无界散射体时域正反散射问题的数值方法研究

发布时间：2018-07-12 12:01

本文选题：时域 + 散射问题　；参考：《吉林大学》2016年博士论文

【摘要】：本文关心几种带特定形状无界散射体的时域正反散射问题,我们分别建立数值方法对正反问题进行求解,并给出相关的分析.散射问题主要研究的是散射体对波场的散射情况,正问题通常是指已知入射波(声波或电磁波)和散射体信息,求解由于散射体存在而产生的散射场或远场,而反问题则是已知入射场和部分散射场或远场数据,来重构散射体的位置和形状.在各类散射问题中,本文关心的是不可穿透散射体对声波的散射.我们的分析在时域进行,即考虑声波为非时谐波.此时时间项不可以忽略,波场满足波动方程.时域问题依赖于时间相关的数据,相对于频域问题,时域问题和地球物理勘探,医学成像以及无损检测等众多应用领域的关系更加密切.并且在实际操作中,时域分析所需的和时间相关的动态数据更容易获得,这样的时域数据所包含的信息量也远多于频域单一频率或者多个离散频率的数据.对正问题,我们使用基于时域位势函数的边界积分方程方法进行求解.时域位势函数的建立基于波动方程的Green函数,通过位势函数在散射体边界上的跃度分析,可以得到时域散射问题解的位势函数表示,进而由此建立边界积分方程.时域位势函数的定义和一个推迟时间相关,因此这样建立的边界积分方程也被叫做推迟势边界积分方程(RPBIE)本文使用基于单层位势的“第一类”RPBIE来求解正问题,对不同问题建立RPBIE的方法也有所不同.在数值计算中,通常不是直接对RPBIE进行离散求解,由于RPBIE关于时间变量的卷积特性,我们使用CQ方法(the convolution quadrature method)实现时间变量的离散,将时域方程变为一组Helmholtz方程,之后进一步进行频域离散实现数值求解.对反散射问题,我们使用时间相关的线性采样法和基于正问题边界积分方程的Newton型迭代法进行求解.线性采样法的基本思想是把非线性不适定的反散射问题转化为一个线性的第一类Fredholm积分方程,在时域中,所得到的积分方程被称为近场方程.线性采样法对散射体的重构基于近场方程的“爆破”性,即近场方程的解在散射体所在区域内部有界,但在穿过散射体边界进入外部时,方程的解出现“爆破”行为,趋向于无穷.迭代法是求解反散射问题经典方法,其重构效果较好,理论上通过迭代数值解可以无限趋近真解.在正问题RPBIE的基础之上,基于Newton法可以建立迭代方程.取定初始数据,通过循环的求解迭代方程和更新初始数据,我们得到的散射体数据将逐渐靠近真实值.本文的几个主要工作如下：1.讨论时域局部扰动半平面正反散射问题求解的数值方法.首先,通过对称延拓,可以将局部扰动半平面问题就转化成等价的全平面中的散射问题.对于正问题,我们把具有对称结构的散射问题限制在半空间内进行分析,利用半空间Green函数重新定义时域位势函数,进而得到半空间上的RPBIE并证明其唯一可解性.之后考虑反散射问题,即通过测量的散射场数据来反演局部扰动.对反问题,使用时域的线性采样法进行求解.为了适应半空间内的数值计算,利用问题的对称性质重新定义近场方程,并证明该方程所具有的“爆破”性质.本文提出的计算策略简单易行,我们给出若干数值算例来证明算法的可行性.2.研究局部扰动半平面问题的三维推广,即三维局部扰动问题.对正问题,我们试图对无界域上散射问题直接分析,并在无界边界建立积分方程进行求解.本文使用基于半空间Green函数的时域单层位势定义,并在此基础上建立边界积分方程,进而证明求解无界边界上的积分方程等价于求解一个定义在积分核有界支集上的积分方程,并对无界边界上的边界积分方程的可解性给出理论分析.对反问题,仍然使用时域线性采样法进行求解,并给出三维问题近场方程的“爆破”性质.3.考虑时域二维开腔体正反散射问题.散射体为带局部凹陷的半平面,此局部凹陷即为我们所说的开腔体.对正问题,通过在洞穴开口处建立透射边界条件,可以得到在有界开腔体区域上的初边值问题,其在开腔体底部和洞穴开口处满足不同的边界条件.通过积分变换的手段进行分析,我们给出弱解意义下正问题的唯一可解性.在边界条件的基础上,利用时域散射问题解的位势表示,我们在开腔体边界上建立RPBIE来求解正问题,并给出其时间离散的CQ方法.对反问题,在正问题RPBIE的基础之上,通过分析RPBIE中各算子的Frechet导数,我们建立反问题的Newton型迭代求解方法.以上是我们近些年的主要研究内容,也构成了本文的主要章节,但带无界散射体的时域正反散射问题的研究不止于此,要做的工作还有很多.此外,我们也对时域的其他散射问题有所涉猎,在本文中也简单提到了一些,做为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.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

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