時(shí)的,文中首先采用漸近波形估計(jì)技術(shù),推導(dǎo)和計(jì)算阻抗矩陣和激勵(lì)矢量的高階導(dǎo)數(shù),然后在給定頻點(diǎn)對電通量密度進(jìn)行展開,用padé逼近進(jìn)一步展開帶寬,可以快速預(yù)估頻帶內(nèi)任意頻點(diǎn)的電通量密度,進(jìn)而獲得非均勻介質(zhì)體的寬帶散射特性,大大減少了計(jì)算時(shí)間.最后通過數(shù)值算例驗(yàn)證了該方法的準(zhǔn)確性與高效性.

體積分方程;漸近波形估計(jì);寬帶散射

現(xiàn)代電子科學(xué)技術(shù)的發(fā)展,對寬頻帶系統(tǒng)性能的要求越來越高.非均勻介質(zhì)體的寬帶電磁散射在目標(biāo)識別、復(fù)雜環(huán)境的波傳播和資源探測等方面具有重要意義.在電磁計(jì)算中,矩量法(Method of Moment, Mo M)[1-3]作為一種對積分方程的精確分析方法,對介質(zhì)體的計(jì)算通常采用PMCHWT[4]方程或體積分方程(Volume Integral Equation,VIE)[5-6]來描述.然而,PMCHWT方程通常用來計(jì)算均勻介質(zhì)體的散射,對于非均勻介質(zhì)體需要處理在不同介質(zhì)分界面處的復(fù)雜邊界條件,并且隨著非均勻程度的增加,關(guān)于邊界條件的處理也更加復(fù)雜[7].而采用體積分方程則可以很好地解決關(guān)于復(fù)雜邊界條件處理的問題.該算法對任意非均勻介質(zhì)體進(jìn)行四面體元剖分(也可以用曲面四面體元,六面體元或者曲面六面體元),可以充分采集其內(nèi)部的非均勻信息,同時(shí)很好地?cái)M合其物理結(jié)構(gòu),此外,用SWG基函數(shù)能夠很好地模擬四面體元內(nèi)的電通量密度的密度,使得計(jì)算結(jié)果可以達(dá)到很高的精度.

然而,對于非均勻介質(zhì)體的寬帶散射計(jì)算,需要在給定頻段的不同頻點(diǎn)進(jìn)行重復(fù)計(jì)算,這不僅需要重復(fù)填充阻抗矩陣,還需要重復(fù)求解線性方程組,使得整個(gè)分析過程非常的耗時(shí).從20世紀(jì)80年代末以來,基于梅利逼近[8-10]、漸近波形估計(jì)(Asymptotic Waveform Evaluation,AWE)[11-14]和柯西逼近[15]等的快速掃頻技術(shù)得到了深入研究和廣泛應(yīng)用.上述方法中,AWE因易于與積分方程的數(shù)值方法相結(jié)合,在電磁領(lǐng)域越來越受到重視.筆者將其與VIE結(jié)合來分析任意非均勻介質(zhì)體的寬帶散射特性,在給定的頻點(diǎn)對電通量密度進(jìn)行泰勒展開,然后通過padé逼近得到任意頻點(diǎn)的電通量密度,從而實(shí)現(xiàn)快速寬帶散射特性分析.在保證計(jì)算精度的前提下,相比于VIE逐點(diǎn)計(jì)算,該方法大大提高了計(jì)算效率.

1 理論分析

1.1 體積分方程矩量法(VIE-MoM)

考慮自由空間中一個(gè)三維任意非均勻介質(zhì)體在平面波Ei照射下,其所在區(qū)域感應(yīng)出的電通量密度產(chǎn)生散射場Es.目標(biāo)的體積為V,表面積為S,介電常數(shù)ε(r)=ε0εr(r),磁導(dǎo)率為μ0.文中的時(shí)間因子均采用exp(-jωt).

VIE可以表示為

其中,Etot和Es代表總電場和散射場,

其中,

其中,r和r′分別為場點(diǎn)和源點(diǎn),Jv(r′)和ρv(r′)分別為體電流密度和體電荷密度,根據(jù)電流連續(xù)性公式可以將兩者聯(lián)系起來,

因?yàn)樵诜蔷鶆蚪橘|(zhì)體的介質(zhì)分界面兩側(cè),電通量密度之密度Jv(r)法向分量并不連續(xù),而電通量密度D(r)法向分量在介質(zhì)分界面兩側(cè)相等,為方便起見,一般采用電通量密度作為體積分方程的未知量,

將式(2)～(7)代入式(1)中,可得

其中,?′· (κ(r′)D(r′))根據(jù)矢量恒等式得

其中,

式(9)中等號右邊第2項(xiàng)涉及非均勻介質(zhì)分界面兩側(cè)的介電常數(shù),正是因此使得VIE對于非均勻介質(zhì)體的分析不需要處理復(fù)雜的邊界條件.

在三維理想導(dǎo)體的數(shù)值計(jì)算中,一般采用RWG基函數(shù)[16]來展開電場積分方程的未知量.與此類似,文中采用SWG基函數(shù)fn(r)[5]來展開體積分方程的未知量D(r),即為

SWG基函數(shù)的表達(dá)式為

N為未知量D(r)的維數(shù).用伽略金法檢驗(yàn)式(8),可得到矩陣方程,即

其中,Z為N×N維的阻抗矩陣,V為N維的激勵(lì)矢量,D為待求解的N維電通量密度矢量.阻抗矩陣Z和激勵(lì)矢量V的元素表達(dá)式為

其中,fm和fn分別代表檢驗(yàn)函數(shù)和基函數(shù),Tm和Tn是第m個(gè)和第n個(gè)基函數(shù)對應(yīng)的四面體元對,Sn為第n個(gè)基函數(shù)中兩個(gè)四面體的公共面,R是場點(diǎn)和源點(diǎn)之間的距離.

1.2 漸近波形估計(jì)

利用矩量法求解體積分方程,通過四面體元剖分非均勻介質(zhì)體所在區(qū)域,未知電通量密度矢量D(k)用SWG基函數(shù)fn(r)展開,得到矩陣方程如式(13)所示,不難發(fā)現(xiàn)Z、V和D均為波數(shù)k的函數(shù),k表示頻點(diǎn)f對應(yīng)的波數(shù),因此式(13)可以寫為

為了得到一定頻域內(nèi)的寬帶散射特性,必須重復(fù)求解式(16),這將耗費(fèi)大量的時(shí)間.而應(yīng)用AWE技術(shù)實(shí)現(xiàn)快速掃頻則很好地解決了這個(gè)問題.AWE將未知電通量密度D(k)在給定中心頻率f0展開成泰勒級數(shù)

其中,k0表示頻點(diǎn)f0對應(yīng)的波數(shù),mq表示未知量的系數(shù),Q表示泰勒級數(shù)的截?cái)嚯A數(shù).在求解Q+1個(gè)未知量系數(shù)mq(q=0,1,2,…,Q)時(shí),根據(jù)以下遞推式:

其中,V(q)(k0)是激勵(lì)矢量V(k)關(guān)于波數(shù)k在k0點(diǎn)的q階導(dǎo)數(shù),Z(i)(k0)是阻抗矩陣Z(k)關(guān)于波數(shù)k在k0點(diǎn)的i階導(dǎo)數(shù),它們的元素表達(dá)式為

其中,

其中,Ei,(q)(r)是入射電場關(guān)于k在k0點(diǎn)的q階導(dǎo)數(shù),和分別是GA(r,r′)和Gφ(r,r′)關(guān)于k在k0點(diǎn)的i階導(dǎo)數(shù).

由于未知電通量密度的泰勒展開會受到帶寬的限制,為了展寬帶寬,應(yīng)用padé逼近將泰勒級數(shù)轉(zhuǎn)化為有理函數(shù),即

其中,L+M=Q.式(25)中對比等號兩邊(k-k0)q的系數(shù),bn,j可以由式(26)求得

進(jìn)一步根據(jù)式(27),可以計(jì)算得到

確定了an,i和bn,j之后,將其帶入式(25),得到一定頻域內(nèi)任意頻點(diǎn)的電通量密度,進(jìn)而得到非均勻介質(zhì)體的寬帶雷達(dá)散射截面(Radar Cross-Section,RCS).

2 數(shù)值結(jié)果與分析

為了驗(yàn)證VIE-AWE算法在任意非均勻介質(zhì)體寬帶散射計(jì)算中的準(zhǔn)確性與高效性,這里給出3個(gè)算例的數(shù)值結(jié)果,所有算例均在主頻2.4 GHz、內(nèi)存4 GB的個(gè)人電腦上完成,數(shù)據(jù)類型都采用雙精度類型存儲.由于算例涉及逐點(diǎn)計(jì)算非常耗時(shí),為方便起見,求解線性方程組采用MKL庫函數(shù)中的并行LU分解法.

算例1計(jì)算圖1所示的3層非均勻介質(zhì)球在1～19 GHz的寬帶單站RCS.目標(biāo)的物理尺寸R1=1.5 mm,R2=2.5 m m,R3=3.5 mm,對應(yīng)的相對介電常數(shù)分別為ε1=7.0,ε2=4.0,ε3=9.0.

圖1 非均勻介質(zhì)球的剖面圖

圖2 非均勻介質(zhì)球的寬帶單站RCS

在頻點(diǎn)10 GHz處對目標(biāo)進(jìn)行剖分,得到820個(gè)四面體元,形成1 742個(gè)未知量.圖2對比了用仿真軟件FEKO、逐點(diǎn)法和AWE掃頻算法得到的寬帶RCS.FEKO的計(jì)算結(jié)果與逐點(diǎn)法相比較,可以看出VIE計(jì)算的準(zhǔn)確性.在給定頻點(diǎn)10 GHz處,用AWE對電通量密度進(jìn)行Q階(Q=8,6,4,2)泰勒展開,以逐點(diǎn)法為標(biāo)準(zhǔn),AWE結(jié)果與之吻合的帶寬分別為:當(dāng)Q=8時(shí),為1.0～17.01 GHz;當(dāng)Q=6時(shí),為1.97～15.75 GHz;當(dāng)Q=4時(shí),為2.57～13.41 GHz;當(dāng)Q=2時(shí),為1.00～12.57 GHz.可以看出,在保證結(jié)果精確的前提下,隨著AWE階數(shù)的增加,帶寬趨于更寬.逐點(diǎn)法計(jì)算頻帶寬度為1～17.01 GHz的RCS,頻率間隔的步長取0.1 GHz需要7 728.65 s,AWE取Q=8計(jì)算同樣帶寬和步長的RCS需要440.72 s,計(jì)算時(shí)間減少了94.30%.因此,在保證精度的前提下,應(yīng)用AWE結(jié)合VIE算法可以大幅度減少計(jì)算時(shí)間.表1給出了AWE取不同階數(shù)Q時(shí)的計(jì)算特性對比.

表1 AWE取不同階數(shù)Q時(shí)的計(jì)算特性

圖3 非均勻介質(zhì)長方體的物理結(jié)構(gòu)

算例2計(jì)算了圖3所示的非均勻介質(zhì)長方體在平面波照射下2～42 GHz的寬頻帶單站RCS.入射波沿θ=45°、φ=0°方向入射.目標(biāo)由7個(gè)同樣的小長方體無間隙地排列在一起,每個(gè)小長方體的大小L=2 mm、H=0.5 mm,相對介電常數(shù)ε1=2.0和ε2=4.0.在頻點(diǎn)20 GHz對目標(biāo)進(jìn)行剖分,得到1 704個(gè)四面體,形成未知量個(gè)數(shù)為3996.圖4同樣對比了FEKO仿真軟件、逐點(diǎn)法和AWE掃頻算法計(jì)算的寬帶RCS.對比FEKO和逐點(diǎn)法計(jì)算結(jié)果,證明了VIE計(jì)算的精確性.AWE在20 GHz對電通量密度進(jìn)行泰勒展開,展開階數(shù)分別取Q=8,6,4,2.以逐點(diǎn)法為參考標(biāo)準(zhǔn),AWE的計(jì)算結(jié)果與之吻合的帶寬分別為:當(dāng)Q=8和6時(shí),為2.0～42.0 GHz;當(dāng)Q=4時(shí),為2.0～32.49 GHz;當(dāng)Q=2時(shí),為13.32～26.07 GHz.可以看出,隨著AWE階數(shù)的增加,帶寬趨于更寬.隨著階數(shù)Q的增長,計(jì)算結(jié)果逐漸收斂.采用逐點(diǎn)法計(jì)算2.0～42.0 GHz帶寬的RCS,步長取0.2 GHz需要44 823.54 s.應(yīng)用AWE計(jì)算同樣帶寬和步長間隔的RCS,當(dāng)Q=8和Q=6時(shí),需要的時(shí)間分別為2 224.51 s和1 739.7 s,計(jì)算時(shí)間分別減少了95.04%和96.12%.由此證明了AWE結(jié)合VIE算法的準(zhǔn)確性和高效性.表2給出了AWE取不同階數(shù)Q時(shí)的計(jì)算特性對比.

圖4 非均勻介質(zhì)長方體的寬帶單站RCS

表2 AWE取不同階數(shù)Q時(shí)的計(jì)算特性

3 結(jié)束語

結(jié)合VIE和AWE分析了任意非均勻介質(zhì)體的寬帶散射特性.對于介質(zhì)體的非均勻結(jié)構(gòu),VIE采用四面體元進(jìn)行剖分可以獲取介質(zhì)體內(nèi)部的非均勻信息,并通過Mo M進(jìn)行了精確的計(jì)算.對于寬帶散射特性的計(jì)算,必須在帶寬內(nèi)各頻點(diǎn)重復(fù)計(jì)算,為了提高計(jì)算效率,筆者結(jié)合AWE大大減少了所需時(shí)間.通過算例分析,驗(yàn)證了VIE-AWE在非均勻介質(zhì)體情況下計(jì)算的準(zhǔn)確性與高效性.

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(編輯:王 瑞)

Fast analysis of wide-band scattering properties of inhomogeneous dielectric bodies

ZHAO Bo,GONG Shuxi,WANG Xing,DONG Hailin
(Science and Technology on Antenna and Microwave Lab.,Xidian Univ.,Xi’an 710071,China)

The asymptotic waveform evaluation(AWE)based on the volume integral equation(VIE)is applied to accelerate wide-band electromagnetic scattering properties of arbitrarily shaped inhomogeneous dielectric bodies.For the analysis over a single frequency,the discontinuity of the normal component of the volume current on both sides of the medium interface should be considered.Nevertheless,the problem can be solved well by adoption of VIE and expansion of the electric flux density with SWG basis functions. However,the wide-band analysis requires repeated radar cross section(RCS)calculations over different frequencies,which takes excessively much time.To acquire the prediction of wide-band RCS rapidly,in this paper,AWE greatly reduces the computation time by deducing and computing the high derivatives of the impedance matrix and excitation vector,expanding the electric flux density with the Taylor series and further broadening the bandwidth using padéapproximation.Finally,numerical examples are displayed to demonstrate the accuracy and efficiency of this method.

volume integral equation;asymptotic waveform evaluation;wide-band scattering

O441.6;TN011

A

1001-2400(2015)05-0080-06

2014-09-30< class="emphasis_bold">網(wǎng)絡(luò)出版時(shí)間:

時(shí)間:2014-12-23

國家高技術(shù)研究發(fā)展計(jì)劃(863計(jì)劃)資助項(xiàng)目(2012AA01A308);國家自然科學(xué)基金資助項(xiàng)目(61301069,61072019);教育部新世紀(jì)優(yōu)秀人才支持計(jì)劃資助項(xiàng)目(NCET-13-0949);陜西省青年科技新星資助項(xiàng)目(2013KJXX-67);中央高?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金資助項(xiàng)目(JB140224)

趙 博(1987-),男,西安電子科技大學(xué)博士研究生,E-mail:m15991342657@163.com.

http://www.cnki.net/kcms/detail/61.1076.TN.20141223.0946.014.html

10.3969/j.issn.1001-2400.2015.05.014