大波数声波散射问题的新型有限差分方法的研究

发布时间：2018-05-19 10:56

本文选题：Helmholtz方程 + 声波散射　；参考：《重庆大学》2016年博士论文

【摘要】：声波散射问题一直受到人们广泛关注,也是数学和物理学中的重要研究领域。相关的研究不仅有理论意义,还有更重要的实用价值,在医学、地球物理、信号图像处理等,尤其是大波数问题。研究大波数散射问题面临的困难主要有两个:解的剧烈震荡性和计算区域的无界性。对于该类问题的模拟逼近,已经存在很多数值方法。但是对于大波数问题产生的“污染效果”,在高维空间中还是没有消除。本文中,利用(Wang et al,2015)提出的新型有限差分方法,我们对一些特殊区域上具有大波数的Helmholtz方程进行了研究,并希望这些工作能促进声波散射问题数值算法研究的发展。主要工作有:1.对于具有不可穿透的圆柱形障碍物的三维浅水波中的散射问题,首先,应用DtN算子将无界区域转化到有界区域。其次,在柱坐标系下对问题进行变量分离将三维问题转化为一系列一维问题,然后利用泰勒展式及方程固有的特性,考虑泰勒展式中无限多项的影响,构造无污染的新型有限差分格式,对问题进行模拟逼近,并给出了一系列的数值算例来验证该格式的可行性。2.基于上述问题的结果,研究具有可穿透的障碍物的声波传输问题。这类问题的数值模拟面临的困难主要有两个:一是波在传输界面上的突变性,另一个是区域的无界性。对于此类无界传输问题,应用Dt N算子,我们将无界区域转化为有界区域,得到一个包含障碍物的具有人工边界的散射体问题,它包含两个区域,即障碍区域与外散射区域。我们对不同区域采用不同剖分,这样既可以降低计算量,同时又能保证数值解在传输界面上的精度。3.对于大波数问题,“污染效果”只能在一维方程中彻底消除,对一般高维维问题却不能完全消除。但是随着有限差分格式收敛阶的提高,在一定程度上可以提高模拟效果。以环形区域上一般Helmholtz方程的大波数问题为研究对象,运用方程解的光滑性与波数的关系,构造新型高阶有限差分方法,提高大波数问题的模拟结果。
[Abstract]:The problem of acoustic scattering has been widely paid attention to. It is also an important research field in mathematics and physics. The related research not only has theoretical significance, but also has more important practical value. In medicine, geophysics, signal image processing and so on, especially the large wave number problem, there are two main difficulties in studying the problem of large wave scattering. There are many numerical methods for the simulation approximation of this kind of problem. But the "pollution effect" produced by the large wave number problem has not been eliminated in the high dimensional space. In this paper, we use the new finite difference method proposed by (Wang et al, 2015) to some special areas. The Helmholtz equation with large wave numbers is studied, and it is hoped that these work can promote the development of numerical algorithm for acoustic scattering problems. The main work is: 1. for the scattering problem in the three-dimensional shallow water wave with an unpenetrable cylindrical obstacle, first, the unbounded region is converted to the bounded region by using the DtN arithmetic. Secondly, The three-dimensional problem is transformed into a series of one-dimensional problems in the column coordinate system, and then the Taylor extension and the inherent characteristics of the equation are used to consider the infinitely many effects of the Taylor's extension. A new finite difference scheme without pollution is constructed to simulate the problem, and a series of numerical examples are given to verify this problem. The feasibility of the format.2. is based on the results of the above problems to study the transmission of sound waves with penetrable obstacles. There are two main difficulties in the numerical simulation of these problems: one is the mutagenicity on the transmission interface and the other is the unbounded domain. For this kind of unbounded transmission problem, we apply the Dt N operator to unbounded areas. The domain is transformed into a bounded region, and a scatterer with an artificial boundary containing obstacles is obtained. It contains two regions, that is, the barrier region and the outer scattering region. We use different sections for different regions, which can reduce the amount of computation and guarantee the accuracy of the numerical solution at the transmission interface.3. for the large wave number problem. The "pollution effect" can only be eliminated completely in one dimensional equation and can not completely eliminate the general high dimensional dimension problem. However, with the increase of the convergence order of the finite difference scheme, the simulation effect can be improved to some extent. The problem of the large wave number of the general Helmholtz equation in the ring region is taken as the research object, and the smoothness and wave of the equation solution are used. A new high-order finite difference method is constructed to improve the simulation results of large wave numbers.
【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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