Studies on adaptive boundary integral equation methods using wavelets

小波自适应边界积分方程方法研究

• 批准号: 07650530
• 负责人: NISHIMURA Naoshi
• 金额: $ 1.34万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07650530/
• 关键词: BIEM; BEM; Wavelets; Numerical Analysis; Fast Solution Methods; 数値解析; 波動

We originally planned to investigate the use of spline wavelets in BIEM for wave equations, both in time and frequency domains. However, we found that these functions do not drastically improve accuracy and efficiency of the solutions of BIE compared to the conventional shape functions. In particular some of wavelet functions are found not to be very convenient as time shape functions as one considers the causality of the problem. We thus concluded that it is more appropriate to concentrate on frequency domain approaches than the time domain ones, and that it would be necessary to return to the more fundamental cases of Laplace's equation. Fortunately, it is found that the multiscale analysis of wavelet functions is closely related to fast solution methods of BIE,which are studied extensively these days. We could thus formulate and test the wavelet-Galerkin BIEM,which, in our opinion, is more effective and useful than what we originally intended to investigate.The wavelet-Galerkin BIEM improves the conventional Galerkin BIEM,which uses only scaling functions, by using Haar's wavelet functions. Since Haar's wavelet functions integrate to zero, the single and double layr potentials, with wavelet density functions decay more quickly than with conventional shape functions. In addition the use of Haar's wavelet functions as test functions further accelerates the decay ; indeed, the rate of decay of off-diagonal terms in the matrix of the wavelet-Galerkin equation is bigger by the order of 2 than that of the conventional Galerkin method. Therefore the proposed method makes the matrix equation more diagonally dominated and makes it possible to replace some of off-diagonal terms by zero without deterioration in the quality of the solution. As we found, replacing even 75% of the components in the matrix by zero was acceptable in a certain problem with approximately 500 DOF.We thus conclude that the wavelet-Galerkin BIEM is very promising as a fast solution method of BIE.

我们最初计划研究BIEM中的样条小波在时间和频域中的波动方程。但是，我们发现与常规形状函数相比，这些功能并不能大大提高BIE解决方案的准确性和效率。特别是发现某些小波函数并不是很方便，因为时间形状函数认为问题的因果关系。因此，我们得出的结论是，专注于频域方法比时域更合适，并且有必要返回到Laplace方程的更基本的情况。幸运的是，发现小波函数的多尺度分析与BIE的快速解决方案方法密切相关，这些方法已被广泛研究。因此，我们可以制定和测试小波 - 盖尔金biem，在我们看来，它比我们最初打算研究的更有效和有用。小波 - 盖尔金biem改善了传统的Galerkin Biem，它仅使用缩放函数，通过使用缩放函数，使用使用缩放函数。 Haar的小波功能。由于HAAR的小波函数集成为零，因此单层和双Layr电势具有比传统形状函数更快的小波密度函数衰减。此外，使用HAAR小波作为测试功能的功能进一步加速了衰减。实际上，小波 - 盖尔金方程基质中非对角线术语的衰减速率比2的阶数大于传统的盖尔金方法。因此，所提出的方法使矩阵方程更具对角度支配，并可以通过零替换一些偏对术语，而不会在溶液质量中替换。正如我们发现的那样，在大约500 dof的某个问题中可以接受矩阵中75％的组件。