本文選題：無網(wǎng)格法　切入點：復變量移動最小二乘近似　出處：《重慶師范大學》2017年碩士論文

【摘要】：無網(wǎng)格法是繼有限元法之后發(fā)展起來的一種新的數(shù)值計算方法。該方法的核心在于形函數(shù)的構造,移動最小二乘近似是當前應用最為廣泛的無網(wǎng)格近似方案之一,然而基于移動最小二乘近似的無網(wǎng)格法計算量較大。復變量移動最小二乘近似是一種基于復變量理論的、針對向量函數(shù)逼近的移動最小二乘近似。在復變量移動最小二乘近似中,二維函數(shù)的近似只需使用一維基函數(shù),導致試函數(shù)中的待定系數(shù)減少,進而所需節(jié)點個數(shù)大大減少。因此復變量型無網(wǎng)格法可以在保障計算精度的情況下,大大減少求解域內(nèi)的節(jié)點個數(shù)。基于復變量移動最小二乘近似的無網(wǎng)格法在工程領域已經(jīng)被廣泛地應用,然而其相應的數(shù)學理論還很不完善,為了更好地促進其應用,分析其誤差就必不可少。本文詳細討論了復變量移動最小二乘近似的誤差,主要內(nèi)容如下:本文第一章介紹了幾種主要的偏微分方程數(shù)值計算方法,無網(wǎng)格法發(fā)展歷史以及研究現(xiàn)狀,第二章詳細介紹了移動最小二乘近似及復變量移動最小二乘近似。第三章是本文的主要工作,在對權函數(shù)以及節(jié)點分布做出假設的基礎上,針對光滑函數(shù),分析了逼近函數(shù)及其偏導數(shù)的誤差估計,分析結果表明誤差與節(jié)點間距密切相關,最后通過算例驗證了理論分析的正確性。第四章是本文的另一個主要工作,對于被逼近函數(shù)光滑性較弱的情形,在對權函數(shù)以及節(jié)點間距做出適當假設的基礎上,詳細推導了復變量移動最小二乘近似在Sobolev空間中的誤差估計并給出了數(shù)值算例。
[Abstract]:Meshless method is a new numerical method developed after finite element method.The core of this method is the construction of shape function. Moving least square approximation is one of the most widely used meshless approximation schemes at present, but the meshless method based on moving least square approximation has a large amount of computation.The moving least squares approximation of complex variables is a moving least squares approximation based on the theory of complex variables.In the moving least square approximation of complex variables, only one wiki function is used in the approximation of two-dimensional functions, which results in the reduction of the undetermined coefficients in the trial function and the reduction of the number of nodes required.Therefore, the complex variable meshless method can greatly reduce the number of nodes in the solution domain under the condition of guaranteeing the calculation accuracy.Meshless method based on moving least-square approximation of complex variables has been widely used in engineering field, but its corresponding mathematical theory is not perfect. In order to promote its application better, it is necessary to analyze its error.In this paper, the error of moving least square approximation of complex variables is discussed in detail. The main contents are as follows: in the first chapter of this paper, several main numerical methods of partial differential equations are introduced, and the development history and research status of meshless method are introduced.In chapter 2, moving least squares approximation and complex variable moving least square approximation are introduced in detail.The third chapter is the main work of this paper. Based on the assumption of weight function and node distribution, the error estimation of approximation function and its partial derivative is analyzed for smooth function. The results show that the error is closely related to the distance between nodes.Finally, the correctness of the theoretical analysis is verified by an example.The fourth chapter is another main work of this paper. In the case of weak smoothness of the approximated function, we make appropriate assumptions about the weight function and the distance between nodes.The error estimation of moving least square approximation of complex variables in Sobolev space is derived in detail and a numerical example is given.
【學位授予單位】：重慶師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.82

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