茹 靜,馬富明

(1.吉林化工學(xué)院 理學(xué)院,吉林 吉林 132022;2.吉林大學(xué) 數(shù)學(xué)研究所,長(zhǎng)春 130012)

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二維多洞穴電磁散射問題的快速算法

茹 靜1,2,馬富明2

(1.吉林化工學(xué)院 理學(xué)院,吉林 吉林 132022;2.吉林大學(xué) 數(shù)學(xué)研究所,長(zhǎng)春 130012)

針對(duì)無界域上具有矩形結(jié)構(gòu)多洞穴電磁散射問題的數(shù)值計(jì)算提出一種快速算法,該算法可快速計(jì)算尺寸較大及高波數(shù)的洞穴散射問題.數(shù)值算例驗(yàn)證了方法的有效性.

散射問題;Helmholtz方程;Fourier級(jí)數(shù)

洞穴散射問題通常指由于平面局部凹陷而產(chǎn)生的散射現(xiàn)象,在軍事領(lǐng)域,尤其是飛機(jī)設(shè)計(jì)中應(yīng)用廣泛.由于飛機(jī)的進(jìn)氣道、尾噴管雷達(dá)艙等均可視為洞穴結(jié)構(gòu),而這種結(jié)構(gòu)會(huì)對(duì)入射場(chǎng)產(chǎn)生較強(qiáng)的散射,因此也是雷達(dá)探測(cè)的主要對(duì)象,目前洞穴散射問題的研究和計(jì)算已得到廣泛關(guān)注.從數(shù)學(xué)角度研究洞穴散射問題,通常假設(shè)洞穴開口與無限地平面一致,從而簡(jiǎn)化洞穴外部區(qū)域的模型.Ammari等[1]對(duì)洞穴內(nèi)部可以是非均勻介質(zhì)的洞穴散射問題進(jìn)行了研究,給出了其解的存在性和唯一性結(jié)果.目前已有很多方法用于研究洞穴散射問題,如有限元方法[2-5]、邊界元方法[6]、雜交有限元方法[7-8]以及模態(tài)匹配法[9-10]等.本文考慮一類無界域上洞穴形狀為矩形的多洞穴散射問題,這類問題有實(shí)際應(yīng)用背景,但目前關(guān)于其計(jì)算方法的研究不多,Li等[11]討論了此類問題.本文采用模態(tài)匹配法進(jìn)行數(shù)值計(jì)算,該算法具有精度好、效率高、可處理高波數(shù)情況的優(yōu)點(diǎn).

1 數(shù)學(xué)模型

設(shè)平面波ui(x,y)=ei(αx-βy)由洞穴上方入射,其中α=k0sinθ,β=k0cosθ,k0是波數(shù),θ∈(-π/2,π/2)是入射角.設(shè)u為Ω中的全場(chǎng),記u|Ωj=uj(j=0,1,2).TM情形下洞穴散射問題可描述為：給定平面入射波ui,求解全場(chǎng)u,使得u滿足Helmholtz方程

圖1 開洞穴Fig.1 Open cavity

(1)

當(dāng)(x,y)∈Ω0時(shí),全場(chǎng)u0(x,y)=ui(x,y)+ur(x,y)+us(x,y),其中:ur=-ei(αx+βy)為反射場(chǎng);us為散射場(chǎng),us滿足輻射條件:

(2)

當(dāng)(x,y)∈Ωj(j=0,1,2)時(shí),全場(chǎng)uj滿足邊界條件:

(3)

其中Γj=(aj,bj)×{0},j=1,2.

在洞穴開口處,即(x,y)∈Γ1∪Γ2時(shí),全場(chǎng)u滿足連續(xù)性條件:

(4)

(5)

2 數(shù)值方法

對(duì)全場(chǎng)u的求解可分為無界域Ω0和有界域Ω1∪Ω2兩部分.當(dāng)(x,y)∈Ωj(j=1,2)時(shí),由方程(1)和邊界條件(3),用分離變量法易求出全場(chǎng)uj(j=1,2)的表達(dá)式如下：

(6)

(7)

根據(jù)連續(xù)性條件(4),

(8)

將式(8)代入式(7),可得

(9)

進(jìn)而可計(jì)算

記M(ξ)=-γ0(ξ),特別地,

再根據(jù)連續(xù)性條件(5),有

結(jié)合式(4),整理可得方程

(10)

(11)

由式(6)知：

(12)

(13)

將式(12),(13)代入式(10)得

(14)

其中:

(15)

(16)

其中:

(17)

將式(16)與式(17)的求和做有限項(xiàng)截?cái)?即取充分大的N.記

并令

(18)

下面給出計(jì)算散射問題(1)的模態(tài)匹配法,步驟如下：

2)解方程(16)和(17)構(gòu)成的方程組(18);

3)計(jì)算u1(x,y),u2(x,y);

4)計(jì)算us(x,y),最后求出u0(x,y).

3 數(shù)值實(shí)驗(yàn)

采用表1所列的3組數(shù)據(jù)進(jìn)行數(shù)值實(shí)驗(yàn),洞穴形狀如圖1所示,選取不同的截?cái)鄶?shù)N、波數(shù)、入射角及洞穴的尺寸進(jìn)行數(shù)值實(shí)驗(yàn),實(shí)驗(yàn)結(jié)果如圖2～圖4所示.其中l(wèi)1,l3,l2分別表示兩個(gè)洞穴開口的尺寸和兩洞穴間的距離.圖2～圖4分別表示表1中3組數(shù)據(jù)取不同截?cái)鄶?shù)N時(shí)對(duì)應(yīng)的|u0(x,0)|在洞穴開口Γ1∪Γ2處的取值.圖中坐標(biāo)系的坐標(biāo)原點(diǎn)取為洞穴開口Γ1的左端點(diǎn),橫坐標(biāo)表示洞穴開口的尺寸,縱坐標(biāo)表示開口處|u0(x,0)|的值.

表1 入射角與洞穴尺寸Table 1 Incident angles and open cavity sizes

3組數(shù)值實(shí)驗(yàn)中,第一組數(shù)據(jù)截?cái)鄶?shù)N分別取不同的數(shù)值,當(dāng)N取很小的數(shù)20和40時(shí),計(jì)算結(jié)果就很好了,與有限元方法相比計(jì)算量小很多,速度也較快.第二組和第三組數(shù)據(jù)分別計(jì)算了洞穴開口尺寸很大及波數(shù)k的取值很大的情況,這兩種情況很多傳統(tǒng)算法都處理不了,本文的計(jì)算結(jié)果較好.

圖2 第一組數(shù)據(jù)對(duì)應(yīng)的|u0(x,0)|Fig.2 |u0(x,0)| of the first data

圖3 第二組數(shù)據(jù)對(duì)應(yīng)的|u0(x,0)|Fig.3 |u0(x,0)| of the second data

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(責(zé)任編輯：趙立芹)

FastAlgorithmforTwo-DimensionalMultipleCavityScatteringProblem

RU Jing1,2,MA Fuming2

(1.CollegeofSciences,JilinUniversityofChemicalTechnology,Jilin132022,JilinProvince,China;2.InstituteofMathematics,JilinUniversity,Changchun130012,China)

A fast algorithm was proposed for the numerical computation of the electromagnetic scattering by multiple rectangular cavities embedded on an infinite ground plane,the algorithm can be used to compute the scattering problems with large cavities and high wave numbers.The numerical experiments confirm the effectiveness of our method.

scattering problem;Helmholtz equation;Fourier series

10.13413/j.cnki.jdxblxb.2015.03.13

2014-09-30.

茹 靜(1978—),女,漢族,博士研究生,講師,從事數(shù)學(xué)物理反問題的研究,E-mail:rujing10@mails.jlu.edu.cn.

國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào)：11371172).

O241.82

：A

：1671-5489(2014)03-0419-05