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小波数值方法及其在计算力学中的应用

杨兵 王记增 刘小靖 周又和 冯永固

杨兵, 王记增, 刘小靖, 周又和, 冯永固. 小波数值方法及其在计算力学中的应用. 力学进展, 待出版 doi: 10.6052/1000-0992-24-009
引用本文: 杨兵, 王记增, 刘小靖, 周又和, 冯永固. 小波数值方法及其在计算力学中的应用. 力学进展, 待出版 doi: 10.6052/1000-0992-24-009
Yang B, Wang J Z, Liu X J, Zhou Y H, Feng Y G. Wavelet-based numerical methods and their applications in computational mechanics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-009
Citation: Yang B, Wang J Z, Liu X J, Zhou Y H, Feng Y G. Wavelet-based numerical methods and their applications in computational mechanics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-009

小波数值方法及其在计算力学中的应用

doi: 10.6052/1000-0992-24-009
基金项目: 感谢国家杰出青年科学基金(11925204)和甘肃省创新研究群体项目的资助(23JRRA1172)
详细信息
    作者简介:

    王记增, 兰州大学萃英学者特聘教授, 教育部长江学者特聘教授, 国家杰出青年基金获得者. 长期从事强非线性力学问题原创普适定量求解方法体系的构建及应用, 生物材料力学问题的建模、 实验及应用拓展研究

    通讯作者:

    jzwang@lzu.edu.cn

  • 中图分类号: O302

Wavelet-based numerical methods and their applications in computational mechanics

More Information
  • 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, 这些能力之中包括了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.

     

  • 图  1  N = 6和M1 = 7 的Coiflet小波. (a)尺度函数和小波函数, (b) 尺度函数和小波函数频谱

    图  3  计算滤波器系数h−1h1节点模板. (a) N = 6 对称小波, (b) N = 5 非对称小波

    图  5  二维椭圆域上小波逼近格式边界延拓所需节点示意图 (红色节点为内部节点, 蓝色节点为域外延拓待插值节点, 黑色节点为对域内小波函数逼近无影响节点)

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  • 录用日期:  2024-05-07
  • 网络出版日期:  2024-05-16

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