楊孝英, 王旖旎

(長(zhǎng)春工業(yè)大學(xué) 基礎(chǔ)科學(xué)學(xué)院, 長(zhǎng)春 130012)

自從Bérenger[1]提出解電磁散射問(wèn)題的PML方法以來(lái), 已構(gòu)造了許多不同的PML方法[2-4]. 文獻(xiàn)[5-6]用PML方法求解了洞穴散射問(wèn)題, 但在實(shí)際計(jì)算中, 若PML的厚度過(guò)大, 則用有限元計(jì)算需要大量的剖分節(jié)點(diǎn), 從而導(dǎo)致數(shù)值計(jì)算耗費(fèi)內(nèi)存和計(jì)算時(shí)間. 因此PML參量的確定是實(shí)際計(jì)算的關(guān)鍵. 本文提出一種帶小參數(shù)的PML, 并證明了如此構(gòu)造的PML吸收效果較好.

1 解散射問(wèn)題的優(yōu)化PML方法

考慮散射問(wèn)題:

(1)

a(u,ψ)=(g,ψ)ΓD, ?ψ∈H1(ΩR),

(2)

其中: (·,·)表示L2內(nèi)積;

(3)

算子T:H1/2(ΓR) →H-1/2(ΓR)定義為

?f∈H1/2(ΓR).

(4)

則問(wèn)題(2)的解存在唯一[8].

針對(duì)上述問(wèn)題, Chen等[7]給出了圓形PML區(qū)域的自適應(yīng)有限元方法; Bermúdez等[9]針對(duì)方形PML區(qū)域給出了解散射問(wèn)題的一種優(yōu)化PML方法, 即選取PML介質(zhì)參量為在PML廣義積分無(wú)界的函數(shù), 這樣選取的參量能保證PML計(jì)算與PML的厚度無(wú)關(guān), 從而節(jié)省了計(jì)算量. 但文獻(xiàn)[9]選取的介質(zhì)參量函數(shù)使PML方程的系數(shù)產(chǎn)生奇異, 因此導(dǎo)致數(shù)值分析及截?cái)郟ML問(wèn)題的收斂性分析存在困難, 并且在數(shù)值計(jì)算中, 邊界層附近的PML解出現(xiàn)不穩(wěn)定現(xiàn)象.

圖1 優(yōu)化PML的幾何構(gòu)造Fig.1 Geometric structure of optimal PML

本文基于文獻(xiàn)[7,9], 針對(duì)圓形PML區(qū)域提出一種新的PML介質(zhì)參量構(gòu)造方法. 如圖1所示, 在區(qū)域ΩPML={x∈R2:R

(5)

其中:ε0為足夠小的正數(shù);m≥2.

令

(6)

(7)

(8)

證明: 由式(8), 有

由于

故

其中δ為PML的厚度. 證畢.

(9)

2 數(shù)值實(shí)例

例2考慮金屬凹槽的散射問(wèn)題. 入射平面波為ui=eikx, 在其邊界上散射場(chǎng)u=-ui.

圖3為δ=1,2,0.5,ε0=0.01,0.005,0.001時(shí)PML解的實(shí)部曲線. 由圖3可見(jiàn), 優(yōu)化的PML方法對(duì)于解金屬凹槽散射問(wèn)題有效.

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