小波方法及其非线性力学问题应用分析

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。本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。

Wavelet Methods and Applications in Nonlinear Mechanics Problems

Wavelet analysis is a mathematical branch developed in the past several decades, which is known as the so-called "numerical microscope". Wavelets have the unique mathematical property of multiresolution analysis. When using wavelet and scaling functions as basis, they have excellent mathematical characteristics of orthogonality, compactness, symmetry, low-pass filter, approximate linear phase and interpolation etc. These properties brought new opportunities to developing advanced numerical techniques on accurately and efficiently solving differential equations in nonlinear mechanics problems. Since the 1990s, numerical methods such as wavelet Galerkin method, wavelet collocation method, wavelet finite element method and wavelet boundary element method etc. have been constructed and successfully applied to the quantitative research of mechanical problems. Most importantly, wavelet analysis provides a totally new way to develop robust and adaptive methods for efficiently solving mechanical problems with large local gradients, and to propose closure algorithms to uniformly solving problems with strong nonlinearity. Problems with these two types of features are usually very difficult to deal with by using most traditional methods. Starting from the review of historical background and theory of multiresolution analysis, this review systematically discusses how specific mathematic property of the wavelets can merit high efficiency and accuracy of the wavelet-based method, and why the Coiflet-based method is a good choice in developing advanced numerical algorithms for solving nonlinear differential equations. Also, this paper analyzes the existing numerical methods related to wavelets and summarizes the advantages, disadvantages and possible development directions of wavelet based methods. Especially, this paper discusses the closed-form numerical algorithm based on the Coiflets for solving nonlinear mechanical problems in detail. An example on the shallow water equation demonstrates that such a method has the ability to capture major pattern characteristics of the solution even under very coarse space-time meshes. Our ultimate goal is to eventually provide a valuable reference to the further development of wavelet methods and their applications in various complex mechanics problems.