声波传输问题和传输特征值问题的数值方法研究

发布时间：2018-02-16 13:13

本文关键词： 声波传输问题 DtN算子有限元方法传输特征值问题 C~0内罚间断伽辽金法　出处：《重庆大学》2016年博士论文　论文类型：学位论文

【摘要】：时谐声波传输问题和传输特征值问题在实际科学和工程领域都有广泛的应用。传输特征值能用来估计散射体材料的性质,并且在逆散射理论中对于证明解的唯一性和重构有着重要的作用。本论文主要研究声波传输问题和传输特征值问题的数值方法。本文主要由两部分组成:第一部分关于声波传输问题的数值方法;第二部分关于传输特征值问题的数值方法。对于声波传输问题,最常用的处理无界区域的方法是边界积分方程方法及其与有限元方法的耦合,比如Hsiao和Xu用边界积分方程方法处理无界区域的计算(Hsiao and Xu,2011)。为了应用有限元方法求解此类问题,最常用的方法是通过引入一个包含障碍物的人工边界将无界区域分解为一个有界区域和一个无界区域。我们根据人工边界外声波满足的散射问题,分别利用傅里叶级数和边界积分算子定义了两个不同的Dirichlet-to-Neumann(DtN)算子。接着,通过在人工边界上引入DtN算子,可以将原无界传输问题转化为等价的非局部边值问题。在适当的索伯列夫空间下,证明了相应的变分问题解的存在唯一性。对于声波传输特征值问题,为了避免直接计算非自伴特征值问题,我们将原传输特征值问题转化为等价的一系列自伴的四阶特征值问题。接着,提出了用拉格朗日元的C~0内部惩罚间断伽辽金(C~0 IPG)方法去研究四阶特征值问题。由于低阶项对离散算子范数收敛性的影响,不能直接应用Babu?ka-Osborn理论证明收敛性。为了克服这个困难,根据Descloux等人在(Descloux et al,1978a)中发展起来的收敛性理论以及Antonietti等人用DG方法求Laplace特征值问题的思想(Antonietti et al,2006),我们首先证明了C~0 IPG方法是谱正确的,接着证明了其最优收敛性。而对于原传输特征值问题转化的一个等价的非自伴的四阶特征值问题,我们给出了离散四阶传输特征值问题的C0 IPG方法,并证明了C~0 IPG方法最优收敛阶。对于高阶椭圆问题,相比于经典的协调有限元方法,C0 IPG方法的数值实现更简单,因为它的基函数简单并且有更少的自由度。对于每一种提出的数值方法,我们给出一些数值例子来验证方法的有效性和准确性。
[Abstract]:The time-harmonic acoustic wave propagation problem and the propagation eigenvalue problem are widely used in practical scientific and engineering fields. The transmission eigenvalues can be used to estimate the properties of scattering materials. And in the inverse scattering theory, it plays an important role in proving the uniqueness and reconstruction of the solution. In this paper, we mainly study the numerical methods of acoustic wave transmission problem and transmission eigenvalue problem. This paper mainly consists of two parts: the first part. The numerical method of acoustic wave transmission; The second part deals with the numerical method of transmitting eigenvalue problem. For acoustic wave transmission problem, the most commonly used method to deal with the unbounded region is the boundary integral equation method and its coupling with the finite element method. For example, Hsiao and Xu use the boundary integral equation method to deal with the computation of unbounded region. The most commonly used method is to decompose the unbounded region into a bounded region and an unbounded region by introducing an artificial boundary containing an obstacle. Two different Dirichlet-to-Neumann-DtN operators are defined by using Fourier series and boundary integral operators, respectively. Then, by introducing DtN operators on artificial boundaries, The original unbounded transmission problem can be transformed into an equivalent nonlocal boundary value problem. The existence and uniqueness of the solution of the corresponding variational problem are proved in the appropriate Soberlev space. In order to avoid directly computing the non-self-adjoint eigenvalue problem, we transform the original transmission eigenvalue problem into an equivalent series of self-adjoint fourth-order eigenvalue problems. In this paper, the fourth order eigenvalue problem is studied by using the Lagrangian C0 internal penalty discontinuous Galerkin C0 IPG method. Because of the influence of low order terms on the convergence of discrete operator norm, Babu? Ka-Osborn theory proves convergence. In order to overcome this difficulty, according to the convergence theory developed by Descloux et al in Descloux et al1978a and the idea of Antonietti et al. to solve Laplace eigenvalue problem by DG method, we first prove that the C0 IPG method is spectral correct. Then we prove its optimal convergence. For an equivalent non-self-adjoint fourth-order eigenvalue problem transformed from the original transmission eigenvalue problem, we give a C0 IPG method for discrete fourth-order transmission eigenvalue problem. It is proved that the optimal convergence order of the C0 IPG method is simpler than that of the classical concordant finite element method (C0 IPG) for higher order elliptic problems. Because its basis function is simple and has less degree of freedom, we give some numerical examples to verify the validity and accuracy of the proposed numerical method.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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