本文選題：時域 + 散射問題��； 參考：《吉林大學(xué)》2016年博士論文

【摘要】：本文關(guān)心幾種帶特定形狀無界散射體的時域正反散射問題,我們分別建立數(shù)值方法對正反問題進行求解,并給出相關(guān)的分析.散射問題主要研究的是散射體對波場的散射情況,正問題通常是指已知入射波(聲波或電磁波)和散射體信息,求解由于散射體存在而產(chǎn)生的散射場或遠場,而反問題則是已知入射場和部分散射場或遠場數(shù)據(jù),來重構(gòu)散射體的位置和形狀.在各類散射問題中,本文關(guān)心的是不可穿透散射體對聲波的散射.我們的分析在時域進行,即考慮聲波為非時諧波.此時時間項不可以忽略,波場滿足波動方程.時域問題依賴于時間相關(guān)的數(shù)據(jù),相對于頻域問題,時域問題和地球物理勘探,醫(yī)學(xué)成像以及無損檢測等眾多應(yīng)用領(lǐng)域的關(guān)系更加密切.并且在實際操作中,時域分析所需的和時間相關(guān)的動態(tài)數(shù)據(jù)更容易獲得,這樣的時域數(shù)據(jù)所包含的信息量也遠多于頻域單一頻率或者多個離散頻率的數(shù)據(jù).對正問題,我們使用基于時域位勢函數(shù)的邊界積分方程方法進行求解.時域位勢函數(shù)的建立基于波動方程的Green函數(shù),通過位勢函數(shù)在散射體邊界上的躍度分析,可以得到時域散射問題解的位勢函數(shù)表示,進而由此建立邊界積分方程.時域位勢函數(shù)的定義和一個推遲時間相關(guān),因此這樣建立的邊界積分方程也被叫做推遲勢邊界積分方程(RPBIE)本文使用基于單層位勢的“第一類”RPBIE來求解正問題,對不同問題建立RPBIE的方法也有所不同.在數(shù)值計算中,通常不是直接對RPBIE進行離散求解,由于RPBIE關(guān)于時間變量的卷積特性,我們使用CQ方法(the convolution quadrature method)實現(xiàn)時間變量的離散,將時域方程變?yōu)橐唤MHelmholtz方程,之后進一步進行頻域離散實現(xiàn)數(shù)值求解.對反散射問題,我們使用時間相關(guān)的線性采樣法和基于正問題邊界積分方程的Newton型迭代法進行求解.線性采樣法的基本思想是把非線性不適定的反散射問題轉(zhuǎn)化為一個線性的第一類Fredholm積分方程,在時域中,所得到的積分方程被稱為近場方程.線性采樣法對散射體的重構(gòu)基于近場方程的“爆破”性,即近場方程的解在散射體所在區(qū)域內(nèi)部有界,但在穿過散射體邊界進入外部時,方程的解出現(xiàn)“爆破”行為,趨向于無窮.迭代法是求解反散射問題經(jīng)典方法,其重構(gòu)效果較好,理論上通過迭代數(shù)值解可以無限趨近真解.在正問題RPBIE的基礎(chǔ)之上,基于Newton法可以建立迭代方程.取定初始數(shù)據(jù),通過循環(huán)的求解迭代方程和更新初始數(shù)據(jù),我們得到的散射體數(shù)據(jù)將逐漸靠近真實值.本文的幾個主要工作如下：1.討論時域局部擾動半平面正反散射問題求解的數(shù)值方法.首先,通過對稱延拓,可以將局部擾動半平面問題就轉(zhuǎn)化成等價的全平面中的散射問題.對于正問題,我們把具有對稱結(jié)構(gòu)的散射問題限制在半空間內(nèi)進行分析,利用半空間Green函數(shù)重新定義時域位勢函數(shù),進而得到半空間上的RPBIE并證明其唯一可解性.之后考慮反散射問題,即通過測量的散射場數(shù)據(jù)來反演局部擾動.對反問題,使用時域的線性采樣法進行求解.為了適應(yīng)半空間內(nèi)的數(shù)值計算,利用問題的對稱性質(zhì)重新定義近場方程,并證明該方程所具有的“爆破”性質(zhì).本文提出的計算策略簡單易行,我們給出若干數(shù)值算例來證明算法的可行性.2.研究局部擾動半平面問題的三維推廣,即三維局部擾動問題.對正問題,我們試圖對無界域上散射問題直接分析,并在無界邊界建立積分方程進行求解.本文使用基于半空間Green函數(shù)的時域單層位勢定義,并在此基礎(chǔ)上建立邊界積分方程,進而證明求解無界邊界上的積分方程等價于求解一個定義在積分核有界支集上的積分方程,并對無界邊界上的邊界積分方程的可解性給出理論分析.對反問題,仍然使用時域線性采樣法進行求解,并給出三維問題近場方程的“爆破”性質(zhì).3.考慮時域二維開腔體正反散射問題.散射體為帶局部凹陷的半平面,此局部凹陷即為我們所說的開腔體.對正問題,通過在洞穴開口處建立透射邊界條件,可以得到在有界開腔體區(qū)域上的初邊值問題,其在開腔體底部和洞穴開口處滿足不同的邊界條件.通過積分變換的手段進行分析,我們給出弱解意義下正問題的唯一可解性.在邊界條件的基礎(chǔ)上,利用時域散射問題解的位勢表示,我們在開腔體邊界上建立RPBIE來求解正問題,并給出其時間離散的CQ方法.對反問題,在正問題RPBIE的基礎(chǔ)之上,通過分析RPBIE中各算子的Frechet導(dǎo)數(shù),我們建立反問題的Newton型迭代求解方法.以上是我們近些年的主要研究內(nèi)容,也構(gòu)成了本文的主要章節(jié),但帶無界散射體的時域正反散射問題的研究不止于此,要做的工作還有很多.此外,我們也對時域的其他散射問題有所涉獵,在本文中也簡單提到了一些,做為今后可能的研究方向.
[Abstract]:In this paper, we are concerned with the time-domain positive and inverse scattering problems of a number of unbounded scatterers with specific shapes. We establish numerical methods to solve the positive and negative problems and give the related analysis. The scattering problem mainly deals with the scattering of the scattering body to the wave field, and the positive problem is usually known as the known incident wave (sound wave or electromagnetic wave) and the scatterer information. To solve the scattering field or far field caused by the existence of the scatterer, the inverse problem is to reconstruct the position and shape of the scatterer by the known incident field and some scattering field or far field data. In all kinds of scattering problems, this paper is concerned with the scattering of the acoustic waves by the non penetrating scatterers. Time term can not be ignored at this time. The wave field satisfies the wave equation. The time domain problem depends on the time dependent data. Relative to the frequency domain problem, the time domain problem is more closely related to many applications such as geophysical exploration, medical imaging and nondestructive testing. And in practical operation, time domain analysis needs to be related to time. The dynamic data is easier to obtain. This time domain data contains much more information than the frequency domain single frequency or multiple discrete frequency data. For the positive problem, we use the boundary integral equation method based on the time domain potential function. The time domain potential function builds the Green function based on the wave equation, through the potential potential. The analysis of the jump on the boundary of the scatterer can obtain the potential function of the solution of the time domain scattering problem and then establish the boundary integral equation. The definition of the potential function of the time domain is related to a delay time, so the boundary integral equation is also called the delayed potential boundary integral equation (RPBIE), and this paper is based on the single layer. The "first class" RPBIE of the potential is used to solve the positive problem, and the method of establishing RPBIE for different problems is also different. In numerical calculation, it is usually not a direct solution to RPBIE. Because of the convolution properties of time variables on RPBIE, we use the CQ method (the convolution quadrature method) to realize the discretization of time variables. The domain equation becomes a group of Helmholtz equations, and then the numerical solution is realized in the frequency domain. For the inverse scattering problem, we use the time dependent linear sampling method and the Newton type iterative method based on the positive problem boundary integral equation. The basic idea of the linear sampling method is to transform the nonlinear unsuitable inverse scattering problem. For a linear first class Fredholm integral equation, the integral equation is called the near field equation in the time domain. The linear sampling method reconstructs the scatterer based on the "blasting" of the near field equation, that is, the solution of the near field equation is bounded within the region of the scatterer, but the solution of the equation appears when the boundary of the scatterer enters the outer region. The "blasting" behavior tends to infinity. The iterative method is a classical method for solving the inverse scattering problem. Its reconstruction effect is good. In theory, the iterative numerical solution can reach the true solution infinitely. On the basis of the positive problem RPBIE, the iterative equation can be established based on the Newton method. The data we obtain will gradually close to the true value. The main work of this paper is as follows: 1. the numerical method to discuss the solution of the semi plane positive and negative scattering problem in the time domain is discussed. First, by the symmetric extension, the local perturbation semi plane problem can be transformed into the equivalent scattering problem in the whole plane. The scattering problem with symmetric structures is restricted in half space, and the half space Green function is used to redefine the time domain potential function, and then the RPBIE in the semi space is obtained and its unique solvability is proved. Then, the inverse scattering problem is considered, that is, the local perturbation is retrieved by the measured scattering field data. In order to adapt to the numerical calculation in the semi space, in order to adapt to the numerical calculation in the half space, the near field equation is redefined by the symmetric property of the problem, and the "blasting" property of the equation is proved. The proposed calculation strategy is simple and easy. We give some numerical examples to prove the feasibility of the algorithm.2. to study the partial disturbance of the local disturbance. In order to solve the problem of three-dimensional local perturbation, we try to analyze the scattering problem on the unbounded domain directly and establish the integral equation on the unbounded boundary. In this paper, we use the definition of the single layer potential in the time domain based on the half space Green function, and establish the boundary integral equation on this basis, and then prove that the solution is not solved. The integral equation on boundary boundary is equivalent to solving an integral equation defined on the bounded support set of the integral kernel, and a theoretical analysis is given for the solvability of the boundary integral equation on the unbounded boundary. On the inverse problem, the time domain linear sampling method is still used to solve the problem, and the "blasting" property of the three dimensional question near field equation.3. is given in the time domain. The scattering problem of a two-dimensional cavity is a positive and negative scattering problem. The scatterer is a half plane with a local sag, and the local sag is the cavity of the cavity we speak. By means of integral transformation, we give the unique solvability of the positive problem in the sense of weak solution. On the basis of the boundary condition, we use the potential representation of the solution of the time domain scattering problem. We establish the RPBIE to solve the positive problem on the opening boundary, and give the CQ method of its time discretization. The inverse problem, in the positive problem RPBIE, is given. On the basis of the analysis of the Frechet derivative of each operator in RPBIE, we establish the Newton type iterative solution for the inverse problem. The above is the main research content of our recent years, and also constitutes the main chapter of this paper, but the study of the time domain positive and negative scattering problem with unbounded scatterers is more than that, and there are many work to be done. We also dabbled in other scattering problems in time domain. In this paper, we also mentioned briefly some of them as possible directions for future research.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O241.8

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