本文選題：Helmholtz方程 + 聲波散射��； 參考：《重慶大學(xué)》2016年博士論文

【摘要】：聲波散射問題一直受到人們廣泛關(guān)注,也是數(shù)學(xué)和物理學(xué)中的重要研究領(lǐng)域。相關(guān)的研究不僅有理論意義,還有更重要的實(shí)用價(jià)值,在醫(yī)學(xué)、地球物理、信號(hào)圖像處理等,尤其是大波數(shù)問題。研究大波數(shù)散射問題面臨的困難主要有兩個(gè):解的劇烈震蕩性和計(jì)算區(qū)域的無界性。對(duì)于該類問題的模擬逼近,已經(jīng)存在很多數(shù)值方法。但是對(duì)于大波數(shù)問題產(chǎn)生的“污染效果”,在高維空間中還是沒有消除。本文中,利用(Wang et al,2015)提出的新型有限差分方法,我們對(duì)一些特殊區(qū)域上具有大波數(shù)的Helmholtz方程進(jìn)行了研究,并希望這些工作能促進(jìn)聲波散射問題數(shù)值算法研究的發(fā)展。主要工作有:1.對(duì)于具有不可穿透的圓柱形障礙物的三維淺水波中的散射問題,首先,應(yīng)用DtN算子將無界區(qū)域轉(zhuǎn)化到有界區(qū)域。其次,在柱坐標(biāo)系下對(duì)問題進(jìn)行變量分離將三維問題轉(zhuǎn)化為一系列一維問題,然后利用泰勒展式及方程固有的特性,考慮泰勒展式中無限多項(xiàng)的影響,構(gòu)造無污染的新型有限差分格式,對(duì)問題進(jìn)行模擬逼近,并給出了一系列的數(shù)值算例來驗(yàn)證該格式的可行性。2.基于上述問題的結(jié)果,研究具有可穿透的障礙物的聲波傳輸問題。這類問題的數(shù)值模擬面臨的困難主要有兩個(gè):一是波在傳輸界面上的突變性,另一個(gè)是區(qū)域的無界性。對(duì)于此類無界傳輸問題,應(yīng)用Dt N算子,我們將無界區(qū)域轉(zhuǎn)化為有界區(qū)域,得到一個(gè)包含障礙物的具有人工邊界的散射體問題,它包含兩個(gè)區(qū)域,即障礙區(qū)域與外散射區(qū)域。我們對(duì)不同區(qū)域采用不同剖分,這樣既可以降低計(jì)算量,同時(shí)又能保證數(shù)值解在傳輸界面上的精度。3.對(duì)于大波數(shù)問題,“污染效果”只能在一維方程中徹底消除,對(duì)一般高維維問題卻不能完全消除。但是隨著有限差分格式收斂階的提高,在一定程度上可以提高模擬效果。以環(huán)形區(qū)域上一般Helmholtz方程的大波數(shù)問題為研究對(duì)象,運(yùn)用方程解的光滑性與波數(shù)的關(guān)系,構(gòu)造新型高階有限差分方法,提高大波數(shù)問題的模擬結(jié)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.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.82

【參考文獻(xiàn)】

相關(guān)期刊論文 前2條

1 ;A TAILORED FINITE POINT METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS IN HETEROGENEOUS MEDIUM[J];Journal of Computational Mathematics;2008年05期

2 ;COMPACT FOURTH-ORDER FINITE DIFFERENCE SCHEMES FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS[J];Journal of Computational Mathematics;2008年01期

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本文編號(hào)：1909836