【作者】 李俊贤；

【导师】 陈一鸣；

【作者基本信息】 燕山大学 ， 计算数学， 2010， 硕士

【摘要】 小波边界元法是近几年发展起来的一种新型的数值计算方法。目前对它的研究还比较少,但这种方法从一开始就展现了它的独特优点和强劲生命力。本文主要是研究小波基函数在自然边界元方法中的应用。目的是解决自然边界元方法中存在的奇异积分的困难,减小计算量,提高计算精度。其基本思想是将微分方程经过自然边界归化得到与之等价的变分问题,再利用Galerkin-wavelet方法或小波插值法将其离散,得到具有独特优点的刚度矩阵,这样就能大大减小计算量。同时对数值解的误差估计作了进一步的分析。给出的算例表明了各种小波边界元方法的有效性。论文共分五章。第一章介绍了边界元方法、自然边界归化理论以及小波分析理论,并给出了Laplace方程在典型域上通过自然边界归化所得的自然边界积分方程和Possion积分公式。第二章针对上半平面Laplace方程的Neumann边值问题在求解过程中遇到的强奇异积分的困难,论文充分利用Shannon小波在频域上有限带宽的优良性质,采用Shannon小波边界元方法求解,使问题简化且得到的数值解比较精确。第三章论文充分利用拟小波基在时域中光滑性高、快速衰减且具有插值性的特点,使其与自然边界元方法相结合,有效解决了解Laplace方程时存在的奇异积分困难。第四章论文采用Hermite三次样条多小波边界元方法对上半平面Laplace方程的Neumann边值问题做了进一步的探讨,并通过算例验证了该方法的有效性。第五章论文将Quak三角小波的尺度函数作为基函数离散自然边界积分方程,得到了仅由一项或两项表出的刚度矩阵系数的计算公式,从而使问题简化,给出的算例也表明了该方法的可行性。更多还原

【Abstract】 In recent years, wavelet boundary element method has developed a new numerical method. Although there is few works about this method, it still shows unique advantages and strong vitality at the beginning. To overcome the shortcoming of singular integral difficulty existed in the natural boundary element method, in this paper, we mainly study the application of the wavelet base function in the natural boundary element method, it not only simplify the calculation process, but also improve the computation accuracy. The basic idea is transforming the differential equations into its equivalent variational problem by the natural boundary naturalization, and then discreting it by the Galerkin-wavelet method or wavelet interpolation to get the corresponding stiffness matrix with unique advantage, it substantially reducing computation. Meanwhile we make the further analysis to the error estimate of the numerical solution. The given example indicates the validity of each kind of wavelet boundary element method.This paper includes five chapters. Chapter 1 introduces the boundary element method, the natural boundary naturalization theory and the wavelet analysis theory. The natural boundary integral equation of Laplace equation and the Possion integral formula by the natural boundary naturalization on the typical domain are also given.In chapter 2, since there is difficulty in solving the Neumann boundary value problem of the Laplace equation on half-plane encounters strongly singular integral. In order to simplify the problem and get more precise numerical solution, we take full advantages of the Shannon wavelet limited bandwidth in frequency domain and use Shannon wavelet boundary element method to solve the problem.In chapter 3, combining the properties of smooth, fast weaken and the interpolation of the wavelet base with the natural boundary element method, it makes the computation reduce and improves accuracy. And it effectively resolves the trouble of the singular integral when solving Laplace equation.In chapter 4, we apply Hermite Cubic Spline Multi-wavelet Natural Boundary Element Method to the Neumann boundary value problem of the Laplace equation on the upper half-plane and make further research. Moreover, the given example proves that the method is effective and flexible.In chapter 5, we applies the criterion of Quak Triangular Wavelet as the base function to discrete Natural Boundary integral equation, and we get the stiffness matrix coefficient formula expressed by one item or two items, which can simply the problem and get more precise solution.更多还原