小波方法及其非线性力学问题应用分析

发布时间：2018-04-02 19:14

本文选题：小波　切入点：微分方程　出处：《固体力学学报》2017年04期

【摘要】：小波分析是近几十年来发展起来的重要数学分支,被誉为"数学显微镜",其独具的多分辨分析和大量可供选择的,可兼具正交性、紧支性、对称性、低通滤波、线性相位及插值性等优良数学品质的小波基函数为强非线性微分方程的数值求解带来了新的契机.自上世纪90年代以来,诸如小波伽辽金法、小波配点法、小波有限单元法和小波边界单元法等数值方法被先后构建出来并成功应用于各类力学问题的定量研究之中.论文从小波提出的历史背景及作为其理论基础的多分辨分析出发,对现有基于小波理论的各类数值方法进行梳理,总结各自的优点、缺点和下一步可能的发展方向,为未来基于小波理论的定量分析方法的发展及其在复杂非线性力学问题中的应用研究提供参考.
[Abstract]:Wavelet analysis is an important branch of mathematics developed in recent decades. It is called "mathematical microscope". Its unique multi-resolution analysis and a large number of alternative, can have orthogonality, compactness, symmetry, low-pass filtering.Wavelet basis functions with good mathematical properties such as linear phase and interpolation bring a new opportunity for the numerical solution of strongly nonlinear differential equations.Since the 1990s, numerical methods such as wavelet Galerkin method, wavelet collocation method, wavelet finite element method and wavelet boundary element method have been constructed successively and successfully applied to the quantitative study of various mechanical problems.Based on the historical background put forward by small wave and the multi-resolution analysis which is the theoretical basis of wavelet theory, this paper combs all kinds of numerical methods based on wavelet theory, summarizes their advantages, disadvantages and possible future development direction.It provides a reference for the development of quantitative analysis method based on wavelet theory and its application in complex nonlinear mechanical problems.
【作者单位】：西部灾害与环境力学教育部重点实验室兰州大学土木工程与力学学院;
【基金】：国家自然科学基金项目(11502103,11421062和11472119)资助
【分类号】：O302;O322

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