麥宏晏,王連堂

(1.長安大學(xué)理學(xué)院,陜西西安 710064;2.西北大學(xué)數(shù)學(xué)系,陜西西安 710069)

利用遠(yuǎn)場模式的不完全數(shù)據(jù)反演聲波
阻尼系數(shù)

麥宏晏1,王連堂2

(1.長安大學(xué)理學(xué)院,陜西西安 710064;2.西北大學(xué)數(shù)學(xué)系,陜西西安 710069)

從阻尼邊界條件聲波散射問題的散射場遠(yuǎn)場模式的部分?jǐn)?shù)據(jù)信息出發(fā)給出了反演聲波阻尼系數(shù)的一種新方法,該問題既是非線性的又是不適定的,這里利用Tikhonov正則化方法將問題轉(zhuǎn)化為一個(gè)最優(yōu)化問題,成功地處理了第一類算子方程的不適定性及該問題的非線性性,給出了具體的數(shù)值方法并對(duì)其收斂性進(jìn)行了嚴(yán)格地證明,數(shù)值結(jié)果表明該方法是非常準(zhǔn)確且簡單易行的.

阻尼邊界條件;遠(yuǎn)場模式;反演;部分?jǐn)?shù)據(jù)

1 引言

文[1-4,6,9]給出一些利用遠(yuǎn)場模式的完全或不完全數(shù)據(jù)反演聲波阻尼系數(shù)和阻尼區(qū)域的結(jié)果.本文給出一種比文[4]更加簡單易行的方法.

考慮在各向同性的介質(zhì)D?R2中傳播的聲波,令D是有界的單連通區(qū)域,?D是C2類的且取入射場為ui(x)=exp[ikx·α],其中k＞0是波數(shù),α是單位向量.散射場記為us,全場記為u=ui+us,于是阻尼邊界條件的聲波正散射問題歸結(jié)為求u∈C2(R2)∩C(R2D),滿足

其中ν表示單位外法向量,λ(x)∈C(?D)是阻尼系數(shù).

2 反演方法

假定k2在?D內(nèi)部不是負(fù)的Lapacian算子的Dirichlet特征值,例如可以選擇?D是半徑為R的圓且kR不是球Bessel函數(shù)jn的零點(diǎn),n=0,1,2,…,由文[8],利用單層位勢

2 收斂性分析

4 數(shù)值算例

下面我們給出具體的算法描述,為了獲得更準(zhǔn)確的反演結(jié)果,取入射波為

表1 例1的反演結(jié)果

表2 例2的反演結(jié)果

圖1 λ(x)=的準(zhǔn)確結(jié)果及反演結(jié)果

圖2 λ(x)=的準(zhǔn)確結(jié)果及反演結(jié)果

這里正問題的參數(shù)n選取的越大,正問題計(jì)算精度越高,由此反演結(jié)果也會(huì)越好；不過參數(shù)n選取的過大,不僅增加了正問題的計(jì)算量,更是大大增加了反問題的計(jì)算量,從而使反問題的計(jì)算速度大大降低.參數(shù)N用來表示入射方向的個(gè)數(shù),反演聲波阻尼系數(shù)和區(qū)域時(shí),該參數(shù)取的太小不能反映真實(shí)情況,取的太大又增加了計(jì)算量.參數(shù)n1,n2與阻尼系數(shù)的逼近有關(guān),取的越大,反演結(jié)果越準(zhǔn)確,當(dāng)然也會(huì)相應(yīng)地增加計(jì)算量.綜合計(jì)算量和計(jì)算精度的考慮,同時(shí)參閱文[7]給出的誤差分析,本文參數(shù)的選取是合適的.

由上面的算例還看出由完全數(shù)據(jù)反演的結(jié)果比部分?jǐn)?shù)據(jù)反演的結(jié)果要好,當(dāng)δ取的較大時(shí)反演結(jié)果較好.

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Recovering the acoustic wave impedance coefficient from the incomplete far field pattern data

MAI Hong-yan1,WANG Lian-tang2
(1.The School of Science,Chang’an University,Xi’an710064,China; 2.Department of Mathematics,Northwest University,Xi’an710069,China)

The method we are considering in this article is a novel method to recover the impedance coefficient from the knowledge of the incomplete far field pattern data of the scattered wave for the time-hormonic acoustic scattering problem.It is transformed into an optimization problem employing Tikhonov regularization method in order to solve the ill-posedness of the equation of the first kind and its nonlinearity successfully.Numerical method is given and the convergence of the method is rigorously proven.Numerical examples are shown that this method is both accurate and simple to use.

impedance boundary condition,far field pattern,recover,incomplete data

O175.29

A

1008-5513(2009)03-0566-07

2008-02-27.

長安大學(xué)科技發(fā)展基金(07J05).

麥宏晏(1972-),講師,研究方向：數(shù)學(xué)物理方程反問題.

2000MSC:31A25