| 小波方法及其力学应用研究进展 |
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| 引用本文： | 刘小靖,周又和,王记增.小波方法及其力学应用研究进展[J].应用数学和力学,2022,43(1):1-13. |
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| 作者姓名： | 刘小靖 周又和 王记增 |
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| 作者单位： | 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 |
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| 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)。 |
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| 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...
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| 关 键 词： | 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 |
| 收稿时间： | 2021-12-09 |
Research Progresses of Wavelet Methods and Their Applications in Mechanics |
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| LIU Xiaojing,ZHOU Youhe,WANG Jizeng.Research Progresses of Wavelet Methods and Their Applications in Mechanics[J].Applied Mathematics and Mechanics,2022,43(1):1-13. |
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| Authors: | LIU Xiaojing ZHOU Youhe WANG Jizeng |
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| Affiliation: | 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 |
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| 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. |
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| Keywords: | wavelet analysis multiresolution analysis mechanics problem irregular domain strong nonlinearity highorder differential equation |
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