小波方法及其力学应用研究进展

刘小靖; 周又和; 王记增

doi:10.21656/1000-0887.420388

小波方法及其力学应用研究进展

doi: 10.21656/1000-0887.420388

兰州大学土木工程与力学学院西部灾害与环境力学教育部重点实验室，兰州 730000

基金项目: 国家自然科学基金（11925204；12172154）；高等学校学科创新引智计划（B14044）；国家重大工程（GJXM92579）

详细信息

作者简介:
刘小靖（1986—），男，副教授，博士（E-mail：liuxiaojing@lzu.edu.cn）

王记增（1974—），男，教授，博士（通讯作者. E-mail：jzwang@lzu.edu.cn）

中图分类号: O302
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出版历程
- 收稿日期: 2021-12-09
- 录用日期: 2021-12-30
- 修回日期: 2021-12-29
- 网络出版日期: 2021-12-31
- 刊出日期: 2022-01-01

Research Progresses of Wavelet Methods and Their Applications in Mechanics

Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education of China, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, P.R.China

摘要

摘要:
小波理论在进行信号处理与函数逼近时体现出非常独特的时频局部性与多分辨分析能力，小波基函数则可兼具正交性、紧支性、低通滤波与插值性等优良的数学性质，这均使得小波分析理论在计算数学与计算力学领域具有很大的应用潜力，也进一步为这些领域的突破性发展带来了新的契机。自20世纪90年代以来，大量的研究已经证明，基于小波理论的数值方法在微分方程求解中具有非常明显的优势，但与此同时也暴露出了一些由小波基函数本身与其特有逼近方式所造成的数值计算应用局限。为了促进小波理论在计算数学与力学领域的创新性应用，给研究人员提供新的研究视角，该文简要梳理了小波分析的发展背景以及基于小波理论的数值方法的研究历史，并着重讨论分析了后者所面临的问题，以及近年来针对这些问题中的基础性难题所取得的研究进展。这些总结与评述有望为后续进一步发展并完善基于小波理论的定量数学求解方法，以及拓展其在力学乃至广泛工程问题求解中的应用提供有意义的参考。
- 小波分析 /
- 多分辨分析 /
- 力学问题 /
- 非规则区域 /
- 强非线性 /
- 高阶微分方程
Abstract:
The wavelet theory shows very unique time-frequency localization and multi-resolution analysis ability in signal processing and function approximation. The wavelet basis function has excellent mathematical properties such as orthogonality, compactness, low-pass filtering and interpolation, which endows the wavelet analysis theory with great application potential in the fields of computational mathematics and computational mechanics, and creates new opportunities for breakthrough development in these fields. Since the 1990s, a large number of studies have proved that the numerical method based on the wavelet theory has very obvious advantages in solving differential equations, but at the same time, have exposed some limitations of numerical calculation application caused by the wavelet basis function itself and its unique approximation method. In order to promote the innovative application of the wavelet theory in the fields of computational mathematics and mechanics and provide researchers with a new research perspective, the development background of the wavelet analysis and the research history of methods based on the wavelet theory were reviewed, and the numerical method problems were emphasized and the research progresses made in recent years discussed. The conclusions and comments may provide a meaningful reference for the further development and improvement of quantitative mathematical solution methods based on the wavelet theory and applications in mechanics as well as solutions of a wide range of engineering problems.
- wavelet analysis /
- multiresolution analysis /
- mechanics problem /
- irregular domain /
- strong nonlinearity /
- high-order differential equation

HTML全文

图 1 广义正交Coiflet小波(γ=6，M₁=7)：(a)尺度函数$\phi (x)$和小波函数$\psi (x)$；(b)频谱

Figure 1. The generalized orthogonal Coiflet with γ=6 and M₁=7: (a) scaling function $\phi (x)$ and wavelet function $\psi (x)$; (b) frequency spectrum

下载: 全尺寸图片幻灯片

图 2 在有限区域上逼近函数所需的尺度基函数及边界延拓示意图

注　为了解释图中的颜色，读者可以参考本文的电子网页版本，后同。

Figure 2. Diagrammatic drawing of the required scaling basis function and the corresponding boundary extension in the approximation of a function in a finite domain

下载: 全尺寸图片幻灯片

图 3 小波多分辨定向插值的节点分布示意图

Figure 3. Diagrammatic drawing of the distribution of nodes for targeted wavelet interpolation

下载: 全尺寸图片幻灯片

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