NUMERICAL SIMULATION ON THEINTERIOR TRANSMISSIONEIGENVALUE PROBLEM

内传输特征值问题数值模拟

齐迎春

11649044

硕士

应用数学

李景治

2018-06-04

2018-07-06

哈尔滨工业大学

深圳

In recent years, the interior transmission eigenvalue problem, denoted by the ITEPin the following context, has become one of the important fields of acoustic waves and electromagnetic waves inverse scattering theory. The reason for this is largely driven by the urgent needs arising from the application of other disciplines and many engineering and technology fields. At the same time, because it has a distinctive novelty and challenge in theory, it has caused many scholars and practical workers at home and abroad to engage in research and application. So far, it has developed into a hot subject direction in the cross- cutting fields of computational mathematics, applied mathematics and system science. Almost all kinds of disciplines involve it. For example, medicine can be applied to the medical imaging industry for non-destructive testing, geology can be used to detect the internal structure of the earth, and so on. Therefore, its application is very extensive, so it is transmitted. The study of the ITEP is of great significance.The ITEP is a boundary value problem in which two partial differential equations coupled with the first and second boundary conditions. According to the inverse scatter- ing theory, the ITE can only constitute at most one discrete set. However, because the ITEP is a nonlinear, non-selfadjoint mathematical problem, using the standard method of elliptic operators can not be solved and this problem is therefore more challenging. At present, there has been many new developments on the ITEs of constant coefficients ellip- tic operators and the eigenvalues of the ITEP for general case. These results cover many sides including the existence, discreteness and infinity of the ITEs. From the practical angle, the property that is ITEs form a discrete set is very vital of studying the acous- tic waves in inverse scattering theory, because if the corresponding wave numbers to the ITEs, two important methods that used to reconstruct the inhomogeneous media region: the linear sampling method and the factorization method will fail, so in the reconstruction of scatterers, we should avoid the ITEs. This result expand the field of non-destructing test. On the contrary, if we select the ITEs as the wave numbers, the phenomenon of invisibility cloaking would happen. In recent years, there have been many studies, such as, if a certain non-transparency condition is satisfied, then there exists an infinite set of incident waves such that the cloaking device is nearly-invisible under the correspondingwave interrogation. In particular, if the obstacle has a pointed angle, it is unlikely to beinvisible, because corners always scatter. That is why we study this problem.So far, researchers have devoted themselves to calculate the ITEs and corresponding interior transmission eigenfunctions. Due to the ITEP is a non-selfadjoint problem, the existence of the ITEs for a non spherical medium is still unresolved, until 2008 it is proved that if the medium contrast function is big enough, then there is at least one ITE. To be more precise, it leads to much development in many fields. In 2010, people provide a more general conclusion on the existence of the ITEs: if the contrast is a nonzero positive constant, then there are more than infinite ITEs. In addition, they obtained that an estimate of the first, that is, the smallest ITE and the scattering data determines of the ITEs. This also consolidates its important role in the field of target configurations, such as radar. The same results also has been proved for anisotropic case. Later, great developments have been made on the existence of the ITEs. And the conditions on the contrasts have been removed. From the above analysis, the whole proof of the ITEs form an infinite discrete set, which started the program of research by using the ITEs to deduce some properties of the scatterers.From now on, all the discussions on the ITEP are only limited to the case of inhomo- geneous media with contrast function. Since the ITEP is neither elliptic nor selfadjoint, a new technique is needed to solve this problem. After the corresponding equivalent re- formulation of the ITEP, a fourth order nonlinear eigenvalue problem can be obtained, this makes it very difficult to calculate the ITEs. Therefore, the development of effective numerical methods of the ITEs has always been a hot subject. Colton et al. firstly pro- posed three finite element methods to calculate the ITEs. The first one is Argyris elements, which is based on a fourth-order formation of the original problem and does not produce pseudo-interior transmission eigenvalues. However, this method is difficult in practical applications because the boundary normal vector does not follow the affine transforma- tion. And a lot of local degrees of freedom lead to the corresponding discrete system is very large. The second is the mixed finite element method, which is relatively easy to apply in practice, but produces pseudo-interior transmission eigenvalues in the calcula- tions. The third is the continuous finite element method, which is the easiest than the previous two finite elements and produce no pseudo-interior transmission eigenvalues. Jiguang Sun applies iterative method to calculate the ITEs, including bisection method and secant method. These two methods convert a fourth-order nonlinear problem into anon-Hermitian eigenvalue problem, and the latter converges faster than the former, butthey require an extra calculations of the approximate range of ITEs. Then a novel method based on the spectral projection was proposed. Other methods have been also proposed recently. And a mixed finite element method using Lagrange elements was developed and the convergence of this method was proved. Cakoni et. al. obtained the ITEs by measur- ing the scattering far field data. However, the precision of these algorithms is not high, which brings great difficulties to the practical applications. Therefore, a more effective numerical method is needed to calculate the ITEs.In this thesis, the finite element method is used to simulate the ITEP, and the ITEs and corresponding interior transmission eigenfunctions are simultaneously obtained. We know that each eigenfunction corresponds to an eigenvalue, and it is no exception to the ITEP. So each interior transmission eigenfunction corresponds to an interior transmission eigenvalue. In particular, for a domain with a corner, it has more complicated properties. There are some have already been known results: the interior properties of the interior transmission eigenfunctions corresponding to the ITEs. An important consequence of the study is the fact that the interior transmission eigenfunction cannot be analytically extend- ed across a corner point to form an entire solution to the Helmholtz equation in R2 or R3. However, we note that due to the interior regularity, the interior transmission eigenfunc- tion is always analytic away from the corner point. Hence, the failure of the analytic ex- tension may indicate that the interior transmission eigenfunction either vanishes or blows up when approaching the corner point. The proof of this conclusion is quite complicated, even in the case of special isotropic media. Therefore, we use the finite element method to solve the ITEs and the corresponding interior transmission eigenfunctions.This thesis discussed on the ITEP not the transmission eigenvalue problem, because the transmission eigenvalue problem includes not only the ITEP but also the exterior trans- mission eigenvalue problem, denoted by ETEP in the following context. It is characterized by finding a pair of nonzero solutions to two elliptic equations in an unbounded domain with Cauchy data on the boundary, both of which satisfy the sommerfeld radiation con- dition. People presently start to study the ETEP, and conclude that for the spherical re- gion, the refractive index satisfies the certain conditions, there also will be infinite real or complex exterior transmission eigenvalues (ETEs). In this thesis, we discuss the interior transmission eigenvalue problem from numerical calculations. The structure is as follows:In Chapter 1, we give a brief history of acoustic waves and electromagnetic wavesscattering problem, the existing work, and some related knowledge about acoustic wavesand electromagnetic waves scattering problem. We also present the research content and structure of this thesis.In Chapter 2, we recall some preliminary knowledge: including the Helmholtz equa- tion, direct and inverse scattering problem, the properties of the ITEs.In Chapter 3, An analytical method for solving the scattering problem of time-harmonic sound waves in a homogeneous media with corners is developed. The analytical methodis based on the separation variable method. we review some analytical formulas of the ITEs and numerical methods to solving the ITEs.In Chapter 4, we use the finite element method to compute the ITEs for a variety of domains with at least one corner whose interior angle is less than 180◦ or interior angle is more than 180◦, and compared with other numerical methods. Moreover, we find that the sharper the corner, the higher the convergence order. And we also obtain the concrete expression of the inverse relation between the convergent order of the interior transmission eigenfunctions and the angle of the sharp angle.The advantage of this thesis is that we can not only solve the ITEs but also the cor- responding interior transmission eigenfunctions. The numerical simulation of this thesis is mainly divided into two types: its interior angle is less than 180◦ and greater than 180◦. And it can not only obtain the ITEs of the scatterer, but also the corresponding interior transmission eigenfunctions. This thesis not only discusses that the refractive index is a constant, but also discusses that it is not a constant value, all these numerical experimental results show that when the interior angle of the obstacle is less than 180◦, the interior trans- mission eigenfunctions will vanish near the sharp angle; When the inner angle is greater than 180◦, the interior transmission eigenfunction will localize near the sharp angle. Fur- thermore, we do so many numerical experiments and observe that the vanishing order is related to the interior angle of the cusp and inversely proportional to the interior angle of the cusp. In this thesis, we obtain the relationship of data fitting: the sharper the corner, the higher the convergence order. The method of curve fitting is also used to obtain the formula of the inverse relation between the convergent order of the interior transmissioneigenfunctions and the angle of the sharp angle.

近年来，内传输特征值问题越来越成为声波、电磁波反散射理论的一个重要领域之一。之所以如此，在很大程度上是受其它学科与众多工程技术领域的 应用中产生的迫切需要所驱动;同时，由于它在理论上又具有鲜明的新颖性和 挑战性，所以引起了国内外许多学者和实际工作者从事研究和应用。迄今, 它 己发展成为具有交叉性的计算数学、应用数学和系统科学中的一个热门学科方 向。几乎各类学科都涉及到它，比如医学上可以应用到医学成像、工业上可以 用来无损检测、地质学上可以用来探测地球内部结构等等，因此它的应用非常 广泛，故而对传输特征值问题的研究具有重要意义。内传输特征值问题是由两个偏微分方程和第一、二类边界条件稠合的边值 问题。根据反散射理论，内传输特征值最多只能构成一个离散的集合。但是因 为内传输特征值问题是一个非线性、非自伴的数学问题，所以用椭圆算子的标 准方法是无法解决的，因此也更具有挑战性。目前，对于常系数椭圆算子的内 传输特征值问题和一般的内传输特征值问题，都取得了很大的进展。这些科研 成果涵盖很多方面，包括内传输特征值的存在性、离散性和无穷性等。从实际 角度出发，内传输特征值离散性这一特性对研究反散射问题是至关重要的，因 为如果将内传输特征值作为相应的波数，用此入射波去照射障碍物的话，那么 用来重构非均匀介质区域的形状的两种重要方法:线性采样法和分解方法就会 失效，所以在重构障碍物的形状时要避免用将内传输特征值作为相应的波数的 入射波来照射散射体。这一理论成果开拓了无损探测等应用领域。相反，如果 我们选择用内传输特征值作为波数的入射波去照射散射体的话，就会出现隐形 的现象。近年来，也有很多关于这方面的研究，如果满足某种非透明条件，则 存在无穷的入射波使得隐形装置在声波照射下近乎隐形。特别地，假如这个障 碍物带有尖角，那么它是不太可能会发生隐形的，因为带角的区域总会发生散 射。本文研究的重点就在于此。目前，学者们致力于计算内传输特征值和对应的传输特征函数。由于内传 输特征值问题是非自伴问题，所以对于非球形介质，内传输特征值问题一直尚 未解决，直到 2008 年，Sylvester 和 Päivärinta 两人证明了假如背景介质与散射 体介质的折射率的差异(介质的差异)足够大的话，那么至少会存在一个内传 输特征值，由此推进了许多领域的发展。2010 年，人们给出了更完整的关于内 传输特征值的存在性的结果:如果假设介质的差异是一个非零的正常数，那么就会有无限多个内传输特征值。并且，他们还得到了第一个也就是最小的内传输特征值的一些估计和散射数据可以决定内传输特征值。这些结果巩固了它在 雷达等目标配置领域的重要地位。同样的结果在各向异性介质的情况下也成 立。后来，内传输特征值的存在性结果得到了进一步发展，并且介质的差异的 限定条件也被去除了。对于上述情况，内传输特征值形成了一个无限离散集得 到了证明，它开启了利用内传输特征值来推断散射体的性质的研究方案。到目前为止，人们对内输特征值问题的研究多数是限于差异函数存在的非 均匀介质的情况。由于内传输特征值问题既不是椭圆的也不是自伴随的，所以 需要新的技巧去解决这个问题，对内传输特征值问题作相应的等价变形之后， 可以得到一个四阶的非线性特征值问题，这使得计算内传输特征值变得非常 困难，因此发展有效的内传输特征值数值计算方法一直是人们所关心的热门 课题。Colton 等人第一次提出了三种有限元方法去计算内传输特征值。第一种 是 Argyris 元，它基于原问题的一种四阶格式，不会产生伪内传输特征值。然而 在实际应用中这种方法是比较困难的，因为边界法向量并不遵从仿射变换。而 且很多的局部自由度导致相应的离散系统是很大的。第二种是混合有限元法， 在 实 际 应 用 中 它 是 相 对 容 易 的，但 在 计 算 中 会 产 生 伪 内 传 输 特 征 值。第 三 种 是连续有限元法，和之前两种有限元相比，它是最容易的。而且不会出现伪内 传输特征值。Jiguang Sun 采用迭代方法计算内传输特征值，包括二分法和迭代 法。这两种方法将一个四阶的非线性问题转化为一个非 Hermitian 的特征值问 题，而且后者比前者收敛的速度要快，但是需要额外计算内传输特征值的大致 范围。后来又有人提出了基于谱投影的新方法，还有一些其他方法可以计算内 传输特征值。随后，人们又推广了拉格朗日元素的混合有限元。而且，这种方 法的一致性也得到了证明。Cakoni 等人还通过测量散射的远场数据得到了内传 输特征值。但是这些算法精度阶不高，这给实际应用带来很大的困难。因此需 要一个更加有效的数值方法去计算内传输特征值。本文将采用有限元方法对内传输特征值问题进行数值模拟，并且同时求解 内传输特征值和相应的传输特征函数。我们知道一个特征函数对应一个特征 值，对于内传输特征值问题也不例外，每个内传输特征函数都对应一个内传输 特征值。特别地，对于一个带角的区域来说，它的性质要更加复杂。目前，人 们已经得到了的一些科研成果:与内传输特征值相对应的内传输特征函数的内 在几何结构。结果表明，如果存在尖峰奇点，那么在尖峰附近，内传输特征函 数会有一定的内在定量行为。一个重要的研究结果是内传输特征函数不能被解 析地延拓到一个尖角点上，从而形成二维或者三维实数空间上的 Helmholtz 方程的一个整体解。然而，我们注意到由于内部正则性，内传输特征函数在远离尖角点的地方总是解析的。因此，解析延拓的失败可能表明在接近尖角点的地 方会消失或爆破。这个结论的证明相当复杂，即使是在特殊的各向同性的情况下。本文讨论的不是传输特征值问题而是内传输特征值问题，因为传输特征值问题不仅包括内传输特征值问题还包括外传输特征值问题，其特征是在边界上 有柯西数据的无界域中找到两个椭圆方程的一对非零的解，这两个解都要满足 Sommerfeld 条件。目前，也有人开始研究外传输特征值问题，并得出了在径向 对称的球形区域，如果折射率满足一定的条件，也会存在无穷多个实或者复的 外传输特征值的结论。本文从数值计算方面对声波的内传输特征值问题进行深 入研究，取得了一些新的结果。本文内容结构如下:第一章简要介绍了声波和电磁波的散射问题，总结前人所做的工作，以及 有关声波和电磁波散射问题的一些相关知识，并且详细给出了本文的研究内容和结构。第二章我们回顾了一些预备知识:包括 Helmholtz 方程、正散射问题和反散射问题的介绍、内传输特征值的一些性质。第二章给出在各性同向的非均匀介质的情况下，求解内传输特征值的解析 表达式和数值方法。第四章，我们利用第二部分介绍的有限元方法计算了对于至少含有一个尖 角的区域的内传输特征值，分为其内角小于 180◦ 和大于 180◦ 两类，并与其他 数值方法比较。另外，我们还发现尖角越尖，收敛阶越高，并得到了内传输特 征函数的收敛阶与尖角之间的反比例关系的具体表达式。本文的数值方法的优势是不仅能得到散射体区域的内传输特征值，而且还 能 得 到 相 应 的 内 传 输 特 征 函 数。我 们 做 了 大 量 的 数 值 实 验，数 值 实 验 结 果 表 明:当障碍物的区域的内角小于 180 度时，内传输特征函数会在接近尖角的地方消失:而当内角大于 180 度时，内传输特征函数会在接近尖角的地方聚集。 此外，本文还讨论了折射率是常数的情况，而且还考虑了不是常值的情况，而 且数值实验的结果均与上述结论吻合。根据内传输特征函数的内在特征，可以 更好的重构障碍物的轮廓，并且通过对内传输特征函数的内在几何结构的研 究，可以运用远场数据来确定内传输特征值和对应的内传输特征函数。另外， 本文通过大量的数值实验得出了一个新结果:内传输特征函数的收敛阶与尖角 的角度有关，并且成反比，即尖角越尖，收敛阶越高。本文采用由线拟合的方法还得到了内传输特征函数的收敛阶与尖角的角度反比例关系的具体表达式。

Interior transmission eigenvalues interior transmission eigenfunctions in-verse scattering objects with corners finite element method

内传输特征值 内传输特征函数 逆散射 带角物体 有限元方法

英语

联合培养

南方科技大学

Qi YC. NUMERICAL SIMULATION ON THEINTERIOR TRANSMISSIONEIGENVALUE PROBLEM[D]. 深圳. 哈尔滨工业大学,2018.