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摘要: 随着力学研究不断向多学科交叉、跨尺度关联及极端环境影响推进, 强非线性、强间断、多场强耦合、多尺度与复杂几何构型已成为解决各类力学问题定量分析所面临的共性特征. 长期的定量研究表明, 实现具有这类特征力学问题有效求解的核心之一, 是构建在多尺度情形、非线性因素作用下具有准确识别、定位、捕获以及分离各个尺度特征尤其是小尺度局部特征能力的数值工具, 这些能力包括大尺度低阶近似解与小尺度高阶微小截断误差的有效分离解耦. 而小波理论固有的多分辨分析和时频局部化特性, 以及丰富的基函数可选择性, 恰好能满足这一数学特征需求, 因此, 可为发展各类复杂力学问题的高效定量求解方法提供有效的理论支撑和丰富多样的技术途径. 基于这一事实, 本文对小波理论进行了全面探讨, 着重介绍了双正交多分辨分析的理论框架和常用小波基的构造方法. 在此基础上, 深入研究了有限区域上小波逼近格式的构建方法, 并系统阐述了各类基于小波理论的数值方法的基本原理、发展历程及其优缺点. 特别关注了近期出现的具有突出性能的几类新型小波方法, 并详细评述了它们在典型力学问题求解中的应用效果. 同时, 本文也指出了当前小波方法在复杂强非线性力学问题求解中所面临的挑战, 旨在为小波数值方法的未来发展及其在复杂力学与工程问题中的应用拓展提供有益的参考, 并为最终实现这些问题的高效、高精度普适定量求解提供新的视角和方法.Abstract: Mechanics research advances towards interdisciplinary research, cross-scale correlations, and extreme environmental impacts. Strong nonlinearity, strong discontinuity, significant multi-physics coupling, multiscale, and complex geometries have become common characteristics in solving various mechanics problems quantitatively. Long-term quantitative research indicates that one of the core ingredients for solving such problems effectively lies in constructing numerical methods that can accurately identify, locate, capture, and separate different scale characteristics especially small-scale local characteristics, under multiscale and nonlinearity circumstances. These numerical methods should also possess the capacity to isolate and decouple the large-scale lower-order approximation from the small-scale higher-order truncation error effectively. The intrinsic multiresolution analysis and time-frequency localization characteristics of wavelet theory, as well as the various selectivity of basis functions, align precisely with the demands of this mathematical feature. Therefore, they can provide fundamental theory and diverse approaches for developing efficient quantitative methods to address various complex mechanics problems. Based on this fact, this paper provides a comprehensive discussion of wavelet theory, focusing on the theoretical framework of biorthogonal multi-resolution analysis and the construction method of frequently used wavelet bases. Furthermore, wavelet approximation of the function defined on a finite domain is presented in detail. The fundamental principles, development, merits, and shortcomings of different wavelet-based numerical methods are systematically elucidated. Several novel wavelet-based methods with outstanding performance developed recently are elaborated especially, and their applications in solving typical mechanics problems are reviewed. Meanwhile, the present paper also points out challenges encountered by the existing wavelet-based numerical methods when addressing complex and strongly nonlinear mechanics problems. This may provide valuable references for the development of wavelet-based numerical methods and their applications in complex mechanics and engineering problems, and introduce new perspectives and methods for solving these problems efficiently, accurately, and universally.
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