幾類問題基于自然邊界歸化的算法研究
發(fā)布時(shí)間：2018-09-04 05:32
【摘要】：自然邊界歸化理論是由馮康教授首創(chuàng),經(jīng)由其及余德浩教授等學(xué)者發(fā)展完善.該理論與有限元方法和辛幾何算法構(gòu)成了馮先生的主要學(xué)術(shù)貢獻(xiàn).自然邊界元法,可以直接用于求解某些無界區(qū)域橢圓邊值問題,其與有限元方法、區(qū)域分解算法和多重網(wǎng)格算法的耦合算法亦是處理無界區(qū)域及凹角、斷裂區(qū)域問題的有效手段之一,并在二維及三維領(lǐng)域內(nèi)取得了許多重要的研究成果.之前的研究通常以圓(二維情況)、球面(三維情況)作為人工邊界,但對于某些特殊區(qū)域,例如,長條型區(qū)域,用長橢球面或橢圓作人工邊界,則可大大減小計(jì)算區(qū)域,從而可以減少計(jì)算量和存儲量.本文主要研究三維各向異性外問題的基于橢球面人工邊界的區(qū)域分解算法(Schwarz交替算法和D-N交替算法)和二維Helmholtz方程外問題的多重網(wǎng)格算法.第一章介紹了兩類正交坐標(biāo)系、幾類特殊函數(shù)和Sobolev空間的相關(guān)概念和定理,作為以后各章進(jìn)行理論分析的重要工具.第二章研究了三維各向異性外問題的基于基于自然邊界歸化的Schwarz交替算法.首先對所研究問題進(jìn)行變量替換,得到相應(yīng)的Laplace方程外問題.進(jìn)一步得到旋轉(zhuǎn)橢球外區(qū)域上問題自然積分方程和Poisson積分公式.然后,給出Schwarz交替算法,分析了該算法的收斂性,并給出了數(shù)值解的誤差估計(jì),通過數(shù)值算例以示算法的可行性與有效性.第三章討論了三維各向異性外問題的基于旋轉(zhuǎn)球面人工邊界的D-N交替算法.根據(jù)第二章相應(yīng)內(nèi)容,給出D-N交替算法和等價(jià)的Richardson迭代算法.其次,分析了該算法的收斂性,給出了等價(jià)的變分形式及其離散形式.然后,對其離散形式進(jìn)行收斂性分析,最后通過數(shù)值算例以示方法的可行性與有效性.第四章研究了二維Helmholtz方程外問題的基于自然邊界元方法的多重網(wǎng)格算法.首先給出了問題等價(jià)的變分形式,其次建立了多重網(wǎng)格算法并分析了該算法的收斂性、收斂速度分析及離散情形的誤差估計(jì).最后,通過數(shù)值算例以示方法的可行性與有效性.
[Abstract]:Naturalization theory of natural boundary was initiated by Professor Feng Kang and developed and perfected by Professor Yu Dehao and other scholars. The theory, the finite element method and the symplectic geometric algorithm constitute Mr. Feng's main academic contribution. The natural boundary element method can be directly used to solve the elliptic boundary value problems in some unbounded regions. The coupling algorithms of the natural boundary element method and the finite element method, the domain decomposition algorithm and the multi-grid algorithm are also used to deal with the unbounded region and the concave angle. One of the effective methods of fault region problem, many important research results have been obtained in two and three dimensions. Previous studies usually use circles (two-dimensional cases) and spherical surfaces (three-dimensional cases) as artificial boundaries, but for some special regions, such as long ellipsoid or ellipse, the computational area can be greatly reduced. Thus, the amount of computation and memory can be reduced. In this paper, the domain decomposition algorithm based on ellipsoidal artificial boundary (Schwarz alternating algorithm and D-N alternating algorithm) and the multigrid algorithm for two-dimensional Helmholtz equation are studied. The first chapter introduces two kinds of orthogonal coordinate systems, some special functions and some related concepts and theorems of Sobolev space, which are important tools for theoretical analysis in the following chapters. In chapter 2, we study the Schwarz alternating algorithm based on natural boundary normalization for three dimensional anisotropic exterior problems. First, the variables are replaced to obtain the corresponding problem of Laplace equation. Furthermore, the natural integral equation and Poisson integral formula for the problem in the outer domain of a rotating ellipsoid are obtained. Then, the Schwarz alternating algorithm is given, the convergence of the algorithm is analyzed, and the error estimation of the numerical solution is given. A numerical example is given to show the feasibility and effectiveness of the algorithm. In chapter 3, the D-N alternating algorithm based on the artificial boundary of rotating sphere is discussed. According to the corresponding contents of the second chapter, D-N alternating algorithm and equivalent Richardson iterative algorithm are given. Secondly, the convergence of the algorithm is analyzed, and the equivalent variational form and its discrete form are given. Then, the convergence of the discrete form is analyzed. Finally, numerical examples are given to show the feasibility and effectiveness of the method. In chapter 4, the multi-grid algorithm based on natural boundary element method for two-dimensional Helmholtz equation is studied. In this paper, the equivalent variational form of the problem is given, and then the multigrid algorithm is established and its convergence, convergence rate analysis and error estimation in discrete case are analyzed. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the method.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O241.8
本文編號：2221161
[Abstract]:Naturalization theory of natural boundary was initiated by Professor Feng Kang and developed and perfected by Professor Yu Dehao and other scholars. The theory, the finite element method and the symplectic geometric algorithm constitute Mr. Feng's main academic contribution. The natural boundary element method can be directly used to solve the elliptic boundary value problems in some unbounded regions. The coupling algorithms of the natural boundary element method and the finite element method, the domain decomposition algorithm and the multi-grid algorithm are also used to deal with the unbounded region and the concave angle. One of the effective methods of fault region problem, many important research results have been obtained in two and three dimensions. Previous studies usually use circles (two-dimensional cases) and spherical surfaces (three-dimensional cases) as artificial boundaries, but for some special regions, such as long ellipsoid or ellipse, the computational area can be greatly reduced. Thus, the amount of computation and memory can be reduced. In this paper, the domain decomposition algorithm based on ellipsoidal artificial boundary (Schwarz alternating algorithm and D-N alternating algorithm) and the multigrid algorithm for two-dimensional Helmholtz equation are studied. The first chapter introduces two kinds of orthogonal coordinate systems, some special functions and some related concepts and theorems of Sobolev space, which are important tools for theoretical analysis in the following chapters. In chapter 2, we study the Schwarz alternating algorithm based on natural boundary normalization for three dimensional anisotropic exterior problems. First, the variables are replaced to obtain the corresponding problem of Laplace equation. Furthermore, the natural integral equation and Poisson integral formula for the problem in the outer domain of a rotating ellipsoid are obtained. Then, the Schwarz alternating algorithm is given, the convergence of the algorithm is analyzed, and the error estimation of the numerical solution is given. A numerical example is given to show the feasibility and effectiveness of the algorithm. In chapter 3, the D-N alternating algorithm based on the artificial boundary of rotating sphere is discussed. According to the corresponding contents of the second chapter, D-N alternating algorithm and equivalent Richardson iterative algorithm are given. Secondly, the convergence of the algorithm is analyzed, and the equivalent variational form and its discrete form are given. Then, the convergence of the discrete form is analyzed. Finally, numerical examples are given to show the feasibility and effectiveness of the method. In chapter 4, the multi-grid algorithm based on natural boundary element method for two-dimensional Helmholtz equation is studied. In this paper, the equivalent variational form of the problem is given, and then the multigrid algorithm is established and its convergence, convergence rate analysis and error estimation in discrete case are analyzed. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the method.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O241.8
【參考文獻(xiàn)】
相關(guān)期刊論文 前3條
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