【摘要】：單側(cè)問(wèn)題是一類含有變分不等式的數(shù)學(xué)物理問(wèn)題。科學(xué)工程中諸如電鍍問(wèn)題、障礙問(wèn)題、自由水壩問(wèn)題等都被統(tǒng)稱為單側(cè)問(wèn)題。在數(shù)值計(jì)算中,無(wú)網(wǎng)格方法不需要生成網(wǎng)格,適用性強(qiáng),它是一種新的數(shù)值求解偏微分方程邊界問(wèn)題的方法。本論文就是用無(wú)網(wǎng)格投影迭代法來(lái)解決單側(cè)問(wèn)題,具體研究工作如下:本論文第1章緒論介紹了單側(cè)問(wèn)題的研究背景和研究現(xiàn)狀,引出了本文的研究目的是用無(wú)網(wǎng)格投影迭代法和無(wú)網(wǎng)格邊界積分方程方法結(jié)合起來(lái),嘗試用無(wú)網(wǎng)格插值邊界無(wú)單元投影迭代法來(lái)解決單側(cè)問(wèn)題。第2章是本文的核心部分,系統(tǒng)的闡述了單側(cè)問(wèn)題的無(wú)網(wǎng)格投影迭代算法。首先,介紹了用改進(jìn)的插值型移動(dòng)最小二乘方法來(lái)構(gòu)造形函數(shù),接著引入投影迭代算子和隱式投影格式,提出了單側(cè)問(wèn)題的插值邊界無(wú)單元法,總結(jié)了相應(yīng)的算法步驟,并對(duì)該算法的收斂性進(jìn)行了推導(dǎo)和證明。第3章采用本文的算法解決分析了三個(gè)數(shù)值算例,經(jīng)過(guò)對(duì)比分析,發(fā)現(xiàn)該算法收斂性好,迭代次數(shù)少,計(jì)算時(shí)間也相對(duì)較短,并且數(shù)值解與精確解很吻合,進(jìn)一步表明本算法具有較好的可行性與有效性。第4章給出了相關(guān)總結(jié),說(shuō)明該無(wú)網(wǎng)格投影迭代法是解決單側(cè)問(wèn)題模型的一種有效方法,有利于工程應(yīng)用。
[Abstract]:Unilateral problems are a class of mathematical and physical problems with variational inequalities. In scientific engineering, such as electroplating problem, obstacle problem, free dam problem and so on are all referred to as unilateral problems. Meshless method is a new numerical method for solving boundary problems of partial differential equations. In this paper, the meshless projection iteration method is used to solve the unilateral problem. The specific research work is as follows: the first chapter of this paper introduces the background and research status of the unilateral problem. The purpose of this paper is to combine the meshless projection iteration method with the meshless boundary integral equation method and try to solve the unilateral problem by using the meshless interpolation boundary element free projection iteration method. Chapter 2 is the core of this paper. Firstly, an improved interpolation moving least square method is introduced to construct the shape function. Then the projection iterative operator and implicit projection scheme are introduced, and the interpolation boundary element free method for the one-sided problem is proposed, and the corresponding algorithm steps are summarized. The convergence of the algorithm is deduced and proved. In chapter 3, three numerical examples are solved and analyzed by using the algorithm presented in this paper. Through comparison and analysis, it is found that the algorithm has good convergence, few iterations and relatively short calculation time, and the numerical solution is in good agreement with the exact solution. It further shows that this algorithm is feasible and effective. In chapter 4, the relevant summary is given, which shows that the meshless projection iterative method is an effective method to solve the one-sided problem model, which is beneficial to engineering application.
【學(xué)位授予單位】：重慶師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O176;O241.8

【參考文獻(xiàn)】

相關(guān)期刊論文 前10條

1 胡齊芽,余德浩;Signorini傳輸問(wèn)題的有限元邊界元耦合[J];中國(guó)科學(xué)(A輯);2001年02期

2 程玉民;;移動(dòng)最小二乘法研究進(jìn)展與述評(píng)[J];計(jì)算機(jī)輔助工程;2009年02期

3 龐作會(huì)，，葛修潤(rùn)，鄭宏，王水林;一種新的數(shù)值方法——無(wú)網(wǎng)格伽遼金法(EFGM)[J];計(jì)算力學(xué)學(xué)報(bào);1999年03期

4 ;A BOUNDARY ELEMENT APPROXIMATION OF A SIGNORINI PROBLEM WITH FRICTION OBEYING COULOMB LAW[J];Journal of Computational Mathematics;1994年02期

5 ;THE NONCONFORMING FINITE ELEMENT METHOD FOR SIGNORINI PROBLEM[J];Journal of Computational Mathematics;2007年01期

6 何炳生;一類求解單調(diào)變分不等式的隱式方法[J];計(jì)算數(shù)學(xué);1998年04期

7 何炳生;求解單調(diào)變分不等式的一類預(yù)測(cè)-校正方法的統(tǒng)一框架(英文)[J];南京大學(xué)學(xué)報(bào)(自然科學(xué)版);2003年04期

8 ZHANG Zan;WANG JianFei;CHENG YuMin;LIEW Kim Meow;;The improved element-free Galerkin method for three-dimensional transient heat conduction problems[J];Science China(Physics,Mechanics & Astronomy);2013年08期

9 童小嬌;何炳生;;一類單調(diào)變分不等式的非精確交替方向法[J];數(shù)學(xué)物理學(xué)報(bào);2006年02期

10 龐作會(huì),葛修潤(rùn),王水林;無(wú)網(wǎng)格伽遼金法(EFGM)在邊坡開挖問(wèn)題中的應(yīng)用[J];巖土力學(xué);1999年01期

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